A game theoretic approach to a two firm bidding problem

In this paper a model is developed for determining optimal strategies for two competing firms which are about to submit sealed tender bids on K contracts. A contract calls for the winning firm to supply a specific amount of a commodity at the bid price. By the same token, the production of that commodity involves various amounts of N different resources which each firm possesses in limited quantities. It is assumed that the same two firms bid on each contract and that each wants to determine a bidding strategy which will maximize its profits subject to the constraint that the firm must be able to produce the amount of products required to meet the contracts it wins. This bidding model is formulated as a sequence of bimatrix games coupled together by N resource constraints. Since the firms' strategy spaces are intertwined, the usual quadratic programming methods cannot be used to determine equilibrium strategies. In lieu of this a number of theorems are given which partially characterize such strategies. For the single resource problem techniques are developed for determining equilibrium strategies. In the multiple resource problem similar methods yield subequilibrium strategies or strategies that are equilibrium from at least one firm's point of view.     

A note on allocating jobs to two machines

A recent article in this journal by Mehta, Chandrasekaran, and Emmons [1] described a dynamic programming algorithm for assigning jobs to two identical parallel processors in a way that minimizes the average delay of these jobs. Their problem has a constraint on the sequence of the jobs such that any group of jobs assigned to a processor must be processed in the order of the sequence. This note has two purposes. First, we wish to point out a relationship between this work and some prior work [2]. Second, we wish to point out that Mehta, Chandrasekaran, and Emmons formulation, slightly generalized, can be used to find the optimum assignment of jobs to two machines in a more general class of problems than they considered including a subclass in which the jobs are not constrained to be processed in a given sequence.     

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.     

The lower bounded and partial upper bounded distribution model

This paper pruvides a smaller equivalelnt bounded variable transportation problem than that in Charnes, Glover, and Klingman [1] for the lower bounded and partial upper bounded distribution model.     

Bounds on the availability function

This paper gives bounds on the availability function for an alternating renewal process with exponential failure and general repair times. A bound on the error is also given. Several of the bounds with greatest practical consequence are worked out and illustrated. Repair distributions for which a lower bound on availability is easily computed are gamma (integer shape parameter), log normal, and Weibull. Finally, some simulation results for log normal repair versus gamma repair are given.     

An algorithm for separable piecewise convex programming problems

We present a branch and bound algorithm to solve mathematical programming problems of the form: Find x =|(x₁,…x_n) to minimize Σ?_i0(x₁) subject to x?G, l≦x≦L and Σ?_i0(x₁)≦0, j=1,…,m. With l=(l₁,…,l_n) and L=(L₁,…,L_n), each ?_ij is assumed to be lower aemicontinuous and piecewise convex on the finite interval [l_i.L_i]. G is assumed to be a closed convex set. The algorithm solves a finite sequence of convex programming problems; these correspond to successive partitions of the set C={x|l ≦ x ≦L} on the bahis of the piecewise convexity of the problem functions ?_ij. Computational considerations are discussed, and an illustrative example is presented.     

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.     

Stochastic duels with nonrepairable weapons

This paper examines the effect of limitation, regarding weapons that are likely to fail during the period of deployment, on the final outcome in a stochastic duel model. Inter-firing times as well as inter-failure times have been assumed to be exponentially distributed.     

Optimal multicommodity network flows with resource allocation

The problem of determining multicommodity flows over a capacitated network subject to resource constraints may be solved by linear programming; however, the number of potential vectors in most applications is such that the standard arc-chain formulation becomes impractical. This paper describes an approach—an extension of the column generation technique used in the multicommodity network flow problem—that simultaneously considers network chain selection and resource allocation, thus making the problem both manageable and optimal. The flow attained is constrained by resource availability and network capacity. A minimum-cost formulation is described and an extension to permit the substitution of resources is developed. Computational experience with the model is discussed.     

On constraint qualifications in nonlinear programming

In this paper we examine the relationship between two constraint qualifications developed by Abadie and Arrow, Hurwicz, and Uzawa. A third constraint qualification is discussed and shown to be weaker than either of those mentioned above.