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doi: 10.3934/jimo.2021118
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The optimization of a multi-period multi-product closed-loop supply chain network with cross-docking delivery strategy

Fatemeh Kangi 1, , Seyed Hamid Reza Pasandideh 2,, , Esmaeil Mehdizadeh 1, and  Hamed Soleimani 1,3,

1. 

Department of Industrial Engineering, Faculty of Industrial and Mechanical Engineering, Qazvin Branch, Islamic Azad University, Qazvin, Iran
2. 

Department of Industrial Engineering, Faculty of Engineering, Kharazmi University, Tehran, Iran
3. 

School of Mathematics and Statistics, University of Melbourne, Melbourne, Parkville, VIC 3010, Australia

* Corresponding author: Seyed Hamid Reza Pasandideh

Received  December 2020 Revised  March 2021 Early access July 2021

The main reason for the development of this research refers to the increased attention of businesses to the CLSC concept due to the social responsibilities, strict international legislations and economic motives. Hence, this study investigates the issue of optimizing a CLSC problem involving multiple manufacturers, a hybrid cross-dock/collection center, multiple retailers and a disposal center in deterministic, multi-product and multi-period contexts. The bi-objective MILP model developed here is to simultaneously minimize total costs and total processing time of CLSC. Both strategic and tactical decisions are considered in the model where retailer demands and capacity constraints are satisfied. Since the presented model is NP-hard, NSGAII and MOPSO are hired to find near-to-optimal results for practical problem sizes in polynomial time.Then, to increase the accuracy of solutions by tuning the algorithms' parameters, the Taguchi method is applied. The practicality of the developed

Keywords: Closed-loop supply chain, hybrid facilities, transportation cost discount, outsourcing, third-party logistics providers, multi-objective optimization, de novo programming.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.
Citation: Fatemeh Kangi, Seyed Hamid Reza Pasandideh, Esmaeil Mehdizadeh, Hamed Soleimani. The optimization of a multi-period multi-product closed-loop supply chain network with cross-docking delivery strategy. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021118
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Figure 1.  General structure for the considered CLSC network
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62]">Figure 2.  The pseudo code of NSGA-II [62]
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Figure 3.  The pseudo code related to product's flow from manufacturer to retailer
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Figure 4.  A sample of crossover operator
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Figure 5.  A sample of mutation operator
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55]">Figure 6.  The pseudo code of MOPSO [55]
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Figure 7.  Average S/N ratio levels for NSGAII's parameters
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Figure 8.  Average S/N ratio levels for MOPSO's parameters
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Figure 9.  Individual 95% CIs for mean based on Pooled StDev
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Figure 10.  Individual 95% CIs for mean based on Pooled StDev
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Figure 11.  Kruskal-Wallis test on computational time
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Figure 12.  Sensitivity analysis of the demand parameter
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Table 1.  A brief review of related literatures
Reference Model Characteristics Decision variables Objective Method
Flow Hybrid fac. Period Product Out. Disc. Cross. Example No. Des.
[50] CLSC Yes Mu Mu Yes No No Test problem Loc/Alloc Si $\downarrow$ total costs LINGO software, Metaheuristic
[59] CLSC No Mu Mu Yes No No Test problem Loc/Alloc Si $\downarrow$ total logistics costs LINGO software, Metaheuristic
[24] CLSC No Si Mu Yes No No Test problem Loc/Alloc Mu $\downarrow$ total costs
$\downarrow$ total tardiness		 Scatter search, Dual simplex, $\epsilon$-constraint
[68] CLSC No Si Si No No No Test problem Loc/Alloc Mu $\downarrow$ total costs
$\uparrow$ responsiveness		 Metaheuristics
[67] CLSC Yes Mu Mu No No No an office document company Loc/Alloc/Inv Mu $\downarrow$ total costs
$\uparrow$ service efficiency		 Goal programming, Compromise programming
[5] CLSC No Si Mu No No No Copier remanufacturing Loc/Alloc Mu $\downarrow$ total costs
$\uparrow$ environmental factors		 Weighted sums, $\epsilon$-constraint
[22] CLSC No Si Si No No No Test problem Loc/Alloc Mu $\downarrow$total costs
$\downarrow$ environmental impacts
$\uparrow$ ↑social benefits		 GAMS software, Metaheuristics
[7] CLSC No Mu Mu No No No Refrigerator industry Loc/Alloc/Inv Mu $\downarrow$total costs
$\downarrow$total travel time		 $\epsilon$-constraint, GAMS software, Metaheuristics
[20] CLSC Yes Si Mu No No No Test problem Loc/Alloc Mu $\uparrow$ total profit
$\downarrow$ total spent energy
$\downarrow$ harmful emissions		 LINGO software, Goal programming
[89] CLSC Yes Si Mu No No No Test problem Loc/Alloc Mu $\downarrow$ total costs
$\downarrow$ waiting time in services		 Interval-stochastic, robust optimization, Metaheuristic, Lower bound procedure, GAMS software
[48] RL No Si Mu No No Yes Test problem Loc/Alloc Si $\downarrow$ total costs GAMS software
[93] CLSC No Si Si No No No Gold industry Loc/Alloc Mu $\downarrow$ total costs
$\uparrow$ total incomes
$\downarrow$ CO2 emissions		 LINGO software, Metaheuristic
[54] CLSC No Si Mu No No No Test problem Loc/Alloc Mu $\downarrow$ total costs
$\downarrow$ environmental impacts		 LINGO software, LP-metrics
[85] CLSC No Si Mu No No No Copiers industry Loc/Alloc Mu $\downarrow$ total costs
$\downarrow$ CO2 emissions		 $\epsilon$-constraint
[92] CLSC No Mu Mu No No No LCD and LED TV Loc/Alloc/ Route/Inv Mu $\downarrow$ total costs
$\downarrow$ environmental impacts
$\uparrow$ social impacts		 stochastic-possibilistic programming, modified game theory, lower bound procedure, GAMS software, Hybrid metaheuristic
[18] CLSC No Si Si No No No Solar cell industry Loc/Alloc Mu $\downarrow$ total costs
$\downarrow$ CO2 emissions		 branch & bound, CPLEX software, Metaheuristic
[94] RL No Si Mu No No Yes Test problem Alloc Si $\downarrow$ total costs CPLEX software
[37] CLSC No Mu Si Yes No No Filter Loc/Alloc/SS/Inv/Price Mu $\uparrow$ total profit
$\downarrow$ CO2 emissions		 Karush–Kuhn–Tucker, conditions possibilistic method, $\epsilon$-constraint, CPLEX software
[75] CLSC Yes Mu Mu No Yes No Test problem Loc/Alloc/SS Mu $\downarrow$total costs
$\downarrow$ CO2 emissions
$\uparrow$ customer satisfaction		 CPLEX software, LP-metrics
[72] CLSC No Si Si No No Yes Test problem Loc/Alloc Mu $\downarrow$total costs
$\downarrow$ environmental impacts
$\uparrow$ social benefits		 GAMS software, Metaheuristics
[40] CLSC Yes Mu Mu No No No Test problem Loc/Alloc/Inv Mu $\uparrow$ increase in the cash flow
$\uparrow$ social responsibility
$\downarrow$ amount of unreliable raw materials		 $\epsilon$-constraint, GAMS software, Metaheuristics
[61] CLSC No Mu Mu Yes No No Battery Loc/Alloc/TPS Mu $\uparrow$ total profit
$\uparrow$ environmental compliance		 Fully fuzzy stochastic programming
[86] CLSC No Mu Si No Yes No CFL light bulb Alloc/Inv/Price Mu $\downarrow$total costs
$\downarrow$ environmental impacts
$\downarrow$ social impacts		 Fuzzy TH approach [88]
[65] CLSC No Si Mu No No No Tanker industry Loc/Alloc/SS Mu $\downarrow$total costs
$\downarrow$ environmental impacts
$\uparrow$ social impacts		 Multi-choice goal programming with utility function
[66] CLSC No Mu Si No No No Test problem Loc/Alloc Mu $\uparrow$supply chain surplus
$\downarrow$ CO2 emissions		 MATLAB software
This paper CLSC Yes Mu Mu Yes Yes Yes Test problem Loc/Alloc/TPS Mu $\downarrow$ total costs
$\downarrow$ total processing times		 $\epsilon$-constraint, LINGO software, Metaheuristics
Notes:
fac. (facility); Out. (outsource); Disc. (discount); Cross. (cross-dock); Des. (description); RL (reverse logistic); CLSC (closed loop supply chain); Si (single); Mu (multi); Loc (location); Alloc (allocation); Inv (inventory); Route (routing); SS (supplier selection); TPS (third party selection); Price (pricing)
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Reference Model Characteristics Decision variables Objective Method
Flow Hybrid fac. Period Product Out. Disc. Cross. Example No. Des.
[50] CLSC Yes Mu Mu Yes No No Test problem Loc/Alloc Si $\downarrow$ total costs LINGO software, Metaheuristic
[59] CLSC No Mu Mu Yes No No Test problem Loc/Alloc Si $\downarrow$ total logistics costs LINGO software, Metaheuristic
[24] CLSC No Si Mu Yes No No Test problem Loc/Alloc Mu $\downarrow$ total costs
$\downarrow$ total tardiness		 Scatter search, Dual simplex, $\epsilon$-constraint
[68] CLSC No Si Si No No No Test problem Loc/Alloc Mu $\downarrow$ total costs
$\uparrow$ responsiveness		 Metaheuristics
[67] CLSC Yes Mu Mu No No No an office document company Loc/Alloc/Inv Mu $\downarrow$ total costs
$\uparrow$ service efficiency		 Goal programming, Compromise programming
[5] CLSC No Si Mu No No No Copier remanufacturing Loc/Alloc Mu $\downarrow$ total costs
$\uparrow$ environmental factors		 Weighted sums, $\epsilon$-constraint
[22] CLSC No Si Si No No No Test problem Loc/Alloc Mu $\downarrow$total costs
$\downarrow$ environmental impacts
$\uparrow$ ↑social benefits		 GAMS software, Metaheuristics
[7] CLSC No Mu Mu No No No Refrigerator industry Loc/Alloc/Inv Mu $\downarrow$total costs
$\downarrow$total travel time		 $\epsilon$-constraint, GAMS software, Metaheuristics
[20] CLSC Yes Si Mu No No No Test problem Loc/Alloc Mu $\uparrow$ total profit
$\downarrow$ total spent energy
$\downarrow$ harmful emissions		 LINGO software, Goal programming
[89] CLSC Yes Si Mu No No No Test problem Loc/Alloc Mu $\downarrow$ total costs
$\downarrow$ waiting time in services		 Interval-stochastic, robust optimization, Metaheuristic, Lower bound procedure, GAMS software
[48] RL No Si Mu No No Yes Test problem Loc/Alloc Si $\downarrow$ total costs GAMS software
[93] CLSC No Si Si No No No Gold industry Loc/Alloc Mu $\downarrow$ total costs
$\uparrow$ total incomes
$\downarrow$ CO2 emissions		 LINGO software, Metaheuristic
[54] CLSC No Si Mu No No No Test problem Loc/Alloc Mu $\downarrow$ total costs
$\downarrow$ environmental impacts		 LINGO software, LP-metrics
[85] CLSC No Si Mu No No No Copiers industry Loc/Alloc Mu $\downarrow$ total costs
$\downarrow$ CO2 emissions		 $\epsilon$-constraint
[92] CLSC No Mu Mu No No No LCD and LED TV Loc/Alloc/ Route/Inv Mu $\downarrow$ total costs
$\downarrow$ environmental impacts
$\uparrow$ social impacts		 stochastic-possibilistic programming, modified game theory, lower bound procedure, GAMS software, Hybrid metaheuristic
[18] CLSC No Si Si No No No Solar cell industry Loc/Alloc Mu $\downarrow$ total costs
$\downarrow$ CO2 emissions		 branch & bound, CPLEX software, Metaheuristic
[94] RL No Si Mu No No Yes Test problem Alloc Si $\downarrow$ total costs CPLEX software
[37] CLSC No Mu Si Yes No No Filter Loc/Alloc/SS/Inv/Price Mu $\uparrow$ total profit
$\downarrow$ CO2 emissions		 Karush–Kuhn–Tucker, conditions possibilistic method, $\epsilon$-constraint, CPLEX software
[75] CLSC Yes Mu Mu No Yes No Test problem Loc/Alloc/SS Mu $\downarrow$total costs
$\downarrow$ CO2 emissions
$\uparrow$ customer satisfaction		 CPLEX software, LP-metrics
[72] CLSC No Si Si No No Yes Test problem Loc/Alloc Mu $\downarrow$total costs
$\downarrow$ environmental impacts
$\uparrow$ social benefits		 GAMS software, Metaheuristics
[40] CLSC Yes Mu Mu No No No Test problem Loc/Alloc/Inv Mu $\uparrow$ increase in the cash flow
$\uparrow$ social responsibility
$\downarrow$ amount of unreliable raw materials		 $\epsilon$-constraint, GAMS software, Metaheuristics
[61] CLSC No Mu Mu Yes No No Battery Loc/Alloc/TPS Mu $\uparrow$ total profit
$\uparrow$ environmental compliance		 Fully fuzzy stochastic programming
[86] CLSC No Mu Si No Yes No CFL light bulb Alloc/Inv/Price Mu $\downarrow$total costs
$\downarrow$ environmental impacts
$\downarrow$ social impacts		 Fuzzy TH approach [88]
[65] CLSC No Si Mu No No No Tanker industry Loc/Alloc/SS Mu $\downarrow$total costs
$\downarrow$ environmental impacts
$\uparrow$ social impacts		 Multi-choice goal programming with utility function
[66] CLSC No Mu Si No No No Test problem Loc/Alloc Mu $\uparrow$supply chain surplus
$\downarrow$ CO2 emissions		 MATLAB software
This paper CLSC Yes Mu Mu Yes Yes Yes Test problem Loc/Alloc/TPS Mu $\downarrow$ total costs
$\downarrow$ total processing times		 $\epsilon$-constraint, LINGO software, Metaheuristics
Notes:
fac. (facility); Out. (outsource); Disc. (discount); Cross. (cross-dock); Des. (description); RL (reverse logistic); CLSC (closed loop supply chain); Si (single); Mu (multi); Loc (location); Alloc (allocation); Inv (inventory); Route (routing); SS (supplier selection); TPS (third party selection); Price (pricing)
Table 2.  Parameters and their levels for NSGAII
Parameters Symbols Levels Value Tuned
Level 1 Level 2 Level 3
Pop Size (A) 100 150 200 100
Iteration (B) 100 150 200 200
Crossover Rate (C) 0.85 0.9 0.95 0.85
Mutation Rate (D) 0.03 0.05 0.1 0.05
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Parameters Symbols Levels Value Tuned
Level 1 Level 2 Level 3
Pop Size (A) 100 150 200 100
Iteration (B) 100 150 200 200
Crossover Rate (C) 0.85 0.9 0.95 0.85
Mutation Rate (D) 0.03 0.05 0.1 0.05
Table 3.  Parameters and their levels for MOPSO
Parameters Symbols Levels Value Tuned
Level 1 Level 2 Level 3
Pop Size (A) 50 100 150 100
Iteration (B) 100 150 200 200
Inertia Weight (C) 0.75 0.8 0.85 0.75
C1 (D) 1.0 1.5 2.0 1.5
C2 (E) 1.0 1.5 2.0 1.5
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Parameters Symbols Levels Value Tuned
Level 1 Level 2 Level 3
Pop Size (A) 50 100 150 100
Iteration (B) 100 150 200 200
Inertia Weight (C) 0.75 0.8 0.85 0.75
C1 (D) 1.0 1.5 2.0 1.5
C2 (E) 1.0 1.5 2.0 1.5
Table 4.  Size and level of problems
Problem levels Problem size (I, C, J, K, M, N, P, T)
Small scale P1. (2, 2, 5, 2, 2, 2, 4, 3) P6. (4, 3, 10, 2, 2, 2, 4, 3)
P2. (2, 2, 7, 3, 2, 2, 4, 3) P7. (5, 2, 5, 2, 2, 2, 4, 3)
P3. (3, 3, 10, 2, 2, 2, 4, 3) P8. (5, 2, 7, 3, 2, 2, 4, 3)
P4. (3, 2, 5, 2, 2, 2, 4, 3) P9. (6, 3, 10, 2, 2, 2, 4, 3)
P5. (4, 2, 7, 3, 2, 2, 4, 3) P10. (6, 3, 10, 3, 2, 2, 4, 3)
Medium scale P11. (7, 4, 15, 3, 2, 3, 4, 5) P16. (11, 5, 30, 3, 3, 3, 4, 5)
P12. (7, 4, 20, 4, 2, 4, 4, 5) P17. (13, 4, 15, 3, 2, 3, 4, 5)
P13. (9, 5, 30, 3, 3, 3, 4, 5) P18. (13, 4, 20, 4, 2, 4, 4, 5)
P14. (9, 4, 15, 3, 2, 3, 4, 5) P19. (15, 5, 30, 3, 3, 3, 4, 5)
P15. (11, 4, 20, 4, 2, 4, 4, 5) P20. (15, 5, 30, 4, 3, 4, 4, 5)
Large scale P21. (16, 6, 50, 5, 3, 5, 4, 10) P26. (20, 9,100, 5, 4, 5, 4, 10)
P22. (16, 6, 75, 7, 3, 7, 4, 10) P27. (22, 6, 50, 5, 3, 5, 4, 10)
P23. (18, 9,100, 5, 4, 5, 4, 10) P28. (22, 6, 75, 7, 3, 7, 4, 10)
P24. (18, 6, 50, 5, 3, 5, 4, 10) P29. (24, 9,100, 5, 4, 5, 4, 10)
P25. (20, 6, 75, 7, 3, 7, 4, 10) P30. (24, 9,100, 7, 4, 7, 4, 10)
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Problem levels Problem size (I, C, J, K, M, N, P, T)
Small scale P1. (2, 2, 5, 2, 2, 2, 4, 3) P6. (4, 3, 10, 2, 2, 2, 4, 3)
P2. (2, 2, 7, 3, 2, 2, 4, 3) P7. (5, 2, 5, 2, 2, 2, 4, 3)
P3. (3, 3, 10, 2, 2, 2, 4, 3) P8. (5, 2, 7, 3, 2, 2, 4, 3)
P4. (3, 2, 5, 2, 2, 2, 4, 3) P9. (6, 3, 10, 2, 2, 2, 4, 3)
P5. (4, 2, 7, 3, 2, 2, 4, 3) P10. (6, 3, 10, 3, 2, 2, 4, 3)
Medium scale P11. (7, 4, 15, 3, 2, 3, 4, 5) P16. (11, 5, 30, 3, 3, 3, 4, 5)
P12. (7, 4, 20, 4, 2, 4, 4, 5) P17. (13, 4, 15, 3, 2, 3, 4, 5)
P13. (9, 5, 30, 3, 3, 3, 4, 5) P18. (13, 4, 20, 4, 2, 4, 4, 5)
P14. (9, 4, 15, 3, 2, 3, 4, 5) P19. (15, 5, 30, 3, 3, 3, 4, 5)
P15. (11, 4, 20, 4, 2, 4, 4, 5) P20. (15, 5, 30, 4, 3, 4, 4, 5)
Large scale P21. (16, 6, 50, 5, 3, 5, 4, 10) P26. (20, 9,100, 5, 4, 5, 4, 10)
P22. (16, 6, 75, 7, 3, 7, 4, 10) P27. (22, 6, 50, 5, 3, 5, 4, 10)
P23. (18, 9,100, 5, 4, 5, 4, 10) P28. (22, 6, 75, 7, 3, 7, 4, 10)
P24. (18, 6, 50, 5, 3, 5, 4, 10) P29. (24, 9,100, 5, 4, 5, 4, 10)
P25. (20, 6, 75, 7, 3, 7, 4, 10) P30. (24, 9,100, 7, 4, 7, 4, 10)
Table 5.  Parameters' range in test problems
Parameter Random generation function
Demand (DE) U [50,150]
Production capacity (MC) U [200*J/I, 200*J/I+275*J/I]
Transportation capacity (VC) U [200*N*J/I, 200*N*J/I+275*N*J/I]
Upper bound (U) Transportation capacity/P+1
Transportation fixed cost I (FCC) U [15, 20]
Transportation fixed cost II (FCR) U [8, 15]
Transportation fixed cost III (FCD) U [5, 10]
Transportation cost I (TIC) U [10, 15]
Transportation cost II (TCJ) U [4, 8]
Transportation cost III (TCD) U [3, 5]
Production cost (PC) U [80,100]
Opening cost of hybrid facility (EC) U [5, 30]
Recovery cost (RC) U [20, 30]
Consolidation time (TA) U [0.08, 0.11]
Inspection time (TI) U [0.03, 0.06]
Transportation time I (TC) U [2, 20]
Transportation time II (TM) U [1.5, 12]
Transportation time III (TD) U [3, 5]
Fraction of returned product ($ \alpha $) U [0.02, 0.04]
Fraction of recoverable product ($ \beta $) U [0.25, 0.8]
Weight/volume of each unit of product ($ \gamma $) U [0.5, 3]
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Parameter Random generation function
Demand (DE) U [50,150]
Production capacity (MC) U [200*J/I, 200*J/I+275*J/I]
Transportation capacity (VC) U [200*N*J/I, 200*N*J/I+275*N*J/I]
Upper bound (U) Transportation capacity/P+1
Transportation fixed cost I (FCC) U [15, 20]
Transportation fixed cost II (FCR) U [8, 15]
Transportation fixed cost III (FCD) U [5, 10]
Transportation cost I (TIC) U [10, 15]
Transportation cost II (TCJ) U [4, 8]
Transportation cost III (TCD) U [3, 5]
Production cost (PC) U [80,100]
Opening cost of hybrid facility (EC) U [5, 30]
Recovery cost (RC) U [20, 30]
Consolidation time (TA) U [0.08, 0.11]
Inspection time (TI) U [0.03, 0.06]
Transportation time I (TC) U [2, 20]
Transportation time II (TM) U [1.5, 12]
Transportation time III (TD) U [3, 5]
Fraction of returned product ($ \alpha $) U [0.02, 0.04]
Fraction of recoverable product ($ \beta $) U [0.25, 0.8]
Weight/volume of each unit of product ($ \gamma $) U [0.5, 3]
Table 6.  The obtained metrics for algorithms' performance (MID, SM, NPS, QM and Time)
Problem size $\sharp$ P. MID SM NPS QM Time
NSGA-II MOPSO NSGA-II MOPSO NSGA-II MOPSO NSGA-II MOPSO NSGA-II MOPSO
Small P1. 0.2965 0.9135 0.52047 0.79424 2 13 0.11743 0.88256 152.2674 83.61384
P2. 0.7985 0.7594 1.44002 1.42646 11 5 0.21714 0.78285 170.5 93.31762
P3. 0.6678 0.8303 0.90528 1.30920 11 20 0.16347 0.83652 170.5483 112.2679
P4. 0.7380 0.8242 0.34031 1.31610 4 12 0.03538 0.96461 169.0048 89.31582
P5. 0.8355 0.9241 0.04645 0.77740 3 13 0.18888 0.81111 170.3636 105.0191
P6. 0.8890 0.6733 0.73833 1.26314 9 11 0.14164 0.85835 170.8992 117.5417
P7. 0.5898 0.8458 0.92847 1.49796 9 19 0.025 0.975 170.4384 115.9775
P8. 0.6611 0.8644 1.26429 1.74167 8 12 0.03760 0.96239 170.6712 132.081
P9. 0.8354 0.8519 0.63885 0.40465 4 7 0 1 170.698 131.0991
P10. 0.9213 0.8708 0.24135 0.98781 9 3 0.01818 0.98181 170.9277 133.0108
Medium P11. 1.2217 0.6809 0.24001 0.59929 2 5 0.08008 0.91991 171.9736 171.6591
P12. 1.0055 0.6922 0.76749 0.72169 7 10 0.25 0.75 174.4272 164.5119
P13. 0.9574 0.6393 0.27271 0.12447 4 7 0.38650 0.61349 173.6264 172.9455
P14. 1.0100 0.8820 0.57146 0.46826 5 8 0.27200 0.72799 172.9297 173.0899
P15. 1.0500 0.7352 0.65799 1.15036 4 4 0.15320 0.84679 173.1716 174.1405
P16. 0.9179 0.6964 0.56273 0.59754 6 10 0.38333 0.61666 178.755 175.1082
P17. 0.8986 0.8084 0.53351 0.79303 2 7 0.05714 0.94285 173.6318 173.4457
P18. 0.9193 0.7470 0.75717 0.54668 7 7 0.28666 0.71333 175.8487 164.4145
P19. 1.0065 0.6942 0.76486 0.44983 3 4 0.24 0.76 172.0987 173.5797
P20. 0.9107 0.6542 0.41413 1.07685 7 10 0.51414 0.48585 171.5135 174.0156
Large P21. 0.9056 0.5901 0.54131 1.23029 6 4 0.64047 0.35952 183.5318 182.9749
P22. 0.7044 0.8123 0.98102 0.64139 4 8 0.375 0.625 196.9311 178.7448
P23. 0.8133 0.5921 0.70779 0.39384 5 9 0.40090 0.59909 189.7017 195.596
P24. 0.8936 0.6095 0.74845 0.32664 7 5 0.60333 0.39666 196.8208 173.8915
P25. 0.8663 0.6634 0.43799 0.39058 7 3 0.61555 0.38444 180.2806 180.9208
P26. 0.8217 0.7219 0.10674 0.63042 7 3 0.31666 0.68333 184.1588 183.5843
P27. 0.7576 0.7515 1.17926 0.91817 9 7 0.23690 0.76309 180.4263 193.2142
P28. 0.9310 0.6332 0.85185 0.95080 8 5 0.59047 0.40952 192.3841 180.048
P29. 0.7122 0.7540 0.77547 0.99490 7 4 0.34381 0.65619 177.788 186.2371
P30. 0.9133 0.7173 0.48094 0.91014 5 5 0.59285 0.40714 211.5774 188.7306
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Problem size $\sharp$ P. MID SM NPS QM Time
NSGA-II MOPSO NSGA-II MOPSO NSGA-II MOPSO NSGA-II MOPSO NSGA-II MOPSO
Small P1. 0.2965 0.9135 0.52047 0.79424 2 13 0.11743 0.88256 152.2674 83.61384
P2. 0.7985 0.7594 1.44002 1.42646 11 5 0.21714 0.78285 170.5 93.31762
P3. 0.6678 0.8303 0.90528 1.30920 11 20 0.16347 0.83652 170.5483 112.2679
P4. 0.7380 0.8242 0.34031 1.31610 4 12 0.03538 0.96461 169.0048 89.31582
P5. 0.8355 0.9241 0.04645 0.77740 3 13 0.18888 0.81111 170.3636 105.0191
P6. 0.8890 0.6733 0.73833 1.26314 9 11 0.14164 0.85835 170.8992 117.5417
P7. 0.5898 0.8458 0.92847 1.49796 9 19 0.025 0.975 170.4384 115.9775
P8. 0.6611 0.8644 1.26429 1.74167 8 12 0.03760 0.96239 170.6712 132.081
P9. 0.8354 0.8519 0.63885 0.40465 4 7 0 1 170.698 131.0991
P10. 0.9213 0.8708 0.24135 0.98781 9 3 0.01818 0.98181 170.9277 133.0108
Medium P11. 1.2217 0.6809 0.24001 0.59929 2 5 0.08008 0.91991 171.9736 171.6591
P12. 1.0055 0.6922 0.76749 0.72169 7 10 0.25 0.75 174.4272 164.5119
P13. 0.9574 0.6393 0.27271 0.12447 4 7 0.38650 0.61349 173.6264 172.9455
P14. 1.0100 0.8820 0.57146 0.46826 5 8 0.27200 0.72799 172.9297 173.0899
P15. 1.0500 0.7352 0.65799 1.15036 4 4 0.15320 0.84679 173.1716 174.1405
P16. 0.9179 0.6964 0.56273 0.59754 6 10 0.38333 0.61666 178.755 175.1082
P17. 0.8986 0.8084 0.53351 0.79303 2 7 0.05714 0.94285 173.6318 173.4457
P18. 0.9193 0.7470 0.75717 0.54668 7 7 0.28666 0.71333 175.8487 164.4145
P19. 1.0065 0.6942 0.76486 0.44983 3 4 0.24 0.76 172.0987 173.5797
P20. 0.9107 0.6542 0.41413 1.07685 7 10 0.51414 0.48585 171.5135 174.0156
Large P21. 0.9056 0.5901 0.54131 1.23029 6 4 0.64047 0.35952 183.5318 182.9749
P22. 0.7044 0.8123 0.98102 0.64139 4 8 0.375 0.625 196.9311 178.7448
P23. 0.8133 0.5921 0.70779 0.39384 5 9 0.40090 0.59909 189.7017 195.596
P24. 0.8936 0.6095 0.74845 0.32664 7 5 0.60333 0.39666 196.8208 173.8915
P25. 0.8663 0.6634 0.43799 0.39058 7 3 0.61555 0.38444 180.2806 180.9208
P26. 0.8217 0.7219 0.10674 0.63042 7 3 0.31666 0.68333 184.1588 183.5843
P27. 0.7576 0.7515 1.17926 0.91817 9 7 0.23690 0.76309 180.4263 193.2142
P28. 0.9310 0.6332 0.85185 0.95080 8 5 0.59047 0.40952 192.3841 180.048
P29. 0.7122 0.7540 0.77547 0.99490 7 4 0.34381 0.65619 177.788 186.2371
P30. 0.9133 0.7173 0.48094 0.91014 5 5 0.59285 0.40714 211.5774 188.7306
Table 7.  ANOVA results for MID criterion
Source DF SS MS F-Test P-Value
Factor 1 0.1517 0.1517 8.09 0.006
Error 58 1.0869 0.0187
Total 59 1.2386
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Source DF SS MS F-Test P-Value
Factor 1 0.1517 0.1517 8.09 0.006
Error 58 1.0869 0.0187
Total 59 1.2386
Table 8.  ANOVA results for QM criterion
Source DF SS MS F-Test P-Value
Factor 1 3.0071 3.0071 75.25 0.000
Error 58 2.3179 0.0400
Total 59 5.3250
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Source DF SS MS F-Test P-Value
Factor 1 3.0071 3.0071 75.25 0.000
Error 58 2.3179 0.0400
Total 59 5.3250
Table 9.  ANOVA results for NPS criterion
Source DF SS MS F-Test P-Value
Factor 1 56.1 56.1 4.35 0.041
Error 58 747.9 12.9
Total 59 803.9
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Source DF SS MS F-Test P-Value
Factor 1 56.1 56.1 4.35 0.041
Error 58 747.9 12.9
Total 59 803.9
Table 10.  ANOVA results for QM criterion
Source DF SS MS F-Test P-Value
Factor 1 3.0071 3.0071 75.25 0.000
Error 58 2.3179 0.0400
Total 59 5.3250
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Source DF SS MS F-Test P-Value
Factor 1 3.0071 3.0071 75.25 0.000
Error 58 2.3179 0.0400
Total 59 5.3250
Table 11.  Results of sensitivity analysis
Objective functions Demand's change interval
-20% -10% 0% 10% 20%
Total cost 431,871 553,901 806,133 849,666 948,245
Total processing time 186 221 240 276 298
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Objective functions Demand's change interval
-20% -10% 0% 10% 20%
Total cost 431,871 553,901 806,133 849,666 948,245
Total processing time 186 221 240 276 298
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