Figure 1.
General structure for the considered CLSC network
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The optimization of a multi-period multi-product closed-loop supply chain network with cross-docking delivery strategy
| 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 |
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 and Management Optimization, doi: 10.3934/jimo.2021118
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Figure 3.
The pseudo code related to product's flow from manufacturer to retailer
Figure 4.
A sample of crossover operator
Figure 5.
A sample of mutation operator
Figure 7.
Average S/N ratio levels for NSGAII's parameters
Figure 8.
Average S/N ratio levels for MOPSO's parameters
Figure 9.
Individual 95% CIs for mean based on Pooled StDev
Figure 10.
Individual 95% CIs for mean based on Pooled StDev
Figure 11.
Kruskal-Wallis test on computational time
Figure 12.
Sensitivity analysis of the demand parameter
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 | LINGO software, Metaheuristic | |
| [59] | CLSC | No | Mu | Mu | Yes | No | No | Test problem | Loc/Alloc | Si | LINGO software, Metaheuristic | |
| [24] | CLSC | No | Si | Mu | Yes | No | No | Test problem | Loc/Alloc | Mu | Scatter search, Dual simplex, |
|
| [68] | CLSC | No | Si | Si | No | No | No | Test problem | Loc/Alloc | Mu | Metaheuristics | |
| [67] | CLSC | Yes | Mu | Mu | No | No | No | an office document company | Loc/Alloc/Inv | Mu | Goal programming, Compromise programming | |
| [5] | CLSC | No | Si | Mu | No | No | No | Copier remanufacturing | Loc/Alloc | Mu | Weighted sums, |
|
| [22] | CLSC | No | Si | Si | No | No | No | Test problem | Loc/Alloc | Mu | GAMS software, Metaheuristics | |
| [7] | CLSC | No | Mu | Mu | No | No | No | Refrigerator industry | Loc/Alloc/Inv | Mu | ||
| [20] | CLSC | Yes | Si | Mu | No | No | No | Test problem | Loc/Alloc | Mu | LINGO software, Goal programming | |
| [89] | CLSC | Yes | Si | Mu | No | No | No | Test problem | Loc/Alloc | Mu | Interval-stochastic, robust optimization, Metaheuristic, Lower bound procedure, GAMS software | |
| [48] | RL | No | Si | Mu | No | No | Yes | Test problem | Loc/Alloc | Si | GAMS software | |
| [93] | CLSC | No | Si | Si | No | No | No | Gold industry | Loc/Alloc | Mu | LINGO software, Metaheuristic | |
| [54] | CLSC | No | Si | Mu | No | No | No | Test problem | Loc/Alloc | Mu | LINGO software, LP-metrics | |
| [85] | CLSC | No | Si | Mu | No | No | No | Copiers industry | Loc/Alloc | Mu | ||
| [92] | CLSC | No | Mu | Mu | No | No | No | LCD and LED TV | Loc/Alloc/ Route/Inv | Mu | 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 | branch & bound, CPLEX software, Metaheuristic | |
| [94] | RL | No | Si | Mu | No | No | Yes | Test problem | Alloc | Si | CPLEX software | |
| [37] | CLSC | No | Mu | Si | Yes | No | No | Filter | Loc/Alloc/SS/Inv/Price | Mu | Karush–Kuhn–Tucker, conditions possibilistic method, |
|
| [75] | CLSC | Yes | Mu | Mu | No | Yes | No | Test problem | Loc/Alloc/SS | Mu | CPLEX software, LP-metrics | |
| [72] | CLSC | No | Si | Si | No | No | Yes | Test problem | Loc/Alloc | Mu | GAMS software, Metaheuristics | |
| [40] | CLSC | Yes | Mu | Mu | No | No | No | Test problem | Loc/Alloc/Inv | Mu | ||
| [61] | CLSC | No | Mu | Mu | Yes | No | No | Battery | Loc/Alloc/TPS | Mu | Fully fuzzy stochastic programming | |
| [86] | CLSC | No | Mu | Si | No | Yes | No | CFL light bulb | Alloc/Inv/Price | Mu | Fuzzy TH approach [88] | |
| [65] | CLSC | No | Si | Mu | No | No | No | Tanker industry | Loc/Alloc/SS | Mu | Multi-choice goal programming with utility function | |
| [66] | CLSC | No | Mu | Si | No | No | No | Test problem | Loc/Alloc | Mu | MATLAB software | |
| This paper | CLSC | Yes | Mu | Mu | Yes | Yes | Yes | Test problem | Loc/Alloc/TPS | Mu | ||
| 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) |
||||||||||||
| 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 | LINGO software, Metaheuristic | |
| [59] | CLSC | No | Mu | Mu | Yes | No | No | Test problem | Loc/Alloc | Si | LINGO software, Metaheuristic | |
| [24] | CLSC | No | Si | Mu | Yes | No | No | Test problem | Loc/Alloc | Mu | Scatter search, Dual simplex, |
|
| [68] | CLSC | No | Si | Si | No | No | No | Test problem | Loc/Alloc | Mu | Metaheuristics | |
| [67] | CLSC | Yes | Mu | Mu | No | No | No | an office document company | Loc/Alloc/Inv | Mu | Goal programming, Compromise programming | |
| [5] | CLSC | No | Si | Mu | No | No | No | Copier remanufacturing | Loc/Alloc | Mu | Weighted sums, |
|
| [22] | CLSC | No | Si | Si | No | No | No | Test problem | Loc/Alloc | Mu | GAMS software, Metaheuristics | |
| [7] | CLSC | No | Mu | Mu | No | No | No | Refrigerator industry | Loc/Alloc/Inv | Mu | ||
| [20] | CLSC | Yes | Si | Mu | No | No | No | Test problem | Loc/Alloc | Mu | LINGO software, Goal programming | |
| [89] | CLSC | Yes | Si | Mu | No | No | No | Test problem | Loc/Alloc | Mu | Interval-stochastic, robust optimization, Metaheuristic, Lower bound procedure, GAMS software | |
| [48] | RL | No | Si | Mu | No | No | Yes | Test problem | Loc/Alloc | Si | GAMS software | |
| [93] | CLSC | No | Si | Si | No | No | No | Gold industry | Loc/Alloc | Mu | LINGO software, Metaheuristic | |
| [54] | CLSC | No | Si | Mu | No | No | No | Test problem | Loc/Alloc | Mu | LINGO software, LP-metrics | |
| [85] | CLSC | No | Si | Mu | No | No | No | Copiers industry | Loc/Alloc | Mu | ||
| [92] | CLSC | No | Mu | Mu | No | No | No | LCD and LED TV | Loc/Alloc/ Route/Inv | Mu | 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 | branch & bound, CPLEX software, Metaheuristic | |
| [94] | RL | No | Si | Mu | No | No | Yes | Test problem | Alloc | Si | CPLEX software | |
| [37] | CLSC | No | Mu | Si | Yes | No | No | Filter | Loc/Alloc/SS/Inv/Price | Mu | Karush–Kuhn–Tucker, conditions possibilistic method, |
|
| [75] | CLSC | Yes | Mu | Mu | No | Yes | No | Test problem | Loc/Alloc/SS | Mu | CPLEX software, LP-metrics | |
| [72] | CLSC | No | Si | Si | No | No | Yes | Test problem | Loc/Alloc | Mu | GAMS software, Metaheuristics | |
| [40] | CLSC | Yes | Mu | Mu | No | No | No | Test problem | Loc/Alloc/Inv | Mu | ||
| [61] | CLSC | No | Mu | Mu | Yes | No | No | Battery | Loc/Alloc/TPS | Mu | Fully fuzzy stochastic programming | |
| [86] | CLSC | No | Mu | Si | No | Yes | No | CFL light bulb | Alloc/Inv/Price | Mu | Fuzzy TH approach [88] | |
| [65] | CLSC | No | Si | Mu | No | No | No | Tanker industry | Loc/Alloc/SS | Mu | Multi-choice goal programming with utility function | |
| [66] | CLSC | No | Mu | Si | No | No | No | Test problem | Loc/Alloc | Mu | MATLAB software | |
| This paper | CLSC | Yes | Mu | Mu | Yes | Yes | Yes | Test problem | Loc/Alloc/TPS | Mu | ||
| 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 |
| 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 |
| 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) |
| 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 ( |
U [0.02, 0.04] |
| Fraction of recoverable product ( |
U [0.25, 0.8] |
| Weight/volume of each unit of product ( |
U [0.5, 3] |
| 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 ( |
U [0.02, 0.04] |
| Fraction of recoverable product ( |
U [0.25, 0.8] |
| Weight/volume of each unit of product ( |
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 | |
| 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 |
| 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 |
| 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 |
| 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 |
| 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 |
| 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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