Effect of Inspection Policies and Residual Value of Collected Used Products: A Mathematical Model and Genetic Algorithm for a Closed-Loop Green Manufacturing System
Abstract
:1. Introduction
1.1. Background
1.2. Problem Description
2. Literature Review
3. Mathematical Model
3.1. Notations
Cins_before | Unit inspection cost before collection |
Cins_after | Unit inspection cost after collection |
Cmanuf | Unit manufacturing cost |
Cdisposal | Unit disposal cost |
Craw | Unit raw material cost for manufacturing |
Cpenalty | Fine for violating obligatory take-back quota |
N | Remanufacturing types |
Ctype_n | Unit remanufacturing cost for type n, n = 1 ,…, N |
Qmin,type_n | Minimum quality level for type n remanufacturing |
Q | Quality level of used products, 0 ≤ q ≤ 1 |
P | Unit sale price for new (or remanufactured) products |
πi | Revenue of green manufacturing company of model i |
D | Demand for new (or remanufactured) products |
R | Amount of CUPs |
fc(Cpb) | Collection rate function |
fq(x) | Quality distribution of used products |
βi | Disposal rate for CUPs for model i |
ai, bi | Systemic parameters for disposal rate function for model i |
δ | Obligatory take-back quota |
dn | An amount of CUP of type n remanufacturing |
Qmin,i | Decision variable and minimum quality level of CUPs for model i |
Cpb,i | Decision variable and unit buy-back cost for model i |
3.2. Assumptions
3.3. Collection Rate and Quality Distribuion of Collected Used Products
3.4. Objective Functions
3.4.1. Profit Functions of Model 1: Inspection Policy 1
3.4.2. Profit Functions of Model 2: Inspection Policy 2
3.5. Constraints
4. Solution Procedure
STEP 1: | Set generation index i = 0 and current best fitness value = −∞. |
STEP 2: | Generate initial population (Nc chromosomes) of ith generation. |
STEP 3: | For each chromosome, calculate its fitness value according to Equations (3) and (4) for model 1 and 2, respectively. If the fitness value of a chromosome in ith generation is greater than the current best fitness value, update the current best fitness value and save the corresponding chromosome. |
STEP 4: | If i < Ng, create next population for (i + 1)th generation by selection, crossover and mutation in Section 4.3. Otherwise, go to STEP 6 |
STEP 5: | i← i + 1 and go to STEP 3. |
STEP 6: | Finish genetic algorithm. |
4.1. Chromosome Structure
4.2. Fitness Function
4.3. Selection, Crossover, and Mutation
STEP 1: | Let Pi be the population of ith generation. Let Cm be the mth chromosome of population Pi,m∈{1,…, Nc}. Let FF(Cm) be the fitness value of chromosome Cm. |
STEP 2: | Calculate |
STEP 3: | Select chromosome c as Parent 1 with the probability of |
STEP 4: | Repeat STEP 3 to obtain Parent 2 |
5. Numerical Experiments
5.1. Validity Analysis of Proposed Models
5.1.1. Systemic Parameters
5.1.2. Computational Results
5.2. Sensitivity Test on the Obligatory Take-Back Quota
5.3. Sensitivity Test on Quality Distribution
6. Concluding Remarks
Author Contributions
Conflicts of Interest
References
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Parameter | Value | Parameter | Value |
---|---|---|---|
P | $10.00/unit | Ctype_3 | 1.60 |
D | 10,000 unit | Qmin, type_4 | 0.60 |
Cins_before | $0.05/unit | Ctype_4 | 2.50 |
Cins_after | $0.03/unit | Qmin, type_5 | 0.50 |
Cmanuf | $3.00/unit | Ctype_5 | 3.50 |
Cdisposal | $0.10/unit | Qmin, type_6 | 0.40 |
Cpenalty | $20.00/unit | Ctype_6 | 4.60 |
Craw | $2.00/unit | Qmin, type_7 | 0.30 |
(a1, a2, b1, b2) | (0.07, 0.07, 0.03, 0.03) | Ctype_7 | 5.90 |
δ | 0.70 | Qmin, type_8 | 0.20 |
Qmin, type_1 | 0.90 | Ctype_8 | 7.20 |
Ctype_1 | 0.30 | Qmin, type_9 | 0.10 |
Qmin, type_2 | 0.80 | Cype_9 | 8.60 |
Ctype_2 | 0.90 | Qmin, type_10 | 0.00 |
Qqmin, type_3 | 0.70 | Ctype_10 | 10.00 |
Cpb,i ($) | Qmin,i | Amount of Remanufactured Products Per Type (Unit) | Total Profit ($) | CPU Time (s) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |||||
Model 1 | 2.52 | 0.09 | 188 | 510 | 752 | 913 | 995 | 995 | 913 | 752 | 510 | 94 | 37,230.1 | 0.235 |
Model 2 | 2.41 | 0.40 | 192 | 521 | 767 | 932 | 1014 | 1014 | 0 | 0 | 0 | 0 | 42,810.4 | 0.192 |
Take-Back Quota | 0.0 | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1.0 | |
Return Rate | M1 | 0.42 | 0.42 | 0.42 | 0.48 | 0.59 | 0.63 | 0.67 | 0.72 | 0.80 | 0.89 | 0.89 |
M2 | 0.27 | 0.27 | 0.27 | 0.30 | 0.40 | 0.50 | 0.60 | 0.70 | 0.80 | 0.89 | 0.89 | |
Buy-Back Cost | M1 | 1.12 | 1.12 | 1.12 | 1.33 | 1.79 | 2.03 | 2.22 | 2.58 | 3.22 | 4.41 | 4.41 |
M2 | 0.64 | 0.64 | 0.64 | 0.72 | 1.03 | 1.39 | 1.84 | 2.41 | 3.22 | 4.40 | 4.40 | |
Min. Quality Level | M1 | 0.50 | 0.50 | 0.50 | 0.42 | 0.38 | 0.30 | 0.20 | 0.11 | 0.01 | 0.00 | 0.00 |
M2 | 0.40 | 0.40 | 0.40 | 0.40 | 0.40 | 0.40 | 0.40 | 0.40 | 0.40 | 0.40 | 0.40 |
Case1) Beta (5, 2) | Cpb,i ($) | Qmin,i | Amount of Remanufactured Products Per Type (Unit) | Total Profit ($) | CPU Time (s) | |||||||||
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |||||
Model 1 | 2.44 | 0.27 | 785 | 1579 | 1612 | 1281 | 849 | 469 | 205 | 29 | 0 | 0 | 52,518.5 | 0.198 |
Model 2 | 2.41 | 0.40 | 783 | 1580 | 1613 | 1282 | 849 | 436 | 0 | 0 | 0 | 0 | 53,202.0 | 0.190 |
Case2) Beta (2, 5) | Cpb,i ($) | Qmin,i | Amount of Remanufactured Products Per Type (Unit) | Total Profit ($) | CPU Time (s) | |||||||||
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |||||
Model 1 | 2.41 | 0.00 | 0 | 10 | 60 | 195 | 445 | 806 | 1217 | 1531 | 1500 | 744 | 20,865.4 | 0.191 |
Model 2 | 2.41 | 0.40 | 0 | 10 | 64 | 206 | 469 | 849 | 0 | 0 | 0 | 0 | 34,729.7 | 0.194 |
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Song, B.D.; Ko, Y.D. Effect of Inspection Policies and Residual Value of Collected Used Products: A Mathematical Model and Genetic Algorithm for a Closed-Loop Green Manufacturing System. Sustainability 2017, 9, 1589. https://doi.org/10.3390/su9091589
Song BD, Ko YD. Effect of Inspection Policies and Residual Value of Collected Used Products: A Mathematical Model and Genetic Algorithm for a Closed-Loop Green Manufacturing System. Sustainability. 2017; 9(9):1589. https://doi.org/10.3390/su9091589
Chicago/Turabian StyleSong, Byung Duk, and Young Dae Ko. 2017. "Effect of Inspection Policies and Residual Value of Collected Used Products: A Mathematical Model and Genetic Algorithm for a Closed-Loop Green Manufacturing System" Sustainability 9, no. 9: 1589. https://doi.org/10.3390/su9091589