Robust scheduling of parallel machines with sequence-dependent set-up costs

Abstract

In this paper we propose a robust approach for solving the scheduling problem of parallel machines with sequence-dependent set-up costs. In the literature, several mathematical models and solution methods have been proposed to solve such scheduling problems, but most of which are based on the strong assumption that input data are known in a deterministic way. In this paper, a fuzzy mathematical programming model is formulated by taking into account the uncertainty in processing times to provide the optimal solution as a trade-off between total set-up cost and robustness in demand satisfaction. The proposed approach requires the solution of a non-linear mixed integer programming (NLMIP), that can be formulated as an equivalent mixed integer linear programming (MILP) model. The resulting MILP model in real applications could be intractable due to its NP-hardness. Therefore, we propose a solution method technique, based on the solution of an approximated model, whose dimension is remarkably reduced with respect to the original counterpart. Numerical experiments conducted on the basis of data taken from a real application show that the average deviation of the reduced model solution over the optimum is less than 1.5%.

Introduction

In the planning of manufacturing production systems, generally, set-up includes the work to prepare machines or position works in the material-processing phase. The production problems that explicitly take into account set-up considerations may be grouped into two different classes. In the first one, set-up time (cost) depends only on the job to be processed and the problem is classified, from the set-up point of view, as sequence-independent. In the latter, set-up depends on both the job to be processed and the immediately preceding one. In this case, the problem is classified as sequence dependent.

Scheduling problems requiring explicit treatment of set-up can be found in several shop environments. For a complete research review on scheduling involving set-up consideration see [1], [17]. In some applications, the set-up costs are directly proportional to set-up times and, consequently, schedules optimal with respect to set-up times are also optimal with respect to set-up costs. This is true when set-up operations are related only to machine idle time. In many other cases, particularly when set-up operations require high-skilled labour, set-up costs are relatively high while set-up time is relatively less. The problem studied in this paper regards the scheduling of independent jobs on identical parallel machines with sequence-dependent set-up costs, in order to minimise the total set-up cost. In the scheduling literature, several authors have addressed such NPhard scheduling problems, investigating the use of heuristic solution able to give a good solution in reasonable computation time [20], [21]. In particular, Sumischrast et al. [20], [21] develop an efficient heuristic, realistic sized problems characterising the industrial applications motivating the study, by exploiting a mathematical model from Dearing and Henderson [4]. The proposed models are based on the assumption that input data are deterministically known. In this way, if some input data change with respect to those used in the computations, the solution given by the scheduler may be not optimal or even feasible anymore and consequently the application to real cases may be compromised. This is one of the reasons for which it is important to consider uncertainty in the modelling problem phase.

There are two major manners to represent uncertainty: random numbers and fuzzy numbers. In order to choice how to incorporate uncertainty in the optimisation problem some knowledge of the uncertainty is required. In the stochastic approach uncertain data are modelled by specifying the probability distributions, for example inferred from historical data. The literature of stochastic scheduling addresses the identical parallel machines problem in several research works, most of them focused on demonstrating the optimality of priority-index policies. In particular, the main performance objectives investigated has been total expected flow-time minimisation as well as expected makespan minimisation. In particular, the shortest expected processing time rule (SEPT) has been shown to be optimal for the flow-time objective in the following cases: all the jobs processing time distributions are exponential [10]; all the jobs have a common general processing time distribution with a no-decreasing hazard rate function [23]; job processing times are stochastically ordered [24]. For the expected makespan objective, the longer expected processing times policy (LEPT) has been shown to be optimal if the following conditions are verified: the processing times distributions are exponential [3]; jobs have a common processing time distribution with a non-increasing hazard rate function [23]. If historical data are not available (e.g. the plants often diversify their products or introduce in the production plans new product typologies), an alternative way to model imprecision is represented by the fuzzy approach. In these cases, the production data, on which the scheduling models are based, are characterised by imprecision and expressed in linguistic terms. Fuzzy sets theory [19] provides a conceptual framework which may efficiently be used for dealing with situations characterised by imprecision and provide a very efficient framework to reduce scheduling computational complexity with respect to the same problem formulated in a probabilistic way. We should make clear here that such an imprecision is due to subjective and qualitative evaluations, rather than the effect of uncontrollable events.

Zimmermann [26] provides an extensive coverage of the theoretical and applied approaches to fuzzy sets. Fuzzy sets concepts enrich traditional Operational Research in various applications. In particular, fuzzy sets theory has been exploited in the scheduling applications to model flexible constraints, and the uncertainty in the definition of time parameters in flow shop, job shop and project problems (see [10], [11], [13], [14], [15], [18], [25]).

In this paper an identical parallel-machine scheduling problem with sequence-dependent set-up costs under the hypothesis of fuzzy processing times knowledge is analysed. In the considered shop environment, since processing requirement of each different lot is expressed in terms of product demand, each lot can be split arbitrarily into sub-lots and processed independently on machines. Therefore, the uncertain processing times are modelled through uncertain machine production speeds. In Section 2, the scheduling problem is described and modelled through an integer linear programming (ILP) model. Section 3 illustrates the mathematical features of the corresponding fuzzy scheduling problem. In particular, the necessity degree measure is used in order to formulate a minimum risk approach, where the optimal solution is a trade-off between total set-up cost and robustness in demand satisfaction. Since the minimum risk approach requires to solve a non-linear mixed integer programming (NLMIP) model, in Section 4, we illustrate how to build an equivalent formulation in which we avoid the non-linearity presented in the preceding formulation. The resulting mixed integer linear programming (MILP) model in real applications could be intractable due to its NP-hardness. Therefore, we propose in Section 5 a specific technique, on the basis of which we are able to reduce the dimension of the model, and, consequently, to increase the number of real applications solvable. From the approximated model, a feasible solution of the original model can easily be found. Numerical experiments conducted on the basis of data taken from a real application show the effectiveness and efficiency of the proposed approach. The results are reported in Section 6. Section 7 is devoted to the conclusion of this work.