ISSN 1854-6250 APEM journal Advances in Production Engineering & Management Volume 18 | Number 2 | June 2023 Published by CPE apem-journal.org Advances in Production Engineering & Management Identification Statement APEM journal ISSN 1854‐6250 | Abbreviated key title: Adv produc engineer manag | Start year: 2006 ISSN 1855‐6531 (on‐line) Published quarterly by Chair of Production Engineering (CPE), University of Maribor Smetanova ulica 17, SI – 2000 Maribor, Slovenia, European Union (EU) Phone: 00386 2 2207522, Fax: 00386 2 2207990 Language of text: English APEM homepage: apem‐journal.org University homepage: www.um.si APEM Editorial Editor‐in‐Chief Miran Brezocnik editor@apem‐journal.org, info@apem‐journal.org University of Maribor, Faculty of Mechanical Engineering Sme‐ tanova ulica 17, SI – 2000 Maribor, Slovenia, EU Desk Editor Website Technical Editor Martina Meh Lucija Brezocnik desk1@apem‐journal.org desk3@apem‐journal.org Janez Gotlih desk2@apem‐journal.org Editorial Board Members Eberhard Abele, Technical University of Darmstadt, Germany Bojan Acko, University of Maribor, Slovenia Joze Balic, University of Maribor, Slovenia Agostino Bruzzone, University of Genoa, Italy Borut Buchmeister, University of Maribor, Slovenia Ludwig Cardon, Ghent University, Belgium Nirupam Chakraborti, Indian Institute of Technology, Kharagpur, India Edward Chlebus, Wroclaw University of Technology, Poland Igor Drstvensek, University of Maribor, Slovenia Illes Dudas, University of Miskolc, Hungary Mirko Ficko, University of Maribor, Slovenia Vlatka Hlupic, University of Westminster, UK David Hui, University of New Orleans, USA Pramod K. Jain, Indian Institute of Technology Roorkee, India Isak Karabegović, University of Bihać, Bosnia and Herzegovina Janez Kopac, University of Ljubljana, Slovenia Qingliang Meng, Jiangsu University of Science and Technology, China Lanndon A. Ocampo, Cebu Technological University, Philippines Iztok Palcic, University of Maribor, Slovenia Krsto Pandza, University of Leeds, UK Andrej Polajnar, University of Maribor, Slovenia Antonio Pouzada, University of Minho, Portugal R. Venkata Rao, Sardar Vallabhbhai National Inst. of Technology, India Rajiv Kumar Sharma, National Institute of Technology, India Katica Simunovic, J. J. Strossmayer University of Osijek, Croatia Daizhong Su, Nottingham Trent University, UK Soemon Takakuwa, Nagoya University, Japan Nikos Tsourveloudis, Technical University of Crete, Greece Tomo Udiljak, University of Zagreb, Croatia Ivica Veza, University of Split, Croatia Subsidizer: The journal is subsidized by Slovenian Research Agency Creative Commons Licence (CC): Content from published paper in the APEM journal may be used under the terms of the Creative Commons Attribution 4.0 International Licence (CC BY 4.0). Any fur‐ ther distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Statements and opinions expressed in the articles and communications are those of the individual contributors and not necessarily those of the editors or the publisher. No responsibility is accepted for the accuracy of information contained in the text, illustrations or advertise‐ ments. 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Advances in Production Engineering & Management is indexed and abstracted in the WEB OF SCIENCE (maintained by Clarivate Analytics): Science Citation Index Expanded, Journal Citation Reports – Science Edition, Current Contents – Engineering, Computing and Technology • Scopus (maintained by Elsevier) • Inspec • EBSCO: Academic Search Alumni Edition, Academic Search Complete, Academic Search Elite, Academic Search Premier, Engineering Source, Sales & Marketing Source, TOC Premier • ProQuest: CSA Engineering Research Database – Cambridge Scientific Abstracts, Materials Business File, Materials Research Database, Mechanical & Transportation Engineering Abstracts, ProQuest SciTech Collection • TEMA (DOMA) • The journal is listed in Ulrich’s Periodicals Directory and Cabell's Directory Chair of Production Engineering (CPE) Advances in Production Engineering & Management Volume 18 | Number 2 | June 2023 | pp 133–266 Contents Scope and topics Real-time scheduling for dynamic workshops with random new job insertions by using deep reinforcement learning 136 137 Genetic algorithm-based approach for makespan minimization in a flow shop with queue time limits and skip-ping jobs 152 Hybrid forecasting modelling of cost and time entities for planning and optimizing projects in the die-cast aluminium industry 163 When core sorting and quality grading is beneficial to remanufacturers: Insights from analytical models 175 Engineering-to-order manufacturing: A criticality analysis of key challenges and solutions based on literature review 187 Ranking dominant losses in small and medium-sized enterprises (SMEs) in the context of the lean concept application 199 An improved discrete particle swarm optimization approach for a multi-objective optimization model of an urban logistics distribution network considering traffic congestion 211 Blockchain-based tripartite evolutionary game study of manufacturing capacity sharing 225 Enhancing automated defect detection through sequential clustering and classification: An industrial case study using the Sine-Cosine Algorithm, Possibilistic Fuzzy c-means, and Artificial Neural Network 237 Simulation and Genetic Algorithm-based approach for multi-objective optimization of production planning: A case study in industry 250 Calendar of events Notes for contributors 263 265 Sun, Z.Y.; Han, W.M.; Gao, L.L. Han, J.H.; Lee, J.Y. Muñoz-Ibañez, C.; Chairez, I.; Jimenez-Martinez, M.; Molina, A.; Alfaro-Ponce, M. Cao, X.H.; Shi, X.L.; Lan, H.J.; Huang, D. Fortes, C.S.; Tenera, A.B.; Cunha, P.F.; Teixeira, J.P. Kondić, V.; Maglić, L.; Runje, L.; Marić, D. Li, K.; Li, D.; Ma, H.Q. Wang, T.Y.; Zhang, H. Nguyen, T.P.Q.; Yang, C.L.; Le, M.D.; Nguyen, T.T.; Luu, M.T. Bojic, S.; Maslaric, M.; Mircetic, D.; Nikolicic, S.; Todorovic, V. Journal homepage: apem-journal.org ISSN 1854-6250 (print) ISSN 1855-6531 (on-line) Published by CPE, University of Maribor. Scope and topics Advances in Production Engineering & Management (APEM journal) is an interdisciplinary refereed international academic journal published quarterly by the Chair of Production Engineering at the University of Maribor. The main goal of the APEM journal is to present original, high quality, theoretical and application-oriented research developments in all areas of production engineering and production management to a broad audience of academics and practitioners. In order to bridge the gap between theory and practice, applications based on advanced theory and case studies are particularly welcome. For theoretical papers, their originality and research contributions are the main factors in the evaluation process. General approaches, formalisms, algorithms or techniques should be illustrated with significant applications that demonstrate their applicability to real-world problems. Although the APEM journal main goal is to publish original research papers, review articles and professional papers are occasionally published. Fields of interest include, but are not limited to: Additive Manufacturing Processes Advanced Production Technologies Artificial Intelligence in Production Assembly Systems Automation Big Data in Production Block Chain in Manufacturing Computer-Integrated Manufacturing Cutting and Forming Processes Decision Support Systems Deep Learning in Manufacturing Discrete Systems and Methodology e-Manufacturing Evolutionary Computation in Production Fuzzy Systems Human Factor Engineering, Ergonomics Industrial Engineering Industrial Processes Industrial Robotics Intelligent Manufacturing Systems Joining Processes Knowledge Management Logistics in Production 136 Machine Learning in Production Machine-to-Machine Economy Machine Tools Machining Systems Manufacturing Systems Materials Science, Multidisciplinary Mechanical Engineering Mechatronics Metrology in Production Modelling and Simulation Numerical Techniques Operations Research Operations Planning, Scheduling and Control Optimisation Techniques Project Management Quality Management Risk and Uncertainty Self-Organizing Systems Smart Manufacturing Statistical Methods Supply Chain Management Virtual Reality in Production Advances in Production Engineering & Management ISSN 1854-6250 Volume 18 | Number 2 | June 2023 | pp 137–151 Journal home: apem-journal.org https://doi.org/10.14743/apem2023.2.462 Original scientific paper Real-time scheduling for dynamic workshops with random new job insertions by using deep reinforcement learning Sun, Z.Y.a,b, Han, W.M.a,*, Gao, L.L.a aSchool bSchool of Economics and Management, Jiangsu University of Science and Technology, Zhenjiang, Jiangsu, P.R. China of Software, Pingdingshan University, Pingdingshan, Henan, P.R. China ABSTRACT ARTICLE INFO Dynamic real-time workshop scheduling on job arrival is critical for effective production. This study proposed a dynamic shop scheduling method integrating deep reinforcement learning and convolutional neural network (CNN). In this method, the spatial pyramid pooling layer was added to the CNN to achieve effective dynamic scheduling. A five-channel, two-dimensional matrix that expressed the state characteristics of the production system was used to capture the state of the real-time production of the workshop. Adaptive scheduling was achieved by using a reward function that corresponds to the minimum total tardiness, and the common production dispatching rules were used as the action space. The experimental results revealed that the proposed algorithm achieved superior optimization capabilities with lower time cost than that of the genetic algorithm and could adaptively select appropriate dispatching rules based on the state features of the production system. Keywords: Real-time scheduling; Machine learning; Deep reinforcement learning (DRL); Spatial pyramid pooling layer; Artificial neural networks (ANN); Convolutional neural networks (CNN) *Corresponding author: wlmh63@163.com (Han, W.M.) Article history: Received 7 December 2022 Revised 6 June 2023 Accepted 25 June 2023 Content from this work may be used under the terms of the Creative Commons Attribution 4.0 International Licence (CC BY 4.0). Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. 1. Introduction With the rapid development of information technology, China is gradually entering intelligent industry 4.0 [1]. With changing market demand, multiple-product small-batch order-type production has become prevalent in the manufacturing industry. Therefore, achieving sustainable and efficient workshop production under complex production conditions, responding quickly to market changes, and satisfying the diverse needs of customers are critical. Although job shop scheduling problems (JSSP) [2, 3] primarily address static scheduling issues, many real-time disruption factors, such as equipment failure, dynamic order arrival, emergency order insertion, in the production process are ignored [4, 5]. The insertion of a new order can drastically change the number and mode of processing tasks. Error accumulation renders existing production scheduling schemes ineffective, which results in the failure of the production planning system [6]. Therefore, dynamic real-time production scheduling on the insertion of new 137 Sun, Han, Gao jobs is crucial for timely response to disruption events and ensuring production requirements are satisfied. Current dynamic scheduling methods under order disturbance include heuristic algorithms [7-11] and dispatching rules [12]. Wang et al. [13] proposed an improved particle swarm optimization (PSO) algorithm to solve the dynamic job shop scheduling problem. Caldeira et al. [14] solved the flexible job shop scheduling problem on the arrival of a new job by using the improved backtracking search optimization algorithm that minimized makespan, energy consumption, and system stability. Ghaleb et al. [15] proposed three heuristic algorithms to address the real-time scheduling problem when new jobs are added and equipment fails. Tang et al. [16] considered minimum energy consumption and makespan as optimization objectives and proposed an improved PSO algorithm to solve the dynamic scheduling problem of flexible flow shops under new job arrival and equipment failure. In most heuristic algorithms, the dynamic scheduling problem is converted into a multi-stage static problem. Short-sightedness appears as the disturbance scale increases. The dispatching rules method can immediately respond to dynamic disturbance events and exhibits short computing time and high solution efficiency. Hundreds of dispatching rules have been proposed for shop scheduling [17, 18]. Zhang et al. [19] proposed a job shop dispatching rule selection system based on semantics to achieve adaptive selection of dispatching rules by scheduling objectives. To reduce the time of job completion and complexity of process design in the conventional dispatching rule design process, Zhang et al. [20] proposed an improved genetic programming algorithm that evolves effective dispatching rules automatically. Ferreira et al. [21] combined machine learning with the problem domain reasoning to generate effective dispatching rules. Although dispatching rules can respond to dynamic disturbance events in real time and exhibit short computing time and high solving efficiency, these methods are prone to local optimum and cannot be adjusted adaptively to respond to various production states. Reinforcement learning (RL) [22] has been widely used in production scheduling because of its excellent optimization ability and high computational speed. The continuous interaction between agent and environment maximizes cumulative rewards [23]. Dispatching rules can be dynamically and flexibly selected based on the real-time production status, which is suitable for the dynamic production scheduling problems. Wang et al. [24] used Q-learning to train a single machine agent and realized the dynamic selection of the three basic dispatching rules with the minimum average tardiness as the optimization objective. Chen et al. [25] proposed a ruledriven method for generating high-quality composite dispatching rules for the multi-objective dynamic job shop scheduling problem by using the Q-learning algorithm. Qu et al. [26] proposed a Q-learning algorithm that solves the dynamic job shop scheduling problem under random job arrival and equipment failure by combining the neighborhood search algorithm with the Qfactor. Although the conventional reinforcement learning algorithm has achieved excellent results in solving dynamic production scheduling problems, the algorithm is limited to situations in which the dimension and scale of the system state space are small and discrete. In the deep reinforcement learning (DRL) [27] algorithm, the perception ability of deep learning is effectively combined with the decision-making ability of reinforcement learning. Thus, DRL can effectively performs complex decision-making in the high-dimensional state space. Zhu et al. [28] proposed a proximal policy optimization (PPO) algorithm to solve the flexible flow shop scheduling problem by minimizing makespan. The PPO algorithm outperformed the conventional heuristic algorithm in terms of the quality of the solution. Luo et al. [29] proposed a deep Q-network (DQN) algorithm to solve the real-time workshop scheduling problem with dynamic job arrival to minimize total tardiness and achieved excellent results in the randomly generated data experiment. Yang et al. [30] used the A2C algorithm to train an intelligent model for the permutation flow job shop scheduling problem. This model outperformed the conventional heuristic algorithm in terms of the solving time and solution quality. Li et al. [31] proposed a hybrid DQN (HDQN) algorithm to solve the dynamic flexible job shop scheduling problem under transportation resource constraints. In most studies, the production system state is expressed through numerical features, which requires special manual design. 138 Advances in Production Engineering & Management 18(2) 2023 Real-time scheduling for dynamic workshops with random new job insertions by using deep reinforcement learning The CNN is used for the feature extraction of system state features expressed by multichannel images to effectively reduce manual design difficulty and exhibits excellent generality. However, because of the limitation of CNN feature extraction on the input size of training images, static scheduling is widely used. Liu et al. [32] designed three channels of two-dimensional data matrices as system state features for job shop scheduling problems and solved these problems using the AC algorithm to achieve excellent results on benchmark examples. Han et al. [33] combined the CNN and DRL to achieve dynamic job shop scheduling. Wang et al. [34] used the PPO algorithm to solve the job shop scheduling problem outperform GA. Subsequently, random fine-tuning was performed on some examples to test the generalization ability of the model. This method is simple and produces excellent results by describing the state features of the production system using multi-channel images. However, because of the structural characteristics of the CNN, applying the method to dynamic variable data of various image sizes is difficult. The static model can only process data of the same size, and it exhibits limited generalization. Multi-channel image system feature representation is yet to be used for dynamic scheduling. Therefore, in this study, an SPP layer [35] was added to the last layer of the CNN so that the neural network can handle any size of the input image information and achieve dynamic scheduling under the state image expression mode of the production system. The contributions of this paper are as follows: (1) This study is the first to use SPP with neural networks to solve the dynamic scheduling problem to allow the system model to handle the input state data of any scale; (2) The state feature expression mode of the production system was improved. The limitations of the conventional three-channel image design was overcome, and the system state feature was represented by five-channel data, considering both the global and local features of system processing state; (3) To assess the effect of the dispatching rules selected at each decision point on the overall scheduling objective, a novel reward function targeting total tardiness was developed; (4) Comprehensive parameter sensitivity experiments and algorithm result comparisons were performed. The effectiveness of the calculation speed and optimization ability of the proposed scheme was demonstrated. The method achieved excellent generalization results. 2. Overall scheduling framework A dueling double DQN (DDDQN) algorithm framework was proposed to achieve dynamic job shop real-time scheduling with constant random new order insertion. The neural network of the DDDQN algorithm consists of a Q-network and a target network. Each network exhibits the same structure and is composed of the convolution layer and a full connection layer. To address the problem of the dynamic image size change caused by the dynamic order arrival, an SPP layer was added between the CNN convolution layer and full connection layer to ensure consistent output size of the CNN convolution layer. Thus, the scheduling problem was transformed into a multi-stage decision-making process by designing a state feature, action space, and reward function. The agent is trained through interaction with the environment, and the trained agent is applied to solve the online problem. The framework includes two parts, namely offline training and online application (Fig. 1). The scheduling environment comprises equipment and order in the production system, which is used for the interaction with the agent and providing the current production system status information. The agent outputs the most appropriate dispatching rule and selects the highest priority operation for processing. The production system then enters the next state. In the offline training phase, intermediate data generated in the learning process is stored in the replay memory, and a minibatch number of sample data are randomly sampled for training. The Q-network and the target network calculate the Q and target values of the system state, respectively. Q-network parameters are updated by the loss function calculated by the target value and Q value. The parameters of the target network are copied from the Q-network after a certain number of steps. The optimal action is selected according to the result of the Q value. Although offline scheduling requires considerable time to train an agent, when an agent learns good policy, it can be widely used in online actual data scheduling to obtain optimal Advances in Production Engineering & Management 18(2) 2023 139 Sun, Han, Gao scheduling results rapidly. The execution process only requires the Q-network to calculate the optimal Q value without updating various network parameters, calculating the reward value, or storing sample data and other operations. Scheduling Environment Queue Q1 Machine Queue Q2 m1 Machine m2 DDDQN Action Action Queue Q3 Machine m3 TD-error sta t e Update parameters 48 Offline Training 6 10 27 0 0 0 0 46 4322 0 5 1 0 20 13 0 0 0 0 0 0 0 49 0 22 0 0 0 0 0 0 0 33 0 0 49 0 Reward 0 0 16 yi Q（s，a;ω） Q-network Q-network 0 0 S P P Action 0 〈 s0 , a0 , r0 , s1〉 Replay memory D Adam 〈 s1, a1, r0 , s2 〉 Minibatch of transitions Target Target network network S P P ︙ 〈 st , at , rt , st +1〉 ︙ ︙ Start Actual production data Online Scheduli ng Current production state Decision point No All operations are finished Dynamic arrival of job Select action at， Obverse the next state Yes End 3. DRL for scheduling Fig. 1 Scheduling framework with the DDDQN 3.1 Problem formulation The JSPP with new order insertions can be described as follows: a processing system has N orders that are processed on M machines. Each order has 𝑛𝑛𝑗𝑗 operations. The objective of production scheduling is to generate an optimal scheduling scheme based on the scheduling objectives and satisfy all constraints. However, when the new order arrives, the operations that were not started in the original scheduling scheme should be combined with operations in the new order for rescheduling. When creating the new scheduling scheme, factors such as the starting processing times and the number of the remaining operations of each order, should be considered. The scheduling problem should satisfy the following assumptions: (1) each order has a sequence constraint on the operations, that is, the next operation can only be processed after the previous operation is completed; (2) each operation of each order can only be processed by one machine; (3) each machine can only perform one operation at the same time; (4) when the new order arrives, the ongoing operation cannot be interrupted; (5) the processing time of each operation on the corresponding machine is known. 3.2 DQN principle The DQN algorithm is used to solve the problem. The DQN is the most classical algorithms of DRL. Based on Q-learning, the deep neural network is used to represent value function 𝑓𝑓. The input of the neural network is the current state s, and the output is the state value func- 140 Advances in Production Engineering & Management 18(2) 2023 Real-time scheduling for dynamic workshops with random new job insertions by using deep reinforcement learning tion Q(s, a). In the DQN, empirical data are used to train the neural network, which is prone to instability and convergence difficulties. To solve these problems, replay memory and target network are used in the DQN. Experience replay stores the intermediate data in a fixed-size storage experience pool in the form of 〈𝑠𝑠𝑡𝑡 , 𝑎𝑎𝑡𝑡 , 𝑟𝑟𝑡𝑡 , 𝑠𝑠𝑡𝑡+1 〉, which is generated in the learning process of Q-learning. The system randomly samples a certain number of small-batch samples from the replay memory for training. This random sampling not only breaks the correlation of training samples but also ensures an independent and homogeneous distribution of training samples. Target network reconstructs a network with the same structure as the original network. The original network is the Q-network, and the generated network is the target network. During training, only Q-network parameters are updated, and the parameters of the target network remain unchanged temporarily. After reaching a certain number of update steps C, the parameters of the Q-network are copied to the target network so that the value of the target network does not change in a certain update step to ensure system stability. The target value is calculated as follows: 𝑌𝑌𝐷𝐷𝐷𝐷𝐷𝐷 ≡ 𝑟𝑟𝑡𝑡+1 + 𝛾𝛾 𝑚𝑚𝑚𝑚𝑚𝑚 𝑄𝑄� (𝑠𝑠𝑡𝑡+1 , 𝑎𝑎; 𝜔𝜔𝑡𝑡− ) (1) 𝑎𝑎 The neural network calculates TD errors through the target network and Q-network to update parameters. In the standard DQN algorithm, the action with the largest Q value is selected, which results in an overestimation of the Q value. Therefore, the Double DQN (DDQN) algorithm is used to separate the action selected from the calculation of the Q value and two value functions are used. The target value of the DDQN is calculated as follows: 𝑌𝑌𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 ≡ 𝑟𝑟𝑡𝑡+1 + 𝛾𝛾𝑄𝑄� (𝑠𝑠𝑡𝑡+1 , 𝑎𝑎𝑎𝑎𝑎𝑎 max 𝑄𝑄(𝑠𝑠𝑡𝑡+1 , 𝑎𝑎; 𝜔𝜔𝑡𝑡 ), 𝜔𝜔𝑡𝑡− ) (2) 𝑎𝑎 By using various value functions to decouple the target action and Q value, the DDQN algorithm mitigates the overestimation of Q values and achieves excellent stability. In the DQN, the network outputs the Q value of action, but in practice, the Q value is associated with the action and state. Therefore, Dueling DQN improves the network structure of the DQN by adding state value function V(s, ω, a) related to the system state and the advantage function A(s, a, ω, β) related to the action before the network output layer and synthesizing these two functions to generate the action function in the final output layer. Thus, we have the following expression: 1 𝑄𝑄(𝑠𝑠, 𝑎𝑎, 𝜔𝜔, 𝛼𝛼, 𝛽𝛽) = 𝑉𝑉(𝑠𝑠; 𝜔𝜔, 𝛼𝛼) + (𝐴𝐴(𝑠𝑠, 𝑎𝑎, 𝜔𝜔, 𝛽𝛽) − � 𝐴𝐴(𝑠𝑠, 𝑎𝑎𝑡𝑡+1 , 𝜔𝜔, 𝛽𝛽)) (3) |𝒜𝒜| 𝑎𝑎𝑡𝑡+1 where ω is a parameter of the common part, α is a parameter of the value function, and β is a parameter of the advantage function. In this study, dueling DQN and double DQN are used to solve the dynamic production scheduling problem. 3.3 CNN with the SPP layer The insertion of new orders dynamically changes the size of multi-channel images expressed by state features. However, the full connection layer in the conventional CNN requires an input of fixed size. An SPP layer was added between the last convolutional layer and the first full connection layer in the CNN to divide the feature graph obtained after convolution into a fixed number of grids of various sizes. The grid is then pooled with mean values. Thus, the feature graph convolved with any size can be changed into the output of a fixed size so that the graph has the same dimension of feature vector with the following full connection layer. Thus, image convolution with any image size input can achieved as follows (Fig. 2). Advances in Production Engineering & Management 18(2) 2023 141 Sun, Han, Gao Convolution Layers 4x80 2x80 spatial pyramid pooling layer 3.4 Scheduling problem transformation Fixed length output full connection layer 1x80 Fig. 2 SPP principle The problem transformation between scheduling and algorithm design is critical for applying DRL to JSPP and involves three aspects, namely state feature, action space, and reward function design. State features expression To improve the state changes of the production system, the following rules should be followed for describing the state features: • State features should be able to reflect the features and changes of the production system, and both global and local state features should be considered. • State features at each moment are represented by a universal feature set. • State features should be represented numerically for easy calculation and standardization for uniform scaling of various features. This study optimized and upgraded the production system state features based on literature [33]. The limitation of three channels in the conventional system state feature expression was overcome. Five channels were designed for characterization. The first channel is represented by the two-dimensional matrix of the order to be processed. The rows of the matrix represent the order, and the columns represent the operation. The initial value is the processing time of the corresponding operation in the order. The value of the corresponding position becomes zero on the completion of an operation. The second channel is represented by the two-dimensional matrix of the completed operations of the order. The initial value is zero. The value of the corresponding position is the processing time on completion of an operation. The third channel is the remaining processing time of the processing operation. The fourth channel is the processing time of each operation in the queue to be processed. The fifth channel is the waiting time of each operation in the waiting queue. The first and second channels express the global state feature, whereas the third, fourth, and fifth channels express the local state feature. All channel data are linearly normalized to the maximum value. Definition of ACTIONS The action selection involves selecting the most suitable operation waiting for processing, and the production scheduling rule can select an appropriate process at each scheduling decision point. In this article, 16 commonly used production dispatching rules are selected as the action space in DRL. The details are as follows: the select the job with the shortest processing time (SPT), select the job with the longest processing time (LPT), select the job with the longest remaining processing time (LWKR), select the job with the shortest remaining processing time (MWKR), select the job with the shortest processing time of subsequent operation (SSO), select 142 Advances in Production Engineering & Management 18(2) 2023 Real-time scheduling for dynamic workshops with random new job insertions by using deep reinforcement learning the job with the longest processing time of subsequent operation (LSO), select the job with the shortest remaining processing time in addition to the current operation (SRM), select the job with the longest remaining processing time in addition to the current operation (LRM), select the job that arrives first (FIFO), select the job with the earliest due date, select the job with the minimum sum of processing time of the current and subsequent operation (SPT+SSO), select the job with the maximum sum of processing time of the current and subsequent operation (LPT+LSO), select the job with the minimum ratio of current processing time to the total working time (SPT/TWK), select the job with the maximum ratio of current processing time to the total working time (LPT/TWK), select the job with the minimum product of the current processing time and total working time (SPT *TWK), select the job with the maximum product of current processing time and total working time (LPT * TWK). The diversity of dispatching rules is increased so that the agent can adaptively select dispatching rules. Reward FUNCTION The reward function is key to DRL and directly affects the direction of learning and is closely related to optimization. The reward function should be designed as follows: (1) the reward function should accurately express the immediate reward of the current action. (2) Cumulative reward should be closely related to the scheduling objective. (3) Reward function should be universal and can be used for problems of various scales. Because the overall scheduling objective is to minimize total tardiness, the following reward function should be designed: 𝑟𝑟𝑘𝑘 = 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑘𝑘−1 − 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑘𝑘, 𝑛𝑛 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑘𝑘 = � 𝛿𝛿𝑗𝑗 (𝜏𝜏) 𝑗𝑗=1 (4) where 𝛿𝛿𝑗𝑗 (𝜏𝜏) represents the tardiness of job J at the current system state, 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑘𝑘 is the total tardiness of all the jobs in the current system. For the job without tardiness, the tardiness is zero. Here, 𝑟𝑟𝑘𝑘 represents the reward received at the decision-making time 𝑡𝑡𝑘𝑘−1 after executing the action, then the system arrives at decision-making time 𝑡𝑡𝑘𝑘 . The derivation process of cumulative reward function R is as follows: 𝐾𝐾 𝐾𝐾 𝑘𝑘=1 𝑘𝑘=1 R = � 𝑟𝑟𝑘𝑘 = � 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑘𝑘−1 − 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑘𝑘 = 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇0 − 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇1 + 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇1 −𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇2 + ⋯ + 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝐾𝐾−1 − 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝐾𝐾 (5) = 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇0 −𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝐾𝐾 = −𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝐾𝐾 The derivation process of the cumulative reward formula reveals that it is inversely proportional to the total tardiness, that is, the smaller the total tardiness is, the larger the cumulative reward value is, which is consistent with the scheduling objective. Training based on DRL The scheduling process is a semi-Markov decision process. When any machine completes an operation or a new order arrives as a decision time point, the agent adaptively selects appropriate dispatching rules, and the highest priority operation is selected for processing. Subsequently, the machine enters the next state after receiving rewards. The cycle continues until all operations are finished, that is, a scheduling scheme is obtained. The process is as follows: Algorithm 1 DDDQN-based training method 1: Initialize replay memory D, minibatch k，learning rate α，target network parameters update every C steps. 2: Initialize Q-network with random weights ω 3: Initialize target network � 𝑄𝑄 with weights 𝜔𝜔− = 𝜔𝜔 4: For episode = 1 : M do 5: Reset the system scheduling status to 𝑠𝑠0 and clear schedule results. 6: while True：(t is the decision time point at which an operation is completed or a new order arrives, the Boolean variable done terminates the loop when all operations are complete) Advances in Production Engineering & Management 18(2) 2023 143 Sun, Han, Gao 7: 8: 9: 10: 11: 12: 13: 14: 15: Select action 𝑎𝑎𝑡𝑡 based on ε-greedy strategy Execute action 𝑎𝑎𝑡𝑡 , calculate the immediate reward 𝑟𝑟𝑡𝑡 ，observe the next state 𝑠𝑠𝑡𝑡+1 Store transition〈𝑠𝑠𝑡𝑡 , 𝑎𝑎𝑡𝑡, 𝑟𝑟𝑡𝑡 , 𝑠𝑠𝑡𝑡+1 , 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑〉 in D Random sampling minibatch of transitions 〈𝑠𝑠𝑖𝑖 , 𝑎𝑎𝑖𝑖, 𝑖𝑖, 𝑠𝑠𝑖𝑖+1 , 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑〉 from D 𝑟𝑟𝑖𝑖 𝑦𝑦𝑖𝑖 = � 𝑟𝑟𝑖𝑖 + 𝛾𝛾𝑄𝑄� (𝑠𝑠𝑖𝑖+1 , 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 (𝑠𝑠𝑖𝑖+1 , 𝑎𝑎; 𝜔𝜔), 𝜔𝜔− ) Compute TD-error(𝑦𝑦𝑖𝑖 − 𝑄𝑄(𝑠𝑠𝑖𝑖 , 𝑎𝑎; 𝜔𝜔))2 to update the parameters of Q-network Every C steps update the parameters of the target network 𝑄𝑄� : 𝜔𝜔− = 𝜔𝜔 end while end for if done = true otherwise The training process is divided into inner and outer loops, the outer loop represents the times of training. After M loops of cyclic training, the agent gradually reaches the ability to adaptively select the optimal dispatching rules at various decision moments. The inner loop represents a complete scheduling scheme generation process, starting from the first operation until all operations are finished, which is an episode, lines 5-14 describe the execution of the inner loop that starts from the initial state of 𝑠𝑠0 , at the decision moment t the agent select and execute the action 𝑎𝑎𝑡𝑡 based on the ε-greedy strategy, the suitable operation is scheduled. The reward 𝑟𝑟𝑘𝑘 and the next state 𝑠𝑠𝑡𝑡+1 are observed, the transition 〈𝑠𝑠𝑡𝑡 , 𝑎𝑎𝑡𝑡, 𝑟𝑟𝑡𝑡 , 𝑠𝑠𝑡𝑡+1 , 𝑑𝑑𝑜𝑜𝑜𝑜𝑜𝑜〉 is stored. Minibatch transitions are randomly sampled from the replay memory for system training. A loss was calculated according to lines 11 and 12, and gradient descent was used to update the parameters of Qnetwork. The parameters of the target network were updated by the parameters of Q-network according to the contents of 13 lines after every C step. 4. Numerical simulation In numerical simulation, multiple groups of data were used to train the DDDQN. The optimal training model is then saved, and the model is tested in the new test data of multiple groups of various production scenarios. The data generation method proposed in a previous study [29] is randomly generated, and the parameters are presented in Table 1. According to the arrival time and number of new orders, the number of machines, and the due date of orders, 81 groups of data of various production scenarios were randomly generated for the test. Assuming that 30 orders exist at the beginning of each production scenario, the arrival time of new jobs follow Poisson distribution. Therefore, the time interval between two consecutive new jobs is subjected to exponential distribution. The due date tightness (DDT) repre𝑛𝑛 sents the urgency of orders. The due date of order i can be calculated as 𝐷𝐷𝑖𝑖 = 𝐴𝐴𝑖𝑖 + (∑𝑗𝑗 𝑖𝑖 𝑡𝑡𝑖𝑖𝑖𝑖 ) ∙ 𝐷𝐷𝐷𝐷𝐷𝐷. Here, 𝐴𝐴𝑖𝑖 is the arrival time of the order, 𝑛𝑛𝑖𝑖 is the number of operations in the order, 𝑡𝑡𝑖𝑖𝑖𝑖 is the processing time of the j operation of the order. The smaller the DDT is, the more urgent the order due date is. Table 1 Parameter settings of various production scenarios Parameter Value Number of machines 5,10,15 Number of initial jobs 30 Number of new job insertions 10,30,50 Processing time of each operation Unif[0,50] Due date tightness 1.0,1.5,2.0 Average value of exponential distribution 25,50,100 between two successive new job arrivals 4.1 Network structure and system parameter setting The CNN structure of the DDDQN consists of four convolution layers and two full connection layers. To solve the problem of variable image size, a SPP layer was added between the convolution layer and the full connection layer. From the first layer to the fourth layer for the convolution layers, the size of the convolution kernel was 6 × 6, 4 × 4, 3 × 3, 2 × 2, the step size was 2, 2, 2, 1, and the number of output channels was 20, 40, 60, and 80. Because each element in the state feature image represents an operation, the pooling layer in the CNN results in incomplete 144 Advances in Production Engineering & Management 18(2) 2023 Real-time scheduling for dynamic workshops with random new job insertions by using deep reinforcement learning scheduling information. Therefore, the pooling layer was not used. The full connection layer consists of two branches of the full connection layer with 512 units. The two branches connect the state value and the advantage value. Finally, the state and advantage values were combined to obtain the final result output. The RELU activation function was used for every layer. The Adam optimizer was used to update the parameters. Parameter setting considerably affected DDDQN performance. Five groups of data were randomly generated for the parameter sensitivity experiment under 10 new order insertions, 10 machines, 50 average time interval between new order arrival, and DDT tardiness coefficient of 1.5. The effects of the training batch, learning rate, replay memory buffer size, and target network parameter updating frequency on algorithm performance were verified. Fig. 3 displays the training effect under various parameter settings, the total number of training was set to 3000 episodes. (a) (b) (c) (d) Fig. 3 Verification results of each hyperparameter: (a) Minibatch size; (b) Learning rate; (c) Replay memory buffer size (d) Target network updating frequency Fig. 3(a) verifies the influence of the training batch on the algorithm. The figure reveals that all parameters exhibits excellent stability. The performance with a batch size of 32 decreased slightly. Fig. 3(b) displays the influence of various learning rates on the algorithm. The higher the learning rate is, the more the training effect is unstable. When the learning rate is 0.001, the algorithm does not even converge. Fig. 3(c) displays the influence of various replay memory buffer sizes on the algorithm. As displayed in the figure, the larger the replay memory is, the better the convergence of the algorithm is, and the replay memory with a capacity of 100000 exhibits superior stability. Fig. 3(d) displays the influence of the target network parameter updating frequency on algorithm performance, and the parameters exhibit an effect under various updating frequencies. According to the verification of various parameters, the final neural network parameter settings are presented in Table 2. Advances in Production Engineering & Management 18(2) 2023 145 Sun, Han, Gao Table 2 Setting of neural network parameters Parameters Values Number of episodes 3000 Explore times steps 3000 · total operation number · 0.3 Epsilon 1 − (1 − 𝜀𝜀𝑚𝑚𝑚𝑚𝑚𝑚 · min (1, current iter /totalsteps)) Replay memory buffer size 100000 Learning rate 0.000001 Minibatch 256 Target network updating frequency 200 Discount factor 1.0 The ε-greedy action selection strategy was implemented according to the method mentioned in a previous study [33]. To ensure the maximum cumulative rewards learned by the agent, the discount factor is selected as 1.0, that is, the cumulative rewards are not discounted. 4.2 Comparison of various status features expression To verify the validity of the state feature expression of the proposed five-channel images, the influence of three state feature expression modes of three-channel images, four-channel images, and five-channel images on the algorithm were compared. In three-channel images, the production system state feature expression method in literature is adopted [33]. Five-channel images were the proposed production system state feature expression method, and four-channel images were separated from five-channel images to remove the waiting time channel of each operation in the waiting queue. Five groups of data were randomly generated for testing under the following production configurations: the number of machines was 10, the average time interval between two consecutive new order arrival was 50, DDT was 1.5, and the number of new order insertion were 10, 30, and 50. The results are displayed in Fig. 4. The figure reveals that with the increase in the new order insertion scale, the expression methods of three-channel and four-channel both fluctuated considerably, whereas the proposed expression method exhibited excellent stability. (a) (b) (c) Fig. 4 Verification results of various state features: (a) 10 new job insertions; (b) 30 new job insertions; (c) 50 new order insertions 146 Advances in Production Engineering & Management 18(2) 2023 Real-time scheduling for dynamic workshops with random new job insertions by using deep reinforcement learning 4.3 Training process of the DDDQN The DDDQN was used to train a model with certain generalization ability. The model was tested with various test data. The model was trained according to the number of machine and the time interval between the dynamic arrival of orders. Each model was trained for 3000 episodes. In the training process, 12 groups of randomly generated data were used according to the number of new orders of 10, 30, and 50 and the DDT of 1.0 and 2.0. Fig. 5 displays the model training process with five machines and 100 times intervals. (a) (b) Fig. 5 Model training process in five machines and 100 times intervals: (a) Average total tardiness during training; (b) Average reward during training Figs. 5(a) and 5(b) display the change process of the total tardiness and reward during the training process. The reward value gradually increases and becomes stable with the increase in training episodes, whereas the total tardiness decreases. After 1000 episodes of training, the model becomes stable, which indicates that the DDDQN has learned to adaptively select the appropriate dispatching rules at the decision time. The curve trend of the average reward value is similar to that of the average total tardiness, which indicates that the designed reward function exhibited a high correlation with the optimization objective with the minimum total tardiness. A small fluctuation was observed after the model convergence. The fluctuation was related to the exploration mechanism of DRL. 4.4 Comparison with conventional dispatching rules To verify whether the training model can select appropriate dispatching rules at various decision moments, 16 dispatching rules were compared on the test data set. Test data were configured for 81 production scenarios according to all parameter configurations in Table 1, and 30 groups of test data were randomly generated for each production scenario. Tables 3-5 displays the comparison results under various number of machines. The data in the table are the average values of test data. The results of the optimal values are displayed in bold for easy identification. The test results indicate that the algorithm model of the DDDQN is superior to the single scheduling rule in most cases, which reveals satisfactory solution solving ability and generalization ability of various problems. Finding a scheduling rule that can perform well in all production scenarios is difficult. Advances in Production Engineering & Management 18(2) 2023 147 Sun, Han, Gao Table 3 Test results compared with dispatching rules under five machines Table 4 Test results compared with dispatching rules under ten machines Table 5 Test results compared with dispatching rules under fifteen machines 148 Advances in Production Engineering & Management 18(2) 2023 Real-time scheduling for dynamic workshops with random new job insertions by using deep reinforcement learning 4.5 Comparison with the GA algorithm To prove the computational speed and optimization ability of the model, the DDDQN was compared with the GA. In the GA, an active decoding approach and an elite retention strategy are used. For the medium- and large-scale problems, the GA first generates an initial scheduling scheme based on the initial order data. When a new order arrives, the GA reschedules to generate a new scheduling scheme. The start processing time of all orders differs considerably, and the number of the remaining operations of each order also differs. The parameters of the GA are set as follows: population size is 50, the crossover rate is 0.9, the mutation rate is 0.1, and the iteration number is 300. Some representative data were selected for verification. Each group of test data consists of 30 randomly generated data. The results are presented in Table 6. The data in the table are the average values of test data. The scheduling results and calculation time of the model are superior to the GA in all cases. The average calculation time of the DDDQN model to generate the scheduling scheme for test data at each decision moment was 0.05 s, which was almost instantaneous. Thus, the model can be used for real-time scheduling. m 5 10 15 Eavg 25 100 25 100 25 100 DDT 1.0 2.0 1.0 2.0 1.0 2.0 1.0 2.0 1.0 2.0 1.0 2.0 5. Conclusion nadd 10 50 10 50 10 50 10 50 10 50 10 50 10 50 10 50 10 50 10 50 10 50 10 50 Table 6 Comparison results of DDDQN and GA Total tardiness CPU times (s) DDDQN GA DDDQN GA 17357 26263 0.04 46.29 45832 64650 0.02 57.38 12571 21354 0.04 46.26 36174 53352 0.02 56.34 15173 22436 0.04 35.42 14768 23195 0.01 8.11 10417 18107 0.04 35.82 9874 17173 0.01 8.10 20981 30552 0.10 92.18 54353 107956 0.05 163.86 11218 20531 0.10 91.82 35443 88138 0.05 164.19 17211 28167 0.10 82.69 19839 36521 0.04 28.24 7743 18470 0.10 82.35 8800 22720 0.04 28.30 22058 32677 0.18 139.78 62010 130030 0.08 266.73 7334 17217 0.15 138.98 32904 98901 0.08 265.63 18289 31864 0.18 131.80 22665 48197 0.08 57.58 4589 17439 0.16 131.53 5889 26732 0.07 58.57 A DRL algorithm, namely the DDDQN, was proposed to solve real-time dynamic job shop scheduling with new order insertions. SPP technology was applied to the neural network structure. A five-channel production system state feature expression method that considered both global and local feature information was considered. As the action space, 16 commonly used dispatching rules were used, and the corresponding reward function was designed to minimize total tardiness. Finally, considerable data from various production scenarios were generated at random to train and test the system model. Compared with conventional dispatching rules and heuristic algorithms, the results revealed that the algorithm outperformed the single scheduling rule method in most cases, which indicated that the algorithm can select dispatching rules adaptively in various production states. ComAdvances in Production Engineering & Management 18(2) 2023 149 Sun, Han, Gao pared with the GA, the computational speed and optimization ability of the trained models were validated, and real-time optimization and online decision were performed in dynamic event disturbance. 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Advances in Production Engineering & Management 18(2) 2023 151 Advances in Production Engineering & Management ISSN 1854-6250 Volume 18 | Number 2 | June 2023 | pp 152–162 Journal home: apem-journal.org https://doi.org/10.14743/apem2023.2.463 Original scientific paper Genetic algorithm-based approach for makespan minimization in a flow shop with queue time limits and skipping jobs Han, J.H.a, Lee, J.Y.b,* aDepartment bDivision of Industrial Engineering, Pusan National University, Republic of Korea of Business Administration & Accounting, Kangwon National University, Republic of Korea ABSTRACT ARTICLE INFO This study investigates a flow shop scheduling problem with queue time limits and skipping jobs, which are common scheduling requirements for semiconductor and printed circuit board manufacturing systems. These manufacturing systems involve the most complex processes, which are strictly controlled and constrained to manufacture high-quality products and satisfy dynamic customer orders. Further, queue times between consecutive stages are limited. Given that the queue times are limited, jobs must begin the next step within the maximum queue time after the jobs in the previous step are completed. In the considered flow shop, several jobs can skip the first step, referred to as skipping jobs. Skipping jobs exist because of multiple types of products processed in the same flow shop. For the considered flow shop, this paper proposes a mathematical programming formulation and a genetic algorithm to minimize the makespan. The GA demonstrated its strengths through comprehensive computational experiments, demonstrating its effectiveness and efficiency. As the problem size increased, the GA's performance improved noticeably, while maintaining acceptable computation times for real-world fab facilities. We also validated its performance in various scenarios involving queue time limits and skipping jobs, to further emphasize its capabilities. Keywords: Scheduling; Flow shop; Makespan; Queue time limits; Skipping jobs; Optimization; Modeling; Genetic algorithm *Corresponding author: Jy.lee@kangwon.ac.kr (Lee, J.Y.) Article history: Received 9 October 2022 Revised 4 July 2023 Accepted 6 July 2023 Content from this work may be used under the terms of the Creative Commons Attribution 4.0 International Licence (CC BY 4.0). Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. 1. Introduction This study focused on the scheduling problem in a three-machine flow shop with queue time limits and skipping jobs. Such scheduling problems are found in semiconductor and printed circuit board (PCB) manufacturing systems, which play key roles in the electronics industry as the major components of most electronic products. Such manufacturing systems generally fabricate multiple types of products. Therefore, meeting customer demand in terms of quality and improving the throughput rate are the most important objectives. Queue time limits and skipping jobs are related to product quality and multiple types of products, respectively. In this paper, we investigate the scheduling problem in the considered flow shop with the objective of minimizing makespan subject to queue time limits and skipping jobs. Semiconductor wafer fabrication is generally a time-critical production environment owing to many highly reactive chemical processes. Thus, the queue times of work-in-process wafers in a fab 152 Genetic algorithm-based approach for makespan minimization in a flow shop with queue time limits and skipping jobs are strictly managed and constrained to satisfy wafer quality requirements [1]. These constraints are common in wafer lot scheduling, especially at Korean semiconductor manufacturing companies that enforce queue time limits across multiple process steps to enhance overall quality. Generally, semiconductor manufacturing involves limited queue times for two main reasons. Firstly, it is crucial to maintain the cleanliness of the wafer surfaces during the waiting period before proceeding to the next stage. Prolonged exposure of wafer surfaces to air heightens the risk of contamination by impurities, leading to significant quality issues. Secondly, it is essential to preserve the chemical efficacy treated in the previous step while awaiting the subsequent stage. Therefore, chemically treated wafers must proceed to the next step within a specified timeframe known as the limited queue time. Failure to initiate the subsequent processing step within this queue time limit may require reworking or discarding the wafers, especially in cases of severe contamination [2]. Consequently, this causes lower productivity and business losses. As semiconductor processes become more complex, the number of processes with queue time limits increases, increasing their importance to semiconductor manufacturing. In addition to semiconductor manufacturing systems, queue time limits can also be found in many manufacturing systems for batteries, food, steel, and crude oil [33]. In a semiconductor manufacturing fab, several types of wafer products are produced simultaneously. Their main process flows are similar, but some specific steps are different depending on the product types. Thus, flow shops modeled in a semiconductor fab are designed to process multiple product types, some skipping unnecessary steps. For example, if the first step is cleaning or metrology, some lots that do not need this step skip it and go straight to the second step. For such jobs, the queue time limits are set between the second and third steps. With an increase in the number of semiconductor products, this feature has become more common. Flow shops with skipping jobs are also found in PCB manufacturing systems for multi-variety low-volume production. Additionally, skipping jobs are found in various manufacturing systems, including pharmaceuticals, molten iron, stainless steel, and flour [1, 27, 30]. A typical flow shop consists of a series of machines or workstations arranged in sequential order and processes every job through the machines in that order. If the orders of all jobs are the same across all machines, this is referred to as a permutation schedule. In general, when job i completes on machine k but job j is still unfinished on machine (𝑘𝑘 + 1), job i can wait until job j completes. However, in a flow shop with queue time limits, job i cannot wait for a long time. In other words, after job i completes on machine k, its subsequent operation on machine (𝑘𝑘 + 1) must begin within a specified limited queue time. On the other hand, Most flow shop scheduling studies have assumed that every job processes at all stages, which is currently not valid because of the increased customization and diversification of products [3]. Using three-field notation in [4] to define theoretically the considered scheduling problem, it is F3|max-wait, skip|Cmax. The first field means a flow shop with three machines. The second field represents problem characteristics, i.e., queue time limits and skipping jobs, respectively. The last field shows the makespan as the objective function to be minimized. As mentioned in [5], a two-machine flow shop with queue time limits is NP-hard, so this scheduling problem is also NPhard. We provide a mixed-integer programming (MIP) formulation to describe the scheduling problem clearly. The MIP can be solved using a commercial optimization solver. However, obtaining optimal solutions using the solver may require a significant computation time or it cannot be achieved within an acceptable time, given that the problem is NP-hard. Therefore, we propose a genetic algorithm to rapidly obtain adequate and effective solutions. The genetic algorithm was evaluated under different cases of computational experiments, and it showed highly effective and efficient performance. The structure of this paper is as follows. Section 2 outlines the assumptions and notation used to describe the proposed algorithms and the mixed-integer programming formulation for the scheduling problem. Section 3 introduces heuristic algorithms, while their evaluation is discussed in Section 4. Lastly, Section 5 provides the study's conclusions and a summary of the findings. Advances in Production Engineering & Management 18(2) 2023 153 Han, Lee 2. Literature survey With reference to the available literature, most flow shop scheduling studies with queue time limits (QTL) have focused on two-machine problems. Yang et al. [5] proved that minimizing the makespan in a two-machine flow shop with QTL is NP-hard and proposed a branch-and-bound (B&B) algorithm. Additionally, advanced lower bounds and dominance properties were developed in [6, 7], while a constructive heuristic algorithm was proposed in [8]. Furthermore, An et al. [9] included not only QTL but also sequence-dependent setup times and developed heuristic algorithms and a B&B algorithm, while Lee [10] suggested a genetic algorithm to minimize the total tardiness for the same flow shop. In addition, Dhouib et al. [11] proposed a MIP and a simulated annealing algorithm to hierarchically minimize the number of tardy jobs and makespan for multi-machine flow shops with QTL, whereas Kim and Lee [12] introduced a three-machine flow shop with overlapping QTL. To minimize the total tardiness, Hamdi and Loukil [13] suggested a lower bound scheme based on Lagrangian relaxation and a heuristic algorithm, and Joo et al. [14] performed simulation experiments with a list scheduling rule. Furthermore, flow shop problems with QTL have been considered in several recent studies [15-23]. In the relevant literature, scheduling problems with skipping jobs have not been extensively investigated in comparison with other types of flow shops. However, a feature of skipping jobs has recently received significant research attention, given that multiple product types with different specifications are produced in the same manufacturing fabrication. Rajendran and Ziegler [24] examined the performance of dispatching rules and a heuristic algorithm to minimize total flow time. Saravanan et al. [25] and Dios et al. [26] proposed a simulated annealing algorithm and various heuristic algorithms, respectively, to minimize the makespan, whereas, Saravanan et al. [27] suggested a genetic algorithm to minimize the mean tardiness. On the other hand, Tseng et al. [28] proposed a heuristic to change a given permutation schedule to an improved nonpermutation schedule for minimizing the makespan, while Li et al. [29] proposed a multiobjective artificial bee colony algorithm with respect to flowtime, earliness, and tardiness in molten iron processing. Recently, in the context of Industry 4.0 production, Rossit et al. [30] introduced a flow shop with skipping jobs. In the literature, QTL and skipping jobs in flow shop scheduling have been separately studied, and their incorporation of them has received limited attention, with only a few studies exploring them together. Notably, for no-wait flow shop problems with skipping jobs, Glass et al. [31] proposed and analyzed several heuristic algorithms for minimizing the makespan in a two-machine, while Smutnicki et al. [32] devised a method to determine the cyclic schedule with minimal cycle time. For the general queue time limits, Ruiz et al. [33] investigated a comprehensive flow shop problem encompassing various features, such as queue time limits, skipping jobs, and the inclusion of machine eligibility, machine release dates, precedence constraints, and sequencedependent setup times, and they evaluated simple dispatching rules and heuristic algorithms. Additionally, Yu et al. [1] focused on a two-machine flow shop with QTL and skipping jobs and they analyzed mathematical properties, explored the reduction of the search space, and developed efficient approximation algorithms for minimizing queue time variations. Furthermore, Han and Lee [34] dealt with minimizing total tardiness in a flow shop scheduling problem that involved queue time limits and skipping jobs. In the existing literature, there is a notable gap in flow shop research regarding F3|max-wait, skip|Cmax. It is important to highlight that the study conducted by Han and Lee [34] focused on minimizing total tardiness in a flow shop scheduling problem specifically within the semiconductor foundry business. This industry places significant emphasis on satisfying customer delivery dates. However, in our present study, we shift our attention to the memory business, where factors such as equipment utilization and throughput take precedence. Thus, the objective of this study is to minimize the makespan. It is worth mentioning that this study represents the first attempt to tackle this particular problem, and the outcomes obtained can serve as a foundational basis for future research and advancements in this field. 154 Advances in Production Engineering & Management 18(2) 2023 Genetic algorithm-based approach for makespan minimization in a flow shop with queue time limits and skipping jobs 3. Problem description In this section, we describe the considered scheduling problem with assumptions, notation, and MIP formulation. The considered flow shop has three stages (k = 1, 2, 3), and each stage has a machine mk. All jobs (i = 1, 2,…, n) are given and are available to start at time zero, and all information for making schedules are provided in advance. That is, for job i, processing time PTik on machine k and queue time limit QTik between mk and m(k+1) are already known. Among the jobs, there are jobs starting at m1, called normal jobs, and skipping jobs starting at m2. For this scheduling problem, the objective is to minimize the makespan which is equal to the completion time of the last processed job on m3. To address this problem, we restrict a solution space only to permutation schedules. In particular, given a specific job sequence, all jobs follow that sequence and are processed on machines in the same order. Although an optimal schedule may not be included in the permutation schedules, many studies with QTL have assumed permutation schedules. This is because lots in semiconductor manufacturing fabs are typically processed in permutation schedules for traceability, manageability, and flexibility in material handling [7]. The followings are additional assumptions made in this study. Pre-emption is not permitted, meaning a job cannot be interrupted once it starts processing. Each job can be processed on only one machine at a time, and each machine can process only one job at a time. The queue time limits must be satisfied without failure. However, the number of jobs queueing between machines is not limited. Additionally, the following symbols in Table 1 are used to describe the MIP and the proposed algorithms. Symbol h [h] xih ST[h]k CT[h]k Table 1 The symbols used in the algorithms Meaning index of the position in a sequence, h = 1,…, n index of a job placed at the h-th position in a sequence = 1 if job i is placed at the h-th position in a sequence; otherwise, 0 the time at which the h-th job starts on mk the time at which the h-th job completes on mk We provide equations for the completion times of all jobs in a given sequence. For the first scheduled job (h = 1), Eqs. 1 to 3 calculate the completion times on the three machines, respectively. Obviously, the first job is processed immediately without waiting. 𝐶𝐶𝐶𝐶[1]1 = 𝑃𝑃𝑃𝑃[1]1 (1) 𝐶𝐶𝐶𝐶[1]3 = 𝐶𝐶𝐶𝐶[1]2 + 𝑃𝑃𝑃𝑃[1]3 (3) 𝐶𝐶𝐶𝐶[1]2 = 𝐶𝐶𝐶𝐶[1]1 + 𝑃𝑃𝑃𝑃[1]2 (2) From the second position onwards (h = 2,…, n), the calculations for the completion times on the first two machines (m1 and m2) differ between the normal jobs starting at m1 and skipping jobs starting at m2. Eqs. 4 and 5 computes the completion times of the normal jobs on m1 and m2, while Eqs. 6 and 7 computes those of the skipping jobs. The completion times of the final machine are computed using Eq. 8. 𝐶𝐶𝐶𝐶[ℎ]1 = 𝑚𝑚𝑚𝑚𝑚𝑚�𝐶𝐶𝐶𝐶[ℎ−1]1 + 𝑃𝑃𝑃𝑃[ℎ]1 , 𝐶𝐶𝐶𝐶[ℎ−1]2 − 𝑄𝑄𝑄𝑄[ℎ]1 , 𝐶𝐶𝐶𝐶[ℎ−1]3 − 𝑄𝑄𝑄𝑄[ℎ]2 − 𝑃𝑃𝑃𝑃[ℎ]2 − 𝑄𝑄𝑄𝑄[ℎ]1 � (4) 𝐶𝐶𝐶𝐶[ℎ]1 = 𝐶𝐶𝐶𝐶[ℎ−1]1 (6) 𝐶𝐶𝐶𝐶[ℎ]2 = 𝑚𝑚𝑚𝑚𝑚𝑚�𝑚𝑚𝑚𝑚𝑚𝑚�𝐶𝐶𝐶𝐶[ℎ]1 , 𝐶𝐶𝐶𝐶[ℎ−1]2 � + 𝑃𝑃𝑃𝑃[ℎ]2 , 𝐶𝐶𝐶𝐶[ℎ−1]3 − 𝑄𝑄𝑄𝑄[ℎ]2 � 𝐶𝐶𝐶𝐶[ℎ]2 = 𝑚𝑚𝑚𝑚𝑚𝑚�𝐶𝐶𝐶𝐶[ℎ−1]2 + 𝑃𝑃𝑃𝑃[ℎ]2 , 𝐶𝐶𝐶𝐶[ℎ−1]3 − 𝑄𝑄𝑄𝑄[ℎ]2 � 𝐶𝐶𝐶𝐶[ℎ]3 = 𝑚𝑚𝑚𝑚𝑚𝑚�𝐶𝐶𝐶𝐶[ℎ]2 , 𝐶𝐶𝐶𝐶[ℎ−1]3 � + 𝑃𝑃𝑃𝑃[ℎ]3 Advances in Production Engineering & Management 18(2) 2023 (5) (7) (8) 155 Han, Lee The next is the MIP formulation to clearly define the considered scheduling problem and to serve a benchmark schedule using an optimization solver. Herein, normal jobs and skipping jobs are not distinguished. In particular, it is assumed that skipping jobs have zero processing time at m1 and sufficiently long queue times between m1 and m2, to render the queue time limits nonsignificant, i.e., PTi1 = 0 and QTi1 = ∞ for skipping jobs. [P] Minimize 𝐶𝐶𝐶𝐶[𝑛𝑛]3 (9) subject to ∑𝑖𝑖 𝑥𝑥𝑖𝑖ℎ = 1, ∑ℎ 𝑥𝑥𝑖𝑖ℎ = 1, 𝑆𝑆𝑆𝑆[ℎ]𝑘𝑘 + ∑𝑖𝑖 𝑃𝑃𝑃𝑃𝑖𝑖𝑖𝑖 𝑥𝑥𝑖𝑖ℎ ≤ 𝑆𝑆𝑆𝑆[ℎ+1]𝑘𝑘 , 𝑆𝑆𝑆𝑆[ℎ]𝑘𝑘 + ∑𝑖𝑖 𝑃𝑃𝑃𝑃𝑖𝑖𝑖𝑖 𝑥𝑥𝑖𝑖ℎ ≤ 𝑆𝑆𝑆𝑆[ℎ]𝑘𝑘+1 , 1 ≤ ℎ ≤ 𝑛𝑛 ∀𝑖𝑖 1 ≤ ℎ ≤ 𝑛𝑛 − 1, 1 ≤ 𝑘𝑘 ≤ 3 1 ≤ ℎ ≤ 𝑛𝑛, 1 ≤ 𝑘𝑘 ≤ 2 𝑆𝑆𝑆𝑆[ℎ]𝑘𝑘 + ∑𝑖𝑖(𝑃𝑃𝑃𝑃𝑖𝑖𝑖𝑖 + 𝑄𝑄𝑄𝑄𝑖𝑖𝑖𝑖 )𝑥𝑥𝑖𝑖ℎ ≥ 𝑆𝑆𝑆𝑆[ℎ]𝑘𝑘+1, 𝑆𝑆𝑆𝑆[ℎ]3 + ∑𝑖𝑖 𝑃𝑃𝑃𝑃𝑖𝑖3 𝑥𝑥𝑖𝑖ℎ ≤ 𝐶𝐶𝐶𝐶[ℎ]3 , 𝑆𝑆𝑆𝑆[ℎ]𝑘𝑘 ≥ 0, 1 ≤ ℎ ≤ 𝑛𝑛, 1 ≤ 𝑘𝑘 ≤ 2 1 ≤ ℎ ≤ 𝑛𝑛 1≤ ℎ ≤ 𝑛𝑛, 1 ≤ 𝑘𝑘 ≤ 3 𝑥𝑥𝑖𝑖ℎ ∈ {0, 1}, ∀𝑖𝑖, 1 ≤ ℎ ≤ 𝑛𝑛 (10) (11) (12) (13) (14) (15) (16) (17) Eq. 9 represents the objective function's makespan, which is the maximum completion time of the jobs. To ensure permutation schedules without preemption, Eqs. 10 and 11 stipulate that each position can accommodate only one job and each job can and must only occupy one position in the sequence, respectively. Eq. 12 defines the relationship between the start times of two consecutive jobs, while Eq. 13 defines the relationship between the start times on two consecutive machines. To satisfy queue time limits, Eq. 14 guarantees that each job's subsequent operation starts within the specified queue time. Eq. 15 calculates the completion times of the jobs, and Eqs. 16 and 17 establish the decision variables' domains. 4. Used Methods: Genetic algorithms Because the problem considered is NP-hard, an effective and efficient heuristic algorithm is required to quickly obtain adequate solutions. Thus, we propose a genetic algorithm (GA), which is a nature-inspired evolutionary optimization method. The GA is one of the most commonly used methods because it has low mathematical requirements and is highly flexible in application [35-38]. The following subsections describe the proposed GA design. 4.1 Solution representation and fitness The solutions to the problem considered are expressed in permutation schedules, which are sequences of job indices that can be directly represented in the chromosome structure of the GA. Each solution has an objective function value, which is the makespan of the schedule and should be minimized. Thus, the makespan to be minimized was used for fitness evaluation. 4.2 Initial population To generate an initial population, a constructive heuristic algorithm referred to as the NEH (Nawaz, Enscore, and Ham) algorithm was used. Since the NEH algorithm has shown good performance in many flow shop scheduling problems, but it does not guarantee an optimal solution. Thus, it has been commonly used to find an initial solution for further improvement using other optimization techniques. The NEH is based on the concept of inserting jobs into a sequence iteratively to minimize the makespan. Initially, it calculates the total processing time for each job by summing up the processing times across all machines. Next, the jobs are sorted in decreasing order based on their 156 Advances in Production Engineering & Management 18(2) 2023 Genetic algorithm-based approach for makespan minimization in a flow shop with queue time limits and skipping jobs total processing times. For each job in the sorted order, it is inserted at all possible positions in the sequence, and the resulting makespan is evaluated. Then, the position that yields the lowest makespan is chosen, and the job is inserted at that position. Herein, for population diversity, we used shortest processing time (SPT)-based list scheduling rules to obtain different solutions. The remaining initial population was randomly generated. • • • • • SPT1 with pi1 SPT2 with pi2 SPT3 with pi3 SPT4 with pi2 + pi3 SPT5 with pi1 + pi2 + pi3 4.3 Selection The selection operation randomly selects chromosomes for the next generation. The chromosomes are selected with a selection probability proportional to their fitness values. This operation is analogous to the survival of the fittest in the theory of evolution and is referred to as the roulette method in the GA. Note that the performance of the roulette method has been demonstrated in recent related studies [14, 21, 22]. Because the objective function should be minimized, the inverse fi of the objective function value for each chromosome i in the population is computed, and the selection probability probi of each chromosome is obtained by probi = fi/∑j fj. The next population is composed based on the selection probabilities of the chromosomes. 4.4 Crossover The crossover operation generates offspring by exchanging information about the selected parents. Various methods can be used for crossover, such as the one-point crossover and two-point crossover methods [14]. Hosseinabadi et al. [39] and Hasançebi and Erbatur [40] evaluated various crossover methods for genetic algorithms and found that a one-point crossover demonstrated high performance in scheduling problems. Lee [14] reported that a one-point crossover demonstrated relatively high performance in a flow shop with queue time limits. Based on the existing research results, a one-point crossover was applied in this study. For each chromosome in the current population, if a randomly generated number (between 0 and 1) is less than the probability of crossover pc, the chromosome is copied into a pool. They are then randomly paired and referred to as parents. For the parents (Parent1 and Parent2), a one-point crossover is applied to generate two offsprings (Offspring1 and Offspring2) using the following procedure: Step 0: Randomly select a crossover point from among the parent genes. Step 1: Offspring1 inherits genes up to the crossover point within Parent1. Step 2: Except for the genes that Offspring1 already contains, Offspring1 inherits genes from Parent2 in the order in which they appear in Parent2. Step 3: Offspring2 is generated in the same manner in Steps 1 and 2. 4.5 Mutation The mutation is an operation that prevents premature convergence and maintains the diversity within a population. This operation partially mutates several genes in a chromosome with low probability. For each chromosome in the population, if a random number (between 0 and 1) is less than the mutation probability, the chromosome is mutated once. The mutated ones are also included in the population. In this study, two common methods were used, namely, insertion and exchange, with the same probability. Insertion selects a job at random and inserts it into a randomly selected position, whereas exchange interchanges two randomly selected jobs 4.6 Local search We used a local search technique to improve the performance of the GA. To prevent a rapid increase in the calculation time owing to the local search, it was only applicable to 10 % of the soAdvances in Production Engineering & Management 18(2) 2023 157 Han, Lee lutions in a population at each generation. In this study, insertion and exchange methods were used. In particular, insertion or exchange with the same probability was applied 3n times to the solution, where n is the number of jobs. If the new solution was superior to the current solution, the current solution was replaced. 4.7 Entire procedure The proposed GA terminates when the maximum number of generations is reached; the overall flowchart is shown in Fig. 1. Fig. 1 Flowchart of the proposed GA 5. Computational experiments To assess the performance of the proposed algorithms, we performed computational experiments using instances generated as follows. The experiments encompassed various levels of the number of jobs, n = (10, 20, 30, 40, 50, 100, 150, and 200). The processing times p were generated from discrete uniform distributions within the range of [1, 50]. Additionally, the queue times were generated from three different distributions with a range of [1, w], where w = (30, 50, and 70). Regarding the proportion of skipping jobs, we considered three levels of the proportion λ of skipping jobs, where λ = (0.3, 0.5, and 0.7), i.e., (0.3n, 0.5n, and 0.7n). For each combination of (n, w, and λ), ten instances were generated, resulting in a comprehensive set of experimental data. We coded the proposed algorithms using the Java programming language and conducted experiments on a personal computer equipped with an Intel Core i7-8700 CPU running at 3.2GHz. To solve the MIP formulation, we used CPLEX 12.10, a commercial solver, with a maximum CPU time of 1,000 seconds to prevent excessive computation time. As for the proposed GA, it terminated once the maximum number of generations was reached and returned the best solution found within the population. For each instance, the GA was run independently 30 times, and a mode value from the 30 runs was used for the evaluation. Before evaluating the GA, we performed experimental calibration to improve its performance. In the calibration, we assumed n = 100, λ = 0.5, and w from [1, 50], which were the medium levels of the considered problem instances. The GA contained four parameter types: population size S, number of generations G, probability of crossover pc, and probability of mutation pm. If the population size is excessively small, diverse solutions are not available. On the other hand, if the population size is excessively large, the time required for crossover and local search increases. Thus, excessive time is required to obtain a solution. Therefore, it is necessary to select an appropriate population size related to the number of jobs to be considered. In this study, the size of the population was set to 158 Advances in Production Engineering & Management 18(2) 2023 Genetic algorithm-based approach for makespan minimization in a flow shop with queue time limits and skipping jobs (𝜌𝜌 ∙ 𝑛𝑛) and a preliminary experiment was conducted for 𝜌𝜌 = 1, 2,…, 5. Considering the trade-off between the algorithm performance and calculation time, 𝜌𝜌 was set to 4. The proposed GA used the maximum number of generations G as the termination condition. Therefore, an appropriate number of generations was required. If G is excessively large, a significant amount of time is required to obtain a solution, whereas, if G is excessively small, it is highly probable that a solution close to the optimal solution will not be obtained. We tested {500, 1000, 1500, 2000, 2500, and 3000} for G. Thereafter, 1000 was set as the appropriate number of generations for the preliminary experiments. In particular, preliminary experiments were conducted on {0.3, 0.5, 0.7, and 0.9} for pc and {0.05, 0.1, 0.2, 0.3, and 0.4} for pm, which were set to 0.7 and 0.2, respectively. Note that because of the extensive range of potential combinations among the four types of parameters examined and the resulting substantial amount of data, this paper does not include the results of the preliminary tests. First, we tested the MIP performance using CPLEX. Table 2 lists the average CPU times of CPLEX with a computation time limit (1,000 s). As seen from the table, CPLEX shows a tendency to work well when the number of jobs skipping the first stage increased and the queue time limits were loose, whereas CPU times increased in the opposite case. The level of difficulty can be estimated based on the problem situation. The CPU time increased significantly as the number of jobs increased. When n = 50, 100, 150, and 200, CLPEX did not obtain an optimal solution for 10, 40, 79, and 89 instances (out of 90) within the time limit. Next, we evaluated the effectiveness of the simple heuristics used to generate the initial population. Table 3 presents the average percentage error (PE), which is defined as 100 × (objA – objCPLEX) / objCPLEX for Algorithm A, where N in the names of the last six algorithms represents the NEH algorithm. As seen from the table, the six list scheduling rules provided solutions with large errors compared with those from CPLEX. However, the proposed NEH algorithm demonstrated high performance. Every NEH algorithm obtained schedules with errors of approximately 2 % or less. Moreover, for instances with n = 150 and 200, schedules superior to those obtained from CPLEX were obtained. In addition, all NEH algorithms required less than 1 s. Thus, it was confirmed that an effective and efficient heuristic algorithm should be developed. In addition, we evaluated the performance of the proposed GA. In addition, to verify the effectiveness of the local search, a GA without a local search (GAl-) was tested. Table 4 presents the performance in comparison with CPLEX, where NI and CPUT indicate the number of instances in which the GA and GAl- obtained a superior schedule to that obtained from CPLEX and the average computation time of the GAs, respectively. The relative deviation index (RDI) is defined as (objA – objmin) / (objmax – objmin), where objA is the objective value for Algorithm A, and objmin and objmax are the minimum and maximum objective values, respectively. As shown in the table, the effectiveness of the GA was demonstrated more clearly as the number of jobs increased. For all instances with n = 150 and 200, the GA obtained a solution that was superior to that obtained by CPLEX. The performance of the GA was improved using local search. Although the calculation time increased owing to the inclusion of the local search, the average CPUT was 87 s for n = 200, which is practicable. λ 0.3 0.5 0.7 w 30 50 70 30 50 70 30 50 70 n=10 0.08 0.06 0.05 0.06 0.04 0.03 0.06 0.03 0.04 20 231.62 0.53 0.14 64.17 0.25 0.13 0.46 0.13 0.09 Table 2 CPU times of CPLEX 30 40 50 378.25 277.15 688.18 3.95 144.09 313.47 0.31 3.31 39.78 4.9 169.36 140.77 3.32 21.56 59.21 0.89 12.95 12.35 2.83 111.82 44.83 0.89 12.95 12.35 0.19 10.08 2.62 100 984.75 547.89 571 818.78 432.6 529.04 411.79 529.04 254.99 150 1000 1000 940.44 1000 933.92 953.12 852.79 953.12 804.98 200 1000 1000 1000 1000 1000 1000 972.94 1000 1000 Finally, we evaluated the performance of the GA concerning different problem parameters (λ and w) for instances with n = 100. The results are summarized in Table 5, which presents the Advances in Production Engineering & Management 18(2) 2023 159 Han, Lee average PE values. When both parameters decreased, the PE values also decreased. This indicates that the GA was effective when the limited queue times were strict and the number of jobs that skipped the first stage decreased. Problems with these features are more difficult than others; thus, CPLEX failed to find an optimal or adequate solution to these problems. It should be noted that if λ and w are zero, the problem transforms into a three-machine no-wait flow shop and into a two-machine flow shop problem if both parameters increase. 6. Conclusion This study investigated a flow shop scheduling problem specifically focusing on the incorporation of queue time limits and skipping jobs, which are critical requirements in semiconductor manufacturing. To address this problem, this study proposed a MIP formulation and GA. Computational experiments were carried out to assess the performance of both the MIP and GA, with a particular emphasis on evaluating the effectiveness and efficiency of the GA. The results demonstrated that as the problem size increased, the GA exhibited improved performance and maintained an acceptable computation time within a real fab facility. Additionally, the GA's performance was verified under various conditions of queue time limits and skipping jobs. However, it should be noted that in certain conditions during the computational experiments, the GA failed to achieve an optimal solution. Thus, future endeavors should be directed toward enhancing the performance of the GA. Potential approaches for improvement include exploring alternative heuristic algorithms such as state-of-the-art metaheuristics, genetic programmingbased algorithms, or machine learning-based algorithms. Furthermore, it is worth considering the integration of bi-objective optimization approaches, incorporating measures based on both makespan and due dates, as the semiconductor business often follows an order-based manufacturing paradigm. Additionally, future research can explore different types of queue time limits, n=10 19.23 22.41 30.00 21.91 17.05 25.60 1.65 2.12 2.33 1.91 1.50 1.31 SPT1 SPT2 SPT3 SPT4 SPT5 LPT SPT1N SPT2N SPT3N SPT4N SPT5N LPTN n GA 0.144 0.106 0.039 0.065 -0.050 -0.554 -2.071 -3.612 10 20 30 40 50 100 150 200 w 160 0.3 0.5 0.7 average 20 22.32 23.11 29.28 20.44 17.02 22.53 1.23 2.00 2.03 1.33 1.06 0.74 PE (%) GAl0.281 0.243 0.144 0.129 0.010 -0.532 -2.054 -3.608 Table 3 Percent errors of heuristic algorithm 30 40 50 100 21.97 23.99 24.72 25.83 22.41 22.99 23.76 24.73 27.43 27.21 27.68 27.53 18.86 17.84 18.54 17.62 14.94 15.54 15.97 16.36 19.93 19.10 18.80 17.72 1.13 1.18 1.19 0.89 1.49 1.69 1.48 0.72 1.55 1.49 1.37 0.72 0.99 0.75 0.64 -0.21 0.72 0.81 0.60 -0.08 0.43 0.35 0.15 -0.43 Table 4 Performance of the proposed GA NI CPUT (s) GA GAlGA GAl75 54 0.0 0.0 81 47 0.2 0.1 82 54 0.5 0.1 75 48 1.0 0.3 82 57 1.8 0.4 89 76 12.1 1.8 90 90 38.0 4.6 90 90 87.3 9.3 150 24.58 23.64 25.59 15.92 15.12 16.03 -0.54 -0.76 -0.86 -1.81 -1.60 -1.98 GA 0.134 0.076 0.052 0.121 0.068 0.011 0.000 0.000 200 22.73 21.55 23.19 14.06 13.30 14.10 -2.14 -2.36 -2.41 -3.35 -3.31 -3.57 RDI GAl0.393 0.476 0.400 0.472 0.393 0.181 0.010 0.001 Table 5 Percent errors of the proposed GA on different parameters (n = 100) λ 30 50 70 -2.377 -0.499 -0.360 -1.185 -0.261 -0.043 -0.097 -0.157 -0.004 -1.220 -0.306 -0.136 average 23.17 23.08 27.24 18.15 15.66 19.23 0.57 0.80 0.78 0.03 -0.04 -0.37 CPLEX 0.000 0.000 0.000 0.022 0.108 0.444 0.878 0.989 average -1.079 -0.496 -0.086 -0.554 Advances in Production Engineering & Management 18(2) 2023 Genetic algorithm-based approach for makespan minimization in a flow shop with queue time limits and skipping jobs including overlapping queue time limits as presented in [11] and generalized skipping jobs that possess the ability to skip any stage. 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Evaluation of crossover techniques in genetic algorithm based optimum structural design, Computers & Structures, Vol. 78, No. 1-3, 435-448, doi: 10.1016/S0045-7949(00)00089-4. 162 Advances in Production Engineering & Management 18(2) 2023 Advances in Production Engineering & Management ISSN 1854-6250 Volume 18 | Number 2 | June 2023 | pp 163–174 Journal home: apem-journal.org https://doi.org/10.14743/apem2023.2.464 Original scientific paper Hybrid forecasting modelling of cost and time entities for planning and optimizing projects in the die-cast aluminium industry Muñoz-Ibañez, C.a, Chairez, I.a, Jimenez-Martinez, M.b, Molina, A.a, Alfaro-Ponce, M.a,* aTecnologico bTecnologico de Monterrey, Institute of Advanced Materials for Sustainable Manufacturing, Mexico City, Mexico de Monterrey, School of Science and Engineering, Puebla, Mexico ABSTRACT ARTICLE INFO The techniques employed to manage an industrial project are based on tools that aim to achieve the objectives set by an organization. Most of these techniques consider the development of operative and predictive models. The difficulty in developing project planning models relies on estimating large sets of parameters and the need to include model sections of poorly identifiable, that increase costs and time. This work develops a hybrid forecasting model for all the phases that make up die-casting projects through a series of parameters and sub-models that contemplate the particularities of each case, thereby achieving greater precision in the forecast. The model identifies the cost and time factors that affect project planning, specifically in the die-casting industry, and intends to predict their future behaviour when certain initially given conditions are modified. To estimate the parameters of the hybrid model, several factors in the processes were considered that interact in this industry, such as primary matter costs and activities associated to the process. The considered processes that have a substantial economic impact on the implementation of the project were selected. The criteria for this selection considered identifying the relevant parts of the design and manufacturing in the diecasting industry. Process factors such as the Cost of aluminium and its related activities, whose processes will be grouped into cost and time entities to build a set of metrics that allow better control over them. Finally, the proposed model is based on analytical, parametric, and analog methods that achieve accuracy greater than 85 % in predicting the time and Cost of the process. Keywords: Hybrid models; Entity modelling; Project planning; Forecasting models; Aluminium die-casting; Cost factors; Time factors; Optimization *Corresponding author: marielalfa@gmail.com (Alfaro-Ponce, M.) Article history: Received 5 May 2023 Revised 17 June 2023 Accepted 19 June 2023 Content from this work may be used under the terms of the Creative Commons Attribution 4.0 International Licence (CC BY 4.0). Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. 1. Introduction Project planning is useful for improving time and costs in company projects involving management techniques. These approaches have been designed and developed since the middle of the 20th century [1-4] intending to improve the outcome of project planning, the development of qualitative [5-7] and quantitative models [8-11] has ben considered. These models have motivated the implementation several optimization strategies that have contributed to reducing operation costs and operative times of processes included in the project. Developing valuable models (usually known as estimation models) in project planning requires analyzing historical information regarding the effects and results of different operative strategies and control techniques implemented in previous projects. Due to the specific conditions of each productive sector where the estimation models could be applied, they must depend 163 Muñoz-Ibañez, Chairez, Jimenez-Martinez, Molina, Alfaro-Ponce on different operative, commercial, and economic variables, including the technology for developing the project, raw materials, workforce, etc. These highly specialized models have been developed for sectors as varied as software [12], construction [13-14], aerospace [15, 16], and automotive [17, 18], machining [19], among others. Specifically, the use of models for estimating manufacturing costs and time, widely used to determine the Cost of forged parts [20], rotating parts [21], manufacturing of die-casting molds [22], and preformed parts employing die cuts [23]. The importance of estimation models in manufacturing processes lies in their ability to accurately predict the costs of primary materials and methods and operative times. In addition, the successful models considered specific needs and practical implications that are part of the environment in which the project is developed. Also, on many occasions, the precision and speed in estimating costs and operative times are associated with the final purchase order, which the customer defines. With the intention to determine goods production times and costs, the manufacturing processes can be calculated using several methods, like the analytical methods that break down manufacturing activities into their elementary parts. However, despite the benefits of these models, their descriptions usually have many parameters. They are very nonlinear and may not consider some problematic aspects of the model. A different technique for developing the estimation model is based on non-parametric methods that could produce a mathematical representation of productive relationships using the information generated by the industrial sector under study. These non-parametric models can be complemented with analogous methods to classify product indices according to their dimensional and quality characteristics [24]. All these methods have been successfully used in estimating time and costs for diverse projects of various industrial sectors, including metal mechanics. In the aluminum die-casting industry, the existing estimation models focused on several factors that affect the manufacturing process. An example of such a method is the model developed by Madan that focuses on determining the optimal Cost and time of production, depending on the geometric conditions of the part; through these data, they determine the number of cavities necessary for the die casting dies [25]. On the other hand, the model developed by Sung focuses on simulating the pores within the die-cast process. Specifically, this model considers factors such as trapped air during the die-casting process caused by the design of the die-casting mold and the properties changes by anisotropic composition due to the die-cast process conditions [26]. Srivastava proposes a model to determine the conditions that cause thermal fatigue, and cracking of casting die [27]. The work presented by Tsoukalas produces a model to determine the porosity of AlSi9 Cu3 aluminum used in die casting. The model is supported by genetic algorithms that validate as main variables the die casting temperature, the die casting mold temperature, and the speed of the phases of die casting [28]. The models described above establish connections between operative conditions in the diecasting process. Nonetheless, they are not formally related to the economic aspects of the final product and its delivery time to the final consumer. Nowadays, many companies apply parametric or probabilistic cost estimation methods. The characteristics and rapid implementation are unique elements of these models. However, these models use synthetic data obtained by approximating the total production cost and overall production time and assuming some desired features for the final product. The product features can be diverse depending on the design requirements and manufacturing activities. The main problem of this production process is the lack of information about the cost structure and the product's manufacturing processes. As a result, it became difficult for the designer to visualize the necessary modifications that should be applied to the model to reduce costs in the quotation activities. Such a fact makes it complex to assign only a cost value to the product, limiting the transparent negotiation of the Cost and consequently causing a delay with the customer. The indirect costs are other factors to consider in the product cost structure, materialized by the support activities. Besides, the causal relationship between cost objects (products and services) and consumable resources is difficult to assess. This evaluation can be solved using trace164 Advances in Production Engineering & Management 18(2) 2023 Hybrid forecasting modelling of cost and time entities for planning and optimizing projects in the die-cast aluminium … ability, which makes cost analysis explicit in a network with its incorporation into products or services, which is difficult to achieve with the traditional cost estimation approach. The application of a hybrid model allows accurate forecasting of the phases that make up the processes of die-casting projects. Hybrid models have significant advantages in their use, as shown in the following key applications: a) the optimization of manufacturing processes for obtaining fiberglass in the automotive industry [29], b) the improvement of pill production processes in the pharmaceutical sector [30], and c) in the construction of metal structures at high temperatures through process planning [31]. Likewise, other critical applications for the application of hybrid models that have advantages in precise parameterization in industrial activities are, for example, for inventory control [32], electrical energy consumption [33], in the handling of the materials [34] and the costing for the manufacture of turbines [35]. The advantages of using hybrid models for process planning lie in the flexibility of these models and their ability to adapt to a significant number of features that they may have, selecting the model that best suits the characteristics to be predicted [36]. However, it is also essential to consider that the main disadvantage of its use lies in the large number of data that must be used to achieve considerable precision of the model. This study presents a data-driven hybrid model to asses costs and process time in the diecast aluminum industry. The model shown in this work uses analytical, parametric, and analogous methods. Among the variables that are considered for the model development are: a) the Cost of the raw material (which is a determining factor in signing long-term contracts with the automotive and aerospace sector), b) the processing time, and c) the Cost of the operations that will be carried out for the manufacture of the piece, including intermediate process such as diecasting, grinding, die-cutting, drilling, shot blasting, and packing, among others. The goal of the modeling process is to generate a projection that allows controlling the project plan of the diecasting process. In addition, the proposed model considers technical and economic aspects essential for developing projects in the die-casting industry. The successful development becomes the model into an auxiliary tool for estimating the costs and delivery times of the customer product. As a result, the problem of estimating the Cost and time in die-casting projects focuses on constructing a model that can be described as an integral formulation. Such completeness is a consequence of considering a series of sub-models that allow its adaptation to the specific characteristics of the casting process to predict. Hence, this work considers three fundamental aspects of the model design: • The changes in the price of aluminum in international markets. • The changes in the production process that have more significant impact on the Cost and the time of the project (die casting, grinding, drilling, and packing, among others). • The model aspects related to the manufacture of the die-casting mould. This work is organized as follows. Section 2 is the mathematical basis and the details of the construction of the model. The description of the data set used for its test is included in this part. Section 3, the simulation results achieved by the model. Finally, Sections 4 and 5 close the study with the model's accuracy and relevant conclusions describing some problems related to the discussed topic. 2. Model construction For the development of the model proposed in this work, it was necessary to estimate the following steps: • Determination of cost and time entities (CTE) divided according to each primary activity. • Development of CTE for the manufacturing of parts and manufacturing tooling. • The CTE for the raw material only considers the Cost, not the time, since it is a complementary value chain activity. • The CTE's information is gathered from various productive projects to carry out the simulations and the relationships between the projects' time, Cost, and weight. Advances in Production Engineering & Management 18(2) 2023 165 Muñoz-Ibañez, Chairez, Jimenez-Martinez, Molina, Alfaro-Ponce • The weight directly determines the production costs and expenses to complement the Cost Entity. • The equations for each CTE are formulated according to the information and the simulation. 2.1 Cost and time entity For die-casting, estimating Cost and time is the crucial combination for predicting how much it will cost and how long it will take to make a certain product or many products through the analysis of the particularities attributed to it [37]. To identify and control these intrinsic particularities of each development, CTE is proposed. Developed for the analysis of the Cost and time of each critical activity, it provides information related to the added value of the project, so the CTE will allow the estimation of the Cost and time between the products, considering the Cost per kilogram and the time of the product manufactured for homogeneity. The CTE (Fig. 1) is composed of two elements. Its first element is the Cost Entity (CE), whose primary function is controlling each project phase's Cost. The Time Entity (TE) is its second element, which determines the time assigned to each activity to be carried out. It's essential to define once the homogeneous resources for the project are stable and interdependent. They are stable because of the imputation of these within the productive chain, 𝛽𝛽 𝛽𝛽 𝛽𝛽 which is given by �i. e., \ , �. Notice that each resource does not need to change depend𝑋𝑋 𝑚𝑚𝑚𝑚𝑚𝑚 𝑘𝑘𝑘𝑘 ing on the product. These components are interdependent since the resources are consumed in the same proportion by one element in any way the product is used. Assuming that 𝑅𝑅𝑖𝑖 = [𝑘𝑘: 𝑘𝑘 ∈ 𝐾𝐾] represents the number of resources consumed by the 𝐶𝐶𝐶𝐶𝐶𝐶𝑖𝑖 , 𝑇𝑇𝑇𝑇𝑖𝑖 or 𝐶𝐶𝐶𝐶𝑖𝑖 to realize the 𝑖𝑖𝑡𝑡ℎ activity gave by 𝐴𝐴𝑖𝑖 . If �𝑦𝑦𝑖𝑖𝑘𝑘 (𝑋𝑋𝑖𝑖 ) = 𝑋𝑋𝑖𝑖 𝛼𝛼𝑘𝑘 � is equivalent to the number of resources 𝑘𝑘 consumed by carrying out the activity 𝐴𝐴𝑖𝑖 under the condition 𝑋𝑋𝑖𝑖 (where 𝛼𝛼𝑘𝑘 is the coefficient with which resources 𝑘𝑘 are consumed) and if 𝐶𝐶𝑘𝑘 = 𝑦𝑦𝑖𝑖𝑘𝑘 (𝑋𝑋𝑖𝑖 ) and 𝑇𝑇𝑘𝑘 = 𝑦𝑦𝑖𝑖𝑘𝑘 (𝑋𝑋𝑖𝑖 ) is the allocation fee corresponding to the Cost and time of the resource units and units 𝑘𝑘. Therefore, Eqs. 1 and 2 are basic for the determination of Cost and time: 𝐶𝐶𝐶𝐶 = � 𝐶𝐶𝑘𝑘 �𝑦𝑦𝑖𝑖𝑘𝑘 (𝑋𝑋𝑖𝑖 )� 𝑘𝑘∈𝑅𝑅𝑖𝑖 𝑇𝑇𝑇𝑇 = � 𝑇𝑇𝑘𝑘 �𝑦𝑦𝑖𝑖𝑘𝑘 (𝑋𝑋𝑖𝑖 )� 𝑘𝑘∈𝑅𝑅𝑖𝑖 The definitions for the distribution of the Cost of a resource would be the following: ℎ 𝑛𝑛𝑛𝑛 (1) (2) 𝛽𝛽 ℎ • Machining (𝑛𝑛𝑛𝑛)𝛼𝛼𝑘𝑘 � � an imputation fee � �. • �(𝑋𝑋𝑖𝑖 )𝛼𝛼𝑘𝑘 � is an imputation fee, determined by Cost based on weight. • 𝑦𝑦𝑖𝑖𝑘𝑘 (𝑋𝑋𝑖𝑖 ) is an imputation fee, determined by the time. The 𝛼𝛼𝑘𝑘 is the coefficient with which resource 𝑘𝑘 is consumed, 𝑛𝑛𝑛𝑛 is the identification of the Cost for machining. The coefficients to determine the degree to which resources are consumed within the project activities were determined through statistical relationships obtained from 20 projects within the die-casting industry. Fig. 1 Graphic representation of the CTE 166 Advances in Production Engineering & Management 18(2) 2023 Hybrid forecasting modelling of cost and time entities for planning and optimizing projects in the die-cast aluminium … Cost and time entity pattern The Cost and Time Entity Pattern (CTEP) is a macro-entity that meets the characteristic of homogeneity of the resources of the CTE. CTEP simplifies the model for determining Cost and time and helps establish an overview of the manufacturing chain. The basic information for estimating costs and time in projects of the die-casting industry is represented in the Scheme of the Cost and Time Entity Patterns (SCTEP) shown in supplementary_file_1. The SCTEP is built based on the product's characteristics, such as; physical specifications of the product (weight and type of aluminum), manufacturing (the type of surface finish, drilling, packing, among others), and the tooling (number of cavities, volume, etc.). For the estimation of the Cost and time in the design as well as in the manufacturing, the purchase order is used. Such application allows us to quantify the flexibility of the order and give an adequate estimation of the phases necessary to develop the project, so the purchase order represents a priority aspect [38]. It is important to establish that the CE and TE maintain the homogeneity condition understood as the similarity of the activities between the parameters; resources, activities, inputs, and outputs. Therefore, the union of the parameters and the sum of the costs and elementary times of the different cost entity patterns (CEP) and time entity patterns (TEP) are described in Eqs. 3 and 4 𝐶𝐶𝐶𝐶𝐶𝐶 = � � 𝐶𝐶𝑘𝑘 �𝑦𝑦𝑖𝑖𝑘𝑘 (𝑋𝑋𝑖𝑖 )� 𝑖𝑖∈𝑁𝑁 𝑘𝑘∈𝑅𝑅𝑖𝑖 𝑇𝑇𝑇𝑇𝑇𝑇 = � � 𝑇𝑇𝑘𝑘 �𝑦𝑦𝑖𝑖𝑘𝑘 (𝑋𝑋𝑖𝑖 )� 𝑖𝑖∈𝑁𝑁 𝑘𝑘∈𝑅𝑅𝑖𝑖 where 𝑁𝑁 is the number of CE that is part of CTEP. (3) (4) General model for determining cost and time Once the elements that will be part of the model structure have been defined, it is important to identify the main activities to be modeled. For this work, the activities taken into consideration are: • The price of the metal. • The manufacturing time of the product. • The Cost of the Product. • The manufacturing time of the die-cast tooling. • The Cost of the die-cast tooling. Considering it, the models proposed for the determination of the total Cost and time of the project can be represented as Eqs. 5 and 6. 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 = 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝑅𝑅𝑅𝑅 + 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 + 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 (5) 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 = 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 + 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 (6) In some cases, there may be the possibility that the customer no longer requests a part of the project; for example, the customer can provide the tooling. In consequence, it would only be necessary to establish the Cost of the raw material and manufacturing. A binary matrix can be constructed to establish this relation, where zero identifies an activity that does not occur and one that occurs in the project. 3. Methodology and primary results CTEP1 of the raw materials In this case, the CTEP1 for the raw material will include only the calculation of the Cost of aluminum, for which the historical data reported by the London Metal Exchange (LME) of the Cost in dollars per ton will be used. The start date of data collection was January 1, 2016, and the end of Advances in Production Engineering & Management 18(2) 2023 167 Muñoz-Ibañez, Chairez, Jimenez-Martinez, Molina, Alfaro-Ponce it on May 25, 2019. The database contains five monthly readings that will give 246 data for the analysis, this information can be consulted at supplementary_file_2. Intending to determine the most accurate forecast model, it is necessary to apply tests to establish the stationarity of the time series, for which the Eq. 7 gives the correlation function of the sample [39]. 𝑇𝑇 rk = ��(𝑥𝑥𝑡𝑡 − 𝑥𝑥̅ )(𝑥𝑥𝑡𝑡−𝑘𝑘 − 𝑥𝑥̅ )� �� 𝑡𝑡=1 T )2 (xt -x� � t=1 −1 (7) Therefore, it is possible to develop the correlogram shown in Fig. 2(a), in which the analyzed series of the LME presents a simple and partial first-order correlation, which determines that the immediately previous data are the ones that have the greatest influence on the value of the following data. 𝑇𝑇 𝑑𝑑 = ��(𝑒𝑒𝑡𝑡 − 𝑒𝑒𝑡𝑡−1 𝑡𝑡=2 )2 𝑇𝑇 −1 � �� 𝑒𝑒𝑡𝑡2 � 𝑡𝑡=2 (8) Fig. 2 (a) Aluminium price correlogram for LME, and (b) Dickey-Fuller test Once the correlogram is obtained, the autocorrelation of the data is determined by the statistical Eq. 8 of Durbin-Watson [39], obtaining a value of 1.869886, establishing no autocorrelation. Δ𝑦𝑦𝑡𝑡 = 𝜙𝜙1 + 𝜙𝜙2𝑡𝑡 𝑝𝑝 + 𝛼𝛼𝑦𝑦𝑡𝑡−1 + � 𝛿𝛿𝑖𝑖 Δ𝑡𝑡−1 + 𝜀𝜀𝑖𝑖 𝑖𝑖=1 (9) On the other hand, the analysis of the stationarity series is carried out using the Dickey-Fuller test as shown in Eq. 9 [39]. The results are shown in Fig. 2(b), setting a decision value of 3.428349 and -1.420186. As a result, it is established that the time series of the LME is not stationary and have a unitary root, so it is possible to model this PEC through an ARIMA model. 168 Advances in Production Engineering & Management 18(2) 2023 Hybrid forecasting modelling of cost and time entities for planning and optimizing projects in the die-cast aluminium … With the main characteristics of the time series determined, such as their autocorrelation and non-stationarity, it is possible to select a model that best adapts to the forecast. In the mentioned above case, an ARMA model (2,2) was chosen, which will be the model for determining the 𝑃𝑃𝑃𝑃𝑃𝑃1 ; this determines the Cost of the raw material expressed with Eq. 10 𝑃𝑃𝑃𝑃𝑃𝑃1 = 𝑌𝑌𝑡𝑡−1 + 𝜃𝜃(𝑌𝑌𝑡𝑡−2 − 𝑌𝑌𝑡𝑡−3 ) + 𝜑𝜑(𝑎𝑎�𝑡𝑡−1 ) (10) where the coefficient values for the self-aggressive part expressed by 𝜃𝜃 is 0.797492 and for the mobile part expressed by 𝜑𝜑 is 0.833451, and 𝛼𝛼𝑡𝑡−1 = 𝑦𝑦𝑡𝑡−1 − 𝑦𝑦�𝑡𝑡−1 . This model adjust precisely to the series of the Cost of aluminum. CTEP2 of Product Manufacturing To build the CTEP2 corresponding to the manufacture of the piece, the first step is to propose the matrix of activities of the processes used in its manufacture so it is possible to establish the parametric relationships. For this, 25 projects were selected from the Maquinados e Inyecciones Tecamac SA de CV client portfolio. This company die-casts more than 45 tons of aluminium annually for clients such as Siemens-Mexico and Donaldson, among others, being a second-order supplier of assemblers such as Nissan and GMC. Therefore, the projects (supplementary_file_3) cover 87 % of the company's annual production. It is important to mention that this Table will become a binary matrix indicating whether the activity is carried out (1) or not (0). Once the project has defined the activities, the times required by activity for its manufacture are established, and the relationship between these times and the weight of the piece is established. To achieve this, it is necessary to calculate the number of cycles to be observed given by Eq. 11 𝑠𝑠𝑠𝑠 2 (11) 𝑛𝑛 = � � 𝑘𝑘𝑋𝑋� where 𝑛𝑛 is the number of cycles to be observed given that 𝑠𝑠 is the standard deviation of the test, 𝑘𝑘 the desired degree of accuracy that is 95 % and 𝑋𝑋� the average time, with this a pilot run is established stating that it is necessary to observe 50 cycles. Hence, the study of times of the 25 projects is carried out according to the activities listed in supplementary_file_3. The study performed for the Optimized Shield in supplementary_file_4 (named as part name no. 2 in supplementary_file_3) is taken as an example. When time and movement studies were carried out for each case, the data fit the best distribution that resembles its behavior. This allows for generating a more significant number of observations artificially. This setting can be seen in supplementary_file_5, in the case of Optimized Shield time's part name no. 2. This process is repeated for each of the projects, as well as for the operations that are necessary for its manufacture. With this, it is possible to determine the appropriate distribution type, resulting in more than 7,000 data supporting the forecasts; this process is illustrated in supplementary_file_6 for the die-cast operation. Construction of the TEP2,1 for the time of part manufacturing activities The data shown in supplementary_file_6 allows us to determine the relevant equations for each process and estimate the time for the manufacture of the piece. The resulting equations are shown in supplementary_file_7, where 𝑌𝑌𝑖𝑖𝑘𝑘 is a binary variable that takes the value of 1 if the activity is carried out and 0 if the activity is not carried out. Also, 𝑋𝑋𝑖𝑖 is the weight of the piece to be manufactured. With the specific equations and metrics that govern each of the 𝑇𝑇𝑇𝑇 of 𝑇𝑇𝑇𝑇𝑇𝑇2,1, the general equation is constructed, which is expressed as follows. 𝑇𝑇𝑇𝑇𝑇𝑇2,1 = � 𝑦𝑦𝑖𝑖𝑘𝑘 𝐸𝐸𝑇𝑇𝑖𝑖 𝑋𝑋𝑖𝑖 = 𝑌𝑌𝑖𝑖𝑘𝑘 (27.36 + 51.87𝑋𝑋𝑖𝑖 ) + 𝑌𝑌𝑖𝑖𝑘𝑘 �0.9068 + 60.65𝑋𝑋𝑖𝑖 − 𝑘𝑘𝑘𝑘𝑅𝑅𝑖𝑖 𝑌𝑌𝑖𝑖𝑘𝑘 (7.94 + 33.47𝑋𝑋𝑖𝑖 ) + 𝑌𝑌𝑖𝑖𝑘𝑘 (0.78 + 26.20𝑋𝑋𝑖𝑖 ) + 𝑌𝑌𝑖𝑖𝑘𝑘 (0.61 + 64𝑋𝑋𝑖𝑖 ) 𝑌𝑌𝑖𝑖𝑘𝑘 �5.15 + 80.63𝑋𝑋𝑖𝑖 − 163.27𝑋𝑋𝑖𝑖2 � + 𝑌𝑌𝑖𝑖𝑘𝑘 (2.22 + 26.33𝑋𝑋𝑖𝑖 ) Advances in Production Engineering & Management 18(2) 2023 89.98𝑋𝑋𝑖𝑖2 � + (12) 169 Muñoz-Ibañez, Chairez, Jimenez-Martinez, Molina, Alfaro-Ponce Construction of the CEP2,2 for the Cost of part manufacturing For the construction of the CEP, a characterization of the Cost is made concerning kilograms manufactured; for this, the following descriptions were generated: • Workforce vs. weight. For this relationship, the direct, indirect, and administrative workforce is considered as a whole, considering that a relationship is kept according to the kilograms produced by the company. • Gas vs. weight. Gas is the fuel oil used to fuse aluminum in melting furnaces, the second most important raw material. It is directly related to the number of kilograms produced by the company. • Electric power vs. weight. Electric power allows the operation of the die-casting equipment, as well as the various finishing and melting equipment; the consumption of electrical energy is determined directly by the number of kilograms produced by the organization. • Expenses vs. weight. The expenses are all those agents necessary for the organization to carry out its productive activity but are not delivered in the final product; these are distributed equally in the kilograms produced by the company. The company history is reviewed for the workforce vs. weight evaluation to establish its correspondence; the data is shown in supplementary_file_8. Once the workforce's Cost per kilogram is obtained, the next step to determine the CE is to make the corresponding adjustment in the behavior of the data shown in supplementary_file_11. In this way, the trend will be analyzed, so we can apply a more suitable model for simulation. This procedure is carried out in the same way for the rest of the costs, leaving the equations as shown in Table 1. Table 1 Cost of the workforce against sales from December 17 to February 19, in Mexican pesos CE Equation Type of distribution Workforce 𝐶𝐶𝐶𝐶2,2,1 = 𝑁𝑁(19.92, 6.40)𝑋𝑋𝑖𝑖 Normal Gas 𝐶𝐶𝐶𝐶2,2,2 = 𝑁𝑁(10.63, 1.38)𝑋𝑋𝑖𝑖 Normal Power Supply 𝐶𝐶𝐶𝐶2,2,3 = 𝑁𝑁(5.77, 3.88, 1.84)𝑋𝑋𝑖𝑖 Triangular Expenses 𝐶𝐶𝐶𝐶2,2,4 = 𝑁𝑁(26.09, 11.89, 7.53)𝑋𝑋𝑖𝑖 Triangular Where 𝑋𝑋𝑖𝑖 is the weight of the piece to be developed, which will have an allocation rate for the Cost according to the distribution that best fits the item. Construction of CTEP3 for the Tooling Manufacturing The complementary activities for the time estimation of machining that help in the tool manufacturing process are the following [40]: • Habilitation of the material. It consists of preparing the steel that will be machined by adjusting the shingles to have a reference cut for the machining; this activity lasts between 5 to 10 percent of the total machining activity. • Preparation of tools. The tools that will be used in the machining operations are selected and prepared; this process lasts 5 to 10 percent of the machining activity. • Generation of operations. The design of the tooling or device is executed according to the specifications and requirements of the client, then the designs are transformed into instructions for the operation of the CNC. The generation of operations is one of the main activities within the manufacture of tooling, and its duration lasts between 30 and 35 percent of machining activities. • Machining. The design is emptied into the material, and a swarf is removed. This activity is carried out with numerical control equipment, which gives precision and certainty. This element gives added value to the manufacturing, so it is the guiding operation within the forecast model. • First Mold Assembly. An initial assembly is carried out to see no interferences inside the cavity and the sealing is adjusted. This operation lasts no more than 2,880 min. 170 Advances in Production Engineering & Management 18(2) 2023 Hybrid forecasting modelling of cost and time entities for planning and optimizing projects in the die-cast aluminium … • Heat treatment. It is subjected to the cavity at high temperatures to subsequently cool it gradually; this allows the cavity to have a hardness on its surface; this operation lasts no more than 2,880 min compared to the standard established by the company. • Second mould assembly. Once the cavities have been treated thermal, these are assembled again to verify if there are deformations or interferences inside the cavity. If these occur, they are readjusted and returned to verify the closure of the cavity, and this operation does not last more than 2,160 minutes for the standard set by the company. • First test run. The die-cast mould is tested inside the die-cast equipment, allowing them to perform the necessary actions for the final adjustment. This operation does not last more than 1,440 min by the standard established by the company. • Final adjustment. The adjustment of all the observations that were presented in the execution of the test is carried out. In general, these operations include the adjustment of the knobs and the adjustment in the closing of the mould; this operation does not last more than 2,880 minutes, according to the standard established by the company. Defined the operations that make up the manufacturing of the injection mould, the times obtained from four documented projects are those described in supplementary_file_9, where the different machining times are established and likewise, the relationship between the machining is sought, the weight and the number of cavities that make up the tool. With these data, it is possible to identify the correlation between the weight removed against the weight of the mould (weight of the piece by the number of cavities) by applying the Pearson linear correlation index according to Eq. 13. Which establishes a correlation of 0.9369 and, subsequently, the correlation between the machining time required against the weight eliminated; with this value, it is possible to obtain the TEP2,2 as follows: 𝑇𝑇𝑇𝑇𝑇𝑇3,1 = � 𝑦𝑦𝑖𝑖𝑘𝑘 𝑇𝑇𝑘𝑘 𝑃𝑃𝑖𝑖 = 𝑌𝑌𝑖𝑖𝑘𝑘 (705.00 + 161.75𝑃𝑃𝑖𝑖 ) (13) 𝑘𝑘𝑘𝑘𝑅𝑅𝑖𝑖 Where 𝑌𝑌𝑖𝑖𝑘𝑘 is a binary variable that takes the value of 1 if the activity is carried out, and 0 if the activity is not carried out, 𝑃𝑃𝑖𝑖 which is then withdrawn weight given by 𝑃𝑃𝑖𝑖 = 0.252 + 155.98𝑋𝑋𝑖𝑖 , and 𝑋𝑋𝑖𝑖 is the weight of the piece to be manufactured. For the estimation of the ECT3,2, it is established that it is directly related to the time invested and the fixed and variable costs of the machining activity. The time costs are shown in the Table 2. When the CTEP3 has been developed for the activity of the manufacture of moulds, it is possible to use the model to forecast the Cost and time within the projects of die casting, as well as to measure its degree of precision. Table 2 CNC hourly costs for the manufacture of injection molds Cost per hour Range in minutes Min 400 0 250 101664 200 177352 4. Model accuracy Max 101663 177352 ∞ To determine the model's accuracy, the test of the 𝜒𝜒 2 represented in Eq. 14 will be carried out to establish that the sample obeys the entity model for the estimation of Cost and time. Where 𝑉𝑉0 is the real value, and 𝑉𝑉𝑒𝑒 is the expected or predicted value, the 𝑘𝑘 degrees of freedom are adjusted according to the number of observations generated for each test. 2 𝜒𝜒(𝑘𝑘−1) 𝑁𝑁 =� 𝑘𝑘=1 (𝑉𝑉0 (𝑘𝑘) − 𝑉𝑉𝑒𝑒 )2 𝑉𝑉𝑒𝑒 (14) The test will be applied to each one of the developed sub-models of the CTEP, so the following elements will be evaluated: Advances in Production Engineering & Management 18(2) 2023 171 Muñoz-Ibañez, Chairez, Jimenez-Martinez, Molina, Alfaro-Ponce • • • • • Cost of raw material. Cost of the piece. Time of manufacture of the piece. Tooling cost. Tool manufacturing time. To determine the degree of precision when estimating the Cost of the raw material, a test with 19 degrees of freedom is applied, as shown in supplementary_file_10. The obtained value of 5.4135 can be interpreted as a 95 % reliability of the model for the Cost of the raw material. The same procedure follows the rest of the CTEP, which we can summarize in Table 3. CTEP Cost of raw material Cost of the piece Time of manufacture of the piece Tooling cost Tool manufacturing time Degree of freedom 19 Table 3 Model sensitivity analysis χ2 Degree of Standard Reliability (%) deviation Jarque-Bera 0.0607 -0.0253 4.3209 95 1.8213 12.9878 80 0.0248 -0.0376 0.2059 -1.1548 12.1588 5 2.2916 5 Asymmetry 5.4135 19 19 Kurtosis 2.2857 85 80 80 0.5015 398.44 0.2871 -1.2481 -0.0230 -0.1643 -0.0665 -0.0580 3.6807 4.6675 9.0323. 8.6379 5. Conclusion The results obtained from the different tests carried out to determine the model's reliability show that it presents a degree of accuracy above 85 percent. This result allows a quick forecast to be made with the minimum information required for the quotation, leading to the construction of an entity-based model to efficiently determine costs and time to provide certainty and a competitive advantage to the die-casting industry. Analyzing the model structure established that the information necessary for estimating Cost and time is introduced through the different entities built. Such analysis facilitates the person in charge of the quotation's fast and efficient handling of all the data required during the quotation process. These entities of Cost and time have in their main structures the Cost of the raw material, the Cost and time of manufacture of the tools, and the Cost and time of manufacture of the part. Such determination is an advantage in the process since it allows the evaluation of the different projects, although these represent other structures. An important innovation that the use of CEP provided was that the model increases accuracy and facilitates user interaction. In the same way, it was possible to gather information to generate a solid and well-structured database on which the model can base its approximations. Finally, in the development of future research, it would be possible to use other modeling systems, such as genetic algorithms or artificial intelligence, to increase the accuracy of the proposed model. Acknowledgment The authors would like to thank the Tecnologico de Monterrey seed founding program, projects IDs IJXT070-22TE60001 and IKXT025-22UI60001, and Consejo Mexiquense de Ciencia y Tecnología grant "PROGRAMA PARA LA VINCULACIÓN DE EMPRESAS CON INSTITUCIONES DE EDUCACION SUPERIOR Y CENTROS DE INVESTIGACIÓN COMECYT-EDOMEX CONVOCATORIA 2023" project number 34544 and Industrias MAR-ROB S.A. de C.V. 172 Advances in Production Engineering & Management 18(2) 2023 Hybrid forecasting modelling of cost and time entities for planning and optimizing projects in the die-cast aluminium … References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] Campero, M., Alarcón, L.F. (2018). 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Injection molding handbook, 3rd edition, Kluwer Academic Publishers, Norwell, Massachusetts, USA, doi: 10.1007/978-1-4615-4597-2. 174 Advances in Production Engineering & Management 18(2) 2023 Advances in Production Engineering & Management ISSN 1854-6250 Volume 18 | Number 2 | June 2023 | pp 175–186 Journal home: apem-journal.org https://doi.org/10.14743/apem2023.2.465 Original scientific paper When core sorting and quality grading is beneficial to remanufacturers: Insights from analytical models Cao, X.H.a, Shi, X.L.a, Lan, H.J.a, Huang, D.a,* aSchool of Economics and Management, Beijing Jiaotong University, Beijing, P.R. China ABSTRACT ARTICLE INFO In this paper, we study the core acquisition and remanufacturing problem in which the remanufactured products are produced from acquired cores with uncertain quality condition, and are used to satisfy customer demand. Decision-making models are developed to examine the potential value of core sorting and quality grading in the remanufacturing system: a single-period model with deterministic demand, and a single-period model with stochastic demand (i.e., a newsvendor-type model). In each model, both the sorting strategy and the non-sorting strategy are discussed and compared. Our theoretical and numerical results show that: (1) In the deterministic demand case, core sorting is cost-effective only when the unit sorting cost is below a threshold value and the unit acquisition cost falls into a specific interval. Furthermore, in the case with two quality grades the adoption of sorting strategy with respect to the expected fraction of high-quality cores may be nonmonotone: an initial increase in the expected fraction of high-quality cores may motivate a switch to core sorting, however, further increase in the expected fraction may motivate a reverse switch; (2) Similarly, in the stochastic demand case, the sorting strategy also becomes unattractive when the unit sorting cost is sufficiently high. In addition, the value of core sorting will be better off under more fluctuating demand for remanufactured products if the sorting strategy is the dominant strategy. Otherwise, it will be worse off. Keywords: Remanufacturing; Product (core) acquisition management; Core sorting; Quality grading; Optimization; Analytical models; Cost-effectiveness *Corresponding author: huangd@bjtu.edu.cn (Huang, D.) Article history: Received 10 April 2023 Revised 12 June 2023 Accepted 17 June 2023 Content from this work may be used under the terms of the Creative Commons Attribution 4.0 International Licence (CC BY 4.0). Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. 1. Introduction Remanufacturing, which refers to the industrial process of restoring used products or parts to like-new condition, has become to be a more and more important part of the circular economy (Mihai et al., 2018), [1]. It also has been applied to various products such as automotive parts, electrical and electronics products, machinery, information and communication equipment, ink and toner cartridges, medical devices and furniture (Yoon and Joung, 2019; Davidavičienė et al., 2019, Yoon and Joung 2021), [2-4]. Many companies, including Daimler, Volvo, Lenovo, Huawei, Hewlett-Packard, Xerox, and IKEA, have engaged in the business of acquiring and remanufacturing used products (referred to as cores) from end-users. For example, the Volvo Group has reported that the total sales of remanufactured components amounted to SEK 10 billion in 2018 with a yearly average increasing rate of 10 %. Through an old-for-new program, Huawei has collected more than 140,000 used cell phones in 2018, and recovered and reused 82.3 % materials of these recycled cell phones. In a “Buy-back service” project launched in 2017, IKEA Japan 175 Cao, Shi, Lan, Huang recycled 1,900 second-hand furniture products in the first six months, and sold 85 % of them to customers after repair and refurbishing. Since the quality condition of acquired cores can be so highly variable that may complicate the remanufacturing process (Guide, 2000; He, 2018), [5, 6], cores are usually sorted and graded into different categories based on their quality conditions by remanufacturers. For example, ReCellular, a cellphone remanufacturer, classified the returned phones into six quality grades based on their functional and physical criteria in which the remanufacturing cost of the lowest quality grade is eight times more than that of the highest quality grade (Guide et al., 2003), [7]. Similarly, a returned Personal Computer may also fall into different conditions: some may be brand new with their package box has never been opened; some may be used for a few times; some may need repair; and some may only be salvaged for parts or materials (Blackburn et al., 2004), [8]. It was reported that Pitney-Bowes, an equipment remanufacturer, categorized the returned mechanical products into three quality grades (i.e., remanufacturing, reassembling or recycling) according to different recovery options (Ferguson et al., 2009), [9]. In the reverse logistics network studied by Sedehzadeh and Seifbarghy (2021), [10], returned products are classified into two categories according to their health security: usable products and recoverable products. The sorting and grading operation resolves the quality uncertainties of acquired cores in advance, and thus allows remanufacturing these cores in a greedy and economical sequence. In practice, core sorting and quality grading are typically implemented through visual inspection by sophisticated workers and usage information monitoring by information systems with tracking and tracing technology such as RFID. As the tasks of sorting and grading are primarily labour-intensive, it is usually costly to make such quality assessment. Furthermore, this prevailing operation helps firms to mitigate uncertainty in quality condition of cores, but requires acquiring more returned products than the demand for remanufactured products. As a result, the additional acquisition cost as well as the sorting cost offsets the remanufacturing cost savings obtained by processing a higher quality unit rather than a lower quality one. The trade-off between these two effects of core sorting and quality grading makes it less attractive in remanufacturing. Managers in remanufacturing firms are aware of the remanufacturing cost difference between different quality grades, but are unsure of the exact value of core quality information and the addition effort required to collecting and utilizing such information (Ferguson et al., 2009), [9]. Some researchers also found that under certain conditions remanufacturers can hardly benefit from core sorting and quality grading (see, e.g., Li et al., 2016; Yanıkoğlu and Denizel, 2021), [11, 12]. The research on core acquisition management, which refers to the management of the core acquisition process by dealing with the uncertainties with respect to return timing, quantity and core quality in reverse logistics, has intensified in recent years. An excellent review can be found in Wei et al. (2015), [13]. Our paper is most relevant to the research work that addressed the complications of quality uncertainty of cores in remanufacturing, thus we mainly review the literature of this research stream below. Guide et al. (2003), [7], first developed a single-period model to determine the optimal core acquisition prices and remanufactured products’ selling price. They assumed that the collectors grade cores and sell them to the remanufacturer in different quality classes at different prices, and within a certain quality class all the cores have the same associated remanufacturing cost. Galbreth and Blackburn (2006), [14], considered the case where the remanufacturer acquires unsorted used products from the third party collectors. The quality condition of cores is highly variable, and the remanufacturer must sort each core into different quality grades. They assumed that the cumulative distribution of cores condition is known exactly. In a single-period model, they derived optimal core acquisition and sorting policies for both deterministic demand case and stochastic demand case. Their sorting policy is defined by the value of remanufacturing cost: cores with remanufacturing cost above a threshold value are scrapped, and those with cost below the threshold value are remanufactured. After that, Galbreth and Blackburn (2010), [15], studied core acquisition decisions when the quality condition of each acquired core is uncertain. Their work is extended to the case where the quality condition of each acquired core follows general distributions by Yang et al. (2015), [16]. By incorporating quantity discount and carbon tax scheme into the core acquisition models dis176 Advances in Production Engineering & Management 18(2) 2023 When core sorting and quality grading is beneficial to remanufacturers: Insights from analytical models cussed in Galbreth and Blackburn (2006, 2010), [14, 15], Yang et al. (2016), [17], analysed the optimal core acquisition policies under the cases with quality variability and with both quality variability and condition uncertainty, respectively. Teunter and Flapper (2011), [18], assumed that there are multiple types of acquired cores, and that the type of a core follows a multinomial distribution. They derived optimal core acquisition and remanufacturing policies for both deterministic and stochastic demand. Mutha et al. (2016), [19], developed a two-period model to study core acquisition decisions when used products can be acquired either in bulk with uncertain quality levels, or in sorted grades with known quality levels, and the remanufacturer can acquire remanufacture cores before or after demand is realized. Lv et al. (2017), [20], discussed the two-period and multi-period manufacturing/remanufacturing models where the collection rate of used products can be influenced by a fixed investment. Mircea et al. (2023), [21] studied a repeated game in remanufacturing where used products are collected by the online/offline recycler and the manufacturer, and derived the optimal recycling prices for each collector. In a newsvendor-type model setting, Li et al. (2016), [11], analyzed the optimal decisions on core acquisition and remanufacturing quantities under both the remanufacturing-to-stock (RMTS) mode and the remanufacturing-to-order (RMTO) mode. They found that sorting may never be adopted in the RMTS mode regardless of the sorting cost. Ferguson et al. (2009), [9], and Yanıkoğlu and Denizel (2021), [12], studied the value of quality grading in the multi-period setting. The numerical results in Ferguson et al. (2009), [9], indicated that the ratio of return rates to demand rates, the number of quality grades, the distribution of the quality of returns and the cost difference between quality grades are the main drivers of the value of quality grading. Based on Ferguson et al. (2009)’s work [9], Yanıkoğlu and Denizel (2021), [12], study the core acquisition and remanufacturing problem under the case where the unit remanufacturing cost and unit resource requirement are uncertain. They found that when considering the unit grading cost, it may cause a significant deterioration in the value of grading, even make the grading totally useless. This work is motivated by academic research (see, e.g., Ferrer and Ketzenberg, 2004, [22]; Ketzenberg et al., 2006, [23]; Ferguson et al., 2009, [9]; Li et al., 2016, [11]), and industrial practice (see, e.g., ReCellular, Pitney-Bowes, PneuLaurent, Caterpillar). We study the core acquisition and remanufacturing problem under a single-period setting in which the remanufactured products used to satisfy customer demand are produced from acquired cores with uncertain quality. Both the deterministic demand case and the stochastic demand case are discussed to examine the potential value of core sorting and quality grading in remanufacturing. In each case, two strategies are discussed and compared: (1) the sorting strategy, i.e., sorting and remanufacturing cores in a greedy sequence, and (2) the non-sorting strategy, i.e., remanufacturing cores in the natural sequence. During these different model settings, we aim to address the following questions: (1) Is core sorting and quality grading a cost-effective operation for remanufacturing firms? (2) Under what conditions does the remanufacturer benefit from core sorting and quality grading? (3) What are the impacts of demand and cost parameters on the value of core sorting and quality grading? (4) Whether these impacts change under different core acquisition and remanufacturing decision-making environments? More specifically, we start with a single period model with deterministic demand, for which we derive the optimal core acquisition quantity for both the sorting and non-sorting strategy. By comparing with these two strategies, we find that core sorting and quality grading is costeffective only when the unit sorting cost is sufficiently small and the unit core acquisition cost falls into a specific interval. Further analysis in a special case with two quality grades show that, an initial increase in the expected fraction of high-quality cores may motivate a switch to core sorting and quality grading, however, further increase may motivate a reverse switch. We then continue our study in a newsvendor-type model and characterize the optimal acquisition, sorting and remanufacturing policies. Similar to the deterministic demand case, core sorting and quality grading is beneficial to the remanufacturer only when the sorting cost is sufficiently low. Another observation is that when the sorting strategy is superior to the non-sorting strategy, it is more valuable under a higher level of demand variation. Advances in Production Engineering & Management 18(2) 2023 177 Cao, Shi, Lan, Huang The rest of this paper is organized as follows. In Section 2, we describe the basic problem setting and introduce the notations and assumptions used throughout the paper. In Section 3 and Section 4, we present a single-period core acquisition and remanufacturing model with deterministic and stochastic demand, respectively. Finally, we conclude the paper in Section 5. 2. Problem description Consider a remanufacturing firm who acquires cores and sells remanufactured products to the market. The cores are acquired in bulk at unit acquisition cost 𝑐𝑐𝑟𝑟 and their quality conditions are uncertain. Without loss of generality, assume that the acquired cores can be categorized into 𝑁𝑁 nominal quality grades based on their quality levels, with grade 1 being the highest quality grade and grade 𝑁𝑁 being the lowest. In general, the higher quality grades are cheaper to remanufacture, that is, 𝑐𝑐1 < 𝑐𝑐2 <. . . < 𝑐𝑐𝑁𝑁 , where 𝑐𝑐𝑖𝑖 denotes the unit remanufacturing cost of grade 𝑖𝑖, 𝑖𝑖 = 1,2, . . . , 𝑁𝑁. The fraction of acquired cores belonging to grade 𝑖𝑖, 𝜃𝜃𝑖𝑖 , is a random variable with the domain [0, 1]. The expectation of 𝜃𝜃𝑖𝑖 is denoted by 𝜇𝜇𝑖𝑖 . The fractions 𝜃𝜃1 , 𝜃𝜃2 ,…, 𝜃𝜃𝑁𝑁 are jointly distributed with p.d.f. 𝑔𝑔𝜃𝜃1 ,𝜃𝜃2 ,...,𝜃𝜃𝑁𝑁 (∙) and c.d.f. 𝐺𝐺𝜃𝜃1 ,𝜃𝜃2 ,...,𝜃𝜃𝑁𝑁 (∙) and their summation is 1. After a quantity of cores, 𝑄𝑄 units, are acquired, the firm has two alternative choices: (1) adopting a sorting procedure to resolve the quality uncertainties by incurring a unit sorting cost 𝑐𝑐𝑔𝑔 , (2) doing nothing. If cores are sorted and graded, the fractions of core quality distribution are revealed before remanufacturing, otherwise they are known to the firm only until all the cores are remanufactured. The market demand for remanufactured products studied in the single-period setting has two alternative forms: (a) deterministic demand, (b) stochastic demand. For the deterministic case, market demand 𝐷𝐷 is known and no shortage is allowed. For the stochastic case, market demand 𝐷𝐷 is a random variable with probability density function 𝑓𝑓(𝐷𝐷) and cumulative distribution function 𝐹𝐹(𝐷𝐷). In addition, we assume that 𝐷𝐷 and (𝜃𝜃1 , 𝜃𝜃2 , . . . , 𝜃𝜃𝑁𝑁 ) are independent. At the beginning of each period, the remanufacturing firm decides remanufacturing quantities from each grade of core before demand is realized. Then demand realization is observed. A unit shortage penalty cost, 𝑏𝑏𝑟𝑟 , is charged for the unsatisfied demand and a unit holding cost, ℎ𝑟𝑟 , is charged for the leftover remanufactured product. At the end of the period, the leftover cores are disposed at unit cost, ℎ𝑢𝑢 . In general, the sequence of events is as follows: acquiring bulk cores, sorting and grading (or doing nothing), remanufacturing, demand realization. The decisions to be made by the remanufacturing firm include the core acquisition quantity and the remanufacturing quantity of each quality grade. The objective is to maximize the total expected profit with respect to core acquisition, remanufacturing, and demand fulfilment. In summary, the following notions and assumptions are employed in this paper: 𝑁𝑁 𝐷𝐷 𝑄𝑄 𝑝𝑝 𝑐𝑐𝑟𝑟 𝑐𝑐𝑖𝑖 𝑐𝑐𝑔𝑔 ℎ𝑢𝑢 𝑏𝑏𝑟𝑟 ℎ𝑟𝑟 𝜃𝜃𝑖𝑖 𝜇𝜇𝑖𝑖 𝑐𝑐𝜇𝜇 𝑔𝑔𝜃𝜃1 ,𝜃𝜃2 ,...,𝜃𝜃𝑁𝑁 (∙) 𝐺𝐺𝜃𝜃1 ,𝜃𝜃2 ,...,𝜃𝜃𝑁𝑁 (∙) 𝑓𝑓(𝐷𝐷) 178 number of nominal quality grades of cores market demand for remanufactured products acquisition quantity of cores unit selling price for a remanufactured product unit acquisition cost unit remanufacturing cost of grade 𝑖𝑖, 𝑖𝑖 = 1,2, . . . , 𝑁𝑁 unit sorting cost unit disposal cost unit shortage penalty cost for unsatisfied demand unit holding cost for the remanufactured product fraction of acquired cores belonging to grade 𝑖𝑖, 𝑖𝑖 = 1,2, . . . , 𝑁𝑁 expected value of 𝜃𝜃𝑖𝑖 , 𝑖𝑖 = 1,2, . . . , 𝑁𝑁 𝑁𝑁 expected unit remanufacturing cost, where 𝑐𝑐𝜇𝜇 = 𝐸𝐸�∑𝑁𝑁 𝑖𝑖=1 𝜃𝜃𝑖𝑖 𝑐𝑐𝑖𝑖 � = ∑𝑖𝑖=1 𝜇𝜇𝑖𝑖 𝑐𝑐𝑖𝑖 joint probability density function of 𝜃𝜃1 , 𝜃𝜃2 , . . . , 𝜃𝜃𝑁𝑁 joint cumulative distribution function of 𝜃𝜃1 , 𝜃𝜃2 , . . . , 𝜃𝜃𝑁𝑁 probability density function of 𝐷𝐷 Advances in Production Engineering & Management 18(2) 2023 When core sorting and quality grading is beneficial to remanufacturers: Insights from analytical models 𝐹𝐹(𝐷𝐷) 𝑞𝑞𝑖𝑖 𝜋𝜋 cumulative distribution function of 𝐷𝐷 quantity of cores of grade 𝑖𝑖 that are remanufactured, 𝑖𝑖 = 1,2, . . . , 𝑁𝑁 total expected profit of the remanufacturing firm 3. The single-period, deterministic demand case We begin our analysis with the single-period, deterministic demand case. Both the sorting scenario and the non-sorting scenario are discussed. For ease of exposition, we put superscript D and subscripts s and ns on some notations to represent these two scenarios, respectively. 3.1 The non-sorting scenario When the manufacturer adopts the non-sorting operation, it is optimal to acquire the exact de𝐷𝐷∗ 𝐷𝐷∗ and 𝜋𝜋𝑛𝑛𝑛𝑛 be the optimal acquisition quantity and the optimal manded quantity of cores. Let 𝑄𝑄𝑛𝑛𝑛𝑛 𝐷𝐷∗ 𝐷𝐷∗ = 𝐷𝐷 and 𝜋𝜋𝑛𝑛𝑛𝑛 = expected profit under the non-sorting scenario, respectively. Then we have 𝑄𝑄𝑛𝑛𝑛𝑛 (𝑝𝑝 − 𝑐𝑐𝑟𝑟 − 𝑐𝑐𝜇𝜇 )𝐷𝐷, where 𝑐𝑐𝜇𝜇 = ∑𝑁𝑁 𝜇𝜇 𝑐𝑐 is the expected unit remanufacturing cost. 𝑖𝑖=1 𝑖𝑖 𝑖𝑖 3.2 The sorting scenario Let 𝑄𝑄𝑠𝑠𝐷𝐷 be the acquisition quantity of cores under the sorting scenario in the deterministic demand case. We define the acquisition ratio as 𝛼𝛼 = 𝐷𝐷⁄𝑄𝑄𝑠𝑠𝐷𝐷 , where 𝛼𝛼 can be interpreted as the fraction of acquired cores that must be remanufactured to satisfy the demand. It is reasonable that the remanufacturer will acquire more cores than demanded under the sorting scenario (Galbreth and Blackburn, 2006), [14]. Thus, we have 0 < 𝛼𝛼 ≤ 1. The optimal remanufacturing policy is straightforward that follows a greedy rule: first processing cores of grade 1, and if the remanufacture-up-to level is still not reached, then processing cores of grade 2, and so on, stop until the remanufacture-up-to level is reached or all the cores are remanufactured. Let 𝑅𝑅0 = 0 and 𝑖𝑖 𝑅𝑅𝑖𝑖 = 𝑚𝑚𝑚𝑚𝑚𝑚 �𝐷𝐷, � 𝜃𝜃𝑗𝑗 𝑄𝑄𝑠𝑠𝐷𝐷 � , 𝑖𝑖 = 1,2, . . . , 𝑁𝑁 (1) 𝑞𝑞𝑖𝑖𝐷𝐷 = 𝑅𝑅𝑖𝑖 − 𝑅𝑅𝑖𝑖−1 , 𝑖𝑖 = 1,2, . . . , 𝑁𝑁 (2) 𝑗𝑗=1 Then the quantity of cores of grade 𝑖𝑖 remanufactured under the sorting scenario, 𝑞𝑞𝑖𝑖𝐷𝐷 , is The expected profit of the remanufacturer, 𝜋𝜋𝑠𝑠𝐷𝐷 , is 𝜋𝜋𝑠𝑠𝐷𝐷 = 𝑝𝑝 × 𝐷𝐷 − (𝑐𝑐𝑟𝑟 + 𝑐𝑐𝑔𝑔 )𝑄𝑄𝑠𝑠𝐷𝐷 𝑁𝑁 − 𝐸𝐸 �� 𝑐𝑐𝑖𝑖 𝑞𝑞𝑖𝑖𝐷𝐷 � − ℎ𝑢𝑢 (𝑄𝑄𝑠𝑠𝐷𝐷 − 𝐷𝐷) which can be rewritten as a function of 𝛼𝛼, where 𝑖𝑖=1 𝜋𝜋𝑠𝑠𝐷𝐷 (𝛼𝛼) = (𝑝𝑝 − 𝑐𝑐𝑁𝑁 + ℎ𝑢𝑢 ) × 𝐷𝐷 − (𝑐𝑐𝑟𝑟 + 𝑐𝑐𝑔𝑔 + ℎ𝑢𝑢 ) 𝑁𝑁−1 where 𝑔𝑔∑𝑖𝑖 𝑗𝑗=1 𝜃𝜃𝑗𝑗 {1,2, . . . , 𝑁𝑁}. (𝑥𝑥) and 𝐺𝐺∑𝑖𝑖 𝛼𝛼 𝐷𝐷 𝛼𝛼 𝑥𝑥 + 𝐷𝐷 × �(𝑐𝑐𝑖𝑖+1 − 𝑐𝑐𝑖𝑖 ) �1 − � (1 − 𝛼𝛼)𝑔𝑔∑𝑖𝑖 𝑗𝑗=1 𝜃𝜃𝑗𝑗 0 𝑖𝑖=1 𝑗𝑗=1 𝜃𝜃𝑗𝑗 (𝑥𝑥)𝑑𝑑𝑑𝑑 � (3) (4) (𝑥𝑥) are the p.d.f. and c.d.f. of 𝑥𝑥, where 𝑥𝑥 = ∑𝑖𝑖𝑗𝑗=1 𝜃𝜃𝑗𝑗 for any 𝑖𝑖 ∈ Theorem 1. If 𝑐𝑐𝑟𝑟 + 𝑐𝑐𝑔𝑔 + ℎ𝑢𝑢 ≥ 𝑐𝑐𝑁𝑁 − 𝑐𝑐𝜇𝜇 , 𝜋𝜋𝑠𝑠𝐷𝐷 (𝛼𝛼) is monotonically increasing in 𝛼𝛼, and 𝛼𝛼 ∗ = 1. Otherwise, 𝜋𝜋𝑠𝑠𝐷𝐷 (𝛼𝛼) is quasi-concave in 𝛼𝛼, and 𝛼𝛼 ∗ is given by 𝑁𝑁−1 𝛼𝛼 ∗ � �(𝑐𝑐𝑖𝑖+1 − 𝑐𝑐𝑖𝑖 ) � 𝑥𝑥𝑔𝑔∑𝑖𝑖 𝑖𝑖=1 0 𝑗𝑗=1 𝜃𝜃𝑗𝑗 (𝑥𝑥)𝑑𝑑𝑑𝑑 � = 𝑐𝑐𝑟𝑟 + 𝑐𝑐𝑔𝑔 + ℎ𝑢𝑢 Proof: Take the first derivate of 𝜋𝜋𝑠𝑠𝐷𝐷 (𝛼𝛼) with respect to 𝛼𝛼, we have Advances in Production Engineering & Management 18(2) 2023 (5) 179 Cao, Shi, Lan, Huang 𝜋𝜋𝑠𝑠,𝐷𝐷 (𝛼𝛼) 𝑁𝑁−1 𝛼𝛼 𝐷𝐷 𝐷𝐷 = (𝑐𝑐𝑟𝑟 + 𝑐𝑐𝑔𝑔 + ℎ𝑢𝑢 ) 2 − 2 � �(𝑐𝑐𝑖𝑖+1 − 𝑐𝑐𝑖𝑖 ) � 𝑥𝑥𝑔𝑔∑𝑖𝑖 𝜃𝜃 (𝑥𝑥)𝑑𝑑𝑑𝑑 � 𝑗𝑗=1 𝑗𝑗 𝛼𝛼 𝛼𝛼 0 𝑖𝑖=1 𝛼𝛼 It is easy to verify that the term ∑𝑁𝑁−1 𝑖𝑖=1 �(𝑐𝑐𝑖𝑖+1 − 𝑐𝑐𝑖𝑖 ) ∫0 𝑥𝑥𝑔𝑔∑𝑖𝑖 𝜋𝜋𝑠𝑠,𝐷𝐷 (𝛼𝛼) is decreasing in 𝛼𝛼. Note that 𝛼𝛼 (i) when 𝛼𝛼 → 0, ∑𝑁𝑁−1 𝑖𝑖=1 �(𝑐𝑐𝑖𝑖+1 − 𝑐𝑐𝑖𝑖 ) ∫0 𝑥𝑥𝑔𝑔∑𝑖𝑖 𝛼𝛼 𝑗𝑗=1 𝜃𝜃𝑗𝑗 (ii) when 𝛼𝛼 = 1, ∑𝑁𝑁−1 𝑖𝑖=1 �(𝑐𝑐𝑖𝑖+1 − 𝑐𝑐𝑖𝑖 ) ∫0 𝑥𝑥𝑔𝑔∑𝑖𝑖 (𝑥𝑥)𝑑𝑑𝑑𝑑 � → 0, 𝑗𝑗=1 𝜃𝜃𝑗𝑗 𝑗𝑗=1 𝜃𝜃𝑗𝑗 (𝑥𝑥)𝑑𝑑𝑑𝑑 � is increasing in 𝛼𝛼, and 𝑖𝑖 (𝑥𝑥)𝑑𝑑𝑑𝑑 � = ∑𝑁𝑁−1 𝑖𝑖=1 �(𝑐𝑐𝑖𝑖+1 − 𝑐𝑐𝑖𝑖 ) ∑𝑗𝑗=1 𝜇𝜇𝑗𝑗 � = 𝑐𝑐𝑁𝑁 − 𝑐𝑐𝜇𝜇 . If 𝑐𝑐𝑟𝑟 + 𝑐𝑐𝑔𝑔 + ℎ𝑢𝑢 ≥ 𝑐𝑐𝑁𝑁 − 𝑐𝑐𝜇𝜇 , then we always have 𝜋𝜋𝑠𝑠,𝐷𝐷 (𝛼𝛼) ≥ 0 for any 𝛼𝛼 ∈ (0,1]. This implies that 𝜋𝜋𝑠𝑠𝐷𝐷 (𝛼𝛼) is increasing in 𝛼𝛼, and attains its maximum at 𝛼𝛼 = 1. If𝑐𝑐𝑟𝑟 + 𝑐𝑐𝑔𝑔 + ℎ𝑢𝑢 < 𝑐𝑐𝑁𝑁 − 𝑐𝑐𝜇𝜇 , with the increase of 𝛼𝛼, 𝜋𝜋𝑠𝑠,𝐷𝐷 (𝛼𝛼) decreases and changes sign at most once from positive to negative. Denote 𝛼𝛼 ∗ as the changing point where 𝜋𝜋𝑠𝑠,𝐷𝐷 (𝛼𝛼 ∗ ) = 0. Thus, 𝜋𝜋𝑠𝑠,𝐷𝐷 (𝛼𝛼) > 0 for any 𝛼𝛼 ∈ (0, 𝛼𝛼 ∗ ) and 𝜋𝜋𝑠𝑠,𝐷𝐷 (𝛼𝛼) < 0 for any 𝛼𝛼 ∈ (𝛼𝛼 ∗ , 1]. This implies that 𝜋𝜋𝑠𝑠𝐷𝐷 (𝛼𝛼) is increasing in 𝛼𝛼 on the interval (0, 𝛼𝛼 ∗ ) and decreasing in 𝛼𝛼 on the interval (𝛼𝛼 ∗ , 1]. Therefore, 𝜋𝜋𝑠𝑠𝐷𝐷 (𝛼𝛼) is quasi-concave in 𝛼𝛼, and attains its maximum at 𝛼𝛼 = 𝛼𝛼 ∗ .□ Theorem 1 suggests that when the unit core processing (i.e., acquisition and sorting and disposal) cost is sufficiently high, the acquired cores are fully remanufactured. Otherwise, they are partially remanufactured. However, it is never optimal to adopt the sorting operation if all the acquired cores are remanufactured (i.e., 𝛼𝛼 ∗ = 1). 3.3 Comparative analysis We examine the value of core sorting by comparing the expected profit of the remanufacturer under both the sorting scenario and the non-sorting scenario. Proposition 1 characterizes the situations when core sorting is beneficial to the remanufacturer. ∗ 𝐷𝐷 Proposition 1. There exist 𝑐𝑐𝑟𝑟𝑚𝑚𝑚𝑚𝑚𝑚 , 𝑐𝑐𝑟𝑟𝑚𝑚𝑚𝑚𝑚𝑚 and 𝑐𝑐𝑔𝑔𝑚𝑚𝑚𝑚𝑚𝑚 such that (i) 𝜋𝜋𝑠𝑠𝐷𝐷 (𝛼𝛼 ∗ ) > 𝜋𝜋𝑛𝑛𝑛𝑛 , if 𝑐𝑐𝑟𝑟𝑚𝑚𝑚𝑚𝑚𝑚 < 𝑐𝑐𝑟𝑟 < 𝑐𝑐𝑟𝑟𝑚𝑚𝑚𝑚𝑚𝑚 ∗ 𝐷𝐷 and 𝑐𝑐𝑔𝑔 < 𝑐𝑐𝑔𝑔𝑚𝑚𝑚𝑚𝑚𝑚 ; (ii) 𝜋𝜋𝑠𝑠𝐷𝐷 (𝛼𝛼 ∗ ) ≤ 𝜋𝜋𝑛𝑛𝑛𝑛 , otherwise. Proof. By Proposition 1, when 𝑐𝑐𝑟𝑟 + 𝑐𝑐𝑔𝑔 + ℎ𝑢𝑢 ≥ 𝑐𝑐𝑁𝑁 − 𝑐𝑐𝜇𝜇 , 𝛼𝛼 ∗ = 1, i.e., all the acquired cores are remanufactured. Then it is never optimal for the remanufacturer to choose the sorting strategy since additional sorting cost is required but no remanufacturing cost savings are obtained from 𝐷𝐷∗ core sorting. Thus, under this case we have 𝜋𝜋𝑠𝑠𝐷𝐷 (𝛼𝛼 ∗ ) < 𝜋𝜋𝑛𝑛𝑛𝑛 . When 𝑐𝑐𝑟𝑟 + 𝑐𝑐𝑔𝑔 + ℎ𝑢𝑢 < 𝑐𝑐𝑁𝑁 − 𝑐𝑐𝜇𝜇 , 𝜋𝜋𝑠𝑠𝐷𝐷 (𝛼𝛼 ∗ ) − 𝐷𝐷∗ 𝜋𝜋𝑛𝑛𝑛𝑛 𝑁𝑁−1 = �𝑐𝑐𝑟𝑟 + 𝑐𝑐𝜇𝜇 − 𝑐𝑐1 + ℎ𝑢𝑢 �𝐷𝐷 − 𝐷𝐷 � �(𝑐𝑐𝑖𝑖+1 − 𝑐𝑐𝑖𝑖 )𝐺𝐺∑𝑖𝑖 Thus, if 𝑐𝑐𝑟𝑟 > 𝑐𝑐1 + ∑𝑁𝑁−1 𝑖𝑖=1 �(𝑐𝑐𝑖𝑖+1 − 𝑐𝑐𝑖𝑖 )𝐺𝐺∑𝑖𝑖 𝐷𝐷∗ . 𝜋𝜋𝑠𝑠𝐷𝐷 (𝛼𝛼 ∗ ) ≤ 𝜋𝜋𝑛𝑛𝑛𝑛 𝑚𝑚𝑚𝑚𝑚𝑚 Let 𝑐𝑐𝑟𝑟 = 𝑐𝑐𝑁𝑁 𝑗𝑗=1 𝜃𝜃𝑗𝑗 𝑗𝑗=1 𝜃𝜃𝑗𝑗 𝑖𝑖=1 (𝛼𝛼 ∗ )� 𝐷𝐷∗ (𝛼𝛼 ∗ )� − 𝑐𝑐𝜇𝜇 − ℎ𝑢𝑢 , then 𝜋𝜋𝑠𝑠𝐷𝐷 (𝛼𝛼 ∗ ) > 𝜋𝜋𝑛𝑛𝑛𝑛 ; otherwise, 𝑁𝑁−1 − 𝑐𝑐𝜇𝜇 − 𝑐𝑐𝑔𝑔 − ℎ𝑢𝑢 , and 𝑐𝑐𝑟𝑟𝑚𝑚𝑚𝑚𝑚𝑚 = 𝑐𝑐1 + ∑𝑖𝑖=1 �(𝑐𝑐𝑖𝑖+1 − 𝑐𝑐𝑖𝑖 )𝐺𝐺∑𝑖𝑖 ∗ 𝑗𝑗=1 𝜃𝜃𝑗𝑗 (𝛼𝛼 ∗ )� − 𝑐𝑐𝜇𝜇 − ℎ𝑢𝑢 . Note 𝐷𝐷 that under this case the necessary condition for 𝜋𝜋𝑠𝑠𝐷𝐷 (𝛼𝛼 ∗ ) > 𝜋𝜋𝑛𝑛𝑛𝑛 is 𝑐𝑐𝑟𝑟𝑚𝑚𝑚𝑚𝑚𝑚 > 𝑐𝑐𝑟𝑟𝑚𝑚𝑚𝑚𝑚𝑚 , i.e., 𝑐𝑐𝑔𝑔 < ∑𝑁𝑁−1 𝑖𝑖=1 �(𝑐𝑐𝑖𝑖+1 − 𝑐𝑐𝑖𝑖 )[1 − 𝐺𝐺∑𝑖𝑖 𝑗𝑗=1 𝜃𝜃𝑗𝑗 (𝛼𝛼 ∗ )]� = 𝑐𝑐𝑔𝑔𝑚𝑚𝑚𝑚𝑚𝑚 . Proposition 1 implies that core sorting in remanufacturing is cost-effective only when the unit sorting cost is smaller than a threshold and the unit core acquisition cost falls into a specific interval. Furthermore, we examine the two-grade case to gain more managerial insights. It is found that the adoption of core sorting with respect to the expected fraction of high-quality cores, 𝜇𝜇1 , may be non-monotone: an initial increase in 𝜇𝜇1 may motivate a switch to core sorting, however, further increase in 𝜇𝜇1 may motivate a reverse switch. Proposition 2. In the case with two quality grades, there exist 𝜇𝜇1𝑚𝑚𝑚𝑚𝑚𝑚 , 𝜇𝜇1𝑚𝑚𝑚𝑚𝑚𝑚 and 𝑐𝑐𝑔𝑔𝑚𝑚𝑚𝑚𝑚𝑚 such that (i) 𝐷𝐷∗ 𝐷𝐷∗ 𝜋𝜋𝑠𝑠𝐷𝐷 (𝛼𝛼 ∗ ) > 𝜋𝜋𝑛𝑛𝑛𝑛 , if 𝜇𝜇1𝑚𝑚𝑚𝑚𝑚𝑚 < 𝜇𝜇1 < 𝜇𝜇1𝑚𝑚𝑚𝑚𝑚𝑚 and 𝑐𝑐𝑔𝑔 < 𝑐𝑐𝑔𝑔𝑚𝑚𝑚𝑚𝑚𝑚 ; (ii) 𝜋𝜋𝑠𝑠𝐷𝐷 (𝛼𝛼 ∗ ) ≤ 𝜋𝜋𝑛𝑛𝑛𝑛 , otherwise. 180 Advances in Production Engineering & Management 18(2) 2023 When core sorting and quality grading is beneficial to remanufacturers: Insights from analytical models Proof. According to the proof of Proposition 1, when 𝑁𝑁 = 2, from 𝑐𝑐𝑟𝑟𝑚𝑚𝑚𝑚𝑚𝑚 < 𝑐𝑐𝑟𝑟 < 𝑐𝑐𝑟𝑟𝑚𝑚𝑚𝑚𝑚𝑚 we have 𝑐𝑐𝑟𝑟 + 𝑐𝑐𝑔𝑔 + ℎ𝑢𝑢 𝑐𝑐𝑟𝑟 + ℎ𝑢𝑢 < 𝜇𝜇1 < 1 − 𝐺𝐺𝜃𝜃1 (𝛼𝛼 ∗ ) + = 𝜇𝜇1𝑚𝑚𝑚𝑚𝑚𝑚 𝜇𝜇1𝑚𝑚𝑚𝑚𝑚𝑚 = 𝑐𝑐2 − 𝑐𝑐1 𝑐𝑐2 − 𝑐𝑐1 Furthermore, from 𝜇𝜇1𝑚𝑚𝑚𝑚𝑚𝑚 > 𝜇𝜇1𝑚𝑚𝑚𝑚𝑚𝑚 we have 𝑐𝑐𝑔𝑔 < (𝑐𝑐2 − 𝑐𝑐1 )[1 − 𝐺𝐺𝜃𝜃1 (𝛼𝛼 ∗ )] = 𝑐𝑐𝑔𝑔𝑚𝑚𝑚𝑚𝑚𝑚 . The results of Proposition 2 are illustrated in Fig. 1, where the parameter settings in the associated numerical example are: 𝐷𝐷 = 50, 𝑝𝑝 = 100, 𝑐𝑐𝑟𝑟 = 2, 𝑐𝑐𝑔𝑔 = 2, ℎ𝑢𝑢 = 1, 𝑐𝑐1 = 5, 𝑐𝑐2 = 30. The fraction of acquired cores with high-quality level, 𝜃𝜃1 , follows a Beta distribution, i.e., 𝜃𝜃1 ~B(𝑎𝑎, 𝑏𝑏). We consider different values of 𝑎𝑎 and 𝑏𝑏 in the numerical study, where (𝑎𝑎, 𝑏𝑏) ∈ {𝑎𝑎 = 0.5𝑖𝑖, 𝑏𝑏 = 10 − 0.5𝑖𝑖, 𝑖𝑖 = 1, . . . ,19}, thus the expectation of 𝜃𝜃1 , 𝜇𝜇1 , changes its value from 0.05 to 0.95 in steps of 0.05. The impacts of the expected fraction of high-quality cores, 𝜇𝜇1 , on the expected profit of the remanufacturer are shown in Fig. 1. The explanation is as follows. When the fraction of highquality cores is excessively high, there is no need to sorting cores as most of the acquired cores are in good quality condition. While when the fraction of high-quality cores is excessively low, core sorting could not help to improve the poor quality condition of acquired cores and thus is useless. Fig. 1 Remanufacturer’s expected profit under sorting and non-sorting strategies at different levels of expected fraction of high-quality cores 4. The single-period, stochastic demand case Then we extend our analysis to the single-period, stochastic demand case. Similarly, we use superscript R and subscripts s and ns to represent this case under sorting and non-sorting scenarios. 4.1 The non-sorting scenario In the non-sorting scenario, without the quality information of cores in advance, the firm will process all the acquired cores to satisfy market demand. The decision-making problem is a newsvendor-type problem. The expected profit of the firm is 𝑅𝑅 𝑅𝑅 𝑅𝑅 + 𝑅𝑅 𝑅𝑅 𝜋𝜋𝑛𝑛𝑛𝑛 = 𝐸𝐸[𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝(𝑄𝑄𝑛𝑛𝑛𝑛 , 𝐷𝐷) − 𝑏𝑏𝑟𝑟 (𝐷𝐷 − 𝑄𝑄𝑛𝑛𝑛𝑛 ) − ℎ𝑟𝑟 (𝑄𝑄𝑛𝑛𝑛𝑛 − 𝐷𝐷)+ ] − (𝑐𝑐𝜇𝜇 + 𝑐𝑐𝑟𝑟 )𝑄𝑄𝑛𝑛𝑛𝑛 𝑅𝑅∗ The optimal acquisition quantity, 𝑄𝑄𝑛𝑛𝑛𝑛 , is ∗ 𝑅𝑅 𝑄𝑄𝑛𝑛𝑛𝑛 = 𝐹𝐹 −1 �1 − 𝑐𝑐𝑟𝑟 + 𝑐𝑐𝜇𝜇 + ℎ𝑟𝑟 � 𝑝𝑝 + 𝑏𝑏𝑟𝑟 + ℎ𝑟𝑟 Advances in Production Engineering & Management 18(2) 2023 (6) (7) 181 Cao, Shi, Lan, Huang 4.2 The sorting scenario Let 𝑄𝑄𝑠𝑠𝑅𝑅 be the acquisition quantity of the cores, and 𝑞𝑞𝑖𝑖𝑅𝑅 the quantity of cores of grade 𝑖𝑖 remanufactured. The expected profit of the firm under the sorting scenario, is 𝜋𝜋𝑠𝑠𝑅𝑅 = 𝐸𝐸[𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝(𝑄𝑄𝑠𝑠𝑅𝑅 , 𝐷𝐷) 𝑁𝑁 − � 𝑐𝑐𝑖𝑖 𝑞𝑞𝑖𝑖𝑅𝑅 𝑖𝑖=1 − 𝑏𝑏𝑟𝑟 (𝐷𝐷 − 𝑁𝑁 � 𝑞𝑞𝑖𝑖𝑅𝑅 )+ 𝑖𝑖=1 − 𝑁𝑁 ℎ𝑟𝑟 (� 𝑞𝑞𝑖𝑖𝑅𝑅 𝑖𝑖=1 𝑁𝑁 − 𝐷𝐷)+ − ℎ𝑢𝑢 (𝑄𝑄𝑠𝑠𝑅𝑅 − � 𝑞𝑞𝑖𝑖𝑅𝑅 )] 𝑖𝑖=1 (8) −(𝑐𝑐𝑟𝑟 + 𝑐𝑐𝑔𝑔 )𝑄𝑄𝑠𝑠𝑅𝑅 𝑠𝑠. 𝑡𝑡. 0 ≤ 𝑞𝑞𝑖𝑖𝑅𝑅 ≤ 𝜃𝜃𝑖𝑖 𝑄𝑄𝑠𝑠𝑅𝑅 , 𝑖𝑖 = 1,2, . . . , 𝑁𝑁 Similarly, let 𝑅𝑅0 = 0 and 𝑅𝑅𝑖𝑖 be the aggregate remanufactured quantity from grade 1 to grade 𝑖𝑖 cores, 𝑖𝑖 = 1,2, . . . , 𝑁𝑁. Then following the greedy rule, 𝑞𝑞𝑖𝑖𝑅𝑅 = 𝑅𝑅𝑖𝑖 − 𝑅𝑅𝑖𝑖−1 . The expected profit (8) can be rewritten as 𝜋𝜋𝑠𝑠𝑅𝑅 = 𝐸𝐸[𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝(𝑄𝑄𝑠𝑠𝑅𝑅 , 𝐷𝐷) 𝑁𝑁 − � 𝑐𝑐𝑖𝑖 (𝑅𝑅𝑖𝑖 − 𝑅𝑅𝑖𝑖−1 ) − 𝑏𝑏𝑟𝑟 (𝐷𝐷 − 𝑅𝑅𝑁𝑁 )+ − ℎ𝑟𝑟 (𝑅𝑅𝑁𝑁 − 𝐷𝐷)+ − ℎ𝑢𝑢 (𝑄𝑄𝑠𝑠𝑅𝑅 − 𝑅𝑅𝑁𝑁 )] 𝑖𝑖=1 −(𝑐𝑐𝑟𝑟 + 𝑐𝑐𝑔𝑔 )𝑄𝑄𝑠𝑠𝑅𝑅 𝑠𝑠. 𝑡𝑡. 0 ≤ 𝑅𝑅𝑖𝑖 ≤ ∑𝑖𝑖𝑘𝑘=1 𝜃𝜃𝑘𝑘 𝑄𝑄𝑠𝑠𝑅𝑅 , 𝑖𝑖 = 1,2, . . . , 𝑁𝑁 (9) The remanufacture-up-to levels to the unconstrained optimization problem (9) are characterized in Proposition 3, which are also newsvendor-type solutions. Proposition 3. The remanufacture-up-to level from grade 1 to grade 𝑖𝑖 core, 𝑅𝑅𝑖𝑖∗, is 𝑐𝑐𝑖𝑖 + ℎ𝑟𝑟 − ℎ𝑢𝑢 𝑅𝑅𝑖𝑖∗ = 𝐹𝐹 −1 �1 − � , 𝑖𝑖 = 1,2, . . . , 𝑁𝑁 𝑝𝑝 + 𝑏𝑏𝑟𝑟 + ℎ𝑟𝑟 where 𝑅𝑅1∗ > 𝑅𝑅2∗ >. . . > 𝑅𝑅𝑁𝑁∗ . (10) Proof. If only cores of grade 1 to grade 𝑖𝑖 (𝑖𝑖 = 1,2, . . . , 𝑁𝑁) are remanufactured, then the remanufacturer’s expected profit (9) can be rewritten as 𝜋𝜋𝑠𝑠𝑅𝑅 𝑖𝑖 = 𝐸𝐸[𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝(𝑅𝑅𝑖𝑖 , 𝐷𝐷) − � 𝑐𝑐𝑙𝑙 (𝑅𝑅𝑙𝑙 − 𝑅𝑅𝑙𝑙−1 ) − 𝑏𝑏𝑟𝑟 (𝐷𝐷 − 𝑅𝑅𝑖𝑖 )+ − ℎ𝑟𝑟 (𝑅𝑅𝑖𝑖 − 𝐷𝐷)+ − ℎ𝑢𝑢 (𝑄𝑄𝑠𝑠𝑅𝑅 − 𝑅𝑅𝑖𝑖 )] − (𝑐𝑐𝑟𝑟 𝑙𝑙=1 + 𝑐𝑐𝑔𝑔 )𝑄𝑄𝑠𝑠𝑅𝑅 Taking the first and second derivative of 𝜋𝜋𝑠𝑠𝑅𝑅 with respect to 𝑅𝑅𝑖𝑖 respectively, we have 𝜕𝜕𝜋𝜋𝑠𝑠𝑅𝑅 = (𝑝𝑝 + 𝑏𝑏𝑟𝑟 + ℎ𝑢𝑢 − 𝑐𝑐𝑖𝑖 ) − (𝑝𝑝 + 𝑏𝑏𝑟𝑟 + ℎ𝑟𝑟 )𝐹𝐹(𝑅𝑅𝑖𝑖 ) 𝜕𝜕𝑅𝑅𝑖𝑖 and 𝜕𝜕 2 𝜋𝜋𝑠𝑠𝑅𝑅 = −(𝑝𝑝 + 𝑏𝑏𝑟𝑟 + ℎ𝑟𝑟 )𝑓𝑓(𝑅𝑅𝑖𝑖 ) < 0 𝜕𝜕𝑅𝑅𝑖𝑖 2 Thus, 𝜋𝜋𝑠𝑠𝑅𝑅 is concave in 𝑅𝑅𝑖𝑖 and the optimal remanufacture-up-to level is given by the first-order conditions. Moreover, since 𝑐𝑐1 < 𝑐𝑐2 <. . . < 𝑐𝑐𝑁𝑁 , it is easy to verify that 𝑅𝑅1∗ > 𝑅𝑅2∗ >. . . > 𝑅𝑅𝑁𝑁∗ .□ Let 𝑦𝑦𝑖𝑖∗ be the optimal aggregate remanufactured quantity from grade 1 to grade 𝑖𝑖 cores after ∗ remanufacturing, then the optimal remanufacturing quantity of cores of grade 𝑖𝑖 is 𝑞𝑞𝑖𝑖𝑅𝑅 = 𝑦𝑦𝑖𝑖∗ − ∗ , where 𝑦𝑦0∗ = 0. Based on the remanufacture-up-to levels derived in Proposition 3, the opti𝑦𝑦𝑖𝑖−1 mal remanufacturing policy under the sorting strategy with stochastic demand is characterized in Theorem 2. Theorem 2. After core sorting, the fractions 𝜃𝜃1 , 𝜃𝜃2 , . . . , 𝜃𝜃𝑁𝑁 are realized, then for a given core acquisition quantity 𝑄𝑄𝑠𝑠𝑅𝑅 , the optimal remanufacturing policy, {𝑦𝑦1∗ , . . . , 𝑦𝑦𝑖𝑖∗ , . . . , 𝑦𝑦𝑁𝑁∗ }, is as follows: (1) if 𝑅𝑅1∗ ≤ 𝜃𝜃1 𝑄𝑄𝑠𝑠𝑅𝑅 , 𝑦𝑦𝑁𝑁∗ =. . . = 𝑦𝑦1∗ = 𝑅𝑅1∗; (2) for 𝑖𝑖 = 2, . . . , 𝑁𝑁, ∗ ∗ ∗ ∗ 𝑖𝑖 𝑙𝑙 𝑅𝑅 𝑅𝑅 ∗ 𝑅𝑅 if ∑𝑖𝑖−1 𝑘𝑘=1 𝜃𝜃𝑘𝑘 𝑄𝑄𝑠𝑠 < 𝑅𝑅𝑖𝑖 ≤ ∑𝑘𝑘=1 𝜃𝜃𝑘𝑘 𝑄𝑄𝑠𝑠 , 𝑦𝑦𝑁𝑁 =. . . = 𝑦𝑦𝑖𝑖 = 𝑅𝑅𝑖𝑖 , 𝑦𝑦𝑙𝑙 = ∑𝑘𝑘=1 𝜃𝜃𝑘𝑘 𝑄𝑄𝑠𝑠 , 𝑙𝑙 = 1,2, . . . , 𝑖𝑖 − 1; ∗ ∗ ∗ ∗ ∗ 𝑖𝑖−1 𝑙𝑙 ∗ 𝑅𝑅 𝑅𝑅 if 𝑅𝑅𝑖𝑖 ≤ ∑𝑘𝑘=1 𝜃𝜃𝑘𝑘 𝑄𝑄𝑠𝑠 < 𝑅𝑅𝑖𝑖−1, 𝑦𝑦𝑁𝑁 =. . . = 𝑦𝑦𝑖𝑖 = 𝑦𝑦𝑖𝑖−1 , 𝑦𝑦𝑙𝑙 = ∑𝑘𝑘=1 𝜃𝜃𝑘𝑘 𝑄𝑄𝑠𝑠 , 𝑙𝑙 = 1,2, . . . , 𝑖𝑖 − 1; (3) if 𝑄𝑄𝑠𝑠𝑅𝑅 < 𝑅𝑅𝑁𝑁∗ , 𝑦𝑦𝑖𝑖∗ = ∑𝑖𝑖𝑘𝑘=1 𝜃𝜃𝑘𝑘 𝑄𝑄𝑠𝑠𝑅𝑅 , 𝑖𝑖 = 1,2, . . . , 𝑁𝑁. 182 Advances in Production Engineering & Management 18(2) 2023 When core sorting and quality grading is beneficial to remanufacturers: Insights from analytical models Proof. According to Proposition 3, when cores of grade 1 to grade 𝑖𝑖 (𝑖𝑖 = 1,2, . . . , 𝑁𝑁) are remanufactured, the optimal remanufacture-up-to level is 𝑅𝑅𝑖𝑖∗, which is given by Eq. (10). However, for a given core acquisition quantity 𝑄𝑄𝑠𝑠𝑅𝑅 and the realized fractions 𝜃𝜃1 , 𝜃𝜃2 , . . . , 𝜃𝜃𝑁𝑁 , the available quantity of grade 1 to grade 𝑖𝑖 cores is ∑𝑖𝑖𝑘𝑘=1 𝜃𝜃𝑘𝑘 𝑄𝑄𝑠𝑠𝑅𝑅 . Therefore, the optimal aggregate remanufactured quantity from grade 1 to grade 𝑖𝑖 cores after remanufacturing, 𝑦𝑦𝑖𝑖∗ , depends on the relationship between 𝑅𝑅𝑖𝑖∗ and ∑𝑖𝑖𝑘𝑘=1 𝜃𝜃𝑘𝑘 𝑄𝑄𝑠𝑠𝑅𝑅 . ∗ 𝑅𝑅 𝑅𝑅 When 𝑅𝑅𝑖𝑖∗ > ∑𝑖𝑖𝑘𝑘=1 𝜃𝜃𝑘𝑘 𝑄𝑄𝑠𝑠𝑅𝑅 , we have 𝑅𝑅1∗ >. . . > 𝑅𝑅𝑖𝑖−1 > 𝑅𝑅𝑖𝑖∗ > ∑𝑖𝑖𝑘𝑘=1 𝜃𝜃𝑘𝑘 𝑄𝑄𝑠𝑠𝑅𝑅 > ∑𝑖𝑖−1 𝑘𝑘=1 𝜃𝜃𝑘𝑘 𝑄𝑄𝑠𝑠 >. . . > 𝜃𝜃1 𝑄𝑄𝑠𝑠 , then it is optimal to make grade 1 to grade 𝑖𝑖 cores fully remanufactured. Otherwise, it is optimal 𝑅𝑅 to remanufacture up to 𝑅𝑅𝑖𝑖∗, however, if 𝑅𝑅𝑖𝑖∗ < ∑𝑖𝑖−1 𝑘𝑘=1 𝜃𝜃𝑘𝑘 𝑄𝑄𝑠𝑠 , then grade 𝑖𝑖 cores will never be remanufactured. Theorem 2 implies that in the optimal remanufacturing policy there exists a certain quality grade so that the acquired cores from grades that are lower than this grade are fully remanufactured, and the acquired cores from grades that are higher than this grade are disposed of, while the acquired cores of this grade are partially remanufactured. Some properties of 𝜋𝜋𝑠𝑠𝑅𝑅 are derived in Proposition 4, which can facilitate us to derive the optimal core acquisition quantity. Proposition 4. 𝜋𝜋𝑠𝑠𝑅𝑅 (𝑄𝑄𝑠𝑠𝑅𝑅 ) has the following properties: (i) 𝜋𝜋𝑠𝑠𝑅𝑅 (𝑄𝑄𝑠𝑠𝑅𝑅 ) is piece-wise concave and continuous with respect to 𝑄𝑄𝑠𝑠𝑅𝑅 ; (ii) 𝜋𝜋𝑠𝑠𝑅𝑅 (𝑄𝑄𝑠𝑠𝑅𝑅 ) has a unique global optimum. Proof. The proof follows along the lines of proof in Brown and Lee (2003), [24]. In Eq. (8), since {𝑄𝑄𝑠𝑠𝑅𝑅 : 𝑄𝑄𝑠𝑠𝑅𝑅 ≥ 0} is a convex set, �𝑞𝑞1𝑅𝑅 , . . . , 𝑞𝑞𝑁𝑁𝑅𝑅 : 0 ≤ 𝑞𝑞𝑖𝑖𝑅𝑅 ≤ 𝜃𝜃𝑖𝑖 𝑄𝑄𝑠𝑠𝑅𝑅 , 𝑖𝑖 = 1,2, . . . , 𝑁𝑁� is a nonempty set, and −𝜋𝜋𝑠𝑠𝑅𝑅 is a convex function on the nonempty convex set, then following Proposition A.4 in Porteus −𝜋𝜋𝑠𝑠𝑅𝑅 (𝑄𝑄𝑠𝑠𝑅𝑅 ) is convex in 𝑄𝑄𝑠𝑠𝑅𝑅 , i.e., max 𝜋𝜋 𝑅𝑅 (𝑄𝑄𝑠𝑠𝑅𝑅 ) is concave in 𝑄𝑄𝑠𝑠𝑅𝑅 . Hence, (2002), [25], min 𝑅𝑅 𝑅𝑅 𝑅𝑅 𝑅𝑅 𝑠𝑠 0≤𝑞𝑞𝑖𝑖 ≤𝜃𝜃𝑖𝑖 𝑄𝑄𝑠𝑠 0≤𝑞𝑞𝑖𝑖 ≤𝜃𝜃𝑖𝑖 𝑄𝑄𝑠𝑠 𝑅𝑅 𝑅𝑅 𝜋𝜋𝑠𝑠 (𝑄𝑄𝑠𝑠 ) has a unique global optimum. Moreover, based on Theorem 𝜋𝜋𝑠𝑠𝑅𝑅 (𝑄𝑄𝑠𝑠𝑅𝑅 ) is also a piece-wise continuous function with respect to 𝑄𝑄𝑠𝑠𝑅𝑅 .□ 2 it is easy to show that Fig. 2 illustrates the results of Proposition 4. The parameter settings of the numerical example are: 𝑁𝑁 = 2 , 𝐷𝐷~𝑁𝑁(100,20), 𝑝𝑝 = 100 , 𝑐𝑐𝑟𝑟 = 2 , 𝑐𝑐𝑔𝑔 = 2 , ℎ𝑟𝑟 = 2 , 𝑏𝑏𝑟𝑟 = 5 , ℎ𝑢𝑢 = 1 , 𝑐𝑐1 = 6 , 𝑐𝑐2 = 30 , 𝜃𝜃1 ~𝐵𝐵(8, 2). It can be seen from Fig. 2 that 𝜋𝜋𝑠𝑠𝑅𝑅 (𝑄𝑄𝑠𝑠𝑅𝑅 ) is divided into three pieces by 𝑄𝑄𝑠𝑠𝑅𝑅 = 𝑅𝑅1∗ and 𝑄𝑄𝑠𝑠𝑅𝑅 = 𝑅𝑅2∗ , and each piece is a concave function. Fig. 2 Remanufacturer’s expected profit with different core acquisition quantities Advances in Production Engineering & Management 18(2) 2023 183 Cao, Shi, Lan, Huang 4.3 Comparative analysis We also examine the value of core sorting under the stochastic demand case. By comparison, it is also found that core sorting in remanufacturing is cost-effective only when the unit sorting cost is below a threshold value and the unit core processing (i.e., acquisition, sorting and disposal) cost is sufficiently low. ∗ ∗ 𝑅𝑅 𝑅𝑅 (𝑄𝑄𝑛𝑛𝑛𝑛 ), if 𝑐𝑐𝑟𝑟 + 𝑐𝑐𝑔𝑔 + ℎ𝑢𝑢 < 𝑐𝑐𝑁𝑁 − 𝑐𝑐𝜇𝜇 and Proposition 5. There exists 𝑐𝑐𝑔𝑔𝑚𝑚𝑚𝑚𝑚𝑚 such that (i) 𝜋𝜋𝑠𝑠𝑅𝑅 (𝑄𝑄𝑠𝑠𝑅𝑅 ) > 𝜋𝜋𝑛𝑛𝑛𝑛 𝑚𝑚𝑚𝑚𝑚𝑚 𝑅𝑅 𝑅𝑅∗ 𝑅𝑅 𝑅𝑅∗ 𝑐𝑐𝑔𝑔 < 𝑐𝑐𝑔𝑔 ; (ii) 𝜋𝜋𝑠𝑠 (𝑄𝑄𝑠𝑠 ) < 𝜋𝜋𝑛𝑛𝑛𝑛 (𝑄𝑄𝑛𝑛𝑛𝑛 ), otherwise. Proof. It follows Theorem 2 that when 𝑄𝑄𝑠𝑠𝑅𝑅 ≤ 𝑅𝑅𝑁𝑁∗ , 𝑦𝑦𝑖𝑖∗ = ∑𝑖𝑖𝑘𝑘=1 𝜃𝜃𝑘𝑘 𝑄𝑄𝑠𝑠𝑅𝑅 , 𝑖𝑖 = 1,2, . . . , 𝑁𝑁. Then, on the interval [0, 𝑅𝑅𝑁𝑁∗ ] we have 𝜋𝜋𝑠𝑠𝑅𝑅 = 𝐸𝐸[𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝(𝑄𝑄𝑠𝑠𝑅𝑅 , 𝐷𝐷) − 𝑏𝑏𝑟𝑟 (𝐷𝐷 − 𝑄𝑄𝑠𝑠𝑅𝑅 )+ − ℎ𝑟𝑟 (𝑄𝑄𝑠𝑠𝑅𝑅 − 𝐷𝐷)+ ] − (𝑐𝑐𝜇𝜇 + 𝑐𝑐𝑟𝑟 + 𝑐𝑐𝑔𝑔 )𝑄𝑄𝑠𝑠𝑅𝑅 Taking the first derivative of 𝜋𝜋𝑠𝑠𝑅𝑅 with respect to 𝑄𝑄𝑠𝑠𝑅𝑅 , we have 𝜋𝜋𝑠𝑠,𝑅𝑅 = (𝑝𝑝 + 𝑏𝑏𝑟𝑟 − 𝑐𝑐𝑟𝑟 − 𝑐𝑐𝑔𝑔 − 𝑐𝑐𝜇𝜇 ) − (𝑝𝑝 + 𝑏𝑏𝑟𝑟 + ℎ𝑟𝑟 )𝐹𝐹(𝑄𝑄𝑠𝑠𝑅𝑅 ) It is easy to verify that 𝜋𝜋𝑠𝑠,𝑅𝑅 is decreasing in 𝑄𝑄𝑠𝑠𝑅𝑅 . Moreover, when 𝑄𝑄𝑠𝑠𝑅𝑅 = 0, 𝜋𝜋𝑠𝑠,𝑅𝑅 (𝑄𝑄𝑠𝑠𝑅𝑅 = 0) > 0; when 𝑄𝑄𝑠𝑠𝑅𝑅 = 𝑅𝑅𝑁𝑁∗ , 𝜋𝜋𝑠𝑠,𝑅𝑅 (𝑄𝑄𝑠𝑠𝑅𝑅 = 𝑅𝑅𝑁𝑁∗ ) = 𝑐𝑐𝑁𝑁 − 𝑐𝑐𝜇𝜇 − (𝑐𝑐𝑟𝑟 + 𝑐𝑐𝑔𝑔 + ℎ𝑢𝑢 ). When 𝑐𝑐𝑟𝑟 + 𝑐𝑐𝑔𝑔 + ℎ𝑢𝑢 ≥ 𝑐𝑐𝑁𝑁 − 𝑐𝑐𝜇𝜇 , 𝜋𝜋𝑠𝑠,𝑅𝑅 (𝑄𝑄𝑠𝑠𝑅𝑅 = 𝑅𝑅𝑁𝑁∗ ) ≤ 0, thus the optimal core acquisition quantity ∗ 𝑄𝑄𝑠𝑠𝑅𝑅 is on the interval [0, 𝑅𝑅𝑁𝑁∗ ]. This implies that all the acquired cores should be remanufactured even with the sorting strategy. As mentioned above, under this condition core sorting has no value to the remanufacturer. When 𝑐𝑐𝑟𝑟 + 𝑐𝑐𝑔𝑔 + ℎ𝑢𝑢 < 𝑐𝑐𝑁𝑁 − 𝑐𝑐𝜇𝜇 , 𝜋𝜋𝑠𝑠,𝑅𝑅 (𝑄𝑄𝑠𝑠𝑅𝑅 = 𝑅𝑅𝑁𝑁∗ ) > 0, thus 𝜋𝜋𝑠𝑠𝑅𝑅 is increasing in 𝑄𝑄𝑠𝑠𝑅𝑅 on the interval ∗ [0, 𝑅𝑅𝑁𝑁∗ ] since 𝜋𝜋𝑠𝑠,𝑅𝑅 is always positive on that interval. This suggests that 𝑄𝑄𝑠𝑠𝑅𝑅 is larger than 𝑅𝑅𝑁𝑁∗ , i.e., at least the acquired cores of grade 𝑁𝑁 are partially remanufactured. Note that if 𝑐𝑐𝑔𝑔 = 0, then for any given quantity of cores, the sorting strategy is not inferior to the non-sorting strategy due to the ∗ 𝑅𝑅∗ 𝑅𝑅 𝑅𝑅∗ potential remanufacturing cost savings. Thus, we have 𝜋𝜋𝑠𝑠𝑅𝑅 (𝑄𝑄𝑠𝑠𝑅𝑅 ) > 𝜋𝜋𝑠𝑠𝑅𝑅 (𝑄𝑄𝑛𝑛𝑛𝑛 ) ≥ 𝜋𝜋𝑛𝑛𝑛𝑛 (𝑄𝑄𝑛𝑛𝑛𝑛 ), where ∗ ∗ the first inequality is valid due to the optimality of 𝑄𝑄𝑠𝑠𝑅𝑅 for 𝜋𝜋𝑠𝑠𝑅𝑅 . With the increase of 𝑐𝑐𝑔𝑔 , 𝜋𝜋𝑠𝑠𝑅𝑅 (𝑄𝑄𝑠𝑠𝑅𝑅 ) 𝑅𝑅 𝑅𝑅∗ decreases but 𝜋𝜋𝑛𝑛𝑛𝑛 (𝑄𝑄𝑛𝑛𝑛𝑛 ) remains the same. Therefore, there exists a value of 𝑐𝑐𝑔𝑔 on the inter∗ 𝑅𝑅 𝑅𝑅∗ val(0, 𝑐𝑐𝑁𝑁 − 𝑐𝑐𝜇𝜇 − 𝑐𝑐𝑟𝑟 − ℎ𝑢𝑢 ), denoted as 𝑐𝑐𝑔𝑔𝑚𝑚𝑚𝑚𝑚𝑚 , so that when0 < 𝑐𝑐𝑔𝑔 < 𝑐𝑐𝑔𝑔𝑚𝑚𝑚𝑚𝑚𝑚 , 𝜋𝜋𝑠𝑠𝑅𝑅 (𝑄𝑄𝑠𝑠𝑅𝑅 ) > 𝜋𝜋𝑛𝑛𝑛𝑛 (𝑄𝑄𝑛𝑛𝑛𝑛 ).□ Proposition 5 implies that when the sorting cost is too high, core sorting is useless to the remanufacturer. It also can be seen from Table 1 that when 𝑐𝑐𝑔𝑔 ≤ 2, the remanufacturer’s expected profit under the sorting strategy is higher, while when 𝑐𝑐𝑔𝑔 ≥ 2.5, the non-sorting strategy makes the remanufacturer more profitable. We further examine the impacts of demand uncertainty on the value of core sorting. Let ∆𝑔𝑔 be the increase in remanufacturer’s expected profit by using the sorting strategy relative to the case 𝜋𝜋 −𝜋𝜋 where the remanufacturer adopts the non-sorting strategy, where ∆𝑔𝑔 = 𝑔𝑔 𝑚𝑚 × 100%. Based on 𝜋𝜋 𝑚𝑚 the same demand and cost parameter settings with that of Fig. 2, we vary the values of 𝑎𝑎 and 𝑏𝑏 (distribution parameters of 𝜃𝜃1 ) where (𝑎𝑎, 𝑏𝑏) ∈ {𝑎𝑎 = 0.5𝑖𝑖, 𝑏𝑏 = 10 − 0.5𝑖𝑖, 𝑖𝑖 = 1, . . . ,19}, and consider three different levels of demand variation: 𝛿𝛿𝐷𝐷 ∈ {10, 20, 30}. Fig. 3 illustrates the increase in remanufacturer’s expected profit by adopting the sorting strategy under different levels of demand variation. The observation is similar to the deterministic demand case that the sorting strategy is valuable to remanufacturer only when the expected fraction of high-quality cores is within a suitable interval. Another observation is that when the sorting strategy is superior to the non-sorting strategy, it is more valuable under a higher level of demand variation. This is because a higher level of demand variation may increase the possibility that a high-quality core is remanufactured rather than being disposed of, which facilitates the adoption of sorting strategy. Table 1 Remanufacturer’s expected profit under sorting and non-sorting strategies at different levels of sorting cost 𝑐𝑐𝑔𝑔 0 0.5 1 1.5 2 2.5 3 3.5 𝑅𝑅 8633.3 8549.0 8466.4 8385.2 8305.4 8226.7 8149.1 8072.5 𝜋𝜋𝑠𝑠 𝑅𝑅 8238.8 8238.8 8238.8 8238.8 8238.8 8238.8 8238.8 8238.8 𝜋𝜋𝑛𝑛𝑛𝑛 Note: Other parameters take the same value as in the numerical example of Fig. 2. 184 4 7996.8 8238.8 4.5 7921.9 8238.8 5 7847.9 8238.8 Advances in Production Engineering & Management 18(2) 2023 When core sorting and quality grading is beneficial to remanufacturers: Insights from analytical models Fig. 3 Increase in Remanufacturer’s expected profit under sorting and non-sorting strategies at different levels of demand variation 5. Conclusion In this paper, we study the core acquisition and remanufacturing problem under a single-period setting in which the remanufactured products are produced from acquired cores with uncertain quality. Both the deterministic demand case and the stochastic demand case are discussed to examine the potential value of core sorting and quality grading in remanufacturing. For deterministic demand, it is found that core sorting is cost-effective only when the sorting cost is below a threshold value and the acquisition cost falls into a specific interval. Furthermore, in the case with two quality grades the sorting strategy is valuable to remanufacturer only when the expected fraction of high-quality cores is within a suitable interval. For stochastic demand, it is observed that the sorting strategy also becomes unattractive when the sorting cost is sufficiently high. In addition, the value of core sorting will be larger with more variable demand for remanufactured products if the sorting strategy is the dominant strategy. 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Advances in Production Engineering & Management 18(2) 2023 Advances in Production Engineering & Management ISSN 1854-6250 Volume 18 | Number 2 | June 2023 | pp 187–198 Journal home: apem-journal.org https://doi.org/10.14743/apem2023.2.466 Original scientific paper Engineering-to-order manufacturing: A criticality analysis of key challenges and solutions based on literature review Fortes, C.S.a,b,*, Tenera, A.B.b,c, Cunha, P.F.a, Teixeira, J.P.b aEscola Superior de Tecnologia de Setúbal (ESTS), Instituto Politécnico de Setúbal, Portugal DEMI, NOVA School of Science & Technology (FCT), Universidade NOVA de Lisboa, Portugal cLASI, Laboratório Associado de Sistemas Inteligentes, Portugal bUNIDEMI, ABSTRACT ARTICLE INFO Engineer-to-Order (ETO) manufacturing companies involve customised production products based on specific customer requirements, and face a significant challenge. Some of those challenges are related with this type of company’s activities in production scheduling, planning and control, efficiency improvement and lead time reduction. The present study was conducted with a systematic literature review and a survey from ETO firms to identify the most frequent and critical problems. Among the most critical issues identified is the difficulty in optimising production performance (P3), with a GCI value of 16, implying that both time and cost share the same critical level. An analysis using a proposed Criticality Matrix was then performed enabling companies to prioritise decision-making and resource allocation. The results highlight the importance of adopting mass customisation strategies, innovative approaches and workflow optimisation. Continuous monitoring and analysis of criticality levels can also help ETO companies identify emerging issues and improve informed decisions. Effective communication and collaboration among stakeholders were also identified as vital. Future research could be done expanding further the study sample and developing decision-support tools for ETO manufacturing companies. This study contributes to the field by providing a new criticality matrix for ETO companies to understand better and address their production challenges, aiding in decision-making and resource allocation. Keywords: Communication and collaboration; Critical factors; Decision-making; Engineer-to-order; Literature review; Production planning and control; Production scheduling; Resource allocation; Workflow optimisation *Corresponding author: Carlos.fortes@estsetubal.ips.pt (Fortes, C.S.) Article history: Received 5 June 2023 Revised 23 June 2023 Accepted 28 June 2023 Content from this work may be used under the terms of the Creative Commons Attribution 4.0 International Licence (CC BY 4.0). Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. 1. Introduction 1.1 Background Engineering-to-Order (ETO) companies typically manufacture highly complex and customised products with significant engineering components that provide high value. As highlighted by Jiang and Xi, these companies utilise an ETO strategy particularly when complex structures are required to be built [1]. In general, and particularly in European Union, such companies are small or medium-sized and play essential roles in specific sectors and regions, primarily operating in international markets. ETO companies' economic development and competitiveness are crucial for their success, as they supply various industrial sectors, such as the automotive, aeronautical, and pharmaceutical/medical device industries. Their business model is centred around their ability to 187 Fortes, Tenera, Cunha, Teixeira design, develop and manufacture products to order, delivering value to their clients [2]. One example of a sector where ETO companies are common is the mould manufacturing industry, especially tools and plastics. ETO companies face many challenges in production planning and shop floor management as product and process diversity and complexity increase system variability. These highly competitive sectors require the efficient execution of numerous activities to meet customer expectations. Therefore, shop floor planning and operational decision-making effectiveness are essential for these companies. In addition, they must fulfil increasingly shorter delivery times defined by the market or clients, who also demand quality and lower costs. ETO companies often must also deal with a workflow that requires the quick generation of product designs, detailed bill of material and manufacturing work instructions for each order, to meet tailored customer specifications. 1.2 Scope and outline In the context of Engineer-to-Order (ETO), companies encounter diverse tasks, high-level customisation of products, and an unpredictable market. These elements call for insightful planning and decision-making, seamlessly integrated with essential operations such as tendering and procurement, to enhance performance across their value chain. Consequently, the impetus for this investigation is threefold: (a) to assist ETO companies in honing their efficiency and effectiveness in operations, augmenting their impact and importance in a specific industrial domain and the broader economy; (b) to introduce novel strategies, methodologies, procedures, or resources to ETO companies to boost their market competitiveness; and (c) to lessen the innate challenges associated with ETO operations, particularly by enhancing their capacity to manage the volatility and intricacy within their entire value chain. A SLR was conducted to identify the main challenges and issues related to ETO companies and the methods and approaches presented in the scientific literature to resolve or mitigate these issues in the value chain of this type of company. The search for scientific literature on the topic aimed to gain knowledge about various viewpoints and holistically interpret the study's theories and models. Additionally, ETO companies, were surveyed to identify the most frequent and critical problems, with a subsequent criticality analysis using a developed criticality matrix for problems in ETO companies, offering valuable insights into industry perspectives and potential avenues for improvement. This paper is structured as follows: Section 1 presents an overview of the study's background and motivation. 2. Materials and methods 2.1 Research methodology An SLR was developed to identify ETO companies’ difficulties and the technical and scientific solutions proposed to address them, followed by a survey of ETO companies to assess the criticality of the identified problems in those organisations. Unlike narrative reviews, the SLR aim to answer a specific research question through a planned and structured approach to identifying, selecting, and critically appraising relevant studies [3]. By analysing and synthesising different authors findings, a comprehensive understanding of the existing body of work is gained, gaps to explore are identified, and conclusions are drawn on what is known and not known [4]. The review followed the PRISMA [5] guidance approach, with specific eligible criteria that included: papers discussing ETO problems, process planning, manufacturing related articles, production planning and control, and articles published in English. Articles that did not meet these criteria were excluded. The relevant studies were identified by searching digital databases, such as Web of Science and Scopus, as shown in Table 1. Articles published between January 2017 and April 2022 were the ones in this study considered, using search terms such as "engineer* to order" AND "variab*" OR "uncertaint*" OR "complex*" AND "produc* control" OR "plan*" OR "schedul*" OR "produc* process" OR "workflow*" OR "shop-floor*" OR "digital*" OR "problem*" OR "issue*" OR "constraint*" OR "inefficient*" NOT "ship*" OR "building*". 188 Advances in Production Engineering & Management 18(2) 2023 Engineering-to-order manufacturing: A criticality analysis of key challenges and solutions based on literature review Table 1 Databases used for the search Database Search Terms Scopus TITLE-ABS-KEY("engineer* to order") AND TITLE-ABS-KEY(variab* OR uncertaint* OR complex*) AND TITLE-ABS-KEY("produc* control" OR plan* OR schedul* OR "produc* process" OR workflow* OR "shop-floor*" OR digital* OR problem* OR issue* OR constraint* OR inefficient*) AND NOT(ship* OR building*) AND (LIMIT-TO(DOCTYPE,"ar") OR LIMIT-TO(DOCTYPE,"cp")) AND (LIMIT-TO(LANGUAGE,"English")) Web of Science TS = ("engineer* to order") AND TS = (variab* OR uncertaint* OR complex*) AND TS = ("produc* control" OR plan* OR schedul* OR "produc* process" OR workflow* OR "shopfloor*" OR digital* OR problem* OR issue* OR constraint* OR inefficient*) NOT TS = (ship* OR building*) The study search yielded 83 WoS publications and 57 Scopus publications in a total of 140 potentially relevant publications, from which 32 were duplicated as shown in Fig. 1. From the initial 108 publications analysed for inclusion, 62 articles did not meet the selection criteria due to their publication dates falling outside of the 2017-2022 range, thus leading to their exclusion. This process yielded 46 potentially most relevant references to undergo a comprehensive indepth analysis, out of which only 18 were deemed pertinent. Further to this rigorous analysis, six articles were identified and subsequently included in the study. Fig. 1 shows the flow of citations through the systematic review process. The final sample size of 24 papers can be considered a robust number for this analysis, as similar sample sizes have been used in other studies such as 19 papers in nursing studies [6], 23 papers in learning management [7], and 23 papers in project management studies [8]. In the following section, the results of the in-depth content analysis of the identified articles are synthesised, focusing on the general framework and competitive environment in which ETO operates and, in the challenges and issues faced by this type of company, including its principal characteristics and organisation. In addition to the SLR, a survey was also conducted among ETO companies to better understand the frequency and the criticality of the identified problems. This survey aimed to check the findings of the SLR and provide any new real-world perspectives on the challenges encountered in ETO environments. The collected data were analysed using a proposed criticality matrix to assess the identified problems potentially severe impact and prioritise potential solutions. This approach allowed for a more comprehensive understanding of all the identified problems and issues faced by ETO companies, providing an informed potential action. Fig. 1 A schematic flow chart diagram of the selected studies 2.2 A concept map approach of ETOs challenges and solutions ETO companies specialise in fabricating high-value-added customised products in low volumes, typically small batches or one-of-a-kind products, as shown in Fig. 2. As project-based organisations, ETO companies focus mainly on designing and manufacturing new products, involving a complex and highly specialised process that demands qualified and flexible workers [9]. Moreover, ETO companies are customer-centric [10], involving strong customer participation in product development. However, due to the initial phase's complexity and uncertainty in customer product characteristics, raw material purchase and product manufacturing only begin after the customer confirms the order quotation. Advances in Production Engineering & Management 18(2) 2023 189 Fortes, Tenera, Cunha, Teixeira Fig. 2 Main attributes of ETO ETO Potential key challenges The demand for increasingly customised products has resulted in the need for better performance in ETO companies [11]. However, the discrete nature of shop floor workflows in ETO companies poses a significant management challenge requiring higher decision-making resilience to workflow performance [12]. The tactical-level management of value chain information can also be considered another critical issue in the dynamic environment of ETO companies [13]. During the contracting stage, uncertainties about costs and delivery date forecasts are typical, resulting in higher manufacturing costs [14], longer lead times, higher inventory costs, transportation costs, and other supply risks. Planning in ETO companies is complex, given the combination of multiple technical responsibility areas and the difficulty in predicting workload resources and lead times [15]. Customers can be involved in all project stages, such as design, production, assembly and testing [16]. These challenges often result in manufacturing process delays due to changes made during the project's progress [17]. Late changes in engineering design can cause costs increases and affect resource allocation along the value chain, according to Gosling et al. [18]. The involvement of customers and the various interactions place a high demand on production planning and control activities [19]. Also, as noted by Akcay et al. [20]. One-of-a-kind products or low-volume batches increase the number of jobs pending in manufacturing. Developing a concept map for scientific and technological solutions A thorough comprehension of ETO organisations and their challenges can be significantly enhanced by adopting innovative and engaging visual representations, such as concept maps. Unlike traditional tables, these maps can provide an intuitive and interactive means of conveying detailed information. Derived from the problem table resulting from the SLR conducted in the research, an ETO domain, concept map can be proven beneficial for identifying and examining ETOs diverse issues, weaknesses, and areas warranting further investigation [21]. From Fortes et al. [21], Fig. 3, depicts a multi-layered concept map, with the ETO companies and their characteristics placed at the centre, encircled by a second ring that contains the surrounding issues. The third ring introduces the authors' proposals, and the outermost ring emphasises the contributions and limitations of the studies. This layout was design to enhance the understanding of problem areas, underline the need for further research, and to promote the development of adaptive strategies to tackle ETO organisations' multifaceted challenges. This concept map employs five distinct colours and three unique shapes to differentiate identified distinct categories. Blue balloons represent the characteristics of ETO companies; white balloons signify the issues to be addressed; yellow balloons indicate approaches; green balloons denote contributions; and red balloons symbolise weaknesses. For example, the item "Bid solution uncertain", as presented in the study by Sylla et al. [22], is represented by a white balloon, signifying the issue that needs to be addressed. Subsequent yellow rectangles illustrate the approach of "a multi-criteria approach to bid solution uncertainty". For example, a green balloon showcases contributions, such as "providing bid solutions with accurate and timely replies", whilst a red balloon points out weaknesses, like "single case study with limited data". This visual approach intends to enable readers to effortlessly identify the categories of each item and their relationships with one another. Furthermore, connections between balloons can be established to illustrate the identified links between items. For instance, a connection between balloons one and two can be drawn, as both pertain to uncertain bid solutions. 190 Advances in Production Engineering & Management 18(2) 2023 Engineering-to-order manufacturing: A criticality analysis of key challenges and solutions based on literature review So, concept maps offer a compelling and effective way of organising and presenting complex information [23] related to ETO organisations’ challenges and potential solutions. By employing a visually appealing and interactive layout, these maps can facilitate a deeper understanding of the issues, approaches, contributions, and weaknesses within this field, ultimately encouraging further investigation and development of comprehensive strategies to address the dynamic challenges faced by ETO companies. Fig. 1 ETO concept diagram [21] Advances in Production Engineering & Management 18(2) 2023 191 Fortes, Tenera, Cunha, Teixeira 3. Results and discussion: Criticality analysis of ETO companies’ problems The systematic literature review (SLR) results and ETOS’s companies problem assessment survey examined the frequency and identified problems impact these organisations encounter, establishing the associated criticality. To ensure a diverse sample survey was developed specifically targeting high-level professionals such as managers, department directors, and project managers from a diverse range of ETO companies, including different industry sectors, sizes, and geographical locations. This investigation uncovered new valuables insights, enabling stakeholders to better understand the pressing concerns affecting ETO businesses. In addition, insights into pressing issues in ETO companies, allowing stakeholders to prioritise efforts, develop effective strategies, and improve performance and competitiveness within the sector can be provided by the proposed "Criticality Analysis of ETO Company Problems" table. 3.1 Problem frequency assessment The analysis involved assessing the relative citation frequencies of each problem identified in the SLR cf and comparing them to the average organisational occurrence rankings reported by the study survey fr to check the theoretical known SLR findings and identify any new current problems. This study’s results discovered no additional problems, and all problems were reported as occurring in practice. Table 2 lists the 14 problems discovered through the SLR, where problem P10 (Production Scheduling) emerged as the most cited, followed by P11 (Production Planning and Control), with P2 (Tender Proposal), P3 (Production Performance), P12 (Product Specification), and P14 (Lead Time) in third place. In terms of reported frequency fr all problems were assessed as existing validating the SLR identified problems and revealing that 12 of the 14 problems (P1, P2, P4, P5, P7, P8, P9, P10, P11, P12, P13, and P14) were equally frequently assessed with P3 (Production Performance) the most frequent problem and P6 (Unstructured Knowledge) the least concerned from the surveyed companies’ perspective. Item Designation Capacity P1 Planning Tender P2 Proposal Production P3 Performance Workflow P4 Visibility Configuration P5 Process Unstructured P6 Knowledge Lot Size P7 Uncertainty P8 P9 P10 P11 P12 P13 P14 Quotation Workflow Production Scheduling Production Planning & Control Product Specification Customer Changes Lead Time Table 2 Criticality analysis of ETO company problems Description pf (%) Ranking fr (%) Ranking Deviation Difficulty in adapting existing capacities to new 7 +2 4 4th 2nd product development and production plans. Limitations in accurately defining tender pro7 +1 8 3rd 2nd posals due to uncertainty. Difficulty in process optimisation. Difficulty in visualising workload and supporting decision-making in a dynamic shop floor. Frequent constraints on workflow configuration due to resource scarcity. Inefficient reuse of design knowledge for new products. Uncertainty of lot size. Difficulty in product design optimisation, influencing quotations. Information flow failures, limiting collaboration and operational performance. Difficulty in defining optimal job sequences for production. Addressing Variability in Operation Execution Times in Manufacturing and Assembly Processes Lack of information and detail in customerprovided product specifications. Late design changes due to customer-imposed alterations. Long execution time. 8 3rd 10 4 4th 7 4th 7 4th 7 17 2nd 7 4 4th 7 4 4 4 4 4 21 8 8 4th 7 4th 5 4th 7 1st 7 3rd 7 3rd 7 1st +2 2nd +2 2nd +2 2nd +2 2nd 0 2nd +2 3rd +1 2nd +2 2nd -1 2nd +1 2nd +1 2nd pf – published frequencies (Scopus, Web of Science); fr – frequency as reported in the survey; Dif – difference: Dif = fp – fr 192 +2 Advances in Production Engineering & Management 18(2) 2023 Engineering-to-order manufacturing: A criticality analysis of key challenges and solutions based on literature review So, upon examining the table, several conclusions can be drawn regarding the criticality of problems ETO companies face. First, comparing published frequencies pf from the Scopus and Web of Science databases with the frequencies reported in the survey fr, which included 12 participating ETOs companies, reveals some discrepancies between the literature and the real-world challenges experienced by these firms. Table 2 results highlight that Production Scheduling (P10) and Production Planning & Control (P11) are the most critical issues for ETO companies, as evidenced by their high published frequencies (21 % and 17 %, respectively) and their shared second positions in the survey. These findings, indicate that these issues are consistently recognised as significant challenges in academic literature and industry practice. However, most of the problems display a positive deviation in their rankings between the published frequencies pf and the frequencies reported in the survey (fr), suggesting that the participating ETO companies perceive these issues to be more critical than what is reflected in the literature. This discrepancy could imply that the existing literature must fully capture ETO companies' unique challenges or that ETO practitioners are more aware of the issues within their sector. Considering these findings, further research is recommended to explore the specific challenges faced by ETO companies in greater depth and foster closer collaboration between practitioners and academics. This collaboration can help ensure that the literature accurately reflects the practical issues experienced in the field. By addressing this gap, ETO companies can better understand and mitigate their challenges, improving their overall performance and competitiveness. Therefore, by focusing on the most frequent concerned issues, such as Production Scheduling (P10) and Production Planning & Control (P11), researchers can contribute to developing more practical solutions and strategies, enhancing the sector's ability to tackle the complex problems inherent in ETO operations. Moreover, the detailed examination of the impact on the field of each problem can provide valuable insights into the prioritization of challenges faced by ETO companies, guiding future research and decision-making in the sector. 3.2 Analysis of ETO problem criticality This session analysis the critical [24] level of the ETO companies’ problems to prioritise ETOs organisations' challenges. Here, the criticality assessment approach is based on the probability measure by the frequency assessment and impact of the identified problems. It is employed to assist companies in better understanding the challenges they face and develop targeted strategies for addressing them. Furthermore, this type of analysis can allow a more effective allocation of resources and informed decision-making, improving the overall performance and success of ETO projects. Table 3 presents the proposed matrix for evaluating the criticality level of the identified ETOs problems. The criticality index CI results from the intersection of the row probability P and the column impact I, where the values are multiplied, thus obtaining the criticality level through the general Eq. 1. 𝐶𝐶𝐶𝐶 = 𝑃𝑃 × 𝐼𝐼 (1) Table 3 Risk matrix (Probability/Impact matrix) 5 4 3 2 1 0 Probability (P) Probability/Impact Non-existent/Very Low [0, 4] Low [4, 6] Moderate [6, 12] 0 0 0 0 0 0 0 5 4 3 2 1 0 1 10 15 20 25 8 12 16 20 6 9 12 15 4 6 8 10 2 3 4 5 0 0 0 0 2 3 4 5 Impact (I) High [12, 16] Very High [16, 25] The criticality level of problems may vary among organisations, resulting in different evaluations based on each company's specific situation, as demonstrated in Table 4. Advances in Production Engineering & Management 18(2) 2023 193 Fortes, Tenera, Cunha, Teixeira Issue ETO1 P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12 P13 P14 ETO4 P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12 P13 P14 ETO7 P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12 P13 P14 ETO10 P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12 P13 P14 P 2 2 3 2 3 1 1 2 1 1 2 3 3 3 P 3 3 4 3 4 4 3 4 4 4 3 4 4 3 P 3 1 5 3 1 1 1 1 3 2 3 2 2 1 P 5 4 5 5 5 2 4 2 2 3 5 2 4 4 cIETO1 2 3 3 1 4 1 1 3 4 1 3 1 4 5 cIETO4 5 3 3 4 4 3 3 4 4 2 3 4 4 4 cIETO7 3 1 4 1 2 1 1 1 1 2 3 3 2 1 cIET10 4 3 5 5 5 1 1 2 2 4 5 2 3 4 tIETO1 4 4 5 4 4 3 2 4 5 2 3 4 5 1 tIETO4 4 3 3 3 3 1 4 3 4 4 4 4 5 5 tIETO7 3 1 3 1 2 1 2 1 1 2 2 1 4 1 tIET10 5 4 5 4 5 1 1 4 2 4 5 2 2 4 Table 4 Specific organisational evaluation results Issue Issue OCI ETO2 P cIETO2 tIETO2 OCI ETO3 8 P1 2 5 4 10 P1 8 P2 2 4 2 8 P2 15 P3 2 5 2 10 P3 8 P4 2 2 3 6 P4 12 P5 3 5 4 15 P5 3 P6 2 1 2 4 P6 2 P7 2 2 2 4 P7 8 P8 2 5 5 10 P8 5 P9 2 3 5 10 P9 2 P10 2 3 1 6 P10 6 P11 2 3 1 6 P11 12 P12 3 5 5 15 P12 15 P13 2 5 5 10 P13 15 P14 3 5 1 15 P14 OCI ETO5 P cIETO5 tIETO5 OCI ETO6 15 P1 3 3 3 9 P1 9 P2 2 3 3 6 P2 12 P3 3 2 3 9 P3 12 P4 3 3 3 9 P4 16 P5 3 3 2 9 P5 12 P6 3 3 3 9 P6 12 P7 3 3 3 9 P7 16 P8 3 2 2 6 P8 16 P9 3 3 3 9 P9 16 P10 3 3 3 9 P10 12 P11 4 3 2 12 P11 16 P12 2 2 3 6 P12 20 P13 2 3 2 6 P13 15 P14 2 3 3 6 P14 OCI ETO8 P cIETO8 tIETO8 OCI ETO9 9 P1 4 4 5 20 P1 1 P2 4 4 5 20 P2 20 P3 4 3 5 20 P3 3 P4 5 5 5 25 P4 2 P5 3 3 3 9 P5 1 P6 1 1 1 1 P6 2 P7 1 1 1 1 P7 1 P8 4 5 4 20 P8 3 P9 4 4 4 16 P9 4 P10 1 1 1 1 P10 9 P11 3 3 5 15 P11 6 P12 3 3 4 12 P12 8 P13 3 3 4 12 P13 1 P14 4 3 5 20 P14 OCI ETO11 P cIET11 tIET11 OCI ETO12 25 P1 4 4 4 16 P1 16 P2 4 5 4 20 P2 25 P3 3 4 5 15 P3 25 P4 4 3 3 12 P4 25 P5 3 4 5 15 P5 2 P6 3 2 2 6 P6 4 P7 4 4 3 16 P7 8 P8 4 4 4 16 P8 4 P9 4 4 4 16 P9 12 P10 4 5 4 20 P10 25 P11 5 5 5 25 P11 4 P12 4 5 4 20 P12 12 P13 4 5 4 20 P13 16 P14 4 5 5 20 P14 P – Probability; cI – Cost Impact; tI – Time Impact; OCI – Organizational Criticality Index 194 P 2 2 4 2 4 3 2 2 2 2 2 2 2 3 P 4 3 5 5 4 2 2 4 2 2 2 5 4 5 P 2 2 3 3 2 1 5 5 5 3 2 2 3 3 P 2 3 3 3 3 3 2 3 3 3 3 2 3 3 cIETO3 4 2 5 4 5 1 4 2 4 5 5 5 5 2 cIETO6 3 2 4 3 2 1 1 2 3 2 2 2 2 3 cIETO9 2 3 4 4 4 4 3 4 4 3 3 2 4 3 cIET12 4 5 5 5 5 5 4 5 5 5 5 5 5 5 tIETO3 4 2 2 3 4 2 2 5 5 1 1 5 5 1 tIETO6 5 4 5 5 5 2 2 2 5 3 4 5 5 4 tIETO9 4 2 5 5 4 5 4 4 5 3 5 2 4 4 tIET12 4 4 4 4 4 4 4 4 4 4 4 4 4 4 OCI 8 4 20 8 20 6 8 10 10 10 10 10 10 6 OCI 20 12 25 25 20 4 4 8 10 6 8 25 20 20 OCI 8 6 15 15 8 5 20 20 25 9 10 4 12 12 OCI 8 15 15 15 15 15 8 15 15 15 15 10 15 15 Advances in Production Engineering & Management 18(2) 2023 Engineering-to-order manufacturing: A criticality analysis of key challenges and solutions based on literature review Therefore, their specific values may differ from the general average trend values. The Organisational Criticality Index OCI results from the multiplication of the problem reported frequency with the highest value among its reported impacts on costs cI(x) or time tI(x). For the analysis of the general criticality index GCI of each identified problem, the approximate average level of reported occurrence frequency of was similarly multiplied by the average impacts on costs cI and time tI for each identified problem using Eqs. 2 to 4. (𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓1 + … + 𝑓𝑓𝑓𝑓 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸) (2) 𝑛𝑛 (𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐1 + … . + 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐) (3) 𝑐𝑐𝑐𝑐 = 𝑛𝑛 (𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡1 + … + 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡) (4) 𝑡𝑡𝑡𝑡 = 𝑛𝑛 Table 5 summarises the criticality levels obtained for each problem, identifying the impact variable that most contributed to its value through time t or cost c. Based on the findings from Table 5, the following conclusions can be drawn in the context of ETO companies. First, our analysis of the GCI highlights the key challenges that significantly impact these organisations in terms of time t and cost c. This understanding can enable ETO companies to develop targeted strategies and address these critical issues effectively, thereby enhancing their overall performance and competitiveness in the market. These results demonstrate that several problems substantially impact ETO companies' performance. As noted by Leksic, the performance of an ETO company is directly tied to the efficiency of its production process, reinforcing the magnitude of the identified problems in this study [25]. Among the most critical issues identified is the difficulty in optimising production performance (P3), with a GCI value of 16tc meaning that both time and cost share the same critical level. This challenge arises from the highly customised nature of ETO products, which necessitates constant adaptation of manufacturing processes to meet the unique requirements of each project. Consequently, achieving the desired quality standards may require increased time and cost. Capacity planning (P1) is another significant challenge that the surveyed ETOs companies face, with a GCI value of 12tc. This issue stems from the difficulty in aligning existing capacities with planning new product development and their introduction into production. As a result, ETO companies must diligently allocate resources and manage production schedules to cater to each customer's specific needs, which may pressure the overall capacity planning process. Moreover, the analysis emphasises the importance of the configuration process (P5), which has a GCI value of 12tc. Frequent constraints in the workflow configuration process may occur due to the scarcity of resources concerning planned and ongoing works. The customer-centric approach of ETO companies exacerbates this issue, as accommodating unique customer requirements may lead to frequent changes and adjustments in the workflow configuration. 𝑜𝑜𝑜𝑜 = 12 ETO Item P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12 P13 P14 fo 3 3 4 3 3 2 3 3 3 3 3 3 3 3 tI - time impact; cI - cost impact Table 5 Level of General Criticality Index (GCI) Average Frequency Index of ETO Survey cI tI 4 4 3 3 4 4 3 4 4 4 2 2 2 3 3 4 3 4 3 3 4 3 3 4 4 4 4 3 Advances in Production Engineering & Management 18(2) 2023 GCI 12tc 9tc 16tc 12t 12tc 4tc 9t 12t 12t 9tc 12c 12t 12tc 12c 195 Fortes, Tenera, Cunha, Teixeira Additionally, the impact of customer-imposed changes on product design (P13) features prominently in our findings, with a GCI value of 12tc. Late changes in product design can disrupt the manufacturing process, resulting in delays and increased costs. Therefore, ETO companies must establish strategies to manage and minimise the impact of these changes on their operations. Our analysis results also indicate that several other challenges possess moderate to high levels of criticality for ETO companies. These include workflow visibility (P4), the uncertainty of batch size (P7), pricing (P8), information flow failures (P9), production scheduling (P10), production planning and control (P11), product specification (P12), and lead time (P14). Each of these challenges is directly or indirectly influenced by the unique characteristics of ETO companies, such as their customer-centric approach, project-based operations, high customisation, low volume, extended lead times, high complexity, and high variability. In conclusion, our examination of the criticality index for various issues faced by ETO companies emphasises the need for these organisations to recognise and address the challenges arising from their distinctive operational characteristics. Furthermore, by comprehending the criticality of these issues concerning time and cost, ETO companies can devise targeted strategies and solutions to improve their overall performance [26]. Potential strategies to tackle these challenges may include adopting lean management practices, enhanced communication and coordination among different departments, investment in employee training and development, and implementing advanced tools and technologies to boost process efficiency [27]. By addressing the critical issues identified in our study and considering the unique characteristics of ETO companies, it is feasible to enhance operational efficiency, reduce costs and lead times, and ultimately increase customer satisfaction and market competitiveness. 4. Conclusion Analysing ETO companies' challenges highlights the importance of understanding and addressing critical factors that influence ETO manufacturing systems. By adopting mass customisation strategies and assessing problem criticality, ETO companies might more efficiently allocate resources, thereby improving time and cost management. Furthermore, focus on resource management, innovative approaches, and workflow optimisation will allow ETO companies to adapt to customer demands, increasing satisfaction and competitiveness. This study conducted an SLR and survey of ETO companies to gain insights into problem frequency and criticality. Consistent issues include production scheduling, planning and control, efficiency, and lead time reduction. The criticality analysis enables ETO companies to understand problem implications and prioritise decision-making and resource allocation. Addressing the significant criticality problems will minimise negative consequences on ETO project success. The cooperation between industry and academia is crucial for bridging the gap between theoretical knowledge and practical implementation. Continuous monitoring and analysis of criticality indices will aid ETO companies in identifying emerging issues and make informed decisions throughout projects. Balancing high and low criticality problems is crucial to avoid unexpected complications. By understanding ETOs problem criticality and considering Industry 4.0 paradigms, ETO companies can develop targeted strategies for risk reduction and workflow resilience. Effective communication and collaboration within ETO companies and with external stakeholders are vital. Continuous investment in research and development, employee training, and advanced technology adoption ensures long-term success. In addition, fostering solid collaborations between industry and academia addresses critical challenges more effectively. This research finding suggests that ETO companies should improve resource efficiency, adopt innovative management methods, and optimise workflows. Methods such as the implementation of Lean Management principles, which are proven to enhance production performance by improving quality, reducing costs and shortening production times [28], could further augment ETO companies' performance. In addition, continuous monitoring and analysis of criticality indices will aid in identifying emerging issues and making more informed decisions throughout the ETO project lifecycle. 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Applicability of the lean concept to the management of small-scale manufacturing enterprises in Serbia, Tehnički vjesnik – Technical Gazette, Vol. 24, No. 6, 1929-1934, doi: 10.17559/TV-20150807194942. 198 Advances in Production Engineering & Management 18(2) 2023 Advances in Production Engineering & Management ISSN 1854-6250 Volume 18 | Number 2 | June 2023 | pp 199–210 Journal home: apem-journal.org https://doi.org/10.14743/apem2023.2.467 Original scientific paper Ranking dominant losses in small and medium-sized enterprises (SMEs) in the context of the lean concept application Kondić, V.a, Maglić, L.b, Runje, L.c, Marić, D.b,* aUniversity North, University Center Varaždin, Croatia Engineering Faculty in Slavonski Brod, University of Slavonski Brod, Slavonski Brod, Croatia cMinistarstvo obrane Republike Hrvatske, Croatia bMechanical ABSTRACT ARTICLE INFO The Lean concept was devised in large business systems and is tailored to this way of conducting business. It is a set of principles, techniques and procedures used to identify and eliminate losses within processes. The results of applying this concept are impressive. Western businesses are delighted with the success of large enterprises that have implemented or have begun to implement the Lean concept. Considering the structures of business systems in transitional and EU countries, a question has arisen as to whether it is possible to apply the Lean concept to small and medium-sized enterprises, as these account for more than 99 % of all business systems. The research which was conducted with the goal of designing a suitable model for the implementation of the Lean concept in small to medium-sized enterprises was based on an analysis of the essential elements of this concept. This article presents part of the conducted research that refers to analysis of losses and identification of the dominant losses according to the opinions of real sector experts and scientists from the academic community. The results of this research were used to define procedures for the elimination of major losses and design a final model for the implementation of the Lean concept in small and medium-sized enterprises. Keywords: Manufacturing; Small and medium-sized enterprises (SMEs); Lean manufacturing; Dominant losses; Ranking; Analysis of losses; Elimination of losses *Corresponding author: dmaric@unisb.hr (Marić, D.) Article history: Received 3 June 2022 Revised 6 June 2023 Accepted 19 June 2023 Content from this work may be used under the terms of the Creative Commons Attribution 4.0 International Licence (CC BY 4.0). Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. 1. Introduction Worldwide experience has shown that the Lean concept can be successfully applied to improve the processes and overall functioning of large corporate systems, and the results of this application have been impressive. The methodology was first developed in a large Japanese company (TPS-Toyota). It was then disseminated to large companies in the United States and Europe. It was developed and adapted for large companies. As with other successful large system solutions, the Lean concept was gradually adopted by small and medium-sized companies. This has been particularly pronounced more recently, when this methodology has been more intensively researched and applied in Europe and other regions where corporate systems are, on average, much smaller than in the United States [1, 2]. 199 Kondić, Maglić, Runje, Marić The lean concept envisions products that adapt to the actual needs and expectations of customers, i.e., meet requirements at the required quality and within the agreed delivery times with minimal use of resources [3, 4]. To achieve these goals, it is necessary to eliminate those activities that do not add value or cause waste due to over-production, delays, unnecessary transportation or movement, warehousing, or the like [5, 6]. The authors of [7] state that companies, for example, have successfully reduced their losses and production of waste by applying the lean concept to identify and eliminate or reduce losses. It is important to emphasise that the Lean concept is the industry standard in the automotive industry and its basis for eliminating losses is increasingly being applied in other manufacturing [8] and service sectors. Authors who have studied the lean concept and its application emphasise its importance and role in identifying losses in processes [9, 10]. Errors in the basic approach to the lean concept, e.g. a partial approach, are often cited as reasons for lower success rates in the application of the lean concept [11-13]. Research by authors [14-18] emphasises that the lean concept is expected to improve the capability of companies and increase value for customers through lower prices and improved quality. The key elements for the success of the lean concept in small and medium enterprises are the efficient identification of processes in relation to their business results, the determination of the top management, the application of the general principles of the lean concept, the reduction of losses, the use of modern lean techniques, continuous improvement, and a good conceptual structure tailored to the particular company [19, 23]. It is important to note that the entire process must begin with questions that identify where resources (time, materials, and the like) are being used and how waste can be reduced or eliminated. This results in the need to identify and rank the dominant losses to finally have an approach for designing an ultimate model for implementing the lean concept in small and medium enterprises [1]. The following chapters briefly describe the most important losses in SMEs, the research and analysis conducted, and the ranking of the losses. Finally, the most important guidelines and actions to eliminate the main losses are recommended to the managers in the production processes. Overview of post-implementation activities. 2. Losses in small and medium-sized enterprises 2.1 Defining losses Losses or wastes are essentially anything that customers are not willing to pay for. The authors of the article define losses (waste) as all actions in a process that are not necessary for the successful completion of the process. Furthermore, these authors state that after such losses are eliminated, only those operations and activities (called 'value added') remain in the process that are necessary for the successful delivery of such a product or service to customers. The Japanese word muda is translated as 'waste' or 'losses' in most other languages, but the core of the word has a much deeper meaning. The work that takes place in a process is realised through discrete stages or operations. In each of these process stages, input elements are transformed into output elements, either adding or not adding value to the product or service. Losses are manifested in errors that need to be corrected, in the production of products for which there is no demand, in excessive inventory, and in residual products that are produced unnecessarily. It can also include activities that are not required in the processes, unnecessary movement of employees, unplanned movement of products from one workstation to another, waiting for work, documentation, inspection, handover, or any other form of delay, including complaints from or refunds to customers or product end users. Taiichi Ohno [24] classified losses in production processes into seven categories: Over-production, unnecessary inventory, defects, unnecessary/superfluous movement, improper processing, waiting, and transportation. Subsequent scientific literature has defined another category of losses, namely losses due to inadequate utilisation of human potential. So, when we analyse losses, we refer to these 7+1 types of losses. 200 Advances in Production Engineering & Management 18(2) 2023 Ranking dominant losses in small and medium-sized enterprises (SMEs) in the context of the lean concept application Today, environmental protection as part of the required business efficiency is gaining more importance than in the past. This is because customers and end users are looking for environmentally friendly products and services due to the increasing scarcity of renewable resources and the imposition of (mandatory) regulations. Therefore, this article includes losses attributable to environmental, energy, and safety management systems (green losses) [25]. 2.2 The place and role of small and medium-sized enterprises in the economy For more than two decades, small and medium-sized enterprises have been the engine of economic development. They hire new employees, adapt to changes more easily than large companies, require fewer corporate resources, have shorter and faster communication channels, etc. But their significant role in generating income and investing in fixed assets should not be overlooked either. In most countries, the size of a company is determined by legislation. Three criteria are considered, total assets, total income, and the number of employees, two of which must reach certain values for a company to be classified in a certain size category. According to the legislation and recommendations of the Commission of the European Union, the main criteria for defining small and medium-sized enterprises are the following [1]: • Medium-sized enterprises are enterprises with fewer than 250 employees and an annual turnover of less than 50 million euros and/or an annual balance sheet total not exceeding 43 million EUR; • Small enterprises are enterprises that employ fewer than 50 people and/or whose annual balance sheet total does not exceed 10 million euros; and • Micro enterprises are enterprises that employ fewer than 10 employees and/or have an annual turnover or annual balance sheet total of less than 2 million euros. Thus, small and medium enterprises account for 99.7 % of the total number of enterprises in Croatia (92.2 % are micro enterprises, 6.3 % are small and 1.2 % are medium enterprises). They account for 68.3 % of total employment (1.03 million), which is higher than the EU average, and generate added value of 20.5 billion euros (59 %), which is equal to the EU average, and 51 % of GDP [1]. Small and medium-sized enterprises create the most jobs. The fundamental characteristic of small and medium-sized enterprises is their ability to adapt quickly to changing economic conditions. 3. Research implementation 3.1 Research sample selection Since the research was partially conducted in small and medium enterprises of the real sector, it was necessary to define a meaningful sample eligible for this research. This was ensured by an appropriate selection of companies, taking into account their activities and their international classification (European Accreditation Classification – EAC). It is important to point out that the studied sample of small and medium enterprises is from the industrial sector, i.e. manufacturing. The researcher's assessment and the opinion of experts on the lean concept led to an approximate number of SMEs in the Republic of Croatia that apply the lean concept. The sample size was 36 SMEs, which corresponds to 34 % of the studied companies. According to the form and characteristics of the sample, it was a quota sample, i.e. an intentional sample, which means that the researchers of this article selected the companies to be studied. One to two respondents from each of the 26 selected small and medium enterprises participated in the survey, making the total number of respondents 30. The sample included companies that were 'reachable' at the time this research was conducted, were willing to provide the required responses, and were at various stages of implementing the lean concept. The criteria used to select the study participants from the selected small and medium enterprises were their knowledge of the Lean concept and their position in the company [1]. In this research, an appropriate opinion of scientists from the academic community was obtained. Care was taken to select scientists from the Republic of Croatia and from other countries. Advances in Production Engineering & Management 18(2) 2023 201 Kondić, Maglić, Runje, Marić The aim was to compare the opinions of experts from practice and academia on the problem of losses in SMEs. A total of 85 surveys were sent out, 30 of which were answered within a reasonable period of time. The opinions of scientists from academic communities outside the Republic of Croatia were also included in this research. Several times, about 220 surveys were sent out, of which 30 were returned within the time limit. The criterion of knowledge of the Lean concept and experience in theoretical and practical work with its implementation was used in the selection of scientists [1]. It is assumed that the sample size is significant for the field in which the research was conducted. Since the opinion of experts from all over the world was also obtained, it is assumed that the results can be generalized to countries that have a similar economic structure as the Republic of Croatia, but taking into account their specifics. 3.2 Research methodology and ranging losses Ranging losses in the real sector Research has been conducted in both the real sector and the academic community to classify the prevailing losses. A method based on Spearman's Rank correlation coefficient was developed to rank the losses in the private sector [26]. The real sector ranking was done using a semi-structured interview. Participants assigned an appropriate rank to each factor based on their opinion: rank 1 (the most important) to rank 9 (the least important). Factors with equal influence were also ranked in the same way. Respondents could assign the same ranks to different factors. Respondents could assign the same ranks to different factors. For these reasons, the ranks had to be redesigned. Therefore, factors with the same rank received a new rank that corresponded to the mean value of the rank that the factors shared among themselves. Multiple factors with the same rank value were assigned a related rank. The average rank was determined as the arithmetic mean of the ranks that the factors would have received if they had not been related ranks. With the rank table redesigned, the null and alternative hypotheses for the Spearman's Rank correlation coefficient were as follows: 𝐻𝐻0 : 𝜌𝜌 = 0 𝐻𝐻1 : 𝜌𝜌 ≠ 0 The probability of a type I error was set an extremely high significance limit, i.e. α = 0.01. The significance limit α denotes an area of rejection of the H0 hypothesis. Calculation of Spearman's Rank correlation coefficient ρ is suitable if the number of pairs in the sequence is less than or equal to 30, and is calculated using Eq. 1. 𝜌𝜌 = 1 − 6 ∙ ∑9𝑖𝑖=1(𝑅𝑅 (1) − 𝑅𝑅(2) )2 𝑁𝑁 ∙ (𝑁𝑁 2 − 1) (1) where R is the squared ranks’ difference of the corresponding variable value pair, and N is the number of influencing factors (N = 9). The verification of the statistical value of ρ was performed using the hypothesis test for the significance of the correlation coefficient tr. The statistical value for ρ was verified using the tdistribution, whose value was calculated with Eq. 2. 𝑡𝑡𝑟𝑟 = 𝜌𝜌 2 �1 − 𝜌𝜌 𝑁𝑁 − 2 (2) where ρ is Spearman's Rank correlation coefficient (ρ = 0.984). Since the calculated value tr (15.04) was higher than the tabular value tt (2.99), it was assumed that the correlation coefficient was significant. With this statement, a one-sided and positive correlation was confirmed. Since the value of Spearman's Rank correlation coefficient reached 1, it could be confirmed with high reliability that the newly designed ranking tables were suitable for initial ranking and could be used for further analysis. 202 Advances in Production Engineering & Management 18(2) 2023 Ranking dominant losses in small and medium-sized enterprises (SMEs) in the context of the lean concept application Since there were several equal ranks in the range of losses defined by the experts, the expression for the Spearman's Rank correlation coefficient had to be corrected. The corrected rank correlation coefficient is called Kendall's concordance coefficient and is usually referred to as Kendall's W in the literature. By its logic, the concordance coefficient W tests the relationship between the reviewer's actual concordance and the maximum possible concordance. Eq. 3 was used for the calculation 𝑊𝑊 = 12 ∑ 𝑆𝑆 2 𝑚𝑚2 (𝑁𝑁 3 − 𝑁𝑁) − 𝑚𝑚 ∑𝑚𝑚 𝑖𝑖=1 𝑇𝑇𝑖𝑖 (3) where m is the number of interlocutors conducting significance ranking factors (m = 30), Ti is the sum of correction factors for related ranks, and S2 is the sum of squared sum of the rank deviations of all respondents’ and the sum of the averages. The significance of Kendall's coefficient of concordance (W) was performed by applying the χ2test, as defined by Eq. 4. This required that the calculated value of this value is greater than that in the table. (4) 𝜒𝜒𝑟𝑟2 = 𝑚𝑚 ∙ (𝑁𝑁 − 1) ∙ W Since the ratio between the calculated and tabulated values was 46.56 > 20.1, the hypothesis test regarding the agreement of the opinions of the respondents showed that the ranks were interdependent, which means that the hypothesis of the agreement of the ranks of the respondents could be accepted. The calculation, i.e. the determination of the strength of the influence of all factors, was performed with the help of the dominance coefficient ϕ, which was calculated with the Eq. 5. ∑𝑚𝑚 𝑖𝑖=1 𝑠𝑠𝑖𝑖𝑖𝑖 ∙ 𝜏𝜏𝑖𝑖 𝜙𝜙 = 𝑚𝑚 𝑘𝑘 (5) ∑𝑖𝑖=1 ∑𝑗𝑗=1 𝜏𝜏𝑖𝑖 𝑠𝑠𝑖𝑖𝑖𝑖 where τ represents the coefficient of participating experts’ competences. The values of the dominance coefficient are shown in Fig. 1. Ranging losses by scientists A suitable questionnaire was developed for the purpose of this research. It was designed according to the previous selection of variables and allowed the assessment of the respondents involved. The form contained 9 statements with 5 response options that corresponded to the Likert scale, allowing the expression of positive or negative attitudes towards each statement. In their responses, respondents expressed the degree of their agreement or disagreement with the position expressed in the statement (disagree at all, agree, do not know/neutral, agree, and strongly agree). After the initial evaluation, a weighted average calculation was performed for each participant, from which an individual ranking was determined for all 9 variables. These ranks were then used in the subsequent analysis. The weighted average, also referred to as the weighted arithmetic mean, is the average in which a weight (the weighting factor) is assigned to each quality item. The difference with the arithmetic mean is that the data that are averaged are not considered important. In this way, these calculated weights determine the relative average importance of each quantity [1]. Table 1 and Fig. 2 show the ranking of losses of scientists within the Republic of Croatia according to the weighted average. An identical procedure was performed with scientists outside the Republic of Croatia. Table 2 and Fig. 3 show their ranking of losses. The statements offered to the participants were related to the impact of losses on business results, and for each of the following 9 statements, the participants' opinion was sought on which aspect is the main cause of losses in the production process: Statement 1 – 'Over-production' is the dominant cause of losses. Statement 2 – 'Inventory' is the dominant cause of losses. Statement 3 – 'Transportation' is the dominant cause of losses. Statement 4 – 'Waiting' is the dominant cause of losses. Statement 5 – 'Excess motion' is the dominant cause of losses. Advances in Production Engineering & Management 18(2) 2023 203 Kondić, Maglić, Runje, Marić Statement 6 – 'Defects' are the dominant cause of losses. Statement 7 – 'Over-processing' is the dominant cause of losses. Statement 8 – 'Utilisation of human potential' is the dominant cause of losses. Statement 9 – 'Green losses' are the dominant cause of losses. 3.3 Brief description of losses Losses related to over-production Over-production means producing more, earlier, or faster than is required for the next stage of the production process. It is neglected by companies that view over-production and larger inventories as a safety rather than a loss. Over-production is often the result of production planning based on sales forecasts rather than specific orders. In addition, over-production can be the result of poor communication and business relationships with suppliers, the use of high performance hardware, extensive labour, long lead times, poor business decisions, and a host of other reasons. In both the technical literature and academia, this is considered a major cause of losses. The cost of initialising machinery and production processes beyond what is required is often a hidden loss and must be accounted for. Combined with the additional cost of inventory or the inability to sell additional inventory after a certain time, this can lead to further losses [27]. The lean concept is based on the pull principle of production, which involves manufacturing products according to customer orders. This means that production must be adjusted to demand, i.e., the goal should be to produce only what is needed when a customer demands it, according to the 'just in time' principle, while many companies use the 'just in case' principle. The statement for the survey participants was: 'Over-production' as a loss represents a dominant impact on the efficiency of the production process. Losses related to inventory Accumulating unnecessary inventories of materials, work in process, tooling, or finished goods creates unnecessary costs. It is important to note that unnecessary inventory adds no value to the final product and represents a net cost to the manufacturer until it is sold to a customer. This cost is a significant problem for SMEs because unnecessary inventory creates additional space problems, additional moves and transportation, and the possibility of damage or reduction in the quality or value of the products. The statement for survey respondents was that 'inventories' as a loss represent a dominant impact on the efficiency of the production process. Losses related to transportation Transportation is the movement of materials and products from one place to another. This is a loss because it does not add value to the product and does not impose a cost that customers are willing to pay for. For manufacturers, it is an unnecessary cost because they must pay employees who are directly or indirectly involved in internal or external transportation, and they must also provide adequate transportation and appropriate equipment and supplies. Regardless of whether transportation is optimally set up or integrated into the technological process, it always represents a loss [28]. In this sense, the lean concept requires an analysis of all technological processes and the elimination or shortening of unnecessary transportation. The statement for the survey participants was: 'Transport' as a loss has a dominant influence on the efficiency of the production process. Losses related to waiting Waiting times are very often the result of machine breakdowns, supply problems, poor material and capacity planning, poor design of technological procedures used in the production process, poor structuring of documented information and poor control structures, and so on. Waiting time is the time lost due to slow or stopped production in a single part of the production process while the previous step is being completed. A classic example of waiting time is any production line with multiple orders or operations. If there is a different technological time between two operations, i.e. if the processing time of the second operation is longer than that of the first or vice versa, waiting times or losses occur. Operations that require more time must be made more efficient or 204 Advances in Production Engineering & Management 18(2) 2023 Ranking dominant losses in small and medium-sized enterprises (SMEs) in the context of the lean concept application production must be balanced by measuring and calculating with production rates. Losses due to waiting lead to a disruption in the flow of work, which is one of the main principles of lean production [29]. The statement for the survey participants was: 'Waiting' as a loss has a dominant influence on the efficiency of the production process. Losses related to excess motion at workplaces Any movement of workers in the workplace that is not directly related to value creation is unproductive and represents a loss. For example, by observing a worker in a processing system, it is possible to determine when value is actually being added. In theory, this is a very short period of time, sometimes only a few seconds. The other movements are actions that do not add value, such as lifting or lowering in the position where processing is taking place. The few seconds lost in such an operation due to unnecessary employee movements become minutes and eventually hours due to repetitive activities. Excessive movements in the workplace are often the result of poor ergonomic solutions in the machines, tools or equipment themselves. The statement for the survey participants was: 'Excessive movement' has a dominant influence on the efficiency of the production process as a loss. Losses related to defects Defects (scrap) in production can occur in items, assemblies, or finished products. It results from quality control measures that identify a specific nonconformance of a product to requirements, specifications, or contracts. For non-conforming products, repairs (rework) of deterioration must be made. If for any reason it is not possible to make such repairs, or if a customer is unwilling to purchase the product at a reduced price, the manufacturer must scrap the product as defective. All actions taken to fix a nonconforming product represent additional costs because time and materials are lost in the manufacturing process. However, the greatest cost to manufacturers comes from nonconforming products reaching the customer. Sometimes these costs can be much higher than expected [30]. This can lead to high reclamation or modification costs or simply the loss of customers. Non-conforming products result from frequent design changes, machine setup errors, decreasing operator concentration, errors in production documentation, and the like. The statement for survey participants was that 'defects' represent a dominant impact on the efficiency of the production process as a loss [31, 32]. Losses related to over-processing Over-processing adds value to a product that customers have neither asked for nor want to pay for. Such activities represent an unnecessary cost to manufacturers. This loss comes in the form of lost time for employers and use of resources. Over time, these are costs that can significantly reduce the efficiency and effectiveness of a given process. Over-production can result from unclear standards and specifications or poor technical documentation. Many manufacturers strive to produce the highest quality product possible, but do not know what truly adds value to a product or what is essential to the end user. Examples of over-processing include painting surfaces that will never be seen or exposed to corrosive processes, or polishing and nickel plating surfaces that are not required by the customer. The statement for survey respondents was, 'Over-machining' as a loss represents a dominant impact on the efficiency of the production process. Losses related to the utilisation of human potential When existing knowledge and entrepreneurial know-how are not fully utilised, this harms companies in various ways: Loss of efficiency, unnecessary new hires, demotivated employees who lack recognition, etc. Employees remain underutilised, and companies fail to realise the full potential of their workforce. Employees are a company's most valuable resource, and without their commitment and loyalty, companies cannot be as competitive as they can be in the marketplace. It is very important that the contribution of all employees is recognised [1]. Most companies do not allow their employees to participate in the production process beyond what is required because they fear that they will become overqualified, demand higher wages, or leave for another company that can offer them better conditions as compensation for their newly acquired Advances in Production Engineering & Management 18(2) 2023 205 Kondić, Maglić, Runje, Marić knowledge and experience. The statement for the survey participants was that 'the non-utilisation of human potential' represents as a loss a dominant impact on the efficiency of the production process. It must be emphasised that this loss is the most difficult to identify and quantify and, in this context, the most difficult to eliminate. Losses related to environmental protection As stated in the introductory part of this article, losses in the production process and in other processes of any system are also associated with losses related to the energy management system, environmental protection, and workplace safety. Seven types of losses have been identified, grouped under the title 'Green Losses'. They are related to green production, minimising process waste and pollution from processes, and from the design and execution of products and services. The statement for survey participants was that 'Green losses' represent a dominant impact on the efficiency of the production process as a loss. 3.4 Results of loss ranging The results of loss ranging by competent experts from the field are shown in Fig. 1, and the results of loss ranging by scientists within the Republic of Croatia (ZRH) are shown in Table 1 and Fig. 2. The results of loss ranging by scientists outside the Republic of Croatia (ZIRH) are presented in Table 2 and Fig. 3. Fig. 1 A presentation of loss ranging as defined by real sector professionals Ordinal Number 1. 2. 3. 4. 5. 6. 7. 8. 9. 206 Table 1 Loss ranging on the basis of weighted sverages (Croatia) Degree of Agreement * Loss Type Weighted -StatementAverage SD D N A SA Over-production 1 12 6 10 1 1.97 Inventories 0 8 8 11 3 2.10 Transportation 0 9 6 13 2 1.90 Waiting 0 1 4 18 7 1.60 Excessive Movements 1 4 11 13 1 1.87 Scrap 0 4 2 15 9 1.77 Over-processing 0 5 7 15 3 1.87 Utilisation of Human 1 4 6 13 6 2.13 Resources Green Losses 0 5 18 5 2 1.70 * SD-Strongly Disagree; D-Disagree; UN-Unknown/Neutral; A-Agree; SA-Strongly Agree Range 7 8 6 1 4 3 5 9 2 Fig. 2 A presentation of loss ranging as defined by experts from within Croatia Advances in Production Engineering & Management 18(2) 2023 Ranking dominant losses in small and medium-sized enterprises (SMEs) in the context of the lean concept application Ordinal Number 1. 2. 3. 4. 5. 6. 7. 8. 9. Table 2 Loss ranging on the basis of weighted averages (outside Croatia) Degree of Agreement * Loss Type Weighted -StatementAverage SD D UN A SA Over-production 0 4 8 14 4 1.93 Inventory 2 6 6 10 6 2.47 Transportation 0 8 4 17 1 1.63 Waiting 0 0 3 16 11 1.56 Excessive Movements 2 4 5 15 4 1.97 Scrap 1 5 6 12 6 2.23 Over-processing 1 6 0 18 5 1.63 Utilisation of Human 3 6 5 11 5 2.43 Resources Green Losses 6 2 21 1 0 1.43 * SD-Strongly Disagree; D-Disagree; Unknown/N-Neutral; A-Agree; SA-Strongly Agree Range 6 9 3 2 5 7 4 8 1 Fig. 3 A Presentation of loss ranging as defined by experts from outside Croatia Statistical data verification Because the data describing the losses are ordinal in nature, the analysis involves ranking the data rather than the data themselves. In nonparametric statistics, the Kruskal-Wallis test is most commonly used to compare the ranks of losses from three or more samples. In this study, three samples were used. The first sample refers to the private sector (RS), the second to academic scientists from the Republic of Croatia, and the third to academic experts from outside the Republic of Croatia. The Kruskal-Wallis test essentially tests the hypotheses that have been established. It is a test of an analysis of variance, except that instead of numerical measurement data (continuous variables), rankings (discrete variables) are used [1]. A test of statistic H is created by the Eq. 6. 12 𝑇𝑇𝑖𝑖 𝐻𝐻 = � − 3(N + 1) (6) 𝑁𝑁(𝑁𝑁 + 1) 𝑁𝑁𝑖𝑖 where N is the total number of observations, and Ti is the sum of the ranks in a single sample. If the samples are large enough (in this case the samples are considered large enough because each sample contains 5 results), H has the same distribution as the HI-squared, so its significance can be read in the χ2 Table with k−1 denoting the degree of freedom and the level of significance α. The results of the Kruskal-Wallis test for the individual samples are given in Tables 3, 4 and 5 for a significance of 5 % (α = 0.05). Test Statistic H 43.40 Test Statistic H 32.08 Test Statistic H 39.38 Table 3 Analysis of losses within the RS sample RS Value of p Test Statistic H Custom Value 0.000 43.43 Table 4 Analysis of losses within the 'Croatia' sample Croatia Value of p Test Statistic H Custom Value 0.000 35.63 Table 5 Analysis of losses within the ZIRH sample Outside Croatia Value of p Test Statistic H Custom Value 0.000 43.40 Advances in Production Engineering & Management 18(2) 2023 Value of p 0.000 Value of p 0.000 Value of p 0.000 207 Kondić, Maglić, Runje, Marić Before determining the final ranking of losses, it is necessary to determine whether there is a statistically significant difference between the three samples in terms of perceptions of the intensity of the impact of these losses on the overall efficiency of small and medium-sized enterprises. To test this, the hypotheses H0 and H1 were established. H0 – there is no significant statistical difference between RS, 'Croatia' and 'Outside Croatia' in terms of perception of the impact of losses on the overall efficiency of small and medium enterprises. H1 – there is a significant statistical difference between RS, 'Croatia' and 'Outside Croatia' in terms of perception of the impact of losses on the overall efficiency of small and medium enterprises. The results of the Kruskal-Wallis test in relation to the perception of the impact of losses on the overall efficiency of small and medium-sized enterprises on the part of RS, 'Croatia' and 'Outside Croatia' are presented in Table 6. These results indicate whether the null hypothesis can be rejected, i.e. an alternative is accepted at the selected significance level. Specifically, these test results show that the ranks defined by the real sector, scientists from the Republic of Croatia and scientists from outside the Republic of Croatia are not statistically significantly different in terms of the perception of the impact of losses on the overall efficiency of SMEs (p-value is 0.084, i.e. p > 0.05). From this it could be concluded that it is statistically correct to accept the H0 hypothesis. Table 6 Results of the Kruskal-Wallis test for the Three Samples (RS, Croatia and Outside Croatia) RS, Croatia and Outside Croatia Test Statistic H Test Statistic H Value of p Value of p Custom Value 13.90 0.084 14.06 0.080 From the analysis and comparison of the individual ranks (see Tables 1 and 2), the following can be deduced. When analysing the ranking determined by scientists from the Republic of Croatia (Croatia), a high ranking (ranking number 2) for 'Green Losses' can be observed. In this particular case, 18 participants or 60 % declared themselves neutral/undecided about this type of loss. This means that they did not have a specific opinion about this type of loss. Similarly, 5 or 16.7 % stated that they disagreed, while 16.7 % expressed agreement. The analysis of the ranking determined by researchers outside the Republic of Croatia (Outside Croatia) showed, as mentioned above, a high ranking (rank 1) for 'Green losses'. In this particular case, 21 participants, or 70 %, stated that they were neutral/undecided about this type of loss, while 5, or 20 %, disagreed with the statement that this is the predominant loss that significantly affects the business efficiency of SMEs. For these reasons, the responses of the participants from 'Croatia' and 'Outside Croatia' regarding 'Green Losses' are excluded from the analysis. Taking into account the results of the Kruskal-Wallis test and the above statement, the rankings were combined and the final results of the loss ranking (Table 7) were presented. There are three 'significant losses' for SMEs, namely: waiting time, transportation, and excessive movement. There are other losses, but these three are prevalent and require more attention in improving the processes. Code G1 G2 G3 G4 G5 G6 G7 G8 G9 208 Table 7 An analysis of individual ranks and a presentation of the final loss rankings RANGE 1 RANGE 2 RANGE 3 SUM OF LOSSES 'RS' 'ZRH' 'ZIRH' RANGES Over-production 4 7 6 17 Inventory 5 8 9 22 Transportation 2 6 3 11 Waiting 1 1 2 4 Excessive Movement 3 4 5 12 Scrap 7 3 7 17 Over-processing 8 5 4 17 Utilisation of Human 6 9 8 23 Resources Green Losses 9 2 1 12 FINAL RANGE 4 5 2 1 3 4 4 6 - Advances in Production Engineering & Management 18(2) 2023 Ranking dominant losses in small and medium-sized enterprises (SMEs) in the context of the lean concept application 4. Conclusion Eliminating and reducing losses is one of the main goals of any improvement system, and this includes the Lean concept. This paper is mainly about identifying the most important loss types from the defined 9 loss types that occur in the production environment. The calculation of the ranks is based on the opinions of the surveyed participants using Spearman's rank correlation coefficient and weighted average method. The results indicate that the top three loss types were identified from the total nine losses. Most respondents indicated that losses related to waiting time, transportation, and excessive movements were the main causes of losses for small and medium enterprises. Statistical analysis of the results confirmed this conclusion. Based on the results of this research, it can be concluded that production managers and professionals responsible for the production process must systematically monitor these three main types of losses so that they can be eliminated or continuously reduced. Elimination of the prevailing losses significantly increases productivity and other indicators of production processes [33, 34]. The activities of those responsible for loss elimination should focus on the following: • • • • • • Identification of losses; Implementation of the proposed loss elimination systems; Training employees in the application of the lean concept to eliminate or reduce losses; Measuring losses; Establishing liability for losses, Analysing risks and opportunities for loss management. Specific procedures for eliminating losses are based on the nature of the losses, their magnitude, and their priorities. Procedures include continuous training of staff and planning of activities of responsible staff with continuous measurement, monitoring and improvement. Various lean techniques can be applied to reduce losses in production systems, such as: 5S, Kanban, Andon, production bottleneck analysis, Kaizen, Jidoka, JIT, TPM, VSM, Gemba, SMED, OEE, PDCA, TQM, SMART goals, Continuous production flow, Tact time, Visual Factory, Poka-Yoke, etc. [35, 36]. The application of various lean techniques is essential in all occasions for the identification, analysis and elimination of losses, with emphasis on numerical indicators (Six Sigma). Well-organised loss management and the lean concept in general help SMEs to withstand unforeseen situations such as a pandemic, political and economic unrest, etc. unforeseen events. References Kondić, V. (2010). Poboljšavanje efikasnosti procesa na načelima Lean koncepcije u malim i srednjim poduzećima, PhD thesis, University of Slavonski Brod, Slovonski Brod, Croatia. [2] Stoijljković, V. (2017). ISO 9001:2015 i Lean, Integracija dva koncepta menadžmenta – ISO 9001:2015 sistem menadžmenta kvalitetom i Lean koncept, Talija Izdavaštvo, Niš, Serbia. [3] Piškor, M., Kondić, V. (2010). Lean production kao jedan od načina povećavanja konkurentnosti hrvatskih poduzeća na globalnom tržištu, Tehnički glasnik, Vol. 4, No. 1-2, 37-41. [4] Saad, D.A., Masoud, M., Osman, H. (2021). 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The importance of subcontracting and its relationship with lean philosophy in automotive industry, International Journal of Industrial Engineering and Management, Vol. 13, No. 3, 186-193, doi: 10.24867/IJIEM-2022-3-311. 210 Advances in Production Engineering & Management 18(2) 2023 Advances in Production Engineering & Management ISSN 1854-6250 Volume 18 | Number 2 | June 2023 | pp 211–224 Journal home: apem-journal.org https://doi.org/10.14743/apem2023.2.468 Original scientific paper An improved discrete particle swarm optimization approach for a multi-objective optimization model of an urban logistics distribution network considering traffic congestion Li, K.a, Li, D.a,*, Ma, H.Q.b aSchool of Business Administration, Shanghai Lixin University of Accounting and Finance, Shanghai, P.R. China Resources Department, Shanghai Institute of Visual Arts, Shanghai, P.R. China bHuman ABSTRACT ARTICLE INFO To optimize urban logistics networks, this paper proposes a multi-objective optimization model for urban logistics distribution networks (ULDN). The model optimizes vehicle usage costs, transportation costs, penalty costs for failing to meet time windows, and carbon emission costs, while also considering the impact of urban road traffic congestion on total costs. To solve the model, a DPSO (Discrete Particle Swarm Optimization) algorithm based on the basic principle of PSO (Particle Swarm Optimization) is proposed. The DPSO introduces multiple populations to handle multiple targets and uses a variable neighbourhood search strategy to improve the search ability of particles, which helps to improve the local search ability of the algorithm. Simulation results demonstrate the effectiveness of the proposed model in avoiding traffic congestion, reducing carbon emissions costs, and time penalty costs. The optimization comparison results between DPSO and PSO also verify the superiority of the DPSO algorithm. The proposed model can be applied to real-world urban logistics networks to improve their efficiency, reduce costs, and minimize environmental impact. Keywords: Urban logistics distribution network; Traffic congestion; Optimization; Modelling; Multi-objective optimization; Vehicle routing problem (VRP); Swarm intelligence; Discrete particle swarm optimization algorithm (DPSO) *Corresponding author: gslidan@lixin.edu.cn (Li, D.) Article history: Received 23 April 2023 Revised 7 June 2023 Accepted 13 June 2023 Content from this work may be used under the terms of the Creative Commons Attribution 4.0 International Licence (CC BY 4.0). Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. 1. Introduction As urban populations continue to grow, the demand for urban logistics services has increased significantly. The logistics industry plays a crucial role in the economic development of cities [1-6]. However, the development of urban economies and the increase in private cars have led to increased traffic congestion, which has added pressure to the urban transportation network. Logistics companies must not only deal with traffic congestion during distribution, but also meet increasingly strict time requirements from customers. To meet the growing demand for timely delivery services under complex and unstable urban traffic conditions, logistics companies must find ways to provide efficient and effective distribution services that meet customer needs. 211 Li, Li, Ma In the process of urban logistics distribution, the shortest distance between two customer nodes is often not the most efficient route, especially under urban traffic conditions where the shortest distance route is often the most congested. To avoid congested sections, shorten delivery times, and respond to customer time needs, logistics companies require an optimized distribution network. Improving the efficiency of logistics service delivery while meeting customer needs and ensuring customer loyalty can not only improve the core competitiveness of enterprises, but also alleviate urban traffic congestion and improve the urban environment. In addition, environmental pollution caused by urban logistics activities has become a critical issue. Low-carbon logistics plays an important role in the low-carbon development of the environment and provides an essential guarantee for economic and social low-carbon development. Therefore, logistics companies must address the environmental problems caused by urban logistics activities. Green distribution is a necessary consideration for cities to optimize the overall logistics facility layout and distribution network. To optimize urban logistics networks, it is essential to consider both traffic congestion and green distribution, as it can improve the economic benefits of logistics enterprises while reducing the carbon emissions generated in the process of logistics distribution. This paper proposes an urban logistics network optimization model that considers the impact of traffic congestion on urban logistics distribution. The model aims to minimize vehicle usage costs, transportation costs, time penalty costs, and carbon emission costs. To solve the model, we introduce an improved discrete particle swarm optimization (DPSO) algorithm that introduces multiple populations to handle multiple targets and uses a variable neighbourhood search strategy to enhance the search ability of particles and improve the local search ability of the algorithm. Simulation experiments verify the effectiveness of the proposed model and the superiority of the DPSO algorithm. 2. Literature review The vehicle routing problem (VRP) is a fundamental problem in the optimization of urban logistics networks. Since its inception in 1945, numerous research studies have been published [7-15]. Gulczynski et al. [16] proposed a vehicle routing problem based on batch delivery and developed a heuristic method to solve the problem. Contardo and Martinelli [17] studied the multi-site vehicle routing problem with capacity and routing length constraints and designed a new exact algorithm to solve the problem. Bertazzi and Secomandi [18] focused on vehicle routing problems with random demand and replenishment and introduced a new method to approximate the expected cost of any VRPSD with replenishment. In recent years, research on VRP has become more diverse, with scholars proposing models to optimize different aspects of logistics transportation. For example, Islam et al. [19] studied the Clustered Vehicle Routing Problem (CluVRP) and proposed a new hybrid meta-heuristic algorithm combining particle swarm optimization (PSO) and variable neighbourhood search (VNS) to solve the model. Solomon and Desrosiers [20] incorporated the concept of time windows into vehicle routing problems, and Jabali et al. [21] proposed a vehicle routing problem with soft and hard time windows. Rodríguez-Martín and Yaman [22] developed a periodic vehicle routing problem with driver consistency, and Yuan et al. [23] studied the generalized vehicle routing problem with time windows. Zhao et al. [24] considered the departure time and the distance between two customers. They proposed a bi-objective mixed integer linear model to optimize the total transportation cost and time cost. The Green Vehicle Routing Problem has gained significant attention in recent years, aiming to promote green development by reducing energy consumption and carbon emissions in logistics activities. Scholars have proposed various models to optimize different aspects of logistics transportation while reducing carbon emissions. For example, Demir et al. [25] compared and analyzed six models for carbon emissions and energy consumption. Kirschstein and Meiselb [26] designed a comprehensive carbon emission calculation model by considering factors such as vehicle speed, load, and road conditions. Naderipur and Alinaghian [27] studied low carbon VRP with the goal of reducing vehicle energy consumption and carbon emissions. Kwon et al. [28] proposed a multi212 Advances in Production Engineering & Management 18(2) 2023 An improved discrete particle swarm optimization approach for a multi-objective optimization model of an urban … vehicle VRP model based on carbon emissions to minimize total cost. Suzuki [29] established a VRP model aimed at minimizing energy consumption and carbon emissions. Li et al. [30] constructed a VRP model that aimed to minimize the sum of vehicle fixed usage costs, fuel consumption, and carbon emissions costs. Wen et al. [31] proposed a multi-site model of vehicle routing to optimize carbon emissions, fuel consumption, vehicle rental, and driver wage costs. They developed an improved adaptive large neighbourhood search (ALNS) algorithm to effectively solve the problem. Guo et al. [32] studied the multi-compartment vehicle routing problem considering carbon emissions and optimized the total transportation cost, including carbon emissions, using a three-dimensional ant colony optimization algorithm (TDACO). Li and Li [33] proposed a multiobjective supply chain network optimization model that aimed to optimize network costs, carbon trading costs, and customer satisfaction losses. They developed a new improved NSGA-II algorithm to solve the model. Zhu et al. [34] established a CVR for multiple warehouses with the goal of minimizing the carbon emissions of the fleet required to deliver the required goods to customers. Despite the significant progress made in the field of the Green Vehicle Routing Problem, there is still a need for more effective and comprehensive models that can address the challenges posed by traffic congestion and low-carbon emission reduction in urban logistics network optimization. Therefore, this paper proposes a new model and algorithm to address these gaps in the literature, which has practical significance for the development of low-carbon logistics and the improvement of urban traffic congestion. Urban traffic networks have time-varying characteristics due to factors such as morning and evening traffic peaks, road speed limits, traffic regulations, and external accidents. To address this, some scholars have studied time-dependent vehicle routing problems (TDVRP) under time-varying road networks. For example, Jabbarpour et al. [35] established a TDVRP model that aimed to minimize driving time and fuel consumption, and designed different traffic congestion scenarios for experiments. Xiao and Konak [36] proposed that highway transportation companies can reduce their CO2 emissions through effective vehicle routing and delivery schedules based on traffic congestion in their service areas. Poonthalir and Nadarajan [37] focused on behavior in variable speed environments and its impact on route costs and fuel consumption. They built a TDVRP model that aimed to minimize vehicle travel distance and fuel consumption and designed an improved particle swarm optimization algorithm to solve the problem. Çimen and Soysal [38] considered the vehicle routing problem under time-dependent and random vehicle speeds. The research results showed that incorporating vehicle speed randomness into the model enabled optimization of the final distribution route in terms of travel duration, carbon emissions, and travel costs. Ehmkea et al. [39] constructed a TDVRP model that aimed to minimize carbon emissions and solved it using a tabu search algorithm. Sarbijan and Behnamian [40] proposed that in the context of congestion in urban transportation networks, there is a higher requirement for fast, flexible, reliable, and low-cost delivery in urban areas. In real-time collaborative regional vehicle routing problems with flexible time windows, a combination of urban logistics transportation and distribution composed of various vehicles can reduce the number of times to return to physical warehouses, reduce costs, and save time. Through a comprehensive literature review, scholars have proposed various models to address the challenges posed by time-varying road networks in urban logistics transportation. However, there is still a need for more effective and comprehensive models that can address the challenges posed by traffic congestion, low-carbon emission reduction, and time-varying road networks in urban logistics network optimization. Therefore, this paper proposes a new model and algorithm to address these gaps in the literature, which has practical significance for the development of low-carbon logistics and the improvement of urban traffic congestion. Through a comprehensive literature review, it was found that some research results have been generated on the Vehicle Routing Problem (VRP) under traffic congestion. However, the existing research is limited to constant vehicle speed, and there is a lack of research on VRP under time-varying road networks. Additionally, there is a shortage of research on the impact of traffic congestion and lowcarbon emission reduction on urban logistics network optimization. To address these gaps in the literature, this paper proposes a new multi-objective urban logistics network optimization model that considers traffic congestion to optimize vehicle usage costs, transportation costs, penalty costs, and carbon emission costs. The proposed model takes into Advances in Production Engineering & Management 18(2) 2023 213 Li, Li, Ma account the impact of time-varying road networks and the need for low-carbon logistics. To solve the model, a DPSO algorithm based on the basic principle of PSO is developed. This research has practical significance for the development of low-carbon logistics and the improvement of urban traffic congestion. By optimizing vehicle routing and reducing carbon emissions, this research can help reduce the negative impact of logistics activities on the environment and promote sustainable development. Moreover, the proposed model and algorithm can provide valuable guidance for logistics companies and transportation departments in urban areas to optimize their logistics network and reduce transportation costs. 3. Optimization model of vehicle routing problem in urban logistics distribution networks 3.1 Problem description In today's low-carbon economic environment, it is crucial to consider not only conventional costs such as vehicle usage costs, transportation costs, and penalty costs for failing to meet the time window in urban logistics distribution but also the carbon emission costs associated with transportation. Additionally, the impact of urban road traffic conditions on total costs should also be considered. Therefore, this paper proposes a new model for the Vehicle Routing Problem (VRP) in Urban Logistics Distribution Networks (ULDN), which takes into account traffic congestion. The problem can be described as a distribution center providing logistics distribution services to multiple customer points within a specified time. The goal of the proposed model is to comprehensively optimize vehicle usage costs, transportation costs, time penalty costs, and carbon emissions costs in ULDN. The customer location, customer demand, and time window are known, and the urban traffic congestion period and traffic congestion status can be obtained from the transportation department. The proposed model and algorithm will provide valuable guidance for logistics companies and transportation departments in urban areas to optimize their logistics network and reduce transportation costs while considering the impact of traffic congestion and low-carbon emissions. By reducing carbon emissions and optimizing vehicle routing, this research can help mitigate the negative impact of logistics activities on the environment and promote sustainable development. 3.2 Assumptions In the proposed model, the following assumptions are made: (1) Only one distribution center with sufficient supply is considered. (2) The delivery vehicles are of the same type and depart from the logistics center at different times as needed, returning to the same logistics center afterward. (3) Customer demand is less than the vehicle capacity, and there is a service time window requirement. (4) During periods of traffic congestion, vehicles travel at a congested speed, while during noncongested periods, vehicles travel at normal speeds. (5) The maximum load capacity of each vehicle is fixed, and each customer is served by only one vehicle. (6) Vehicles generate carbon emissions during driving time and do not generate carbon emissions during the rest of the time. By considering these assumptions, the proposed model provides a practical and realistic approach for logistics companies and transportation departments to optimize their logistics network while reducing carbon emissions and transportation costs. The proposed model and algorithm can be used to guide logistics companies in making informed decisions on vehicle routing and scheduling, ultimately improving the efficiency and sustainability of urban logistics distribution networks. 214 Advances in Production Engineering & Management 18(2) 2023 An improved discrete particle swarm optimization approach for a multi-objective optimization model of an urban … 3.3 Notations I I’ J K N fk tijkn O D Q qi cti Tik vijkn tijkn βijkn dijkn δ g θijkn RTik LTik [Ej, Lj] αde αdl λij Set of customers Set of all nodes in the urban logistics distribution network Set of routes Set of vehicles Set of road section Fixed departure cost of vehicles k The time consumed by the vehicle k traveling on road segment n in road (i, j) Unit time cost of vehicle usage Unit human resource cost of vehicles Maximum vehicle load capacity The demand for customer i; Unloading service time at customer i Waiting time for vehicle k to arrive at customer i in advance The traveling speed of the vehicle k in the section n of the road (i, j) The time consumed by the vehicle k traveling on road segment n in road (i, j) Carbon emission rate of vehicle k in section n of road (i, j) (kg/km) The distance travelled by vehicle k in section n of road (i, j) Unit carbon emission cost (yuan/kg) Unit fuel consumption cost (yuan/L); The fuel consumption rate of the vehicle k in the section n of the road (i, j) (L/km) The time when the vehicle k arrives at the customer i The time when the vehicle k leaves the customer i The service time window for customer i The penalty factor when the vehicle arrives early The penalty factor when the vehicle arrives late Traffic congestion coefficient 𝑥𝑥𝑖𝑖𝑖𝑖𝑖𝑖 = � 𝑌𝑌𝑗𝑗𝑗𝑗 = � 1 If vehicle k travel in the path (i,j) 𝑖𝑖 ∈ 𝐼𝐼, 𝑗𝑗 ∈ 𝐼𝐼, 𝑘𝑘 ∈ 𝐾𝐾 0 otherwise 1 0 𝑈𝑈𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 = � 𝑟𝑟𝑘𝑘 = � 1 0 1 0 If vehicle k serves consumer j 𝑗𝑗 ∈ 𝐼𝐼, 𝑘𝑘 ∈ 𝐾𝐾 otherwise If vehicle k travel in the path n i ∈ 𝐼𝐼,j ∈ 𝐽𝐽,k ∈ 𝐾𝐾, 𝑛𝑛 ∈ 𝑁𝑁 otherwise If vehicle k is used 𝑘𝑘 ∈ 𝐾𝐾 otherwise 3.4 Mathematical model With the rapid increase in urban cars, traffic congestion has become a common phenomenon in urban areas. The impact of traffic congestion on logistics delivery efficiency and quality makes it necessary to consider the space-time effect in ULDN. The shortest path between two customers should not be based on the shortest spatial distance but rather on the shortest time, which varies due to different road congestion conditions and vehicle speeds during different periods of time. To quantify the degree of road congestion, the paper introduces a traffic congestion coefficient λij. The speed on the road during congestion vc = vf/λij, where vf is the vehicle speed when the road is clear. When the vehicle travels on a sub-road section with a sufficiently short distance, the speed can be considered constant based on the actual situation and driving rules. In the proposed urban logistics network optimization model, the carbon emission cost, vehicle usage cost, transportation cost, and time penalty cost are considered. They are calculated as follows: Advances in Production Engineering & Management 18(2) 2023 215 Li, Li, Ma (1) Carbon emission cost Carbon emissions are mainly generated during vehicle transportation, and the carbon emission coefficient is used to calculate the carbon emissions during transportation. The carbon emission coefficient represents the unit carbon emissions of the logistics distribution, which quantifies the total carbon dioxide content in the logistics distribution. The carbon emission cost C1 is calculated using Eq. 1. (2) Vehicle usage cost 𝐶𝐶1 = � � � � 𝛽𝛽𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑈𝑈𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑣𝑣𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑡𝑡𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 (1) 𝑖𝑖∈𝐼𝐼 𝑗𝑗∈𝐼𝐼 𝑘𝑘∈𝐾𝐾 𝑛𝑛∈𝑁𝑁 Vehicle usage costs mainly include vehicle departure costs, vehicle rental costs, and labour costs. The vehicle rental cost and labour cost are the product of the total travel time and unit cost. The total travel time is the sum of road travel time, customer service unloading time, and waiting time at the customer point. Therefore, the vehicle management usage cost C2 is calculated using Eq. 2. 𝐶𝐶2 = � 𝑟𝑟𝑘𝑘 𝑓𝑓𝑘𝑘 + � � � � 𝑈𝑈𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑡𝑡𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 (𝑂𝑂 + 𝐷𝐷) + (� � 𝑦𝑦𝑗𝑗𝑗𝑗 𝑐𝑐𝑐𝑐𝑗𝑗 𝑘𝑘∈𝐾𝐾 (3) Transportation cost 𝑖𝑖∈𝐼𝐼 𝑗𝑗∈𝐼𝐼 𝑘𝑘∈𝐾𝐾 𝑛𝑛∈𝑁𝑁 𝑗𝑗∈𝐼𝐼 𝑘𝑘∈𝐾𝐾 + � � 𝑦𝑦𝑗𝑗𝑗𝑗 𝑇𝑇𝑗𝑗𝑗𝑗 )(𝑂𝑂 + 𝐷𝐷) 𝑗𝑗∈𝐼𝐼 𝑘𝑘∈𝐾𝐾 (2) Transportation cost refers to the fuel consumption cost generated during vehicle transportation. The transportation cost C3 is calculated using Eq. 3. (4) Time penalty cost 𝐶𝐶3 = � � � � 𝑔𝑔𝑔𝑔𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑣𝑣𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑡𝑡𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝜃𝜃𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 (3) 𝑖𝑖∈𝐼𝐼 𝑗𝑗∈𝐼𝐼 𝑘𝑘∈𝐾𝐾 𝑛𝑛∈𝑁𝑁 In the process of urban logistics distribution, scheduling errors, and low delivery efficiency may cause distribution vehicles to miss the specified delivery time. Such delays may negatively impact customers, such as supermarkets and shopping malls, which have their own business hours. Delayed delivery times may increase the cost of the enterprise due to decreased customer satisfaction, and penalties should be imposed accordingly. The penalty cost in the vehicle distribution is calculated using Eq. 4. 𝐶𝐶4 = 𝛼𝛼𝑑𝑑𝑑𝑑 � �(𝐸𝐸𝑖𝑖𝑖𝑖 − 𝑅𝑅𝑅𝑅𝑖𝑖𝑖𝑖 , 0) + 𝛼𝛼𝑑𝑑𝑑𝑑 � �(𝑅𝑅𝑅𝑅𝑖𝑖𝑖𝑖 − 𝐿𝐿𝑖𝑖𝑖𝑖 , 0) (4) 𝑚𝑚𝑚𝑚𝑚𝑚 𝑍𝑍1 = � � � � 𝛽𝛽𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑈𝑈𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑣𝑣𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑡𝑡𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 (5) 𝑖𝑖∈𝐼𝐼 𝑘𝑘∈𝐾𝐾 𝑖𝑖∈𝐼𝐼 𝑘𝑘∈𝐾𝐾 By considering these costs, the proposed model and algorithm can provide valuable guidance for logistics companies and transportation departments in urban areas to optimize their logistics network while reducing transportation costs and carbon emissions. The 𝐸𝐸𝑖𝑖𝑖𝑖 − 𝑅𝑅𝑅𝑅𝑖𝑖𝑖𝑖 is the waiting time for vehicle k to arrive at customer i in advance, and 𝑅𝑅𝑅𝑅𝑖𝑖𝑖𝑖 − 𝐿𝐿𝑖𝑖𝑖𝑖 is the waiting time for customer i due to vehicle lateness. The objective function of the proposed VRP model in ULDN is constructed as follows: 𝑖𝑖∈𝐼𝐼 𝑗𝑗∈𝐼𝐼 𝑘𝑘∈𝐾𝐾 𝑛𝑛∈𝑁𝑁 𝑚𝑚𝑚𝑚𝑚𝑚 𝑍𝑍2 = � 𝑟𝑟𝑘𝑘 𝑓𝑓𝑘𝑘 + � � � � 𝑈𝑈𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑡𝑡𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 (𝑂𝑂 + 𝐷𝐷) + (� � 𝑦𝑦𝑗𝑗𝑗𝑗 𝑐𝑐𝑐𝑐𝑗𝑗 𝑘𝑘∈𝐾𝐾 𝑖𝑖∈𝐼𝐼 𝑗𝑗∈𝐼𝐼 𝑘𝑘∈𝐾𝐾 𝑛𝑛∈𝑁𝑁 + � � 𝑦𝑦𝑗𝑗𝑗𝑗 𝑇𝑇𝑗𝑗𝑗𝑗 )(𝑂𝑂 + 𝐷𝐷) 𝑗𝑗∈𝐼𝐼 𝑘𝑘∈𝐾𝐾 216 𝑗𝑗∈𝐼𝐼 𝑘𝑘∈𝐾𝐾 (6) Advances in Production Engineering & Management 18(2) 2023 An improved discrete particle swarm optimization approach for a multi-objective optimization model of an urban … 𝑚𝑚𝑚𝑚𝑚𝑚 𝑍𝑍3 = � � � � 𝑔𝑔𝑔𝑔𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑣𝑣𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑡𝑡𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝜃𝜃𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑖𝑖∈𝐼𝐼 𝑗𝑗∈𝐼𝐼 𝑘𝑘∈𝐾𝐾 𝑛𝑛∈𝑁𝑁 Subject to (7) 𝑚𝑚𝑚𝑚𝑚𝑚 𝑍𝑍4 = 𝛼𝛼𝑑𝑑𝑑𝑑 � �(𝐸𝐸𝑖𝑖𝑖𝑖 − 𝑅𝑅𝑅𝑅𝑖𝑖𝑖𝑖 , 0) + 𝛼𝛼𝑑𝑑𝑑𝑑 � �(𝑅𝑅𝑅𝑅𝑖𝑖𝑖𝑖 − 𝐿𝐿𝑖𝑖𝑖𝑖 , 0) (8) � 𝑥𝑥𝑖𝑖𝑖𝑖𝑖𝑖 = 1 , ∀(𝑖𝑖, 𝑗𝑗) ∈ 𝐽𝐽 (9) 𝑖𝑖∈𝐼𝐼 𝑘𝑘∈𝐾𝐾 𝑖𝑖∈𝐼𝐼 𝑘𝑘∈𝐾𝐾 𝑘𝑘∈𝐾𝐾 � 𝑦𝑦𝑗𝑗𝑗𝑗 = 1 , ∀𝑗𝑗 ∈ 𝐼𝐼 (10) 𝑥𝑥𝑖𝑖𝑖𝑖𝑖𝑖 ≤ ∑𝑛𝑛∈𝑁𝑁 𝑈𝑈𝑖𝑖𝑖𝑖𝑖𝑖 , ∀(𝑖𝑖, 𝑗𝑗) ∈ 𝐽𝐽, 𝑘𝑘 ∈ 𝐾𝐾 (12) 𝑘𝑘∈𝐾𝐾 𝑥𝑥𝑖𝑖𝑖𝑖𝑖𝑖 ≥ 𝑈𝑈𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 , ∀(𝑖𝑖, 𝑗𝑗) ∈ 𝐽𝐽, 𝑘𝑘 ∈ 𝐾𝐾, ℎ ∈ 𝐻𝐻 � � 𝑑𝑑𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑈𝑈𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 = 𝑥𝑥𝑖𝑖𝑖𝑖𝑖𝑖 𝑑𝑑𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 , ∀(𝑖𝑖, 𝑗𝑗) ∈ 𝐽𝐽, 𝑘𝑘 ∈ 𝐾𝐾, 𝑛𝑛 ∈ 𝑁𝑁 𝑘𝑘∈𝐾𝐾 𝑛𝑛∈𝑁𝑁 � 𝑥𝑥0𝑗𝑗𝑗𝑗 ≤ 1 , ∀𝑘𝑘 ∈ 𝐾𝐾 𝑗𝑗∈𝐼𝐼′ 𝑑𝑑𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 ≤ 𝑑𝑑𝑖𝑖𝑖𝑖 𝑈𝑈𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 , ∀(𝑖𝑖, 𝑗𝑗) ∈ 𝐽𝐽, 𝑘𝑘 ∈ 𝐾𝐾, 𝑛𝑛 ∈ 𝑁𝑁 � 𝑞𝑞𝑗𝑗 𝑦𝑦𝑗𝑗𝑗𝑗 ≤ 𝑄𝑄, ∀𝑘𝑘 ∈ 𝐾𝐾 𝑗𝑗∈𝐽𝐽 𝑥𝑥𝑖𝑖𝑖𝑖𝑖𝑖 ∈ {0,1} 𝑦𝑦𝑗𝑗𝑗𝑗 ∈ {0,1} 𝑈𝑈𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 ∈ {0,1} 𝑟𝑟𝑘𝑘 ∈ {0,1} (11) (13) (14) (15) (16) (17) (18) (19) (20) Eq. 9 indicates that only one vehicle is allowed to drive on the selected road. Eq. 10 indicates that each customer can only be served by one vehicle, and all customers must be served. Eqs. 11 and 12 represent the limiting relationship between variables xijk and Uijkn. Eq. 13 indicates that the vehicle should travel the entire road as long as the road is selected. Eq. 14 indicates that each vehicle is only used once. Eq. 15 represents the restriction relationship between dijkn and dij. Eq.16 represents vehicle capacity constraints. Eqs. 17-20 represent variable value constraints. By considering these constraints and the objective function, the proposed model and algorithm can optimize vehicle usage costs, transportation costs, time penalty costs, and carbon emissions costs in ULDN while considering traffic congestion. This approach can help logistics companies and transportation departments in urban areas to optimize their logistics network and reduce transportation costs and carbon emissions, ultimately promoting sustainable development. 4. Improved Discrete Particle Swarm Optimization The Particle Swarm Optimization (PSO) is a global optimization algorithm based on swarm intelligence, which was proposed by American scholars Kennedy and Eberhart [41]. The PSO algorithm simulates the social behaviour of animal groups, such as flocks of birds and fish, by following three typical rules: 1) Fly away from the nearest individual to avoid collisions; 2) Fly towards a predetermined goal; 3) Fly to the center of the group. For example, a flock of birds usually determines its flight direction and speed based on its own flight experience, which leads to consistent flock behavior. However, when one bird in the group changes direction and flies to a new habitat, other Advances in Production Engineering & Management 18(2) 2023 217 Li, Li, Ma birds will also be affected and fly to the new habitat, causing the remaining birds to imitate this behaviour until they all fall into the new habitat. In the PSO algorithm, each possible solution in a population is represented as a particle without volume or mass. All such particles fly at a certain speed in the search space, and their speed is derived from past flight experience. The PSO algorithm enables the entire population to develop towards global optimization through information sharing among particles. It has the ability to search multiple points and can obtain multiple Pareto optimal solutions through one operation. Therefore, the advantages of the PSO algorithm are suitable for solving multi-objective optimization problems in urban logistics networks. To further improve the PSO algorithm, the paper proposes an improved discrete particle swarm optimization algorithm (DPSO) to solve the VRP model in ULDN. The DPSO algorithm uses multiple populations to process multiple targets and develops a variable neighbourhood search strategy to improve the search ability of particles. The DPSO conducts a randomized deep search to improve the "premature convergence" problem of the PSO algorithm, which improves the quality of understanding. (1) Coding of the DPSO To implement the DPSO algorithm for solving the VRP model in ULDN, the solution space of VRP in ULDN is represented by a directed complete graph, denoted as G = (V, E), where each potential solution is a generated subgraph of G. The search space of the entire particle swarm is the arc set E in the complete graph G. The position of each particle is represented by a set consisting of arcs, forming a subset A. The search space of the particle swarm is the edge set of the directed complete graph of urban logistics distribution customer nodes. The position of a particle is a subset of the edge set of a complete graph, and the edges in this subset are connected end-to-end to form a directed Hamilton loop, serving as the distribution path for logistics vehicles. The speed of a particle is a collection of all nodes, and edges in the speed collection may be selected to build a new location for the particle. Each element in the individual is converted to a number in the floating point interval [0,1]. The velocities of all particles are calculated, and then the element is converted to an integer based on the relative position index. (2) Particle position update The DPSO algorithm for solving the VRP model in ULDN uses n candidate solutions to find the optimal solution in the search space. The position vector corresponding to the particle is represented by 𝑋𝑋𝑖𝑖 = [𝑥𝑥𝑖𝑖0 , 𝑥𝑥𝑖𝑖1 , . . . , 𝑥𝑥𝑖𝑖𝑛𝑛 ], where each dimension of X is represented by 𝑥𝑥𝑖𝑖𝑑𝑑 = [(𝑚𝑚, 𝑑𝑑), (𝑑𝑑, 𝑘𝑘)], 𝑚𝑚, 𝑑𝑑 ∈ {0,1, . . . , 𝑑𝑑 − 1, 𝑑𝑑 + 1, 𝑛𝑛}, 𝑚𝑚 ≠ 𝑘𝑘. Each individual in the population represents a feasible solution, and 𝑥𝑥𝑖𝑖𝑑𝑑 is composed of two arcs connected by customer d. n is the total dimension (total number of customers), d represents the current dimension index, m is the predecessor node of customer t (the customer served before customer t), and k is the successor node of customer t (the customer served after customer t). Eq. 21 is used to update the position of particles. 𝑣𝑣𝑖𝑖 𝑑𝑑 (𝑡𝑡 + 1) = 𝑣𝑣𝑖𝑖𝑖𝑖 (𝑡𝑡) + 𝑐𝑐1 × 𝑟𝑟1 × (𝑝𝑝𝑏𝑏𝑖𝑖𝑖𝑖 (𝑡𝑡) − 𝑥𝑥𝑖𝑖𝑖𝑖 (𝑡𝑡)) + 𝑐𝑐2 × 𝑟𝑟2 × (𝑔𝑔𝑏𝑏𝑑𝑑 (𝑡𝑡) − 𝑥𝑥𝑖𝑖𝑖𝑖 (𝑡𝑡)) (21) The velocity of particle i in the d-th dimension at the t-th iteration is represented as vid(t)∈R, and PBi(t) = (pbi1(t), pbi2(t),… , pbid(t),… , pbiD(t)) is the position of the individual historical extreme value of particle i up to the t iteration. GB(t) = (gb1(t), gb2(t),… , gbd(t),… , gbD(t)) is the best position experienced by all particles in the population up to the t iteration. Each particle updates its own velocity and location based on individual historical extremum and global optimal value. The t represents the t-th search, Vi(t) represents the speed of the i-th particle, Xi(t) is the current position of the i-th particle, r1 and r2 are random numbers between (0,1), and the constants c1 and c2 are learning factors, usually taking the same value between 0 and 2. (3) Particle Speed Update In the DPSO algorithm for solving the VRP model in ULDN, when the optimization value of particle i changes very little, the speed of the particle is updated according to Eq. 22, where αi defines the historical optimal value of particle i. For each dimension of each particle in the D-dimensional 218 Advances in Production Engineering & Management 18(2) 2023 An improved discrete particle swarm optimization approach for a multi-objective optimization model of an urban … space, the corresponding dimension of the historical optimal value of one particle is selected from all particles according to a certain probability to learn. Thereby, a learning particle is constructed randomly for each particle i. Each dimension of each particle in the population learns from the optimal solution set with a probability p. The particle speeds are updated by Eqs. 22 and 23, where Gbest is the optimal solution set, and G is the number of candidate solutions in the optimal solution set, which is equal to the number of population sizes. 𝑉𝑉𝑖𝑖𝑑𝑑 = 𝑤𝑤 × 𝑉𝑉𝑖𝑖𝑑𝑑 + 𝑐𝑐 × 𝑟𝑟𝑟𝑟𝑟𝑟𝑑𝑑𝑖𝑖𝑑𝑑 × �𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝑡𝑡𝑓𝑓𝑑𝑑𝑖𝑖(𝑑𝑑) − 𝑋𝑋𝑖𝑖𝑑𝑑 � 𝜌𝜌𝐶𝐶𝑝𝑝 � 𝜕𝜕𝜕𝜕 + 𝑢𝑢 ∙ 𝛻𝛻𝛻𝛻� = 𝛻𝛻(𝑘𝑘𝑘𝑘𝑘𝑘) + 𝜂𝜂𝛾𝛾̇ 2 𝜕𝜕𝜕𝜕 (4) Variable neighbourhood search (22) (23) During the operation of PSO, particle swarm optimization may quickly approach the current optimal position of the population, which can lead to weak global search ability and the "premature convergence" phenomenon. To address this issue, the DPSO algorithm utilizes variable neighbourhood search local search operators, which can expand the local search space and improve the quality of individual search, thereby improving the group optimization ability. In the DPSO algorithm for solving the VRP model in ULDN, the vehicle route that passes through the least number of customers is selected to determine the individual and global extrema. Then, under certain hypothetical conditions, the customer closest to the current route is selected to be inserted, and it is determined whether at least that customer exists in another vehicle route. If it exists, the location where the total logistics cost of the new route is the smallest is selected, and if it does not exist, it will not be inserted. The optimal solution set is updated after completing the above operation. Fig. 1 Algorithm steps of DPSO The specific steps of variable domain search are as follows: For L1(s): Step 1: Select a path R randomly. Step 2: Select two nodes N1 and N2 in path R randomly. Step 3: Exchange N1 and N2. Advances in Production Engineering & Management 18(2) 2023 219 Li, Li, Ma For L2(s)： Step 1: Select two paths R1 and R2 randomly. Step 2: Select a node N1 randomly in path R1, and select a node N2 randomly in path R2. Step 3: Exchange N1 and N2. For L3(s)： Step 1: Select the path R with the lowest bearing capacity. Step 2: Insert the customers in the distribution path R into other distribution paths. (5) Algorithm Steps The DPSO steps is as follows: Step 1: Initialize the particle population and initialize each particle randomly. Step 2: Update the speed and position of each particle. Step 3: Local search on the optimal solution set. Step 4: Update the optimal solution set. If the stop condition is not satisfied, return to Step 2 to continue optimizing the population. Step 5: Terminate the entire algorithm and obtain the optimal solution set. The specific steps are shown in Fig. 1. 5. Simulation Based on the urban logistics distribution data of a logistics enterprise in Shanghai, the DPSO algorithm is applied to optimize the logistics network. The logistics enterprise distributes products to various consumers in Shanghai, and has one logistics distribution center that needs to meet the distribution needs of 50 stores and supermarkets (customers). The number 0 refers to the logistics distribution center, and the numbers 1-50 refer to the consumers. The initial time is set as 6:00 a.m., and this time is set as 0 o'clock in the model. According to urban traffic laws, the time periods between 7:00 to 9:00 and 17:00 to 19:00 are considered as traffic congestion periods, and the rest of the time periods are considered as normal driving periods. Under a time-varying network, the normal driving speed vf is 60 km/h, and the speed vc during congestion is 30 km/h. The vehicle has an unloaded weight of 5000 kg and a capacity of 2t. The values of other parameters are shown in Table 1. The parameters of the DPSO algorithm are set as follows: the population size is 50, the number of iterations is 100, w = 0.7, c1 = c2 = 1.5, r1 = r2 = 0.5. The simulation experiment using the DPSO algorithm to optimize the logistics network of a logistics enterprise in Shanghai took 180.2 s to execute. Seven vehicles were used for distribution, and the total distribution cost was 5240 yuan. The vehicle usage costs (vehicle startup and rental fees) were 2305 yuan, the transportation cost was 2638 yuan, the carbon emission cost was 103 yuan, and the time penalty cost was 194 yuan. The optimization results are shown in Table 2. The optimization results indicate that the DPSO algorithm can obtain optimal routes in a relatively short time. Vehicles 3 and 5 completely avoided the congestion period, while vehicles 1, 4, and 6 had two sections in the morning and evening peak congestion periods. Vehicle 2 had two sections in the morning and evening peak congestion period, and vehicle 7 had three sections in the morning and evening peak congestion period. This indicates that the proposed DPSO algorithm can reasonably avoid traffic congestion periods and improve vehicle delivery efficiency. Parameters fk O D δ g αde αdl vf vc 220 Table 1 Parameters Value 130yuan/vehicle 6yuan/h 10yuan/h 0.5yuan/kg 6yuan/L 10yuan/h 10yuan/h 60km/h 30 km/h Advances in Production Engineering & Management 18(2) 2023 An improved discrete particle swarm optimization approach for a multi-objective optimization model of an urban … Route 1 2 3 4 5 6 7 Table 2 Optimization results of urban logistics distribution network Number of routes Carbon Vehicle TransportDistribution route during congestion emission usage ation cost periods cost costs 0-4-5-10-12-151 11 286 325 20-21-0 0-1-6-7-25-41-82 15 355 410 16-44-0 0-32-45-28-0 0 8 167 203 0-14-2-17-19-491 14 290 348 38-26-0 0-42-31-23-3-90 12 348 322 18-43-0 0-22-11-13-341 19 365 402 47-27-50-36-0 0-29-46-24-3335-30-39-40-483 24 494 628 37-0 Time penalty cost Total cost 45 825 26 0 23 0 21 79 648 378 675 682 807 1225 Based on the comprehensive distribution route optimization results, it can be found that due to the constraints of urban congestion and customer service time windows, there are significant differences in the distribution routes. The maximum number of customers served by vehicle 7 is 10, while the number of customers served by vehicle 3 is the lowest, with only 3. This is due to the different time windows at each customer, indicating that logistics enterprises should consider time dependence when planning their routes. Logistics enterprises should plan their routes scientifically based on the actual conditions such as road network conditions and customer time windows. In addition, the impact of traffic congestion on the optimization results of VRP in ULDN is illustrated in the paper. The traffic congestion coefficient λij is set to 1.5, 2.0, 2.5, and 3.0, with corresponding congestion speeds of 40 km/h, 30 km/h, 24 km/h, and 20 km/h, while the normal driving speed vf remains unchanged at 60 km/h. The optimization results are shown in Table 3. Traffic congestion coefficient 1.5 2.0 2.5 3.0 Table 3 Optimization results for different traffic congestion coefficient Number of delivery vehicles 7 7 8 9 Carbon emission cost 89 106 134 165 Vehicle usage costs 2260 2302 2377 2456 Transportation cost 2554 2644 2880 3012 Time penalty cost 181 199 221 247 Total cost 5084 5251 5612 5880 From the perspective of the optimization process of the DPSO algorithm, the optimal evolution iterations corresponding to different traffic congestion coefficients fluctuate slightly between 30 and 45 generations after 100 iterations. Although the change in vehicle usage costs is not significant, the optimal values of transportation costs, carbon emissions costs, and time penalty costs increase as the traffic congestion coefficient increases, leading to an increase in total costs. As the congestion coefficient increases, fuel consumption and carbon emissions increase, indicating that congestion conditions can affect the greenness of logistics delivery routes. The cost of time penalty increases, which indicates that traffic congestion can affect vehicle speed and affect the service time. Furthermore, the effectiveness of the DPSO algorithm is verified by comparing the optimization results of DPSO with the basic PSO algorithm. The convergence of the two algorithms within 100 iterations is obtained under the same parameter settings. The parameters of the algorithm are: the population size is 50, the number of iterations is 100, w = 0.7, c1 = c2 = 1.5, r1 = r2 = 0.5. The optimization results of DPSO and PSO are shown in Figure 2. The optimization results of the DPSO algorithm show that the minimum value of the total cost maintains an overall downward trend with the increase of genetic iterations. Meanwhile, the convergence of the DPSO algorithm is significantly better than that of the PSO algorithm. The DPSO algorithm basically reaches the optimal solution around the 35th generation, while the PSO Advances in Production Engineering & Management 18(2) 2023 221 Li, Li, Ma algorithm converges to the optimal solution in the 80th generation, and its overall optimization cost is greater than that of the DPSO algorithm. This indicates the effectiveness of the DPSO algorithm in optimizing the logistics network in ULDN and improving the efficiency of vehicle delivery. 10 1.8 4 DPSO PSO 1.6 1.4 Total cost 1.2 1 0.8 0.6 0.4 0 20 40 60 80 100 Iterations 6. Conclusions Fig. 2 Comparison of the optimal total cost In this paper, we proposed a model of vehicle routing problem in urban logistics distribution under traffic congestion, taking into account vehicle management costs, transportation costs, carbon emissions costs, and penalty costs comprehensively. To solve this complex multi-objective optimization problem, we developed an improved DPSO algorithm based on the PSO, which uses multiple populations to process multiple targets, and incorporates a variable neighbourhood search strategy to improve the search ability of particles. The randomized deep search was also conducted to improve the "premature convergence" problem of PSO, which improves the quality of optimization results. The simulation results showed that our proposed model can obtain the lowest cost and optimal delivery route, effectively avoiding traffic congestion, reducing carbon emissions costs and penalty costs, and improving customer satisfaction. Comparative analysis of optimization results between the DPSO and the PSO showed that the proposed DPSO algorithm has better convergence and effectiveness in VRP of urban logistics distribution. Our proposed model and algorithm provide effective solutions and references for solving practical logistics distribution network optimization problems in urban areas with complex traffic conditions. By optimizing delivery routes, logistics enterprises can reduce transportation costs, carbon emissions, and penalties, while improving customer satisfaction and promoting sustainable development in urban logistics. In summary, our study contributes to the field of urban logistics distribution and provides a valuable reference for future research in this area. The types of vehicles and the dynamic changes in customer demand are not considered in the study. Future research could introduce different types of delivery vehicles and dynamic changes in customer demand in the delivery paths optimization. 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Computational Cybernetics and Simulation, Orlando, Florida, USA, 4104-4108, doi: 10.1109/ICSMC.1997.637339. 224 Advances in Production Engineering & Management 18(2) 2023 Advances in Production Engineering & Management ISSN 1854-6250 Volume 18 | Number 2 | June 2023 | pp 225–236 Journal home: apem-journal.org https://doi.org/10.14743/apem2023.2.469 Original scientific paper Blockchain-based tripartite evolutionary game study of manufacturing capacity sharing Wang, T.Y.a, Zhang, H.a,* aJiangsu University of Science and Technology, School of Economics and Management, Zhenjiang, P.R. China ABSTRACT ARTICLE INFO In the context of the new round of manufacturing innovation, the sharing economy drives the transformation of manufacturing industry to accelerate the integration and development. However, there are some problems in the process of manufacturing capacity sharing, such as information privacy and security, and difficulty in tracing the sharing process, etc. The application of blockchain technology can effectively solve these problems. To explore the capacity sharing behaviour of manufacturing enterprises from the perspective of blockchain, the article combines evolutionary game theory and constructs a tripartite game model of manufacturing capacity sharing. The replication dynamics and evolutionary stability of the model are analysed using evolutionary game theory, and numerical simulations are carried out using MATLAB software to analyse the impact of parameter changes on the evolutionary outcome. The research results show that the incentive and penalty coefficients under blockchain technology have a facilitating effect on enterprises to carry out sharing, and the enhancement of reputation gain coefficient and loss can promote positive services on the platform. Keywords: Blockchain; Manufacturing; Capacity sharing; Tripartite evolutionary game; Simulation; MATLAB *Corresponding author: haozh168@aliyun.com (Zhang, H.) Article history: Received 17 November 2022 Revised 15 June 2023 Accepted 29 June 2023 Content from this work may be used under the terms of the Creative Commons Attribution 4.0 International Licence (CC BY 4.0). Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. 1. Introduction With the development of the Internet of Things, big data, artificial intelligence and a new round of manufacturing reform, the sharing economy has driven the transformation of the manufacturing industry to accelerate its development and provided a transformation direction for the manufacturing industry [1-5]. Manufacturing capacity sharing is an important element in deepening the integration and development of manufacturing and the Internet, with broad development prospects, and it is particularly important to realize the sustainable development of manufacturing capacity sharing. The imbalance between supply and demand in manufacturing capacity is a common problem in the market, and capacity sharing can alleviate the mismatch between supply and demand [6]. Capacity sharing can only be achieved by "the platform, the companies that demand manufacturing capacity and the companies that own the manufacturing capacity". The strategic choices of platforms and enterprises play an important role in the capacity utilisation of the manufacturing industry and the development of the economy [7, 8]. However, in the process of manufacturing capacity sharing operation, there will be some problems, such as information privacy and security, the sharing process is difficult to trace and transaction supervision difficulties. Blockchain is the 225 Wang, Zhang underlying technology for many digital cryptocurrencies, and its features such as decentralisation, open ledger, hashing algorithm, and asymmetric encryption [9, 10] can avoid the risk of information leakage during transactions and make them more secure. These features of blockchain coincide with the demand problems that exist in manufacturing capacity sharing, which can improve the efficiency of the platform and effectively solve the problems that exist in the sharing platform [11, 12]. In previous research, some scholars have applied games to production control on the shop floor to effectively deal with production control problems involving multiple production lines or production goals [13]. Xiao M and Tian Z Y proposed a framework for cloud manufacturing capacity sharing based on a cooperative game algorithm and using MATLAB to analyse the evolutionary outcome [14]. Some scholars have applied blockchain technology to agricultural supply chains, supply chain management, and the financial industry [15-18], but in the manufacturing industry it is mostly applied to manufacturing supply chains, industry 4.0 sustainability, and SCQM [19-21], and few scholars have applied blockchain technology to manufacturing capacity sharing. Accordingly, the article will combine the characteristics of blockchain to construct a blockchain-based manufacturing capacity sharing model and use evolutionary games to study the capacity sharing behaviour of the manufacturing industry in the blockchain environment. 2. Blockchain-based tripartite evolutionary game analysis of manufacturing capacity sharing 2.1 Main principles of blockchain technology (1) Distributed ledger technology. A data storage technology is a decentralised distributed databased. The data in distributed ledger technology is shared, replicated, and synchronized among the nodes, and it records the transactions between the nodes without the involvement of third parties. Each piece of data in a distributed ledger is signed with a complete and unique timestamp and digital cryptography and cannot be tampered with. (2) Asymmetric encryption. Asymmetric encryption refers to the encryption of data using different ciphers, i.e. a public key and a private key. Blockchain uses asymmetric encryption algorithms to improve the reliability of data. The public key is a cipher that everyone knows and can be used to encrypt data information, while the private key is a cipher used to decrypt data information and only the recipient of the data information has the private key. (3) Smart contracts. A smart contract is essentially a program whose content is infinitely scalable and is fully distributed. If both or more parties to the contract meet the triggering conditions, the contract will be automatically triggered and irrevocable, and the execution of the contract will be published to the whole network, with all information immutable, i.e. the transaction is traceable, transparent and irreversible. 2.2 Problem analysis As shown in the left diagram of Fig. 1., without blockchain technology, enterprises need to publish their information to the platform, and then the platform will match the enterprises for transactions, and the symmetry and openness of information between the two enterprises in the transaction matching process are relatively low, and there is a risk of information privacy being leaked. In the right figure, when blockchain technology is introduced, the decentralized feature of blockchain can reduce the role of platform domination, and the information symmetry and openness between the two sides of enterprises in the transaction matching process is higher, which avoids privacy leakage and improves sharing efficiency [22, 23]. 226 Advances in Production Engineering & Management 18(2) 2023 Blockchain-based tripartite evolutionary game study of manufacturing capacity sharing Fig. 1 Before and after introducing blockchain technology 2.3 Coupling analysis of blockchain manufacturing capacity sharing Coupling analysis refers to the process of considering the interaction or cross-influence of multiple disciplines in a finite analysis. The article constructs a blockchain-based manufacturing capacity sharing model as a fusion innovation for the development of traditional manufacturing capacity sharing platform model, and conducts a coupling analysis between blockchain and manufacturing capacity sharing from three aspects: resource utilization, data trust, and benefit optimization, as shown in Fig. 2. Manufacturing Capacity Sharing Blockchain Coupling Analysis Resource Utilisation Data Trust Optimal benefit Fig. 2 Analysis of the coupling of blockchain and manufacturing capacity sharing (1) The coupling of blockchain and manufacturing capacity sharing resource utilization. The capacity provider uploads the redundant capacity information to the platform, and the capacity demander can seek the capacity they need from the platform to maximize resource utilization. The platform data under blockchain technology are all public, and all nodes can query information through the public interface. (2) The coupling of blockchain and manufacturing capacity sharing data trust. Blockchain technology uses consensus-based specifications and protocols (e.g., open and transparent algorithms) to allow all nodes of the system to exchange data freely and securely in a trusted environment, shifting from trust in the "enterprise" to trust in the "technology". (3) Optimization of the coupling benefits of blockchain and manufacturing capacity sharing. The purpose of manufacturing capacity sharing is to maximize the utilization of equipment, machines, etc. by integrating and allocating unused capacity. The demanders of capacity can use the capacity, while the providers of capacity can gain revenue and the platform can gain reputation through good service to achieve the best overall benefits for all three parties. One of the features of blockchain is decentralization, where each node is independent and nodes interact with each other without paying additional fees, thus increasing the overall benefits. 2.4 Model assumptions and construction The article uses evolutionary game theory to investigate the capacity sharing behaviour of manufacturing firms from a blockchain perspective, with each assumption as follows. Assumption 1. In the game of manufacturing capacity sharing behaviour, there are three nodes of interest, ownership of the capacity, demand for the capacity and the platform of the participant. Advances in Production Engineering & Management 18(2) 2023 227 Wang, Zhang Assumption 2. All three nodes have two possible strategies, and will continuously adjust their strategies according to the gains and losses they obtain. Node a has a strategy space of a = (a1, a2) = (shared, unshared) and chooses a1 with probability 𝑥𝑥 and a2 with probability (1 − 𝑥𝑥). Node b has a strategy space of b = (b1, b2) = (shared, unshared) and chooses b1 with probability 𝑦𝑦 and b2 with probability (1 − 𝑦𝑦). Platform o has a strategy space of o = (o1, o2) = (active service , negative service), with probability 𝑧𝑧 of choosing o1 and probability (1 − 𝑧𝑧) of choosing o2 where, 0 ≤ 𝑥𝑥, 𝑦𝑦, 𝑧𝑧 ≤ 1. Assumption 3. When only one party shares, the sharing party pays the corresponding cost 𝐶𝐶𝑗𝑗 , but the digitization level of its enterprise is also improved; the digitization level coefficient 𝑑𝑑, which brings the benefit 𝑒𝑒𝑗𝑗 , then the digitization benefit obtained by the enterprise is 𝑑𝑑𝑒𝑒𝑗𝑗 [24]. The manufacturing alliance nodes under blockchain technology have an incentive coefficient 𝑚𝑚 for the sharing party and a penalty coefficient 𝑛𝑛 for the non-sharing party. Assumption 4. When both nodes share, they receive additional synergy gains under blockchain technology with a synergy gain coefficient θ; the number of shared enterprises is 𝑄𝑄𝑗𝑗 , and quantifying the capacity sharing level of enterprises [25], the sharing level is 𝐺𝐺𝑗𝑗 . When both nodes do not share, the base gain of enterprises is 𝑅𝑅𝑗𝑗 . Assumption 5. The horizontal revenue coefficient of the platform under blockchain technology is 𝑅𝑅ℎ h when the platform is actively serving, and 𝑅𝑅𝑙𝑙 when it is negatively serving, and 𝑅𝑅ℎ > 𝑅𝑅𝑙𝑙 . The fixed cost of the platform is 𝐶𝐶ℎ when it is actively serving, and 𝐶𝐶𝑙𝑙 when it is negatively serving, and 𝐶𝐶ℎ > 𝐶𝐶𝑙𝑙 . The platform will bring good reputation revenue for itself when it is actively serving, and the revenue coefficient The probability of matching AB when the firm shares both AB when serving positively is 𝐾𝐾ℎ h, the probability of matching AB when serving negatively is 𝐾𝐾𝑙𝑙 , and 𝐾𝐾ℎ > 𝐾𝐾𝑙𝑙 . Constructing a revenue matrix is in Table 1: Node o active service 𝑧𝑧 Node o negative service 1-𝑧𝑧 Table 1 Game payoff matrix Node a shares 𝑥𝑥 Node a does not share 1 − 𝑥𝑥 Node b does not share Node b does not share Node b shares 𝑦𝑦 Node b shares 𝑦𝑦 1 − 𝑦𝑦𝑦𝑦 1 − 𝑦𝑦 (𝑅𝑅ℎ + 𝐹𝐹ℎ )(𝑄𝑄𝑎𝑎 + 𝑄𝑄𝑏𝑏 ) − 𝐶𝐶ℎ (𝑅𝑅ℎ + 𝐹𝐹ℎ )𝑄𝑄𝑎𝑎 − 𝐶𝐶ℎ (𝑅𝑅ℎ + 𝐹𝐹ℎ )𝑄𝑄𝑏𝑏 − 𝐶𝐶ℎ −𝐶𝐶ℎ 𝑑𝑑𝑒𝑒𝑎𝑎 − 𝐶𝐶𝑎𝑎 + 𝑚𝑚𝐺𝐺𝑎𝑎 + 𝜃𝜃𝐾𝐾ℎ 𝐺𝐺𝑎𝑎 𝑑𝑑𝑒𝑒𝑎𝑎 − 𝐶𝐶𝑎𝑎 + 𝑚𝑚𝐺𝐺𝑎𝑎 𝑅𝑅𝑎𝑎 − 𝑛𝑛𝐺𝐺𝑏𝑏 𝑅𝑅𝑎𝑎 𝑑𝑑𝑒𝑒𝑏𝑏 − 𝐶𝐶𝑏𝑏 + 𝑚𝑚𝐺𝐺𝑏𝑏 + 𝜃𝜃𝐾𝐾ℎ 𝐺𝐺𝑏𝑏 𝑅𝑅𝑏𝑏 − 𝑛𝑛𝐺𝐺𝑎𝑎 𝑑𝑑𝑒𝑒𝑏𝑏 − 𝐶𝐶𝑏𝑏 + 𝑚𝑚𝐺𝐺𝑏𝑏 𝑅𝑅𝑏𝑏 (𝑅𝑅𝑙𝑙 − 𝐹𝐹𝑙𝑙 )(𝑄𝑄𝑎𝑎 + 𝑄𝑄𝑏𝑏 ) − 𝐶𝐶𝑙𝑙 (𝑅𝑅𝑙𝑙 − 𝐹𝐹𝑙𝑙 )𝑄𝑄𝑎𝑎 − 𝐶𝐶𝑙𝑙 (𝑅𝑅𝑙𝑙 − 𝐹𝐹𝑙𝑙 )𝑄𝑄𝑏𝑏 − 𝐶𝐶𝑙𝑙 −𝐶𝐶𝑙𝑙 𝑑𝑑𝑒𝑒𝑎𝑎 − 𝐶𝐶𝑎𝑎 + 𝑚𝑚𝐺𝐺𝑎𝑎 + 𝜃𝜃𝐾𝐾𝑙𝑙 𝐺𝐺𝑎𝑎 𝑑𝑑𝑒𝑒𝑎𝑎 − 𝐶𝐶𝑎𝑎 + 𝑚𝑚𝐺𝐺𝑎𝑎 𝑅𝑅𝑎𝑎 − 𝑛𝑛𝐺𝐺𝑏𝑏 𝑅𝑅𝑎𝑎 𝑑𝑑𝑒𝑒𝑏𝑏 − 𝐶𝐶𝑏𝑏 + 𝑚𝑚𝐺𝐺𝑏𝑏 + 𝜃𝜃𝐾𝐾𝑙𝑙 𝐺𝐺𝑏𝑏 𝑅𝑅𝑏𝑏 − 𝑛𝑛𝐺𝐺𝑎𝑎 𝑑𝑑𝑒𝑒𝑏𝑏 − 𝐶𝐶𝑏𝑏 + 𝑚𝑚𝐺𝐺𝑏𝑏 𝑅𝑅𝑏𝑏 3. Model analysis 3.1 Analysis of replication dynamics and evolutionary stabilization strategies for node a Based on the gain matrix in the table above, the gains when node a is shared and when it is not shared can be derived as: 𝑈𝑈𝑎𝑎1 = 𝑦𝑦𝑦𝑦(𝑑𝑑𝑒𝑒𝑎𝑎 − 𝐶𝐶𝑎𝑎 + 𝑚𝑚𝐺𝐺𝑎𝑎 + 𝜃𝜃𝐾𝐾ℎ 𝐺𝐺𝑎𝑎 ) + 𝑦𝑦(1 − 𝑧𝑧)(𝑑𝑑𝑒𝑒𝑎𝑎 − 𝐶𝐶𝑎𝑎 + 𝑚𝑚𝐺𝐺𝑎𝑎 + 𝜃𝜃𝐾𝐾𝑙𝑙 𝐺𝐺𝑎𝑎 ) (1) + 𝑧𝑧(1 − 𝑦𝑦)(𝑑𝑑𝑒𝑒𝑎𝑎 − 𝐶𝐶𝑎𝑎 + 𝑚𝑚𝐺𝐺𝑎𝑎 ) + (1 − 𝑦𝑦)(1 − 𝑧𝑧)(𝑑𝑑𝑒𝑒𝑎𝑎 − 𝐶𝐶𝑎𝑎 + 𝑚𝑚𝐺𝐺𝑎𝑎 ) 𝑈𝑈𝑎𝑎2 = 𝑦𝑦𝑦𝑦(𝑅𝑅𝑎𝑎 − 𝑛𝑛𝐺𝐺𝑏𝑏 ) + 𝑦𝑦(1 − 𝑧𝑧)(𝑅𝑅𝑎𝑎 − 𝑛𝑛𝐺𝐺𝑏𝑏 ) + 𝑧𝑧(1 − 𝑦𝑦)𝑅𝑅𝑎𝑎 + (1 − 𝑦𝑦)(1 − 𝑧𝑧)𝑅𝑅𝑎𝑎 (2) 𝑑𝑑𝑥𝑥 = 𝑥𝑥�𝑈𝑈𝑎𝑎 − 𝑈𝑈𝑎𝑎1 � = 𝑥𝑥(1 − 𝑥𝑥)�𝑈𝑈𝑎𝑎1 − 𝑈𝑈𝑎𝑎2 � 𝑑𝑑𝑡𝑡 = 𝑥𝑥(1 − 𝑥𝑥)[𝑦𝑦𝑦𝑦𝑦𝑦𝐺𝐺𝑎𝑎 (𝐾𝐾ℎ − 𝐾𝐾𝑙𝑙 ) + 𝑦𝑦𝑦𝑦𝐾𝐾𝑙𝑙 𝐺𝐺𝑎𝑎 + 𝑦𝑦𝑦𝑦𝐺𝐺𝑏𝑏 + 𝑑𝑑𝑒𝑒𝑎𝑎 − 𝐶𝐶𝑎𝑎 + 𝑚𝑚𝐺𝐺𝑎𝑎 − 𝑅𝑅𝑎𝑎 ] (4) Therefore, the average expected return of node a and the replication dynamic equation are: 𝑈𝑈𝑎𝑎 = 𝑥𝑥𝑥𝑥𝑎𝑎1 + (1 − 𝑥𝑥)𝑈𝑈𝑎𝑎2 𝐹𝐹(𝑥𝑥) = (3) The evolutionary strategy of node a must satisfy 𝐹𝐹(𝑥𝑥) = 0 and 𝐹𝐹 ′ (𝑥𝑥) < 0. For 𝐹𝐹(𝑥𝑥) = 0, the so𝐶𝐶 −𝑑𝑑𝑒𝑒 +𝑅𝑅𝑎𝑎 −𝑚𝑚𝐺𝐺𝑎𝑎 −𝑦𝑦𝑦𝑦𝐾𝐾𝑙𝑙 𝐺𝐺𝑎𝑎 −𝑦𝑦𝑦𝑦𝐺𝐺𝑏𝑏 lution is 𝑥𝑥 ∗ = 0 , 𝑥𝑥 ∗ = 1 and 𝑧𝑧 ∗ = 𝑎𝑎 𝑎𝑎 𝑦𝑦𝑦𝑦𝐺𝐺 . 𝐹𝐹 ′ (𝑥𝑥) = (1 − 2𝑥𝑥)[𝑦𝑦𝑦𝑦𝑦𝑦𝐺𝐺𝑎𝑎 (𝐾𝐾ℎ − (𝐾𝐾 −𝐾𝐾 ) 228 𝑎𝑎 ℎ 𝑙𝑙 Advances in Production Engineering & Management 18(2) 2023 Blockchain-based tripartite evolutionary game study of manufacturing capacity sharing 𝐾𝐾𝑙𝑙 ) + 𝑦𝑦𝑦𝑦𝐾𝐾𝑙𝑙 𝐺𝐺𝑎𝑎 + 𝑦𝑦𝑦𝑦𝐺𝐺𝑏𝑏 + 𝑑𝑑𝑒𝑒𝑎𝑎 − 𝐶𝐶𝑎𝑎 + 𝑚𝑚𝐺𝐺𝑎𝑎 − 𝑅𝑅𝑎𝑎 ], make 𝑆𝑆(𝑦𝑦) = 𝑦𝑦𝑦𝑦𝑦𝑦𝐺𝐺𝑎𝑎 (𝐾𝐾ℎ − 𝐾𝐾𝑙𝑙 ) + 𝑦𝑦𝑦𝑦𝐾𝐾𝑙𝑙 𝐺𝐺𝑎𝑎 + 𝑦𝑦𝑦𝑦𝐺𝐺𝑏𝑏 + 𝑑𝑑𝑒𝑒𝑎𝑎 − 𝐶𝐶𝑎𝑎 + 𝑚𝑚𝐺𝐺𝑎𝑎 − 𝑅𝑅𝑎𝑎 . Since 𝑆𝑆 ′ (𝑦𝑦) > 0, 𝑆𝑆(𝑦𝑦) is an increasing function with respect to 𝑦𝑦. Therefore, when 𝑧𝑧 = 𝑧𝑧 ∗ , 𝑆𝑆(𝑦𝑦) = 0, at which point 𝐹𝐹 ′ (𝑥𝑥) = 0, no stable strategy can be determined; when 𝑧𝑧 < 𝑧𝑧 ∗ , 𝑆𝑆(𝑦𝑦) < 0, at which point 𝑥𝑥 = 0 satisfies the stability condition; when 𝑧𝑧 > 𝑧𝑧 ∗ , 𝑆𝑆(𝑦𝑦) > 0, at which point 𝑥𝑥 = 1 is ESS, satisfying the stability condition, and the evolutionary phase diagram is shown in Fig. 3. Fig. 3 Digital phase diagrams of node a 3.2 Analysis of replication dynamics and evolutionary stabilization strategies for node b The gains when node b is shared versus not shared are: 𝑈𝑈𝑏𝑏1 = 𝑥𝑥𝑥𝑥(𝑑𝑑𝑒𝑒𝑏𝑏 − 𝐶𝐶𝑏𝑏 + 𝑚𝑚𝐺𝐺𝑏𝑏 + 𝜃𝜃𝐾𝐾ℎ 𝐺𝐺𝑏𝑏 ) + 𝑥𝑥(1 − 𝑧𝑧)(𝑑𝑑𝑒𝑒𝑏𝑏 − 𝐶𝐶𝑏𝑏 + 𝑚𝑚𝐺𝐺𝑏𝑏 + 𝜃𝜃𝐾𝐾𝑙𝑙 𝐺𝐺𝑏𝑏 ) + 𝑧𝑧(1 − 𝑥𝑥)(𝑑𝑑𝑒𝑒𝑏𝑏 − 𝐶𝐶𝑏𝑏 + 𝑚𝑚𝐺𝐺𝑏𝑏 ) + (1 − 𝑥𝑥)(1 − 𝑧𝑧)(𝑑𝑑𝑒𝑒𝑏𝑏 − 𝐶𝐶𝑏𝑏 + 𝑚𝑚𝐺𝐺𝑏𝑏 ) 𝑈𝑈𝑏𝑏2 = 𝑥𝑥𝑥𝑥(𝑅𝑅𝑏𝑏 − 𝑛𝑛𝐺𝐺𝑎𝑎 ) + 𝑥𝑥(1 − 𝑧𝑧)(𝑅𝑅𝑏𝑏 − 𝑛𝑛𝐺𝐺𝑎𝑎 ) + 𝑧𝑧(1 − 𝑥𝑥)𝑅𝑅𝑏𝑏 + (1 − 𝑥𝑥)(1 − 𝑧𝑧)𝑅𝑅𝑏𝑏 Therefore, the average expected return of node b and the replication dynamic equation are: 𝑈𝑈𝑏𝑏 = 𝑦𝑦𝑈𝑈𝑏𝑏1 + (1 − 𝑦𝑦)𝑈𝑈𝑏𝑏2 𝑑𝑑𝑦𝑦 𝐹𝐹(𝑦𝑦) = = 𝑦𝑦�𝑈𝑈𝑏𝑏 − 𝑈𝑈𝑏𝑏1 � = 𝑦𝑦(1 − 𝑦𝑦)�𝑈𝑈𝑏𝑏1 − 𝑈𝑈𝑏𝑏2 � 𝑑𝑑𝑡𝑡 = 𝑦𝑦(1 − 𝑦𝑦)[𝑥𝑥𝑥𝑥𝑥𝑥𝐺𝐺𝑏𝑏 (𝐾𝐾ℎ − 𝐾𝐾𝑙𝑙 ) + 𝑥𝑥𝑥𝑥𝐾𝐾𝑙𝑙 𝐺𝐺𝑏𝑏 + 𝑥𝑥𝑥𝑥𝐺𝐺𝑎𝑎 + 𝑑𝑑𝑒𝑒𝑏𝑏 − 𝐶𝐶𝑏𝑏 + 𝑚𝑚𝐺𝐺𝑏𝑏 − 𝑅𝑅𝑏𝑏 ] (5) (6) (7) (8) The evolutionary strategy of node b must satisfy 𝐹𝐹(𝑦𝑦) = 0 and 𝐹𝐹 ′ (𝑦𝑦) < 0. For 𝐹𝐹(𝑦𝑦) = 0, the so𝐶𝐶 −𝑑𝑑𝑒𝑒 +𝑅𝑅 −𝑚𝑚𝐺𝐺 −𝑥𝑥𝑥𝑥𝐾𝐾𝑙𝑙 𝐺𝐺𝑏𝑏 −𝑥𝑥𝑥𝑥𝐺𝐺𝑎𝑎 lution is𝑦𝑦 ∗ = 0, 𝑦𝑦 ∗ = 1 and 𝑧𝑧 ∗ = 𝑏𝑏 𝑏𝑏 𝑏𝑏 (𝐾𝐾𝑏𝑏 . 𝐹𝐹 ′ (𝑦𝑦) = (1 − 2𝑦𝑦)[𝑧𝑧𝑧𝑧𝐺𝐺𝑏𝑏 (𝐾𝐾ℎ − 𝐾𝐾𝑙𝑙 ) + ) 𝑥𝑥𝑥𝑥𝐺𝐺𝑏𝑏 ℎ −𝐾𝐾𝑙𝑙 𝑥𝑥𝑥𝑥𝐾𝐾𝑙𝑙 𝐺𝐺𝑏𝑏 + 𝑥𝑥𝑥𝑥𝐺𝐺𝑎𝑎 + 𝑑𝑑𝑒𝑒𝑏𝑏 − 𝐶𝐶𝑏𝑏 + 𝑚𝑚𝐺𝐺𝑏𝑏 − 𝑅𝑅𝑏𝑏 ] , make 𝑆𝑆(𝑥𝑥) = 𝑧𝑧𝑧𝑧𝐺𝐺𝑏𝑏 (𝐾𝐾ℎ − 𝐾𝐾𝑙𝑙 ) + 𝑥𝑥𝑥𝑥𝐾𝐾𝑙𝑙 𝐺𝐺𝑏𝑏 + 𝑥𝑥𝑥𝑥𝐺𝐺𝑎𝑎 + 𝑑𝑑𝑒𝑒𝑏𝑏 − 𝐶𝐶𝑏𝑏 + 𝑚𝑚𝐺𝐺𝑏𝑏 − 𝑅𝑅𝑏𝑏 . Since 𝑆𝑆 ′ (𝑥𝑥) > 0, 𝑆𝑆(𝑥𝑥) is an increasing function with respect to 𝑥𝑥. Therefore, when 𝑧𝑧 = 𝑧𝑧 ∗ , 𝑆𝑆(𝑥𝑥) = 0, at which point 𝐹𝐹 ′ (𝑦𝑦) = 0, no stable strategy can be determined; when 𝑧𝑧 < 𝑧𝑧 ∗ , 𝑆𝑆(𝑥𝑥) < 0, at which point 𝑦𝑦 = 0 satisfies the stability condition; when 𝑧𝑧 > 𝑧𝑧 ∗, 𝑆𝑆(𝑥𝑥) > 0, at which point 𝑦𝑦 = 1 is ESS, satisfying the stability condition, and the evolutionary phase diagram is shown in Fig. 4. Fig. 4 Digital phase diagrams of node b Advances in Production Engineering & Management 18(2) 2023 229 Wang, Zhang 3.3 Analysis of replication dynamics and evolutionary stabilization strategies for node c The gains when node o is shared versus not shared are: 𝑈𝑈𝑜𝑜1 = 𝑥𝑥𝑥𝑥[(𝑅𝑅ℎ + 𝐹𝐹ℎ )(𝑄𝑄𝑎𝑎 + 𝑄𝑄𝑏𝑏 ) − 𝐶𝐶ℎ ] + 𝑥𝑥(1 − 𝑦𝑦)[(𝑅𝑅ℎ + 𝐹𝐹ℎ )𝑄𝑄𝑎𝑎 − 𝐶𝐶ℎ ] + 𝑦𝑦(1 − 𝑥𝑥)[(𝑅𝑅ℎ + 𝐹𝐹ℎ )𝑄𝑄𝑏𝑏 − 𝐶𝐶ℎ ] + (1 − 𝑥𝑥)(1 − 𝑦𝑦)(−𝐶𝐶ℎ ) (9) 𝑈𝑈𝑜𝑜2 = 𝑥𝑥𝑥𝑥[(𝑅𝑅𝑙𝑙 − 𝐹𝐹𝑙𝑙 )(𝑄𝑄𝑎𝑎 + 𝑄𝑄𝑏𝑏 ) − 𝐶𝐶𝑙𝑙 ] + 𝑥𝑥(1 − 𝑦𝑦)[(𝑅𝑅𝑙𝑙 − 𝐹𝐹𝑙𝑙 )𝑄𝑄𝑎𝑎 − 𝐶𝐶𝑙𝑙 ] + 𝑦𝑦(1 − 𝑥𝑥)[(𝑅𝑅𝑙𝑙 − 𝐹𝐹𝑙𝑙 )𝑄𝑄𝑏𝑏 (10) − 𝐶𝐶𝑙𝑙 ] + (1 − 𝑥𝑥)(1 − 𝑦𝑦)(−𝐶𝐶𝑙𝑙 ) Therefore, the average expected return of node o and the replication dynamic equation are: 𝑈𝑈𝑜𝑜 = 𝑧𝑧𝑈𝑈𝑜𝑜1 + (1 − 𝑧𝑧)𝑈𝑈𝑜𝑜2 (11) 𝑑𝑑𝑧𝑧 𝐹𝐹(𝑧𝑧) = = 𝑧𝑧�𝑈𝑈𝑜𝑜 − 𝑈𝑈𝑜𝑜1 � = 𝑧𝑧(1 − 𝑧𝑧)�𝑈𝑈𝑜𝑜1 − 𝑈𝑈𝑜𝑜2 � 𝑑𝑑𝑡𝑡 (12) = 𝑧𝑧(1 − 𝑧𝑧)[𝑥𝑥𝑄𝑄𝑎𝑎 (𝑅𝑅ℎ + 𝐹𝐹ℎ − 𝑅𝑅𝑙𝑙 + 𝐹𝐹𝑙𝑙 ) + 𝑦𝑦𝑄𝑄𝑏𝑏 (𝑅𝑅ℎ + 𝐹𝐹ℎ − 𝑅𝑅𝑙𝑙 + 𝐹𝐹𝑙𝑙 ) − 𝐶𝐶ℎ + 𝐶𝐶𝑙𝑙 ] The evolutionary strategy of node o must satisfy 𝐹𝐹(𝑧𝑧) = 0 and 𝐹𝐹 ′ (𝑧𝑧) < 0. For 𝐹𝐹(𝑧𝑧) = 0, the so𝐶𝐶 −𝐶𝐶 −𝑦𝑦𝑄𝑄 (𝑅𝑅 +𝐹𝐹 −𝑅𝑅 +𝐹𝐹 ) lution is 𝑧𝑧 ∗ = 0 , 𝑧𝑧 ∗ = 1 and 𝑥𝑥 ∗ = ℎ 𝑄𝑄𝑙𝑙 (𝑅𝑅 𝑏𝑏+𝐹𝐹 ℎ−𝑅𝑅 ℎ+𝐹𝐹 )𝑙𝑙 𝑙𝑙 . 𝐹𝐹 ′ (𝑧𝑧) = (1 − 2𝑧𝑧)[𝑥𝑥𝑄𝑄𝑎𝑎 (𝑅𝑅ℎ + 𝐹𝐹ℎ − 𝑅𝑅𝑙𝑙 + 𝑎𝑎 ℎ ℎ 𝑙𝑙 𝑙𝑙 𝐹𝐹𝑙𝑙 ) + 𝑦𝑦𝑄𝑄𝑏𝑏 (𝑅𝑅ℎ + 𝐹𝐹ℎ − 𝑅𝑅𝑙𝑙 + 𝐹𝐹𝑙𝑙 ) − 𝐶𝐶ℎ + 𝐶𝐶𝑙𝑙 ], make 𝑆𝑆(𝑦𝑦) = 𝑥𝑥𝑄𝑄𝑎𝑎 (𝑅𝑅ℎ + 𝐹𝐹ℎ − 𝑅𝑅𝑙𝑙 + 𝐹𝐹𝑙𝑙 ) + 𝑦𝑦𝑄𝑄𝑏𝑏 (𝑅𝑅ℎ + 𝐹𝐹ℎ − 𝑅𝑅𝑙𝑙 + 𝐹𝐹𝑙𝑙 ) − 𝐶𝐶ℎ + 𝐶𝐶𝑙𝑙 . Since 𝑆𝑆 ′ (𝑦𝑦) > 0, 𝑆𝑆(𝑦𝑦) is an increasing function with respect to 𝑦𝑦. Therefore, when 𝑥𝑥 = 𝑥𝑥 ∗ , 𝑆𝑆(𝑦𝑦) = 0, at which point 𝐹𝐹 ′ (𝑧𝑧) = 0, no stable strategy can be determined; when 𝑥𝑥 < 𝑥𝑥 ∗ , 𝑆𝑆(𝑦𝑦) < 0, at which point 𝑧𝑧 = 0 satisfies the stability condition; when 𝑥𝑥 > 𝑥𝑥 ∗ , 𝑆𝑆(𝑦𝑦) > 0, at which point 𝑧𝑧 = 1 is ESS, satisfying the stability condition, and the evolutionary phase diagram is shown in Fig. 5. Fig. 5 Digital phase diagrams of node o 4. Stability analysis of the equilibrium point of a tripartite evolutionary game system Letting the replicated dynamic equations equal zero, it is known that there are the following system equilibria E1(0,0,0), E2(0,0,1), E3(0,1,0), E4(0,1,1), E5(1,0,0), E6(1,0,1), E7(1,1,0) and E8(1,1,1), where the Jacobian matrix of the tripartite evolutionary game system is 𝜕𝜕𝜕𝜕(𝑥𝑥) 𝜕𝜕𝜕𝜕(𝑥𝑥) 𝜕𝜕𝜕𝜕(𝑥𝑥) ⎡ ⎤ 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 ⎥ ⎢ 𝐽𝐽1 𝐽𝐽2 𝐽𝐽3 ⎢𝜕𝜕𝜕𝜕(𝑦𝑦) 𝜕𝜕𝜕𝜕(𝑦𝑦) 𝜕𝜕𝜕𝜕(𝑦𝑦)⎥ 𝐽𝐽 = �𝐽𝐽4 𝐽𝐽5 𝐽𝐽6 � = ⎢ 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 ⎥ 𝐽𝐽7 𝐽𝐽8 𝐽𝐽9 ⎢ ⎥ ⎢ 𝜕𝜕𝜕𝜕(𝑧𝑧) 𝜕𝜕𝜕𝜕(𝑧𝑧) 𝜕𝜕𝜕𝜕(𝑧𝑧) ⎥ ⎣ 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 ⎦ 𝐽𝐽1 = (1 − 2𝑥𝑥)[𝑦𝑦𝑦𝑦𝑦𝑦𝐺𝐺𝑎𝑎 (𝐾𝐾ℎ − 𝐾𝐾𝑙𝑙 ) + 𝑦𝑦𝑦𝑦𝐾𝐾𝑙𝑙 𝐺𝐺𝑎𝑎 + 𝑦𝑦𝑦𝑦𝐺𝐺𝑏𝑏 + 𝑑𝑑𝑒𝑒𝑎𝑎 − 𝐶𝐶𝑎𝑎 + 𝑚𝑚𝐺𝐺𝑎𝑎 − 𝑅𝑅𝑎𝑎 ] 𝐽𝐽2 = 𝑥𝑥(1 − 𝑥𝑥)[𝑧𝑧𝑧𝑧𝐺𝐺𝑎𝑎 (𝐾𝐾ℎ − 𝐾𝐾𝑙𝑙 ) + 𝜃𝜃𝐾𝐾𝑙𝑙 𝐺𝐺𝑎𝑎 + 𝑛𝑛𝐺𝐺𝑏𝑏 ] 𝐽𝐽3 = 𝑥𝑥(1 − 𝑥𝑥)[𝑦𝑦𝑦𝑦𝐺𝐺𝑎𝑎 (𝐾𝐾ℎ − 𝐾𝐾𝑙𝑙 )] 230 Advances in Production Engineering & Management 18(2) 2023 Blockchain-based tripartite evolutionary game study of manufacturing capacity sharing 𝐽𝐽4 = 𝑦𝑦(1 − 𝑦𝑦)[𝑧𝑧𝑧𝑧𝐺𝐺𝑏𝑏 (𝐾𝐾ℎ − 𝐾𝐾𝑙𝑙 ) + 𝜃𝜃𝐾𝐾𝑙𝑙 𝐺𝐺𝑏𝑏 + 𝑛𝑛𝐺𝐺𝑎𝑎 ] 𝐽𝐽5 = (1 − 2𝑦𝑦)[𝑥𝑥𝑥𝑥𝑥𝑥𝐺𝐺𝑏𝑏 (𝐾𝐾ℎ − 𝐾𝐾𝑙𝑙 ) + 𝑥𝑥𝑥𝑥𝐾𝐾𝑙𝑙 𝐺𝐺𝑏𝑏 + 𝑥𝑥𝑥𝑥𝐺𝐺𝑎𝑎 + 𝑑𝑑𝑒𝑒𝑏𝑏 − 𝐶𝐶𝑏𝑏 + 𝑚𝑚𝐺𝐺𝑏𝑏 − 𝑅𝑅𝑏𝑏 ] 𝐽𝐽6 = 𝑦𝑦(1 − 𝑦𝑦)[𝑥𝑥𝑥𝑥𝐺𝐺𝑏𝑏 (𝐾𝐾ℎ − 𝐾𝐾𝑙𝑙 )] 𝐽𝐽7 = 𝑧𝑧(1 − 𝑧𝑧)[𝑄𝑄𝑎𝑎 (𝑅𝑅ℎ + 𝐹𝐹ℎ − 𝑅𝑅𝑙𝑙 + 𝐹𝐹𝑙𝑙 )] 𝐽𝐽8 = 𝑧𝑧(1 − 𝑧𝑧)[𝑅𝑅𝑏𝑏 (𝑅𝑅ℎ + 𝐹𝐹ℎ − 𝑅𝑅𝑙𝑙 + 𝐹𝐹𝑙𝑙 )] 𝐽𝐽9 = (1 − 2𝑧𝑧)[𝑥𝑥𝑄𝑄𝑎𝑎 (𝑅𝑅ℎ + 𝐹𝐹ℎ − 𝑅𝑅𝑙𝑙 + 𝐹𝐹𝑙𝑙 ) + 𝑦𝑦𝑄𝑄𝑏𝑏 (𝑅𝑅ℎ + 𝐹𝐹ℎ − 𝑅𝑅𝑙𝑙 + 𝐹𝐹𝑙𝑙 ) − 𝐶𝐶ℎ + 𝐶𝐶𝑙𝑙 ] Using Lyapunov first method, the stability of each equilibrium point is analysed as shown [26]. Table 2 Equilibrium point analysis Equilibrium λ1, λ2, λ3 point E1(0,0,0) 𝑑𝑑𝑒𝑒𝑎𝑎 − 𝐶𝐶𝑎𝑎 + 𝑚𝑚𝐺𝐺𝑎𝑎 − 𝑅𝑅𝑎𝑎 ，𝑑𝑑𝑒𝑒𝑏𝑏 − 𝐶𝐶𝑏𝑏 + 𝑚𝑚𝐺𝐺𝑏𝑏 − 𝑅𝑅𝑏𝑏 ，−𝐶𝐶ℎ + 𝐶𝐶𝑙𝑙 E2(0,0,1) 𝑑𝑑𝑒𝑒𝑎𝑎 − 𝐶𝐶𝑎𝑎 + 𝑚𝑚𝐺𝐺𝑎𝑎 − 𝑅𝑅𝑎𝑎 ，𝑑𝑑𝑒𝑒𝑏𝑏 − 𝐶𝐶𝑏𝑏 + 𝑚𝑚𝐺𝐺𝑏𝑏 − 𝑅𝑅𝑏𝑏 ， (−1)(−𝐶𝐶ℎ + 𝐶𝐶𝑙𝑙 ) 𝜃𝜃𝐾𝐾𝑙𝑙 𝐺𝐺𝑎𝑎 + 𝑛𝑛𝐺𝐺𝑏𝑏 + 𝑑𝑑𝑒𝑒𝑎𝑎 − 𝐶𝐶𝑎𝑎 + 𝑚𝑚𝐺𝐺𝑎𝑎 − 𝑅𝑅𝑎𝑎 , (−1)(𝑑𝑑𝑒𝑒𝑏𝑏 − 𝐶𝐶𝑏𝑏 + 𝑚𝑚𝐺𝐺𝑏𝑏 − 𝑅𝑅𝑏𝑏 ), E3(0,1,0) 𝑄𝑄𝑏𝑏 (𝑅𝑅ℎ + 𝐹𝐹ℎ − 𝑅𝑅𝑙𝑙 + 𝐹𝐹𝑙𝑙 ) − 𝐶𝐶ℎ + 𝐶𝐶𝑙𝑙 𝜃𝜃𝐺𝐺𝑎𝑎 (𝐾𝐾ℎ − 𝐾𝐾𝑙𝑙 ) + 𝜃𝜃𝐾𝐾𝑙𝑙 𝐺𝐺𝑎𝑎 + 𝑛𝑛𝐺𝐺𝑏𝑏 + 𝑑𝑑𝑒𝑒𝑎𝑎 − 𝐶𝐶𝑎𝑎 + 𝑚𝑚𝐺𝐺𝑎𝑎 − 𝑅𝑅𝑎𝑎 , (−1)(𝑑𝑑𝑒𝑒𝑏𝑏 − 𝐶𝐶𝑏𝑏 + E4(0,1,1) 𝑚𝑚𝐺𝐺𝑏𝑏 − 𝑅𝑅𝑏𝑏 ), (−1)[𝑄𝑄𝑏𝑏 (𝑅𝑅ℎ + 𝐹𝐹ℎ − 𝑅𝑅𝑙𝑙 + 𝐹𝐹𝑙𝑙 ) − 𝐶𝐶ℎ + 𝐶𝐶𝑙𝑙 ] (−1)(𝑑𝑑𝑒𝑒𝑎𝑎 − 𝐶𝐶𝑎𝑎 + 𝑚𝑚𝐺𝐺𝑎𝑎 − 𝑅𝑅𝑎𝑎 ), 𝜃𝜃𝐾𝐾𝑙𝑙 𝐺𝐺𝑏𝑏 + 𝑛𝑛𝐺𝐺𝑎𝑎 + 𝑑𝑑𝑒𝑒𝑏𝑏 − 𝐶𝐶𝑏𝑏 + 𝑚𝑚𝐺𝐺𝑏𝑏 − E5(1,0,0) 𝑅𝑅𝑏𝑏 ,𝑄𝑄𝑎𝑎 (𝑅𝑅ℎ + 𝐹𝐹ℎ − 𝑅𝑅𝑙𝑙 + 𝐹𝐹𝑙𝑙 ) − 𝐶𝐶ℎ + 𝐶𝐶𝑙𝑙 (−1)(𝑑𝑑𝑒𝑒𝑎𝑎 − 𝐶𝐶𝑎𝑎 + 𝑚𝑚𝐺𝐺𝑎𝑎 − 𝑅𝑅𝑎𝑎 ), 𝜃𝜃𝐺𝐺𝑏𝑏 (𝐾𝐾ℎ − 𝐾𝐾𝑙𝑙 ) + 𝜃𝜃𝐾𝐾𝑙𝑙 𝐺𝐺𝑏𝑏 + 𝑛𝑛𝐺𝐺𝑎𝑎 + 𝑑𝑑𝑒𝑒𝑏𝑏 − 𝐶𝐶𝑏𝑏 + E6(1,0,1) 𝑚𝑚𝐺𝐺𝑏𝑏 − 𝑅𝑅𝑏𝑏 , (−1)[𝑄𝑄𝑎𝑎 (𝑅𝑅ℎ + 𝐹𝐹ℎ − 𝑅𝑅𝑙𝑙 + 𝐹𝐹𝑙𝑙 ) − 𝐶𝐶ℎ + 𝐶𝐶𝑙𝑙 ] (−1)(𝜃𝜃𝐾𝐾𝑙𝑙 𝐺𝐺𝑎𝑎 + 𝑛𝑛𝐺𝐺𝑏𝑏 + 𝑑𝑑𝑒𝑒𝑎𝑎 − 𝐶𝐶𝑎𝑎 + 𝑚𝑚𝐺𝐺𝑎𝑎 − 𝑅𝑅𝑎𝑎 ), (−1)(𝜃𝜃𝐾𝐾𝑙𝑙 𝐺𝐺𝑏𝑏 + 𝑛𝑛𝐺𝐺𝑎𝑎 + 𝑑𝑑𝑒𝑒𝑏𝑏 − E7(1,1,0) 𝐶𝐶𝑏𝑏 + 𝑚𝑚𝐺𝐺𝑏𝑏 − 𝑅𝑅𝑏𝑏 ), 𝑄𝑄𝑎𝑎 (𝑅𝑅ℎ + 𝐹𝐹ℎ − 𝑅𝑅𝑙𝑙 + 𝐹𝐹𝑙𝑙 ) + 𝑄𝑄𝑏𝑏 (𝑅𝑅ℎ + 𝐹𝐹ℎ − 𝑅𝑅𝑙𝑙 + 𝐹𝐹𝑙𝑙 ) − 𝐶𝐶ℎ + 𝐶𝐶𝑙𝑙 (−1)( 𝜃𝜃𝐺𝐺𝑎𝑎 𝐾𝐾ℎ + 𝑛𝑛𝐺𝐺𝑏𝑏 + 𝑑𝑑𝑒𝑒𝑎𝑎 − 𝐶𝐶𝑎𝑎 + 𝑚𝑚𝐺𝐺𝑎𝑎 − 𝑅𝑅𝑎𝑎 ), (−1)(𝜃𝜃𝐺𝐺𝑏𝑏 𝐾𝐾ℎ + 𝑛𝑛𝐺𝐺𝑎𝑎 + 𝑑𝑑𝑒𝑒𝑏𝑏 − E8(1,1,1) 𝐶𝐶𝑏𝑏 + 𝑚𝑚𝐺𝐺𝑏𝑏 − 𝑅𝑅𝑏𝑏 ), (−1)[𝑄𝑄𝑎𝑎 (𝑅𝑅ℎ + 𝐹𝐹ℎ − 𝑅𝑅𝑙𝑙 + 𝐹𝐹𝑙𝑙 ) + 𝑄𝑄𝑏𝑏 (𝑅𝑅ℎ + 𝐹𝐹ℎ − 𝑅𝑅𝑙𝑙 + 𝐹𝐹𝑙𝑙 ) − 𝐶𝐶ℎ + 𝐶𝐶𝑙𝑙 ] (-,-,-) (-,-,+) Equilibrium results ESS Unstable (?,+,?) Unstable Symbol (?,+,?) (+,?,?) (+,?,?) (?,?,?) (?,?,?) Unstable Unstable Unstable Unstable Unstable (1) E1(0,0,0): λ1= 𝑑𝑑𝑒𝑒𝑎𝑎 − 𝐶𝐶𝑎𝑎 + 𝑚𝑚𝐺𝐺𝑎𝑎 − 𝑅𝑅𝑎𝑎 , λ2= 𝑑𝑑𝑒𝑒𝑏𝑏 − 𝐶𝐶𝑏𝑏 + 𝑚𝑚𝐺𝐺𝑏𝑏 − 𝑅𝑅𝑏𝑏 , λ3=−𝐶𝐶ℎ + 𝐶𝐶𝑙𝑙 . Scenario 1: 𝑑𝑑𝑒𝑒𝑎𝑎 − 𝐶𝐶𝑎𝑎 + 𝑚𝑚𝐺𝐺𝑎𝑎 < 𝑅𝑅𝑎𝑎 , 𝑑𝑑𝑒𝑒𝑏𝑏 − 𝐶𝐶𝑏𝑏 + 𝑚𝑚𝐺𝐺𝑏𝑏 < 𝑅𝑅𝑏𝑏 , 𝐶𝐶𝑙𝑙 < 𝐶𝐶ℎ . Then for nodes a and b, the benefit when only one node shares is less than the benefit when neither node shares; when neither node shares, the cost of negative platform service is less than the cost of positive platform service. At this point λ1, λ2 and λ3 are all less than zero, and the point is the ESS stability point. (2) E7(1,1,0): λ1= (−1)(𝜃𝜃𝐾𝐾𝑙𝑙 𝐺𝐺𝑎𝑎 + 𝑛𝑛𝐺𝐺𝑏𝑏 + 𝑑𝑑𝑒𝑒𝑎𝑎 − 𝐶𝐶𝑎𝑎 + 𝑚𝑚𝐺𝐺𝑎𝑎 − 𝑅𝑅𝑎𝑎 ) , λ2= (−1)(𝜃𝜃𝐾𝐾𝑙𝑙 𝐺𝐺𝑏𝑏 + 𝑛𝑛𝐺𝐺𝑎𝑎 + 𝑑𝑑𝑒𝑒𝑏𝑏 − 𝐶𝐶𝑏𝑏 + 𝑚𝑚𝐺𝐺𝑏𝑏 − 𝑅𝑅𝑏𝑏 ), λ3=𝑄𝑄𝑎𝑎 (𝑅𝑅ℎ + 𝐹𝐹ℎ − 𝑅𝑅𝑙𝑙 + 𝐹𝐹𝑙𝑙 ) + 𝑄𝑄𝑏𝑏 (𝑅𝑅ℎ + 𝐹𝐹ℎ − 𝑅𝑅𝑙𝑙 + 𝐹𝐹𝑙𝑙 ) − 𝐶𝐶ℎ + 𝐶𝐶𝑙𝑙 , at this point the values of λ1, λ2 and λ3 are all uncertain and are discussed by case. Scenario 2: If 𝜃𝜃𝐾𝐾𝑙𝑙 𝐺𝐺𝑎𝑎 + 𝑛𝑛𝐺𝐺𝑏𝑏 + 𝑑𝑑𝑒𝑒𝑎𝑎 − 𝐶𝐶𝑎𝑎 > 𝑅𝑅𝑎𝑎 − 𝑚𝑚𝐺𝐺𝑎𝑎 , 𝜃𝜃𝐾𝐾𝑙𝑙 𝐺𝐺𝑏𝑏 + 𝑛𝑛𝐺𝐺𝑎𝑎 + 𝑑𝑑𝑒𝑒𝑏𝑏 − 𝐶𝐶𝑏𝑏 > 𝑅𝑅𝑏𝑏 − 𝑚𝑚𝐺𝐺𝑏𝑏 , (𝑄𝑄𝑎𝑎 + 𝑄𝑄𝑏𝑏 )(𝑅𝑅ℎ + 𝐹𝐹ℎ ) − 𝐶𝐶ℎ < (𝑄𝑄𝑎𝑎 + 𝑄𝑄𝑏𝑏 )(𝑅𝑅𝑙𝑙 − 𝐹𝐹𝑙𝑙 ) − 𝐶𝐶𝑙𝑙 , then in the case of platform negative service, the gain when both nodes share is greater than the gain when only one side shares; when both sides share, the gain of platform positive service is less than the gain of platform negative service. At this point λ1, λ2 and λ3 are all less than zero, and the point is the ESS point. Scenario 3: If 𝜃𝜃𝐾𝐾𝑙𝑙 𝐺𝐺𝑎𝑎 + 𝑛𝑛𝐺𝐺𝑏𝑏 + 𝑑𝑑𝑒𝑒𝑎𝑎 − 𝐶𝐶𝑎𝑎 > 𝑅𝑅𝑎𝑎 − 𝑚𝑚𝐺𝐺𝑎𝑎 , 𝜃𝜃𝐾𝐾𝑙𝑙 𝐺𝐺𝑏𝑏 + 𝑛𝑛𝐺𝐺𝑎𝑎 + 𝑑𝑑𝑒𝑒𝑏𝑏 − 𝐶𝐶𝑏𝑏 > 𝑅𝑅𝑏𝑏 − 𝑚𝑚𝐺𝐺𝑏𝑏 , (𝑄𝑄𝑎𝑎 + 𝑄𝑄𝑏𝑏 )(𝑅𝑅ℎ + 𝐹𝐹ℎ ) − 𝐶𝐶ℎ > (𝑄𝑄𝑎𝑎 + 𝑄𝑄𝑏𝑏 )(𝑅𝑅𝑙𝑙 − 𝐹𝐹𝑙𝑙 ) − 𝐶𝐶𝑙𝑙 , then in the case of platform negative service, the gain when both nodes share is greater than the gain when only one side shares; when both sides share, the gain from platform positive service is greater than the gain from platform negative service. At this point there is an eigenvalue greater than zero in λ1, λ2 and λ3, which is not an ESS point. (3) E8(1,1,1): λ1=(−1)( 𝜃𝜃𝐺𝐺𝑎𝑎 𝐾𝐾ℎ + 𝑛𝑛𝐺𝐺𝑏𝑏 + 𝑑𝑑𝑒𝑒𝑎𝑎 − 𝐶𝐶𝑎𝑎 + 𝑚𝑚𝐺𝐺𝑎𝑎 − 𝑅𝑅𝑎𝑎 ), λ2=(−1)(𝜃𝜃𝐺𝐺𝑏𝑏 𝐾𝐾ℎ + 𝑛𝑛𝐺𝐺𝑎𝑎 + 𝑑𝑑𝑒𝑒𝑏𝑏 − 𝐶𝐶𝑏𝑏 + 𝑚𝑚𝐺𝐺𝑏𝑏 − 𝑅𝑅𝑏𝑏 ), λ3=(−1)[𝑄𝑄𝑎𝑎 (𝑅𝑅ℎ + 𝐹𝐹ℎ − 𝑅𝑅𝑙𝑙 + 𝐹𝐹𝑙𝑙 ) + 𝑄𝑄𝑏𝑏 (𝑅𝑅ℎ + 𝐹𝐹ℎ − 𝑅𝑅𝑙𝑙 + 𝐹𝐹𝑙𝑙 ) − 𝐶𝐶ℎ + 𝐶𝐶𝑙𝑙 ], at this point the values of λ1, λ2 and λ3 are all uncertain and are discussed by case. Advances in Production Engineering & Management 18(2) 2023 231 Wang, Zhang Scenario 4: If 𝜃𝜃𝐺𝐺𝑎𝑎 𝐾𝐾ℎ + 𝑛𝑛𝐺𝐺𝑏𝑏 + 𝑑𝑑𝑒𝑒𝑎𝑎 − 𝐶𝐶𝑎𝑎 > 𝑅𝑅𝑎𝑎 − 𝑚𝑚𝐺𝐺𝑎𝑎 , 𝜃𝜃𝐺𝐺𝑏𝑏 𝐾𝐾ℎ + 𝑛𝑛𝐺𝐺𝑎𝑎 + 𝑑𝑑𝑒𝑒𝑏𝑏 − 𝐶𝐶𝑏𝑏 > 𝑅𝑅𝑏𝑏 − 𝑚𝑚𝐺𝐺𝑏𝑏 , (𝑄𝑄𝑎𝑎 + 𝑄𝑄𝑏𝑏 )(𝑅𝑅ℎ + 𝐹𝐹ℎ ) − 𝐶𝐶ℎ > (𝑄𝑄𝑎𝑎 + 𝑄𝑄𝑏𝑏 )(𝑅𝑅𝑙𝑙 − 𝐹𝐹𝑙𝑙 ) − 𝐶𝐶𝑙𝑙 , then, in the case of positive platform service, the gain when both nodes share is greater than the gain when only one side shares; when both sides share, the gain from positive platform service is greater than the gain from negative service, when λ1, λ2 and λ3, are all less than zero, and the point is the ESS point. Scenario 5: If 𝜃𝜃𝐺𝐺𝑎𝑎 𝐾𝐾ℎ + 𝑛𝑛𝐺𝐺𝑏𝑏 + 𝑑𝑑𝑒𝑒𝑎𝑎 − 𝐶𝐶𝑎𝑎 < 𝑅𝑅𝑎𝑎 − 𝑚𝑚𝐺𝐺𝑎𝑎 , 𝜃𝜃𝐺𝐺𝑏𝑏 𝐾𝐾ℎ + 𝑛𝑛𝐺𝐺𝑎𝑎 + 𝑑𝑑𝑒𝑒𝑏𝑏 − 𝐶𝐶𝑏𝑏 < 𝑅𝑅𝑏𝑏 − 𝑚𝑚𝐺𝐺𝑏𝑏 , (𝑄𝑄𝑎𝑎 + 𝑄𝑄𝑏𝑏 )(𝑅𝑅ℎ + 𝐹𝐹ℎ ) − 𝐶𝐶ℎ < (𝑄𝑄𝑎𝑎 + 𝑄𝑄𝑏𝑏 )(𝑅𝑅𝑙𝑙 − 𝐹𝐹𝑙𝑙 ) − 𝐶𝐶𝑙𝑙 , then, in the case of platform positive service, the gain when both nodes share is less than the gain when only either one shares; when both share, the gain from platform positive service is less than the gain from negative service. At this point there is an eigenvalue greater than zero in λ1, λ2 and λ3, which is not an ESS point. 5. MATLAB simulation analysis To verify the validity of the evolutionary stability analysis, the article incorporates a three-way evolutionary game model, gives the model initial values and uses MATLAB for simulation. The initial values are given in the following table. The initial values were evolved 50 times over time from different combinations of policies, respectively. As can be seen in Fig. 6, regardless of the initial probability of policy selection for the tripartite nodes, the evolutionary result tends to be (1,1,1), with the corresponding evolutionary policy being (shared, shared, active service), at which point the evolutionary result satisfies Scenario four. Variables 𝐶𝐶𝑎𝑎 𝐶𝐶𝑏𝑏 d 𝑒𝑒𝑎𝑎 𝑒𝑒𝑏𝑏 𝑚𝑚 𝑛𝑛 𝜃𝜃 𝐺𝐺𝑎𝑎 𝐺𝐺𝑏𝑏 𝑅𝑅𝑎𝑎 Table 3 Initial values of variables Initial value Variables 5 𝑅𝑅𝑏𝑏 5 𝑅𝑅ℎ 0.5 𝑅𝑅𝑙𝑙 6 𝐶𝐶ℎ 6 𝐶𝐶𝑙𝑙 0.6 𝐹𝐹ℎ 0.2 𝐹𝐹𝑙𝑙 0.4 𝐾𝐾ℎ 20 𝐾𝐾𝑙𝑙 20 𝑄𝑄𝑎𝑎 10 𝑄𝑄𝑏𝑏 Initial value 10 0.7 0.5 6 3 0.6 0.2 0.8 0.5 15 15 Fig. 6 Evolutionary results for scenario four Change the initial values so that 𝑅𝑅𝑎𝑎 = 20 and 𝑅𝑅𝑏𝑏 = 20, and evolve the values 50 times over time from different strategy combinations respectively. As can be seen from Fig. 7, there is only one evolutionary stable strategy combination for the evolving system at this point, and the result satisfies case one, i.e. it eventually converges to (0,0,0) and the corresponding evolutionary strategy is (no share, no share, negative service). 232 Advances in Production Engineering & Management 18(2) 2023 Blockchain-based tripartite evolutionary game study of manufacturing capacity sharing Fig. 7 Evolutionary results for scenario one Change the value of 𝐶𝐶ℎ to satisfy case two. The platform is reluctant to provide positive services because the cost of positive services is too high, at which point the system evolves to a stable point (shared, shared, negative services). With blockchain technology, a reasonably set incentive and penalty for the platform to fulfil its duty to provide good service when both parties share, promoting the sharing of manufacturing capacity and avoiding negative service, as shown in Fig 8. Fig. 8 Evolutionary results for scenario two To explore the effects of changes in the initial probabilities and other parameters on the evolutionary results, 𝑃𝑃 was adjusted to 0.25, 0.45 and 0.65; 𝑚𝑚 was adjusted to 0.2, 0.6 and 1; 𝑛𝑛 was adjusted to 0, 0.3 and 0.6; 𝑑𝑑 was adjusted to 0, 0.5 and 1; 𝐹𝐹ℎ was adjusted to 0.2, 0.6 and 1; and 𝐹𝐹𝑙𝑙 was adjusted to 0, 0.2, 0.4. 0.4. Observe the dynamic course of the evolutionary results over time, as shown in Figs. 9 to 14. We can see that: as 𝑃𝑃 increases, the probability that the tripartite nodes tend to share, share and actively serve increases, and the speed of evolution gradually increases. When the incentive coefficient m of blockchain technology for nodes is too low, both nodes are unwilling to share, then the platform will pay more costs for positive service than negative service, and eventually the platform will gradually choose to provide negative service; when 𝑚𝑚 increases, the probability of nodes a and b sharing and platform positive service increases, and the convergence speed gradually accelerates to (1,1,1). As the penalty coefficient 𝑛𝑛 increases, the evolution of both nodes choosing to share and platform positive services converge to 1 and the evolution speed gradually accelerates. Penalties under blockchain technology have a positive effect on promoting inter-node behaviour and can effectively facilitate capacity sharing between enterprises. When the digital revenue 𝑑𝑑 is too low, it will lead to the revenue when nodes share is less than the revenue when they do not share, and the nodes gradually evolve to (0,0,0). A reasonable 𝑑𝑑 will promote the behaviour of sharing among enterprise nodes and active service of platform nodes, and the larger 𝑑𝑑 is, the more the node evolution converges to (1,1,1) the faster the rate of convergence. The positive service of the platform can bring good reputational benefits, and as 𝐹𝐹ℎ gets higher, the rate of evolution of individual nodes to 1 gradually increases faster. Likewise, negative services also bring reputational losses to the platform, and the higher the 𝐹𝐹𝑙𝑙 , the faster each node converges to 1. With blockchain technology, reasonable and effective penalties and incentives can promote positive service of platform nodes and capacity sharing of enterprise nodes. Advances in Production Engineering & Management 18(2) 2023 233 Wang, Zhang Fig. 9 Impact of 𝑃𝑃 Fig. 10 Impact of 𝑚𝑚 Fig. 11 Impact of 𝑛𝑛 Fig. 12 Impact of 𝑑𝑑 Fig. 13 Impact of 𝐹𝐹ℎ Fig. 14 Impact of 𝐹𝐹𝑙𝑙 6. Conclusion and management recommendations 6.1 Conclusion From the results of the above analysis, it can be seen that: • The game subjects will not change the evolutionary results under the initial value setting, no matter what the proportion of strategy selection is, if the relevant parameters are changed, it will have an impact on the strategy selection of the three subjects. • Incentive and penalty coefficients are set under blockchain technology for rewarding the sharing party and punishing the non-sharing party. The incentive coefficient under blockchain technology has a positive impact on the sharing behaviour of manufacturing enterprises, and the penalty coefficient has a negative impact on the sharing behaviour of manufacturing enterprises. • The digital benefit coefficient affects the behaviour of manufacturing enterprises, too high or too low is not conducive to the choice of manufacturing enterprises, and an appropriate digital benefit coefficient will promote the sharing behaviour of manufacturing enterprises. 234 Advances in Production Engineering & Management 18(2) 2023 Blockchain-based tripartite evolutionary game study of manufacturing capacity sharing • The platform under blockchain technology introduces a regulatory mechanism combined with smart contracts to constrain the behaviour of enterprises and the platform. Both positive regulation and negative regulation coefficients have a positive impact on the behaviour of the three parties. 6.2 Recommendation Based on the above research findings, the following recommendations are made. On the one hand, the alliance nodes under blockchain technology can appropriately optimize the incentive mode and penalty mode, and find suitable allocation coefficients to maximize the incentive for the capacity-sharing behaviour of both supply and demand sides and promote the cooperation between them; at the same time, the third-party platform should actively supervise and strengthen its supervisory capacity to avoid the emergence of malicious pigeonholing behaviour, improve the effective cooperation between supply and demand sides and promote the long-term development of the manufacturing industry. In addition, both supply and demand sides should maintain good integrity to help create a good market environment, thus promoting the benign development of the manufacturing industry. 6.3 Shortcomings The article uses an evolutionary game approach to investigate the capacity sharing behaviour of manufacturing firms and uses numerical simulations in MATLAB to analyse the effect of changes in different parameters on the evolutionary outcome. However, the article does not consider the effects of other aspects such as product quality, delivery time, human emotion, social development and environment on the game behaviour of capacity sharing, which can be further explored by introducing prospect theory in the future, and I am currently conducting related research. Acknowledgement Thanks to the National Social Science Foundation of China for funding (2022BJY021). References Dai, H.-N., Zheng, Z., Zhang, Y. (2019). 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Three-party evolutionary game and simulation analysis of drug quality supervision under the government reward and punishment mechanism, Chinese Journal of Management Science, Vol. 2021, No. 11, 55-67, doi: 10.16381/j.cnki.issn1003-207x.2019.0481. 236 Advances in Production Engineering & Management 18(2) 2023 Advances in Production Engineering & Management ISSN 1854-6250 Volume 18 | Number 2 | June 2023 | pp 237–249 Journal home: apem-journal.org https://doi.org/10.14743/apem2023.2.470 Original scientific paper Enhancing automated defect detection through sequential clustering and classification: An industrial case study using the Sine-Cosine Algorithm, Possibilistic Fuzzy c-means, and Artificial Neural Network Nguyen, T.P.Q.a,*, Yang, C.L.b, Le, M.D.c, Nguyen, T.T.d, Luu, M.T.e aFaculty of Project Management, The University of Danang – University of Science and Technology, Danang, Vietnam of Industrial Management, National Taiwan University of Science and Technology, Taipei, Taiwan (ROC) cFaculty of Transportation Mechanical Engineering, The University of Danang – University of Science and Technology, Danang, Vietnam dFaculty of Computer Science and Engineering, University of Technology, Vietnam eNghia Nippers Corporation, Vietnam bDepartment ABSTRACT ARTICLE INFO Most existing inspection models solely classify defects as either good or bad, focusing primarily on separating flaws from perfect ones. The sequential clustering and classification technique (SCC) is used in this work to not only identify and categorize the defects but also investigate their root causes. Conventional clustering techniques like k-means, fuzzy c-means, and self-organizing map are employed in the first stage to find the defects in the finished products. Then, a novel clustering method, that combines a sine-cosine algorithm and possibilistic fuzzy c-means (SCA-PFCM), is proposed to classify the detected defects into multiple groups to identify the defect categories and analyze the root causes of failures. In the second stage, the ground truth labels taken from the clustering technique are used to construct an automated inspection system using back propagation neural networks (BPNN). The proposed approach is applicable for detecting and identifying the causes of errors in manufacturing industry. This study applies a case study in nipper manufacture. The SCA-PFCM algorithm can detect 97 % of defects and classify them into four types while BPNN shows a predicted accuracy of up to 96 %. Additionally, an automated inspection system is developed to reduce the time and cost of the inspection process. Keywords: Back Propagation Neural Network; Clustering; Classification; Combined SCA-PFCM; Defect detection; Nipper manufacturing; Possibilistic Fuzzy c-means (PFCM); Root cause analysis; Sine-Cosine Algorithm (SCA) *Corresponding author: ntpquyen@dut.udn.vn (Nguyen, T.P.Q) Article history: Received 24 April 2023 Revised 12 June 2023 Accepted 23 June 2023 Content from this work may be used under the terms of the Creative Commons Attribution 4.0 International Licence (CC BY 4.0). Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation, and DOI. 1. Introduction The rapid rise of both information and operational technology is accelerating the transition from traditional manufacturing to smart factories. A smart factory typically relies on modern information and communication technology, with all of the factory's components being smartly integrated and operated [1]. A smart factory needs to optimize production conditions with a minimum of resources and time. Defects are one of the primary causes of high production costs. Therefore, 237 Nguyen, Yang, Le, Nguyen, Luu the inspection system for defect detection is crucial in smart manufacturing, which necessitates the quick detection of defects and categorizes defect types. Effective defect categorization aids in determining and analyzing the type of failure that occurs based on the process conditions [2]. Besides, the defect types can be used to diagnose equipment failure and analyze its root causes. Moreover, an effective inspection system can raise customer satisfaction levels by preventing them from obtaining substandard goods. Many manufacturing factories still operate the visual and sampling inspection manually. Sampling inspections cannot catch all the defects like a thorough inspection can. Manual inspection may lead to inconsistent results, expensive costs, and time-consuming [3]. Enterprises expect quick defect detection and root cause investigation to provide timely regulation of production line failures [4]. A more effective and reliable automated defect inspection system is required to overcome the drawbacks of sampling and manual inspection in manufacturing. Generally, a data analysis framework to detect the defects and investigate their causes is necessary. However, the majority of the research that has been done so far has solely focused on identifying defects or defect prediction models, which can result in dividing the finished products into two groups: 1) defects, and 2) non-defective ones [5]. In practice, the manager may not only need to detect failure or an abnormal state in the manufacturing process but also identify the causes of defects in the routine execution of the production plan to make timely adjustments and corrections [6]. This process is known as root cause analysis which aims to determine the underlying cause of an issue and the measures required to solve it. A novel model of sequential clustering and classification-based genetic algorithm (NSGAII-SCC) has recently been proposed by Yang and Quyen to investigate the hidden structure of data and to identify the features correlated with the explored patterns [7]. The NSGAII-SCC framework is applied to analyze the point of sale (POS) data for a chain of bakeries in China [8]. The bakery store is partitioned into several clusters using a clustering technique and classification is then used to investigate the factors that contribute to the partition process of each store cluster. Kuo et al. presented an extension of the NSGAII-SCC by combining a deep learning and multi-objective sinecosine algorithm (Deep MOSCA-SCC) [9]. The Deep MOSCA-SCC algorithm employed a deep clustering technique that combined auto-encoder and k-means to improve clustering performance. Generally, the SCC approach is useful for defect detection and classification, analyzing its root causes as well as establishing an automatic detect defection model in the above analysis. Thus, this study focuses on developing a model-based SCC method to detect the defects and analyze their root causes. In the first stage, the proposed model-based SCC method first employs clustering methods to detect the defects in the finished product. Thereafter, the defects are classified into several groups with similar properties to identify the defect categories. Most of the research used traditional clustering techniques like k-means algorithm[10] or fuzzy c-means (FCM) algorithm [11] for industrial applications since they are simple to implement and interpret the result. However, the clustering result was also comparable. In the first stage, several conventional clustering techniques, i.e., k-means, FCM, self-organizing map (SOM) [12], and DBSCAN [13], are employed to detect the defects from the flawless ones since this process is not very complicated due to the predetermined number of clusters k is known as two labels (defect or non-defect). To determine the defect categories and analyze the root causes of failures, a novel clustering method that combines a sine-cosine algorithm and possibilistic fuzzy c-means (SCA-PFCM), is proposed. In the second stage, a back propagation neural network (BPNN) [14] is employed to develop an automatic defect detection model based on the ground truth labels adopted from the clustering technique. In contrast to existing SCC methods, the proposed inspection model-based SCC contributes by combining innovative clustering and classification techniques, namely SCA-PFCM and BPNN, whereas the original NSGAII-SCC suggested an integration of traditional k-means and decision tree classifier. The research result is then applied to an inspection system of a nipper manufacturing factory in Vietnam. The paper is arranged as follows. The review of sequential clustering and classification method is shown in Section 2. Section 3 describes the production and inspection process in nail nipper manufacturing. The methodology is presented in Section 4. Section 5 illustrates the result analysis of a case study. The concluding remarks and future research direction come in Section 6. 238 Advances in Production Engineering & Management 18(2) 2023 Enhancing automated defect detection through sequential clustering and classification: An industrial case study using … 2. Methods and materials 2.1 Review of sequential clustering and classification method Since clustering is frequently carried out without knowledge regarding the membership of data instances to predetermined labels, it is commonly referred to as a technique for unsupervised learning [15]. The goal of clustering is exploratory the structure of the dataset. The clustering process divides objects in a given dataset into distinct groups or clusters, with each cluster consisting of objects that are similar to each other in that cluster but dissimilar to objects in other clusters. On the contrary, classification is a supervised approach that assigns categories or labels to data records based on previously collected information [16]. The classification process consists of two steps. The first step is to develop a training model based on the classification rules using the given data, which contains a set of attributes and their corresponding outcomes. The training model is then used in the second step to forecast the labels for incoming unknown data. The complexity of the large number of features utilized as a classifier is what makes these processes challenging [17]. Yang and Nguyen [7] proposed the SCC framework, which combines clustering and classification sequentially on two different datasets. The SCC framework can investigate the hidden structure of data and discover the features that are relevant to the explored patterns. Fig. 1 illustrates the SCC framework. From the original dataset, the target interests were identified and these features were then used for discovering data patterns by the clustering algorithm. The remaining dataset contained the relevant features that are correlated with the target features. A clustering algorithm was implemented on the dataset containing target features to explore the data patterns. The result of clustering algorithm was embedded into the training process of the classification model. Two popular clustering algorithms, k-means, and hierarchical clustering, were used to perform clustering. The classification algorithm employed decision trees [18], artificial neural networks (ANN) [19], k-nearest neighbor (KNN) [20], and support vector machine (SVM) [21]. As a result, there are eight clustering-classification algorithm combinations built from two clustering methods and four classification methods. Besides, as combining heuristic techniques and machine learning become more prevalent [22, 23], the SCC framework combined with NSGAII to deal with two objective functions from clustering and classification tasks. Because the number of clusters (k) is not predetermined, the SCC framework implemented the various values of k (from 2 to 10) and finally selected the solution based on the Pareto front. For each value of k, multiple solutions are selected from the first Pareto front to balance two objective functions of clustering and classification tasks. Performance was superior to the others when k-means clustering and decision tree classification were combined. The Deep MOSCA-SCC algorithm [9] is also an SCC approach that employed deep clustering to enhance the clustering compactness and classification accuracy. Herein, MOSCA [24], which is a simple and easy-to-implement method, was applied to exploit the optimal result for the SCC approach. The utilization of MOSCA in the Deep MOSCA-SCC algorithm was expected to speed up computation compared to NSGAII. However, the experimental results demonstrated that there was no benefit in Deep MOSCA-SCC computational time due to the use of autoencoder, which increased computational time. Target features Original dataset Feature selection Class labels Splitting data Correlated features Clustering algorithm Feature selection Solutions’ evaluation End Classification algorithm Fig. 1 The SCC approach Advances in Production Engineering & Management 18(2) 2023 239 Nguyen, Yang, Le, Nguyen, Luu 2.2 Proposed methodology The manual examination procedure necessitates a significant amount of human labor and expenditure. Industry 4.0 is having an impact on nail manufacturing, with automated defect detection reducing inspection time and human labor. Automatic visual inspection, which plays a key part in the quality inspection process of smart manufacturing, is a promising technology in this scenario. The SCC technique is used for algorithm development of an automatic visual inspection in nipper manufacturing in this study. This study utilizes the SCC method [7] to not only detect the defects from the flawless ones but also classify the defects into several groups with similar attributes for root cause analysis in the first stage. An automated defect detection system is developed in the second stage. The result of the first stage is used to train the classification model for automatic defect detection. The methodology framework is shown in Fig. 2. Fig. 2 Methodology Framework Stage 1: Defect detection and root cause analysis using clustering technique Detecting defects from the finished products In this stage, the clustering techniques are employed for defect detection and root cause analysis in nipper manufacturing. The dataset is collected from a nipper factory in Vietnam. The Department of quality control provides historical data on finished items, which is only divided into two categories: defective products and non-defective products. Thus, this study first uses the clustering technique to detect the defects from the flawless ones. The number of clusters is defined as 𝑘𝑘 = 2 and the class labels of all data instances are predetermined based on the historical data. kmean, FCM, SOM, and DBSCAN are employed to classify the defects and non-defected items in the first stage since they are in the group of popular clustering methods applied for industry applications [25, 26]. Clustering accuracy (𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶_𝐴𝐴𝐴𝐴𝐴𝐴) is used to evaluate the result of this step. The 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶_𝐴𝐴𝑐𝑐𝑐𝑐 is calculated as follows [27]: ∑𝑘𝑘 𝑏𝑏 𝑖𝑖 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶_𝐴𝐴𝐴𝐴𝐴𝐴 = 𝑖𝑖=1 , (1) 𝑛𝑛 where 𝑏𝑏𝑖𝑖 is the number of instances that are classified in a corrected cluster, 𝑛𝑛 is the number of total data instances. Proposed SCA-PFCM for identifying the defect categories The defects revealed in the first step are classified into several types of defects that have similar properties to analyze the root causes of defects. Herein, k is unknown. Thus, this study proposes a novel clustering method, i.e., SCA-PFCM, which not only automatically determines the optimal k but also simultaneously partitions the clusters to reveal data patterns corresponding to the selected k. The proposed SCA-PFCM inherited the approach of identifying the optimal number of clusters in an automatic fuzzy clustering method proposed by Nguyen and Kuo [28], which deals with automatic clustering for categorical data. However, the dataset from the nipper industry contains numerical features. Thus, the distance measure in the clustering procedure is changed to be appropriate for the data features. Besides, SCA is employed since it is a simple and straightforward algorithm with competitive performance as compared with GA, and PSO [29]. The sine and cosine functions are utilized to update the particle’s position in the SCA algorithm as follows: (𝑡𝑡+1) 𝑋𝑋𝑖𝑖 (𝑡𝑡+1) 𝑋𝑋𝑖𝑖 240 = 𝑋𝑋𝑖𝑖𝑡𝑡 + 𝑟𝑟1 × sin(𝑟𝑟2 ) × �𝑟𝑟3 𝑃𝑃𝑖𝑖𝑡𝑡 − 𝑋𝑋𝑖𝑖𝑡𝑡 �, 𝑟𝑟4 < 0.5 = 𝑋𝑋𝑖𝑖𝑡𝑡 + 𝑟𝑟1 × cos(𝑟𝑟2 )× �𝑟𝑟3 𝑃𝑃𝑖𝑖𝑡𝑡 − 𝑋𝑋𝑖𝑖𝑡𝑡 �, 𝑟𝑟4 ≥ 0.5 (2) (3) Advances in Production Engineering & Management 18(2) 2023 Enhancing automated defect detection through sequential clustering and classification: An industrial case study using … (𝑡𝑡+1) where 𝑋𝑋𝑖𝑖𝑡𝑡 𝑎𝑎𝑎𝑎𝑎𝑎 𝑋𝑋𝑖𝑖 is the position of individual i at iteration t and 𝑡𝑡 + 1, respectively. 𝑃𝑃𝑖𝑖𝑡𝑡 is the optimal position at the current iteration 𝑡𝑡. 𝑟𝑟1 is a control number that is calculated as: µ 𝑟𝑟1 = µ − 𝑡𝑡 (4) 𝑇𝑇 while 𝑟𝑟2 is a number randomly chosen between 0 and 2π, whereas 𝑟𝑟3 , 𝑟𝑟4 ∈ (0,2). µ is a constant. The SCA-PFCM algorithm's process is explained as follows: Step 1: Identify the maximum k (𝑘𝑘𝑚𝑚𝑚𝑚𝑚𝑚 ) A local density-based approach named RECOME [30] is employed to determine 𝑘𝑘𝑚𝑚𝑚𝑚𝑚𝑚 . The cluster center typically has higher local density and is surrounded by neighbors with lower local densities. If the number of high-density centers is discovered, the 𝑘𝑘𝑚𝑚𝑚𝑚𝑚𝑚 can be obtained. The procedure to determine 𝑘𝑘𝑚𝑚𝑚𝑚𝑚𝑚 is shown in Fig. 3. Step 2: Set up parameters for SCA and PFCM algorithms. Step 3: Initialization: Randomly generate the initial population. A solution representation consists of two parts. The first part is used to define the k. A control element 𝐶𝐶𝑝𝑝 is used to determine k in each particle where 𝑘𝑘𝑚𝑚𝑚𝑚𝑚𝑚 ≤ 𝐶𝐶𝑝𝑝 ≤ 𝑘𝑘𝑚𝑚𝑚𝑚𝑚𝑚 . 𝑘𝑘𝑚𝑚𝑚𝑚𝑚𝑚 is set at 2. To determine 𝐶𝐶𝑝𝑝 , we randomly generate a vector 𝐶𝐶 = 𝐶𝐶1 , 𝐶𝐶2 , … , 𝐶𝐶𝑗𝑗 , … , 𝐶𝐶𝑘𝑘𝑚𝑚𝑚𝑚𝑚𝑚 in the range of [0, 1]. 𝐶𝐶𝑝𝑝 is determined by counting the elements in C that are greater than 0.5: 𝐶𝐶𝑝𝑝 = count (C|𝐶𝐶𝑗𝑗 ≥ 0.5). The second part of a particle is the cluster center corresponding to the value of k in the first part. Step 4: Calculate fitness: The PFCM algorithm is implemented for each particle in the population. The fitness value is obtained through the objective function of PFMC algorithm. Step 5: Identify the optimal solution 𝑃𝑃𝑖𝑖𝑡𝑡 at iteration t. Step 6: Compute 𝑟𝑟1 using Eq. 4 and randomly generate 𝑟𝑟2 𝑟𝑟3 , and 𝑟𝑟4 . Step 7: Update the cluster centers using Eq. 2 and Eq. 3. Step 8: Return to step 4 until the stopping condition is met. Fig. 3 Procedure to determine 𝑘𝑘𝑚𝑚𝑚𝑚𝑚𝑚 [28] The result in this stage, which includes the class labels and the number of clusters for defects, is used to train the model in the second stage. Advances in Production Engineering & Management 18(2) 2023 241 Nguyen, Yang, Le, Nguyen, Luu Stage 2: Automatic defect detection using back propagation neural network (BPNN) BPNN [14] is a popular branch of ANN that contains a multi-layer feedforward network trained by the error BP technique. A BPNN consists of three different layer types: input, hidden, and output layers. The hidden layer acts as a link for signal forward propagation connecting the input and output layers. If the output layer is unable to produce the required results, the error BP will transmit an error signal from the output nodes back via the hidden layer to the input nodes before repeating the signal forward propagation. This study employs BPNN to develop an automated defect prediction model. The labels that are revealed by clustering techniques in the first stage are used to train the BPNN model. The number of neurons in the input layer is identified by the number of data features that significantly impact the output. The output layer nodes depend on the defect categories that are partitioned in the first stage. It is calculated by multiplying the number of defect types by one to represent the defect categories and good products. The performance of the BP network is significantly influenced by the number of hidden layers and hidden layer nodes. A single hidden layer or two hidden layers can be considered in the design of BPNN model depending on its predictive performance [31]. Besides, better performance can be achieved with more hidden layer nodes, although a lengthy training period may result. Typically, an empirical formula is used to estimate the number of hidden layer nodes. The empirical formula is displayed as follows [32]: 𝐻𝐻 = √𝐼𝐼 + 𝑂𝑂 + 𝑎𝑎, (5) where I, H, and O are the number of nodes in the input, hidden, and output layers, respectively, 𝑎𝑎 is a constant number in the range [0, 10]. Cross-entropy is selected as the loss function, which is calculated as follows [33]: 𝐸𝐸 = − ∑𝑁𝑁 �𝑖𝑖 ) + (1 − 𝑦𝑦𝑖𝑖 ) ∗ 𝑙𝑙𝑙𝑙𝑙𝑙(1 − 𝑦𝑦�𝑖𝑖 )), (6) 𝑖𝑖=1(𝑦𝑦𝑖𝑖 ∗ 𝑙𝑙𝑙𝑙𝑙𝑙(𝑦𝑦 where 𝑦𝑦𝑖𝑖 is the target value, 𝑦𝑦�𝑖𝑖 is predicted value. The structure of BPNN that contains two hidden layers for automated defect prediction is shown in Fig. 4. The proposed method is applicable for detecting and identifying the causes of errors in industrial manufacturing. There are two ways that the input data can be gathered from the industry: sensor signal data or image data. The proposed method in its current form can work well for signal data. If the input is the image data, it can be preprocessed to extract the features from the image before the proposed method is implemented. This paper analyzes a case study in nipper manufacturing. An automatic defect detection system is developed in Section 3.3 in which the product images are taken on the production line. The user can adjust the model to match the type of input data. The nipper manufacturing industry in Vietnam, which is selected as a case study for this research, will be covered in the next subsection, along with the production and inspection processes. 242 Fig. 4 The structure of BPNN model for automated defect prediction Advances in Production Engineering & Management 18(2) 2023 Enhancing automated defect detection through sequential clustering and classification: An industrial case study using … 2.3 Production and inspection processes in nipper manufacturing This section introduces the production and inspection processes in nipper manufacturing. There are two types of nippers: cuticle nippers and nail nippers. The cuticle nipper usually has a smaller jaw and handle, as compared to the nail nipper. Besides, the cuticle nipper also has a lighter weight than the nail nipper. A nipper consists of five main components: jaw, joint, spring, spring pin, and handles. The jaw is designed with exact blade alignment and precision sharpening. For the cuticle nipper, there is different jaw size such as J12, J14, and J16 which are corresponding to the blade sizes are 4.5-5.5 mm, 5.5-6.5 mm, and 6.5-7.5 mm, respectively. In contrast, the nail nipper only has one jaw size of 6.5-8.0 mm. The joints aim to transmit the cutting force from the handles to the two jaws when the user squeezes them. Box joint is designed to warranty it is durable. Springs are used to assist (push and elastic) the user when squeezing the two handles. The pin is used to hold the spring with the handle. Besides, the company name or logo is also etched on the handle. The manufacturing process of a nipper is described in Fig. 5. There are four main stages in the whole production process. The first stage is to make a mold from the mold blueprint. In the pressing stage, several processes, such as cutting, stamping for shaping, bending the blade edges, annealing mill, and grinding the surface, are continuously carried out. The output of this stage is a shaped nipper. The process is continued in the pre-processing stage which includes the stamping and drilling assembly, thermal process, shaping of the jaw, polishing of the handle, and plating. Thereafter, the preliminary nipper is transferred to the finishing stage with the following processes: installing springs, printing laser, polishing the jaw, sharpening the blade edge, and cleaning the finished product. There is an inspection process to check product quality at each stage. Then, the inspection process is applied to the finished product. The criteria for inspection consist of sharpness, the shape of the jaw, logo printing, length of the blade, and handle length. The inspection process is performed manually and is costly. Moreover, the number of products per day is also huge. Thus, sampling inspection is currently used with a sample of 10 % to 20 % of finished products randomly taken for inspection. The sampling inspection process cannot guarantee the quality of all the finished products. Fig. 5 The manufacturing process of a nipper 3. Case study: Result and discussion 3.1 Data collection and parameter setting The experiment uses the historical data gathered from a nipper manufacturing factory in Vietnam. According to the company's current production needs, a total of 15000 nippers of various types must be produced per day. As a result, the amount of data recorded day by day becomes very huge to process. This study analyzes three different types of nail nippers, coded D555, D401, and D363, respectively. D555 includes 84320 data instances, whereas D401 and D363 have 48384 and 33690 data instances. Each data instance has the 12 features such as nipper length, handle, blade length, width and thickness of nipper handles, thickness of two nipper bearings, and so on. In the first stage, k-means, FCM, SOM, and DBSCAN are used to classify the defects from the flawless ones. The number of clusters is predetermined as 𝑘𝑘 = 2. Clustering accuracy is employed to evaluate the result. Then the discovered defects are used to perform clustering to determine Advances in Production Engineering & Management 18(2) 2023 243 Nguyen, Yang, Le, Nguyen, Luu the defect types by using the proposed SCA-PFCM algorithm. To compare the optimal k obtained from the SCA-PFCM, the elbow method-based k-means is employed in which the sum of squared error (SSE) is considered as a performance indicator. SSE is calculated as follows: 𝑆𝑆𝑆𝑆𝑆𝑆 = ∑𝑘𝑘𝑗𝑗=1 ∑𝑥𝑥∈𝐶𝐶𝑗𝑗 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑(𝑥𝑥, 𝑚𝑚𝑗𝑗 )2 (7) The parameter setting for BPNN model will be discussed in more detail in the next subsection after getting the result of the first stage. 3.2 Result analysis First-stage experimental findings Clustering methods are first used to detect the defects in the finished product. The k-means, FCM, SOM, and DBSCAN algorithms are implemented on the three datasets. Each algorithm is executed 30 times for each tested dataset. Table 1 shows the clustering accuracy based on the average values of 30 runs and the standard deviation. It is not much different from the clustering accuracy by k-means and FCM. The DBSCAN algorithm is slightly better since its accuracy dominates the other algorithms for all tested data. Thus, the clustering result of DBSCAN is used in the next step. Table 2 shows the number of defective and non-defective products detected by DBSCAN method. Dataset D555 D401 D363 Table 1 Clustering accuracy of the process to classify defects from flawless ones k-means FCM SOM DBSCAN 0.912 ± 0.001 0.904 ± 0.003 0.896 ± 0.003 0.913 ± 0.001 0.938 ± 0.001 0.940 ± 0.002 0.913 ± 0.002 0.942 ± 0.002 0.953 ± 0.001 0.951 ± 0.001 0.948 ± 0.002 0.965 ± 0.001 Dataset D555 D401 D363 Table 2 Clustering results in terms of class labels Number of instances Total Non-defective 84320 46873 48484 28245 33690 18970 Defective 37447 20239 14720 To analyze the root cause of failures, the defects are classified based on their features. The number of clusters, which is represented by how many types of defects, is unknown. Elbow method-based k-means clustering is applied. Fig. 6 shows the SSE plot for various k in the D555, D401, and D363 datasets, respectively. The result is quite consistent since the optimal k is selected at 𝑘𝑘 = 4, which means that there are 4 types of defects. Fig. 6 Identify the number of defect types based on the elbow method Besides, the number of defect types is also determined using the proposed SCA-PFCM. The result is quite similar to the elbow-based k-means since four types of defects are determined. However, the performance in terms of SSE of the SCA-PFCM is smaller than that of the elbowbased k-means, as shown in Table 3. It means that the clustering result provided by SCA-PFCM is more compact. The number of defect types and the label of each defect instance obtained from the 244 Advances in Production Engineering & Management 18(2) 2023 Enhancing automated defect detection through sequential clustering and classification: An industrial case study using … SCA-PFCM are used in the second stage. The defect types can be listed as follows. The first type of defect is untight jaws since there is a clearance between two blades of the nipper that exceeds the gap standard. In the blade sharpening process, the craftsman leaves a gap that is allowed according to the standard. If they sharpen it a lot, there is a gap between the two blades of the nipper, and the length of the blade may be shorter. Most of the defects derive from this cause. The second and third types of defect are large gaps in the upper and lower gills of the nipper caused by the assembly process of the box joint. The four defect type belongs to the length of the nipper blade as it can be shorter or longer than its standard length due to the blade grinding process. Table 3 Comparison of SCA-PFCM and elbow-based k-means in terms of SSE SSE Dataset Optimal k SCA-PFCM Elbow method D555 4 14233.89 15784.78 D401 4 10602.53 11649.34 D363 4 8321.94 9157.63 Experimental results in the second stage The number of defect types and the label for each defect are employed to train the model in the second stage. This stage aims to not only detect the defects from the finished products but also classify the defects into certain groups that are identified in the first stage. Hence, there are five labels representing four types of defects and good product and their corresponding number of instances are presented in Table 4. The parameter for BPNN model with a single hidden layer (denoted as BPNN-1) is set up as follows. First, the input layer nodes equal to 12 which are the significant non-linear relationship variables to represent product features. The output layer nodes are 5 including the defect types and good product. Based on Eq. 5, the number of neurons in the hidden layer is from 4 to 14. After several trial experiments, the hidden layer nodes are selected as 10, initial weight 𝑤𝑤 = 0.2, learning rate 𝑟𝑟 = 0.1, and inertia weight 𝑐𝑐 = 0.6. Logsig and purelin are the activation functions for the hidden and output layers, respectively [31]. Regarding the BPNN model with two hidden layers (BPNN-2), the input and output layers are set up similarly to the BPNN-1 model. The number of neurons in the first hidden layer is also from 4 to 14 as determined by Eq. 5. This study eliminates half of the constant number in Eq. 5, which now ranges from 0 to 5, to shorten the computation time. Thus, there are 4 to 9 neurons in the second hidden layer. The activation functions and other parameters are similar to the BPNN-1 model. After a series of experimental trials, the first and second hidden layers are chosen to have 12 and 6 neurons, respectively, based on their predictive performance. Table 5 shows the selection of key parameters for BPNN models. Label 1 2 3 4 5 Model BPNN-1 BPNN-2 Table 4 Data properties for classification Types of defect Non-defective Untight jaws Large gap in the upper gill of nipper Large gap in the lower gill of nipper Length of nipper blade D555 46873 19326 8768 6293 3060 Number of instances D401 28245 11017 3790 4032 1400 D363 18970 6793 3278 3480 1169 Table 5 Setting for BPNN model Number of neurons Activation function Input layer Hidden layer Output layer Hidden layer Output layer 12 10 5 logsig purelin 12 12-6 5 logsig purelin To evaluate how well the BPNN model predicts, the confusion matrix is utilized [34]. Four predictive performance metrics are obtained from the confusion metrics, as follows: Advances in Production Engineering & Management 18(2) 2023 245 Nguyen, Yang, Le, Nguyen, Luu 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 =(TP+TN)/(TP+FP+TN+FN) (8) 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 = TP/(TP+FN) (9) 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = 𝑇𝑇𝑇𝑇/(𝑇𝑇𝑇𝑇 + 𝐹𝐹𝐹𝐹) (10) 𝐹𝐹_𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 2 · 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 · 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃/(𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 + 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃) (11) where 𝑇𝑇𝑇𝑇, 𝑇𝑇𝑇𝑇, 𝐹𝐹𝐹𝐹, and 𝐹𝐹𝐹𝐹 represent true positives, true negatives, false positives, and false negatives, respectively. Table 6 shows the experimental result in terms of the confusion matrix of dataset D555 implemented by the BPNN-1 model. The overall accuracy is 94.6 % which is calculated by taking the average of the correctly predicted values in each class. The overall error rate correspondingly is 5.4 %. Besides, Kappa coefficient [35] is also used to evaluate the agreement between predicted values and truth class values. The kappa coefficient is computed as 93.2 % to represent a perfect agreement. Similarly, the confusion matrix of dataset D555 implemented by the BPNN-2 model is displayed in Table 7. The overall accuracy, overall error rate, and Kappa coefficient achieved by BPNN-2 are 96 %, 4 %, and 94.8 %, respectively, which are relatively higher than those achieved by BPNN-1 model. Table 8 summarizes the result comparison of the BPNN-1 and BPNN-2 models in the three tested datasets. The BPNN-2 model is significantly better than the BPNN-1 model in terms of accuracy and Kappa coefficient. Besides, the computational time on the product group of each tested dataset is also presented in Table 8. The BPNN-2 takes more time for the training model. Therefore, compared to the BPNN-1, its computing time is significantly longer. However, the result in Table 8 is shown for the product group that includes numerous objects. The highest computational times, while taking into account the average computational time for each product, are 0.052 and 0.045 for BPNN-2 and BPNN-1, respectively. The average computation time per product provided by the two models does not differ considerably while the accuracy of BPNN-2 is significantly better. Thus, the BPNN-2 model is used to develop an automatic defect inspection system presented in the next subsection. Predicted Table 6 Confusion matrix of dataset D555 implemented by the BPNN-1 model BPNN-1 Actual Dataset D555 Non-de- Untight Large gap in Large gap in Over-length of fective jaws the upper gill the lower gill nipper blade (%) (%) (%) (%) (%) Non-defective 100 0 0 0 0 Untight jaws 0 100 2 13 0 Large gap in the upper gill 0 0 88 0 0 Large gap in the lower gill 0 0 8 87 2 Over-length of nipper blade 0 0 2 0 98 Table 7 Confusion matrix of dataset D555 implemented by the BPNN-2 model BPNN-2 Dataset D555 Predicted Non-defective Untight jaws Large gap in the upper gill Large gap in the lower gill Over-length of nipper blade Dataset Overall accuracy (%) Overall error rate (%) Kappa Coefficient (%) Computational time (s) 246 Non-defective (%) 100 0 0 0 0 Untight jaws (%) 0 100 0 0 0 Actual Large gap in Large gap in the upper gill the lower gill (%) (%) 0 0 0 10 93 0 5 90 2 0 Table 8 Result comparison D555 BPNN-1 BPNN-2 94.6 96 5.4 4 93.2 94.8 2440.5 2916.2 D401 BPNN-1 BPNN-2 93.5 95.2 6.5 4.8 93.1 94.7 1795.5 2091.9 Over-length of nipper blade (%) 0 0 0 3 97 D363 BPNN-1 BPNN-2 92.8 96.3 7.2 3.7 92.1 95.9 1529.2 1751.6 Advances in Production Engineering & Management 18(2) 2023 Enhancing automated defect detection through sequential clustering and classification: An industrial case study using … 3.3 Automated defect inspection system The BPNN-2 model whose results were validated in Section 5.2.2 is used in the automatic inspection system for nipper manufacturing. To transfer from manual inspection to the automatic inspection process, an advanced sensing system with smart camera sensors is installed. The system includes two industry cameras that are installed perpendicular to the conveyor. One camera is used to check the overall sizes of a nipper. Another one is put closer for enlarging to check the criteria related to the nipper’s jaw. The cameras take pictures of the product from different angles. Then the system will analyze the images to extract the related feature to evaluate the product quality. There are a total of 12 features extracted from the image captured by from sensor camera, such as the nipper length, handle, blade length, the clearance between two nipper blades, width, and thickness of nipper handles, the thickness of two nipper bearings, and so on. Besides, the optical illumination system is installed under a transparent conveyor and shines in the opposite direction through the conveyor towards the camera. The automatic visual inspection system in nipper manufacturing is displayed in Fig. 7. The system is operated as follows. Operator 1 puts the nippers on the conveyor in the correct direction. The system will adjust to make the nipper on the conveyor alignment. The sensor camera takes pictures automatically when the nipper passes through the inspection position. The image is transmitted to the processing system and the product features are extracted. Fig. 7 The automatic inspection system in nipper manufacturing The previous system at the company only detects a nipper to a defect or non-defective product. If a tested nipper is determined as a non-defective one, the product sorting mechanism is not active. Thus, the nipper will automatically transfer to the finished product tray. Otherwise, the product sorting mechanism is active to push a nipper on the other side of the conveyor and transfer it to the defect tray. However, this study considers not only detecting the defects but also finding the root cause of defects. Thus, the defects are classified into four groups based on their features as the result of the experiment. A classification model based on the clustering result in the first stage is developed to classify the tested nipper and determine exactly what type of defect the tested nipper is. The result is not only displayed on the screen but also has a sound warning to remind Operation 2 of the types of defects. Operator 2 has a responsibility to take the classified defect into the correct tray of defect types. 4. Conclusion Defect detection is critical for quality control in the company. By detecting the defects in the finished products, the company can prevent sending out defects to customers. Classifying the defects into different categories based on their features also helps to determine the defect’s causes. The management can utilize this information to make better decisions about addressing the defects or Advances in Production Engineering & Management 18(2) 2023 247 Nguyen, Yang, Le, Nguyen, Luu eliminating the factors that caused defects. Thus, the SCC method is utilized in this study to perform these works for the inspection process in the nipper factory in Vietnam. The DBSCAN clustering method is used to detect the defects in the finished products. Besides, a novel SCA-PFCM algorithm is proposed for defect categories. Then BPNN is employed to make an automatic inspection system. The result showed that the clustering and classification accuracy are all high to prove that the model is robust. The defects in the nipper were detected and classified into four different types based on the features of the defective products. Based on this result, an automatic defect detection model-based visual inspection was developed to help the factory improve the inspection process in terms of quality, time, and cost. There are some limitations in the proposed method that can be enhance for further research. First, the clustering stage is critical important and affect the result of the classification stage since its result is embedded in the classification process. Thus, improving the clustering result is necessary. The proposed SCA-PFCM only considered within cluster distance as the objective function in the clustering process. Future research can investigate mul-objective function This research can be extended to detecting the failures in the whole manufacturing process, not only in the finishing stage, as the recommendation by the manager. Besides, developing an algorithm to improve the quality of combining clustering and classification is also necessary. Acknowledgment This work was funded by Vingroup Joint Stock Company (Vingroup JSC) and supported by Vingroup Innovation Foundation (VINIF) under project code VINIF.2020.DA19. The support is much appreciated. The authors also thank Nghia Nippers Corp. for providing a dataset and support during the project. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] 248 Yun, J.P., Shin, W.C., Koo, G., Kim, M.S., Lee, C., Lee, S.J. (2020). Automated defect inspection system for metal surfaces based on deep learning and data augmentation, Journal of Manufacturing Systems, Vol. 55, 317-324, doi: 10.1016/j.jmsy.2020.03.009. Wang, T., Chen, Y., Qiao, M., Snoussi, H. (2018). A fast and robust convolutional neural network-based defect detection model in product quality control, International Journal of Advanced Manufacturing Technology, Vol. 94, No. 9, 3465-3471, doi: 10.1007/s00170-017-0882-0. Ko, T., Lee, J.H., Cho, H., Cho, S., Lee, W., Lee, M. (2017). 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Advances in Production Engineering & Management 18(2) 2023 249 Advances in Production Engineering & Management ISSN 1854-6250 Volume 18 | Number 2 | June 2023 | pp 250–262 Journal home: apem-journal.org https://doi.org/10.14743/apem2023.2.471 Original scientific paper Simulation and Genetic Algorithm-based approach for multi-objective optimization of production planning: A case study in industry Bojic, S.a, Maslaric, M.a,*, Mircetic, D.b, Nikolicic, S.a, Todorovic, V.c aFaculty of Technical Sciences, University of Novi Sad, Novi Sad, Serbia for Artificial Intelligence Research and Development of Serbia, Novi Sad, Serbia cScience and Technology Park Novi Sad, Novi Sad, Serbia bInstitute ABSTRACT ARTICLE INFO To stay competitive on the constantly changing and demanding market, production systems need to optimize their performance daily. This is particularly challenging in labour-intensive industries, which is characterized by highly volatile customer demand and significant daily variability of available workers. The Uncertainty related to the key production parameters in the industry is causing disruptions in long-term production planning and optimization, which leads to the long lead production times, operational risks and accumulation of inventory. To address these challenges, production systems need to ensure adequate operational production planning and optimization of all variables that are influencing the productivity of their systems on a daily basis. To tackle the problem, this study elaborates the application of discrete event simulations and genetic algorithm, using the Tecnomatix Plant Simulation software, to support decision-making and operational production planning and optimization in the industry. The simulation model developed for this purpose considers: customers demand changes, variable production times, operationally available resources and production batch size, to provide an optimal production sequence with the highest number of produced pieces and the lowest total work in process (WIP) inventory per day. To demonstrate the efficiency of the methodology and prove the benefits of the selected optimization approach, a case study is conducted in the textile factory. Keywords: Discrete event simulation (DES); Genetic algorithm (GA); Production planning; Multi-objective optimization; Textile industry; Tecnomatix Plant Simulation software *Corresponding author: marinko@uns.ac.rs (Maslaric, M.) Article history: Received 18 May 2023 Revised 8 June 2023 Accepted 28 June 2023 Content from this work may be used under the terms of the Creative Commons Attribution 4.0 International Licence (CC BY 4.0). Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. 1. Introduction To address the constantly growing customer demands the retailers put great pressure on the textile industry requesting frequent delivery of small quantities of a large number of different products [1, 2]. Satisfying such a stochastic demand with a high level of efficiency and low level of inventory requires effective operational planning and optimization of a production system, using new technologies that can support not only strategical, but also real-time, operational decision-making [3]. Many researchers investigated different influencing parameters which influence the efficiency of a production system, such as production process organization [4], product type [5], lot size scheduling [4] and operator’s skill level [2]. However, there has been little research and even less practical implementation related to the relationship between these parameters and their 250 Simulation and Genetic Algorithm-based approach for multi-objective optimization of production planning: A case study … common joint influence on the daily operational planning and optimization of a production system in the textile industry. In today’s industrial practice decision-making and operational planning still heavily rely on the production planners and their experience (expert knowledge) and future projections based on the data from the past. In those circumstances, without a proper digital tool, it is not possible to anticipate the impact of each influencing parameter and ensure optimal operational production planning and decision-making [6]. Considering the number, stochasticity and complexity of the influencing parameters, many researchers have used computer simulation, particularly discrete event simulation, to investigate the performance of production systems and to perform their optimization [2, 7-15]. High complexity of the optimization problem in production systems is also the reason why researchers are using genetic algorithms to reduce the computational time and ensure the quality of the obtained solution [16-18]. According to [19, 20], optimization solutions for production planning which satisfy the demand for increased volumes with an increased number of styles and personalized products, are a very actual topic in the textile industry. More specifically, over the past decades, several business and market trends have emerged that have reshaped the way the garment and textile industry is organized. This means that the textile manufacturers must be able to respond to a significant number of small and irregular orders, which would only be possible in the case of a more efficient and agile production system. This demonstrates the significant need for practical simulation models that would simultaneously consider the impacts of all the typical parameters relevant for the daily operational production planning, while supporting real-time decision-making with the minimal required computational time. This study presents an application of the discrete event simulations and genetic algorithm, using the Tecnomatix Plant Simulation software, to support decision-making and production planning and optimization in textile industry. The discrete event simulations enable complex systems representations including all relevant parameters, while the GA provides variation and sequencing of all influencing parameters in a short computational time required for the efficient operational planning. Although the approach is demonstrated in the textile industry, it can be equally efficiently used in other industry sectors. Of special interest are the labour intensive industries with the high variability of demand. To show the benefits of the proposed approach, a case study was analysed based on the validated simulation model of a textile factory. The simulation model considers following factors: the changing customers demand (variable in terms of different product types and quantity); the production times in relation to the product type and quantity; the operationally available resources and their distribution within the production system; as well as the production batch size; in order to provide the optimal production sequence which will result in the highest number of produced pieces per day with the lowest total work in process (WIP) inventory. In contrast to the papers using similar approach [19] for the strategical and tactical planning, this study investigates the benefits of the dynamic daily operational production planning, simultaneously considering the influence of all relevant variables. To fulfil this objective and contribute new knowledge, the paper is structured as follows. The next chapter provides a brief introduction into the research approach and literature review within the chosen topic. Chapter 3 formulates the case study and outlines simulation models for the company under consideration. Chapter 4 details the benefits of using genetic algorithm (GA) optimization in daily production planning, while in the fifth chapter a final discussion is presented, offering an overall summary and key findings. 2. Materials and methods 2.1 Research method In this study we used literature review not only to provide an overview of the use of optimization tools in operational production planning, but also to highlight a specific type of industry in Advances in Production Engineering & Management 18(2) 2023 251 Bojic, Maslaric, Mircetic, Nikolicic, Todorovic which additional research is needed. The literature review is followed by a case study as the research method suitable to understand how and why some phenomenon takes place in specific situations [21]. This review has shown that the textile industry still lacks sufficient research related to operational production planning optimization. Moreover, our case study showed that the textile industry certainly represents a type of industry in which the application of simulation models in operational production planning can contribute significant impact. A detailed analysis of the case study method can be found in [22]. Credibility of the case study is based on the transparency of the applied research, processes, and procedures, which allows verification by other researchers [23]. 2.2 Literature review Simulation is becoming one of the most used technologies to understand and analyse the dynamics of manufacturing systems [24]. It can be used as a descriptive tool to: realistically represent real-world systems, increase understanding of the relationships between the various components of a system; predict the performance of the system under new operating conditions; support the decision-making etc., without disrupting the ongoing production activities [25]. Simply, simulation is a problem-identification and solving tool that is flexible and less costly than physical prototyping and experimentation [26]. Computer simulation has been successfully used by many authors for planning and optimization of particular production lines or entire production and logistics systems in the textile industry. Simulation has been used for the shirt manufacturing production line design [27], for balancing the trousers production line [28], for balancing a sweat-shirt sewing line [29], for balancing the garment production line, in the five different scenarios, based on the production process times collected through RFID [10]. Some other authors used simulation for monitoring of the garment production line, comparing the production times gathered through the manual time taking and through the installed sensing [9]. The model-driven decision support system that is fed by real-time data, using data simulation and communication technologies was developed to improve the productivity of the manufacturing process of a garment manufacturing line [30]. Discrete event simulation (DES) model was proposed as the first step towards sustainable production scheduling in the textile manufacturing industry [6]. As DES is widely used to analyse activities of planning, implementation and operation of manufacturing and logistics systems, the commercially available DES software tools are commonly used for decision-making in different manufacturing systems [19]. Some studies that compared the different DES software tools recognized the Tecnomatix Plant Simulation by Siemens as the one with good visual aspects and the ability to integrate with other software [31]. Other studies have proven that Plant Simulation is a useful tool for optimal resource utilization [32]. Optimization problems which require minimization or maximization of functions with several variables and constraints can be very complex and difficult to solve with the use of conventional optimization methods. This is why they are commonly solved using algorithms based on the principle of evolution [2]. Some of the best-known meta-heuristic techniques, such as simulated annealing, the genetic algorithm (GA), and tabu search, were reviewed by some authors [33, 34] who concluded that these methods are remarkably effective in solving many types of optimization problems, particularly in finding near-optimal solutions for problems in complex multi-dimensional search spaces. Modified steady state GA was used for batch sizing and production scheduling in a hybrid flow shop with a limited buffer [35]. The genetic algorithm GA has been increasingly used for production optimization and operational management in the textile industry, such as loading, facility layout design, line balancing and lot sizing [2]. A genetic algorithm (GA) was successfully applied by different authors for optimizing the assignment of operatives in an assembly line [36] and for balancing workers’ walking along the assembly line [37]. The unrelated parallel machines scheduling problem, in the textile industry, with machine and sequence-dependent setup times and limited resources, was addressed by applying the genetic algorithm GA [38]. Some researchers proposed a multi-objective GA model to schedule multiple products to different assembling stages considering the total costs and manufacturing capacities. Based on the experimental results they concluded that the GA is more powerful than any exist- 252 Advances in Production Engineering & Management 18(2) 2023 Simulation and Genetic Algorithm-based approach for multi-objective optimization of production planning: A case study … ing methods for solving production planning and scheduling problems [39]. Other authors combined simulation with the GA for stochastic lot size scheduling problems to define sequencing and lot size rules, which maximize the expected profit per time unit, demonstrating that GA simulation is the right approach to be used in complex environments [40]. 3. Case study To demonstrate the GA-supported DES approach to daily operational production planning, with short computation time, the simulation model for a textile company was developed using the Tecnomatix Plant Simulation software. The model is developed and implemented in the company that produces high-quality shirts. The company was established over 20 years ago as a private family business and today is recognized as one of the best shirts manufacturing companies in the Republic of North Macedonia. The company produces more than 300,000 shirts annually and exports over 95 % of its production. The company’s production system includes following four departments: (1) Tailoring and cutting; (2) Preparation 1 (collars and cuffs); (3) Preparation 2 (front, back and sleeves); (4) Shirt assembly and final control. Each of these departments has a different number of workers and performs a different list of operations. The sequence of operations, as well as the process times, depend on the model of the shirt. The material flow within the departments is organized with the conveyors ensuring the first in first out (FIFO) principle. The material flows between the last operation in one department and the first operation in the next department are organized with the production carts which also play the role of the WIP inventory holders between the departments. Production is realized in batches whereas the batch size depends on the model of the shirt and the customer order quantity, however, the most common batch size contains 10 product pieces. 3.1 Previous reality Before the simulation model was developed, production planning in the company was performed by the production manager, using Excel and the data on the customer orders from their Enterprise Resource System (ERP). Based on the customer orders and the operational lists for each model of the shirt, the production work orders were prepared and printed out to physically follow the material flows through the production system. The work orders include the data about the product (type, colour, size, etc.), batch size, list of operations that the product needs to go through, and process time for each of these operations. There is no real-time tracking of the products status or location through the system before the production is completed. The operational production scheduling in each department is performed by the department managers, based on their knowledge and experience. However, it is not coordinated among the departments, which slows the material flow and creates an uncontrolled WIP inventory accumulation. The deep dive analysis of the production system revealed the following: unbalanced production between the departments; long production lead times, high WIP inventory, efficiency highly dependent on qualified labour availability and its distribution between the departments, no evidence or analysis on the influence of the production batch size and the production sequencing. The mentioned challenges are greatly conditioned with the daily variation of: product types and therewith the required production operations, demand-quantity of each product that needs to be produced and therewith the required production times per product, number of available workers in each department. Therefore, the simulation model was developed, as a supporting tool for the operational daily production planning that will ensure the highest possible production effectiveness while at the same time considering the influence of all relevant influential parameters, in a short computation time. 3.2 Way forward: Using simulation model Modelling and simulation start with the proper identification of the problem and specification of simulation objectives. Once the objectives are set, it is required to determine which fixed system parameters and variables are relevant for the observed system [22]. Advances in Production Engineering & Management 18(2) 2023 253 Bojic, Maslaric, Mircetic, Nikolicic, Todorovic The customer demands (type and number of products, as well as the delivery times) represent the main input data for this simulation model. The data are gained from the real production system, for the period of ten weeks between June and August 2022. The company collects the customer orders on a weekly basis and sets the production plan based on the FIFO principle without the optimization of production sequencing according to the type or the quantity of the demanded products. The input of these data in the simulation was modelled through the delivery table called Production Plan (Fig. 1). In the table Production Plan, the first column defines the time when the products are entering the production system. Here, the times in all rows are set to one second, meaning that the products will enter the production system according to the PUSH principle and the FIFO sequencing. The customer demands, representing the number and the type of products that need to be produced, are defined in columns three and four, respectively. Special attributes of the products are defined in column five where the main attribute is the batch size of the product. Fig. 1 Production plan representing weekly customer demands The production operations, including the real system process times, for every potential product type in each department, are defined in the table Production Times (Fig. 2). The production times are defined based on the real system measurements considering the type of the product, operations that the product is going through during production and process times of each operation. These production times are further recalculated in the model, in each simulation run, relative to the currently available number of workers in each department. Fig. 2 Production times for each product in each department 254 Advances in Production Engineering & Management 18(2) 2023 Simulation and Genetic Algorithm-based approach for multi-objective optimization of production planning: A case study … The numbers of available operators in each department are defined as variables (i, j, k, d) which are replicated in the simulation model according to the real system, as well as the variable s – representing the total number of available workers in all departments. The initial number of workers in the department preparation 1, was set to I = 20, the number of workers in the department preparation 2, was set to j = 40, the number of workers in the assembly and final control department was set to k = 40, while the number of workers in the tailoring and cutting department was set to d = 4. Consequently, s was initially set to 104 and changed in some simulation runs according to the exact number of available workers on a particular day. Fig. 3 Simulation model of the case study textile factory The simulation model developed for this case study (Fig. 3) is based on the steps proposed by [21] and followed by other researchers [6, 15]. This approach ensures that the simulation model has all the necessary characteristics which allow it to be used as a decision-making tool. In the first step, to ensure that the simulation model can be used as a production planning and decision-making tool, it was important to verify the modelling logic and validate the model results with the real production data. Verification, in this case, considered the use of low-speed graphic animation within the Tecnomatix Plant Simulation software, ensuring that the simulation model has the same logic of movement as the real system. Validation was performed by comparing the simulation results of the selected production operating days (the first day in each of the ten observed weeks) with the existing analytical data of the real system performance in those exact operating days. The simulation results of all simulation runs were within the 95 % confidence interval of the real system performance (Table 1) thus validating the accuracy and reliability of the simulation approach. In the second step, the genetic algorithm was introduced in the model to perform optimization and increase production efficiency with adequate daily operational production planning and sequencing in comparison to the FIFO principle used in the real system. 3.3 New reality: Operational production planning and optimization with the simulation model using GA Once the verification and validation of the simulation model were performed, a GA Wizard was implemented in the model, to provide optimization of the system operational production planning. Two opposite objectives were set for the GA optimization: • Maximize the number of produced products per day, considering the customer weekly demand and currently available number of workers, and • Minimize the total WIP inventory, in order to reduce the product lead-time and to reduce the operational risks of having high inventory levels. Advances in Production Engineering & Management 18(2) 2023 255 Bojic, Maslaric, Mircetic, Nikolicic, Todorovic The optimization with the GA Wizard considers: • Production sequence of work orders generated according to the customer demand, considering the weekly demand and daily available production resources; • Distribution of available workers between the departments, considering the distribution limits that are defined based on the worker's skills; • Optimal batch size, considering the defined limits that are defined based on the literature and the company’s production planners’ experience; • Total WIP inventory level, considering the capacity limits that are defined based on the targeted inventory levels, to achieve the highest production efficiency and throughput with the lowest possible WIP inventory. It is hereby anticipated that the increased throughput in the system with the low WIP inventory will positively influence the product lead-time. Fig. 4 Defined limits for the distribution of available workers and for the buffer capacities in each department The total number of available workers in the real system is 104, however, it varies daily. Therefore, the variable s was initially set to 104, and adjusted in each simulation run to the number of workers that came to work on that exact date. The limits for the distribution of the available workers in each of the departments (variables i, j, k, d) were set to: 15 < i < 25; 35 < j < 45; 35 < k < 45; 2 < d < 4; with an increment of one (Fig. 4). The limit for the batch size was set 1 < batch < 10, with an increment of one. The limit for the buffer after the department of tailoring and cutting was set to 2000 with an increment of 100, while for all other buffers between the other departments was set to 450, with an increment of 50 (Fig. 4). The GA Wizard performed work orders (representing customer demands) sequencing while varying the number of workers in each department and varying the capacity of the available buffers, within the defined limits, to find the right sequence which will ensure the best fitness value considering the defined weightings of each of the defined objectives. To model the opposing objectives, a variable called MinWIP was introduced and defined in an EndSim Method (Fig. 5). In the method, the MinWIP is calculated as the deviation between 10.000 and total WIP inventory between all production departments. Fig. 5 EndSim Method, calculating the MinWIP and penalizing solutions that are not good enough to be considered 256 Advances in Production Engineering & Management 18(2) 2023 Simulation and Genetic Algorithm-based approach for multi-objective optimization of production planning: A case study … This formulation enables the minimization of the total WIP inventory to be reformulated into the maximization of the MinWIP variable (the higher the deviation, the lower the total WIP inventory) which enables the search for the optimal solution by maximizing both variables (the number of finished products and MinWIP) in the GA Wizard (Fig. 6). Additionally, the EndSim method is penalizing the solutions in which the number of produced products per day is less than 800, by simply setting the value of the MinWIP variable to one. In that way, the search for the optimal solution from the point of both objectives is facilitated. The parameters used during the Tecnomatix GA Wizard optimization were: (i) one hundred generations; (ii) the size of generation is ten individuals; (iii) the elitist system was used for the selection, where the best solutions were used to generate offspring for the next generation; (iv) multi-objective optimization, and (v) weighting of the optimization objectives (Fig. 6). The best optimization solution was searched as the best fines value expressed through the weighting of the two defined objectives, where the weights of the first and second objectives were based on the preferences of the company’s production manager and were set to 0.7 and 0.3, respectively (Fig. 6). Fig. 7 shows the evolution of optimized solutions against the number of generations, confirming that one hundred generations are sufficient for obtaining the optimization convergence in these simulation runs. The developed model was tested for the daily operation of the first working day in each of the ten weeks. The main output of each simulation run consisted of the sequence of the production work orders (Fig. 8), distribution of workers to the production departments (Table 2) and the optimal batch size (Table 3). Fig. 8 shows the GA optimization results (of the 5. simulation run): production buffers’ required capacity, workers’ distribution, batch size and production sequence. This case study does not belong to computationally complex ones. Therefore, an average computation time of the ten simulation runs, for the defined parameters and limits, was 25 seconds. The computation time increases significantly with the problem complexity (in relation to the number and type of variables analysed in the system, their range and increment, as well as in relation to the number and size of generations of the GA). However, it can be expected that this approach can perform production optimisation and operational production planning of medium complex problems in just a few minutes. Fig. 6 Optimization parameters set in the GA Wizard of the Tecnomatix Plant Simulation Advances in Production Engineering & Management 18(2) 2023 257 Bojic, Maslaric, Mircetic, Nikolicic, Todorovic a) b) c) d) Fig. 7 Typical performance graphs of the ten performed simulation runs (a: 1, b: 5, c: 7, d: 10) Fig. 8 GA optimization results, workers distribution, production buffers' capacity and production sequencing 4. Discussion The simulation results of both developed models are presented in Table 1, where the basic one was used for the verification and validation of the simulation approach, while the second was integrated with the GA. Comparison of the real system data and the simulation results without GA, undoubtedly verify the relevance and accuracy of the simulation model. Comparison of the real system and the simulation results with the GA demonstrate great optimization potential of simulation and genetic algorithm for the operational production planning and sequencing (Fig. 9). 258 Advances in Production Engineering & Management 18(2) 2023 Simulation and Genetic Algorithm-based approach for multi-objective optimization of production planning: A case study … Table 1 Real system vs. simulation results – daily production and WIP in the selected days, in number of pieces Real system Simulation Simulation with GA Days Production WIP Production WIP Production WIP 1 1098 6654 1110 6588 1212 499 2 1000 4889 980 4958 1073 300 3 1153 8185 1130 8243 1310 391 4 1104 5154 1110 5128 1195 315 5 1245 5048 1260 4988 1268 434 6 1137 6253 1120 6348 1164 541 7 1117 5899 1100 5989 1160 409 8 1202 7742 1188 7758 1283 350 9 1031 6553 1020 6566 1204 537 10 1135 9235 1150 9099 1300 360 Fig. 9 Daily production – comparison of the real system analytics and simulation results Fig. 10 Effects of the GA optimization As shown in Fig. 10, the use of GA for the production sequencing can lead to an average increase in productivity between 8.5 % and 17 %. However, even more significant savings can be achieved through the reduction of WIP (Fig. 11). According to the simulation results for the selected days, the operational production planning and sequencing with the use of GA and limitations of the WIP inventory, can lead to an average reduction of the total WIP between 93 % and 96 % (Fig. 10). This reduction is the result of the adequate production sequencing as well as the optimal distribution of the available workers between the departments and the optimization of batch size. In the real system, the available workers were always distributed in the same way. Contrastingly, in the simulation model, the GA optimization proposed different workers’ distribution between the departments in each simulation run (Table 2). Fig. 11 Total WIP – comparison of the real system analytics and simulation results Advances in Production Engineering & Management 18(2) 2023 Fig. 12 Comparison of the mean production lead times for the simulated products (min) 259 Bojic, Maslaric, Mircetic, Nikolicic, Todorovic Table 2 Distribution of available workers between the departments in real system and in each simulation run Real system Simulation with GA s i j k d i j k d 1 104 20 40 40 4 22 37 43 2 2 96 18 36 40 2 20 37 36 3 3 104 20 40 40 4 22 39 39 4 4 104 20 40 40 4 25 40 36 3 5 104 20 40 40 4 21 37 44 2 6 99 20 38 37 4 21 37 39 2 7 95 17 39 35 4 22 35 36 2 8 104 20 40 40 4 24 38 39 3 9 104 20 40 40 4 24 38 39 3 10 104 20 40 40 4 25 35 41 3 Day 1 2 3 4 5 6 7 8 9 10 Table 3 GA optimization of batch size and its influence on assortment size Real system GA optimization Batch size Assortment size Batch size Assortment size 10 5 1 6 10 3 1 6 10 5 6 4 10 5 5 6 10 7 1 7 10 7 1 6 10 1 1 4 10 4 1 6 10 5 3 5 10 3 10 4 Table 3 shows GA optimization of production batch size in comparison to the constant batch size that is determined in the real system. The simulation results of the assortment size (Table 3) prove that the increase of productivity through the GA optimization is not due to the assortment size reduction, as one might suppose, but a consequence of the optimal sequencing, as well as the resource and material flow distribution. Furthermore, the simulation results proved that such an extreme reduction of the WIP and optimization of the batch size, as expected, positively influenced the product lead time. Based on the simulation results, the reduction of product lead time was more than 3 times (from average 318 min to average 95 min). The mean value of the lead time reduction, per day for all products, was between 70 % and 78 % (Fig. 12). These results prove and verify the effectiveness of the DES and GA approach for operational production planning and optimization in the case study. 5. Conclusions The textile industry is challenged by highly stochastic and unpredictable demand, as well as specifically stochastic production performance caused by the predominantly manual operation and low level of digitalization, which are resulting in high levels of WIP inventory and long product lead-time. At the same time, the production planning and resource utilization optimization is based on outdated data and the experience of the production planners. This is not specific to just the textile industry, but is common in other industries as well, particularly the labour intensive ones with the low level of automation and digitalization. The approach proposed within this case study investigates the possibility to improve production performance in industry by improving operational production planning and optimization. The approach considers the application of discrete event simulation and genetic algorithm, to perform the operational production sequencing of the customer orders, distribute the available workers to the work departments, and determine the optimal batch size, in a way to maximize productivity and minimize the WIP inventory and product lead time. The proposed approach was tested using the Tecnomatix Plant Simulation software to eliminate the weaknesses of the manual scheduling procedures based on the judgment and experience of the production planners. Although it was applied to the case 260 Advances in Production Engineering & Management 18(2) 2023 Simulation and Genetic Algorithm-based approach for multi-objective optimization of production planning: A case study … study of the shirts production company, the proposed approach is not limited to the textile industry but could be efficiently applied in other industry sectors. The results of this case study validated the ascendency of the simulation approach in operational planning and optimization, as well as the high efficiency of the genetic algorithm. Future steps in the research and practice should consider integration of the simulation software with the existing ERP system to ensure real-time data consideration, as well as with the potential MES (Manufacturing Execution System) which would transform the simulation model into the production system digital shadow. In that way, the simulation model would become a real-time dynamic operational planning tool with the capacity to perform not only the daily but also the real-time production scheduling and optimization. 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APEM Advances in Production Engineering & Management journal Chair of Production Engineering (CPE) University of Maribor APEM homepage: apem-journal.org Volume 18 | Number 2 | June 2023 | pp 133-266 Contents Scope and topics 136 Real-time scheduling for dynamic workshops with random new job insertions by using deep reinforcement learning Sun, Z.Y.; Han, W.M.; Gao, L.L. 137 Genetic algorithm-based approach for makespan minimization in a flow shop with queue time limits and skip-ping jobs Han, J.H.; Lee, J.Y. 152 Hybrid forecasting modelling of cost and time entities for planning and optimizing projects in the die-cast aluminium industry Muñoz-Ibañez, C.; Chairez, I.; Jimenez-Martinez, M.; Molina, A.; Alfaro-Ponce, M. 163 When core sorting and quality grading is beneficial to remanufacturers: Insights from analytical models Cao, X.H.; Shi, X.L.; Lan, H.J.; Huang, D. 175 Engineering-to-order manufacturing: A criticality analysis of key challenges and solutions based on literature review Fortes, C.S.; Tenera, A.B.; Cunha, P.F.; Teixeira, J.P. 187 Ranking dominant losses in small and medium-sized enterprises (SMEs) in the context of the lean concept application Kondić, V.; Maglić, L.; Runje, L.; Marić, D. 199 An improved discrete particle swarm optimization approach for a multi-objective optimization model of an urban logistics distribution network considering traffic congestion Li, K.; Li, D.; Ma, H.Q. 211 Blockchain-based tripartite evolutionary game study of manufacturing capacity sharing Wang, T.Y.; Zhang, H. 225 Enhancing automated defect detection through sequential clustering and classification: An industrial case study using the Sine-Cosine Algorithm, Possibilistic Fuzzy c-means, and Artificial Neural Network Nguyen, T.P.Q.; Yang, C.L.; Le, M.D.; Nguyen, T.T.; Luu, M.T. 237 Simulation and Genetic Algorithm-based approach for multi-objective optimization of production planning: A case study in industry Bojic, S.; Maslaric, M.; Mircetic, D.; Nikolicic, S.; Todorovic, V. 250 Calendar of events 263 Notes for contributors 265 Published by CPE, University of Maribor ISSN 1854-6250 9 771854 625008 apem-journal.org