Elsevier

Journal of Cleaner Production

Volume 135, 1 November 2016, Pages 201-213
Journal of Cleaner Production

Advanced cross-entropy in closed-loop supply chain planning

https://doi.org/10.1016/j.jclepro.2016.04.006Get rights and content

Abstract

Developing new methodologies for nondeterministic polynomial (NP-hard) problems such as supply chain network design is always a major consideration for academia and practitioners. In this paper a cross-entropy (CE) based solution methodology is developed in order to cope with complex combinatorial problems. The NP-hard problem of designing and planning a closed-loop supply chain (CLSC) is considered. Furthermore, a multi-product multi-period CLSC network in a mixed-integer programming structure is regarded. On the other side, cross-entropy is one of the newly developed and successful metaheuristic algorithms. Thus, in order to achieve better solutions in comparison with current solution methodologies, a cross-entropy algorithm is developed for the first time in CLSC design and planning. Then, the capabilities of the cross-entropy algorithm are elevated, in order to achieve solutions that are more robust. Therefore, an algorithm, which is called “advanced cross-entropy” (ACE) is proposed. Finally, two presented CE-based algorithms are compared with a developed genetic algorithm (GA) for the same problem. GA is the most well-known metaheuristic algorithm, which has been abundantly developed in CLSC. Results prove that both of proposed CE-based algorithms dominate current methodologies. Both can find acceptable solutions in comparison with GA. Furthermore, the proposed advanced cross-entropy performs even better than CE in the quality of solutions and time.

Introduction

In order to cope with complex combinatorial problems, traditional exact methods like the branch and bound or branch and cut, and some approximation methods, are reliable but are generally better suited for small-size problem instances (Kannan et al., 2014). Indeed, they are inefficient techniques when we deal with real-world instances. On the other hand, in metaheuristic issues (Govindan et al., 2015a), exploiting and elevating new approaches instead of older and conventional ones (as genetic algorithm (Chen et al., 2013a, Chen et al., 2013b, Kannan et al., 2009b), tabu search (Caballero et al., 2007), simulated annealing (Khan and Govindan, 2011, Raj Mohan et al., 2009), etc.) will be an important step to the future. Cross-entropy is one of the recently developed metaheuristics that has been utilized successfully in NP-hard problems such as the Knapsack problem (Caserta et al., 2008), max cut, traveling salesman problem (Rubinstein and Kroese, 2004), vehicle routing problem (Chepuri and Homem-de-Mello, 2005, Govindan et al., 2014), and network reliability (Hui et al., 2005). Rubinstein proposed this search algorithm in 1997–1999 and named it cross-entropy algorithm (Rubinstein, 1997 and Rubinstein, 1999). CE is a recent generic Monte Carlo technique, which has similarities with the ant colony algorithm and has been successfully extended for various simulation and optimization problems.

Due to various reasons such as environmental concerns and increasing awareness of customers, reverse logistics (RLs) and Closed-Loop Supply Chain (CLSC) have become interesting areas of research and practice (Sasikumar and Kannan, 2009, Govindan et al., 2015a, Govindan et al., 2015b, Govindan et al., 2015c, Soleimani and Kannan, 2015, Kannan et al., 2012 Govindan et al., 2015b, Govindan and Soleimani, 2016, Pochampally et al., 2009, Soleimani et al., 2014, Wei et al., 2015, Bouzon et al., 2015, Bouzon et al., 2016, Govindan et al., 2012 Govindan et al., 2013, Govindan and Popiuc, 2014; Kannan, Diabat, Alrefaei, Kumar et al., 2014, Mangla et al., 2016, Sasikumar et al., 2010, Sasikumar and Kannan, 2009, Sasikumar and Kannan, 2008a, Sasikumar and Kannan, 2008b). Reverse alternatives, such as collecting, recycling, disassembling, and remanufacturing, have shifted traditional supply chain management practices into becoming more effective (Akçalı et al., 2009, Pochampally et al., 2009, Xia et al., 2015). Therefore, designing and planning a CLSC is one of the most practical and necessary areas of research (Soleimani et al., 2014, Wei et al., 2015). We cannot solve such a complex problem with an exact method for any practical-sized instance. Some metaheuristic algorithms have been utilized in this area but we cannot judge the quality of their solutions. Consequently, attempts to achieve solutions that are more robust should be continued (Lalmazloumian et al., 2013, Govindan et al., 2014). Reviewing the acceptable performance of the cross-entropy algorithm in different combinatorial problems leads us to exploit its capabilities in CLSC design and planning.

To the best of our knowledge, CE has not been utilized in the design and planning problem of CLSC/RLs. Previous attempts generally regard other types of metaheuristic algorithms such as genetic algorithm (Ko and Evans, 2007, Min and Ko, 2008, Soleimani et al., 2014), tabu search (Aras and Aksen, 2008), simulated annealing (Pishvaee et al., 2010), memetic algorithm (Hasani et al., 2015), and particle swarm optimization (Chen et al., 2015, Soleimani and Kannan, 2015). Recent trends in metaheuristic algorithms show the necessity of developing such algorithms and the importance of continuing the process of elevating current solution methodologies with new advanced algorithms such as cross-entropy.

Finally, in this paper, a cross-entropy algorithm is developed in order to cope with design and planning a CLSC. A multi-period multi-product multi-echelon CLSC network is considered. To the best of our knowledge, it is the only paper to utilize the cross-entropy algorithm as a solution methodology in this field. In order to evaluate this method, a genetic algorithm, the most well-known metaheuristic in this field, is exploited (Soleimani et al., 2013). We then go further and try to elevate the capabilities of the cross-entropy algorithm in order to achieve more robust solutions (called advanced cross-entropy). Two presented CE-based algorithms are compared with a developed genetic algorithm for the same problem. As mentioned, CLSC design and planning is a problem of magnitude that belongs in the class of NP-hard problems (Krarup and Pruzan, 1983, Schrijver, 2003). Therefore, proposing new and efficient methodologies which can deal with real-size problems is completely necessary.

The rest of the paper is organized as follows. In the next section, a comprehensive literature review is offered. A model description and formulation is detailed in Section 3. In Section 4, the cross-entropy algorithm and the proposed advanced cross-entropy are explained. Computational studies and the performance evaluation of the proposed algorithms are presented in Section 5. Finally, in Section 6, conclusions and further developments are briefly reviewed.

Section snippets

Literature review

The attempts to solve NP-hard problems have a long history. Researchers have tried to solve complex problems utilizing exact, approximation, heuristics, and metaheuristic algorithms. Due to their time limitations, exact algorithms are inefficient when we want to cope with real-size instances. A complete review of exact methodologies in reverse logistics networks is illustrated in Bostel et al. (2005). An annotated bibliography of various networks and solution methodologies in RLs/CLSC network

Model description and formulation

The proposed CLSC network is a single-objective multi-echelon multi-product model that includes seven echelons. A schematic view of the network is presented in Fig. 1. In the forward path, suppliers aim to provide raw materials for manufacturing centers in order to produce the new products. Then, these products are delivered to the customers through wholesalers. In the reverse network, the returned products are collected from customers at collection centers. Here, the end of life products are

Solution methodologies

In this section, two new proposed solution methodologies will be completely explained and discussed. Cross-entropy algorithm and a new hybrid of cross-entropy and genetic algorithm are two developed methodologies.

Computational study

In order to evaluate two developed algorithms (CE and ACE), a genetic algorithm (which is the most well-known algorithm in this field) is presented. The developed GA is based on the general procedure of this algorithm. At first, the initial populations (chromosomes) are generated randomly. Then by exploiting cross over (based on Radcliffe, 1992) operator, children are generated. The mutation rate also affects some chromosomes based on a limited predetermined rate (here 0.2). Then, fitness

Conclusion and future research

In order to improve current solution approaches in the design and planning of CLSC, a new cross-entropy based solution methodology is developed and elevated. Roughly, a multi-period multi-product CLSC network is considered; then, a cross-entropy algorithm and an advanced cross-entropy method are proposed to elevate current solution methodologies, including especially the genetic algorithm. To the best of our knowledge, it is the first paper that develops a CE, and surely an ACE, algorithm for

Acknowledgment

The authors sincerely thank the editor and the anonymous reviewers for their constructive and important comments on the paper. This research was supported in part by National Nature Science Foundation of China (Grant No. 71302005, E060202), Major Program of the National Social Science Fund of China (Grant No. 13&ZD147), and the Key Technologies R & D Program of Tianjin (No. 16YFZCGX00090).

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