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

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.

Abstract: In 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

45 citations

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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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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18 citations

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

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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177 citations

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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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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108 citations