A simple Markov model for the analysis of multiple air combat

It is proposed to describe multiple air-to-air combat having a moderate number of participants with the aid of a stochastic process based on end-game duels. A simple model describing the dominant features of air combat leads to a continuous time discrete-state Markov process. Solution of the forward Kolmogorov equations enables one to investigate the influence of initial force levels and performance parameters on the outcome probabilities of the multiple engagement. As is illustrated, such results may be useful in the decision-making process for aircraft and weapon system development planning. Some comparisons are made with Lanchester models as well as with a semi-Markov model.     

Distribution of a linear function and the ratio of two independent linear functions of independent generalized gamma variables

In this paper the exact distribution of a linear function and the ratio of two independent linear functions of independent generalized gamma variables is given.     

A saddle-point theorem for a class of infinite games

In this paper, the existence of a saddle point for two-person zero-sum infinite games of a special type is proved. The games have continuous bilinear payoff functions and strategy sets which are convex, noncompact subsets of an infinite-dimensional vector space. The closures of the strategy sets are, however, compact. The payoff functions satisfy conditions which allow the use of dominance arguments to show that points in the closure of a strategy set are dominated by or are strategically equivalent to points in the strategy set itself. Combining the dominance arguments with a well-known existence theorem produces the main result of the paper. The class of games treated is an extension of a class studied by J. D. Matheson, who obtained explicit solutions for the saddle points by using necessary conditions.     

A general treatment of upper unbounded and bounded hitchcock problems

This paper is designed to treat (a) the problem of the determination of the absolute minimum cost, with the associated assignments, when there is no limit, N, on the number of parcels available for shipment in a modified Hitchcock problem. This is accomplished with the use of a transformed cost matrix. C*, to which the so-called transportation paradox does not apply. The general Hitchcock solution using C* gives the cost T*, which is the absolute minimum cost of the original problem, as well as sets of assignments which are readily transformed to give the general assignments of the original problem. The sum of these latter assignments gives the value of N_u, the unbounded N for minimum cost. In addition, this paper is designed to show (b) how the method of reduced matrices may be used, (c) how a particular Hitchcock solution can be used to determine a general solution so that one solution using C* can provide the general answer, (d) how the results may be modified to apply to problems with fixed N, and hence (e) to determine the function of the decreasing T as N approaches N_u, and finally (f) to provide a treatment when the supplies at origin i and/or the demands at destination j, are bounded.     

Control variable methods in the simulation of a model of a multiprogrammed computer system

One approach to the evaluation of the performance of multiprogranmed computer systems includes the development of Monte Carlo simulations of transitions of programs within such systems, and their strengthening by control variable and concomitant variable methods. An application of such a combination of analytical, numerical, and Monte Carlo approaches to a model of system overhead in a paging machine is presented.     

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.     

War reserve spares kits supplemented by normal operating assets

In peacetime, base stock levels of spares are determined on the assumption of normal resupply from the depot. In the event of war, however, a unit must be prepared to operate from stock on hand for a period of time without being resupplied from the depot. This paper describes a mathematical model for determining such war reserve spares (WRS) requirements. Specifically, the model solves the following kind of optimization problem: find the least-cost WRS kits that will keep the probability of a stockout after K cannibalizations less than or equal to some target objective α. The user of the model specifies the number of allowable cannibalizations, and the level of protection that the kit is supposed to provide. One interesting feature of this model is that in the probability computation it takes into account the possiblility of utilizing normal base operating assets. Results of a sensitivity analysis indicate that if peacetime levels were explicitly taken into account when designing a WRS kit, a cost saving of nearly 40 percent could be effected without degrading base supply performance in wartime.     

Optimal policies for a multi-product inventory system with negotiable lead times

The objective of this paper is to determine the optimum inventory policy for a multi-product periodic review dynamic inventory system. At the beginning of each period two decisions are made for each product. How much to “normal order” with a lead time of λ_n periods and how much to “emergency order” with a lead time of λ_e periods, where λe = λ_n - 1. It is assumed that the emergency ordering costs are higher than the normal ordering costs. The demands for each product in successive periods are assumed to form a sequence of independent identically distributed random variables with known densities. Demands for individual products within a period are assumed to be non-negative, but they need not be independent. Whenever demand exceeds inventory their difference is backlogged rather than lost. The ordering decisions are based on certain costs and two revenue functions. Namely, the procurement costs which are assumed to be linear for both methods of ordering, convex holding and penalty costs, concave salvage gain functions, and linear credit functions. There is a restriction on the total amount that can be emergency ordered for all products. The optimal ordering policy is determined for the one and N-period models.     

Cyclical job sequencing on multiple sets of identical machines

The problem posed in this paper is to sequence or route n jobs, each originating at a particular location or machine, undergoing r?1 operations or repairs, and terminating at the location or machine from which it originated. The problem is formulated as a 0-1 integer program, with block diagonal structure, comprised of r assignment subproblems; and a joint set of constraints to insure cyclical squences. To obtain integer results the solutions to each subproblem are ranked as required and combinations thereof are implicitly enumerated. The procedure may be terminated at any step to obtain an approximate solution. Some limited computational results are presented.