# Robust Multi-Objective Sustainable Reverse Supply Chain Planning: An Application in the Steel Industry

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## Abstract

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## 1. Introduction

- Identifying and categorizing the necessary processes to implement the reverse logistics network of the steel industry;
- Maximizing the operating profit of the SC so as to meet economic requirements;
- Minimizing the adverse environmental impacts to meet environmental requirements;
- Maximizing the satisfaction of suppliers and customers to meet social requirements.

- Using a robust optimization approach and NSGA-II algorithm for the multi-objective modeling of the reverse SCN, including the flows of materials and transportation planning in conditions of uncertain demand;
- Evaluating environmental indicators based on CO
_{2}emissions as one of the most important greenhouse gas emissions in the environment; - Evaluating customer service levels (CSLs) based on maximizing the received products returned from suppliers/previous customers and selling new products to customers;
- Defining different scenarios for dealing with uncertainty of demand and quantifying them according to expert opinion.

- Most research focuses on the design of a new SC, and there exists a shortage of network redesign;
- The impact of the number and location of facilities on the environment is not considered;
- There are very few models that consider reconstructing reverse SC with a simultaneous analysis of social, economic, and environmental goals;
- Uncertainty about the number of resources and demand for recycled products, along with the management of diverse materials, are issues that require investigation in the future.

- The use of a robust optimization approach for redesigning a recycling SC network, including multiple flows of materials and uncertainties regarding the waste products used as raw materials, and the final demand for recycled products;
- The structure of the expected functional index for evaluating a configuration for a new SC considering the economic and environmental objectives in different scenarios.

## 2. Methods and Materials

#### 2.1. Assumptions

- Uncertainty in the demand parameter has been considered;
- The studied SC consists of four levels when acting in a single period;
- The capacity of the gathering centers is unlimited, and the capacity of the recycling plants is limited;
- The numbers of gathering centers and recycling plant candidates are limited;
- Fixed and variable costs (gathering, recycling and transportation) and the number and capacity of the transportation modes are determined;
- The flow of material between two non-consecutive levels is not allowed;
- The numbers of suppliers and customers are fixed and are five and three, respectively;
- The nominated locations for selecting gathering centers and recycling plants are five and three, respectively. These will determine the exact number of centers during the process of solving the model [22].

#### 2.2. Model Notations

**Indexes**

$i\in I$ | A set of renewable waste suppliers |

$j\in J$ | A set of location candidates for gathering centers |

$k\in K$ | A set of location candidates for recycling plants |

$l\in L$ | A set of customers |

$m\in M$ | A set of transportation modes |

$p\in P$ | A set of final products |

$s\in S$ | A set of scenarios |

**Parameters**

G_{ip} | The amount of product supply (p) by the supplier (i), in tons per month |

CT_{mp} | The capacity of the transportation mode (m) for the transfer of product (p), in tons, on the trip |

C_{kp} | The capacity of the recycling plant (k) to produce the product (p), in tons per month |

D_{lps} | The amount of product (p) which is demanded by the customer (l) under the scenario (s), in tons per month |

NV_{m} | Total number of trips available for each mode (m) |

IT_{m} | The environmental impact of moving materials in the transportation mode (m) on the environmental index, per ton-km |

IE | The environmental effect of the total gas consumption of the system on the environmental index, per normal cubic meter per hour |

IP | The environmental impact of infrastructure in the environmental index |

IA | The environmental impact generated by water consumption on the whole system in the environmental index, per cubic meter |

Cfe_{k} | Stable gas consumption at the recycling plant (k) in relation to normal condtions, in cubic meters per hour |

Cfe_{j} | Fixed gas consumption of the gathering center (j) in normal conditions, in cubic meters per hour |

Cve_{p} | Variable gas consumption to produce a unit of product (p) in normal conditions, in cubic meters per hour |

Cvap | Variable water consumption to obtain a unit of product (p), in cubic meters |

${\alpha}_{k}\xb7{\beta}_{j}$ | Gathering center (j) and recycling plant (k) capacity ratio |

${d}_{ij}^{SR}$ | Distance between supplier (i) and the gathering center (j), in km |

${d}_{jk}^{RP}$ | Distance between gathering center (j) and recycling plant (k), in km |

${d}_{kl}^{PC}$ | Distance between recycling plant (k) and customer (l), in km |

CUR_{jp} | The cost of production (p) in the gathering center (j), in rials per ton |

CUP_{kp} | The cost of production (p) in the recycling plant (k), in rials per ton |

CUT_{m} | The variable cost of the transport mode (m), in rials per km |

CFR_{j} | Fixed cost of using the gathering center (j), in rials |

CFP_{k} | Fixed cost of using a recycling plant (k), in rials |

PS_{s} | Probability of scenario (s) |

PRI_{p} | Product sales price (p) |

**Variables**

QSR_{ijmps} | The amount of product supply (p) that is transmitted in the transportation mode (m) between the waste supplier (i) and the gathering center (j) under the scenario (s), in tons |

QRP_{jkmps} | The amount of product (p) that is transmitted in the transportation mode (m) between the gathering center (j) and the recycling plant (k) under the scenario (s), in tons |

QPC_{klmps} | The amount of product (p) transmitted in the transportation mode (m) between the recycling plant (k) and customer (l) under the scenario (s), in tons |

VSR_{ijms} | The number of trips between the waste supplier (i) and the gathering center (j) using the transportation mode (m) under the scenario (s) |

VRP_{jkms} | The number of trips between the gathering center (j) and the recycling plant (k) using the transportation mode (m) under the scenario (s) |

VPC_{klms} | The number of trips between the recycling plant (k) to the customer (l) using the transportation mode (m) under the scenario (s) |

HSR_{ijms}HRP _{jkms}HPC _{klms} | Variables that indicate the number of trips (excess or defect) to balance between the transportation modes |

R_{j} | Variable; 1 if the gathering center (j) is used, otherwise it is zero |

P_{k} | Variable; 1 if the recycling plant (k) is used, otherwise it is zero |

#### 2.3. Model Objective Functions

#### 2.4. Model Constraints

- Equations (4) to (6) guarantee the flow of materials through the SCN. The output from each center is, at most, equal to the inputs from different centers at the previous level of the SC;$$\sum _{j}{\displaystyle \sum _{m}{{\displaystyle QSR}}_{ijmps}\le {{\displaystyle G}}_{ip}}}\hspace{1em}\forall i,p,s$$$$\sum _{k}{\displaystyle \sum _{m}{{\displaystyle QRP}}_{jkmps}\le {\displaystyle \sum _{i}{\displaystyle \sum _{m}{{\displaystyle QSR}}_{ijmps}}}}}\hspace{1em}\forall j,p,s$$$$\sum _{l}{\displaystyle \sum _{m}{{\displaystyle QPC}}_{klmps}\le {\displaystyle \sum _{j}{\displaystyle \sum _{m}}}}}{{\displaystyle QRP}}_{jkmps}\hspace{1em}\forall k,p,s$$

- Equations (7) and (8) respectively guarantee that the flow of materials rate does not exceed the maximum capacity of the recycling plants and the product demand;$$\sum _{l}{\displaystyle \sum _{m}{{\displaystyle QPC}}_{klmps}\le {{\displaystyle C}}_{kp}}}\hspace{1em}\forall k,p,s$$$$\sum _{k}{\displaystyle \sum _{m}{{\displaystyle QPC}}_{klmps}\le {{\displaystyle D}}_{lps}}}\hspace{1em}\forall l,p,s$$

- Equations (9) to (11) maintain the balance between two facilities concerning the number of transportation. Since the number of transport must be an integer value, a series of inactive variables have been suggested to maintain the model’s probability;$$\sum _{p}\frac{{{\displaystyle QSR}}_{ijmps}}{{{\displaystyle CT}}_{mp}}+{{\displaystyle HSR}}_{ijms}={{\displaystyle VSR}}_{ijms}}\hspace{1em}\forall i,j,m,s$$$$\sum _{p}\frac{{{\displaystyle QRP}}_{jkmps}}{{{\displaystyle CT}}_{mp}}+{{\displaystyle HRP}}_{jkms}={{\displaystyle VRP}}_{jkms}}\hspace{1em}\forall j,k,m,s$$$$\sum _{p}\frac{{{\displaystyle QPC}}_{klmps}}{{{\displaystyle CT}}_{mp}}+{{\displaystyle HPC}}_{klms}={{\displaystyle VPC}}_{klms}}\hspace{1em}\forall k,l,m,s$$

- Equations (12) to (14) guarantee that ineffective variables focus only on differences in the number of transport;$${{\displaystyle VSR}}_{ijms}+{{\displaystyle HSR}}_{ijms}\ge 0\begin{array}{l}\forall i,j,m,s,\\ (-1{{\displaystyle HSR}}_{ijms}1)\end{array}$$$${{\displaystyle VRP}}_{jkms}+{{\displaystyle HRP}}_{jkms}\ge 0\begin{array}{l}\forall j,k,m,s,\\ (-1{{\displaystyle HRP}}_{jkms}1)\end{array}$$$${{\displaystyle VPC}}_{klms}+{{\displaystyle HPC}}_{klms}\ge 0\begin{array}{l}\forall k,l,m,s,\\ (-1{{\displaystyle HPC}}_{klm}1)\end{array}$$

- Equation (15) limits the number of trips per transportation mode [21];$$\sum _{i}{\displaystyle \sum _{j}{{\displaystyle VSR}}_{ijms}+{\displaystyle \sum _{j}{\displaystyle \sum _{k}{{\displaystyle VRP}}_{jkms}}}+{\displaystyle \sum _{k}{\displaystyle \sum _{l}{{\displaystyle VPC}}_{klms}\le {{\displaystyle NV}}_{m}}}}}\hspace{1em}\forall m,s$$

- According to Equations (16) and (17), binary variables should be assumed, such that if a gathering center or recycling plant is used in the model, then the value is 1, and otherwise it is zero;$$\sum _{k}{\displaystyle \sum _{m}{\displaystyle \sum _{p}{{\displaystyle QRP}}_{jkmps}\le M\text{}{R}_{j}}}}\hspace{1em}\forall j,s$$$$\sum _{l}{\displaystyle \sum _{m}{\displaystyle \sum _{p}{{\displaystyle QPC}}_{klmps}\le M{{\displaystyle P}}_{k}}}}\hspace{1em}\forall k,s$$

- Finally, Equations (18) and (19) show the nature of the variables.$$\begin{array}{l}{{\displaystyle QRP}}_{jkmps},{{\displaystyle QPC}}_{klmps},{{\displaystyle QSR}}_{ijmps},{{\displaystyle VPC}}_{klms},\\ {{\displaystyle VRP}}_{jkms},{{\displaystyle VSR}}_{ijms}\ge 0\end{array}$$$${{\displaystyle R}}_{j},{{\displaystyle P}}_{k}\in \{0,1\}$$

## 3. Results

#### 3.1. Definition of Chromosomes in the NSGA-II Algorithm

#### 3.2. NSGA-II Operator Selection

#### 3.3. Initial Model Solving Results

#### 3.4. Model Validation

#### 3.5. The Parameter Adjustment of the NSGA-II Algorithm

#### 3.6. Taguchi Design of Experiment

#### 3.7. Results of Solving the Main Model (Case Study)

#### 3.8. Sensitivity Analysis

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Beamon, B.M. Supply chain design and analysis: Models and methods. Int. J. Prod. Econ.
**1998**, 55, 281–294. [Google Scholar] [CrossRef] - Elzen, B.; Geels, F.W.; Green, K.E. System Innovation and the Transition to Sustainability: Theory, Evidence, and Policy; Edward Elgar Publishing: Cheltenham, UK, 2004; pp. 154–160. [Google Scholar]
- Fleischmann, M.; Bloemhof-Ruwaard, J.M.; Dekker, R.; Van Der Laan, E.; Van Nunen, J.A.E.E.; Van Wassenhove, L.N. Quantitative models for reverse logistics: A review. Eur. J. Oper. Res.
**1997**, 103, 1–17. [Google Scholar] [CrossRef] [Green Version] - Santoso, T.; Ahmed, S.; Goetschalckx, M.; Shapiro, A. A stochastic programming approach for supply chain network design under uncertainty. Eur. J. Oper. Res.
**2005**, 167, 96–115. [Google Scholar] [CrossRef] - Pishvaee, M.S.; Rabbani, M.; Torabi, S.A. A robust optimization approach to closed-loop supply chain network design under uncertainty. Appl. Math. Modell.
**2011**, 35, 637–649. [Google Scholar] [CrossRef] - Razavi Hajiagha, S.H.; Mahdiraji, H.A.; Zavadskas, E.K.; Hashemi, S.S. Fuzzy multi-objective linear programming based on compromise VIKOR method. Int. J. Inf. Technol. Decis. Mak.
**2014**, 13, 679–698. [Google Scholar] [CrossRef] - Mahdiraji, H.A.; Zavadskas, E.K.; Razavi Hajiagha, S.H. Game-theoretic approach for coordinating unlimited multi-echelon supply chains. Transf. Bus. Econ.
**2015**, 14, 35. [Google Scholar] - Pishvaee, M.S.; Razmi, J. Environmental supply chain network design using multi-objective fuzzy mathematical programming. Appl. Math. Modell.
**2012**, 36, 3433–3446. [Google Scholar] [CrossRef] - Jia, P.; Mahdiraji, H.A.; Govindan, K.; Meidutė, I. Leadership selection in an unlimited three-echelon supply chain. J. Bus. Econ. Manag.
**2013**, 14, 616–637. [Google Scholar] [CrossRef] [Green Version] - Mahdiraji, H.; Arabzadeh, M.; Ghaffari, R. Supply chain quality management. Manag. Sci. Lett.
**2012**, 2, 2463–2472. [Google Scholar] [CrossRef] - Mahdiraji, H.A.; Govindan, K.; Zavadskas, E.K.; Razavi Hajiagha, S.H. Coalition or decentralization: A game-theoretic analysis of a three-echelon supply chain network. J. Bus. Econ. Manag.
**2014**, 15, 460–485. [Google Scholar] [CrossRef] [Green Version] - Beheshti, M.; Mahdiraji, H.A.; Zavadskas, E.K. Strategy portfolio optimisation: A COPRAS G-MODM hybrid approach. Transf. Bus. Econ.
**2016**, 15, 500–519. [Google Scholar] - Wang, F.; Lai, X.; Shi, N. Multi-Objective optimization for green supply chain network design. Decis. Support Syst.
**2011**, 51, 262–269. [Google Scholar] [CrossRef] - Jamalnia, A.; Mahdiraji, H.A.; Sadeghi, M.R.; Razavi Hajiagha, S.H.; Feili, A. An integrated fuzzy QFD and fuzzy goal programming approach for global facility location-allocation problem. Int. J. Inf. Technol. Decis. Mak.
**2014**, 13, 263–290. [Google Scholar] [CrossRef] - Garg, K.; Kannan, D.; Diabat, A.; Jha, P.C. A multi-criteria optimization approach to manage environmental issues in a closed-loop supply chain network design. J. Clean. Prod.
**2015**, 100, 297–314. [Google Scholar] [CrossRef] - Razavi Hajiagha, S.H.; Akrami, H.; Hashemi, S.S.; Amoozad, H. An integer grey goal programming for project time, cost and quality trade-off. Inzinerine Ekon.-Eng. Econ.
**2015**, 26, 93–100. [Google Scholar] [CrossRef] [Green Version] - Razavi Hajiagha, S.H.; Hashemi, S.S.; Mahdiraji, H.A.; Azaddel, J. Multi-period data envelopment analysis based on Chebyshev inequality bounds. Expert Syst. Appl.
**2015**, 42, 7759–7767. [Google Scholar] [CrossRef] - Razavi Hajiagha, S.H.; Mahdiraji, H.A.; Hashemi, S.S.; Turskis, Z. Determining weights of fuzzy attributes for multi-attribute decision-making problems based on a consensus of expert opinions. Technol. Econ. Dev. Econ.
**2015**, 21, 738–755. [Google Scholar] [CrossRef] - Azadfallah, M.; Azizi, M. A new approach in group decision-making based on pairwise comparisons. J. Int. Bus. Entrep. Dev.
**2016**, 8, 159–165. [Google Scholar] [CrossRef] - Rezaee, A.; Dehghanian, F.; Fahimnia, B.; Beamon, B. Green supply chain network design with stochastic demand and carbon price. Ann. Oper. Res.
**2017**, 250, 463–485. [Google Scholar] [CrossRef] - World Steel Association. World Steel in Figures; World Steel Association: Brussels, Belgium, 2019. [Google Scholar]
- Feitó-Cespón, M.; Sarache, W.; Piedra-Jimenez, F.; Cespón-Castro, R. Redesign of a sustainable reverse supply chain under uncertainty: A case study. J. Clean. Prod.
**2017**, 151, 206–217. [Google Scholar] [CrossRef] - Guo, S.; Aydin, G.; Souza, G.C. Dismantle or remanufacture? Eur. J. Oper. Res.
**2014**, 233, 580–583. [Google Scholar] [CrossRef] - Niknejad, A.; Petrovic, D. Optimization of integrated reverse logistics networks with different product recovery routes. Eur. J. Oper. Res.
**2014**, 238, 143–154. [Google Scholar] [CrossRef] - Boukherroub, T.; Ruiz, A.; Guinet, A.; Fondrevelle, J. An integrated approach for sustainable supply chain planning. Comput. Oper. Res.
**2015**, 54, 180–194. [Google Scholar] [CrossRef] - Sarrafha, K.; Rahmati, S.H.A.; Niaki, S.T.A.; Zaretalab, A. A bi-objective integrated procurement, production, and distribution problem of a multi-echelon supply chain network design: A new tuned MOEA. Comput. Oper. Res.
**2015**, 54, 35–51. [Google Scholar] [CrossRef] - Santibanez-Gonzalez, E.D.; Diabat, A. Modeling logistics service providers in a non-cooperative supply chain. Appl. Math. Modell.
**2016**, 40, 6340–6358. [Google Scholar] [CrossRef] - Inderfurth, K. Impact of uncertainties on recovery behavior in a remanufacturing environment. A numerical analysis. Int. J. Phys. Distrib. Logist. Manag.
**2005**, 35, 318–336. [Google Scholar] [CrossRef] - Azadeh, A.; Raoofi, Z.; Zarrin, M. A multi-objective fuzzy linear programming model for optimization of the natural gas supply chain through a greenhouse gas reduction approach. J. Nat. Gas Sci. Eng.
**2015**, 26, 702–710. [Google Scholar] [CrossRef] - Memari, A.; Rahim, A.R.A.; Ahmad, R.B. An integrated production-distribution planning in the green supply chain: A multi-objective evolutionary approach. Procedia Corp.
**2015**, 26, 700–705. [Google Scholar] [CrossRef] [Green Version] - Pishvaee, M.S.; Khalaf, M.F. Novel robust fuzzy mathematical programming methods. Appl. Math. Modell.
**2016**, 40, 407–418. [Google Scholar] [CrossRef] - Mirmajlesi, S.R.; Shafaei, R. An integrated approach to solve a robust forward/reverse supply chain for short lifetime products. Comput. Ind. Eng.
**2016**, 97, 222–239. [Google Scholar] [CrossRef] - Dondo, R.G.; Méndez, C.A. Operational planning of forwarding and reverse logistics activities on multi-echelon supply-chain networks. Comput. Chem. Eng.
**2016**, 88, 170–184. [Google Scholar] [CrossRef] - Musavi, M.; Rayat, F. A bi-objective green truck routing and scheduling problem in a cross dock with the learning effect. Iranian J. Oper. Res.
**2017**, 8, 2–14. [Google Scholar] [CrossRef] [Green Version] - Rafie-Majd, Z.; Pasandideh, S.H.R.; Naderi, B. Modeling and solving the integrated inventory-location-routing problem in a multi-period and multi-perishable product supply chain with uncertainty: Lagrangian relaxation algorithm. Comput. Chem. Eng.
**2018**, 109, 9–22. [Google Scholar] [CrossRef] - Mogale, D.G.; Kumar, M.; Kumar, S.K.; Tiwari, M.K. Grain silo location-allocation problem with dwell time for optimization of food grain supply chain network. Transp. Res. Part E Logist. Transp. Rev.
**2018**, 111, 40–69. [Google Scholar] [CrossRef] - Dai, Z.; Aqlan, F.; Zheng, X.; Gao, K. A location-inventory supply chain network model using two heuristic algorithms for perishable products with fuzzy constraints. Comput. Ind. Eng.
**2018**, 119, 338–352. [Google Scholar] [CrossRef] - Seifbarghy, M.; Hasanzadeh, H. Designing a three-layer supply chain considering different transportation channels and delivery time-dependent demand. Int. J. Ind. Eng. Theory Appl. Pract.
**2018**, 253, 370–386. [Google Scholar] - Khodaparasti, S.; Bruni, M.E.; Beraldi, P.; Maleki, H.R.; Jahedi, S. A multi-period location-allocation model for nursing home network planning under uncertainty. Oper. Res. Health Care
**2018**, 18, 4–15. [Google Scholar] [CrossRef] - Doolun, I.S.; Ponnambalam, S.G.; Subramanian, N.; Kanagaraj, G. The data-driven hybrid evolutionary analytical approach for multi-objective location-allocation decisions: Automotive green supply chain empirical evidence. Comput. Oper. Res.
**2018**, 98, 265–283. [Google Scholar] [CrossRef] - Tsao, Y.C.T.; Thanh, V.V.; Lu, J.C.L.; Yu, V. Designing sustainable supply chain networks under uncertain environments: Fuzzy multi-objective programming. J. Clean. Prod.
**2018**, 174, 1550–1565. [Google Scholar] [CrossRef] - Sadeghi Rad, R.; Nahavandi, N. A novel multi-objective optimization model for the integrated problem of green closed-loop supply chain network design and quantity discount. J. Clean. Prod.
**2018**, 196, 1549–1565. [Google Scholar] [CrossRef] - Shen, J. An environmental supply chain network under uncertainty. Phys. A Stat. Mech. Appl.
**2019**, 12, 12–34. [Google Scholar] [CrossRef] - Falsafi, M.; Fornasiero, R.; Terkaj, W. Performance evaluation of stochastic forward and reverse supply networks. Procedia CIRP
**2019**, 81, 1342–1347. [Google Scholar] [CrossRef] - Alizadeh, M.; Makui, A.; Paydar, M.M. Forward and reverse supply chain network design for consumer medical supplies considering biological risk. Comput. Ind. Eng.
**2020**, 140, 106229. [Google Scholar] [CrossRef] - Debb, K.; Agrawal, S.; Pratap, A.; Meyarivan, T. A Fast Elitist Non-dominated Sorting Genetic Algorithm for Multi-objective Optimization: NSGA-II. In Proceedings of the International Conference on Parallel Problem Solving from Nature, Paris, France, 18–20 September 2000; pp. 849–858. [Google Scholar]
- Deb, K.; Pratap, A.; Agrawal, S.; Meyarivan, T. A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans. Evol. Comput.
**2002**, 6, 182–197. [Google Scholar] [CrossRef] [Green Version] - Gao, X.; Chen, B.; He, X.; Qiu, T.; Li, J.; Wang, C.; Zhang, L. Multi-objective optimization for periodic operation of the naphta pyrolysis process using a new parallel hybrid algorithm combining NSGA-II with SQP. Comput. Chem. Eng.
**2008**, 32, 2801–2811. [Google Scholar] [CrossRef] - Rabbani, M.; Farrokhi-Asl, H.; Asgarian, B. Solving a bi-objective location routing problem by a NSGA-II combined with clustering approach: Application in waste collection problem. J. Ind. Eng. Int.
**2017**, 13, 13–27. [Google Scholar] [CrossRef] [Green Version] - Wang, Y.; Zhang, S.; Guan, X.; Peng, S.; Wang, H.; Liu, Y.; Xu, M. Collaborative multi-depot logistic network design with time window assignment. Expert Syst. Appl.
**2020**, 140, 112910. [Google Scholar] [CrossRef] - Karimi, N.; Zandieh, M.; Karamooz, H.R. Bi-objective group scheduling in hybrid flexible flowshop: A multi-phase approach. Expert Syst. Appl.
**2010**, 37, 4024–4032. [Google Scholar] [CrossRef] - Zitzler, E.; Thiele, L. Multi-objective optimization using evolutionary algorithms a comparative case study. In Proceedings of the Fifth International Conference on Parallel Problem Solving from Nature (PPSN-V), Amsterdam, The Netherlands, 27–30 September 1998; pp. 292–293. [Google Scholar]
- Sarrafha, K.; Kazemi, A.; Alinezhad, A. A multi-objective evolutionary approach for integratd production-distribution planning problem in a supply chain network. J. Optim. Ind. Eng.
**2014**, 14, 89–102. [Google Scholar] - Sharifi, M.; Pourkarim Guilani, P.; Shahriari, M. Using NSGA-II algorithm for a three-objective redundancy allocation problem with k-out-of-n sub-systems. J. Optim. Ind. Eng.
**2016**, 19, 87–95. [Google Scholar] - Peace, G.S. Taguchi Methods; Addison-Wesley Publishing Company: Boston, MA, USA, 1993. [Google Scholar]

Solution Method | Network Flow | Network Design | Attributes of the Mathematical Model | Objective Function | Researcher | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Reverse | Forward | Collecting and Distribution | Transport | Facilities | Repair and Recovery | Recycling | Limited Capacity | Multi-Period | Multi-Product | Uncertainty | Customer Service Level | Environmental Issues | Minimizing the Cost | ||

Mathematical Programming | ✓ | ✓ | ✓ | ✓ | [23] | ||||||||||

New Optimization Model | ✓ | ✓ | ✓ | ✓ | ✓ | [24] | |||||||||

Mathematical Programming and Goal Programming Technique | ✓ | ✓ | ✓ | ✓ | ✓ | [25] | |||||||||

Genetic Algorithm | ✓ | ✓ | ✓ | [26] | |||||||||||

Multi-Objective Programming | ✓ | ✓ | ✓ | ✓ | [27] | ||||||||||

Metaheuristic Method | ✓ | ✓ | ✓ | ✓ | ✓ | [28] | |||||||||

Multi-Objective Linear Fuzzy Programming | ✓ | ✓ | ✓ | ✓ | [29] | ||||||||||

Multi-Objective Genetic Algorithm | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | [30] | |||||||

Fuzzy Optimization | ✓ | ✓ | ✓ | ✓ | ✓ | [31] | |||||||||

Mixed Integer Nonlinear Programming Model | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | [32] | |||||||

Column Generation Paradigm | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | [33] | |||||||

Robust Optimization | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | [22] | ||||||||

Mathematical Programming | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | [34] | ||||||||

Two-Stage Stochastic Programming | ✓ | ✓ | ✓ | ✓ | ✓ | [20] | |||||||||

Mathematical Programming and Lagrange Algorithm | ✓ | ✓ | ✓ | ✓ | ✓ | [35] | |||||||||

Mathematical Programming | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | [36] | ||||||||

Complex Integer Nonlinear Programming, HGA and HHS | ✓ | ✓ | ✓ | ✓ | [37] | ||||||||||

Single Objective Programming, Genetic and Neighborhood Search | ✓ | ✓ | ✓ | [38] | |||||||||||

Mathematical Programming | ✓ | ✓ | ✓ | ✓ | ✓ | [39] | |||||||||

Mathematical Programming | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | [40] | ||||||||

Mathematical Programming, Two-Phase Stochastic Programming | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | [41] | ||||||||

Mathematical Programming, Lp-Metric Based Method | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | [42] | |||

Multi-Objective | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | [43] | |||||||

Queueing Network Model | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | [44] | ||||||

De Novo Programming Method | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | [45] | ||||||

Robust Optimization | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | Current Study |

X3 | L1 | L2 | L3 |

K1 | 3 | 2 | 1 |

K2 | 0 | 0 | 0 |

K3 | 0 | 0 | 0 |

X4 | K1 | K2 | K3 |

J1 | 0 | 0 | 0 |

J2 | 0 | 0 | 0 |

J3 | 0 | 0 | 0 |

J4 | 0 | 0 | 0 |

J5 | 4 | 0 | 0 |

X5 | J1 | J2 | J3 | J4 | J5 |

I1 | 0 | 0 | 0 | 0 | 3 |

I2 | 0 | 0 | 0 | 0 | 2 |

I3 | 0 | 0 | 0 | 0 | 5 |

I4 | 0 | 0 | 0 | 0 | 1 |

I5 | 0 | 0 | 0 | 0 | 1 |

Number of Suppliers | 2 | Number of Gathering Center | 2 | Number of products | 2 |

umber of Recycling Plant | 2 | Number of Customers | 2 | ||

Number of Transportation modes | 2 | Number of Scenarios | 2 |

Answer No. | The Value of the First Objective Function | The Value of the Second Objective Function | The Value of the Third Objective Function |
---|---|---|---|

1 | −128.17 | 2271.64 | 0.8 |

2 | −247.05 | 1135.82 | 0.6 |

3 | −92.47 | 801.17 | 0.41 |

4 | 67.78 | 1135.82 | 0.2 |

5 | 87.77 | 2271.64 | 0.6 |

Answer No. | The Value of the First Objective Function | The Value of the Second Objective Function | The Value of the Third Objective Function |
---|---|---|---|

1 | −21.51 | 1961.65 | 0.4876 |

2 | −8.1 | 1944.2 | 0.4819 |

3 | 3.65 | 1920.59 | 0.46 |

4 | 3.85 | 1896.37 | 0.4706 |

5 | 13.4 | 1860.68 | 0.4607 |

**Table 7.**Comparison of indices for five examples with NSGA-II algorithms and the Epsilon Constraint method.

Item | Epsilon Constraint | NSGA-II | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

MID | Spacing | Diversity | NoS | Time(s) | MID | Spacing | Diversity | NoS | Time(s) | |

1 | 8556.59 | 51.73 | 2562.22 | 14 | 8 | 7283.27 | 13.05 | 2858.35 | 96 | 55.96 |

2 | 7574.8 | 73.37 | 3820.57 | 16 | 13 | 7749.36 | 79.91 | 2247.99 | 97 | 63.73 |

3 | 5383.46 | 181.73 | 6709.34 | 23 | 48 | 6720.7 | 30.82 | 3616.23 | 98 | 58.43 |

4 | 6317.78 | 181.41 | 5209.74 | 16 | 93 | 7150.89 | 28.77 | 2423.52 | 99 | 57.68 |

5 | 7109.71 | 242.98 | 5219.62 | 18 | 407 | 7903.34 | 13.88 | 1650.52 | 95 | 58.53 |

NSGA-II Parameters | Low Level (1) | Middle Level (2) | High Level (3) |
---|---|---|---|

MaxIt | 60 | 80 | 100 |

nPop | 50 | 70 | 100 |

Pc | 0.7 | 0.8 | 0.9 |

Pm | 0.15 | 0.25 | 0.35 |

Exam No. | MaxIt | nPop | Pc | Pm |
---|---|---|---|---|

1 | 1 | 1 | 1 | 1 |

2 | 1 | 2 | 2 | 2 |

3 | 1 | 3 | 3 | 3 |

4 | 2 | 1 | 2 | 3 |

5 | 2 | 2 | 3 | 1 |

6 | 2 | 3 | 1 | 2 |

7 | 3 | 1 | 3 | 2 |

8 | 3 | 2 | 1 | 3 |

9 | 3 | 3 | 2 | 1 |

No. | MID | Spacing | Diversity | NoS | Time(s) |
---|---|---|---|---|---|

1 | 7636.4 | 23.4 | 1869.3 | 50 | 27.4 |

2 | 7725.7 | 19.6 | 2329.9 | 68 | 48.4 |

3 | 7677.5 | 22.4 | 2453.4 | 100 | 130.4 |

4 | 7745.1 | 70.8 | 2677.1 | 49 | 42.7 |

5 | 7615.1 | 58.1 | 3041.4 | 69 | 69.5 |

6 | 7801.7 | 91.8 | 3019.3 | 97 | 125.9 |

7 | 7640.1 | 32.1 | 1924.5 | 48 | 48.7 |

8 | 7670.5 | 39.2 | 2856.5 | 70 | 85.5 |

9 | 7678.4 | 30.6 | 2598.7 | 98 | 152.9 |

**Table 11.**Normalized results and the calculation of responses for setting the parameter of the NSGA-II algorithm.

No. | MID | Spacing | Diversity | Nos | Time(s) | Response |
---|---|---|---|---|---|---|

1 | 0. 11 | 0.05 | 1.00 | 0.96 | 0.00 | 22.46 |

2 | 0.59 | 0.00 | 0.61 | 0.62 | 0.17 | 65.89 |

3 | 0.33 | 0.04 | 0.5 | 0.00 | 0.82 | 39.22 |

4 | 0.7 | 0.71 | 0.31 | 0.98 | 0.12 | 81.30 |

5 | 0.00 | 0.53 | 0.00 | 0.6 | 0.34 | 6.24 |

6 | 1.00 | 1.00 | 0.02 | 0.06 | 0.79 | 111.05 |

7 | 0.13 | 0.17 | 0.95 | 1.00 | 0.17 | 25.37 |

8 | 0.3 | 0.26 | 0.16 | 0.58 | 0.46 | 35.24 |

9 | 0.34 | 0.15 | 0.38 | 0.04 | 1.00 | 40.34 |

MaxIt | nPop | Pc | Pm | |
---|---|---|---|---|

NSGA-II | Level 3 | Level 2 | Level 3 | Level 1 |

100 | 70 | 0.9 | 0.15 |

Number of Suppliers | 5 | Number of Gathering Centers | 5 | Number of Products | 5 |

Number of Recycling Plants | 3 | Number of Customers | 3 | ||

Number of Transportation Modes | 225 | Number of Scenarios | 5 |

Answer No. | The Value of the First Objective Function | The Value of the Second Objective Function | The Value of the Third Objective Function |
---|---|---|---|

1 | 1,168,678,032,301 | 17,570,971,633 | 0.4738 |

2 | 1,535,360,428,421 | 18,258,944,734 | 0.5864 |

3 | 1,532,610,197,259 | 18,179,813,506 | 0.5912 |

4 | 1,226,611,751,948 | 17,585,580,845 | 0.3662 |

5 | 1,534,765,140,449 | 18,251,470,312 | 0.5337 |

MaxIt | nPop | Pc | Pm | |
---|---|---|---|---|

NSGA-II | 100 | 100 | 0.6 | 0.3 |

**Table 16.**A sample of optimal decision variables for the first Pareto solution of Table 14.

Variable | Value | Variable | Value |
---|---|---|---|

QSR_{1,5,1,1,1} | 2.4605 | QSR_{1,5,2,1,1} | 7.9593 |

QSR_{2,5,1,1,1} | 2.9064 | QSR_{2,5,2,1,1} | 7.7476 |

QSR_{3,5,1,1,1} | 6.1607 | QSR_{3,5,2,1,1} | 11.3636 |

QSR_{4,5,1,1,1} | 8.0460 | QSR_{4,5,2,1,1} | 7.8387 |

QSR_{5,5,1,1,1} | 4.5488 | QSR_{5,5,2,1,1} | 6.7933 |

No. | Before Changing the Demand Parameter | After Changing the Demand Parameter | ||||
---|---|---|---|---|---|---|

Amount of 1st O.F. | Amount of 2nd O.F. | Amount of 3rd. O.F. | Amount of 1st O.F. | Amount of 2nd O.F. | Amount of 3rd. O.F. | |

1 | −128.17 | 2271.64 | 0.8 | −170.57 | 2117.32 | 0.8 |

2 | −247.05 | 1135.82 | 0.6 | −92.47 | 801.17 | 0.41 |

3 | −92.47 | 801.17 | 0.41 | 42.57 | 1058.66 | 0.2 |

4 | 67.78 | 1135.82 | 0.2 | 49.15 | 2117.32 | 0.6 |

5 | 87.77 | 2271.64 | 0.6 | 158.82 | 1058.66 | 0.08 |

**Table 18.**Changes to the first objective function for the initial model induced by changing the amount of demand.

Item | The Amount of Demand Average | The Amount of First Objective Function |
---|---|---|

1 | 9.57 | 459.08 |

2 | 17.76 | 423.237 |

3 | 30.765 | 369.45 |

4 | 51.994 | 276.155 |

5 | 82.643 | 90.683 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Antucheviciene, J.; Jafarnejad, A.; Amoozad Mahdiraji, H.; Razavi Hajiagha, S.H.; Kargar, A.
Robust Multi-Objective Sustainable Reverse Supply Chain Planning: An Application in the Steel Industry. *Symmetry* **2020**, *12*, 594.
https://doi.org/10.3390/sym12040594

**AMA Style**

Antucheviciene J, Jafarnejad A, Amoozad Mahdiraji H, Razavi Hajiagha SH, Kargar A.
Robust Multi-Objective Sustainable Reverse Supply Chain Planning: An Application in the Steel Industry. *Symmetry*. 2020; 12(4):594.
https://doi.org/10.3390/sym12040594

**Chicago/Turabian Style**

Antucheviciene, Jurgita, Ahmad Jafarnejad, Hannan Amoozad Mahdiraji, Seyed Hossein Razavi Hajiagha, and Amir Kargar.
2020. "Robust Multi-Objective Sustainable Reverse Supply Chain Planning: An Application in the Steel Industry" *Symmetry* 12, no. 4: 594.
https://doi.org/10.3390/sym12040594