Invited Review
A concise survey of scheduling with time-dependent processing times

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Abstract

We consider a class of machine scheduling problems in which the processing time of a task is dependent on its starting time in a schedule. On reviewing the literature on this topic, we provide a framework to illustrate how models for this class of problems have been generalized from the classical scheduling theory. A complexity boundary is presented for each model and related existing results are consolidated. We also introduce some enumerative solution algorithms and heuristics and analyze their performance. Finally, we suggest a few interesting areas for future research.

Introduction

Machine scheduling problems with time-dependent processing times have received increasing attention in recent years. For many years, most scheduling research has focused on problems with deterministic parameters. As mentioned in Rinnooy Kan (1976), the traditional restrictive assumptions may correspond to a somewhat over-simplified picture of reality, though they can take great advantage of computational convenience. In real-life applications, many systems exhibit dynamic behaviors characterized by a set of dynamic parameters. This fact is commonly recognized in control theory, systems engineering and many other areas. But scheduling problems with dynamic parameters have been studied only in a few papers. It should, however, be noted that a considerable body of literature has existed for stochastic scheduling that deals with scheduling problems in an environment of uncertainty, see, Mohring and Rademacher (1985), Righter (1994) and Pinedo (1995).

Based on some scheduling problems with dynamic parameters considered by Gupta et al. (1987) or some earlier Russian papers (e.g. Tanaev et al., 1994), Gupta and Gupta (1988) introduced an interesting scheduling model in which the processing time of a task is a polynomial function of its starting time. From a modeling perspective, however, the makespan scheduling problem with quadratic time-dependent processing times is already very intricate. For this reason, most subsequent research along this line has concentrated on problems with linear or piecewise linear time-dependent processing times.

This model reflects some real-life situations in which the expected processing time of a task increases/decreases linearly or piecewise linearly with its starting time. Examples can be found in financial management, steel production, resource allocation and national defense, where any delay in tackling a task may result in an increasing/decreasing effort (time, cost, etc.) to accomplish the task. The reader is referred to Kunnathur and Gupta (1990), Mosheiov, 1994, Mosheiov, 1996a and Sundararaghavan and Kunnathur (1994) for a list of applications. Moreover, it seems that in other cases, for example, fire fighting, learning effect and maintenance scheduling, a linear or piecewise linear function is a close approximation of the actual nonlinear phenomenon.

Research on time-dependent problems has spawned a new area in the scheduling field. It has uncovered many new properties neglected in the classical scheduling theory and led to efficient methodological approaches to algorithm design and NP-complete reduction. For example, techniques based on reductions from a multiplicative type NP-complete problem, such as Subset Product, are crucial to the NP-completeness proofs for many time-dependent scheduling problems. Regarding the development of polynomial time algorithms, a very interesting phenomenon is related to the existence of algorithms with time complexities of O(n5) or O(n6logn), which is not so common in the deterministic scheduling literature. Thus, research on these problems is significant in both practical and theoretical senses.

Alidaee and Womer (1999) presented a review on scheduling problems with time-dependent processing times. Our study aims not only at surveying recent developments in this line of research but also at investigating several unsolved problems. Based upon state-of-the-art status of research on scheduling problems with time-dependent processing times, we discuss the relationships of different models and explicate how they are generated from a basic linear model. A complexity boundary is presented for each model and existing and new results are consolidated. For the intractable problems, we also introduce some enumerative solution algorithms and heuristics and analyze their performance. Finally, we give some insights into scheduling problems of this type, which reveal several potential future directions of research for this exciting field of study.

In Section 2, we introduce a notation-and-model system for the scheduling problem with time-dependent processing times. In Section 3, we present a set of complexity results for each model. In Section 4, we illustrate a series of polynomial and pseudo-polynomial algorithms. In Section 5, we discuss some enumerative and heuristic solution algorithms in the literature. Some concluding remarks and suggestions for future research are given in Section 6.

Section snippets

Notation and models

Since most of the time-dependent scheduling problems are a natural generalization of their classical counterpart, we adopt the notation, definitions and assumptions prevalent in classical scheduling theory, see the survey of Graham et al. (1976).

Research on time-dependent problems has mainly dealt with the single machine model, with only a few exceptions dealing with the parallel machine and flow shop situations. The objective has been confined to the minimization of a handful of traditional

NP-complete problems

The NP-complete results for the class of classical scheduling problems have been well researched and surveyed in several papers and books, for example, Rinnooy Kan (1976), Graham et al. (1976) and Lawler et al. (1993). However, the time-dependent scheduling problems present quite different boundaries for the computational complexities of the problems. In this section, we introduce the new scheduling results, starting from the simple models and gradually progressing to their generalizations.

In

Polynomially and pseudo-polynomially solvable problems

In the light of the complexity boundaries of the time-dependent problems, based on the results reported in the last section, we attempt to develop polynomial or pseudo-polynomial solution algorithms for those tractable and semi-tractable problems in this section. Now we introduce the solvable cases, beginning with the simplest models and gradually progressing to their generalizations.

First, we consider the model pi=bis. Mosheiov (1994) presented several solvable cases as follows: The makespan C

Enumerative and heuristic algorithms

There are a number of enumerative and heuristic solution algorithms for some intractable problems reported in the literature. Mosheiov, 1996a, Mosheiov, 1996b, Mosheiov, 1998 and Chen (1996) presented heuristic algorithms for the model pi=bisi. Gupta et al. (1987), Gupta and Gupta (1988), Mosheiov (1991) and Hsu and Lin (2002) investigated heuristic rules, dynamic programming procedures and branch and bound algorithms for the model pi=ai+bisi. Kunnathur and Gupta (1990) considered heuristic

Conclusions

In this paper, we consider a class of machine scheduling problems in which the processing time of a task is dependent on its starting time in a schedule. On reviewing the literature on this topic, we provide a framework to illustrate how the models have been generalized from the classical scheduling theory. A complexity boundary is presented for each model and many existing results are consolidated. We also introduce some enumerative and heuristic solution algorithms and analyze their

Acknowledgements

This research was supported in part by The Hong Kong Polytechnic University under grant number G-S818. We are grateful for three anonymous referees for their constructive comments on an earlier version of this paper.

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