Since production systems involve various important aspects such as cost, time and available (human and material) resources, multi-objective planning prob-lems are of growing practical relevance. As mentioned in [78], multi-objective scheduling problems can be tackled using five different approaches: the lexico-graphic, utility, goal programming, simultaneous orinteractive approaches. No perfect approach comes out and each one has its own advantages and draw-backs. Some approaches require more parameters whereas others are unable to generate efficient solutions under certain conditions. See the survey [32] on multi-objective combinatorial optimization, including theoretical results as well as exact and heuristic methods, and [64] on multi-objective metaheuristics. A good reference book on this topic can be found in [31].
Multi-objective scheduling problems often involve minimizing the makespan while considering setup costs and times. Various approaches have been pro-posed to tackle makespan minimization in the literature (e.g., [97]). For a sur-vey on scheduling techniques accounting for setup issues, the reader is referred to [1]. Single machine scheduling methods to minimize a regular objective func-tion under setup constraints are proposed in [6]. In [37], heuristic methods and a branch and bound procedure are described for parallel machines subject to setup constraints. For the same problem, a computationally intensive non-linear mixed integer model and an approximation method for real-world sized instances are proposed in [2]. In [88], a hybridization of a particle swarm and local search algorithms is proposed to solve the flexible multi-objective job-shop problem. Whereas in a job-shop problem a set of jobs must be scheduled on a set of different machines and each job has a specific routing on the machines, a flexible job-shop problem is an extension where each job can be processed by a set of different machines along different routes.
1.1. INTRODUCTION 21 Nowadays, new constraints, known as smoothing constraints, are attracting a growing attention in the area of job scheduling (see the survey on smoothing constraints known as “balancing in assembly line” in [10]). Smoothing con-straints (or costs) allow to schedule the jobs with a well-balanced consumption of the production resources. These constraints are now widely used, in par-ticular for car sequencing problems (see for example [8]), where cars must be scheduled before production in an order respecting various constraints (colors, optional equipment, due dates, etc.), while avoiding overloading some important resources. As an example, if the yellow cars with air-conditioning are scheduled first, the unlucky customer who ordered a grey car without air-conditioning may wait for a long time. For the car plant, balancing between optional equipment and colors allows to respect customers deadlines and to prevent overloading some resources (machines or employees), which has an impact on cost reduc-tion. As mentioned in [126], there is a complex tradeoff at the core of many practical scheduling problems, which involves balancing the benefits of long pro-duction runs of a similar product against the costs of completing work before it is needed (and potentially causing other work to be tardy).
In this paper, we address a multi-objective production scheduling problem with smoothing costs inspired by the problem proposed by the car manufacturer Re-nault in the ROADEF 2005 Challenge (http://challenge.roadef.org/2005/
en/). In the Renault problem, car families are defined so that two cars of the same family contain the same optional equipment. Each car optioni is associ-ated with api/qiratio constraint requiring that at mostpivehicles with option ican be scheduled in any subsequence ofqivehicles, otherwise a penalty occurs.
This penalty was artificially created by Renault in order to penalize the pos-sible congestion (which corresponds to an unbalanced use of some production resources) that might occur in the assembly line. Since another goal consists in minimizing the number of color changes in the production sequence, the overall objective is to minimize a weighted function involving the numbers of ratio con-straint violations and color changes. The resulting problem isN P-hard and no exact algorithm can be competitive because the instances involve hundreds of cars. A survey of the above challenge can be found in [120]. The winning team proposed a very fast local search which combines a standard local search with a tuned transformation step [33]. The team ranked second developed a variable neighborhood search based on an iterated local search procedure, along with in-tensification and diversification strategies [108]. Efficient tabu search methods
were also devised (e.g., [26, 130]).
The multi-objective scheduling problem we propose and investigate here is a variant of the Renault problem. Unlike in the Renault problem, we consider several machines (resources) which are not identical, eligibility constraints (a job cannot necessarily be performed on all the machines), setup constraints, and three different objectives functions. Adopting a realistic priority among the objectives, we aim at minimizing in a lexicographic order the overall makespan, smoothing costs, and setup costs. Mathematically such a lexicographic order al-lows to convert a multi-objective problem into a problem with a single objective.
Preliminary experiments showed the relevance of the proposed lexicographical approach, as many solutions with the same first objective values are very likely to occur. In the studied problem, we use a pi/qi ratio constraint withpi = 2 andqi= 3. In other words, a penalty occurs if more than two jobs of the same family are sequentially produced on the same machine. As confirmed in [90], the problem considered in this paper is of obvious interest from a modern car production point of view. Indeed, car manufacturers often face problems where smoothing different resources is mandatory. As stated above, color changes is of important matter. In addition, resources such as the workforce must often be smoothed, for example by alternatively performing tasks that requires more workers with less constrained workforce tasks. The problem we describe here is interestingly complete as it intends to include different type of real-world constraints, which were never jointly considered before.
Building on our preliminary work [106], this paper adds: an extended literature review, a practical motivation, an accurate test of a mixed integer linear pro-gramming (MILP) formulation, new and more efficient metaheuristics (namely, a greedy randomized adaptive search procedure (GRASP), a refined tabu search with intensification and diversification procedures, and an adaptive memory al-gorithm), and computational results for a much larger set of instances.
The remainder of the paper is organized as follows. In Section 1.2, we describe the problem under consideration, pointing out the differences with respect to theRenault problem proposed in the ROADEF 2005 Challenge and we give a MILP formulation. In Section 1.3 we provide a network interpretation. Since the MILP formulation is very challenging even for small-size instances, in Section 1.4, we describe three advanced metaheuristic methods, namely a GRASP, a
1.2. PROBLEM DESCRIPTION AND MILP FORMULATION 23