Optimizing a multiproduct closedloop supply chain using NSGAII, MOSA, and MOPSO metaheuristic algorithms
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Abstract
This study aims to discuss the solution methodology for a closedloop supply chain (CLSC) network that includes the collection of used products as well as distribution of the new products. This supply chain is presented on behalf of the problems that can be solved by the proposed metaheuristic algorithms. A mathematical model is designed for a CLSC that involves three objective functions of maximizing the profit, minimizing the total risk and shortages of products. Since three objective functions are considered, a multiobjective solution methodology can be advantageous. Therefore, several approaches have been studied and an NSGAII algorithm is first utilized, and then the results are validated using an MOSA and MOPSO algorithms. Prioritybased encoding, which is used in all the algorithms, is the core of the solution computations. To compare the performance of the metaheuristics, random numerical instances are evaluated by four criteria involving mean ideal distance, spread of nondominance solution, the number of Pareto solutions, and CPU time. In order to enhance the performance of the algorithms, Taguchi method is used for parameter tuning. Finally, sensitivity analyses are performed and the computational results are presented based on the sensitivity analyses in parameter tuning.
Keywords
Closedloop logistics NSGAII MOPSO MOSA Taguchi Mathematical model PrioritybasedIntroduction
Classification of papers related to CLSC
Case study  No.  Articles 

Without case study  14  Govindan et al. (2013); Özkır and Başlıgil (2013); Nikolaou et al. (2013); Bai and Sarkis (2013); Rashid et al. (2013); Dwivedy and Mittal (2012); Pialot et al. (2012); Chen and Sheu (2013); Gerrard and Kandlikar (2007); Hatcher et al. (2011); Binnemans et al. (2013); Qiang (2015); Corum et al. (2014); Chen et al. (2014) 
Car parts suppliers  11  Subramoniam et al. (2009); Matsumoto (2010); Subramoniam et al. (2010); Hojas et al. (2011); Subramoniam et al. (2013); Blume and Walther (2013); Amelia et al. (2009); Abdulrahman et al. (2014); Xia et al. (2014); Wang et al. (2014); Tian et al. (2014) 
Vehicle manufacturer/remanufacturer  8  Seitz (2007); Saavedra et al. (2013); McKenna et al. (2013); Forslind (2005); Go et al. (2011); Demirel et al. (2014); Kurdve et al. (2015); Ziout et al. (2014) 
Electronics  8  Georgiadis and Besiou (2008); Mafakheri and Nasiri (2013); Rathore et al. (2011); White et al. (2003); El korchi and Millet (2011); Ravi (2012); Low et al. (2014); JiménezParra et al. (2014) 
Steel  3  Giannetti et al. (2013); Schultmann et al. (2004); Salmi and Wierink (2011) 
Tannery  2  Hu et al. (2011); Gutterres et al. (2010) 
Power  2  Ortegon et al. (2013); Jiang et al. (2011) 
Printing  2  Kerr and Ryan (2001); Cristóbal Andrade et al. (2012) 
Palm oil  2  Ng et al. (2012); Mumtaz et al. (2010) 
Machine tools  2  Silva et al. (2013); Du et al. (2012) 
Nutrition  2  Huysveld et al. (2013); Jefferies et al. (2012) 
Others  13  Fahimnia et al. 2013; Simpson (2012); Zhu and Geng (2013); Matsumoto (2009); Zeng et al. (2013); Östlin et al. (2009); Pigosso et al. (2010); Queiruga et al. (2012); Shaharudin et al. (2014); Murakami et al. (2014); Goodall et al. (2014); Gelbmann and Hammerl (2014); Song et al. (2014) 
Oil involves many derivatives such as kerosene, lubricant, gasoline, and tar. In this study, we concentrate on the lubricants. Engine oil that is one of the important products which is used in cars for lubrication of engine parts. Since oil is one of the expensive and nonrenewable products, designing an optimized reverse logistics will be advantageous both financially and environmentally (Özceylan and Paksoy 2013).
This paper aims to design a closedloop reverse logistics for collecting used engine oil that will be used in manufacturing centers. Most of realworld supply chain (SC) problems are complex due to high number of indices which increase the dimension of the problem and it may lead to inefficiency of routine solution approaches (Fahimnia et al. 2013). Regarding increase in the size of problem, exponential growth in complexity makes the model become NPhard (Park et al. 2007; Jolai et al. 2011). To deal with this problem, metaheuristic solution methodologies are applied in this study. The algorithms that have been employed in this study are NSGAII, MOSA, and MOPSO. In order to validate the results generated by each algorithm, three metaheuristics are used.
The paper is composed of the following sections. In “Literature review,” a comprehensive study is performed to examine earlier researches and introduce the present research path. The problem is described and the mathematical model is represented in “Problem statement” section. Metaheuristics are presented in the “Solution methodology”. The criteria are introduced in “Performance measure.” Taguchi approach is discussed in “Parameter setting,” and finally, brief results of the study and future research recommendations are provided in “Conclusion and future research” section.
Literature review
In this section, solution methodologies for solving mathematical models are presented specifically supply chain network models which were proposed by recent researchers. Since the current work focused on the metaheuristic solution approaches the field of the supply chain, the concerned papers are discussed as follows.
A brief of reviewed metaheuristic algorithm
Author(s)  Implemented metaheuristic algorithm  Field of the research  

Memetic  GA  NSGAII  NRGA  ABC  Hybrid GA  SA  MOPSO  MOGA  MOSA  
Bandyopadhyay et al. (2008)  *  Theoretical discussion  
Pishvaee et al. (2009)  *  Reverse logistics  
Lee et al. (2009)  *  Reverse logistics  
Mahdavi et al. (2009)  *  Cellular manufacturing  
Altiparmak et al. 2009  *  Forward logistics  
Pishvaee et al. 2010  *  Forward/reverse logistics  
Umar et al. (2012)  *  Vehicle routing  
Lotfi and TavakkoliMoghaddam (2013)  *  Transportation problem  
Yang et al. (2013)  *  Line balancing model  
Subramanian et al. (2013)  *  Closedloop supply chain  
Ghasimi et al. (2014)  *  Forward logistics  
Yadegari et al. (2015)  *  Forward/reverse logistics  
*  Forward logistics  
Santosa and Kresna (2015)  *  Facility location problem  
Liu et al. (2015)  *  Oil–gas production  
Sarrafha et al. (2015)  *  Forward logistics  
Kumar et al. (2015)  *  Reverse logistics  
Mousavi et al. (2016)  *  *  *  Inventory model  
Current study  *  *  *  Closedloop supply chain 
The main classification of supply chain papers considering prioritybased representation
Author(s)  Objective function  Product (s)  Period(s)  Type of supply chain  

Single objective  Multiobjective  Single product  Multiple products  Single period  Multi period  Open  Closedloop  
Forward  Reverse  Forward & reverse  
Pishvaee et al. (2009)  *  *  *  *  
Lee et al. (2009)  *  *  *  *  
Altiparmak et al. (2009)  *  *  *  *  
Pishvaee et al. (2010)  *  *  *  *  
Subramanian et al. (2013)  *  *  *  *  
Yadegari et al. (2015)  *  *  *  *  
*  *  *  *  
Sarrafha et al. (2015)  *  *  *  *  
Current study  *  *  *  * 
Based on the reviewed papers, none of the studies have been performed on the closedloop supply chains considering multiobjective, multiproduct, and multiperiod models. Current paper considers a CLSC for collecting used engine oil that is multiproduct, multiperiod, and multiobjective. In this study, the NSGAII, MOSA, and MOPSO multiobjective solution methods are utilized to get the optimal solution for the proposed multiobjective mathematical model.
Problem statement
Assumptions, notations, and parameters

Manufacturing centers are two types. The first one is manufacturing type one that produces products type one using original oil and the second one is manufacturing type two which produces products by used engine oil.

Some hybrid centers are defined which function as distribution and collection centers.

Used engine oil is supplied by vendors and original oil is provided by the suppliers.

The capacity of containers that is used for transferring the used engine oil from vendors to collection centers is PO1.

Vehicles which transfer oil from collection centers to manufacturing centers have a capacity of PO2.
Indices  
c  Index of the first type of products; c = 1,…, C 
d  Index of the second type of products; d = 1,…, D 
r  Index of products; r = c ∪ d 
i  Index of the first type of manufacturing plants (MPs); i = 1,…, I 
j  Index of the second type of MPs; j = 1,…, J 
p  Index of MPs; p = i ∪ j 
e  Index of distribution centers (DCs); e = 1,…, E 
f  Index of collection centers (CCs); f = 1,…, F 
h  Index of hybrid (multipurpose) centers; h = 1,…, H 
k  Index of distribution and multipurpose centers; k = d ∪ h 
k′  Index of collection and multipurpose centers; k′ = c ∪ h 
v  Index of vendors; v = 1,…, V 
t  Index of periods; t = 1,…,T 
Parameters  
C _{ rpt }  Production cost related to product r which is produced in MP p in period t 
WEI_{ r }  Weight of product r 
TC1_{ rpkt }  Cost of shipping product r from MP p to DC k in period t 
TC2_{ rkvt }  Cost of shipping product r from DC k to vendor v in period t 
\( TC3_{{vk^{\prime}t}} \)  Cost of shipping one unit of used engine oil from vendor v to CC k ^{′} in period t 
\( TC4_{{k^{\prime}pt}} \)  Cost of shipping one unit of used engine oil from CC k ^{′} to MP p in period t 
Cap _{ rpt }  Capacity of MP r for product p in period t 
Cap′_{ kt }  Capacity of DC k in period t 
\( Cap_{{k^{\prime}t}}^{{\prime \prime }} \)  Number of available PO2 containers in CC k ^{′} in period t 
Cap _{ vt } ^{′″}  Capacity of vendor v to supply used oil in period t 
BE _{ vt }  Purchase cost for each PO1 container of used oil from vendor v in period t 
PP _{ rpt }  Price of product r in MP p in period t 
\( {\text{B}}_{\text{rt}} \)  Shortage cost of product r in period t 
τ _{ r }  Amount of oil to produce one unit of product r 
PO1  Weight of small container to ship used oil from vendors to collection and hybrid centers 
PO2  Weight of each tanker of used engine oil shipped from CC to MP 
D _{ rvt }  Demand of vendor v for product r in period t 
RI _{ k′v }  Rate of risk of supplying used oil by vendor v to CC k′ 
EC _{ e }  Establishment cost of DC of e 
EC _{ f } ^{ ′ }  Establishment cost CC of f 
EC _{ h } ^{″}  Establishment cost of h 
M  A big number 
Decision variables  
Q _{ rpt }  Quantity of production of product r in MP p in period t 
X _{ rpkt }  Quantity of product r shipped from MP p to DC k in period t 
Y _{ rkvt }  Quantity of product r shipped from DC k to vendor v in period t 
\( Y^{\prime}_{{vk^{\prime}t}} \)  Amount of used oil transferred from vendor v to CC k ^{′} in period t 
\( X^{\prime}_{{k^{\prime}pt}} \)  Amount of used oil transferred from CC k′ to MP p in period t 
L _{ rvt }  Shortage of product r for vendor v in period t 
DES_{ e }  If DC e is open 1, otherwise 0 
CES_{ f }  If CC f is open 1, otherwise 0 
HES_{ h }  If HC h is open 1, otherwise 0 
Mathematical model
Constraints
Solution methodology
Different solution methods are utilized in order to solve problems. In case of simple problems, routine methods can be used. However, by increasing the problem size, the routine solution methods would not work efficiently (Leu and Yang 1999). In these cases, using optimization techniques that are called evolutionary techniques are advantageous. Genetic algorithm (GA), particle swarm optimization (PSO), and simulated annealing (SA) are of these methods.
In this paper, three metaheuristic algorithms are utilized for optimization. Using different algorithms can validate the accuracy of results. Since the model involves three objective functions, using multiobjective solution methodology is necessary. Nondominated sorting genetic algorithm (NSGAII), multiobjective simulated annealing (MOSA), and multiobjective particle swarm optimization (MOPSO) are used for this purpose. Two encoding schemes are proposed which are prioritybased and permutation encoding. In the prioritybased representation, the priorities of the nodes are shown by the values in a list. According to this method, the nodes with higher values of priorities are selected earlier (Peng and Wei 2008; Taleizadeh et al. 2010). In this paper, prioritybased encoding is utilized due to its compatibility with the problem.
Chromosome representation
NSGAII
Crossover and mutation operators
MOSA
Neighborhood procedure
Neighborhood procedure is a mechanism which generates new solutions based on the initial solution. It works like crossover or mutation, but their procedures are not the same. To apply this procedure, some mechanisms such as swap, reversion and insertion are defined. By applying these mechanisms, new solutions are generated. In this paper, each gene contains a priority. Swap changes the positions of two positions. Reversion is a mechanism which generates new solution by reversing the positions. In the third method, called “insertion,” two positions are selected randomly. The first position is omitted and moved after that position (Sarrafha et al. 2015). The illustrations of the procedures are shown in “Appendix” Figs. 17, 18 and 19.
MOPSO
Particle swarm optimization (PSO) works with particles that is originally taken from the bird and fish movements (James and Russell 1995). PSO searches the solution space using particles that are updated in each iteration. Each particle is specified by its position and velocity. A swarm of particles is initialized randomly which will be modified according to two experiences: the local and the global best. The local best (personal best (Pbest)) is the best experience for each particle that is ever found and the global best (Gbest) is the best position among all the particles. Considering these two best sets, the velocity is computed for each particle by subtracting the local and global best from the current position (Prasanna Venkatesan and Kumanan 2012). The mechanism of updating each particle is defined by changing the velocity randomly.
Velocity update
Position update
 1.Round the velocity and number of dimensions.$$ v_{ik}^{t + 1} = \text{int} (v_{ik}^{t + 1} ),{\text{ s}}_{ik}^{t + 1} = \text{int} (s_{ik}^{t + 1} ) $$(32)
 2.New positions of a vector are calculated according to the following procedure. Formula (33) generates values in interval [1,n]. When a value in the permutation changes from λ _{1} to λ _{2}, another change should be done to keep the values in permutation exclusive. This change is replacing λ _{2} by λ _{1}.$$ x_{ik}^{t + 1} = \left\{ {\begin{array}{ll} {\frac{{x_{ik}^{t} + v_{ik}^{t + 1} }}{n}},&\quad v_{ik}^{t + 1} \ge 0 \\ \frac{{x_{ik}^{t} + v_{ik}^{t + 1} }}{n} + n,&\quad v_{ik}^{t + 1} < 0 \\ \end{array} } \right. $$(33)
Performance measure
There are some measures defined for evaluating multiobjective metaheuristics algorithms. In this study, four performance metrics are considered for assessing the proposed algorithms.
Number of Pareto solutions (NPS): This measure presents the number of Pareto optimal solutions in each algorithm.
Parameter settings
Two strategies are defined for performing the experiments: (1) changing one factor against other fixed factors. This method is called “one factor at time (OFAT). (2) Changing some factors simultaneously, so that the impact of each factor and their interactions will be specified. The second one is the socalled “DOE”.
Algorithm parameters and their levels
Parameters  Level 1  Level 2  Level 3  

NSGAII  Population  15  25  50 
Rate of crossover  0.7  0.8  0.95  
Rate of mutation  0.2  0.25  0.3  
Iteration  50  80  100  
MOSA  Population  5  10  15 
Neighbor  3  8  13  
Initial temperature  95  200  350  
Final temperature  5  10  15  
Swap probability  0.4  0.45  0.5  
Reversion probability  0.6  0.55  0.5  
Subiteration  5  25  50  
Iteration  50  80  120  
MOPSO  Local learning coefficient  0.7  0.85  0.95 
Global learning coefficient  0.95  0.85  0.75  
Inertia weight  0.25  0.5  0.75  
Grid number  5  10  15  
Iteration  50  80  100  
Population  10  15  20  
Repository  5  10  15 
Dimension of each instance
Indices/instance  1  2  3  4  5  6  7  8  9  10 

Number of periods  4  4  4  4  4  6  6  6  6  6 
Number of products type one  6  6  6  6  6  6  6  6  6  6 
Number of products type two  6  6  6  6  6  6  8  8  8  10 
Number of vendors  10  10  15  20  25  30  30  32  35  35 
Number of hybrid centers  4  8  10  10  10  10  10  10  10  10 
Number of collection centers  4  6  8  10  12  12  12  12  12  12 
Number of distribution centers  8  10  10  12  12  12  14  14  14  14 
Number of manufacturing center type two  2  4  4  6  8  8  8  8  8  10 
Number of manufacturing center type one  2  4  6  8  10  10  11  12  12  14 
Ranges of parameters for generating random instances
Parameter  Parameter range  Parameter  Parameter range  Parameter  Parameter range 

OP _{ st }  [8000,11,000]  tcw _{ cwt }  [300,2000]  C _{ kwt }  [12,000,20,000] 
pc _{ vt }  [300,2000]  thw _{ hwt }  [300,2000]  twd _{ kwdt }  [300,2000] 
pr _{ kwt }  [11 × 10^{5}, 16 × 10^{5}]  Cap _{ kwt }  [1000,4000]  twh _{ kwht }  [300,2000] 
Cap _{ vt } ^{’’’}  [1000,7000]  Cap’_{ ot }  [7000,40,000]  tdv _{ kdvt }  [300,2000] 
Cap _{ vt } ^{’’’’}  [22,000,90,000]  \( Cap_{{o^{\prime}t}}^{''} \)  [0,2]  thv _{ khvt }  [300,2000] 
O _{ h } ^{’’}  [7 × 10^{7}, 9 × 10^{7}]  tvh _{ vht }  [300,2000]  tvc _{ vct }  [300,2000] 
O _{ c } ^{′}  [7 × 10^{7}, 8 × 10^{7}]  D _{ kvt }  [50,700]  O _{ d }  [5 × 10^{7}, 7 × 10^{7}] 
The normalized outputs for different approach
MOPSO  MOSA  NSGA  

MID  NPS  SNS  MID  NPS  SNS  MID  NPS  SNS 
0.066  0  0.996  0.026  0.200  0.201  0  0.359  0.022 
0.372  0  0.718  0.042  0.000  0.216  0.053  0.500  0 
0.057  0  1.000  0.107  0.400  0  0.059  0.400  0 
0.090  0  0.992  0.073  0  0.019  0.063  0.400  0 
0.034  0  0.995  0.035  0.666  0  0  0.148  0.639 
0  0  0.993  0.014  0.666  0  0  0.269  0 
0.016  0  0.996  0.040  0.000  0.128  0.025  0.250  0 
0.015  0  0.999  0.217  0.666  0  0  0.452  0 
0.025  0  0.999  0.021  0.333  0  0.011  0.583  0 
0.361  0  0.677  0.026  0.200  0  0.037  0.091  0 
Decision matrix
w (score)  CPU time (sec.)  

MOPSO  0.422  1003.272 
NSGAII  0.105  507.507 
MOSA  0.109  2416.977 
Conclusion and future research
In this study, a CLSC of the used engine oil is optimized using different metaheuristics. Optimizing smallsize problems can be performed using optimization software. However, in case the dimensions of the problems increase, the new optimization tools dominate the routine optimization software. The Metaheuristic algorithms are one of these methods that assists engineers to solve the bigsize problems faster. In this paper, three metaheuristic algorithms are utilized to optimize the proposed mathematical model. Since a multiobjective model is discussed, a multiobjective metaheuristic should be applied in this case. The NSGAII, MOSA and MOPSO are three algorithms implemented to deal with the multiobjective model. The feasibility control of the constraints is very essential, so it encouraged us to concentrate on the prioritybased encoding and position update very carefully. In all the algorithms, a prioritybased encoding is used. Instances are generated randomly to compare the capability of three algorithms. To compare three algorithms, MID, NPS, SNS, and CPU time are used as measures. Considering these criteria, NSGAII outperformed other algorithms for the closedloop supply chain of engine oil. This algorithm works better than other proposed algorithms for sophisticated problems such as closedloop used engine oil logistics. Taguchi approach that is used in this study helps enhance the performance of the present algorithm and determine the best present of operational parameters.
This study could be pursued by the direction of solution methodology and parameter settings. Other algorithms can be provided to compare the performance of the proposed algorithms and other algorithms. For enthusiasts of supply chain management, there would be an option to modify the mathematical model to cover the requirement in used engine oil collection. Vehicle routing for distribution and collection part of the proposed supply chain can be complementary to the proposed mathematical model. In addition, other objective functions could be added to the present objective functions to cover realworld problems. Considering the uncertainty of realworld problems can be helpful for this purpose.
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