Warranty analysis and renewal function estimation

Estimation of the expected cost of a warranty for a stochastically failing unit is closely tied to estimation of the renewal function. The renewal function is a basic tool also used in probabilistic models arising in other areas such as reliability theory, inventory theory, and continuous sampling plans. In these other areas, estimation of a straight line approximation of the renewal function instead of direct estimation of the renewal function has proved successful. This approximation is based on a limit expression for large values of the argument, say t, of the renewal function. However, in warranty analusis, typically t is small compared to the mean failure time of the unit. Hence, alternative methods for renewal function estimation, both parametric and nonparametric, are presented and discussed. An important aspect of this paper is to discuss the performance of the renewal function estimators when only a small number of failed units is available. A Monte Carlo study is given which suggests guidelines for choosing an estimator under various circumstances.     

An application of empirical bayes techniques to the simultaneous estimation of many probabilities

Consider the following situation: Each of N different combat units is presented with a number of requirements to satisfy, each requirement being classified into one of K mutually exclusive categories. For each unit and each category, an estimate of the probability of that unit satisfying any requirement in that category is desired. The problem can be generally stated as that of estimating N different K-dimensional vectors of probabilities based upon a corresponding set of K-dimensional vectors of sample proportions. An empirical Bayes model is formulated and applied to an example from the Marine Corps Combat Readiness Evaluation System (MCCRES). The EM algorithm provides a convenient method of estimating the prior parameters. The Bayes estimates are compared to the ordinary estimates, i.e., the sample proportions, by means of cross validation, and the Bayes estimates are shown to provide considerable improvement.     

Book Reviews

National Security Concepts of States: New Zealand. By Kennedy Graham. Taylor & Francis, London (1989), ISBN 0-8448-1614-0, £27.00

Power and Prestige in the British Army. By R.G. L. von Zugbach. Gower, Aldershot (1988), ISBN 0-566-05561-9, £22.50

The American Civil War and the Origins of Modern Warfare—Ideas, Organization, and Field Command. By Edward Hagerman. Indiana University Press, Bloomington, IN (1988), ISBN 0-253-30546-2, $37.50 (£23.29)

British Seapower and Procurement between the Wars: a Reappraisal of Rearmament. By G. A. H. Gordon. Macmillan, London (1988), ISBN 0-333-42332-1. £29.50

Armies in Low-intensity Conflict: a Comparative Analysis. Edited by David A. Charters and Maurice Tugwell. Brassey's Defence Publishers, London (1989), ISBN 0-08-036253-2, £25.00 ($45.00); Deadly Paradigms: the Failure of U.S. Counterinsurgency Policy. By Michael Shafer, Leicester University Press, Leicester (1988), ISBN 0-7185-1311-8, £28.00

British Defence Policy Striking the Right Balance. By J. Baylis. Macmillan, London (1989), ISBN 0-333-49133-5, £29.50 or £9.99     

A note on a model for evasive vehicle movement along a fixed path

A model for a vehicle moving evasively along a fixed path is defined in terms of a two- state semi-Markov process. An important feature of this model is the continuous movement of the vehicle as a function of time. One potential application of this model is the development of a strategy for the deployment of long-range missiles on long underground tracks.     

The discrete max-min problem

The historic max-min problem is examined as a discrete process rather than in its more usual continuous mode. Since the practical application of the max-min model usually involves discrete objects such as ballistic missiles, the discrete formulation of the problem seems quite appropriate. This paper uses an illegal modification to the dynamic programming process to obtain an upper bound to the max-min value. Then a second but legal application of dynamic programming to the minimization part of the problem for a fixed maximizing vector will give a lower bound to the max-min value. Concepts of optimal stopping rules may be applied to indicate when sufficiently near optimal solutions have been obtained.     

An application of servomechanisms to inventory

In recent years, some attention has been devoted to the application of techniques of control theory to inventory management. In particular, H. Vassian (1955) developed a model for a periodic review inventory system utilizing techniques of discrete variable servomechanisms to analyze the system in a cost-free structure. The resulting model is inherently deterministic, however, and emphasizes the control of inventory fluctuation about a safety level by selecting an appropriate order policy. Such an order policy is defined only up to an arbitrary method of forecasting customer demands. The present paper is a continuation of the model developed by Vassian in which exponential smoothing is used as a specific forecasting technique. Full recognition of the probabilistic nature of demand is taken into account and the requirement of minimizing expected inventory level is imposed. In addition, explicit formulas for the variance in inventory are derived as functions of the smoothing constant and the tradeoff between small variance and rapid system response is noted. Finally, in an attempt to remove the bias inherent in exponential smoothing, a modification of that technique is defined and discussed as an alternate forecasting method.     

Relationships among potential sorties,ground support,and aircraft reliability

A methodology is developed for assessing tactical airfield/aircraft system effectiveness, and for evaluating effectiveness changes resulting from incremental investments in ground support resources and/or aircraft reliability. Two categories of ground support functions–turnaround and maintenance–are distinguished. The measure of effectiveness is the maximum potential sortie rate achievable by the system. The methodology enables empirical derivation of the general equation of the tactical airfield/aircraft system. It also enables graphical presentation of the system tradeoffs in the form of a System Analysis Chart.     

Suboptimal algorithms for the quadratic assignment problem

The idea of combining relatively simple continuous methods with discrete procedures is used for the construction of suboptimal algorithms for quadratic assignment problems. Depending on the nature of the special problem these steps may vary in complexity. The simplest procedures require minimum storage space and result in tolerable computation times. Different choices of parameters and random variations may be used in order to obtain statistical distributions of suboptimal solutions. Computational results for sample problems indicate improvements on results of Steinberg, Gilmore, and Hillier and Connors.