A multi-objective production scheduling case study solved by simulated annealing
Introduction
Metaheuristics, like simulated annealing [19], tabu search and genetic algorithms [4], [14] have demonstrated their ability to solve combinatorial problems such as vehicle routing, production scheduling, time tabling, etc. So, some authors suggested to adapt metaheuristics in order to solve Multi-Objective COmbinatorial (MOCO) problems. Ulungu et al. [17], [18] conceived a Multi-Objective Simulated Annealing (MOSA) algorithm for solving combinatorial optimization problems and an interactive version was also designed by Teghem et al. [13].
Surveys of this stream of research can be found in [2], [7].
In the literature concerning multi-objective optimization problems we can distinguish five main approaches:
- (a)
Hierarchical approach: the objectives considered are ranked in a priority order and optimized in this order.
- (b)
Utility approach: a utility function or weighting function—often a weighted linear combination of the objectives—is used to aggregate the considered objectives in a single one.
- (c)
Goal programming (or satisficing approach): all the objectives are taken into account as constraints which express some satisficing levels (or goals) and the objective is to find a solution which provides value as close as possible of the pre-defined goal for each objective. Sometimes one objective is chosen as the main objective and is optimized under the constraint related to other objectives.
- (d)
Simultaneous (or Pareto) approach: the aim is to generate—or to approximate in case of a heuristic method—the complete set of efficient solutions.
We recall that for a multi-objective optimization problema solution X★ ∈ S is efficient or Pareto optimal (or non-dominated) if there is no other solution X ∈ S such that zk(X) ⩽ zk(X★) ∀k with at least one strict inequality.
- (e)
Interactive approach: at each step of the procedure, the decision-maker expresses his preferences in regard to one (or several) solutions proposed so that the method will progressively converge to a satisfying compromise between the considered objectives.
The MOSA method is designed to tackle a MOCO problem (P) following the approach (d) above. The aim is to generate a good approximation of the set of efficient solutions E(P). The procedure is valid for any number K ⩾ 2 of objectives. Similarly to a single objective heuristic in which a potentially optimal solution emerges, in the MOSA method the set will contain potentially efficient solutions, i.e. solutions which are not dominated by any other solution generated during the procedure.
Since scheduling problems are also combinatorial problems, applying the metaheuristics to production scheduling with multiple criteria is suitable.
In fact, a scheduling problem has often a multi-objective nature; moreover, it is well known that the optimal solution can be quite different if the objective chosen changes.
Often, each decision-maker wants to minimize a given criterion. For example in a company, the commercial manager is interested in satisfying customers and thus, minimizing the tardiness. On the other hand, the production manager wishes to minimize the use of the machines by minimizing the makespan or the work in process by minimizing the maximum flow time. Since these objectives are conflictual, a solution may perform well for one criterion, but gives bad results for others.
Therefore, any proposed scheduling approach has to find a compromise between them. Such a compromise may give the decision-maker the more satisfying solution or a list of efficient solutions.
Despite their importance, scant attention has been given to multiple criteria scheduling problems, especially in the case of multiple machines [15], [11]. This is due to the extreme complexity of these combinatorial problems.
It appears from the analysis of the literature that often the methods proposed are
- •
either very sophisticated (complex branch and bound, dynamic programming, dominance relations, etc.) and complex to implement;
- •
or only able to solve small size problems and with two objectives;
- •
or completely dependant of the model treated: the methods are no more valid if any change in the constraints or a fortiori a change in the objectives is introduced.
So there is a need for a general method able to treat a large class of models—even with large scale instances—and independent of the considered objectives. This is specially important when either no dedicated algorithm exists for the problem to solve or possibly if such algorithm appears too complex to be implemented in the firm concerned by the problem. MOSA aspires to be a such method.
Recently, we analyzed particular multi-objective scheduling problems—one machine model, parallel machines model and permutation flow-shop model—with the MOSA method [9].
The aim of the present paper is to show how MOSA methods can be used to solve complex multi-criteria scheduling problems in flexible job shops by generation of a list of potentially efficient solutions. We will present the use of MOSA method through a case study performed at a Tunisian firm.
Even if the problem treated here has not been studied before, related papers (see for instance [1], [6], [8], [12]) exist in the large literature on general or flexible job-shop problems.
This paper is organized as follows. Section 2 describes the case study. The MOSA method is presented in Section 3 as well as its adaptation to the particular multi-objective production scheduling problem. Section 4 reports some computational results.
Section snippets
The case study
The problem treated is inspired by a real case study concerning a Tunisian firm specialized in brass-products, like cocks, water-gates, water-counters, etc. There are 50 different products. An order-book for a given time period (like a week) corresponds to different products i to be produced in quantity qi (i = 1, … , I); these quantities must be produced in one batch.
The final products are obtained by the assembly of different sub-products. So two steps can be distinguished:
- (i)
the production of each
The multi-objective approach
In this section, we describe the proposed method to tackle the problem described in Section 2, recalling the principles of the MOSA method (for more details see [18]).
The application of the MOSA method to the case study
We first present the adaptation of the MOSA method to the particular multiple objective case study and then the numerical results obtained.
Conclusions
As usual some experiments have been realized, to fix adequate values to the different parameters of the simulated annealing procedure (initial temperature, cooling factor, etc.). Even if more experiments should still be made to improve the quality of the approximation set , it appears already that the heuristic approach described in this paper is an efficient tool to tackle multi-objective production scheduling problems. Effectively, the main advantage of the approach is its flexibility:
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