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.     

A generalization of the entropy model for brand purchase behavior

Hemiter's entropy model for brand purchase behavior has been generalized for Renyi's measure of entropy which is a more general concept than Shannon's measure of entropy used by Herniter and which includes Shannon's measure as a limiting case. The generalized model considered here is more flexible than Herniter's model since it can give different marketing statistics for different products and it can give these statistics even when only some of the brands are considered.     

Statistical tests for exponential service from m/g/1 waiting-time data

From an original motivation in quantitative inventory modeling, we develop methods for testing the hypothesis that the service times of an M/G/1 queue are exponentially distributed, given a sequence of observations of customer line and/or system waits. The approaches are mostly extensions of the well-known exponential goodness-of-fit test popularized by Gnedenko, which results from the observation that the sum of a random exponential sample is Erlang distributed and thus that the quotient of two independent exponential sample means is F distributed.     

Experiments with linear fractional problems

In this paper we present the results of a limited number of experiments with linear fractional problems. Six solution procedures were tested and the results are expressed in the number of simplex-like pivots required to solve a sample of twenty problems randomly generated.     

Hybrid heuristics for minimum cardinality set covering problems

Minimum cardinality set covering problems (MCSCP) tend to be more difficult to solve than weighted set covering problems because the cost or weight associated with each variable is the same. Since MCSCP is NP-complete, large problem instances are commonly solved using some form of a greedy heuristic. In this paper hybrid algorithms are developed and tested against two common forms of the greedy heuristic. Although all the algorithms tested have the same worst case bounds provided by Ho [7], empirical results for 60 large randomly generated problems indicate that one algorithm performed better than the others.     

On estimating population characteristics from record-breaking observations. i. parametric results

Consider an experiment in which only record-breaking values (e.g., values smaller than all previous ones) are observed. The data available may be represented as X₁,K₁,X₂,K₂, …, where X₁,X₂, … are successive minima and K₁,K₂, … are the numbers of trials needed to obtain new records. We treat the problem of estimating the mean of an underlying exponential distribution, and we consider both fixed sample size problems and inverse sampling schemes. Under inverse sampling, we demonstrate certain global optimality properties of an estimator based on the “total time on test” statistic. Under random sampling, it is shown than an analogous estimator is consistent, but can be improved for any fixed sample size.     

Multi-item service constrained (s,S) policies for spare parts logistics systems

Many organizations providing service support for products or families of products must allocate inventory investment among the parts (or, identically, items) that make up those products or families. The allocation decision is crucial in today's competitive environment in which rapid response and low levels of inventory are both required for providing competitive levels of customer service in marketing a firm's products. This is particularly important in high-tech industries, such as computers, military equipment, and consumer appliances. Such rapid response typically implies regional and local distribution points for final products and for spare parts for repairs. In this article we fix attention on a given product or product family at a single location. This single-location problem is the basic building block of multi-echelon inventory systems based on level-by-level decomposition, and our modeling approach is developed with this application in mind. The product consists of field-replaceable units (i.e., parts), which are to be stocked as spares for field service repair. We assume that each part will be stocked at each location according to an (s, S) stocking policy. Moreover, we distinguish two classes of demand at each location: customer (or emergency) demand and normal replenishment demand from lower levels in the multiechelon system. The basic problem of interest is to determine the appropriate policies (s_i S_i) for each part i in the product under consideration. We formulate an approximate cost function and service level constraint, and we present a greedy heuristic algorithm for solving the resulting approximate constrained optimization problem. We present experimental results showing that the heuristics developed have good cost performance relative to optimal. We also discuss extensions to the multiproduct component commonality problem.     

The LP/POMDP marriage: Optimization with imperfect information

A new technique for solving large‐scale allocation problems with partially observable states and constrained action and observation resources is introduced. The technique uses a master linear program (LP) to determine allocations among a set of control policies, and uses partially observable Markov decision processes (POMDPs) to determine improving policies using dual prices from the master LP. An application is made to a military problem where aircraft attack targets in a sequence of stages, with information acquired in one stage being used to plan attacks in the next. © 2000 John Wiley & Sons, Inc., Naval Research Logistics 47: 607–619, 2000