Efficient computational devices for the capacitated transportation problem

This paper presents the details for applying and specializing the work of Ellis Johnson [10] and [11] to develop a primal code for the well-known capacitated transportation problem. The code was developed directly from the work of Johnson, but is similar to codes developed by Glover, Karney, Klingman, and Napier [6] and Srinivasan and Thompson [14]. The emphasis in the presentation is the use of the graphical representation of the basis to carry out the revised simplex operations. This is a means of exploiting the special structure and sparseness of the constraint matrix to minimize computational effort and storage requirements. We also present the results of solving several large problems with the code developed.     

Assembly of systems having maximum reliability

The first problem considered in this paper is concerned with the assembly of independent components into parallel systems so as to maximize the expected number of systems that perform satisfactorily. Associated with each component is a probability of it performing successfully. It is shown that an optimal assembly is obtained if the reliability of each assembled system can be made equal. If such equality is not attainable, then bounds are given so that the maximum expected number of systems that perform satisfactorily will lie within these stated bounds; the bounds being a function of an arbitrarily chosen assembly. An improvement algorithm is also presented. A second problem treated is concerned with the optimal design of a system. Instead of assembling given units, there is an opportunity to “control” their quality, i.e., the manufacturer is able to fix the probability, p, of a unit performing successfully. However, his resources, are limited so that a constraint is imposed on these probabilities. For (1) series systems, (2) parallel systems, and (3) k out of n systems, results are obtained for finding the optimal p's which maximize the reliability of a single system, and which maximize the expected number of systems that perform satisfactorily out of a total assembly of J systems.     

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.     

The single server queue in discrete time-numerical analysis II

This paper deals with the numerical problems arising in the computation of higher order moments of the busy period for certain classical queues of the M|G|I type, both in discrete and in continuous time The classical functional equation for the moment generating function of the busy period is used. The higher order derivatives at zero of the moment generating function are computed by repeated use of the classical differentiation formula of Fá di Bruno. Moments of order up to fifty may be computed in this manner A variety of computational aspects of Fá di Bruno's formula, which may be of use in other areas of application, are also discussed in detail.     

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.     

Allocation of resources to offensive strategic weapon systems

In this paper a very versatile game model is developed for use in the long range planning of our strategic force posture. This highly aggregate model yields optimal force mixes for the triad (land- and sea-based missile systems and bombers) under a variety of constraints. The model described here is a survivability model; however, it is shown how the model can still be used as a measure of overall system effectiveness. Constraints imposed on the problem include both SALT and budget limitations.     

The bilinear programming problem

The paper deals with bilinear programming problems and develops a finite algorithm using the “piecewise strategy” for large-scale systems. It consists of systematically generating a sequence of expanding polytopes with the global optimum within each polytope being known. The procedure then stops when the final polytope contains the feasible region.     

Confidence bounds and propagation of uncertainties in system availability and reliability computations

This paper develops bounds on the uncertainties in system availabilities or reliabilities which have been computed from structural (series, parallel, etc.) relations among uncertain subsystem availabilities or reliabilities. It is assumed that the highly available (reliable) subsystems have been tested or simulated to determine their unavailabilities (unreliabilities) to within some small percentages of uncertainty. It is shown that series, parallel and r out of n structures which are nominally highly available will have unavailability uncertainties whose percentages errors are of the same order as the subsystem uncertainties. Thus overall system analysis errors, even for large systems, are of the same order of magnitude as the uncertainties in the component probabilities. Both systematic (bias type) uncertainties and independent random uncertainties are considered.