Optimality conditions for convex semi-infinite programming problems

This paper gives characterization of optimal Solutions for convex semiinfinite programming problems. These characterizations are free of a constraint qualification assumption. Thus they overcome the deficiencies of the semiinfinite versions of the Fritz John and the Kuhn-Tucker theories, which give only necessary or sufficient conditions for optimality, but not both.     

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.     

A branch-bound algorithm for the capacitated facilities location problem

This paper discusses a mixed integer programming method for solving the Facilities Location Problem with capacities on the facilities. The algorithm uses a Decomposition technique to solve the dual of the associated continuous problem in each branch-bound iteration. The method was designed to produce the global optimum solution for problems with up to 100 facilities and 1,000 customers. Computational experience and a complete example are also presented in the appendix.     

Locating facilities in three-dimensional space by convex programming

We consider the problem of simultaneously locating any number of facilities in three-dimensional Euclidean space. The criterion to be satisfied is that of minimizing the total cost of some activity between the facilities to be located and any number of fixed locations. Any amount of activity may be present between any pair of the facilities themselves. The total cost is assumed to be a linear function of the inter-facility and facility-to-fixed locations distances. Since the total cost function for this problem is convex, a unique optimal solution exists. Certain discontinuities are shown to exist in the derivatives of the total cost function which previously has prevented the successful use of gradient computing methods for locating optimal solutions. This article demonstrates the use of a created function which possesses all the necessary properties for ensuring the convergence of first order gradient techniques and is itself uniformly convergent to the actual objective function. Use of the fitted function and the dual problem in the case of constrained problems enables solutions to be determined within any predetermined degree of accuracy. Some computation results are given for both constrained and unconstrained problems.     

An empirical evaluation of further approximations to an approximate continuous review inventory model

This paper describes an empirical evaluation of several approximations to Hadley and Whitin's approximate continuous review inventory model with backorders. It is assumed that lead time demand is normally distributed and various exponential functions are used to approximate the upper tail of this distribution. These approximations offer two important advantages in computing reorder points and reorder quantities. One advantage is that normal tables are no longer required to obtain solutions, and a second advantage is that solutions may be obtained directly rather than iteratively. These approximations are evaluated on two distinct inventory systems. It is shown that an increase in average annual cost of less that 1% is expected as a result of using these approximations. The only exception to this statement is with inventory systems in which a high shortage cost is specified and ordering costs are unusually low.     

A note on sojourn times in M/G/1 queues with instantaneous,bernoulli feedback

Queueing systems which include the possibility for a customer to return to the same server for additional service are called queueing systems with feedback. Such systems occur in computer networks for example. In these systems a chosen customer will wait in the queue, be serviced and then, with probability p, return to wait again, be serviced again and continue this process until, with probability (1 – p) = q, it departs the system never to return. The time of waiting plus service time, the nth time the customer goes through, we will call his nth sojourn time. The (random) sum of these sojourn times we will call the total sojourn time (abbreviated, sojourn time when there is no confusion which sojourn time we are talking about). In this paper we study the total sojourn time in a queueing system with feedback. We give the details for M/G/1 queues in which the decision to feedback or not is a Bernoulli process. While the details of the computations can be more difficult, the structure of the sojourn time process is unchanged for the M/G/1 queue with a more general decision process as will be shown. We assume the reader is familiar with Disney, McNickle and Simon [1].     

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.     

An efficient primal approach to bottleneck transportation problems

This article addresses bottleneck linear programming problems and in particular capacitated and constrained bottleneck transportation problems. A pseudopricing procedure based on the poly-ω procedure is used to facilitate the primal simplex procedure. This process allows the recent computational developments such as the Extended Threaded Index Method to be applied to bottleneck transportation problems. The impact on problem solution times is illustrated by computational testing and comparison with other current methods.     

The traveling salesman problem: An update of research

During the course of the last few years, attacks on the traveling salesman problem have resulted in a variety of often innovative and rather powerful computational procedures. In this article, we present a review of these results for problems defined on weighted and unweighted graphs. Some account of computational behavior for exact algorithms is provided; however, the primary coverage deals with the strategy of particular procedures. In addition, we include some aspects of nonexact algorithms with major interest confined to the establishment of worst-case bounds.     

Testing goodness of fit to the increasing failure rate family

One branch of the reliability literature is concerned with devising statistical procedures with various nonparametric “restricted family” model assumptions because of the potential improved operating characteristics of such procedures over totally nonparametric ones. In the single-sample problem with unknown increasing failure rate (IFR) distribution F, (1) maximum-likelihood estimators of F have been calculated, (2) upper or lower tolerance limits for F have been determined, and (3) tests of the null hypothesis that F is exponential have been constructed. Barlow and Campo proposed graphical methods for assessing goodness of fit to the IFR model when the validity of this assumption is unknown. This article proposes several analytic tests of the IFR null hypothesis based on the maximum distance and area between the cumulative hazard function and its greatest convex minorant (GCM), and the maximum distance and area between the total time on test statistic and its GCM. A table of critical points is provided to implement a specific test having good overall power properties.