An open expanding economy model

The authors extend the generalized von Neumann model they developed (with J. G. Kemeny) in 1956 to an open model by assuming that there are exogeneously determined export and import prices and that any amount can be exported or imported at these prices. The open model is then characterized by means of seven axioms. It is shown, by applying the theory of linear programming, that if four economically reasonable assumptions hold, the open model has at least one solution in which at least one good with positive export price is exported and at least one good with positive import price is imported. It is also shown that, in general, a continuum of expansion rates can be achieved by varying certain control variables. The choice of these expansion rates gives indirectly the choice of a suitable sub-economy and also determines the exports and imports of the economy. Other results and examples are discussed.     

Time-minimizing transportation problems

A method is given for finding those solutions of a transportation problem which minimize the total time necessary for transporting goods from the suppliers to the consumers. Several extensions of the model are presented.     

A sequential bayes procedure for reliability demonstration

A common problem in life testing is to demonstrate that the mean time to failure, θ, exceeds some minimum acceptable value, say θ₁, with a given confidence coefficient γ. When this is true, it is said that “θ₁ has been demonstrated with a confidence γ”. In this paper a Sequential Bayes Procedure (SBP) for demonstrating (by means of. a probability statement) that θ exceeds θ₁ is presented. The SBP differs from the classical procedure in the sense that a prior distribution is assumed on the parameter θ, calling for a Bayesian approach. The procedure is based on the sequence of statistics.     

Interception in a network

This paper presents an algorithm for determining where to place intercepting units in order to maximize the probability of preventing an opposing force from proceeding from one particular node in an undirected network to another. The usual gaming assumptions are invoked; namely, the strategy for placing the units is known to the opponent and he will choose a path through the network which, based on this knowledge, maximizes his probability of successful traverse. As given quantities, the model requires a list of the arcs and nodes of the network, the number of intercepting units available to stop the opposing force, and the probabilities for stopping the opposition at the arcs and nodes as functions of the number of intercepting units placed there. From these quantities, the algorithm calculates the probabilities for placing the unit at the arcs and nodes when one intercepting unit is available, and the expected numbers of units to place at the arcs and nodes when multiple intercepting units are available.     

The relationship between decision variables and penalty cost parameter in (Q,R) inventory models

This paper is concerned with the optimum decision variables found using order quantity, reorder point (Q, R) inventory models. It examines whether the optimum variables (Q* and R*) are necessarily monotonic functions of the backorder cost parameter (or equivalently of the performance objective). For a general class of models it is proved that R* must increase as the performance objective is raised, and an inequality condition is derived which governs how Q* will change. Probability distributions of lead time demand are cited or found for which Q* increases, Q* decreases, and Q* is independent of increases in performance objectives or backorder cost parameter.     

A solution for queues with instantaneous jockeying and other customer selection rules

This paper presents a general solution for the M/M/r queue with instantaneous jockeying and r > 1 servers. The solution is obtained in matrices in closed form without recourse to the generating function arguments usually used. The solution requires the inversion of two (Z^r?1) × (2^r?1) matrices. The method proposed is extended to allow different queue selection preferences of arriving customers, balking of arrivals, jockeying preference rules, and queue dependent selection along with jockeying. To illustrate the results, a problem previously published is studied to show how known results are obtained from the proposed general solution.     

Values for optimum reject allowances

Industrial situations exist where it is necessary to estimate the optimum number of parts to start through a manufacturing process in order to obtain a given number of completed good items. The solution to this problem is not straightforward when the expected number of rejects from the process is a random variable and when there are alternative penalties associated with producing too many or too few items. This paper discusses various aspects of this problem as well as some of the proposed solutions to it. In addition, tables of optimum reject allowances based on a comprehensive model are presented.