Elsevier

Applied Mathematics and Computation

Volume 242, 1 September 2014, Pages 309-314
Applied Mathematics and Computation

Single-machine bicriterion group scheduling with deteriorating setup times and job processing times

https://doi.org/10.1016/j.amc.2014.05.048Get rights and content

Abstract

This paper considers a group scheduling problem with two ordered criteria where both setup times and job-processing times are increasing functions of their starting times. It is assumed that the jobs be classified into several groups and the jobs of the same group have to be processed contiguously. We consider two objectives where the primary criterion is the total weighted completion time and the secondary criterion is the maximum cost. A polynomial time algorithm is presented to solve this bicriterion group scheduling problem with deteriorating setup times and job-processing times. This algorithm can also solve single-machine group scheduling problems with deteriorating setup times and job-processing times in several ordered maximum cost and arbitrary precedence.

Introduction

In classical group scheduling problems, most research assumes that the processing time of a job and the setup time of a group be a constant. However, there are many situations in which a job or a group that is processed later consumes more time than the same job or group when processed earlier. For example, in fire fighting when the time and effort required to control a fire increases if there is a delay in the start of the fire-fighting effort. Hence, there is a growing interest in the literature to study scheduling problems involving deteriorating jobs, i.e. jobs whose processing times are increasing functions of their starting times. We refer the reader to the book by Gawiejnowicz [1] for more details on single-machine, parallel-machine and dedicated-machine scheduling problems with deteriorating jobs.

On the other hand, scheduling problems with group technology have been studied by Baker [2], Cheng et al. [3], Ham et al. [4], Mitrofanov [5], Ozden et al. [6], Webster and Baker [7], to name a few. To the best of our knowledge, only a few results concerning scheduling problems with deteriorating jobs and group technology simultaneously are known. Since longer setup or preparation might be necessary as food quality deteriorates or a patient’s condition worsens, Wu et al. [8] considered a situation where the group setup times and job processing times are both described by a simple linear deterioration function. They showed that the makespan and the total completion time problems remain polynomially solvable under the proposed model. Wang et al. [9] considered the same model with Wu et al. [8], but with proportional deterioration. They proved that the makespan minimization problem and total weighted completion time minimization problem can be solved in polynomial time. Wang et al. [10] considered the same model with Wu et al. [8], but with a more general linear deterioration. For single machine group scheduling, they proved that the makespan minimization problem can be solved in polynomial time. Wang et al. [11] considered a single machine scheduling problem with deteriorating jobs, ready times and group technology, in which the group setup times are assumed to be known and fixed. For a special case, they showed that the makespan minimization problem can be polynomially solvable. More recent papers which have considered scheduling jobs with deteriorating jobs and group technology include Yang [12], Yang and Yang [13], Wei and Wang [14], Cheng et al. [15], Bai et al. [16], Lee and Lu [17], Huang and Wang [18], Wang et al. [19], Xu et al. [20], Lu et al. [21], Wang and Wang [22], and Yin et al. [23].

However, traditional research on the scheduling problem with deteriorating jobs assume that all jobs to be processed have a single criterion. In real production settings and service environments, scheduling decisions are made with respect to bicriterion (multicriteria) performance rather than a single criterion (T’kindt and Billaut [24]). In this paper we consider a bicriterion scheduling problem with deteriorating setup times and deteriorating job processing times. This bicriterion model was proposed by Cheng et al. [3]. The remainder of this paper is organized as follows. In Section 2 we provide the notation and formulation of the problem. Section 3 deals with the makespan minimization problem. Concluding remarks are given in the last section.

Section snippets

Notation and problem statement

In this section, the notation that is used throughout the paper will be introduced first, followed by the formulation of the problem.

Notation

Gthe number of groups (G2)
Gigroup i,i=1,2,,G
nithe number of jobs in Gi,i=1,2,,G
nthe total number of jobs i.e., (n1+n2++nG=n)
Jijjob j in Gi,i=1,2,,G,j=1,2,,ni
δithe deterioration rate of setup time for Gi
αijthe deterioration rate for the jth job in Gi
sithe actual setup time of Gi
pijthe actual processing time of Jij
ωijthe relative importance for the jth job in G

Formulating the bicriterion problem as a single criterion problem

We introduce precedence constraints on a set of groups and on the set of jobs for each group and formulate our bicriterion problem as a single criterion problem to minimize fmax subject to these precedence constraints. First of all, we give two lemmas.

Lemma 1

If the sequence of groups is fixed, then a schedule minimizes the total weighted completion time is obtained when αijωij(1+αij) of each job in Gi(i=1,2,,G) is increasing.

Proof

Let the sequence of groups be fixed. Consider an optimal schedule π in Gi.

Conclusions

In this paper we considered single-machine bicriterion group scheduling problems with deteriorating setup times and deteriorating job processing times. We transform the primary criterion which the total weighted completion time is minimal to two precedence constraints condition. Then we formulate the bicriterion problem as a single criterion problem to minimize fmax subject to two specially constructed precedence constraints. An effective algorithm is given in this paper. Certainly, the

Acknowledgements

The authors are grateful for two anonymous referees for their helpful comments on earlier version of the article. This research was supported by the National Natural Science Foundation of China (Grant Nos. 11001181 and 71271039), New Century Excellent Talents in University (NCET-13-0082), Changjiang Scholars and Innovative Research Team in University (IRT1214), the Fundamental Research Funds for the Central Universities (DUT14YQ211).

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