Discrete Optimization Robust optimization for the cyclic hoist scheduling problem Ada Che a , Jianguang Feng a,⇑ , Haoxun Chen b , Chengbin Chu c a School of Management, Northwestern Polytechnical University, Xi’an 710072, PR China b Laboratoire LOSI, Université de Technologie de Troyes, Troyes 10010, France c Laboratoire Génie Industriel, Ecole Centrale Paris, Grande Voie des Vignes, 92295 Châtenay-Malabry Cedex, France article info Article history: Received 8 December 2013 Accepted 28 June 2014 Available online 7 July 2014 Keywords: Cyclic scheduling Hoist scheduling Robust scheduling Mixed integer linear programming abstract This paper deals with the robust optimization for the cyclic hoist scheduling problem with processing time window constraints. The robustness of a cyclic hoist schedule is defined as its ability to remain stable in the presence of perturbations or variations of certain degree in the hoist transportation times. With such a definition, we propose a method to measure the robustness of a cyclic hoist schedule. A bi-objective mixed integer linear programming (MILP) model, which aims to optimize cycle time and robustness, is developed for the robust cyclic hoist scheduling problem. We prove that the optimal cycle time is a strictly increasing function of the robustness and the problem has infinite Pareto optimal solutions. Furthermore, we derive the so-called ideal point and nadir point that define the lower and upper bounds for the objective values of Pareto front. A Pareto optimal solution can be obtained by solving a single-objective MILP model to minimize the cycle time for a given value of robustness or maximize the robustness for a specific cycle time. The single-objective MILP models are solved using commercial optimization software CPLEX. Computational results on several benchmark instances and randomly generated instances indicate that the proposed approach can solve large-scale problems within a reasonable amount of time. Ó 2014 Elsevier B.V. All rights reserved. 1. Introduction Many production systems employ a computer-controlled hoist to perform the transportation of parts between processing stages, such as automated electroplating lines in the manufacture of printed circuit boards (PCBs). Such a system generally consists of a sequence of tanks with a variety of chemical solutions for part surface processing, and a single computer-controlled hoist in charge of material handling between the tanks. The parts are treated successively in the tanks according to their processing routine. There is no intermediate storage buffer between the tanks. After the processing in a tank has been completed, the part should be unloaded by the hoist, then transported to the next tank accord- ing to its processing routine, and loaded into that tank for the next processing. Due to specificity of chemical processing, the process- ing time in each tank must be within its prescribed time window defined by a lower bound and an upper bound. If the processing time window in any tank is violated, i.e., the processing time in a tank is shorter than the given lower bound or longer than the upper bound, defective parts would be produced. As all the material handling operations between tanks are performed by the hoist, the efficiency of such a production system is strongly dependent on the schedule of the hoist operations. This problem is the so-called hoist scheduling problem in the literature (Chauvet, Levner, Meyzin, & Proth, 2000; Che & Chu, 2007; Chen, Chu, & Proth, 1998; Kats, Lei, & Levner, 2008; Kats & Levner, 2011; Leung, Zhang, Yang, Mak, & Lam, 2004; Levner, Kats, de Pablo, & Cheng, 2010; Liu, Jiang, & Zhou, 2002; Phillips & Unger, 1976; Shapiro & Nuttle, 1988). As a large amount of parts is processed in the production line, the system commonly runs in a cyclic mode. That is to say, the hoist is programmed in advance to perform a specified sequence of operations repeatedly, referred to as a cyclic hoist schedule. Each repetition of such a sequence of hoist operations is called a cycle, and its duration is called cycle time. The cyclic hoist scheduling problem has been well examined over the past three decades. Phillips and Unger (1976) developed its first mixed integer linear programming (MILP) model. The problem was proven to be strongly NP-hard by Lei and Wang (1989). Numerous researchers, such as Shapiro and Nuttle (1988), Lei and Wang (1994), Ng (1996), Chen et al. (1998) and Yan, Chu, Yang, and Che (2010), put forward various branch-and-bound http://dx.doi.org/10.1016/j.ejor.2014.06.047 0377-2217/Ó 2014 Elsevier B.V. All rights reserved. ⇑ Corresponding author. Tel.: +86 2988431805; fax: +86 2988431781. E-mail addresses: ache@nwpu.edu.cn (A. Che), fengjianguangnpu@gmail.com (J. Feng), haoxun.chen@utt.fr (H. Chen), chengbin.chu@ecp.fr (C. Chu). European Journal of Operational Research 240 (2015) 627–636 Contents lists available at ScienceDirect European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor