A branch and bound algorithm to minimize completion time variance on a single processor

doi:10.1016/S0305-0548(02)00062-X

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A branch and bound algorithm to minimize completion time variance on a single processor

Journal Article•DOI•

A branch and bound algorithm to minimize completion time variance on a single processor

G. Viswanathkumar¹, Geetha Srinivasan¹•Institutions (1)

Indian Institute of Technology Madras¹

01 Jul 2003-Computers & Operations Research (Elsevier Science Ltd.)-Vol. 30, Iss: 8, pp 1135-1150

TL;DR: A branch and bound algorithm is presented and the same algorithm is extended to generate epsilon optimal solutions for large sized problems (i.e., number of jobs > 30) and results of simulated annealing are compared.

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About: This article is published in Computers & Operations Research.The article was published on 2003-07-01. It has received 37 citations till now. The article focuses on the topics: Branch and bound & Branch and cut.

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Journal Article•DOI•

A single-machine deteriorating job scheduling problem with a non-regular criterion

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Chin-Chia Wu¹, Wen-Chiung Lee¹, Yau-Ren Shiau¹•Institutions (1)

Feng Chia University¹

20 Nov 2011-Asia-Pacific Journal of Operational Research

TL;DR: This study considers the variance of job completion time minimization problem on a single machine and shows that an optimal schedule is V-shaped with respect to the job deteriorating rates.

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Abstract: In scheduling environments with deterioration, a job processed later consumes more time than that same job when processed earlier. The deteriorating job scheduling problems have been widely studied in the last two decades. However, no result of the completion time variance has been reported. In this study, we consider the variance of job completion time minimization problem on a single machine. It is assumed that the job processing time is a simple linear function of its starting time. We show that an optimal schedule is V-shaped with respect to the job deteriorating rates. A heuristic algorithm utilized the V-shaped property is then proposed, and a computational experiment shows that the proposed heuristic algorithm is quite accurate.

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12 citations

Journal Article•DOI•

Completion time variance minimization in single machine and multi-machine systems

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B. Srirangacharyulu¹, Geetha Srinivasan¹•Institutions (1)

Indian Institute of Technology Madras¹

01 Jan 2010-Computers & Operations Research

TL;DR: A new heuristic and a method based on genetic algorithms to solve the problem of minimizing the completion time variance of n jobs on a single machine with deterministic processing times are proposed.

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12 citations

Cites background or methods from "A branch and bound algorithm to min..."

...We also present a lower bound used in Viswanathkumar and Srinivasan [12] as Result 3, which we use to compare the goodness of the proposed heuristic method....
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...Viswanathkumar and Srinivasan [12] developed a branch and bound algorithm for the CTV problem and tested it with problems up to 30 jobs....
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...For problem sizes >40, the lower bound of Viswanathkumar and Srinivasan [12] is used for evaluating the goodness of the SMGA....
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...To explain Result 3, we define the jobs as in Viswanathkumar and Srinivasan [12]....
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Journal Article•DOI•

Scheduling in static jobshops for minimizing mean flowtime subject to minimum total deviation of job completion times

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Viswanath Kumar Ganesan, Appa Iyer Sivakumar¹•Institutions (1)

Nanyang Technological University¹

01 Oct 2006-International Journal of Production Economics

TL;DR: In this paper, a simulated annealing algorithm using the concept of backward scheduling is proposed for minimizing mean flowtime subject to optimal TADC, and the proposed algorithm is evaluated using 82 jobshop scheduling problems taken from literature, of size varying from 6-jobs 6-machines to 50-jobs 20-maches.

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11 citations

Journal Article•DOI•

A Lower Bound for Weighted Completion Time Variance

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Rabia Nessah¹, Chengbin Chu²•Institutions (2)

Lille Catholic University¹, École Centrale Paris²

16 Dec 2010-European Journal of Operational Research

TL;DR: A heuristic and a lower bound based on job splitting and the Viswanathkumar and Srinivasan procedure are proposed and the test on more than 2000 instances shows that this lower bound is very tight and the heuristic yields solutions very close to optimal ones.

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11 citations

Cites background or methods from "A branch and bound algorithm to min..."

...The following lemmas are due to Viswanathkumar and Srinivasan [24] and Kubiak et al....
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...Let LBVS be the lower bound obtained by the Viswanathkumar–Srinivasan procedure applied to the problem P2....
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...The lower bound is based on job splitting and the procedure of Viswanathkumar and Srinivasan....
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...We have also proved a lower bound based on splitting of jobs and the Viswanathkumar and Srinivasan procedure, which were proved to be very efficient to prune the search tree....
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...The following lemmas are due to Viswanathkumar and Srinivasan [24] and Kubiak et al. [15]....
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Journal Article•DOI•

Minimizing job completion time variance for service stability on identical parallel machines

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Xueping Li¹, Yuerong Chen¹, Yang Sun²•Institutions (2)

University of Tennessee¹, California State University, Sacramento²

01 May 2010-Computers & Industrial Engineering

TL;DR: This paper addresses a job scheduling problem on multiple identical parallel machines so as to minimize job completion time variance (CTV) and proves several dominant properties about the optimal solution to the problem.

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11 citations

Cites methods from "A branch and bound algorithm to min..."

...…and Kumar (1993), a simulated annealing method in Mittenthal, Raghavachari, and Rana (1993), a tabu search method in Al-Turki, Fediki, and Andijani (2001), a branch and bound method in Viswanathkumar and Srinivasan (2003), and an ant-colony optimization algorithm in Gajpal and Rajendran (2006)....
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Journal Article•DOI•

Minimizing Variation of Flow Time in Single Machine Systems

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John J. Kanet¹•Institutions (1)

University of Georgia¹

01 Dec 1981-Management Science

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.

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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.

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199 citations

"A branch and bound algorithm to min..." refers background in this paper

...For further references and applications in various contexts, see Kanet [5], Vani and Ragavachari [6], Bagchi et al....
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...For further references and applications in various contexts, see Kanet [5], Vani and Ragavachari [6], Bagchi et al. [7], Gupta et al. [8], Mittenthal et al. [9], Gupta et al. [10], Ventura and Weng [11], Manna and Prasad [12]....
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Journal Article•DOI•

Minimising Waiting Time Variance in the Single Machine Problem

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Samuel Eilon¹, I. G. Chowdhury¹•Institutions (1)

Imperial College London¹

01 Feb 1977-Management Science

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.

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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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187 citations

Journal Article•DOI•

Variance Minimization in Single Machine Sequencing Problems

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Alan G. Merten, Mervin E. Muller

01 May 1972-Management Science

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.

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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.

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169 citations

"A branch and bound algorithm to min..." refers background or methods in this paper

...It can be observed from expressions (3) and (14) that (C[1] − Pa1)(2) + (C[2] − Pa1)(2) + (C[n−1] − Pa1)(2) + (C[n] − Pa1)(2)¿V; (15) where V represents the sum of squared deviations of four completion times from their average Pa and hence it is smaller than the squared deviations about Pa1....
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...where, C[1]; C[2]; C[3]; : : : ; C[n] are the completion times of jobs scheduled in positions 1; 2; 3; : : : ; n in a given partial sequence....
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...Let C∗ [1]; C∗ [2]; C∗ [3]; : : : ; C∗ [n] be the completion times of jobs in positions 1; 2; 3; : : : ; n in the optimal sequence (S∗), and MCT ∗ and CTV ∗ be the mean completion time and completion time variance of the jobs, respectively, in the optimal sequence....
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...The sum of squares of deviations of the six completion times from the average Pa1 is given by V1 = (C[1] − Pa1)(2) + (C[2] − Pa1)(2) + (C[3] − Pa1)(2) + (C[n−2] − Pa1)(2) + (C[n−1] − Pa1)(2) + (C[n] − Pa1)(2); (14)...
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...The objective of variance minimization was introduced by Merten and Muller [1] in 7le organization problems....
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Journal Article•DOI•

On the minimization of completion time variance with a bicriteria extension

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Prabuddha De, Jay B. Ghosh, Charles E. Wells

01 Nov 1992-Operations Research

TL;DR: It is shown that the dynamic programming solution is easy to extend to a bicriteria version of the problem in which it is desired to simultaneously minimize the mean completion time and a fully polynomial approximation scheme is proposed.

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Abstract: We discuss a single-machine scheduling problem where the objective is to minimize the variance of job completion times. To date, the problem has not been solved in polynomial time. This paper presents a dynamic programming algorithm that is pseudopolynomial in complexity. We also propose a fully polynomial approximation scheme and derive a lower bound that is useful in its implementation. Furthermore, we show that the dynamic programming solution is easy to extend to a bicriteria version of the problem in which it is desired to simultaneously minimize the mean completion time.

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122 citations

"A branch and bound algorithm to min..." refers background or methods in this paper

...[13] in terms of the number of solutions evaluated for various problem sizes....
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...[13] presented a dynamic programming algorithm that is pseudo-polynomial in complexity to minimize the variance of job completion times with bi-criteria extension and derived a lower bound that is useful in its implementation....
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...[13], which has a pseudo-polynomial complexity....
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...[13] using a simple proof and tested the bound on randomly generated problems....
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Journal Article•DOI•

Minimizing the Time-in-System Variance for a Finite Jobset

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Linus Schrage

01 Jan 1975-Management Science

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.

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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.

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113 citations