ISSN 1854-6250

APEM
journal

Advances in
Production
Engineering &
Management
Volume 18 | Number 4 | December 2023

Published by CPE
apem-journal.org

Advances in Production Engineering & Management
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APEM
journal

ISSN 1854‐6250 | Abbreviated key title: Adv produc engineer manag | Start year: 2006
ISSN 1855‐6531 (on‐line)
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Chair of Production Engineering (CPE)

Advances in Production Engineering & Management
Volume 18 | Number 4 | December 2023 | pp 399–514

Contents
Scope and topics
A combined genetic algorithm and A* search algorithm for the electric vehicle routing
problem with time windows

402
403

Optimizing rock breaking performance: The influence of chamfer on polycrystalline
diamond compact (PDC) cutters

417

Dynamic price competition market for retailers in the context of consumer learning behavior
and supplier competition: Machine learning-enhanced agent-based modeling and simulation

434

IoT-based Deep Learning Neural Network (DLNN) algorithm for voltage stability control and
monitoring of solar power generation

447

Comparing Fault Tree Analysis methods combined with Generalized Grey Relation Analysis:
A new approach and case study in the automotive industry

462

Optimizing smart manufacturing systems using digital twin

475

Incentive modeling analysis in engineering applications and projects with
stochastic duration time

486

Reduction of surface defects by optimization of casting speed using genetic programming:
An industrial case study

501

Calendar of events
Notes for contributors

512
513

Wang, D.L.; Ding, A.; Chen, G.L.; Zhang, L.

Ju, P.

Deng, G.F.

Shweta, R.; Sivagnanam, S.; Kumar, K.A.

Shi, J.L.; Lu, Z.C.; Xu, H.H.; Ren, M.M.; Shu, F.L.
Ojstersek, R.; Javernik, A.; Buchmeister, B.

Zhao, J.; Su, J.F.

Kovacic, M.; Zuperl, U.; Gusel, L.; Brezocnik, M.

Journal homepage: apem-journal.org
ISSN 1854-6250 (print)
ISSN 1855-6531 (on-line)
Published by CPE, University of Maribor.

Scope and topics
Advances in Production Engineering & Management (APEM journal) is an interdisciplinary refereed international academic journal published quarterly by the Chair of Production Engineering at
the University of Maribor. The main goal of the APEM journal is to present original, high quality,
theoretical and application-oriented research developments in all areas of production engineering and production management to a broad audience of academics and practitioners. In order to
bridge the gap between theory and practice, applications based on advanced theory and case
studies are particularly welcome. For theoretical papers, their originality and research contributions are the main factors in the evaluation process. General approaches, formalisms, algorithms
or techniques should be illustrated with significant applications that demonstrate their applicability to real-world problems. Although the APEM journal main goal is to publish original research papers, review articles and professional papers are occasionally published.

Fields of interest include, but are not limited to:
Additive Manufacturing Processes
Advanced Production Technologies
Artificial Intelligence in Production
Assembly Systems
Automation
Big Data in Production
Block Chain in Manufacturing
Computer-Integrated Manufacturing
Cutting and Forming Processes
Decision Support Systems
Deep Learning in Manufacturing
Discrete Systems and Methodology
e-Manufacturing
Evolutionary Computation in Production
Fuzzy Systems
Human Factor Engineering, Ergonomics
Industrial Engineering
Industrial Processes
Industrial Robotics
Intelligent Manufacturing Systems
Joining Processes
Knowledge Management
Logistics in Production

402

Machine Learning in Production
Machine-to-Machine Economy
Machine Tools
Machining Systems
Manufacturing Systems
Materials Science, Multidisciplinary
Mechanical Engineering
Mechatronics
Metrology in Production
Modelling and Simulation
Numerical Techniques
Operations Research
Operations Planning, Scheduling and Control
Optimisation Techniques
Project Management
Quality Management
Risk and Uncertainty
Self-Organizing Systems
Smart Manufacturing
Statistical Methods
Supply Chain Management
Virtual Reality in Production

Advances in Production Engineering & Management

ISSN 1854-6250

Volume 18 | Number 4 | December 2023 | pp 403–416

Journal home: apem-journal.org

https://doi.org/10.14743/apem2023.4.481

Original scientific paper

A combined genetic algorithm and A* search algorithm for
the electric vehicle routing problem with time windows
Wang, D.L.a, Ding, A.a, Chen, G.L.b, Zhang, L.a,c,*
aSchool

bSchool

of Economics and Management, China University of Geosciences (Beijing), Haidian District, Beijing, P.R. China
of Geophysics and Information Technology, China University of Geosciences (Beijing), Haidian District, Beijing, P.R.

cFaculty

of Business, City University of Macau, Macau, P.R. China

China

ABSTRACT

ARTICLE INFO

With growing environmental concerns, the focus on greenhouse gases (GHG)
emissions in transportation has increased, and the combination of smart
microgrids and electric vehicles (EVs) brings a new opportunity to solve this
problem. Electric vehicle routing problem with time windows (EVRPTW) is an
extension of the vehicle routing problem (VRP) problem, which can reach the
combination of smart microgrids and EVs precisely by scheduling the EVs.
However, the current genetic algorithm (GA) for solving this problem can
easily fall into the dilemma of local optimization and slow iteration speed. In
this paper, we present an integer hybrid planning model that introduces time
of use and area price to enhance realism. We propose the GA-A* algorithm,
which combines the A* algorithm and GA to improve global search capability
and iteration speed. We conducted experiments on 16 benchmark cases,
comparing the GA-A* algorithm with traditional GA and other search algorithms, results demonstrate significant enhancements in searchability and
optimal solutions. In addition, we measured the grid load, and the model
implements the vehicle-to-grid (V2G) mode, which serves as peak shaving and
valley filling by integrating EVs into the grid for energy delivery and exchange
through battery swapping. This research, ranging from model optimization to
algorithm improvement, is an important step towards solving the EVRPTW
problem and improving the environment.

Keywords:
Vehicle routing problem (VRP);
Electric vehicle;
Optimization;
Time windows;
Spatiotemporal electricity price;
Smart microgrids;
Genetic algorithm (GA);
A* search algorithm;
GA-A* algorithm
*Corresponding author:
zhanglongdragon@hotmail.com
(Zhang, L.)
Article history:
Received 25 September 2023
Revised 17 November 2023
Accepted 21 November 2023

Content from this work may be used under the terms of
the Creative Commons Attribution 4.0 International
Licence (CC BY 4.0). Any further distribution of this work
must maintain attribution to the author(s) and the title of
the work, journal citation and DOI.

1. Introduction
With global warming and environmental problems intensifying, more and more attention has
been paid to reducing greenhouse gas emissions in transportation. New energy vehicles have
emerged globally as an environmentally friendly and energy-efficient alternative [1], but face
challenges in practical applications due to limitations in battery technology and charging facilities [2]. To overcome these challenges, smart microgrids have emerged as a new type of energy
management system, which can improve power sharing and power quality improvement and
promote the sustainable development of new energy vehicles [3, 4]. Meanwhile, AI plays an important role in solving energy management problems in smart microgrids by applying various
optimization methods and developing optimal energy management strategies, which can help to
increase energy efficiency, reduce the total cost, and improve power quality [5].
403

Wang, Ding, Chen, Zhang

EVRPTW as an optimization problem, aims to efficiently plan the routes of electric vehicles to
meet the time window constraints and minimize the total cost or maximize the benefits. It optimizes the routes and charging strategies of electric vehicles to maximize the use of renewable
energy and grid power for efficient energy management. By rationally arranging the use of
charging stations and charging periods, it can reduce the dependence on the traditional grid and
improve energy utilization. Meanwhile, by rationally planning the routes of electric vehicles,
EVRPTW can reduce pollutant emissions and carbon footprint. Introducing EVs into smart microgrids and the whole grid, can promote the use of green energy and reduce dependence on oil
and other limited resources.
Keskin and Çatay first started working on the EVRPTW problem in 2016 and specifically studied the allowed partial charging problem (EVRPTW-PR), where they formulated the problem as
0-1 mixed integer linear programming, developed an adaptive large neighborhood search
(ALNS) algorithm, and used benchmark instances to test those solutions that can effectively find
high quality solutions, As the result shows, the partial tolling scheme can substantially improve
the routing decisions [6]. Keskin and Çatay reformulated the problem as mixed integer linear
programming in 2018, where they built on the original by combining an ALNS method with an
exact method equipped with various destroy-repair algorithms to efficiently explore neighborhoods and using CPLEX to strengthen the obtained routes [7]. In 2019, Wang et al. constructed a
mathematical model to minimize the total cost based on EVRPTW considering time windows
and battery swapping stations (2E-EVRPTW-BSS) for the two-stage vehicle path problem of electric vehicles and verified the validity of the model [8]. Gocken and Yaktubay solved the VRPTW
problem by applying first clustering and then using genetic algorithm solution for planning.
Meanwhile different clustering algorithms are compared and finally the superiority of K-means
algorithm for initialising the population is concluded [9]. In 2020, Raeesi and Zografos developed a program utilizing a two-phase hybrid of dynamic programming and integer programming algorithms. The resulting program serves as the cornerstone of a robust, large neighborhood search algorithm, designed for the rapid resolution of instances related to EVRPTW-SMBS
[10]. In 2021, Zhu et al. applied the elitist genetic algorithm to the EV path problem with a time
window [11]. Deng et al. studied the EV path problem with a time window and nonlinear charging constraints (EVRPTW-NL) and proposed an improved hybrid algorithm combining an improved differential evolution (IDE) and several heuristics [12]. Bac and Erdem developed a series of neighborhood operators for the EVRPTW problem in the Variable Neighborhood Search
(VNS) and Variable Neighborhood Drop (VND) heuristics for the local search process [13]. Lin et
al. came out with EVRPTW (EVRPTW-TP) under time-varying tariffs, formulating it as an optimal problem, proposing a Lagrangian relaxation method and a mixed-variable neighborhood
search/tabu search heuristic to obtain high-quality lower bounds and feasible solutions, respectively [14]. Lin et al. proposed an end-to-end deep reinforcement learning framework for solving
the EVRPTW [15]. In 2022, Erdelić and Carić implemented the Adaptive Large Neighborhood
Search (ALNS) meta-heuristic algorithm, utilizing the ruin-recreate strategy. This algorithm integrates a novel initial solution heuristic, partial search, path removal, and an exact procedure,
resulting in the achievement of an optimal layout for charging stations. The results show that
ALNS can find 38 new optimal solutions on the benchmark EVRPTW instance and that the advantages and disadvantages of using a partial charging strategy compared to a full charging
strategy are evident [16]. In 2022, Niu et al. proposed the idea of consumers being able to choose
multiple delivery addresses and used a large-scale neighbourhood search algorithm to facilitate
further matching of logistics and distribution companies with customer needs [17]. In 2022, Liu
et al. developed a hybrid Genetic Algorithm that combines the 2-opt algorithm with GA [18].
Ding designed an adaptive particle swarm optimization algorithm for the driving cycle based
time window electric vehicle routing problem (EVRPTW-DC) to solve the problem [19]. In 2023,
Kumar et al. proposed a firefly and ant colony algorithm with a new pad heuristic avoiding local
optimums [20]. Kempton and Tomić first proposed electric vehicle-to-grid (V2G) technology,
which utilizes large amounts of electric vehicle energy storage through interaction between electric vehicles and the grid to act as a buffer between the grid and renewable energy sources [21].
404

Advances in Production Engineering & Management 18(4) 2023

A combined genetic algorithm and A* search algorithm for the electric vehicle routing problem with time windows

In 2018, Shao et al. solved the EVRP model based on a hybrid genetic algorithm and used the
dynamic Dijkstra algorithm to make some improvements to the classical Dijkstra algorithm [22].
In the same year, Wang et al. proposed to solve complex multi-objective problems based on a
heuristic algorithm (ST-VNSGA) consisting of a variable neighbourhood search method and GA
considering the spatiotemporal distance [23]. Zhu et al. investigated the path algorithm, and
then used the elitist genetic algorithm and proposed an improved neighbour path initialization
method to solve the EV routing problem [24]. Hien et al. proposed a greedy search algorithm
GSGA inspired by clustering [25]. Wang et al. used Montecarlo tree search algorithm to improve
the genetic algorithm, taking into account the flexibility of the paths and saving allocation time.
The more complex the composition of the road network, the better the results obtained by the
algorithm [26]. In 2023, Amiri et al. proposed developed two meta-heuristic algorithms including Non-dominated Sorting Genetic Algorithm II (NSGA-II) and Adaptive Large Neighbourhood
Search (ALNS), and combined with multi-objective solution methods (e.g. weighted and epsilon
constraints and hybrid methods) [27].
In this paper, based on the previous research, firstly, established an integer hybrid planning
model for the EVRPTW problem that comprehensively considers multiple factors, including vehicle fixed cost, energy cost, time cost, and penalty cost for violating constraints, which is closer
to the actual engineering background. Meanwhile, the Time of use Pricing and Area price are
introduced, and the EVs into the grid based on the V2G model realize the formation of a smart
microgrid, which effectively acquires and delivers energy by means of the EVs swap under the
dual perspectives of time and space. In particular, the use of the A* algorithm in the initializing
population step allows for a better initial solution, which avoids the problem of slow iteration
speed and satisfies the stochastic nature of genetic algorithms. Experiments demonstrate that
the GA-A* algorithm performs well in solving the EVRPTW problem, applies to different sizes of
arithmetic cases, and outperforms the traditional GA algorithm and other path planning algorithms. In addition, it helps the smart microgrid not only achieves the successful reduction of the
system's load peak-to-valley difference rate and mean square deviation based on the V2G model
by incorporating EVs into the grid, effectively utilizing EVs to reduce the grid's volatility and
improve the quality of power but also enables the EVs to rationally distribute energy in the grid
by prioritizing the delivery of energy during peak hours of power consumption and acquiring
energy during the trough hours of power consumption, which achieves a reasonable distribution
of power.

2. Methodology
2.1 Math model

The main problem investigated in this paper is the Electric-Vehicle Routing Problem with Time
Windows (EVRPTW), which is described as a number of EVs transporting the required supplies
from a distribution center to various customer nodes, where they can be charged or discharged
at a charging station, which adopts battery swapping. On the one hand, swapping batteries simplifies the model complexity, on the other hand, the replaced batteries can be used as part of a
microgrid for charging and discharging operations, bringing profit to the grid operator. Verma
confirmed that battery swapping leads to better path planning options for electric vehicles [28].
The objective function is to minimize the total cost including vehicle fixed cost, energy cost
which considers the temperature coefficient, time cost, penalty cost for violating the load constraint, penalty cost for violating the time window, penalty cost for violating the power constraint, and charging and discharging cost.
EVs are subject to constraints:
• Each customer has a fixed amount of demand and is allowed to be served by only one vehicle and only once;
• EVs exist that the maximum load cannot exceed the maximum load capacity;
• EVs have a battery capacity that cannot exceed the maximum battery capacity or a battery
capacity that is less than zero;

Advances in Production Engineering & Management 18(4) 2023

405

Wang, Ding, Chen, Zhang

• EVs depart from the distribution center in a fully charged state and eventually return to
the charging center;
• EVs are allowed to visit the charging station multiple times;
• EVs are required to arrive by the latest time window or pay a penalty cost;
• EVs arriving early at the customer node are not required to pay penalty costs, only time
costs according to time.

In the problem, the Time of use Pricing and area price are introduced to charge different prices for 24 hours of the day according to different periods and in different areas. Higher prices are
charged during the periods and areas with excessive electricity demand; conversely, the prices
are lowered. The base load is referred to the data provided by Xiao et al. [29]. Fig. 1 shows the 24
hours area price and the grid load at each moment, and the area price is set with reference to the
data provided by Fu et al. [30]. Fig. 2 shows the area price and the grid load in different areas.
Huo has demonstrated that ambient temperature has a great influence on the energy consumption of EVs, and Fig. 3 shows the temperature curve fitted according to the data of this article
[31]. Fig. 4 shows the annual time-averaged temperature profile for Beijing in 2022, with data
from the National Oceanic and Atmospheric Administration (NOAA) National Center for Environmental Information (NCEI).

Fig. 1 Time of use pricing and grid loads

Fig. 2 Area price adjustment

Fig. 3 Temperature-energy consumption per 100 km curve Fig. 4 Beijing's annual average temperature time curve

𝑦𝑦 = 0.000052307𝑥𝑥 3 + 0.0069952𝑥𝑥 2 − 0.37395𝑥𝑥 + 18.276

(1)

Eq. 1 shows the fitted temperature-energy consumption per 100 km curve.
Due to the short driving range of EVs, they need to be recharged at the charging station during the driving process. In this paper, electric energy is recharged by swapping batteries and
connected to the power grid to deliver/acquire energy. According to the impact of the Time of
use Pricing and the area price, electricity is sold to the power grid for profit. V2G technology is
introduced by simulating V2G based on the Time of use Pricing and the area price. This paper
divides a day into 24 time periods, each with different electricity prices. During periods of high
electricity demand, it is the peak period of electricity consumption, and the Time of use Pricing is
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Advances in Production Engineering & Management 18(4) 2023

A combined genetic algorithm and A* search algorithm for the electric vehicle routing problem with time windows

higher. Conversely, it is a low valley stage, and the Time of use Pricing is lower. Divide the entire
map into different regions based on different datasets. In areas of peak electricity consumption,
area prices are higher; conversely, area prices are lower. When the vehicle is charging, replace
the battery with a fully charged battery at the charging station; When the vehicle is discharged,
replace the battery at the charging station with enough power to support the next charging station or distribution center.
The EVRPTW model studied in this paper is, given a graph 𝐺𝐺 = (𝑁𝑁 ∪ 𝐸𝐸, 𝐴𝐴), where the point
set 𝑁𝑁 = {0,1, ⋯ , 𝑛𝑛} represents a set of 𝑛𝑛 customers' points, where 0 represents a distribution
center, 𝐸𝐸 = {𝑛𝑛, 𝑛𝑛 + 1, ⋯ , 𝑛𝑛 + 𝑚𝑚} represents m charging stations, 𝐴𝐴 = {(𝑖𝑖, 𝑗𝑗)|𝑖𝑖, 𝑗𝑗 ∈ 𝑁𝑁 ∪ 𝐸𝐸, 𝑖𝑖 ≠ 𝑗𝑗|}
represents all connected arcs in 𝑁𝑁 ∪ 𝐸𝐸.
The symbol descriptions in the model are shown in Table 1.
Symbol
𝑂𝑂
𝑁𝑁
𝐾𝐾
𝐸𝐸
𝑉𝑉
𝑑𝑑𝑖𝑖𝑖𝑖
𝐷𝐷𝑖𝑖
𝐿𝐿mc
𝐿𝐿𝑖𝑖𝑖𝑖
𝐵𝐵𝑄𝑄
𝐵𝐵𝑎𝑎𝑎𝑎𝑎𝑎
𝐵𝐵𝑙𝑙𝑙𝑙𝑙𝑙
𝐸𝐸𝑖𝑖
𝐿𝐿𝑖𝑖
𝑡𝑡𝑎𝑎𝑎𝑎𝑎𝑎
𝑡𝑡𝑙𝑙𝑙𝑙𝑙𝑙
𝑡𝑡𝑤𝑤𝑤𝑤𝑤𝑤
𝑡𝑡𝑚𝑚𝑚𝑚𝑚𝑚
𝑡𝑡𝑠𝑠𝑠𝑠𝑠𝑠
𝑡𝑡𝑖𝑖𝑖𝑖𝑖𝑖
𝑣𝑣
𝜇𝜇
𝑇𝑇𝑇𝑇𝑇𝑇γ𝑘𝑘
𝐶𝐶𝑑𝑑
𝐶𝐶1
𝐶𝐶2
𝐶𝐶3
𝐶𝐶𝑡𝑡
𝑆𝑆𝛾𝛾𝛾𝛾
𝑃𝑃𝐿𝐿𝐿𝐿𝐿𝐿

Table 1 Symbol description
Instructions
Distribution center
Customer node set
Set of the number of EVs used
Charging Station set
All node set
Node 𝑖𝑖 to node 𝑗𝑗 traveling distance
Customer node 𝑖𝑖 demand
Max. loading capacity of the vehicle
Remaining load at arrival of vehicle 𝑘𝑘 at node 𝑖𝑖
Max. battery capacity of vehicle
Remaining capacity at arrival of vehicle 𝑘𝑘 at node 𝑖𝑖
Remaining capacity at left of vehicle 𝑘𝑘 at node 𝑖𝑖
Customer node 𝑖𝑖's earliest service time, 𝑖𝑖 ∈ 𝑁𝑁
Customer node 𝑖𝑖's latest service time, 𝑖𝑖 ∈ 𝑁𝑁
Time at arrival of vehicle 𝑘𝑘 at node 𝑖𝑖, 𝑖𝑖 ∈ 𝑁𝑁
Time at leaving of vehicle 𝑘𝑘 at node 𝑖𝑖, 𝑖𝑖 ∈ 𝑁𝑁
Waiting Time of vehicle 𝑘𝑘 at node 𝑖𝑖, 𝑖𝑖 ∈ 𝑁𝑁
Missing Time of vehicle 𝑘𝑘 at node 𝑖𝑖, 𝑖𝑖 ∈ 𝑁𝑁
Service/Swap Time of vehicle 𝑘𝑘 at node 𝑖𝑖, 𝑖𝑖 ∈ 𝑁𝑁
Time at driving of vehicle 𝑘𝑘 from node 𝑖𝑖 to node 𝑗𝑗, 𝑖𝑖, 𝑗𝑗 ∈ 𝑁𝑁
velocity of vehicles traveling in distribution
Battery consumption rate
𝛾𝛾 time vehicle 𝑘𝑘 traveling ambient temperature coefficient
Cost per distance
Penalty costs for violating load constraints
Penalty costs for violating the time window
Penalty costs for violating electricity constraints
Cost per time
𝛾𝛾 time area 𝜃𝜃 price
No EVs in the original grid 𝛾𝛾 time area 𝜃𝜃 power

The EVRPTW model proposed in this paper is improved from the mathematical model proposed by Li et al. [32], and on this basis, increases the Time of use Pricing, Area price, and ambient temperature coefficient. On the one hand, it is closer to the real working environment, and
on the other hand, it can help regulate and store electricity in space, which can better improve
energy quality and play a role in energy distribution. According to the description of the
EVRPTW problem, the objective function includes vehicle fixed cost, energy cost which considers the temperature coefficient, time cost, penalty cost for violating load constraints, penalty
cost for violating time windows, penalty cost for violating energy constraints, and charging and
discharging cost. A mathematical model for this problem is established.
Decision variables:
1, Vehicle 𝑘𝑘 from customer 𝑖𝑖 to customer 𝑗𝑗
𝑋𝑋𝑖𝑖𝑖𝑖𝑖𝑖 = �
0, otherwise
Advances in Production Engineering & Management 18(4) 2023

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Wang, Ding, Chen, Zhang

1,
𝑌𝑌𝑘𝑘 = �
0,

vehicle 𝑘𝑘 violated the power constraints
otherwise

𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 𝐶𝐶𝑜𝑜 � 𝑋𝑋𝑘𝑘 + 𝐶𝐶1 � 𝑚𝑚𝑚𝑚𝑚𝑚[0, 𝐷𝐷𝑘𝑘 − 𝐿𝐿𝑚𝑚𝑚𝑚 ] + 𝐶𝐶2 � � 𝑡𝑡𝑚𝑚𝑚𝑚𝑚𝑚
𝑘𝑘∈𝐾𝐾

𝑘𝑘∈𝐾𝐾

24

𝑖𝑖∈𝑉𝑉 𝑘𝑘∈𝐾𝐾

+𝐶𝐶3 � 𝑌𝑌𝑘𝑘 + 𝐶𝐶𝑑𝑑 � � � � 𝑇𝑇𝑇𝑇𝑇𝑇𝛾𝛾𝛾𝛾 𝑑𝑑𝑖𝑖𝑖𝑖 𝑥𝑥𝑖𝑖𝑖𝑖𝑖𝑖
𝑘𝑘∈𝐾𝐾

𝑖𝑖∈𝑉𝑉 𝑗𝑗∈𝑉𝑉 𝑘𝑘∈𝐾𝐾 𝛾𝛾=1
24 𝑛𝑛

(2)

+𝐶𝐶𝑡𝑡 � � �(𝑡𝑡𝑖𝑖𝑖𝑖𝑖𝑖 + 𝑡𝑡𝑠𝑠𝑠𝑠𝑠𝑠 ) + � � ��𝑃𝑃𝛾𝛾𝛾𝛾 �𝑆𝑆𝛾𝛾𝛾𝛾
𝑖𝑖∈𝑉𝑉 𝑗𝑗∈𝑉𝑉 𝑘𝑘∈𝐾𝐾

𝛾𝛾=1 𝜃𝜃=1 𝑘𝑘∈𝐾𝐾

� 𝑋𝑋𝑖𝑖𝑖𝑖𝑖𝑖 = 1, ∀𝑗𝑗 ∈ 𝑁𝑁, 𝑘𝑘 ∈ 𝐾𝐾

𝑖𝑖∈𝑁𝑁,𝑖𝑖≠𝑗𝑗

�

𝑗𝑗∈𝑉𝑉，𝑖𝑖≠𝑗𝑗

𝑋𝑋0𝑗𝑗𝑗𝑗 ≤ 1 , 𝑘𝑘 ∈ 𝐾𝐾

� 𝑋𝑋𝑖𝑖𝑖𝑖𝑖𝑖 − � 𝑋𝑋𝑗𝑗𝑗𝑗𝑗𝑗 = 0, ∀𝑗𝑗 ∈ 𝑉𝑉, 𝑘𝑘 ∈ 𝐾𝐾

𝑖𝑖∈𝑉𝑉𝑉𝑉≠𝑗𝑗

𝑖𝑖∈𝑉𝑉𝑉𝑉≠𝑗𝑗

0 ≤ 𝐷𝐷𝑗𝑗 ≤ 𝐿𝐿𝑖𝑖𝑖𝑖 − 𝐷𝐷𝑖𝑖 , 𝑖𝑖 ∈ 𝑉𝑉, 𝑗𝑗 ∈ 𝑉𝑉, 𝑘𝑘 ∈ 𝐾𝐾
0 ≤ � 𝐷𝐷𝑖𝑖𝑖𝑖 ≤ 𝐿𝐿𝑚𝑚𝑚𝑚 , ∀𝑘𝑘 ∈ 𝐾𝐾
𝑖𝑖∈𝑁𝑁

0 ≤ 𝐵𝐵𝑎𝑎𝑎𝑎𝑎𝑎 ≤ 𝐵𝐵𝑙𝑙𝑙𝑙𝑙𝑙 + 𝑇𝑇𝑇𝑇𝑇𝑇𝛾𝛾𝛾𝛾 𝑑𝑑𝑖𝑖𝑖𝑖 𝑋𝑋𝑖𝑖𝑖𝑖𝑖𝑖 𝜇𝜇, ∀𝑖𝑖 ∈ 𝑁𝑁, 𝑗𝑗 ∈ 𝑁𝑁, 𝑘𝑘 ∈ 𝐾𝐾
� 𝑇𝑇𝑇𝑇𝑇𝑇𝛾𝛾𝛾𝛾 𝑑𝑑𝑖𝑖𝑖𝑖 𝑋𝑋𝑖𝑖𝑖𝑖𝑖𝑖 𝜇𝜇 ≤ 𝐵𝐵𝑙𝑙𝑙𝑙 ≤ 𝐵𝐵𝑄𝑄 , 𝑖𝑖 ∈ 𝐺𝐺

𝑖𝑖,𝑗𝑗∈𝑉𝑉

(4)

(5)
(6)
(7)
(8)
(9)

0 ≤ 𝐵𝐵𝑎𝑎𝑎𝑎𝑎𝑎 ≤ 𝐵𝐵𝑄𝑄 , ∀𝑖𝑖 ∈ 𝑉𝑉, 𝑘𝑘 ∈ 𝐾𝐾

(10)

𝑡𝑡𝑤𝑤𝑤𝑤𝑤𝑤 = 𝑚𝑚𝑚𝑚𝑚𝑚[0, 𝑒𝑒𝑖𝑖 − 𝑡𝑡𝑎𝑎𝑎𝑎𝑎𝑎 ], ∀𝑖𝑖 ∈ 𝑁𝑁, 𝑘𝑘 ∈ 𝐾𝐾

(12)

𝐵𝐵𝑎𝑎𝑎𝑎𝑎𝑎 = 𝐵𝐵𝑙𝑙𝑙𝑙𝑙𝑙 , ∀𝑖𝑖 ∈ 𝑉𝑉, 𝑘𝑘 ∈ 𝐾𝐾

𝑡𝑡𝑚𝑚𝑚𝑚𝑚𝑚 = 𝑚𝑚𝑚𝑚𝑚𝑚[0, 𝑡𝑡𝑎𝑎𝑎𝑎𝑎𝑎 − 𝑙𝑙𝑖𝑖 ], ∀𝑖𝑖 ∈ 𝑁𝑁, 𝑘𝑘 ∈ 𝐾𝐾
𝑡𝑡𝑙𝑙𝑙𝑙𝑙𝑙 = 𝑡𝑡𝑎𝑎𝑎𝑎𝑎𝑎 + 𝑡𝑡𝑠𝑠𝑠𝑠𝑠𝑠 + 𝑡𝑡𝑤𝑤𝑤𝑤𝑤𝑤 , ∀𝑖𝑖 ∈ 𝑁𝑁 ∪ 𝐺𝐺

𝑡𝑡𝑎𝑎𝑎𝑎𝑎𝑎

(3)

𝑑𝑑𝑖𝑖𝑖𝑖𝑖𝑖
, ∀𝑖𝑖 ∈ 𝑉𝑉, 𝑗𝑗 ∈ 𝑉𝑉, 𝑘𝑘 ∈ 𝐾𝐾
𝑣𝑣
= (𝑡𝑡𝑙𝑙𝑙𝑙𝑙𝑙 + 𝑡𝑡𝑖𝑖𝑖𝑖𝑖𝑖 )𝑋𝑋𝑖𝑖𝑖𝑖𝑖𝑖 , ∀ 𝑖𝑖 ∈ 𝑉𝑉, 𝑗𝑗 ∈ 𝑁𝑁 ∪ 𝐺𝐺, 𝑘𝑘 ∈ 𝐾𝐾
𝑡𝑡𝑖𝑖𝑖𝑖𝑖𝑖 =

𝑋𝑋𝑖𝑖𝑖𝑖𝑖𝑖 = {0,1}, ∀ 𝑖𝑖 ∈ 𝑉𝑉, 𝑗𝑗 ∈ 𝑉𝑉, 𝑘𝑘 ∈ 𝐾𝐾
𝑌𝑌𝑘𝑘 = {0,1}, ∀𝑘𝑘 ∈ 𝐾𝐾

(11)
(13)
(14)
(15)
(16)
(17)
(18)

The objective function (Eq. 2) represents the minimum total cost, including vehicle fixed cost,
energy consumption cost, time cost, charging and discharging cost, and penalty cost. Among
them 𝑆𝑆𝛾𝛾𝛾𝛾 are the Time of use Pricing and Area price. When the battery is discharged, 𝑆𝑆𝛾𝛾𝛾𝛾 is negative, i.e. when the cost of charging and discharging is negative, it means that the charging and
discharging process is generally profitable. Constraint Eq. 3 indicates that each customer is
served only once. In the cost of time penalty, waiting time is not included in the penalty, but rather placed within the cost of time. We believe that this can improve the efficiency of resource
allocation. Constraint Eq. 4 indicates that only one transport vehicle is arranged for each distribution route. Constraint Eq. 5 indicates equal number of vehicles entering and exiting. Con408

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A combined genetic algorithm and A* search algorithm for the electric vehicle routing problem with time windows

straint Eq. 6 indicates that the demand of the vehicle at the next node cannot be higher than the
remaining demand of the vehicle at the current node minus the demand at the current node.
Constraint Eq. 7 indicates that all demands of a transportation route must not exceed the maximum load capacity. Constraint Eq. 8 indicates that the remaining power upon reaching the next
node must not be less than the current node's power minus the power consumption from that
node to the next node. Constraint Eq. 9 indicates that the energy consumption after charging and
discharging at the charging station is not less than the energy consumption of the sub path and
not higher than the maximum energy consumption. Constraint Eq. 10 indicates that the remaining electricity at any point is not negative. Constraint Eq. 11 indicates that the remaining electricity is unchanged before and after the vehicle visits the customer node. Constraint Eq. 12 represents the calculation method of waiting time. Constraint Eq. 13 represents the calculation
method of late time. Constraint Eq. 14 indicates that the time of a vehicle leaves the node is the
sum of arrival time, waiting time, and service time. Constraint Eq. 15 represents the calculation
method of the time taken by the vehicle from node 𝑖𝑖 to node 𝑗𝑗. Constraint 16 indicates that the
time for the vehicle to reach the next node is equal to the sum of the time to leave the current
node and the time spent on the journey. Constraints Eqs. 17 and 18 are binary 0-1 variables.
2.2 Algorithm

To solve the Electric-Vehicle Routing Problem with Time Windows, this paper proposes an improved genetic algorithm GA-A*. Genetic algorithm has the advantages of starting from the
population, having potential for parallelism, using evaluation functions for inspiration, simple
processes, probability mechanisms for iteration, and randomness. However, genetic algorithms
have certain dependencies in initial population selection, which may lead to the problem of falling into local optimal solutions.
To overcome these problems, this paper introduces the A* algorithm as part of the improved
genetic algorithm. The A* algorithm is a heuristic search algorithm that guides the search process by evaluating the cost function and heuristic function of nodes. It has high efficiency and
accuracy in path search. The purpose of introducing the A* algorithm is to optimize the initial set
of points by solving for the optimal path and cutting it into seed nodes. Then, by randomly creating other nodes as genes, the initialized population is formed. Such an improvement measure
aims to improve the search efficiency and effectively avoid the occurrence of local optimal solutions.
This paper chooses to introduce the A* algorithm. Firstly, it can evaluate nodes through heuristic functions to quickly find the optimal path. Secondly, the A* algorithm has good performance and accuracy in solving the optimal path problem. By applying the A* algorithm to the
optimization of the initial point set, the quality of the initial population can be improved, providing a better starting point for the search process of genetic algorithms. Finally, this improvement
by introducing the A* algorithm helps to improve the overall algorithm's ability to solve the
EVRPTW and provides better results.
The pseudo-code of the GA-A* algorithm is shown in Fig. 5. Through this improvement, this
paper aims to overcome the limitations of traditional genetic algorithms and improve the efficiency and quality of solving path planning problems.
Firstly, based on the A* algorithm, the optimal path passing through all nodes is obtained and
the path is divided into sub-paths according to the needs of the customer nodes. Then sort the
remaining unincorporated nodes by time window order, and insert randomly signed charging
station nodes to generate multiple genes as the initial population. Using decimal signed encoding
to represent genes, including information such as customer nodes, charging nodes, and distribution centers. The fitness function considers vehicle costs, energy consumption costs, time costs,
and penalty costs for constraint violations. Selection, using the roulette selection method. Crossover, using the first 1/3 encoding after crossover to replace the last 1/3 encoding before crossover. Variation, using an elitist strategy to retain elitist individuals, varies general genes, generates multiple genes, and calculates the fitness to retain the highest term. Multiple iterations are
performed to obtain the final path encoding.
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Wang, Ding, Chen, Zhang

2.3 Experimental design

Fig. 5 GA-A* pseudo-code

To evaluate the performance of the model and algorithm. In this paper, 16 test cases were selected. Test cases are derived from a shared database [33]. In these cases, each customer node
has different requirements, and each example has a different charging station and number of
customers. For example, the case "R-2-C-40" means that there are two charging stations and
forty customers in the case. To test the performance of the GA-A* algorithm, experiments were
designed to test it against the conventional GA, and referred to the data made by Li et al using
adaptive large neighbourhood search (ALNS), large neighbourhood search (LNS), and variable
neighbourhood search (VNS) algorithms [32] for comparison. The algorithm is written in Python
and the test platform is Intel Xeon E5-2680 v4.

3. Result

3.1 Simulation results
The algorithm was used to test the optimal path under 16 groups of different cases, Some of the
initial maps are shown in Fig. 6 for 30, 50, 70 and 120 charging station nodes.
Fig. 7 shows the paths plotted from the optimal paths derived from the GA-A* operation with
other methods.
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A combined genetic algorithm and A* search algorithm for the electric vehicle routing problem with time windows

Fig. 6 Initial map

Fig. 7 Route map

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Wang, Ding, Chen, Zhang

Fig. 8 Grid load comparison

As shown in Fig. 7, the algorithm can find the route that fulfils the conditions and ensures the
correctness of the solution. After that, the paper tests the effect of the GA-A* algorithm on the
power grid at different times, and the experimental results show that the GA-A* algorithm has a
good effect on power sharing and power quality.
Fig. 8 shows better power distribution under the GA-A* algorithm scheduling. Most of the
charging time is concentrated between 21:00 and 3:00 the next day, which is in the off-peak period of electricity consumption. The discharge time is concentrated from 9:00 to 19:00, which is
in the peak period of power consumption, the peak load decreases from the original 11403.73
kW to 11365.10 kW, and the peak differential rate decreases from the original 41.50-41.27 %.
Charging when the electricity price is low and discharging when the electricity price is high, fully
shows that the scheduling function of the GA-A* algorithm can cut peak and fill the valley and
reduce the charging cost. As shown in Table 2.
Due to the existence of area price, the area price can be used to regulate loads in different areas of the grid. As shown in Fig. 9, if the charging station is in a region with low electricity prices,
the route is more concerned with charging in this region; if it is in a region with high electricity
prices, the route is more concerned with discharging in this region.
scenarios
original grid
GA-A*

412

Table 2 Load characteristic index in the different scenarios
Max. load (kW)
Min. load (kW)
mean square
peak-to-valley ratio (%)
11403.73
6670.81
1846.76
41.50
11365.10
6675.15
1825.69
41.27

Fig. 9 Zonal grid load changes

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A combined genetic algorithm and A* search algorithm for the electric vehicle routing problem with time windows

3.2 Comparison with other methods
Fig. 10 shows the comparison between the GA-A* algorithm and the traditional GA algorithm. As
shown in the figure, the iteration speed of GA-A* is significantly faster than that of the GA algorithm, and it enters the low value region 89 times and completes the iteration quickly. The
shortest distance produced by GA-A* is also significantly smaller than the optimal solution produced by GA.

Fig. 10 GA-A* versus GA

Table 3 shows the comparison results between GA-A* and the other three algorithms. It is
found that the GA-A* algorithm has a better optimization effect than the other three algorithms.
In comparison to LNS, VNS, and ALNS, the average reduction is 281.41, -67.43, and 207.80 km,
respectively. The results show that the optimization effect of GA-A* is significantly higher than
that of LNS and ALNS.
case

r-8-c-120
r-8-c-70
r-8-c-50
r-8-c-30
r-6-c-120
r-6-c-70
r-6-c-50
r-6-c-30
r-4-c-80
r-4-c-70
r-4-c-50
r-4-c-30
r-2-c-90
r-2-c-70
r-2-c-50
r-2-c-30

4. Discussion

Table 3 Load characteristic index in the different scenarios
optimal solution /km
GA-A*
LNS
VNS
3282.46
3617.30
3160.50
2067.05
2382.70
1938.40
1401.98
1672.30
1419.90
947.25
1333.30
983.60
2428.78
2132.90
1841.20
1791.31
2196.80
1767.40
1536.47
2417.70
1556.90
820.92
1469.90
828.50
2249.17
2369.50
2181.10
1975.49
2305.20
2117.10
1469.40
1532.80
1327.90
849.50
1087.90
881.10
2455.17
2722.80
2285.50
1934.04
2054.20
1842.20
1369.70
1575.50
1357.60
873.24
1083.70
884.20

ALNS
3602.10
2242.20
1646.70
1145.50
2479.10
2057.80
1784.40
1009.10
2465.90
2310.90
1583.60
1043.10
2707.10
2099.90
1569.80
1029.50

In the process of solving the EVRPTW problem, in order to reduce the total cost in the transportation process and make the model closer to the actual engineering background, we comprehensively consider several factors and establish the model by using integer mixed programming.
These factors include the fixed cost of the vehicle, energy cost which considers the temperature
coefficient, the cost of time, the penalty cost of violating the load constraint, the penalty cost of
violating the time window, and the penalty cost of violating the power constraint and the charge
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Wang, Ding, Chen, Zhang

and discharge cost. Through such comprehensive consideration, we obtained a model of
EVRPTW that is more in line with the actual engineering context.
By introducing the Time of use Pricing and Area price, we have reached the V2G model, integrating EVs into the grid, and realizing the transportation and exchange of energy is realized. By
means of power swapping for electric vehicles, we have achieved the purpose of forming a smart
microgrid. In this mode, EVs can choose to deliver energy during peak hours of electricity consumption and obtain energy during low hours of electricity consumption according to the demand of the power grid, rationalizing the distribution of electric energy, reducing the peaks and
valleys and volatility of the load on the power grid, and improving the quality of electric energy.
V2G mode makes EVs become part of the power grid, participate in the transportation and exchange of energy, and make full use of the energy storage capacity and flexibility of EVs through
intelligent energy scheduling and control.
In terms of algorithm, we improve the steps of initializing the population and introduce the
A* algorithm to obtain a better initial solution. By combining GA and A* algorithm, we not only
avoid the problem of slow iteration speed of the genetic algorithm but also meet the randomness
requirement of GA. Experimental results show that this GA-A* algorithm can provide correct
path planning, and has better iteration speed and optimization quality when solving EVRPTW
problem. Compared with the traditional genetic algorithm, our algorithm shows obvious advantages in iteration speed and optimal solution quality. Compared with other different types of
path planning algorithms, such as LNS, ALNS, and VNS algorithms, our algorithm also shows
better performance and can get good results in different scale examples. In some specific cases
as shown in Table 3 it can be seen that the distance of GA-A* is less than the solution derived by
VNS for small scale cases with eighty client nodes, but in large scale cases VNS shows better performance. This paper argues that the implementation of the VNS algorithm searches a wider
space of solutions in variable neighbourhoods, no matter the size of the arithmetic case can find
a better solution by means of global search. VNS follows this feature for solutions at different
scales, whereas GA-A*, which has fixity in its initialisation, shows better performance in small
scale solving. In this paper, we argue that GA-A* has significantly outperformed other algorithms
under the path length perspective for small-scale problems, and some algorithms for large-scale
problems; it outperforms other algorithms under the perspective for spatio-temporal deployment of energy consumption. The algorithm has been initialised in such a way as to find the optimal solution passing through all nodes, so that there is a certain degree of similarity between
the initialised population and the final solution for the small-scale example in the global solution; the algorithm outperforms other algorithms in terms of energy consumption because it
takes into account the spatio-temporal tariffs. There are also some weaknesses in that the initialised population is still somewhat fixed even with the addition of random values, so larger than
large-scale arithmetic cases lack global search capability. So introduction of the A* algorithm as
part of the initialization allows the genetic algorithm to start searching for a better initial solution. This combination can fully utilize the advantages of the two algorithms and improve the
search efficiency and solution quality of the algorithm.
In summary, by comprehensively considering multiple practical factors and establishing an
integer mixed programming model, introducing Time of use Pricing and Area price as well as the
V2G model, and combining the genetic algorithm and A* algorithm for optimization, we have
made remarkable progress in solving EVRPTW problem. Our model and algorithm can better
conform to the actual engineering background, obtain a better solution, improve the efficiency
and quality of power grid energy management, and bring a positive impact on the operation of
smart microgrid and the entire power grid. Such as Wang et al. used heuristics and collision
avoidance algorithms for collaborative scheduling planning of multiple AGVs [34]. GA-A* can be
generalised to AGVs in industrial manufacturing for logistics optimisation tasks such as optimal
routing through improvements.

414

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A combined genetic algorithm and A* search algorithm for the electric vehicle routing problem with time windows

5. Conclusion
To solve the EVRPTW, this paper comprehensively considers multiple factors in reality and
takes the minimum total cost as the model proposes the GA-A* algorithm which introduces the
A* algorithm, and tests 16 cases of four groups of different scales. Experiments show that compared with the traditional GA algorithm and other algorithms, it has a certain optimization effect. The algorithm itself can provide a solution to the np-hard problem and get the right route.
At the same time, based on the V2G mode, the smart microgrid can reduce fluctuations and improve power quality, and the reasonable spatio-temporal allocation of EVs plays a role in peaking and valley filling for the entire power grid.
The GA-A* algorithm introduced in this paper can provide better path planning in the search
process, but it may face the problem of too long computation time when dealing with large-scale
problems. Therefore, for larger scale EVRPTW problems, it may be necessary to further optimize
the efficiency of the algorithm. At the same time, in practical problems, there are often multiple
conflicting objectives that need to be optimized, The GA-A* algorithm can be further extended.
Therefore, to solve the above problems, we can consider further optimizing the efficiency of the
algorithm, such as introducing a pruning strategy or parallel computing. Multi-Objective Optimization can be supported by improvement, such as considering the problem of simultaneously
optimizing total cost and total time.

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Advances in Production Engineering & Management 18(4) 2023

Advances in Production Engineering & Management

ISSN 1854-6250

Volume 18 | Number 4 | December 2023 | pp 417–433

Journal home: apem-journal.org

https://doi.org/10.14743/apem2023.4.482

Original scientific paper

Optimizing rock breaking performance: The influence of
chamfer on polycrystalline diamond compact (PDC) cutters
Ju, P.a,*
aCCTEG

Xi'an Research Institute, Xi’an, P.R. China

ABSTRACT

ARTICLE INFO

Research on the rock-breaking performance of the Polycrystalline Diamond
Compact (PDC) cutter has primarily focused on sharp cutters, often overlooking the influence of chamfer. Notably, the design of chamfer parameters has
been largely unreported. In this study, we established a theoretical model of
cutting force that takes chamfer into account. We analysed the primary and
secondary relationships of four factors – back rake angle, depth of cut, chamfer
angle, and chamfer length – on the force of the PDC cutter. This was done
through a pseudo-level orthogonal level test. A numerical simulation, based on
the Smooth Particle Hydrodynamic (SPH) method, was conducted to analyse
the rock-breaking force and stress distribution characteristics of PDC cutters
with different chamfer angles. Combined with a drop hammer impact test, we
provided an optimized design of chamfer parameters. Our findings revealed
that while the chamfer had a relatively minor influence on the force of the PDC
cutter, it contributed to the optimal distribution of stress on the PDC cutter.
This effectively protected the cutting edge and prevented early cracks and
spalls of the cutter. When the chamfer angle was less than or equal to the back
rake angle, the resultant force of the PDC cutter increased with the increase of
the chamfer angle. However, when the chamfer angle was greater than the back
rake angle, the resultant force of the PDC cutter first increased and then slightly
decreased with the increase of the chamfer angle. Additionally, the resultant
force of the PDC cutter increased approximately linearly with the increase of
chamfer length. When the chamfer angle of the PDC cutter was between 30°
and 45°, the fluctuation of the cutting force was relatively smooth, the rockbreaking process was stable, and the cutter’s impact resistance energy was relatively higher. These findings will provide valuable guidelines for the design of
chamfered PDC cutters.

Keywords:
Polycrystalline diamond compact
(PDC) cutter;
Chamfer parameters;
Optimization;
Cutting force;
Theoretical analysis;
Numerical simulation;
Smooth Particle Hydrodynamic
(SPH);
Stress characteristics
*Corresponding author:
jpnt2005@163.com
(Ju, P.)

Article history:
Received 25 October 2023
Revised 14 December 2023
Accepted 15 December 2023
Content from this work may be used under the terms of
the Creative Commons Attribution 4.0 International Licence (CC BY 4.0). Any further distribution of this work
must maintain attribution to the author(s) and the title of
the work, journal citation and DOI.

1. Introduction
In the field of petroleum and coal geological drilling, PDC bit is widely used in soft to medium hard
formations, thanks to its strong rock cutting ability and high drilling efficiency. As the direct rock
breaking component of PDC bit, the performance of polycrystalline diamond compact (PDC) cutter determines the drilling effect and service life of PDC bit. The PDC cutter is synthesized by polycrystalline diamond layer and cemented carbide substrate at high temperature and high pressure environment, its shape is generally cylindrical, and it has the advantages of strong self-sharpening ability, good thermal stability and strong wear resistance. However, when drilling into hard
and complex formation, PDC cutters will withstand serious impact and vibration, and the problems of cutter breakage and collapse occur frequently, which seriously restrict the performance
of PDC bit.
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In view of the above problems, a lot of studies have been carried out. Regarding the influence
of back rake angle on the performance of PDC cutter, Vusal, Zhang et al. pointed out that the mechanical specific energy increases as back rake angle increases [1, 2]; Adzis et al. raised that the
larger the back rake angle, the smaller the Von Mises stress induced in the cutter, and the cutter
wear rate decreases as the back rake angle increases [3]. In view of the influence of depth of cut
on the rock breaking force of PDC cutter, Dai et al. said that at shallow depth of cut, only powdery
rock chips are produced, and the fluctuation amplitude of cutting force is small, while at deep
cutting depth, massive rock breakage occurs, and the fluctuation amplitude of cutting force increases significantly [4]; Li et al. proposed that an overall positive correlation is observed between
the maximum value of the cutting force and depth of cut, the cuttings size appears to increase with
the increase of depth of cut [5]; Rahmani et al. stated that as the increase of depth of cut, the force
and force area increase gradually, and the stress on the cutter changes from tensile to compressive
[6]; Joodi et al. put forward that with the increase of depth of cut, fractures extend deeper on the
rock, and the failure mode of the rock gradually changes from ductile failure to brittle failure [7].
As for the influence of wear on the performance of PDC cutter, Iman et al. indicated that at a wide
range of depth of cut, the frictional force on the wear flat of a blunt cutter was positively related
to the angle of wear flat [8]; Zhang, Yang et al. stated that the cutting force, temperature change
and large volume cuttings of the worn cutters are more obvious compared with the new cutter,
and under the same depth, the greater the wear height is, the more obvious the force fluctuation
appears[9, 10]; Liu et al. proposed that the wear reduces the aggressiveness of cutter, and the
grinding effect of cutter on cuttings is enhanced [11].
Through previous studies, the comprehensive performance and rock breaking ability of PDC
cutter have been continuously improved. In particular, it is found that although the sharp cutting
edge of PDC cutter can ensure its strong attack performance, when drilling into hard and heterogeneous rock, the polycrystalline diamond layer is prone to collapse. To solve this problem, some
scholars put forward the scheme of pre-chamfering the edge of polycrystalline diamond layer,
with the newly added chamfered surface, the stress on the cutter is extended to a larger area, thus
reducing the stress gradient on the cutting edge and enhancing the impact and wear resistance of
PDC cutter [12]. Shao et al. pointed out that the cutting efficiency can be greatly enhanced by reducing the chamfer size, but the small chamfer size makes PDC cutters vulnerable to the premature failure [13]. Fu et al. proposed that the chamfer has a more significant influence on the normal
than the tangential cutting force, as the chamfer becomes larger, the aggressiveness of the PDC
cutter decreases considerably [14]. Akbari et al. put forward that at shallow cuts, a chamfered PDC
cutter behaves like a cutter with higher back rake angle, the deeper the cut gets the less pronounced chamfer effect [15]. Jamaludin et al. expressed that the significance of the cutter geometry to the wear rate is chamfer angle, back rake angle, side rake angle and cutter diameter accordingly, the chamfer angle has the strongest correlation to the wear characteristic of PDC cutter [16].
At present, the research on the influence of chamfer on the performance of PDC cutter are
mostly carried out from a macro perspective, while the specific design of the chamfer parameters
has not been discussed in detailed. In this work, a theoretical cutting force model of PDC cutter
considering the influence of chamfer was established; based on orthogonal design test, the primary and secondary relationship of back rake angle, depth of cut, chamfer angle and chamfer
length was analysed, and the influence law of chamfer parameters on cutting force were summarized. The rock cutting simulation of PDC cutters with different chamfer angles was carried out,
the influence of chamfer angle on the cutting force and stress of PDC cutter were evaluated. At the
same time, the drop hammer impact test of PDC cutter was carried out, so as to guide the optimization design of chamfer parameters of PDC cutter.

2. Theoretical model of cutting force considering chamfer

Decomposing the force of PDC cutter with chamfer, the resultant force F of chamfered PDC cutter
is composed of the resultant force Fc on the front surface of cutter and the resultant force 𝐹𝐹 𝑐𝑐ℎ on
the chamfered surface of cutter. The force decomposition diagram of PDC cutter considering the
chamfer is shown in Fig. 1.
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Optimizing rock breaking performance: The influence of chamfer on polycrystalline diamond compact (PDC) cutters

Fig. 1 Cutting force decomposition diagram of chamfered PDC cutter

According to Fig. 1, there are:

⎧
⎪
⎪

𝐹𝐹 = �𝐹𝐹ℎ2 + 𝐹𝐹𝑣𝑣2

𝐹𝐹 𝑐𝑐 = �𝐹𝐹ℎ𝑐𝑐2 + 𝐹𝐹𝑣𝑣𝑐𝑐2

(1)

⎨
⎪
⎪𝐹𝐹 𝑐𝑐ℎ = �𝐹𝐹 𝑐𝑐ℎ2 + 𝐹𝐹 𝑐𝑐ℎ2
𝑣𝑣
ℎ
⎩

F is the resultant force of chamfered PDC cutter (N); Fh and Fv denote the horizontal force and
normal force of chamfered PDC cutter (N); Fc is the resultant force on the front surface of chamfered PDC cutter (N); Fch and 𝐹𝐹𝑣𝑣𝑐𝑐 denote the horizontal force and normal force on the front surface
of chamfered PDC cutter (N); 𝐹𝐹 𝑐𝑐ℎ is the resultant force on the chamfered surface of chamfered
PDC cutter (N); 𝐹𝐹ℎ𝑐𝑐ℎ and 𝐹𝐹𝑣𝑣𝑐𝑐ℎ denote the horizontal force and normal force on the chamfered surface of chamfered PDC cutter (N).
The force of a chamfered PDC cutter can be obtained in horizontal and vertical directions respectively:
𝐹𝐹 = 𝐹𝐹ℎ𝑐𝑐 + 𝐹𝐹ℎ𝑐𝑐ℎ
� ℎ
(2)
𝐹𝐹𝑣𝑣 = 𝐹𝐹𝑣𝑣𝑐𝑐 + 𝐹𝐹𝑣𝑣𝑐𝑐ℎ
According to reference [17], it can be obtained:
𝐹𝐹ℎ𝑐𝑐 = 𝜀𝜀𝐴𝐴𝑐𝑐
(3)
� 𝑐𝑐
𝐹𝐹𝑣𝑣 = tan (𝜃𝜃 + 𝜑𝜑)𝜀𝜀𝐴𝐴𝑐𝑐
𝐹𝐹 𝑐𝑐ℎ = 𝜀𝜀𝐴𝐴𝑐𝑐ℎ
(4)
� 𝑐𝑐ℎ ℎ
𝐹𝐹𝑣𝑣 = tan (𝜃𝜃 + 𝜑𝜑)𝜀𝜀𝐴𝐴𝑐𝑐ℎ
Ac is the contact area between the front surface and the rock (m2); 𝐴𝐴𝑐𝑐ℎ is the contact area between
the chamfered surface and the rock, (m2); θ is the back rake angle (rad); ε is the intrinsic specific
energy of rock (Pa); φ is the internal friction angle of rock (rad).
Substituting Eq. 3 and Eq. 4 into Eq. 1 and Eq. 2, the forces on the front surface and chamfered
surface can be obtained, and the resultant force of chamfered PDC cutter can be calculated.
It is important to note that, the contact area 𝐴𝐴𝑐𝑐 and 𝐴𝐴𝑐𝑐ℎ are related to the relationship between
chamfer angle and back rake angle, the calculation of the contact area needs to be considered from
the following three cases respectively.
(1) Chamfer angle is greater than back rake angle

When the chamfer angle is greater than back rake angle, the chamfered surface of the PDC cutter
is in full contact with the rock. The contact state and contact area between the PDC cutter and the
rock are shown in Fig. 2.
According to Fig. 2, it can obtained:
1

1

𝐴𝐴𝑐𝑐 = 𝑆𝑆𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 − 𝑆𝑆𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 = 2 (∠𝐸𝐸𝐸𝐸𝐸𝐸) ∙ |𝑂𝑂𝑂𝑂|2 − 2 |𝑂𝑂𝑂𝑂| ∙ |𝑂𝑂𝑂𝑂| ∙ 𝑠𝑠𝑠𝑠𝑠𝑠(∠𝐸𝐸𝐸𝐸𝐸𝐸)
�
1
1
𝐴𝐴𝑐𝑐ℎ = 𝑆𝑆𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 − 𝑆𝑆𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 = 2 (∠𝐸𝐸𝐸𝐸𝐸𝐸) ∙ |𝑂𝑂𝑂𝑂|2 − 2 (∠𝐸𝐸𝐸𝐸𝐸𝐸) ∙ |𝑂𝑂𝑂𝑂|2

Advances in Production Engineering & Management 18(4) 2023

(5)

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Ju

Fig. 2 Diagram of the contact state and contact area between the PDC cutter and the rock when chamfer angle is
greater than back rake angle

The ∠𝐸𝐸𝐸𝐸𝐸𝐸 is expressed in radians.

|𝑂𝑂𝑂𝑂|
|𝑂𝑂𝑂𝑂| − |𝐵𝐵𝐵𝐵| − |𝐵𝐵𝐵𝐵|
∠𝐸𝐸𝐸𝐸𝐸𝐸 = 2 arc cos �
� = 2 arc cos �
�
|𝑂𝑂𝑂𝑂|
|𝑂𝑂𝑂𝑂|
ℎ
ℎ
|𝑂𝑂𝑂𝑂| − |𝐵𝐵𝐵𝐵| − �
|𝑂𝑂𝑂𝑂| − |𝐵𝐵𝐵𝐵| − �
− |𝐵𝐵𝐵𝐵|�
− (|𝐵𝐵𝐵𝐵| − |𝐷𝐷𝐷𝐷|)�
cos 𝜃𝜃
cos 𝜃𝜃
= 2 arc cos �
� = 2 arc cos �
�
|𝑂𝑂𝑂𝑂| − |𝐵𝐵𝐵𝐵|
|𝑂𝑂𝑂𝑂| − |𝐵𝐵𝐵𝐵|
= 2 arc cos �

ℎ
ℎ
𝑟𝑟 − 𝑙𝑙 𝑐𝑐ℎ tan 𝛽𝛽 − �
− (𝑙𝑙 𝑐𝑐ℎ tan 𝛽𝛽 − 𝑙𝑙 𝑐𝑐ℎ tan 𝜃𝜃)�
𝑟𝑟 −
− 𝑙𝑙 𝑐𝑐ℎ tan 𝜃𝜃
cos 𝜃𝜃
cos𝜃𝜃
�
=
2
arc
cos
�
�
𝑟𝑟 − 𝑙𝑙 𝑐𝑐ℎ tan 𝛽𝛽
𝑟𝑟 − 𝑙𝑙 𝑐𝑐ℎ tan 𝛽𝛽

|𝑂𝑂𝑂𝑂| = |𝑂𝑂𝑂𝑂| = |𝑂𝑂𝑂𝑂| = |𝑂𝑂𝑂𝑂| − |𝐵𝐵𝐵𝐵| = 𝑟𝑟 − 𝑙𝑙 𝑐𝑐ℎ tan 𝛽𝛽

(6)
(7)

where 𝑟𝑟 is the radius of chamfered PDC cutter (m); ℎ is the depth of cut (m); 𝛽𝛽 is the chamfer angle
(rad); 𝑙𝑙 𝑐𝑐ℎ is the chamfer length (m).
Substituting Eq. 6 and Eq. 7 into Eq. 5, Ac and 𝐴𝐴𝑐𝑐ℎ can be expressed:
ℎ
ℎ
𝑟𝑟 −
− 𝑙𝑙 𝑐𝑐ℎ tan 𝜃𝜃
𝑟𝑟 −
− 𝑙𝑙 𝑐𝑐ℎ tan 𝜃𝜃
⎧ 𝑐𝑐
1
cos 𝜃𝜃
cos 𝜃𝜃
𝑐𝑐ℎ
2
(𝑟𝑟
�
�
�2
�
���
𝐴𝐴
=
−
𝑙𝑙
tan
𝛽𝛽)
∙
�arc
cos
−
sin
arc
cos
⎪
𝑟𝑟 − 𝑙𝑙 𝑐𝑐ℎ tan 𝛽𝛽
2
𝑟𝑟 − 𝑙𝑙 𝑐𝑐ℎ tan 𝛽𝛽
⎪
⎨
⎪
⎪
⎩

𝐴𝐴𝑐𝑐ℎ = arc cos �

𝑟𝑟 −

ℎ
− 𝑙𝑙 𝑐𝑐ℎ tan 𝜃𝜃
cos𝜃𝜃
� ∙ [𝑟𝑟 2 − (𝑟𝑟 − 𝑙𝑙 𝑐𝑐ℎ tan 𝛽𝛽)2 ]
𝑟𝑟 − 𝑙𝑙 𝑐𝑐ℎ tan 𝛽𝛽

(8)

Substituting Eq. 8 into Eq. 3 and Eq. 4, the horizontal force and normal force acting on the front
surface and chamfered surface of PDC cutter can be obtained.
(2) Chamfer angle is equal to back rake angle

Fig. 3 shows the contact state and contact area between the PDC cutter and the rock when the
chamfer angle is equal to back rake angle, at this time, the chamfered line BG showed in Fig. 3 is
parallel to the horizontal direction.

Fig. 3 Diagram of the contact state and contact area between the PDC cutter and rock when chamfer angle is
equal to back rake angle

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Optimizing rock breaking performance: The influence of chamfer on polycrystalline diamond compact (PDC) cutters

According to Fig. 3, it can obtained:

𝐹𝐹 𝑐𝑐ℎ = 𝜇𝜇𝐹𝐹𝑣𝑣𝑐𝑐ℎ
� 𝑐𝑐ℎ ℎ
𝐹𝐹𝑣𝑣 = tan(𝜃𝜃 + 𝜑𝜑)𝜀𝜀𝐴𝐴𝑐𝑐ℎ

where μ is the friction coefficient of rock.
�

1

(9)
1

𝐴𝐴𝑐𝑐 = 𝑆𝑆𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 − 𝑆𝑆𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 = (∠𝐸𝐸𝐸𝐸𝐸𝐸) ∙ |𝑂𝑂𝑂𝑂|2 − |𝑂𝑂𝑂𝑂| ∙ |𝑂𝑂𝑂𝑂| ∙ sin(∠𝐸𝐸𝐸𝐸𝐸𝐸)
2
2
1
2

1
2

𝐴𝐴𝑐𝑐ℎ = 𝑆𝑆𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 − 𝑆𝑆𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 = (∠𝐸𝐸𝐸𝐸𝐸𝐸) ∙ |𝑂𝑂𝑂𝑂|2 − (∠𝐸𝐸𝐸𝐸𝐸𝐸) ∙ |𝑂𝑂𝑂𝑂|2

|𝑂𝑂𝑂𝑂|
|𝑂𝑂𝑂𝑂| − |𝐵𝐵𝐵𝐵| − |𝐵𝐵𝐵𝐵|
∠𝐸𝐸𝐸𝐸𝐸𝐸 = 2 arc cos �
� = 2 arc cos �
�
|𝑂𝑂𝑂𝑂|
|𝑂𝑂𝑂𝑂|

ℎ
𝑟𝑟 − 𝑙𝑙 𝑐𝑐ℎ tan 𝛽𝛽 −
|𝑂𝑂𝑂𝑂| − |𝐵𝐵𝐵𝐵| − |𝐵𝐵𝐵𝐵|
cos
𝜃𝜃 �
= 2 arc cos �
� = 2 arc cos �
|𝑂𝑂𝑂𝑂| − |𝐵𝐵𝐵𝐵|
𝑟𝑟 − 𝑙𝑙 𝑐𝑐ℎ tan 𝛽𝛽

|𝑂𝑂𝑂𝑂| = |𝑂𝑂𝑂𝑂| = |𝑂𝑂𝑂𝑂| = |𝑂𝑂𝑂𝑂| − |𝐵𝐵𝐵𝐵| = 𝑟𝑟 − 𝑙𝑙 𝑐𝑐ℎ tan 𝛽𝛽

Substituting Eq. 11 and Eq. 12 into Eq. 10, Ac and 𝐴𝐴𝑐𝑐ℎ can be expressed:

ℎ
ℎ
𝑟𝑟 − 𝑙𝑙 𝑐𝑐ℎ tan 𝛽𝛽 −
𝑟𝑟 − 𝑙𝑙 𝑐𝑐ℎ tan 𝛽𝛽 −
⎧ 𝑐𝑐
1
cos
𝜃𝜃
cos
𝜃𝜃 ���
𝑐𝑐ℎ
2
𝐴𝐴 = (𝑟𝑟 − 𝑙𝑙 tan 𝛽𝛽) ∙ �arc cos �
� − sin �2 arc cos �
⎪
𝑟𝑟 − 𝑙𝑙 𝑐𝑐ℎ tan 𝛽𝛽
2
𝑟𝑟 − 𝑙𝑙 𝑐𝑐ℎ tan 𝛽𝛽
⎪
⎨
⎪
⎪
⎩

𝐴𝐴𝑐𝑐ℎ = arc cos �

ℎ
cos 𝜃𝜃 � ∙ [𝑟𝑟 2 − (𝑟𝑟 − 𝑙𝑙 𝑐𝑐ℎ tan 𝛽𝛽)2 ]
𝑟𝑟 − 𝑙𝑙 𝑐𝑐ℎ tan 𝛽𝛽

𝑟𝑟 − 𝑙𝑙 𝑐𝑐ℎ tan𝛽𝛽 −

(10)
(11)
(12)

(13)

Substituting Eq. 13 into Eq. 3 and Eq. 9, the horizontal force and normal force acting on the
front surface and chamfered surface of PDC cutter can be obtained.
(3) Chamfer angle is less than back rake angle

When the chamfer angle is less than back rake angle, in an ideal state, the chamfered surface of
the PDC cutter does not contact with the rock. The contact state and contact area between the PDC
cutter and the rock are shown in Fig. 4.

Fig. 4 Diagram of the contact state and contact area between the PDC cutter and rock when chamfer angle is
equal to back rake angle

According to Fig. 4, it can obtained:

𝐹𝐹ℎ𝑐𝑐ℎ = 𝐹𝐹𝑣𝑣𝑐𝑐ℎ = 0

1

1

𝐴𝐴𝑐𝑐 = 𝑆𝑆𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 − 𝑆𝑆𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 = (∠𝐸𝐸𝐸𝐸𝐸𝐸) ∙ |𝑂𝑂𝑂𝑂|2 − |𝑂𝑂𝑂𝑂| ∙ |𝑂𝑂𝑂𝑂| ∙ sin(∠𝐸𝐸𝐸𝐸𝐸𝐸)
2

2

|𝑂𝑂𝑂𝑂|
|𝑂𝑂𝑂𝑂| − |𝐵𝐵𝐵𝐵| − |𝐵𝐵𝐵𝐵|
∠𝐸𝐸𝐸𝐸𝐸𝐸 = 2 arc cos �
� = 2 arc cos �
�
|𝑂𝑂𝑂𝑂|
|𝑂𝑂𝑂𝑂|

ℎ
𝑟𝑟 − 𝑙𝑙 𝑐𝑐ℎ tan 𝛽𝛽 −
|𝑂𝑂𝑂𝑂| − |𝐵𝐵𝐵𝐵| − |𝐵𝐵𝐵𝐵|
cos
𝜃𝜃�
= 2 arc cos �
� = 2 arc cos �
|𝑂𝑂𝑂𝑂| − |𝐵𝐵𝐵𝐵|
𝑟𝑟 − 𝑙𝑙 𝑐𝑐ℎ tan 𝛽𝛽

Advances in Production Engineering & Management 18(4) 2023

(14)

(15)

(16)
421

Ju

|𝑂𝑂𝑂𝑂| = |𝑂𝑂𝑂𝑂| = |𝑂𝑂𝑂𝑂| = |𝑂𝑂𝑂𝑂| − |𝐵𝐵𝐵𝐵| = 𝑟𝑟 − 𝑙𝑙 𝑐𝑐ℎ tan 𝛽𝛽

Substituting Eq. 16 and Eq. 17 into Eq. 15, Ac can be expressed:
𝑐𝑐

𝑐𝑐ℎ

2

𝐴𝐴 = �𝑟𝑟 − 𝑙𝑙 tan 𝛽𝛽� ∙ �arc cos �

𝑟𝑟−𝑙𝑙 𝑐𝑐ℎ tan 𝛽𝛽−

ℎ
cos 𝜃𝜃

𝑟𝑟−𝑙𝑙 𝑐𝑐ℎ tan 𝛽𝛽

�−

1
sin �2 arc
2

cos �

(17)

𝑟𝑟−𝑙𝑙 𝑐𝑐ℎ tan 𝛽𝛽−

ℎ
cos 𝜃𝜃

𝑟𝑟−𝑙𝑙 𝑐𝑐ℎ tan 𝛽𝛽

��� (18)

Substituting Eq. 18 into Eq. 3, the horizontal force and normal force acting on the front surface
of PDC cutter can be obtained.

3. Primary and secondary factors affecting the force of cutter

Based on previous research, it is known that back rake angle and depth of cut have significant
influence on the force of cutter [18-23]. Therefore, combined with the focus of this work, the orthogonal design test of four factors, i.e. back rake angle, depth of cut, chamfer angle and chamfer
length, was established, the influence degree of the four factors on the force of cutter was analysed
according to the above established equations.
3.1 Pseudo-level orthogonal test design

Table 1 is the orthogonal design level table of the factors affecting the force of cutter.
According to Table 1, orthogonal test table 𝐿𝐿81 (49 ) was selected for the pseudo-level design of
the four factors. According to experience, the designed back rake angle is generally between 10
and 20° [24, 25], the depth of cut that the bit drill into the rock per turn is generally between 2
and 3 mm [26, 27], therefore, in the pseudo-level orthogonal test table, the second, third and
fourth levels (10°, 15° and 18°) of the back rake angle were repeated once respectively, and the
third, fourth and fifth levels (2 mm, 2.5 mm, 3 mm) of the depth of cut were repeated once respectively. Because the influence of chamfer length on the force of cutter is unknown, a random repetition of the chamfer length level was performed. Based on this, the pseudo-level orthogonal test
table was designed, and the parameters in the table were substituted into the equations derived
in Section 2, the results are shown in Table 2.
Level

No.

1
2
3
4
5
6
7
8
9
10
11
12
13

422

Factor
1
2
3
4
5
6
7
8
9

Table 1 Orthogonal design level table of the factors affecting the force of cutter
A
B
C
D
Back rake angle (°)
Cutting depth (°)
Chamfer angle (°)
Chamfer length (°)
5
1
5
0.2
10
1.5
10
0.3
15
2
15
0.4
18
2.5
18
0.5
20
3
20
25
3.5
25
30
45
60

Table 2 Pseudo-level orthogonal test level table of the force influencing factors and the results

Back rake
angle (°)
25
5
10
18
15
18
5
15
18
15
18
15
15

Depth of
cut (mm)
1.5
2.5
3
1.5
3.5
2
2
2
2
1
2.5
2.5
1.5

Chamfer
angle (°)
60
45
45
25
18
60
15
45
18
10
15
20
18

Chamfer
Resultant force on
length (mm) front surface (N)
0.3
878.381
0.4
1695.374
0.2
1603.419
0.4
737.354
0.2
2378.183
0.4
978.008
0.3
828.478
0.3
963.713
0.3
1169.110
0.5
388.434
0.2
1618.022
0.3
1457.285
0.2
700.775

Resultant force on
chamfered surface (N)
241.014
256.402
130.957
130.366
70.499
297.613
53.342
172.976
80.727
0.000
0.000
97.073
44.704

Resultant force of
PDC cutter (N)
1119.395
1951.776
1734.376
867.719
2448.682
1275.620
881.820
1136.689
1249.067
388.434
1618.022
1554.358
745.479

Advances in Production Engineering & Management 18(4) 2023

Optimizing rock breaking performance: The influence of chamfer on polycrystalline diamond compact (PDC) cutters

No.

14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79

Back rake
angle (°)
20
10
18
18
18
25
18
18
18
10
5
10
15
25
20
25
10
18
18
25
18
10
15
15
15
10
20
25
25
20
10
10
15
15
10
10
10
20
5
18
20
10
18
5
10
15
10
20
5
5
15
5
18
20
5
15
25
10
15
20
10
15
15
25
10
18

Depth of
cut (mm)
3
2
2.5
1
3
3.5
3
3
3
2
3
1.5
2.5
2.5
3.5
2
3.5
2.5
3.5
1
2.5
1
3
3
3.5
3
2.5
2.5
3
2
2.5
3.5
2.5
3
3
1
3
3
3
1
2.5
2.5
2
2.5
2
2
2.5
2
1
1.5
3
2
3.5
1
3.5
1
3
2
2
1.5
1.5
2
1.5
2
2.5
1.5

Chamfer
angle (°)
5
60
5
20
5
5
10
25
60
25
60
5
5
10
10
20
20
45
15
45
20
15
10
30
45
18
15
25
18
45
10
60
60
30
45
18
15
20
20
30
18
30
10
18
10
5
25
25
5
10
25
30
30
60
25
25
15
5
15
30
20
20
15
30
30
45

Table 2 (continuation)

Chamfer
Resultant force on
length (mm) front surface (N)
0.2
2245.877
0.2
848.894
0.5
1619.379
0.3
420.950
0.3
2102.773
0.3
3363.261
0.5
2091.386
0.2
2073.736
0.4
1847.717
0.5
855.799
0.5
1225.042
0.3
625.856
0.5
1482.830
0.4
2083.419
0.3
2771.957
0.4
1502.580
0.5
2021.925
0.5
1414.735
0.4
2593.402
0.2
511.444
0.2
1612.969
0.4
326.428
0.4
1918.500
0.3
1844.863
0.4
2180.744
0.5
1632.464
0.2
1727.147
0.5
2054.058
0.5
2676.756
0.5
1081.393
0.3
1310.027
0.2
1976.607
0.3
1312.954
0.4
1816.806
0.3
1552.690
0.4
316.120
0.3
1676.193
0.3
2227.834
0.4
1458.881
0.3
391.643
0.4
1715.850
0.2
1257.302
0.2
1175.170
0.3
1140.615
0.3
949.537
0.4
1075.850
0.4
1225.452
0.3
1230.262
0.2
313.160
0.2
558.197
0.2
1889.151
0.5
724.446
0.5
2474.827
0.5
296.278
0.3
1829.778
0.3
361.907
0.3
2696.483
0.4
950.940
0.5
1067.024
0.4
774.430
0.5
569.941
0.2
1061.781
0.5
702.100
0.2
1501.723
0.3
1228.481
0.3
682.543

Advances in Production Engineering & Management 18(4) 2023

Resultant force on
chamfered surface (N)
0.000
126.749
0.000
64.812
0.000
0.000
0.000
95.201
377.403
155.301
348.676
0.000
0.000
0.000
0.000
0.000
173.535
348.157
0.000
108.000
69.961
53.884
0.000
155.515
314.904
144.143
0.000
291.706
0.000
324.940
36.491
174.213
237.843
206.570
194.604
63.660
72.944
121.871
116.273
92.531
0.000
84.328
0.000
71.677
32.402
0.000
141.568
121.130
6.234
20.674
88.195
164.589
304.077
252.482
117.601
72.841
0.000
0.000
98.846
162.056
108.467
57.489
84.981
112.019
125.641
159.383

Resultant force of
PDC cutter (N)
2245.877
975.643
1619.379
485.761
2102.773
3363.261
2091.386
2168.936
2225.120
1011.099
1573.718
625.856
1482.830
2083.419
2771.957
1502.580
2195.459
1762.892
2593.402
619.444
1682.929
380.312
1918.500
2000.378
2495.649
1776.607
1727.147
2345.709
2676.756
1406.333
1345.110
2150.819
1550.797
2023.375
1747.294
379.781
1749.138
2349.031
1575.154
484.174
1715.850
1341.630
1175.170
1212.292
980.695
1075.850
1367.020
1351.392
318.981
578.871
1977.347
889.035
2778.904
548.761
1947.379
434.748
2696.483
950.940
1164.149
936.486
678.408
1119.269
785.639
1613.742
1354.122
841.926

423

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3.2 Multi-factor variance analysis
In the drilling process, rock breaking force is simultaneously affected by four factors: back rake
angle, depth of cut, chamfer angle and chamfer length. Therefore, taking the resultant force on the
front surface, the resultant force on the chamfered surface, and the resultant force of PDC cutter
as targets respectively, the data in Table 2 were analysed by multi-factor variance method [28].
Taking the resultant force on the front surface of PDC cutter as object, the varication analysis results of each factor are shown in Table 3.
Table 3 Multi-factor variance analysis with the object of the resultant force on the front surface of PDC cutter
Source of variation
Sum of deviation square
Freedom
Mean square
Value of F
Factor A (back rake angle)
12303791.19
5
2460758.239
2.145812091
Factor B (depth of cut)
53507230.5
5
10701446.1
9.331795407
Factor C (chamfer angle)
1508953.542
8
188619.1928
0.164478305
Factor D (chamfer length)
3924539.023
3
1308179.674
1.140749107
Factor E (error)
9174179.787
8
1146772.473

According to Table 3, from high to low, the influence degree of each factor on the resultant
force of PDC cutter is as follows: depth of cut, back rake angle, chamfer length and chamfer angle.
The depth of cut shows the most significant effect on the resultant force of front surface.
Taking the resultant force on the chamfered surface of PDC cutter as object, the varication analysis results of each factor are shown in Table 4.
Table 4 Multi-factor variance analysis with the object of the resultant force on the chamfered surface of PDC cutter
Source of variation
Sum of deviation square
Freedom
Mean square
Value of F
Factor A (back rake angle)
99076.91606
5
19815.38321
1.331109649
Factor B (depth of cut)
109763.2665
5
21952.6533
1.474681984
Factor C (chamfer angle)
551868.6539
8
68983.58174
4.634011378
Factor D (chamfer length)
69317.13208
3
23105.71069
1.552139271
Factor E (error)
119090.9147
8
14886.36434

According to Table 4, from high to low, the influence degree of each factor on the resultant
force of PDC cutter is as follows: chamfer angle, chamfer length, depth of cut and back rake angle.
The chamfer angle shows the most significant effect on the resultant force of chamfered surface.
Taking the resultant force of PDC cutter as object, the varication analysis results of each factor
are shown in Table 5.
Table 5 Multi-factor variance analysis with the object of the resultant force of PDC cutter
Source of variation
Sum of deviation square
Freedom
Mean square
Value of F
Factor A (back rake angle)
14082530.17
5
2816506.033
2.320851882
Factor B (depth of cut)
57888672.68
5
11577734.54
9.540262536
Factor C (chamfer angle)
568633.273
8
71079.15912
0.058570512
Factor D (chamfer length)
4160716.972
3
1386905.657
1.142835331
Factor E (error)
9708524.89
8
1213565.611

According to Table 5, from high to low, the influence degree of each factor on the resultant
force of PDC cutter is as follows: depth of cut, back rake angle, chamfer length and chamfer angle.
The depth of cut shows the most significant effect on the resultant force of PDC cutter, while the
chamfer angle and chamfer length have no significant effect.
Through multi-factor variance analysis, it can be obtained that, the depth of cut plays a major
role in the force of PDC cutter, followed by the back rake angle. The chamfer shows the significant
effect on the resultant force of chamfered surface, it mainly affects the stress distribution of the
PDC cutter, that is, the chamfer mainly affects the wear and impact resistance of the cutter.

424

Advances in Production Engineering & Management 18(4) 2023

Optimizing rock breaking performance: The influence of chamfer on polycrystalline diamond compact (PDC) cutters

4. Influence of chamfer parameters on the force of PDC cutter
Although the chamfer contributes minimally to the force of the PDC cutter, it is essential to examine the influence of the chamfer parameters on the force. This examination will provide guidance
for the design of chamfer parameters. Based on the equations established in Section 2, the influence
of the chamfer angle and chamfer length on the force of the PDC cutter was analysed separately.
4.1 Influence of chamfer angle on the force of PDC cutter

Based on previous research, fixing the back rake angle 15° and the depth of cut 0.3 mm unchanged
[29, 30], the influence law of chamfer angle on the force of PDC cutter with different chamfer
lengths was analysed. Figs. 5, 6 and 7 show the variation characteristics of the resultant force on
front surface, the resultant force on chamfered surface and the resultant force of PDC cutter with
chamfer angle respectively.
Resultant force on front surface / N

2200

2000
1800
1600

chamfer length 0.2mm

1400

chamfer length 0.3mm
chamfer length 0.4mm

1200

chamfer length 0.5mm

1000
800

0

10

20

30

40

50

60

70

40

50

60

70

Chamfer angle / degree

Fig. 5 Variation curve of the resultant force on front surface of PDC cutter with chamfer angle
500

Resultant force on chamfered surface / N

chamfer length 0.2mm
chamfer length 0.3mm

400

chamfer length 0.4mm
chamfer length 0.5mm

300
200
100

0

0

10

20

30

Chamfer angle / degree

Fig. 6 Variation curve of the resultant force on chamfered surface of PDC cutter with chamfer angle
Resultant force of the cutter / N

2200
2100
2000
1900

chamfer length 0.2mm
chamfer length 0.3mm

1800

chamfer length 0.4mm

1700
1600

chamfer length 0.5mm
0

10

20

30

40

Chamfer angle / degree

50

60

70

Fig. 7 Variation curve of the resultant force of PDC cutter with chamfer angle

Advances in Production Engineering & Management 18(4) 2023

425

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According to Figs. 5, 6 and 7, when the chamfer angle is small (chamfer angle not greater than
back rake angle), the chamfer angle appears almost no influence on the resultant force on both
front and chamfered surface, while the resultant force of PDC cutter increases significantly with
the increase of chamfer angle. When the chamfer angle is greater than back rake angle, with the
increase of chamfer angle, the resultant force on the front surface decreases linearly, while the
resultant force on chamfered surface increases approximately linearly, and the larger the chamfer
length, the faster the reduction or increase rate. The resultant force of PDC cutter increases first
and then decreases slightly with the increase of chamfer angle, the decreasing trend can be ignored when the chamfer length is small.
4.2 Influence of chamfer length on the force of PDC cutter

Fixing the back rake angle 15° and the depth of cut 0.3 mm unchanged, the influence of chamfer
length on the force of PDC cutter with different chamfer angles was analysed. Figs. 8, 9 and 10
show the variation characteristics of the resultant force on front surface, the resultant force on
chamfered surface and the resultant force of PDC cutter with chamfer length respectively.
According to Figs. 8, 9 and 10, when the chamfer angle is small (chamfer angle not greater than
back rake angle), the chamfer length appears almost no influence on both the resultant force on
front surface, chamfered surface and the PDC cutter. When chamfer angle is greater than back
rake angle, with the increase of chamfer length, the resultant force on the front surface decreases
linearly, while the resultant force on chamfered surface increases approximately linearly, and the
greater the chamfer angle, the faster the reduction or increase rate. The resultant force of PDC
cutter increases linearly with the increase of chamfer length, and the amplitude of linear increase
is almost unchanged under different chamfer angles.
Resultant force on front surface / N

2400

2200
2000
1800
1600
1400
1200
1000
800
600

0,1

chamfer angle 5 degree
chamfer angle 15 degree
chamfer angle 20 degree
chamfer angle 30 degree
chamfer angle 45 degree
chamfer angle 60 degree

0,2

0,3

0,4

0,5

0,6

0,4

0,5

0,6

Chamfer length / mm

Fig. 8 Variation curve of the resultant force on front surface of PDC cutter with chamfer length
Resultant force on chamfered surface / N

800

700

chamfer angle 5 degree

500

chamfer angle 30 degree

chamfer angle 15 degree

600

chamfer angle 20 degree
chamfer angle 45 degree

400

chamfer angle 60 degree

300
200
100

0

0,1

0,2

0,3

Chamfer length / mm

Fig. 9 Variation curve of the resultant force on chamfered surface of PDC cutter with chamfer length

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Optimizing rock breaking performance: The influence of chamfer on polycrystalline diamond compact (PDC) cutters
2400

Resultant force of the cutter / N

2200
2000
1800

chamfer angle 5 degree

1600

chamfer angle 15 degree

1400

chamfer angle 20 degree
chamfer angle 30 degree

1200

chamfer angle 45 degree

1000

800

0,1

0,2

chamfer angle 60 degree

0,3
0,4
Chamfer length / mm

0,5

0,6

Fig. 10 Variation curve of the resultant force of PDC cutter with chamfer length

5. Rock breaking simulation of PDC cutters with different chamfer angles
It is beneficial to improve the stress distribution by chamfering the PDC cutter, but it is difficult to
observe the stress state of the PDC cutter through theoretical and experimental analysis. So in this
work, numerical simulation based on Smooth Particle Hydrodynamic (SPH) method was used to
reveal the rock breaking characteristics and stress state of PDC cutters with different chamfer
angles.
5.1 Simulation model

The SPH is a meshless continuum mechanics method, in which the computation domain is discretized into a series of interacting particles. It’s good adaptive characteristics make it can deal with
the problems of large deformation and post-instability well, so can effectively avoid the mesh distortion caused by rock breaking [31, 32]. In this work, the following assumptions were made: (1)
the rocks were continuous, homogeneous and isotropic; (2) the influence of temperature and fluid
was ignored; (3) regardless of repeated breaking, the rock unit was deleted immediately after
being broken.
Drucker-Prager model was used to simulate the constitutive relation of rock [33]. In this model,
the formation and peeling process of cuttings can be represented by setting shear failure criterion,
and the shear failure of rock can be simulated well. The failure equivalent plastic strain was defined as the criterion of rock damage. When the equivalent plastic strain of rock was equal to the
failure plastic strain, the rock element was broken and deleted [24]. The cutter was represented
by an elastic model, which can consider the wear of the cutter. Table 6 shows the material parameters of the cutter and the rock.
The size of rock was 40 × 30 × 30 mm, and it was modelled by SPH particles, the particle size
was 0.8 mm. The size of cutter was Φ 13.44 × 8 mm, and it was modelled by hexahedral element,
the mesh size was 0.4 mm. The rock was completely fixed, the cutter cut the rock linearly along
the Y axis at a speed of 100 m/h, and the depth of cut was set 3 mm unchanged. Fig. 11 shows the
3D model of linear cutting rock by PDC cutter.
The idea of unit erosion algorithm was taken for reference to deal with the failure of SPH rock
particles, the rock particles were deemed to be invalid when the damage variable of rock particles
reach the critical value, the rock particles were deleted, but the mass and momentum of the failed
particles were still retained, thus ensuring the conservation of mass and momentum of the system.

Cutter
Rock

Density
(kg·m-3)
3560
2500

Table 6 Material parameters of the cutter and rock
Elastic
Poisson's
Compressive
Shear
Cohesion
modulus
ratio
strength
strength
(MPa)
(GPa)
(MPa)
(MPa)
850
0.07
40
0.27
75
10.0
27.2

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Internal friction
angle (°)
35.0

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Fig. 11 3D model of linear cutting rock by PDC cutter

According to the analysis in Section 2, the relationship between the chamfer angle and back
rake angle determines the contact state between PDC cutter and the rock. Therefore, in this simulation, the rock breaking characteristics of PDC cutters with different chamfer angles under different back rake angles were analysed. Table 7 shows the parameters of PDC cutter in the simulation.
Cutter size (mm)

Φ 13.44 × 8

Table 7 Parameters of PDC cutter
Chamfer length (mm)
Back rake angle (°)
15

0.3

18
20

5.2 Rock breaking force

Chamfer angle (°)
15
30
45
60
15
30
45
60
15
30
45
60

Fig. 12 shows the stress nephogram of rock and force curve of PDC cutter during rock breaking
process. As can be observed, the rock particles were broken under the shear and extrusion actions
of PDC cutter, an arc-shaped crushing pit was formed, the maximum stress on the rock was located
in the front surface of the cutting edge, which took the shape of an arc belt. When the rock breaking
process was stable, the cutting force, axial force and lateral force all fluctuated with time, and the
collapse failure of rock particles was reflected in the sudden decrease of cutting force. In this work,
all the analysis are based on the stable cutting stage, and the lateral force is relatively small, so it
is ignored.
Fig. 13 shows the force curves of PDC cutters with different chamfer angles under different
back rake angles. As can be observed, both the cutting force and the axial force increase with the
increase of back rake angle, which is consistent with the conclusion in reference 34 [34]. Under
all the back rake angles, the cutting force and axial force present the same changing law with the
increase of chamfer angle, that is, a trend of first increasing and then slightly decreasing, and the
decrease amplitude is very small (which is consistent with the conclusion obtained in Fig. 7).
Moreover, the larger the back rake angle is, the more significant the change is. When the back rake
angle is 15°, the cutting force and axial force show little difference with the chamfer angle. When
the chamfer angle is less than back rake angle, the influence of chamfer angle on cutting force and
axial force is more obvious, while when the chamfer angle is larger than back rake angle, the influence of chamfer angle on cutting force and axial force is very small, which can be almost ignored.
According to Fig. 12, the cutting force fluctuates up and down around a certain value in the
process of rock breaking, the more severe the fluctuation, the greater the possibility of vibration
damage to the PDC cutter. The cutting force difference coefficient is used to evaluate the severity
of the fluctuation, which is defined as the ratio between the standard deviation of cutting force
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Optimizing rock breaking performance: The influence of chamfer on polycrystalline diamond compact (PDC) cutters

and the mean value of cutting force. The smaller the cutting force difference coefficient is, the gentler the fluctuation of the cutting force is. Fig. 14 shows the cutting force difference coefficient
curves of PDC cutters with different chamfer angles under different back rake angles.
It can be seen from Fig. 14, when the chamfer angle is between 30° and 45°, the cutting force
difference coefficient is small, and the rock breaking process is relatively stable. Similar conclusions can be drawn at different back rake angles.
3,0E+03

Cutting force

2,5E+03

Axial force

Lateral force

Force / N

2,0E+03
1,5E+03
1,0E+03
5,0E+02

0,0E+00
0,0E+00

2000

2200

6,0E+04

Time / us

Acial force / N

Cutting force / N

1600

1800

1400

1600
1400
1200

0

10

20

back rake angle 15 degree
back rake angle 18 degree
back rake angle 20 degree
30

40

50

Chamfer angle / degree

60

8,0E+04

1,0E+05

back rake angle 15 degree
back rake angle 18 degree
back rake angle 20 degree

1800

2000

800

4,0E+04

Fig. 12 Stress nephogram of rock and force curve of PDC cutter

2400

1000

2,0E+04

1200
1000
800
600
400

70

0

10

20

30

40

50

Chamfer angle / degree

60

70

cutting force difference coefficient

Fig. 13 Force curves of PDC cutters with different chamfer angles under different back rake angles
0,1

0,09

back rake angle 15 degree

0,08

back rake angle 18 degree

0,07

back rake angle 20 degree

0,06
0,05
0,04
0,03
0,02
0,01
0

0

10

20

30

40

Chamfer angle / degree

50

60

70

Fig. 14 Cutting force difference coefficient curves of PDC cutters with different chamfer angles

5.3 Stress distribution characteristic of PDC cutter

Fig. 15 shows the stress distribution nephogram of PDC cutters with different chamfer angles at a
back rake angle of 25°. With the increase of chamfer angle, the maximum stress on the PDC cutter
decreases gradually, and the stress distribution area expands from the cutting edge to a larger
area. Specifically, when the chamfer angle is 15°, the surface stress on the PDC cutter is concentrated in the part where the lower edge of the PDC cutter contacts with the rock. At this time, the
stress on the lower of the cutting edge is concentrated, and the collapse failure of the cutting edge
is easy to occur. When the chamfer angle is increased to 60°, the surface stress on the PDC cutter
spreads to a larger area, which effectively protects the cutting edge, prevents the occurrence of
early cracks and spalls of the polycrystalline diamond layer.
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(a) Chamfer angle 15°

(b) Chamfer angle 30°

(c) Chamfer angle 45°

(d) Chamfer angle 60°

Fig. 15 Stress distribution nephogram of PDC cutters with different chamfer angles

6. Impact resistance test of PDC cutters with different chamfer angles
Drop hammer impact tests of PDC cutters with chamfer angles of 15°, 30°, 45° and 60° were carried out to further verify the influence of chamfer on the impact resistance of PDC cutter. Fig. 16
shows the photos of PDC cutters with different chamfer angles, and the cutter sizes are Φ 13.44 ×
8 mm.
The drop hammer impact device was used in the test (shown as Fig. 17), PDC cutters were fixed
on the punch head with a back rake angle of 20°. The mould steel was continuously impacted by
the PDC cutter with a given energy, the hardness of the mould steel was HRC58-62. Every time the
punch impacted, the mould steel rotated by 10°, so to ensure the PDC cutter can impact the undamaged mould steel plane every time.
The impact energy was loaded in a step-by-step energy mode. Starting from 10 J impact energy
for 10 times, the energy was increased step-by-step for 20 J impact energy for 10 times, 30 J impact energy for 10 times and 40 J impact energy for 10 times, the test was stopped once the polycrystalline diamond layer was damaged. The total impact energy that the PDC cutter can withstand was calculated according to the sum of the single impact energy. Two pieces of each chamfered cutter were tested, the average of the two test results was taken as the final impact resistance energy of the chamfered cutter. Table 8 shows the cumulative average impact energy of
the PDC cutter with different chamfer angles.

(a) Chamfer angle 15°

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(b) Chamfer angle 30°

(c) Chamfer angle 45°

Fig. 16 Photos of PDC cutters with different chamfer angles

(d) Chamfer angle 60°

Fig. 17 Photo of drop hammer impact device
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Optimizing rock breaking performance: The influence of chamfer on polycrystalline diamond compact (PDC) cutters

PDC cutter
Chamfer angle
15°
Chamfer angle
30°
Chamfer angle
45°
Chamfer angle
60°

Table 8 Impact test data of PDC cutters
Number of impact
Cumulative average impact energy/ J
10J
20J
30J
40J
10

10

10

10

10
10
10
10
10
10

2

/

10

10

1

10

9

/

10

1

10

10

2

10

10

10

10
10

10
10

5
6

Damage morphology

500
605
740
920

According to Table 8, when the impact energy was less than 20 J, all the PDC cutters were intact.
When the impact energy was increased to 30 J, the whole polycrystalline diamond layer collapsed
on one PDC cutter with chamfer angle of 15°, and the side edge of polycrystalline diamond layer
broke on one PDC cutter with chamfer angle of 30°. When the impact energy was increased to 40
J, the main damage forms of all the PDC cutters were the whole collapsing of polycrystalline diamond layer, the upper surface of polycrystalline diamond layer cracked on one PDC cutter with
chamfer angle of 45°, and the side edge of polycrystalline diamond layer broke on one PDC cutter
with chamfer angle of 60°. Considering the impact energy and damage morphology comprehensively, the impact resistance of PDC cutters with chamfer angle of 45° and 60° were superior, and
this was in good agreement with the simulation results.

7. Conclusion

In this study, theoretical model of cutting force considering chamfer was established, the primary
and secondary relationship of four factors – back rake angle, depth of cut, chamfer angle and
chamfer length – on the force of PDC cutter was analysed through pseudo-level orthogonal level
test. The depth of cut played a major role in the resultant force on front surface and the PDC cutter,
followed by the back rake angle. The chamfer angle showed the most significant effect on the resultant force on chamfered surface.
The influence of the chamfer parameters on the force of PDC cutter was analysed. When the
chamfer angle was small (chamfer angle not greater than back rake angle), the chamfer angle and
chamfer length appeared almost no influence on the resultant force on both front surface and
chamfered surface of PDC cutter, while the resultant force of PDC cutter increased significantly
with the increase of chamfer angle. When chamfer angle was greater than back rake angle, with
the increase of chamfer angle and chamfer length, the resultant force on the front surface decreased linearly, while the resultant force on the chamfered surface increased approximately linearly; for the resultant force of PDC cutter, it increased first and then decreased slightly with the
increase of chamfer angle, also it increased linearly with the increase of chamfer length.
Numerical simulation based on SPH method was carried out to analyse the rock breaking characteristics of PDC cutters with different chamfer angles. With the increase of chamfer angle, both
cutting force and axial force increased first and then decreased slightly, also the decrease amplitude was very small. When the chamfer angle was less than back rake angle, the influence of chamfer angle on cutting force and axial force was more obvious. When the chamfer angle was between
30° and 45°, the rock breaking process was relatively stable. With the increase of chamfer angle,
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Ju

the maximum stress on the PDC cutter decreased gradually, and the distribution area of stress on
the PDC cutter was wider, which was beneficial to protect the cutting edge, and prevent the occurrence of early cracks and spalls in the polycrystalline diamond layer of PDC cutter. The same
results were obtained through drop hammer impact tests.
This work can provide a theoretical basis for the optimal design of chamfer parameters, future
research will focus on studying the rock breaking effect of chamfered PDC cutters with different
sizes (Φ 16 mm, Φ 19 mm), aiming to offer more comprehensive and perfect guidance for the
optimization design of chamfer parameters. At the same time, field drilling test of PDC bit with
different chamfered PDC cutters can be carried out, combined with the analysis of micro-morphology, the micro-mechanism of the chamfer on improving the impact resistance of the PDC cutter
can be further revealed.

Acknowledgements

This research was supported by Key Research and Development project in Shaanxi Province, grant number 2022GY146, Science and Technology Innovation Fund Project of CCTEG Xi'an Research Institute, grant number 2021XAYJCQ02,
and the National Key Research and Development project for young scientists, grant number 2021YFC2900200. I appreciate their support very much.

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Advances in Production Engineering & Management

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Volume 18 | Number 4 | December 2023 | pp 434–446

Journal home: apem-journal.org

https://doi.org/10.14743/apem2023.4.483

Original scientific paper

Dynamic price competition market for retailers in the
context of consumer learning behavior and
supplier competition: Machine learning-enhanced
agent-based modeling and simulation
Deng, G.F.a,*
a

National Taichung University of Science and Technology, College of Languages, Applied Information & Japanese Program,
Taichung, Taiwan (R.O.C.)

ABSTRACT

ARTICLE INFO

This study analyzes the impact of consumer learning behavior and supplier
price competition on retailer price competition in a complex adaptive system.
Using machine Learning-enhanced agent-based modeling and simulation, the
study applies fuzzy logic and genetic algorithms to model price decisions, and
reinforcement learning and swarm intelligence to model consumer behavior.
Simulations reveal that different learning behaviors result in different retailer
competition patterns, and that supplier price competition affects the strength
of retailer price competition. Simulation results demonstrate that consumer
learning behavior influences retailer competition, with self-learning consumers leading to higher-priced partnerships, and collective-learning consumers
leading to a shift in price competition among retailers. In contrast, perfect
rationality consumers result in low-price competition and the lowest average
margin and profit. Additionally, the competitive price behavior of suppliers
impacts retailers' price competition patterns, with supplier price competition
reducing retailer price competition in the perfect rationality consumer market and enhancing it in the self-learning and collective-learning consumer
markets, leading to lower average prices and profits for retailers. This study
presents a simulated market for price competition among suppliers, retailers,
and consumers that can be expanded by subsequent scholars to test related
hypotheses.

Keywords:
Pricing competitive model;
Complex adaptive system (CAS);
Agent-based modeling and
simulation (ABMS);
Machine learning (ML);
Genetic algorithms (GA);
Fuzzy logic (FL);
Reinforcement learning (RL);
Swarm intelligence (SW);
Consumer learning behavior
*Corresponding author:
raymalddeng@gmail.com
(Deng, G.F.)

Article history:
Received 10 April 2023
Revised 14 November 2023
Accepted 16 November 2023
Content from this work may be used under the terms of
the Creative Commons Attribution 4.0 International
Licence (CC BY 4.0). Any further distribution of this work
must maintain attribution to the author(s) and the title of
the work, journal citation and DOI.

1. Introduction
Competitive markets are often complex because they are the emergent result of many individual
agents (e.g., consumers, retailers, supplier) whose motivations and actions combine so that even
simple behavioral rules can result in surprising patterns [1, 2]. Agent-based modeling and simulation (ABMS) is a rich platform for studying complex evolving systems to test behavioral economics theory and bridge micro and macro models. Many studies are beginning to apply ABMS
to investigate how individual agents interact during the competitive process and achieve a balanced outcome from the perspective of a dynamic evolutionary game [3-8]. ABMS frameworks
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Dynamic price competition market for retailers in the context of consumer learning behavior and supplier competition: …

are applied to design artificial adaptive agents to simulate the decision-making and learning
behaviors of real-world individuals and their interactions with incomplete information and partially rational decisions. In this way, the payoffs and final equilibrium outcomes of individuals
can be observed over time to support and assist marketing strategies [9]. The advantage of
ABMS is that it can produce results similar to those of the analytic model under the same assumptions and can be further extended and relaxed to analyze dynamic simulation results in
more complex situations, test and develop theories, and provide strategic implications [10-14].
The earliest assumptions of the price-competitive model were to explore the equilibrium of
the final price-competitive situation when oligopolistic competitors sell homogeneous products
without regard to their capacity, when they have the same cost function, when demand is certain
and known, and when consumers choose only the lower-priced products [15]. Hunt’s general
theory of competition describes the complexity and evolutionary nature of competitive markets
from a different perspective [16]. The theoretical view is that competition in the market is
caused by market imbalances. Competitive behavior stems from the endogenous learning behavior of suppliers. Tay and Lusch applied ABMS to construct a producer competitive market and
observed the price competitive dynamics and equilibrium of suppliers. The results validate
Hunt’s general theory of competition [17].
The subsequent development of the price competition model includes a number of considerations, such as consumer different cognitive decisions [18], heterogeneous consumers with objective supplier preferences [19] and switching behavior [20], consumers’ sensitivity to price information [21], consumers’ demonstrated loyalty [22], consumers’ social network word-ofmouth [23], consumers’ different levels of learning ability [24], consumers’ ability to demonstrate strategic purchasing behavior etc. [25]. In summary, previous studies that have examined
the price competitive market from a dynamic perspective have focused on price competition
among suppliers and incorporated consumer learning behavior to construct an ABMS market to
observe the dynamic effects of learning behavior on overall market prices but have neglected the
role of retailers in the price competitive market [26].
In a price competitive market characterized by independent suppliers and retailers, the dynamics of price setting and adjustment are influenced by three main competitive forces. First,
there is the competition that occurs at the supplier level, as each supplier seeks to offer the most
attractive price for the product they produce relative to other suppliers in the market. Second,
there is the competition that occurs at the retailer level, where retailers seek to set the most
attractive prices for the set of goods they offer, taking into account the prices set by their competitors. Third, there is the vertical interaction competition that occurs between the suppliers
and retailers, where the two parties negotiate and adjust prices in response to each other's decisions and actions. Together, these three forces shape the competitive landscape of the market
and ultimately influence the price decisions of the firms involved [15, 27]. The price decisions of
retailers nowadays have a significant degree of influence on the market. The impact of retailer
competition on the price competitive market is an issue worth clarifying [14, 28].
In this study, ABMS is applied to construct a price competition simulation market involving
retailers, suppliers, and consumers, where each retailer, supplier, and consumer can establish
their behavioral decision rules as adaptive agents. Individuals will interact with each other to
maximize their rewards and be given learning behaviors to observe over time in a complex
adaptive system (CAS) simulation to understand the intricate competitive relationships between
retailers, suppliers, and consumers [29].
Evolutionary game theory has been applied to price competition, emulating the behavior of
human deductive reasoning and inductive reasoning, where suppliers and retailers have price
decision rules and learned behavior mechanisms that apply fuzzy logic (FL) to their price decisions and genetic algorithms (GA) to the price rules adjustment [27, 30]. In constructing consumer purchase learning behavior, reinforcement learning (RL) from psychology was used to
model self-purchase learning behavior [12, 30], while swarm intelligence from biology was used
to model collective learning consumers [32].
This study will answer the question: from a price competition perspective, is it possible to
observe how different consumer learning behavior and supplier price competition behavior affect
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the price competition process of retailers and the eventual price equilibrium outcome that may
result? This study will examine the following:

• The impact of three different consumer learning behaviors, namely perfect rationality, selflearning, and collective learning on retailers’ dynamic co-opetition strategies.
• The impact of the non-price competitive and price competitive behavior of suppliers on retailers’ dynamic co-opetition strategies.

2. Pricing competition model

The price competitive market in this study comprises a number of suppliers, retailers, and consumers. The suppliers produce a single product, and their price decisions are primarily setting
the wholesale price of that product. The suppliers’ products are homogeneous, and there is price
competition between suppliers. Retailers are responsible for selling the products produced by
their respective suppliers. The retailer’s pricing decisions primarily determine the margins of
the supplier’s products, which are also price-competitive with each other. The supplier’s wholesale price and the retailer’s margins are added together to form the retail price. There is still a
so-called price competition between suppliers and retailers. In this price-competitive interaction, suppliers and retailers make price decisions with the aim of maximizing their own profit. In
addition, they follow human deductive reasoning and inductive reasoning behavior, apply fuzzy
logic as the basis for their price decisions and genetic algorithms as a way of adjusting price
rules through empirical learning, resulting in an evolutionary game of price competition between retailers, between suppliers, and between suppliers and retailers.
In competitive markets, consumers engage in purchase learning behavior. In a market, the
price of each retailer is a form of incomplete information to the consumer. As a result, consumers judge which retailer they can get the best price from based on their past purchasing experience. Once in the retailer’s shop, the consumer can directly compare prices between suppliers
and choose the lower price. Consumers can use the rewards of this purchase to form an experience and adjust their choice of retailer for the next time through learned behavior. In addition,
consumers can model self-learning behavior about purchases through reinforcement learning in
psychology [12] and the learning behavior of a group of consumers through swarm intelligence
in biology [32].
Price competition means that suppliers will consider the wholesale prices of other suppliers
and margins of downstream retailers when making price decisions. Retailers will consider the
margins of other retailers and wholesale prices of upstream suppliers when making price decisions. No price competition means that there is peaceful co-existence between suppliers and between suppliers and retailers. As a result, suppliers’ price decisions remain fixed throughout the
simulation period. At the same time, the retailer’s price decisions are made without regard to the
supplier’s price.
2.1 Price competitive market

The price competitive market in this study consists of supplier 𝑆𝑆𝑖𝑖 (𝑖𝑖 = 1, … , 𝑖𝑖 ∗ ) that each produc𝑓𝑓
es its own supplier’s product and has the same fixed cost 𝑐𝑐𝑖𝑖 . Retailer 𝑅𝑅𝑗𝑗 (𝑗𝑗 = 1, … , 𝑗𝑗 ∗ ) sells each
𝑓𝑓
supplier 𝑆𝑆𝑖𝑖 ’s product with the same fixed cost 𝑐𝑐𝑗𝑗 . At each point in time 𝑡𝑡, supplier 𝑆𝑆𝑖𝑖 determines
the wholesale price 𝑤𝑤𝑖𝑖 for the product. The 𝑤𝑤𝑖𝑖 set for the period applies to all retailers 𝑅𝑅𝑗𝑗 (𝑗𝑗 =
1, … , 𝑗𝑗 ∗ ). Next, retailer 𝑅𝑅𝑗𝑗 determines the margin 𝑚𝑚j for that product. The 𝑚𝑚𝑗𝑗 set for the period
applies to all suppliers 𝑆𝑆𝑖𝑖 (𝑖𝑖 = 1, … , 𝑖𝑖 ∗ ). The sum of 𝑚𝑚𝑗𝑗 and 𝑤𝑤𝑖𝑖𝑖𝑖 determines the retail price of 𝑆𝑆𝑖𝑖 at
𝑅𝑅𝑗𝑗 , 𝑅𝑅𝑖𝑖𝑖𝑖 . In the market, there are many heterogeneous consumers 𝐶𝐶𝑘𝑘 (𝑘𝑘 = 1, … , 𝑘𝑘 ∗ ). Consumers
have different price sensitivities, forgetting rate, degree of rationality and different propensities
towards retailers 𝜃𝜃𝑘𝑘,𝑗𝑗 (𝑗𝑗 = 1, … , 𝑗𝑗 ∗ ). The profit of 𝑆𝑆𝑖𝑖 (𝑖𝑖 = 1, … , 𝑖𝑖 ∗ ) and 𝑅𝑅𝑗𝑗 (𝑗𝑗 = 1, … , 𝑗𝑗 ∗ ), as well as
the accumulated capital profiti, profitj, can be calculated based on the demand 𝑞𝑞𝑖𝑖𝑖𝑖 for retailer 𝑅𝑅𝑗𝑗 ’s
product 𝑆𝑆𝑖𝑖 generated by the consumer’s purchase behavior.
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The price competitive market simulation works as follows: At each point in time 𝑡𝑡, the interactive steps described below are included:

Step 1: In the mode, the simulation time is 𝑡𝑡 ∗ , which contains multiple time points 𝑡𝑡. Each time
point 𝑡𝑡 contains multiple rounds 𝑟𝑟 ∗ . Each round 𝑟𝑟 contains multiple encounters 𝜀𝜀 ∗ .
Step 2: For each round 𝑟𝑟, the supplier and retailer select a price decision rule from a library of
price decision rules to conduct an 𝜀𝜀 ∗ encounters test.
Step 3: For each round 𝜀𝜀, the supplier refers to the previous price interaction with other competing suppliers and retailers and applies fuzzy logic to determine its own wholesale price,
and proposes its wholesale price to each retailer.
Step 4: The retailer takes into account its margin with other competing retailers and the supplier’s wholesale price in the previous round and applies fuzzy logic to determine its own
margin.
Step 5: Consumers will be able to evaluate and choose a retailer based on their purchasing decisions. The consumer does not have access to the current prices set by the retailer for
each supplier’s product but must enter the retailer’s shop in order to obtain price information. After deciding on a retailer, the consumer can directly choose the supplier with
the lower price, which in turn generates the demand 𝑞𝑞𝑖𝑖𝑖𝑖 for the supplier 𝑀𝑀𝑖𝑖 among the
retailers 𝑅𝑅𝑗𝑗 and the profit of each retailer and supplier in that transaction.
Step 6: Consumers use purchase learning behavior to adjust their purchase decisions based on
the rewards generated by this epsilon purchase decision. The learning behavior is modified using self- and collective learning.
Step 7: When 𝜀𝜀 < 𝜀𝜀 ∗ , return to step 3 and 𝜀𝜀 = 𝜀𝜀 + 1. Otherwise, determine if 𝑟𝑟 < 𝑟𝑟 ∗ is valid, and
if it is, go back to step 2 with 𝑟𝑟 = 𝑟𝑟 + 1, and if it is not, go to step 8.
Step 8: Retailers and suppliers evaluate the performance of their price decisions based on profit
and apply genetic algorithms to adjust their price decisions.
Step 9: Determine whether 𝑡𝑡 reaches the maximum simulation time 𝑡𝑡 ∗ , and stop if it does. Otherwise, go back to step 2, 𝑡𝑡 = 𝑡𝑡 + 1.

2.2 Price competitive behavior of supplier and retailer

The study explores the modeling of human deductive and inductive reasoning through the continuous adjustment of pricing strategies by suppliers and retailers using fuzzy decision rules,
with the ultimate goal of enhancing their survival. The FL-GA theoretical framework provides an
excellent opportunity to showcase the effectiveness of ABMS in this context. A schematic representation of the FL-GA architecture is illustrated in [17]. The process of fuzzy rules adjustment
for improved decision-making involves two methods: exploitation, which involves the recombination of existing genetic material in novel ways via crossover, and exploration, which involves
the adoption of new genetic material via mutation. Selection and reproduction are used to keep
and replicate successful decision-making approaches while discarding those that are ineffective.
Both exploitation and exploration strategies rely on the use of selection and reproduction to
optimize decision-making performance.
These rules are constantly evaluated and adjusted based on their predictive accuracy in forecasting market behavior. Successful rules are retained and acted upon while poorly performing
rules are discarded. The discarded rules are replaced with new hybrid rules generated from the
effective ones. Additionally, new rules are generated and tested in response to emerging market
information. This iterative process of learning and adaptation allows suppliers and retailers to
continually adapt to the constantly evolving market conditions. Please refer to the study by Tay
and Lusch for details on the FL-GA theoretical framework [17].
2.3 Consumer learning behavior

Each consumer is an artificial adaptive agent and will exhibit bounded rationality learning behavior in the face of incomplete information about prices. After each purchase, the individual
will calculate the price difference based on his or her past purchase experience or observation of
other people’s purchases as a reward for the purchase and adjust his or her propensity to buy

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from each retailer. The aim is to get the same product at a lower price the next time. The study
distinguishes three types of purchase learning behavior as follows:
Type I

Perfect Rationality: This type was designed as a control group. The consumer is provided with all price information prior to purchase. There is no price information search
cost for the consumer and the lowest price for each purchase, so there is no learning
behavior, and it can be used as a basis for comparison with other learning behaviors.

Type II Self-Learning: Because this type of consumer has the highest price search costs, they
will only adjust their purchase decisions after each purchase by comparing them to
their last purchase price. The consumer decides which retailer to go to this time to get a
better price based on his or her past purchase experience and decides which retailer to
go to this time. The self-learning behavior is based on reinforcement learning (RL) algorithms.

Type III Collective-Learning: Collective-Learning emphasizes a way of learning by comparing
purchasing experiences with those of others as a basis for future revision decisions.
Consumers use the swarm intelligence algorithm to compare prices between groups after making a purchase decision to see if the purchase is more expensive or cheaper, and
then adjust their propensity for the retailer to facilitate a higher return on their next
purchase decision.

2.3.1 Self learning behavior

RL means that actions that produced good results in the past will be reinforced, making them
more likely to be taken again in the future. Actions that produced bad results in the past will be
weakened, making them less likely to be taken again in the future. Action means that the consumer decides which retailer to buy from. The result is a price differential in the price paid for
the product purchased. With incomplete information, the current price is only known after the
consumer has chosen the retailer, and the price information for other retailers is still not available and must be recalled from past memory. When the return on price differential is perceptually positive, the likelihood of choosing that retailer next time increases, and vice versa. Such a
learning model is based entirely on one’s own experience and fully expresses the spirit of selfadaptive learning. Therefore, RL algorithms were used to model this self-learning behavior. The
relevant behavioral patterns and algorithm flow are as follows.
Step 1: Initial action

RL is the conversion of past experience into a propensity to act on that decision. The level of
propensity is seen as a preference for a particular Action. A consumer’s propensity is his or her
preference for a retailer 𝜃𝜃𝑗𝑗 . For example, a consumer who chooses between two retailers and
whose purchase action is either to shop at retailer 𝑅𝑅1 or to shop at retailer 𝑅𝑅2 , is given an 𝜃𝜃𝑗𝑗=1
and 𝜃𝜃𝑗𝑗=2 to indicate the consumer’s propensity for each of the two actions. Parameters are set
for each consumer, including their initial propensity to shop at the retailer, upper and lower
limits of propensity, rationality, price sensitivity, etc.
Step 2: Rules of selection

According to the rules of selection for retailers, the rules of selection are used to determine the
retailers that consumers decide to buy from at this stage, based on their current propensity for
each retailer. The higher an individual’s propensity for a particular action, the more likely that
individual is to choose that action. The choice of retailer is determined by the ratio of each retailer’s propensity to the total propensity. This ratio indicates the probability of each retailer
being selected. The formula for the choice of action rule is as follows:
𝑥𝑥𝑗𝑗 (𝑡𝑡) = �

438

𝑚𝑚𝑚𝑚𝑚𝑚

𝜃𝜃𝑗𝑗 (𝑡𝑡)

𝜃𝜃𝑗𝑗 (𝑡𝑡)
∑𝑛𝑛
𝑗𝑗=1� 𝜃𝜃𝑗𝑗(𝑡𝑡)�

if 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 < 𝛽𝛽

else

(1)

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The variable 𝑥𝑥𝑗𝑗 (𝑡𝑡) is the consumer’s choice of retailer 𝑗𝑗 at time point 𝑡𝑡, and 𝛽𝛽 is the degree of
rationality of the consumer’s purchase decision. A higher 𝛽𝛽 value indicates a higher probability
of choosing a retailer with a higher propensity to buy. Random variable 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟: between 0 and 1.
When 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 < 𝛽𝛽, the consumer directly chooses the retailer with the highest propensity; otherwise, the consumer is given a chance to choose the retailer with the highest propensity according to the ratio of each retailer’s propensity and then directly chooses the supplier with the lowest price.
Step 3: Calculation of reward

The price at which the consumer purchases at this point in time 𝑡𝑡 is compared to the price of the
previous purchase (𝑡𝑡 − 1) to calculate the return on the consumer’s choice of retailer for this
purchase. The formula for calculating the reward at time point 𝑡𝑡 is as follows.
𝑡𝑡
𝑡𝑡
𝑢𝑢𝑘𝑘,𝑗𝑗
= 𝛼𝛼𝑘𝑘 (𝑝𝑝𝑗𝑗𝑡𝑡−1 − 𝑝𝑝𝑘𝑘,𝑗𝑗
)

(2)

𝑡𝑡
The variables are defined as follows. 𝑢𝑢𝑘𝑘,𝑗𝑗
is the reward from retailer 𝑗𝑗 arising from consumer 𝑘𝑘’s
choice to buy from retailer 𝑗𝑗. 𝛼𝛼𝑘𝑘 is price sensitivity of the consumer 𝑘𝑘. The greater the price sen𝑡𝑡
sitivity, the greater the effect on the price differential. 𝑃𝑃𝑘𝑘,𝑗𝑗
is the price at which consumer 𝑘𝑘 buys
𝑡𝑡−1
𝑡𝑡
at this time at retailer 𝑗𝑗. 𝑝𝑝𝑗𝑗 is the price of the consumer 𝑘𝑘’s last purchase. 𝑝𝑝𝑗𝑗𝑡𝑡−1 − 𝑝𝑝𝑘𝑘,𝑗𝑗
is price
differential.

Step 4: The propensity to update decisions

The reward generated by a consumer action is the key factor that allows for the reinforcement of
learning. Each time a decision is made, RL updates the reward to a propensity for that strategy.
The remaining strategies that are not selected are not updated because of the learning effect.
Past propensities are partially lost over time. Therefore, through repeated purchases, each propensity will increase or decrease after each purchase, making it more or less likely to be chosen
next time. The propensity update method for each retailer is:
𝜃𝜃𝑠𝑠 (𝑡𝑡+1) = (1 − 𝛿𝛿)𝜃𝜃𝑠𝑠 (𝑡𝑡) + 𝑢𝑢𝑠𝑠 (𝑡𝑡)
𝜃𝜃𝑢𝑢 (𝑡𝑡 + 1) = (1 − 𝛿𝛿)𝜃𝜃𝑢𝑢 (𝑡𝑡)

(3)

𝜃𝜃𝑠𝑠 (𝑡𝑡) indicates the propensity to update for the selected retailer 𝑠𝑠 and 𝜃𝜃𝑢𝑢 (𝑡𝑡) is the propensity to
update for the unselected retailer 𝑢𝑢. 1 ≧ 𝛿𝛿 ≧ 0 is a memory parameter (recency), which indicates that past experiences or memories are forgotten over time. A larger 𝛿𝛿 value indicates a
greater emphasis on the most recent memory. Over time, consumers will have a relatively high
propensity to buy from retailers that generate high returns. The probability of being selected is
relatively high. In the end, the consumer’s action set tends to be simple, and learning tends to be
stable.
2.3.2 Collective learning behavior

Collective learning is generally conceptualized as a dynamic and cumulative process that results
in the production of knowledge. Such knowledge is institutionalized in the form of structures,
rules, routines, norms, discourse, and strategies that guide future action. This study uses Swarm
Intelligence to model consumer collective-learning behavior. Swarm Intelligence primarily mimics the process of a bird’s collective flight in search of a food location. Each individual modifies
the intensity of the flight by taking into account both their own past experience and the experience of other individuals in the flock to determine the direction of the next flight.
In collective purchasing learning behavior, consumers also decide which retailer to buy from
based on the rules of retailer choice. Under the retailer choice rule, consumers decide which
retailer to choose at this stage based on their current propensity for each retailer through the
choice action rule. The higher an individual’s propensity for a particular action, the more likely
that individual is to choose that action. The choice of retailer is determined by the ratio of each
retailer’s propensity to the total propensity. The ratio indicates how likely each retailer is to be
selected, where the rules of selection are the same as for self-learning. After each purchase, conAdvances in Production Engineering & Management 18(4) 2023

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sumers adjust their propensity to buy from a particular retailer by calculating rewards based on
their past purchasing experience and the experience of others, as expressed in the following
equation:
𝑡𝑡+1
𝑡𝑡−1
𝑡𝑡
𝑡𝑡
𝑢𝑢𝑘𝑘,𝑗𝑗
= 𝛼𝛼𝑘𝑘 × 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟()�(𝑝𝑝𝑘𝑘,𝑗𝑗
− 𝑝𝑝𝑘𝑘,𝑗𝑗
) + (𝑝𝑝𝑔𝑔𝑡𝑡 − 𝑝𝑝𝑘𝑘,𝑗𝑗
)�
(4)
𝑡𝑡+1
𝑡𝑡
𝑡𝑡+1
𝜃𝜃𝑘𝑘,𝑗𝑗 = 𝜃𝜃𝑘𝑘,𝑗𝑗 + 𝑢𝑢𝑘𝑘,𝑗𝑗

𝑡𝑡+1
𝑢𝑢𝑘𝑘,𝑗𝑗
is the reward generated by the consumer 𝑘𝑘’s choice of retailer 𝑖𝑖, which can be used as a
basis for adjusting the consumer’s propensity for each retailer. 𝛼𝛼𝑘𝑘 is the price sensitivity of consumer 𝑘𝑘. 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 is The degree of randomness indicates an extraneous environmental variable, a
𝑡𝑡−1
random number between 0 and 1. 𝑝𝑝𝑘𝑘,𝑗𝑗
is the retailer price at which consumer 𝑘𝑘 purchased in
𝑡𝑡
the previous purchase. 𝑝𝑝𝑘𝑘,𝑗𝑗 is the retailer price at which consumer 𝑘𝑘 purchased this purchase. 𝑝𝑝𝑔𝑔𝑡𝑡 is
𝑡𝑡
the retailer whose price is the lowest among the group at time point t. 𝜃𝜃𝑘𝑘,𝑗𝑗
is consumer 𝑘𝑘’ propen𝑡𝑡
𝑡𝑡
sity for various retailers 𝑗𝑗 at this stage. 𝑝𝑝𝑔𝑔 − 𝑝𝑝𝑘𝑘,𝑗𝑗 is consumer perception of the price difference.

3. Experimental results

3.1 Setup for price competitive market simulation
In this study, the Matlab programming language was used to implement the ABMS system. The
parameters of the simulation were set as shown in the table 1. The number of retailers 𝑛𝑛𝑟𝑟 is set
to 2, the number of suppliers 𝑛𝑛𝑚𝑚 is set to 2, and the number of consumers 𝑛𝑛𝑐𝑐 is set to 100. The
simulation was conducted 25 times in each market environment setting. Simulation time 𝑡𝑡 ∗ =
1000. A time point 𝑡𝑡 consists of 10 rounds (𝑟𝑟 ∗ ). Each retailer competes on 4 encounters (ε*) per
round, and the average margin, average profit, and the cumulative profit of the two retailers are
recorded for each time point 𝑡𝑡. Finally, the average of the 25 experiments is taken to produce the
experimental data. This study found that equilibrium was reached after 1000 simulation time 𝑡𝑡 ∗ .
Retailers’ price competition patterns are repeated. It was therefore decided that 1000 simulation time 𝑡𝑡 ∗ would be the time for the simulation.

Retailer/
Supplier

Consumer

System settings

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Table 1 Parameter settings for the price competitive market
Initial Settings for the Variables/Parameters
Setup Values/Range
Number of retailers 𝑛𝑛𝑟𝑟 / Number of suppliers 𝑛𝑛𝑚𝑚
2／2
𝑐𝑐
)
400
Fixed costs (𝑐𝑐𝑖𝑖 & 𝑗𝑗
Initial assets
1000
Initial wholesale price (𝑤𝑤𝑖𝑖𝑡𝑡=0 )
8
𝑟𝑟
(𝑡𝑡 = 0)
Initial margin 𝑚𝑚𝑖𝑖,𝑗𝑗
8
Upper and lower price limits
6-10
Number of rules for the library (𝑛𝑛𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 )
16
Number of semantic values of input variables under fuzzy rules
4
Number of semantic values of output variables under fuzzy rules
8
Mating rate
0.8
Mutation rate
0.2
Code
Binary encoding
Fitness function
Total sales
Mating operator
Two-point mating
Number of individual consumers (𝑛𝑛𝑐𝑐 )
100
Initial retailer propensity of consumers (𝜃𝜃𝑗𝑗 ): Normal distribution
𝑢𝑢� = 8 𝜎𝜎 = 0.5
Upper bound of consumer propensity to retailer
10
Lower bound of consumer propensity to retailer
2
Price sensitivity (𝛼𝛼)
1
Degree of rationality (𝛽𝛽)
0.8
Consumer forgetting rate (𝛿𝛿)
0.2
System implementation language
Matlab
Simulation time (𝑡𝑡 ∗ )
1000
Number of consumer purchases per cycle (𝑟𝑟 ∗ )
40
Number of experiments per market environment
25 times

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Dynamic price competition market for retailers in the context of consumer learning behavior and supplier competition: …

The retailer determines the unit margin for the product. The unit margin is summed with the
wholesale price of the product and becomes the selling price. The unit margin is set at a maximum of $10 and a minimum of $6 per unit. The retailer’s fixed cost per operation is (𝑐𝑐𝑖𝑖 ) $400.
𝑟𝑟
(𝑡𝑡 = 0) is set at $8. In a non-competitive situation, the wholesale price
The initial unit profit 𝑚𝑚𝑖𝑖,𝑗𝑗
is set at a fixed price of $8 per unit of product. In the case of a competitive supplier, the wholesale price is determined using fuzzy logic. The wholesale price per unit is also set at a maximum
of $10 and a minimum of $6 per unit. The supplier’s fixed cost (𝑐𝑐𝑖𝑖 ) per production run is $400.
The initial wholesale price 𝑤𝑤𝑖𝑖 (𝑡𝑡 = 0) is set at $8. The initial propensity of consumers toward
retailers presents 𝜃𝜃𝑖𝑖𝑘𝑘 a normal distribution, where 𝑢𝑢� = 8 and 𝜎𝜎 = 0.5. The lower bound of consumer propensity towards retailers ranges from 2 to 10. Consumer price sensitivity α is designed to be 1, and consumer forgetting rate 𝛿𝛿 is 0.2. The degree of rationality of consumers’
purchase decisions was set at 0.8.
3.2 Impact of price competition among retailers

In this study, three different consumer learning behaviors and two different suppliers
competitive behaviors were simulated to obtain a total of six market environment settings. Table 2 presents that the different consumer learning behavior and supplier price compe-

tition behavior affect the overall average margin, the overall cumulative profit, and the profit gap
at the retailer end.
First, we looked at the impact of three different consumer learning behaviors on retailers’ coopetition relationship. Market 1 shows that the overall average margin (6.7842) and cumulative
profit (-2,845,861) at the retailer end are the lowest when consumers demonstrate perfect rationality and suppliers do not engage in price competition. It can be inferred that retailers are
more likely to make higher profits through low-price strategies when consumers have immediate access to price information, therefore, more likely to engage in low-price competition among
retailers. As a result of low-price competition, the average cumulative profit of retailers is also
the lowest.
If we further analyze the price competition pattern in Fig. 1, we can see that during the price
competition between the two retailers, margin tend to fall quickly to the lowest price ($6) under
low price competition. After that, both parties have the opportunity to see the benefits of cooperation, and prices are gradually raised to $8, but only for a relatively short period of time (50
rounds). The retailer learns that it is more rewarding to set lower price than its rival. As a result,
the retailer will begin to compete at a lower price. The cycle goes on and on, with no equilibrium
of low-price convergence. The validity of the basic price competition model can be verified by
the results of the experiment in which consumers exhibit perfect rationality behavior, which is
the same as the Bertrand model of competition, as mentioned in the previous literature. Under
the same assumptions, it can be found that price competition among producers eventually approaches the equilibrium price of the lowest price.
Market 2 shows that the overall average margin price ($9.7216) and cumulative profit
($3,435,891) at the retailer end are highest when consumers exhibit self-learning behavior and
suppliers do not engage in price-competitive behavior. According to the company, consumers
who do not have immediate access to price information must rely on their past purchasing experience to make purchasing decisions. As a result, retailers set prices high for long periods of time
and only occasionally run low-price promotions to retain consumers and reduce price competition among retailers.
Further analysis of the price competition pattern in Fig. 2 shows that in the early stages of
price competition between the two retailers, those who adopt a low-price strategy are able to
make higher profits. The retailer with a high-price strategy continues to lose money but does not
compete at lower prices. After a certain period, the retailer with the low-price strategy finds that
higher prices will lead to higher profits and gradually raises its prices. Thereafter, the two retailers positioned their prices at a high level. Even when prices are adjusted, only sporadic lowprice fluctuations occur.
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Market
environment
1
2
3
4
5
6

Consumer
learning
behavior

Perfect
rationality
Self-learning
Collectivelearning
Perfect
rationality
Self-learning
Collectivelearning

Table 2 Experimental results
Competitive
Overall averOverall cumubehavior of
age margin
lative profit
suppliers
on the
on the
retailer side
retailer side

Overall cumulative profit
gap on the
retailer side

Chart coding

N/A

3,686,571

Fig. 2

N/A

6.7842

-2,845,861

N/A

9.6558

3,282,325

Yes
Yes
Yes

9.7216
7.1787
9.3794
9.3208

3,435,891

-2,060,160
2,633,278
2,472,801

1,932,295

Fig. 1

2,776,079

Fig. 3

4,451,600
7,582,923
6,780,293

Fig. 4
Fig. 5
Fig. 6

Market 3 shows that with consumers exhibiting collective-learning behavior and suppliers not
engaging in price-competitive behavior, the overall average margin ($9.6558) and cumulative
profit ($3,435,891) at the retailer end is slightly lower than self-learning behavior but not significantly different. This is a more interesting result. From a practical point of view, if consumers
adjust their purchasing propensity for retailers by making inter-group comparisons after making a purchase, this should result in more intense competition among retailers for lower prices.
Further analysis in Fig. 3 reveals that the difference in competitive patterns between Markets 2
and 3 is that: Whilst retailers’ prices are positioned at a high level, and there is sporadic and very
short-lived low-price competition (2-5 rounds) between retailers in self-learning, collectivelearning is likely to experience more frequent and short-lived medium-price competition (lasting 20-50 rounds). However, the modeling results do not suggest that long-term low-price competition will occur.
Comparing Fig. 2 and Fig. 3 on the comparison of consumer propensity and cumulative profit
patterns at the retailer end, it can be seen that there is little difference between the propensity of
consumers who can engage in self-learning behavior towards the two retailers and the change in
propensity is relatively gentle. This means that consumers show a more gradual adjustment
when their propensity for retailers changes. In contrast, consumers who are able to engage in
collective-learning have a high frequency of change and a steeper curve of change in their propensity for retailers. This means that consumers tend to adjust their propensity for both retailers in a wide range of ways, immediately, quickly, and frequently. Then we looked at the average
profit pattern of retailers per round. The self-learning behavior presented shows that retailers
take longer (more rounds) to make a lead transition per round. Collective-learning shows that
retailers often have short lead transitions. This is due to post-facto group price comparison by
consumers. Price information can still be spread over a short period of time. A one-time lowprice strategy can sometimes have an effect.
Next, we looked at the impact of the non-price competitive and price competitive behavior of
suppliers on retailers’ co-opetition relationship. The study found that there were similarities
with the non-competitive behavior of suppliers. For example, the retailer with the lowest average
price and cumulative profit ($7.18) was found in the case of perfect rationality behavior by consumers. Retailers presented an overall average margin price ($9.35 vs. $9.32) and cumulative
profit ($2,633,278 vs. $2,472,801) that were fairly similar under self-learning and collectivelearning, with retailers under self-learning still being slightly higher than retailers under collective-learning.
Market 4 shows that overall average margin and cumulative profit at the retailer end are
lower than the overall average margin and overall profit, given the perfect rationality of consumers and the competitive pricing practices of suppliers. Looking further at Markets 1 vs 4, it can be
seen that price competition by suppliers increases the overall average margin and overall cumulative profit at the retailer end. However, when comparing Markets 2 vs 5 and Markets 6 vs 3, it
can be seen that the overall average margin price and cumulative profit at the retailer end decreases when suppliers engage in price competition. The reason for this is inferred to be that if
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Dynamic price competition market for retailers in the context of consumer learning behavior and supplier competition: …

there is price competition from suppliers, this will result in more drastic price changes in the
market, which in turn will require longer purchase learning behavior on the part of consumers
to recognize the pattern of price competition among retailers. As a result, retailers would need
to increase the learning rewards for consumers by offering more low prices.
If we further analyze Figs. 1 and 4, we can see that if suppliers are competing on price, the
price competition pattern does not drop in the short term and remain locked at the lowest price
($6) for a long period of time. Instead, the price will fluctuate in a relatively gentle cycle between
$6 ~$8. This pattern is also reflected in the profit patterns of retailers in each round. Price competition from suppliers has resulted in less lead shifting and wider lead gap for retailers.
If we further analyze Figs. 2 and 5, we can see that if suppliers are competing on price, the
price competition pattern does not rise in the short term and remain locked at the higher price
($10) for a long period of time. Instead, the price will fluctuate in a relatively gentle cycle between $9 ~$10. In terms of consumer propensity and retailers’ profit patterns in each round,
consumer propensity to swap leadership becomes more frequent, the gap becomes wider, and
the speed of adjustment in propensity increases if suppliers engage in price competition. Retailers’ profit patterns show an increase in the likelihood of mutual and significant profit leads,
which is unlike the high price levels over time, fewer lead swaps, and less of a profit gap seen in
Market 2.
If we further analyze Figs. 3 and 6, we can see that if there is competition from suppliers, the
price competition pattern does not produce the same short-term increase in price competition
and long-term lock-in at high prices that would occur if suppliers did not compete. Instead, there
would be several short-term price competitions at prices falling between $8.50 and $9.

Fig. 1 Market 1: Non-competitive suppliers & perfect
rationality consumers

Fig. 3 Market 3: Non-competitive suppliers &
collective-learning consumers

Advances in Production Engineering & Management 18(4) 2023

Fig. 2 Market 2: Non-competitive suppliers &
self-learning behavior consumers

Fig. 4 Market 4: Competitive suppliers & perfect
rationality consumers

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Fig. 5 Market 5: Competitive suppliers &
self-learning consumers

Fig. 6 Market 6: Competitive suppliers &
collective-learning consumers

In terms of consumer propensity and the profit pattern of retailers in each round, the propensity and profit of retailers in Market 3 changes most sharply in the absence of competitive
behavior by suppliers. The adjustment in propensity is steep, and there is a high degree of frequent switching of leadership. Where suppliers compete, the change in propensity and profit
tends to moderate. Retailers with a profitable lead remain steadily and significantly ahead for a
period of time, but lead changes can still occur in very short periods of time.
Overall, there is competition for the lowest price at the retailer’s end as consumers display
perfect rationality behavior. In the case of self-learning behavior, if suppliers do not compete on
price, retailers will cooperate, and all adopt a high price strategy. The overall average price and
overall cumulative profit at the retailer end will be the highest, with the least amount of interchange between retailers and the least change in profit. However, if suppliers were to compete
on price, this would increase the intensity of market change. In the absence of price competition
by suppliers, the most dramatic changes in retailer propensity and profit patterns occur in the
absence of Collective-Learning behavior, although retailer price competition can result in high
prices, including large leads, the most frequent lead switching, and the steepest lead transitions.
Price competition by suppliers can help to mitigate the intensity of market change.

4. Conclusions

The contribution of this study is to present a simulated market for price competition involving
suppliers, retailers, and consumers and observe how different ‘consumer learning behavior’ and
supplier price competition behavior affect the price competition process of retailers and the eventual price equilibrium outcome.
For the retailer side, this study found that consumers engaged in self-learning behaviors that
were most beneficial to the retailer side of the competition. Collective-learning has the potential
to become a major learning method at a time when e-commerce and online communities are
becoming more prevalent. The impact of this could also bring the overall profit of the retailer
side closer to that of the consumer market where self-learning is present. It is important to note,
however, that if consumers engage in collective-learning, this will lead to increased competition
for the retailer as a whole. Therefore, individual retailers need to be cautious in dealing with
collective-learning consumers and be aware of market dynamics and be flexible in adjusting
their pricing strategies to avoid becoming loss-making retailers due to rigid pricing strategies.
In the face of price competition from suppliers, only in the control group (i.e., the perfect rationality consumer market) can the overall retailer side see an increase in profit. Other environments will result in lower profit. Therefore, in a market where consumers are likely to selflearning and collective-learning, it is more advantageous for retailers as a whole to stabilize the
wholesale price of suppliers. From a retailer’s perspective, where suppliers do not compete, if
consumers engage in self-learning behavior, this will result in the highest overall average price
for the retailer and maximize overall cumulative profit. Therefore, from a practical point of view,
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Dynamic price competition market for retailers in the context of consumer learning behavior and supplier competition: …

a retailer can achieve higher profits if it can keep supplier prices stable and induce consumers to
engage mainly in self-learning behavior.
The price competitive market proposed in this study also models the price competitive behavior of retailers under the basic assumptions of the Bertrand model (no competition among
suppliers, perfect rationality among consumers). The experimental results are not only identical
to the Bertrand model of competition but also show the dynamic process of price competition. In
the further, the price competitive market can be extended or expanded by subsequent scholars
to include specific factors to observe the results of the interaction and test related hypothesis.

Acknowledgement

The authors are grateful to the MOE Teaching Practice Research Program , ROC, for the partial financial support of this
work under the grant PGE1120918 .

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Advances in Production Engineering & Management

ISSN 1854-6250

Volume 18 | Number 4 | December 2023 | pp 447–461

Journal home: apem-journal.org

https://doi.org/10.14743/apem2023.4.484

Original scientific paper

IoT-based Deep Learning Neural Network (DLNN)
algorithm for voltage stability control and monitoring of
solar power generation
Shweta, R.a,*, Sivagnanam, S.b, Kumar, K.A.c
aAnnamalai

University, Tamilnadu, India
University, Tamilnadu, India
cBHEL, Tamilnadu, India
bAnnamalai

ABSTRACT

ARTICLE INFO

Today, Solar Photovoltaic (SPV) energy, an advancing and attractive clean
technology with zero carbon emissions, is widely used. It is crucial to pay
serious attention to the maintenance and application of Solar Power Generation (SPG) to harness it effectively. The design was more costly, and the automatic monitoring is not precise. The main objective of the work related to
designed and built up the Internet of Things (IoT) platform to monitor the
SPV Power Plants (SPVPP) to solve the issue. IoT platform designing and Data
Analytics (DA) are the two phases of the proposed methodology. For building
the IoT device in the IoT platform designing phase, diverse lower-cost sensors
with higher end-to-end delivery ratio, higher network lifetime, throughput,
residual energy, and better energy consumption are considered. Then, Sigfox
communication technology is employed at the Low-Power Wireless Area
Network (LPWAN) communication layer for lower-cost communication.
Therefore, in the DA phase, the sensor monitored values are evaluated. In the
analysis phase, which is the most significant part of the work, the input data
are first pre-processed to avoid errors. Next, to monitor the Energy Loss (EL),
the fault, and Potential Energy (PE), the solar features are extracted as of the
pre-processed data. The significance of utilizing the Transformation Search
centered Seagull Optimization (TSSO) algorithm, the significant features are
chosen as of the extracted features. Therefore, the computational time of the
solar monitoring has been decreased by the Feature Selection (FS). Next, the
features are input into the Gaussian Kernelized Deep Learning Neural Network (GKDLNN) algorithm, which predicts the faults, PE, and EL. In the experimental evaluation, solar generation is assessed based on Wind Speed (WS),
temperature, time, and Global Solar Radiation (GSR). The systems are satisfactory and produce more power during the time interval from 12:00 PM to 1:00
PM. The performance of the proposed method is evaluated based on performance metrics and compared with existing research techniques. When compared to these techniques, the proposed framework achieves superior results
with improved precision, accuracy, F-measure, and recall.

Keywords:
Solar photovoltaic (SPV);
Internet of things (IoT);
Data analytics;
Sigfox communication technology;
Low-power wireless area network
(LPWAN);
Energy loss;
Machine learning;
Transformation search centered
seagull optimization
algorithm (TSSO);
Gaussian kernelized deep learning
Neural Network (GKDLNN)
*Corresponding author:
shwetaraj434@gmail.com
(Shweta, R.)

Article history:
Received 3 April 2023
Revised 13 December 2023
Accepted 15 December 2023
Content from this work may be used under the terms of
the Creative Commons Attribution 4.0 International
Licence (CC BY 4.0). Any further distribution of this work
must maintain attribution to the author(s) and the title of
the work, journal citation and DOI.

1. Introduction
Owing to the economic and sustainable characteristics, the RE is drawing considerable attention
in research [1]. Solar energy has several benefits despite more RE being available. Here, via a
Photovoltaic System (PVS), electrical energy is acquired. The solar panels which can directly
convert sunlight or solar energy into electricity are included in this system [2]. Temperature,
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Shweta, Sivagnanam, Kumar

irradiance, voltage, and current are the fundamental parameters that affect the solar panel’s
efficiency. Owing to the disparities in the temperature, solar irradiance, weather conditions, together with several other factors, the power produced from SPV installations is vulnerable [3].
Therefore, a real-time solar monitoring system is necessary for augmenting the PV panel’s performance by contrasting it with the experimental outcome to start preventative action [4, 5]. The
PV Monitoring System (PVMS), which is based on both wired and wireless networks, is designed
to transmit parameters to a remote coordinator. This coordinator provides a web-based application for remote access [6]. Recently, the SPVMS has been incorporated with a wireless platform
that involves data acquisition as of diverse sensors along with nodes via wireless data transmission [7, 8]. In the project data’s practical verification and in the optimization of power plants’
conversion efficacy, data acquisition is extremely significant [9-12]. The IoT facilitates objects to
be recognized along with managed remotely over the prevailing system, developing new doors
for the physical world’s unadulterated integration into computer-centered systems and resulting
in enhanced accuracy, economic benefits, together with productivity [13, 14]. For transmitting
the data to the cloud, sensors and microcontrollers play an extremely vital part in relation to WiFi module [14, 15-17]. Amassing data, a network layer for the data’s transmission, a data processing layer for processing essential data, and finally, an application layer that operates as an
interface betwixt end devices and the network all are the four-layer IoT structure in the solarcentered IoT monitoring system [18]. Wireless sensor networks, GPS, 2G/3G/4G, GPRS, GSM,
WI-FI, microcontroller, RFID, microprocessor, etc. accomplish this IoT [19, 20]. On a range of
parameters like energy extracted, energy potential, historical generation analysis, fault identification, and related ELs, the PVMS’s intention is to offer accurate data [21]. Tracking the panel
voltage, temperature, current, and the solar system’s real data is to be synchronized as of time to
time are the numerous difficulties faced by the solar PVS’s remote monitoring [22]. There is an
assortment of advantages in utilizing IoT, like enhanced accuracy, efficacy, lesser human involvement, along with cost reduction, which is exhibited by numerous investigations [23, 24]. On
the monitor, the historical data and extracted energies are effortlessly presented. The manual
computation takes a longer time when faults, PE, and ELs are not manually computed. Therefore,
for SPVPP’s DA, this work exhibits the design together with the development of an IoT platform.

2. Literature review

Kandimalla and Kishore [25] designed a prototype for the implementation of a cost-efficient
technique based on IoT. This prototype checks an SPVPP for performance assessment using
open-source tools and resources such as Thingspeak and Arduino. Thingspeak, a Software as a
Service (SaaS) platform, provides web space for monitoring parameters. On Website designing
and maintenance, Thingspeak offered every service for free of cost, which saved many investments. Thus, for the ordinary people who set up rooftop solar plants, it concentrated on a lowercost system with a simple interface so that they could monitor the solar plants effortlessly, devoid of depending on service providers. Besides real-time monitoring, it facilitated Fault Detection (FD), preventive maintenance, along with plant’s historical analysis.
Almonacid-Olleros et al. [26] examined the output power generation’s evaluation by humancrafted characteristics with numerous temporal windows along with Deep Learning (DL) methodologies to acquire relative outcomes concerning the PV system’s analytical models regarding
error metrics along with learning time. In a PVS with IoT capacities created within the Opera
Digital Platform underneath the Univer Project, the surrounding data along with ground truth of
energy production had amassed, which was installed for two decades on the Campus of the University of Jaén (Spain). When analogized to the existing analytical model, the machine learning
models provided enhanced outcomes, with considerable dissimilarities in learning time together
with performance. For enriching the performance, the multiple temporal windows’ utilization
had exhibited as an appropriate tool to model temporal features.
Ramamurthi and Nadar [27] concentrated on an SPP’s interdisciplinary field in India’s southern region. It was integrated with the latest digital technology of IoT that bolstered to check realtime data of temperature, solar irradiation, panel voltage, peak power, current, together with the
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tilt angle of solar panels. Also, to manage the solar panel’s optimum inclination angle for augmenting PV power utilizing the Genetic Algorithm (GA). For enhancing the system’s tracking
behaviour, an integrated SPP with IoT had constructed with hardware setup along with it was
virtually tested. The performance of monitoring together with maintenance of power plant parameters with cost-efficiency had considerably improved by the IoT-centered control of SPPs
[28-31].
Shapsough et al. [32] proffered an IoT-centered architecture, which employs IoT hardware,
software, along with communication technologies to facilitate real-time monitoring together
with the management of SPVMS at huge scales. This system facilitated stakeholders to remotely
regulate and observe the PVSs along with to inspect the diverse environmental factor’s effects
like soiling, air quality, along with the weather. Regarding the network delay along with resource
consumption, the system was executed and examined. For wide-ranging real-time communication, Message Queueing Telemetry Transport (MQTT) was utilized. Since the average network
delay was lesser than 1 s, it was verified that the architecture was perfect for solar along with
smart grid monitoring systems. The assessment exhibited that the hardware consumes about 3 %
of the panel’s output for resource consumption, whilst the application as well used an extremely
small percentage of the CPU [33]. Since the lower-cost constrained edge devices deployed the
architecture in an excellent way where the incorporation of IoT-centered paradigm, effective
MQTT communication, along with lower-resources consumption made this system cost-efficient
and scalable.
Pulungan et al. [34] explained the design, manufacture along with testing of a single axis
online solar tracker monitoring system with the IoT. In real-time, the monitoring outcome’s data
retrieval occurred and was exhibited in the graphic data’s form. Next, the data acquired as of
sensors was linked to a microcontroller, and then linked to a WIFI module [8]. Samkria et al. [30]
presented IoT along with LabVIEW-centered automatic FD of 3 × 3 solar array systems for controlling and observing internet connectivity distantly [35-36]. In the PVS, to produce a panel
alert for damaged panels, the handling of the GUI indicator aided the monitoring system.
Through internet connectivity, node MCU in the receiver segment facilitated fault status’ transmission of PV arrays. Intended for visualizing the 3 × 3 PV array’s fault status, the IoT-centered
Blynk app was utilized. The Blynk’s dashboard visualized each array with the status.

3. Methodology

3.1 Design and development of IOT platform for analytics of solar power
A larger percentage of the energy produced as of renewable resources is shared by the SPV,
which is one of the well-known sustainable energy sources. Monitoring technologies have attained substantial focus corresponding to the performance advancement since the necessity for
solar energy has increased immensely in recent decades. Presently, via wireless data transmission, the SPV-MS has been incorporated with the wireless platform, which includes data acquisition as of several sensor nodes. Nevertheless, signal interference, larger data management, security, and long-range data transmission are the challenges that influence the performance of SPVMS. Therefore, with the DA of S-PVPP monitoring, the IoT platform has been developed here
regarding various sensor devices along with varied communication technologies. IoT platform
designing along with DA of SPV data was encompassed in this proposed methodology. Sensors,
solar panel model, Wi-Fi module, and authentication are explicated in the IoT platform design. In
DA, pre-processing, feature extraction, FS, and analysis steps are performed to assess the data.
By wielding the TSSO model, the FS is conducted. Then, the GKDLNN is employed for the assessment. Fig. 1 exhibits the proposed methodology’s block diagram.

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Fig. 1 A Block diagram for the proposed research methodology

3.2 Designing IoT platform

Initially, to obtain the SPV data, the IoT platform is designed. Firstly, the SPV model is structured
for the design. Next, to sense the values with higher end-to-end delivery ratio, lower cost, higher
network lifetime, residual energy, better energy consumption, throughput, along with better
performance, the sensor of IoT devices is chosen. Then, the authentication phase is executed.
The device is permitted to forward the data if it is an authorized device. The Wi-Fi module was
structured to transfer the data. In this, the Sigfox technology at the LPWAN communication layer
is employed for communication purposes. A greater performance was offered by the Sigfox
technology at a lower cost.
3.3 Solar PV model

The sunlight is converted into DC electrical energy with a set of parallel along with series PV
cells, which are included in the PV panels. At the PV modules’ terminals, the observation of the
characteristics desires the accumulation of extra parameters to the basic equation:
𝐿𝐿+𝑅𝑅𝑠𝑠 𝐻𝐻
�−
𝜆𝜆𝐿𝐿𝑡𝑡

𝐻𝐻 = 𝐻𝐻𝑃𝑃𝑃𝑃 − 𝐻𝐻𝑜𝑜 �exp �

1� −

𝐾𝐾+𝑅𝑅𝑠𝑠 𝐻𝐻
𝑅𝑅𝑝𝑝

(1)

The PV and saturation currents are specified as 𝐻𝐻𝑃𝑃𝑃𝑃 and 𝐻𝐻𝑜𝑜 , the diode’s ideality factor is signified as 𝜆𝜆, the series and parallels resistances are defined as 𝑅𝑅𝑠𝑠 and 𝑅𝑅𝑝𝑝 , and the thermal voltage is
proffered as 𝐿𝐿𝑡𝑡 . The 𝐻𝐻𝑃𝑃𝑃𝑃 is expressed as:
𝐻𝐻𝑃𝑃𝑃𝑃 = �𝐻𝐻𝑃𝑃𝑃𝑃,𝑛𝑛 + 𝑀𝑀𝑙𝑙 𝛥𝛥𝛥𝛥�

𝑄𝑄
𝑄𝑄𝑛𝑛

(2)

The light generated current is notated as 𝐻𝐻𝑃𝑃𝑃𝑃,𝑛𝑛 , the short circuit current-temperature coefficient
is proffered as 𝑀𝑀𝑙𝑙 , the temperature difference betwixt the actual and nominal values is illustrated as 𝐸𝐸, the solar irradiation and its nominal value are described as 𝑄𝑄 and 𝑄𝑄𝑛𝑛 . 𝐻𝐻𝑃𝑃𝑃𝑃,𝑛𝑛 in conjunction with 𝐻𝐻𝑜𝑜 is formulated as:
450

𝐻𝐻𝑃𝑃𝑃𝑃,𝑛𝑛 =

𝑅𝑅𝑝𝑝 +𝑅𝑅𝑠𝑠
𝑅𝑅𝑝𝑝

𝐻𝐻𝑠𝑠𝑠𝑠,𝑛𝑛

(3)

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𝐻𝐻𝑜𝑜 =

𝐻𝐻𝑠𝑠𝑠𝑠,𝑛𝑛 +𝑀𝑀𝑙𝑙 𝛥𝛥𝛥𝛥

(4)

𝐿𝐿
+𝑀𝑀 𝛥𝛥𝛥𝛥
exp� 𝑜𝑜𝑜𝑜,𝑛𝑛 𝑣𝑣 �−1
𝜆𝜆𝐿𝐿𝑡𝑡

The short circuit current in conjunction with open-circuit voltages underneath the nominal condition is represented as 𝐻𝐻𝑠𝑠𝑠𝑠,𝑛𝑛 , 𝐿𝐿𝑜𝑜𝑜𝑜,𝑛𝑛 , the open-circuit voltage per temperature coefficient is indicated as 𝑀𝑀𝑣𝑣 . To generate the desired output power, the larger PVPP contains numerous PV modules, which are linked in series-parallel. This PVPP is modelled as:
𝑂𝑂
𝐿𝐿+𝑅𝑅𝑠𝑠 � 𝑚𝑚 �𝐻𝐻

𝐻𝐻 = 𝑂𝑂𝑝𝑝 𝐻𝐻𝑃𝑃𝑃𝑃 − 𝑂𝑂𝑝𝑝 𝐻𝐻𝑜𝑜 �exp �

𝑂𝑂𝑝𝑝

𝑂𝑂𝑚𝑚 𝜆𝜆𝐿𝐿𝑡𝑡

� − 1� −

𝑂𝑂
𝐿𝐿+𝑅𝑅𝑠𝑠 � 𝑚𝑚�𝐻𝐻
𝑂𝑂𝑝𝑝
𝑂𝑂𝑚𝑚
𝑅𝑅𝑝𝑝 � �
𝑂𝑂𝑝𝑝

(5)

The number of series-connected modules in a string is specified as 𝑂𝑂𝑚𝑚 ; in addition, the number
of parallel-connected strings is signified as 𝑂𝑂𝑝𝑝 . The DC-DC Buck-boost converter is wielded in the
SPV module; similarly, to maximize the power, the Maximum Power Point Tracking (MPPT) is
utilized.
3.4 Measurement of sensors

Various sensors are utilized to gauge the PV model’s characteristics. In this, six varied sensors
are considered. The solar irradiance SP110, ambient temperature DS18B20, dust GP2Y1010AU0F,
humidity DHT22, WS anemometer, and the PV module surface temperature sensor PT100 are
read by the six sensors. The terms in the sensor device 𝑆𝑆𝑆𝑆 are denoted as:
𝑆𝑆𝑆𝑆 = {𝑇𝑇𝑇𝑇, 𝐼𝐼𝐼𝐼, 𝐻𝐻𝐻𝐻, 𝐷𝐷𝐷𝐷, 𝑊𝑊𝑊𝑊, 𝑆𝑆𝑆𝑆}

(6)

The temperature sensor device is specified as 𝑇𝑇𝑇𝑇, the irradiance device is signified as 𝐼𝐼𝐼𝐼, the humidity device is illustrated as 𝐻𝐻𝐻𝐻, the dust device is proffered as 𝐷𝐷𝐷𝐷, the wind device is defined
as 𝑊𝑊𝑊𝑊, and the surface temperature sensor device is symbolized as 𝑆𝑆𝑆𝑆.
Temperature sensor: The DS18B20 sensor is utilized to gauge the temperature. The DS18B20 is a
1-wire programmable Temperature sensor. With a decent accuracy of ± 5 °C, an extensive range
of temperature can be measured from -55 °C to +125 °C. The values in the hard along with soft
environments are gauged by this sensor.

Irradiance sensor: The SP110 sensor, which is a self-powered, analog sensor with a 0-400 mV
output, is employed to measure solar irradiance. An effortless connection of dataloggers along
with controllers is achieved by this sensor, which integrates a silicon-cell photodiode with a rugged, self-cleaning sensor housing design, together with a higher-quality cable terminating in a
pre-tinned pigtail.

Humidity sensor: The DHT22 sensor, which is an essential, lower-cost digital temperature along
with a humidity sensor, is employed for gauging the humidity. To gauge the surrounding air, a
capacitive humidity together with a thermistor is utilized; subsequently, it discharges a digital
signal on the data pin (analog input pins are not required). It has several benefits like having 0100 % humidity readings with 2-5 % accuracy. In addition, it is better for -40 °C to 80 °C temperature readings with ± 0.5 °C accuracy.
Dust sensor: A simple air monitoring module with onboard Sharp GP2Y1010AU0F is termed a
Dust Sensor. Fine particles, which are greater than 0.8 μm in diameter, even like cigarette smoke,
are discovered by this sensor. The sensor’s analog voltage output is linear with dust density. To
sustain an extensive range of power supplies, an embedded voltage boost circuit has been utilized by this module.

Wind speed sensor: For utilizing this module, the black wire is linked to the power along with the
signal ground, the brown wire is connected to 7-24VDC (9 V is utilized with success); in addition,
the analog voltage is measured on the blue wire. The voltage ranges from 0.4 V (0 m/s wind) up
to 2.0 V (for 32.4 m/s WS).
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PV surface temperature sensor: The PT100 sensor is employed for gauging the PV surface temperature. Therefore, the sensor is steady, fast, along with precise. The IP65 field, which is an
onsite 2-point calibration, has exceptional long-term stability.

3.5 Authentication

The sensor nodes must be registered on the server to transfer the data. The data as of the authorized sensor devices alone is received by the server. Initially, the sensor devices are registered. Details about sensor devices are provided in the registration phase. Therefore, in the database, the details are amassed. The device login to the server at the time device along with personal password is provided to transmit the data. In the verification phase, the device is permitted to broadcast the data to the server only if the details presented in the login phase are existed
in the database or else the device is specified as an unauthorized device. The authentication factors are signified as:
𝑅𝑅𝑅𝑅 → (𝐷𝐷𝐷𝐷, 𝑝𝑝𝑝𝑝𝑝𝑝, 𝐺𝐺𝐺𝐺)
(7)
𝐿𝐿𝐿𝐿 → (𝐷𝐷𝐷𝐷, 𝑝𝑝𝑝𝑝𝑝𝑝)

Allowed, if 𝐿𝐿𝐿𝐿 = 𝑅𝑅𝑅𝑅
𝑉𝑉𝑉𝑉 → �
Not allowed, otherwise

(8)

(9)

The authentication’s registration, verification, and verification process are notated as 𝑅𝑅𝑅𝑅 𝐿𝐿𝐿𝐿 and
𝑉𝑉𝑉𝑉, the sensor device’s device name and general characteristics are symbolized as 𝐷𝐷𝐷𝐷 and 𝐺𝐺𝐺𝐺,
and the password is exhibited as 𝑝𝑝𝑝𝑝𝑝𝑝.

Wi-Fi Module: Prior to amassing the computed data on an IoT server or cloud, it is processed by
the AT Mega 328 (ESP8266) utilizing a Wi-Fi module. The renowned Thingspeak IoT platform is
used for regular, weekly, and monthly data analysis.
Data storage: The sensed data are amassed in the cloud server with the aid of the Wi-Fi module.
Petabytes of data, which are required to be amassed, processed, along with evaluated, are produced by sensor networks. In storage services, an effectual role is performed by cloud computing.
Data Analytics: The data being gathered are evaluated here. On the monitor, the energy along
with historical power generation is displayed. The automatic monitoring process is desired since
the prediction fault, EL, and PE are not conducted effortlessly in the manual process. The procedures in automatic DA are explicated below.
Pre-processing: Firstly, the pre-processing is performed. Some missing values along with repeated values might occur in the sensor data. An error is caused at the end of the evaluation if there
occurs any missing along with repeated value in the dataset. The mean value of before together
with after values of the missing place is computed to replace the missing values. The mean computation is expressed as:
𝑇𝑇 +𝑇𝑇
𝑇𝑇𝑖𝑖 = 𝑖𝑖+1 2 𝑖𝑖−1
(10)

The after value as of the missing value is represented as 𝑇𝑇𝑖𝑖+1 and the before value is indicated as
𝑇𝑇𝑖𝑖−1 . By employing the Hadoop-MapReduce (MR) function, the repeated data are eliminated. For
writing applications, Hadoop-MR, which is a programming along with software framework, is
utilized. Here, an enormous amount of data is processed speedily on larger clusters of computed
nodes. Map phase and reduce phase are the two phases utilized here. Every single phase contains key-value pairs as input as well as output. Regarding the data structured (key, value) in
pairs, the map along with reduce functions of MR are proffered. A set of input key-value pairs is
taken during computation; thus, producing a set of output key-value pairs. The map along with
reduce functions in Hadoop-MR is formulated as:
𝑀𝑀𝑝𝑝 (𝛼𝛼1 , 𝑇𝑇1 ) → 𝑙𝑙𝑡𝑡 (𝛼𝛼2 , 𝑇𝑇2 )

452

𝐶𝐶𝑒𝑒 �𝛼𝛼2 , 𝑙𝑙𝑡𝑡 (𝑇𝑇2 )� → 𝑙𝑙𝑡𝑡 (𝑇𝑇2 )

(11)
(12)

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The map function is specified as 𝑀𝑀𝑝𝑝 , the key values are signified as 𝛼𝛼1 and 𝛼𝛼2 , the reduce function is defined as 𝐶𝐶𝑒𝑒 and the list of MR functions is proffered as 𝑙𝑙𝑡𝑡 .
3.6 Feature extraction

Subsequent to pre-processing, from the pre-processed data, the features are extracted. Relative
Humidity, Irradiance, Temperature, PV surface Temperature, WS, and Dust Accumulation are the
features being extracted. The feature terms are notated as:
𝑍𝑍𝑠𝑠 = {𝑧𝑧1 , 𝑧𝑧2 , 𝑧𝑧3 , . . . . . , 𝑧𝑧𝑛𝑛 }, or 𝑧𝑧𝑖𝑖 , 𝑖𝑖 = 1,2, . . . . , 𝑛𝑛

The feature set is indicated as 𝑍𝑍𝑠𝑠 and the n-number of features is depicted as 𝑧𝑧𝑛𝑛 .

3.7 Feature selection

(13)

Here, to alleviate the monitoring process’s computational time, as of the feature set being extracted, the significant features are chosen. The TSSO algorithm is deployed in the proposed
model. The migration along with the behaviours of seagulls in nature is inspired for the development of the Seagull Optimization (SO) algorithm. Seagulls are a type of seabird. They encompass various species that primarily feed on fish, insects, amphibians, reptiles, and earthworms.
This version is more direct and clearer. They are extremely intelligent and use their intelligence
to find food and attack prey. For instance, they use breadcrumbs to attract fish, and to catch
earthworms hidden underground, they mimic the sound of rain with their feet. A better performance was offered by the traditional SO algorithms. However, the issue of poor exploration
along with exploitation searching capacity is faced them. Thus, in this paper, the transformation
search is applied in the SO algorithm to address the aforementioned problem. The most crucial
behaviours of seagulls are migration along with attack. The source food for seagulls is proffered
as migration behaviours. The seagulls’ attack towards the migrating birds at sea is mentioned as
attack behaviour. Firstly, the populations are initialized. In this, the features being selected are
regarded as the population. By the transformation search function, the population is initialized.
It is represented as:
(14)
𝐴𝐴𝑖𝑖 = 𝜛𝜛(𝑙𝑙𝑙𝑙 + 𝑢𝑢𝑢𝑢) − (𝑧𝑧𝑖𝑖 − 𝑧𝑧̃𝑖𝑖 )

The initialized population set is proffered as 𝐴𝐴𝑖𝑖 , the elastic factor is notated as 𝜛𝜛, the upper and
lower bound values are denoted as 𝑙𝑙𝑙𝑙 and 𝑢𝑢𝑢𝑢. Next, the Fitness Value (FV) is computed. In this,
the accurate evaluation is pondered as a FV. Following initialization, the migration along with
the attacking behaviours is explicated individually.
(a) Migration behaviour

The algorithm imitates the seagulls’ movement from one place to another during the migration
process. Avoiding collisions, the best position’s direction, and reaching the best position are the
three criteria that must be satisfied by the seagulls. These criteria are explicated as follows. Initially, an extra variable 𝐹𝐹 is presented to avoid collisions. Thus, betwixt the neighbours, the collision is prevented. For the new Search Agent (SA) position, the new variable is utilized.
𝐿𝐿𝑂𝑂𝑠𝑠 = 𝐹𝐹 𝑊𝑊𝑠𝑠 (𝑡𝑡)

(15)

The SA’s location that doesn’t collide with other SAs is specified as 𝐿𝐿𝑂𝑂𝑠𝑠 , the SA’s current location
is indicated as 𝑊𝑊𝑠𝑠 , the current iteration time is denoted as 𝑡𝑡, and the SA’s mobile behaviour in the
provided search space is signified as 𝐹𝐹.
𝜁𝜁

𝐹𝐹 = 𝜁𝜁𝑐𝑐 − �𝑡𝑡 �𝑎𝑎𝑥𝑥𝑐𝑐 �� , 𝑡𝑡 = 0,1,2, . . . . , 𝑎𝑎𝑥𝑥𝑖𝑖𝑖𝑖𝑖𝑖
𝑖𝑖𝑖𝑖

(16)

To regulate the utilization frequency of variable 𝐹𝐹, the parameter 𝜁𝜁𝑐𝑐 is introduced. The variable
𝐹𝐹 is linearly reduced from 𝜁𝜁𝑐𝑐 to 0. After satisfying the condition of not colliding with other individuals, the seagull moves towards the best position. It is illustrated as:
𝑆𝑆𝐴𝐴𝑠𝑠 = 𝛿𝛿 �𝑊𝑊𝑏𝑏𝑏𝑏 (𝑡𝑡) − 𝑊𝑊𝑠𝑠 (𝑡𝑡)�

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The position of current location Search Agent is indicated SA W𝑠𝑠 towards the best-fit SA 𝑊𝑊𝑏𝑏 (that
is to say, the best seagull with a smaller FV) is represented as 𝑆𝑆𝑆𝑆𝑠𝑠. The behaviour of 𝛿𝛿, which is
accountable for appropriate balancing betwixt exploration and exploitation, is randomized. It is
gauged as:
𝛿𝛿 = 2 𝐹𝐹 2 𝑟𝑟

(18)

A random number which lies in the range of [0, 1] is specified as 𝑟𝑟. Finally, regarding the best SA,
the SA can update its position.
(19)
𝜑𝜑𝑠𝑠 = |𝐿𝐿𝑂𝑂𝑠𝑠 + 𝑆𝑆𝐴𝐴𝑠𝑠 |

The distance between the SA and best-fit SA (that is to say, best seagull whose FV is less) is denoted as 𝜑𝜑𝑠𝑠 .
(b) Attack behaviour

The seagulls make a spiral motion in the air whilst attacking the prey. This behaviours in 𝑢𝑢, 𝑣𝑣,
and 𝑥𝑥 planes are elucidated as:
𝑢𝑢′ = 𝑑𝑑 cos(𝑦𝑦)
(20)
(21)
𝑣𝑣 ′ = 𝑑𝑑 sin(y)
(22)
𝑥𝑥 ′ = 𝑑𝑑 𝑦𝑦
(23)
𝑑𝑑 = 𝑝𝑝 𝑒𝑒 𝑦𝑦𝑦𝑦

The radius of every single turn of the spiral is specified as 𝑑𝑑, a random number in the range
[0 ≤ 𝑦𝑦 ≤ 2𝜋𝜋] is defined as 𝑦𝑦, constants to proffer the spiral shape are notated as 𝑝𝑝 and 𝑞𝑞, and the
natural logarithm’s base is represented as 𝑒𝑒. The SA’s updated position is computed by wielding
the Eqs. 15-23.
𝑊𝑊𝑠𝑠 (𝑡𝑡) = �𝜑𝜑𝑠𝑠 𝑢𝑢′ 𝑣𝑣 ′ 𝑥𝑥 ′ � + 𝑊𝑊𝑏𝑏𝑏𝑏 (𝑡𝑡)
(24)
The best solution is saved together with the position of other SAs is updated by the 𝑊𝑊𝑠𝑠 (𝑡𝑡).

Fig. 2 Pseudo code for TSSO

Fig. 2 depicts the TSSO’s pseudo-code. Fig. 3 displays the flowchart for the TSSO.
The features being selected are modelled as:
𝑍𝑍�𝑠𝑠 = {𝑧𝑧̃1 , 𝑧𝑧̃2 , 𝑧𝑧̃3 , . . . . . , 𝑧𝑧̃𝑛𝑛 }, or 𝑧𝑧̃𝑖𝑖 , 𝑖𝑖 = 1,2, . . . . , 𝑛𝑛

The feature set is notated as 𝑍𝑍�𝑠𝑠 and the n-number of features is symbolized as 𝑧𝑧̃𝑛𝑛 .

454

(25)

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IoT-based Deep Learning Neural Network (DLNN) algorithm for voltage stability control and monitoring of solar power …

3.8 DLLN analysis

Fig 3 Flowchart for TSSO algorithm

Subsequent to FS, the features being selected are inputted to the analysis phase. Faults, PE, and
EL are forecasted by this evaluation. In numerous applications, the Artificial Neural Network
(ANN), which is an intelligent system, is utilized to address complex issues. Three layers are encompassed in the ANN structure. They are: (i) an Input Layer (IL), which includes the collected
data, (ii) an Output Layer (OL), which generates computed data, along with (iii) one or more
Hidden Layers (HLs), which are apt for linking the IL along with the OL. Collecting input and
generating output are the two functions performed by a neuron, which is a fundamental processing unit of a NN. To obtain an output, every single input is multiplied by connection weights;
the products and biases are added. Afterwards, they are passed via an Activation Function (AF).
However, for the deep analysis, a single HL is possessed by the traditional NN. A more number of
HLs is utilized in this proposed framework. A random weight initialization methodology is possessed by the baseline algorithm. In the output, a minor alteration in weight initialization can
lead to a critical change, thereby worsening the outcome. Regarding the Gaussian neighbourhood function, the weight values are altered in the proposed methodology to conquer the aforementioned complication. Additionally, an effective AF is desired for the deep analysis. Consequently, the kernelized AF is utilized here. With this alteration, the NN is renamed as the
GKDLNN algorithm. Fig. 4 displays the GKDLNN’s structure.
To start with, the selected features 𝑧𝑧̃𝑖𝑖 are inputted to the IL. Subsequently, the HL is provided
with the IL’s output. Regarding the weights, which is the sum of weighted synapse connections,
the output is computed by the HL. The input is filtered by eliminating the redundant data and is
forwarded to the subsequent HL for further processing. The HL process is mathematically formulated as:
(26)
ℎ𝑑𝑑𝑖𝑖 = 𝑖𝑖𝑏𝑏𝑠𝑠 + ∑𝑛𝑛𝑖𝑖=1 𝑧𝑧̃𝑖𝑖 . 𝜀𝜀𝑖𝑖

Here, the bias value is notated as 𝑖𝑖𝑏𝑏𝑠𝑠 , the weight value is symbolized as 𝜀𝜀𝑖𝑖 , and the input feature
is signified as 𝑧𝑧̃𝑖𝑖 . By employing the Gaussian neighbourhood function, the weight adjustment is
derived as:
(27)
𝜀𝜀𝑖𝑖+1 = 𝜀𝜀𝑖𝑖 + 𝑁𝑁𝑁𝑁𝑖𝑖 [𝑧𝑧̃𝑖𝑖 − 𝜀𝜀𝑖𝑖 ]
The Gaussian neighbourhood function is modelled as 𝑁𝑁𝑁𝑁𝑖𝑖 .

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Shweta, Sivagnanam, Kumar

Fig. 4 Structure of the DLNN algorithm

𝑁𝑁𝑁𝑁𝑖𝑖 = 𝜏𝜏𝑖𝑖 𝑒𝑒

�−

�𝑐𝑐𝑝𝑝𝑐𝑐 −𝑐𝑐𝑝𝑝𝑝𝑝 �
2𝜎𝜎2
𝑖𝑖

2

�

(28)

The learning rate is exhibited as 𝜏𝜏𝑖𝑖 , the coordinate positions of the current neuron and the previous neuron are depicted as 𝑐𝑐𝑝𝑝𝑐𝑐 and 𝑐𝑐𝑝𝑝𝑝𝑝 , and the neighbourhood radius’s width is proffered as
𝜎𝜎𝑖𝑖 . Consequently, the HL output is inputted to the OL, in which by pondering the kernelized function, the AF is obtained as:
(29)
𝑜𝑜𝑙𝑙𝑖𝑖 = 𝑖𝑖𝑏𝑏𝑠𝑠 + ∑𝑛𝑛𝑖𝑖=1 𝛾𝛾𝑖𝑖 (ℎ𝑑𝑑𝑖𝑖 𝜀𝜀𝑖𝑖 )

Herein, the output unit is specified as 𝑜𝑜𝑙𝑙𝑖𝑖 and the kernelized function is signified as 𝛾𝛾𝑖𝑖 . Finally,
the loss function is measured as:
(30)
𝑙𝑙𝑙𝑙 = (𝑡𝑡𝑡𝑡 − 𝑜𝑜𝑙𝑙𝑖𝑖 )

The loss function is exhibited as 𝑙𝑙𝑙𝑙, the output unit is illustrated as 𝑜𝑜𝑙𝑙𝑖𝑖 and the network’s target
output is depicted as 𝑡𝑡𝑡𝑡. The obtained loss is verified whether it is matched with the specified
threshold value or not. If it is matched, then the output is mentioned as the final output or else
the weight value is altered again.

4. Result and discussion

Here, the proposed research technique’s performance is examined. MATLAB has been employed
to execute the IoT platform design and SPVPP DA. The proposed model utilizes publicly obtainable data for the performance investigation.

4.1 Performance analysis of displayed values
The sensed values as of the sensor are examined here. Regarding temperature, time, GSR, along
with WS, the SPG is evaluated.
Regarding diverse time instants, Fig. 5 exhibits the SPG. From 6:00 to 19:00, it can perceive.
For SPP monitoring, it is one of the other vital parameters. The solar generation level is higher at
12:00 and 13:00. The SPG is lesser for the remaining time. Regarding the temperature variation,
Fig. 6 presents the SPG. It is noticed that as the temperature increases, the SPG also increases.
The generated solar power is 3500 kW if the temperature is 270 °C. The solar power is 1725 kW
if the temperature is 25.50 °C. Therefore, it exhibits that the temperature change offers a large
effect on SPG.
Regarding WS changes, Fig. 7 displays the SPG. For power plant monitoring, it is a significant
parameter. For the WS’s variation, the vast difference is not exhibited. SPG is varied slightly.
However, the higher SPG is acquired for the WS of 16.3 km/h. Fig. 8 exhibits the SPG regarding
GSR. It is observed that the SPG is increased as the average GSR is increased. The generated solar
power is 3900 kW when the average GSR is 5. Conversely, the solar power generated is 995 kW
when the average GSR is 1.5. Hence, on SPG, there is a larger effect of average GSR.
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IoT-based Deep Learning Neural Network (DLNN) algorithm for voltage stability control and monitoring of solar power …

Fig. 5 Analysis of SPG at different time instants

Fig. 7 Analysis of SPG with respect to WS

Fig. 6 Analysis of SPG with respect to temperature

Fig. 8 Power generation analysis with respect to the
global solar radiation changes

4.2 Performance analysis of solar power plant monitoring

Here, regarding precision, accuracy, F-measure, and recall metrics, the performance of the SPP
monitoring by the GKDLNN is evaluated with the prevailing Support Vector Machine (SVM),
ANN, Convolutional NN (CNN), and DLNN algorithms. Furthermore, regarding fitness vs. iteration analysis, the TSSO centered FS process’s performance is analogized with prevailing GA, Particle Swarm Optimization (PSO), Gray Wolf Optimization (GWO), and SO.
Regarding precision, accuracy, F-measure, and recall metrics, the GKDLNN centered SPP
monitoring along with the prevailing algorithm-centered monitoring system’s performance is
exhibited in Table 1. When analogized to the other prevailing algorithms and the GKDLNN algorithm, the prevailing SVM acquires extremely poor performance in this investigation. On considering the prevailing method, the GKDLNN algorithm acquires higher-level performance. Here,
the prevailing DLNN surpassed the other prevailing algorithms; however, it is also lesser than
the GKDLNN algorithm.
Table 1 Performance analysis of proposed GKDLNN with the existing algorithm based on performance metrics
Performance
Proposed
DLNN
CNN
ANN
SVM
Metrics
GKDLNN
Accuracy
96.5
91
88
83.2
79
Precision
94
89.3
85
81.3
78
Recall
94.5
90.3
86
81.5
79.2
F-Measure
94.24
89.79
85.49
81.39
78.59

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Fig. 9 Graphical representation of accuracy analysis for the proposed and existing algorithms

In Fig. 9, the pictorial depiction of accuracy investigation for the GKDLNN centered SPP monitoring is analogized with the prevailing algorithm-centered monitoring like SVM, ANN, CNN, and
DLNN algorithms. The common assessment metric for the predictions is the accuracy metric. It
is the number of correct predictions made as a ratio of every prediction done. Accuracy of 96.5 %
is acquired by the GKDLNN methodology, which is higher than the prevailing study techniques.
For DLNN, CNN, ANN, and SVM, the prevailing technique’s accuracy values are 91 %, 88 %, 83.2 %,
and 79 %, respectively.

(a)
(b)
Fig. 10 Demonstrate the performance of the proposed and existing algorithms in terms of
(a) precision and (b) recall metrics

Regarding (a) precision and (b) recall metrics, the GKDLNN method's performance investigation with the prevailing algorithms is displayed in Fig. 10. The two very vital model evaluation
metrics are recall and precision. Precision implies the percentage of the outcomes which is pertinent, recall implies the percentage of total pertinent outcomes precisely classified by the algorithm. The GKDLNN algorithm accomplished 94 % of precision and 94.5 % of recall; however,
the prevailing algorithms like DLNN, CNN, ANN, and SVM have 89.3 % and 90.3 %, 85 % and 86 %,
81.3 %, and 81.5 %, and 78 % and 79.2 %, respectively. When analogized to the prevailing study
techniques, the GKDLNN accomplishes higher performance in this analysis.

Fig. 11 F-Measure analysis

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Fig. 12 Fitness vs. iteration analysis
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IoT-based Deep Learning Neural Network (DLNN) algorithm for voltage stability control and monitoring of solar power …

The GKDLNN and prevailing technique’s F-measure are exhibited in Fig. 11. F-Measure is the
harmonic mean value of the precision and the recall metric. A higher value is also accomplished
by the F-measure. The GKDLNN method accomplishes better performance centered on the precision together with recall metrics, so the F-measure-centered analysis is better for the GKDLNN
method. Here, an F-measure of 94.24 % is acquired by the GKDLNN technique. The prevailing
algorithms’ F-measures are 89.79 % for DLNN, 85.49 % for CNN, 81.39 % for ANN, and 78.59 %
for SVM. Therefore, the GKDLNN technique’s performance surpassed the prevailing techniques
as per the analysis.
The TSSO algorithm’s fitness vs. iteration investigation with the prevailing GWO, GA, SO, and
PSO algorithms are displayed in Fig. 12. The TSSO algorithm’s FV is 85 when the iteration count
is 15; however, the prevailing techniques namely SO, GWO, PSO, and GA have 76, 73.89, 69.9, and
65.1, respectively. When analogized to the prevailing technique, the TSSO has a greater fitness
value for the other iterations also.

5. Conclusion

In this work DA of SPVPPs, an IoT platform’s design and development are presented. IoT design and DA are the two phases of this study method. The LPWAN communication layer is
employed with Sigfox communication technology, and diverse sensors are employed in the
IoT designing phase. The PE, EL, and the fault are evaluated in the DA phase. The GKDLNN
algorithm is primarily utilized for analytics. The publicly obtainable dataset is wielded for the
performance study. In the outcome analysis phase, the sensed values as of the sensor are
plotted, and regarding the recall, precision, F-measure, and accuracy, the GKDLNN algorithm’s
performance is verified with the prevailing DNN, ANN, DLNN, and SVM models. The GKDLNN
methodology acquires the greatest accuracy of 96.5 %. Furthermore, regarding fitness, the
TSSO’s performance is scrutinized with the prevailing GA, PSO, GWO, and SO. Better performance is achieved by the TSSO algorithm. Therefore, for the SPVPP monitoring, the recommended technique-centered monitoring and the chosen devices are highly cooperative. For
ameliorating the system’s performance, the proposed mechanism can be expanded encompassing more sensors, the latest devices, along with advanced algorithms.

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Advances in Production Engineering & Management

ISSN 1854-6250

Volume 18 | Number 4 | December 2023 | pp 462–474

Journal home: apem-journal.org

https://doi.org/10.14743/apem2023.4.485

Original scientific paper

Comparing Fault Tree Analysis methods combined with
Generalized Grey Relation Analysis: A new approach and
case study in the automotive industry
Shi, J.L.a, Lu, Z.C.a,*, Xu, H.H.a, Ren, M.M.a, Shu, F.L.a
aSchool

of Mechanical Engineering and Automation, Dalian Polytechnic University, Dalian, P.R. China

ABSTRACT

ARTICLE INFO

The failure modes of products gradually show a diversified trend with the precision and complexity of the product structure. The combination of fault tree
analysis and generalized grey relational analysis is widely used in the fault diagnosis of complex systems. In this study, we utilize a method that combines
fault tree analysis and generalized grey relational analysis. This method is applied to diagnose the Expansion Adhesive Debonding fault of automobile doors.
Then, we analyse and compare the differences in actual fault diagnosis results.
The comparison involves three analysis methods: Fault Tree Analysis combined with Absolute Grey Relation Analysis (F-AGRA), Fault Tree Analysis combined with Relative Grey Relation Analysis (F-RGRA), and Fault Tree Analysis
combined with Comprehensive Grey Relation Analysis (F-CGRA). Subsequently, we compare the findings with actual production results. This comparison allows us to discuss the differences between the three methods in the fault
diagnosis of complex systems. We also discuss the application occasions of
these methods. This study will provide a new method for fault analysis and
fault diagnosis in the actual production of the automobile manufacturing industry. This method can eliminate faults effectively and accurately and improve product quality and productivity.

Keywords:
Fault tree analysis (FTA);
Generalized Grey Relation Analysis
(GGRA);
Failure mode;
Fault diagnosis;
Complex system;
Fault Tree Analysis
combined with Absolute Grey
Relation Analysis (F-AGRA);
Fault Tree Analysis
combined with Relative Grey
Relation Analysis (F-RGRA);
Fault Tree Analysis combined with
Comprehensive Grey Relation
Analysis (C-GRA)
*Corresponding author:
577345879@qq.com
(Lu, Z.C.)

Article history:
Received 5 November 2023
Revised 19 December 2023
Accepted 21 December 2023
Content from this work may be used under the terms of
the Creative Commons Attribution 4.0 International Licence (CC BY 4.0). Any further distribution of this work
must maintain attribution to the author(s) and the title of
the work, journal citation and DOI.

1. Introduction
The diversified needs of customers and the increasingly improved automation and intelligent production have brought about product function diversity and fast-paced production. However, the
diagnosis and analysis of the complicated and diversified failure modes have led to new challenges. The products in the automobile manufacturing industry exhibit more complex failure
modes, more diverse fault causes, and a stronger correlation between fault causes than those in
other industries. The simple fault analysis method cannot solve the complex fault problem.
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Therefore, a fault analysis method for complex system analysis that can accurately, quickly, and
efficiently determine the cause of product failure and take corresponding troubleshooting
measures is urgently needed.
Complex systems are generally composed of various interacting components with complex relationships. This scenario also leads to system failures that typically involve multiple levels of organization and interrelated factors. As the system’s complexity increases, the interactions between components and the complexity of the hierarchical structure and interrelationships between components increase. Thus, the complexity of analysis increases with the increase in system complexity.
Fault Tree Analysis (FTA) is currently recognized as an effective technique for determining and
analysing failure modes of complex products or systems [1]. This kind of inverted-tree logical
cause-and-effect diagram is established for studying the system’s functional failure [2]. This analysis method has been widely used in the mechanical engineering [3], medicine [4], chemistry [5],
transportation [6], construction [7], aviation [8] and others for qualitative and quantitative fault
diagnoses. It plays an important role in system reliability analysis, safety analysis, and risk assessment [9]. However, the correlation between the basic and top events is ignored in the practical
application of FTA. The components’ potential and relative failure events in complex systems often exist simultaneously. Thus, the system failure and the resulting failure are usually uncertain
[10]. Moreover, the correlation information between them is lacking. Thus, inaccurate results easily occur when FTA is used to analyse complex systems. Therefore, the correlation between fault
events and system failures must be analysed by other effective methods combined with FTA. Grey
relational analysis (GRA) is one of the most widely used methods.
Based on the sample data of various factors, GRA uses grey relational degrees to describe the
strength, size, and order of the relationships between factors [11-12]. The basic idea is that if the
changing trends of the two factors are consistent, they can be considered relatively related. Otherwise, the correlation between them is small. Therefore, according to the basic idea of GRA, it can
be applied to comprehensive evaluation. The evaluated object and the ideal object are regarded
as two systems, and the advantages and disadvantages of the evaluated object can be obtained by
calculating the correlation degree of the two systems. The greater the correlation degree of the
two systems is, the better the evaluated object is. Alternatively, the evaluated object and the negative ideal object can be regarded as two systems. The greater the correlation between the two
systems is, the more similar the failure mode of the evaluated object is to the negative ideal solution [13].
The combination of FTA and GRA has been extensively researched in recent years. Pang calculated the relationship between basic event and top event failure rate trend in the electromagnetic
machining process by combining T-S FTA and GRA. In this way, they realized the targeted repair
of the fault system in the electromagnetic machining process and achieved the purpose of quality
control [14]. Tien optimized the process parameters by combining fuzzy GRA and particle swarm
optimization based on the Taguchi experiment, considerably improving the performance indicators of materials [15]. Applying this machine learning method to fault diagnosis enables effective
handling of large-scale, high-dimensional data and the discovery of fault features and trends hidden behind extensive datasets. This method is crucial for fault diagnosis in complex systems. It
can analyse and learn from data in real time. Thus, it has the potential for real-time fault diagnosis.
It can also provide objective, data-driven analyses, thereby reducing subjective interventions.
Wang et al. introduced grey theory and fuzzy theory into the FTA to analyse the wind turbine.
They proposed the gearbox transmission system based on grey theory and T-S fuzzy FTA to solve
uncertain failure probability [16]. Wang introduced the method of combining GRA and FTA in the
analysis and diagnosis of the loosening accident in a mine hoist. He analysed the possibility of
various fault modes in the FTA of excessive looseness in a mine hoist [17]. Zhang et al. proposed
a failure mode analysis method to improve the generalized grey relational fault tree for mine car
fall accidents. They used this method to determine and rank the possibility of occurrence of various mine car fall failure modes [18]. Chen et al. proposed the LSSC reliability diagnosis method
for hazardous materials by improving the generalized grey relation fault tree. This method can
quickly identify the accident factors of the key chemical logistics operation system [19].
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The works above show that FTA is mainly used for deterministic fault analysis, whereas GRA
is primarily used for uncertain fault analysis. Deterministic methods have low data requirements
and may be fast because they focus on deterministic faults. On the contrary, uncertain fault analysis methods are comprehensive and can handle the interrelationships among multiple faults in a
system. They usually have high data requirements to capture the relationships and uncertainties.
This comprehensiveness can enhance problem identification accuracy but may require increased
computational resources and time.
Therefore, combining FTA and GRA methods can leverage the advantages of deterministic and
nondeterministic methods for fault analysis and diagnosis. This approach allows for the rapid and
effective analysis of the interrelationships among multiple fault causes in system failures. Therefore, applying the combination of FTA and GRA to solve product failures encountered in industrial
product manufacturing has a great generalization value. Chen et al. determined the weak link in
the LSSC system for a hazardous chemical by combining generalized GRA with a fault tree [19].
This method can be extended to product failure analysis in industrial product manufacturing.
At present, the analysis methods combining generalized GRA with fault trees commonly include FTA combined with Absolute Grey Relation Analysis (called F-AGRA), FTA combined with
Relative Grey Relation Analysis (called F-RGRA), and FTA combined with Comprehensive Grey
Relation Analysis (called F-CGRA). These three methods have their respective advantages, disadvantages, and applications. However, the current study has no special research and focused discussion on the applicability and accuracy of the three methods in the failure mode of complex
products. As a result, the failure analysis using the combination of generalized GRA and FTA cannot quickly and accurately choose among the three methods. Thus, the production efficiency is
affected.
The present study established a fault tree by taking the typical automobile doors’ Expansion
Adhesive Debonding fault encountered in the production of G Group as the research object to make
up for the shortcomings of the studies above. The automobile doors’ Expansion Adhesive Debonding fault was also analysed based on F-AGRA, F-RGRA, and F-CGRA methods. We followed the actual problem-solving process in the production site to understand the causes and final solutions
of the failure in the actual production. In this study, the theoretical analysis of the three methods
was compared with that of the troubleshooting methods in the actual production. We studied and
analysed the reasons for the differences between the three methods to choose the suitable fault
analysis method. This study can provide an efficient and accurate idea of analysing the causes of
faults for enterprise fault improvement to solve system faults quickly.

2. Expansion Adhesive Debonding fault analysis

In the disassembly and self-inspection processes for the exit door assembly of a certain model of
G Group, the expansion adhesive filled between the inner and outer plates of the automobile doors
did not fit the surface of the sheet metal parts. Moreover, the expansion adhesive section was neat.
No attachment was found between the door plate and the door plate, and a large gap exists. According to the “Specification for Complete Destruction of Body in White Connection Quality” of the
group, the expansion adhesive should have adhesion traces on all the panels on both sides of the
split. Otherwise, the expansion adhesive may have insufficient filling, leading to noise caused by
the vibration during driving. The noise becomes increasingly obvious with the increase in speed,
seriously affecting user comfort. Therefore, the Expansion Adhesive Debonding fault of the automobile door is defined as a system fault, and the F-AGRA, F-RGRA, and F-CGRA methods are
adopted for analysis. The results are compared with the actual practice to analyse the reasons for
the differences and applicable occasions of the three methods.
2.1 Fault analysis process based on Fault Tree Analysis and Generalized Grey Relation Analysis

The Fault Tree and Grey Relation Analysis are used to analyse the problem. The specific process
is as Fig. 1.
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Determine
Failure Mode
Build a fault
tree analysis

Promotion and
application

Analyze the probability of
system failures and their
causes

Applicability of
analytical methods

System Fault Analysis
Based on F-AGRA Method
System Fault Analysis
Based on F-RGRA Method

Summarize
experience and
form a document

Yes
Comparison between
fault diagnosis results
and actual practice

System Fault Analysis
Based on F-GGRA Method

Is theory
consistent with
practice

Applicability of
analytical methods

No
Exploring the reasons
for differences

Fig. 1 Fault analysis process based on Fault Tree Analysis and Generalized Grey Relation Analysis

2.2 Constructing the Fault Tree Analysis construction

FreeFta software is used to construct the fault tree of automobile doors’ Expansion Adhesive
Debonding, as shown in Fig. 2. T represents the fault tree top event, Mi represents the intermediate
event, and Ni represents the basic event. In this case, the Expansion Adhesive Debonding fault of
the automobile door is defined as the top event, which is a system fault. The settings of top events,
intermediate events, and basic events are shown in Table 1.
2.3 Failure probability assessment

G Group established a fault analysis team comprising field technicians and engineers to solve the
Expansion Adhesive Debonding fault of automobile doors by evaluating the possible risk probability of the basic events of the fault tree. Moreover, Table 2 is the standard table of the problem level
and probability assessment found in the production process investigation, and Table 3 is the probability assessment table of the basic event N1-N15 of the Expansion Adhesive Debonding fault of
automobile doors.

Fig. 2 Fault Tree Analysis of Expansion Adhesive Debonding of automobile doors
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Event Type
Top event

Intermediate
events

Elementary
event

Table 1 Events represented by letters in Fault Tree Analysis
Event Code Fault event
T
Automobile doors Expansion Adhesive Debonding
M1
Excessive spacing between sheet metal parts
M2
Gluing equipment error
M3
The properties of the expansion adhesive do not meet the requirements
M4
Improper transportation and storage methods for assembly parts
M5
Part size deviation exceeding standard
M6
Excessive deformation of parts
M7
Part position deviation exceeds the standard
M8
Part deformation caused by the process of transferring parts
The properties of the expansion adhesive are not sufficient to meet the needs of sea
M9
transportation
N1
Dimensional deviation of outer door panel
N2
Dimensional deviation of door inner panel
N3
Dimensional deviation of reinforced beams
N4
Deformation of the reinforced beam caused by taking it
N5
Deformation of parts caused by handling after welding completion
N6
Overpressure of fixture pressure arm
N7
Poor consistency of solder joints leads to fluctuations in part position
N8
Excessive wear of fixture positioning pins
N9
Deviation of gluing point position
N10
Insufficient glue application
N11
Insufficient viscosity of expansion adhesive
N12
Deterioration of expanded adhesive due to prolonged exposure time
N13
Expansion adhesive deteriorates under high temperature and humidity conditions
N14
The coated expansion adhesive is shipped by sea to M country for baking
N15
Improper storage method for expansion adhesive during transportation

Table 2 Standard table of problem level and probability assessment found in production process investigation
Probability of failure occurrence
Level
Possibility of occurrence
Probability of occurrence value
one
great
0.9
two
more
0.8
three
secondary
0.6
four
less
0.2
five
very small
0.1

Table 3 Probability assessment table of basic event N1-N15 of Expansion Adhesive Debonding failure of automobile
Elementary event Ni
N1
N2
N3
N4
N5
N6
N7
N8
Probability P (Ni)
0.05
0.05
0.2
0.05
0.1
0.4
0.3
0.2
Elementary event Ni
N9
N10
N11
N12
N13
N14
N15
Probability P (Ni)
0.2
0.3
0.05
0.7
0.7
0.5
0.4

3 Analysis based on Fault Tree Analysis of Expansion Adhesive Debonding
3.1 Top event probability

If the basic events in the fault tree are independent of each other, then the probability of the top
event can be obtained from Eqs. 1 and 2 [17].
The probability of occurrence of the output event of the and gate structure：
𝑛𝑛

𝑃𝑃(𝑇𝑇) = � 𝑃𝑃𝑖𝑖
𝑖𝑖=1

The probability of occurrence of the output event of the or gate structure：
𝑛𝑛

𝑃𝑃(𝑇𝑇) = 1 − �(1 − 𝑃𝑃𝑖𝑖 )
𝑖𝑖=1

(1)
(2)

According to Eqs. 1 and 2, the probability of top event occurrence can be calculated as P (T) =
0.9750.
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3.2 Minimum cut set of fault tree analysis
If a set of basic events occur in the fault tree, the top event is bound to occur. The set of basic
events is called the cut set; the minimum cut set represents a minimum failure mode that causes
the top event of the fault tree to occur. The study of the minimum cut set helps identify the fault
tree’s weaknesses [18].
This study adopts the upward method to calculate the minimum cut sets in the fault tree of the
automobile doors Expansion Adhesive Debonding. The calculation process is shown in the Eq. 3
[19].
𝑇𝑇 = 𝑀𝑀1 + 𝑀𝑀2 + 𝑀𝑀3 + 𝑀𝑀4
= 𝑀𝑀5 + 𝑀𝑀6 + 𝑀𝑀7 + 𝑁𝑁9 + 𝑁𝑁10 + 𝑁𝑁11 + 𝑀𝑀9 + 𝑁𝑁14 + 𝑁𝑁15
(3)
= 𝑁𝑁1 𝑁𝑁2 𝑁𝑁3 + 𝑁𝑁6 𝑀𝑀8 + 𝑁𝑁7 𝑁𝑁8 + 𝑁𝑁9 + 𝑁𝑁10 + 𝑁𝑁11 + 𝑁𝑁12 + 𝑁𝑁13 + 𝑁𝑁14 + 𝑁𝑁15
= 𝑁𝑁1 𝑁𝑁2 𝑁𝑁3 + 𝑁𝑁6 𝑁𝑁4 + 𝑁𝑁6 𝑁𝑁5 + 𝑁𝑁7 𝑁𝑁8 + 𝑁𝑁9 + 𝑁𝑁10 + 𝑁𝑁11 + 𝑁𝑁12 + 𝑁𝑁13 + 𝑁𝑁14 + 𝑁𝑁15
It should be noted that 𝑁𝑁1 𝑁𝑁2 𝑁𝑁3 is a simplified expression of {𝑁𝑁1 , 𝑁𝑁2 , 𝑁𝑁3 }, representing a fault
event, and the rest are the same. According to Eq. 3, the corresponding minimum cut sets labelled
C1-C11 are: C1 = {N1, N2, N3}, C2 = {N4, N6}, C3 = {N5, N6}, C4 = {N7, N8}, C5 = {N9}, C6 = {N10}, C7 = {N11},
C8 = {N12}, C9 = {N13}, C10 = {N14}, C11 = {N15}.

3.3 Minimum cut sets probability

Based on the probability of each basic event, each minimum cut set probability is calculated using
Eq. 4 [20].
𝑃𝑃(𝐶𝐶𝑖𝑖 ) = � 𝑃𝑃�𝑥𝑥𝑗𝑗 �
𝑗𝑗∈𝐶𝐶𝑖𝑖

(4)

According to the calculation of Eq. 4, the minimum cut sets probability of the automobile doors
Expansion Adhesive Debonding fault is obtained, as shown in Table 4.
Table 4 Minimum cut sets probability of Expansion Adhesive Debonding fault in automobile doors
Min. cut set Ci
C1
C2
C3
C4
C5
C6
Probability P (Ci)
0.0005
0.02
0.04
0.06
0.2
0.3
Min. cut set Ci
C7
C8
C9
C10
C11
Probability P (Ci)
0.05
0.7
0.7
0.5
0.4
0.05

4. Generalized Grey Relation Analysis
4.1 Importance of basic event

The basic event importance indicates the degree of influence of the basic event on the top event
and is calculated by Eq. 5 [19].
∑𝑗𝑗∈𝐶𝐶𝑖𝑖 𝑃𝑃(𝐶𝐶𝑖𝑖 )
(5)
𝐼𝐼𝑗𝑗 =
𝑃𝑃(𝑇𝑇)
According to the calculation of Eq. 5, the importance of basic events of Expansion Adhesive
Debonding failure of automobile doors is shown in Table 5.
Table 5 Importance of basic events of Expansion Adhesive Debonding failure of automobile doors
Elementary event Ni
N1
N2
N3
N4
N5
N6
N7
N8
Importance Ij (N0)
0.0005
0.0005
0.0005
0.0205
0.0410
0.0615
0.0615
0.0615
Elementary event Ni
N9
N10
N11
N12
N13
N14
N15
Importance Ij (N0)
0.2051
0.3077
0.0513
0.7179
0.7179
0.5128
0.4103

4.2 Fault Analysis based on F-AGRA method

The F-AGRA method can reflect each factor’s degree of influence on the system. The fault tree
concept indicates that this degree of influence is the longitudinal difference between the initial
feature matrix of the minimum cut sets and the vector of patterns to be examined consisting of
the importance of the underlying events. It ultimately reflects the fault patterns’ degree of contribution represented by each minimum cut set to the top event fault.
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Xi represents the basic event probability of the i-th minimum cut set, 𝑋𝑋 = {𝑋𝑋1 , 𝑋𝑋2 , ⋯ , 𝑋𝑋𝑖𝑖 , ⋯,
𝑋𝑋𝑚𝑚 }𝑇𝑇 , 𝑋𝑋𝑖𝑖 = {𝑥𝑥𝑖𝑖 (1), 𝑥𝑥𝑖𝑖 (2), ⋯ , 𝑥𝑥𝑖𝑖 (𝑛𝑛)} , here m is the number of minimum cut sets, and n is the number of elementary events in the fault system. The characteristic matrix X consisting of subsequences can be expressed as the Eq. 6.
𝑋𝑋1
𝑥𝑥1 (1) 𝑥𝑥1 (2) ⋯ 𝑥𝑥1 (𝑛𝑛)
⎧ 𝑋𝑋 ⎫ ⎧ 𝑥𝑥 (1) 𝑥𝑥 (2) ⋯ 𝑥𝑥 (𝑛𝑛) ⎫
2
2
⎞⎪
⎪ 2 ⎪ ⎪⎛ 2
…
⋯
⋯
⋯
⋯
⎟
𝑋𝑋 =
= ⎜
(6)
⎨ 𝑋𝑋𝑖𝑖 ⎬ ⎨⎜ 𝑥𝑥𝑖𝑖 (1) 𝑥𝑥𝑖𝑖 (2) ⋯ 𝑥𝑥𝑖𝑖 (𝑛𝑛) ⎟⎬
⋯
⋯
⋯
⋯
⎪…⎪ ⎪
⎪
⎩𝑋𝑋𝑚𝑚 ⎭ ⎩⎝𝑥𝑥𝑚𝑚 (1) 𝑥𝑥𝑚𝑚 (2) ⋯ 𝑥𝑥𝑚𝑚 (𝑛𝑛)⎠⎭
In the characteristic matrix X, if a basic event is not in some minimum cut sets, then the vector
value for that position is 0. Otherwise, the vector value for that position is the probability of this
basic event. Therefore, the characteristic matrix of the basic event probability column can be expressed as the Eq. 7.
0.05 0.05 0.2 0
0 0
0
0
0 0.05 0 0.4
⎛
0
0
0
0 0.2 0.4
⎜
0
0
0
0
0 0
⎜
0
0
0
0
0 0
⎜
0
0
0
0
0 0
⎜
0
0
0
0
0 0
⎜
0
0
0
0 0
⎜ 0
⎜ 0
0
0
0
0 0
0
0
0
0
0 0
⎝ 0
0
0
0
0 0

0
0
0
0.3
0
0
0
0
0
0
0

0
0
0
0.2
0
0
0
0
0
0
0

0
0
0
0
0.2
0
0
0
0
0
0

0
0
0 0 0 0
0
0
0 0 0 0
⎞
0
0
0 0 0 0
⎟
0
0
0 0 0 0⎟
0
0
0 0 0 0⎟
0.3 0
0 0 0 0⎟
0 0.05 0 0 0 0 ⎟
0
0 0.7 0 0 0 ⎟
0
0
0 0.7 0 0 ⎟
0
0
0 0 0.5 0
0
0
0 0 0 0.4⎠

(7)

Define the 𝑋𝑋0 = {𝑥𝑥0 (1), 𝑥𝑥0 (2), … , 𝑥𝑥0 (𝑛𝑛)} as the primary sequence, the vector values corresponding to x0(1), x0(2), ..., x0(n) are the importance of each basic event respectively, and get pending pattern vector X0 = {0.0005, 0.0005, 0.0005, 0.0205, 0.0410, 0.0615, 0.0615, 0.0615, 0.2051,
0.3077, 0.0513, 0.7179, 0.7179, 0.5128, 0.4103}.
Assuming that 𝑋𝑋0 = {𝑥𝑥0 (1), 𝑥𝑥0 (2), … , 𝑥𝑥0 (𝑛𝑛)} is the primary sequence and X0 and Xi have the
same length, the absolute Grey Relation Degree 𝜀𝜀0𝑖𝑖 can be derived from Eq. 8 [21].
𝜀𝜀0𝑖𝑖 =

1 + |𝑆𝑆0 | + |𝑆𝑆𝑖𝑖 |
， 𝑖𝑖 = 1,2, ⋯ , 𝑛𝑛
1 + |𝑆𝑆0 | + |𝑆𝑆𝑖𝑖 | + |𝑆𝑆𝑖𝑖 − 𝑆𝑆0 |
𝑛𝑛−1

1
|𝑆𝑆0 | = �� 𝑥𝑥0 (𝑘𝑘) + 𝑥𝑥0 (𝑛𝑛)�
2
𝑘𝑘=1
𝑛𝑛−1

1
|𝑆𝑆𝑖𝑖 | = �� 𝑥𝑥𝑖𝑖 (𝑘𝑘) + 𝑥𝑥𝑖𝑖 (𝑛𝑛)�
2

⎫
⎪
⎪
⎪
⎪

⎬
⎪
𝑘𝑘=1
⎪
𝑛𝑛−1
⎪
1
|𝑆𝑆𝑖𝑖 − 𝑆𝑆0 | = ��(𝑥𝑥𝑖𝑖 (𝑘𝑘) − 𝑥𝑥0 (𝑘𝑘)) + (𝑥𝑥𝑖𝑖 (𝑛𝑛) − 𝑥𝑥0 (𝑛𝑛))�⎪
2
⎭
𝑘𝑘=1

(8)

∣S0∣ is the fluctuation amplitude sum of the primary feature series, ∣Si∣ is the fluctuation amplitude sum of the feature series composed of the factors in the system, and ∣Si-S0∣ is the difference
between the amplitude of fluctuations of factors in the main series and subseries. According to the
equation, the absolute grey relational degree of the minimum cut sets for the Expansion Adhesive
Debonding fault of automobile doors is shown in Table 6.
According to Table 6, we get the Importance Ranking of the minimum cut sets for Expansion
Adhesive Debonding fault of automobile doors based on F-AGRA method, as:
C8 = C9 > C3 > C4 = C10 > C2 > C11 > C1 = C6 > C5 > C7

Fig. 3 is histogram of Absolute Grey Relation of minimum cut sets.

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Table 6 Absolute Grey Relation Degree of minimum cut sets for Expansion Adhesive Debonding
fault of automobile doors
Min cut set Ci
C1
C2
C3
C4
C5
Absolute Grey Relation Degree 𝜀𝜀0𝑖𝑖
0.6154
0.6371
0.6587
0.6443
0.6010
Min cut set Ci
C7
C8
C9
C10
C11
Absolute Grey Relation Degree 𝜀𝜀0𝑖𝑖
0.5794
0.6731
0.6731
0.6443
0.6299

C6
0.6154

0,7
0,65
0,6
0,55
0,5

C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11

Fig. 3 Histogram of Absolute Grey Relation of minimum cut sets

4.3 Fault Analysis based on F-RGRA method

The F-RGRA method is used to analyse the differences in the longitudinal variation of the subsequence and the primary sequence. Under the fault tree concept, it can be understood as the degree
of conformity of the change in each value within each row of the initial feature matrix of the minimum cut sets with the change in the value of the vector of factors of the pattern to be examined
consisting of the importance of the basic events. It can eventually reflect the degree of influence
of the change in the failure mode of each minimum cut set on the change of the top event.
In the calculation of the relative grey relational degree, the initial value is generally calculated
by dividing the first term of the sequence, as shown in the Eq. 9.
𝑋𝑋𝑖𝑖′ = {𝑥𝑥𝑖𝑖′ (1), 𝑥𝑥𝑖𝑖′ (2), … , 𝑥𝑥𝑖𝑖′ (𝑛𝑛)} = �

𝑥𝑥𝑖𝑖 (1) 𝑥𝑥𝑖𝑖 (2)
𝑥𝑥𝑖𝑖 (𝑛𝑛)
,
,…,
�
𝑥𝑥𝑖𝑖 (1) 𝑥𝑥𝑖𝑖 (1)
𝑥𝑥𝑖𝑖 (1)

(9)

However, this approach becomes problematic if the first term in the data set is 0. Therefore,
the equation can express the definition of the initial value by replacing the first term with the
largest value in the data set (Eq. 10),
𝑋𝑋𝑖𝑖′ = {𝑥𝑥𝑖𝑖′ (1)，𝑥𝑥𝑖𝑖′ (2), … , 𝑥𝑥𝑖𝑖′ (𝑛𝑛)} = �

𝑥𝑥𝑖𝑖 (1) 𝑥𝑥𝑖𝑖 (2)
𝑥𝑥𝑖𝑖 (𝑛𝑛)
,
,…,
�
𝑥𝑥𝑖𝑖 (𝑑𝑑) 𝑥𝑥𝑖𝑖 (𝑑𝑑)
𝑥𝑥𝑖𝑖 (𝑑𝑑)

(10)

where xi (d) = max {xi}. The characteristic matrix of the initial value image can be expressed as Eq.
11.
𝑥𝑥1 (1) 𝑥𝑥1 (2)
⎧
⎪⎛ 𝑥𝑥1 (𝑑𝑑) 𝑥𝑥1 (𝑑𝑑)
𝑋𝑋1′
⎪
⎧ ′ ⎫ ⎪⎜ 𝑥𝑥2 (1) 𝑥𝑥2 (2)
𝑋𝑋
2
⎪ ⎪ ⎪⎜ 𝑥𝑥2 (𝑑𝑑) 𝑥𝑥2 (𝑑𝑑)
⎜
…
⋯
𝑋𝑋 =
= ⎜ ⋯
′
(1)
𝑥𝑥
𝑥𝑥
𝑋𝑋
𝑖𝑖 (2)
⎨ 𝑖𝑖 ⎬ ⎨⎜ 𝑖𝑖
…
⎪ ⎪ ⎪⎜ 𝑥𝑥𝑖𝑖 (𝑑𝑑) 𝑥𝑥𝑖𝑖 (𝑑𝑑)
′
⋯
⎩𝑋𝑋𝑚𝑚
⎭ ⎪⎜ ⋯
⎪ 𝑥𝑥 (1) 𝑥𝑥 (2)
𝑚𝑚
⎪ 𝑚𝑚
⎩⎝𝑥𝑥𝑚𝑚 (𝑑𝑑) 𝑥𝑥𝑚𝑚 (𝑑𝑑)

⋯
⋯

⋯

⋯

⋯

⋯

𝑥𝑥1 (𝑛𝑛)
⎫
𝑥𝑥1 (𝑑𝑑) ⎞⎪
𝑥𝑥2 (𝑛𝑛) ⎟⎪
⎪
𝑥𝑥2 (𝑑𝑑) ⎟⎪
⋯ ⎟
𝑥𝑥𝑖𝑖 (𝑛𝑛) ⎟⎬
⎟
𝑥𝑥𝑖𝑖 (𝑑𝑑) ⎟⎪
⋯ ⎟⎪
𝑥𝑥𝑚𝑚 (𝑛𝑛) ⎪
⎪
𝑥𝑥𝑚𝑚 (𝑑𝑑)⎠⎭

(11)

The characteristic matrix of the data after the initial image processing:

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469

Shi, Lu, Xu, Ren, Shu

0.25
0
⎛
0
⎜
⎜ 0
⎜ 0
⎜ 0
⎜ 0
⎜ 0
⎜ 0
0
⎝ 0

0.25
0
0
0
0
0
0
0
0
0
0

1
0
0
0
0
0
0
0
0
0
0

0
0.125
0
0
0
0
0
0
0
0
0

0
0
0.5
0
0
0
0
0
0
0
0

0
1
1
0
0
0
0
0
0
0
0

0
0
0
1
0
0
0
0
0
0
0

0
0
0
0.67
0
0
0
0
0
0
0

0
0
0
0
1
0
0
0
0
0
0

0
0
0
0
0
1
0
0
0
0
0

0
0
0
0
0
0
1
0
0
0
0

0
0
0
0
0
0
0
1
0
0
0

0
0
0
0
0
0
0
0
1
0
0

0
0
0
0
0
0
0
0
0
1
0

0
0
⎞
0
⎟
0⎟
0⎟
0⎟
0⎟
0⎟
0⎟
0
1⎠

(12)

Similarly, in the primary sequence 𝑋𝑋0′ = {𝑥𝑥0′ (1), 𝑥𝑥0′ (2), ⋯ , 𝑥𝑥0′ (𝑛𝑛)}, 𝑥𝑥0′ (1), 𝑥𝑥0′ (2), ⋯ , 𝑥𝑥0′ (𝑛𝑛) can be
obtained from Eq. 13.
𝑥𝑥0 (1) 𝑥𝑥0 (2)
𝑥𝑥0 (𝑛𝑛)
𝑋𝑋0′ = {𝑥𝑥0′ (1), 𝑥𝑥0′ (2), … , 𝑥𝑥0′ (𝑛𝑛)} = �
,
,…,
�
(13)
𝑥𝑥0 (𝑑𝑑) 𝑥𝑥0 (𝑑𝑑)
𝑥𝑥0 (𝑑𝑑)
The pending pattern vector 𝑋𝑋0′ = {0.0007, 0.0007, 0.0007, 0.0256, 0.0571, 0.0857, 0.0857,
0.0857, 0.2857, 0.4286, 0.0715, 1, 1, 0.7143, 0.5715}.
If 𝑋𝑋0′ and 𝑋𝑋𝑖𝑖′ are the initial value images of 𝑋𝑋0 and 𝑋𝑋𝑖𝑖 , then the absolute grey relational degree
between 𝑋𝑋0′ and 𝑋𝑋𝑖𝑖′ is the relative relation degree between 𝑋𝑋0 and 𝑋𝑋𝑖𝑖 , denoted as 𝑟𝑟0𝑖𝑖 ,which can be
obtained from the Eq. 14 [22].
1 + |𝑆𝑆0′ | + |𝑆𝑆𝑖𝑖′ |
𝑟𝑟0𝑖𝑖 =
， 𝑖𝑖 = 1,2, ⋯ , 𝑛𝑛 ⎫
1 + |𝑆𝑆0′ | + �𝑆𝑆𝑖𝑖′ � + �𝑆𝑆𝑖𝑖′ − 𝑆𝑆0′ �
⎪
𝑛𝑛−1
⎪
1 ′
′
′
⎪
|𝑆𝑆0 | = �� 𝑥𝑥0 (𝑘𝑘) + 𝑥𝑥0 (𝑛𝑛)�
2
⎪
𝑘𝑘=1
𝑛𝑛−1
(14)
1
⎬
|𝑆𝑆𝑖𝑖′ | = �� 𝑥𝑥𝑖𝑖′ (𝑘𝑘) + 𝑥𝑥𝑖𝑖′ (𝑛𝑛)�
⎪
2
𝑘𝑘=1
⎪
𝑛𝑛−1
⎪
1
|𝑆𝑆𝑖𝑖′ − 𝑆𝑆0′ | = ��(𝑥𝑥𝑖𝑖′ (𝑘𝑘) − 𝑥𝑥0′ (𝑘𝑘)) + (𝑥𝑥𝑖𝑖′ (𝑛𝑛) − 𝑥𝑥0′ (𝑛𝑛))�⎪
2
⎭

𝑘𝑘=1
′
|𝑆𝑆0 | is the sum of the rates of change of the volatility of the total characteristic sequence, |𝑆𝑆i′ | is

the sum of the rates of change of the volatility of the characteristic sequence of the factors in the
system, and |𝑆𝑆i′ − 𝑆𝑆0′ | is the sum of the differences in the rates of change in the volatility of the
corresponding factors in the primary sequence and the subseries. According to the calculation of
the Eq. 14, the relative grey relational degree of the minimum cut sets for the Expansion Adhesive
Debonding fault of automobile doors is shown in Table 7.
Table 7 Relative Grey Relation Degree of minimum cut sets for Expansion Adhesive Debonding
faults in automobile doors
Min cut set Ci
C1
C2
C3
C4
C5
Relative Grey Relation Degree 𝑟𝑟0𝑖𝑖
0.7161
0.6756
0.7161
0.7345
0.6621
Min cut set Ci
C7
C8
C9
C10
C11
Relative Grey Relation Degree𝑟𝑟0𝑖𝑖
0.6621
0.6621
0.6621
0.6621
0.6080

C6
0.6621

According to Table 7, we get the Importance Ranking of the minimum cut sets for Expansion
Adhesive Debonding fault of automobile doors based on F-RGRA method, as:
C4>C1=C3>C2=C5=C6=C7=C8=C9=C10>C11

Fig. 4 is histogram of Relative Grey Relation of minimum cut sets.

470

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Comparing fault tree analysis methods combined with generalized grey relation analysis: A new approach and case study …
0,8
0,6
0,4
0,2
0

C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11

Fig. 4 Histogram of Relative Grey Relation of minimum cut sets

4.4 Fault Analysis based on F-CGRA method

The F-CGRA method is a combination of the F-AGRA method and the F-RGRA method. It integrates
the longitudinal differences and rates of change of the selected indicators (called: primary sequence) and related factors (called: subseries). Finally, it realizes the relation analysis of the selected indicators and system-related factors.
In general, the Comprehensive Grey Relation Degree 𝜌𝜌0𝑖𝑖 is defined by combining the Absolute
Grey Relation Degree and the Relative Grey Relation Degree as shown in Eq. 15 [23].
𝜌𝜌0𝑖𝑖 = 𝜃𝜃𝑖𝑖 𝜀𝜀0𝑖𝑖 + (1 − 𝜃𝜃𝑖𝑖 )𝑟𝑟0𝑖𝑖

(15)

In the equation, 𝜃𝜃𝑖𝑖 is the distribution coefficient, which can be determined by the maximum
deviation method proposed by Sun et al. as in Eq. 16 [24].
𝐵𝐵𝜀𝜀𝜀𝜀
𝜃𝜃𝑖𝑖 =
(16)
𝐵𝐵𝜀𝜀𝜀𝜀 + 𝐵𝐵𝑟𝑟𝑟𝑟
Total deviation of Absolute Grey Relation Degree [25]:
𝑛𝑛

𝐵𝐵𝜀𝜀𝜀𝜀 = �|𝜀𝜀0𝑖𝑖 − 𝜀𝜀0𝑘𝑘 |

(17)

𝑘𝑘=1

Total deviation of Relative Grey Relation Degree [25]:
𝑛𝑛

𝐵𝐵𝑟𝑟𝑟𝑟 = �|𝑟𝑟0𝑖𝑖 − 𝑟𝑟0𝑘𝑘 |

(18)

𝑘𝑘=1

According to the calculation of Eqs. 16, 17, 18, the Comprehensive Grey Relation Degree of each
minimum cut set of automobile doors Expansion Adhesive Debonding fault is shown in Table 8.
Table 8 Comprehensive Grey Relation Degree of minimum cut sets for Expansion Adhesive Debonding
faults in automobile doors
Min cut set Ci
C1
C2
C3
C4
C5
C6
Total dispersion of Absolute Grey
0.3031
0.2524
0.3316
0.2596
0.4039
0.3031
Relation Degree 𝐵𝐵𝜀𝜀𝜀𝜀
Total dispersion of Relative Grey
0.4910
0.2885
0.4910
0.6566
0.2480
0.2480
Relation Degree 𝐵𝐵𝑟𝑟𝑟𝑟
partition coefficient 𝜃𝜃𝑖𝑖

Comprehensive Grey Relation
Degree 𝜌𝜌0𝑖𝑖
Min cut set Ci
Total dispersion of Absolute Grey
Relation Degree 𝐵𝐵𝜀𝜀𝜀𝜀
Total dispersion of Relative Grey
Relation Degree𝐵𝐵𝑟𝑟𝑟𝑟
partition coefficient 𝜃𝜃𝑖𝑖

Comprehensive Grey Relation
Degree 𝜌𝜌0𝑖𝑖

0.3817

0.4666

0.4031

0.2833

0.6196

C7

C8

C9

C10

C11

0.6738

0.6576

0.6930

0.7089

0.6242

0.5983

0.4342

0.4342

0.2596

0.2596

0.7070

0.6365

0.6365

0.5114

0.2610

0.2480
0.6036

0.2480
0.6691

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0.2480
0.6691

0.2480
0.6530

0.5500
0.6364

0.7349
0.6137

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Shi, Lu, Xu, Ren, Shu

According to Table 8, we get the Importance Ranking of the minimum cut sets for Expansion
Adhesive Debonding fault of automobile doors based on F-CGRA method, as:
C4 > C3 > C1 > C8 = C9 > C2 > C10 > C6 > C5 > C11 > C7

Fig. 5 is minimum cut sets Comprehensive Grey Relation Degree histogram.
0,7500
0,7000
0,6500
0,6000
0,5500

C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11

Fig. 5 Minimum cut sets Comprehensive Grey Relation Degree histogram

5. Result analysis and discussion

The FTA results of the three methods are compared. The results show that if a basic event occurs
in a different minimum cut set, the probability of occurrence of the top event is greatly reduced
after that event is excluded, even if the probability of occurrence of that event is low. This finding
is also proved by the practice in the actual production. Unlike the FTA method alone, the combination of FTA and GRA can effectively analyse the relationship between system fault characteristics and intuitively reflect the likelihood magnitude of each basic event fault in the system.
In the actual production, the fault analysis team of G Group found the Expansion Adhesive
Debonding fault in the automobile doors after 13 days through several checks and experiments,
such as measurement of sheet metal parts, adjustment of the spacing between sheet metal parts,
adjustment of glue application position and glue application amount, and high-temperature exposure of expansion adhesive placement, using inspection tools. The group found that the main
cause of the problem is the intermediate events M1 (the nature of expansion adhesive was not
enough to meet the shipping demand) and M3 (the spacing between the sheet metal parts was
excessive), which include the basic events N1-N8, N11-N13. After the improvement of the basic
events N5, N6, N7, N8, N12, N13, N14, the Expansion Adhesive Debonding fault of the automobile doors
was prevented from occurring.
The FTA in the third part shows that the basic events N5, N6, N7, N8, N12, N13 and N14 constitute
the basic event combination {N5, N6}, {N7, N8}, {N12}, {N13} and {N14} corresponding to the five failure modes C8, C9, C3, C4 and C10. This order is consistent with the F-AGRA method’s ranking of fault
levels in this study. C8 = C9 > C3 > C4 = C10 > C2 > C11 > C1 = C6 > C5 > C7 indicates a high degree of
conformity. Therefore, the F-AGRA method is the most consistent with the actual production situation. It is followed by the F-CGRA method and the F-RGRA method in the analysis of the Expansion Adhesive Debonding fault of automobile doors.
The comparison result of the three methods indicates that F-AGRA focuses on factors with a
great influence on the system. The contribution of the failure modes represented by each minimum cut set to the top event is calculated. F-RGRA obtains the initial value of the vector set 𝑋𝑋0′ −
′
by dividing the largest term in the vector max {xi}, which amplifies the influence of the failure
𝑋𝑋11
mode changes in each minimum cut set on the top event. The degree of influence on the change of
the top event is focused on reflecting the trend of factors and system changes. However, the direct
influence of the failure mode represented by its minimum cut sets on the top event is not as obvious as F-AGRA. F-CGRA calculates the total deviation of the two correlations’ balance reflects the
direct contribution and variable influence of the fault mode represented by the minimum cut set
to the top event.
For the direct contribution and change in the impact of the ranking, the unilateral accuracy and
embodiment of the trend in the combined method are not as obvious as those in the two separate
methods. Therefore, the method failure analysis between the above two methods is conducted.
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The characteristics and applicable scenarios of the three methods are summarized in Table 9.
Method
F-AGRA
F-RGRA
F-CGRA

Table 9 Characteristics and applicable occasions of F-AGRA, F-RGRA, and F-CGRA methods
Characteristic
Applicable occasions
Visually reflect the contribution of the fault
The fault tree analysis is mainly composed of or
modes represented by each minimum cut set to
gate structure events. and there are few and gate
the top event
structure events, the interaction between elementary event is less
The focus is on reflecting which factors are conThe fault tree analysis is mainly composed of and
sistent with the system change trend, but the digate structure events, or there are few or gate
rect impact of the fault mode represented by the
structure events
minimum cut set on the top event is not obvious
Balance reflects the direct contribution and
The fault tree analysis has many levels and is comchanging impact of the fault mode represented by plex, and there are and gate events close to the
the minimum cut set on the top event
root of the fault tree analysis

6. Conclusion

This study analyses the occasional and difficult-to-diagnose fault problems in the automotive industry. It uses three methods for fault analysis, explores the applicable scenarios of the three
methods, improves the efficiency of fault analysis, and provides the automotive manufacturing
industry with new ideas for fault analysis and fault diagnosis. Thus, the causes of product failure
can be efficiently targeted for improvement, and the efficiency of improvement can be improved.
This study can also be extended to the failure analysis of other complex products or systems by
providing an efficient, accurate, and scientific theoretical basis for handling accident priorities,
controlling the occurrence of accidents, and improving the reliability and safety of systems.
In this study, the method is only used to study the Expansion Adhesive Debonding fault of automobile doors. The method will be verified using other cases in the future. The analysis of the top
event relies on the accurate construction of the fault tree and the comprehensive investigation of
the cause of the failure. The probability of failure of each basic event must be accurately determined; otherwise, accurate conclusions may not be drawn. The fault diagnosis analysis method
based on fault tree and generalized grey theory remains complicated in the calculation process
when used for complex fault tree structures. Future research should focus on the sensitivity analysis of fault events while designing the program software using the three methods matching the
operation to solve the problem of complex operation.

Acknowledgment

The authors gratefully acknowledge the support of Liaoning Province Education Department Project (NO.JYTMS20230430: Research on environmental friendly mechanism of machinery and equipment remanufacturing process and
industrial chain under the background of "double carbon") and Zhejiang Geely Holding Group Co., LTD. The authors
would like to thank the editor and reviewers for their constructive suggestions of the paper.

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Advances in Production Engineering & Management

ISSN 1854-6250

Volume 18 | Number 4 | December 2023 | pp 475–485

Journal home: apem-journal.org

https://doi.org/10.14743/apem2023.4.486

Original scientific paper

Optimizing smart manufacturing systems using digital twin
Ojstersek, R.a,*, Javernik, A.a, Buchmeister, B.a
aUniversity

of Maribor, Faculty of Mechanical Engineering, Maribor, Slovenia

ABSTRACT

ARTICLE INFO

Presented paper investigates the application of digital twins for the optimisation of intelligent manufacturing systems and focuses on the comparison between simulation modelling results and real-world production conditions. A
digital twin was created in the Simio software environment using a data-driven
simulation model derived from a real-world production system. Running the
digital twin in real time, which was displayed graphically, facilitated the analysis of key parameters, including the number of finished products, average flow
time, workstation utilization and product quality. The discrepancies were attributed to the use of random distributions of input data in the dynamic digital
twin, as opposed to the long-term measurements and averages in the realworld system. Despite the limitations in the case study, the results underline
the financial justification and predictive capabilities of digital twins for optimising production systems. Real-time operation enables continuous evaluation
and tracking of parameters and offers high benefits for intelligent production
systems. The study emphasises the importance of accurate selection of input
data and warns that even small deviations can lead to inaccurate results. Finally, the paper highlights the role of digital twins in optimising production
systems and argues for careful consideration of input data. It highlights the importance of analysing real-world production systems and creating efficient
simulation models as a basis for digital twin solutions. The results encourage
extending the research to different types of production, from job shop to mass
production, in order to obtain a comprehensive optimisation perspective.

Keywords:
Smart manufacturing;
Digital twin;
Optimisation;
Simulation modelling;
Simio;
Case study

*Corresponding author:
robert.ojstersek@um.si
(Ojstersek, R.)

Article history:
Received 19 July 2023
Revised 17 December 2023
Accepted 20 December 2023
Content from this work may be used under the terms of
the Creative Commons Attribution 4.0 International Licence (CC BY 4.0). Any further distribution of this work
must maintain attribution to the author(s) and the title of
the work, journal citation and DOI.

1. Introduction
With the transition to Industry 4.0, digitization brings special challenges and concerns for production. One way to deal with these challenges is to use a digital twin. Digital Twin has emerged as a
central paradigm in the field of smart manufacturing and Industry 4.0, with profound implications
for production management, control frameworks and the broader landscape of modern manufacturing. The digital twin is not just a technological artefact, but also a multi-layered construct that
connects physical entities with their virtual counterparts. It encompasses various dimensions, including its connotation, its reference model, its applications, and the associated research questions [1]. In the context of complex product assembly shop floors, a digital twin-based smart production management and control is emerging. This framework uses the concept of the digital twin
to improve the agility and responsiveness of manufacturing processes. It provides a dynamic, realtime representation of the physical system that enables predictive maintenance, optimized resource allocation and adaptive production scheduling [2]. The interplay between digital twin and
smart manufacturing are explored and their importance for transforming traditional manufacturing processes is highlighted [3]. This approach enables real-time monitoring, data-driven deci475

Ojstersek, Javernik, Buchmeister

sion-making and improved operational efficiency [4]. The study focuses on optimising manufacturing takt time within a supply chain, with an emphasis on the strategic placement of a decoupling point. While the study provides a basis for improved operational efficiency, it is important
to discuss the assumptions and generalisability to different production environments. Applications range from predictive maintenance to quality control, highlighting the versatility and transformative potential of digital twin-driven smart manufacturing [5]. The correlation and comparison between digital twins and cyber–physical systems (CPS) form a critical discourse. Both concepts have similarities in bridging the physical and digital worlds. While CPS emphasizes the integration of computational algorithms with physical processes, digital twins extend this integration to a comprehensive virtual representation [6]. This study looks at the central role of planning
algorithms in smart manufacturing and addresses the associated challenges. The study not only
highlights the advances in algorithms, but also emphasises the need for adaptive algorithms that
can cope with dynamic production environments. Understanding their correlation helps to refine
the conceptual framework for smart manufacturing [7]. The fusion of digital twin and big data [8]
is explored in a 360-degree comparison. Big data analyses enrich the concept of the digital twin
by incorporating extensive data sets into the virtual representation, thus enabling improved analyses and well-founded decisions. The synergy between the digital twin and big data serves as a
cornerstone in the evolution towards smarter manufacturing processes [9]. In exploring the practical applications of digital twins in production logistics, this study presents a testbed that uses
real-time location data [10]. Although the study is promising, it provides ideas for thought on the
challenges of scalability and the need for further validation in complex production ecosystems
[11]. By examining the enabling technologies, challenges and open research areas related to digital twins, a comprehensive understanding of the current landscape is achieved. From advanced
sensor technologies to the intricacies of data interoperability, the path to realizing the full potential of digital twins is fraught with challenges. However, these challenges also present opportunities for further exploration and innovation that push the field into uncharted territories [12]. The
focus on dynamics in manufacturing presents a new paradigm [13]. This reference highlights the
transformative potential of the digital twin in redefining shop-floor operations by providing realtime insights and adaptive capabilities critical to managing the complexity of modern manufacturing [13]. By integrating the digital twin and improved bacterial foraging, job scheduling becomes more adaptable in dynamic production environments [14]. However, the limits of algorithmic scalability and generalisability of the proposed approach underline the need for further investigation [15]. Researchers look at the values digital twin brings to the modelling landscape,
navigates the challenges faced in implementation, and identifies enablers that are critical to successful modelling [16]. Anticipating future developments broadens the discourse. This study [17]
examines the enabling technologies and potential applications and provides a forward-looking
perspective on industrial development beyond Industry 4.0. The comprehensive review provides
an examination of the concepts, technologies, and industrial applications of the digital twin [18].
This synthesis enriches our understanding by providing a panoramic view of the historical development, current technological landscape, and practical applications of the digital twin in various
industry sectors. By integrating the digital twin and improved bacterial foraging, job scheduling
becomes more adaptable in dynamic production environments.
The digital twin plays a central role in the vision of the smart factory. It enables the transition
from analysing the past to predicting the future. The concept of the digital twin represents a transformative force in the landscape of smart manufacturing and Industry 4.0. Its multidimensional
nature, as explored through the systematic literature review and research, highlights its versatility and the need for further exploration. The intersection of digital twins with smart manufacturing frameworks, cyber-physical systems, big data, and their associated challenges points to a
promising path for the future of manufacturing paradigms.

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2. Methods
2.1 Digital twin
In the rapidly evolving landscape of modern manufacturing, the integration of digital technologies
has revolutionised traditional approaches, giving rise to the concept of digital twins and paving
the way for the era of smart manufacturing. At the centre of this transformative paradigm is the
seamless connection between the physical system and its virtual counterpart, creating a symbiotic
relationship that increases operational efficiency, productivity, and overall manufacturing system
performance. The physical system, as the cornerstone of the production environment, embodies
the tangible entities and processes that are essential to production. In parallel, sensing devices,
consisting of sensors, internet of things devices and other data sources, play a central role by actively collecting real-time data from the physical system. This data, which is often voluminous and
diverse, serves as the lifeblood of the digital twin ecosystem and requires a robust communication
infrastructure for its transmission to the data processing unit. The communication infrastructure
is an important link in the data flow chain that enables the transmission of the collected data to
the data processing unit. This unit, which is responsible for pre-processing and organising the
incoming data, plays a crucial role in ensuring data quality and relevance. The importance of this
step cannot be overestimated, as the effectiveness of subsequent analyses and decision-making
processes is highly dependent on the integrity and accuracy of the processed data. Subsequently,
the focus is on data collection, in which the processed data is recorded and stored for further analysis. This stock of information forms the basis for the creation of the digital twin model. The digital
twin, a virtual representation of the physical system, contains current states, behaviours and other
relevant attributes derived from the collected data. In the field of intelligent manufacturing, the
digital twin model goes beyond mere simulation and becomes an intelligent entity capable of making informed decisions [19]. Control systems capitalise on the insights offered by the digital twin
by using it as a virtual mirror of the physical system, enabling precise decision-making and issuing
commands to the control units [20]. This synergy between the digital and physical worlds creates
a closed loop in which the digital twin continuously refines its understanding and responses based
on feedback from the real world. As we delve into the scientific study of digital twins in smart
manufacturing, the intricate interplay between the physical system, sensing devices, communication infrastructure, data processing, data acquisition, modelling of the digital twin and control systems underpinning this transformative paradigm becomes apparent. By deciphering the complexity and dynamics within this interconnected framework, we can unlock unimagined potential for
efficiency, adaptability, and innovation in the manufacturing landscape, as shown in Fig. 1.

Fig. 1 Digital twin proposed block diagram structure

• Physical system: Represents the actual physical entity or process in the manufacturing environment.
• Sensing Devices: Collect data from the physical system. These can be sensors, IoT devices or
other data sources.
• Communication infrastructure: Transmits the collected data to the data processing unit.

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• Data processing: Includes the pre-processing and organisation of the data. This step is crucial to ensure the quality and relevance of the data.
• Data acquisition: Captures and stores the processed data for further analysis.
• Digital twin model: Uses the captured data to create a virtual representation of the physical
system. This representation includes the current state, behaviour and other relevant attributes.
• Control systems: Use the digital twin model to make decisions and send commands to the
control units.

2.2 Simio

Simulation modelling is a central aspect of decision-making processes that relies on simulation
languages with different prerequisites for the programming language. This paper describes the
Simio programming environment, which is characterised by its simulation language SIMIO, developed in the last decade. This language is characterised by the integration of 3D animations and
graphical representations, which increase the visual appeal. The Simio software environment facilitates the creation and use of dynamic 3D animation models and offers a comprehensive library
of standard elements. This library includes elements such as sources, sinks, servers, combiners,
separators, resources, vehicles, workers, basic nodes, transfer nodes, connectors, paths, time
paths and conveyors.

• Source: Serves as the entry point for new workpieces/orders and creates entities with specific types and arrival patterns.
• Sink: Represents the ending point for workpieces and is responsible for terminating entities
that have completed processing.
• Server: Represents a workstation where processes are executed on workpieces.
• Combiner: Integrates individual entities, e.g. by assembling a product from different components.
• Separator: Works in the opposite way to a combiner by splitting one entity into several or
creating several copies.
• Resource: Represents any intermittent resource that is required for a job and then returns
to its intended location.
• Vehicle: Represents a transport mechanism that either has a predetermined route or is executed in response to specific requests.
• Worker: A mobile resource that is activated for specific tasks or to facilitate entity transfers
between node locations.
• Basic node: Represents a connection point with no special functions, typically used upon
arrival at the workstation.
• Transfer node: A dedicated node that is used to implement transport logic and is primarily
used when leaving the workstation.
• Connector: Establishes connections between individual nodes; it is characterised by negligible time expenditure and is suitable for cases in which travel time is irrelevant.
• Path: Represents a classic transport path in which the user specifies both the distance and
the speed of the element being traversed, thereby influencing the travel times.
• Time path: A special path in which the user specifies the duration of time that the element
travels through.
• Conveyor: Represents a perpetual transport system, e.g. a constantly moving belt.

Each element is characterised by properties, states, events, appearance, and logic. Properties
include user-defined input values, while states represent dynamic values that can change during
the simulation. Events are temporal triggers that notify other elements of subsequent events. The
appearance is displayed as a 3D graphical representation of the element within the model. Logic
encapsulates a model within a model by prescribing the element's reactions to certain events and
thus specifying its behavioural dynamics. Once the model is finalised, the Simio software environment offers the possibility to define different scenarios. These scenarios allow different parame478

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ters to be analysed simultaneously and their combined effects on the system to be observed. Individual scenarios facilitate the modification of model properties and provide insights into their isolated effects on the dynamics of the system in general.
2.3 Case study description

In the first phase of creating the simulation model, which was based on the architectural layout of
the company's production facility in the Simio software environment, we meticulously described
the spatial allocation of the machine installations while maintaining realistic dimensions. The existing transport and transit routes, warehouses and ancillary rooms were fully taken into account.
It was decided to start the simulation model with an authentic workplace layout. The specific locations of the operational workstations were determined from the floor plan of the production
hall and then integrated into our model using the dimensions initially determined. This preparatory step simplifies the positioning of the subsequent simulation elements, where several constraints were identified and carefully considered when creating the model:
• Element constraint: the specification that the model must not contain more than 30 elements required a pragmatic solution for the source and sink of entities, so these components
had to be merged into a single source and sink.
• Time specification: The time frame of the simulation was limited to 10 working days. In
order to adapt to this time parameter, the processing and setup / tear-down times were
divided proportionally by 10 to ensure the temporal congruence of the simulation.

Before the actual construction of the model, overarching goals were defined to capture the
planned attributes of the simulation:

• Different attributes: Different sequences of orders, processing times and setup and teardown times depending on the type of entity created.
• Variable entity frequency: Entities of different types are generated with different frequencies.
• Type-dependent entity generation: The number of entities generated simultaneously depends on the type of entity generated.
• Trigger mechanism: The generation of new entities is triggered by the release of product A.
• Operational monitoring: The simulation interface provides a continuous overview of the
length of the queues at the individual work centres.

In line with these constraints and the formulated objectives, three different entities were defined in the model, using the Model Entity element and symbolising the workpieces with the designation’s product A, product B and product C. To increase the clarity of the model, each entity
was clearly colour-coded. In addition, a travelling speed of 2 m/s was defined between the individual workstations, which represents product transport with forklift trucks. Assuming that an
entity corresponds to a product batch, the intrinsic value of the entity was determined by multiplying the product cost and the batch size.

2.4 Digital twin model in Simio

The Simio simulation environment not only facilitates the creation of digital twins, but also offers
a significant advantage when utilising input data from real-world production systems. The datadriven simulation model, complemented by a user-friendly graphical interface, not only enables
efficient and robust numerical calculations, but also ensures a fast response time. This fast calculation capability is crucial for the successful implementation of a digital twin. In addition, the short
calculation times in Simio enable the development of a digital twin where the numerical results
can be seamlessly integrated into the real-time simulation within a visual virtual reality (VR) interface. This integration improves the overall experience of the simulation and provides a dynamic and interactive representation of the system behaviour. The use of a visual VR interface
contributes to a deeper understanding of the simulated processes and facilitates in-depth analysis
and decision making to optimise the real production system. The input data comes from a realworld production system consisting of seven workstations. Table 1 shows the main input data

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that drives the data-driven simulation model and describes the financial, time, energy, location,
and efficiency parameters of the real-world production system.
In the production system shown in Fig. 2, our analysis revolves around the dynamic flow of
three different products that pass through a series of seven strategically positioned workstations.
Each workstation is supervised by a dedicated employee who ensures that the production processes run smoothly. The intricate choreography of the production cycle involves the skilful use
of a forklift to enable the smooth transport of raw materials at the beginning of the system, navigating through the intermediate stages and finally delivering the finished products. The simulation model ingeniously visualises the operation of the system by assuming a two-shift operation,
with each shift lasting twelve hours, over the course of a conventional working week. This scheduling strategy is a fundamental part of the simulation as it provides a realistic representation of
the daily operation of the system. Crucially, the efficiency of this production system is quantified
through theoretical calculations of its production capacity. Based on these calculations, the system
has a planned utilization rate of 85 %, which emphasises its ability to effectively convert input
resources into products. This utilization metric serves as a valuable benchmark for evaluating and
optimising the performance of the production system and provides insights for potential improvements and operational refinements.
Workplace

Operating cost (EUR/h)
Idle cost (EUR/h)
xloc (m)
yloc (m)
Setup time (min)
Machine operation
energy consumption
(kWh)
Machine idle energy
consumption (kWh)
Machine scrap (%)

3. Results

Table 1 Digital twin input parameters

Cut 1

Cut 2

Machining 1

Machining 2

Machining 3

Assembly 1

Assembly 2

12

12

8

8

8

1.5

1.5

54
36
9.6
11.4
12

1.5
2

54
36
9.6
5.4
12

1.5
1.5

42
21
15
1.2
18

1
4

42
21
22.2
1.2
18

1

4.5

42
21
29.4
1.2
18

1
5

38
13
36.6
1.2
9.6

0.2
1.5

38
13
43.2
6.6
9.6

0.2
2

Fig. 2 Digital twin model in Simio

The section dedicated to results is primarily concerned with highlighting the discrepancy between
the results of a digital twin created in the Simio software environment and the results that emerge
from the empirical reality of the real-world production system. The scientific literature emphasises the acceptance of results generated using traditional simulation methods. A discrepancy of
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less than 3 % between the simulation results and the empirical reality is considered satisfactory.
Conversely, any escalation beyond this threshold represents an increased risk of inefficiency of
the simulation model, especially in terms of economic, safety and other operational constraints.
Table 2 presents the numerical results of the comparative analysis between the authentic production system and its digital counterpart, specifying the parameters for quantifying the final
products and determining the average throughput intervals. The collected comparative results
show deviations between the results of the digital twin and the real-world production environment. It is worth noting that a decadal average, derived from ten iterations of the simulation model
and representing a normal five-day working week, shows a recognisable discrepancy in four out
of seven parameters examined. For three parameters, however, the discrepancy remains below
the defined threshold value of 3 %. The limited validity of the results emphasises the fact that the
analysis embedded in the presented digital twin shortens its focus in time, while the input data of
the authentic production system has a longer temporal extension.
Fig. 3 shows the graphical representation of the empirical production system contrasted with
the digital twin, focussing on the parameter describing the quantity of finished products. The empirical evidence shows that the digital twin exactly fulfils the allocation rule for the proportional
distribution of the workload depending on the composition of products A, B and C.
The number of completed products under the updated production system exceeds the corresponding yield of the digital twin. This discrepancy is due to the contrast between the use of stochastic distribution functions for the input parameters and the imposition of static values assumed by the real-world production system. The strategic integration of stochastic distribution
functions into the simulation model of the digital twin serves to mimic authentic diurnal fluctuations inherent in the operating cadence of the empirical production system.
Table 2 Real-world vs digital twin numerical results comparison
Real-world results
Digital twin results
Number of finished products
163
158
Number of finished product A
95
92
Number of finished product B
46
46
Number of finished product C
22
20
Average flow time product A (h)
171.67
175.90
Average flow time product B (h)
149.80
152.70
Average flow time product C (h)
104.70
122.16

Difference [%]
-3.07
-3.16
0
-9.09
+2.46
+1.94
+16.68

Fig. 3 Number of finished products comparisons

The parameter of an average flow time, as shown in Fig. 4, plays a central role in production
system analysis. This graphical representation includes a quantitative representation of the average flow time taken during the entire production process. The average flow time calculated in this
way is a key performance indicator that provides sophisticated insights into the complex dynamics that determine the efficiency, effectiveness, and overall vitality of the production system. This

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metric is a fundamental benchmark that serves not only as an evaluation tool, but also as a catalyst
for continuous improvement measures. Its importance stems from its ability to recognise the average flow time efficiency of the production system and to decipher crucial information about the
progression of products or services over time from inception to completion. It also serves as a
measurement tool for the systematic evaluation of the production process and provides a quantitative basis for identifying areas for optimisation. Average flow time plays a central role in the
pursuit of the dual objectives of customer satisfaction and overarching corporate goals. By quantifying the time subtleties of production, it facilitates informed decision-making aimed at harmonising the production process with customer expectations and strategic business objectives. Consequently, the metric serves as a fundament for adaptive strategies that enable companies to proactively address challenges, optimise resource allocation and improve overall operational efficiency.
For the average flow time parameter, we find that the deviation between the results of the realworld production system and the result of the digital twin for products A and B is acceptable.
However, for product C, where the production quantity is low, larger deviations occur. This is
since a small number of products has a proportionally large influence on the deviation of the result
comparison.
Table 3 and Fig. 5 show the comparative results of the workstations utilization and the percentage of well-made products. In this case, the comparative results illustrate the difference between the values of the real-world production system and the results of the digital twin. As the
case under consideration is an order-related production, the results show that the system has a
high-capacity utilization. The order planning and the utilisation of the available capacities are effective, so that there are only minor delays due to micro-interruptions and irregularities in the
transport of raw materials and intermediate stocks. It is noteworthy that the utilisation in the realworld production system is lower than indicated by the digital twin. This emphasises the need for
effective management in order-related production, which poses considerable challenges for the
long-term sustainable operation of the production system due to its highly dynamic nature.
There are also differences in the parameter of the appropriate number of finished products;
the results show that there are relatively higher values of insufficiently manufactured products in
the real production system. Errors in production occur in individual operations, during transport
and in the completion of the product. The discrepancy between the comparative results of the
real-world production system and the digital twin can be attributed to the complexity of tracking
and detailed monitoring of irregularities in the production/processing of semi-finished products
within the digital twin, which occur continuously at all stages of production in the real-world production system.

Fig. 4 Average flow time comparison

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Table 3 Average utilization and quality products numerical results comparison
Real-world results
Digital twin results
Difference (%)
Average utilization (%)
83.6
87.8
+4.2
Quality products (%)
91.5
95.3
+3.8

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4. Discussion

Fig. 5 Average utilization and quality products graphical presentation

In line with the original research question of whether digital twins can be used to optimise production systems and whether simulation results are comparable with the real state of production,
this article presents the construction of a digital twin within the Simio software environment. The
digital twin is based on a data-driven simulation model of a real-world production system. The
input data describes the digital twin in a complex way and enables its execution in real time
through a graphical representation of the task execution in production. The numerical and graphical results obtained show the description of the production system in terms of parameters such
as the number of finished products, the average flow time, the average utilization of the work
centres and the appropriate quality of finished products.
In a comparative study of the results of the real-world production system and the digital twin,
an average deviation of 4.9 % is found. This deviation exceeds the threshold value of 3 % recommended in the literature. From the point of view of exceeding the threshold, there are obvious
overruns in parameters describing infrequently repeated measurements where individual measurements account for a significant common proportion. To address this, it would be advisable to
extend the operation and calculation of the simulation model beyond the current five-day working
week to a longer time frame, e.g., to a monthly or quarterly operation of the production system.
The observed discrepancies are due to the use of random distributions of input data by the
digital twin, which is essential in highly dynamic production environments, in contrast to the realworld production system, where the parameters are measured over a longer period and the data
represent averages of months. The research presented is limited by the case study considered,
which indicates the need for extension and, in certain aspects, generalisation of the study. The
results shown within the given limitations show that the implementation of digital twins enables
the analysis of the performance of the production system with the aim of optimisation [21]. The
main advantage of a digital twin in this regard is its financial justification, as it allows us to predict
in advance, what will happen in the production system. The operation of a digital twin is also possible in real time during the continuous operation of the production system to evaluate and track
certain parameters of the production system.
The presented approach shows the high benefit of digital twins in intelligent production systems. Given the high market dynamics in production systems, continuous optimisation is encouraged, and digital twins enable this by allowing the investigation of all important parameters in
terms of efficiency, finances, environment, etc. When using digital twins, the results show that the
right choice of input data is crucial, which must accurately represent the production system being
analysed. Even a slight deviation in the data can lead to completely wrong results or a major deviation from the real-world situation. In this case, detailed collection, analysis, filtering and understanding of the input data is of utmost importance. Only in this way we can ensure the accuracy
of the numerical and graphical results, whereby the final evaluation of the constructed digital twin
also plays a decisive role.
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5. Conclusion
In this paper, we explore the use of digital twins as an optimisation method for intelligent manufacturing systems. By using the digital twin, together with intelligent algorithms, companies can
achieve data monitoring, improve the operation of the production system, and develop innovative
products and services. We emphasise the importance of both studying the real-world production
system and creating an efficient simulation model as a basis for a digital twin solution. The results
show that digital twins can be used as a basis for the optimisation of production systems where
the input data is one of the most important factors.
The experiments conducted were designed in a laboratory environment using input parameters from a real-world production system. In order to fully utilise the potential of the proposed
digital twin, the scope of the studied problem needs to be extended to different types of production systems, from job shop to mass production. The proposed concept offers production companies a more comprehensive optimisation perspective for their production systems with the aim
of increasing overall efficiency and global competitiveness. Despite the laboratory environment,
this work emphasises the importance of using digital twin solutions in real-world production systems. In the current research landscape, our work represents a cutting-edge contribution to the
investigation of the possibilities of the digital twin in intelligent production systems. The growing
presence of the digital twin in the real world emphasises the importance of our findings.
This study provides new and fascinating insights into the field of digital twin methods as an
optimisation tool for intelligent manufacturing systems. Our study lays the foundation for future
research and recognises that the results presented here only capture certain findings that cannot
be conclusively refuted. To increase the authenticity of our findings, we plan to extend our experiment in the future by expanding the types of manufacturing systems and investigating the usefulness of digital twins depending on the characteristics of the manufacturing systems. In further
research steps, we plan to transfer the results of the digital twins to a real-world application
where a comparison of real-time optimisation can be performed.
The implications of the presented research go beyond the presented framework and offer potential avenues for further investigation and practical applications in the field of digital twins and
smart manufacturing. Through a comprehensive analysis of several parameters (average flow
time, total number of finished products, average utilisation of workstations and quality of products), this paper contributes to the current understanding of the optimisation possibilities of
smart manufacturing systems and lays the foundation for future research.

Acknowledgement

The authors gratefully acknowledge the support of the Slovenian Research Agency (ARIS), Research Core Funding No.
P2-0190. The authors acknowledge the use of research equipment system for development and testing cognitive production approaches in Industry 4.0: Collaborative robots with equipment and Sensors, hardware and software for ergonomic analysis of a collaborative workplace, procured within the project "Upgrading national research infrastructures – RIUM", which was cofinanced by the Republic of Slovenia, the Ministry of Education, Science and Sport and the
European Union from the European Regional Development Fund.

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Advances in Production Engineering & Management

ISSN 1854-6250

Volume 18 | Number 4 | December 2023 | pp 486–500

Journal home: apem-journal.org

https://doi.org/10.14743/apem2023.4.487

Original scientific paper

Incentive modeling analysis in engineering applications and
projects with stochastic duration time
Zhao, J.a,*, Su, J.F.b,*
aSchool

of Management and Economics, University of Electronic Science and Technology of China, Chengdu, P.R. China
College, Krirk University, Bangkok, Thailand

bInternational

ABSTRACT

ARTICLE INFO

Incentives are quite common to be utilized in engineering applications such as
some infrastructure development projects or construction projects. Due to the
increasing complexity of construction management and the continuing trend
towards outsourcing of component or engineering outsourcing activities, we
focus on the issue of incentive design. Time collaboration is one of the main
focuses of random project duration time in parallel projects. In this article, we
consider a setting where a manufacturer outsources two parallel subtasks to
two different suppliers, and the manufacturer is time sensitive. On the premise that the project completion time follows the exponential distribution,
some models are established to compare the proposed incentives and we get
the comparative analysis of the proposed incentives. This paper puts forward
three kinds of time-based incentive mechanisms, namely, deadline incentive
mechanism, competition mechanism and mixed incentive mechanism. We do
modeling analysis for all incentive mechanisms. We get the optimal work
rates determined by suppliers and compare various incentive mechanisms to
maximize manufacturers' profits.

Keywords:
Engineering applications;
Incentive mechanism;
Modeling analysis;
Stochastic duration time;
Deadline incentive model;
Competition model;
Parallel subtasks;
Mixed model
*Corresponding author:
15982405709@163.com
(Zhao, J.)
jiafu.su@hotmail.com
(Su, J.F.)

Article history:
Received 12 July 2023
Revised 13 December 2023
Accepted 15 December 2023
Content from this work may be used under the terms of
the Creative Commons Attribution 4.0 International
Licence (CC BY 4.0). Any further distribution of this work
must maintain attribution to the author(s) and the title of
the work, journal citation and DOI.

1. Introduction
In recent years, with the increasing complexity of cross-industry projects, scholars have paid
more and more attention to the importance and significance of engineering and project management research. In traditional construction or modern product development management,
there is an increasing tendency to explore the role of incentive mechanisms in different environments. At the same time, project management and engineering applications have expanded
dramatically from traditional building and infrastructure management to include new product
development, information technology, pharmaceuticals, and service development. Project-based
organizations are increasingly dependent on suppliers, and as a result, supply chain risks can
arise. These new applications create different characteristics, and we need to develop new and
innovative ways to solve the problems that may arise in these areas.
486

Incentive modeling analysis in engineering applications and projects with stochastic duration time

Project management includes planning and controlling the uncertainty in the duration of various components or tasks. Tools such as CPM, PERT and cost-time trade analysis are often used
for project management. Therefore, the project owner can have a detailed overview when managing projects. These tools are effective and efficient when there is little uncertainty about the
project completion time or operating costs. However, uncertainty arises in industries such as
architecture, hardware and software development, national defense and management due to the
increasing complexity of project management. For example, new power station, aircraft, and
telecommunications network are all characterized by long construction period, high risks, and
key links with engineering and commercial teams. We know little about how to deal with and
manage the risks arising from project uncertainty.
In the field of consumer electronics, for example, more and more enterprises cooperate with
Foxconn, Flextronics and other electronic manufacturing service providers to conduct research,
development, design and production of electronic products. In a setting where the manufacturer
cannot begin to work until all parts are delivered, the manufacturer's schedule depends on the
slowest delivery. Project delivery delays can have a significant negative impact on the interests
of manufacturers, so manufacturers need to devise mechanisms to encourage suppliers to work
faster so as to speed up project completion time.
This article considers a setting where a manufacturer outsources two parallel subtasks to two
suppliers with random project completion times. The manufacturer (she) plays the role of
Stackelberg leader, making decisions about the best price she can offer to the suppliers. Each
supplier (he) must determine the best work rate to obtain the best expected discount profit.
When the manufacturer is time urgent to launch new products, she is very sensitive to the completion time of the entire project. We assume that the supplier's work rate is not adjustable during the whole period due to some objective factors or external conditions. We mainly focus on
time-based incentive mechanisms in this paper.
Time-based incentive mechanism is common in practice and is a tool for manufacturers to
motivate suppliers to complete tasks on time while managing random projects. Construction
projects involve complex processes, so avoiding delays during the construction phase is very
important. For example, after the Northridge earthquake in 1994, the city of Los Angeles (the
project manager) wanted to repair the Santa Monica Freeway within 180 days. The city of Los
Angeles quickly offered Clint Meyers (the contractor) an incentive contract. Specifically, if the
project is completed 180 days ahead of schedule, the contractor will be paid an additional
$200,000 per day; However, a penalty of $200,000 per day will be paid if the contractor is late
[1, 2]. We propose three types of time-based incentives in our article.
In the case of deadline incentive mechanism, the project contractor is usually given a deadline
in traditional project management and is responsible for the penalty charged by the project
manager if the deadline is not met. The project owner often provides incentives/disincentives to
the contractors. Deadlines are also commonly used in product development processes [3, 4]. In
an empirical study of the global computer industry, Eisenhardt and Tabrizi [5] found that frequent deadlines help accelerate product development.
Due to the randomness of the task completion time, the two suppliers may end the tasks with
different task duration times. The result could be that one supplier finishes earlier (hereinafter
referred as faster supplier) and one later (hereinafter referred as slower supplier), and this
asynchronization may harm the manufacturer’s profit. Thus, we propose a competition incentive
mechanism, which means that the faster supplier will be awarded, and the slower supplier will
be punished.
On the problem of designing incentives for parallel random subprojects, our paper is the first
one to integrate reward and punishment incentives design together with project structure. We
also incorporate the criteria for rewards and penalties in those different incentive mechanisms,
and we do a more comprehensive comparative analysis of proposed incentive mechanisms.
We then combine the above two incentive mechanisms together, thus getting a mixed incentive
mechanism. We then examine its influence on manufacturer’s benefits. By comparing those
three different incentive mechanisms, we get some illuminating results to shed light on project
management and our analyses solve the following questions:
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Q1: What is the supplier's optimal work rate under different incentive mechanisms?
Q2: What incentives will result in higher profits for the manufacturer?

The remainder of this paper is organized as follows: in Section 2, we provide a related literature review on project supply chain management. Section 3 presents our base model settings. In
Section 4, we compare three kinds of incentive mechanisms together with the benchmark case.
In Section 5, by some numerical analysis, we give manufacturer’s optimal choice of incentive
mechanisms under different conditions. In Section 6, we summarize our work and give a conclusion of this paper. All proofs are relegated to Appendix.

2. Literature review

Scholars and practitioners have extensively studied project management. In traditional project
management, scheduling, and planning activities are the main concerns, and optimization is the
main way to solve these problems. We recommend that readers refer to the study [6, 7] for further information. Hall [8] provides a comprehensive review and discussion of research and
teaching opportunities in this area. In this section, we mainly focus on the problem of project
contracting and incentive mechanism design within multi-agents. We review some important
contributions that researchers have achieved. This paper mainly involves the following four research directions.
2.1 Deadlines in project

Deadlines are often used in product development processes and construction projects. A deadline means that the project must be stopped when the timeline reaches the stop point, or agents
who do not meet the deadline will be penalized. Zhang [9] examines the value of deadlines from
the agency theory perspective and considers that a company pays an agent to lead product development activities. It mainly focuses on the trade-off between the project's return and the project's labor cost. The paper concludes that deadlines are good, because they diminsh the agent's
incentive to procrastinate dynamically. Bordley et al. [10] consider a deadline problem with uncertainty. Traditional project management deals with unexpected changes in project deadlines
through an external change control process. They introduce a new counter-intuitive idea of recognizing uncertainty in project deadlines and show that it can greatly enhance the value of managers' decisions. Du et al. [11] studied the optimal timing for occasional placement of 'fulcrum' in
crowdfunding. The project will be funded successfully only if the target is met within the specified deadline. They evaluated three strategies in detail, namely, seeding, feature upgrading and
time-limited offering, to increase the likelihood of turning the random pledge process from failure to success.
Our paper differs from the study [9, 11], in which the deadline serves as a stopping point of
the project. However, ours mainly focuses on the reward or penalty associated with the specific
deadline.
2.2 Time-based incentive contract

In the traditional supply contract, the uncertainty of demand is the main focus, and the quantity
of order is the main decision variable. However, in project contracting, time uncertainty is the
main focus, and the contractor's work rate decision is the main decision variable. Therefore,
various incentive contracts based on time are studied by scholars in the field of project management. Gupta et al. [12] studied A+B infrastructure procurement mechanism, which state
transportation agencies use to provide incentives for faster completion. Early completion will be
rewarded, late will be penalized. Managers use time-based incentive contracts to encourage contractors to make greater efforts to complete tasks faster. Tang et al. [13] compared two different
time-dependent project management contracts (C1 and C2) when managers conducted reverse
auctions. In each incentive contract, completion earlier than the due date will be incentivized
and completion later than the due date will be discouraged.
488

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Incentive modeling analysis in engineering applications and projects with stochastic duration time

Chen and Lee [14] consider a coordination problem where the manufacturer carries out a series of tasks, and each task needs to procure a certain material from a supplier. They propose a
delivery-schedule-based incentive contract to mitigate incentive misalignment between firms in
a project supply chain. The delivery delay than the due date will be punished and early delivery
will be awarded. Bayiz and Corbett [2] provide a framework to integrate contracting and asymmetric information into a project management context and study the value of incentive contracts. They derived the optimal incentive contracts the project manager offers to the subcontractors when the two subprojects are conducted in parallel or in serial, respectively. Kwon et al.
[15] consider a coordination problem when managing a project with uncertain completion and
an unobservable contractor’s work rate. Fixed price, time-based, cost-based, and rate-based incentive contracts are examined in their work. Zhao and Mu [16] consider a situation where a
manufacturer outsources two parallel tasks to two different suppliers, and they propose nine
kinds of time-based incentive contracts a manufacturer can offer to the suppliers.
2.3 Reward-penalty mechanism

A project usually consists of a set of activities that introduce delays or expeditions related to
penalty or reward issues when the planned completion time is incorrectly estimated. Bergantiños and Lorenzo [17] consider a situation where a planner wants to execute a project involving
several companies. If the deadline is missed, the company will be penalized. They discussed two
ways to impose penalties: they would apply only if the entire project was delayed; Even if the
project is completed on time, the penalties apply to every company that causes delays. EstévezFernández [18] analyzed the situation where a project with multiple activities was not realized
as planned. If the project is accelerated, there is a reward. Similarly, if the project is delayed,
there is a penalty. In this paper, we consider the existence of any non-decreasing reward function and penalty function for total exploration and delay, respectively, focusing on the distribution of total reward (penalty) function between activities. Chen et al. [19] proposed an incentive
payment contract for a series of random items. Their proposed contract reflects the convex timecost trade-off that is well known in the project literature. In contrast to fixed-price contracts,
such incentive contracts imply penalties for suppliers.
2.4 Parallel projects

Project structure is a key focus when conducting product development activities or outsourcing
activities. These sub-projects are usually performed in parallel (tasks have to be performed in
parallel), or in serial (tasks have to be performed sequentially) or in network (tasks have to be
performed in parallel-serial). Our work mainly considers a scenario where the sub-projects are
conducted in parallel. Kwon et al. [20] studied a delayed payment scheme where multiple concurrent subprojects were outsourced to different suppliers. Under the delayed payment scheme,
each supplier will be paid after all tasks are completed, that is, faster supplier payments are delayed. Song et al. [21] explored incentives for firms under risk-sharing partnerships in the context of project management. In this partnership, each partner pays its own costs and shares project completion results (rewards or losses) between faster and slower suppliers. They examine
the project network in serial, parallel, and assembled ways. Dawande et al. [22] studied the coordination problems faced by enterprises when undertaking projects consisting of multiple
tasks. Under the assumption of exponential completion time, the optimal coordination contract
of parallel and sequential tasks is studied.
The above literatures only consider the incentive mechanism design problem under partial
factors. However, our paper considers the time-based incentive mechanism design problem
faced by a manufacturer who carries out outsourcing activities in more complicated cases. Our
proposed incentive mechanisms take deadlines in projects issue and reward-penalty incentives
into consideration, and we also consider the time synergy problem in parallel projects. Our paper is a more comprehensive one that considers the incentive mechanisms design issue by the
project owner.
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3. Model settings
In our context, we consider a scenario where the manufacturer outsources two parallel subtasks with stochastic project completion time to two different suppliers. The duration of each
subtask 𝑋𝑋𝑖𝑖 (𝑖𝑖 = 1, 2) is exponentially distributed with parameter 𝜇𝜇𝑖𝑖 , where the work rate 𝜇𝜇𝑖𝑖 > 0
is selected by supplier 𝑖𝑖 at the beginning of the project. Throughout our analysis, we assume that
the two subtasks are of the same workloads and difficulties. The suppliers cannot change the
work rates once selected at the beginning, due to some technical or practical reasons. The manufacturer cannot start to work until the suppliers deliver all parts.
We consider the time value of revenue gained and cost incurred by the supplies into our context. Thus, our model takes the discounting issue into our analysis. Let the continuous-time discount rate be 𝛼𝛼 > 0, which captures the fact that the suppliers want to receive his payment earlier and incur costs later.
We consider a situation where the manufacturer is time sensitive to the completion time of
the total project. For example, while developing some new products, the manufacturer is urgent
to launch or release new products to occupy the market earlier. The total completion time of the
project satisfies that 𝑇𝑇 = max(𝑋𝑋1 , 𝑋𝑋2 ). For ease of exposition, we assume that the project’s value
𝑉𝑉(𝑇𝑇) to the manufacturer is a linear, decreasing function of the project completion time, specifically, we define 𝑉𝑉(𝑇𝑇) = 𝐴𝐴 − 𝐵𝐵𝐵𝐵 (𝐴𝐴, 𝐵𝐵 > 0).
By the properties of exponential distributions, we have that the probability density function
of random variable 𝑋𝑋𝑖𝑖 (𝑖𝑖 = 1,2) is 𝜇𝜇𝑖𝑖 𝑒𝑒 −𝜇𝜇𝑖𝑖𝑡𝑡 and the cumulative distribution function of a random
variable 𝑋𝑋𝑖𝑖 is 𝐹𝐹𝑖𝑖 (𝑡𝑡) = 1 − 𝑒𝑒 −𝜇𝜇𝑖𝑖𝑡𝑡 . We can infer that the discount factor satisfies that 𝐸𝐸[𝑒𝑒 −𝛼𝛼𝑋𝑋𝑖𝑖 ] =
∞
𝜇𝜇
∫0 𝜇𝜇𝑖𝑖 𝑒𝑒 −𝛼𝛼𝛼𝛼 𝑒𝑒 −𝜇𝜇𝑖𝑖𝑡𝑡 𝑑𝑑𝑑𝑑 = 𝑖𝑖 .
𝛼𝛼+𝜇𝜇𝑖𝑖

While working at the rate of 𝜇𝜇𝑖𝑖 (𝑖𝑖 = 1,2), each supplier incurs an operating cost of 𝜅𝜅(𝜇𝜇𝑖𝑖 ) per
unit time. We assume that the suppliers’ operating cost 𝜅𝜅(𝜇𝜇𝑖𝑖 ) per unit time associating with the
work rate 𝜇𝜇𝑖𝑖 is a convex-increasing function, and it is given by 𝜅𝜅(𝜇𝜇𝑖𝑖 ) = 𝑘𝑘𝜇𝜇𝑖𝑖2 with 𝑘𝑘 > 0. Furthermore, we have that supplier 𝑖𝑖 ’s expected discounted operating costs equal that
𝑇𝑇

∞

𝑡𝑡

𝑘𝑘𝜇𝜇2

𝐸𝐸 �∫0 𝑖𝑖 𝜅𝜅(𝜇𝜇𝑖𝑖 )𝑒𝑒 −𝛼𝛼𝛼𝛼 𝑑𝑑𝑑𝑑� = ∫0 [∫0 𝑖𝑖 𝜅𝜅(𝜇𝜇𝑖𝑖 )𝑒𝑒 −𝛼𝛼𝛼𝛼 𝑑𝑑𝑑𝑑] 𝜇𝜇𝑖𝑖 𝑒𝑒 −𝜇𝜇𝑖𝑖𝑡𝑡𝑖𝑖 𝑑𝑑𝑡𝑡𝑖𝑖 = 𝛼𝛼+𝜇𝜇𝑖𝑖 .
𝑖𝑖

We compare four kinds of mechanisms manufacturer can offer to the suppliers (three timedbased incentive models together with the benchmark case). For the sake of perceptual intuition,
we list some characteristics among the four mechanisms. Table 1 shows the specific elements
contained in these four kinds of models.
In the base model, we consider a setting where the manufacturer does not offer any incentive
to the suppliers. The two suppliers start to work simultaneously and make the work rate
decisions 𝜇𝜇𝑖𝑖 (𝑖𝑖 = 1,2). Each supplier can get a payment of 𝜔𝜔 after his own subtask is finished, and
the two suppliers make their decision independently.
In the deadline incentive model, the manufacturer imposes a specific due date, which we also
referred as a deadline 𝐷𝐷. In addition, to paying a base price 𝜔𝜔, the manufacturer awards the
supplier a reward at 𝑟𝑟 per unit time of early completion than the deadline, and charges the
supplier a penalty at 𝑝𝑝 per unit time of delay than the deadline.
In the competition incentive model, the manufacturer awards the faster supplier a reward of
𝑟𝑟 per unit time earlier than the slower supplier; she also charges the slower supplier a penalty of
𝑝𝑝 per unit time later than the faster supplier, together with a base price 𝜔𝜔.
Table 1 Classification of different mechanisms
Deadline Incentive Model
• A base price
• A specific due date.
• An incentive/disincentive scheme
Competition Model
Mixed Model
• A base price
• A base price
• A comparison of the faster and slower
• A specific due date.
• An incentive/disincentive scheme
• A comparison of the faster and slower
• An incentive/disincentive scheme
Base Model
• A base price

490

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Incentive modeling analysis in engineering applications and projects with stochastic duration time

In the mixed incentive model, the manufacturer imposes a specific due date 𝐷𝐷. The manufacturer awards the faster supplier a reward of 𝑟𝑟 per unit time earlier than the deadline and charges the slower supplier a penalty of 𝑝𝑝 per unit time later than the deadline, together with a base
price 𝜔𝜔.

4. The model

4.1 Base model
In our base model, there is no incentive mechanisms offered to the suppliers by the manufacturer. The manufacturer outsources two subtasks to two different suppliers. The two suppliers
start to work simultaneously, and each supplier does not take the other’s decision into his own
consideration. Thus, each supplier makes his work rate decision 𝜇𝜇𝑖𝑖 independently. When supplier 𝑖𝑖 finishes his own subtask, he will get a payment of 𝜔𝜔, and his expected discounted profit can
be expressed as:
𝑘𝑘𝜇𝜇2

Π𝑖𝑖 = 𝜔𝜔𝜔𝜔[𝑒𝑒 −𝛼𝛼𝑋𝑋𝑖𝑖 ] − 𝛼𝛼+𝜇𝜇𝑖𝑖 (𝑖𝑖 = 1,2)

(1)

𝑖𝑖

From the concavity of Eq. 1, we can get:
Proposition 1. Both suppliers’ optimal work rates are the same in equilibrium, and they are
characterized by
𝜇𝜇∗ = −𝛼𝛼 + �𝛼𝛼 2 +

𝜔𝜔𝜔𝜔
𝑘𝑘

(2)

We have that 𝜇𝜇∗ is strictly increasing with 𝜔𝜔.
When both subtasks are finished and delivered to the manufacturer, the outsourcing activity
1
1
1
ends. Because we have that 𝐸𝐸[𝑇𝑇] = 𝐸𝐸[max(𝑋𝑋1 , 𝑋𝑋2 )] = +
− +𝜇𝜇 , thus, in equilibrium, we
can get that 𝐸𝐸[𝑇𝑇] =

𝜇𝜇𝑖𝑖

3
.
2𝜇𝜇∗

𝜇𝜇−𝑖𝑖

𝜇𝜇𝑖𝑖

−𝑖𝑖

The manufacturer makes an optimal price decision 𝜔𝜔, and her profit can be expressed as:
Π𝑚𝑚 = 𝐴𝐴 − 𝐵𝐵𝐵𝐵[𝑇𝑇] − 2𝜔𝜔.
By the concavity of Eq. 3, we can obtain:

Proposition 2. The manufacturer’s optimal price decision satisfies that:
�𝛼𝛼 2 +

𝜔𝜔∗ 𝛼𝛼
��𝛼𝛼 2
𝑘𝑘

+

𝜔𝜔∗ 𝛼𝛼
𝑘𝑘

2

− 𝛼𝛼� =

3𝐵𝐵𝐵𝐵
8𝑘𝑘

(3)
(4)

The specific certification process is in Appendix.
The above proposition provides the manufacturer's optimal price decision, and we can obtain
the manufacturer’s optimal profit under the benchmark case.
4.2 Deadline incentive mechanism
In this case, we consider a situation where the manufacturer offers a deadline incentive mechanism to the suppliers. The manufacturer decides optimal price 𝜔𝜔 paid to the suppliers, and the
suppliers make the optimal work rate decision 𝜇𝜇𝑖𝑖 (𝑖𝑖 = 1,2). Unlike sub-section 4.1, the two suppliers’ decisions are no longer independent of each other under the incentive mechanisms.
Consistent with the above analysis, this incentive mechanism contains the following three
components:
A base price. The price manufacturer pays to the supplier is 𝜔𝜔, which is irrelevant to the performance of supplier’s delivery time.
A specific due date. The manufacturer sets a pre-determined delivery date, which we referred as
a deadline 𝐷𝐷.
An incentive/disincentive scheme. The supplier will be awarded a reward at 𝑟𝑟 per unit time of
early completion than the deadline and will be charged a penalty at 𝑝𝑝 per unit time of delay than
the deadline.
Advances in Production Engineering & Management 18(4) 2023

491

Zhao, Su

Under the deadline incentive mechanism, one supplier makes his optimal work rate decision
regardless of the other supplier’s decision. After the completion of his own subtask, supplier
𝑖𝑖 (𝑖𝑖 = 1,2) will get an expected payment of 𝑊𝑊 𝑑𝑑 (𝑖𝑖), which satisfies that:
𝑊𝑊 𝑑𝑑 (𝑖𝑖) = 𝜔𝜔 + 𝑟𝑟𝑟𝑟(𝐷𝐷 − 𝑋𝑋𝑖𝑖 )+ − 𝑝𝑝𝑝𝑝(𝑋𝑋𝑖𝑖 − 𝐷𝐷)+

(5)

By some calculations, we can get the following:
𝐷𝐷
𝐷𝐷
𝑊𝑊 𝑑𝑑 (𝑖𝑖) = 𝜔𝜔 + 𝑝𝑝𝑝𝑝 − 𝑝𝑝𝑝𝑝(𝑋𝑋𝑖𝑖 ) + (𝑟𝑟 − 𝑝𝑝) �𝐷𝐷 ∫0 𝑓𝑓(𝑡𝑡𝑖𝑖 )𝑑𝑑𝑡𝑡𝑖𝑖 − ∫0 𝑡𝑡𝑖𝑖 𝑓𝑓(𝑡𝑡𝑖𝑖 )𝑑𝑑𝑡𝑡𝑖𝑖 �

Lemma 1. The expected payment 𝑊𝑊 𝑑𝑑 (𝑖𝑖) satisfies that:
1

1

𝑊𝑊 𝑑𝑑 (𝑖𝑖) = 𝜔𝜔 + 𝑟𝑟𝑟𝑟 − 𝑟𝑟 𝜇𝜇 + (𝑟𝑟 − 𝑝𝑝) 𝜇𝜇 𝑒𝑒 −𝜇𝜇𝑖𝑖𝐷𝐷
𝑖𝑖

(6)

𝑖𝑖

See Appendix for the specific certification process.
Furthermore, we can express supplier 𝑖𝑖’s expected discounted profit as follows:
Π𝑖𝑖𝑑𝑑 = 𝑊𝑊 𝑑𝑑 (𝑖𝑖)𝐸𝐸[𝑒𝑒 −𝛼𝛼𝑋𝑋𝑖𝑖 ] −

𝑘𝑘𝜇𝜇𝑖𝑖2
𝛼𝛼+𝜇𝜇𝑖𝑖

(7)

By some calculations, we can get the following proposition.

Proposition 3. The suppliers’ optimal work rates are the same in equilibrium and it is characterized by:
𝑘𝑘(𝜇𝜇𝑑𝑑∗ )2 + 2𝑘𝑘𝑘𝑘𝜇𝜇𝑑𝑑∗ + (𝑟𝑟 − 𝑝𝑝)𝑒𝑒 −𝜇𝜇

𝑑𝑑∗ 𝐷𝐷

�1 + 𝐷𝐷𝐷𝐷 + 𝐷𝐷𝜇𝜇𝑑𝑑∗ � = 𝜔𝜔𝜔𝜔 + 𝑟𝑟𝑟𝑟𝑟𝑟 + 𝑟𝑟

(8)

The specific certification process is in Appendix.
We define 𝐺𝐺 = 𝑟𝑟 − 𝑝𝑝. From proposition 3, we have that if 𝐺𝐺 ≤ 0, we can get that 𝜇𝜇𝑑𝑑∗ > 𝜇𝜇∗ . If
𝑑𝑑∗
𝑑𝑑∗
𝑟𝑟𝑟𝑟𝑟𝑟+𝑟𝑟
𝐺𝐺 > 0, when 𝑒𝑒 −𝜇𝜇 𝐷𝐷 �1 + 𝐷𝐷𝐷𝐷 + 𝐷𝐷𝜇𝜇𝑑𝑑∗ � <
, we have 𝜇𝜇𝑑𝑑∗ > 𝜇𝜇∗ . When 𝑒𝑒 −𝜇𝜇 𝐷𝐷 �1 + 𝐷𝐷𝐷𝐷 +
𝑑𝑑∗

𝐷𝐷𝜇𝜇 � >

𝑟𝑟𝑟𝑟𝑟𝑟+𝑟𝑟
, we have 𝜇𝜇𝑑𝑑∗
𝑟𝑟−𝑝𝑝

𝑟𝑟−𝑝𝑝

∗

< 𝜇𝜇 . In equilibrium, we have 𝑊𝑊 𝑑𝑑 (1) = 𝑊𝑊 𝑑𝑑 (2) = 𝑊𝑊 𝑑𝑑 .

We then make some comparative static analyses concerning on the supplier’s optimal work
rate under the deadline incentive mechanism. Table 2 shows the relationship between supplier’s
optimal work rate and the pre-determined specific due date.
We then show the relationship between supplier’s optimal work rate and the disincentive
mechanism factor, and the relationship between supplier’s optimal work rate and the base price
in Table 3.
As to the relationship between supplier’s optimal work rate and the incentive mechanism
factor, the results are given in Table 4.
𝑑𝑑

𝑑𝑑𝑑𝑑

if

𝑑𝑑

𝜇𝜇𝑑𝑑∗

𝜇𝜇𝑑𝑑∗

𝑑𝑑𝑑𝑑

Table 2 The relationship between 𝜇𝜇𝑑𝑑∗ and 𝐷𝐷
𝐺𝐺 = 0
𝐺𝐺 > 0

𝐺𝐺 < 0

𝑑𝑑

𝑑𝑑𝑑𝑑

> 0,

<

1

2𝑘𝑘
− ln 2 (𝑟𝑟−𝑝𝑝)
𝐷𝐷
𝐷𝐷

𝜇𝜇𝑑𝑑∗ < 0,

1

if 𝜇𝜇𝑑𝑑∗ > − ln
𝐷𝐷

𝑑𝑑

𝑑𝑑𝑑𝑑

2𝑘𝑘

ii.

𝑑𝑑

𝑑𝑑𝑑𝑑

i.

𝐷𝐷2 (𝑟𝑟−𝑝𝑝)

𝑑𝑑

𝑑𝑑𝑑𝑑
𝑑𝑑

𝐺𝐺 ≤ 0

𝑑𝑑𝑑𝑑

492

𝜇𝜇𝑑𝑑∗ > 0

i.

𝜇𝜇𝑑𝑑∗

>0

𝜇𝜇𝑑𝑑∗ > 0

𝜇𝜇𝑑𝑑∗ > 0,

𝛼𝛼

𝛼𝛼 2

if 𝜇𝜇𝑑𝑑∗ < − + �
2

𝛼𝛼

𝛼𝛼 2

if 𝜇𝜇𝑑𝑑∗ > − + �

𝜇𝜇𝑑𝑑∗ < 0,

2

𝛼𝛼

4

𝛼𝛼 2

if 𝜇𝜇𝑑𝑑∗ > − + �
2

4

4

𝛼𝛼

+ .
𝐷𝐷
𝛼𝛼

𝑑𝑑∗ 𝐷𝐷

(𝐷𝐷𝐷𝐷𝜇𝜇𝑑𝑑∗ + 𝐷𝐷(𝜇𝜇𝑑𝑑∗ )2 − 𝛼𝛼) <

𝑝𝑝−𝑟𝑟

𝛼𝛼

𝑑𝑑∗ 𝐷𝐷

(𝐷𝐷𝐷𝐷𝜇𝜇𝑑𝑑∗ + 𝐷𝐷(𝜇𝜇𝑑𝑑∗ )2 − 𝛼𝛼) >

𝑝𝑝−𝑟𝑟

+ and 𝑒𝑒 −𝜇𝜇
𝐷𝐷

+ and 𝑒𝑒 −𝜇𝜇
𝐷𝐷

Table 3 The relationship between 𝜇𝜇𝑑𝑑∗ and 𝑝𝑝, 𝜔𝜔
𝑑𝑑

𝐺𝐺 > 0
1
𝜇𝜇𝑑𝑑∗ > 0, if 𝜇𝜇𝑑𝑑∗ < − ln

𝑟𝑟𝑟𝑟
𝑟𝑟𝑟𝑟

2𝑘𝑘

𝐷𝐷2 (𝑟𝑟−𝑝𝑝)
2𝑘𝑘
𝑑𝑑∗
𝑑𝑑∗
𝜇𝜇 < 0, if 𝜇𝜇 > − ln 2 (𝑟𝑟−𝑝𝑝)
𝑑𝑑𝑑𝑑
𝐷𝐷
𝐷𝐷
𝑑𝑑 𝑑𝑑∗
1
2𝑘𝑘
𝜇𝜇 > 0, if 𝜇𝜇𝑑𝑑∗ < − ln 2(𝑟𝑟−𝑝𝑝)
𝑑𝑑𝑑𝑑
𝐷𝐷
𝐷𝐷
𝑑𝑑 𝑑𝑑∗
1
2𝑘𝑘
𝜇𝜇 < 0, if 𝜇𝜇𝑑𝑑∗ > − ln 2(𝑟𝑟−𝑝𝑝).
𝑑𝑑𝑑𝑑
𝐷𝐷
𝐷𝐷
𝑑𝑑𝑑𝑑
𝑑𝑑

𝐷𝐷
1

Advances in Production Engineering & Management 18(4) 2023

Incentive modeling analysis in engineering applications and projects with stochastic duration time
Table 4 The relationship between 𝜇𝜇𝑑𝑑∗ and 𝑟𝑟

𝐺𝐺 ≤ 0

if

𝜇𝜇𝑑𝑑∗

if 𝜇𝜇

𝑑𝑑∗

𝑑𝑑 𝑑𝑑∗
𝜇𝜇 > 0,
𝑑𝑑𝑑𝑑

< −𝛼𝛼

+ �𝛼𝛼 2

+

𝑑𝑑 𝑑𝑑∗
𝜇𝜇 < 0,
𝑑𝑑𝑑𝑑

> −𝛼𝛼

+ �𝛼𝛼 2

+

𝑑𝑑

𝑑𝑑𝑑𝑑
𝑝𝑝+𝐷𝐷𝐷𝐷𝐷𝐷+𝜔𝜔𝜔𝜔
𝑘𝑘

i.

ii.

𝑑𝑑

𝑑𝑑𝑑𝑑

𝑝𝑝+𝐷𝐷𝐷𝐷𝐷𝐷+𝜔𝜔𝜔𝜔
𝑘𝑘

i.

ii.

𝜇𝜇𝑑𝑑∗ > 0,

𝐺𝐺 > 0

if 𝜇𝜇𝑑𝑑∗ < min �−𝛼𝛼 + �𝛼𝛼 2 +

if 𝜇𝜇𝑑𝑑∗ > max �−𝛼𝛼 + �𝛼𝛼 2 +

𝜇𝜇𝑑𝑑∗ < 0,

if 𝜇𝜇𝑑𝑑∗ ~ �−𝛼𝛼 + �𝛼𝛼 2 +
1

if 𝜇𝜇𝑑𝑑∗ ~ �− ln
𝐷𝐷

2𝑘𝑘

𝐷𝐷2 (𝑟𝑟−𝑝𝑝)

𝑝𝑝+𝐷𝐷𝐷𝐷𝐷𝐷+𝜔𝜔𝜔𝜔
𝑝𝑝+𝐷𝐷𝐷𝐷𝐷𝐷+𝜔𝜔𝜔𝜔

𝑝𝑝+𝐷𝐷𝐷𝐷𝐷𝐷+𝜔𝜔𝜔𝜔
𝑘𝑘

𝑘𝑘

𝑘𝑘

1

1

, − ln

𝐷𝐷

, −𝛼𝛼 + �𝛼𝛼 2 +

�

𝐷𝐷2 (𝑟𝑟−𝑝𝑝)

𝐷𝐷

𝐷𝐷2 (𝑟𝑟−𝑝𝑝)

1

, − ln

, − ln

2𝑘𝑘

𝐷𝐷

2𝑘𝑘

2𝑘𝑘

�

𝐷𝐷2 (𝑟𝑟−𝑝𝑝)

𝑝𝑝+𝐷𝐷𝐷𝐷𝐷𝐷+𝜔𝜔𝜔𝜔
𝑘𝑘

�

�

After the delivery of subtasks by the two suppliers, the outsourcing activities ends. The
manufacturer’s profit can be expressed as
𝑑𝑑
Π𝑚𝑚
= 𝐴𝐴 − 𝐵𝐵𝐵𝐵[𝑇𝑇] − 2𝑊𝑊 𝑑𝑑 .

(9)

In order to solve the above problem, we transfer (8) as 𝐿𝐿(𝜇𝜇𝑑𝑑∗ , 𝜔𝜔) = 0. Manufacturer’s profit
maximizing problem in Eq. 9 can be redeemed as an optimization problem, which can be recognized as follows:
𝑑𝑑
= 𝐴𝐴 − 𝐵𝐵𝐵𝐵[𝑇𝑇] − 2𝑊𝑊 𝑑𝑑
max𝜇𝜇𝑑𝑑∗ ,𝜔𝜔 Π𝑚𝑚
s.t. 𝐿𝐿(𝜇𝜇𝑑𝑑∗ , 𝜔𝜔) = 0
Because of the implicit function form of Eq. 8, it’s hard to get the analytical expression of optimal price decision made by the manufacturer. We do some numerical examples in Section 5 to
determine if deadline incentive mechanisms dominate the others. We use optimization algorithms to determine the optimal (𝜇𝜇𝑑𝑑∗ , 𝜔𝜔). We do some comparative analyses concerning supplier’s optimal work rate decision in this sub-section and discuss the other two incentive mechanisms in the following analysis.
4.3 Competition mechanism

In this section, we consider a situation where the manufacturer offers a competition mechanism
to the suppliers. The manufacturer makes a decision of optimal price 𝜔𝜔 paid to the suppliers, and
the suppliers make the optimal work rate decision 𝜇𝜇𝑖𝑖 (𝑖𝑖 = 1,2). The two suppliers also have to
take the other’s decision into his own consideration under this incentive mechanisms.
This incentive mechanism manufacturer offers to the suppliers contains the following three
components:
A base price. The manufacturer pays each supplier a base price of 𝜔𝜔.

A comparison of the faster and slower. Because of the stochasticity of project duration times, the
result ex post can be distinguished from a faster supplier and a slower supplier.

An incentive/disincentive scheme. The faster supplier will be awarded a reward of 𝑟𝑟 per unit time
earlier than the slower supplier; the slower supplier will be charged a penalty of 𝑝𝑝 per unit time
later than the faster supplier.

For supplier 𝑖𝑖 (𝑖𝑖 = 1,2), we can get that his expected payment gained from the manufacturer
after his completion of the subtask satisfies:
𝑊𝑊 𝑐𝑐 (𝑖𝑖) = 𝜔𝜔 + 𝑟𝑟𝑟𝑟[𝑋𝑋−𝑖𝑖 − 𝑋𝑋𝑖𝑖 ]+ − 𝑝𝑝𝑝𝑝[𝑋𝑋𝑖𝑖 − 𝑋𝑋−𝑖𝑖 ]+
(10)

Lemma 2. With simple calculations, we can infer that:
𝑊𝑊 𝑐𝑐 (𝑖𝑖) = 𝜔𝜔 + 𝑟𝑟 𝜇𝜇

𝜇𝜇𝑖𝑖
(𝜇𝜇
−𝑖𝑖 𝑖𝑖 +𝜇𝜇−𝑖𝑖 )

𝜇𝜇

−𝑖𝑖
− 𝑝𝑝 𝜇𝜇 (𝜇𝜇 +𝜇𝜇
𝑖𝑖

𝑖𝑖

−𝑖𝑖 )

The specific certification process is in Appendix.
Furthermore, we have that supplier 𝑖𝑖’s expected discounted profit can be expressed as:
Advances in Production Engineering & Management 18(4) 2023

493

Zhao, Su
𝑘𝑘𝜇𝜇2

Π𝑖𝑖𝑐𝑐 = 𝑊𝑊 𝑐𝑐 (𝑖𝑖)𝐸𝐸[𝑒𝑒 −𝛼𝛼𝑋𝑋𝑖𝑖 ] − 𝛼𝛼+𝜇𝜇𝑖𝑖

(11)

4𝑘𝑘(𝜇𝜇𝑐𝑐∗ )4 + 8𝑘𝑘𝑘𝑘(𝜇𝜇𝑐𝑐∗ )3 − (3𝑝𝑝 + 4𝜔𝜔𝜔𝜔)(𝜇𝜇𝑐𝑐∗ )2 + (𝑟𝑟 − 𝑝𝑝𝑝𝑝)𝜇𝜇𝑐𝑐∗ = 𝑟𝑟𝑟𝑟

(12)

𝑖𝑖

Proposition 4. In equilibrium, supplier’s optimal work rate is given by:

See Appendix for the specific certification process.
From the above result, we can see that if 𝑝𝑝 = 𝑟𝑟 = 0, we have 𝜇𝜇𝑐𝑐∗ = 𝜇𝜇∗ . Furthermore, we have
the following:

7−4√3
𝑝𝑝
7+4√3
< <
, we have 𝜇𝜇𝑐𝑐∗ > 𝜇𝜇∗ .
𝛼𝛼
𝛼𝛼
𝑟𝑟
𝑝𝑝
7−4√3
𝑝𝑝
7+4√3
𝑟𝑟−𝑝𝑝𝑝𝑝−�𝑝𝑝2 𝛼𝛼2 +𝑟𝑟 2 −14𝑝𝑝𝑝𝑝𝑝𝑝
If
< 𝛼𝛼
or
> 𝛼𝛼 , we have when 𝜇𝜇𝑐𝑐∗ <
or 𝜇𝜇𝑐𝑐∗ >
6𝑝𝑝
𝑟𝑟
𝑟𝑟
𝑟𝑟−𝑝𝑝𝑝𝑝+�𝑝𝑝2 𝛼𝛼2 +𝑟𝑟 2 −14𝑝𝑝𝑝𝑝𝑝𝑝
𝑟𝑟−𝑝𝑝𝑝𝑝−�𝑝𝑝2 𝛼𝛼2 +𝑟𝑟 2 −14𝑝𝑝𝑝𝑝𝑝𝑝
𝑟𝑟−𝑝𝑝𝑝𝑝+�𝑝𝑝2 𝛼𝛼2 +𝑟𝑟 2 −14𝑝𝑝𝑝𝑝𝑝𝑝
, then 𝜇𝜇𝑐𝑐∗ > 𝜇𝜇∗ ; when
< 𝜇𝜇𝑐𝑐∗ <
,
6𝑝𝑝
6𝑝𝑝
6𝑝𝑝
𝑐𝑐∗
∗
𝑐𝑐 (1)
𝑐𝑐 (2)
𝑐𝑐

If

then 𝜇𝜇 < 𝜇𝜇 . And in equilibrium, we have 𝑊𝑊
= 𝑊𝑊
= 𝑊𝑊 .
We then make some comparative static analyses under the competitive mechanism, and
show the relationship between supplier’s optimal work rate and the base price, the incentive
factor and the disincentive factor in Table 5.
Table 5 comparative static analyses under competition model
𝑑𝑑

𝑑𝑑

𝜇𝜇𝑐𝑐∗

𝑑𝑑

𝜇𝜇𝑐𝑐∗

𝑑𝑑𝑑𝑑

𝑑𝑑𝑑𝑑
𝑑𝑑

𝑑𝑑𝑑𝑑

𝜇𝜇𝑐𝑐∗

𝜇𝜇𝑐𝑐∗ > 0,
i. if 𝜇𝜇𝑐𝑐∗ < 𝛼𝛼 and 16𝑘𝑘(𝜇𝜇𝑐𝑐∗ )3 + 24𝑘𝑘𝑘𝑘(𝜇𝜇𝑐𝑐∗ )2 − 8𝜔𝜔𝜔𝜔𝜇𝜇𝑐𝑐∗ − 6𝑝𝑝𝜇𝜇𝑐𝑐∗ > 𝑝𝑝𝑝𝑝 − 𝑟𝑟
ii. if 𝜇𝜇𝑐𝑐∗ > 𝛼𝛼 and 16𝑘𝑘(𝜇𝜇𝑐𝑐∗ )3 + 24𝑘𝑘𝑘𝑘(𝜇𝜇𝑐𝑐∗ )2 − 8𝜔𝜔𝜔𝜔𝜇𝜇𝑐𝑐∗ − 6𝑝𝑝𝜇𝜇𝑐𝑐∗ < 𝑝𝑝𝑝𝑝 − 𝑟𝑟
𝑑𝑑 𝑐𝑐∗
𝜇𝜇 < 0,
𝑑𝑑𝑑𝑑
i. if 𝜇𝜇𝑐𝑐∗ < 𝛼𝛼 and 16𝑘𝑘(𝜇𝜇𝑐𝑐∗ )3 + 24𝑘𝑘𝑘𝑘(𝜇𝜇𝑐𝑐∗ )2 − 8𝜔𝜔𝜔𝜔𝜇𝜇𝑐𝑐∗ − 6𝑝𝑝𝜇𝜇𝑐𝑐∗ < 𝑝𝑝𝑝𝑝 − 𝑟𝑟
ii. if 𝜇𝜇𝑐𝑐∗ > 𝛼𝛼 and 16𝑘𝑘(𝜇𝜇𝑐𝑐∗ )3 + 24𝑘𝑘𝑘𝑘(𝜇𝜇𝑐𝑐∗ )2 − 8𝜔𝜔𝜔𝜔𝜇𝜇𝑐𝑐∗ − 6𝑝𝑝𝜇𝜇𝑐𝑐∗ > 𝑝𝑝𝑝𝑝 − 𝑟𝑟
𝑑𝑑 𝑐𝑐∗
𝜇𝜇 > 0, if 16𝑘𝑘(𝜇𝜇𝑐𝑐∗ )3 + 24𝑘𝑘𝑘𝑘(𝜇𝜇𝑐𝑐∗ )2 − 8𝜔𝜔𝜔𝜔𝜇𝜇𝑐𝑐∗ − 6𝑝𝑝𝜇𝜇𝑐𝑐∗ > 𝑝𝑝𝑝𝑝 − 𝑟𝑟
𝑑𝑑𝑑𝑑

𝑑𝑑𝑑𝑑
𝑑𝑑
𝑑𝑑𝑑𝑑
𝑑𝑑

𝑑𝑑𝑑𝑑
𝑑𝑑
𝑑𝑑𝑑𝑑

𝜇𝜇𝑐𝑐∗ < 0, if 16𝑘𝑘(𝜇𝜇𝑐𝑐∗ )3 + 24𝑘𝑘𝑘𝑘(𝜇𝜇𝑐𝑐∗ )2 − 8𝜔𝜔𝜔𝜔𝜇𝜇𝑐𝑐∗ − 6𝑝𝑝𝜇𝜇𝑐𝑐∗ < 𝑝𝑝𝑝𝑝 − 𝑟𝑟

𝜇𝜇𝑐𝑐∗ > 0, if 16𝑘𝑘(𝜇𝜇𝑐𝑐∗ )3 + 24𝑘𝑘𝑘𝑘(𝜇𝜇𝑐𝑐∗ )2 − 8𝜔𝜔𝜔𝜔𝜇𝜇𝑐𝑐∗ − 6𝑝𝑝𝜇𝜇𝑐𝑐∗ > 𝑝𝑝𝑝𝑝 − 𝑟𝑟
𝜇𝜇𝑐𝑐∗ < 0, if 16𝑘𝑘(𝜇𝜇𝑐𝑐∗ )3 + 24𝑘𝑘𝑘𝑘(𝜇𝜇𝑐𝑐∗ )2 − 8𝜔𝜔𝜔𝜔𝜇𝜇𝑐𝑐∗ − 6𝑝𝑝𝜇𝜇𝑐𝑐∗ < 𝑝𝑝𝑝𝑝 − 𝑟𝑟

After both subtasks are completed, the outsourcing activity ends. We can express the
manufacturer’s profit as follows:
𝑐𝑐
Π𝑚𝑚
= 𝐴𝐴 − 𝐵𝐵𝐵𝐵[𝑇𝑇] − 2𝑊𝑊 𝑐𝑐

(13)

We redefine Eq. 12 as 𝐿𝐿(𝜇𝜇𝑐𝑐∗ , 𝜔𝜔) = 0. Furthermore, the manufacturer’s problem can be redeemed as:
𝑐𝑐
= 𝐴𝐴 − 𝐵𝐵𝐵𝐵[𝑇𝑇] − 2𝑊𝑊 𝑐𝑐
max𝜇𝜇𝑐𝑐∗ ,𝜔𝜔 Π𝑚𝑚
s.t. 𝐿𝐿(𝜇𝜇𝑐𝑐∗ , 𝜔𝜔) = 0

Due to the complexity of Eq. 12, the analytical expression of the optimal price decision made
by the manufacturer is hard to determine. Some numerical expression is provided in Section 5.
We seek to compare the results among different models with respect to manufacturer’s profit
function.
4.4 Mixed mechanism

In this case, we consider an incentive mechanism combining the above analyzed deadline and
competition incentives. The manufacturer makes a decision of optimal price 𝜔𝜔 paid to the suppliers, and the suppliers make the optimal work rate decision 𝜇𝜇𝑖𝑖 (𝑖𝑖 = 1,2).
The mixed incentive mechanism manufacturer can offer to the suppliers contains the following four components:
A base price. The manufacturer pays a base price of 𝜔𝜔 to the suppliers.

A specific due date. The manufacturer sets a pre-determined delivery due date 𝐷𝐷.
494

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Incentive modeling analysis in engineering applications and projects with stochastic duration time

A comparison of the faster and slower. Due to the randomness of project completion times, the
result ex post can be distinguished from a faster supplier and a slower supplier.

An incentive/disincentive scheme. The faster supplier will be awards a reward of 𝑟𝑟 per unit time
earlier than the deadline; the slower supplier will be charged a penalty of 𝑝𝑝 per unit time later
than the deadline.

For supplier 𝑖𝑖 (𝑖𝑖 = 1,2), after his completion of his own subtasks, he receives the pre-agreed
base price 𝜔𝜔. We can get that his expected payment gained from the manufacturer satisfies:
𝑊𝑊 𝑚𝑚 (𝑖𝑖) = 𝜔𝜔 + 𝑟𝑟𝑟𝑟[(𝐷𝐷 − 𝑋𝑋𝑖𝑖 )+ |𝑋𝑋𝑖𝑖 < 𝑋𝑋−𝑖𝑖 ] − 𝑝𝑝𝑝𝑝[(𝑋𝑋𝑖𝑖 − 𝐷𝐷)+ |𝑋𝑋𝑖𝑖 > 𝑋𝑋−𝑖𝑖 ]

Lemma 3. With some algebra calculations, we can get the following:
𝑊𝑊 𝑚𝑚 (𝑖𝑖) = 𝜔𝜔 + 𝑟𝑟𝑟𝑟 − 𝑟𝑟

1
𝜇𝜇𝑖𝑖 +𝜇𝜇−𝑖𝑖

+ 𝑟𝑟

1
𝑒𝑒 −(𝜇𝜇𝑖𝑖+𝜇𝜇−𝑖𝑖)𝐷𝐷
𝜇𝜇𝑖𝑖 +𝜇𝜇−𝑖𝑖

− 𝑝𝑝

1 𝜇𝜇𝑖𝑖 +𝜇𝜇−𝑖𝑖 −𝜇𝜇 𝐷𝐷
� 𝜇𝜇 𝑒𝑒 𝑖𝑖
𝜇𝜇−𝑖𝑖
𝑖𝑖

𝜇𝜇

(14)

− 𝜇𝜇 +𝜇𝜇𝑖𝑖 𝑒𝑒 −(𝜇𝜇𝑖𝑖+𝜇𝜇−𝑖𝑖)𝐷𝐷 �
𝑖𝑖

−𝑖𝑖

See Appendix for the certification process.
Furthermore, supplier 𝑖𝑖’s expected discounted profit can be expressed as follows:
Π𝑖𝑖𝑚𝑚 = 𝑊𝑊 𝑚𝑚 (𝑖𝑖)𝐸𝐸[𝑒𝑒 −𝛼𝛼𝑋𝑋𝑖𝑖 ] −

𝑘𝑘𝜇𝜇𝑖𝑖2
.
𝛼𝛼+𝜇𝜇𝑖𝑖

Proposition 5. The first-order condition of supplier’s profit satisfies that:
4𝑘𝑘(𝜇𝜇𝑚𝑚∗ )4 + Φ(𝜇𝜇𝑚𝑚∗ )3 − Ψ(𝜇𝜇𝑚𝑚∗ )2 − Υ𝜇𝜇𝑚𝑚∗ = 𝑟𝑟𝑟𝑟 − 𝑟𝑟𝑟𝑟𝑒𝑒 −2𝜇𝜇
−𝜇𝜇𝑚𝑚∗ 𝐷𝐷

−2𝜇𝜇𝑚𝑚∗ 𝐷𝐷

𝑚𝑚∗ 𝐷𝐷

(15)
(16)

in which Φ = 8𝑘𝑘𝑘𝑘 − 8𝑝𝑝𝑝𝑝𝑒𝑒
+ 2𝑝𝑝𝑝𝑝𝑒𝑒
,
𝑚𝑚∗
𝑚𝑚∗
𝑚𝑚∗
−2𝜇𝜇𝑚𝑚∗ 𝐷𝐷
−𝜇𝜇𝑚𝑚∗ 𝐷𝐷
Ψ = 4𝜔𝜔𝜔𝜔 + 4𝑟𝑟𝑟𝑟𝑟𝑟 − 2𝑟𝑟𝑟𝑟𝑒𝑒
+ 8𝑝𝑝𝑝𝑝𝑝𝑝𝑒𝑒
− 2𝑝𝑝𝑝𝑝𝑝𝑝𝑒𝑒 −2𝜇𝜇 𝐷𝐷 + 4𝑝𝑝𝑒𝑒 −𝜇𝜇 𝐷𝐷 + 𝑝𝑝𝑒𝑒 −2𝜇𝜇 𝐷𝐷 , and
𝑚𝑚∗
𝑚𝑚∗
𝑚𝑚∗
𝑚𝑚∗
Υ = 3𝑟𝑟 − 3𝑟𝑟𝑒𝑒 −2𝜇𝜇 𝐷𝐷 − 2𝑟𝑟𝑟𝑟𝑟𝑟𝑒𝑒 −2𝜇𝜇 𝐷𝐷 − 4𝑝𝑝𝑝𝑝𝑒𝑒 −𝜇𝜇 𝐷𝐷 + 3𝑝𝑝𝑝𝑝𝑒𝑒 −2𝜇𝜇 𝐷𝐷 .

The specific certification process is in Appendix.
If 𝑟𝑟 = 𝑝𝑝 = 0, we have that 𝜇𝜇𝑚𝑚∗ = 𝜇𝜇∗ . In equilibrium, we have 𝑊𝑊 𝑚𝑚 (1) = 𝑊𝑊 𝑚𝑚 (2) = 𝑊𝑊 𝑚𝑚 . We
can infer that if formula Eq. 15 has a maximum value, then supplier’s optimal work rate satisfies
Eq. 16; otherwise, we can find the maximum value of Eq. 15 by limiting the scope of parameter 𝜔𝜔.
Furthermore, we have that the manufacturer’s profit satisfies that:
𝑚𝑚
Π𝑚𝑚
= 𝐴𝐴 − 𝐵𝐵𝐵𝐵[𝑇𝑇] − 2𝑊𝑊 𝑚𝑚

(17)

In the next section, we also provide the manufacturer’s optimal price decision through some
numerical examples. Section 4 provides three kinds of incentive mechanisms to compare with
the benchmark case. Deadline, competition, and mixed incentive mechanisms are analyzed and
the comparison results are given in the following analysis.

5. Numerical analysis

Due to the complexity of the manufacturer’s profit function, we cannot get the analytical expression of manufacturer’s optimal price decision under the three proposed incentive mechanism
models. We compare the results among different models through some numerical examples.
In our numerical examples, we set 𝛼𝛼 = 1, 𝑘𝑘 = 1, 𝐴𝐴 = 40 and 𝐵𝐵 = 1. We use some abbreviations to represent different incentive models, i.e., BM is short for base model, DM is short for
deadline incentive model, CM is short for competition model and MM short for mixed model.
Fig. 1 shows the results among different models when parameter 𝑝𝑝 varies from 0 to 10 given
parameter 𝐷𝐷 = 0.5. With different values of parameter 𝑟𝑟, i.e., 𝑟𝑟 = 0, 2, 4, 6, 8, 10, we get six numerical examples, and manufacture’s optimal profit among different incentive mechanisms differs under different circumstances.
Observation 1. The comparison results among different incentive models differ concerning different absolute value and relative relationship between parameter 𝑟𝑟 and 𝑝𝑝. All four mechanisms
may be in the domination position under different circumstances.

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Zhao, Su

Fig. 1 Numerical examples with varying parameter 𝑟𝑟 with given 𝐷𝐷 = 0.5

Fig. 2 Numerical examples with varying parameter 𝐷𝐷 with given 𝑟𝑟 = 4

We then show the comparison results among different models with a different specific deadline. Fig. 2 gives the specific results with given parameter 𝑟𝑟 = 4, and 𝑝𝑝 varies from 0 to 10. By
choosing parameter 𝐷𝐷 among 0.35, 0.5 and 0.65, we get three numerical examples.

Observation 2. Given the absolute and relative value of parameters 𝑟𝑟 and 𝑝𝑝, the influence of
specific due date 𝐷𝐷 has a different impact on the comparison results among different incentive
models. A smaller 𝐷𝐷 has a same impact on the results with a smaller 𝑟𝑟 when other parameters
are the same; a larger 𝐷𝐷 has a same impact on the results with a lager 𝑟𝑟 when other parameters
are the same.

496

Fig. 3 Numerical examples with varying parameter 𝑟𝑟 and 𝑝𝑝

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Incentive modeling analysis in engineering applications and projects with stochastic duration time

Fig. 3 shows the results with varying parameter 𝐷𝐷 from 0 to 2. With 3 different pairs of parameters 𝑟𝑟 and 𝑝𝑝, we get different numerical results on manufacturer’s optimal choice from
choosing different incentive mechanisms.

Observation 3. Different combination pairs of parameters 𝑟𝑟 and 𝑝𝑝 has a bigger influence on the
performance of incentive model CM. The performance of incentive models DM and MM decreases with the specific due date 𝐷𝐷. Model MM dominates DM when parameter 𝐷𝐷 is smaller and
model DM nominates MM when parameter 𝐷𝐷 is larger.

In this Section, we show the comparative results through some numerical examples, and the
meaning results can shed light on some managerial implications for the real-world business
practice.

6. Conclusion

We consider an incentive mechanism design problem faced by a manufacturer when conducting
outsourcing activities. Three different kinds of incentive mechanisms are proposed in our article.
We get some comparative results and management implications for the real-world business
practice.
6.1 Findings

We proposed three kinds of incentive mechanisms. The first one is deadline incentive mechanism, in which the manufacturer impose a specific deadline and when the supplier completes
earlier than the deadline will be rewarded and will be penalized when completes later than the
deadline.
The second incentive mechanism is competition mechanism. In our setting where the manufacturer outsources two parallel subtasks to two different suppliers, the two suppliers may end
the corresponding subtask with different duration times due to the stochasticity of the project.
The third incentive mechanism combines the above two mechanisms, which we referred to as
mixed incentive mechanism. It considers both the deadline in projects and competitiveness in
parallel projects. Only the faster supplier is awarded when earlier than the deadline and only the
slower supplier is penalized when later than the deadline.
We find that the results of the comparison of the three mechanisms depend on the related parameters settings, and are all likely to be in a dominant position. We give the specific results
through some numerical examples.
We assume that the supplier can get a base price 𝜔𝜔 in the above analysis, and different base
price the manufacturer paid to the supplier is related to different project size. Thus, our findings
can be generalized to projects of different magnitudes, which is meaningful to the real-world
business practice.
6.2 Future research

There are some extensions that we can continue to work on in the future. We list some possible
research schemes in the following content.
In our work, we consider a two parallel sub-projects structure, however, what happens when
there are more than two tasks in parallel? It could be a lot more complicated and needs further
study by scholars. We can figure that the deadline incentive mechanism can gain the same result
as we show in sub-section 4.2. Regarding to competition incentive mechanism, different from
two parallel sub-projects structure, when there are 𝑛𝑛 (𝑛𝑛 ≥ 2) tasks in parallel, we can set that
the fastest supplier can get a reward and the slowest supplier will get a penalty. The specific
incentive result needs our further study. As to the mixed incentive mechanism, we can also consider that the fastest supplier can get a reward if his completion time is earlier than the deadline
and vice versa. However, in this case, the modeling process may be very complicated and whether we can gain some insightful results is unclear.
Serial project structure is another direction we can pay attention to. Also, we are wondering
that what happens when there are more than two tasks in a general assembly network? Do any

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Zhao, Su

of the results reported in this paper continue to hold? When involving network structure, the
study will be more complicated and difficult.
The above discussion prompts us to pay more attention to the problems in project supply
chain management and try to adopt new methods to solve these problems. In addition, emerging
applications in project management will certainly identify further important research questions
and opportunities in addition to the issues we discuss here. With the increasing complexity of
project supply chain and the increasing subcontracting activities, how to solve this problem is a
great challenge. Therefore, it requires the attention and time of our scholars to conduct research
and demonstrate some management implications to guide guidance into the real world.
We believe that in the future, scholars will make great progress in academic research in this
field, which will further guide the practical business application of project management. In the
next several years we will have more and more important results, which are of course the basis
for future research.

Acknowledgement

This work was supported by the Technology Research Program of Chongqing Municipal Education Commission (Grant
No. KJQN202200808).

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Appendix

Proof of Proposition 2. By taking the first-order condition of manufacturer’s profit function, we
have:
𝑑𝑑
3𝐵𝐵
𝛼𝛼
Π𝑚𝑚 = −2 +
.
2 ∙
𝜔𝜔𝜔𝜔
𝑑𝑑𝑑𝑑

2�−𝛼𝛼+�𝛼𝛼 2 +

𝜔𝜔𝜔𝜔
�
𝑘𝑘

2𝑘𝑘�𝛼𝛼 2 +

By taking the second-order condition of Π𝑚𝑚 , we can obtain:
𝑑𝑑 2

𝑑𝑑𝜔𝜔2

Π𝑚𝑚 =

3𝐵𝐵𝐵𝐵
4𝑘𝑘

𝑑𝑑 2

∙

−1

3
𝜔𝜔𝜔𝜔
2�−𝛼𝛼+�𝛼𝛼 2 + �
𝑘𝑘

∙�

𝛼𝛼

𝜔𝜔𝜔𝜔
2𝑐𝑐�𝛼𝛼 2 +
𝑘𝑘

2

� +

3𝐵𝐵𝐵𝐵
4𝑘𝑘

∙

𝑘𝑘

1

2
𝜔𝜔𝜔𝜔
�−𝛼𝛼+�𝛼𝛼 2 + 𝑘𝑘 �

∙

−1

3
𝜔𝜔𝜔𝜔
2�𝛼𝛼2 + �2
𝑘𝑘

.

We can get that 2 Π𝑚𝑚 < 0. Thus, we can infer that Π𝑚𝑚 is a concave function with respect to
𝑑𝑑𝜔𝜔
parameter 𝜔𝜔.
Proof of Lemma 1. From 𝑊𝑊 𝑑𝑑 = 𝜔𝜔 + 𝑟𝑟𝑟𝑟(𝐷𝐷 − 𝑋𝑋𝑖𝑖 )+ − 𝑝𝑝𝑝𝑝(𝑋𝑋𝑖𝑖 − 𝐷𝐷)+ , we can get:
𝐷𝐷
∞
𝑊𝑊 𝑑𝑑 = 𝜔𝜔 + 𝑟𝑟 ∫0 (𝐷𝐷 − 𝑡𝑡𝑖𝑖 )𝑓𝑓(𝑡𝑡𝑖𝑖 )𝑑𝑑𝑡𝑡𝑖𝑖 − 𝑝𝑝 ∫𝐷𝐷 (𝑡𝑡𝑖𝑖 − 𝐷𝐷)𝑓𝑓(𝑡𝑡𝑖𝑖 )𝑑𝑑𝑡𝑡𝑖𝑖 = 𝜔𝜔 + 𝑝𝑝𝑝𝑝 − 𝑝𝑝𝑝𝑝(𝑋𝑋𝑖𝑖 ) + (𝑟𝑟 −
𝐷𝐷

𝐷𝐷

𝑝𝑝) �𝐷𝐷 ∫0 𝑓𝑓(𝑡𝑡𝑖𝑖 )𝑑𝑑𝑡𝑡𝑖𝑖 − ∫0 𝑡𝑡𝑖𝑖 𝑓𝑓(𝑡𝑡𝑖𝑖 )𝑑𝑑𝑡𝑡𝑖𝑖 �.

Thus, we have 𝑊𝑊 𝑑𝑑 = +𝑟𝑟𝑟𝑟 − 𝑟𝑟

1
𝜇𝜇𝑖𝑖

+ (𝑟𝑟 − 𝑝𝑝)

1 −𝜇𝜇 𝐷𝐷
𝑒𝑒 𝑖𝑖 , and the lemma is
𝜇𝜇𝑖𝑖

proved.

Proof of Proposition 3. By some calculations, we can obtain:
𝑑𝑑
1
[𝜔𝜔𝜔𝜔 + 𝑟𝑟𝑟𝑟𝑟𝑟 + 𝑟𝑟 − (𝑟𝑟 − 𝑝𝑝)𝑒𝑒 −𝜇𝜇𝑖𝑖𝐷𝐷 (1 + 𝐷𝐷𝐷𝐷 + 𝐷𝐷𝜇𝜇𝑖𝑖 ) − 𝑘𝑘(𝜇𝜇𝑖𝑖 )2 − 2𝑘𝑘𝑘𝑘𝜇𝜇𝑖𝑖 ].
Π 𝑑𝑑 =
𝑑𝑑𝜇𝜇 𝑖𝑖
(𝛼𝛼+𝜇𝜇 )2
𝑑𝑑 2
Π 𝑑𝑑
𝑑𝑑(𝜇𝜇𝑖𝑖 )2 𝑖𝑖

𝑖𝑖

=

𝑖𝑖

1
[(𝑟𝑟
(𝛼𝛼+𝜇𝜇𝑖𝑖 )3

− 𝑝𝑝)𝑒𝑒 −𝜇𝜇𝑖𝑖𝐷𝐷 (𝐷𝐷𝐷𝐷 + 𝐷𝐷𝜇𝜇𝑖𝑖 )𝐷𝐷(𝛼𝛼 + 𝜇𝜇𝑖𝑖 ) + 2(𝑟𝑟 − 𝑝𝑝)𝑒𝑒 −𝜇𝜇𝑖𝑖𝐷𝐷 (1 + 𝐷𝐷𝐷𝐷 + 𝐷𝐷𝜇𝜇𝑖𝑖 ) − 2𝑐𝑐𝛼𝛼 2 −

2(𝜔𝜔𝜔𝜔 + 𝑟𝑟𝑟𝑟𝑟𝑟 + 𝑟𝑟)].
is a concave function. Thus, we can get the result by taking the

that Π𝑖𝑖𝑑𝑑

If 𝑟𝑟 ≤ 𝑝𝑝, we can get
first order condition of (7).
If 𝑟𝑟 > 𝑝𝑝, for sufficiently large 𝜇𝜇𝑖𝑖 , we have
1
[𝜔𝜔𝜔𝜔
𝛼𝛼 2

𝑑𝑑
Π 𝑑𝑑
𝑑𝑑𝜇𝜇𝑖𝑖 𝑖𝑖

𝑑𝑑
Π𝑖𝑖𝑑𝑑
𝑑𝑑𝜇𝜇
𝜇𝜇𝑖𝑖 →0
𝑖𝑖

< 0. And we can infer that lim

=

+ 𝑟𝑟𝑟𝑟𝑟𝑟 + 𝑟𝑟 − (𝑟𝑟 − 𝑝𝑝)(1 + 𝐷𝐷𝐷𝐷)] = 𝜔𝜔𝜔𝜔 + 𝑝𝑝 + 𝑝𝑝𝑝𝑝𝑝𝑝 > 0.
Therefore, we can infer that there exists a maximum value of the supplier’s profit function
and the proposition is proved.
Proof of Lemma 2. Let 𝑍𝑍 = 𝑋𝑋𝑖𝑖 − 𝑋𝑋−𝑖𝑖 , thus, the probability density function of random variable 𝑍𝑍
when 𝑍𝑍 > 0 satisfies that
∞
𝜇𝜇 𝜇𝜇
𝑓𝑓(𝑧𝑧) = ∫𝑡𝑡 =𝑧𝑧 𝜇𝜇𝑖𝑖 𝑒𝑒 −𝜇𝜇𝑖𝑖𝑡𝑡𝑖𝑖 𝜇𝜇−𝑖𝑖 𝑒𝑒 −𝜇𝜇−𝑖𝑖(𝑡𝑡𝑖𝑖−𝑧𝑧) 𝑑𝑑𝑡𝑡𝑖𝑖 = (𝜇𝜇 𝑖𝑖 −𝑖𝑖 ) 𝑒𝑒 −𝜇𝜇𝑖𝑖𝑧𝑧 .
+𝜇𝜇
𝑖𝑖

𝑖𝑖

Therefore, we can get that
∞
𝜇𝜇−𝑖𝑖
𝐸𝐸[𝑋𝑋𝑖𝑖 − 𝑋𝑋−𝑖𝑖 ]+ = ∫0 𝑧𝑧𝑧𝑧(𝑧𝑧)𝑑𝑑𝑑𝑑 = 𝜇𝜇 (𝜇𝜇 +𝜇𝜇
𝑖𝑖

Thus, the lemma is proved.

𝑖𝑖

−𝑖𝑖

−𝑖𝑖 )

.

Proof of Proposition 4. From formula Eq. 12, we have
𝑑𝑑
1
𝜇𝜇
Π𝑖𝑖𝑐𝑐 =
�𝜔𝜔𝜔𝜔(𝜇𝜇𝑖𝑖 + 𝜇𝜇−𝑖𝑖 )2 + 𝑟𝑟(𝛼𝛼 + 𝜇𝜇𝑖𝑖 ) − 𝑟𝑟 𝜇𝜇 𝑖𝑖 (𝜇𝜇𝑖𝑖 + 𝜇𝜇−𝑖𝑖 ) + 𝑝𝑝𝜇𝜇−𝑖𝑖 (𝛼𝛼 + 2𝜇𝜇𝑖𝑖 + 𝜇𝜇−𝑖𝑖 ) −
)2 (𝜇𝜇 +𝜇𝜇 )2
𝑑𝑑𝜇𝜇𝑖𝑖

(𝛼𝛼+𝜇𝜇𝑖𝑖

𝑖𝑖

−𝑖𝑖

�𝑘𝑘𝜇𝜇𝑖𝑖2

−𝑖𝑖

+ 2𝑘𝑘𝑘𝑘𝜇𝜇𝑖𝑖 �(𝜇𝜇𝑖𝑖 + 𝜇𝜇−𝑖𝑖 )2 �.

Advances in Production Engineering & Management 18(4) 2023

499

Zhao, Su

For sufficiently large 𝜇𝜇𝑖𝑖 , we have

𝑑𝑑
Π 𝑐𝑐
𝑑𝑑𝜇𝜇𝑖𝑖 𝑖𝑖

< 0. And we can get that lim

𝑑𝑑

𝜇𝜇𝑖𝑖 →0 𝑑𝑑𝜇𝜇𝑖𝑖

Π𝑖𝑖𝑐𝑐 =

1
2
2 �𝜔𝜔𝜔𝜔𝜇𝜇−𝑖𝑖
𝛼𝛼 2 𝜇𝜇−𝑖𝑖

+

𝑟𝑟𝑟𝑟 + 𝑝𝑝𝜇𝜇−𝑖𝑖 (𝛼𝛼 + 𝜇𝜇−𝑖𝑖 )� > 0.
Thus, we can infer that there exists a maximum value of the supplier’s profit function and the
proposition is proved.
Pro(𝑋𝑋𝑖𝑖 <𝑡𝑡,𝑋𝑋𝑖𝑖 <𝑋𝑋−𝑖𝑖 )
.
Pro(𝑋𝑋𝑖𝑖 <𝑋𝑋−𝑖𝑖 )
𝑡𝑡
∞
𝜇𝜇
∫0 𝜇𝜇𝑖𝑖 𝑒𝑒 −𝜇𝜇𝑖𝑖𝑡𝑡𝑖𝑖 ∫𝑡𝑡 𝜇𝜇−𝑖𝑖 𝑒𝑒 −𝑢𝑢−𝑖𝑖𝑡𝑡−𝑖𝑖 𝑑𝑑𝑡𝑡−𝑖𝑖 𝑑𝑑𝑡𝑡𝑖𝑖 = 𝜇𝜇 +𝜇𝜇𝑖𝑖 �1 −
𝑖𝑖
𝑖𝑖
−𝑖𝑖

Proof of Lemma 3. The conditional distribution function F(𝑋𝑋𝑖𝑖 < 𝑡𝑡|𝑋𝑋𝑖𝑖 < 𝑋𝑋−𝑖𝑖 ) =
We first examine that Pro(𝑋𝑋𝑖𝑖 < 𝑡𝑡, 𝑋𝑋𝑖𝑖 < 𝑋𝑋−𝑖𝑖 ) =
𝑒𝑒 −(𝜇𝜇𝑖𝑖+𝜇𝜇−𝑖𝑖)𝑡𝑡 �.

𝑡𝑡

∞

𝜇𝜇

By calculation, we can infer that Pro(𝑋𝑋𝑖𝑖 < 𝑋𝑋−𝑖𝑖 ) = ∫0 𝜇𝜇−𝑖𝑖 𝑒𝑒 −𝜇𝜇−𝑖𝑖𝑡𝑡−𝑖𝑖 ∫0 −𝑖𝑖 𝜇𝜇𝑖𝑖 𝑒𝑒 −𝜇𝜇𝑖𝑖𝑡𝑡𝑖𝑖 𝑑𝑑𝑡𝑡𝑖𝑖 𝑑𝑑𝑡𝑡−𝑖𝑖 = 𝜇𝜇 +𝜇𝜇𝑖𝑖 .
𝑖𝑖

−𝑖𝑖

Thus, we have F(𝑋𝑋𝑖𝑖 < 𝑡𝑡|𝑋𝑋𝑖𝑖 < 𝑋𝑋−𝑖𝑖 ) = 1 − 𝑒𝑒 −(𝜇𝜇𝑖𝑖+𝜇𝜇−𝑖𝑖)𝑡𝑡 .
𝐷𝐷
Thus, we can get that 𝐸𝐸[(𝐷𝐷 − 𝑋𝑋𝑖𝑖 )+ |𝑋𝑋𝑖𝑖 < 𝑋𝑋−𝑖𝑖 ] = (𝜇𝜇𝑖𝑖 + 𝜇𝜇−𝑖𝑖 ) ∫0 (𝐷𝐷 − 𝑡𝑡) 𝑒𝑒 −(𝜇𝜇𝑖𝑖+𝜇𝜇−𝑖𝑖)𝑡𝑡 𝑑𝑑𝑑𝑑 = 𝐷𝐷 +

1
�𝑒𝑒 −(𝜇𝜇𝑖𝑖+𝜇𝜇−𝑖𝑖)𝐷𝐷
𝜇𝜇𝑖𝑖 +𝜇𝜇−𝑖𝑖

− 1�.

Pro(𝑋𝑋𝑖𝑖 <𝑡𝑡,𝑋𝑋𝑖𝑖 >𝑋𝑋−𝑖𝑖 )
. Furthermore, we have
Pro(𝑋𝑋𝑖𝑖 >𝑋𝑋−𝑖𝑖 )
𝑡𝑡
𝑡𝑡
𝜇𝜇−𝑖𝑖
− 𝑒𝑒 −𝜇𝜇𝑖𝑖𝑡𝑡 +
that
Pro(𝑋𝑋𝑖𝑖 < 𝑡𝑡, 𝑋𝑋𝑖𝑖 > 𝑋𝑋−𝑖𝑖 ) = ∫0 𝜇𝜇𝑖𝑖 𝑒𝑒 −𝜇𝜇𝑖𝑖𝑡𝑡𝑖𝑖 ∫0 𝑖𝑖 𝜇𝜇−𝑖𝑖 𝑒𝑒 −𝜇𝜇−𝑖𝑖𝑡𝑡−𝑖𝑖 𝑑𝑑𝑡𝑡−𝑖𝑖 𝑑𝑑𝑡𝑡𝑖𝑖 = 𝜇𝜇 +𝜇𝜇
𝑖𝑖
−𝑖𝑖
𝜇𝜇𝑖𝑖
𝜇𝜇−𝑖𝑖
𝑒𝑒 −(𝜇𝜇𝑖𝑖+𝜇𝜇−𝑖𝑖)𝑡𝑡 , and we can infer that Pro(𝑋𝑋𝑖𝑖 > 𝑋𝑋−𝑖𝑖 ) = 𝜇𝜇 +𝜇𝜇
.
𝜇𝜇𝑖𝑖 +𝜇𝜇−𝑖𝑖
𝑖𝑖
−𝑖𝑖
∞
𝜇𝜇
Thus, we can get that 𝐸𝐸[(𝑋𝑋𝑖𝑖 − 𝐷𝐷)+ |𝑋𝑋𝑖𝑖 > 𝑋𝑋−𝑖𝑖 ] = 𝜇𝜇 𝑖𝑖 (𝜇𝜇𝑖𝑖 + 𝜇𝜇−𝑖𝑖 ) ∫𝐷𝐷 (𝑡𝑡 − 𝐷𝐷) �𝑒𝑒 −𝜇𝜇𝑖𝑖𝑡𝑡 −
−𝑖𝑖
1 𝜇𝜇𝑖𝑖 +𝜇𝜇−𝑖𝑖 −𝜇𝜇 𝐷𝐷
𝜇𝜇𝑖𝑖
−(𝜇𝜇𝑖𝑖 +𝜇𝜇−𝑖𝑖 )𝐷𝐷
𝑖𝑖 −
𝑒𝑒 −(𝜇𝜇𝑖𝑖+𝜇𝜇−𝑖𝑖)𝑡𝑡 �𝑑𝑑𝑑𝑑 =
𝑒𝑒
𝑒𝑒
�
�.
𝜇𝜇𝑖𝑖
𝜇𝜇𝑖𝑖 +𝜇𝜇−𝑖𝑖
𝜇𝜇−𝑖𝑖

By the same way, we can get that F(𝑋𝑋𝑖𝑖 < 𝑡𝑡|𝑋𝑋𝑖𝑖 > 𝑋𝑋−𝑖𝑖 ) =

Thus, the lemma is proved.

Proof of Proposition 5. From the supplier’s profit function, we can get
𝑑𝑑
Π 𝑚𝑚
𝑑𝑑𝜇𝜇𝑖𝑖 𝑖𝑖

1
�(𝜔𝜔𝜔𝜔+𝑟𝑟𝑟𝑟𝑟𝑟)(𝜇𝜇𝑖𝑖 + 𝜇𝜇−𝑖𝑖 )2 − 𝑟𝑟(𝛼𝛼 + 𝜇𝜇𝑖𝑖 )�𝑒𝑒 −(𝜇𝜇𝑖𝑖+𝜇𝜇−𝑖𝑖)𝐷𝐷 − 1� − 𝑟𝑟(𝜇𝜇𝑖𝑖 +
(𝛼𝛼+𝜇𝜇𝑖𝑖 )2 (𝜇𝜇𝑖𝑖 +𝜇𝜇−𝑖𝑖 )2
𝛼𝛼
𝜇𝜇 +𝜇𝜇
(𝜇𝜇𝑖𝑖 + 𝜇𝜇−𝑖𝑖 )2 � 𝑖𝑖 −𝑖𝑖 𝑒𝑒 −𝜇𝜇𝑖𝑖𝐷𝐷 −
𝜇𝜇−𝑖𝑖 )�𝑒𝑒 −(𝜇𝜇𝑖𝑖+𝜇𝜇−𝑖𝑖)𝐷𝐷 − 1� − 𝑟𝑟𝑟𝑟(𝛼𝛼 + 𝜇𝜇𝑖𝑖 )(𝜇𝜇𝑖𝑖 + 𝜇𝜇−𝑖𝑖 )𝑒𝑒 −(𝜇𝜇𝑖𝑖+𝜇𝜇−𝑖𝑖)𝐷𝐷 − 𝑝𝑝
�
𝜇𝜇
𝜇𝜇

=

−𝑖𝑖

𝑖𝑖

𝜇𝜇𝑖𝑖
𝜇𝜇
𝜇𝜇 +𝜇𝜇
𝜇𝜇
𝑒𝑒 −(𝜇𝜇𝑖𝑖+𝜇𝜇−𝑖𝑖)𝐷𝐷 � − 𝑝𝑝 𝑖𝑖 (𝛼𝛼 + 𝜇𝜇𝑖𝑖 )(𝜇𝜇𝑖𝑖 + 𝜇𝜇−𝑖𝑖 )2 �−𝐷𝐷 𝑖𝑖 −𝑖𝑖 𝑒𝑒 −𝜇𝜇𝑖𝑖𝐷𝐷 − −𝑖𝑖2 𝑒𝑒 −𝜇𝜇𝑖𝑖𝐷𝐷 +
𝜇𝜇𝑖𝑖 +𝜇𝜇−𝑖𝑖
𝜇𝜇−𝑖𝑖
𝜇𝜇𝑖𝑖
𝜇𝜇𝑖𝑖
𝜇𝜇𝑖𝑖
𝜇𝜇
𝐷𝐷
𝑒𝑒 −(𝜇𝜇𝑖𝑖+𝜇𝜇−𝑖𝑖)𝐷𝐷 − (𝜇𝜇 +𝜇𝜇−𝑖𝑖 )2 𝑒𝑒 −(𝜇𝜇𝑖𝑖+𝜇𝜇−𝑖𝑖)𝐷𝐷 � − �𝑘𝑘𝜇𝜇𝑖𝑖2 + 2𝑘𝑘𝑘𝑘𝜇𝜇𝑖𝑖 �(𝜇𝜇𝑖𝑖 + 𝜇𝜇−𝑖𝑖 )2 �.
𝜇𝜇𝑖𝑖 +𝜇𝜇−𝑖𝑖
𝑖𝑖
−𝑖𝑖

By letting 𝜇𝜇𝑖𝑖 = 𝜇𝜇−𝑖𝑖 = 𝜇𝜇𝑚𝑚∗ , we can get formula Eq. 16.
Thus, the proposition is proved.

500

Advances in Production Engineering & Management 18(4) 2023

Advances in Production Engineering & Management

ISSN 1854-6250

Volume 18 | Number 4 | December 2023 | pp 501–511

Journal home: apem-journal.org

https://doi.org/10.14743/apem2023.4.488

Original scientific paper

Reduction of surface defects by optimization of casting
speed using genetic programming: An industrial case study
Kovacic, M.a,b,c,*, Zuperl, U.d, Gusel, L.d, Brezocnik, M.d
aŠTORE

STEEL, d.o.o., Research and Development, Štore, Slovenia
of Ljubljana, Faculty of Mechanical Engineering, Laboratory for Fluid Dynamics and Thermodynamics, Ljubljana,
Slovenia
cCollege of Industrial Engineering, Celje, Slovenia
dUniversity of Maribor, Faculty of mechanical engineering, Maribor, Slovenia
bUniversity

ABSTRACT

ARTICLE INFO

Štore Steel Ltd. produces more than 200 different types of steel with a continuous caster installed in 2016. Several defects, mostly related to thermomechanical behaviour in the mould, originate from the continuous casting process. The same casting speed of 1.6 m/min was used for all steel grades. In
May 2023, a project was launched to adjust the casting speed according to the
casting temperature. This adjustment included the steel grades with the highest number of surface defects and different carbon content: 16MnCrS5, C22,
30MnVS5, and 46MnVS5. For every 10 °C deviation from the prescribed casting temperature, the speed was changed by 0.02 m/min. During the 2-month
period, the ratio of rolled bars with detected surface defects (inspected by an
automatic control line) decreased for the mentioned steel grades. The decreases were from 11.27 % to 7.93 %, from 12.73 % to 4.11 %, from 16.28 %
to 13.40 %, and from 25.52 % to 16.99 % for 16MnCrS5, C22, 30MnVS5, and
46MnVS5, respectively. Based on the collected chemical composition and
casting parameters from these two months, models were obtained using
linear regression and genetic programming. These models predict the ratio of
rolled bars with detected surface defects and the length of detected surface
defects. According to the modelling results, the ratio of rolled bars with detected surface defects and the length of detected surface defects could be
minimally reduced by 14 % and 189 %, respectively, using casting speed
adjustments. A similar result was achieved from July to November 2023 by
adjusting the casting speed for the other 27 types of steel. The same was predicted with the already obtained models. Genetic programming outperformed
linear regression.

Keywords:
Continuous casting of steel;
Surface defects;
Automatic control;
Machine learning;
Modelling;
Optimization;
Prediction;
Linear regression;
Genetic programming

*Corresponding author:
miha.kovacic@store-steel.si
(Kovacic, M.)

Article history:
Received 3 November 2023
Revised 15 December 2023
Accepted 21 December 2023
Content from this work may be used under the terms of
the Creative Commons Attribution 4.0 International
Licence (CC BY 4.0). Any further distribution of this work
must maintain attribution to the author(s) and the title of
the work, journal citation and DOI.

1. Introduction
Modern steel production is hard to imagine without continuous casting. However, due to the
thermomechanical behaviour during continuous casting, especially in the mould, several types of
defects can occur on the cast material. These defects can also manifest on the rolled material.
They can be reduced or eliminated with several approaches:

• Optimization of casting equipment (e.g., tundish, submerged entry nozzles, mould or water
sprays geometry, casting powder, tundish powder),
• Optimization of secondary metallurgy (e.g., deoxidation, refinement, homogenization, stirring), and
501

Kovacic, Zuperl, Gusel, Brezocnik

• Optimization of casting parameters (e.g., casting temperature, speed, cooling water flow
and pressure).

In the literature, adjustments to the casting speed have been discussed in relation to casting
defects (e.g., surface defects [1-13], breakouts [9, 14, 15]) and productivity [1, 3, 14-16]. These
discussions have involved individual steel grades [1, 9] or have been more general [3-5, 10-12,
15-17]. Additional factors such as the melt level in the mould [17, 18], melt level fluctuations
[12, 17, 18], and melt flow [3] have also been examined. Artificial intelligence approaches have
been utilized as well [4, 7, 10, 16, 17]. Unfortunately, these attempts have only been practically
implemented in an industrial environment a few times [11, 15, 17].
This article investigates the reduction of surface defects, detected during the examination
(via an automatic control line) of the rolled material, using casting speed adjustments based on
the casting temperature. The average casting speed adjustments were calculated for several different grades using an in-house developed solidification model [19, 20]. The primary objective
during simulations was to maintain the same metallurgical length (i.e., the distance from the
mould to the location where the entire melt solidifies) by adjusting the casting speed when the
casting temperature changed. Based on several simulations, for every 10 °C deviation from the
prescribed casting temperature, the speed was adjusted by 0.02 m/min. The casting temperature was determined based on the content of carbon, sulphur, and aluminium, the liquidus temperature (calculated using the Wensel equation [21]), and the number of cast sequences (several
batches – individual melts are continuously cast without interruption).
The materials and methods section discusses the significance of data collection. Subsequently, this data is utilized to predict the ratio of rolled bars with detected surface defects and their
lengths using linear regression and genetic programming methods. The results are then analysed and implemented in practice. Finally, the general results are analysed and conclusions are
drawn.

2. Materials and methods

Production at the Štore Steel plant begins with scrap melting in an electric arc furnace, followed
by tapping, ladle treatment (i.e., secondary metallurgy), and continuous casting of billets measuring 180 mm × 180 mm. The billets can undergo additional heat treatment or be cooled under
hoods. Before rolling, they are reheated. The rolled bars can then be straightened, examined for
inner soundness and surface quality, cut, sawn, chamfered, drilled, and peeled.
Since March 2016, a new two-strand continuous casting machine with a radius of 9 m has
been in operation. The solidification process involves a water-cooled copper mould for primary
cooling, water sprays for secondary cooling, and air cooling for tertiary cooling.
During primary cooling, the solidified shell is subjected to thermomechanical stresses, which
can lead to numerous casting defects. In the case of square billets, non-uniform shell solidification (i.e., non-uniform heat removal in the mould) can result in rhombic distortion (i.e., rhomboidity). This distortion can cause off-corner cracks, which manifest as longitudinally opened
surface defects on the rolled material. These defects are typically detected during an automatic
control line examination.
The origin of open surface defects on rolled material can be confirmed based on metallurgical
reports, which include analyses of billets and both flat and round bars, dating back to the initiation of the continuous caster in 2016. Fig. 1 displays a macro-etched sample of the billet’s crosssection. A typical bright macrostructure is observable due to the operation of the mould’s electromagnetic stirrers (indicated by arrows). Subsurface cracks outside the corner are also visible
in the upper left and lower right corners. The same billet macrostructure and open cracks outside the corners can be seen in the cross-section of the macro-etched round rolled bar, as shown
in Fig. 2.

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Reduction of surface defects by optimization of casting speed using genetic programming: An industrial case study

Fig. 1 A macro-etched sample of the billet’s cross-section reveals a typical bright macrostructure, which can be observed due to the operation of the mould’s electromagnetic stirrers (indicated by arrows). Subsurface cracks outside
the corner are also visible in the upper left and lower right corners.

Fig. 2 The black square signifies the orientation of the billet. Both defects (on the left and right) are located at the
corner of the billet, indicating the presence of off-corner cracks in the billet, which is a cast semi-product. The operation of the mould’s electromagnetic stirrers results in a typically bright macro-structure, as indicated by the arrows.

In May 2023, a project was started where the casting speed was adjusted according to the
casting temperature. This adjustment involved steel grades with the highest number of surface
defect occurrences (i.e., the average ratio of rolled bars with detected surface defects and their
lengths) and varying carbon contents: 16MnCrS5, C22, 30MnVS5, and 46MnVS5. For every 10 °C
deviation from the prescribed casting temperature, the speed was altered by 0.02 m/min. The
number of cast batches, the average ratio of rolled bars with detected surface defects, and their
lengths for the one-year period from May 2022 to May 2023 are presented in Table 1.
In an effort to reduce casting defects on the rolled material, the following parameters were
collected for the batches cast from May 2023 to June 2023. This two-month period involved adjustments to the casting speed based on the casting temperature. The adjustments were specifically applied to steel grades with the highest occurrences of surface defects, namely 16MnCrS5,
C22, 30MnVS5, and 46MnVS5:

• Chemical composition: Content of carbon, silicon, manganese, sulphur, chromium, nickel,
aluminium and vanadium. Chemical composition influence on material properties also
during solidification (e.g., shrinkage, ductility, mechanical properties).
• Casting parameters:
− Average casting temperature (in °C). Casting temperature influences the thermal field
in the mould, which influences the heat removal and solidification.
− Changes of the casting speed based on deviation from the prescribed casting temperature (m/min). For every 10 °C deviation from the prescribed casting temperature, the
speed was changed by 0.02 m/min. Casting speed influences the heat removal and solidification.

Advances in Production Engineering & Management 18(4) 2023

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Kovacic, Zuperl, Gusel, Brezocnik

− The average mould water flow (in l/min). The highest heat removal occurs in the
mould, where thermomechanical behaviour influences on shell solidification.
− The average cooling water pressure (in bar) and flow (in l/min) are observed in the
first zone of secondary cooling, which is directly below the mould. The melt primarily
solidifies in the mould. After exiting the mould, the strand is cooled by water sprays.
The water flux can be automatically adjusted, varying the water pressure and flow.
Secondary cooling also influences the thermomechanical behaviour during solidification.
• Average ratio of rolled bars with detected surface defects (in %).
• Average length of detected surface defects (in mm/mm).

Rolled bars are examined using an automatic control line equipped with a flux leakage inspection system, which has a surface defect depth detection limit of 0.15 mm. Data on defect
depths and lengths for each examined bar are available. It’s important to note that scrap is considered when the maximum permissible depth of surface defects is exceeded. The maximum
permissible depth is defined by the customer or international standards (e.g., ISO 9443, EN
10221, EN 10277-1) and can be significantly (i.e., several times) larger than the detection limit.
Accordingly, this study uses data on detectable defects that are deeper than the detection limit of
0.15 mm.
Table 1 The number of cast batches, average ratio of rolled bars with detected surface defects and their lengths for
individual steel grade for the one-year period from May 2022 and May 2023
Steel grade
Number of
Average ratio of
The length of
cast batches
rolled bars
the detected
with the detected
surface defects, mm/m
surface defects, %
C45S
255
12.92
10.57
46MNVS5
67
25.52
18.08
16MNCRS5
70
11.27
1.41
30MNVS6
29
16.28
12.29
42CRMOS4
26
13.90
3.73
28MNCRNIB
43
6.97
4.88
20MNCRS5
43
5.65
0.30
C45
36
6.36
3.07
20MNV6
83
2.62
0.57
C22
17
12.73
1.425
S355J2
28
7.59
1.46
42CRMO4
23
8.79
1.55
51CRV4
22
8.52
3.48
20NICRMOS2-2
16
8.26
0.63
16MNCR5
28
3.81
0.47
23MNNICRMO5-2
15
7.04
8.80
18CRNIMO7-6
18
5.84
0.07
C35S
7
11.38
32.62
25CRMOS4
6
10.91
9.88
28MNCRB7
6
9.15
1.80
25CRMO4
8
6.30
0.06
100CR6
9
5.02
2.13
C60
19
2.25
0.26
20MNCR5
6
6.04
0.27
31CRMOV9
4
6.67
2.71
38MNVS6
7
3.34
0.50
16NICRS4
7
3.06
6.62
17NICRMOS6-4
4
2.09
0.08
30CRNIMO8
3
2.11
2.67
15CRNI6
3
2.06
0.40
P460NH
3
1.38
0.09

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Reduction of surface defects by optimization of casting speed using genetic programming: An industrial case study

Before implementing changes to the casting speed based on deviations from the prescribed
casting temperature into the continuous casting process, models were developed using linear
regression and genetic programming. These models predict the ratio of rolled bars with detected
surface defects and the length of these defects. By comparing the calculated ratios of rolled bars
with detected surface defects and their lengths, with or without changes in casting speed, further
measures were taken.
Average ratio of rolled
bars with the detected
surface defects, %

16MNCR5
16MNCR5
16MNCR5
16MNCR5
30MNVS6
30MNVS6
30MNVS6
30MNVS6
46MNVS5
46MNVS5
46MNVS5
46MNVS5
46MNVS5
46MNVS5
46MNVS5
46MNVS5
46MNVS5
46MNVS5
C22
C22
C22
C22
C22
C22

0.16
0.16
0.16
0.17
0.29
0.30
0.29
0.30
0.47
0.47
0.47
0.47
0.47
0.47
0.47
0.47
0.47
0.47
0.22
0.22
0.22
0.22
0.23
0.22

1.11
1.12
1.12
1.12
1.43
1.40
1.41
1.37
1.15
1.17
1.15
1.15
1.15
1.15
1.15
1.15
1.15
1.14
0.43
0.41
0.44
0.43
0.43
0.46

1.00
1.02
1.00
0.99
0.12
0.12
0.20
0.19
0.26
0.26
0.24
0.24
0.24
0.25
0.25
0.24
0.24
0.24
0.20
0.14
0.10
0.08
0.11
0.14

0.021
0.022
0.022
0.022
0.013
0.016
0.013
0.013
0.006
0.006
0.005
0.005
0.005
0.005
0.005
0.005
0.005
0.005
0.020
0.019
0.018
0.024
0.020
0.020

0.00
0.00
0.02
0.00
0.02
0.00
0.02
0.00
0.00
0.02
0.02
0.02
0.00
0.02
0.02
0.00
0.00
0.00
0.00
0.02
0.02
0.02
0.02
0.00

2.42
2.42
2.55
2.43
2.41
2.44
2.49
2.47
2.30
2.28
2.00
2.08
2.03
1.99
2.04
1.98
2.01
2.11
3.50
3.49
3.48
3.49
3.48
3.48

The length of the detected surface defects,
mm/m

The average cooling
water flow in the first
zone of secondary
cooling
FLUXZ1, l/min
The average cooling
water pressure in the
first zone of secondary
cooling PRESS, bar

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24

The average casting
temperature TEMP, °C

The average cooling
water flow in the mould
FLUXM, l/min

0.58
1.80
0.59
2.27
2.91
2.44
5.56
0.97
3.45
3.52
2.45
11.16
21.10
18.07
12.48
17.58
16.28
2.22
2.43
1.45
1.39
1.82
1.55
2.18

Changes of the casting
speed
SPEED, m/min

V
V, %

35.21
35.19
34.99
34.84
35.09
35.01
35.14
34.97
32.26
32.11
31.94
32.53
32.10
31.81
32.25
31.72
31.96
31.90
42.22
42.16
42.15
42.20
42.18
42.17

Al
Al, %

Ni
Ni, %

1780.03
1780.01
1780.04
1780.05
1780.01
1780.03
1779.99
1780.03
1780.03
1780.02
1779.97
1780.03
1780.01
1780.01
1780.02
1780.00
1780.00
1780.00
1780.03
1779.98
1779.95
1779.99
1780.02
1780.00

Cr
Cr, %

S
S, %

0.01
0.01
0.01
0.01
0.09
0.08
0.09
0.10
0.11
0.11
0.11
0.11
0.11
0.11
0.11
0.11
0.11
0.11
0.00
0.00
0.00
0.00
0.00
0.00

Mn
Mn, %

0.19
0.17
0.13
0.16
0.09
0.07
0.12
0.10
0.15
0.16
0.16
0.16
0.16
0.16
0.17
0.17
0.16
0.16
0.11
0.10
0.07
0.08
0.08
0.09

C
C, %

0.009
0.008
0.008
0.007
0.027
0.033
0.033
0.026
0.062
0.063
0.068
0.063
0.063
0.066
0.060
0.067
0.064
0.063
0.006
0.007
0.006
0.003
0.004
0.008

Steel grade

0.24
0.24
0.25
0.25
0.58
0.60
0.56
0.60
0.65
0.63
0.64
0.65
0.64
0.66
0.66
0.64
0.64
0.66
0.21
0.24
0.23
0.25
0.24
0.25

Batch number

Si
Si, %

Table 2 Data from May 2023 to June 2023 where the casting temperature has been adjusted, including steel grades
with the highest number of surface defect occurrences – 16MnCrS5, C22, 30MnVS5 and 46MnVS5

1553.0
1552.0
1564.0
1551.0
1542.0
1539.0
1542.0
1538.0
1517.0
1524.0
1513.0
1525.0
1514.0
1516.0
1534.0
1526.0
1513.0
1516.0
1554.0
1558.0
1555.0
1552.0
1553.0
1555.0

0.12
0.11
0.12
0.17
0.67
1.03
0.21
0.09
1.22
1.32
0.31
1.41
2.16
2.36
4.85
9.21
19.68
2.07
0.14
0.04
0.12
0.64
0.18
0.12

3. Results and discussion
Based on the collected data (Table 2), the prediction of the average ratio of rolled bars with detected surface defects, as well as the average length of detected surface defects, was conducted
using linear regression and genetic programming. The fitness function was defined as the average deviation between the predicted and experimental data. It is defined as follows:
∆=

∑𝑛𝑛𝑖𝑖=1|𝑋𝑋𝑖𝑖 − 𝑋𝑋 ′ 𝑖𝑖 |
𝑛𝑛

(1)

where n is the size of the monitored data and X’i and Xi are the actual and the predicted (i.e. calculated) values, respectively.
3.1 Modelling of the ratio of rolled bars with detected surface defects and their lengths using linear
regression

Linear regression is a statistical method used to model the relationship between a dependent
variable and one or more independent variables. It provides a way to predict the dependent variable’s value based on the values of the independent variables, making it a valuable tool in fields
such as machine learning, economics, engineering, and biology.
Based on the linear regression results, it can be concluded that the model significantly predicts the average ratio of rolled bars with detected surface defects (p < 0.05, ANOVA). It is found
that 72.78 % of total variances can be explained by the variances of independent variables (Rsquare). The only significantly influential parameter is the average cooling water pressure in the
first zone of secondary cooling (PRESS) (p < 0.05).
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Kovacic, Zuperl, Gusel, Brezocnik

The linear regression model for predicting the average ratio of rolled bars with detected surface defects is as follows:
−0.789 ∙ 𝐶𝐶 − 0.450 ∙ 𝑆𝑆𝑆𝑆 − 1.357 ∙ 𝑀𝑀𝑀𝑀 + −0.235 ∙ 𝑆𝑆 + 3.577 ∙ 𝐶𝐶𝐶𝐶 − 0.302 ∙ 𝑁𝑁𝑁𝑁 − 0.576 ∙ 𝐴𝐴𝐴𝐴 + 2.822
∙ V + 3.772 ∙ 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 + 0.711 ∙ 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 + 0.015 ∙ 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹1 − 0.455 ∙ 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃
+ 0.002 ∙ 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 − 1268.557.

(2)

The average deviation from experimental data is 2.32 %.
Similarly, the second model significantly predicts the average length of detected surface defects (p < 0.05, ANOVA). However, in this case, only 50.17 % of total variances can be explained
by the variances of independent variables (R-square). Interestingly, in this context, there are no
significantly influential parameters (p < 0.05).
The linear regression model for predicting the average length of detected surface defects is as
follows:
−179.589 ∙ 𝐶𝐶 + 44.737 ∙ 𝑆𝑆𝑆𝑆 − 116.991 ∙ 𝑀𝑀𝑀𝑀 − 7.780 ∙ 𝑆𝑆 − 99.020 ∙ 𝐶𝐶𝐶𝐶 − 32.913 ∙ 𝑁𝑁𝑁𝑁 − 39.153 ∙ 𝐴𝐴𝐴𝐴 +
409.933 ∙ 𝑉𝑉 + 188.939 ∙ 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 − 22.826 ∙ 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 − 2.781 ∙ 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹1 − 17.145 ∙ 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 + 0.228 ∙
𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 + 40476.188.

(3)

The average deviation from the experimental data is 5.44 mm/m.
Based on the data gathered in Table 2 and the developed linear regression models, the average ratio of rolled bars with detected surface defects and the average length of detected surface
defects were also calculated in the scenario where changes to the casting speed were not made.
According to calculations from both linear regression models, changes in casting speed contributed to a decrease in the average ratio of rolled bars with detected surface defects and the
average length of detected surface defects, from 6.47 % to 5.68 % and from 3.81 mm/m to 2.01
mm/m, respectively.
3.2 Modelling of the ratio of rolled bars with detected surface defects and their lengths using
genetic programming

Genetic programming is an evolutionary algorithm, similar to genetic algorithms, used for automatic generation of computer programs to solve problems. It involves evolving a population of
computer programs over several generations, using genetic operators like crossover and mutation to produce new candidate solutions. The programs are represented as tree structures, allowing for the evolution of complex solutions. In genetic programming, the representation of
solutions as tree structures enables the evolution of diverse and complex programs, allowing for
the exploration of a broad solution space. Unlike genetic algorithms which typically evolve fixedlength strings, which are intended to solve a very broad spectrum of problems [22-28], genetic
programming evolves variable-sized structures, allowing for more flexibility in representing
solutions of varying complexity [29-31]. Genetic programming can automatically discover both
the structure and parameters of a solution, making it suitable for problems where the optimal
solution's form is not known a priori. The genetic programming operates on variable-length
structures, making it more suitable for evolving complex solutions, especially in symbolic regression and automatic code generation.
The genetic programming method was used several times in Štore Steel Ltd. [32-36]. For the
purposes of this study, organisms that underwent adaptation were indeed represented as mathematical expressions, i.e., models for predicting the average ratio of rolled bars with detected
surface defects, and models for predicting the average length of detected surface defects. These
models consist of the selected functions, i.e., basic arithmetical functions of addition, subtraction,
multiplication and division, and terminal genes, i.e., independent input parameters, and random
floating-point constants.
The LISP based in-house genetic programming system was run 200 times to develop independent civilizations. In each run we obtain either the model for prediction of the average ratio
of rolled bars with detected surface defects or the average length of detected surface defects.
After the modelling phase, we analysed the results and selected the two best prediction models.
506

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Reduction of surface defects by optimization of casting speed using genetic programming: An industrial case study

The best genetically developed mathematical model for predicting the average ratio of rolled
bars with detected surface defects obtained from 100 runs of genetic programming system is:
𝑆𝑆

Si

+

Si

PRESSZ1

𝑆𝑆

+ 2SPEED + 𝑉𝑉 � +
Al

Si

PRESSZ1

+

Si

MnPRESSZ1(−5.45733+Si)(−𝑆𝑆+SPEED+(Si+𝐶𝐶(6.32902+𝐶𝐶)Si )SPEED )

+ (6.32902 + PRESSZ1 +

6.32902Si)𝑉𝑉 �𝑆𝑆 / �𝐶𝐶(−𝑆𝑆 + SPEED + (Si + 𝐶𝐶(6.32902 + 𝐶𝐶)Si )SPEED ) �−5.45733 + 1 / �(−𝑆𝑆 + SPEED + (Si +

(4)

𝐶𝐶(6.32902 + Mn)Si )SPEED ) �−5.45733 + 1 / �Mn(−𝑆𝑆 + SPEED + (Si + Si2 )SPEED )�−5.45733 + (−5.45733 +

Si)Si (−𝑆𝑆 + SPEED + (Si + 𝐶𝐶(6.32902 + 𝐶𝐶)Si )SPEED )������ − �0.1832�Si + 𝐶𝐶(6.32902 + Si)�� / �MnPRESSZ1Si �−𝑆𝑆 +
𝑆𝑆

Si2 + SPEED + �𝐶𝐶 + Si /�MnPRESSZ1(−5.45733 + Si)(−𝑆𝑆 + SPEED + (Si + 𝐶𝐶(6.32902 + 𝐶𝐶)Si )SPEED )�� 𝑉𝑉����. .

The average deviation from experimental data is 1.59 %. The model obtained by genetic programming is 1.47-times better than the one obtained using linear regression.
The best mathematical model for predicting the average length of detected surface defects
obtained from 100 runs of genetic programming system is:
𝑆𝑆+

𝑆𝑆
�−Mn+𝐶𝐶(PRESSZ1+𝑆𝑆)�SPEED

−Mn+PRESSZ1+

SPEED

SPEED�−Mn+PRESSZ1+
�
𝑆𝑆
⎛
⎞
Mn
+
𝑆𝑆+
𝑆𝑆
Mn+𝑆𝑆
PRESSZ1+2𝐶𝐶𝐶𝐶−SPEED
⎜
⎟
−Mn+PRESSZ1+
𝑆𝑆
Mn
⎜
⎟
Mn+
(−1+PRESSZ1+2𝐶𝐶𝐶𝐶)�PRESSZ1−𝑆𝑆+𝐶𝐶Mn(𝐶𝐶+𝑆𝑆)�
SPEED ⎟.
⎜
SPEED�−Mn+PRESSZ1+ Mn �
𝑆𝑆
𝑆𝑆+
SPEED+
Mn+𝑆𝑆
⎜
⎞⎟
−Mn+PRESSZ1+
SPEED ⎛
Mn
PRESSZ1
+𝐶𝐶𝐶𝐶−
⎟
⎜(−1+PRESSZ1+2𝐶𝐶𝐶𝐶)�−Mn+PRESSZ1+ Mn+𝑆𝑆 �⎜
SPEED
⎜
⎟⎟
(−1+PRESSZ1)𝑆𝑆�PRESSZ1−𝑆𝑆+
�

⎝

⎝

𝐶𝐶2Mn

⎠

(5)

⎠

The average deviation from experimental data is 0.97 mm/m. The model obtained by genetic
programming is 5.63-times better than the one obtained using linear regression.
Based on calculations using the two obtained genetic programming models, changes in casting speed contributed to a decrease in the average ratio of rolled bars with detected surface defects and the average length of detected surface defects, from 6.1 % to 6.0 % and 8.87 mm/m to
1.57 mm/m.
3.3 Discussion and validation of modelling results

Based on modelling results the average ratio of rolled bars with detected surface defects and the
average length of detected surface defects could be decreased in the best case scenario from 6.47 %
to 5.68 % (for 1.14-times) and from 8.87 mm/m to 1.57 mm/m (for 5.65-times), if the casting
speed would be adjusted based on deviations from the prescribed casting temperature, respectively.
Accordingly, the casting speed was adjusted based on the casting temperature for all casting
batches from July 2023 to November 2023. The data obtained was compared to a one-year period (May 2022 to May 2023) where no casting speed adjustments were made. These results are
collected in the Table 3.
Based on actual data the average ratio of rolled bars with detected surface defects and the
average length of detected surface defects decreased statistically significantly (t-test, p < 0.05)
from 10.63 % to 8.82 % and from 6.20 mm/m to 3.40 mm/m in the period without (from May
2022 to May 2023) and with casting speed adjustments (from July 2023 to November 2023),
respectively.
Out of 31 different steel grades the average ratio of rolled bars with detected surface defects
and the average length of detected surface defects statistically significantly decreased in 11 and
15 steel grades, respectively, while the rest remained statistically significantly the same (t-test,
p < 0.05).
Advances in Production Engineering & Management 18(4) 2023

507

Kovacic, Zuperl, Gusel, Brezocnik

The average ratio of rolled bars with detected
surface defects – significance (t.test, p<0.05)

The average length of detected surface defects –
significance (t.test, p < 0.05)

3
2
6
23
3
2
5
0
4
1
5
16
3
4
1
3
4
4
2
2
2
1
2
9
2
9
65
3
5
5
4

The average length of detected surface defects
from July to November 2023 (period with
casting speed adjustments), mm/m

Number of batches from July to November 2023
(period with casting speed adjustments), where
the casting speed was actually adjusted

3
2
9
39
4
6
12
2
7
2
8
18
4
8
2
9
10
4
5
4
4
15
3
39
4
18
141
9
10
12
35

The average length of detected surface defects
from May 2022 to May 2023 (period without
casting speed adjustments), mm/m

Number of batches from July to November 2023
(period with casting speed adjustments)

4
3
19
83
7
7
28
9
18
6
8
36
4
43
3
15
28
22
23
6
3
43
6
70
7
17
255
16
26
29
67

The average ratio of rolled bars with detected
surface defects from July to November 2023
(period with casting speed adjustments), %

Number of batches from May 2022 to May 2023
(period without casting speed adjustments)

17NICRMOS6-4
30CRNIMO8
C60
20MNV6
16NICRS4
38MNVS6
16MNCR5
100CR6
18CRNIMO7-6
20MNCR5
25CRMO4
C45
31CRMOV9
28MNCRNIB
15CRNI6
23MNNICRMO5-2
S355J2
51CRV4
42CRMO4
28MNCRB7
P460NH
20MNCRS5
25CRMOS4
16MNCRS5
C35S
C22
C45S
20NICRMOS2-2
42CRMOS4
30MNVS6
46MNVS5

The average ratio of rolled bars with detected
surface defects from May 2022 to May 2023
(period without casting speed adjustments), %

Steel grade

Table 3 Data from May 2023 to June 2023 where the casting temperature has been adjusted, including steel grades
with the highest number of surface defect occurrences - 16MnCrS5, C22, 30MnVS5 and 46MnVS5

2.09
2.11
2.25
2.62
3.06
3.34
3.81
5.02
5.84
6.04
6.30
6.36
6.67
6.97
6.98
7.04
7.59
8.52
8.79
9.15
9.23
10.59
10.91
11.27
11.38
12.73
12.92
13.45
13.90
16.28
25.52

4.62
0.56
3.47
1.29
2.79
6.88
4.01
10.43
5.89
3.19
4.84
1.95
7.46
5.40
2.06
6.50
3.78
12.18
10.54
16.04
1.38
5.65
6.00
7.93
5.61
4.11
13.00
8.26
9.71
13.40
16.99

0.18
2.67
0.26
0.57
6.62
0.50
1.31
2.13
0.07
0.27
0.06
3.07
2.71
4.87
1.48
8.80
1.46
3.48
1.55
1.80
0.75
1.12
9.88
1.41
32.62
1.43
10.57
1.83
3.73
12.29
18.08

0.08
0.02
0.83
0.19
0.21
1.03
0.47
1.98
0.05
0.35
0.09
0.32
2.82
1.27
0.40
0.51
0.76
0.62
0.96
1.55
0.09
0.30
0.57
1.29
6.00
0.15
6.46
0.63
3.28
4.57
9.98

NO
NO
NO
YES
NO
NO
NO
NO
NO
NO
NO
YES
NO
YES
YES
NO
YES
NO
NO
NO
YES
YES
NO
YES
YES
YES
NO
NO
NO
NO
YES

NO
NO
NO
YES
YES
NO
YES
NO
NO
NO
NO
YES
NO
YES
YES
YES
NO
YES
NO
NO
YES
NO
YES
NO
NO
YES
YES
NO
NO
NO
YES

4. Conclusion
In the article the reduction of surface defects, detected during examination (automatic control
line) of the rolled material, with casting speed adjustments based on the casting temperature, is
presented. For all produced steel grades, the same casting speed of 1.6 m/min was used before.
The average casting speed adjustments were calculated for several different grades based on inhouse developed solidification model. For every 10 °C deviation from the prescribed casting
temperature, the speed was changed by 0.02 m/min.
The reduction of surface defects was designed as follows:
• Period with casting speed adjustments based on the casting temperature involving steel
grades with highest occurrences of surface defects and with various carbon content:
16MnCrS5, C22, 30MnVS5 and 46MnVS5.
• Modelling of the average ratio of rolled bars with detected surface defects and the average
length of detected surface defects using linear regression and genetic programming.
• Implementing of modelling results based on four most problematic steel grades
(16MnCrS5, C22, 30MnVS5 and 46MnVS5) into practice – the casting speed adjustments
were used for all steel grades from July 2023 to November 2023.

508

Advances in Production Engineering & Management 18(4) 2023

Reduction of surface defects by optimization of casting speed using genetic programming: An industrial case study

• Analysis of the average ratio of rolled bars with detected surface defects and the average
length of detected surface defects for batches cast from July 2023 to November 2023.
• Comparison of the batches produced prior (from May 2022 to May 2023) and after (from
July 2023 to November 2023) the implementation of casting speed adjustments.

Chemical composition (content of carbon, silicon, manganese, sulphur, chromium, nickel,
aluminium and vanadium) and casting parameters (casting speed adjustments, the average
mould cooling water flow, the average cooling water flow and pressure in the first zone of the
secondary cooling, casting temperature) were gathered for the batches cast from May 2023 to
June 2023.
Based on gathered data modelling was performed using linear regression and genetic programming. For the fitness function, the average deviation between predicted and experimental
data was selected.
The average deviation of the linear regression model for predicting the average ratio of bars
with detected surface defects from experimental data is 2.32 %. The average deviation of the
linear regression model for predicting the average length of detected surface defects from experimental data is 5.44 mm/m. Based on calculations from both linear regression models, the
changes in casting speed contributed to a decrease in the average ratio of rolled bars with detected surface defects and the average length of detected surface defects, from 6.47 % to 5.68 %
and 3.81 mm/m to 2.01 mm/m, respectively.
The average deviation of the genetic programming model for predicting the average ratio of
bars with detected surface defects from experimental data is 1.59 %. The model obtained by
genetic programming is 1.47 times better than the one obtained using linear regression. The
average deviation of the linear regression model for predicting the average length of detected
surface defects from experimental data is 0.97 mm/m. The model obtained by genetic programming is 5.63 times better than the one obtained using linear regression. Based on calculations
using both obtained genetic programming models, the changes of casting speed contributed to
decreasing the average ratio of rolled bars with detected surface defects and the average length
of detected surface defects, from 6.1 % to 6.0 % and 8.87 mm/m to 1.57 mm/m, respectively.
Accordingly, the casting speed has been adjusted based on the casting temperature for all cast
batches from July 2023 to November 2023. The obtained data were compared with a one-year
period (from May 2022 to May 2023) where adjustments to the casting speed were not made.
Based on actual data, the average ratio of rolled bars with detected surface defects and the average length of detected surface defects decreased statistically significantly (t-test, p < 0.05) from
10.63 % to 8.82 % and from 6.20 mm/m to 3.40 mm/m in the period without (from May 2022 to
May 2023) and with casting speed adjustments (from July 2023 to November 2023), respectively. Out of 31 different steel grades, the average ratio of rolled bars with detected surface defects
and the average length of detected surface defects statistically significantly decreased in 11 and
15 steel grades, respectively, while the rest remained statistically insignificantly the same (t-test,
p < 0.05).
Until June 2024, detailed analyses of the possible casting speed adjustments for individual
steel grades will be conducted with the in-house developed solidification model. The geometry
of the mould and the geometry, pumps, and nozzles of the secondary cooling will be changed.

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Calendar of events
• European Simulation and Modelling Conference (ESM 2023), October 24-26, 2023, Toulouse,
France.

• 17th International Conference on Industrial and Manufacturing Systems Engineering, November 27-28, 2023, London, United Kingdom.
• International Conference on Digital Manufacturing and Industrial Design ICDMID on December 13-14, 2023, Rome, Italy.

• International Conference on Digital Manufacturing and Automation ICDMA 2023, December
13-15, 2023, Singapore, Singapore.
• International Conference on Multiple Machines and Manufacturing Systems ICMMMS, December 18-19, 2023, Bangkok, Thailand.
• International Conference on Design and Development of Reconfigurable Manufacturing Systems ICDDRMS, December 20-21, 2023, Honolulu, USA.

• International Conference on Digital Manufacturing, Modeling and Design ICDMMD, December
20-21, 2023, Dubai, United Arab Emirates.
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2023, Dubai, United Arab Emirates.

• International Conference on Intelligent Manufacturing and Automation Engineering ICIMA,
December 22-24, 2023, Changsha, China.

• 17. International Conference on Rapid Manufacturing December 25-26, 2023, Paris, France.

• International Conference on Digital Manufacturing Systems ICDMS, January 7-8, 2024, Tokyo,
Japan.

• International Conference on Manufacturing and Intelligent Machining ICMIM, January 14-15,
2024, Zurich, Switzerland.
• 18th International Conference on Advanced Manufacturing Engineering and Technologies,
January 15-16, 2024, Montevideo, Uruguay.

• International Conference on Cyber Manufacturing Systems ICCMS, January 18-19, 2024, Sydney, Australia.
• International Conference on Manufacturing and Industrial Technologies ICMIT, January 2527, 2024, Budapest, Hungary.

• International Conference on Mechatronics and Manufacturing ICMM, February 2-4, 2024,
Penang, Malaysia.

• International Conference on Manufacturing and Optimization ICMO, February 25-26, 2024,
Buenos Aires, Argentina.

• International Conference on Mechanical and Intelligent Manufacturing Technologies ICMIMT,
March 7-9, 2024, Cape Town, South Africa.

• 2024 Annual Modeling and Simulation Conference (ANNSIM 2024), May 20-23, 2024, Washington D.C., USA.

• North American manufacturing research conference (NAMRC) 52, June 17-21, 2024, Knoxville, TN, USA.

• 18th International Conference on Industrial and Manufacturing Systems Engineering, August
9-10, 2024, Lagos, Nigeria.

512

Advances in Production Engineering & Management 18(4) 2023

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. Manuscript should
be written in English. Responsibility for the contents of the paper rests upon the authors and not
upon the editors or the publisher. The content from published paper in the APEM journal may be
used under the terms of the Creative Commons Attribution 4.0 International Licence (CC BY 4.0).
For most up-to-date information please see the APEM journal homepage apem-journal.org.

Submission of papers
A submission must include the corresponding author's complete name, affiliation, address,
phone and fax numbers, and e-mail address. All papers for consideration by Advances in Production Engineering & Management should be submitted by e-mail to the journal Editor-in-Chief:
Miran Brezocnik, Editor-in-Chief
UNIVERSITY OF MARIBOR
Faculty of Mechanical Engineering
Chair of Production Engineering
Smetanova ulica 17, SI – 2000 Maribor
Slovenia, European Union
E-mail: editor@apem-journal.org

Manuscript preparation
Manuscript should be prepared in Microsoft Word 2010 (or higher version) word processor.
Word .docx format is required. Papers on A4 format, single-spaced, typed in one column, using
body text font size of 11 pt, should not exceed 12 pages, including abstract, keywords, body text,
figures, tables, acknowledgements (if any), references, and appendices (if any). The title of the
paper, authors' names, affiliations and headings of the body text should be in Calibri font. Body
text, figures and tables captions have to be written in Cambria font. Mathematical equations and
expressions must be set in Microsoft Word Equation Editor and written in Cambria Math font. For
detail instructions on manuscript preparation please see instruction for authors in the APEM
journal homepage apem-journal.org.

The review process
Every manuscript submitted for possible publication in the APEM journal is first briefly reviewed
by the editor for general suitability for the journal. Notification of successful submission is sent.
After initial screening, and checking by a special plagiarism detection tool, the manuscript is
passed on to at least two referees. A double-blind peer review process ensures the content's
validity and relevance. Optionally, authors are invited to suggest up to three well-respected experts in the field discussed in the article who might act as reviewers. The review process can
take up to eight weeks on average. Based on the comments of the referees, the editor will take a
decision about the paper. The following decisions can be made: accepting the paper, reconsidering the paper after changes, or rejecting the paper. Accepted papers may not be offered elsewhere for publication. The editor may, in some circumstances, vary this process at his discretion.
Proofs
Proofs will be sent to the corresponding author and should be returned within 3 days of receipt.
Corrections should be restricted to typesetting errors and minor changes.

Offprints
An e-offprint, i.e., a PDF version of the published article, will be sent by e-mail to the corresponding author. Additionally, one complete copy of the journal will be sent free of charge to the corresponding author of the published article.

APEM

Advances in
Production
Engineering &
Management

journal

Chair of Production Engineering (CPE)
University of Maribor
APEM homepage: apem-journal.org

Volume 18 | Number 4 | December 2023 | pp 399-514
Contents
Scope and topics

402

A combined genetic algorithm and A* search algorithm for the electric vehicle routing problem with time windows
Wang, D.L.; Ding, A.; Chen, G.L.; Zhang, L.

403

Optimizing rock breaking performance: The influence of chamfer on polycrystalline diamond compact (PDC) cutters
Ju, P.

417

Dynamic price competition market for retailers in the context of consumer learning behavior and supplier competition:
Machine learning-enhanced agent-based modeling and simulation
Deng, G.F.

434

IoT-based Deep Learning Neural Network (DLNN) algorithm for voltage stability control and monitoring of
solar power generation
Shweta, R.; Sivagnanam, S.; Kumar, K.A.

447

Comparing Fault Tree Analysis methods combined with Generalized Grey Relation Analysis: A new approach and
case study in the automotive industry
Shi, J.L.; Lu, Z.C.; Xu, H.H.; Ren, M.M.; Shu, F.L.

462

Optimizing smart manufacturing systems using digital twin
Ojstersek, R.; Javernik, A.; Buchmeister, B.

475

Incentive modeling analysis in engineering applications and projects with stochastic duration time
Zhao, J.; Su, J.F.

486

Reduction of surface defects by optimization of casting speed using genetic programming: An industrial case study
Kovacic, M.; Zuperl, U.; Gusel, L.; Brezocnik, M.

501

Calendar of events

512

Notes for contributors

513

Published by CPE, University of Maribor
ISSN 1854-6250

9 771854 625008

apem-journal.org