# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2021107
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## A robust multi-objective model for managing the distribution of perishable products within a green closed-loop supply chain

 1 Department of Industrial Engineering, Nour Branch, Islamic Azad University, Nour, Iran 2 Department of Industrial Engineering and Quality Research Centre, Nour Branch, Islamic Azad University, Nour, Iran 3 Department of Industrial Engineering, Qaemshahr Branch, Islamic Azad University, Qaemshahr, Iran 4 Innovation and Management Research Center, Ayatollah Amoli Branch, Islamic Azad University, Amol, Iran

* Corresponding author: Fatemeh Harsej

Received  December 2020 Revised  April 2021 Early access June 2021

The required processes of supply chain management include optimal strategic, tactical, and operational decisions, all of which have important economic and environmental effects. In this regard, efficient supply chain planning for the production and distribution of perishable productsis of particular importance due to its leading role in the human food pyramid. One of the main challenges facing this chain is the time when products and goods are delivered to the customers and customer satisfaction will increase through this.In this research, a bi-objective mixed-integer linear programming (MILP)model is proposedto design a multi-level, multi-period, multi-product closed-loop supply chain (CLSC) for timely production and distribution of perishable products, taking into account the uncertainty of demand. To face the model uncertainty, the robust optimization (RO) method is utilized. Moreover, to solve and validate the bi-objective model in small-size problems, the $\epsilon$-constraint method (EC) is presented. On the other hand, a Non-dominated Sorting Genetic Algorithm (NSGA-II) is developed for solving large-size problems. First, the deterministic and robust models are compared by applying the suggested solutions methods in a small-size problem, and then, the proposed solution methods are compared in large-size problems in terms of different well-known metrics. According to the comparison, the proposed model has an acceptable performance in providing the optimal solutions and the proposed algorithm obtains efficient solutions.Finally, managerial insights are proposed using sensitivity analysis of important parameters of the problem.

Citation: Maedeh Agahgolnezhad Gerdrodbari, Fatemeh Harsej, Mahboubeh Sadeghpour, Mohammad Molani Aghdam. A robust multi-objective model for managing the distribution of perishable products within a green closed-loop supply chain. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021107
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The proposed supply chain network
Flowchart of the proposed NSGA-II (Rabbani et al., 2021)
The Pareto solution obtained by two methods
The comparison of NSGA-II and EC at uncertainty level 0.2
The comparison of NSGA-II and EC at uncertainty level 0.4
The comparison of NSGA-II and EC at uncertainty level 0.5
The comparison of solution time for NSGA-II and EC
A brief comparison between previously-performed studies and our study
 Reference Year Levels of Network Features Objectives Solution methods Supply centers Production centers Collection centers Recycling centers Distribution centers Recovery centers Repair center Disposal center Uncertainty Perishable products Responsiveness Environmental Social Economic Pishvaee et al. 2014 * * * * * * * * LINGO Govindan et al. 2014 * * * * Benders decomposition Devika et al. 2014 * * * * * * * * * * LINGO Azadeh et al. 2015 * * * * * * * $\epsilon$-constraint Wu et al. 2017 * * * * * NSGA-II Keshavarz Ghorabaee et al. 2017 * * * * * * * GAMS Cheraghalipour et al. 2018 * * * * * * NSGA-II Kayvanfar et al. 2018 * * * * * * * * Benders decomposition Dai et al. 2018 * * * * LINGO Yavari andGeraeli 2019 * * * * * * * * Heuristics Parsa et al. 2020 * * * * * * Branch-and-bound (B & B) algorithm Lotfi et al. 2021 * * * * * * * * * * LP-Metric method Current work 2021 * * * * * * * * * * $\epsilon$-constraint and NSGA-II
 Reference Year Levels of Network Features Objectives Solution methods Supply centers Production centers Collection centers Recycling centers Distribution centers Recovery centers Repair center Disposal center Uncertainty Perishable products Responsiveness Environmental Social Economic Pishvaee et al. 2014 * * * * * * * * LINGO Govindan et al. 2014 * * * * Benders decomposition Devika et al. 2014 * * * * * * * * * * LINGO Azadeh et al. 2015 * * * * * * * $\epsilon$-constraint Wu et al. 2017 * * * * * NSGA-II Keshavarz Ghorabaee et al. 2017 * * * * * * * GAMS Cheraghalipour et al. 2018 * * * * * * NSGA-II Kayvanfar et al. 2018 * * * * * * * * Benders decomposition Dai et al. 2018 * * * * LINGO Yavari andGeraeli 2019 * * * * * * * * Heuristics Parsa et al. 2020 * * * * * * Branch-and-bound (B & B) algorithm Lotfi et al. 2021 * * * * * * * * * * LP-Metric method Current work 2021 * * * * * * * * * * $\epsilon$-constraint and NSGA-II
An example of chromosome
 First Part 0.41 0.72 0.93 Second Part Customer 1 Customer 2 Customer 3 Customer 4 Customer 5 Distribution Center 1 0.61 0.29 0.43 0.27 0.35 Distribution Center 2 0.45 0.73 0.28 0.34 0.19 Distribution Center 3 0.35 0.91 0.73 0.58 0.39
 First Part 0.41 0.72 0.93 Second Part Customer 1 Customer 2 Customer 3 Customer 4 Customer 5 Distribution Center 1 0.61 0.29 0.43 0.27 0.35 Distribution Center 2 0.45 0.73 0.28 0.34 0.19 Distribution Center 3 0.35 0.91 0.73 0.58 0.39
Interpretation of chromosome
 First Part 0 1 1 Second Part Customer 1 Customer 2 Customer 3 Customer 4 Customer 5 Distribution Center 1 0 0 0 0 0 Distribution Center 2 0.52 0.44 0.27 0.37 0.32 Distribution Center 3 0.48 0.56 0.73 0.63 0.68
 First Part 0 1 1 Second Part Customer 1 Customer 2 Customer 3 Customer 4 Customer 5 Distribution Center 1 0 0 0 0 0 Distribution Center 2 0.52 0.44 0.27 0.37 0.32 Distribution Center 3 0.48 0.56 0.73 0.63 0.68
The value of parameters for NSGA-II
 Parameter Value Npop 50 80 100 Max iteration 100 200 300 Cross rate 0.5 0.7 0.9 Mut rate 0.5 0.3 0.1
 Parameter Value Npop 50 80 100 Max iteration 100 200 300 Cross rate 0.5 0.7 0.9 Mut rate 0.5 0.3 0.1
The optimal value of parameters of NSGA-II
 Parameter Value Npop 100 Max iteration 200 Cross rate 0.7 Mut rate 0.3
 Parameter Value Npop 100 Max iteration 200 Cross rate 0.7 Mut rate 0.3
The small-size instance for the supply chain network
 Set Number Suppliers 4 Production centers 3 Distribution centers 3 Customers 5 Collection centers 2 Recovery centers 2 Disposal centers 2 Products 2 Raw materials 2 Time periods 1 Technology levels 2 Transportation modes 2
 Set Number Suppliers 4 Production centers 3 Distribution centers 3 Customers 5 Collection centers 2 Recovery centers 2 Disposal centers 2 Products 2 Raw materials 2 Time periods 1 Technology levels 2 Transportation modes 2
Value of parameters
 Parameter Value Parameter Value Customer demand U(1,200) Quantity of raw material U(50,350) Capacity of suppliers U(1000, 2500) Capacity of production centers U(500, 2000) Capacity of distribution centers U(1000, 2500) Capacity of collection centers U(1000, 2500) Capacity of disposal centers U(1000, 2000) Capacity of recovery centers U(1000, 2000) Cost of transporting raw materials from the supply center to the production center U(50,150) Cost of transporting products from production center to distribution center U(50,150) Cost of transporting from distribution center to customer U(50,150) Cost of transporting from customer to collection center U(50,150) Cost of transporting from the collection center to the disposal center U(50,150) Cost of transporting from the collection center to the recovery center U(50,150) Cost of transporting from the recovery center to the production center U(50,150) Volume of $CO_2$ emission released to transport raw material from the supply center to the production center U(50,100) Volume of $CO_2$ emission released to transport products from the production center to the distribution center U(50,100) Volume of $CO_2$ emission released to transport from the distribution center to the customer U(50,100) Volume of $CO_2$ emission released to transport from customer to the collection center U(50,100) Volume of $CO_2$ emission released to transport from the collection center to the disposal center U(50,100) Volume of $CO_2$ emission released to transport from the collection center to the recovery center U(50,100) Volume of $CO_2$ emission released to transport from the recovery center to the production center U(50,100) Preparation time for transportation of raw material from the supply center to production center U(10, 20) Preparation time for the transportation of products from distribution center to customers U(10, 20) Distance between supply center and production center U(500, 1500) Distance between production center and distribution center U(500, 1500) Distance between the distribution center and customer U(500, 1500) Distance between customer and collection center U(500, 1500) Distance between collection center and disposal center U(500, 1500) Distance between collection center and recovery center U(500, 1500) Distance between recovery center and production center U(500, 1500) Production cost in production centers U(50,100) Processing cost in distribution centers U(50,100) Processing cost in production centers U(50,100) Processing cost in disposal centers U(50,100) Processing cost in recovery centers U(50,100) Fixed cost of establishing a production center U(5000, 15000) Fixed cost of establishing a distribution center U(5000, 15000) Fixed cost of establishing a collection center U(5000, 15000) Fixed cost of establishing a disposal center U(5000, 15000) Fixed cost of establishing a recovery center U(5000, 15000) Inventory holding cost U(100,200) Inventory shortage cost U(100,150) Consumption coefficient of raw materials U(0.3, 0.7) Recovery coefficient of products U(0.1, 0.4) Flow rate of retuned products U(0, 0.5) Flow rate of disposable products U(0, 0.3)
 Parameter Value Parameter Value Customer demand U(1,200) Quantity of raw material U(50,350) Capacity of suppliers U(1000, 2500) Capacity of production centers U(500, 2000) Capacity of distribution centers U(1000, 2500) Capacity of collection centers U(1000, 2500) Capacity of disposal centers U(1000, 2000) Capacity of recovery centers U(1000, 2000) Cost of transporting raw materials from the supply center to the production center U(50,150) Cost of transporting products from production center to distribution center U(50,150) Cost of transporting from distribution center to customer U(50,150) Cost of transporting from customer to collection center U(50,150) Cost of transporting from the collection center to the disposal center U(50,150) Cost of transporting from the collection center to the recovery center U(50,150) Cost of transporting from the recovery center to the production center U(50,150) Volume of $CO_2$ emission released to transport raw material from the supply center to the production center U(50,100) Volume of $CO_2$ emission released to transport products from the production center to the distribution center U(50,100) Volume of $CO_2$ emission released to transport from the distribution center to the customer U(50,100) Volume of $CO_2$ emission released to transport from customer to the collection center U(50,100) Volume of $CO_2$ emission released to transport from the collection center to the disposal center U(50,100) Volume of $CO_2$ emission released to transport from the collection center to the recovery center U(50,100) Volume of $CO_2$ emission released to transport from the recovery center to the production center U(50,100) Preparation time for transportation of raw material from the supply center to production center U(10, 20) Preparation time for the transportation of products from distribution center to customers U(10, 20) Distance between supply center and production center U(500, 1500) Distance between production center and distribution center U(500, 1500) Distance between the distribution center and customer U(500, 1500) Distance between customer and collection center U(500, 1500) Distance between collection center and disposal center U(500, 1500) Distance between collection center and recovery center U(500, 1500) Distance between recovery center and production center U(500, 1500) Production cost in production centers U(50,100) Processing cost in distribution centers U(50,100) Processing cost in production centers U(50,100) Processing cost in disposal centers U(50,100) Processing cost in recovery centers U(50,100) Fixed cost of establishing a production center U(5000, 15000) Fixed cost of establishing a distribution center U(5000, 15000) Fixed cost of establishing a collection center U(5000, 15000) Fixed cost of establishing a disposal center U(5000, 15000) Fixed cost of establishing a recovery center U(5000, 15000) Inventory holding cost U(100,200) Inventory shortage cost U(100,150) Consumption coefficient of raw materials U(0.3, 0.7) Recovery coefficient of products U(0.1, 0.4) Flow rate of retuned products U(0, 0.5) Flow rate of disposable products U(0, 0.3)
Results of the solution methods for the robust model
 No. NSGA-II EC Objective 1 Objective 2 Objective 1 Objective 2 1 860870 26703 860824 26700 2 861031 26288 861029 26277 3 864920 25883 864875 25854 4 869473 25445 869468 25441 5 874268 25014 874115 25010
 No. NSGA-II EC Objective 1 Objective 2 Objective 1 Objective 2 1 860870 26703 860824 26700 2 861031 26288 861029 26277 3 864920 25883 864875 25854 4 869473 25445 869468 25441 5 874268 25014 874115 25010
Results of the solution methods for deterministic and robust models
 Model NSGA-II EC Objective 1 Objective 2 CPU time Objective 1 Objective 2 CPU time Deterministic 858651.6 25271.3 12.94 858242.1 24971.3 31.84 Robust 866112.4 25866.6 14.04 866062.2 25856.4 40.56
 Model NSGA-II EC Objective 1 Objective 2 CPU time Objective 1 Objective 2 CPU time Deterministic 858651.6 25271.3 12.94 858242.1 24971.3 31.84 Robust 866112.4 25866.6 14.04 866062.2 25856.4 40.56
Information of the problem instances in medium- and large- size
 Sets P1 P2 P3 P4 Suppliers 8 15 20 40 Production centers 5 10 15 25 Distribution centers 5 10 15 30 Customers 8 20 35 50 Collection centers 4 6 10 15 Recovery centers 4 6 10 15 Disposal centers 4 6 10 15 Products 4 6 10 15 Raw materials 4 6 10 15 Time periods 3 4 5 8 Technology levels 3 4 5 8 Transportation modes 3 4 5 8
 Sets P1 P2 P3 P4 Suppliers 8 15 20 40 Production centers 5 10 15 25 Distribution centers 5 10 15 30 Customers 8 20 35 50 Collection centers 4 6 10 15 Recovery centers 4 6 10 15 Disposal centers 4 6 10 15 Products 4 6 10 15 Raw materials 4 6 10 15 Time periods 3 4 5 8 Technology levels 3 4 5 8 Transportation modes 3 4 5 8
The average value of criteria for the two algorithms in the uncertainty level of 0.2
 Criteria DM MID SM NPS Problem/ Method EC NSGA-II EC NSGA-II EC NSGA-II EC NSGA-II 1 1.09 1.13 0.82 0.88 1.12 1.01 4 8 2 1. 23 1.2 1.13 1.09 1.07 0.99 2 14 3 0.92 0.95 0.94 0.9 0.71 0.66 3 23 4 - 1.13 - 1.55 - 2.39 - 32
 Criteria DM MID SM NPS Problem/ Method EC NSGA-II EC NSGA-II EC NSGA-II EC NSGA-II 1 1.09 1.13 0.82 0.88 1.12 1.01 4 8 2 1. 23 1.2 1.13 1.09 1.07 0.99 2 14 3 0.92 0.95 0.94 0.9 0.71 0.66 3 23 4 - 1.13 - 1.55 - 2.39 - 32
The average value of criteria for the two algorithms in the uncertainty level of 0.4
 Criteria DM MID SM NPS Problem/ Method EC NSGA-II EC NSGA-II EC NSGA-II EC NSGA-II 1 1.03 1.09 0.86 0.74 2.33 2.14 3 7 2 1.16 1.21 0.93 0.82 2.02 1.91 2 16 3 0.75 0.69 0.79 0.83 0.72 0.92 2 28 4 - 1.31 - 1.02 - 1.24 - 43
 Criteria DM MID SM NPS Problem/ Method EC NSGA-II EC NSGA-II EC NSGA-II EC NSGA-II 1 1.03 1.09 0.86 0.74 2.33 2.14 3 7 2 1.16 1.21 0.93 0.82 2.02 1.91 2 16 3 0.75 0.69 0.79 0.83 0.72 0.92 2 28 4 - 1.31 - 1.02 - 1.24 - 43
The average value of criteria for the two algorithms in the uncertainty level of 0.5
 Criteria DM MID SM NPS Problem/ Method EC NSGA-II EC NSGA-II EC NSGA-II EC NSGA-II 1 1.17 1.24 0.59 0.69 1.25 1.03 2 6 2 0.84 0.93 0.32 0.23 1.97 1.51 4 15 3 1.2 1.31 0.43 0.31 0.9 0.87 2 37 4 - 0.71 - 0.92 - 2.34 - 52
 Criteria DM MID SM NPS Problem/ Method EC NSGA-II EC NSGA-II EC NSGA-II EC NSGA-II 1 1.17 1.24 0.59 0.69 1.25 1.03 2 6 2 0.84 0.93 0.32 0.23 1.97 1.51 4 15 3 1.2 1.31 0.43 0.31 0.9 0.87 2 37 4 - 0.71 - 0.92 - 2.34 - 52

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