Some limiting distributions associated with sequences of multinomial trials

A series of independent trials is considered in which one of k ≥ 2 mutually exclusive and exhaustive outcomes occurs at each trial. The series terminates when m outcomes of any one type have occurred. The limiting distribution (as m → ∞) of the number of trials performed until termination is found with particular attention to the situation where a Dirichlet distribution is assigned to the k vector of probabilities for each outcome. Applications to series of races involving k runners and to spares problems in reliability modeling are discussed. The problem of selecting a stopping rule so that the probability of the series terminating on outcome i is k^?1 (i.e., a “fair” competition) is also studied. Two generalizations of the original asymptotic problem are addressed.     

An operator theory of parametric programming for the generalized transportation problem—IV—global operators

This paper investigates the effect of the optimal solution of a (capacitated) generalized transportation problem when the data of the problem (the rim conditions—i.e., the available time of machine types and demands of product types, the per unit production costs, the per unit production time and the upper bounds) are continuously varied as a linear function of a single parameter. Operators that effect the transformation of optimal solution associated with such data changes, are shown to be a product of basis preserving operators (described in our earlier papers) that operate on a sequence of adjacent basis structures. Algorithms are furnished for the three types of operators—rim, cost, and weight. The paper concludes with a discussion of the production and managerial interpretations of the operators and a comment on the “production paradox”.     

Isoquants of continuous production correspondences

Bol has discussed consequences of the continuity of production correspondences in connection with relations between efficient input and output vectors. Isoquants of continuous production correspondences are used here to extend this work. Simplifications to existing theory are discussed.     

A proportional defense model

This paper considers the problem of defending a set of point targets of differing values. The defense is proportional in that it forces the offense to pay a price, in terms of reentry vehicles expended, that is proportional to the value of the target. The objective of the defense is to balance its resources so that no matter what attack is launched, the offense will have to pay a price greater than or equal to some fixed value for every unit of damage inflicted. The analysis determines which targets should be defended and determines the optimal firing doctrine for interceptors at defended targets. A numerical example is included showing the relationship between the total target damage and the size of the interceptor force for different values of p, the interceptor single shot kill probability. Some generalizations are discussed.     

Applications of Mills' Differential

Using Mills Differential a game value function ?, used in the Kemeny-Morgenstern-Thompson growth model to show the existence of solutions, is discussed. In doing so, several proofs become shorter, in addition known results are seen in a new context. Finally connections between the existence of a so-called economic solution and the differentiability or non-differentiability of function ? are analyzed.     

Target selection in Lanchester combat: Linear-law attrition process

We develop the solution to a simple problem of target selection in Lanchester combat against two enemy force types each of which undergoes a “linear-law” attrition process. In addition to the Pontryagin maximum principle, the theory of singular extremals is required to solve this problem. Our major contribution is to show how to synthesize the optimal target selection policies from the basic optimality conditions. This solution synthesis methodology is applicable to more general dynamic (tactical) allocation problems. For constant attrition-rate coefficients we show that whether or not changes can occur in target priorities depends solely on how survivors are valued and is independent of the type of attrition process.     

Optimal inventory policy when stockouts alter demand

An inventory model in which future demand is affected by stockouts has been considered recently by B. L. Schwartz. Some generalizations of Schwartz's model are presented in this paper and properties of the optimal policies are determined. In the case of deterministic demand, a set-up cost is included and a mixture of backlogged and nonbacklogged orders is allowed during stockout. It is proved that the optimal policy entails either no stockout or continual stockout, depending on the values of three parameters. For stochastic demand, the effect of stockouts on demand density is postulated, the resulting optimal inventory policy is discussed, and an example involving an exponential density function is then analyzed in detail.     

Determining adjacent vertices on assignment polytopes

To rank the solutions to the assignment problem using an extreme point method, it is necessary to be able to find all extreme points which are adjacent to a given extreme solution. Recent work has shown a procedure for determining adjacent vertices on transportation polytopes using a modification of the Chernikova Algorithm. We present here a procedure for assignment polytopes which is a simplification of the more general procedure for transportation polytopes and which also allows for implicit enumeration of adjacent vertices.     

Armor configurations via dynamic programming

The problem of selecting materials, their thicknesses and order for armor designed for the defeat of shaped charge threats, has been formulated as a constrained optimization problem. The mathematical model provides an optimal order and thickness of each layer of material such that the resulting armor configuration will be of minimum mass per unit area subject to constraints on total thickness and shaped charge jet tip exit velocity.