A branch and bound algorithm to minimize completion time variance on a single processor
Introduction
In this study we consider the problem of scheduling n given independent jobs, each having a known processing time pk (k=1 to n) to be processed one at a time in sequence on a single machine to minimize Completion Time Variance (CTV) given bywhere Ci is the completion time of job i, MCT the mean completion time=(1/n)∑i=1nCi.
Completion Time Variance minimization is especially important to any environment where all jobs require the same treatment i.e., each job should spend same amount of time in the system as every other job for service. The objective of variance minimization was introduced by Merten and Muller [1] in file organization problems. In computing systems with large data files, often the response time to a user's request is strongly dependent on the time required to access or retrieve the data files referenced by the user. Minimizing the variance of response time by minimizing the variance of access time often provides a uniform response time to the users. This problem is very much relevant when a manufacturing/service system places emphasis on Just-in-Time philosophy.
Merten and Muller [1] proved an interesting type of duality theorem. They showed that the schedule that minimizes the variance of waiting times is antithetical to the schedule that minimizes the completion time variance, and the completion time variance of any schedule is equal to the waiting time variance of its antithetic schedule. Schrage [2] proved that the sequence for minimizing the variance of the completion times must have the largest job processed first, and proposed that in the optimal sequence second largest job should be placed last and the third largest job be placed in the second position which was later verified by Hall and Kubiak [3]. Eilon and Chowdhury [4] introduced the concept of V-shaped sequences, (i.e., the sequence in which all jobs prior to the shortest processing time job are in descending order and all jobs after the shortest processing time job are in ascending order) and provided heuristic methods for determining the optimal schedule for waiting time variance minimization, when the number of jobs to be scheduled is relatively small. They had shown that V-shapedness is a necessary condition for the optimality of a sequence i.e., one may restrict the search to only 2n−1 V-shaped sequences. 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].
De et al. [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. Prasad et al. [14] corrected the above lower bound of De et al. [13] using a simple proof and tested the bound on randomly generated problems. Federgruen and Mosheiov [15] addressed the multi-machine scheduling problem with earliness and tardiness costs, and derived an easily computable lower bound for the minimum cost value. They extended its applicability by considering the problems of minimizing the sum of squared deviations from a common due-date and the completion time variance minimization as special cases.
In this paper, we define a lower bound on similar lines with Federgruen and Mosheiov [15] and extend the developed lower bound expression to the computation of bounds for a given partial sequence with more than 3 jobs fixed. A branch and bound algorithm has been developed using the above bounding scheme. We further extend the branch and bound algorithm to generate epsilon optimal solutions and compare the performance of the developed branch and bound algorithm with the modified dynamic programming algorithm of De et al. [13] in terms of the number of solutions evaluated for various problem sizes.
Section snippets
Lower bound for CTV
Let the processing times of n jobs be p1,p2,…,pn where p1⩽p2⩽⋯⩽pn. Placing the two jobs, n and n−2 in the first and second position, and job n−1 in the last position [2], we have the following partial sequence S=(n,n−2,…,n−1). The completion time of the last job is given byand the completion time of the last but one job is given by M−pn−1.
The average of the four completion times isThe sum of squares of deviations of the four completion times from the average is
Branch and bound algorithm
The following notations are defined for the branch and bound algorithm:
- S0
initial partial sequence with three jobs fixed
- L
lower bound for the sequence S0
- Si+1,Si+2
partial sequences obtained by fixing the next unscheduled job to the left and right available positions from a given partial sequence
- Li+1,Li+2
lower bound values for the partial sequences Si+1 and Si+2
- nfix
number of jobs fixed in the partial sequence
- forward
number of jobs fixed from the first position
- back
number of jobs fixed from the last position
- tforward
number of jobs temporarily
Computational study
The above branch and bound algorithm has been computationally tested by choosing problems of sizes from six to thirty jobs. For each problem size, 30 problems have been randomly generated. The processing times of the individual jobs have been drawn at random from a uniform distribution U(1,100). In order to determine the accuracy of the epsilon optimal solutions generated using the above branch and bound algorithm for large problems, 16 different problem sets of sizes (n=25,26,27,…,100) have
Summary
In this paper, we present a branch and bound algorithm to minimize single machine Completion Time Variance problem. A CTV lower bound is defined on similar lines with Federgruen and Mosheiov [15] and it has been extended to the development of lower bound expression for the computation of bounds for a given partial sequence with more than 3 jobs fixed. The performance of the developed branch and bound algorithm is compared with the modified dynamic programming algorithm of De et al. [13] in
Acknowledgements
The authors thank the referees for the constructive suggestions and comments to improve the earlier version of the paper.
G. Viswanathkumar is a Ph.D. Scholar in the Indian Institute of Technology Madras working in the area of production scheduling. He completed his Masters degree in Industrial Engineering from PSG College of Technology, Coimbatore before joining IIT Madras.
References (17)
- et al.
Proof of a conjecture of Schrage about the completion time variance problem
Operations Research Letters
(1991) - et al.
A hybrid simulated annealing approach for single machine scheduling problems with non-regular penalty functions
Computers & Operations Research
(1993) - et al.
Minimizing the flow time variance in single machine systems using genetic algorithms
European Journal of Operational Research
(1993) - et al.
Bounds for the position of the smallest job in completion time variance minimization
European Journal of Operational Research
(1999) - et al.
Variance minimization in single machine sequencing problems
Management Science
(1972) Minimizing the Time-in-System Variance for a Finite Jobset
Management Science
(1975)- et al.
Minimizing the waiting time variance in the single machine problem
Management Science
(1977) Minimizing variance of flow time in single machine systems
Management Science
(1981)
Cited by (39)
Time-flexible min completion time variance in a single machine by quadratic programming
2024, European Journal of Operational ResearchA multi-machine scheduling solution for homogeneous processing: Asymptotic approximation and applications
2022, International Journal of Production EconomicsCitation Excerpt :Driven by the interest in just-in-time production systems and project management (Li et al., 2006; Van de Vonder et al., 2008; Balouka and Cohen 2019; Romero-Silva and Hernández-López 2020; Yu and Han 2021), the completion time earliness, lateness and variance have been used as non-regular performance measures to target homogeneity in scheduling problems. This has given rise to important theoretical results in both single and multiple machines systems by the last half a century (Merten and Muller 1972; Eilon and Chowdhury 1977; Cai and Cheng 1998; Viswanathkumar and Srinivasan 2003; Li et al., 2010; Srirangacharyulu and Srinivasan 2010; Nasini and Nessah 2021). In the case of multiple machines (namely, where n jobs have to be processed on m parallel machines), the min-sum and the min-max are standard aggregation rules, laying in the two extreme points of the multi-criteria solution preferences (T'kindt & Billaut 2006). (
An almost exact solution to the min completion time variance in a single machine
2021, European Journal of Operational ResearchA branch and price algorithm for single-machine completion time variance
2019, Computers and Operations ResearchCitation Excerpt :possible ones, which minimizes the completion time variance. This problem finds applications in Just-in-Time services (Manna and Prasad, 1999; Nessah and Chu, 2010; Rajkanth et al., 2017; Srirangacharyulu, 2017; Viswanathkumar and Srinivasan, 2003) and in manufacturing and service industries where customer classes are not differentiable and have similar job properties (Mehta et al., 2012), i.e., with the requirement of approximately the same level of service. In these applications, the manufacturer or service provider must provide a fair treatment in terms of order fulfillments.
Completion time variance minimisation on two identical parallel processors
2017, Computers and Operations ResearchAn exact algorithm to minimize mean squared deviation of job completion times about a common due date
2013, European Journal of Operational Research
G. Viswanathkumar is a Ph.D. Scholar in the Indian Institute of Technology Madras working in the area of production scheduling. He completed his Masters degree in Industrial Engineering from PSG College of Technology, Coimbatore before joining IIT Madras.
G. Srinivasan is an Associate Professor of Production and Operations Management at the Indian Institute of Technology Madras. He completed his Ph.D. at IIT Madras. He has published numerous articles in refereed international journals. His research interests include cellular manufacturing and production scheduling.