Minimising Waiting Time Variance in the Single Machine Problem
TL;DR: A heuristic method is proposed for solving the problem where n is large; this method requires very little computing and was found to produce very good results for a sample of problems of varying size.
Abstract: The paper considers the problem of n given jobs to be processed on a single machine where it is desirable to minimise the variance of job waiting times. A theorem is presented to the effect that the optimal sequence must be V-shaped i.e., the jobs must be arranged in descending order of processing times if they are placed before the shortest job, but in ascending order of processing times if placed after it, and an algorithm for determining the optimal solution is given. A heuristic method is proposed for solving the problem where n is large; this method requires very little computing and was found to produce very good results for a sample of problems of varying size. The concept of the “efficient set” is examined and heuristic methods for generating this set are given.
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TL;DR: A framework to show how results have been generalized starting with a basic model that contains symmetric penalties, one machine and a common due date is provided and such features as parallel machines, complex penalty functions and distinct due dates are added.
Abstract: We consider the problem of scheduling n jobs to minimize the total earliness and tardiness penalty. We review the literature on this topic, providing a framework to show how results have been generalized starting with a basic model that contains symmetric penalties, one machine and a common due date. To this base we add such features as parallel machines, complex penalty functions and distinct due dates. We also consolidate many of the existing results by proving general forms of two key properties of earliness/tardiness models.
979 citations
TL;DR: A critical review of a particular segment of scheduling research in which the due to date assignment decision is of primary interest is presented, observing that while the static single- machine problem with constant or common due dates has been well researched, very little or no work has been done on the dynamic multi-machine problem with sophisticated due date assignment methods.
498 citations
01 Jan 1998
TL;DR: This work focuses on deterministic machine scheduling for which it is assumed that all data that define a problem instance are known with certainty.
Abstract: The scheduling of computer and manufacturing systems has been the subject of extensive research for over forty years. In addition to computers and manufacturing, scheduling theory can be applied to many areas including agriculture, hospitals and transport. The main focus is on the efficient allocation of one or more resources to activities over time. Adopting manufacturing terminology, a job consists of one or more activities, and a machine is a resource that can perform at most one activity at a time. We concentrate on deterministic machine scheduling for which it is assumed that all data that define a problem instance are known with certainty.
336 citations
TL;DR: In this article, a single-machine scheduling problem in which penalities occur when a job is completed early or late is considered, where the objective is to minimize the total penalty subject to restrictive assumptions on the due dates and penalty functions.
Abstract: This paper considers a single-machine scheduling problem in which penalities occur when a job is completed early or late. The objective is to minimize the total penalty subject to restrictive assumptions on the due dates and penalty functions for jobs. A procedure is presented for finding an optimal schedule.
331 citations
TL;DR: It is proved that the recognition version of this problem is NP-complete in the ordinary sense, and a computationally efficient dynamic programming algorithm is presented that is polynomially solvable.
Abstract: This paper and its companion Part II concern the scheduling of jobs with cost penalties for both early and late completion. In Part I, we consider the problem of minimizing the weighted sum of earliness and tardiness of jobs scheduled on a single processor around a common due date, d. We assume that d is not early enough to constrain the scheduling decision. The weight of a job does not depend on whether the job is early or late, but weights may vary between jobs. We prove that the recognition version of this problem is NP-complete in the ordinary sense. We describe optimality conditions, and present a computationally efficient dynamic programming algorithm. When the weights are bounded by a polynomial function of the number of jobs, a fully polynomial approximation scheme is given. We also describe four special cases for which the problem is polynomially solvable. Part II provides similar results for the unweighted version of this problem, where d is arbitrary.
307 citations
References
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01 Jan 1967
TL;DR: Reading theory of scheduling as one of the reading material to finish quickly to increase the knowledge and happiness in your lonely time.
Abstract: Feel lonely? What about reading books? Book is one of the greatest friends to accompany while in your lonely time. When you have no friends and activities somewhere and sometimes, reading book can be a great choice. This is not only for spending the time, it will increase the knowledge. Of course the b=benefits to take will relate to what kind of book that you are reading. And now, we will concern you to try reading theory of scheduling as one of the reading material to finish quickly.
2,356 citations
TL;DR: While the two mean performance measures attain their minimums at the same job sequence, it is shown that the sequence that minimizes the variance of flow-time is antithetical to the sequenceThat minimizesThe variance of waiting-time, and the minimum values of the two variance measures are equal.
Abstract: The variance of flow-time and variance of waiting-time performance measures are analyzed for the single machine sequencing problem. These measures are compared and contrasted to the performance measures of mean flow-time and mean waiting-time. In particular, while the two mean performance measures attain their minimums at the same job sequence, it is shown that the sequence that minimizes the variance of flow-time is antithetical to the sequence that minimizes the variance of waiting-time. However, the minimum values of the two variance measures are equal. Relationships are also derived for the special problems where either all the job processing-times are equal or all the job weights are equal.
169 citations
TL;DR: In this article, a finite number of jobs are scheduled on a single machine and the objective is to sequence the jobs so that the time-in-system (or equivalently, the completion time) variance is minimized.
Abstract: There are a finite number of jobs to be scheduled on a single machine. All jobs are available from the start and the objective is to sequence the jobs so that the time-in-system (or equivalently, the completion time) variance is minimized. A number of necessary conditions for an optimal sequencing (which for small jobsets turn out to be sufficient) are presented.
113 citations
"Minimising Waiting Time Variance in..." refers background in this paper
...For every sequence there is a "dual" that yields the same variance (see [2] and [3])....
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...Merten and Muller [2] argue that the variance performance measure is important in file organisation problems where uniform response time to users is often desirable; they show that the sequence that minimises the variance of waiting times is antithetical to the sequence that minimises the variance of flow times; Schrage [3] proves that the sequence for the former must have the longest job processed last and goes on to analyse the problem up to n = 5....
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...The optimal sequence has the longest job last (this is Schrage's Theorem 3 [3])....
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