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Journal ArticleDOI

Minimizing Variation of Flow Time in Single Machine Systems

01 Dec 1981-Management Science (INFORMS)-Vol. 27, Iss: 12, pp 1453-1459
TL;DR: This paper addresses the problem of n jobs to be scheduled on a single machine in such a way that flow time variation is minimized and a heuristic method for scheduling is proposed.
Abstract: This paper addresses the problem of n jobs to be scheduled on a single machine in such a way that flow time variation is minimized. When the measure of variation is total absolute difference of completion times TADC the problem is shown to be quite simple. Sufficient conditions are shown for minimal TADC and a simple method for generating an optimal solution is provided. When the measure of variation is variance of flow time the problem is much more difficult. For this case a heuristic method for scheduling is proposed. The heuristic is simple and provides solutions which compare favorably with others found in the literature.
Citations
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Journal ArticleDOI
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

Journal ArticleDOI
TL;DR: In this paper, the authors consider a single machine scheduling problem in which the objective is to minimize the mean absolute deviation of job completion times about a common due date and present an algorithm for determining multiple optimal schedules under restrictive assumptions about the due date, and an implicit enumeration procedure when the assumptions do not hold.
Abstract: We consider a single machine scheduling problem in which the objective is to minimize the mean absolute deviation of job completion times about a common due date. We present an algorithm for determining multiple optimal schedules under restrictive assumptions about the due date, and an implicit enumeration procedure when the assumptions do not hold. We also establish the similarity of this problem to the two parallel machines mean flow time problem.

165 citations

Journal ArticleDOI
TL;DR: This paper considers bicriterion scheduling problems involving job completion times and job waiting times in a nonpreemptive single machine environment and presents an efficient algorithm for minimizing the variation of waiting time.
Abstract: In this paper, we consider bicriterion scheduling problems involving job completion times and job waiting times in a nonpreemptive single machine environment. As an example, we seek to determine a schedule that yields both a low mean and a low variation of completion time. Such a schedule is attractive when the system has the dual objectives of minimizing in-process inventory and providing different jobs the same quality of service. Similarly, we seek a schedule that performs well on both the mean and variation of waiting time. As the measure of variation, we use the sum of absolute pairwise differences. We present an efficient algorithm for minimizing the variation of waiting time. Next, we assume that the total scheduling cost is a function of the mean and variation of completion time alone, and is linear. We give an efficient algorithm for minimizing this total cost. Finally, we consider parametric analysis of this cost function and present simple procedures for generating all or some of the optimal sc...

154 citations

Journal ArticleDOI
TL;DR: In this paper it is proved that the CTV problem is NP-hard.

133 citations


Cites methods from "Minimizing Variation of Flow Time i..."

  • ...Kanet [8] showed the model as being applicable to any service and manufacturing setting where it is desirable to provide customers or jobs with approximately the same service....

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Journal ArticleDOI
TL;DR: It is shown that the standard single-machine scheduling problem with p-s-d setup times and any of the above objective functions can be solved in O(nlog n) time (where n is the number of jobs) by a sorting procedure.

131 citations


Cites background from "Minimizing Variation of Flow Time i..."

  • ...Kanet (1981), provides a detailed discussion on the number of optimal solutions for the 1kTADC problem....

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  • ...As was first shown by Kanet (1981), TADC ¼ Xn i¼1 Xn j¼i jCj Cij ¼ Xn r¼1 ðr 1Þðn r þ 1Þðs½r þ p½r Þ ¼ Xn r¼1 " ðr 1Þðn r þ 1Þ þ c Xn j¼rþ1 ðj 1Þðn jþ 1Þ # p½r : ð4Þ As in the case of Eqs....

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References
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Journal ArticleDOI
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.

187 citations


"Minimizing Variation of Flow Time i..." refers background or methods in this paper

  • ...The sample problems used were the seven problems from [1] for which data were available....

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  • ...In addition, a new heuristic is described for minimizing flow time variance and its performance is compared to the heuristics of Eilon and Chowdhury [1]....

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  • ...The procedure compares favorably to the best alternatives given by Eilon and Chowdhury [1]....

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  • ...Eilon and Chowdhury [1] strengthen the results of Schrage by showing that optimal flow time variance schedules must be V-shaped....

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  • ...1 of Eilon and Chowdhury [1]....

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Journal ArticleDOI
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


"Minimizing Variation of Flow Time i..." refers background in this paper

  • ...As Merton and Muller [3] indicate, in computer systems with large data sets the response time to a user's request is often strongly dependent on the file-access time....

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  • ...In the earliest paper on this topic, Merton and Muller [3] analyze the relation between flow time variance and waiting time variance....

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  • ..., [1], [3], and [4]) and no theoretical breakthrough is claimed here....

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Journal ArticleDOI
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


"Minimizing Variation of Flow Time i..." refers background or methods in this paper

  • ...The solutions indicated by an asterisk (*) have been shown by Schrage [4] to be optimal....

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  • ...In [4], Schrage demonstrates that for n = 6 one simply cannot specify the optimal solution based on some order of the job processing times....

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  • ...It is known from Schrage [4] that property I is necessary for minimum variance of completion times....

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  • ..., [1], [3], and [4]) and no theoretical breakthrough is claimed here....

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  • ...Method E&C is based on the conjecture by Schrage [4] that for n jobs, there exists an optimal sequence of the form S= {1,3,4,???,2}....

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