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A robust optimization model for sustainable and resilient closed-loop supply chain network design considering conditional value at risk

  • * Corresponding author: Yahia Zare Mehrjerdi

    * Corresponding author: Yahia Zare Mehrjerdi 
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  • One of the challenges facing supply chain designers is designing a sustainable and resilient supply chain network. The present study considers a closed-loop supply chain by taking into account sustainability, resilience, robustness, and risk aversion for the first time. The study suggests a two-stage mixed-integer linear programming model for the problem. Further, the robust counterpart model is used to handle uncertainties. Furthermore, conditional value at risk criterion in the model is considered in order to create real-life conditions. The sustainability goals addressed in the present study include minimizing the costs, $ \text{CO}_2 $ emission, and energy, along with maximizing employment. In addition, effective environmental and social life-cycle evaluations are provided to assess the associated effects of the model on society, environment, and energy consumption. The model aims to answer the questions regarding the establishment of facilities and amount of transported goods between facilities. The model is implemented in a car assembler company in Iran. Based on the results, several managerial insights are offered to the decision-makers. Due to the complexity of the problem, a constraint relaxation is applied to produce quality upper and lower bounds in medium and large-scale models. Moreover, the LP-Metric method is used to merge the objectives to attain an optimal solution. The results revealed that the robust counterpart provides a better estimation of the total cost, pollution, energy consumption, and employment level compared to the basic model.

    Mathematics Subject Classification: Primary: 90B15, 90C29.

    Citation:

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  • Figure 1.   

    Figure 2.  Life cycle stages and corresponding inventories

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    Figure 9.   

    Table 1.  Survey on CLSC

    Reference Kind of CLSC Resilience Disruption Uncertainty Risk Objectives Industry Method
    Talaei et al.[46] Reliable Both partial, complete disruption Probabilistic mixed programming P-robust Economic Numerical example Epsilon-constraint
    Ghomi Avili et al. [17] Reliable and resilient Extra inventory Lateral transshipment Reliable and unreliable suppliers Complete disruption Two-stage probabilistic mixed programming Supply risk Economic Numerical example *CS
    Tavakkoli Moghaddam et al. [47] Possibilistic fuzzy approach Economic Numerical example CS
    Mari et al. [28] Sustainable and resilient Probabilistic disruption Probabilistic Economic and emissions of Carbon footprints Disruption costs Textile industry CS
    Amin and Baki [1] Fuzzy programming Economic Electronics industry CS
    Amin et al. [2] Scenario Scenario tree Economic Tire marketing CS
    Soleimani and Govindan [42] Two-stage scenario CVaR Economic Numerical example CS
    Cardoso et al. [7] Stochastic Variance, *VI, *DR, and CVaR Economic (ENPV) Numerical example *AEC
    Subulan et al. [44] Stochastic-fuzzy and possibilistic VaR, CVaR, and downside risk Economic and the average of the collected volume of the used products Lead-acid battery CS
    Prakash et al. [38] Robust and reliable Scenario Stochastic Worst risk case Economic Electronics trade industry CS
    Prakash et al. [37] Waiting times Economic Hospital beds CS
    Sahebjamnia et al. [40] Sustainable and resilient Economic, environmental, and social Tire industry *MH
    Behzadi et al. [3] Resilient Diversified demand market, backup demand market, and flexible rerouting Scenario Robust optimization Two-stage stochastic Economic Kiwifruit CS
    Brandenburg [5] Scenario Stochastic Economic and environmental FMCG manufacturer *WGP
    Brandenburg [6] Green Simulation VaR Economic and environmental Numerical example CS
    The present study Robust, sustainable resilient, and reliable Capacity Partial disruption Stochastic CVaR Economic, Environmental, social and energy Car manufacturing industry CS NEOS
    *CS: Commercial Solver, AEC: Augmented epsilon constraint, MH: RDA and SA algorithm, GA and WWO algorithm, WGP: Weighted goal programming, VI: Variability index, DR: Downside risk, NA: Not Applicable.
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    Table 2.  Optimal value of the robust objective function and the value of the global criterion objective function

    Objective The optimal value of proposed objective function The optimal value of base model objective function *Avg. Gap
    Cost Pollutant (CO2) Energy Employ Cost Pollutant (CO2) Energy Employ.
    Min Z1(Cost) 71470.14 1989597.20 2274555.62 1749 71357.80 1901777.18 2181635.25 1788 1.7%
    Min Z2(CO2) 174731.64 1250941.04 1953758.20 4399 171286.39 1217249.59 1882470.29 4499 1.6%
    Min Z3(Energy) 78459.12 1317174.17 1591575.21 2100 76899.32 1258753.29 1556561.01 2150 1.6%
    Max Z4(Employ) 176760.32 1734074.55 2358201.83 4505 173265.9 1650207.07 2263490.75 4520 2.7%
    Min Lp 76688.59 1285769.68 1594682.21 2141 76589.90 1251038.15 1559701.11 2151 1.2%
    GAP 0.1% 2.8% 2.2% -0.45% 1.2%
    * Avg. GAP = Average ((Proposed model objective- base model objective)/ base model objective)
     | Show Table
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    Table 3.  Weight variations versus objectives

    $ X_1 $ $ X_2 $ $ X_3 $ $ X_4 $ Cost Pollutant $ (\text{CO}_2) $ Energy Employment
    0 0.33 0.33 0.33 78143.63 1285793 1594659 2141.56
    0.5 0.16 0.16 0.16 76688.59 1285770 1594682 2141.56
    1 0 0 0 71470.15 1989597 2274556 1749.06
    0.33 0 0.33 0.33 76689.36 1316802 1591633 2141.56
    0.16 0.5 0.16 0.16 79603.18 1274957 1612078 2214.48
    0 1 0 0 174731.6 1250941 1953758 4399.22
    0.33 0.33 0 0.33 81873.39 1270004 1672336 2340.66
    0.16 0.16 0.5 0.16 76688.97 1289052 1592359 2141.56
    0 0 1 0 78459.12 1317174 1591575 2100.21
    0.33 0.33 0.33 0 76688.59 1285770 1594682 2100.75
    0.16 0.16 0.16 0.5 76688.59 1285770 1594682 2141.56
    0 0 0 1 176760.3 1734075 2358202 4505.85
    0.25 0.25 0.25 0.25 76688.59 1285769.68 1594682.21 2141.55
     | Show Table
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    Table 4.  Medium and large scale problems

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    Table 5.  Comparison of the main model with the lower bound and worst possible case

    Problem Lower bound Main model Worst-case GAP1 GAP2
    LP-Relax $ 0\le X\le 1 $ (A) Time GAMS Main model $ X\in\{0,1\} $(B) Time Relaxation $ X=1 $(C) Time GAMS
    P1 10862.19 2.00 76688.59 8.40 173172.68 2.41 -86% 126%
    P2 15720.97 3.83 90009.11 93.68 239036.43 3.41 -83% 166%
    P3 21307.43 11.33 111813.32 1082.85 301446.39 8.33 -81% 170%
    P4 44956.51 843.11 *127011.40 *3705.6 457454.73 521.33 -65% 260%
    P5 74585.42 2967.01 *165745.37 *28810 668562.73 1530.36 -55% 303%
    P6-P8 No solution was found
    * Solved by NEOS-Server, GAP1= (B-A)/A, GAP2=(C-B)/B.
     | Show Table
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    Table A2-1.  Model parameters for medium and large scale problems

    Parameters Value Description
    $ de{{m}_{rpt{s}'}} $ ($ \left| {{s}'} \right| $-1)*1000 + uniform(1000, 2000) Demand for various scenarios
    $ f{{s}_{s}} $ uniform (1000, 2000)
    $ f{{m}_{m}} $ uniform(40000, 50000)
    $ f{{d}_{d}} $ uniform(3000, 4000
    $ f{{r}_{r}} $ uniform(1000, 2000) Fixed costs (opening) (Thousand dollar)
    $ f{{m}_{m}} $ uniform(2000, 3000)
    $ f{{k}_{k}} $ uniform(2000, 3000)
    $ f{{e}_{e}} $ uniform(1000, 2000)
    $ Vsm_{smpts'} $ uniform(3, 4)
    $ Vsm_{mdpts'} $ uniform(3, 4)
    $ Vdr_{drpts'} $ uniform(3, 4)
    $ Vrc_{rcpts'} $ uniform(3, 4) Variable costs (Dollar)
    $ Vck_{ckpts'} $ uniform(3, 4)
    $ Vke_{kepts'} $ uniform(3, 4)
    $ Vksc_{kscpts'} $ uniform(3, 4)
    $ Vkm_{kmpts'} $ uniform(3, 4)
    $ Ems_{sts'} $ uniform(100,200)
    $ Emm_{mts'} $ uniform(1000, 2000)
    $ Emd_{dts'} $ uniform(100,200)
    $ Emr_{rts'} $ uniform(100,200) Fixed pollution (opening) (carbon dioxide) (Centiton)
    $ Emc_{cts'} $ uniform(100,200)
    $ Emk_{kts'} $ uniform(100,200)
    $ Eme_{ets'} $ uniform(100,200)
    $ Emsm_{smpts'} $ uniform(4, 5) Variable pollution (carbon dioxide) (Centiton)
    $ Emmd_{mdpts'} $ uniform(4, 5)
    $ Emdr_{drpts'} $ uniform(4, 5)
    $ Emrc_{rcpts'} $ uniform(4, 5)
    $ Emck_{ckpts'} $ uniform(4, 5)
    $ Emke_{kepts'} $ uniform(4, 5)
    $ Emksc_{kscpts'} $ uniform(4, 5)
    $ Emkm_{kmpts'} $ uniform(4, 5)
    $ Es_{sts'} $ uniform(4000, 5000)
    $ Em_{mts'} $ uniform(40000, 50000)
    $ Ed_{dts'} $ uniform(4000, 5000)
    $ Er_{rts'} $ uniform(4000, 5000) Fixed consumed energy (opening) (MJ)
    $ Ec_{mts'} $ uniform(4000, 5000)
    $ Ek_{kts'} $ uniform(4000, 5000)
    $ Ee_{ets'} $ uniform(4000, 5000)
    $ Esm_{smpts'} $ uniform(4, 5) Variable pollution (MJ)
    $ Eemd_{mdpts'} $ uniform(4, 5)
    $ Edr_{drpts'} $ uniform(4, 5)
    $ Erc_{rcpts'} $ uniform(4, 5)
    $ Eck_{ckpts'} $ uniform(4, 5)
    $ Eke_{kepts'} $ uniform(4, 5)
    $ Eksc_{kscpts'} $ uniform(4, 5)
    $ Ekm_{kmpts'} $ uniform(4, 5)
    $ Os_{sts'} $ uniform(40, 50)
    $ Om_{mts'} $ uniform(300,400)
    $ Od_{dts'} $ uniform(40, 50)
    $ Or_{rts'} $ uniform(5, 10) Fixed employment (person)
    $ Om_{mts'} $ uniform(20, 30)
    $ Ok_{kts'} $ uniform(10, 15)
    $ Oe_{ets'} $ uniform(5, 10)
    $ VOs_{st'} $ uniform(1000, 1100)
    $ VOm_{mt'} $ uniform(1000, 1100)
    $ VOd_{dt'} $ uniform(1000, 1100)
    $ VOr_{rt'} $ uniform(1000, 1100) Salary Cost (Dollars)
    $ VOc_{ct'} $ uniform(1000, 1100)
    $ VOk_{kt'} $ uniform(1000, 1100)
    $ VOe_{et'} $ uniform(1000, 1100)
    $ prs_{s'} $ uniform(0.95, 0.98);
    $ prm_{m'} $ uniform(0.95, 0.98);
    $ prd_{d'} $ uniform(0.95, 0.98);
    $ prr_{r'} $ uniform(0.95, 0.98); Availability probability (percent)
    $ prm_{m'} $ uniform(0.95, 0.98);
    $ prk_{k'} $ uniform(0.95, 0.98);
    $ pre_{e'} $ uniform(0.95, 0.98);
    $ CapS_{spts'} $ uniform(50000, 60000)*(($ s' $-1)*0.5+1)
    $ CapM_{mpts'} $ uniform(100000, 110000)*(($ s' $-1)*0.5+1)
    $ CapD_{dpts'} $ uniform(20000, 22000)*(($ s' $-1)*0.5+1)
    $ CapR_{rpts'} $ uniform(3000, 3300)*(($ s' $-1)*0.5+1) Capacity (facility)
    $ CapC_{cpts'} $ uniform(20000, 22000)*(($ s' $-1)*0.5+1)
    $ CapK_{kpts'} $ uniform(5000, 5500)*(($ s' $-1)*0.5+1)
    $ CapE_{epts'} $ uniform(3000, 3300)*(($ s' $-1)*0.5+1)
    $ p^{s'} $ 0.33 Scenario occurrence probability
    $ \beta $ uniform(0, 0.2) Expectation value weight
    $ \omega $ uniform(0, 0.1) Fine associated with demand dissatisfaction
    $ \lambda $ uniform(0, 0.1) CVaR index importance
    $ \alpha $ uniform(0, 0.05) 95% Confidence level in CVaR
    $ k_{s'1} $ 0.05 Fine coefficient of demand dissatisfaction for quadruple objective
    $ k_{s'2} $ 0.05
    $ k_{s'3} $ 0.05
    $ k_{s'4} $ 0.05
    $ \rho_{rpts'} $ uniform(0, 1) Return percentage
    $ \rho_{1pts'} $ uniform(0.7, 0.71)
    $ \rho_{2pts'} $ uniform(0.2, 0.21)
    $ \rho_{3pts'} $ uniform(0.1, 0.11)
    $ W_i $ 0.25 Objective weight
     | Show Table
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