Completion Time Variance Research Articles

A solvable case of the variance minimization problem

An algorithm is derived, which solves the completion time variance (CTV) problem with equal processing times in O( n log n) time. This result indicates that the special case formulated by Merten and Muller [1] is well solvable.

Completion time variance minimization on a single machine is difficult

The complexity of the completion time variance minimization (CTV) problem on a single machine has been open since the paper by Merten and Muller (Management Science 18 (1972)). In this paper we prove that the CTV problem is NP-hard.

Selecting Product Development Projects: Pioneering versus Incremental Innovation Strategies

In this paper, we investigate project selection choices of duopolists facing two alternatives: undertaking a “pioneering” type project (Type A) aimed to develop a highly innovative product, or an “incremental innovation” type project (Type B) aimed to develop a less innovative product such as the modification of an existing product. A key objective of our research is to examine how firm characteristics such as their differential efficiencies in completing projects, differences in the degree of substitutability between Type A and B products, and first mover advantages affect product development strategies. We develop a game-theoretic model to obtain insights into the project selection problem taking into account competitive reactions to a firm's choice of project development strategies and technical uncertainties associated with project completion times. We report model findings on recommended project selection strategies for efficient and disadvantaged firms. Further, we examine how a firm's choice of a Type A project is affected by an increase in the variance of the project completion time of a Type A project relative to that of a Type B project, while the ratio of their mean completion times is held constant.

On the Minimization of Completion Time Variance with a Bicriteria Extension

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.

Heuristic Estimation of the Efficient Frontier for a Bi‐Criteria Scheduling Problem

ABSTRACTWe examine a single‐machine scheduling problem where the objective is to minimize the mean and the variance of the job completion times simultaneously. We seek to identify the efficient frontier which is obtained by parametrically solving a weighted combination of the two criteria. The identification of the true efficient frontier for this problem is notoriously difficult. To estimate the frontier, we propose a heuristic procedure which is quite general and can be applied to other bi‐criteria problems as well. It involves repeated applications of a relatively new technique called beam search in an adaptive manner. To evaluate the proposed procedure, we introduce two measures of performance and conduct a computational study. The results of the study indicate that the procedure is highly effective.

Proof of a conjecture of Schrage about the completion time variance problem

The problem of minimizing the completion time variance of n jobs on a single machine has been studied by several authors. We prove the correctness of a 1975 conjecture due to L. Schrage about the position of the third longest job in an optimal schedule.

Minimizing Mean Squared Deviation of Completion Times About a Common Due Date

This paper addresses a nonpreemptive single machine scheduling problem where all jobs have a common due date and have zero ready time. The scheduling objective is to minimize mean squared deviation (MSD) of job completion times about the due date. This nonregular measure of performance is appropriate when earliness and tardiness are both penalized, and when large deviations of completion time from the due date are undesirable. A special case of the MSD problem, referred to as the unconstrained MSD problem, is shown to be equivalent to the completion time variance problem (CTV) studied by Merten and Muller (Merten, A. G., M. E. Muller. 1972. Variance minimization in single machine sequencing problems. Management Sci. 18(September) 518–528.) and Schrage (Schrage, L. 1975. Minimizing the time-in-system variance for a finite jobset. Management Sci. 21(May) 540–543.). Strong results for this latter problem are combined with several new propositions to develop a reasonably efficient procedure for solving the unconstrained MSD problem. This enables us to improve the existing procedures for the CTV problem. We also propose a branching procedure for the constrained MSD problem and present computational results.

Deterministic and Random Single Machine Sequencing with Variance Minimization

We discuss the problem of scheduling several jobs on a single machine so as to minimize the variance of the jobs' completion times. By exploiting the well-known pooled variance formula, we obtain a general formula for the change in variance due to the interchange of two specified jobs whose positions have not been fixed in the sequence. This result permits us to obtain explicit optimal sequences for problems with 6 or 7 jobs, and to show that Schrage's conjecture on positioning the job with the third largest processing time is true for problems with up to 18 jobs. We then develop a heuristic procedure based on a variance interchange formula and compare it with other known heuristic methods. We also consider a problem never before examined, namely, the case when processing times are not known deterministically but are random variables. We obtain some general results, under certain assumptions on the parameters of the distributions of processing times, and extend to the random situation many results applicable to the deterministic case. Finally, we give some results for the exponential distribution of processing times.

Single‐ and multiple‐processor models for minimizing completion time variance

AbstractThis article concerns the scheduling of n jobs around a common due date, so as to minimize the average total earliness plus total lateness of the jobs. Optimality conditions for the problem are developed, based on its equivalence to an easy scheduling problem. It seems that this problem inherently has a huge number of optimal solutions and an algorithm is developed to find many of them. The model is extended to allow for the availability of multiple parallel processors and an efficient algorithm is developed for that problem. In this more general case also, the algorithm permits great flexibility in finding an optimal schedule.

Stochastic pert networks: OP diagrams, critical paths and the project completion time

Fisher and Goldstein [5–7] have recently derived closed form expressions for the expectation, variance and distribution of the project completion time of stochastic PERT networks when the durations of the tasks in the network can be written as mutually independent, probability mixtures of general-gamma [8] random variables (a more general result is described in [9]). In this article, it is shown that their methods can also be used to obtain the probability that a particular path in the PERT network is critical [6]. One possible criticism of their approach is that computation of both the probability that a particular path is critical and the summary characteristics of the reaction time distribution require the translation of a PERT network into an OP (Order-of-ftocessing) diagram. Currently, this translation can only be done manually, a very laborious process for even moderately complex networks. In this article, an algorithm is developed which can be used to generate the OP diagram for a given PERT network. The algorithm has been implemented in FORTRAN. Once the OP diagram is constructed, computation of both the probability that a particular path is critical and the expectation, variance and distribution of the project completion time requires only simple matrix operations when the durations of the tasks in the network are mutually independent probability mixtures of general-gamma random variables. More complex calculations are required when other distributions of the task durations are involved.

The Completion Time of PERT Networks

This paper proposes an approximation for the completion time mean and variance of PERT networks in which the durations of individual activities are Normally and independently distributed. The proposed approximation involves the manipulation of distribution parameters only and hence can be computed manually, even for large networks.

Minimizing the Time-in-System Variance for a Finite Jobset

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.