Scheduling machine-dependent jobs to minimize lateness on machines with identical speed under availability constraints

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Abstract

In this paper we consider the problem of scheduling n preemptive jobs on m machines with identical speed under machine availability and eligibility constraints when minimizing maximum lateness (Lmax). The lateness of a job is defined to be its completion time minus its due date, and Lmax is the maximum value of lateness among all jobs. We assume that each machine is not continuously available at all time throughout the planning horizon and each job is only allowed to be processed on specific machines. Network flow technique is used to formulate this scheduling problem into a series of maximum flow problems. We propose a polynomial time two-phase binary search algorithm to verify the feasibility of the problem and to solve the scheduling problem optimally if a feasible schedule exists. Finally, we show that the time complexity of the algorithm is O((n+(2n+2x))3log(UB-LB)). Most literature in parallel machine scheduling assume that all machines are continuously available for processing and all jobs can be processed at any available machine throughout the planning horizon. But both assumptions might not be true in some practical environment, such as machine preventive maintenance and machines that have different capabilities to process jobs. This type of scheduling problem is seldom studied in the literature. The purpose of this paper is to examine the scheduling problem with machines with identical speed under machine availability and eligibility constraints. The objective is to minimize maximum lateness. We formulate this scheduling problem into a series of maximum flow problems with different values of Lmax. A polynomial time two-phase binary search algorithm is proposed to verify the feasibility of the problem and to determine the optimal Lmax.

Introduction

We consider the problem of scheduling n preemptive jobs on m machines with identical speed under machine availability and eligibility constraints. The objective is to minimize the maximum lateness (Lmax). Machine availability constraint means that each machine is available for processing only in some intervals of time called availability intervals. In practice, such availability intervals may appear in machine preventive maintenance [1], [2], multitasking computer systems [3], [4], [5], [6] and the case where machines may be partially occupied by a specific set of jobs that must be scheduled at particular time intervals due to various inevitable reasons. Eligibility constraint means that each machine may have the different capability to process jobs; in other words, each job should be processed on specific machines. The application of such an eligibility constraint setting can be found in the semiconductor manufacturing environment [7], [8] and service industry [9].

Minimization of the maximum lateness, Lmax, is the objective considered in this paper. Lmax is important for the cases where the impact of a single large delay is more harmful to a manufacturer than many small delays. In addition, Lmax may be used as an aid for solving other problems. For example, minimizing the maximum flow time can be obtained by setting the due date equal to the release time and then minimizing Lmax.

Below we review the related research on the parallel machine scheduling with machine availability or eligibility constraint. Ullman [10] was the first to address the scheduling problem with the availability constraint from the complexity point of view. Ullman proved the strong NP-completeness of the problem while considering the precedence constraint on an arbitrary number of identical machines. Schmidt [2] investigated the preemptive scheduling problem where each job has a deadline and each machine has different availability. He presented an O(nmlogn) time algorithm for finding a feasible preemptive schedule whenever one exists. Blazewicz et al. [3] analyzed several preemptive scheduling problems where availability is assumed as staircase pattern and jobs have precedence constraints. Later, Blazewicz et al. [4] considered multiprocessor scheduling problems that are subject to the machine availability constraint and proposed polynomial time algorithms to solve the problems. For non-preemption scheduling, parallel machine problems with availability constraints have been studied in Hwang and Chang [11], Kellerer [12], Lee [13], Lee [14], Lee et al. [15], etc., in which heuristic algorithms were proposed and the worse-case ratios were provided. Comprehensive surveys in this area we refer the readers to Liu and Sanlaville [16] and Schmidt [6].

For the problem with the eligibility constraint, Pinedo [17] stated that the least flexible job first (LFJ) rule, which first selects the job that can be processed on the smallest number of machines, is optimal for the problem Pm|pj=1,Mj|Cmax when the Mj sets are nested, where Mj denotes the set of machines that can process job j. Centeno and Armacost [7] investigated problem Pm|rj,Mj|Lmax when due dates are equal to release date plus a constant. The authors first developed a heuristic algorithm based on LFJ rule for the problem, and then they developed few heuristic algorithms for problem Pm|rj,Mj|Cmax and showed that the LPT rule is superior to the LFJ rule in performance when the machine eligibility sets are not nested [8]. Lin and Li [18] considered the problems where jobs have unit process times. They developed polynomial time algorithms for the problem Pm|pj=1,Mj|Cmax, Qm|pj=1, Mj|Cmax, and the problem Pm|pj=1,Mj|Cmax where the subsets of Mj is convex. Hwang et al. [9] investigated the problem where all jobs and machines are specified by service grades, namely a job can only be processed on a machine with a service grade less than the job's service grade. They proposed LPT-based algorithm with a tight bound.

We observe that the foregoing published works did not consider machine availability and eligibility constraints simultaneously. Almost all of those studies dealing with the machine scheduling with availability and eligibility constraints developed LPT- or LFJ-based heuristic algorithm for solving the problems. In this paper, we consider a more practical situation where each machine has arbitrary availability intervals within the planning horizon and each job requires a specific subset of machines for processing.

Based on the well-known three-field, α|β|γ, classification scheme suggested in Pinedo [17], our scheduling issue can be classified as the problem Pm,NCwin|prmp, rj, Mj|Lmax. NCwin in the α field denotes the arbitrary pattern of availability intervals of machines, and Mj in the field β denotes a specific subset of machines that can process job j. In this scheduling problem, we extend the network flow technique used in [3], [19], [20], [21] to formulate the problem into a series of maximum network problems. We propose a two-phase binary search algorithm to determine the optimal Lmax by solving these maximum network problems or to determine the infeasibility of the problem.

The rest of the paper is organized as follows. In Section 2 we define the notations used. In Section 3 we present the two-phase binary search algorithm for the problem. We address the conclusions in Section 4.

Section snippets

Notations

Index set
Athe direct arc set of G
Gthe network
ithe machine index
jthe job index
kthe availability interval index
lthe distinct time epoch index
N1the job node set of G
N2the event interval node set of G
Sthe source of G
Tthe sink of G
wthe critical value index
Parameters
[bik,fik]the kth availability interval on machine i, where bik and fik are the start
time and the end time of this availability interval, respectively, k=1,,N(i)
elthe distinct time epoch, l=1,,|E|
Ethe event epoch set
mthe total number of

Network flow model for problem Pm,NCwin|prmp,rj,Mj|Lmax

Given a planning horizon, there are n preemptive jobs to be scheduled on m machines. Each job j, j=1,,n, associates with a release time rj, an processing time pj and a due date dj. Job j can only be processed on a specific subset of machines, Mj. Each machine i,i=1,,m, is only available for processing within each of N(i) specified availability intervals [bik,fik],k=1,,N(i) throughout the planning horizon [0,maxi=1,,m{fiN(i)}]. Without loss of generality, we assume that Mjφ for j=1,,n; bik+

Conclusion

In this paper, we have analyzed the scheduling problem for machines with identical speed under machine availability and eligibility constraints, while minimizing maximum lateness. The problem arises in the environment where a job requires to be processed on specific machines and machines are not continuously available. This research is the first to consider machine availability and eligibility constraints simultaneously in parallel machine scheduling. We formulate the base problem with respect

Acknowledgements

The authors express their gratitude to an anonymous referee for his/her detailed comments and valuable suggestions to improve the exposition of this paper. This research was supported in part by the National Science Council under Grant NSC 90-2218-E-008-002.

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