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Journal Article•DOI•

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

G. Viswanathkumar1, Geetha Srinivasan1•
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.
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.
Citations
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Journal Article•DOI•
Discrete particle swarm optimisation algorithms for minimising the completion-time variance of jobs in flowshops

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K. Rameshkumar1, Chandrasekharan Rajendran2, K. M. Mohanasundaram3•
Amrita Vishwa Vidyapeetham1, Indian Institute of Technology Madras2, Coimbatore Institute of Technology3
11 Mar 2011-International Journal of Industrial and Systems Engineering
TL;DR: In this article, two particle swarm optimisation algorithms (PSOAs) are proposed and analyzed for the problem of scheduling in the permutation flowshop scheduling problem with the objective of minimizing the completion-time variance of jobs.
Abstract: In this paper, the problem of scheduling in the permutation flowshop scheduling problem is considered with the objective of minimising the completion-time variance of jobs (CTV). Two particle swarm optimisation algorithms (PSOAs) are proposed and analysed. The first PSOA is inspired from the solution construction procedures that are used in ant colony optimisation algorithms. The second algorithm is a newly developed one. The proposed algorithms are applied to a set of benchmark flowshop scheduling problems, and performances of the algorithms are evaluated by comparing the obtained results with the results published in the literature. The performance analysis demonstrates the effectiveness of the proposed algorithms in solving the permutation flowshop sequencing problem with the CTV objective.

10 citations

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

  • ..., 2001), branch and bound techniques (Gowrishankar et al., 2001; Viswanathkumar and Srinivasan, 2003) and apart from the attempts by Eilon and Chowdhury (1977), Kanet (1981), Vani and Raghavachari (1987), Raghavachari (1988) and Manna and Prasad (1997, 1999)....

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Journal Article•DOI•
Heuristics to minimize the completion time variance of jobs on a single machine and on identical parallel machines

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Raju Rajkanth1, Chandrasekharan Rajendran1, Hans Ziegler2•
Indian Institute of Technology Madras1, University of Passau2
01 Feb 2017-The International Journal of Advanced Manufacturing Technology
TL;DR: In this paper, both versions of parallel-machine scheduling problem (restricted and unrestricted) are considered and a good upper bound is obtained using a genetic algorithm, to evaluate the performance of the proposed heuristics for the parallel- machine scheduling problem.
Abstract: This paper addresses the problem of scheduling n jobs on a single machine and on m identical parallel machines to minimize the completion time variance of jobs. This problem of scheduling jobs on parallel machines is motivated by a case study in an automobile ancillary unit. First, a heuristic to solve the single-machine scheduling problem is proposed. The parallel-machine scheduling problem is solved in two phases: job-allocation phase and job-sequencing phase. Two heuristics are proposed in the job-allocation phase, whereas in the job-scheduling phase, the single-machine scheduling approach is used. In this paper, both versions of parallel-machine scheduling problem (restricted and unrestricted) are considered. A good upper bound is obtained using a genetic algorithm, to evaluate the performance of the proposed heuristics for the parallel-machine scheduling problem. An extensive computation evaluation of the proposed heuristics is presented for both single-machine scheduling problem and the parallel-machine scheduling problem (especially considering the case study), along with the comparison of performances with the existing heuristics in the literature.

8 citations

Journal Article•DOI•
An exact algorithm to minimize mean squared deviation of job completion times about a common due date

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B. Srirangacharyulu1, Geetha Srinivasan2•
Indian Institute of Management Tiruchirappalli1, Indian Institute of Technology Madras2
16 Dec 2013-European Journal of Operational Research
TL;DR: This work considers a deterministic n-job, single machine scheduling problem with the objective of minimizing the Mean Squared Deviation of job completion times about a common due date, and proposes a dominance rule and a lower bound on MSD.

8 citations

Journal Article•DOI•
An almost exact solution to the min completion time variance in a single machine

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Stefano Nasini1, Stefano Nasini2, Rabia Nessah2, Rabia Nessah1•
Centre national de la recherche scientifique1, Lille Catholic University2
16 Oct 2021-European Journal of Operational Research
TL;DR: A necessary and sufficient condition is proved for a novel bounding condition for a characterization of an optimal sequence for the characterization of five scheduling positions in the optimal sequence and a new approach is proposed to derive the highest lower bound for the minimal completion time variance.

7 citations

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

  • ...Section 3 uses these novel positions to build a lower bound that improves the current benchmark from Viswanathkumar and Srinivasan (2003). In Section 4, a comprehensive computational analysis is undertaken....

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Journal Article•DOI•
Optimal sequencing of a set of positive numbers with the variance of the sequence's partial sums maximized

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Li Wei, Wangdong Qi, Dingxing Chen, Peng Liu, En Yuan 
01 Aug 2013-Optimization Letters
TL;DR: The optimal sequence is shown to have a nice structure and the symmetric problem which aims at minimizing the variance of the same partial sums is proved to be NP-complete in the literature.
Abstract: We consider the problem of sequencing a set of positive numbers. We try to find the optimal sequence to maximize the variance of its partial sums. The optimal sequence is shown to have a nice structure. It is interesting to note that the symmetric problem which aims at minimizing the variance of the same partial sums is proved to be NP-complete in the literature.

7 citations

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

  • ...nequality (cn+2−im −cim)(ck −cn+2−k)>0 when k ∈I and (n+2−2k)>0 when k ≤u, we finally obtain f C(p) − f C(0) >0 Thus the proof is complete. ⊔ Example 1. Let us regard a sequence C(0) =[1,6,2,3,4,8,7,5], then C(p) =C(2) =[1,6,7,8,4,3,2,5]is its sum-‘n+2’ transform. Note that the transform consists of (3,7)-interchangeand (4,6)-interchange. So we get the set I = i1,i2,...,ip |2 ≤i1 <i2 ··· <ip ...

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References
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Journal Article•DOI•
Minimizing Variation of Flow Time in Single Machine Systems

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John J. Kanet1•
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.
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.

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•
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.
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

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.
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

"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.
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.

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.
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

Related Papers (5)
Variance Minimization in Single Machine Sequencing Problems

[...]

01 May 1972-Management Science
Alan G. Merten, Mervin E. Muller
Completion time variance minimization on a single machine is difficult

[...]

01 Aug 1993-Operations Research Letters
Wieslaw Kubiak
Minimising Waiting Time Variance in the Single Machine Problem

[...]

01 Feb 1977-Management Science
Samuel Eilon, I. G. Chowdhury
Minimizing Variation of Flow Time in Single Machine Systems

[...]

01 Dec 1981-Management Science
John J. Kanet
Minimizing the Time-in-System Variance for a Finite Jobset

[...]

01 Jan 1975-Management Science
Linus Schrage