Completion Time Variance Research Articles

Probabilistic analysis of an asymptotically optimal solution for the completion time variance problem

Scheduling a set of n jobs on a single machine so as to minimize the completion time variance is a well-known NP-hard problem. In this paper, we propose a sequence, which can be constructed in O(n log n) time, as a solution for the problem. Our primary concern is to establish the asymptotical optimality of the sequence within the framework of probabilistic analysis. Our main result is that, when the processing times are randomly and independently drawn from the same uniform distribution, the sequence is asymptotically optimal in the sense that its relative error converges to zero in probability as n increases. Other theoretical results are also derived, including: (i) When the processing times follow a symmetric structure, the problem has 2⌊(n−1)/2⌋ optimal sequences, which include our proposed sequence and other heuristic sequences suggested in the literature; and (ii) when these 2⌊(n−1)/2⌋ sequences are used as approximate solutions for a general problem, our proposed sequence yields the best approximation (in an average sense) while another sequence, which is commonly believed to be a good approximation in the literature, is interestingly the worst. © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 373–398, 1999

Probabilistic analysis of an asymptotically optimal solution for the completion time variance problem

Scheduling a set of n jobs on a single machine so as to minimize the completion time variance is a well-known NP-hard problem. In this paper, we propose a sequence, which can be constructed in O(n log n) time, as a solution for the problem. Our primary concern is to establish the asymptotical optimality of the sequence within the framework of probabilistic analysis. Our main result is that, when the processing times are randomly and independently drawn from the same uniform distribution, the sequence is asymptotically optimal in the sense that its relative error converges to zero in probability as n increases. Other theoretical results are also derived, including: (i) When the processing times follow a symmetric structure, the problem has 2⌊(n−1)/2⌋ optimal sequences, which include our proposed sequence and other heuristic sequences suggested in the literature; and (ii) when these 2⌊(n−1)/2⌋ sequences are used as approximate solutions for a general problem, our proposed sequence yields the best approximation (in an average sense) while another sequence, which is commonly believed to be a good approximation in the literature, is interestingly the worst. © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 373–398, 1999

An Approximation Scheme for Minimizing Agreeably Weighted Variance on a Single Machine

We consider the problem of minimizing the weighted variance of job completion times on a single machine with job-dependent, agreeable weights. In 1995, Cai derived a fully polynomial time approximation scheme for the special case where the weights of the jobs are polynomially bounded in the number n of jobs. In this article, we completely settle the approximability status of this scheduling problem. We construct a fully polynomial time approximation scheme for the general case, without putting any restrictions on the weights of the jobs.

Bounds for the position of the smallest job in completion time variance minimization

The problem of scheduling jobs on a single machine so as to minimize the variance of job completion times is considered. We develop bounds on the position of the smallest job in an optimal sequence, which is vital due to the fact that an optimal sequence is necessarily a V-shaped sequence. Results of numerical experimentation indicate that the bounds are very effective if the job processing times are homogeneous. Using these bounds, we develop a new heuristic method to solve the problem. Based on numerical investigation, it is shown that performance of this heuristic compares favorably with existing but older heuristics.

Algorithms to minimize completion time variance in a two machine flowshop

This paper presents algorithms to minimize the variance of completion time (CTV) when n jobs have to be processed on two machines. All the jobs visit the machines in the same order or technological sequence. These algorithms are applicable in manufacturing systems where both the facilities could be within the factory or situations where one facility represents the bottleneck machine within the shop floor while the other represents the subcontractor facility. The problem is also applicable in file handling systems where two individuals have to process files in a fixed sequence. The paper presents heuristic algorithms that identify the sequence of jobs that minimizes completion time variance. The solutions are compared with a lower bound and are found to give results in the range of 5–10% deviation from the lower bound. In addition, the simulated annealing algorithm developed in this paper gives the optimum solution for all the problems tested up to 11 jobs.

Multi-machine scheduling with variance minimization

This paper addresses the problem of scheduling n jobs on m identical parallel machines so as to minimize the completion time variance. Properties of optimal solutions are derived first. Then, complexity results are obtained, which show that the problem is NP-complete in the strong sense when m is arbitrary, and NP-complete in the ordinary sense when m is fixed. Two algorithms are proposed. The first algorithm can generate an optimal solution in time O( n 2mP m(P − P m) m−1 [m m(m − 1)!] 2 ) , where P is the sum of all the processing times and P m is the sum of the first m largest processing times. The second algorithm can find a near-optimal solution in time O( nP m(P − P m) m−1 m m(m − 1)! ) . It is further shown that the relative error of the near-optimal solution is guaranteed to approach zero at a rate O( n −2) as n increases.

Pseudopolynomial algorithms for CTV minimization in single machine scheduling

The problem of scheduling jobs on a single machine so as to minimize completion time variance (CTV) is considered. In this article, we have derived two dominance criteria and used them in the development of a new pseudopolynomial algorithm. This algorithm is an implicit enumeration scheme based on a binary branching strategy, with larger jobs fixed at early stages, and is superior to those of De, Ghosh and Wells [1] and Kubiak [2] in terms of computational complexity. It is observed that this algorithm is very good when the job processing times are quite heterogeneous, while the algorithm of De, Ghosh and Wells [1] is excellent for homogeneous processing times. By making use of these contrasting merits, another pseudopolynomial algorithm is then proposed. Results of extensive numerical investigation on the performances of the algorithms are also reported.

A Note on Lower Bound for Completition Time Variance in Single Machine Scheduling

This paper is concerned with the single machine scheduling problem where the objective is to minimize the variance of the job completion times. A lower bound on minimum completion time variance given by De et al. (1992) is corrected using a simple proof. Computational results on the lower bound are also provided.

Minimization of expected variance of completion times on single machine for stochastic jobs

This article deals with the problem of scheduling jobs with random processing times on single machine in order to minimize the expected variance of job completion times. Sufficient conditions for the existence of V-shaped optimal sequences are derived separately for general and ordered job processing times. It is shown that when coefficient of variation of random processing times are bounded by a certain value, an optimal sequence is V-shaped. © 1997 John Wiley & Sons, Inc.

Sequencing jobs with random processing times to minimize weighted completion time variance

We examine the problem of sequencing a set of jobs on a single machine, where each job has a random processing time and is associated with a known, job-dependent weight. The objective is to minimize the expectation of the weighted variance of job completion times. We establish the NP-completeness of this problem, and further show that the problem under some compatible conditions is NP-complete in the ordinary sense. We introduce the concept of a W-shaped solution for the problem and find that an optimal W-shaped sequence exists under the compatible conditions. We propose an exact algorithm, based on this W-shaped property, which can generate an optimal solution in pseudopolynomial time.

A New Model For Completion Time Variance Problem With Job-Dependent Weights And Controllable Processing Times

We consider a new model for scheduling n jobs on a single machine. Each job requires a processing time pi, which may be compressed at an additional cost, and is associated with a weight Wi. The objective is to minimize the weighted variance of completion times and the total cost incurred from the compression of processing times. This is the WCTV problem with controllable processing times. The WCTV problem with equal weight and with noncontrollable processing times has been shown to be NP-hard. In this paper, we describe optimality properties, and we propose a heuristic algorithm to solve the problem. Computational results are reported that demonstrate the effectiveness of the algorithm.

A Class of Single Machine Central Tendency-Dispersion Bicriteria Problems

In this paper, a number of non-regular bicriteria penalty functions of completion times on a single machine are studied. In each function, one criterion is a measure of central tendency (CT) of completion times, for example, mean or median, and the other is a measure of completion time dispersion (DSP). Five different DSP measures are considered in the paper. We discuss the characterizations of optimal sequences for some new functions and summarize previous results when appropriate for the pure bicriteria problem as well as for the convex combination of CT and DSP. Further, in the context of minimizing a dispersion measure, a comparison is made between the two dispersion measures of variance of completion times and total absolute deviations of completion times (TADC).

A Class of Single Machine Central Tendency-Dispersion Bicriteria Problems

In this paper, a number of non-regular bicriteria penalty functions of completion times on a single machine are studied. In each function, one criterion is a measure of central tendency (CT) of completion times, for example, mean or median, and the other is a measure of completion time dispersion (DSP). Five different DSP measures are considered in the paper. We discuss the characterizations of optimal sequences for some new functions and summarize previous results when appropriate for the pure bicriteria problem as well as for the convex combination of CT and DSP. Further, in the context of minimizing a dispersion measure, a comparison is made between the two dispersion measures of variance of completion times and total absolute deviations of completion times (TADC).

Heuristics for Multimachine Scheduling Problems with Earliness and Tardiness Costs

We consider multimachine scheduling problems with earliness and tardiness costs. We first, analyze problems in which the cost of a job is given by a general nondecreasing, convex function F of the absolute deviation of its completion time from a (common) unrestrictive due-date, and the objective is to minimize the sum of the costs incurred for all N jobs. (A special case to which considerable attention is given is the completion time variance problem.) We derive an easily computable lower bound for the minimum cost value and a simple “Alternating Schedule” heuristic, both of which are computable in O(N log N) time. Under mild technical conditions with respect to F, we show that the worst case optimality (accuracy) gap of the heuristic (lower bound) is bounded by a constant as well as by a simple function of a single measure of the dispersion among the processing times. We also show that the heuristic (bound) is asymptotically optimal (accurate) and characterize the convergence rate as O(N−2) under very general conditions with respect to the function F. In addition, we report on a numerical study showing that the average gap is less than 1% even for problems with 30 jobs, and that it falls below 0.1% for problems with 90 or more jobs. This study also establishes that the empirical gap is almost perfectly proportional with N−2, as verified by a regression analysis. Finally, we generalize the heuristic to settings with a possibly restrictive due date and general asymmetric, and possibly nonconvex, earliness and tardiness cost functions and demonstrate its excellent performance via a second numerical study.

Variance Minimization – Relationship between Completion-Time Variance and Waiting-Time Variance

AbstractThe completion-time variance (CTV) and the waiting-time variance (WTV) are two performance measures which are commonly used in optimization of single-machine scheduling systems. This paper shows that when the number of jobs is large the two measures are nearly equivalent in a probabilistic environment.

A tight lower bound for the completion time variance problem

We consider the completion time variance problem. Our main result is a tight lower bound for the mean completion time of an optimal sequence. This result can be applied to reduce the time required to solve the problem.

V-shape property for job sequences that minimize the expected completion time variance

We consider the problem of sequencing n jobs with random processing time on a single machine so as to minimize the expected variance of job completion times. Our main result is a new sufficient condition for an optimal sequence to be V-shaped in terms of the mean processing times when n ⩾ 3. We show that this condition is satisfied by a wide variety of problem instances, including those in which the processing times follow different patterns of distributions. This result relaxes a condition proposed before.

Note: ?Simultaneous optimization of efficiency and performance balance measures in single-machine scheduling problems?

n independent jobs are to be scheduled nonpreemptively on a single machine so as to minimize some performance measure. Federgruen and Mosheiov [2] show that a large class of such scheduling problems can be optimized by solving either a single instance or a finite sequence of instances of the so-called SQC problem, in which all the jobs have a fixed or controllable common due date and the sum of general quasiconvex functions of the job completion times is to be minimized. In this note we point out that this is not always true. In particular, we show that the algorithm proposed in [2] does not always find a global optimal schedule to the problem of minimizing the weighted sum of the mean and variance of job completion times.

Minimizing Single-Machine Completion Time Variance

In this article the problem of minimizing the completion time variance in n-job, single-machine scheduling is considered. The release times for all jobs are assumed to be zero. A new quadratic integer programming formulation is introduced. A Lagrangian relaxation (LR) procedure is developed to find a lower bound (LB) to the optimal objective value. When the number of jobs is between 100 and 500, our computational study shows that the lower bounds obtained by the LR procedure are very close to the best known objective values. A new heuristic algorithm is also described. The first phase of the heuristic algorithm is a construction procedure whose purpose is to identify a good initial sequence. The second phase is an improvement procedure based on pairwise interchanges. The new heuristic algorithm provides improved solutions compared to the best known heuristic.

New results on the completion time variance minimization

A single machine scheduling problem where the objective is to minimize the variance of job completion times is considered. The model is applicable to many environments where it is desirable to provide jobs, customers or computer files, with approximately the same service. It is shown that the problem can be formulated as a problem of maximizing a zero-one quadratic function which is a submodular function with a special cost structure. It immediately follows from the cost structure that value of the function for a sequence that minimizes total absolute deviation about a common unrestrictive due date is at most 100% smaller than the one for an optimal sequence for the completion time variance problem. The Monge property holds for the costs. Other simple properties of the function are also presented. A pair of dynamic programs for maximizing the function is proposed. The worst case time complexity of the best of the pair is O( n 2spt), where spt is the mean flow time of an SPT-schedule for all jobs except the longest one.