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Journal ArticleDOI

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

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.

AbstractIn this paper we discuss a single machine scheduling problem with the objective of minimizing the variance of job completion times. The CTV problem has been proved to be NP hard (Oper. Res. Lett. 14 (1993) 49) and no polynomial time algorithm exists to find an optimal solution for CTV minimization on single machine. Hence enumerative techniques and heuristics are used to get optimal and near optimal solutions, respectively. We present a branch and bound algorithm and extend the same algorithm to generate epsilon optimal solutions for large sized problems (i.e., number of jobs > 30). The algorithm has been computationally tested, with randomly generated problems involving up to 100 jobs, using a personal computer (PC) with a 64 MB RAM capacity. The computational time required for generating optimal solutions are in few seconds for problems with jobs between 25 and 30. The performance of the branch and bound algorithm is compared with the pseudo-polynomial algorithm (Oper. Res. 40 (1992) 1148) for small sized problems. For problems with greater number of jobs, the epsilon optimal solutions obtained using branch and bound algorithm are compared with results of simulated annealing (Single machine scheduling with some non-regular objectives, M.S. Thesis, IIT Madras, 1997), tabu search (Proceedings of Operations Management Conference, IIT Madras, 2000) and heuristic proposed by Manna and Prasad (Eur. J. Oper. Res. 114 (1999) 411).

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Citations
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Journal ArticleDOI
TL;DR: A consistent pattern in the plot of WTV over mean of all possible sequences for a set of jobs is discovered, which can be used to evaluate the sacrifice of mean waiting time while pursuing WTV minimization.
Abstract: Minimizing Waiting Time Variance (WTV) is a job scheduling problem where we schedule a batch of n jobs, for servicing on a single resource, in such a way that the variance of their waiting times is minimized. Minimizing WTV is a well known scheduling problem, important in providing Quality of Service (QoS) in many industries. Minimizing the variance of job waiting times on computer networks can lead to stable and predictable network performance. Since the WTV minimization problem is NP-hard, we develop two heuristic job scheduling methods, called Balanced Spiral and Verified Spiral, which incorporate certain proven properties of optimal job sequences for this problem. We test and compare our methods with four other job scheduling methods on both small and large size problem instances. Performance results show that Verified Spiral gives the best performance for the scheduling methods and problems tested in this study. Balanced Spiral produces comparable results, but at less computational cost. During our investigation we discovered a consistent pattern in the plot of WTV over mean of all possible sequences for a set of jobs, which can be used to evaluate the sacrifice of mean waiting time while pursuing WTV minimization.

45 citations

Journal ArticleDOI
TL;DR: Carrier assignment models used in winner determination auctions for transportation procurement are extended to include shipper non-price objectives and carrier transit point costs, and solutions are developed for the new models.
Abstract: This paper extends carrier assignment models used in winner determination auctions for transportation procurement to include shipper non-price objectives and carrier transit point costs. The models are unlike traditional carrier assignment models which incorporate only carrier lane bids, and different from combinatorial auction models which focus on packets of lanes without considering transit point costs. We develop solutions, including metaheuristics, for the new models and through computational experimentation show that the algorithms work well and can be easily implemented.

33 citations


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

  • ...A branch-and-bound (B&B) complete search can be used to examine all possible assignments of the lanes to carriers for small test sizes, where the time performance of the algorithm depends largely on the bounding function used (Viswanathkumar and Srinivasan, 2002)....

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Journal ArticleDOI
TL;DR: The proposed spreadsheet-based genetic algorithm approach is found to perform better than the previous GA, simulated annealing (SA) and Tabu search (TS) approaches for variance minimization in that it produces an optimal solution for all the test problems.
Abstract: This paper addresses an n-job, single-machine scheduling problem with an objective to minimize flow-time variance, a non-regular performance measure. A spreadsheet-based genetic algorithm (GA) approach is presented to solve the problem. A domain-independent general-purpose GA is used, which is an add-in to the spreadsheet software. The paper demonstrates an adaptation of the proprietary GA software to the problem of variance minimization. To test the performance of the proposed approach, test problems are taken from already published literature. The proposed approach is found to perform better than the previous GA, simulated annealing (SA) and Tabu search (TS) approaches for variance minimization in that it produces an optimal solution for all the test problems. Empirical analysis was carried out to study the effect of GA parameters, namely, the crossover rate, mutation rate, and population size.

20 citations


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

  • ...Viswanathkumar and Srinivasan [ 12 ] develop a branch-andbound algorithm to solve the CTV class of problems....

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Journal ArticleDOI
TL;DR: A lower bound on CTV is developed for the static jobshop problem and a backward scheduling approach is proposed using the observations on the development of lower bound for hierarchical minimization of CTV and makespan.
Abstract: This paper addresses the problem of minimizing makespan for a given set of n jobs to be processed on each of m machines in a static jobshop, subject to the minimum completion time variance (CTV). A lower bound on CTV is developed for the static jobshop problem. A backward scheduling approach is proposed using the observations on the development of lower bound for hierarchical minimization of CTV and makespan. A lower bound on makespan subject to minimum CTV is also presented for this problem. Finally, we present two simulated annealing heuristic approaches using the concepts of forward and backward scheduling. Their performances are compared against each other through the use of the lower bounds established in this work. The simulated annealing heuristic based on backward scheduling is shown to perform well by evaluating the developed heuristics on 82 jobshop problems taken from literature.

18 citations

Journal ArticleDOI
TL;DR: Two job scheduling algorithms, Weighted Verified Spiral and Weighted Simplified Spiral, are developed for the WWTV problems, and numerical testing shows that WVS and WSS significantly outperform existing WWTV algorithms.
Abstract: This study considers the job scheduling problem of minimizing the weighted waiting time variance (WWTV) of jobs. It is an extension of WTV minimization problems in which we schedule a batch of n jobs, for servicing on a single resource, in such a way that the variance of their waiting times is minimized. WWTV minimization finds its applications for job scheduling in manufacturing systems with earliness and tardiness (E/T) penalties, in computer and networks systems for the stabilized QoS, and in other fields where it is desirable to minimize WWTV of jobs with different weights for priorities. We formulate a WWTV problem as an integer programming problem, prove the V-shape property for agreeably weighted WWTV problems and the nondelay property for general WWTV problems, and discover the strong V-Shape tendency of the optimal job sequences for this problem. Two job scheduling algorithms, Weighted Verified Spiral (WVS) and Weighted Simplified Spiral (WSS), are developed for the WWTV problems. Numerical testing shows that WVS and WSS significantly outperform existing WWTV algorithms.

14 citations


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

  • ...A branch and bound algorithm to minimize CTV is given in (Viswanathkumar & Srinivasan, 2003) and a tabu search-based solution is developed in (Al-Turki, 2001)....

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References
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Journal ArticleDOI
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.

187 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 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.

177 citations

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.

159 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 ArticleDOI
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.

116 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 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.

108 citations