A linear programming approach to geometric programs

A cutting plane method, based on a geometric inequality, is described as a means of solving geometric programs. While the method is applied to the primal geometric program, it is shown to retain the geometric programming duality relationships. Several methods of generating the cutting planes are discussed and illustrated on some example problems.     

Coordinated replenishments of items under time-varying demand: Dynamic programming formulation

We consider a group (or family) of items having deterministic, but time-varying, demand patterns. The group is defined by a setup-cost structure that makes coordination attractive (a major setup cost for each group replenishment regardless of how many of the items are involved). The problem is to determine the timing and sizes of the replenishments of all of the items so as to satisfy the demand out to a given horizon in a cost-minimizing fashion. A dynamic programming formulation is illustrated for the case of a two-item family. It is demonstrated that the dynamic programming approach is computationally reasonable, in an operational sense, only for small family sizes. For large families heuristic solution methods appear necessary.     

A complete importance ranking for components of binary coherent systems,with extensions to multi-state systems

Means of measuring and ranking a system's components relative to their importance to the system reliability have been developed by a number of authors. This paper investigates a new ranking that is based upon minimal cuts and compares it with existing definitions. The new ranking is shown to be easily calculated from readily obtainable information and to be most useful for systems composed of highly reliable components. The paper also discusses extensions of importance measures and rankings to systems in which both the system and its components may be in any of a finite number of states. Many of the results about importance measures and rankings for binary systems are shown to extend to the more sophisticated multi-state systems. Also, the multi-state importance measures and rankings are shown to be decomposable into a number of sub-measures and rankings.     

A new bivariate negative binomial distribution

A new bivariate negative binomial distribution is derived by convoluting an existing bivariate geometric distribution; the probability function has six parameters and admits of positive or negative correlations and linear or nonlinear regressions. Given are the moments to order two and, for special cases, the regression function and a recursive formula for the probabilities. Purely numerical procedures are utilized in obtaining maximum likelihood estimates of the parameters. A data set with a nonlinear empirical regression function and another with negative sample correlation coefficient are discussed.     

The simple multiserver loss system with time-varying intensities

For the loss system with time-varying intensities and a finite number of identical servers we derive formulas for the expected number of refused customers in a given time interval and for the expected number of servers busy at a given time instant. © 1994 John Wiley & Sons, Inc.     

A new method to solve steady state queueing equations

There are a great number of queueing systems, including the M^X/M^Y/c, the Gl^X/M/c and the discrete Gl/G/1 queue in which the state probabilities are determined by repeated queue equations. This paper gives a simple, efficient and numerically stable algorithm to caiculate the state probabilities and measure of performance for such systems. The method avoids both complex arithmetric and matrix manipulations.     

A note on a computational model for a data/voice communication queueing system

Certain types of communication nodes can be viewed as multichannel queueing systems with two types of arrival streams. Data arrivals are characterized by high arrival and service rates and have the ability to queue if all service channels are busy. Voice arrivals have small arrival and service rates and do not have the ability to wait when the channels are full. Computational procedures are presented for obtaining the invariant probabilities associated with the queueing model.     

A two-machine repair model with variable repair rate

A two-unit cold standby production system with one repairman is considered. After inspection of a failed unit the repairman chooses either a slow or a fast repair rate to carry out the corresponding amount of work. At system breakdown the repairman has an additional opportunity to switch to the fast rate. If there are no fixed costs associated with system breakdowns, then the policy which minimizes longrun average costs is shown to be a two-dimensional control limit rule. If fixed costs are incurred every time the system breaks down, then the optimal policy is not necessarily of control limit type. This is illustrated by a counterexample. Furthermore, we present several performance measures for this maintenance system controlled by a two-dimensional control limit rule. © 1993 John Wiley & Sons, Inc.     

Scheduling semiconductor test operations: Minimizing maximum lateness and number of tardy jobs on a single machine

We examine a class of single-machine scheduling problems with sequence-dependent setup times that arise in the context of semiconductor test operations. We present heuristics for the problems of minimizing maximum lateness with dynamic arrivals and minimizing number of tardy jobs. We exploit special problem structure to derive worst-case error bounds. The special problem structure also enables us to derive dynamic programming procedures for the problems where all jobs are available simultaneously.