An extended study on an open-shop scheduling problem using the minimisation of the sum of quadratic completion times

Abstract

This paper addresses an open-shop scheduling problem with the objective of minimising the total quadratic completion time. A solution approach based on Lagrangian relaxation is presented to handle small-scale problems. Additionally, optimal properties are determined for two special cases of this problem. Finally, numerous experiments demonstrate the effectiveness of the Lagrangian relaxation method.

Introduction

In an open-shop scheduling model, each available job has to be processed on m machines on a route decided by a decision-maker to optimise an objective, such as makespan (the completion time of the last job) or the sum of completion times. Since the pioneering work of Gonzalez and Sahni [1], many researchers have been interested in this area of study. For a comprehensive survey of open-shop scheduling problems, readers can refer to Chen et al. [2] or Potts and Strusevich [3].

We consider an open-shop scheduling problem to minimise the sum of quadratic completion times. Open-shop scheduling problem can arise in many practical applications, such as in a network of diagnostic testing facilities in a hospital, automobile repair, satellite communications and quality control centers. The classical formulations of these problems assume that parameters, such as the processing time, are constant values known in advance. This assumption is inadequate when processing times vary. For example, in a hospital, the treatment cost and time for an emergency patient can increase significantly if the patient’s condition becomes more serious due to waiting for diagnostic testing facilities and operation rooms. Studies have concluded that the sum of quadratic completion time is a suitable objective function for the aforementioned cases [4], i.e., the more time it takes a job to finish, the greater the cost per unit of elapsed time, which motives a study of open-shop scheduling to minimise the sum of quadratic completion times.

Using the three-field notation of Graham et al. [5], the open shop problem to minimise the sum of quadratic completion times can be denoted by $\mathit{Om}\Vert \sum C_j^2$, where m is the number of machines. The objective of minimising total quadratic completion time was first presented by Townsend [6], who showed that using this objective, the single-machine problem can be solved by the Shortest Processing Time first (SPT) rule in polynomial time, and he also provided a branch-and-bound algorithm for the weighted version. Cheng and Liu [4] showed that the quadratic objective is more appropriate for cases where the more time it takes a job to finish, the greater the cost per unit of elapsed time. Additionally, the sum of quadratic completion times as the objective function is a trade-off between the makespan and the total completion time, which are two important objective functions in scheduling problems. The authors proved the strong NP-hardness of $\mathit{Pm}||\sum C_j^2$ and the asymptotical optimality of the SPT heuristic for the problem. Koulamas and Kyparisis [7] showed that $\mathit{Fm}||\sum C_j^2$ is strongly NP-hard and proved that the SPT heuristic with worst case ratio $m^2$ is asymptotically optimal for the problem. Koulamas and Kyparisis [8] introduced a general objective, minimisation of the total loss of job values, $\sum w_j{C}_j^{a_j}$, where the job value $C_j^{a_j}$ is a power function of the completion time job j (clearly, if $w_j=1$ and $a_j=2$ for all jobs, the general objective becomes the quadratic objective). The authors established the NP-hardness for $1||\sum w_j{C}_j^{a_j}$ and strong NP-hardness for $\mathit{Pm}||\sum w_j{C}_j^{a_j}$ and also solved several special cases of this general objective in single-machine and parallel-machine settings in polynomial time, respectively, and constructed a branch-and-bound algorithm and several heuristic algorithms for $\mathit{Pm}||\sum w_j{C}_j^{a_j}$. Janiak et al. [9] presented a job-scheduling problem with deteriorating job processing times and decreasing job values. The objective was to minimise the weighted sum of quadratic completion times in a parallel processor environment. Moreover, the other types of general objective functions dependent on job completion times are also extensively considered in scheduling problems. The interested reader can refer to the work of Raut et al. [10], Janiak et al. [11], Janiak and Krysiak [12], [13], Janiak et al. [14] and Janiak and Krysiak [15].

Thus far, the related studies on open-shop scheduling problem are rare. In this paper, we extend a previous study on the open-shop problem to minimise the sum of quadratic completion times. To avoid duplication, the effectiveness of the quadratic objective and the NP-hardness of this problem can be found in a companion paper of Bai and Zhang [16]. Although Bai and Zhang [16] proved the asymptotic optimality of the SPTB heuristic for large-sized problems, practical tests showed inferior performance for small-sized problems.

In our work, a Lagrangian relaxation method is proposed to obtain the near optimal solution for small-sized instances. In addition, we present optimal properties for two special cases of the problem. Numerical experiments show the superior performance of the Lagrangian relaxation method.

The remainder of the paper is arranged as follows. The formulation of the open-shop problem is presented in Section 2. The Lagrangian relaxation method and the optimal properties are provided in Sections 3 A Lagrangian relaxation based solution approach for small-scale instances, 4 The optimal properties for two special cases, respectively. Section 5 presents the computational results, and the discussion and conclusions appear in Section 6.