Machine scheduling with earliness, tardiness and non-execution penalties

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Abstract

The study of scheduling problems with earliness–tardiness (E/T) penalties is motivated by the just-in-time (JIT) philosophy, which supports the notion that earliness, as well as tardiness, should be discouraged. In this work, we consider several scheduling problems. We begin by generalizing a known polynomial time algorithm that calculates an optimal schedule for a given sequence of tasks, on a single machine, assuming that the tasks have distinct E/T penalty weights, distinct processing times and distinct due dates. We then present new results to problems, where tasks have common processing times. We also introduce a new concept in E/T scheduling problems, where we allow the non-execution of tasks and consequently, are penalized for each non-executed task. The notion of task's non-execution, coincides with the JIT philosophy in that every violation or a breach of an agreement, should be penalized. We develop polynomial time algorithms for special cases in E/T scheduling problems with non-execution penalties.

Introduction

The study of scheduling problems with earliness and tardiness (E/T) penalties is relatively recent. For many years, the research of scheduling problems focused on minimizing measures such as mean flow-time, maximum tardiness, and makespan, all non-decreasing in the completion times of tasks. For these measures, delaying execution of tasks results in a higher cost. However, the current emphasis in industry on the just-in-time (JIT) philosophy, which supports the notion that earliness, as well as tardiness, should be discouraged, has motivated the study of scheduling problems in which tasks are preferred to be ready just at their respective due dates, and both early and tardy products are penalized.

In this paper, we consider several E/T scheduling problems. We are given a set T̃={T1,…,TN} of tasks. Task Ti has an integer processing time pi>0 and a target starting time ai⩾0 (or equivalently, a due date di, where dipi). There are m parallel machines, {M1,…,Mm}. Our notation follows that of Garey et al. [1].

A solution for , is an assignment of each task Ti to a machine Mj and a schedule corresponding to that assignment, which determines a starting time si for Ti on Mj. The scheduling of starting times, must satisfy that no two tasks assigned to the same machine overlap in their execution time, and that the tasks are to be scheduled non-preemptively; once started, a task Ti must be executed to its completion, pi time units later.

A sequence defines the order in which, tasks are to be processed, whereas a schedule is a sequence with starting times calculated for each task. We assume nonnegative earliness penalty weight αi and nonnegative tardiness penalty weight βi, associated with task Ti. Ti incurs the earliness penalty αi(aisi) if si<ai and it incurs the tardiness penalty βi(siai) if si>ai. We define ei=max{0,aisi}≡(aisi)+ and ti=max{0,siai}≡(siai)+ and thus, the penalty incurred by Ti is αiei+βiti. The overall cost of a solution, which we wish to minimize, is the sum of the individual penalties, i.e., i=1Nieiiti). We refer to this cost function, as the Total Weighted Earliness and Tardiness problem (TWET—see [2]). In general, we denote the cost of solution by cost(T̃).

In Section 4, we introduce a new type of penalty to the E/T scheduling problems. Assume we are allowed to not-execute one or more of the tasks. Denote by γi the penalty incurred if Ti is not executed (processed). Thus, a modified TWET problem, is to minimize ∑i=1N[(1−xi)(αiei+βiti)+xiγi] where xi=0 if Ti is executed and xi=1, otherwise. The notion of task's non-execution, fits the JIT philosophy in that every violation or a breach of an agreement, should be penalized.

Section snippets

Literature review

The research can be classified into two main categories, which reflect the due date specifications:

  • 1.

    Problems with common due date {di=d}, which we denote CDD.

  • 2.

    Problems with distinct due dates {di}, which we denote DDD.

The problems can be further categorized with respect to other criteria such as, number of machines and cost functions.

Generalizations of Algorithm GTW

Garey et al. [1], consider the 1|di,pi|MAD problem under a given sequence of tasks and present an O(NlogN) time scheduling algorithm (Algorithm GTW). They also prove that if the tasks are not pre-ordered but, have a common length of processing time p⩾0, it is optimal to sequence the tasks such that aiai+1 for 1⩽i<N. The solution generated by the algorithm to this optimal sequence, is a minimum cost schedule in which sisi+1 for 1⩽i<N. We start this section by presenting Algorithm GTW after

Non-execution penalty

In this section we remove the requirement that all tasks must be executed. Denote by γi the penalty incurred if Ti is not executed. For example, a modified TWET problem is to minimize i=1N[(1−xi)(αieiiti)+xiγi] where xi=0 if Ti is executed and xi=1, otherwise. We will consider different cost functions and present polynomial time algorithms.

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    Present address: 59 Ben-Gurion St., Kfar Saba 44204, Israel.

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