Parallel machine scheduling to minimize the makespan with sequence dependent deteriorating effects

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Abstract

A new unrelated parallel machine scheduling problem with deteriorating effect and the objective of makespan minimization is presented in this paper. The deterioration of each machine (and therefore of the job processing times) is a function of the sequence of jobs that have been processed by the machine and not (as considered in the literature) by the time at which each job is assigned to the machine or by the number of jobs already processed by the machine. It is showed that for a single machine the problem can be solved in polynomial time, whereas the problem is NP-hard when the number of machines is greater or equal than two. For the last case, a set of list scheduling algorithms and simulated annealing meta-heuristics are designed and the effectiveness of these approaches is evaluated by solving a large number of benchmark instances.

Introduction

Research that addresses the scheduling of deteriorating jobs has gained significant popularity in the last two decades. The tenant of problems with deteriorating jobs is that the processing time of the jobs is a function of their start time or the number of jobs since the start of the schedule (or since a maintenance activity), which is again related to the time since the start of the schedule. This paper addresses a variant of the job deterioration problem that considers the case where the deterioration of the processing time for a job depends on the specific jobs that have been previously processed by the machine. This perspective is in line with Yang [1] and Yang et al. [2], where the jobs are not per se deteriorating, but instead the machines are the ones deteriorating, although this differentiation is not made in most models. In our model the deterioration of the machines (and therefore of the job processing time) is a function of the sequence of jobs that have been processed by the same machine and not a function of the two approaches reported in the literature: the time at which the job is assigned to the machine or the number of jobs already processed. Our version of the problem is not yet addressed, and is highly relevant in many practical cases.

Two examples of the proposed relationship between deterioration and job assignment are presented next. The first is the assignment of construction jobs to “work gangs” during a shift. Each job has a baseline processing time, related to the time when all the workers are “fresh”. As the workers perform each job they become increasingly tired and therefore their processing speed deteriorates, but this deterioration depends on the particular job sequence. Let us say there are four independent non-sequential jobs, each taking a baseline time of 2 h (each would take 2 h if done first thing in the morning): dig a trench in a hard ground, demolish a shed, clean a storage area, and paint a wall (ordered by effort). While performing the jobs from hardest to easiest may require 9.4 h, performing the jobs from easiest to hardest may require 8.5 h. In the first sequence, the workers may get tired from having performed the first two jobs and therefore take longer time in performing the easy ones. On the other hand, when performing the easy tasks first, they will be “fresh” to complete the exhausting ones.

The second example, similar to that described by Yang et al. [2], considers a shop where machines are used to process a material, for example cutting stock or shredding wood. It can be assumed that depending on the material hardness the tools deteriorate differently. If the jobs with the “softer” material are processed first, the tools will deteriorate less, therefore the tools will maintain a higher level of performance. On the other hand, if the “hard” material jobs are performed first, the tools will deteriorate “faster” and completing the tasks on the softer material jobs take longer, for example if the machine has to be run slower to assure it properly performs the shredding process.

The remaining of the paper is organized as follows. In the next section, we discuss the recent literature on deteriorating jobs in the parallel machine environment. In Section 3, we formulate the problem for the unrelated machines, show some properties, provide an illustrative example, and finally we present a special case of the problem. In Section 4 we present some heuristics, based on the proprieties showed in Section 3, for the unrelated machines case. An experimental analysis is presented in Section 5. Section 6 concludes the paper and provides suggestions for future research.

Section snippets

Literature review

There has been considerable interest in the problem of deteriorating jobs since the seminal work by Gupta and Gupta [3] and Browne and Yechiali [4]. Reviews of the literature for deteriorating job problems have been completed by Alidaee and Womer [5] and by Cheng et. al. [6]. In this section we focus on recent papers in the deteriorating job problem that consider parallel machines environment.

A research stream in the literature characterizes processing time as a function of the job's start

The problem

The problem under consideration can be stated as follows. There are n independent jobs, N={1,…,j,…n}, to be processed on m parallel machines, M={1,…,k,…m}. All the jobs are non-preemptive and available for processing at time zero. Each machine can process only one job at a time and cannot stand idle until the last job assigned to it has been finished. There are g possible positions in each machine, g=n, and let G be the set of positions. Let pjk be the baseline processing time of job j on

Solution approaches

Since an exact approach would be impractical for our problem, in this section we first present several List Schedule heuristics, and then design some Simulated Annealing meta-heuristics to solve it.

Computational experiments

This section evaluates the performance of the described heuristics to generate optimal or efficient solutions to the problem. As in Gupta and Ruiz-Torres [22] we use two experimental sets: OB and BB. OB uses the optimal solution as the benchmark point, whereas BB uses the best solution found by the set of heuristics as benchmark since no optimal solution is available because of the size of the instances.

Summary and future work

This paper proposes a new unrelated parallel machine scheduling problem that considers the minimization of the makespan when the deteriorating effect depends on the sequence of the jobs in the machines, and designs a set of list scheduling algorithms and simulated annealing meta-heuristics. An extensive computational investigation serves to evaluate the performance of the proposed algorithms against optimal solutions and the best solutions found by the set of heuristics. The results of this

Acknoweldgments

This research was supported by a grant from the Facultad de Administración de Empresas of the University of Puerto Rico at Rio Piedras.

References (22)

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