A network flow-based method to solve performance cost and makespan open-shop scheduling problems with time-windows

Abstract

This paper deals with several bicriteria open-shop scheduling problems where jobs are pre-emptable and their corresponding time-windows must be strictly respected. The criteria are a performance cost and the makespan. Network flow approaches are used in a lexmin procedure with a bounded makespan and the considered bicriteria problems are solved. Finally, the computational complexity of the algorithm and a numerical example are reported.

Introduction

The paper deals with specialized open-shop scheduling problems where jobs are pre-emptable (i.e. can be interrupted). The jobs are divided into operations and each operation must be performed on a designated specialized machine. The paper focuses on those problems in which a time-window is considered for each job: that is, a release time and a due date are specified, and all the operations of the corresponding job must be completely done in such time interval. Feasibility implies that release times are always satisfied and lateness is not permitted (i.e. due dates are, in fact, deadlines). Job processing requires qualified personnel and also resource consumption (water, energy, material, specialized technology, etc.) Personnel and resource consumption generate performance costs. These costs are usually calculated according to different work shifts, where each shift has its own rate. Different rates are based on actual labour conditions: rates differ for overtime, night shift, and weekend hours. Energy and other consumption follow a similar pattern.

In this paper, we model performance costs by a performance cost function, f. The other criterion considered is the completion time of all the jobs, makespan, denoted by C_max. The main aim of the paper is to solve the bicriteria minimization problem which can be denoted by O∣pmtn,r_j,d_j∣(f,C_max) extending the three parameters classification of scheduling problems given by Graham et al. [12] and Lawler et al. [15]. This problem is solved by selecting suitable values of a bound-parameter C, and the resolution of single criterion scheduling problems O∣pmtn,r_j,d_j,C_max ⩽ C∣f in sequence. The paper approaches this last problem for the function f considered by solving a min cost flow problem in a specific network. Finally, a suitable variation of the bound-parameter C allows us to solve the initial bicriteria scheduling problem.

In other words, the way to reach the important goal of solving the bicriteria optimization scheduling problem O∣pmtn,r_j,d_j∣(f,C_max) is through the consideration and solution of some other relevant open-shop scheduling problems, for instance, the nuclear problem O∣pmtn,r_j,d_j,C_max ⩽ C∣f, (hereinafter problem P) and the lexmin optimization problem O∣pmtn,r_j,d_j,C_max ⩽ C∣(f,C_max).

As far as we know, there are no so many papers that consider bicriteria pre-emptable open-shop scheduling problems where time-windows have to be strictly respected, even fewer use network flow approaches. Several papers apply network flow approaches to solve certain scheduling problems with only one single optimization criterion. See, for example, Federgruen and Groenevelt [6], Chen [3], [4], Serafini [21] and McCormick [16]. In this last case, the author uses the parametric push-relabel max flow method originally proposed by Gallo et al. [7].

A second group of papers includes Cho and Sahni [5] (who present a linear programming formulation similar to that obtained by Lawler and Labetoulle [14]), Pinedo [18], González [10] and González and Sahni [11]. This piece of the literature deals with pre-emptable open-shop scheduling problems where time-windows are strictly respected. However, they are still only related to feasibility or single criterion optimization scheduling problems. They do not use network flow approaches.

A network flow approach to solve the feasibility problem O∣pmtn,r_j,d_j∣-, and also the optimization problem O∣pmtn,r_j,d_j∣C_max (that is, feasibility and makespan optimization for the pre-emptable open-shop scheduling problem with time-windows strictly respected) was proposed by Sedeño-Noda et al. [19]. This last paper is different from the first group of the previously cited papers because it already considers specialized machines (open-shop) and time-windows. In addition, the approach proposed by [19] is based on network flow algorithms, which sets it apart from the second group of references. However, [19] still belongs to the feasibility and single criterion optimization cases. As it is already commented above, and differently from the others, the present paper proposes a network flow approach to solve bicriteria open-shop scheduling problems, representing a contribution to the literature.

This paper is structured as follows: Section 2 is devoted to describing problem P, the nuclear problem O∣pmtn,r_j,d_j,C_max ⩽ C∣f. Mathematical models are developed to characterize the feasible region and the performance cost function which models personnel and resource consumption costs. An improvement of the x_ijk-flow procedure given in [19] is also proposed in this section. Section 3 deals with the lexmin optimization problem O∣pmtn,r_j,d_j, C_max ⩽ C∣(f,C_max). Section 4 summarizes the proposed approach to the bicriteria optimization problem O∣pmtn,r_j,d_j∣(f,C_max) and its computational complexity is also reported. A numerical example is included in Section 5 to illustrate the approach. Finally, an overview of the paper and conclusions are offered in Section 6.