In consideration of capacity constraints, fuzzy defect ratio, and fuzzy transport loss ratio, this paper attempted to establish an optimized decision model for production planning and distribution of a multiphase, multiproduct reverse supply chain, which addresses defects returned to original manufacturers, and in addition, develops hybrid algorithms such as Particle Swarm Optimization-Genetic Algorithm (PSO-GA), Genetic Algorithm-Simulated Annealing (GA-SA), and Particle Swarm Optimization-Simulated Annealing (PSO-SA) for solving the optimized model. During a case study of a multi-phase, multi-product reverse supply chain network, this paper explained the suitability of the optimized decision model and the applicability of the algorithms. Finally, the hybrid algorithms showed excellent solving capability when compared with original GA and PSO methods.
Many scholars have been devoted to the study of positive supply chains, for instance [
Shih [
Thus, this paper focuses on the reverse supply chain and constructs an optimized decision model for the selection of supply chain partners as well as determination of production and distribution quantities. Multiechelon logistics issues could be regarded as a Knapsack of a multiple selection portfolio, and resource distribution as a NP-hard issue [
GAs have been widely used in solving real-world complex optimization problems [
Fuzzy theory, first proposed by Zaden in 1965 as an extension of a general set, is a numerical control methodology for imitating human thought and addressing the inaccuracy of all physical systems. According to Fuzzy Theory, the thought logic of human begins as fuzzy and is intended for judgment, even if conditions and data are uncertain, while modern computers feature bipolar logic, that is 0 or 1, different from the logic of humans. However, fuzzy logic theory can represent the degree of fuzzy concepts with values between 0 and 1, namely, “membership function.”
Karkowski [
The triangular fuzzy number can be represented as
Membership functions of a triangular fuzzy number.
In addition, it is essential for practical applications that a fuzzy number should be transformed to a numerical value. The transform process is called “defuzzification.” Associated ordinary number (AON) is a simple method in defuzzification and many researches have employed it directly and effectively [
This paper analyzed the reverse distribution activities of defective products in a complex supply chain network, and explored a method of feeding these products back to the manufacturing partners, with consideration of the capacity constraints of suppliers, and the demands of multi-phase, multi-product production and planning. Moreover, the partners shall be selected based on total objective functions (minimized production, transport, inspection, rework costs, and optimized production quality), in addition, the fuzzified production defect ratio and transport loss ratio are taken into account. Given the fact of numerous suppliers in a supply chain network, with different production characteristics and resource allocation capabilities covering their capacity and yield, it is crucial to select proper suppliers in a complex supply chain network.
Positive logistics are highlighted in a traditional supply chain, but unavoidable defects arising from production or transportation processes are overlooked. In practice, the defective products must be returned to the manufacturers. Relevant original data in this paper were subjected to
The mathematical symbols of a reverse supply chain one listed in the Symbols of Mathematical Model section.
The mathematical model of a multi-phase plan of a reverse supply chain is detailed as follows:
Equation (
In this paper, GA, PSO, PSO-GA, GA-SA, and PSO-SA were applied to determine the best approximate solution, with minimum objective functions, in a reverse supply chain.
Structure of chromosome.
Single-point crossover.
Mutation.
where
In a complex supply chain network, even a leading manufacturer cannot guarantee 100% yield during the production process, or prevent any defect during the transport process. However, the defect ratio and loss ratio are not fixed; thus, fuzzy defect ratio and fuzzy transport loss ratio are applied in this paper.
Based on a supply chain network of {4-4-3-3}, this paper simulated rework activity of returned defective products through a multi-product, multi-phase production plan. Assuming that the initial inventory is zero, the defective products arising from the production process of upstream manufacturers, and from the transport process, are returned to original manufacturers for rework; the suppliers of 1st–4th hierarchy have no fixed production defect ratio or transport loss ratio, meanwhile the multi-product, multi-phase production is planned into three phases, with defective products only returned during the second phase. Moreover, assuming that the production defect ratio and check costs of the first-hierarchy suppliers are not considered, only the defective products from the partners of 2nd to the 4th are returned for rework; in addition, assume that the rework process is the same as the production process, then the rework costs and production costs are the same. According to first-phase production planning,
{4-4-3-3} reverse supply chain network.
PSO, GA, PSO-GA, GA-SA, and PSO-SA were applied for solving the distribution issues of a reverse supply chain network, the experimental design was conducted under the parameters of solving performance, of which every combination of parameters was implemented 20 times to determine an optimal combination, as listed in Table
Combination of experimental parameters.
PSO | Population size | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 |
Generations | 500 | 500 | 1000 | 1000 | 500 | 500 | 1000 | 1000 | |
Max velocity | 0.95 | 1.25 | 0.95 | 1.25 | 0.95 | 1.25 | 0.95 | 1.25 | |
Initial weight | 1.25 | 2.15 | 1.25 | 2.15 | 1.25 | 2.15 | 1.25 | 2.15 | |
|
2.05 | 2.05 | 2.05 | 2.05 | 2.05 | 2.05 | 2.05 | 2.05 | |
Avg. fitness | 24462289 | 24461153 | 24460103 | 24458116 | 24459087 | 24460841 | 24458923 | 24457531 | |
Avg. execution time (sec.) | 6.731 | 6.896 | 12.468 | 12.391 | 12.842 | 12.546 | 21.391 | 21.016 | |
Avg. convergence time (sec.) | 4.766 | 4.275 | 8.791 | 8.437 | 7.986 | 8.311 | 17.694 | 16.972 | |
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GA | Population size | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 10 |
Generations | 500 | 500 | 1000 | 1000 | 500 | 500 | 1000 | 1000 | |
Crossover rate | 0.75 | 0.8 | 0.75 | 0.8 | 0.75 | 0.8 | 0.75 | 0.8 | |
Mutation rate | 0.08 | 0.07 | 0.08 | 0.07 | 0.08 | 0.07 | 0.08 | 0.07 | |
Avg. fitness | 24465387 | 24464085 | 24462752 | 24461924 | 24463297 | 24463159 | 244603191 | 24459450 | |
Avg. execution time (sec.) | 19.373 | 18.859 | 67.693 | 68.047 | 47.375 | 47.734 | 225.836 | 217.512 | |
Avg. convergence time (sec.) | 17.716 | 16.827 | 62.549 | 62.764 | 44.507 | 44.642 | 219.675 | 211.741 | |
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Population size | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 |
Generations | 500 | 500 | 1000 | 1000 | 500 | 500 | 1000 | 1000 | |
Max velocity | 0.95 | 1.25 | 0.95 | 1.25 | 0.95 | 1.25 | 0.95 | 1.25 | |
Initial weight | 1.25 | 2.15 | 1.25 | 2.15 | 1.25 | 2.15 | 1.25 | 2.15 | |
|
2.05 | 2.05 | 2.05 | 2.05 | 2.05 | 2.05 | 2.05 | 2.05 | |
Mutation rate | 0.08 | 0.07 | 0.08 | 0.07 | 0.08 | 0.07 | 0.08 | 0.07 | |
Avg. fitness | 24790482 | 24692938 | 24662117 | 24579829 | 24507556 | 24453060 | 24697893 | 24662084 | |
Avg. execution time (sec.) | 6.758 | 6.579 | 11.864 | 12.714 | 12.898 | 12.685 | 23.257 | 22.934 | |
Avg. convergence time (sec.) | 3.147 | 3.038 | 5.342 | 6.182 | 6.379 | 5.824 | 12.681 | 12.507 | |
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GA-SA | Population size | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 10 |
Generations | 500 | 500 | 1000 | 1000 | 500 | 500 | 1000 | 1000 | |
Crossover rate | 0.75 | 0.8 | 0.75 | 0.8 | 0.75 | 0.8 | 0.75 | 0.8 | |
Mutation rate | 0.08 | 0.07 | 0.08 | 0.07 | 0.08 | 0.07 | 0.08 | 0.07 | |
Initial temperature | 300 | 300 | 400 | 400 | 300 | 300 | 400 | 400 | |
Markov Chain Length | 50 | 50 | 100 | 100 | 50 | 50 | 100 | 100 | |
Cooling rate | 0.9 | 0.9 | 0.99 | 0.99 | 0.9 | 0.9 | 0.99 | 0.99 | |
Final temperature | 1 | 1 | 5 | 5 | 1 | 1 | 5 | 5 | |
Avg. fitness | 24459076 | 24458365 | 24458351 | 24458067 | 24457247 | 24456391 | 24457425 | 24457544 | |
Avg. execution time (sec.) | 27.477 | 26.041 | 81.462 | 79.039 | 68.671 | 66.993 | 279.145 | 274.338 | |
Avg. convergence time (sec.) | 22.374 | 20.768 | 74.744 | 73.412 | 63.467 | 62.522 | 265.074 | 263.862 | |
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Population size | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 |
Generations | 500 | 500 | 1000 | 1000 | 500 | 500 | 1000 | 1000 | |
Max velocity | 0.95 | 1.25 | 0.95 | 1.25 | 0.95 | 1.25 | 0.95 | 1.25 | |
Initial weight | 1.25 | 2.15 | 1.25 | 2.15 | 1.25 | 2.15 | 1.25 | 2.15 | |
|
2.05 | 2.05 | 2.05 | 2.05 | 2.05 | 2.05 | 2.05 | 2.05 | |
Initial temperature | 300 | 300 | 400 | 400 | 300 | 300 | 400 | 400 | |
Markov Chain Length | 50 | 50 | 100 | 100 | 50 | 50 | 100 | 100 | |
Cooling rate | 0.9 | 0.9 | 0.99 | 0.99 | 0.9 | 0.9 | 0.99 | 0.99 | |
Final temperature | 1 | 1 | 5 | 5 | 1 | 1 | 5 | 5 | |
Avg. fitness | 24457588 | 24457887 | 24455842 | 24455147 | 24455895 | 24456068 | 24456457 | 24456778 | |
Avg. execution time (sec.) | 14.073 | 13.422 | 26.087 | 25.971 | 26.479 | 25.706 | 38.926 | 37.433 | |
Avg. convergence time (sec.) | 6.102 | 5.674 | 11.479 | 10.347 | 11.887 | 12.624 | 19.211 | 16.708 |
It is learnt from Table
To compare the advantages and disadvantages of these algorithms, ANOVA was used to judge if convergence value, execution time, and convergence time differed considerably, then the algorithms were compared using Scheffe’s multiple comparison method [
ANOVA verification of fitness values.
Hypothesize | ||||||
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H0: |
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H1: otherwise | ||||||
Algorithm | Numbers | Sum | Average | Variance | ||
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PSO | 30 | 734139577 | 24471319.23 | 1120394034 | ||
GA | 30 | 734406462 | 24480215.4 | 910941494 | ||
PSO-GA | 30 | 733498321 | 24449944.03 | 1372449396 | ||
GA-SA | 30 | 733929774 | 24464325.8 | 752499584 | ||
PSO-SA | 30 | 733666393 | 24455546.43 | 3467767352 | ||
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Source | SS | Freedom | MS |
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Critical value |
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Intergroup |
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4 | 4389633912 | 2.87880644 | 0.02486 | 2.434065 |
Intragroup |
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145 | 1524810372 | |||
Sum |
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149 | ||||
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Result: reject H0 |
ANOVA verification of execution time.
Hypothesize | ||||||
---|---|---|---|---|---|---|
H0: |
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H1: otherwise | ||||||
Algorithm | Numbers | Sum | Average | Variance | ||
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PSO | 30 | 372.022 | 12.40073 | 0.076341375 | ||
GA | 30 | 6524.661 | 217.4887 | 0.165604217 | ||
PSO-GA | 30 | 380.424 | 12.6808 | 0.076438303 | ||
GA-SA | 30 | 2004.847 | 66.82823 | 0.187643495 | ||
PSO-SA | 30 | 775.495 | 25.84983 | 0.192059868 | ||
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Source | SS | Freedom | MS |
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Critical value |
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Intergroup | 908155.7 | 4 | 227038.9 | 1626150.094 | 0.00000 | 2.434065 |
Intragroup | 20.24453 | 145 | 0.139617 | |||
Sum | 908176 | 149 | ||||
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Result: reject H0 |
ANOVA verification of convergence time.
Hypothesize | ||||||
---|---|---|---|---|---|---|
H0: |
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H1: otherwise | ||||||
Algorithm | Numbers | Sum | Average | Variance | ||
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PSO | 30 | 246.605 | 8.220167 | 0.0913266 | ||
GA | 30 | 6375.408 | 212.5136 | 0.7256669 | ||
PSO-GA | 30 | 171.189 | 5.7063 | 0.014925 | ||
GA-SA | 30 | 1863.747 | 62.1249 | 0.4336888 | ||
PSO-SA | 30 | 326.895 | 10.8965 | 0.2385541 | ||
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Source | SS | Freedom | MS |
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Critical value |
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Intergroup | 939149 | 4 | 234787.2 | 780458.96 | 0.00000 | 2.434065 |
Intragroup | 43.62068 | 145 | 0.300832 | |||
Sum |
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89 | ||||
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Result: reject H0 |
In Tables
where
Table
Multiple comparison on fitness value.
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(−, +) | |||
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(+, +) | (+, +) | ||
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(+, +) | (+, +) | (−, +) | |
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(+, +) | (+, +) | (−, +) | (−, +) |
Multiple comparison on execution time.
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(−, −) | |||
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(−, −) | (+, +) | ||
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(−, −) | (+, +) | (−, −) | |
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(−, −) | (+, +) | (−, −) | (+, +) |
Multiple comparison on convergence time.
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(−, −) | |||
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(+, +) | (+, +) | ||
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(−, −) | (+, +) | (−, −) | |
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(−, −) | (+, +) | (−, −) | (+, +) |
Distribution plan by PSO-GA (first period).
From | To | Echelon 2 | Echelon 3 | Echelon 4 | |||||||
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Echelon 1 |
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10a, 6b, 49c | 73, 3, 6 | 42, 110, 59 | 24, 253, 5 | ||||||
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142, 47, 4 | 174, 26, 33 | 34, 18, 2 | 102, 10, 234 | |||||||
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157, 24, 57 | 256, 38, 1 | 7, 29, 56 | 22, 11, 55 | |||||||
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74, 39, 6 | 76, 17, 21 | 31, 8, 13 | 74, 14, 1 | |||||||
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Echelon 2 |
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177, 3, 83 | 116, 8, 0 | 82, 103, 25 | |||||||
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458, 2, 37 | 109, 14, 17 | 8, 67, 6 | ||||||||
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53, 100, 60 | 41, 3, 3 | 17, 58, 65 | ||||||||
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111, 236, 9 | 10, 29, 244 | 92, 11, 30 | ||||||||
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Echelon 3 |
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170, 0, 0 | 100, 0, 174 | 91, 0, 255 | |||||||
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95, 0, 0 | 47, 0, 0 | 123, 0, 0 | ||||||||
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139, 253, 101 | 213, 155, 32 | 245, 204, 0 |
Distribution plan by PSO-GA (second period).
From | To | Echelon 1 | Echelon 2 | Echelon 3 | Echelon 4 | ||||||||||
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Echelon 1 |
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100, 76, 12 | 69, 48, 80 | 61, 349, 47 | 18, 55, 0 | ||||||||||
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34, 19, 22 | 0, 27, 0 | 214, 50, 47 | 91, 10, 14 | |||||||||||
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95, 334, 95 | 131, 13, 51 | 8, 17, 40 | 28, 5, 126 | |||||||||||
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111, 11, 0 | 12, 15, 100 | 9, 91, 0 | 125, 2, 15 | |||||||||||
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Echelon 2 |
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130, 320, 67 | 189, 27, 59 | 10, 85, 0 | |||||||
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155, 37, 213 | 52, 38, 17 | 0, 29, 0 | ||||||||
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82, 344, 12 | 123, 62, 108 | 81, 94, 10 | ||||||||
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55, 0, 51 | 139, 7, 0 | 55, 63, 90 | ||||||||
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Echelon 3 |
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41, 303, 0 | 134, 361, 0 | 117, 0, 0 | |||||||
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130, 0, 0 | 121, 0, 0 | 60, 396, 0 | ||||||||
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226, 0, 151 | 143, 0, 304 | 64, 6, 150 | ||||||||
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Echelon 4 |
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Demand | 400, 300, 150 | 400, 350, 300 | 250, 400, 150 |
Distribution plan by PSO-GA (third period).
From | To | Echelon 1 | Echelon 2 | Echelon 3 | Echelon 4 | ||||||||||
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Echelon 1 |
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86, 18, 4 | 406, 15, 108 | 30, 12, 53 | 24, 83, 13 | ||||||||||
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34, 60, 36 | 137, 6, 28 | 165, 137, 6 | 35, 159, 30 | |||||||||||
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2, 7, 24 | 30, 1, 217 | 47, 116, 0 | 278, 28, 5 | |||||||||||
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56, 0, 10 | 7, 8, 31 | 86, 102, 22 | 17, 46, 98 | |||||||||||
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Echelon 2 |
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26, 9, 15 | 33, 23, 49 | 113, 55, 11 | |||||||
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430, 3, 62 | 37, 27, 318 | 105, 0, 4 | ||||||||
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217, 282, 38 | 8, 22, 38 | 93, 64, 6 | ||||||||
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216, 0, 61 | 22, 183, 48 | 99, 104, 12 | ||||||||
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Echelon 3 |
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190, 0, 354 | 213, 0, 0 | 58, 0, 193 | |||||||
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93, 253, 0 | 165, 0, 0 | 113, 0, 0 | ||||||||
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166, 0, 0 | 123, 302, 87 | 230, 24, 6 | ||||||||
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Echelon 4 |
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Demand | 450, 250, 350 | 500, 300, 100 | 400, 200, 200 |
This paper focused on analyzing the issues of returning defective products to original manufacturers in a reverse supply chain system. Fuzzy defect and fuzzy transport loss ratios were considered, and an optimized mathematical model was developed. This model combined cost and quality with T-technology and described how to plan a reverse supply chain on the precondition of meeting customer demands and realizing the capacity of partners. To solve the problems efficiently, three hybrid algorithms were applied to this reverse model, including PSO-GA, PSO-SA, and GA-SA; then, the performances of these algorithms were compared. The experimental results show that if the fitness value, execution time, and convergence time are considered, PSO-GA has the minimal value, which means that PSO-GA has the qualities and capabilities for dealing with the production planning and distribution of a multi-phase, multi-product reverse supply chain.
Number of products,
Total number
Hierarchy of supply chain,
Total number of hierarchy of supply chain
Production phase,
Total production phase
Serial number of partner of
Serial number of partner of
Check cost of
Production cost of
Transport cost from
Fuzzy transport loss ratio from
Fuzzy defect ratio of
Production quantity of
Transport quantity from
Quantity of defective products transported from
Maximum capacity of
Minimum capacity of
Quality level of
Check cost of
Production cost of
Transport cost from
Standard deviation of quality of
Standard deviation of check cost of
Standard deviation of production cost of
Standard deviation of transport cost from
Integer.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors would like to thank Mr. K. Hsiao for supporting writing programs and the National Science Council of Taiwan for their partial financial support (Grants nos. NSC 102-2410-H-027-009 and NSC 101-2410-H-027-006). The authors would also like to acknowledge the editors and anonymous reviewers for their helpful comments and suggestions, which greatly improved the presentation of this paper.