A multi-objective optimization model for sustainable supply chain network with using genetic
algorithm
Reza Ehtesham Rasi
Department of Industrial Management, Qazvin Branch, Islamic Azad University, Qazvin, Iran, and
Mehdi Sohanian
Department of Industrial Management, Science and Research Branch, Islamic Azad University, Tehran, Iran
Abstract
Purpose–The purpose of this paper is to design and optimize economic and environmental dimensions in a sustainable supply chain (SSC) network. This paper developed a mixed-integer linear programing (MILP) model to incorporate economical and environmental data for multi-objective optimization of the SSC network.
Design/methodology/approach–The overall objective of the present study is to use high-quality raw materials, at the same time the lowest amount of pollution emission and the highest profitability is achieved.
The model in the problem is solved using two algorithms, namely, multi-objective genetic and multi-objective particle swarm. In this research, to integrate sustainable supplier selection and optimization of sustainability performance indicators in supply chain network design considering minimization of cost and time and maximization of sustainability indexes of the system.
Findings–The differences found between the genetic algorithms (GAs) and the MILP approaches can be explained by handling the constraints and their various logics. The solutions are contrasted with the original crisp model based on either MILP or GA, offering more robustness to the proposed approach.
Practical implications–The model is applied to Mega Motor company to optimize the sustainability performance of the supply chain i.e. economic (cost), social (time) and environmental (pollution of raw material). The research method has two approaches, namely, applied and mathematical modeling.
Originality/value–There is limited research designing and optimizing the SSC network. This study is among thefirst to integrate sustainable supplier selection and optimization of sustainability performance indicators in supply chain network design considering minimization of cost and time and maximization of sustainability indexes of the system.
Keywords Sustainability, Supply chain management, Multi-objective, Optimization, Meta-heuristic algorithms, Algorithms
Paper typeResearch paper
Notation used in the MILP model of the paper Sets
T = period,t= 1,2,. . ., T;
J = Type of product produced, j = 1,2,. . ., J;
I = Type of raw material, i = 1,2,. . ., I;
Q = Quality of the raw material type,q= 1,2,. . ., Q;
K = raw material supplier, k = 1,2,. . ., K;
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Received 4 March 2019 Revised 23 July 2020 Accepted 3 August 2020
Journal of Modelling in Management Vol. 16 No. 2, 2021 pp. 714-727
© Emerald Publishing Limited 1746-5664
DOI10.1108/JM2-06-2020-0150
The current issue and full text archive of this journal is available on Emerald Insight at:
https://www.emerald.com/insight/1746-5664.htm
M = Factory of producing, m = 1,2,. . ., M;
D Distributor, d = 1,2,. . ., D; and Z retailer,z= 1,2,. . ., Z Parameters
Ctiqkm= The cost of purchasing, shipping and transporting raw materials I of Q quality from supplier K to factory M in the period T;
Ctjmd = The cost of producing, storing and delivering the product J from factory M to distributor D during period T;
Ctjdz = The cost of maintaining and delivering the product J from the D distributor to the retailer Z in the T period;
Ctjd = Maintenance of each unit of the product J from the distributor D in the period T;
DAtjz = The demand for product J from retailer Z in period T;
CAtik = Supplier’s capacity K for raw materials I in the period T;
CAtjm= Capacity of factory M for product J in period T;
CAtjd = Capacity of Wholesale D for product J in period T;
Uij = The rate of use of each unit of product type J from the primary substance of type I;
Riq = The amount of contamination caused by the primary substance of type I with the quality Q;
and
Ptjz = The product price J in-retailer Z in period T.
Among the above-mentioned parameters,Riqrefers to the mass of non-renewable raw material used in the production of each product and its unit is optional.
Variables
Stjz = The number of product J sold by retailer Z during the T period;
Xtiqkm= The quantity of raw material I, with quality Q shipped from supplier K to the factory M dur- ing the period T;
Xtjmd = The quantity of product J shipped from factory M to distributor D during the period T;
Xtjdz = The quantity of product J shipped from the distributor D to the retailer Z in the period T;
Xtjd = The quantity of product J in the distributor’s warehouse D in the period T; and
Wtjdz = The zero-wake variable for sending a product from a distributor D to a retailer Z in the period T.
1. Introduction
In the recent two decades, the carbon pollution policy and its implementation in the supply chain network have received extensive publicity. With increasing environmental pressure, enterprises have to think about sustainability. Minimizing carbon emissions while minimizing costs are of great importance (Li et al., 2020). In the current circumstance, resources are underneath the standard used in business and greenhouse gas emissions are very high. The design and management of supply chain operations have been observed as a prime factor in fostering environmental sustainability (Rajesh, 2020; Kumaret al., 2017).
Supply chain networks will help companies obtain more profits with low environmental impact, high ecological and energy efficiency and keep the lowest costs within a determined range(Su and Sun, 2019). Intensifying the global competitive landscape in a constantly evolving environment would determine the need for adequate responses from organizations and industrial manufacturing companies, and stress resilience with the volatile external environment (Rasi, 2020). According to Yadlapalliet al. (2018), the outcome is that the introduction of ways of sustainability in the supply chain has been proved to improve the profitability of a business. Most of the sustainability emphasis in supply chain management
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(SCM) is on the environmental and economic aspects of sustainability and less tends to be paid to social sustainability. Sustainability is described as the ability of an organization to make current decisions in such a way that such decisions do not have any negative impact on the future climate, society and industry. Sustainable supply chain management (SSCM) is a new area in both research and industry. It is considered a highly relevant subject where companies to improve their competitiveness through the implementation of sustainable activities in their goods and services (Vivaset al., 2020).
This definition describes that sustainability has three dimensions that can be classified as the environment, economy and society dimensions. Social sustainability focuses on human rights, education and justice, whereas economic efficiency seeks to increase sales by using a minimum of assets and services (Molamohamadi and Ismail, 2013). On the other hand, environmental sustainability is about the extent of the destruction of renewable and non-renewable sources of pollution. All dimensions help to ensure that the needs of the current generation are effectively met without compromising the ability of future generations to meet their needs (Zamaninanet al., 2020). With the rising of the sustainability concept, companies tended to provide sustainability reports. Delivering sustainability reports allows companies to highlight the success of their programs and clarify their attitude toward customers, shareholders, employees and their counterparts (Rajeevet al., 2017). As a result, sustainability in the supply chain and sustainability measurement attracted much attention. SSCM is integrating SCM with environmental and social requirements at all stages of product design, selection and supply of raw materials, production and fabrication, distribution and transfer processes, customer delivery and management of recycling and re-utilization to maximize the amount of energy efficiency and resources associated with improving the overall supply chain performance. The design of the supply chain network as the most important decision at the strategic level plays an important role in the sustainability of the supply chain network.
The purpose of this paper is to provide a mixed-integer linear programing (MILP) model to incorporate economical and environmental data for multi-objective optimization of a SSC network. In this paper, researchers use high-quality raw materials at the same time with the lowest amount of pollution emission and will be achieved the highest profitability.
Main contribution of this study is:
To design a SSC with a proposed mathematical model approach.
To propose a solution methodology for multi-objective mixed-integer linear mathematical model supported with the metaheuristic method.
To integrate sustainable supplier selection and optimization of sustainability performance indicators in supply chain network design considering minimization of cost and time and maximization of sustainability indexes of the system.
To apply to the real-life data obtained from one of the manufacturing companies in the household industry and to assess obtained Pareto solution sets and share with stakeholders.
After the sustainability concept, the SSC concept, general information about the SSC concept are given in Sections 1 and 2, details of the mixed-integer linear problem, solution methodology of the proposed approach, problem definition, genetic algorithm (GA) and model formulation are explained in detail in Section 3. In Section 4, the proposed approach is explained with an illustrative example in Mega Motor Company and the mathematical model performed in Matlab gives the feasible solution set for the objective function. The data of Mega Motor Company operating in manufacturing is used in this example. The
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results of the mixed-integer linear mathematical model have discussed the concluding remarks and future research are presented in Section 5.
2. Literature review
SCM is a set of methods used to effectively and efficiently integrate suppliers, manufacturers, warehouses and vendors, in such a way that to minimize system costs and fulfill the needs of goods and services, produce and distribute goods and services at the right place and at the right time. In a supply chain, all steps, directly or indirectly, engage in the fulfillment of customer demands. The whole chain of supply is a single unit, in such a way that the spirit of co-operation as an insight exists in all its parts and all groups participate in profit and loss. In the supply chain, each organization specializes in what works best for the whole of the supply chain. The supply chain is the new battlefield for competition. Inventory is the latest reason for eliminating imbalances among different categories of suppliers. SCM isflow management between different stages in a chain with the goal of maximizing overall profitability. In this chain, suppliers usually are simultaneously members of several chains, and therefore often play different roles. Thus, SCM is a multiplier system, which uses information technology to moderate communications between some of the key business processes of companies, as well as suppliers, consumers and their business partners.
The evaluation and selection of conventional supply chains have been based solely on economic criteria (Ahi and Searcy, 2013). Price, product quality andflexibility have been some of the traditional economic criteria (Momeni and Nategh, 2010). Nevertheless, in recent years there has been global pressure on supply chains to focus not only on economic criteria but also on social and environmental standards. SSCM is a combination of new criteria that combine environmental and social criteria with economic criteria, according to Torkert and Dietl Research. Today, SSCM criteria are very important (Seuring, 2013). Many companies seek to improve their working conditions and produce environmentally friendly goods (Hugo and Pistikopoulos, 2005). According to the presented articles, several definitions of the SSC are presented, which may be described in a comprehensive way: material, information and capitalflow management; cooperation among companies along the supply chain in the direction integrating goals from all three economic, environmental and social dimensions; as well as SSCM considers to be a management philosophy and as a set of management processes (Subrata and ParthaPriya, 2013). The Brightland Report can be considered as the starting point for papers on the SSC. Subsequently, in the years 2001 and 2003, seven papers were published in the journal “Greener International Management.” Various articles are also featured in magazines such asOperations Management Magazine, International Journal of Production Research, International Journal of Production Economics and Cleaner Magazine. This demonstrates the broad acceptance of this issue among researchers. Published articles are distributed equally in journals related to sustainable management and environmental management with magazines in the field of traditional operations and SCM. While this issue in articles related to ethical and social issues remains in the minority. Also, a series of articles in this area are in journals that are mainly technical in nature but include topics related to natural resources and politics. Environmental literature consists of three journals that have provided many articles in this field. The magazine“Cleaner Production”is in thefirst place, then the magazines“Green Management International” and “Business and Environment Strategy” rank second and third, respectively.
In the operational texts, more journals have been active in this field, including International Journal of Operations and Production Management, International Journal of Supply Chain Management, Operations Management Magazine, Production Management
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and Operations Management, International Journal of Production Research and Supply Chain Management Magazine(formerly International Journal of Purchasing and Materials Management) (Seuring and Muller, 2008).
In a statistical study of research on the SSC and its dimensions in 2016, it is seen that studies in this area are relatively small and further research is needed. The following table shows the thematic distribution of research on the SSC from 2000–2015, which is extensively extracted from 1,068filtered articles (Honget al., 2018).
In general, by looking at the related articles or studies in thisfield, we can see that different dimensions of sustainability are studied to some extent and sometimes these dimensions are considered simultaneously. In this study, by applying changes in problem levels, the type of pollutant emissions, problem constraints, superstructural algorithms and comparison criteria, we tried to represent and analyze a model in different from the previous models.
The GA is the method of optimization inspired by the nature of the living organism (living organisms), and in the classifications can be called as a numerical method, direct and random searches. This algorithm is a repetition-based algorithm and its original principles have been adapted from genetic science and invented by imitation of a number of processes observed in natural evolution, and effectively uses the old knowledge of a population, to render new solutions and improvements. The algorithm is used in a variety of issues such as system optimization, identification and control, image processing and hybrid problems, topology determination and training of artificial neural networks and decision-making systems. This algorithm is used to optimize, search and learning the system. In summary, the GA consists of the following operators, namely, coding, evaluation, combining, mutation and decoding.
3. Problem description and formulation
SCM covers a wide range of topics. Customer orientation, marketing, distribution, production planning and procurement in organizations are working independently and in parallel in the supply chain. Although each of these organizations has its own goals and often these goals are in contradiction with another, so there needs to be a way to align these different goals. SCM is a method that can create this coordination.
3.1 Assumptions
The Assumption of the model would be as following:
The problem is a multi-product, multi-cycle and three-tier layout.
The location and number of suppliers, manufacturers, distribution centers and retailers are specific.
Theflow of materials is only between two successive levels of network layers.
The capacity of each facility (supplier, manufacturer, distributor and retailer) is specific.
The amount of demand and the rest of the existing parameters are definitive.
Each retailer can receive a product from only one wholesaler for each product.
In this research, we are looking for the following outputs for the issue:
Selecting suppliers, manufacturers and distribution centers.
Determining the optimal transportation path for the transfer of raw materials to the factory.
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Moving goods to distribution centers and carrying goods, and delivery to the customers.
An optimization approach to maximize profits and pricing products with definite and time-consuming demand, and minimizing time.
Efforts have been made to model two-stage and three-stage supply chain networks with different assumptions, each of which has been resolved in a different way.Chenet al.(2019) examined these issues in NP-Hard problems, such as multi-choice backpack problems and location-allocation issues. Amiri (2006) presented an innovative method based on the Lagrange method to minimize the cost of the network for a two-stage model. GA has been used to solve supply chain design problems effectively. Thefirst attempt was made byZhu and Sarkis (2006)to use the GA to solve the supply chain design model for the problem of balancing allocations to customers with a disorderly and multiple distributions.Chenet al.
(2019)solved the multi-criteria single-product two-stage supply chain model, using GA and hierarchical analysis method. He provided a GA to minimize the total cost of transport for a three-stage distribution network.
In this paper, we offered a solution to the problem of designing a multi-objective SSC in a three-stage supply chain network by GA and particle swarm and the use of priority axis coding. To provide a mathematical model of the problem, wefirst recall the assumptions of the problem: the number of customers and their demand is definite; the numbers of suppliers, factories and distribution centers, as well as their maximum capacities, are definite. Each supplier of raw materials, factory and distribution center has the ability to supply all kinds of raw materials, produce various types of products and distribute their products, respectively, and according to their maximum capacity.
3.2 Mathematical model
The mathematical model of MILP for SSCM is as follows:
maxZ1¼ X
T
X
J
X
Z
Ptjz:Stjz X
T
X
I
X
K
X
Q
X
M
Ctiqkm:Xtiqkm X
T
X
J
X
M
X
D
Ctjmd:Xtjmd X
T
X
J
X
D
Ctjd:Xtjd X
T
X
J
X
D
X
Z
Ctjdz:Xtjdz (1)
minZ2¼ X
T
X
I
X
K
X
Q
X
M
Riq:Xtiqkm (2)
In this research, we have tried to solve the problem in such a way as close as possible to the real world. In fact, we have tried to strike a balance between the degree of proximity of the problem to the reality and the complexity of its solution. Also, because the desired problem has been investigated from the sustainability dimension, the second objective function has been added to the problem model by reducing the level of pollutant production by selecting raw materials with lower emission rates.
The total profit is equal to the total sales revenue, minus the cost of procuring, purchasing and transporting raw materials from suppliers to factories, minus the cost of producing, storing and delivering products from factories to distributors, minus the cost of
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maintaining the product in the distributor’s warehouse minus the cost of delivering the product from the distributors to retailers.
Xtjdz#EWtjdz (3)
Stjz#DAtjz 8t;j;z (4)
Stjz# X
D
Xtjzd
8t;j;z (5)
X
z
Xtjdz ¼ Xtjd 8t;j;d (6)
X
d
Wtjdz #1 8t;j;z (7)
X
z
Xtjdz#CAtjd 8t;j;d (8)
X
m
Xtjmd ¼ Xtjd 8t;j;d (9)
Xtjd # CAtjd 8t;j;d (10)
X
D
Xtjmd#CAtjm 8t;j;m (11)
X
D
X
J
Uij:Xtjmd¼ X
K
X
Q
Xtiqkm 8t;i;m (12)
X
M
X
Q
Xtiqkm#CAtik 8t;i;k (13)
Ptjz 0; Int 8t;j;z (14)
Xtiqkm 0 8t;i;q;k;m (15)
Xtjmd 0; Int 8t;j;m;d (16)
Xtjd 0; Int 8t;j;d (17)
Xtjdz 0; Int 8t;j;d;z (18)
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Wtjdz ¼ f0;1g (19) Equation (3)ensures that Xtjdzcan be set when the Wtjdzbinary variable has a value of 1. E is a large positive number. The quantity of products sold in each smaller period equals the amount of demand in the same period inequation (4). The quantity of products sold in each smaller period equals the number of products available from the distributor to the retailer in equation (5).Equation(6)shows the amount of product shipped from each distributor to each retailer is equal to the inventory of each distributor’s stock. Each retailer can receive a product from only a wholesaler for each product inequation (7). Inequation (8), the amount of product shipped from each distributor to each retailer is less than the capacity of each distributor. The amount of product shipped from the factory to each distributor is equal to the amount of product available in the distributor’s warehouse in each period inequation (9).
Equation (10)shows the quantity of product inventory in each distributor is less than the distributor’s storage capacity. The amount of product shipped from each factory to smaller distributors equals the capacity of each plant inequation (11).Equation (12)shows that the amount of raw material consumed in each plant is equal to the amount of raw material shipped from the supplier to the factory. The amount of raw material shipped from each supplier to smaller factories is equal to the capacity of each supplier in equation (13).
Equations (14)–(19) shows that constraints denote nonnegative variables, and the last limitation is related to the binary variable (zero wik).
4. Comparative evaluation of Methahursitic algorithms
Several evolutionary algorithms have been developed to find unanswered responses to multi-dimensional problems, including a non-dominated sorting genetic algorithm (NSGA- II). In multi-objective optimization, there are several different objective functions that tend to be minimized or maximally related to each other simultaneously. Often these goal functions are located at the opposite point so that one of them improves if the other deteriorates.
Therefore, in such problems, sets of optimal responses are obtained that are called optimal Pareto points or Pareto curves. As the model presented in this study is multi-objective and has many uncertain parameters, and because of its high complexity and also its NP- hardness; the metaheuristic algorithm has been used to solve it. To compare the algorithms in this study, four indices are used, which are presented below.
4.1 Mean ideal distance index
This criterion is used to calculate the mean distance of the Pareto answers of the best answers obtained for each objective function. As we can see, the lower the benchmark, the more efficient the algorithm will be.
MID¼ Xn
i¼1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
f1ifbest1 fMax1 totalfMin1 total
2
þ fMaxf2ifbest2 2 totalfMin2 total
2
s
n (20)
In the above equation,nis equal to the number of Pareto points fmaxi:totaland fmini:totalequal to the maximum and minimum values of the target functions among all the algorithms being compared, respectively. Also, fbest2 ;fbest1
is the coordination of the ideal point.
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4.2 Spacing metric index
This indicator shows the uniform distribution of Pareto’s solutions in the solution space.
This index is calculated as follows:
SM¼ Xn1
i¼1jdidj
dðn1Þ (21)
Wherenis equal to the number of Pareto points, diis equal to the Euclidean distance between the two Pareto-side solutions in the solution space andd equal to the mean of di distances. With the spacing metric index (SM) index being smaller, the algorithm will perform better.
4.3 Quality metric
The index of quality is such that all of the Pareto responses obtained by each of the algorithms are considered for each problem set, and then the selection of the non-dominated sorting for all the solutions is done. Finally, the quality of each algorithm is equal to the Pareto response specific to that algorithm of the whole Pareto’s response. Higher quality means the better performance of the algorithm.
Algorithm of each nondominated sortings
total number of nondominated sortings (22)
4.4 Diversification metric
This parameter shows the breadth of Pareto’s responses to an algorithm and can be calculated by the following equation. The higher the diversification metric (DM) index, the better the algorithm.
DM¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi maxif1iminif1i
fMax1 totalfMin1 total
!2
þ maxif2iminif2i
fMax2 totalfMin2 total
!2
vu
ut (23)
4.4.1 Analysis, problem-solving and numerical results.So the definition of the problem and its related mathematical model was implemented in this step to obtain the results, the algorithm-related codes were implemented in MATLAB software and all the results were extracted in Excel tables. The number of Runes is very high and therefore some of the results are presented in the following table:
InTable 1,tis the time period andzis the number of retailers. The input of the time period is independent and the other inputs are dependent on thezvalue so that they increase or decrease with Z changing. The time spent implementing each algorithm at each stage is calculated by the software and the comparison indices (mentioned earlier) are also represented separately for each algorithm.
In the following, the computational process of the mean values of the previous table is calculated in a separate table. This is done to validate and to decide which algorithm has a better performance index. The results are visible below (Figures 1to7).
In the following, we have a sample of the Pareto front graphs related to the GA (t= 5,z= 10) and a sample of Pareto front graphs related to the particle swarm optimization (PSO) algorithm (t= 5,z= 10).
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5. Conclusion and future work directions
According toTable 2, the NSGA-II algorithm performs better and has a shorter time in terms of computational time, but the MOPSO algorithm is more efficient in the mean ideal distance (MID), SM, quality metric (QM) and DM indices.
In this study, a two-objective MILP model was proposed to optimize the SSC. The model’s objectives are three dimensions, namely, economic, social and environmental and each one’s functions not only do not harm the others but also complement them. Although these dimensions are naturally related to one another, and affect the other dimension’s performance; but, it is necessary to make an integrated decision and manage how each of
Table 1.
Numerical results
t 1 1 1 1 1 1 . . .
z 6 7 8 9 10 11 . . .
MOPSO Time 44.12667 50.13481 49.24927 52.27444 55.20541 62.00993 . . . NSGA-II Time 43.73261 44.12795 48.9481 46.72015 53.59358 59.87906 . . . MOPSO MID 0.564915 0.680308 0.573867 0.667967 0.692436 0.403532 . . . MOPSO SM 0.704628 0.540872 0.604202 0.532658 0.523195 0.63315 . . .
MOPSO QM 1 1 1 1 1 1 . . .
MOPSO DM 1.414 1.414 1.414 1.414 1.414 1.414 . . .
NSGA-II MID 0.564915 0.680308 0.573867 0.667967 0.727058 0.443885 . . . NSGA-II SM 0.704628 0.540872 0.604202 0.63977 0.546462 0.789634 . . .
NSGA-II QM 1 1 1 0.904681 0.901125 0.988663 . . .
NSGA-II DM 1.414 1.414 1.414 1.4107 1.39305 1.414214 . . .
Figure 1.
Calculation time for each algorithm 0
5,000 10,000
1 13 25 37 49 61 73 85 97 109 121 133 145 157 169 181 193 205
Time
MOPSO Time NSGA-II Time
Figure 2.
SM index
0 0.5 1 1.5 2 2.5
1 13 25 37 49 61 73 85 97 109 121 133 145 157 169 181 193 205
SM
MOPSO SM NSGA-II SM
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these dimensions is implemented in a way that yields the highest returns. To assess sustainability, an indie was considered for each dimension. For the economic dimension, the total cost index; for the social dimension, the delivery time index; and for the environmental dimension, the indicator of the amount of pollution generated by the raw material was considered. The model was chosen for the reason that in the literature, economic, social and environmental dimensions are rarely considered simultaneously, and most studies were presented as single objective research. Customers are looking for a commodity at a low price, but not at any price, that is, they are looking for a low-cost commodity but without any negative social and environmental impacts, which is why the sustainability debate has become a competitive advantage for businesses. The problem model was optimized using Figure 4.
DM index
0 0.5 1 1.5
1 13 25 37 49 61 73 85 97 109 121 133 145 157 169 181 193 205
DM
MOPSO DM NSGA-II DM
Figure 5.
QM index
0 0.2 0.4 0.6 0.8 1 1.2
1 13 25 37 49 61 73 85 97 109 121 133 145 157 169 181 193 205
QM
MOPSO QM NSGA-II QM Figure 3.
MID index
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
1 13 25 37 49 61 73 85 97 109 121 133 145 157 169 181 193 205
MID
MOPSO MID NSGA-II MID
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Figure 7.
Pareto front genetic algorithm
Table 2.
Average indices for each algorithm
Parameter Mean
MOPSO Time 916.0618
NSGA-II Time 840.9007
MOPSO MID 0.568205
MOPSO SM 0.810167
MOPSO QM 0.740543
MOPSO DM 1.21045
NSGA-II MID 0.658108
NSGA-II SM 0.962056
NSGA-II QM 0.673221
NSGA-II DM 1.174201
Figure 6.
Pareto particle swarm algorithm
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the MOPSO algorithm and the NSGA-II algorithm, and by changing the parameter numbers, the values of the target functions and related Pareto fronts were obtained. By comparing the results of two algorithms in tables and charts, it can be concluded that the performance of the MOPSO algorithm is generally better in terms of comparison indices.
The model used in this study is a four-level model in the SSC including supplier, producer, distributor and retailer. Future research may increase or decrease the number of supply chain levels. In this research, the cost and time were considered as two independent factors; in future research, the cost factor can be considered as time-dependent. In future studies, a fast-ahead index for shipping machines can be considered. In future studies, for the purpose of analyzing and optimizing the problem, we may use other metaheuristic algorithms or consider combining two metaheuristic algorithms.
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Corresponding author
Reza Ehtesham Rasi can be contacted at:[email protected]
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