Some branch-and-bound procedures for fixed-cost transportation problems

In this article we present three properties that will improve the performance of branch-and-bound algorithms for fixed-cost transportation problems. By applying Lagrangian relaxation we show that one can develop stronger up and down penalties than those traditionally used and also develop a strengthened penalty for nonbasic variables. We also show that it is possible to “look ahead” of a particular node and determine the solution at the next node without actually calculating it. We present computational evidence by comparing our developments with existing procedures.     

An efficient heuristic for large set covering problems

A heuristic solution procedure for set covering is presented that works well for large, relatively dense problems. In addition, a confidence interval is established about the unknown global optimum. Results are presented for 30 large randomly generated problems.     

Scheduling stochastic jobs on a single machine subject to breakdowns

A single machine is available to process a collection of stochastic jobs. There may be technological constraints on the job set. The machine sometimes breaks down. Costs are incurred and rewards are earned during processing. We seek strategies for processing the jobs which maximize the total expected reward earned.     

Limit laws in the best of 2n - 1 bernoulli trials

A series of independent Bernoulli trials is considered in which either an outcome of type A or type B occurs at each trial. The series terminates when n outcomes of one type have occurred. Two observable random variables of interest are the total number of outcomes in the series and the number of outcomes of the “losing kind.” Two methods of approximation of the expectations of these random variables for large n are obtained and compared. The limiting distribution of the number of outcomes of the “losing kind” is considered when a beta distribution is assigned to p.     

On the moments of gamma order statistics

A recurrence relation between the moments of order statistics from the gamma distribution having an integer parameter r is obtained. It is shown that if the negative moments of orders -(r?1), …, ?1 of the smallest order statistic in random samples of size n are known, then one can obtain all the moments. Tables of negative moments for r = 2 (1) 5 are also given.     

Simulation of nonhomogeneous poisson processes by thinning

A simple and relatively efficient method for simulating one-dimensional and two-dimensional nonhomogeneous Poisson processes is presented The method is applicable for any rate function and is based on controlled deletion of points in a Poisson process whose rate function dominates the given rate function In its simplest implementation, the method obviates the need for numerical integration of the rate function, for ordering of points, and for generation of Poisson variates.     

Design of a process control scheme for defects per 100 units based on aoql

A process control scheme is developed in which decisions as to the frequency of sampling are made based upon the choice of an Average Outgoing Quality Limit. The scheme utilizes plotted points on a U-control chart for defects and the theory of runs to determine when to switch among Reduced, Normal, Tightened, and 100 percent inspection. The scheme is formulated as a semi-Markov process to derive steady stale equations for the probabilities of being in Reduced, Normal, Tightened, or 100 percent inspection and for Average Outgoing Quality and Average Fraction Inspected. The resulting system and the computer programs used to derive it are discussed.     

A single server queue with arrival rate dependent on server breakdowns

This paper considers a single server queueing system that alternates stochastically between two states: operational and failed. When operational, the system functions as an M/E_k/1 queue. When the system is failed, no service takes place but customers continue to arrive according to a Poisson process; however, the arrival rate is different from that when the system is operational. The durations of the operating and failed periods are exponential with mean 1/cβ and Erlang with mean 1/cβ, respectively. Generating functions are used to derive the steady-state quantities L and W, both of which, when viewed as functions of c, decrease at a rate inversely proportional to c². The paper includes an analysis of several special and extreme cases and an application to a production-storage system.     

Damage calculations for unreliable warheads

For large numbers of perfectly reliable, optimally targeted warheads the square-root law approximates the expected fraction damage achieved on an area target. In this paper a more exact expression is derived for this damage fraction which Holds for all numbers of warheads. This expression is shown to converge to the square-root law when a large number of warheads are fired. The more exact expression is used in a procedure to calculate expected damage when warheads are unreliable, and this procedure is shown to be superior to a modified square-root approximation which has been used previously.     

An algorithm (GIPC2) for solving integer programming problems with separable nonlinear objective functions

This paper presents an algorithm for solving the integer programming problem possessing a separable nonlinear objective function subject to linear constraints. The method is based on a generalization of the Balas implicit enumeration scheme. Computational experience is given for a set of seventeen linear and seventeen nonlinear test problems. The results indicate that the algorithm can solve the nonlinear integer programming problem in roughly the equivalent time required to solve the linear integer programming problem of similar size with existing algorithms. Although the algorithm is specifically designed to solve the nonlinear problem, the results indicate that the algorithm compares favorably with the Branch and Bound algorithm in the solution of linear integer programming problems.