ISSN 1854-6250 APEM journal Advances in Production Engineering & Management Volume 16 | Number 2 | June 2021 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. 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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 16 | Number 2 | June 2021 | pp 141–264 Contents Scope and topics Hybrid ANFIS‐Rao algorithm for surface roughness modelling and optimization in electrical discharge machining 144 145 Agarwal, N.; Shrivastava, N.; Pradhan, M.K. A multi‐objective optimal decision model for a green closed‐loop supply chain under uncertainty: A real industrial case study 161 Fang, I.W.; Lin, W.‐T. Improved Genetic Algorithm (VNS‐GA) using polar coordinate classification for workload balanced multiple Traveling Salesman Problem (mTSP) 173 Wang, Y.D.; Lu, X.C.; Shen, J.R. Change impact analysis of complex product using an improved three‐parameter interval grey relation model 185 Yang, W.M.; Li, C.D.; Chen, Y.H.; Yu, Y.Y. Bone drilling with internal gas cooling: Experimental and statistical investigation of the effect of cooling with CO2 on reduction of temperature rise due to drill bit wear 199 Shakouri, E.; Haghighi Hassanalideh, H.; Fotuhi, S. Joint distribution models in fast‐moving consumer goods wholesale enterprise: Comparative analysis and a case study 212 Wang, L.; Chen, X.Y.; Zhang, H. Designing a warehouse internal layout using a parabolic aisles based method 223 Zhang, Z.Y.; Liang, Y.; Hou, Y.P.; Wang, Q. Optimization of disassembly line balancing using an improved multi‐objective Genetic Algorithm 240 Wang, Y.J.; Wang, N.D.; Cheng, S.M.; Zhang, X.C.; Liu, H.Y.; Shi, J.L.; Ma, Q.Y.; Zhou, M.J. Modelling and optimization of sulfur addition during 70MnVS4 steelmaking: An industrial case study 253 Kovačič, M.; Lešer, B.; Brezocnik, M. Calendar of events Notes for contributors Journal homepage: apem‐journal.org ISSN 1854‐6250 (print) ISSN 1855‐6531 (on‐line) Published by CPE, University of Maribor. 262 263 Scope and topics Advances in Production Engineering & Management (APEM journal) is an interdisciplinary refer‐ eed 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 engineer‐ ing 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 contribu‐ tions are the main factors in the evaluation process. General approaches, formalisms, algorithms or techniques should be illustrated with significant applications that demonstrate their applica‐ bility to real‐world problems. Although the APEM journal main goal is to publish original re‐ search 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 144 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 16 | Number 2 | June 2021 | pp 145–160 Journal home: apem‐journal.org https://doi.org/10.14743/apem2021.2.390 Original scientific paper Hybrid ANFIS‐Rao algorithm for surface roughness modelling and optimization in electrical discharge machining Agarwal, N.a,*, Shrivastava, N.a, Pradhan, M.K.b a Department of Mechanical Engineering, UIT, Rajiv Gandhi Proudyogiki Vishwavidyalaya, Bhopal, India Department of Mechanical Engineering, Maulana Azad National Institute of Technology, Bhopal, India b ABSTRACT ARTICLE INFO Advanced modeling and optimization techniques are imperative today to deal with complex machining processes like electric discharge machining (EDM). In the present research, Titanium alloy has been machined by considering different electrical input parameters to evaluate one of the important surface integrity (SI) parameter that is surface roughness Ra. Firstly, the response surface methodology (RSM) has been adopted for experimental design and for generating training data set. The artificial neural network (ANN) model has been developed and optimized for Ra with the same training data set. Finally, an adaptive neuro‐fuzzy inference system (ANFIS) model has been developed for Ra. Optimization of the developed ANFIS model has been done by applying the latest optimization techniques Rao algorithm and the Jaya algorithm. Different statistical parameters such as the mean square error (MSE), the mean absolute error (MAE), the root mean square error (RMSE), the mean bias error (MBE) and the mean absolute percentage error (MAPE) elucidate that the ANFIS model is better than the ANN model. Both the optimization algorithms results in considerable improvement in the SI of the machined surface. Comparing the Rao algorithm and Jaya algorithm for optimization, it has been found that the Rao algorithm performs better than the Jaya algo‐ rithm. Keywords: Electrical‐discharge machining (EDM); Titanium alloy; Surface roughness; Modelling; Optimization; Artificial neural networks (ANN); Adaptive neuro fuzzy inference system (ANFIS); Rao algorithm; Jaya algorithm *Corresponding author: neeraj.bhopal@gmail.com (Agarwal, N.) Article history: Received 27 October 2021 Revised 14 March 2021 Accepted 15 May 2021 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 Electrical discharge machining (EDM) is used to machine materials that are difficult to machine [1‐2]. In EDM, sparking takes place between the workpiece and tool electrode in the machining zone and due to sparking the removal of material takes place by melting and evaporation [3‐4]. Shrivastava et al. reviewed the EDM‐based hybrid machining process to enhance the different machining performance [5]. Titanium alloy possesses excellent mechanical properties, but it has poor machinability towards conventional machining [6]. Qudeiri et al. did an extensive review on EDM of the Titanium alloy and provided valuable research directions [7]. Kumar et al. pre‐ sented an extensive review of the EDM machining of Titanium alloy and discussed important qualitative and quantitative aspects [8]. To control the EDM process, a wide range of control parameters are available. Control parameters and their range significantly affect the quality re‐ 145 Agarwal, Shrivastava, Pradhan sponse and hence the appropriate selection of control parameters is necessary to improve the process efficiency [9]. The most important consistency indicator of the machining method is sur‐ face roughness. The reliability, wear, and corrosion resistance and the substance are enhanced by good surface integrity [10‐11]. After review of the literature, four important control parame‐ ters, peak‐current (X1), pulse‐on‐time (X2), duty‐factor (X3), and voltage (X4) selected for the re‐ search [12‐13]. Response surface methodology (RSM) offers a minimum number of experiments required and inherits enormous information [14‐15]. It is also used as an optimization tool for the process. RSM uses a sequence for the design of experiments (DOE) to gather information and establish a relation between input and output with a second‐degree polynomial regression model [16‐17]. A model depicts the relation between control parameters with one or more quality responses [18]. RSM is the most popular tool for the development of the regression model and as an opti‐ mization tool, but the regression model is not suitable for noisy data. This leads to an inefficient model with higher prediction errors [19]. Artificial intelligence is widely adopted in manufactur‐ ing industries nowadays. The ANN is a popular modeling technique [20]. Researchers concluded that the ANN model has higher performance as compared to the regression model [21‐22]. The ANFIS model combines both the fuzzy set and the ANN model. The ANFIS is gaining popularity as a modeling tool [23]. Buragohain et al. used a full factorial design to train the ANFIS model [24]. The ANFIS model is trained by gradient descent or least square estimation. Now a day, heuristic optimization algorithms like the Genetic Algorithm (GA), and many other tools are widely used to train the model to minimize error. It is found that a heuristic‐based algorithm produces better results [25‐26]. The ANN model and the ANFIS model have a better prediction efficiency as com‐ pared to the regression model [27]. The ANFIS and the ANN model have approximately similar accuracy [28]. Generally, large experimental data is used as training data. To minimize the train‐ ing data requirement, RSM is adopted as an experimental design, and for generating the training data set. Model of any process generally used for prediction purposes. Researchers are interested to find an optimum solution of the process output, to effectively utilize the resource [29]. A predict‐ able model is used in optimization where one or more response is optimized with an optimiza‐ tion technique. Classical optimization is not able to solve complex problems [30]. To overcome the limitation of classical optimization, metaheuristic or advanced optimization is used. Particle swarm optimization (PSO), GA, ACO, is the most popular metaheuristic optimization algorithm [31]. Klancnik ae al. proposed a new gravitational search algorithm for optimization [32]. Jaya algorithm is a new advanced optimization tool. In this, algorithm‐specific parameters are not required. The solution moves towards the best solution iteratively [33‐34]. Rao et al. proposed MO Jaya algorithm for multi‐response optimization [35]. Singh et al. applied MO Jaya algorithm for simultaneous optimization of three responses [36]. Al‐Refaie et al. used pentagonal fuzzy regression modeling to optimize multiple process measures [37]. Payal et al. did the parametric optimization investigation for EDM of Inconel 825 with seven controllable parameters [38]. Daneshmand et al. investigate the effect of input parameters on surface roughness during EDM of CK45 steel [39]. Rao optimization is a newly proposed, simple, and powerful optimization tool for technique engineering optimization. Rao optimization is a metaphor‐less algorithm [40]. RSM is a widely used optimization tool, but for noisy data, it is not appropriate. Rao reviewed an ad‐ vanced optimization technique used in modern machining processes [41]. Recently, most re‐ searchers used RSM as a modeling and optimization tool. The ANN model is also optimized by advanced optimization by researchers with a better result. In the literature review, advanced optimization is limited to train the ANFIS model only [42]. Singh et al. used an integrated GA‐ ANFIS model where GA is adopted for the training of the ANFIS network [43]. There is no work reported on the ANFIS model optimization using advanced optimization for surface roughness or EDM machining. This is a major research gap. After an exhaustive literature survey, the authors concluded that there is no work reported in the EDM machining process where the ANFIS model is optimized for quality measures by an advanced optimization technique. So, in this work, the ANFIS trained model is optimized for one of the major quality response i.e. surface roughness (Ra). The ANFIS offers an accurate, predicta‐ 146 Advances in Production Engineering & Management 16(2) 2021 Hybrid ANFIS‐Rao algorithm for surface roughness modelling and optimization in electrical discharge machining ble model for complex machining processes like EDM. This ANFIS model is optimized by the Rao algorithm and Jaya Algorithm for better optimum results. The Rao algorithm is suitable to opti‐ mize the ANFIS model under 50 iterations. The ANN model for the EDM process is also devel‐ oped and optimized using Rao optimization. ANFIS model offers better prediction results as compared to the ANN model. 2. Experimental setup For the study, a plate of titanium alloy 685 was chosen. As an electrode, a copper tool with a di‐ ameter of 10 mm is picked. Minitab 18 software is used to implement the RSM design of experi‐ ments. Four important control parameters, peak current (X1), pulse on time (X2), duty factor (X3), and voltage (X4) are selected as a control parameter, surface roughness (Ra) is selected as pro‐ cess outcome. The thirty machining operation is performed as per DOE. Table 1 depicts parame‐ ters for process control and their level. Table 2 displays the experimental observation. Surface roughness is measured with profilometer “TESA‐rugosurf 10‐G” having 0.001 µm accuracy. Parameters Level 1 Level 2 Level 3 No. 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 26 27 28 29 30 Table 1 Process control parameters and their level X1 (A) X2 (µs) X3 (%) 4.0 50.0 25.0 6.0 100.0 37.5 8.0 150.0 50.0 X1 (A) 8 8 4 8 6 6 6 4 4 4 4 4 4 8 6 8 8 8 4 8 6 6 6 6 6 6 6 8 6 4 Table 2 Experimental observations X2 (µs) X3 (%) 150 50 50 25 50 25 50 50 100 37.5 100 37.5 100 37.5 50 50 150 50 150 25 150 25 50 50 50 25 50 50 100 37.5 150 50 150 25 150 25 150 50 50 25 50 37.5 100 37.5 100 37.5 100 50 100 25 100 37.5 150 37.5 100 37.5 100 37.5 100 37.5 X4 (V) 40 40 40 40 70 70 70 40 40 100 40 100 100 100 70 100 100 40 100 100 70 40 100 70 70 70 70 70 70 70 X4 (V) 40.0 70.0 100.0 Ra (µm) 5.134 3.152 2.952 5.294 3.784 3.815 3.746 3.267 4.202 3.954 4.468 4.138 3.274 6.637 3.812 5.260 4.480 3.993 3.880 4.267 3.581 3.716 4.092 4.334 3.543 3.727 3.632 4.129 3.872 3.506 3. Modeling Model is the relation between control parameters and process outcomes. Nowadays the intelli‐ gent model is popular for modeling complex processes. An artificial neural network (ANN) is an interconnected network of neurons. The adaptive neuro‐fuzzy inference system (ANFIS) model combines a fuzzy set into ANN, thus it inherits the advantages of fuzzy systems and the ANN. Advances in Production Engineering & Management 16(2) 2021 147 Agarwal, Shrivastava, Pradhan 3.1 The ANFIS model The ANFIS model is developed to predict the surface roughness (Ra). Fig. 1 depicts the ANFIS architecture. y Fig. 1 The adaptive network‐based fuzzy interface system architecture There are two inputs in the model: x, y, and one output. Both fuzzy logic and ANN are merged into the ANFIS scheme. Sugeno fuzzy model first‐order, there are two If‐then laws used as Rule 1: If x is A1 and y is B1, then Rule 2: If x is A2 and y is B2, then The (lz, mz, and nz) is a set of parameters related to the process. Layer 1 is known as the fuzzified layer and it calculates the membership function (MF) value of the premise parameter. Every node is squarely in this layer. ‘Trimf’ and ‘tarpmf’ are popular MFs for the experimental data. μ for i = 1,2 or , μ for i = 3,4 , μ (1) The (ai, bi, and ci) is a set of parameters. Layer 2 is considered as a layer of law. The second layer reflects the shooting force of the rules. Every node is a circle in this layer and labeled as П. T‐norms operator (like min, product, fuzzy AND) rule applies in this layer. The response of the second layer is given in Eq. 2. μ , ∗μ for i = 1, 2 (2) Layer 3 is known as normalizing firing strength. Every node is a circle in this layer and la‐ beled as N. The i‐th node calculates the ratio of the i‐th rule's firing strength to the sum of all rules firing strengths. It is also known as normalized firing strength as follows. where i = 1,2 , (3) Layer 4: Every node i is an adaptive node with a function as follows (4) , Where i = 1, 2 and wi is a normalized firing strength from layer 3 and (li, mi, ni) is the set of pa‐ rameters for this particular node. This layer computes the contribution of each i‐th rules to‐ wards the total output Layer 5 is known as overall output. This is labeled as ∑. By adding up all incoming signals, de‐ termines the total performance. Overall output is given below , 148 ∑ ∑ ∑ where i = 1, 2. (5) Advances in Production Engineering & Management 16(2) 2021 Hybrid ANFIS‐Rao algorithm for surface roughness modelling and optimization in electrical discharge machining Matlab 2015a is used to form the ANFIS model. There are several MFs and MF type is availa‐ ble for training for the ANFIS model. Table 3 shows the ANFIS training parameters. These pa‐ rameters are decided after trial and error to minimize the error. Fig. 2 shows The ANFIS model for this study. Fig. 3 depicts the error curve for the ANFIS model training. All input is connected to ‘inputmf’. There is a rule layer between ‘inputmf’ and ‘outputmf’. All output signals from ‘out‐ putmf’ is summed at the junction to produce output. For X1 and X3, Fig. 4 displays the surface plot. The surface plot for X1 and X4 is depicted in Fig. 5. The surface plot for X3 and X4 is shown in Fig. 6. Table 3 ANFIS training parameters selected ANFIS parameter Selected parameter Membership function (MFs) trimf (Triangular membership function) Numbers of Mfs 2222 Numbers of epochs 1000 Output MF function Linear Training function Hybrid Fig. 2 The ANFIS architecture with four inputs and one output Fig. 3 Error curve for the training of the ANFIS model Advances in Production Engineering & Management 16(2) 2021 149 Agarwal, Shrivastava, Pradhan Fig. 4 The surface plot for surface roughness where output is Ra, input1 is X1, input3 is X3 (hold values X2 = 100 µs and X4 = 70 V) Fig. 5 Surface plot for surface roughness where output is Ra, input1 is X1, input4 is X4 (hold values X2 = 100 µs and X3 = 37.5 %) Fig. 6 Surface plot for surface roughness where output is Ra, input3 is X3, input4 is X4 (hold values of X2 is 100 µs and X1 = 6A) 3.2 The ANN Model Several interconnected neurons form an ANN network. There are an input layer, an output layer and a hidden layer exist in an ANN network. The input layer and output layer are connected by few hidden layers. The cumulative number of neurons in the input layer equals to the number of input. There are four input parameters, hence input layer has four neurons. The output layer's number of neurons is equal to the model's output. There is one output, hence output layer has a single neuron. After experimenting with various combinations, it is found that a single hidden layer with the five neurons generates a better predictive model. The activation function 'logsig' is used between the input layer and hidden layer. The ‘purelin’ transfer function is used for the output layer. The activation function ‘purelin’ is used with the output layer. To train the network, the back‐ propagation algorithm is adopted. It is based on gradient descent. It updates the weight of the layer to minimize the mean square error (MSE). The network is trained using the experimental data from Table 2. Fig. 7 shows the ANN architecture. 150 Advances in Production Engineering & Management 16(2) 2021 Hybrid ANFIS‐Rao algorithm for surface roughness modelling and optimization in electrical discharge machining Fig. 7 The ANN architecture Input layer: There are four inputs in the model, hence four neurons present in this layer. Hidden Layer: Every neuron of the hidden layer receives input from all neurons of the input lay‐ er. It is multiplied by weight wij with addition to bias bj. The net input received by each neuron as the following Eq. 6. (6) Where j is a number of neurons in the hidden layer, xi is input quantity, wij is weight of i‐th input to j‐th neuron, bj is bias of j‐th neuron. The output from the input layer’s neurons is fed into the activation function ‘logsig’ which converts the input signal into the output of the given neuron. The net summation of the j‐th neu‐ ron is fed into activation function ‘logsig’ and the output of j‐th neuron is given in the following Eq. 7. 1 (7) 1 Output layer: There is 1 neuron in this layer. The output from each of the hidden layer's neu‐ rons is now multiplied by the individual weight. The signal from all neurons goes to the summa‐ tion junction with an addition of the bias, then fed into activation function ‘purelin’ to generate the final output (Ra). The output is represented in the following Eq. 8, where b is the bias for the output. (8) 3.3 The evaluation criteria There is a statistical tool used to calculate the error and efficiency of the model. MSE measures the average of the square of the errors, MAE is the average of the absolute errors, RMSE is the square root of MSE, MBE is the average value of errors, MAPE is the mean absolute percentage error, whereas NSE is the efficiency of the model. ∑ (9) (10) ∑ ∑ | Advances in Production Engineering & Management 16(2) 2021 | (11) 151 Agarwal, Shrivastava, Pradhan (12) ∑ ∑ 1 (13) | | ∑ ∑ (14) Where Ei is observed value, Pi is predicted value from the model, n is total number of experi‐ ments, is mean of the observed value. 4. Optimization 4.1 Rao optimization for the ANFIS model The ANFIS model is made with Matlab software. A Matlab program is written to optimize the ANFIS model. The flow chart is given in Fig. 8. The objective is to minimize the surface roughness (Ra). Step 1: All variables are randomly generated within the upper bound and lower bound as an initial solution Constraints are 4 ≤ X1 ≤ 8, 50 ≤ X2 ≤ 150, 25 ≤ X3≤ 50, 40 ≤ X4 ≤ 100. Initial solu‐ tions are simulated in the ANFIS model and corresponding Ra is recorded in Table 4. Candidate 1 2 3 4 5 Table 4 Initial solution for Rao optimization. X2 (µs) X3 (%) X4 (V) 95.14 42.6479 80.5815 106.4608 32.7404 42.4912 100.0375 49.1606 94.5473 144.7569 46.2096 73.229 51.6323 27.4826 90.5408 X1 (A) 7.6672 5.6282 7.6384 7.4458 4.9393 Ra (µm) 4.3819 3.7440 4.8520 4.6217 3.0694 Status worst best Step 2: New value of the variable is calculated as follows the formula, Eq. 15. ′, , , , , , , , , , (15) There are a number of iterations required to solve this optimization problem. If i is current it‐ eration, j represents for j‐th variable, k represents for candidate number. Where X’j,k,i is new val‐ ue of variable , Xj,k,i is old value of variable, r1,j,i is random number of variable j, Xj,best,i is best value j‐th variable k‐th candidate at i‐th iteration, Xj,worst,i is worst value j‐th variable k‐th candidate at i‐ th iteration. Candidate 5 is chosen as the best candidate since it has the lowest Ra value and all variables are marked as best variable as X1_best is 4.9393, X2_best is 51.6323, X3_best is 27.4826, X4_best is 90.5408. Candidate 3 has highest value of objective function Ra, hence it selected as worst candidate and corresponding variables are marked as worst candidate as X1_worst is 7.6384, X2_worst is 100.0375, X3_worst is 49.1606, X4_worst is 94.5473. There are four random number r1 = 0.3135, r2 = 0.5200, r3 = 0.1954, r4 = 0.0131 for X1, X2, X3, and X4 respectively. The new calculated value of variables are shown in Table 5. The new value of the variable for candidate 1 is: 6.8211, X1 for candidate 1: 7.6672 0.3135 4.9393 7.6384 69.9679, X2 for candidate 1: 95.14 0.5200 51.6323 100.0375 38.413, X3 for candidate 1: 42.6479 0.1954 27.4826 49.1606 80.5289. X4 for candidate 1: 80.5815 0.0131 90.5408 94.5473 Similarly, candidates 2, 3, 4, and 5 are calculated. All candidates to the ANFIS model simulated for Ra value and the corresponding Ra value is recorded in Table 5. 152 Advances in Production Engineering & Management 16(2) 2021 Hybrid ANFIS‐Rao algorithm for surface roughness modelling and optimization in electrical discharge machining Table 5 New value of variables for Rao optimization X1 (A) X2 (µs) X3 (%) X4 (V) 6.8211 69.9679 38.413 80.5289 4.7821 81.2887 28.5056 42.4386 6.7922 74.8654 44.9257 94.4947 6.5996 119.5847 41.9748 73.1764 4.0931 50 25 90.4882 Candidate 1 2 3 4 5 Ra (µm) 4.8138 4.0047 5.6586 3.8493 3.0212 Step 3: There is a comparison between Tables 4 and 5 for a better value of Ra. Since candidate 1 of Table 4 has a lower Ra value as compare to Table 5, hence candidate 1 is selected from Table 4 and placed as nominee 1 in Table 6. Similarly, members 2, 3 are selected from Table 4 and in‐ serted into Table 6 as candidates 2 and 3. Similarly, candidate 4 and candidate 5 have better ob‐ jective values in Table 5 hence candidate 4 and candidate 5 are selected from table 5 and insert‐ ed into Table 6 as candidate 4 and candidate 5 respectively. Table 6 shows a modified variable value for the Rao algorithm. Step 4: Repeat 100 iteration and report the optimum solution. Table 7 shows the iteration‐wise best results. Candidate 1 2 3 4 5 X1 (A) 7.6672 5.6282 7.6384 6.5996 4.0931 Table 6 The modified variable value for the Rao algorithm X2 (µs) X3 (%) X4 (V) Ra (µm) 95.14 42.6479 80.5815 4.3819 106.4608 32.7404 42.4912 3.7440 100.0375 49.1606 94.5473 4.8520 119.5847 41.9748 73.1764 3.8493 50 25 90.4882 3.0212 Table 7 The best result selected from each iteration for Rao optimization X1 (A) X2 (µs) X3 (%) X4 (V) 4.9393 51.6323 27.4826 90.5408 4.0931 50 25 90.4882 4 50 25 88.6559 4 50 25 87.0529 4 50 25 87.0529 4 50 25 76.1071 4 50 25 85.2707 4 50 25 80.0359 4 50 25 80.9771 4 50 25 80.9771 4 50.5715 25 81.5514 4 50 25 81.5721 4.0097 50.0018 25.0032 81.6323 4.0008 50.0008 25 81.6348 4 50 25 81.6445 Iteration No. 1 2 3 4 5 6 7 8 9 10 11 12 24 27 30 Status From Table 4 From Table 4 From Table 4 From Table 5 From Table 5 Ra (µm) 3.0694 3.0212 2.9679 2.9246 2.9246 2.9158 2.8766 2.8181 2.7949 2.7949 2.7859 2.7804 2.7793 2.7789 2.7787 4.2 Jaya algorithm for the ANFIS model optimization With the Jaya algorithm, a Matlab program is written to optimize the ANFIS model. In Fig. 8, the flow diagram of the process is given. ′, , , , , , ∗ , , , , , , ∗ , , | , , | (16) Where If i is current iteration, j represents for j‐th variable, k represents for candidate number. Where X’j,k,i is new value of variable , Xj,k,i is old value of variable, r1,j,i , and r2,j,i are random num‐ bers for j‐th variable , Xj,best,i is best value j‐th variable k‐th candidate at i‐th iteration, Xj,worst,i is worst value j‐th variable k‐th candidate at i‐th iteration. Step 1: Inside the upper and the lower boundaries, all variables are created randomly as an ini‐ tial solution. Constraints are 4 ≤ X1 ≤ 8, 50 ≤ X2 ≤ 150, 25 ≤ X3≤ 50, 40 ≤ X4 ≤ 100. A set of varia‐ bles (candidate) are communicated to the ANFIS model and a corresponding response (Ra) of the ANFIS model is recorded in Table 8. Since if surface roughness is lower, the criteria will be Advances in Production Engineering & Management 16(2) 2021 153 Agarwal, Shrivastava, Pradhan better, hence candidate 1 is marked as the best candidate (marked best variable as X1_best = 7.1251, X2_best = 109.6631, X3_best = 29.36, X4_best = 44.8277) while candidate 2 is marked as the worst candidate because it has the highest value of Ra (hence marked worst variables are as X1_worst = 7.315, X2_worst = 96.5559, X3_worst = 44.8495, and X4_worst = 52.0834). Fig. 8 Flow diagram of the ANFIS/ANN model optimization Candidate 1 2 3 4 5 X1 (A) 7.1251 7.315 4.6139 4.2459 6.3595 Table 8 The initial solution for Jaya algorithm X2 (µs) X3 (%) X4 (V) 109.6631 29.3623 44.8277 96.5559 44.8495 52.0834 96.2368 41.8183 55.3064 140.9336 47.9749 77.332 65.9438 39.0437 62.5608 Ra (µm) 3.5789 5.0244 3.6694 3.5931 4.4153 Status best worst Step 2: Eq. 16 is used to calculate the value of new variables. Consider the random number r1 = 0.3590, r2 = 0.0287 for X1, r3 = 0.7674, and r4 = 0.2374 for X2, r5 = 0.6659 r6 = 0.7485 for X3, r7 = 0.7284 r8 = 0.8143 for X4. X1 for candidate 1: 7.1251 0.3590 7.1251 |7.1251| 0.0287 7.315 |7.1251| 7.1197 X2 for candidate 1: 109.6631 0.7674 109.6631 |109.6631| 0.2374 96.5559 |109.6631| 154 112.7745 Advances in Production Engineering & Management 16(2) 2021 Hybrid ANFIS‐Rao algorithm for surface roughness modelling and optimization in electrical discharge machining X3 for candidate 1: 29.3623 0.6659 29.3623 |9.3623| 0.7485 44.8495 |29.3623| (= 25 lower bound) X4 for candidate 1: 44.8277 0.7284 44.8277 |44.8277| 0.8143 52.0834 |44.8277| (= 40 lower bound) 17.7701 38.9193 These new variables are communicated to the ANFIS model and corresponding Ra records in Table 9, similar calculation is done for candidates 2, 3, 4 and 5 and recorded in the table. Step 3: The Ra of Tables 8 and 9 are compared for better value and select the candidate with better Ra and insert into Table 10. Step 4: The first iteration is over, now repeat a similar procedure for a few iterations. A total of 67 iterations are required to achieve the solution. Table 9 New value of variables for Jaya algorithm Candidate X1 (A) X2 (µs) X3 (%) 1 7.1197 112.7745 25* 2 7.2469 106.614 34.5364 3 5.4379 106.464 31.2549 4 5.1915 127.472 37.9199 5 6.607 92.226 28.2512 * Means adjusted to lower/upper bound Candidate 1 2 3 4 5 Iteration No. 1 2 3 4 5 40 55 59 62 63 64 65 66 67 68 X4 (V) 40* 46.7981 50.2978 74.2143 58.175 Table 10 The updated value of the variable for the Jaya algorithm X1 (A) X2 (µs) X3 (%) X4 (V) Ra (µm) 7.1197 112.7745 25 40 3.4345 7.2469 106.614 34.5364 46.7981 3.9293 4.6139 96.2368 41.8183 55.3064 3.6694 5.1915 127.472 37.9199 74.2143 3.3248 6.607 92.226 28.2512 58.175 4.2568 Table 11 Best result selected from each iteration for Rao optimization X1 (A) X2 (µs) X3 (%) X4 (V) 7.12 109.6631 29.3623 44.8277 5.1915 127.472 37.9199 74.2143 5.1915 127.472 37.9199 74.2143 4.4527 136.1197 36.8609 81.7775 4.4868 121.3322 35.2794 77.0596 4 120.5003 36.1024 74.6033 4 92.1362 29.0386 84.1491 4 94.2808 28.7777 81.9607 4 62.3788 27.9143 81.973 4 54.0164 27.605 82.1781 4 52.8637 27.5728 81.6957 4 50.4888 27.5587 81.5481 4.0061 50 27.0561 81.5907 4.007 50 26.6052 81.6279 4.0041 50 26.6336 81.6432 Ra (µm) 3.4345 3.9293 3.8770 3.3248 4.2568 Status From Table 9 From Table 9 From Table 8 From Table 9 From Table 9 Ra (µm) 3.5789 3.3248 3.3248 3.2419 3.1456 3.0936 3.0655 3.0152 2.8921 2.8256 2.8026 2.7831 2.7784 2.7779 2.7774 4.3 Optimization of the ANN model A predictable ANN model of Ra is developed. The Jaya algorithm is used to optimize this model. Step 1: Population size 5 is considered. Table 12 shows the initial solution. Random numbers within the selected range serve as initial variables. The ANN model is simulated for this input set and corresponding responses of Ra are recorded. The flow chart is shown in Fig. 8. Since surface roughness is better for smaller values, hence candidate 3 is marked as the best candidate while candidate 2 is marked as the worst candidate. Advances in Production Engineering & Management 16(2) 2021 155 Agarwal, Shrivastava, Pradhan Table 12 Initial solution and corresponding predicted Ra value Candidate X1 (A) X2 (µs) X3 (%) X4 (V) 1 2 3 4 5 7.2589 6.5294 4.3902 5.114 6.1875 140.5792 145.7507 146.4889 65.7613 147.0593 28.1747 48.9292 37.1344 45.007 28.5472 94.8026 65.3057 94.9441 87.5324 97.5695 Predicted Ra (µm) 4.5768 5.1702 3.7971 4.061 4.603 Status worst best Eq. 15 is used to calculate the new value of parameters. These new values of parameters are used to simulate Ra from the ANN model and corresponding Ra values are recorded in Table 13. Matlab 2015 software is used for the calculation. Step 2: Two random numbers r1= 0.82 and r2= 0.86 are considered for calculation. Variables cor‐ responding to the best candidate 3 are marked as the best variables, hence Marked X1_best = 4.3902, X2_best = 146.4889, X3_best = 37.1344, and X4_best = 94.9441. Corresponding to candidate 2, all variable have been marked as worst variables as X1_worst = 6.5294, X2_worst = 145.7507, X3_worst = 48.9292 and X4__worst = 65.3057. The calculation for the new value of the variables is as per the Eq. 15 as follows. X1 for candidate 1: 7.2589 0.82 4.3902 |7.2589| 0.86 6.5294 |7.2589| 5.533936 X2 for candidate 1: 140.5792 0.82 146.4889 |140.5792| 0.86 145.7507 |140.5792| 140.977664 X3 for candidate 1: 28.1747 0.82 37.1344 |28.1747| 0.86 48.9292 |28.1747| 17.672784 (=25 as lower limit) X4 for candidate 1: 94.8026 0.82 94.9441 |94.8026| 0.86 65.3057 |94.8026| 120.285964 (=100 upper limit) A similar calculation was done for X1, X2, and X3, X4 for candidates 2, 3 4, and 5. Step 3: Tables 12 and 13 are compared, select the candidate with better Ra, and insert into Table 13. Step 4: This process continues for some iterations. A total of 17 iterations of the Jaya algorithm is used to achieve an optimum solution. Table 15 shows the iteration‐wise best result. Table 13 New value of the variable with predicted Ra Candidate X1 (A) 1 5.5339 2 4.7752 3 4* 4 4* 5 4.4196 *Adjusted to lower or upper limit. X2 (µs) X3 (%) X4 (V) 140.9776 146.3560 147.1237 63.1670 147.7169 25* 39.2574 26.9908 35.1783 25* 100* 89.6091 100* 100* 100* Predicted Ra (µm) 4.9376 3.7629 3.9144 3.6689 4.1232 Table 14 Updated value of the variable with corresponding Ra value Candidate X1 (A) X2 (µs) X3 (%) X4 (V) Predicted Ra (µm) Status 1 2 3 4 5 7.2589 4.7752 4.3902 4 4.4196 140.5792 146.3560 146.4889 63.1670 147.7169 28.1747 39.2574 37.1344 35.1783 25* 94.8026 89.6091 94.9441 100 100* 4.5768 3.7629 3.7971 3.6689 4.1232 From Table 12 From Table 13 From Table 12 From Table 13 From Table 13 156 Advances in Production Engineering & Management 16(2) 2021 Hybrid ANFIS‐Rao algorithm for surface roughness modelling and optimization in electrical discharge machining Table 15 Iteration‐wise best result. Iteration No. X1 (A) X2 (µs) X3 (%) X4 (V) Ra 1 2 3 4 5 6 7 8 9 17 100 4.3902 4 4 4 4.4139 4 4.0315 4 4 4 4 146.4889 63.1670 74.8268 76.4593 78.2127 50 52.6894 53.1526 50 50 50 37.1344 35.1783 40.9319 39.5315 39.6985 36.3944 30.3658 30.5158 31.795 28.5 28.5 94.9441 100 94.8336 95.6589 58.0534 66.47 56.4976 56.4976 40 52.1 52.1 3.7971 3.6689 3.4462 3.3627 3.2436 3.1736 2.9378 2.9972 2.8705 2.8427 2.8427 5. Result and discussion The ANFIS model has a very low MSE 4.5×10‐4 and the regression coefficient is ‘R’= 0.9995. Fig. 9 compares the experimental values of quality characteristics Ra with the ANFIS model predicted values. This indicates a good prediction model. A confirmatory experiment is performed with an optimum set of input variables. The result of confirmatory experiments is given in Table 16. Since the limitation of the EDM machine setting, the value of X4 is selected as 81 instead of 81.6445. Therefore, the experimental value is rather similar to the expected value, validating the ANFIS model. The ANN model is developed for quality measure (Ra) and the mean square error of Ra is calculated as 0.0122 and the regression coefficient is ‘R’ = 0.98862 which is closer to 1. Fig. 10 shows the comparison between experimental versus the ANN model predicted data for Ra. The ANN model is optimized using the Jaya algorithm. The result of confirmatory experiments is giv‐ en in Table 16. Comparison between the ANN and ANFIS model: Regression coefficient R and Nash Sutcliffe Efficiency coefficient are higher for the ANFIS model. The error MSE, RMSE, MAE, and MBE are lower for the ANFIS model. Therefore the ANFIS model has better prediction accuracy. Table 17 shows the comparison between the ANN model and the ANFIS model. Fig. 9 Comparison result of experimental vs. the ANFIS model predicted data for Ra Response Model Table 16 Confirmatory experiments result Process variables Predicted X1 (A) X2 (µs) X3 (%) X4 (V) 4 50 25 81* 2.790 µm Surface ANFIS roughness (Ra) Surface ANN 4 50 28* roughness (Ra) *Adjusted to the nearest available setting of the EDM machine Advances in Production Engineering & Management 16(2) 2021 52* 2.843 µm Experimental 2.861 µm Error (%) 2.48 2.945 µm 3.59 157 Agarwal, Shrivastava, Pradhan Fig. 10 Comparison results of experimental vs. the ANN model predicted data for Ra Property Model error Model efficiency 6. Conclusion Table 17 Comparison between the ANN and ANFIS model Measurement ANFIS MSE 4.5 × 10-4 RMSE 0.021394 MAE 0.00807701 MBE 4.5169 × 10-05 MAPE 0.212868 % Nash Sutcliffe efficiency coeffi0.999149 cient (NSE) Regression coefficient 0.9995 ANN 0.012205 0.110474 0.050963 0.009038 1.388436 % 0.97732 0.9886 Successful machining of Titanium alloy has been done in the present research by using EDM process and performance comparison of Rao Optimization Algorithm and Jaya Algorithm has been done for one of the important surface integrity parameters. Followings are the important findings of the present research work: • The Rao optimization algorithm demonstrates better performances than the Jaya algorithm as 68 iterations were required by the Jaya algorithm for optimization of the ANFIS model, whereas Rao optimization required only 30 iterations for the same. • Both the algorithms successfully optimized the ANFIS model and predicted the identical set of input control factors. 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Integration of GA and neuro-fuzzy approaches for the predictive analysis of gas-assisted EDM responses, SN Applied Sciences, Vol. 2, Article No. 137, doi: 10.1007/s42452019-1533-x. 160 Advances in Production Engineering & Management 16(2) 2021 Advances in Production Engineering & Management ISSN 1854‐6250 Volume 16 | Number 2 | June 2021 | pp 161–172 Journal home: apem‐journal.org https://doi.org/10.14743/apem2021.2.391 Original scientific paper A multi‐objective optimal decision model for a green closed‐loop supply chain under uncertainty: A real industrial case study Fang, I.W.a, Lin, W.‐T.b a National ChengChi University, Department of Management Information Systems, Taipei, Taiwan (R.O.C) National ChengChi University, Department of Management Information Systems, Taipei, Taiwan (R.O.C) b ABSTRACT ARTICLE INFO Green closed‐loop supply chain management is an important topic for busi‐ ness operations today because of increasing resource scarcity and environ‐ mental issues. Companies not only have to meet environmental regulations, but also must ensure high quality supply chain operation as a means to secure competitive advantages and increase profits. This study proposes a multi‐ objective mixed integer programming model for an integrated green closed‐ loop supply chain network designed to maximize profit, amicable production level (environmentally friendly materials and clean technology usage), and quality level. A scenario‐based robust optimization method is used to deal with uncertain parameters such as the demand of new products, the return rates of returned products and the sale prices of remanufactured products. The proposed model is applied to a real industry case example of a manufac‐ turing company to illustrate the applicability of the proposed model. The result shows a robust optimal resource allocation solution that considers multiple scenarios. This study can be a reference for closed‐loop supply chain related academic research and also can be used to guide the development of a green closed‐loop supply chain model for better decision making. Keywords: Green closed‐loop supply chain; Sustainability; Modelling; Robust optimization; Mixed integer programming model; Supply chain management; Uncertainty; LP‐metric method *Corresponding author: 102356508@nccu.edu.tw (Fang, I.W.) Article history: Received 5 June 2021 Revised 23 June 2021 Accepted 24 June 2021 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 Due to increasing awareness of environmental issues, governments’ legislation, and natural re‐ source limitation, related research on the closed‐loop supply chain which integrate the forward and reverse supply chain are increasingly growing [1‐5]. According to Govindan et al. [6], the research gaps of closed‐loop supply chain include the discussion of green/ sustainable issues, the utilization of robust optimization method, the consideration of uncertain factors such as re‐ turn rate, and multi‐objective functions with environmental indicators. The economic and environmental factors are mostly considered in the multiple objective functions [7‐16], while other evaluation indicators of supply chain performance are seldom in‐ cluded. Promoting better quality of the supply chain operations is the method to sustain the long‐term competition for earning the market share [17]. This study proposes a multi‐objective 161 Fang, Lin mixed‐integer programming model of the closed‐loop supply chain network to maximize profit, amicable production level, and quality level. In order to describe real industry environments, uncertain parameters are considered in the mathematical model. With sensitivity analysis, the demand of new products, the return rates of returned products and the sale prices for remanufactured products which are highly sensitive to the objective functions are selected. A scenario‐based robust optimization method is utilized for solving the uncertain problem. After the uncertainty problem is solved, the multi‐objective prob‐ lem is solved by LP‐metric method. A real industry case example of a manufacturing company is applied to illustrate the applicability of the proposed model. The goal of this study is to develop the multi‐objective mathematical model for supporting better decision making in the green closed‐loop supply chain network management. Besides, the impact of the economic, environ‐ mental, and quality factors to the green closed‐loop supply chain model is discussed in this study. This paper advances current research into closed‐loop supply chain models in two ways. First, the amicable production level and quality level are considered in the objective functions for ana‐ lysing simultaneously the impacts of three objective functions in different parameter settings. Second, sensitivity analysis is used to determine the multiple uncertain parameters for fully con‐ sideration of the important factors in this model. The rest of this paper is organized as follows. Section 2 reviews related literature. Section 3 is devoted to the proposed multi‐objective mixed‐ integer linear programming model. Section 4 shows an illustrative case example, and section 5 discusses conclusions. 2. Literature review The objectives of closed‐loop supply chain models include carbon emissions, costs, profits, envi‐ ronmental influence/ cost, environmentally friendly materials, and clean technology etc. Zhao et al. [8] proposed a multi‐objective optimization model for minimizing the inherent risk occurred by hazardous materials, associated carbon emission, and economic cost. Three scenarios are discussed in this study for analyzing the green impact in the supply chain management. Talaei et al. [9] proposed a bi‐objective mixed‐integer linear programming model for minimizing the total network costs and the rate of carbon dioxide emission. The effects of uncertainties of the varia‐ ble costs and demand rate are considered in the closed‐loop supply chain network design. In order to improve economic and environmental performance in an environmental closed‐loop supply chain for enterprises’ competitiveness, Ma et al. [10] proposed a robust multi‐objective mixed integer nonlinear programming model to minimize two conflicting objectives simultane‐ ously, the economic cost and the environmental influence. Also, the uncertain cost parameters and demand fluctuations are considered in the supply chain network design. To illustrate the relationships between supply chain management policies and natural environmental impacts, Altmann and Bogaschewsky [11] proposed a robust multi‐objective closed‐loop supply chain design model which minimizes expected total costs as well as carbon dioxide equivalents. Cus‐ tomer demand and used‐product return rate are considered to be the uncertain parameters. For considering the environmental factors in the closed‐loop supply chain facility‐location model, Amin and Zhang [12] proposed a mixed‐integer linear programming model to minimize the total costs and to maximize the environmentally friendly materials and clean technology usage. The impact of uncertain demand and return is investigated on the network configuration. Pourjavad and Mayorga [16] developed a fuzzy multi‐objective mixed integer linear programming model to minimize costs and environmental impacts and maximize social impacts. Three fuzzy parame‐ ters such as return rates of products from customer centers, the capacity of all facilities, and product demand are considered. The related literatures are summarized in Table 1. Some performance indicators such as quality in the traditional supply chain model are less frequently included in the green closed‐loop supply chain model for comparing the impacts of performance indicators simultaneously (shown as Table 1). Quality is an important concern in response to the performance in supply chain operations [5,14, 17‐20]. Liu et al. [13] proposed a bi‐objective mathematical programming model considering uncertain demand in a green closed 162 Advances in Production Engineering & Management 16(2) 2021 A multi‐objective optimal decision model for a green closed‐loop supply chain under uncertainty: A real industrial case study loop supply chain network. Two objectives are minimizing the total costs including production cost, operation cost, transportation cost, and construction cost, while maximizing the satisfac‐ tion of customers which includes shipping time, product quality, and recovery quantity. Liu et al. [13] only considers the recovery quantity as the environmental indicator that can’t effectively and suitably reveal the impact of green manufacturing to the whole supply chain performance, and only treat demand as the uncertain parameter. For fulfilling the research gap, this study considers the economic, amicable production level, and quality simultaneously for optimizing the green closed‐loop supply chain model which is referenced by Fang and Lin [5]. For in‐depth analysis, this study utilizes the sensitivity analysis to determine the uncertain parameters such as the demand of new products, the return rates of returned products, and the sale prices of re‐ manufactured products which are highly sensitive to the three objective functions. Besides, the comparison between the infeasibility weight and model robustness is made to provide more helpful information for decision making. For solving stochastic problems in closed‐loop supply chain, Govindan et al. [6] suggested that two‐stage stochastic method or robust optimization method can be considerable solutions in the future directions of the related researches. Compared with stochastic programming meth‐ od, the robust optimization method has the advantage of implementing without the known probability distributions of uncertain parameters. Besides, the robust optimization is easier than stochastic programming method for finding the optimal solution [24]. In terms of the suggestions mentioned above, the multi‐objective stochastic problem of this study is solved by robust optimization method and LP‐metrics method referenced by Altmann and Bogaschewsky [11], Ma et al. [10] and Fang and Lin [5] to find the final optimal solution. Sensitivity analysis is utilized to determine the selected uncertain parameters and the AHP method is used to determine the weights of three objective functions for being combined into a single objective function. Papers Amin and Zhang (2013) Ramezani et al. (2013) Altmann and Bogaschew‐ sky (2014) Das and Rao Posinasetti (2015) Saffar et al. (2015) Ma et al. (2016) Talaei et al. (2016) Mohammed et al. (2017) Zhao et al. (2017) Liu et al. (2018) Pourjavad and Mayorga (2018) Karimi et al. (2019) Valizadeh et al. (2020) This study Table 1 Summary of related literature Uncertain Objective function indicators Solutions/Methods parameters Single Multiple Environmental Economic Quality Multi‐objective approach    Stochastic programming Pareto‐optimal solutions    Robust optimization                                Advances in Production Engineering & Management 16(2) 2021   Bi‐objective Pareto optimal solu‐ tions, Goal programming Jimenez approach Robust optimization Robust fuzzy optimization Robust optimization Big data analytic approach, Sce‐ nario analysis Approximation, ‐constraint method, MOSA, NSGA‐II NSGA‐II, NRGA NSGA‐II, NRGA Bertsimas and Sim stable optimi‐ zation approach Multi‐objective approach, Robust optimization 163 Fang, Lin 3. Model formulation 3.1 Problem definition Referenced by Fang and Lin [5], the green closed‐loop supply chain network model in this study (depicted as Fig. 1) includes manufacturing centers, customers, collection centers, and custom‐ ers of the other market. The new products are manufactured by manufacturing centers and sent to customers. The returned products are purchased from customers and sent to collection cen‐ ters. After the returned products are dismantled by the collection centers, some reused materi‐ als are sent back to the manufacturing centers for new products, and some reused materials are manufactured by the collection centers for remanufactured products. The remanufactured products are sent from collection centers to customers of the other market. The goal of this study is to maximize total profit, amicable production level, and quality of products with three uncertain parameters. The assumptions of this research are referenced by Fang and Lin [5]. Fig. 1 Green closed‐loop supply chain network model in this study 3.2 Model description The sets, indices, parameters, decision variables, and scenario variables for the model formula‐ tion are shown in Table 2. Objective functions: F1 = Revenue ∑ Purchase Cost ∑ Processing Cost ∑ ∑ ∑ ∑ ∈ ∑ ∑ ∑ ∑ 3 164 ∈ ∑ ∑ ∑ (4) ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∈ ∑ (5) ∑ ∑ ∑ ∑ ∑ (2) ∑ (3) ∑ ∑ ∑ ∑ ∑ ∑ 2 ∑ ∑ (1) ∑ ∑ ∑ ∈ Transportation Cost ∑ ∑ ∑ ∑ ∈ ∑ ∑ ∑ ∑ ∑ ∈ ∑ ∈ ∑ ∑ ∑ (6) ∑ ∑ ∑ ∑ (7) Advances in Production Engineering & Management 16(2) 2021 A multi‐objective optimal decision model for a green closed‐loop supply chain under uncertainty: A real industrial case study Table 2 The descriptions of variables and parameters Sets & indices C index of fixed locations of customers, c = 1, 2,…, C F index of objective functions, f = 1, 2,…, F L index of potential locations of collection centers, l = 1, 2,…, L R index of reused materials, R = RM  RE, RM is for manufacturing centers, RE is for customers of the other markets P index of new products, p = 1, 2,…, P Parameters PCpm unit production cost of product p from manufactur‐ ing center m CCcp purchasing cost of product p from customer c DEet DPcp CPpm Pet DRle E I M T RCle FC demand of customer t of the other market for re‐ manufactured product e demand of customer c for product p capacity of manufacturing center m for product p CMlp unit sale price of remanufactured product e for customer t of the other market defect rate of remanufactured product e from col‐ lection center l the rate of reused material r (r  rm) dismantled from returned product p weight factor for importance of product p weight factor for importance of remanufactured product e weight factor of using clean technology Ppc RAp CEle index of remanufactured products, e = 1, 2,…, E index of scenarios, i = 1, 2,…, I index of potential locations of manufacturing centers, m = 1, 2,…, M index of fixed locations of customers of the other market, t = 1, 2,…, T unit remanufacturing cost of remanufactured product e from collection center l fixed cost for opening manufacturing center and collection center capacity of collection center l for product p the return rate of product p capacity of collection center l for remanufac‐ tured product e unit sale price of product p for customer c DRmp defect rate of product p from manufacturing center m RMrp RErp the rate of reused material r (r  re) disman‐ tled from returned product p Wp Wf weight factor of objective function f We Wem weight factor of using environmentally friendly materials Wct EMpm the rate of using environmentally friendly materials by manufacturing center m for prod‐ uct p EMle the rate of using environmentally friendly materials CTM the rate of using clean technology for produc‐ by collection center l for remanufactured product e ing product p and remanufactured product e TCpmc unit distribution cost for product p shipped CTS the rate of using clean technology for shipping product p and remanufactured product e between from manufacturing center m to customer c facilities TCprlm unit distribution cost for reused material r (rrm) TCelt unit distribution cost for remanufactured product e shipped from collection center l to of product p shipped from collection center l to customer t of the other market manufacturing center m Decision variables QMpmc quantity of product p produced from manufacturing QLplc quantity of product p returned from customer c center m to customer c to collection center l QRrlmp quantity of reused material r (r  rm) for product p QErelt quantity of reused material r (r  re) for pro‐ ducing remanufactured product e from collec‐ from collection center l to manufacturing center m tion center l to customer t of the other market IMm 1 if manufacturing center m is opened, otherwise 0 ILl 1 if collection center l is opened, otherwise 0 Scenario variables QUpc unfulfilled quantity of product p for customer c QUet unfulfilled quantity of remanufactured product e for customer t of the other market SPi the occurrence probability of scenario i the infeasibility weight  the linearization variable of the first objective the weight of risk  1i function under scenario i the linearization variable of the second objective the linearization variable of the third objective 2i 3i function under scenario i function under scenario i The first objective function F1 is to maximize the total profit which is computed by subtract‐ ing purchase cost, processing cost, and transportation cost from total revenue. The total revenue includes the sale revenue of new products and remanufactured products. The purchase cost in‐ cludes purchasing cost of returned products from customers. The processing cost includes fixed cost of manufacturing centers, production cost of new products, fixed cost of collection centers, and production cost of remanufactured products. The transportation cost is for shipping prod‐ ucts or materials between facilities in the closed‐loop supply chain network. The second objec‐ Advances in Production Engineering & Management 16(2) 2021 165 Fang, Lin tive function F2 is to maximize the amicable production level which means the level of using environmental parameters such as environmentally friendly materials or clean technology in the supply chain operations. The third objective function F3 is to maximize the quality level by min‐ imizing the defect rate of new products and remanufactured products. Constraints: ∑ ∀ , (8) ∀ , (9) ∑ ∀ , (10) ∑ ∀, (11) ∀, (12) ∀ ,c (13) ∀ , l, r rm (14) ∀p, l, r re (15) QNnspm, QMpmc, QLplc, QRrlmp , QErelt, QUpc, QUet, 1i, 2i, 3i  0 ∀r,s,p,c,m,l,n,t,e (16) IMm, ILl  {0,1} ∀ , (17) ∑ ∑ ∑ ∈ ∑ ∑ ∑ ∑ ∑ ∑ ∑ Eq. 8 ensures that the sum of the produced quantity of each product for each customer can meet customer demand. Eq. 9 ensures that the sum of the produced quantity of each remanufac‐ tured product for each customer of the other market can meet the customer demand of the other market. Eq. 10 states that the sum of each product produced for customers by each manufactur‐ ing center does not exceed the capacity of this manufacturing center. Eq. 11 presents that the sum of each returned product collected by each collection center does not exceed the capacity of this collection center. Eq. 12 presents that the sum of each remanufactured product produced by each collection center does not exceed the capacity of this collection center. Eq. 13 ensures the returned quantity of each product from customers to collection centers. Eq. 14 ensures the quantity of each reused material which is dismantled from each returned product in each collec‐ tion center supplied to manufacturing centers for manufacturing products. Eq. 15 ensures the quantity of each reused material which is dismantled from each returned product in each collec‐ tion center used for producing remanufactured products. Eq. 16 preserves the non‐negativity restriction on the decision variables, and Eq. 17 imposes the binary restriction on the decision variables. 3.3 The transformation to a robust optimization model The proposed multi‐objective mixed integer programming model can be transformed to a robust optimization model by the approach proposed by Mulvey et al. [25] and a more suitable formula‐ tion for the first term of the objective function introduced by Yu and Li [26]. The three objective functions and additional constraints of the proposed model can be expressed as follows [5]: 1 ∑ 1 ∑ 1 ∑ 1 2 ∑ (18) 2 ∑ 2 ∑ 2 ∑ 2 2 ∑ (19) 3 ∑ 3 ∑ 3 ∑ 3 2 ∑ (20) s.t. 166 1 ∑ 1 0 ∀ ∈ (21) 2 ∑ 2 0 ∀ ∈ (22) 3 ∑ 3 0 ∀ ∈ (23) Advances in Production Engineering & Management 16(2) 2021 A multi‐objective optimal decision model for a green closed‐loop supply chain under uncertainty: A real industrial case study ∑ ∑ ∑ ∑ ∀ , (24) ∀ , (25) ∀ , (26) Where F1i denotes the first objective function in the proposed model as Eq. 1, F2i denotes the second objective function as Eq. 6, and F3i denotes the third objective function as Eq. 7. Eqs. 21‐ 23 can be interpreted that if the Fxi is greater than ∑ , then xi is equal to 0. If the ∑ ∑ is greater than Fxi, then . Eqs. 10‐12 and Eqs.14‐17 in section 3.2 are still considered in the robust optimization model. 3.4 The transformation to single objective function model The LP‐metrics method is utilized for solving the multi‐objective problem in this study refer‐ enced by Fang and Lin [5]. Firstly, each objective function of the proposed multi‐objective model is solved separately and the objective value ob1*, ob2* and ob3* is gained respectively. Then, the objective function can be formulated as Eq. 27. W1, W2, and W3, the weights of the three compo‐ nents in the Eq. 27, are determined by the AHP method and W1+W2+W3=1. ∗ ∗ ∗ ∗ ∗ ∗ (27) 4. Results and discussion: A case study 4.1 Example description A real case example of a manufacturing company referenced by Fang and Lin [5] is provided to illustrate the applicability of the proposed green closed‐loop supply chain model. According to the supply chain operations of this company, the closed‐loop supply chain network includes two products, two manufacturing centers, five customers, two collection centers, two kinds of reused materials (the first one is for new products, the second one is for remanufactured products), two remanufactured products, and five customers of the other market. The two new products can be manufactured by the two manufacturing centers and sent to the five customers. The returned products can be purchased and sent to collection centers for recycling use. The returned prod‐ ucts can be dismantled into several kinds of reusable materials in collection centers. The re‐ turned product 1 can be dismantled into two kinds of reused materials, namely, R11 and R12. R11 will be sent back to the manufacturing centers for new product 1, R12 will be input material for remanufactured product 1. The returned product 2 can be dismantled into three kinds of reused materials, namely, R21, R22, and R23. R21 and R22 will be sent back to the manufactur‐ ing centers for new product 2, and R23 will be input material for remanufactured product 2. The two remanufactured products can be produced by the two collection centers and sent to the five customers of the other markets. 4.2 Parameters setting In order to consider the confidentiality of the company data, the real company data is modified in different scenarios. The parameter data is shown in Table 3‐8. The AHP method is used for assigning the weights of the three objective functions and the weights of the environmentally friendly materials and clean technology usage. The weight of ob1, ob2 and ob3 is 0.36, 0.36, and 0.28. The weight of the environmentally friendly materials and clean technology usage is 0.5 and 0.5. The weight factor for importance of new product 1 and 2 is 0.3 and 0.2, the same as remanu‐ factured product 1 and 2 is. The return rate of new products is 0.9. The defect rate of new prod‐ ucts and remanufactured products is 0.1. The fixed cost is 30. For the sake of computation con‐ venience, the rates of using environmentally friendly materials and clean technology will be transformed to the corresponding scores as follows: the rate which is less than or equal to 25 % is set to be 25, the rate which is 26‐50 % is set to be 50, the rate which is 51‐75 % is set to be 75, and the rate which is greater than 75 % is set to be 100. Advances in Production Engineering & Management 16(2) 2021 167 Fang, Lin Parameters Table 3 New products and customer data New products Customers 1 2 3 1/2 80/55 85/60 90/65 1/2 90/80 90/80 90/80 4 95/70 90/80 5 100/75 90/80 Table 4 Remanufactured products and customers of the other market data Parameters Remanufactured Customers of the other market products 1 2 3 4 Demand amount (DEet) 1/2 55/154 55/110 110/55 66/55 5 110/154 Unit selling price (Ppc) Purchase cost (CCcpr) Parameters Production cost (PCmp) Capacity limitation (CPpm) Table 5 Manufacturing centers and new products data Manufacturing centers New product 1 1/2 80/85 1/2 800/600 Table 6 Collection centers and remanufactured products data Parameters Collection centers Remanufactured product 1 Production cost (RCle) 1/2 80/60 Capacity limitation (CEle) 1/2 500/500 New product 2 40/35 1000/800 Remanufactured product 2 45/35 300/300 Table 7 The rates of reused materials dismantled from the returned products Returned products The rates of reused materials (RMrp) The rates of reused materials (RErp) R11 R21 R22 R12 R23 1 0.5 ‐ ‐ 0.5 ‐ 2 ‐ 0.55 0.05 ‐ 0.4 Table 8 Other parameters data Parameters Values Parameters The rate of clean technology use (CTmp, CTle) 0.1 Distribution cost (TCpmc) The rate of clean technology use (CTprlm) 0.2 Distribution cost (TCprlm) The rate of clean technology use (CTelt) 0.1 Distribution cost (TCelt) The rate of environmentally The rate of clean technology use (CTpmc) 0.1/m1,0.2/m2 friendly material use (EMle) 0.2/p1,0.9/p2 The rate of environmentally friendly material use (EMpm) 600/p1,700/p2 Capacity limitation(CMlp) Values 2 30 20 1 4.3 Scenarios setting Sensitivity analysis is used for determining which uncertain parameters are highly sensitive to the three objective functions in this study. Referencing the result of sensitivity analysis, the pa‐ rameters including the demand of new products, the return rates of returned products, and the sale prices of remanufactured products are taken into account for expressing uncertain situa‐ tions in the real business environment. In this study, each uncertain parameter has 2 scenarios (high and low), so the combination of three uncertain parameters provides eight scenarios. The probability of each scenario is assumed to be the same. The high and low product return rates are 0.9 and 0.7 for new product 1 and 0.9 and 0.8 for new product 2. The setting of the new product demand and the sale prices of remanufactured products are shown in Table 9. Table 9 The setting of the new product demand and the sale prices of remanufactured products New products demand Customers 1 2 3 4 5 1 High/Low 220/180 220/180 220/180 220/180 220/180 2 High/Low 344/296 344/296 384/336 124/76 124/76 Customers of the other market The sale prices of remanufactured products 1 2 3 4 5 1 High/Low 1400/1000 1400/1000 1500/1100 1500/1100 1600/1200 2 High/Low 680/280 700/300 700/300 700/300 720/320 168 Advances in Production Engineering & Management 16(2) 2021 A multi‐objective optimal decision model for a green closed‐loop supply chain under uncertainty: A real industrial case study 4.4 Computing results To solve the proposed model using the test case data, the Lingo 12.0 software was employed.  is set to be 1 and  is set to be 90. The computing results are shown in Table 10‐13. Using the ro‐ bust optimization approach to solve the uncertain problems, multiple scenarios are considered and the infeasibility in the control constraints can be allowed by means of penalties. Through the solution solving procedures, we have the following findings:  From the sensitivity analysis, the demand of new products, the return rates of returned products, and the sale prices of remanufactured products was highly sensitive to the three objective functions of the proposed model. It can be inferred that the three parameters are relatively important elements to the total performance of this proposed model. The com‐ pany manager should pay more attention to evaluating the impacts of the three parame‐ ters to keep good performance of the green closed‐loop supply chain.  Using the AHP method, the weight of the economic objective is the same with the weight of the environmental objective. Also, the weights of the economic and environmental objec‐ tives are higher than the weight of the quality objective. It can be inferred that the envi‐ ronmental issues have gotten much attention in the closed‐loop supply chain operations, and are treated as important as the company’s profit. Table 10 The allocation of new products Manufacturing Customers centers 1 2 3 1/2 57.5/122.5 43.6/136.4 108.1/71.9 1/2 243.6/52.4 127.3/168.7 0/336 New products 1 2 Returned products 1 2 New products 1 2 Table 11 The allocation of returned products Collection centers Customers 1 2 3 1/2 0/126 15.5/110.5 0/126 1/2 0/236.8 17.4/219.4 95/173.8 4 0/126 25.9/34.9 5 529.1/150.9 629.1/166.9 5 14.5/111.5 25.6/35.2 Table 12 The allocation of reused materials offered to manufacturing centers Manufacturing Reused Manufacturing Collection centers Collection centers centers materials centers 1 1 2 2 1 2 R11 0 0 15 300 R21/R22 0/0 0/0 90.2/8.2 385/35 Remanufactured products 1 2 4 61.7/118.3 0/76 Table 13 The allocation of remanufactured products Collection centers Customers of the other market 1 2 3 4 1/2 0/0 0/29 0/110 15/51 1/2 0/0 18.7/91.3 20.9/19.7 20.9/20.1 5 0/110 5/149 4.5 Trade‐off between the infeasibility weight and model robustness The infeasibility weight  is applied as the model infeasibility under the scenarios of having the unfulfilled demand. Fig. 2 shows the comparison between the infeasibility weight and the unful‐ filled demand of remanufactured products. We can find that when the value of  increases to 90, we get the lowest unfulfilled demand of remanufactured products. Therefore, the best value for  in the test case is 90. We can find the relations between objective functions and infeasibility weights shown as Fig. 3. As the infeasibility weight increases, the value of objective function 1 and 2 decreases but the value of objective function 3 increases. The decreased percentage of objective function 1 and 2 are 25 % and 30 % respectively, and the increased percentage of objective function 3 is 199 %. It is shown that the value of objective function 3 is influenced most appreciably by the change of the infeasibility weight, while the value of objective function 1 and 2 is influenced a little by the Advances in Production Engineering & Management 16(2) 2021 169 Fang, Lin change of the infeasibility weight. From this analysis, the decision maker may consider the dif‐ ferences between the three objective functions generated by the infeasibility weight to find a suitable decision for the green closed‐loop supply chain management. Fig. 2 Comparison between the infeasibility weight and model robustness Fig. 3 Value of objective functions in different infeasibility weights 5. Conclusion Given resource limitation and environmental issues, companies not only have to strive to stay competitive to make more profit, but also have to consider the effective recycling utilization for environmental protection and social legislations. In order to have the holistic view to discuss the optimization of the supply chain, the closed‐loop supply chain network model is adopted in this study. Referenced by Fang and Lin [5], this study proposes a multi‐objective model for a green closed‐loop supply chain network to maximize profit, amicable production level, and quality level. The uncertain parameters such as the demand of new products, the return rates of re‐ turned products, and the sale prices of remanufactured products are selected by sensitivity analysis. The multi‐objective stochastic problem of this study is solved by robust optimization method and LP‐metrics method. A real case example is provided for demonstrating the applica‐ bility of this proposed model. Through the computation result, we can find a robust optimal re‐ source allocation solution for the proposed multi‐objective mixed integer programming model. There are several important managerial insights for managers. First, in order to gain competi‐ tive advantages and sustainable development, considering the multiple objectives in the closed‐ loop supply chain help managers obtain more completed and precise information to make better decision. Second, uncertain factors will affect the performance of the closed‐loop supply chain operations. Utilizing sensitivity analysis to determine the critical uncertain parameters in this proposed model help managers pay more attention to evaluating the impacts of the uncertain parameters to keep good performance of the green closed‐loop supply chain. Third, economic factors and environmental factors have the same importance for optimizing green closed‐loop supply chain operations in this study. It is found that green manufacturing should be noticed more seriously for supply chain management. In future research, regarding environmental issues, other environmental metrics, such as Oršič et al. [27] mentioned, can be considered in the environmental objective function. The im‐ 170 Advances in Production Engineering & Management 16(2) 2021 A multi-objective optimal decision model for a green closed-loop supply chain under uncertainty: A real industrial case study pacts of green closed-loop supply chain performance by the environmental factors can be further evaluated and discussed. 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Measuring sustainable performance among logistic service providers in supply chains, Tehnički Vjesnik – Technical Gazette, Vol. 26, No. 5, 1478-1485, doi: 10.17559/TV-20180607112607. 172 Advances in Production Engineering & Management 16(2) 2021 Advances in Production Engineering & Management ISSN 1854‐6250 Volume 16 | Number 2 | June 2021 | pp 173–184 Journal home: apem‐journal.org https://doi.org/10.14743/apem2021.2.392 Original scientific paper Improved Genetic Algorithm (VNS‐GA) using polar coordinate classification for workload balanced multiple Traveling Salesman Problem (mTSP) Wang, Y.D.a, Lu, X.C.b,*, Shen, J.R.c a Beijing Jiaotong University, Shangyuan Village, Haidian District, Beijing, P.R. China Beijing Jiaotong University, Shangyuan Village, Haidian District, Beijing, P.R. China c Beijing Capital Agribusiness & Food Group Co., Ltd., Xicheng District, Beijing, P.R. China b ABSTRACT ARTICLE INFO The multiple traveling salesman problem (mTSP) is an extension of the travel‐ ing salesman problem (TSP), which has wider applications in real life than the traveling salesman problem such as transportation and delivery, task alloca‐ tion, etc. In this paper, an improved genetic algorithm (VNS‐GA) that uses polar coordinate classification to generate the initial solutions is proposed. It integrates the variable neighbourhood algorithm to solve the multiple objec‐ tive optimization of the mTSP with workload balance. Aiming to workload balance, the first design of this paper is about generating initial solutions based on the polar coordinate classification. Then a distance comparison insertion operator is designed as a neighbourhood action for allocating paths in a targeted manner. Finally, the neighbourhood descent process in the vari‐ able neighbourhood algorithm is fused into the genetic algorithm for the expansion of search space. The improved algorithm is tested on the TSPLIB standard data set and compared with other genetic algorithms. The results show that the improved genetic algorithm can increase computational effi‐ ciency and obtain a better solution for workload balance and this algorithm has wild applications in real life such as multiple robots task allocation, school bus routing problem and other optimization problems. Keywords: Multiple traveling salesman problem (mTSP); Workload balance; Variable neighbourhood search algorithm (VNS); Genetic algorithm (GA); Polar coordinates; Classification *Corresponding author: xclu@bjtu.edu.cn (Lu, X.C.) Article history: Received 4 June 2021 Revised 15 June 2021 Accepted 17 June 2021 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 Multiple Traveling Salesmen Problem (mTSP) is an extension of the classic Traveling Salesman Problem (TSP)[1]. The mTSP with workload balance is different from the basic TSP. In some scenarios, the balanced distribution of workload is an important consideration. For example, school bus routing problem, route arrangement problem, postman distribution problem, express delivery area division, etc., can all be abstracted as mTSP with balanced workload. Omar et al. [2] proposed that mTSP can be used to solve vehicles, robots, and UAVs routing problems. In single‐ objective optimization of mTSP, the total cost is the most commonly used optimization objective. It causes the problem of an unbalanced distribution of distance among traveling salesmen, which is directly reflected in the path as the distance traveled by one traveler is longer than oth‐ er travelers. This situation is hoped to be improved in some industries in reality because the 173 Wang, Lu, Shen unbalanced workload distribution results in operational difficulties. Moreover, avoiding crossing paths with one another is the most significant problem in practical work. For instance, in school bus routing problem, if a certain route is too long during planning school bus route, it may cause the first arrival point to arrive too early, which will affect students’ study and other perfor‐ mance. In postmen routing problem, avoiding crossing paths can improve the efficiency of work. Therefore, our main focus is on the mTSP with workload balance. To solve this problem, we build a multiple objective optimization model and propose an improved genetic algorithm that aims to minimize the total cost and balancing the workload in this paper. 2. Literature review Research on TSP and mTSP is of long historical making. The original solution to this type of problem mainly used accurate algorithms. Ali et al. [3] used the branch and bound method to solve the symmetrical mTSP with 59 cities. Gavish et al. [4] described that limiting the lower bound of the branch improved the branch‐and‐bound method, which can solve the mTSP with 10 traveling salesmen and 100 cities. However, as the scale of the problem expands, the scale of the solution space also increases exponentially, making it difficult to use accurate algorithms. Therefore, heuristic algorithms have become the choice of more and more researchers. Russell [5] was the first one to apply the heuristic algorithm to solve the mTSP, which trans‐ formed mTSP into TSP and then solved it with the Lin‐Kernighan algorithm. It also obtained a more accurate approximate solution. For solving the mTSP with the shortest total distance and the shortest sub‐path, Carter et al. [6] designed a two‐stage chromosome coding scheme and designed a genetic operator to improve the solution speed. Singh et al. [7] used the invading weed algorithm to obtain a comparative optimal solution. Zhou and Li [8] used a mutation oper‐ ator combined with a 2‐opt search operator to improve the genetic algorithm, avoiding the premature phenomenon of genetic algorithm, and used simulation methods to verify the effec‐ tiveness of the algorithm. Xu et al. [9] also combined the 2‐opt operator with a genetic algorithm to solve the mTSP. Hu et al. [10] proposed an improved genetic algorithm that integrates the reproduction mechanism of the weed algorithm and the locally optimized mutation operator to solve the mTSP with balanced workload. Guo [11] used a genetic algorithm to analyze and verify the design of chromosome coding scheme for the genetic algorithm to solve mTSP. Lu et al. [12] adopted a two‐stage algorithm to solve mTSP while avoiding the path‐crossing problem between traveling salesmen. Bostanci et al. [13] used the integer linear programming method to solve the problem of finding the shortest route on networks with GIS. This study provided a new method to solve the optimization problems, however, only TSP was solved by this method, problems more complicated such as mTSP are difficult to find the optimal solution. Recently, with the rise of various heuristic algorithms and machine learning, in addition to genetic algorithms, ant colony optimization algorithm (ACO) [14], artificial bee colony algorithm (ABC)[15], tabu search algorithm (TS)[16] and other heuristic algorithms have also been tried to solve mTSP. Song [17] used the simulated annealing algorithm (SA) to solve the problem of three traveling salesmen in 400 cities, but the algorithm took a long time. Hu [18] constructed an architecture composed of a shared graph neural network and a distributed strategy network to generate an approximate optimal solution for mTSP and used reinforcement learning to train the model, and this method shows good results in large‐scale examples. Justus [19] abstracted the problems in the path of staff guiding tourists during the Mecca pilgrimage as multiple objec‐ tive mTSP with time window and proposed an interactive method for providing a solution. Aim‐ ing the mTSP under different goals, Liu et al. [20] proposed the ForestTraversal algorithm and the Retrace algorithm to solve the maximum‐minimum mTSP and the mTSP that minimizes travel costs, respectively. Genetic algorithm (GA) is a heuristic algorithm that is often used when solving mTSP. This method mainly focuses on coding design [5] and how to avoid genetic algorithms prematurely falling into local optimality. The variable neighbourhood search algorithm (VNS) has the ability to expand the search range by systematically changing the neighbourhood structure. It guaran‐ tees the local search capability while ensuring solutions for having good diversity [21]. There‐ 174 Advances in Production Engineering & Management 16(2) 2021 Improved Genetic Algorithm (VNS‐GA) using polar coordinate classification for workload balanced multiple Traveling … fore, this paper designs a genetic algorithm based on polar coordinates to quickly generate high‐ quality initial solutions, and combines VNS to retain the characteristics of neighbourhood diver‐ sity, designs different neighbourhood actions to improve the algorithm's search ability to solve mTSP with workload balance. 3. Mathematical model and problem solving This paper proposes an improved genetic algorithm, which is improved by fusing the variable neighbourhood algorithm (Variable neighbourhood search genetic algorithm), and VNS‐GA is referred too, as an abbreviation in the text. 3.1 Problem description and mathematical model The mTSP studied in this paper can be described as given n cities, m traveling salesmen and the distance matrix between nodes Dij = (dij)n·n, travelers start from the same node (source node), visit a certain number of nodes and then return to the source node. All nodes are required to be accessed, and the rest of the nodes except the source node can only be accessed once. The aim is to find a Hamiltonian circuit that makes total distance shortest and workload balance. Assuming the problem is symmetrical mTSP, that is, dij = dji, the multiple objective optimization model can be described as follows: 1,2, … , max min , , ∈ , (2) 0,1, … , ; 1,2, … , (3) ∀ 0,1, … , ; 1,2, … , (4) 1,2, … , 0 (5) ∈ (6) , 1, ∈ ∈ ∀ 1, | (1) ⊂ 0,1,2, … , ,∀ ∉ 1 traveler went through arc 0 otherwise 1 traveler visited node 0 otherwise 1,2, … , (7) (8) (9) Where V = {0,1,2,...,n} is the node set, n is the number of nodes; T = {1,2,...,m} is the travelers set, m is the traveling salesman number; A = {( i, j, k)|i, j = 0,1,2,…,m, i≠j, ∀k = 1,2,…,m} is the arc set, which represents the set of routes that the travelers may pass; D = {dij |i, j ∈ A} is the distance matrix, representing the distance from node i to node j. Eqs. 1 and 2 are the objectives of these functions. Eq. 1 minimizes the total path distance; Eq. 2 minimizes the difference between the longest path and the shortest path in m paths. Eq. 3 means that for any downstream node, only one upstream node is allowed for its connection, and Eq. 4 means that in each upstream node, only one downstream node is allowed to be connected; Eq. 5 means that except for the source node, all other nodes are only visited once, and all travel‐ ers start from the source node. In Eq. 6, S is the branch elimination constraint, and Eq. 7 is an expression of S [22], which means that for any node in the travelers’ paths, any of its subsets must be connected to the other subsets in the solution. Advances in Production Engineering & Management 16(2) 2021 175 Wang, Lu, Shen 3.2 Fitness function In multiple objective optimization problems, the goals often conflict with each other. The most common method is the generation method (including weighting method, constraint method, etc.), interactive method, and hybrid method to solve multiple objectives. The weighting method assigns different weights to multiple goals which can convert the multiple objective optimization problem into a single‐objective optimization problem and makes the simple and easy implemen‐ tation of the problem. Therefore, the weighting method is used in this paper to transform the two objectives optimization problem into a single‐objective optimization problem. (10) ∑ ∑ Where , 1,2, … , . w1, w2 are the weight coefficients, which belong to hyperparameters and the values of coefficients vary with the data set. It can be determined by expert evaluation method or comparison method, etc. Therefore, the fitness function in the genetic algorithm is: 1/ (11) It can be seen from Eq. 11 that the greater the fitness, the stronger the individual adaptability, and the better the represented feasible solution. 3.3 Chromosome coding In the genetic algorithm, the design of the chromosome encoding needs to reflect the genetic characteristics of the individual and should be easy to operate. To express the path more effec‐ tively, a one‐piece decimal coding scheme is adopted in this paper. A chromosome should in‐ clude at least one node except the source node, as shown below: Take ten nodes and three trav‐ elers as an example, use 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 as it respectively represents ten nodes, of which 0 is the starting node of the three travelers. Then the path of a traveler can be expressed as: 0‐1‐ 2‐3‐4‐5‐6‐7‐8‐9. To represent multiple travelers, virtual nodes 10 and 11 are inserted to represent the starting node of the route, which is also the source node. Then, the new chromosome can be formed as: 0‐1‐2‐3‐4‐5‐10‐6‐7‐11‐8‐9. Taking virtual nodes 10 and 11 as the path dividing nodes, then the three paths can be ex‐ pressed as: (1) 0‐1‐2‐3‐4‐5‐0, (2) 0‐6‐7‐0, (3) 0‐8‐9‐0. 3.4 Polar coordinate classification method initialize population The quality of the initial population plays an important role in the intelligent group optimization algorithm. For mTSP with a single starting node and closed cycle, it is a common optimization method to firstly use the clustering method to optimize grouping, and then use a heuristic algo‐ rithm to optimize the order within the group. In 2019, Min et al. [23] proposed a method using MRISA (multi‐restart‐iteration sweeping) and tabu search to solve vehicle routing problem. This two‐stage algorithm also can be described as clustering by distance in polar coordinates and optimization by tabu search. Then in the same year, they improved this algorithm by adding an adjust operation named put in & put out. This improved algorithm has three stages: cluster the customer points, use put in & put out operations to adjust the load demand and use tabu search to solve the TSP [24]. Traditional clustering algorithms such as K‐means clustering or fuzzy C‐ means clustering and other clustering methods based on centroids (although they can quickly cluster the set of nodes that need to be visited) can’t well represent the starting point as the origin where each traveler’s sub‐path is distributed around the same starting point. Therefore, a classification method based on polar coordinates is proposed in this paper. The specific operation process of the polar coordinate classification algorithm to generate the initial population is as follows. First, establish a polar coordinate system with the starting node as the origin and map node‐set (n nodes) into the polar coordinate system then sort each node by the angle in the polar coordinate system. Then calculate the two nodes with the furthest an‐ gular distances and select one of them to connect with the origin to establish a new polar coor‐ 176 Advances in Production Engineering & Management 16(2) 2021 Improved Genetic Algorithm (VNS‐GA) using polar coordinate classification for workload balanced multiple Traveling … dinate system axis. After that use the new axis as the starting position to rotate clockwise, and use the traveler number (m travelers) to equally divide the number of nodes that the travelers need to visit. The nodes in the area swept during the rotation are the node‐set to be visited by a traveler, and m initial paths (that is, every traveler roughly needs to visit n/m nodes) constitute an initial feasible solution. Finally, using chromosome fragment flipping as neighborhood shak‐ ing to generate enough chromosomes as the initial population. The process of the polar coordi‐ nate classification algorithm is as shown in Algorithm1: Algorithm1：Polar Coordinate initialize population Input: The number of population p, the number of salesmen n, cartesian coordinates of the cities (xi,yi) and the number of cities m Output: Initialized population k=1 (θi, di): change cartesian coordinates (xi, yi) to polar coordinates and sort the cities by the angle θi max(θ, d): calculate the maximum angle difference among the cities and build new polar coordinate by joining the origin city and one of maximum angle difference cities s0: generate a solution by spinning clockwise and each salesman visits n/m cities repeat s: neighborhood shaking s0 k=k+1 until k>p end 3.5 Neighbourhood actions in VNS Cross‐recombination and mutation operators are often used in genetic algorithm to generate new populations. To enable the algorithm in finding a more efficient combination that makes the travelers’ workload roughly balanced, the distance comparison insertion operator cooperates with cross‐recombination (neighbourhood action 2) and mutation operater (neighbourhood action 3) is designed in this paper to produce the next generation more efficiently. Neighbourhood action 1: Distance comparison operator to find a better solution for making the workload balance faster. The neighbourhood action adopted in this paper is to randomly select a node from the sub‐path with the longest distance and insert it into the sub‐path with the shortest distance. As shown in Fig. 1: Assume that path 0‐1‐2‐3‐4‐5‐0 has the longest distance among all sub‐paths, and 0‐6‐7‐0 is the path with the shortest distance among all sub‐paths; In the domain action, randomly select a node in the longest path (select node 3) and insert it into the shortest path to form a new chromosome. If the fitness of the new chromosome increases after inserting the selected node, the new chromosome will then be retained. Otherwise, the paternal chromosome will be retained and this inferior solution will be recorded to prevent du‐ plication. If the fitness of the new chromosome increases after inserting the selected node, suggests that the new offspring chromosomes generated after the neighbourhood action are: 0‐1‐2‐4‐5‐ 10‐6‐3‐7‐11‐8‐9. Fig. 1 Schematic of gene insertion in neighbourhood action 1 Advances in Production Engineering & Management 16(2) 2021 177 Wang, Lu, Shen To avoid repeated calculations, only the difference between the distance of the sequence is calculated each time in the calculation process. Assuming that the selected node is the node at the i‐th position in the n‐th path, and the insertion position is the node at the j‐th position in the m‐th path, then actual distance to be calculated is d = dn – (di‐1,i + di,i+1) + di‐1,i+1 – (dm + dj‐1,j + dj,j+1 – dj‐1,j+1). Where dn, dm are the path length before path n and m to perform the neighbourhood ac‐ tion, i – 1 and i + 1 are the previous and next nodes of i in path n, the same as j – 1, and j + 1 is the previous and next nodes of j in path m, respectively. d ≥ 0 means that the fitness is not increased after performing the neighbourhood action, otherwise, it also means that the fitness is increased, and the current solution is better than the parent solution. Neighbourhood action 2: local crossover operator. Neighbourhood action 1 can make the off‐ spring develop in the direction of workload balance, but it is easy to fall into the local optimum. Therefore, an improved crossover operator to avoid the offspring from falling into the local op‐ timum is proposed in this paper. To avoid repeated fragments or missing fragments in the offspring that are generated after the crossover and for quickly completing the crossover, we first randomly select a sample gene fragment from the parent 2 and inserts it into the offspring at the original position. Then traverses parent 1 to find the genes that are different from the sample gene fragment and insert‐ ed them into the offspring in sequence, the genes that are the same as the sample gene fragment is skipped. The crossover process is shown in Fig. 2. Neighbourhood action 3: mutation operator. Judging from the characteristics of the traveling salesman problem, there should be no cross edges in the route of the optimal solution. Based on this feature, as shown in Fig. 3 (a is before mutation, b is after mutation), the destruction & re‐ construction method is used to mutate individuals to reduce crossover between paths. Suppose the path set is S = {s1, s2,...,si,...,sk,...,sn}, then sort the distance matrix by distance to get sorted node pairs and randomly selecting a node p in a path si and interchange it with the nearest neighbour node m in the nearest neighbour route sk to reduce the intersection between paths. If the fitness increases, update the current solution; otherwise, it will not update and record the node pair to node pair distance matrix for avoiding repeated exchanges. Fig. 2 Cross process diagram Fig. 3 Mutation operator diagram 3.6 Generate offspring We introduced the idea of searching the neighbourhood structure alternately in VNS to expand the search space of GA, along with the elite retention strategy which is used for retaining the dominant individuals to improve the convergence speed. The specific process of generating the next population is firstly for calculating the fitness of each chromosome in the current population and record the optimal individual. Then, use the rou‐ 178 Advances in Production Engineering & Management 16(2) 2021 Improved Genetic Algorithm (VNS‐GA) using polar coordinate classification for workload balanced multiple Traveling … lette to select separate chromosomes and search in the neighbourhood structure Nk (k = 1,2,...,n) to find a better solution. The better solution will be added to the progeny population and k will be set to 1. If any better solution cannot be found, skip to the next neighbourhood, and repeat the process of selection and neighbourhood descent until the size of the population reaches max‐ imum. At this time, individuals with greater fitness can be selected more in the selection process and the individual’s neighbourhood space can also be fully searched. The pseudo‐code of the generating next population algorithm is shown in Algorithm2: Algorithm2: Variable neighbourhood descent to generate next generation Input: origin population S0 Output: new population S s0: optimal solution of origin population k =1 add s0 to S Sn = 1 repeat s: roulette a solution from S0 repeat s’: find best neighbour s’ if f(s’) > f(s) then add s’ to S k =1 else k = k +1 until k = kmax until Sn = Smax end 3.7 Algorithm flow The VNS‐GA algorithm process is as below: 1) Use polar coordinate classification algorithm to generate initial population S0; defines n neighbourhoods and denoted as Nk (k = 1,2,...,n). 2) Verify the feasibility of the initial population, and use neighbourhood shaking to eliminate infeasible solutions. 3) Calculate the fitness of individuals in the population, and select the current optimal solu‐ tion s. 4) The variable neighbourhood descent process: search optimal solutions in neighbourhood structure N1, if it finds better solution s' than s, then let s = s'. 5) If no better solution can be found in the neighbourhood structure N1, then let k + 1 and go to step 4. 6) Add the current optimal solution to the next‐generation population; 7) Repeat steps 4‐6 until the number of individuals in the population reaches the maximum value. 8) Repeat step 7 until the maximum number of iterations is reached, then output the current optimal solution. 4. Results and discussion To verify the performance of VNS‐GA, first, we quoted the data of the Chinese Traveling Sales‐ man Problem (CTSP) in literature [25]. Use Python code to test it on a computer with the operat‐ ing system Windows10, I5‐8250, 8G RAM. The algorithm runs ten times independently on each case set. For the sake of fairness, the same genetic algorithm parameters as in literature [25] are Advances in Production Engineering & Management 16(2) 2021 179 Wang, Lu, Shen used, and the population size is set to 50, the heredity is 1000 generations, the crossover proba‐ bility is 0.8, and the mutation probability is 0.15. Because the results obtained using SA in litera‐ ture [25] are wrong. According to the detailed path diagram obtained by the simulated annealing algorithm in the literature, the results are revised as shown in Table 1. In the comparison of the three algorithms, the traditional genetic algorithm performs the worst among the three algorithms because of the problem easy falling into the local optimal so‐ lution. After using the variable neighbourhood algorithm to improve the genetic algorithm, the effect is promoted. Compared with the detailed path diagram given in [25] (Fig. 4), it can be stat‐ ed that the improved genetic algorithm is consistent with the nodes set assigned by the simulat‐ ed annealing algorithm in the literature. But the improved genetic algorithm has better perfor‐ mance than the genetic algorithm on a single path, so, the total obtained path is shorter. In addition, this paper selects three data sets: eil51, kroA100, and kroB150 in TSPLIB to test the performance of the algorithm. The performance of the algorithm under the conditions of 3, 5, 10, and 20 travelers is tested against the scale of three data sets, and the parameter settings used in each data set are shown in Table 2 through experiments. We compared the measured data set results with the test results of other algorithms (data from literature [14] and [26]). The results are shown in Table 3, where n is the number of nodes and m is the number of travelers. GA1C is single chromosome coding genetic algorithm, GA2C is the two‐chromosome coding genetic algorithm, GA2PC is two‐segment chromosome coding ge‐ netic algorithm, GGA‐SS is steady‐state grouping genetic algorithm, TCX is improved two‐ segment chromosome coding genetic algorithm, and RGA is belt Genetic algorithm with repro‐ duction mechanism, RLGA is a genetic algorithm that integrates the reproduction mechanism of the weed algorithm and local optimization, IWO is the invasive weed algorithm, and VNS‐GA is the improved genetic algorithm proposed in this paper. Table 1 Comparison results between VNS‐GA and other algorithms (CTSP data set) Algorithm Total distance(km) Sub‐path distance(km) 6106 GA 20225 6477 7643 Origin Modified Origin Modified data data data data 5527 5527 SA 17731 17404 5616 4909 6578 6968 6968 4658 VNS‐GA 17153 5527 Variance 643261 Modified variance 1116314 1361290 Fig. 4 The specific solution path of VNS‐GA and SA Standard data set eil51 kroA100 kroB150 180 Table 2 Parameter setting of different examples Number of travel‐ Population size Cross rate ers 3, 5, 10 60 0.8 3, 5, 10, 20 80 0.8 3, 5, 10, 20 80 0.8 Mutation rate 0.2 0.2 0.2 Advances in Production Engineering & Management 16(2) 2021 Improved Genetic Algorithm (VNS‐GA) using polar coordinate classification for workload balanced multiple Traveling … Table 3 Comparison of the longest path (VNS‐GA with other genetic algorithms) Data 51 100 150 n 51 51 51 100 100 100 100 150 150 150 150 m 3 5 10 3 5 10 20 3 5 10 20 GA1C 234 173 140 14722 11193 9960 9235 19875 15229 12154 10206 GA2C 275 220 165 16229 11606 10200 9470 21067 15450 12382 10338 GA2PC 203 164 123 13556 10589 9463 8388 19687 14748 11158 10044 GGA‐SS 161 119 112 8542 6852 6370 6359 13268 8660 5875 5252 TCX 203 154 113 12726 10086 7064 6402 18019 12619 8054 5673 RGA 174 125 122 10231 7895 6234 6233 14886 8998 5723 5372 RLGA 167 118 112 10115 7812 6225 6211 14629 8927 5613 5251 IWO 160 118 112 8509 6767 6358 6358 13168 8479 5594 5246 VNS‐GA 165 121 112 8613 6445 5764 5395 10878 7711 5937 5750 D (%) 3.13 2.54 0.00 1.22 4.76 7.41 13.14 17.39 9.06 6.13 9.61 The data given in Table 3 are the values of the longest sub‐path. Under the three data sets and different traveler numbers circumstances, the results of VNS‐GA are far superior to the GA1C, GA2C, and GA2PC algorithms. The comparison with the GGA‐SS algorithm shows that only in the case of 3 travelers in the kroA100 data set, the results are slightly worse and the remaining re‐ sults are better than the results of GGA‐SS. When compared with the TCX algorithm, the results are slightly worse only in the case of 20 travelers in the kroB150 data set, and the other results are better than the TCX algorithm. In the case of the kroB150 data set of 10 and 20 travelers, the results are slightly worse than RGA and RLGA algorithms. We use D to visually demonstrate the improvement or lack of results where D = (best_result_among_other_genetic_algorithms – VNS‐ GA_result)/best_result_among_other_genetic_algorithms × 100 %. Thus, D > 0 means the result is improved and is the best result among all genetic algorithms while D < 0 means the result is worse than the best result. To visualize how the performance of the algorithm changes with the number of travelers increase, we introduce the parameter . is the average of different num‐ bers of travelers (i is the number of travelers and i = 3, 5, 10, 20). As can be seen from Fig. 5, with the increase of travelers’ number, the performance of the algorithm decrease, but in general, the performance of the algorithm is improved compared with other genetic algorithms. Fig. 5 VNS‐GA performance variation with the number of travelers Table 4 Comparison of total path (VNS‐GA with other genetic algorithms) Data 51 100 150 n 51 51 51 100 100 100 100 150 150 150 150 m 3 5 10 3 5 10 20 3 5 10 20 GA1C 529 564 801 27036 29753 36890 62471 46111 49443 59341 94291 GA2C 570 627 879 30972 44062 65116 95568 48108 51101 64893 100037 GA2PC 543 586 723 26653 30408 31227 54700 47418 49947 54958 73934 Advances in Production Engineering & Management 16(2) 2021 GGA‐SS 449 479 584 22051 23678 28488 40892 38434 39962 44274 56412 TCX 492 519 670 26130 28612 30988 44686 44674 47811 51326 62400 IWO 448 478 583 21941 23319 27072 38357 38055 38881 42462 53612 VNS‐GA 495 552 1100 25524 31222 49606 101235 31960 35754 55505 106917 181 Wang, Lu, Shen The data in Table 4 are the values of the total path. The balance degree needs to be compared with the longest sub‐path value and the total path value. Although on some data sets, the value of the longest path in the solution obtained by the VNS‐GA algorithm is slightly higher than other algorithms, literature [14]is not multiple objective optimization and the objective of RGA and RLGA algorithms in the literature [26] is to make the longest sub‐path the shortest, neither pro‐ vide the total path length and the balance degree. Therefore, we propose the balance ratio (R) to measure the workload balance. The calculation formula is (longest sub‐path length ‐ shortest sub‐ path length)/average length of each sub‐path × 100%, given the total path length of each stand‐ ard data set in the case of different traveler numbers and the calculation results of balance de‐ gree are shown in Table 5. It can be seen from Tables 3 to 5 that in comparison with other algorithms, the VNS‐GA can ensure that the value of the total path is within an acceptable range while ensuring that the val‐ ue of the longest sub‐path is shortest. Thus, it can be seen as VNS‐GA can find solutions with bet‐ ter balance and shorter total path length. Fig. 6 is a detailed path diagram of the optimal solution of the VNS‐GA algorithm after running on each data set. It can be intuitively seen from the figure that the number of nodes contained in each path is roughly equivalent, and the travel distance required for each traveling salesman is roughly the same. In the case of different numbers of travelers, the VNS‐GA has achieved good results when ex‐ perimenting on several standard data sets. It indicates that the VNS can better solve the situation where GA is easy to fall into local optimality. VNS has a deeper search of the solution space, and the insertion operator designed in the neighbourhood action can optimize the workload balance problem so that the solution can be developed in the direction of equilibrium and the total path is shorter. Data 51 100 150 n 51 51 51 100 100 100 100 150 150 150 150 Table 5 Balance of VNS‐GA on standard data set m Total path Longest path 3 495 165 5 552 121 10 1100 112 3 25524 8613 5 31222 6445 10 49606 5764 20 101235 5395 3 31960 10878 5 35754 7711 10 55505 5937 20 102650 5754 R 0% 29.04 % 4.5 % 2.60 % 9.40 % 22.48 % 26.90 % 4.76 % 17.98 % 11.03 % 14.50 % Fig. 6 Path diagram of VNS‐GA on standard data set 182 Advances in Production Engineering & Management 16(2) 2021 Improved Genetic Algorithm (VNS-GA) using polar coordinate classification for workload balanced multiple Traveling … 5. Conclusion In current times, the research on TSP has been relatively mature, but the mTSP of workload balance involves more constraints and the problem is more complicated. Thus, there are relatively few studies. This paper proposes an improved genetic algorithm (VNS-GA) to solve mTSP with workload balance. Aiming at the goal of workload balance in the mTSP, firstly, the polar coordinate classification algorithm is designed to reduce the intersection between paths and quickly obtain better initial solutions. Then, a distance comparison insertion operator is designed, which specifically takes out the node in the longest path and inserts it into the shortest path to achieve the goal of workload balance faster and to improve the efficiency of the algorithm. To avoid the genetic algorithm falling into the local optimal solution prematurely, the variable neighbourhood descent process is introduced to generate offspring, and different neighbourhood actions are used to search alternately the neighbourhood solution space adequately. Finally, the algorithm is tested on the standard data set of TSPLIB. The experimental results showed that the improved genetic algorithm (VNS-GA) is very competitive. VNS-GA is superior to other improved genetic algorithms on small and medium-sized example sets especially when the number of travelers is small. At the same time, there are still a lot of crossed paths in the detailed path graph, which indicates that the algorithm still has room for improvement. In the future, other neighbourhood actions with better performance can be introduced to increase the space for the search of the algorithm and to improve the search efficiency. Secondly, how to find a better solution when the number of travelers is large remains to be studied further. In addition, the performance of the algorithm on very large data sets still needs further study. Acknowledgement We thank the Editor and two anonymous referees for their many helpful comments on an earlier version of our paper. 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Research on multiple traveling salesman problem based on improved genetic algorithm, Master Thesis, Jiangnan University, Jiangnan, China. 184 Advances in Production Engineering & Management 16(2) 2021 Advances in Production Engineering & Management ISSN 1854‐6250 Volume 16 | Number 2 | June 2021 | pp 185–198 Journal home: apem‐journal.org https://doi.org/10.14743/apem2021.2.393 Original scientific paper Change impact analysis of complex product using an improved three‐parameter interval grey relation model Yang, W.M.a, Li, C.D.a,*, Chen, Y.H.b, Yu, Y.Y.a,* a School of Management Jinan University,Guangzhou, P.R. China International Business School, Jinan University, Zhuhai, P.R. China b ABSTRACT ARTICLE INFO Change impact evaluation of complex product plays an important role in controlling change cost and improving change efficiency of engineering change enterprises. In order to improve the accuracy of engineering change impact evaluation, this paper introduces three‐parameter interval grey num‐ ber to evaluate complex products according to the data characteristics. The linear combination of BWM and Gini coefficient method is used to improve the three‐parameter interval grey number correlation model. It is applied to the impact evaluation of complex product engineering change. This paper firstly constructs a multi‐stage complex network for complex product engi‐ neering change. Then the engineering change impact evaluation index system is determined. Finally, a case analysis was carried out with the permanent magnet synchronous centrifugal compressor in a large permanent magnet synchronous centrifugal unit to verify the effectiveness of the proposed method. Keywords: Manufacturing; Engineering; Complex product; Change impact analysis; Three‐parameter interval grey number; Grey relational model; BWM method (best‐worst model); Gini weighting method *Corresponding author: licd@jnu.edu.cn (Li, C.D.) 3035905183@qq.com (Yu, Y.Y.) Article history: Received 10 April 2021 Revised 23 May 2021 Accepted 5 June 2021 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 There are increasingly fierce competition among complex product manufacturing enterprises in the rapidly changing market environment. In order to improve competitiveness and meet the changing needs of customers for engineering change, companies inevitably face more and more complex engineering changes. When engineering changes occur, many structures of complex products will be affected. The management of engineering change is roughly divided into four stages: engineering change application, engineering change process impact analysis and evalua‐ tion, engineering change decision and approval, and engineering change implementation. In these four stages, the analysis of the engineering change impact can not only be used to deter‐ mine the necessity of change implementation, but also provide guidance for the formulation of change decision and strategies [1‐4]. It is of great significance to control the cost of change and 185 Yang, Li, Chen, Yu improve the efficiency of change, especially to consider the multi‐stage impact when evaluating the impact of engineering change. Many studies have done some research on the production of complex products [5‐9]. Howev‐ er, due to the complexity of parts, the complexity of disciplines and the heterogeneity of knowledge, as well as the difficulty of data acquisition, there is great opacity in the process of change impact evaluation. Therefore, this article improved three‐parameter interval grey rela‐ tional and applied to the evaluation of the impact of engineering changes. Many scholars have studied the impact evaluation of engineering change. It mainly includes the evaluation of change impact scope and the change impact degree. Based on the weighted network theory, (Cheng and Chu, 2012) proposed three variable indexes (degree variable, reachable variable and interval variable) [9]. The degree variability is used to calculate the im‐ pact of direct change by Ahmad et al. (2013) studied a cross domain approach to decompose design and identify possible change propagation links, supplemented by an interactive tool to functions, components and detailed design process [10]. Chen et al. (2015) proposed an assess the impact of changes. This method considered the information domain of requirements, object‐ oriented method, and described its components and related requirements by attributes and links, so as to model the integrated content of products and perform CIA tasks in variant product design [11]. Maazoun et al. (2016) proposed an automatic method to analyze the evolution of feature change model, tracked their impact on SPL design, and provided a set of suggestions to ensure the consistency of the two models [12]. Gong et al. (2021) analyzed the problems existing in modern product packaging and its impact on the ecological environment, and summarized the design methods of modern green packaging [13]. Zheng et al. (2020) put forward the evaluation method of change propagation probability based on grey comprehensive relational analysis and the evaluation method of change propagation impact probability based on configuration change value analysis [14]. (Li and Zhao, 2014) proposed an engineering change scheduling method which combined change propagation simulation with optimization algorithm in complex prod‐ uct development process [15]. Ma et al. (2016) established an engineering change analysis mod‐ el based on the design attribute network and defined the influence of change propagation on the intensity of change propagation through the quantification of change propagation influence fac‐ tors[16]. Maldini et al. (2019) proposed methods to assess the impact of such approaches and applied them to the specific case of "product personalisation” [17]. (Zhang and Yang, 2019) con‐ structed a complex product design structure‐task network evolution model under the influence of engineering changes, and analyzed the impact of changes on design tasks[18]. Maurya et al. (2017)targeted such dependencies and non‐creative hindrances at concept generation stage through a mixed reality implementation. They established requirements for creating a suitable design‐tool and presents a proof‐of‐concept use‐case [19]. Palumbo et al. (2018) presented a method of achieving accurate Life cycle assessment results, which helps with decision‐making and provides support in the selection of building products and materials. [20]. Li et al. (2020) established an engineering change risk propagation model based on load capacity [21]. Some of the existing studies have studied the change risk of change evaluation, and the change propagation. Then based on the multi‐stage complex network model, this paper analyzes the multifaceted change propagation impact from the aspect of change propagation path. In ad‐ dition, there are many parts in complex products and their relationship is complex. The relation‐ ship of each stage and parts is dynamic under the influence of engineering changes. Engineering change involves a series of activities such as product design or process, related documents, components or assembly, self‐made or purchased parts, production process and even suppliers. The acquisition of engineering change data for complex products is more difficult and the data is poor, with greater ambiguity. Therefore, this paper adopts the three‐parameter grey relational model based on BWM and Gini coefficient method to evaluate and analyze the impact of engi‐ neering change propagation. Many scholars have studied three‐parameter interval grey number decision model [22‐27]. There are also some researches on the index weight of grey relational model. (Yin and Ren, 2018) and Liu et al. (2020) respectively introduced entropy weight method into grey relation analysis to study the risk evaluation of tunnel, the representative volume evaluation of concrete 186 Advances in Production Engineering & Management 16(2) 2021 Change impact analysis of complex product using an improved three‐parameter interval grey relation model and the comprehensive analysis of the influencing factors of gas outburst [28, 29]. Based on en‐ tropy TOPSIS grey relational method, Gu et al. (2020) studied the path selection of the evalua‐ tion of the opening level of coastal cities in China and the evaluation of the implementation effect of TCM standards [30]. (Li and Zhu, 2019) studied the grey relational decision model based on AHP and DEA [31]. Based on the sensitivity and grey relational degree, Zhou et al. (2017) pro‐ posed a model based on combining weights and gray correlation analysis [32]. So the index weights of different schemes should be different. Therefore, this article combines the advantages of subjective and objective to comprehensively empower it and apply it to the evaluation of complex product engineering changes. Complex products have the characteristics of many parts, a wide range of technical fields, complex component interfaces, one‐off or even small batch production, and many supporting suppliers. It is different from general mass‐manufactured parts and technical fields. The engi‐ neering change process of complex products need huge human, material and financial resources. If a predictive analysis of the possible impact scope can be made, manufacturing enterprise can avoid the waste of cost, and further accelerate the product development and production cycle. In this paper, the multi‐stage complex network topology is firstly established for complex product engineering change, then the engineering change impact evaluation system is established. Final‐ ly, the proved three‐parameter interval grey relational model is used to evaluate the impact of engineering change. The framework is shown in the Fig. 1. Fig. 1 Framework of the proposed complex product engineering change impact evaluation method Advances in Production Engineering & Management 16(2) 2021 187 Yang, Li, Chen, Yu 2. Multi‐stage complex network Engineering changes occur in product design, process, manufacturing and other stages. When these changes occur, it is necessary to respond at any time to achieve real‐time change design response, rather than follow a fixed process of production. All processes will be affected when the change occurs. It is very vital about how to effectively collect, organize and manage scattered product engineering change knowledge, and use existing domain knowledge to ensure the integ‐ rity of product assembly structure. As an important part of the re‐generation of product design schemes after product changes, it is critical to timely feed back the engineering change infor‐ mation to the design department. This article builds a multi‐stage complex network based on design libraries, knowledge bases, and case libraries. The propagation characteristic of design change will make the simple parameter change of any part may cause the chain change, and even lead to the avalanche effect of change propaga‐ tion, which will bring various negative effects to the enterprise. A reasonable and effective com‐ munication path of design change can provide decision support for designers to implement de‐ sign change, help improve product quality, shorten R & D time and reduce design cost. In addi‐ tion, there are many professional categories of complex products, difficult processing technology, long manufacturing process and complex supporting relationship of parts. Complex products involve multiple processes in the process of engineering change. While continuously shortening the development cycle and improving the product development quality, it tends to the close co‐ ordination of design, process and manufacturing process. We integrate the complex network relationship of design, process and manufacturing process in the production process of complex products, so as to comprehensively consider the changes in each stage. The network of each stage is constructed according to its process knowledge and knowledge base. The construction process is shown in Fig. 2. , , , 1,2, . . . . If there The single stage network is represented as: 1, else, , 0. When the multi‐stage are connecting edges between parts knowledge, the , network is calculated, the connected edges and edge weights of its indexes are added and pro‐ ∑ cessed. For the same connected edge, the weight value is , , . The schematic diagram of the multi‐stage network is shown in Fig. 3. The high speed permanent magnet synchronous variable frequency centrifugal high power chiller of G enterprise is taken as an example. Fig. 2 Multi‐stage complex network of complex product Fig. 3 Multi‐stage complex knowledge network 188 Advances in Production Engineering & Management 16(2) 2021 Change impact analysis of complex product using an improved three‐parameter interval grey relation model 3. Construction of engineering change evaluation index 3.1 Engineering change propagation intensity evaluation The propagation intensity of engineering change defined in this paper includes node proximity, edge betweenness and propagation probability. Node proximity: Node proximity is the reciprocal of total distance from the node to all other nodes: ∑ . Where is the number of edges in the shortest path from the start node to the end node , and is the total number of nodes. Node proximity describes the degree a node is to the center of a network. The larger the value, the more important the node is. Edge betweenness: Edge betweenness is defined as the ratio of the number of paths passing through the edge to the total number of shortest paths in the network. Edge betweenness test is an important index to measure the role of connected edges in the whole network. The edge be‐ tweenness is expressed as: , ∑ ∑ ℎ, , ℎ, / 1,2, . . . , , ℎ ℎ, , (1) Propagation probability of connected edges: , is the probability of propagation from node to node . If node j doesn't belong to the next connected edge, the propagation probability is 0. It is easier to pass through this connecting edge when the propagation probability of the edge is greater. It can be expressed as: | ∩ | (2) Then, the change propagation intensity can be expressed as: 1 ， 0 0, 0 (3) 3.2 Engineering change cost Engineering changes of different parts will require different change costs. Changes in compo‐ nents can be mapped to changes in nodes in the network model (node addition and deletion). Therefore, we can evaluate the impact of customer demand change on complex product change by calculating the change cost of node change (node addition and deletion) in the network, the ∑ , where is the change cost of node change in a network can be expressed as: cost of node change, is the total number of change nodes. The cost details involved in the product production process include: (1) Material cost: It re‐ fers to the cost of product standard consumption, supporting raw materials, product accessories and various materials used for production or providing services. It mainly includes the purchase price, related taxes, freight, loading and unloading fees, insurance premiums and other costs that can be directly attributable to the acquisition of materials. (2) Labor cost: It refers to the remu‐ neration and other expenses paid to employees which in order to obtain the services provided by employees. It mainly includes the salary, bonus, allowance, welfare, education fund and so on. (3) Manufacturing cost: It refers to energy consumption, manufacturing accessories, labor insur‐ ance, office and fixed expenses. (4) Others: Some consumption including fuel cost, power cost, office cost and depreciation consumed by each production unit. 3.3 Engineering change time In the process of engineering, change of different parts needs different change time. The node change time in a network can be expressed as: ∑ Where is the change time of node change. , (4) is the total number of change nodes. Advances in Production Engineering & Management 16(2) 2021 189 Yang, Li, Chen, Yu 3.4 Engineering change profit It refers to the positive impact obtained in the change process, such as customer satisfaction, product performance improvement and so on. The engineering change profit can be expressed ∑ . Where is the engineering change profit of node change. is the total num‐ as: ber of change nodes. The impact evaluation indexes are as shown in Table 1. Table 1 Evaluation indexes of engineering change impact Primary index Secondary index Tertiary indicators Node betweenness Node degree Node proximity Edge betweenness Propagation probability of connected edges Material cost Labor cost Manufacturing cost Manufacturing cost Node change time Edge propagation time improvement of customer satisfaction improvement of product quality Change propagation intensity Engineering change impact evaluation Engineering change cost Engineering change time Engineering change profit 4. Grey relational evaluation model based on three‐parameter interval grey number 4.1 Three‐parameter interval grey number From the definition of three‐parameter interval grey number, it can be known that it refers to the interval grey number where the center of gravity point with the greatest possible value is ̄ , 、 ̄ are the upper and lower , , ̄ , where known. It can be marked as ⊗ limits of the interval respectively. is called the "center of gravity" point (Li and Zhang, 2020) [37]. When two of the three parameters , , ̄ are the same, the three‐parameter interval grey ̄ ,the three parameter interval number degenerates to the interval grey number. When grey number degenerates to the real number. In fact, the interval grey number and the real number are special cases of the three‐parameter interval grey number. Its algorithm is similar to interval grey number. Let three‐parameter interval grey number , , ̄ , ⊗ , , , then ⊗ ⊗ ⊗ ⊗ ∈ , ⊗ =[ ̄ , , , ⊗ ̄ , ̄ ̄ , , , ̄] , , , , ̄ (5) (6) (7) 4.2 Three‐parameter interval grey number grey relational model Suppose that there are n alternative engineering change schemes. They constituted by evalua‐ tion schemes set , ,⋯, , ,⋯, . The index set is composed of m attrib‐ ⊗ utes. The index value of scheme under the evaluation index can be expressed as , , ̄ ̄ , 1,2, ⋯ , ; 1,2, ⋯ , . The effect evaluation vector of each ⊗ ⊗ , ⊗ ,⋯, ⊗ , 1,2, ⋯ , . The weight of index under each scheme is ∑ 1 1,2, . . . , . There are different attribute indexes scheme is , ,⋯, , and ij with different dimensions and measurement standards. In order to increase the comparability of 190 Advances in Production Engineering & Management 16(2) 2021 Change impact analysis of complex product using an improved three‐parameter interval grey relation model alternatives, it is necessary to normalize the effect evaluation vector of decision alternatives. In this paper, we use the range transformation method to normalize the decision matrix. For profitable attribute values: , ̄∗ ̄ , ̄ ̄∗ (8) ̄∗ For cost attribute values: ̄∗ ̄ ̄∗ Where ̄ ∗ , ̄ ̅ 1 ̄∗ , ̄∗ , 1 ̄∗ , ̄ ̄ (9) ̄∗ 1,2, … Let the normalized effect evaluation vector be: ⊗ Where ⊗ ∈ Recorded that ̄ ̄ , , ̄ 1,2, ⋯ , ⊗ , ⊗ ,⋯, ⊗ , 1,2, ⋯ , (10) is a three‐parameter interval grey number in 0,1 . ̄ , , , ̄ , , . Then the m‐dimensional three‐parameter non negative interval grey number vectors ⊗ ⊗ , ⊗ ,⋯, ⊗ , ⊗ ⊗ , ⊗ ,⋯, ⊗ (11) are called ideal optimal scheme effect evaluation vectors and critical scheme effect evaluation vectors respectively. We assume that the grey interval relational degree of the normalized effect evaluation vector ⊗ of scheme with respect to the ideal optimal scheme effect evaluation vector ⊗ , ⊗ . And the grey interval relational degree of critical scheme effect eval‐ ⊗ is ⊗ , ⊗ . Assume that the weights of two grey relational de‐ uation vector ⊗ is grees are ,  2 1 . Then, ⊗ ⊗ , ⊗ ⊗ , 1 ⊗ , 1,2, ⋯ , (12) is the three‐parameter grey interval linear relational degree of the effect evaluation vector ⊗ . (13) ⊗ ⊗ , ⊗ ⊗ , ⊗ 1 , 1,2, ⋯ , is the three‐parameter grey interval product relational degree of the effect evaluation vector ⊗ . The distribution probability of barycenter point with the highest probability of taking the ⊗ ∈ , , ̄ is . Normally, value of three‐parameter interval grey number 60 %. If 60 % it indicates that the decision is wrong, and the most likely value needs to be determined again. Based on the center of gravity, we can build a three‐parameter interval grey number relational degree evaluation model. ⊗ ∈ Definition 1: For three‐parameter interval grey number , , ̄ , then ̄ 1 (14) with respect to is called the three‐parameter grey interval relational coefficient of sub factor. ideal factor . ∈ 0,1 is the resolution coefficient. ∈ 0,1 is the decision preference coeffi‐ cient. Where, , , Advances in Production Engineering & Management 16(2) 2021 ̄ ̄ , 1,2, ⋯ , ; 1,2, ⋯ , 191 Yang, Li, Chen, Yu , , ̄ , , ⊗ , ∑ ⊗ , 1,2, ⋯ , (15) is called the three‐parameter grey interval relational degree of the effect evaluation vector ⊗ about the ideal optimal scheme effect evaluation vector ⊗ . ⊗ ∈ Definition 2: For three‐parameter interval grey number , , ̄ , ̄ 1 (16) is called the three‐parameter grey interval relational coefficient of sub factor. with respect to ideal factor . ∈ 0,1 is the resolution coefficient. ∈ 0,1 is the decision preference coeffi‐ cient. Where, , , ⊗ , ̄ ̄ , , , ̄ , , ⊗ ∑ , 1,2, ⋯ , ; 1,2, ⋯ , 1,2, ⋯ , is called the three‐parameter grey interval relational degree of the effect evaluation vector ⊗ . about the critical scheme effect evaluation vector (17) ⊗ 4.3 Determination of weight At present, scholars attach great importance to the development and application of subjective and objective empowerment methods in the research of evaluation. The subjective weight re‐ flects the subjective willingness of the evaluation subject, and highlights the degree of distinction between the evaluation objects through index data information. The combination of them will make the result more objective. In this paper, the simplified BWM subjective weighting method and the Gini coefficient weighting method which can better reflect the data difference infor‐ mation are selected for combination weighting. 4.3.1 Determination of weight based on BWM BWM(best‐worst method) is a new method to determine the subjective weight of index pro‐ posed by Rezaei in 2014. The most frequently used method in the multiple indexex evaluation is AHP method. In AHP method, any two indexes are usually compared with each other to get the evaluation matrix of indexes, which needs 1 /2 times of comparison. The calculation pro‐ cess of it is complicated and will cause certain errors. However, BWM only needs 2 3 calcula‐ tions by selecting the best and the worst indexes and comparing them with other indexes. It simplifies the complicated process of AHP, greatly reduces the amount of data, reduces the mis‐ takes caused by too much data, makes it easier to pass the consistency test, and improves the reliability. The calculation steps are as follows (Behzad et al. (2020))[33]:  The best index and the worst index set , ,... . are selected according to experts’ opinions in index  Experts use 1‐9 point scale to score and determine the importance of other indexes relative to the optimal indexes. We construct the comparison vector , ..., . rep‐ 192 Advances in Production Engineering & Management 16(2) 2021 Change impact analysis of complex product using an improved three‐parameter interval grey relation model resents the importance of the optimal index compared with index . 1 means is extremely important than . equally important. 9 means and are  We need to determine the unimportance of other indexes relative to the worst indexes and C ,C ,...,C . Where represents the least construct a comparison vector and are equally unim‐ importance of the worst index compared with index j. 1 means portant. 9 means and are extremely unimportant.  From the goal programming model, a mathematical programming formula is established and ∗ solved to obtain the optimal index weight ∗ , ∗, . . . , ∗ . ∑ 0, . . 1,2, . . . 1 (18) Where is the weight of , is the criterion vector. is the weight of . is the weight . represents the importance of to ; . It of represents the importance of to k can be transformed to min , . . ∑ 1,2, . . . , (19) 1 0  Calculate the consistency ratio. The obtained can be represented by ∗ . ratio CR (C1 is the given value) can be obtained from ∗ , and the consistency The closer of the value is to 0, the better the consistency. When it is 0, it is consistent. If there are experts participate in the judgment, the final weight will be calculated by weighted av‐ ∑ erage, and the final weight is ̄ ∗ . 4.3.2 Weight determination method based on Gini coefficient Principle of Gini coefficient weighting method Gini coefficient weighting method is an objective weighting method by calculating Gini coeffi‐ cient of evaluation index and normalizing Gini coefficient of each index. First of all, the different data of evaluation objects of a specific evaluation index can be regarded as the income of differ‐ ent levels people. Then the Gini coefficient of a certain index can be calculated. The value of Gini coefficient can reflect the data difference between different evaluation objects. Then, In order to ensure that weight of all indexes are in the range of 0 to 1 and the sum is 1, the Gini coefficient value of each index will be normalized to get the Gini coefficient weight of the evaluation index. Zahng et al. (2020) [34]. Gini coefficient weight’ calculation of evaluation index is the ith data of the kth index, and μ We assume that is the Gini coefficient of the kth index, K. is the expected value of all data of the kth index. Then the Gini coefficient of the k‐th index is shown as follows: ∑ ∑ ∑ ∑ Advances in Production Engineering & Management 16(2) 2021 /2 / (20) (21) 193 Yang, Li, Chen, Yu Especially, when the mean value of index data is not 0, the Gini coefficient is calculated by the improved formula (13). When the mean value of the index data is 0, the Gini coefficient of the index is calculated by the original formula (14). Gini coefficient of the index truly reflects the data changes of different evaluation objects of the index. Gini coefficient weight of the k‐th index can be obtained by normalizing the Gini coefficient value of each index: / ∑ (22) Where is Gini coefficient weight of the kth index, is Gini coefficient value of the k‐th index, and is the number of indexes. The advantages of Gini coefficient weighting method are as follows: first, the weight calcula‐ tion is not affected by the unit dimension of the index, the definition of Gini coefficient itself eliminates the dimensional influence. Second, Gini coefficient value of the evaluation index re‐ flects the difference between any two evaluation objects. Gini coefficient weight reflects the dif‐ ference between the data of different evaluation objects of an index. And the weight reflects the data information of the index, which meets the requirements of the objective weighting method. 4.3.3 Combination weighting method based on BWM‐Gini coefficient The BWM method determines the index weight according to the subjective preference of the evaluator, and the method of Gini coefficient determines the objective index weight. In order to fully combine the advantages of the two methods, from the subjective and objective point of view, this paper combines BWM method and Gini coefficient method to determine the compre‐ hensive weight of the evaluation index by linear weighting: ∗ 1 ∗ , ∗ ,⋯, ∗ (23) Where ∗ is the comprehensive weight of the decision unit , is the subjective preference coef‐ ficient, 1 is the objective preference coefficient ( ∈ 0,1 ), and the specific value of is given by the decision maker according to personal preference. 5. Case study The high‐speed permanent magnet synchronous centrifugal unit of G enterprise is a high‐tech, high value‐added and complex mechanical product involving multi‐disciplinary and multi do‐ main knowledge. It has high requirements for continuous innovation ability. Centrifugal com‐ pressor is an important part of it, which determines many functions. The product organization diagram and component composition are shown in Fig. 4 and Table 2. The continuous innova‐ tion knowledge of full capacity DC high‐speed permanent magnet synchronous frequency con‐ version centrifugal unit involves many aspects within the enterprise, within the industry and across fields. It has the characteristics of multi domain, high frequency, massive, heterogeneous and complex. Combined with the historical case of common engineering change innovation mode of large capacity full DC high‐speed permanent magnet synchronous variable frequency centrifugal unit and its design process manufacturing process knowledge base, its multi process network is analyzed to determine the evaluation index value of change impact. Fig. 4 Large permanent magnet synchronous centrifugal unit and permanent magnet synchronous centrifugal compressor 194 Advances in Production Engineering & Management 16(2) 2021 Change impact analysis of complex product using an improved three‐parameter interval grey relation model Table 2 Main parts and node name of permanent magnet synchronous centrifugal compressor Node Parts Node Parts V1 mainshaft V13 bend casing V2 impeller rim 1 V14 curved separator V3 roulette 1 V15 refluxer separator refluxer flow V4 blade 1 V16 channels V5 shrink‐ring V17 volute V6 fixed collar V18 impeller rim 2 V7 balance disc V19 roulette 2 V8 reinforcement on the back of impeller V20 blade 2 V9 thrust disc V21 stator winding V10 axle sleeve V22 stator core V11 suction chamber V23 foundation V12 diffuser V24 p‐m rotor It is known that part 4 needs to be improved due to increased customer demand. There are 4 changed routes, and the impact evaluation of the changed routes is carried out. The four routes are as follows: Engineering change node route 1:4‐3‐2‐6‐5‐7‐9; Engineering change node route 2:4‐3‐2‐5‐8‐1; Engineering change node route 3:4‐3‐15‐16‐17‐24; Engineering change node route 4:1‐2‐3‐4‐14‐22. The physical schematic diagram of the change routes are shown in Fig. 5. Fig. 5 Physical schematic diagram of the change routes First of all, we analyze the relationship between process, design and manufacturing network of the direct drive variable frequency centrifugal compressor. The multi‐stage complex network diagram can be referred to Fig. 3. Through the calculation of index system，we can get the three parameter interval grey num‐ ber of the evaluation index as follows: ⊗ 3.38,3.41,3.46 2.97,3.21,3.04 3.23,3.34,4.47 3.18,3.41,3.57 11.21,11.32,11.35 10.71,11.24,11.46 11.05,11.10,11.16 11.28,11.31,11.36 5421,5423,5425 5275,5279,5283 5865,5868,5871 5903,5932,5952 5.98,6.05,6.11 5.76,6.14,6.17 6.03,6.17,6.21 6.11,6.15,6.46 The normalized three‐parameter interval grey number evaluation matrix is: ⊗ 0.59, 0.63, 0.65 0.67, 0.85,1.00 0.00,0.73, 0.74 0.53, 0.55, 0.59 0.61, 0.63, 0.65 0.80, 0.82, 0.83 0.60, 0.65, 0.70 0.67, 0.70,1.00 0.73, 0.81,1.00 0.00,0.73, 0.75 0.72,0.76,0.79 0.00,0.62,0.63 0.64, 0.76,0.79 0.00,0.63,0.69 0.57, 0.58, 0.60 0.70, 0.74,1.00 According to formula (4), the effect evaluation vectors of ideal optimal scheme and critical scheme are obtained: Advances in Production Engineering & Management 16(2) 2021 195 Yang, Li, Chen, Yu ⊗ ⊗ 0.67, 0.85, 1.00 , 0.72, 0.76, 1.00 , 0.80, 0.82,1.00 , 0.70, 0.76, 1.00 0.00,0.55, 0.59 , 0.00,0.62, 0.65 , 0.00, 0.57, 0.60 , 0.00,0.65,0.70 The weight matrix obtained by expert BWM method is as follows: 0.37, 0.16, 0.32, 0.14 The weight obtained from Gini coefficient is as follows: , , , , , , 0.33, 0.11, 0.31, 0.25 Then we can calculate the comprehensive weight. this paper takes the preference coefficient 0.4 can be obtained from formula (17). Then we can get the comprehen‐ 0.4. ∗ 0.6 0.35, 0.14, 0.31, 0.18 , , , sive weight: According to Eq. 8 and Eq. 10, the grey interval relational degree of each scheme with ideal optimal scheme and critical scheme is obtained as follows: ⊗ , ⊗ ⊗ , ⊗ ⊗ , ⊗ ⊗ , ⊗ ⊗ , ⊗ ⊗ , ⊗ ⊗ , ⊗ ⊗ , ⊗ 0 0.88 0.72 0.67 0.80 0.54 0.74 0.80 The three‐parameter grey interval linear relational degree of each scheme is calculated by Eq. 5: ⊗ 0.53, ⊗ 0.67, ⊗ 0.48, ⊗ 0.43. These relational de‐ gree can be expressed as shown in the Fig. 6. According to the linear relational degree of three‐parameter interval grey number, we can find that the most relevant to the ideal optimal scheme is scheme 2. Change node route2 is 4‐3‐ 2‐5‐8‐1:blade1‐roulette1‐impeller rim1‐shrink‐ring‐reinforcement on the back of impeller‐ mainshaft. The physical schematic diagram of the change route 2 is shown in Fig. 7. We can find that the choice of engineering change route is in line with the reality. Fig. 6 Relational degree results Fig. 7 Physical schematic diagram of change route 2 196 Advances in Production Engineering & Management 16(2) 2021 Change impact analysis of complex product using an improved three-parameter interval grey relation model 6. Conclusion In this paper, an improved three parameter interval grey number is introduced into the impact assessment of complex product engineering change, and the knowledge characteristics in the change process are fully considered. Firstly, considering the influence of multi production process, a multi-stage network model considering design, process and manufacturing process is constructed based on complex network. Then the evaluation index system of engineering change impact is constructed. The evaluation method is improved based on the linear weighting method combined with BWM and Gini coefficient. In this paper, the improved three parameter interval correlation model is introduced to evaluate the impact of complex product change, which improves its utilization of uncertain knowledge. This method makes up for the deficiency of the impact evaluation method based on real numbers. In addition, this paper uses the linear weighting method to reasonably balance the subjective and objective weight proportion, so as to make the weight determination more reasonable. In the future research, we can further study the subjective and objective comprehensive weight allocation problem, and apply this method to other research aspects of complex products. On the other hand, we can also consider the behavior, structure and function factors in the design process to improve the multi-stage complex product network. 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Use of fuzzy analytic hierarchy process and environmental gini coefficient for allocation of regional flood drainage rights, International Journal of Environmental Research and Public Health, Vol. 17, No. 6, Article ID 2063, doi: 10.3390/ijerph17062063. 198 Advances in Production Engineering & Management 16(2) 2021 Advances in Production Engineering & Management ISSN 1854‐6250 Volume 16 | Number 2 | June 2021 | pp 199–211 Journal home: apem‐journal.org https://doi.org/10.14743/apem2021.2.394 Original scientific paper Bone drilling with internal gas cooling: Experimental and statistical investigation of the effect of cooling with CO2 on reduction of temperature rise due to drill bit wear Shakouri, E.a, b,*, Haghighi Hassanalideh, H.a, Fotuhi, S.b a Faculty of Engineering, Islamic Azad University‐Tehran North Branch, Tehran, Iran Iranian Tajhiz Sina, its. Co., Tehran, Iran b ABSTRACT ARTICLE INFO Bone drilling is a major stage in immobilization of the fracture site. During bone drilling operations, the temperature may exceed the allowable limit of 47 °C, causing irrecoverable damages of thermal necrosis and seriously threatening the fracture treatment. One of the parameters affecting the tem‐ perature rise of the drilling site is the frequency of applying the drill bit and its extent of wear. The present study attempted to mitigate the effect of drill bit wear on the bone temperature rise through the internal gas cooling meth‐ od via CO2 and to reduce the risk of incidence of thermal necrosis. To this end, drilling tests were conducted at three rotational speeds 1000, 2000, and 3000 r·min‐1 in two states of without cooling and with internal gas cooling by CO2 through an internal coolant carbide drill bit, along with six drill bit states (new, used 10, 20, 30, 40, and 50 times) on a bovine femur bone. The results indicated that in the internal gas cooling state, as the number of drill bit appli‐ cations increased from the new state to more than 50 times, the temperature of the hole site increased on average by ΔT = 2‐3 °C (n = 1000 r·min‐1), ΔT = 5‐ 8 °C (n = 2000 r·min‐1), and ΔT = 5‐7 °C (n = 3000 r·min‐1). Furthermore, the internal gas cooling method was able to significantly reduce the effect of the drill bit wear on the temperature rise of the drilling site and to resolve the risk of incidence of thermal necrosis regardless of the process parameters for drilling operations. Keywords: Bone; Drilling; Thermal necrosis; Tool wear; Internal gas cooling *Corresponding author: e_shakouri@iau‐tnb.ac.ir (Shakouri, E.) Article history: Received 15 March 2021 Revised 18 May 2021 Accepted 19 May 2021 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 Bone drilling is a major operation which is extensively utilized in orthopedic surgery. Since in complicated cases of bone fracture, the internal fixation of fracture site should be performed through screws, wires, and plates, thus creation of a hole in the bone through the drilling pro‐ cess is essential [1]. Drilling operation is considered a type of machining process in which a hole is created through physical contact between the cutting edges of a rotational drill bit and apply‐ ing axial force. As with other machining processes, these operations lead to heat generation in the cutting zone [2]. The major sources of heat generation at the hole site during bone drilling operations are as follows:  The energy exerted to the bone‐tool interface to create a plastic deformation in the bone as well as chip formation;  The friction between the drill bit and hole wall;  The friction between chips and the hole wall. 199 Shakouri, Haghighi Hassanalideh, Fotuhi The heat generated during the bone drilling is evacuated in four different ways [3]:     Part of the heat is cleared off the hole site through the blood and interstitial fluids; Some heat is carried away through bone chips; A portion of the heat enters the tool and elevates is temperature (Fig. 1); Finally, some heat enters the hole site of the bone, causing its temperature elevation (Fig. 1). Fig. 1 Samples of thermographic images of bone drilling: heat generation in the drilling site and temperature rise of bone and drill bit During metal drilling, around 85 % of the heat generated in the process is carried away from the system by chips [4]. However, in orthopedic surgery, as the bone has poor heat capacity and thermal conductivity, the share of the output heat by the chips is far smaller. This means that the heat generated during the bone drilling operations cannot be cleared off the system immediate‐ ly, causing prolongation of the heat retention time in the bone, and eventually local temperature rise. Heat accumulation at the drilling site can cause irrecoverable damages to orthopedic sur‐ gery and patient improvement. The reason is that the local temperature rise can lead to altered nature of the alkaline phosphatase of the bone, incidence of thermal necrosis, cell death, and diminished mechanical strength at the drilling site [5]. The reduction in the mechanical strength of the bone at the hole site may dramatically reduce the success of internal fixation of the frac‐ ture site. It is because in the post‐drilling stage, self‐tapering screws are embedded inside these holes to perform immobilization of the fracture site. Incidence of thermal necrosis, diminished mechanical strength of the drilling site, as well as weakened interaction between the screw and bone, prevent fracture treatment in the desired direction and angle. Thermal necrosis is de‐ pendent on two factors: the extent of temperature rise of the bone and the duration of exposure to that temperature. Some researchers have determined a thermal threshold for osteonecrosis, below which no considerable damage is incurred to the bone tissue. However, beyond this tem‐ perature, the bone cells are affected; this thermal threshold for incidence of necrosis is exposure to 47 °C for 1 min. Henriques has presented a model for thermal damage based on Arrhenius rela‐ tionship. The time‐dependent relation of thermal damage is as follows (Eq. 1): . 273 (1) Where, Ω(t) is the thermal damage, A represents the frequency factor (3.1×1098 s‐1), Ea shows the activation energy (627×103 J.mol‐1), R denotes the universal constant of gases (8.314 J.mol‐1.K‐1), T is the temperature (°C), and t shows the time (s). According to Henriques equation (Eq. 1), per every 1 °C temperature rise, the tolerable time range for the bone diminishes by half, such that beyond 53 °C, thermal necrosis occurs almost immediately [1, 4‐8]. Accordingly, controlling the extent of temperature rise in bone drilling operations is a signifi‐ cant issue of interest to researchers in orthopedic surgery and biomedical engineering. Some researchers have tried to determine the optimum machining conditions to ensure the minimum extent of temperature rise through examining the effect of cutting parameters of drilling opera‐ tions (including the drill bit diameter, drill bit geometry, rotational speed, and feed rate) [9‐12]. In spite of the extensive research so far and their valuable results, it has been found that during 200 Advances in Production Engineering & Management 16(2) 2021 Bone drilling with internal gas cooling: Experimental and statistical investigation of the effect of cooling with CO2 on … conventional drilling operations of the bone, despite the possibility of applying suitable process conditions to minimize the temperature rise, thermal necrosis cannot be prevented definitely. This conclusion has prompted researchers to search for alternative methods for conventional drilling in orthopedic surgical operations. Study on modern techniques including high‐speed machining of bone [3, 13‐15], ultrasonic assisted drilling of bone [16‐21], and abrasive water jet machining of bone [22] has been able to mitigate extreme temperature rise in the cutting zone of the bone to some extent and to minimize the risk of thermal necrosis. Nevertheless, the point is that so far none of the alternative methods of traditional machining for the bone have become commonplace and been used in real surgical operations. This can be due to the following rea‐ sons:  Most of these alternative methods are dependent on special equipment, which currently have only applications, and their usage in orthopedic surgery suffers technical, health and safety, and financial constraints (the equipment includes high speed spindle for high‐ speed machining; transducers and ultrasonic power supply for ultrasonic assisted machin‐ ing; high‐pressure pump and jet nozzle for water jet machining).  In spite of achieving relatively desirable results by alternative machining methods, since heat generation is considered an intrinsic property of machining operations, none of the alternative processes have been able to control temperature rise precisely around the al‐ lowable range (temperature rise <10 °C) and completely eliminate the risk of thermal ne‐ crosis. Thermodynamically, the most effective method of controlling temperature rise in machining operations is applying coolant fluids; in addition to preventing excessive temperature rise (in the tool and workpiece), these fluids help in transferring the heat from the machining system. This has prompted researchers to examine the effect of applying coolant fluids (water, normal saline) on the temperature rise of drilling site during bone drilling operations [23]. For example, in a research by Augustin et al. on porcine femur drilling, it was observed that application of internal water cooling (through the two‐step internal coolant drill) caused the hole temperature not to exceed the critical value of 47 °C and grow to at most 40.5 °C [24]. This achievement clear‐ ly indicated the effective performance of the coolant fluid in controlling the temperature rise of the bone drilling operation. Given the remarkable results of applying coolant fluids in controlling the temperature rise of bone drilling, the idea of employing inert gas cooling methods was developed for drilling opera‐ tions of the bone. In this regard, having created a special drill machine with an internal gas cool‐ ing potential (N2 or CO2) through internal coolant drill bits, Shakouri et al. examined the effect of internal gas cooling on the temperature rise of the bone drilling process. By comparing the ob‐ tained results with the observations of conventional drilling as well as the results of drilling with external normal saline cooling, they found that internal gas cooling can constrain the extent of temperature rise at the drilling site during the bone drilling operations within the allowable range (maximum temperature rise as much as ΔT = 10 °C), thereby completely eliminating the risk of thermal necrosis [25]. The internal flow of the inert gas at the site of drilling first provides adequate cooling for both the drill bit and the bone, while also supporting effective evacuation of heat and bone chips. The only notable point in this method is its dependence on the drill appa‐ ratus with the ability of gas flow passage plus the internal coolant drill bit. Concerning bone drilling, the important point which has been observed in previous research is that with increase in the frequency of applying the bit in orthopedic surgery, it gradually wears and becomes blunt over time. This wear can be considered from different aspects:  Bluntness of cutting edges and the flute edges of the bit;  Changes in the point angle of the bit;  Adhesion of a coating made of the bone mineral matrix to the cutting edges and flutes of the bit, and even clogging the flutes of the bit. Since the extent of sharpness of the drill bit is one of the most important factors affecting the plastic deformation of the material, chip formation, and the cutting efficiency, thus blunt and Advances in Production Engineering & Management 16(2) 2021 201 Shakouri, Haghighi Hassanalideh, Fotuhi worn drill bits require exerting more force for the cutting action, causing excessive frictional heating, and eventually generation of more heat during the drilling [7]. Since the mechanical strength of the bone is far lower than that of metal workpieces, thus the forces required for bone drilling operations are not very considerable. Hence, the first two states of drill bit wear men‐ tioned above occur at a very low rate for the drill bits used in bone drilling, and their effects do not emerge at the low frequencies of applying the drill bit. However, concerning the third state of drill bit wear, considering the temperature rise of the drilling site and evaporation of the bone tissue fluids, incidence of adhesion of a coating made of the bone mineral matrix to the cutting edges and flutes of the bit will not be inevitable even at low frequency of applying the drill bit. In a research conducted by Allan et al. on the effect of drill bit wear on temperature rise of conven‐ tional bone drilling operations, it was found that as the frequency of applying the drill bit in‐ creased, so did the extent of temperature rise at the drilling site [8]. In this regard, Staroveski et al. monitored drill wear during cortical bone drilling, and found that drill wear resulted in in‐ creased temperature and cutting forces [26]. Accordingly, in different studies, to control the course of temperature rise at the site of the drilling and to prevent incidence of thermal necrosis, the maximum allowable frequencies of applying bits has been mentioned as 40 [8, 27, 28]. In some cases, the bit wear challenge has been considered even more critical, and drilling bits have been replaced after 15 times of usage [5]. Since the desirable effect of applying internal gas cooling on significant reduction of tempera‐ ture rise during the bone drilling operations has been observed [25], now this question arises whether the above method can reduce the wear rate of the drill bit and its resulting temperature rise. The aim of the present research is to investigate the effect of frequency of applying drill bit and its resulting wear on the temperature rise of the drilling site and to test the effect of internal gas cooling on the extent of temperature rise resulting from the drill bit wear (from the aspect of adhesion of a coating made of the bone mineral matrix to the cutting edges and flutes of the bit). Another goal is to determine whether internal gas cooling during the bone drilling operations can reduce the drill bit wear‐induced temperature rise. Furthermore, once this internal cooling method is used, the maximum allowable times of applying the drill bit should be tested. The in‐ novation of this research is that so far no documented report has been published regarding the effect of internal gas cooling on the drill bit wear and its resulting temperature rise. 2. Materials and methods Due to the similarity between the mechanical properties of bovine bone and human bone, as well as its frequent usage in previous studies [1, 3, 9, 10, 12, 13, 15, 16, 22, 25], the present re‐ search has used fresh bovine femurs, which had been obtained from a local abattoir. Note that no animal was sacrificed specifically for conducting the tests in this research. Considering the necessity of creating the same conditions for all experimental states, the desirable samples with a width of 15 mm and thickness of around 8 mm have been chosen from the mid‐shaft of the bovine femur diaphysis (Fig. 2). The initial temperature (T0) of the specimens has been 27 °C before the experimental tests. Since the method proposed in this research intends to resolve the effect of drill bit wear on drilling temperature rise through internal gas cooling, thus it was nec‐ essary facilitate the gas passage through the spindle and delivering it to the internal coolant drill bit by making changes to the drill machine (Fig. 3). This tool could be made through modifying Bosch drill GSB 16 RE according to Fig. 4. Note that the efficiency of this drill machine in effective cooling of drilling site through gas has been demonstrated previously [9, 25]. As a tool for mate‐ rial removal as well as a path for concurrent transference of coolant gas to the drilling site, an internal coolant drill bit (Mitsubishi materials MVS0320X05S060 MVS series solid Carbide drill, internal coolant) has been used with a diameter of 3.2 mm possessing two internal channels (diameter is 0.5 mm) for gas passage (Fig. 5). According to the common protocol for tempera‐ ture measurement at the drilling site in bone drilling operations, a thermocouple has been in‐ stalled at the depth of 3 mm with a distance of 0.5 mm away from the hole wall [9, 13, 16, 25, 27], whereby the thermal changes have been recorded via Thermometer Lutron TM‐925. Table 1 presents the different conditions of drilling operations employed in the present research. Based 202 Advances in Production Engineering & Management 16(2) 2021 Bone drilling with internal gas cooling: Experimental and statistical investigation of the effect of cooling with CO2 on … on this table, it is observed that the drilling tests have been totally performed in 36 experimental states for drilling conditions: without cooling and with internal gas cooling, with three rotational speeds (1000, 2000, and 3000 r·min‐1), for six states of drill bit (new, after using 10, 20, 30, 40, and 50 times). Note that in order to ensure the accuracy and replicability of the results, every experimental state has been repeated at least three times. Fig. 2 Bone specimens after cutting Fig. 3 A schema of drill made with internal gas cooling Fig. 4 A schema of drilling machine: The section shows the pathway of gas coolant Fig. 5 Internal coolant drill bit. Table 1 Bone drilling operational parameters Drill bit diameter (mm) Drill bit type Rotational speed (r·min‐1) Feed rate Gas coolant type Coolant flow rate (L·min‐1) Cylinder pressure (bar) Gas temperature during discharge of drill tip (°C) Degree of Drill bit wear Depth of drilling (mm) Modes of drilling Number of iteration of tests 3.2 Mitsubishi materials MVS0320X05S060 MVS series solid Carbide drill, internal coolant 1000‐2000‐3000 Manual CO2 5 2 Advances in Production Engineering & Management 16(2) 2021 ‐11 New drill bit‐& After 10‐20‐30‐40‐50 holes 8 Without cooling‐ With internal gas cooling At least 3 times 203 Shakouri, Haghighi Hassanalideh, Fotuhi 3. Results and discussion Figs. 6‐8 present the results obtained from the bone drilling tests in two states, without drilling and along with internal gas cooling for the rotational speeds of 1000, 2000, and 3000 r·min‐1. Based on the above figures, it can be observed that:  In case of not using any coolants, as the frequency of applying the drill bit increased from the new state to more than 50 times, the hole site temperature rise has grown from ΔT = 24 °C to 34 °C for rotational speed of 1000 r·min‐1, from ΔT = 29 °C to 37 °C for rotational speed of 2000 r·min‐1, and from ΔT = 20 °C to 33 °C for rotational speed of 3000 r·min‐1 (on average from the baseline level, T0 = 27 °C). This suggests that an increase in the fre‐ quency of applying drill bit and its resulting wear significantly influence the temperature rise of the hole site. This temperature rise occurred more quickly for the drill bit utilized more than 40 times (which has been mentioned as the last allowable limit for applying drill bits in different references [8, 27, 28]).  The minimum temperature rise for the case of no coolant utilization was ΔT = 20 °C be‐ longing to the new drill bit at 1000 r·min‐1, which has been far beyond the allowable limit of temperature rise (ΔT < 10 °C). This means that in conventional drilling, it is not possible to control the extent of temperature rise and prevent the incidence of thermal necrosis. As the frequency of applying the drill bit increases, the conditions become further critical. Although in previous studies, the maximum allowable frequency of applying drill bits has been mentioned as 40, the above figures indicate that when no coolant is used, even new drill bits may lead to excessive temperature rise and incidence of thermal necrosis.  In the case of drilling with internal gas cooling, as the frequency of applying the drill bit in‐ creased from new conditions to beyond 50, the hole site temperature rise (ΔT) has grown from ΔT = 2 °C to 3 °C for rotational speed of 1000 r·min‐1, from ΔT = 5 °C to 8 °C for rota‐ tional speed of 2000 r·min‐1, and from ΔT = 5 °C to 7 °C for rotational speed of 3000 r·min‐1 (from the baseline level, T0 = 27 °C). This highlights that when internal gas cooling is used, the increase in the frequency of applying the drill bit has had a minor impact on the tem‐ perature rise of the hole site.  In case of the drilling with internal gas cooling, the temperature rise of the hole site (ΔT) lied within the allowable range at all rotational speeds and the drill bit wear conditions, and did not exceed the threshold of 10 °C. This suggests that bone drilling with internal gas cooling enjoys the ability of controlling temperature changes and preventing the inci‐ dence of thermal necrosis for both new and worn drill bits. Note that the extent of temper‐ ature rise has approached the maximum allowable limit only for the drill bit with more than 50 times of usage at 2000‐3000 r·min‐1. Thus, it can be concluded that bone drilling with internal gas cooling guarantees controlled temperature rise at the hole site and no in‐ cidence of thermal necrosis for new and even 50‐time used drill bits.  Comparing the results obtained from Figs. 6‐8, it can be seen that in both conventional drilling and drilling with internal gas cooling, across all new and worn drill bit states, as the drill bit rotational speed increased from 1000 r·min‐1 to 2000 r·min‐1, the temperature rise (ΔT) was intensified; however, with further increase in the rotational speed from 2000 r·min‐1 to 3000 r·min‐1, the extent of temperature rise of the hole site diminished. Note that, in the drilling without cooling, the minimum extent of temperature rise was ob‐ tained at 3000 r·min‐1, while in the bone drilling with internal gas cooling, the minimum degree of temperature rise occurred at the rotational speed of 1000 r·min‐1. 204 Advances in Production Engineering & Management 16(2) 2021 Bone drilling with internal gas cooling: Experimental and statistical investigation of the effect of cooling with CO2 on … Fig. 6 Temperature change during bone drilling for different modes of drill bit wear (Comparison of drilling without cooling and drilling with internal gas cooling, n = 1000 r·min‐1) Fig. 7 Temperature change during bone drilling for different modes of drill bit wear (Comparison of drilling without cooling and drilling with internal gas cooling, n = 2000 r·min‐1) Fig. 8 Temperature change during bone drilling for different modes of drill bit wear (Comparison of drilling without cooling and drilling with internal gas cooling, n = 3000 r·min‐1) Based on Figs. 6‐8, it is observed that in the bone drilling without internal gas cooling, as the frequency of applying the drill bit increased, so did the extent of temperature rise of the hole site. This is justified considering the gradual increase in the drill bit wear and weakened cutting capability of the drill bit edges. Nevertheless, the important point in the above figures is that the maximum extent of temperature rise has occurred at 2000 r·min‐1, while the minimum has been obtained at 3000 r·min‐1. As a justification, the factors affecting the temperature rise during the bone drilling operations should be considered:  Reduction of the energy required for chip formation and machine forces in response to el‐ evated rotational speed of the drill bit;  Increased friction and frictional heating in response to augmented rotational speed of the cutting tool;  Increased speed of the chip evacuation in response to the elevated rotational speed of the drill bit. Advances in Production Engineering & Management 16(2) 2021 205 Shakouri, Haghighi Hassanalideh, Fotuhi As the rotational speed of the drill bit was augmented from 1000 r·min‐1 to 2000 r·min‐1, the machining force decreased while the chip evacuation speed increased, causing reduction of heat generation in the process to some extent. Nevertheless, since the effect of friction and frictional heating is dominant, the temperature of the hole site increased. These conditions hold for all states of the drill bit wear. In the bone drilling with internal gas cooling, two important points are notable:  Application of internal gas cooling has caused the temperature rise (ΔT) to become con‐ strained within the range less than 10 °C, and allowed 50 times of drill bit usage because of different reasons. They include increased chip evacuation speed, effective cooling of the drill bit and bone, along with diminished effects of the drill bit wear (including bluntness of the cutting edges, adhesion of the bone mineral matrix to the edges and flutes of the drill bit, and clogging of drill bit flutes).  Since cooling with gas is far more effective than increasing the rotational speed of the drill bit for elevating the chip evacuation speed, the rise in the rotational speed from 1000 to 3000 r·min‐1 did not significantly contribute to faster chip evacuation and only caused an intensified effect of friction and frictional heating. Thus, in the case of bone drilling with internal gas cooling, the minimum extent of temperature rise was obtained for both new and worn drill bits at 1000 r·min‐1. To investigate the signs of drill bit wear, some images of it have been presented in Fig. 9 after 50 times of usage for bone drilling operations without cooling and with internal gas cooling. These images have been prepared after cleaning and rinsing the drill bits, and captured under Stereo Microscope ST1740 with maximum 50× magnification. By comparing these images with the image of the new drill bit, it can be found that:  After 50 times of applying the drill bit for bone drilling (in both without and with internal gas cooling), no notable signs of the bluntness of cutting edges and the flute edges of the bit are observed. This is due to the low mechanical strength of the bone, negligible machin‐ ing forces, and high hardness of the carbide bit used in the present research. This causes minor state of wear. In case of further increasing the frequency of applying the drill bit (beyond 50 times), or using standard surgical drill bit (made of stainless steel), which has a lower hardness compared to carbide drill bit, the signs of this wear will emerge more rapidly.  Considering adhesion of a coating made of the bone mineral matrix to the cutting edges and flutes of the bit, and concerning the substantial impacts on the bits, it is clear that the extent of this type of tool wear has been notable in the drilling without cooling, and some evident signs have remained on the tool surfaces. On the other hand, when using drilling with internal gas cooling, the signs of adhesion of a coating made of the bone mineral ma‐ trix to tool surface are very trivial and negligible. This improvement in the qualitative sta‐ tus of the tool and reduction of its wear result from the effect of internal gas cooling on preventing severe temperature rise as well as the effective role of the gas flow in acceler‐ ating the chip evacuation rate. Fig. 9 Images of the abrasion of drill bits in various states (Bottom image is side view of drill bits with10× magnifica‐ tion; Top images are the tool tip section with 25× magnification): The left image is of the new drill bit; the center image is of the drill that had drilled 50 holes (with internal gas cooling), and the right image is of the drill that had drilled 50 holes (without cooling). 206 Advances in Production Engineering & Management 16(2) 2021 Bone drilling with internal gas cooling: Experimental and statistical investigation of the effect of cooling with CO2 on … Table 2 reports the set of results of the bone temperature rise in drilling operation, obtained in other studies similar to our research. To validate the results of the present research, a com‐ parison has been made between these results and the findings of other studies. Note that in some cases, some differences existed between the drilling conditions of other studies [8, 24, 25] and the process conditions applied in the present research regarding the type of bone, rotational speed, drill bit diameter, and hole depth. Nevertheless, Table 2 indicates that, according to the results of the present research, more frequent use of the drill bit and thus the increase in its de‐ gree of wear have led to bone temperature rise [8]. Further, clearly application of internal cool‐ ing methods (with liquid or gas) has resulted in a significant reduction of the bone temperature rise. Although the liquid coolant (water or normal saline) outperformed the gas coolant (N2 or CO2), both cooling modes (either liquid or gas) have been sufficiently able to effectively cool the hole site and prevent exceedance of temperature rise beyond the allowable limit of ΔT < 10 °C [24, 25]. Reference Allan et al. [8] Augustin et al. [24] Shakouri et al. [25] Present study Table 2 Comparison the results of the present research with other studies Rotational Cooling Degree of Hole Bone type Drill bit speed condition depth diameter Drill bit (r·min‐1) wear (mm) (mm) New Dry Pig mandible 1.5 20000 5 After 600 Dry holes Dry 1000 Water Porcine femur 3.4 New 4‐5 Dry 3150 Water Dry CO2 1100 N2 Normal saline Dry CO2 1920 Bovine femur 3.2 New 7‐8 N2 Normal saline Dry CO2 3200 N2 Normal saline Dry 1000 CO2 Dry New 2000 CO2 Dry 3000 CO2 Bovine femur 3.2 8 Dry 1000 CO2 Dry After 50 2000 holes CO2 Dry 3000 CO2 Temperature changes (°C) 7.5 (0.6‐20.5) 13.4 (5.7‐28.3) 8 0.4 9.2 0.6 11.5 3.3 4.9 2.9 18.1 2.5 5.4 1.9 19 5.6 4.8 5.1 24 ±1.5 2 ±0.5 29 ±3 5 ±1 20 ±2 5 ±0.5 34 ±3 3 ±1 37 ±3.5 8 ±1 33 ±1 7 ±0.5 In this section, to investigate the influence of each of the input parameters (rotational speed and the frequency of using the drill bit) on the bone temperature rise, and to determine the sta‐ tistical model for predicting temperature rise, statistical analysis of the results has been per‐ formed using Statistica software. The relationship between the output variable (temperature rise) and input variables (rotational speed and the frequency of using the drill bit) has been ex‐ amined using different regression models. Based on the maximum regression coefficient (R) and coefficient of determination (R2), the most suitable regression model has been chosen, and the Advances in Production Engineering & Management 16(2) 2021 207 Shakouri, Haghighi Hassanalideh, Fotuhi coefficients related to the regression model as well as p‐value at the level of α = 0.05 have been estimated for them. The results of statistical analysis of the bone temperature rise for the modes of bone drilling without cooling and drilling with internal gas cooling are outlined in Table 3. In the bone drilling without cooling, considering the relation obtained for predicting the bone tem‐ perature rise (growth regression), it is observed that temperature changes are only dependent on the frequency of applying the drill bit. This means that in this mode, the drill bit wear plays a significant role in the bone temperature rise, while the rotational speed of the drill bit does not have a significant impact on such rise. The independence of the bone temperature rise of the rotational speed when not using the cooling further highlights the effect of the tool wear on the bone temperature rise during drilling. However, for the drilling with internal gas cooling, con‐ sidering the type of model obtained (linear regression), it can be stated that rotational speed and the drill bit wear have a direct relationship with temperature. Nevertheless, considering the co‐ efficients obtained for these input parameters, it is evident that the tool wear has a greater im‐ pact on the temperature rise as compared to the rotational speed. Table 3 Statistical analysis of temperature rise results. Drilling without cooling Drilling with internal gas cooling Regression model Growth Linear Regression formula ΔT = exp(b1·n + b2·B + a) ΔT = (b1·n) + (b2·B) + a Regression coefficient (R) 0.803 0.794 Determination coefficient (R2) 0.645 0.631 b2 a b1 b2 a b1 Regression parameters 0.00001 0.0067 3.210 0.0014 0.0371 0.96 p‐value (α = 0.05) 0.691 0.0001 0.000 0.00047 0.027 0.241 Resultant regression equation ΔT = exp(0.0067·B + 3.210) ΔT = (0.0014·n) + (0.0371·B) Possibly, one of the major concerns which may challenge the application of internal gas cool‐ ing in orthopedic surgery is the probability of incidence of hypothermia or other similar damag‐ es to the bone in response to exposure to the cool gas flow. To resolve this challenge, three points should be considered:  To create a hole with a diameter of 3.2 mm, a major part of the bone material exposed to CO2 gas coolant at 11 °C detaches off the bone as chips and discharges to the outside of the hole.  Similar to the case for thermal necrosis, where temperature rise and duration of exposure to high‐temperature were influential factors, for hypothermia, again time is a direct influ‐ ential factor. This fact cannot be neglected that during drilling with internal coolant drill bit, the bone is exposed to the coolant gas at 11 °C. Nevertheless, since the duration of drilling operation is only some second, the bone tissue of the hole wall is at risk of expo‐ sure to the cold gas only for a very short duration, and incidence of hypothermia in the bone is very unlikely.  The overall effect of thermal factors, including: a) temperature rise resulting from drilling, and b) temperature fall due to cool gas flow determine the final temperature of the bone tissue of the hole wall. As also indicated by the results of the present research, cooling with CO2 at 11 °C in the best mode resulted in temperature rise of 2 °C. This is the tem‐ perature measured by a contact thermocouple at the depth of 3 mm and distance of 0.5 mm away from the hole wall. It implies no incidence of over‐cooling in the bone site. This means that in the best performance state of the gas coolant, again the bone tissue temper‐ ature has not declined in relation to its initial temperature, and has experienced a mini‐ mum temperature rise of 2 °C. The results of the current study revealed that the application of CO2 gas cooling in the bone drilling operations caused reduced extent of temperature rise and prevented incidence of ther‐ mal necrosis. Furthermore, upon the decrease in the blunting rate of the cutting edges, reducing the extent of adhesion of the bone mineral matrix to the drill bit edges and flutes, and by pre‐ venting clogging of drill bit flutes, it slows down the drill bit wearing and reduces its impact on 208 Advances in Production Engineering & Management 16(2) 2021 Bone drilling with internal gas cooling: Experimental and statistical investigation of the effect of cooling with CO2 on … the temperature rise. The very important point in the present research was independence of the desired results of internal gas cooling of the drilling process parameters (including rotational speed and feed rate). Unlike high-speed drilling and ultrasonic assisted drilling methods, which offer desired results and reduction of temperature rise only within special ranges of rotational speed and feed rate, which in case of neglecting the recommended process ranges have less satisfactory results even compared to conventional drilling [13, 15, 16], the present research indicated that the significant decrease in the temperature rise and elimination of the risk of thermal necrosis in the drilling with internal gas cooling were dependent on neither the process conditions nor even the drill bit wear, and can be easily implemented in orthopedic surgical operations. 4. Conclusion The present research examined the effect of internal gas cooling through CO2 on drill bit wear in bone drilling operations and its resulting temperature rise. The following results were obtained: • In the bone drilling operations without cooling, as the frequency of applying the drill bit increased, the extent of temperature rise of the hole site was intensified considerably. The extent of elevation was more dramatic for the drill bits utilized more than 40 times. This suggested that the increase in the frequency of applying the drill bit is followed by its severe wear. Also, the tool wear significantly influences the temperature rise of the bone. • In the bone drilling operations with internal gas cooling via CO2, at different rotational speeds, the extent of temperature rise remained within the allowable range (ΔT<10 °C), and thermal necrosis was no longer probable to occur. Secondly, internal gas cooling could significantly reduce the effect of the drill bit wear on the bone temperature rise, and guaranteed no incidence of thermal necrosis up to 50 times of drill bit usage. Finally, the desired results of the internal gas cooling were not dependent on cutting parameters and can be adapted to orthopedic surgical operational conditions. Declaration of conflicting interests The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article. Ethical approval All procedures and methods performed in the current study were in accordance with the Declaration of Helsinki 1964 and its later amendments. The research design and protocol were approved by the ethical standards of Research Committee of the Islamic Azad University-Tehran North Branch. Informed consent This article does not contain any studies with human participants/patient data. Funding This work was supported by Iranian Tajhiz Sina, its. Co. Grant Number 990072. References [1] [2] [3] Hillery, M.T., Shuaib, I. (1999). 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Appendix 1 Notation A B Ea R R R2 T T0 a, b1, b2 n t ΔT Ω Arrhenius frequency factor (3.1×1098 s-1) Frequency of using the drill bit Arrhenius activation energy (627×103 J·mol-1) Universal constant of gases (8.314 J·mol-1·K-1) Coefficient of regression Determination coefficient Temperature (°C) Initial temperature (baseline level) (°C) Regression parameters Rotational speed (r·min-1) Time (s) Temperature rise (°C) Arrhenius thermal damage Advances in Production Engineering & Management 16(2) 2021 211 Advances in Production Engineering & Management ISSN 1854‐6250 Volume 16 | Number 2 | June 2021 | pp 212–222 Journal home: apem‐journal.org https://doi.org/10.14743/apem2021.2.395 Original scientific paper Joint distribution models in fast‐moving consumer goods wholesale enterprise: Comparative analysis and a case study Wang, L.a, Chen, X.Y.a,*, Zhang, H.a,b a School of E‐Business and Logistics, Beijing Technology and Business University, Beijing, P.R. China Beijing Food Safety Research Base, Beijing, P.R. China b ABSTRACT ARTICLE INFO Joint distribution means multiple clients were provided distribution services together by only one third‐party logistics company. It is a unified plan and implementation used in distribution centres and a distribution activity im‐ plemented by multiple consortia. Many problems in distribution can be solved through the joint use of distribution warehouse, vehicles and reasonable logistics business, so as to optimize the overall logistics node and route ar‐ rangement. This paper mainly discusses the model of joint distribution of fast moving consumer goods, proposes three types of the optimization model of joint distribution system with Chaopi as an example. We draw the conclusion that Chaopi Trading Co., Ltd. is a joint distribution system optimization busi‐ ness model. This paper puts forward several basic distribution models and analyzes them in combination with practical applications, which has strong practical significance. Although the development of public distribution in China is not very fast, it is an inevitable trend. Through the efforts and explo‐ rations of the governments of various countries, there will be more and more choices of public distribution models. Keywords: Logistics; Joint distribution; Wholesale enterprise; Fast‐moving consumer goods; Distribution models; Optimization *Corresponding author: 845190504@qq.com (Chen, X.Y.) Article history: Received 27 March 2021 Revised 23 May 201 Accepted 15 June 2021 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 continuous improvement of China's economy and the rapid development of people's living standard, the sales volume of fast‐moving consumer goods (FMCG) increase year by year, which shows the good growth and extensive market demands. Practice has proved that the key factor for the success of fast‐moving consumer goods enterprises is market power, and logistics is an important factor for the formation of it. Therefore, efficient logistics system is one of the key factors to support the success of FMCG enterprises. In order to create competitive ad‐ vantages, FMCG distribution enterprises need to improve their profit level and ensure their sur‐ vival and development. In recent years, with the continuous expansion of FMCG sales, the inte‐ gration of FMCG sales channels and the acceleration of innovation, FMCG sales outlets are ex‐ panding rapidly. In order to meet the needs of enterprise scale expansion and market competi‐ tion, they strive to reduce operating costs and improve the reaction speed of stores and consum‐ ers. In order to improve the efficiency of operation and logistics and create important material conditions, FMCG distribution enterprises have improved the logistics infrastructure, storage conditions and transport capacity, and they have actively used advanced logistics technology. 212 Joint distribution models in fast‐moving consumer goods wholesale enterprise: Comparative analysis and a case study Joint distribution, also known as third‐party logistics (3PLs) service sharing, refers to that on‐ ly one 3PLs company provides distribution service for multiple customers. It is a unified plan and implementation for the use of distribution centers and a distribution activity implemented by multiple consortia. Generally speaking, enterprises are usually decentralized and unable to integrate logistics resources. Due to the different logistics objects, many enterprises often have the problems of unbalanced utilization rate of distribution warehouse, unreasonable lines and high vacancy rate of vehicles. But it can be realized through the joint use of logistics, distribution warehouse and vehicles, so as to optimize the overall logistics node and route arrangement, so as to make an efficient and green logistics through unified commodity distribution. From the perspective of enterprises, the first is to reduce the cost of distribution and the capi‐ tal occupation, and the sharing of resources among enterprises can reduce the capital invest‐ ment and cost of logistics chain. Secondly, it is to increase the coverage of the network service. The enterprises may choose the distribution with far distance and limited transportation, if without the network service, which can be improved by joint distribution, because it can improve the service level and visibil‐ ity with the help of network service. Then, core business of the enterprise can be strengthened. As companies no longer invest too much energy in the distribution sectors, trivial problems no longer occur in the distribution pro‐ cess, therefore it will not take any time to resolve, so that we can concentrate on our core busi‐ ness, have more investment in human resource, put more financial resources on the core issues, and enhance the competitiveness of enterprises. In addition, it can also improve the technological content. In order to optimize the joint dis‐ tribution process, it is necessary to unify the identification and packaging, reduce the manual operation, improve the utilization of high‐tech, and prepare for the future development. Finally, in order to improve the degree of specialization of distribution, joint distribution is to complete the detailed planning of time, place and route according to the requirements of various enterprises through professional distribution companies, so that all enterprises can accept and implement network integration and realize the social sharing of resources. In this way, we can make the distribution process more efficient and professional. From a social point of view, the first is the unified management of joint distribution vehicles, which can achieve a high loading rate of vehicles and avoid repeating the previous transporta‐ tion routes, cross transportation, circuitous transportation and occupation of road resources. So we can easily relieve the pressure of traffic congestion. Secondly, joint distribution can reduce the waste of vehicles, with the reduction of the pollu‐ tion caused by vehicle exhaust and vehicle noise. Finally, joint distribution means resource centralized management. Based on the conditions of joint distribution, enterprises will build more warehouses. They will not blindly build all kinds of enterprises according to their own interests. From the overall consideration, the problem of scattered and large‐scale construction of small warehouses will be solved, and the abuse of land resources can be reduced. In addition, the concept of joint distribution breaks the traditional single model of transport. It can integrate resources and optimize logistics social system. Joint distribution is a more reasonable way to meet the needs of customers. In this way, all kinds of resources of the enterprise can be integrated. From an economic or social point of view, its advantages are obvious. 2. Literature review Foreign scholars call the scheduling problems of distribution vehicle as Vehicle Scheduling Prob‐ lem (VSP), which was first proposed in 1959 by the Dantzig and Ramser, and was soon attracted by the operations, applied mathematics, combinatorial mathematics, graph theory and network analysis, logistics science, computer applications, and other subject matter experts, transport planners, and managers of great importance, as the field of operations research and combinato‐ rial optimization of front and hotspot issues [1]. In a retailer case study, Claudia and Laura Advances in Production Engineering & Management 16(2) 2021 213 Wang, Chen, Zhang (2014) mentioned that the importance of 3PLs, aggregation and multi‐user platforms must be recognised by transport planners in supporting the use of intermodal transport by retailers and other large shippers [2]. As for 3PLs and self‐delivery logistics firms, Chun and Jim(2013) inves‐ tigated which delivery technique is more efficient by comparing logistics capabilities, logistics services, and logistics performances based upon transaction cost analysis and resource‐based theory [3]. The empirical results in this study could be strategic logistics management guidelines based upon the theoretical relationships among logistics capabilities, logistics service, and logis‐ tics performance for 3PL users and self‐delivery logistics firms. Liu (2015) proposed a concep‐ tual model that delineates the determinants of consumers’ perceived service quality and tested the model in the Chinese logistic companies [4]. Based on the former analysis, he presented sep‐ arately the activated measures of domestic 3PLs by corporate side, the supply side, and the gov‐ ernment's policy support. Conducting two in‐depth case studies of 3PLs, Cabigiosu et al. (2015) showed that 3PLs extensively rely on service modularity with standard procedures as their con‐ stitutive element [5]. A bi‐objective model is developed to minimize the total shipping cost and time simultaneously. Lopes (2020) evaluated the relationship between the service capabilities and performance of the 3PLs providers in UK and Taiwan, China [6]. The range of service provi‐ sion offered by 3PLs does not directly influence the 3PLs' financial performance. However, 3PLs providers with service capabilities that correspond to the key priorities of customers will gain superior financial performance through a better operational performance. In terms of criteria and methods, Aguezzoul (2014) presented a literature review on 3PLs selection decision and revealed that 3PLs selection is empirical in nature and is related to a region/country, industrial sector, and logistics activities outsourced [7]. Cost is the most widely adopted criterion, followed by relationship, services, and quality. From the perspective of 3PLs providers, Arthanari (2016) found that uncertainty, order frequency, and transaction size, but not asset specificity, are signif‐ icantly associated with 3PLs service, which in turn is significantly associated with value‐to‐client and benefit‐to‐3PLs provider [8]. However, there are still some difficulties in the development of 3PLs. As for the design of logistics scheme and route, many scholars have made many contributions to improve the efficiency. Moutaoukil, Neubert and Derrouiche (2015) focused on the consolida‐ tion of goods flows by using a Distribution Center to redesign the flow of goods inside the city while not increase the cost, reduce pollution and make the city more attractive [9]. Freile and Mula (2020) considered the topics of reverse logistics, waste management and vehicle routing and scheduling to introduce the area of Green Logistics [10]. Hu et al. (2019) provided the met‐ rics of logistics service suppliers used for environmental performance measurement and the barriers and drivers that may hinder or facilitate the adoption of these initiatives, which con‐ tributes to the knowledge of environmental sustainability for logistics and transportation [11]. In order to evaluate the smart logistics solutions, simulation techniques is proposed by Alberto and Anna (2014) to present the current state of practice in modelling smart logistics solutions, provides a roadmap in simulation techniques for urban freight transport solutions and improves the knowledge around the patterns currently followed [12]. A global enterprise must continu‐ ously improve the efficiency of logistic operations between supply chain collaborators. Integrat‐ ing logistic services, resources, and necessary information flows in the supply chain to ensure efficiency and efficacy is critically important to these companies. So Trappey et al. (2016) sys‐ tematically designed, analyzed, and evaluated an improved framework for one‐stop logistic ser‐ vices [13]. Collaborative two‐echelon logistics joint distribution network was organized by Li et al.(2020) through a negotiation process via logistics service providers or participants existing in the logistics system, which can effectively reduce the crisscross transportation phenomenon and improve the efficiency of the urban freight transportation system[14]. Awaga et al (2020) investi‐ gated the role of logistics service providers in the implementation of a differentiated supply chain [15]. In addition, De Marco et al. (2014) researched the factors that influence logistics ser‐ vice provider’s efficiency in urban distribution systems [15]. In conclusion, there are many scholars in the study of logistics distribution model, and now no one can be suitable for China's joint distribution model, so this paper is to introduce a com‐ mon distribution model focuses on in‐depth analysis of its characteristics, and hope that joint distribution of development has played a certain role. 214 Advances in Production Engineering & Management 16(2) 2021 Joint distribution models in fast‐moving consumer goods wholesale enterprise: Comparative analysis and a case study 3. Joint distribution introduction and comparative analysis The purpose of joint distribution is to integrate the resources reasonably and effectively and make the best use of the resources. The joint distribution of working modes can be considered globally and can be represented by the following diagram (Fig. 1). It can be said that in order to realize the joint distribution from suppliers to consumers, there must be good coordination, such as transportation system, information technology, personnel capacity, etc. Only when all parties make concerted efforts to carry out the distribution opera‐ tion can they be unified. In addition, each company has its own characteristics. Because the pro‐ ject cannot be fully implemented, there are many joint distribution models. Different companies need to choose their own common distribution mode in order to operate effectively. According to the utilization degree of logistics resources, distribution can be divided into three models: joint distribution system optimization model, joint distribution field connection and sharing and distribution facilities use type. Fig. 1 Joint distribution operation model 3.1 Model 1: Joint distribution system optimization Professional logistics companies provide the best solution through customer demand, plan and arrange for each user, and make the best processing system in terms of delivery time, quantity, frequency, route, etc., in the premise that all users can accept, comprehensive planning, and rea‐ sonable layout. Main technical conditions: Firstly, all customers have the same conditions. We cannot deviate too much from their delivery time or quantity. The second one is the support of transportation system. Transportation is the basis of distribution system. Once the transportation system is integrated, it can provide guarantee for other aspects of the logistics system. The third is the support of information technology. With the continuous development of society, information technology, network technology and e‐commerce technology will continue to innovate, and get the inevitable trend of logistics informatization. If there is no other supports, the joint distribu‐ tion system will be difficult to carry on. The pattern is shown as Fig. 2. Fig. 2 Optimized joint distribution system schematic Advances in Production Engineering & Management 16(2) 2021 215 Wang, Chen, Zhang 3.2 Model 2: Joint distribution field connection and sharing The purpose of joint distribution is to unite multiple users to share a distribution field. In gen‐ eral, it can be relatively concentrated in the customer's location. The traffic, roads and sites in the area are relatively crowded, and too many vehicles will cause trouble. Moreover, it is very difficult to go to each customer to prepare for their individual needs. Therefore, multiple users can set up distribution receiving points or cargo handling points together. Main technical conditions: Firstly, it is the capacity of staff. Logistics and distribution now re‐ quires more than mere labour force, besides the IT staff needs to have knowledge of the equip‐ ment. Secondly, it is the organization, coordination and management of the various joint distri‐ bution models. We may need to organize and manage, and common distribution with warehouse sharing emphasizes this, because it brings the various organizations together in order. Finally, it is the site requirements, which may be connected to a unique requirement yard ground Shared common distribution model. Because the first site needs to be suitable for all customers pick from, and the area should be large enough, or we may have the phenomenon of congestion. In this way, it will have the same effect with direct distribution to the customer. The reason why this model is to the problems of bringing together distribution via vehicles, siting and establish‐ ing pick point. Fig. 3 shows the diagram of location sharing type common distribution business model. After the production is completed, suppliers A, B, C need to make delivery to the customers according to their demands. In the distribution, each supplier completes the process, but the last step is that goods are concentrated on a common place ‐ the yard to build a connection. When the goods delivered, the customers A, B, C will go to the freight yard to pick up their own goods. Fig. 3 Shared‐off location joint distribution schematic 3.3 Model 3: Distribution facility use type When many different distribution companies are located in the same city or region, in order to save the investment cost of distribution centres and improve the efficiency of distribution and transportation, many enterprises will establish joint venture partnership and establish joint distribution center, or some enterprises will use the existing distribution centres and distribu‐ tion facilities to implement distribution jointly for different enterprise users. Main technical conditions: Firstly, the implementation of joint distribution needs to be guar‐ anteed by perfect logistics facilities, such as transportation equipment, storage equipment, shelves, loading and unloading machinery, office equipment, etc. Secondly, for employees, they must use the appropriate logistics facilities. Then, they need to choose the use and type of stand‐ ard distribution facilities according to their own characteristics. As can be seen from Fig. 4, this is a common type of distribution facilities. In the left column are distribution com‐ panies A, B, C, which can also be manufacturers. Their purpose is to use together distribution centres or other facilities, such as sorting machines and warehouses. As mentioned earlier, some companies sell in two seasons, the sales season and the off‐season. In the off‐season, enterprises do not want to leave warehouses idle, and if they can share facilities and equipment, they will reduce costs. Vehicles can also be used together during distribution. When distribution enter‐ 216 Advances in Production Engineering & Management 16(2) 2021 Joint distribution models in fast‐moving consumer goods wholesale enterprise: Comparative analysis and a case study prises A and B have customers who need to deliver goods, the vehicles can be used together if the vehicles and locations permit, so as to avoid the distribution enterprises A and B from deliv‐ ering by themselves. Fig. 4 Distribution facilities using type common schematic distribution model 3.4 Comparative analysis of three models The three joint distribution of operating models introduced earlier have different characteristics and advantages, and the uses of them are different. We put the following three models here and make a comparison (Table 1). In fact, each model has its own advantages, and no one is a panacea. Not all companies can apply to the three joint distribution model, and not all business needs a joint distribution. If the cost of a single distribution is less than the implementation of a joint distribution, we do not need to implement it. In the implementation process, the corresponding conditions require more attention paid to personnel management, vehicle scheduling, warehouse distribution, etc. It can be said that joint distribution as a whole, each part of the nodes is not disconnected. There are gains and losses. Table 1 Comparative analysis of three models Business model M1: System optimized Contrast species Applications in the Distribution throughout the appli‐ distribution system cation, from the manufacturer to the customer all have to use the whole distribution process Advantages 1. It can control the entire distribu‐ tion process, easy to manage 2. To the greatest possible to meet the needs of suppliers and custom‐ ers 3. Out significantly lower risk 4. You can cross out the whole distribution process, and focus on strengthening the core competen‐ cies 1. Sound distribution system The main technical conditions required 2. Professional distribution logis‐ tics companies 3. Supporting transport system 4. Support information technology Suitable business Basically applies to all businesses, hoping to outsource logistics com‐ panies M2: Field connection and sharing M3: Distribution facilities use type Applied to a particular aspect of Finally, the application delivery the distribution, use common process, only the last link use facilities needed pick 1. Deliver to the destination can save time and improve distribu‐ tion efficiency 2. Self‐built warehouse cost savings 3. In terms of customers, facili‐ tate their management 1. Saving the cost of purchasing the facility 2. Easy to use, and you can make a joint application distri‐ bution channel 1. Relatively concentrated cus‐ tomer location 2. There must be space to build pick point 3. Access to professional yard land management, more custom‐ ers would easily lead to confu‐ sion The geographical distribution of the relative concentration of customers, such as supermar‐ kets, centralized stores, schools, etc. 1. Sales of the best seasonal difference 2. To produce a similar product attributes 3. To complete logistics equip‐ ment Advances in Production Engineering & Management 16(2) 2021 There are seasonal or a distin‐ guished enterprise logistics links and common use of simi‐ lar businesses can benefit 217 Wang, Chen, Zhang 4. Beijing Chaopi Trading Co., Ltd joint distribution pattern 4.1. Beijing Chaopi Trading Co., Ltd Beijing Chaopi Trading Company Limited (Chaopi Company) is a subsidiary of Beijing Jingkelong Group Co., Ltd. The registered capital of this company is 384 million yuan, and the total asset is 1.3 billion yuan. This company engages in FMCG distribution, brand agency, terminal services, logistics and distribution business of commercial wholesale and comprehensive service com‐ bined with 3PLs enterprises. The company engages in food, drinks and other FMCG. Operators and agents at home and abroad are more than 300 commercial brands, operating more than 9,000 kinds of commodities, has a long‐term and stable sales channels. 4.2 Chaopi Company joint distribution pattern analysis The business model of Chaopi Company is based on the demand of end customers (Fig. 5). It makes the distribution according to the inventory and product management of the whole supply chain. "Super dispatch model" makes arrangements for delivery time, path planning and quanti‐ ty. According to the working standards of commodity nature, the utilization rate of cargo hold volume is generally more than 90 % of that of freight cars, and the cost can be greatly reduced. This application model is a typical application of general distribution system optimization in actual distribution. The joint distribution of the company makes 200‐300 suppliers' goods centralized in the lo‐ gistics distribution centre, and the distribution is arranged in the city according to the custom‐ ers' orders. Customers can also order in small quantities and in multiple batches according to their own requirements, so as to reduce the bundling of customers' funds and improve the mar‐ ket share of commodity sales. Chaopi distribution centre integrates the orders of different cus‐ tomers into one vehicle, with an average of 10‐30 suppliers and 4‐5 terminal customers. The distribution and transportation costs are shared by the suppliers, end users and Chaopi Compa‐ ny, so as to reduce the inventory and transportation costs of upstream and downstream enter‐ prises with a huge export and achieve a win‐win situation. At the same time, it can also ensure the timeliness of distribution and commodity turnover rate. According to the characteristics that the recovery rate of fast moving consumer goods outside the city is more than 70 %, the distri‐ bution solution of "planned replenishment" is adopted, with an average of 150,000 containers per day, of which 750 sets are ordered by customers, 200 containers per day, and 4,500 kinds of products per set; 203 daily working vehicles, with an average of 4000 containers per vehicle. Because the scheme is powerful, and the powerful information platform has set up the "early warning" function, so that suppliers have enough time to allocate goods, so as to minimize the loss. Therefore, the tidal skin model is worth learning from. If Chaopi do not implement joint distribution model, suppliers A, B and C distribute apart in the previous model. As a result, it is inevitable that there will be no delivery or high vacancy rate, and the distribution model of small batch and multi batch is not suitable for fast‐moving con‐ sumer goods. Fig. 5 Chaopi Company joint distribution 218 Advances in Production Engineering & Management 16(2) 2021 Joint distribution models in fast‐moving consumer goods wholesale enterprise: Comparative analysis and a case study Compared to the previous model, it can be seen that Chaopi optimizes the entire middle part of logistics chain and collects all the suppliers and distributors. In this way, the distributor's or‐ der is issued to the supplier, and then the delivery plan is implemented to the final delivery. This is a complete system, which reduces duplication and distribution, vacancy rate, etc., and the out rate will be significantly improved. This is good for both suppliers and distributors. Based on "Chaopi Model" research and investigation, we can make a conclusion that a joint distribution model has significant benefits. In terms of economic or social development, this model is desirable. The advantages are:  Saving input costs Select joint distribution model can solve the problems of self‐distribution appears to rent the library (or self‐built warehouse), vehicle distribution efficiency and management, especially in the season without having to hire more warehouse costs. Variable cost manufacturers choose joint distribution of freight costs just been brought, there will be no investment in fixed costs, which they can use to save money to further strengthen the core competitiveness for the sake of the company's development.  Avoiding shortage cost Manufacturers consider criteria for the distribution of cost and time from different aspects, but in tight supply, the case of a small amount of orders, there is no choice. Supply not on time will be out of stock, and will affect the sales. Since the joint distribution plan is strong, and there are warnings out, so that manufacturers have "enough" time for stocking ready information plat‐ form, so that we can effectively prevent the occurrence of shortage cost.  Increasing turnover If manufacturers are faced with hundreds of distribution points, sales and inventory re‐ quirements are not the same. In order to ensure timely delivery, manufacturers will choose large‐tonnage truck delivery, shipping time will be very long, commodity turnover will be in‐ creased, and the liquidity of turnover will be affected. In this way, it will cause customers a great impact. Adoption of a joint distribution model, Chaopi Company will integrate each customer's order. In the city, they use the unified distribution, and transportation costs are apportioned to the three parties, greatly reducing inventory costs and transportation costs downstream, while timeliness and turnover rate can be guaranteed.  Improving the overall efficiency of delivery Import and distribution of trucks loaded cargo compartment, full rate is often less than 50 %, which is a tremendous waste. On the distribution lines, the phenomenon often repeats, and sometimes a small order needs a delivery. Chaopi's joint distribution model is the overall ar‐ rangements for the delivery time, times, travel routes and the quantity of goods, delivery route optimization, consolidation work carried out according to the nature of goods, specifications, usually a lorry can be done 4‐5 home delivery the amount of cargo tank capacity utilization up to 90 %, an average transport truck supplier 10‐30, 3‐4 stores 750 containers of goods, delivery vehicles from "floating warehouses" into a "replenishment train". Logistics and distribution system relies to a large extent complete and convenient urban transport system, starting from the current situation and development trend of Beijing city transportation, logistics and distribution system must take intensive, efficient path, otherwise it is difficult to adapt to the construction needs of the world of the city. Logistics industry with significant economies of scale characteristics, only through continuous integration of various logistics resources to achieve a high degree of intensification and scale, in order to reduce logis‐ tics costs of enterprises and society, and joint distribution is not only the realization of highly intensive and large‐scale the best choice, but also the construction and development direction of metropolitan distribution system. "Chaopi model" had a very good practice, and achieved good results. It’s worth further research and promotion of Chaopi’s success. Advances in Production Engineering & Management 16(2) 2021 219 Wang, Chen, Zhang 5. Fast‐moving consumer goods joint distribution problems and counter‐ measures encountered 5.1 Diversification may bring difficulty of management Management is essential for a business. Generally good management means good development. The inevitable thing of joint distribution of Chaopi is that the ownership relations are too com‐ plicated. There exist some disadvantages: different circulation levels, inefficient distribution, and diverse requirements. Each client will have their own distribution requirements, and no one will put the common interests first, so that the management will be difficult. Organization and coor‐ dination will be very difficult. FMCG achieves unified logistics and distribution decisions on the distribution of this plat‐ form. It’s unnecessary for the companies to hire many managers, and what they should do is have clear division of labour, to prevent each manager being responsible for only one aspect of the mission. Only in this way, will we ensure few errors and more harmonization of all corporate merchandise storage, distribution and other aspects. 5.2 The distribution of benefits may cause disagreement When doing the joint distribution, what various vendors required the number of items or attrib‐ utes are not the same. For example, they are used in the same vehicle distribution, some bulky, but lightweight, and some heavy weight, but occupy less area, so items are on the same car, how to count the costs of distribution, we need to have them clearly defined. So when the joint distribu‐ tion assets to strengthen the internal organizational structure put into use, the division of trans‐ parency need monitor efforts to improve the cost‐sharing mechanisms and public distribution of benefits principle, need to sign joint distribution agreement for those participating in a more stable unit must, need to focus strictly on divided responsibilities, rights and benefits, members must abide by every cooperation, and need to assume their respective responsibilities and equal treatment. Only by doing so, can each company successfully implement joint distribution. 5.3 The logistics standardization is difficult If you want to have a greater joint distribution costs of exploitation, logistics standardization is a factor which cannot be unconsidered. Due to differences between each enterprise product pack‐ aging, volume, weight, etc., the goods stacked, placed on the link will face great difficulties. Logis‐ tics will involve the issues of choice and reference, which are obstacles the joint distribution need to overcome. Enterprises can start from the inside of the logistics system, overall, the development of tech‐ nical standards for its various subsystems of facilities, equipment, special tools, etc., and opera‐ tions standards. Secondly, studies between each subsystem technology standards and business standards will work. According to the requirements, the standard of the entire logistics system is uniform. After studying logistics systems and other systems related to seek unified standard sys‐ tem of logistics, making the logistics standardization can be easy to promote the implementation. 5.4 It’s easy to reveal business secrets Joint distribution is likely to result in the disclosure of business secrets, since the implementa‐ tion of a joint distribution pattern should be unified management, unified planning and schedul‐ ing. Since each company wants unity, we need to discuss and communicate. However, in the dis‐ tribution process, the expansion list may leak. In the same industry, due to the sharing of infor‐ mation and open, competitive strategy, business development will adversely affect, which is one of the reasons why many companies do not participate in joint distribution. Therefore, enterprises should pay attention to this aspect, and strengthen management, so as to avoid supplier information leakage. The implementation of a common distribution is to the trust as a precondition. In addition, we need to take effective prevention, surveillance measures to strictly control systems, and make the best‐marked confidential matters cooperation agree‐ ment, specifying the rights and obligations. 220 Advances in Production Engineering & Management 16(2) 2021 Joint distribution models in fast‐moving consumer goods wholesale enterprise: Comparative analysis and a case study 6. Conclusion In this paper, we did an in‐depth research in a joint distribution business model, including joint distribution of conceptual meaning, with three business models and future development, and so on. Based on this, the paper focuses on the logistics of joint distribution, with an example of Bei‐ jing Chaopi Trading Company, who uses joint distribution system optimization model to bring the maximum benefit for its producers and sellers, and what the company should pay attention to. On the other two models, joint distribution field connection and sharing and distribution fa‐ cilities use models, the paper analyzes their processes, features and technical support. Then the paper tells the future direction of development of joint distribution. The purpose of the paper is to let more people know the joint distribution, more in‐depth understand the joint distribution, and to choose their own model of operation of enterprises. Possible shortcomings of this paper are: First, there may exist some cognitive bias or insuffi‐ ciency of the joint distribution due to most respect and regard of Beijing Chaopi Trading Compa‐ ny, and lack of enough information may have an impact on research. Second, the paper will inev‐ itably have some flaws limited by the individual's knowledge and ability, and we look forward to your corrections. In fact, China's logistics market potential is huge, and the implementation of joint distribution management will become the development trend of modern logistics. Realization of joint distri‐ bution has a long way to go, and the process is bound to encounter a lot of frustration. However, no matter how hard the process, joint distribution will be widely used for the majority of enter‐ prise logistics efficiency, comprehensive, and competitiveness enhancement. From the interna‐ tional market, Japan and other countries are in very common use of joint distribution model. I believe, in the near future, our country will see the benefits brought by joint distribution and use it in a wide range. No matter from the economic level or social level, the joint distribution model is reasonable. However, in the actual operation, joint distribution companies have many problems to resolve, like different commodity business or attributes. From the perspective of competition, some large companies are reluctant to share with others, because it will increase the cost of other distribu‐ tion enterprises and enhance the rival’s competitiveness. From the perspective of interests, eve‐ ry enterprise does not want to disclose its business secrets, but sometimes it can be seen as "se‐ cret" from the orders of some enterprises. So many companies are making some "inconsistency", which requires the guidance of the government. It is understood that China's current policies, no matter in law, have many things to do for a normative system, while some developed countries have established relatively sound laws. The core of joint distribution is to organize distribution uniformly, save cost, save social resources and improve transportation. Therefore, many coun‐ tries are actively promoting this model, coordinating the logistics distribution capacity and im‐ proving the logistics environment. For example, they are trying to solve the problem of distrust among enterprises. Because there is no sound legal system now, I don't know which party the responsibility belongs to, nor which makes some enterprises dare not try. When the government intervenes, there are clear legal provisions and the implementation of supervision can effective‐ ly solve these problems. Therefore, we expect the government to attach importance to the logis‐ tics industry, and through the joint efforts of relevant departments, the city will have a substan‐ tial joint distribution development pattern. Foundation    2018 Beijing Talents foundation of organization department of Beijing Municipal Committee of the CPC (2018000026833ZS09). Beijing Philosophy and Social Science (17GLB013). Science and technology innovation service capacity provincial (19008021111) (19008021171) (19002020217). Advances in Production Engineering & Management 16(2) 2021 221 Wang, Chen, Zhang References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] 222 Vieira, J.G.V., Fransoo, J.C. (2015). How logistics performance of freight operators is affected by urban freight distribution issues, Transport Policy, Vol. 44, 37-47, doi: 10.1016/j.tranpol.2015.06.007. Gómez, S.C.G., Cruz-Reyes, L., González, B.J.J., Fraire, H.H.J., Pazos, R.R.A., Martínez, P.J.J. (2014). 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(2020). Evolutionary game of green manufacturing mode of enterprises under the influence of government reward and punishment, Advances in Production Engineering & Management, Vol. 15, No. 4, 416-430, doi: 10.14743/apem2020.4.375. De Marco, A., Cagliano, A.C., Mangano, G., Perfetti, F. (2014). Factor influencing logistics service providers efficiency in urban distribution systems, Transportation Research Procedia, Vol. 3, 499-507, doi: 10.1016/j.trpro. 2014.10.031. Advances in Production Engineering & Management 16(2) 2021 Advances in Production Engineering & Management ISSN 1854‐6250 Volume 16 | Number 2 | June 2021 | pp 223–239 Journal home: apem‐journal.org https://doi.org/10.14743/apem2021.2.396 Original scientific paper Designing a warehouse internal layout using a parabolic aisles based method Zhang, Z.Y.a,*, Liang, Y.b, Hou, Y.P.a, Wang, Q.c a School of Logistics, Beijing Wuzi University, Beijing, P.R. China Beijing Chaoyang District Committee of the Revolutionary Committee of the Chinese Kuomintang, Beijing, P.R. China c Manufacturing Engineering and Order Delivery Center, Beiqi Foton Motor Co., Ltd, Beijing, P.R. China b ABSTRACT ARTICLE INFO Refined layout is a basis of warehousing efficiency. Straight aisle is a typical feature of current warehouse internal layouts. The purpose of this paper is to explore the possibility of using curve aisles for warehouse layout. By Choosing typical non‐traditional layouts and transforming their inclined cross‐aisle trajectory into parabola, two parabolic aisle layouts, parabolic Flying‐V and parabolic Fishbone, are constructed. For unit‐load warehouses, based on the morphological characteristic analysis and the parabolic types selection, the picking distance model and the cross‐aisle length formula are presented. Interval Numerical Simulation Method (INSM) and Genetic Algorithms (GA) are adopted to solve the model respectively in order to verify the results. This research breaks through the realistic situation of straight aisle leading ware‐ house layout, and enriches the relevant layout theory. The calculation results of 100 warehouses with different sizes show that the picking distance of par‐ abolic Flying‐V could be reduced by 0.22‐0.62 % compared with the straight layout, and the theoretical possible improvement space has been compressed by 2.42‐12.26 %. Its length of cross‐aisle is shortened by 0.03‐3.10 %. The picking distance of parabolic Fishbone could be only reduced by 0.02‐0.04 %. The theoretical possible improvement space has been compressed by 1.27‐ 1.83 %. But its length of cross‐aisle will increase by 4.63‐19.50 % significant‐ ly. We believe that the layout of non‐rectangular complex special‐shaped warehouses based on curve trajectory aisles would become an important research topic. In addition, after some necessary modifications to the objec‐ tives and constraints, the proposed method in this paper may also be used for the arrangement of machines and devices in a workshop in principle. Keywords: Layout design; Warehouse internal layout; Parabolic aisle layout; Layout efficiency; Simulation; Optimization; Interval numerical simulation method (INSM); Genetic algorithms (GA) *Corresponding author: zyfzzy@263.net (Zhang, Z.Y.) Article history: Received 8 April 2021 Revised 26 April 2021 Accepted 5 May 2021 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 Warehouse is an important social logistics infrastructure. The level of warehouse system plan‐ ning and design directly affects the overall logistics efficiency of a country. We know that the travel distance of a picking tour is an imperative factor in improving warehouse operation effi‐ ciency [1]. The efficiency of warehouse processes could be improved by reducing travel time and cost in replenishment and order picking [2]. The layout of internal aisle in warehouses is the basis of storage space planning and warehouse operation scheduling management, which direct‐ ly affects the area utilization rate, the access efficiency, the difficulty of layout adjustment, the overall operation energy consumption and the total cost of warehousing. With the rapid devel‐ opment of warehouse operation and management technology, warehouse enlargement and au‐ 223 Zhang, Liang, Hou, Wang tomation are the future development trends. The benefit of lean warehouse layout research will be more fully reflected in the construction of more and more intelligent super‐large warehouses. Traditional warehouse layouts lack detailed quantitative research on the efficiency mecha‐ nism of aisle layout itself. In practice, it is usually based on the qualitative analysis to choose a basic layout mode, and then to match the specification parameters such as aisle width, function area size and shelf size according to relevant design specifications. For the common rectangular warehouses, most of them are based on experience and intuition, using the layout of parallel shelves and orthogonal straight track aisles (as shown in Fig. 1) [3]. In 2009, Gue and Meller break away from the traditional conventions, and innovatively put forward two new warehouse internal layouts with inclined cross‐aisles: Flying‐V and Fishbone (as shown in Fig. 2) [4]. Numerical simulation results show that, for the unit‐load warehouse, these two new non‐traditional layouts based on the cross‐aisle angle modelling and optimization can reduce the average total picking distance by about 10 % and 20 % respectively compared with the traditional layouts based on empiricism. Literature review shows that both the traditional layouts based on parallel orthogonal aisle mode selection and the non‐traditional layouts based on aisle angle modelling and optimization have a problem or shortcoming. The aisle trajectory is basically a straight line or piecewise straight line by default. The layout design revolves around the direction and distribution of the cross and picking aisles of the straight‐line trajectory. There is no research on warehouse aisle layout based on curved line trajectory has been found in the literature. Choosing straight aisle has become a default research paradigm of warehouse internal layout design. Aiming at the problem or shortcoming, this research widens the view of aisle trajectory selec‐ tion and proposes the conception of exploring the curve path layout method. Focusing on the curve aisle layout problem, we selected two typical non‐traditional layouts, Flying‐V and Fish‐ bone, to carry out the curve transformation of the parabolic cross‐aisle trajectory. The straight trajectory of the inclined cross‐aisles in the original layout is changed into a parabola, but the straight characteristics of the horizontal or vertical cross‐aisle remain unchanged (as shown in Fig. 3). The parabolic aisle layout expands the trajectory shape of the cross‐aisle, and could obtain a layout scheme with higher picking efficiency (Because it has enlarged the feasible region of the optimization model.) without significantly affecting the utilization ratio of warehouse area (Be‐ cause it does not change the number of the cross‐aisles.). In order to balance the contradiction between picking efficiency and utilization ratio of warehouse area better, we introduce the cross‐aisle length index in our model. Fig. 1 Two typical kinds of traditional warehouse internal layouts Fig. 2 Flying‐V and Fishbone internal layouts of warehouse 224 Advances in Production Engineering & Management 16(2) 2021 Designing a warehouse internal layout using a parabolic aisles based method This paper continues with the following content. The next section is a literature review. The third section describes the modelling process of the parabolic layout. The fourth section pro‐ vides the model of the corresponding straight layout. The fifth section gives out the methodology of the model solving. The sixth section presents our numerical results with discussions. The last section concludes this paper. Fig. 3 Parabolic Flying‐V and parabolic Fishbone internal layouts of warehouse 2. Literature review The new non‐traditional layout of Gue and Meller attracted wide attention from academia, and became a research hotspot soon. Pohl et al. studied the optimization of Fishbone layout under the dual‐command [5]. The results show that the background of dual‐command significantly reduces the advantage of Fishbone layout compared with traditional layout. Pohl et al. investi‐ gated the impact of different warehousing strategies on Flying‐V and Fishbone layout under sin‐ gle and dual‐command [6]. The results show that the optimal warehouse design parameters under stochastic strategy also perform well under turnover rate strategy. Gue et al. presented the aisle layout optimization problem in multiple P&D (Pickup and Deposit) points and unit‐load warehouses [7]. Flying‐V and Inverted‐V warehouse models are established respectively. The numerical results show that the advantage of Flying‐V aisle design is not as significant as that of the single P&D point. Cardona et al. made an in‐depth analysis of the cross‐aisle angle in Fish‐ bone layout [8]. According to the fixed and uncertain length and width of warehouse, the deter‐ mination of optimal aisle angle and its robust stability are discussed, which provides a theoreti‐ cal basis for the flexible decision‐making of actual warehouse layout design. Ö.Öztürkoğlu et al. based on the idea of increasing the number of cross‐aisles in traditional warehouse layout and optimizing the angles of its cross and picking aisles, presented three non‐traditional warehouse layouts, Chevron, Leaf and Butterfly [9]. Angular modelling and optimization analysis show that these three non‐traditional warehouse layouts can reduce the total picking distance by 19.53 %, 21.72 % and 22.52 % respectively compared with traditional warehouse layout. Some practical applications of non‐traditional warehouse layout are also discussed. Clark and Meller introduced the vertical travel parameter to modify the picking time model, and investigated the impact of vertical movement on Flying‐V and Fishbone layout on multi‐storey shelves [10]. The results show that the vertical operation of shelves has a great impact on Flying‐V layout, but not on Fishbone layout. Jiang et al. improved Fishbone layout by considering the characteristics of drive‐in racking, and obtained a Leaf‐like layout [11]. Liu et al. based on Fishbone layout, accord‐ ing to the general storage principle, studied the optimization of storage location allocation by using of the genetic Algorithms and MATLAB tool [12]. Ö.Öztürkoğlu et al. considered the rela‐ tionship between the location of a single pallet and the aisle, constructed a constructive multiple P&D points unit‐load warehouse aisle design network‐based model [13]. Particle swarm optimi‐ zation is used to optimize the aisle design scheme. Cardona et al. proposed a very practical three‐dimensional detailed design method of Fishbone layout [14]. This method considers the influence of aisle width, forklift speed, rental fee and maintenance cost, etc. Marco Bortolini et al. proposed a layout of inserting one or more straight non‐orthogonal cross‐aisles into rectangular warehouses [15]. The picking distance and warehouse area loss of this kind of layout under unit‐ load are also analysed. The average picking time and warehouse parameters under random warehousing strategy and individual operation instructions are obtained. The proposed layout is Advances in Production Engineering & Management 16(2) 2021 225 Zhang, Liang, Hou, Wang essentially a straight aisle Flying‐V and its expansion. When the optimal layout is adopted, the picking distance can be reduced by 7‐17 %. Liu et al. studied the problem of solving the optimal Fishbone layout by means of the genetic Algorithms [16]. Akhilesh Mesa discussed the non‐ traditional layout method of multiple P&D points and unit‐load warehouses with multiple cross aisles [17]. Unlike other designs in the past, the aisles are arranged in diamonds. In essence, the layout proposed in this paper can still be regarded as a combination of two Fishbone layouts. Mowrey et al. applied the idea of non‐orthogonal inclined aisle to the internal layout of retail stores [18]. In order to make full use of the limited space in the store and show more goods to customers, this research expands the practical application scope of non‐traditional layout. Zhang et al. proposed the Twin Leaf method of warehouse aisle layout in view of the deficiencies of Leaf layout [19]. Three basic characteristics of the Twin Leaf layout are analysed. Compared with Leaf layout, Twin Leaf layout can reduce the picking distance by 1.02 % on average, and the theoretical possible improvement space can be compressed by 45.45 %. To sum up, in the past ten years, non‐traditional warehouse layout methods have been deeply studied by scholars all over the world and applied by some American enterprises. These re‐ searches can be roughly classified into two major categories: the first one is the innovation of basic layout methods. Five non‐traditional layout methods, Flying‐V, Fishbone, Chevron, Leaf and Butterfly are presented. The second one is the efficiency analysis of basic layouts in specific warehousing operating environment. For example, the impact of different practical operating environments, such as single‐command, dual‐command, multiple P&D points warehouses, spe‐ cific warehousing strategy requirements, location assignment optimization, etc. In fact, the budding idea of non‐orthogonal design of warehouse aisles could be traced back to the 1960s. Moder and Thornton analysed the influence of pallet placement angle and aisle width on the utilization ratio of warehouse area through mathematical modelling [20]. Francis studied the optimal layout of rectangular warehouses based on the assumption of straight path and the consideration of picking and construction cost [21, 22]. Berry put forward a proposal to arrange pallets around a diagonal aisle into different roadways according to the characteristics of inven‐ tory units [23]. White studied the Euclidean efficiency estimation of radial aisles in non‐ rectangular warehouses [24]. Under the assumption of continuous space, Euclidean efficiency of four and six radial aisles is estimated. Bassan et al. based on Francis's research, considered the aisle structure parameters, and analysed the impact of the internal layout of warehouse on the overall operating cost of warehouse [25]. The total cost function is constructed to optimize the warehouse layout. In the early stage, these scattered discussions on the design ideas and meth‐ ods of non‐orthogonal warehouses layout were relatively shallow. Their basic limitation is that they are only preliminary theoretical discussions, no practical design specifications, lack of prac‐ tical application case testing, and thus they failed to arouse attention. Since then, relevant re‐ search has entered a relatively quiet period of about 30 years. 3. Research models Based on the analysis of morphological characteristics, the selection of parabolic type and the efficiency modelling of layout, the research is carried out in turn. For the convenience of analysis and comparison, the idea of continuous space modelling is adopted to build the efficiency model of the parabolic layout in unit‐load warehouses [9]. The cross‐aisle length is introduced as a supplementary evaluation index to the optimization of the layout efficiency. 3.1. Assumptions and symbols There are many factors affecting the layout of warehouse aisles. For the convenience of discus‐ sion, the following premise assumptions and parameter symbols are specially made. Assumptions： 1) The warehouse is rectangular. 2) Neglect the influence of warehouse height on the layout of warehouse aisle. 3) Only one storage or picking operation is carried out for goods at a certain location in the warehouse, and one‐way moving distance is used. 226 Advances in Production Engineering & Management 16(2) 2021 Designing a warehouse internal layout using a parabolic aisles based method 4) One‐time completion the operation of goods in the whole warehouse. 5) Consider the situation of a single P&D point, and every warehousing operation passes through the P&D point. 6) The widths of the cross and picking aisles in warehouse are neglected in our optimization models. For the convenience of comparison, we adopted the same assumption described in Ö.Öztürkoğlu et al. [9]. Although this continuous model with zero width aisles is not completely consistent with the reality, it is close enough for our research purposes, espe‐ cially for super‐large warehouses. 7) The volume of goods and the size of pallets or shelves in the warehouse are all neglected when modelling and analysing. Symbols： P&D: , , : : : , : : ∗ : ∗ : : Pickup and Deposit point, the entrance of goods; O is the origin of the coordinate system. The origin is located at the P&D point. The length direction of the warehouse is set to x axis and the width direction is set to y axis. Half of the length of warehouse horizontal direction; Total vertical width of warehouse; The location coordinates of any storage point in the warehouse: , 0 y h; The cross‐aisle angle of the right half of the warehouse under the straight Flying‐ V layout; The cross‐aisle angle of the right half of the warehouse under the straight Fish‐ bone layout; The optimal angle of the cross‐aisle when the picking distance of the straight Fly‐ ing‐V layout is the smallest; The optimal angle of the cross‐aisle when the picking distance of the straight Fishbone layout is the smallest. 3.2. Morphological characteristics of the parabolic layout These two parabolic layouts are both bilateral symmetrical, so we only need to take the discus‐ sion on the right half of the warehouse. According to the different possible intersection positions of the parabolic cross‐aisle and the warehouse edge, they are divided into two different sub‐ morphologies (as shown in Fig. 4 and Fig. 5). In the sub‐form 1 of parabolic Flying‐V as shown in Fig. 4(a), the right half warehouse is di‐ vided into two picking sub‐areas, A and B, by the parabolic cross‐aisle. In the sub‐form 2 of par‐ abolic Flying‐V as shown in Fig. 4(b), the right half warehouse is divided into three picking sub‐ areas, A, B and C, by the parabolic cross‐aisle and the line segment. Similarly, in the two sub‐ forms of parabolic Fishbone shown in Fig. 5(a) and Fig. 5(b), their right half warehouses are both divided into two picking sub‐areas, A and B, by the parabolic cross‐aisle respectively. (a) (b) Fig. 4 Two sub‐forms of parabolic Flying‐V layout Advances in Production Engineering & Management 16(2) 2021 227 Zhang, Liang, Hou, Wang (a) (b) Fig. 5 Two sub‐forms of parabolic Fishbone layout 3.3. Efficiency model for the parabolic layout The parabolic layout efficiency model consists of two parts. The first part is the total picking distance model. The second is the length formula of the cross‐aisle. Picking distance is the core index reflecting the efficiency of picking operation. The cross‐aisle length is the main factor af‐ fecting the utilization rate of warehouse area, which can be used as a supplementary index for layout efficiency evaluation. The parabolic type of the parabolic cross‐aisle should be deter‐ mined first, and the basic picking distance formula of each picking sub‐region is derived. Then the average total picking distance model and the length formula of the cross‐aisle are obtained by integration. Determination of the parabolic type Because our discussions are all taken in the right half of the warehouse, the parabolic cross‐aisle must pass through the P&D point (the coordinate origin O) of the warehouse, so the selected parabola should pass through the first quadrant and the origin. There are two basic types of pa‐ rabola passing through the origin, they take x axis and y axis as the axis of symmetry respective‐ and . The opening direction of the parabola is deter‐ ly. That are mined by the positive and negative of parameter a. Obviously, the parabola corresponding to a < 0 does not meet this basic requirement and should be discarded. Therefore, the only optional , 0 , 0. On this basis, combined with the parabola is: characteristics of two kinds of parabolic layout (especially, the dual equivalence of thin‐high warehouse and flat warehouse under Fishbone layout), through the trial calculation of typical , 0, 0 is syn‐ thin‐high, square and flat warehouse, the parabola of type thetically determined as the basis of modeling. In addition, because the study only involves the first quadrant, there are, x > 0, y > 0, see Eq. 1. 0, 0, 0, (1) 0 Based on the selected basic parabola type, according to the different integral variables, two expressions of the corresponding parabolic arc length (S) formula are given incidentally, as shown in Eq. 2, Eq. 3 and Eq. 4. (2) √4 1 √4 √4 √ 2 1 √ 228 √4 √ 1 (3) 1 2 2 1 1 1 2 √ 1 (4) Advances in Production Engineering & Management 16(2) 2021 Designing a warehouse internal layout using a parabolic aisles based method Picking distance model of the parabolic layout For convenience of comparison, the optimization model of average total picking distances of the parabolic layouts under unit‐load are constructed by using the idea of continuous space model‐ ling [9]. 1) Picking distance of parabolic Flying‐V. Based on the results of morphological analysis in Fig. 4. and the selected parabolic equation type (1), we can obtain that: √  Picking distance , (5) in A region of sub‐form 1. , , , (6) , (7) , (8) (9) (10) ,  The basic picking distance , in B region of sub‐form 1. (11) , (12) (13)  Picking distance of region A in sub‐form 2 , , , , , . , (14) , (15) 0 (16) ,  Picking distance of region B in sub‐form 2 , , , . (17) 0  Picking distance of area C in sub‐form 2 , , , , . (18) , (19) , 2 1 2 1 √ 2 2 1 1 √ (20) (21) (22) (23) ,  Based on the above continuous space modelling assumption, the optimal average total picking distance of parabolic Flying‐V is ∗ as shown in Eq. 24. ∗ (24) , , , , , , Advances in Production Engineering & Management 16(2) 2021 (25) (26) 229 Zhang, Liang, Hou, Wang 2) Picking distance of parabolic Fishbone. From the analysis of morphological features shown in Fig. 5, it is found that the basic picking distance formulas for the two sub‐forms of para‐ bolic Fishbone layout are the same. They are recorded as , and , . , (27) (28) (29) (30) , , Based on the assumption of continuous modelling, the optimal average total picking dis‐ tance of parabolic Flying‐V layout is ∗ see Eq. 31. ∗ (31) , , (32) , , , (33) Main aisle length formula for parabolic layout From Figs. 4 and 5, the optimal parabolic cross‐aisle lengths bone are respectively: ∗ ∗ ∗ ∗ ， ， ∗ ∗ ∗ ∗ ， ， ∗ ∗ ∗ ∗ and of Flying‐V and Fish‐ (34) (35) 4. Reference models To analyse the efficiency of parabolic layout, the corresponding straight layout should be taken as a reference. However, in the historical literature, the layout efficiency analysis of straight Fly‐ ing‐V and straight Fishbone does not consider the cross‐aisle length, and is discrete modelling. To unify the comparison benchmark, it is necessary to reconstruct the picking distance model of straight Flying‐V and straight Fishbone and the length formula of their cross‐aisle. 4.1 Picking distance model of straight layout Similarly, according to the different intersection positions between the cross‐aisle and the edge of the warehouse, the straight Flying‐V layout also has two different sub‐form, as shown in Fig. 6. In sub‐form 1, the right half warehouse is divided into two regions, A and B, by the cross‐aisle; in sub‐form 2, the right warehouse is divided into three sub‐regions A, B and C, by the cross‐aisle and line segment MM . Similarly, the straight Fishbone layout also has two different sub‐forms, as shown in Fig. 7. The right half warehouse is divided into two picking sub‐areas, A and B, by the cross‐aisle. (a) (b) Fig. 6 Two sub‐forms of straight Flying‐V layout 230 Advances in Production Engineering & Management 16(2) 2021 Designing a warehouse internal layout using a parabolic aisles based method (a) (b) Fig. 7 Two sub‐forms of straight Fishbone layout Picking distance for straight Flying‐V , and , respectively represent the basic picking dis‐ In the following passage, tances of A and B sub‐regions in sub‐form 1; , ， , and , represent the basic picking distances of A, B and C sub‐regions in sub‐form 2, respectively. , , , (36) , (37) , (38) , , (39) , , , (40) , (41) , (42) , , (43) , , , (44) , (45) , , (46) Based on the assumption of continuous modelling, the optimal average total picking distance of straight Flying‐V is ∗ shown in Eq. 47. ∗ , (47) , , , , (48) , (49) Picking distance for straight fishbone From Figure 7, we know that the basic picking distance formula of A and B sub‐regions in the two sub‐forms of straight‐line Fishbone layout are the same respectively. So, we can use , and , to express the basic picking distance formulas of A and B picking area respectively. , (50) Advances in Production Engineering & Management 16(2) 2021 231 Zhang, Liang, Hou, Wang , (51) Based on the continuous modelling assumption mentioned above, the optimal average total picking distance of straight‐line Fishbone layout ∗ can be obtained by the double integral of the above basic picking distance formulas. ∗ (52) , , , , , (53) (54) 4.2 Main aisle length for straight layout From Fig. 6 and Fig. 7, the optimal inclined cross‐aisle lengths and Fishbone are Eq. 55 and Eq. 56 respectively. ∗ / ∗， ∗ ， ∗ / ∗ ， ∗ / ， ∗ / ∗ and of straight Flying‐V (55) (56) 5. Model solving Since the double integral of the average total picking distance models established in this paper are all extremely difficult to obtain the analytical expressions, the analytical optimization analy‐ sis process which has been successfully used in aisle angle optimization by Öztürkoğlu et al. couldn’t be carried out [9]. In order to solve this problem and to ensure the reliability of the re‐ sults, we decided to adopt the Interval Numerical Simulation Method (INSM) and the Genetic Algorithms (GA) respectively to solve this model. INSM which has been successfully applied by Zhang Zhiyong et al. [19]. GA is widely used to solve this type of model. Both methods require differential discretization of the model first. 5.1 Differential discrete processing According to the general principle of differential discretization, w and h of the right half ware‐ house are discretized by m and n meshes respectively. （ , ）is used to represent the coordi‐ nates of the central points of each grid block. 0.5 1,2, … , 0.5 1,2, … , (57) In order to calculate and interpret the results directly and conveniently, the pallet can be un‐ derstood as a square with a side length of 1, and the length and width of the warehouse can be set as an integral multiple of the side length of the square pallet. See Eq. 58. It is worth pointing out that this setting will not affect the conclusions of relevant analysis. 1 (58) 5.2 Interval numerical simulation method Based on the above discretization results, INSM could be carried out. It should be noted that the parameters a and b of parabolic trajectory equation have a non‐closed theoretical range, and they also have a certain compensation effect on the layout efficiency of warehouse aisles with each other. The so‐called mutual compensation effect of parabolic aisle trajectory parameters means that when one parameter value is fixed and unchanged, the layout efficiency will change 232 Advances in Production Engineering & Management 16(2) 2021 Designing a warehouse internal layout using a parabolic aisles based method regularly with the change of another parameter value. Therefore, reasonable determination of bounded numerical simulation interval fixing method for a and b needs to be specially dealt with in conjunction with specific problems. After a lot of trial calculations and analysis, the simula‐ tion‐based optimization intervals selected for the 100 warehouses in this paper are: ∈ 0, 2 , ∈ 0.1, 100 . 5.3 Genetic algorithms The specific parameters of applying Genetic Algorithms (GA) to solve the model in this paper are as follows: The individual coded with natural numbers. The individual consisted of a and b. The value ranges are： ∈ 0, 2 , ∈ 0.1, 100 , same as the optimization interval selected in INSM above. The size of the population is 200. The fitness function by taking the inverse of the objective func‐ tion (Eq. 59). ； 2 ； 4 1 ∗ ； 3 ∗ ∗ ∗ (59) According to the adaptive priority determined by the weighted roulette method, individuals with higher fitness values have a higher probability of contributing to the next generation for one or more offspring [26]. Once individuals were selected, they were subjected to a two‐point crossover operation. After a lot of trial calculations and analysis, the crossover probability is 0.8 in this paper. The single point mutation method has been adopted. After a lot of trial calculations and analysis, the mutation probability is 0.1. The maximum number of generations is 200 in this paper. 6. Results and analysis 6.1 Efficiency analysis indicators As a reference, we also calculated the minimum total picking distance of the 100 sample ware‐ houses using the traditional orthogonal straight aisle layout and the ideal direct flight operation: ∗ and ∗ . They are regarded as the upper and lower bounds of the total picking distance of various non‐traditional layout, respectively, as shown in Eq. 60 and Eq. 61. ∗ ∗ ∑ ∑ ∑ (60) (61) ∑ Other efficiency analysis indicators are: ∗ ∗ 100 ∗ 100 ∗ ∗ (62) 100 ∗ (The theoretical possible improvement space of parabolic Flying‐V and straight Flying‐V) 100 ∗ ∗ 100 ∗ 100 ∗ (The theoretical possible improvement space of parabolic Fishbone and straight Fishbone) Advances in Production Engineering & Management 16(2) 2021 (63) (64) (65) (66) 233 Zhang, Liang, Hou, Wang 100 (67) 6.2 Numerical results With the help of MATLAB tools, the minimum total picking distance and the length of cross‐aisle of 100 warehouses with lengths and widths ranging from 10 to 100 (interval 10) were analysed and solved. The data range covers the size of conventional warehouses, and the results are rep‐ resentative. If we ignore the calculated error, the results of INSM and GA are basically consistent with each other. Among them, the percentage values of picking distance of two kinds of straight‐ line layout and direct‐flying operation of square warehouse are about 85.29, 80.47 and 76.52 respectively, which are basically consistent with the results of the relevant references [4, 7]. This shows that the accuracy of the calculation results in this paper is satisfactory. We draw Figs. 8 to 13 with the data. 6.3 Characteristics of parabolic Flying‐V Parabolic Flying‐V layout has three characteristics compared with straight Flying‐V layout in rectangular warehouses with unit‐load.  Semi‐square warehouses ( ⁄ 1). The total picking distance can be reduced by about 0.35‐0.37 %, and the theoretical possible improvement space has been compressed by about 3.38‐3.57 %. The picking efficiency is improved slightly. The length of the cross‐aisle could be shortened by 2.01‐2.62 %. The area utilization rate is slightly improved.  Semi‐flat warehouses ( ⁄ 1). The total picking distance can be reduced by about 0.38‐ 0.62 %, the theoretical possible improvement space has been compressed by 3.66‐12.26 %. The picking efficiency is improved obviously, and the change rule is roughly gradually greater, the flatter the warehouse, the bigger the value. The length of the cross‐aisle could be shortened by 1.92‐0.03 %, and the change rule is roughly gradually smaller, the flatter the warehouse, the smaller the value. That is to say, the more significant the picking effi‐ ciency improved, the less the warehouse area utilization rate increased. ⁄  Semi‐thin and tall warehouses ( ⁄ 1). Only when (2/3 1), the total picking distance could be reduced by about 0.22‐0.31 %, the theoretical possible improvement space has been compressed by 2.42‐3.12 %, and the picking efficiency is improved slightly. The length of cross‐aisle could be shortened by 2.38‐3.10 %, and the area utilization rate is also improved slightly. For other warehouses with ⁄ 2/3 (ignoring 7/10 outliers), straight Flying‐V layouts are better. 6.4 Characteristics of parabolic fishbone Parabolic Fishbone layout also has three characteristics compared with straight‐line Fishbone layout in rectangular warehouses with unit‐load.  Semi‐square warehouses ( ⁄ 1). The optimal parabola Fishbone degenerates to the optimal straight Fishbone. That is to say, the straight Fishbone layout is better. It is not dif‐ ficult to understand that this feature is determined by the symmetrical distribution of the aisles in Fishbone layout.  Semi‐flat warehouses ( ⁄ 1). Only when ⁄ 3(ignoring calculation error), the total picking distance could be reduced by about 0.02‐0.04 %, the theoretical possible im‐ provement space has been compressed by about 1.27‐1.83 %, and the picking efficiency is improved slightly. However, its length of the cross‐aisle will increase by 4.63‐19.50 %, which means the reduction of the utilization rate of warehouse area. For other ware‐ ⁄ houses with 1 3 (ignoring calculation error), straight Fishbone layouts are bet‐ ter. 234 Advances in Production Engineering & Management 16(2) 2021 Designing a warehouse internal layout using a parabolic aisles based method Fig. 8 Optimal main aisle trajectory map of parabolic Flying‐V of 100 warehouses Fig. 9 Optimal main aisle trajectory map of straight Flying‐V of 100 warehouses Advances in Production Engineering & Management 16(2) 2021 235 Zhang, Liang, Hou, Wang Fig. 10 Optimal main aisle trajectory map of parabolic Fishbone of 100 warehouses Fig. 11 Optimal main aisle trajectory map of straight Fishbone of 100 warehouses 236 Advances in Production Engineering & Management 16(2) 2021 Designing a warehouse internal layout using a parabolic aisles based method (a) TSSFV‐TSPFV (b) TSSFB‐TSPFB Fig. 12 Distribution diagrams of TSSFV‐TSPFV and TSSFB‐TSPFB and TSSFB‐TSPFB of 100 warehouses (a) TGB Fig. 13 Distribution diagrams of TGB (b) TGB and TGB of 100 warehouses  Semi‐thin and tall warehouses ( ⁄ 1). They have the same characteristics as the semi‐ flat warehouse. For Fishbone layout, thin and tall warehouses and flat warehouses with inverted aspect ratio are dual layouts. Let the optimal layout of the flat warehouse rotated 90 degrees counter clockwise with the P&D point as the centre first, and then turn 180 de‐ grees around the longitudinal axis, the optimal layout of thin and tall warehouse with in‐ verted width‐height ratio can be obtained. The optimal parabolic trajectory equation of another warehouse can be obtained by interchanging x and y in the optimal parabolic tra‐ jectory equation of one warehouse. 7. Conclusion Parabolic layout has its own characteristics and advantages. Although the non‐closed theoretical range of parabolic trajectory equation parameters a and b and the mutual compensation effect in layout efficiency bring some technical difficulties to the related analysis, the parabolic transfor‐ mation effect of two typical straight non‐traditional layouts preliminarily verifies the scientifici‐ ty and feasibility of the idea. This research enriches the relevant theory of warehouse aisle lay‐ out, and provides certain theoretical and technical basis for further tapping the efficiency poten‐ tial of warehouse layout, breaking through the realistic situation of straight aisle leading ware‐ house layout. Through the analysis of layout characteristics, the selection of parabolic type and the intro‐ duction of cross‐aisle length index, the efficiency optimization models of two parabolic layouts under unit‐load are constructed. The calculation results of 100 warehouses with different sizes show that the picking distance of parabolic Flying‐V could be reduced by 0.22%‐0.62% com‐ pared with the straight layout, and the theoretical possible improvement space has been com‐ pressed by 2.42‐12.26 %. Its length of cross‐aisle is shortened by 0.03‐3.10 %. The picking Advances in Production Engineering & Management 16(2) 2021 237 Zhang, Liang, Hou, Wang distance of parabolic Fishbone could be only reduced by 0.02-0.04 %. The theoretical possible improvement space has been compressed by 1.27-1.83 %. But its length of cross-aisle will increase by 4.63-19.50 % significantly. The results contribute to the solution of manufacturing logistics problems, and the formation of green production mode. The objective of logistics solutions is to influence and manage material flows in the right way which is very important for manufacturing systems. The solution of manufacturing logistics problems in specific market area requires the utilization of the means of algorithmization, heuristics, mathematical statistics, modelling and computer simulation [27]. Green production mode is an advanced manufacturing mode. From the perspective of enterprises, reducing the operating cost of green manufacturing mode through scientific and technological innovation is a very good decision-making scheme for enterprises [28]. Future research directions include: based on the parabolic layout, combined with the actual needs of warehousing operation and management, putting forward more scientific and reasonable target of multi-attribute comprehensive measurement of efficiency; investigating the effects of task interleaving (dual-command), multiple P&D points, and different warehousing strategies; and further studying the warehouse design guidelines to enrich and perfect the layout principle of curve trajectory aisle [29]. Based on the analysis of efficiency mechanism, the practical design rules of the corresponding layout will be studied, which can provide reference for the formulation of non-linear warehouse layout design specifications. Also, many practical warehousing problems, such as the order assignment, the order batching and the picker routing of large wave picking warehouses which have been modelled by Ardjmand et al. [30], can be integrated with the non-linear warehouse layout. With the continuous development of social economic technology, intelligent control units have become the trend of manufacturing enterprises [31], we believe that the layout of non-rectangular complex special-shaped warehouses based on curve trajectory aisle could become an important research topic. In addition, the proposed method in this paper may also be used in principle for the arrangement of machines and devices in a workshop. However, compared with the internal layout of a warehouse, the layout in a workshop is much more complex. For example, there are many big differences in sizes, shapes, site occupations, handling tools and process relations of different machines and devices. The specific processing technology relationship between machines and devices and materials and personnel must also be considered. Therefore, although the optimization methods can be used to finely study the workshop layout problem, the objectives and constraints of the corresponding mathematical model will be much more complex. However, we do believe that with the popularization of modern large-scale intelligent factories, the fine workshop layout using mathematical models and simulation optimization technology will become an important research direction. Acknowledgement The research is partly supported by the Key Project of National Social Science Foundation of China (grant no. 20AJY016). 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China ABSTRACT ARTICLE INFO Disassembly activities take place in various recovery operations including remanufacturing, recycling, and disposal. Product disassembly is an effective way to recycle waste products, and it is a necessary condition to make the product life cycle complete. According to the characteristics of the product disassembly line, based on minimizing the number of workstations and bal‐ ancing the idle time in the station, the harmful index, the demand index, and the number of direction changes are proposed as new optimization objectives. So based on the analysis of the traditional genetic algorithm into the preco‐ cious phenomenon, this paper constructed the multi‐objective relationship of the disassembly line balance problem. The disassembly line balance problem belongs to the NP‐hard problem, and the intelligent optimization algorithm shows excellent performance in solving this problem. Considering the charac‐ teristics of the traditional method solving the multi‐objective disassembly line balance problem that the solution result was single and could not meet many objectives of balance, a multi‐objective improved genetic algorithm was pro‐ posed to solve the model. The algorithm speeds up the convergence speed of the algorithm. Based on the example of the basic disassembly task, by com‐ paring with the existing single objective heuristic algorithm, the multi‐ objective improved genetic algorithm was verified to be effective and feasible, and it was applied to the actual disassembly example to obtain the balance optimization scheme. Two case studies are given: a disassembly process of the automobile engine and a disassembly of the computer components. Keywords: Assembly; Disassembly; Line balancing; Multi‐objective optimization; Remanufacturing; Product recovery; Product life cycle; NP‐hard problem; Improved genetic algorithm *Corresponding author: wangyj@dlpu.edu.cn (Wang, Y.J.) Article history: Received 13 May 2021 Revised 23 May 2021 Accepted 4 June 2021 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 continuous development of science and technology, the demand for new products is increasing, and the number of waste products and components is increasing, which will inevita‐ bly cause pollution to the environment. In order to solve the shortage of resources and realize sustainable development, the enterprise must pay attention to the recycling and reuse of waste products. Disassembly is the basic action of product recycling, extracting useful parts and recy‐ cling harmful parts to achieve a circular economy and green manufacturing. In recent years, more and more researchers have devoted themselves to the disassembly line balancing problem (DLBP). Gungor and Gupta [1] analyzed and described the DLBP problem and put forward the influencing factors. Avikal et al. [2] used heuristic algorithm to solve DLBP prob‐ lem, but there are some limitations. Altekin and Akkan [3] used a linear programming method to optimize DLBP to achieve the purpose of a balanced disassembly line. The Genetic Algorithm was used to solve the disassembly line balancing problem and the optimal solution was obtained 240 Optimization of disassembly line balancing using an improved multi‐objective Genetic Algorithm in an effective time. Kailayci et al. adopted a simulated annealing algorithm, which had strong processing capacity and better local search capability than other algorithms. However, it took a long time and had a weak ability to obtain a globally optimal solution [4, 5]. Nikola et al. solved the DLBP problem under multi‐objective conditions in actual production [6‐10]. Ding et al. [11] proposed an ant colony algorithm based on Pareto to optimize the four objectives. Cao et al. solved disassembly line balancing problems with different algorithms [12‐22]. By analyzing the above works of literature and methods, it is easy to see intelligent optimiza‐ tion algorithms such as Genetic Algorithm, which have excellent performance in solving multi‐ objective optimization problems. Given the shortcomings of the above researches, an improved genetic algorithm for solving the disassembly line balancing problem was proposed, and its advantages for solving this kind of problem were analyzed and verified through specific problems and examples. The paper is organized as follows. The mathematical model is summarized in Section 2. Op‐ timization and analysis with multiple objectives are also described in this section. Section 3 is the presentation of the solution. The case analysis and discussion are reported in Section 4. Fi‐ nally, the conclusions are reported in Section 5. 2. Mathematical models 2.1 Basic assumptions There are many uncertainties in the actual disassembly process, and these uncertainties are bound to affect the production beat of the disassembly online operation. Considering the appli‐ cation scope of the model in practice, this paper ignores extreme phenomena in the process of building the U‐shaped DLBP model, so the following assumptions are made. 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) The production beat is unchanged. The disassembly resources are the same or similar in structure. The resource is completely disassembled, and the operation stops after the disassembly to the last part. All disassembly tasks are performed on the disassembly line with no missing parts dur‐ ing the operation. Operator proficiency and experience are the same, that is, the disassembly time of each part does not vary from operator to operator During the disassembly, the required parts are guaranteed to be intact, that is, the disas‐ sembly process can bring economic benefits. The disassembly time of the parts is not affected by external factors such as product quality, and there is no dependence between the parts. The disassembly process is carried out according to the disassembly task and cannot be disassembled. Operating time is normally distributed ~ , . The constraints are as follows. a) The total operation time of each workstation shall not exceed the production beat. b) The same disassembly operation shall not be carried out in two or more workstations. 2.2 Basic parameters Assuming that each unassembled part is a disassembly task, the number of parts is (also known as the number of disassembly tasks). is the set of all the tasks, = 1,2,3, … , . is the number of workstations. is the production beat. The parameter represents the sum of activity time of all disassembly tasks allocated on the workstation . 2.3 Decision variables The variable x represents the relationship between task 0. otherwise Advances in Production Engineering & Management 16(2) 2021 and workstation , 1, 241 Wang, Wang, Cheng, Zhang, Liu, Shi, Ma, Zhou 1，task is assigned to workstation 0， else The balanced optimization of the disassembly line should not only consider the balanced distri‐ bution of work tasks but also include the environment and resources. The products to be dis‐ mantled may contain harmful substances, such as heavy metals and chemical poisons, which should be given priority in the disassembly operation. The main purpose of disassembly is to recycle and use the spare parts with surplus value, and the valuable parts should be dismantled as soon as possible. To minimize the disassembly time of the product to be disassembled, the number of changes of disassembly orientation is also included in the optimization space to shorten the disassembly time. In this paper, five objectives of disassembly line balancing are considered and optimized. 1) 2) 3) 4) 5) Minimum number of workstations. Balance the free time of each workstation. Disassemble high‐demand components as soon as possible. Disassemble hazardous parts as soon as possible. The least change of direction for disassembling. (1) (2) =∑ (3) (4) (5) (6) where 1 needed , 0 else 1 harmful . 0 else The direction indicators are introduced to evaluate the solution sequence. The smaller the value, the less the change of direction in the disassembly process, the better the solution will be. The following relation is used to represent each direction of the disassembly process relative to the parts and workstations. 3 Z direction Y direction 2 1 X direction (7) 1 X direction Y direction 2 3 Z direction The direction indicators are expressed in the form of decision variables. ∑ , 1, 0,  (8) The multi‐objective disassembly line balance is represented as the following model by the above balance objectives. 242 Advances in Production Engineering & Management 16(2) 2021 Optimization of disassembly line balancing using an improved multi‐objective Genetic Algorithm , , , (9) s.t. / ⋅ ⋅ (10) , ∀ ∈ 1,2, . . . , ⋅ ,∀ , (11) 1 (12) The objective function (Eq. 1) represents the minimum number of workstations. The objec‐ tive function (Eq. 2) means to balance the free time of each workstation. The objective function (Eq. 5) represents the index value of disassembling parts with high demand first. The objective function (Eq. 6) represents the index value of priority disassembly of hazardous parts. The ob‐ jective function (Eq. 8) represents the least change of direction for disassembling. The constraint function (Eq. 10) means that the number of workstations should be no less than the number of theoretical workstations and no more than the number of disassembly tasks. The constraint function (Eq. 11) represents that the disassembly time in each workstation shall not exceed the production beat. The constraint function (Eq. 12) means the disassembly se‐ quence must meet the disassembly priority relationship. 3. Presentation of solution and used genetic operations Genetic Algorithm (GA) was proposed in 1967 by a scientific research team led by John Holland of the University of Michigan [23]. It is a natural evolution‐based algorithm for intelligent opti‐ mization search based on the concept of biological evolution and the laws of biogenetics and natural selection. GA relies on the information exchange of the individuals in the community and population search, to encode the parameters of the solution as genes, and several genes consti‐ tute a chromosome, as an individual, many chromosomes experience generation genetic by crossover, selection and mutation operation, the search results gradually converge to the region where the optimal solution is located until the optimal solution is found. The advantage of GA in solving the problem of activity order optimization is that the optimal solution can be found without going through the space of all solutions, and the solution effect for the multi‐objective optimization problem is significant. 3.1 Coding Encoding methods include binary encoding, float‐point encoding, real number encoding, and so on. Given the disassembly line activity tasks, by using a chromosome encoding rules based on activity order successively, n disassembly elements are arranged in a row, corresponding to a gene site, and according to the priority diagram of activity order, these disassembly elements are assigned to each workstation and are coded according to the sequence of processes in the work‐ station. The weighted time value of the process shall not be higher than the scheduled produc‐ tion rhythm in the workstation. Zero insertion [24] is used to represent the start (or end) posi‐ tion of the activity element of the current workstation. There are activities and 1 zeros. The activity element between two adjacent zeros is the same workstation. 3.2 Initial population generation The quality of the initial population has an obvious impact on the evolution process and the effi‐ ciency of the algorithm. In order to ensure the diversity of individuals and solutions in the initial population, the topological sorting random search is used to generate the initial population. The process is as follows. Advances in Production Engineering & Management 16(2) 2021 243 Wang, Wang, Cheng, Zhang, Liu, Shi, Ma, Zhou  According to the diagram of activity priority order, the task without pre‐task is found in the complete set of tasks (population) and put into the new set , and the task and its re‐ lated sequences are deleted in the operation sequence diagram.  Repeat the above operations until the task set the empty set, and finish the task assign‐ ment. The selected task is put into the corresponding gene position in each step, and the obtained sequence is the initial feasible disassembly sequence. 3.3 Decoding The coding adopts a one‐dimension group solution sequence based on the task, which cannot determine the individual's merits and demerits. The solution sequence needs to be allocated to each workstation [25, 26]. The operation is as follows.  Start the first workstation j = 1.  Initialize the current workstation time and remaining time.  Find task i in the solution sequence. When the task time allocated exceeds the current re‐ maining time, the assignment fails. Open a new workstation randomly and initialize the current workstation time and remaining time. Else assign task i to the current work‐ station, update the workstation time and remaining time, and cycle until permutation se‐ quence. By decoding each individual and inserting zeros between workstations, each workstation and task assignment in the population can be determined, and the visualization of the algorithm can be improved. 3.4 Fitness Fitness function plays an important role in the evaluation of individuals in GA search evolution. Only the objective function can be used to optimize the system in the solution space. In the Genetic Algorithm space, the objective function is transformed into individual fitness according to certain rules, and the fitness value is evaluated to realize the judgment of the feasible solution in the solution space. 3.5 Genetic operators Selection Roulette is the most commonly used selection method. The sampling idea is that the probability of the selected individual inheriting the next generation is directly proportional to the fitness value. The higher the evaluation of the fitness function of the individual, the greater the proba‐ of an individual selected in bility of inheriting to the next generation. This is the probability function (13). / , 1~ (13) Crossover Crossover is an important way to form new individuals. Two chromosomes are selected from the selected population, and some genes are exchanged with specific rules to form new individuals after recombination [27]. Because the random crossover method in traditional methods often leads to a large number of repetitions and conflicts, results in infeasible solutions and affects the operational efficiency of the algorithm, this process uses two‐point mapping crossover method, randomly determines two crossover points on the parent chromosome, sorts some genes between the two chromatids of the father generation and adopts partial mapping to save the genes on both sides of the cross‐ over point and put them into a new individual, thus producing two new offspring individuals. Assuming that the third and fifth gene locus are randomly selected as the crossover points, the sequence {6, 3, 7}, {2, 4, 9} before the two crossover points in the parent generation can be pre‐ served, and the sequence {8, 5, 1} between the two crossover points in the parent 2 is {1, 5, 8}. The specific process is shown in Fig. 1. 244 Advances in Production Engineering & Management 16(2) 2021 Optimization of disassembly line balancing using an improved multi‐objective Genetic Algorithm Fig. 1 The diagram of crossover Mutation Similar to the crossover operation, the mutation will produce infeasible solutions due to the constraints of the activity priority relationship. In this process, single‐point variation is used. A mutation point (position 4 in Fig. 2) is randomly generated on the chromosome of the parent generation. According to the activity priority diagram, the pre‐process and post‐process of the mutation point are found out, so that the pre‐process and its previous gene position, post‐ process and its subsequent gene position can be retained. Position 4 is randomly inserted into the nearest gene position between the pre‐process and the post‐process in the chromosome, and then the above genes are integrated to generate new progeny chromosomes. If the selected mu‐ tation position does not have an optional insertion position, the mutation point can be re‐ selected. Fig. 2 The diagram of mutation 3.6 Termination conditions As a search tool of repeated iterations, GA approximates the optimal value infinitely through multiple evolutionary cycles, instead of just calculating the optimal solution. Therefore, it is nec‐ essary to determine the termination condition and the generations of genetic iteration. At the beginning of the algorithm, the number of iterations should be set as small as possible, and then increases iterations as appropriate. When the number of iterations exceeds the required maxi‐ mum number of generations the algorithm stops. Advances in Production Engineering & Management 16(2) 2021 245 Wang, Wang, Cheng, Zhang, Liu, Shi, Ma, Zhou 3.7 Implementation process Although the individuals of the initial population are feasible, it is impossible to determine whether the optimal individuals appear in the early stage of the algorithm. Therefore, the prob‐ ability of crossover and mutation can be increased to enhance the optimization ability of the algorithm. In the later stage of iteration, when the algorithm converges gradually, the individual fitness value becomes higher. In order not to destroy the excellence of genes in individuals, the probability of crossover and mutation is often reduced to improve the operation efficiency of the algorithm. In this paper, the adaptive crossover and mutation probability are used. The symbols used to run the algorithm are as follows: is the maximum number of iterations, is the num‐ is the mutation probability. is the min‐ ber of iterations, is the crossover probability. is the maximum probability. is the minimum mutation imum crossover probability, is the maximum mutation probability. probability. (14) (15) Step1: Determination of parameters. Select the value of , , . Step2: Initialization population. Order 0, the initial population P 0 with individuals are generated under the condition of the beat constraint and the priority of tasks. Step3: Fitness assessment. The fitness value of each individual in the generation population is calculated. Step4: Selection. individuals are selected from the generation population and cloned into the 1 generation. Step5: Crossover. Step6: Mutation. Step7: Optimal preservation strategy. Step8: Repeat. Order t← 1, when the termination condition is met, it ends. Else, turn to Step3. 4. Case studies: The practical application and analysis 4.1 Disassembly of the computer components The disassembly information of a computer component with 8 parts is shown in Table 1. The disassembly of the parts is shown in Fig. 3. The improved Genetic Algorithm is used to solve the problem. Matlab R2012b software is used to realize the algorithm program on the windows10 system platform, and the above examples are solved [28]. The minimum number of workstations is 4, the hazard index is 7, the demand is 211, the direction index is 7. The optimal solu‐ tion is given in Table 2. Table 3 shows the optimal disassembly series solution and the work‐ station after balancing. Among them, disassembly tasks 1 and 5 are assigned to workstation 1, and workstation 2 is mainly responsible for disassembly tasks 3, 6, 2, and so on. In addition, the optimal solution removes high demand parts 3, 6, 2, and hazardous parts 7 earlier, allowing sev‐ en directions to change, and the calculation time of the algorithm is less than 1s. The equilibrium and hazard objectives of the solution obtained are the same as those in reference [29], while the demand index is better than that in reference [29], and there is one more direction index than reference [29]. Therefore, the overall performance of the solution obtained is better than that in reference [29]. The optimal solutions obtained by these two single objective algorithms are shown in Table 2, and the parameter design in reference [29] is combined with the DLBP problem. After repeat‐ ed tests on the quality of the solution and the efficiency of the algorithm, the parameters in this 0.5, 0.95, 0.005, paper are set as follows: 1, 100, 0.01. After calculation, the optimal value is obtained and the operation time is analyzed. 246 Advances in Production Engineering & Management 16(2) 2021 Optimization of disassembly line balancing using an improved multi‐objective Genetic Algorithm The optimal disassembly series solutions are shown in Table 3. Comparing the improved Genetic Algorithm with the traditional Genetic Algorithm, it can be seen that the test results are more prominent than the traditional algorithm in dealing with this problem, and the running time is shorter. Disassembly tasks 1 2 3 4 5 6 7 8 Table 1 The disassembly information for eight components Disassembly operation Time Demands Harmful Disassembly PC top cover (TC) 14 360 N Disassembly floppy disk driver (FD) 10 500 N Disassembly hard disk driver (HD) 12 620 N Disassembly bottom plate (BP) 18 480 N Disassembly PCI card 23 540 N 16 750 N Disassembly RAM modular Y Disassembly power supply (PU) 20 295 Disassembly mother board (MB) 14 360 N Direction ‐x +x ‐x ‐x +y +z ‐x ‐x Fig. 3 The activity priority relationship of components Algorithm Reference [29] GA Improved GA Table 2 The comparison of optimization results Number of Balance index Demand quantity Harmful index workstations S F D H 4 211 19275 7 4 211 19025 7 Advances in Production Engineering & Management 16(2) 2021 Direction index R ‐ 7 247 Wang, Wang, Cheng, Zhang, Liu, Shi, Ma, Zhou Table 3 The optimal disassembly series solution of DLBP Workstation i S1 S2 S3 S4 1 14 5 23 3 12 Free time 0 s 6 16 Disassembly series 2 10 8 36 7 20 4 18 Total time 37 38 36 38 Idle time 3 2 4 2 4.2 Disassembly of the automobile engine Taking the automobile engine disassembly example in reference [30] as the research object, the engine is completely disassembled, and Fig. 4 shows the engine disassembly diagram. The origi‐ nal enterprise did not consider the damage and demand of disassembly, and the variability is poor, so it cannot adapt to the change of disassembly task in time. Now the improved Genetic Algorithm is used to improve the engine cylinder block disassembly line. The relevant data are shown in Table 4. The bearing, toothed belt, belt, connection key, connection pin, and other parts in the assem‐ bly are simply removed and replaced with a group of assembly and fasteners to reduce the number of disassembly parts. In addition, the disassembly operation was investigated to obtain detailed disassembly data such as operation time, disassembly priority relationship, constraints, disassembly demands, hazards, and direction changes of each part. Standard operating time (SST) on line disassembly was measured. Standard operating time is the time taken by a skilled worker to complete a process at a normal speed under normal operation. The disassembly oper‐ ation time is measured by the stopwatch method in industrial engineering, and the optimization goal is taken into account the damage of parts, demand, and the change of operation direction. Fig. 4 The engine disassembly diagram 248 Advances in Production Engineering & Management 16(2) 2021 Optimization of disassembly line balancing using an improved multi‐objective Genetic Algorithm No. 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 26 27 28 29 30 31 32 Table 4 The engine part elements Name Time Demand Camshaft coverscrew 5×12 5 Camshaft cover 3 65 Camshaft cover plate screw 4×20 15 Camshaft cover plate 2×8 15 Starting claw 6.5 35 Pulley 20 40 Side cylinder head screw 6.5 45 Side cylinder head 5.5 65 Chain restraint screw 4.5×4 8 Chain restraint mechanism 3 8 Chain 10 15 Camshaft 3.2 60 Cylinder head cover screw 4×8 70 Top cylinder head cover 21 30 Rocker fastening screw 8×16 45 Rocker 3×16 95 Valve 4×16 50 spark plug 28×4 40 Cylinder block screw 5.5 40 Cylinder Block 20 95 Connecting rod cover 10×4 65 Connecting rod 5×4 40 Connecting rod screw 7×4 40 Crankshaft bearing cover screw 5×10 25 Crankshaft bearing cover 3×5 15 Crankshaft 20 95 Cylinder Black 23 30 Flywheel nut 15 30 Flywheel screw 5.5×6 45 Flywheel 30 50 Seat screw 5×12 65 Seat 6 30 Harmful N N N N N N N N N N N N N N N N N Y N N N N N N N N N N N N N N Direction +z +z +z +z +x +x +x +x +x +x +z +z +z +z +z +z +z +z +z +z +z +z +z +z +z +z +z ‐x ‐x ‐x +z +z Input the operation sequence matrix of the automobile engine of the disassembly line and program it on Matlab2012b. The given beat is 120s. The parameters of the algorithm are as follows: = 50, MaxGen = 150, = 0.9. By optimizing multiple objective functions, the algo‐ rithm in this paper achieves the prediction results, jumps out of the local optimal solution to obtain a better solution, and the searchability is significantly improved. The optimization results are shown in Fig. 5. The zeroing operation in encoding and decoding can isolate each workstation. After the oper‐ ation, the balancing result of linear layout and U‐shaped layout is shown in Fig. 6. The number of workstations in linear layout optimization is 15, while the number of U‐shaped layouts is 9. Therefore, the U‐shaped layout can minimize the number of workstations and has a certain economy. The station allocation and operating time range volatility are shown in Table 5 and Table 6. The range volatility is the ratio of idle time to total operating time. All the data fluctuate within the 10 % range. It can be concluded that the assignment and optimization of work elements of both linear and U‐shaped workstations in DLBP problem are reasonable, which proves the scien‐ tific and effectiveness of the model. The utilization rate of each workstation in the disassembly line under the U‐shaped layout is higher than that of the traditional straight line, and the idle waiting time in the workstation is smaller, which shows a significant difference between the two types of disassembly lines. It is proved that the U‐shaped layout is better and can achieve the balance effect. The algorithms can find out a better solution, the number of workstations, balance index, re‐ source index got better optimization, such as the direction indicators have improved, can achieve better solving the performance. The feasibility of the proposed improved genetic algo‐ rithm in solving the balance problem of a multi‐objective U‐shaped disassembly line is verified. The optimization results of straight layout and U‐shaped layout are compared, and it is proved that U‐shaped layout is more suitable for disassembly line layout. Advances in Production Engineering & Management 16(2) 2021 249 Wang, Wang, Cheng, Zhang, Liu, Shi, Ma, Zhou Fig. 5 The optimal results 5 1 14 16 17 19 29 28 30 6 2 7 18 8 12 4 11 10 3 13 9 15 20 24 21 22 26 27 23 32 31 25 (a) The balancing result of linear disassembly line layout 5 30 28 29 1 3 2 13 15 18 32 19 14 31 12 27 23 10 9 27 20 8 7 6 16 4 21 24 26 25 11 22 (b) The balancing result of U‐shped disassembly line layout Fig. 6 The balancing results Table 5 The workstation allocation of liner disassembly line layout Workstation number Station time (s) Idle time (s) 1 2 3 4 5 6 7 8 9 10 114.5 59.5 117.5 72.5 109 114 119 110 55 117 5.5 60.5 2.5 47.5 11 6 1 10 65 3 250 Range fluctuation Range volatility (%) 0.56 6.13 0.25 4.85 1.11 0.61 0.10 1.01 6.58 0.3 Advances in Production Engineering & Management 16(2) 2021 Optimization of disassembly line balancing using an improved multi-objective Genetic Algorithm Table 6 The workstation allocation of U-shaped disassembly line layout Workstation number 1 2 3 4 5 6 7 8 9 5. Conclusion Station time (s) 105.5 108 108 112 119 119 116.5 11.35 89.2 Idle time (s) 14.5 12 12 8 1 1 3.5 6.5 30.8 Range fluctuation (%) 1.47 1.21 1.21 0.81 0.10 0.10 0.35 0.66 3.12 In this study, an improved Genetic Algorithm is used to solve the disassembly line balancing problem. On the premise of ensuring balance, considering the influence of other targets on balance, such as hazard, demand, and disassembly direction index, adopting real coding strategy and topological ordering, the optimal solution of the problem is obtained by improving the crossover and mutation operation. The examples show that DLBP-GA is feasible to solve practical problems, has good adaptability to multi-objective optimization problems, and can better balance operating time and improve the utilization rate of the equipment. Compared with the traditional linear layout, the U-shaped disassembly line layout has more compact space, which can reduce the material handling and walking distance of personnel, and the operators can make faster adaptation to the changes of external demand, which is in line with the concept of flexible production. The U-shaped disassembly line is a line balance form that is more in line with the lean production theory, and the specific engine disassembly problem is solved. In this paper, there are some directions for further research on the disassembly line, including the following contents. In the optimization goal, new optimization can be combined with cost, profit, scrap parts, and other issues. Other representative intelligent optimization algorithms can be selected for fusion, and the clustering optimization method based on Pareto can also be used for multi-objective optimization problems to increase the feasibility of the algorithm to search for the optimal solution. This paper mainly studies the balance of disassembly lines of a single variety, and the subsequent research can be extended to the study of mixed-flow disassembly lines of multiple varieties, and establish a disassembly line model based on complex random operation time. The optimization objective of this paper is to optimize the total number of workstations when the production beat is known. Subsequent researchers can consider optimizing the production beat and establishing a new disassembly line balance model under the premise of a certain number of workstations. Acknowledgments The research was funded by the National Natural Science Foundation of China (No.71603035), Natural Science Foundation of Liaoning province (LJKZ0532), and Natural Science Foundation of Liaoning province (No. J2020108). References [1] [2] [3] [4] Güngör, A., Gupta, S.M. (2002). 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Automobile engine structure and disassembly, Beijing Institute of Technology press, Beijing, China. 252 Advances in Production Engineering & Management 16(2) 2021 Advances in Production Engineering & Management ISSN 1854‐6250 Volume 16 | Number 2 | June 2021 | pp 253–261 Journal home: apem‐journal.org https://doi.org/10.14743/apem2021.2.398 Original scientific paper Modelling and optimization of sulfur addition during 70MnVS4 steelmaking: An industrial case study Kovačič, M.a,b,c,*, Lešer, B.a, Brezocnik, M.d a Štore Steel, d.o.o., Štore, Slovenia University of Ljubljana, Faculty of Mechanical Engineering, Ljubljana, Slovenia c College of Industrial Engineering Celje, Celje, Slovenia d University of Maribor, Faculty of Mechanical Engineering, Maribor, Slovenia b ABSTRACT ARTICLE INFO Štore Steel Ltd. is one of the major flat spring steel producers in Europe. Among several hundred steel grades, 70MnVS4 steel is also produced. In the paper optimization of steelmaking of 70MnVS4 steel is presented. 70MnVS4 is a high‐strength microalloyed steel which is used for forging of connecting rods in the automotive industry. During 70MnVS4 ladle treatment, the sulfur addition in the melt should be conducted only once. For several reasons the sulfur is repeatedly added and therefore threatening clogging during continu‐ ous casting and as such influencing surface defects occurrence and steel cleanliness. Accordingly, the additional sulfur addition was predicted using linear regression and genetic programming. Following parameters were col‐ lected within the period from January 2018 to December 2018 (78 conse‐ quently cast batches): sulfur and carbon cored wire addition after chemical analysis after tapping, carbon, manganese and sulfur content after tapping, time between chemical analysis after tapping and starting of the casting, fer‐ romanganese and ferrosilicon addition and additional sulfur cored wire addi‐ tion. Based on modelling results it was found out that the ferromanganese is the most influential parameter. Accordingly, 12 consequently cast batches (from February 2019 to October 2019) were produced with as lower as pos‐ sible addition of ferromanganese. The additional sulfur addition in all 12 cases was not needed. Also, the melt processing time, surface quality of rolled material and sulfur cored wire consumption did not change statistically signif‐ icantly after reduction of ferromanganese addition. The steel cleanliness was statistically significantly better. Keywords: Metallurgy; Steelmaking; High‐strength steel 70MnVS4; Microalloyed steel; Modelling; Optimization; Evolutionary algorithms; Genetic programming; Multiple linear regression *Corresponding author: miha.kovacic@store‐steel.si (Kovačič, M.) Article history: Received 3 April 2021 Revised 5 June 2021 Accepted 12 June 2021 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 Producing melted steel is commonly called primary steelmaking (i.e., primary metallurgy). The melted steel (made from ore or scrap) can be additionally treated – typically in ladles. These essential processes in modern steelmaking are called ladle treatment or secondary steelmaking (i.e., secondary metallurgy). They are slag formation, deoxidation, alloying, inclusions modifica‐ tion, desulfurization, dephosphorization, analyses of chemical composition of steel and slag, heating (i.e., temperature adjustment), stirring, refining (i.e., melt purification), homogenization and degassing [1‐3]. The addition of alloys is preferably conducted during secondary steelmaking. They can be added also during casting, into the tundish, using bulk material or as wired products. The disso‐ lution of alloys in liquid steel is influenced by their physical and chemical properties, melt su‐ 253 Kovačič, Lešer, Brezocnik perheat, location of addition and stirring. Most important are melting point and density which determine either the additive will float (and entrained in melt or slag) or sink during assimila‐ tion. Consequently, for efficient alloying following areas have been developing: scrap (input ma‐ terial) design [4‐6], alloys design [7‐10], and interactions with the liquid bath [11, 12]. The price of produced steel is mostly influenced by – beside the electric energy consumption – the added alloys which are at high temperatures and presence of oxygen prone to burning‐off [13, 14]. Accordingly, the alloys consumption could be increased together with the need of add‐ ing the alloys several times threatening the production pace, melt homogenization, purification and further melt solidification (e.g. timing, temperature, clogging). The burn‐off of alloys predic‐ tion is aggravated due to the diversity of steelmaking technologies and equipment. In this study sulfur addition (i.e., sulfur burn‐off) during 70MnVS4 steelmaking in Štore Steel Ltd. was modeled. During ladle treatment, instead of only one sulfur addition, the sulfur was repeatedly added several times threatening clogging during continuous casting and as such in‐ fluencing surface defects occurrence and steel cleanliness. Accordingly, the additional sulfur addition was predicted using multiple linear regression and the genetic programming. The ge‐ netic programming has been used several times in Štore Steel Ltd. for modelling and optimiza‐ tion (e.g., [15‐19]). In the beginning of the paper the problem regarding repeatedly added sulfur is presented to‐ gether with the steelmaking technology. Afterwards, the sulfur addition prediction using multi‐ ple linear regression and genetic programming is presented including the implementation of findings in the actual steelmaking process. At the end of the paper, the conclusions are drawn and future work is emphasized. 2. Materials, methods and execution of experiment 70MnVS4 is a high‐strength microalloyed steel which is used for forging of connecting rods in the automotive industry. In Štore Steel Ltd., which is one of Europe’s major flat spring steel pro‐ ducers, 70MnVS4 steel is produced from scrap that is melted using an electric arc furnace. After melting the first chemical composition analysis is conducted. After reaching tapping temperature, the melt is discharged into the ladle. The ladle is trans‐ ported to the ladle furnace. The average batch weighs 50 t. The slag is formed using dolomite, quartz and fluorite. The melting bath is deoxidized using ferromanganese and ferrosilicon. Also alloying using ferrovanadium and homogenization (i.e., argon stirring) are carried out. Then the second chemical composition analysis is conducted. Based on this analysis the sulfur is added for the first time using sulfur cored wire. The melt is homogenized again and also the third chemical composition is conducted. Based on the chemical composition slight adjustments of alloying elements can be made using ferrosilicon, ferromanganese and ferrovanadium. Also for several reasons, the sulfur cored wire should be added again. It is well known that the sulfur forms in‐ clusions which cause clogging of tundish submerged entry nozzles during continuous casting and as such influencing surface defects occurrence and steel cleanliness. After chemical compo‐ sition adjustments the fourth chemical composition analysis is performed. The ladle is transported to the continuous caster. The melt pours into the tundish after the ladle sliding gate is opened, with continuous casting being established throughout a casting sys‐ tem with impact pod, stoppers, submerged entry nozzles and water‐cooled copper molds. Dur‐ ing casting also the final chemical composition is determined which is also stated on the inspec‐ tion certificate. For casting of the 180 mm square billets, a two strand continuous caster with 9 m radius is used. The solidification is conducted throughout primary cooling in the mold and secondary cooling using water sprays. The billets are cooled down on turnover cooling bed. The billets are reheated up to rolling temperature and rolled into round bars with a diameter of up to 50 mm. The same rolled bar surface is also examined using the automatic control line. The surface control is based on the flux leakage method, meaning that the surface of the material is locally magnetized and that deviations of magnetic flux (i.e., flux leakage) at the opened sur‐ face defects are detected. During surface control the data on number of examined bars, bars with defects and defects length are stored in the informational system. 254 Advances in Production Engineering & Management 16(2) 2021 Modelling and optimization of sulfur addition during 70MnVS4 steelmaking: An industrial case study Batch Carbon content after the second chemical composition analysis after tapping (C2), (%) Manganese content after the second chemical composition analysis after tapping (MN2), (%) Silicon content after the second chemical composition analysis after tapping (SI2), (%) Sulfur content after the second chemical composition analysis after tapping (S2), (%) Carbon cored wire addition after the second chemical composition analysis after tapping (CW2), (m) Sulphur cored wire addition after the second chemical composition analysis after tapping (SW2), (m) Time between the second chemical analysis after tapping and final chemical composition analysis at starting of the casting (T2F), (min) Ferro‐manganese (FEMN), (kg) Ferrosilicon (FESI), (kg) Additional sulfur cored wire addition (SWA), (m) Table 1 Parameters collected within the period from January 2018 to December 2018 for 78 consequently cast batches of 70MnVS6 steel 76124 76487 76488 76516 76517 76518 76519 76520 76561 76668 76669 76670 76671 76672 76673 76674 76675 76676 76767 76768 77048 77049 77076 77077 77078 77082 77083 77085 77086 77161 77162 77409 77410 77411 77412 77413 77414 77415 77416 77501 77766 77767 77768 77769 77770 77771 77772 77773 77783 77784 77792 77978 77979 77980 77981 77982 77983 77984 78224 78225 78376 78420 78421 78681 78682 78683 78903 78904 78905 78906 78907 78908 78909 78910 79233 79234 79604 79605 0.56 0.54 0.54 0.61 0.61 0.59 0.59 0.61 0.55 0.58 0.55 0.57 0.62 0.60 0.62 0.63 0.61 0.60 0.57 0.56 0.53 0.57 0.52 0.56 0.57 0.54 0.61 0.60 0.63 0.58 0.60 0.61 0.55 0.56 0.61 0.54 0.59 0.54 0.60 0.58 0.58 0.60 0.60 0.63 0.63 0.60 0.60 0.61 0.61 0.61 0.60 0.69 0.59 0.56 0.56 0.54 0.56 0.55 0.58 0.66 0.55 0.53 0.57 0.63 0.61 0.67 0.53 0.59 0.54 0.57 0.56 0.56 0.61 0.66 0.62 0.62 0.57 0.59 0.80 0.74 0.79 0.76 0.72 0.74 0.73 0.78 0.71 0.79 0.77 0.80 0.78 0.79 0.82 0.79 0.75 0.77 0.74 0.76 0.81 0.84 0.79 0.76 0.76 0.77 0.81 0.77 0.74 0.79 0.78 0.73 0.78 0.76 0.78 0.81 0.81 0.80 0.84 0.76 0.75 0.78 0.77 0.78 0.74 0.77 0.79 0.78 0.76 0.73 0.79 0.78 0.70 0.74 0.76 0.71 0.75 0.71 0.78 0.81 0.74 0.73 0.75 0.74 0.73 0.79 0.70 0.77 0.72 0.73 0.75 0.75 0.78 0.80 0.77 0.70 0.81 0.76 0.20 0.15 0.16 0.14 0.18 0.20 0.20 0.23 0.18 0.18 0.19 0.18 0.20 0.22 0.24 0.24 0.19 0.20 0.19 0.20 0.16 0.19 0.18 0.19 0.19 0.20 0.20 0.20 0.20 0.17 0.19 0.19 0.17 0.17 0.19 0.19 0.22 0.20 0.19 0.20 0.18 0.17 0.16 0.18 0.17 0.17 0.18 0.18 0.17 0.17 0.18 0.17 0.16 0.18 0.16 0.15 0.16 0.16 0.19 0.19 0.22 0.15 0.16 0.17 0.16 0.18 0.18 0.21 0.21 0.21 0.17 0.18 0.18 0.18 0.19 0.15 0.16 0.17 0.022 0.019 0.020 0.023 0.030 0.028 0.030 0.023 0.021 0.026 0.022 0.024 0.022 0.024 0.030 0.024 0.023 0.018 0.024 0.021 0.027 0.021 0.023 0.018 0.022 0.018 0.023 0.019 0.024 0.020 0.025 0.012 0.017 0.023 0.024 0.018 0.017 0.023 0.018 0.030 0.021 0.025 0.022 0.027 0.013 0.022 0.026 0.018 0.019 0.025 0.018 0.019 0.024 0.027 0.026 0.029 0.025 0.030 0.022 0.013 0.029 0.014 0.024 0.020 0.025 0.029 0.017 0.022 0.027 0.026 0.015 0.018 0.022 0.025 0.210 0.035 0.001 0.022 30 80 180 50 120 80 40 40 40 50 70 0 0 120 0 20 0 30 90 0 270 0 200 80 60 40 70 90 100 0 60 80 160 70 50 200 40 0 120 0 210 200 40 40 120 160 340 80 200 80 150 0 110 130 0 30 120 180 60 0 100 100 100 50 0 0 80 40 40 120 160 90 0 80 80 0 40 40 237 223 223 250 200 180 170 200 230 215 193 193 228 187 173 220 192 200 200 210 193 197 240 220 200 243 200 237 190 227 190 190 220 207 193 237 227 218 213 197 237 190 190 185 217 200 183 215 233 190 237 227 207 175 190 190 197 180 213 227 200 227 200 237 197 182 250 224 200 210 248 237 207 200 213 150 247 232 106 103 82 82 90 95 97 83 81 84 79 80 87 73 93 101 81 89 90 24 85 83 104 82 91 87 81 97 82 107 87 98 77 87 85 92 68 76 87 90 88 87 95 84 86 87 102 113 90 91 83 88 74 106 97 93 86 83 101 90 86 95 79 89 83 92 83 91 95 90 75 94 127 92 93 90 100 105 25 75 23 54 76 64 75 32 95 27 47 23 28 29 0 21 51 41 55 59 18 0 25 56 56 45 22 42 72 25 36 84 35 44 32 16 16 16 0 58 65 32 44 34 70 44 23 35 54 88 38 36 115 77 67 104 74 106 40 22 75 95 57 73 84 33 105 52 94 83 62 63 45 19 41 113 11 52 54 44 107 79 53 117 122 90 75 34 128 133 44 79 73 0 77 73 124 97 98 88 64 130 54 44 118 33 110 46 119 51 97 91 94 45 89 81 102 54 65 117 128 104 50 119 109 107 64 129 44 62 135 116 116 63 116 117 52 123 122 64 126 65 132 107 56 113 110 114 78 120 118 103 46 132 57 107 11 80 27 0 25 25 10 0 0 0 15 0 0 30 15 14 0 10 45 60 0 27 50 0 45 0 13 0 190 0 0 28 0 38 0 0 0 0 22 12 0 0 0 0 22 27 0 0 9 0 0 32 37 27 19 30 0 8 50 63 80 12 32 17 0 7 8 30 30 24 20 20 0 20 0 0 30 32 The following parameters were collected within the period from January 2018 to December 2018 for 78 consequently cast batches of 70MnVS6 (Table 1): Advances in Production Engineering & Management 16(2) 2021 255 Kovačič, Lešer, Brezocnik  Carbon (C2), manganese (MN2), silicon (SI2) and sulfur (S2) content after the second chemical composition analysis after tapping in weight percentage (%). Carbon, manganese and sulfur are required according to technical delivery conditions. Manganese and silicon also help deoxidization. Manganese and sulfur form the manganese sulfide inclusions in the steel which improve machinability and enable cracking during connection rod production.  Sulphur (SW2) and carbon (CW2) cored wire addition after the second chemical composi‐ tion analysis after tapping in meters (m). Their addition depends on their actual content in the melt and also final chemical composition required by technical delivery conditions.  Time between the second chemical analysis after tapping and final chemical composition analysis at starting of the casting (T2F) in minutes (min). This time is related with ladle treatment time – from tapping until continuous casting where based on slag formation, al‐ loying, refining and homogenization the chemical reactions took place including sulfur burn‐off.  Ferromanganese (FEMN) and ferrosilicon (FESI) addition in kilograms (kg). Ferromanga‐ nese and ferrosilicon are used as deoxidizers and also alloys.  Additional sulfur cored wire addition (SWA) in meters (m). Due to possibility of undesira‐ ble clogging of tundish submerged entry nozzles during continuous casting this additional sulfur cored wire addition should be minimized. For the purpose of this research, we used two methodological approaches: a multiple linear regression method and the genetic programming method. In multiple linear regression, the linear relationship between a scalar response (i.e., depend‐ ent output variable) and one or more explanatory variables is established (i.e., input variables) [19]. Conventional linear regression method is based on a deterministic approach. A multiple linear regression method is widely used technique in different engineering fields [20]. In contrast to linear regression, however, the genetic programming is a non‐deterministic evolutionary optimization approach that mimics a biological evolution [21]. The genetic pro‐ gramming is similar to a very well‐known method of genetic algorithm. Both methods are evolu‐ tionary computation techniques frequently used for complex optimization tasks in various fields (see for example [19, 21‐25]). The genetic programming usually involves very complex structures (i.e., organisms and/or potential solutions of the problem) that are manipulated during simulated evolution [19]. The shapes of the organisms depend on the problem to be solved. Organisms in the genetic pro‐ gramming are composed of functional and terminal genes. Functional genes are most often basic mathematical operations (e.g., addition, subtraction, multiplication, division, power function, exponential function). Terminal genes are usually explanatory variables of the system under study. A set of constants can be added to a set of terminal genes. The goal of the genetic pro‐ gramming is to find an individual organism (a mathematical model) that best solves the problem we deal with [19]. 3. Modelling of additional sulfur cored wire addition On the basis of the collected data in Table 1, the prediction of additional sulfur cored wire addi‐ tion was conducted using linear regression and genetic programming. For the fitness function, the average absolute deviation between predicted and experimental data was selected. It is de‐ fined as: ∆ ∑ (1) where n is the size of the monitored data and Q’i and Qi are the actual and the predicted addi‐ tional sulfur cored wire addition in meters, respectively. 256 Advances in Production Engineering & Management 16(2) 2021 Modelling and optimization of sulfur addition during 70MnVS4 steelmaking: An industrial case study 3.1 Modelling of additional sulfur cored wire addition using multiple linear regression On the basis of the multiple linear regression results, it is possible to conclude that the model does not significantly predict the additional sulfur cored wire addition (p > 0.05, ANOVA) and that only 7.21 % of total variances can be explained by independent variables variances (R‐ squared). Accordingly, there are also no significantly influential parameters (p>0.05). SWA 45.73 ∙ 2 265.49 ∙ 2 143.37 ∙ 2 0.112 ∙ T2F 0.389 ∙ FEMN 0.049 ∙ 114.30 ∙ 2 188.14 0.024 ∙ 2 0.042 ∙ 2 (2) The average absolute deviation from experimental data is 17.25 meters (m). Regardless ANOVA results, the influences of individual parameters on the additional sulfur cored wire addition while separately changing individual parameter within the individual pa‐ rameter range were calculated (Fig. 1). It is possible to conclude that according to multiple linear regression results the most influential is ferromanganese addition (FEMN). 35 m 30 m 25 m 20 m 15 m 10 m 5m FESI FEMN T2F SW2 CW2 S2 SI2 MN2 C2 0m Fig. 1 Calculated influences of individual parameters on additional sulfur cored wire addition using the multiple linear regression model 3.2 Modelling of additional sulfur cored wire addition using genetic programming For the purpose of this research, we used the basic arithmetical operations of addition, subtrac‐ tion, multiplication and division (i.e., function genes), as well as independent variables (i.e., ter‐ minal genes) of the process to construct a potential successful solution. Each organism in each generation is evaluated for all fitness cases (i.e., for all combinations of input variables) and compared with the corresponding experimental values of dependent output variable according to the Eq. 1. The processes of genetic altering and evaluating of organisms is repeated until the successful solution is obtained [19]. We used in‐house genetic programming system developed in AutoLISP programming lan‐ guage with the following evolutionary parameters: population size 2000, maximum number of generations 500, reproduction probability 0.3, crossover probability 0.7, maximum permissible depth of organisms in the creation of the population 6, maximum permissible depth after the operation of crossover of two organisms 30. Genetic operations of reproduction and crossover were used. We implemented tournament selection method with the tournament size of 7. Model‐ ling experiment involved 200 runs. The best mathematical model for prediction of additional sulfur cored wire addition obtained from 200 runs of genetic programming system is given in Eq. 3. Its average absolute deviation from experimental data is 10.80 m. Similarly, as in case of multiple linear regression, we calculated the influence of individual pa‐ rameter on the additional sulfur cored wire addition while separately changing individual pa‐ rameter value within its range (Fig. 2). It is possible to conclude that according to the genetic programming results the most influential input variable is ferromanganese (FEMN) and ferrosil‐ icon (FESI) additions. Advances in Production Engineering & Management 16(2) 2021 257 Kovačič, Lešer, Brezocnik SWA . . ∙ . ∙ . ∙ ∙ ∙ 2 . 5 ∙ ∙ (3) . 2 . ∙ ∙ ∙ ∙ . . ∙ . ∙ ∙ ∙ ∙ ∙ S2 SW2 . ∙ . . . ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ 25 m 20 m 15 m 10 m 5m 0m C2 SI2 FEMN FESI ‐5 m ‐10 m Fig. 2 Calculated influences of individual parameters on additional sulfur cored wire addition using the genetic programming model 4. Results and discussion Regardless ANOVA results obtained using linear regression, the steelmaking process was changed. In the period from February 24, 2019 to October 17, 2019, a total of 12 batches of 70MnVS6 were produced with minimal ferromanganese additions (FEMN). Please bear in mind that during steelmaking, both ferromanganese and ferrosilicon were used for deoxidization. The results are gathered in the Table 2. It is possible to conclude that the additional sulfur cored wire addition was not necessary. The average absolute deviation from experimental data gathered within period of changed steelmaking process is 19.54 m and 3.99 m at linear regression model and genetically obtained model, respectively. The genetic programming model outperformed the linear regression model for 4.90‐times. Also the role of other parameters which were not changed should be clarified. Carbon, man‐ ganese and silicon content after the second chemical composition analysis after tapping varies due to different scrap chemical composition (i.e., input material chemical composition) and addi‐ tions and alloys which are added during tapping. They are also affected by later carbon, manga‐ nese and silicon addition. The same is with time between the second chemical analysis after tap‐ ping and final chemical composition analysis at starting of the casting which is influenced by technological and maintenance delays and also peak electricity period [17, 26]. The only possi‐ ble changes could be attributed to ferromanganese and ferrosilicon additions. After implementation of changes into production the addition of ferromanganese significantly decreased for 233.03 %, i.e., from 50.29 kg to 21.58 kg (t‐test, p < 0.05). It must be emphasized that micro cleanliness before and after changes in production was also analyzed. According to technical delivery conditions, the K3 and K4 values without taking into account sulfur type of inclusions, determined according to DIN 50 602 were required. Micro cleanliness has been im‐ proved statistically significantly after changing of steelmaking process (t‐test, p < 0.05). K3 and K4 values decreased from 6.49 to 4.58 and from 3.49 to 1.58, respectively. 258 Advances in Production Engineering & Management 16(2) 2021 Modelling and optimization of sulfur addition during 70MnVS4 steelmaking: An industrial case study Batch Carbon content after the second chemical composition analysis after tapping (C2), (%) Manganese content after the second chemical composition analysis after tapping (MN2) , (%) Silicon content after the second chemical composition analysis after tapping (SI2), (%) Sulfur content after the second chemical composition analysis after tapping (S2), (%) Carbon cored wire addition after the second chemical composition analysis after tapping (CW2), (m) Sulphur cored wire addition after the second chemical composition analysis after tapping (SW2), (m) Time between the second chemical analysis after tapping and final chemical composition analysis at starting of the casting (T2F), (min) Ferro‐manganese (FEMN), (kg) Ferrosilicon (FESI), (kg) Additional sulfur cored wire addition (SWA), (m) Table 2 Parameters collected within the period from February 2019 to October 2019 for 12 consequently cast batches of 70MnVS6 using minimal ferromanganese additions 80210 80211 80212 80214 80215 80595 81393 81394 81692 81881 81882 82076 0.018 0.032 0.032 0.033 0.029 0.027 0.033 0.028 0.026 0.007 0.016 0.028 0.57 0.55 0.56 0.62 0.65 0.57 0.53 0.58 0.63 0.58 0.64 0.60 0.82 0.76 0.81 0.79 0.80 0.79 0.80 0.82 0.80 0.81 0.83 0.80 0.19 0.24 0.25 0.19 0.19 0.21 0.20 0.18 0.21 0.21 0.25 0.19 100 30 0 0 40 0 160 120 200 60 140 60 240 180 160 163 180 180 213 239 178 259 220 170 88.00 74.00 119.00 87.00 81.00 97.00 85.00 94.00 93.00 79.00 78.00 75.00 46 47 11 32 23 29 21 0 21 12 0 17 53 107 93 119 124 99 98 52 96 38 82 106 0 0 0 0 0 0 0 0 0 0 0 0 Time between the second chemical analysis after tapping and final chemical composition analysis at starting of the casting, scrap rate of rolled material after automatic control line in‐ spection, and casting, based on evaluation score, have not been changed statistically significantly after changing of steelmaking process (t‐test, p < 0.05). Please mind that casting evaluation score is obtained using in‐house software which evaluate casting based on casting parameters (e.g., stopper rod movements, vibrators, melt level in the mold). 5. Conclusion In this paper the prediction of additional sulfur addition (i.e., sulfur burn‐off) during 70MnVS4 steelmaking in Štore Steel Ltd. was presented. During ladle treatment, instead of only one sulfur addition, the sulfur was repeatedly added several times threatening clogging during continuous casting and as such influencing surface defects occurrence and steel cleanliness. Accordingly, following parameters were collected within the period from January 2018 to De‐ cember 2018 for 78 consequently cast batches of 70MnVS6:  carbon, manganese, silicon and sulfur content after the second chemical composition anal‐ ysis after tapping,  sulphur and carbon cored wire addition after the second chemical composition analysis af‐ ter tapping,  time between the second chemical analysis after tapping and final chemical composition analysis at starting of the casting,  ferromanganese and ferrosilicon addition,  additional sulfur cored wire addition. Based on these data additional sulfur addition was predicted using linear regression and ge‐ netic programming. On the basis of the linear regression results, it is possible to conclude that the model does not significantly predict the additional sulfur cored wire addition (p > 0.05, ANOVA) and that only 7.21 % of total variances can be explained by independent variables vari‐ ances (R‐squared). Similarly, additional sulfur addition was predicted using genetic programming system. Also the influences of individual parameters on the additional sulfur cored wire addition while sepa‐ rately changing individual parameter within the individual parameter range were calculated. It is possible to conclude that the most influential are ferromanganese and ferrosilicon addition. Based on modelling results the steelmaking process was changed. In the period from Febru‐ ary 2019 to October 2019 a total of 12 batches of 70MnVS6 were produced with minimal ferro‐ manganese additions. The additional sulfur cored wire addition was not necessary. Advances in Production Engineering & Management 16(2) 2021 259 Kovačič, Lešer, Brezocnik After implementation of changes into production, the addition of ferromanganese significantly decreased, micro cleanliness has been improved statistically significantly after changing of steelmaking process. 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Advances in Production Engineering & Management 16(2) 2021 261 Calendar of events • 6th International Conference on 3D Printing Technology and Innovations, May 24-25, 2021, Berlin, Germany. • 38th Global Summit on Nanoscience and Technology, June 21-22, 2021, Osaka, Japan. • International Conference on Manufacturing Models and Cloud Manufacturing, July 15-16, 2021, Stockholm, Sweden. • International Conference on Industrial Production Methods and Flow Production, August 5-6, 2021, Montreal, Canada. • 15th International Conference on Micromachining Technology, October 18-19, 2021, Dubai, United Arab Emirates. • 35th Annual European Simulation and Modelling Conference, October 27-29, 2021, Rome, Italy. • 32nd DAAAM International Symposium, October 28-29, 2021, Virtual conference. • International Mechanical Engineering Congress and Exposition, November 1-4, 2021, Virtual conference.CTURING • 15th International Conference on Robotics and Smart Manufacturing, November 15-16, 2021, Copenhagen, Denmark. 262 Advances in Production Engineering & Management 16(2) 2021 Notes for contributors General Articles submitted to the APEM journal should be original and unpublished contributions and should not be under consideration for any other publication at the same time. 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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 16 | Number 2 | June 2021 | pp 141-264 Contents Scope and topics 144 Hybrid ANFIS-Rao algorithm for surface roughness modelling and optimization in electrical discharge machining Agarwal, N.; Shrivastava, N.; Pradhan, M.K. 145 A multi-objective optimal decision model for a green closed-loop supply chain under uncertainty: A real industrial case study Fang, I.W.; Lin, W.-T. 161 Improved Genetic Algorithm (VNS-GA) using polar coordinate classification for workload balanced multiple Traveling Salesman Problem (mTSP) Wang, Y.D.; Lu, X.C.; Shen, J.R. 173 Change impact analysis of complex product using an improved three-parameter interval grey relation model Yang, W.M.; Li, C.D.; Chen, Y.H.; Yu, Y.Y. 185 Bone drilling with internal gas cooling: Experimental and statistical investigation of the effect of cooling with CO2 on reduction of temperature rise due to drill bit wear Shakouri, E.; Haghighi Hassanalideh, H.; Fotuhi, S. 199 Joint distribution models in fast-moving consumer goods wholesale enterprise: Comparative analysis and a case study Wang, L.; Chen, X.Y.; Zhang, H. 212 Designing a warehouse internal layout using a parabolic aisles based method Zhang, Z.Y.; Liang, Y.; Hou, Y.P.; Wang, Q. 223 Optimization of disassembly line balancing using an improved multi-objective Genetic Algorithm Wang, Y.J.; Wang, N.D.; Cheng, S.M.; Zhang, X.C.; Liu, H.Y.; Shi, J.L.; Ma, Q.Y.; Zhou, M.J. 240 Modelling and optimization of sulfur addition during 70MnVS4 steelmaking: An industrial case study Kovacic, M.; Lešer, B.; Brezocnik, M. 253 Calendar of events 262 Notes for contributors 263 Published by CPE, University of Maribor ISSN 1854-6250 9 771854 625008 apem-journal.org