Damage calculations for unreliable warheads

For large numbers of perfectly reliable, optimally targeted warheads the square-root law approximates the expected fraction damage achieved on an area target. In this paper a more exact expression is derived for this damage fraction which Holds for all numbers of warheads. This expression is shown to converge to the square-root law when a large number of warheads are fired. The more exact expression is used in a procedure to calculate expected damage when warheads are unreliable, and this procedure is shown to be superior to a modified square-root approximation which has been used previously.     

Duality for quasi-concave programs with application to economics

A duality theory is developed for mathematical programs with strictly quasi-concave objective functions to be maximized over a convex set. This work broadens the duality theory of Rockafellar and Peterson from concave (convex) functions to quasi-concave (quasi-convex) functions. The theory is closely related to the utility theory in economics. An example from economic planning is examined and the solution to the dual program is shown to have the properties normally associated with market prices.     

Computational results with a branch-and-bound algorithm for the general knapsack problem

In this paper, a branch-and-bound procedure is presented for treating the general knapsack problem. The fundamental notion of the procedure involves a variation of traditional branching strategies as well as the incorporation of penalties in order to improve bounds. Substantial computational experience has been obtained, the results of which would indicate the feasibility of the procedure for problems of large size.     

The simplex method for integral multicommodity networks

The simplex method is interpreted as a labeling procedure for certain classes of multicommodity flow problems in a manner similar to that for single commodity networks. As opposed to general multicommodity algorithms, no explicit matrix inversion is required; all simplex operations are performed graph-theoretically.     

Differential-game examination of optimal time-sequential fire-support strategies

Optimal time-sequential fire-support strategies are studied through a two-person zero-sum deterministic differential game with closed-loop (or feedback) strategies. Lanchester-type equations of warfare are used in this work. In addition to the max-min principle, the theory of singular extremals is required to solve this prescribed-duration combat problem. The combat is between two heterogeneous forces, each composed of infantry and a supporting weapon system (artillery). In contrast to previous work reported in the literature, the attrition structure of the problem at hand leads to force-level-dependent optimal fire-support strategies with the attacker's optimal fire-support strategy requiring him to sometimes split his artillery fire between enemy infantry and artillery (counterbattery fire). A solution phenomnon not previously encountered in Lanchester-type differential games is that the adjoint variables may be discontinuous across a manifold of discontinuity for both players' strategies. This makes the synthesis of optimal strategies particularly difficult. Numerical examples are given.     

A differential equation model of search for randomly arriving and departing targets

In this paper we analyze optimal search strategies in an environment in which multiple, independent targets arrive and depart at random. The analysis revolves around a continuous time differential equation model which captures the time dependent nature of the search process. We explore the impact on optimal strategies of nonzero travel times between regions as well as differing target arrival rates. We derive simple closed form expressions for determining if only one region should be searched.     

An F approximation for two-parameter weibull and log-normal tolerance bounds based on possibly censored data

An approximation suggested in Mann, Schafer and Singpurwalla [18] for obtaining small-sample tolerance bounds based on possibly censored two-parameter Weibull and lognormal samples is investigated. The tolerance bounds obtained are those that effectively make most efficient use of sample data. Values based on the approximation are compared with some available exact values and shown to be in surprisingly good agreement, even in certain cases in which sample sizes are very small or censoring is extensive. Ranges over which error in the approximation is less than about 1 or 2 percent are determined. The investigation of the precision of the approximation extends results of Lawless [8], who considered large-sample maximum-likelihood estimates of parameters as the basis for approximate 95 percent Weibull tolerance bounds obtained by the general approach described in [18]. For Weibull (or extreme-value) data the approximation is particularly useful when sample sizes are moderately large (more than 25), but not large enough (well over 100 for severely censored data) for asymptotic normality of estimators to apply. For such cases simplified efficient linear estimates or maximum-likelihood estimates may be used to obtain the approximate tolerance bounds. For lognormal censored data, best linear unbiased estimates may be used, or any efficient unbiased estimators for which variances and covariances are known as functions of the square of the distribution variance.     

Force-annihilation conditions for variable-coefficient lanchester-type equations of modern warfare

This paper develops a mathematical theory for predicting force annihilation from initial conditions without explicitly computing force-level trajectories for deterministic Lanchester-type “square-law” áttrition equations for combat between two homogeneous forces with temporal variations in fire effectivenesses (as expressed by the Lanchester attrition-rate coefficients). It introduces a canonical auxiliary parity-condition problem for the determination of a single parity-condition parameter (“the enemy force equivalent of a friendly force of unit strength”) and new exponential-like general Lanchester functions. Prediction of force annihilation within a fixed finite time would involve the use of tabulations of the quotient of two Lanchester functions. These force-annihilation results provide further information on the mathematical properties of hyperbolic-like general Lanchester functions: in particular, the parity-condition parameter is related to the range of the quotient of two such hyperbolic-like general Lanchester functions. Different parity-condition parameter results and different new exponential-like general Lanchester functions arise from different mathematical forms for the attrition-rate coefficients. This theory is applied to general power attrition-rate coefficients: exact force-annihilation results are obtained when the so-called offset parameter is equal to zero; while upper and lower bounds for the parity-condition parameter are obtained when the offset parameter is positive.     

A bayesian approach to evaluating military investments in product improvement and testing

Two issues of frequent importance in new product development are product improvement and reliability testing. A question often faced by the developer is: Should the product be distributed in its present state, or should it be improved further and/or tested before distribution? A more useful statement of the question might be: What levels of investment in further improvement and testing are economically permissible? Products for which this question is relevant may vary widely in type and intended use. This paper presents a model for determining these levels for one such product—an equipment modification procedure. The model presented makes use of present value analysis to compare cost streams and of Bayesian statistics to relate the costs to various outcomes under conditions of uncertainty. The model is applied to an actual military problem and a method is described for examining the sensitivity of the results to changes in the prior probabilities and discount rate.     

A Markovian model for process setup and improvement

We examine the setup and improvement policies for a production process with multiple performance states. Assume that the production process deteriorates randomly over time, following a Markovian process with known transition probabilities. In order to reduce the production cost incurred because of process deterioration, the process is inspected at the end of each period. Then one of three actions may be taken: do nothing, perform routine process setup, or perform routine setup and process improvement. The routine setup operation returns the process to its best performance state, whereas the process improvement action may reduce future production and setup costs and improve the process-state transition probabilities. A discounted Markovian model is formulated to find the strategy that minimizes the total cost of operating the production process. © 1997 John Wiley & Sons, Inc. Naval Research Logistics 44: 383–400, 1997