Optimal Manufacturing-Reconditioning Decisions in a Reverse Logistic System under Periodic Mandatory Carbon Regulation
Abstract
:1. Introduction
2. Materials and Methods
2.1. Description of the System
2.2. Mathematical Model
- Equal to the maximum rate production when the state of the machine is up and the stock level capacity is lower than its maximum level.
- Equal to difference between the stock level capacity and its maximum level, when the state of the machine is up and the stock level at time t is between zero and its maximum level.
- Equal to the demand when the state of the machine is up or the stock level at time t has reached its maximum.
- Is null when the state of the machine is down.
- In order to determine the production quantity for manufacturing and reconditioning products under carbon regulation, we have provided a method to determine the quantity that respects the limited carbon emission and answers to the demand according to the determined production quantity under hedging point policy. We recall qpm and qpr present the carbon emissions of manufacturing and reconditioning respectively, and ql is the limit quantity of carbon emission to not exceed. We have developed two algorithms (please see Algorithms 1 and 2 that are given in Appendix A and Appendix B) that build a function system capable to calculate the proportion of the production rate of the nearest manufactured and reconditioned products under the constraint to not exceed the limit of carbon emissions. The steps are as follows: We calculate the proportion of the production for new and reconditioned products:
- We search, with two counters i and j running from 1 to the upper bound (uA(t) and uAr(t)), to respect the constraint of not exceeding the limit of carbon emission:i.qpm + j.qpr < = ql
- We calculate the proportion:
3. Optimization Method
4. Numerical Results
- The time simulation, T = 107 periods
- The life cycle of new product, α = 100 periodsThe carbon emission quantity for producing new products, qpm = 100 carbon units
- The carbon emission quantity for producing reconditioned products, qpr = 10 carbon units
- The quantity limit of carbon emission, ql = 1000 carbon units
- The unit carbon emission cost, ct = 0.01 monetary unit (for example, dollars or euros)
- The maximum production rate, UA = 2500 products/period
- The maximum production rate, UAr = 2300 products/period
- The unit selling price for new products, cvA = 400 monetary units
- The unit selling price for reconditioned products, cvAr = 180 monetary units
- The unit storage cost for new products, csA = 0.0005 monetary units
- The unit storage cost for reconditioned products, csAr = 0.0005 monetary units
- The unit lost sales cost for new products, cpA = 1200 monetary unit
- The unit lost sales cost for reconditioned products, cpAr = 875 monetary units
- The unit production cost of new products, cuA = 50 monetary unit
- The unit production cost of reconditioned products, cuAr = 20 monetary unit
- The unit storage cost for returned product cr = 0.0003 monetary unit.
4.1. Impact of the p on SA*, SAr*, and F(T)
4.2. Impact of the W on SA*, SAr*, p*, and F(T)
4.3. Impact of the ql on SA*, SAr*, p*, and F(T)
4.4. Impact of the ct on SA*, SAr*, p*, and F(T)
5. Conclusions
6. Discussion
Author Contributions
Funding
Conflicts of Interest
Appendix A
Algorithm 1: Function near. |
Near (M[n], x, S[n], y) 01: M[n], x, S[n], y 02: Int i, result; Double D; 03: D = TT [0] − x + SS [0] − y 04: Result = 0; 05: For (i = 1; i < n; i++){ 06: If (TT[i] − x + SS[i] − y; <= D) 07: D= TT[i] − x+SS[i] − y 08: Result = i; 09: End if 10: End for 11: Return Result |
Appendix B
Algorithm 2: Production planning under mandatory carbon emission. |
Production plan (uA(t), uAr(t)) 01: k = 0; 02: If (uA(t)! 0 and uAr(t)! = 0) 03: PSC = uA(t)/(uA(t) + uAr(t)) 04: PAC = uAr(t)/(uA(t) + uAr(t)) 05: For (i = 1 to uA(t)) 06: For (j = 1 to uAr(t)) 07: If (i·qpm + j·qpr < = qL) 08: Pm2co = i/(i + j) 09 PR2CO = j/(i + j) 10: h[k] = Pm2co 11: B[k] = PR2CO 12: v[k] = i 13: c[k] = j 14: k++ 15: End if 16: End for 17: End for 18: lignenear = Near (h, PSC,B, PAC) 19: i = v[lignenear], j = c[lignenear] 20: End if 21: uA(t) = i 22: uAr(t) = j |
Appendix C
Algorithm 3: Pseudo-code of couple of coordinates, iterative neighborhood. |
Iterative_Neighborhood (Vres; Delta) 01: Do 02: Vtest = Vres 03: C_Min_Temp = Cost_Function (Vtest) 04: For (i = 1 to nb_var-1) 05: For (j = i + 1; j < nb_var; j++) 06: Switch (l = 0; l < 4; l++) 07: l = 0: 08: Positive_Variation (Vtest [i]) 09: Positive_Variation (Vtest [j]) 10: l = 1: 11: Positive_Variation (Vtest [i]) 12: Negative_Variation (Vtest [j]) 13: l = 2: 14: Negative_Variation (Vtest [i]) 15: Positive_Variation (Vtest [j]) 16: l = 3: 17: Negative_Variation (Vtest [i]) 18: Negative_Variation (Vtest [j]) 19: End Switch//on l 20: If (Constraints_Respected (V)) 21: C_Temp = Cost_Function (V) 22: If (C_Temp < C_Min_Temp) 23: Vres = V 24: End if 25: End if 26: End for//j 27: End for//i 28: While (Vres! = Vtest) |
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t | instant time. |
∆t | time period length. |
T | total simulation time. |
SA | stock capacity for new products. |
SA* | optimal stock capacity for new products. |
SAr | stock capacity for reconditioned products. |
SAr* | optimal stock capacity for reconditioned products. |
sA(t) | stock level of new products at time t. |
sAr(t) | stock level of reconditioned products at time t. |
sr(t) | stock level of worn products at time t. |
uA(t) | production rate of the machine M1 at time t. |
UA | maximum production rate of the machine M1. |
uAr(t) | production rate of the machine M2 at time t. |
UAr | maximum production rate of the machine M2. |
dA(t) | demand for new products at time t. |
dAr(t) | demand for reconditioned products at time t. |
PA(t) | quantity of unmet demand for new products at time t. |
PAr(t) | quantity of unmet demand for reconditioned products at time t. |
VA(t) | quantity of new products sold at time t. |
VAr(t) | quantity of reconditioned products sold at time t. |
R(t) | quantity of worn products that are collected and returned from market 1 at time t. |
p | percentage of the returned products (worn products). |
p* | optimal percentage of the returned products. |
α | lifetime of products. |
θ(t) | state of the machine M1 at time t. |
β(t) | state of the machine M2 at time t. |
cvA | unit selling price for new products. |
cvAr | unit selling price for reconditioned products. |
cuA | unit production cost of new products. |
cuAr | unit production cost of reconditioned products. |
csA | unit storage cost for new products. |
csAr | unit storage cost for reconditioned products. |
cr | unit storage cost for returned product. |
cpA | unit lost sales cost for new products. |
cpAr | unit lost sales cost for reconditioned products. |
ct | unit carbon emission cost. |
ql | quantity limit of carbon emission. |
qpm | carbon emission quantity of new products. |
qpr | carbon emission quantity of reconditioned products. |
W | length of the mandatory carbon period. |
F(t) | total profit function. |
F(t)* | optimal value of total profit. |
p | SA* | SAr* | F(T) |
---|---|---|---|
10% | 1026 | 5 | 2.0804929 × 108 |
20% | 1019 | 167 | 1.57228 × 1010 |
30% | 1032 | 188 | 2.9439 × 1010 |
40% | 1025 | 253 | 4.5063 × 1010 |
50% | 1029 | 411 | 6.0738 × 1010 |
60% | 1021 | 468 | 4.07471 × 1010 |
70% | 1028 | 517 | 2.06977 × 1010 |
80% | 1033 | 532 | −1.8739222 × 1010 |
W | Lower | Upper | Standard Deviation | SA* | SAr* | p* | F(T) |
---|---|---|---|---|---|---|---|
50 | 25 | 75 | 25 | 1029 | 411 | 50% | 6.0735 × 1010 |
100 | 50 | 150 | 50 | 1241 | 617 | 50% | 6.07324 × 1010 |
500 | 250 | 750 | 250 | 1641 | 747 | 50% | 5.11676 × 1010 |
750 | 375 | 1025 | 375 | 2394 | 783 | 50% | 4.57892 × 1010 |
ql | SA* | SAr* | p* | F(T) |
---|---|---|---|---|
100 | 1578 | 517 | 50% | 6.06718 × 1010 |
250 | 1401 | 483 | 50% | 6.06987 × 1010 |
500 | 1292 | 458 | 50% | 6.07126 × 1010 |
750 | 1173 | 431 | 50% | 6.07297 × 1010 |
1000 | 1029 | 411 | 50% | 6.07350 × 1010 |
ct | SA* | SAr* | p* | F(T) |
---|---|---|---|---|
0.01 | 1135 | 434 | 50% | 6.15698 × 1010 |
1 | 922 | 471 | 50% | 4.65295 × 1010 |
4 | 883 | 502 | 50% | 9.42485 × 108 |
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Turki, S.; Sahraoui, S.; Sauvey, C.; Sauer, N. Optimal Manufacturing-Reconditioning Decisions in a Reverse Logistic System under Periodic Mandatory Carbon Regulation. Appl. Sci. 2020, 10, 3534. https://doi.org/10.3390/app10103534
Turki S, Sahraoui S, Sauvey C, Sauer N. Optimal Manufacturing-Reconditioning Decisions in a Reverse Logistic System under Periodic Mandatory Carbon Regulation. Applied Sciences. 2020; 10(10):3534. https://doi.org/10.3390/app10103534
Chicago/Turabian StyleTurki, Sadok, Soulayma Sahraoui, Christophe Sauvey, and Nathalie Sauer. 2020. "Optimal Manufacturing-Reconditioning Decisions in a Reverse Logistic System under Periodic Mandatory Carbon Regulation" Applied Sciences 10, no. 10: 3534. https://doi.org/10.3390/app10103534