Applied Soft Computing 62 (2018) 328–346 Contents lists available at ScienceDirect Applied Soft Computing journal homepage: www.elsevier.com/locate/asoc Full length article A tri-level location-allocation model for forward/reverse supply chain Amir Mohammad Fathollahi Fard, Mostafa Hajaghaei-Keshteli ∗ Department of Industrial Engineering, University of Science and Technology of Mazandaran, Behshahr, Iran a r t i c l e i n f o Article history: Received 24 January 2017 Received in revised form 11 May 2017 Accepted 2 November 2017 Available online 8 November 2017 Keywords: Tri-level programming Location-allocation problem Stackelberg game Tri-level metaheuristic Keshtel algorithm Water wave optimization a b s t r a c t The design of supply chain network usually directly influences the performance of location-allocation of facilities, especially for the main parties. This paper firstly addresses the tri-level location-allocation design problem which considers the forward and reverse network, simultaneously. The proposed prob- lem is formulated on the static Stackelberg game between the Distribution Centers (DCs), Customer Zones (CZs) and Recover Centers (RCs) in the framework. The literature reports that most of previous works have utilized the various exact approaches which are not efficient and are so complex. In this study, three old and successful methods consist of Variable Neighborhood Search (VNS), Tabu Search (TS) and Particle Swarm Optimization (PSO), as well as two recent nature-inspired algorithms; Keshtel Algorithm (KA) and Water Wave Optimization (WWO) are utilized. Besides, according the nature of the problem, this study proposes a simple nested approach named as tri-level metaheuristic for the first time in order to solve the large scale problems. The performances of the algorithms are probed by using Taguchi experi- mental method to set the proper values for the parameters. Eventually, the efficiency of the algorithms is compared by different criteria and validated through a real case study. The obtained results show that tri-level metaheuristics are effective approaches to solve the underlying tri-level models in large scale network. © 2017 Elsevier B.V. All rights reserved. 1. Introduction The last decade has seen the rapid development of designing the location-allocation models in the term of Supply Chain Man- agement (SCM) which utilizes some new approaches to uniform the different parts of supply chain network [1–3]. The main process of a forward/reverse supply chain is happened on the allocation of customers to distribution centers, the allocation of recovering cen- ters to customers in order to guarantee the used products, and also the location of these facilities. All of them follow a distinguished manner to reduce the transportation cost in each section [4]. In the real world, the management of these mentioned sections are sep- arated and determined in different levels [5]. Customers select the services of DCs from their benefit and behavior. So, the multi-level programming is the useful way to simulate these parts of network, simultaneously. Multi-level programming (or multi-level decision-making) is addressed by the game theory of Van Stackelberg [6]. It aims to apply comprising between the different decision levels which are distributed through a hierarchy. In the preliminary of multi-level ∗ Corresponding author. E-mail address: mostafahaji@mazust.ac.ir (M. Hajaghaei-Keshteli). programming as bi-level programming, the upper level is known as a leader and the lower level is defined as a follower. Their individual decision is made in sequence with the goal of optimization [6]. Usu- ally, one of the important assumptions in these models is that the follower has full knowledge about the leader decisions. In this way, the reacts of follower depends on the used strategy of leader. The bi-level and tri-level programming models are two famous, typical and special of multi-level programming which have motivated a lot of researchers to use the application of these models in different topics, such as, location-allocation design, reliability, vulnerability of power system, interdiction facilities and military applications. Table 1 summarizes the related papers in the application of multi- level programming models. This table is developed according the Mahmoodjanloo et al. [44]. As can be seen in the table, these NP-hard problems have been considered and highlighted in the recent decade by researchers. As mentioned earlier, this paper firstly addresses the tri-level location- allocation design problem which considers the forward and reverse network, simultaneously. All parts of a supply chain act in a coop- erative manner to decide the best strategy [50]. As illustrated in Fig. 1, the Distribution Centers (DCs) as the top-level leader takes decisions to receive products from manufacturers and to select the suitable facilities to be located among all potential facilities. This level is mainly focused on the forward network. According to the https://doi.org/10.1016/j.asoc.2017.11.004 1568-4946/© 2017 Elsevier B.V. All rights reserved.