Heuristics for a multi-machine multi-objective job scheduling problem with smoothing costs Proceeding for the GOL 2012 conference Jean Respen HEC - University of Geneva Switzerland jean.respen@unige.ch Nicolas Zufferey ∗ HEC - University of Geneva Switzerland nicolas.zufferey-hec@unige.ch Edoardo Amaldi DEI - Politecnico di Milano Italy amaldi@elet.polimi.it ABSTRACT We propose a new multi-objective job scheduling problem on non-identical machines involving job and machine dependent setup costs and times, as well as smoothing costs. Smooth- ing issues are very important in several settings, such as for example car production, since they allow to balance resource utilization over an assembly line. In this paper, we describe the problem, give a mixed integer linear programming for- mulation, and propose several heuristics: three greedy pro- cedures, two descent approaches, and a tabu search. Ex- periments, performed on realistic and challenging instances with up to 500 jobs and 8 machines, show that tabu search is a powerful method: it gives the best results for the large instances and is very competitive on the small instances. General Terms Metaheuristics, job scheduling, multi-resource 1. INTRODUCTION Multi-objective scheduling problems often involve minimiz- ing the makespan while considering setup costs and times. Various approaches have been proposed to tackle makespan minimization in the literature (see [8] for a good reference book). For a survey on scheduling techniques accounting for setup issues, the reader is referred to [1]. Nowadays, new constraints, known as smoothing constraints, are attracting a growing attention in the area of job schedul- ing (see the survey on smoothing constraints known as ”bal- ancing in assembly line” in [2]) and in particular for car sequencing problems, where cars must be scheduled before production in an order respecting many constraints (colors, options, due dates, etc.), while avoiding overloading some important resources. As an example, if the yellow cars with air-conditioning are scheduled first, the unlucky costumer ∗ Corresponding author. who ordered a grey car without air-conditioning may wait for a long time. For the car plant, balancing between op- tional equipments and colors allows to respect customers deadlines and to prevent overloading resources (machines or employees), which has an impact on cost reduction. As mentioned in [12], there is a complex tradeoff at the core of many practical scheduling problems, which involves balanc- ing the benefits of long production runs of a similar product against the costs of completing work before it is needed (and potentially causing other work to be tardy). Part of the above problem was the subject of the ROADEF 2005 Challenge (http://challenge.roadef.org/2005/en/) proposed by the car manufacturer Renault, where instances involve hundreds of cars and thus no exact algorithm can be competitive. In the Renault problem, car families are de- fined so that two cars of the same family contain the same optional equipments. Each car option i is associated with a pi /qi ratio constraint, meaning that at most pi vehicles with option i can be scheduled in any subsequence of qi ve- hicles, otherwise a penalty occurs. Another goal consists in minimizing the number of color changes in the produc- tion sequence. Thus, the overall objective is to minimize a weighted function involving the numbers of ratio constraint violations and color changes. A simplified version of this problem was proved to be NP-hard in [6], and a survey of the above challenge can be found in [11]. The winner team pro- posed a local search algorithm called very fast local search (VFLS), and the second team proposed a variable neighbor- hood search (VNS) in combination with an iterated local search (ILS) procedure. The VFLS heuristic is described in [4] and is the conjunction of a standard local search with a tuned transformation step. [9] described a set of heuris- tics, based on the paradigms of VNS and ILS metaheuristics, along with intensification and diversification strategies. The tabu search proposed in [3] was ranked 7th. In this work, we define and investigate a multi-objective pro- duction problem (P) with smoothing costs inspired by the Renault problem. Unlike the latter problem, we consider several non-identical machines (resources), eligibility con- straints (a job cannot necessarily be performed on all the machines), and setup constraints. We aim at minimizing in a lexicographic way the overall makespan, setup costs and smoothing costs. The remainder of the paper is organized as follows. In Sec-