Jsun Y. Wong

On Solving A One-Dimensional Space Allocation Problem With Integer Programming

AbstractThis paper considers the location of n departments on one line. These departments are of different lengths and the material ilow between each pair of departments is known. The objective is to minimize total transportation costs given by the sum of all distance-flow products. The distance between two departments is the separation between their centroids. A binary mixed integer programming formulation is presented to solve this problem. The formulation involves ½n(n – 1) binary variables. Computational results are presented.

Simultaneously estimating the three Weibull parameters from progressively censored samples

Abstract To estimate the three Weibull parameters simultaneously, this paper uses the two common, alternative approaches to maximum likelihood estimation: one approach of directly maximizing the log likelihood function and another approach of solving the system of equations obtained from differentiation. Specifically, with a progressively censored sample from a Weibull population, one of the three equations from differentiation gives a two-unknown expression for the other unknown. Putting this expression into the log likelihood function helps solve this estimation problem. Computer programs and their outputs for the numerical examples from an article are presented here. The computational results shown in these outputs are compared with those from the article.

A note on solving a system reliability problem

Abstract This paper is concerned with solving a constrained system reliability problem from the literature, which is to maximize a nonlinear reliability function subject to two linear constraints. The variables are positive integers, representing the allocations of the stages. By running on an ordinary personal computer the computer program listed in this paper, the best known solution for a widely-known 15-stage example from the literature was obtained thirteen times among the first eighty candidate solutions of one computer run. The computational results presented in this paper suggest that the computer program listed here can be useful as a model for solving the constrained system reliability problem.

Computer programs for non‐linear programming

In this paper we consider the problem of maximizing a non‐linear or linear objective function subject to non‐linear and/or linear constraints. The approach used is an adaptive random search with some non‐random searches built‐in. The algorithm begins with a given point which is replaced by another point if the latter satisfies each of the constraints and results in a bigger functional value. The process of moving from one point to a better point is repeated many times. The value of each of the coordinates of the next point is determined by one of several ways; for example, a coordinate is sometimes forced to have the same value as the value of the corresponding coordinate of the current feasible point. In this algorithm, a candidate point receives no further computational considerations as soon as it is found to be unfeasible; this makes the algorithm general. Computer programs illustrating the details of the new algorithm are given and computational results of two numerical test problems from the literat...

A note on optimization in integers

This note is concerned with solving optimization problems with integers and related problems. For the convenience of other interested researchers, four numerical examples, two computer programs written in BASIC, and the computational results are included in this paper. The author considers the computational results of the present algorithm to be encouraging.

Multiple-step method for solving nonlinear systems of equations

The present method has several steps. The first step starts for each unknown with a random value in the interval for the unknown. The second step starts at a point near the best point obtained in step one; specifically, for each unknown variable, the second step starts with a value which is, say, the first four digits of the value obtained in step one. Going from step two to step three is like going from step one to step two, but more digits, say, the first five digits of the value obtained in step two, are used for step three. Subsequent steps, if needed for higher accuracy than the accuracy already obtained, are similar. For a certain ill‐conditioned nonlinear system of nine equations with nine variables, this method yields in step one a solution with a sum of residuals of 8.61···D‐05 and yields in step four a solution with a sum of residuals of 1.30···D‐06 and with the worst residual of 3.88···D‐07.

Readily obtaining the maximum likelihood estimates of the three parameters of the generalized gamma distribution of Stacy and Mihram

Abstract This paper presents two approaches for obtaining the maximum likelihood estimates of the three parameters of the generalized gamma distribution of Stacy and Mihram, which is more general than the generalized gamma distribution of Stacy. The first approach attempts to maximize the log likelihood function with the help of two of the three equations derived from equating to zero the three partial derivatives with respect to the three parameters. The second approach attempts to find all solutions to the system of three nonlinear equations. For one numerical example from the literature, three solutions to the nonlinear system were obtained by running the second computer program listed here. One of these three solutions to the system gives the maximum likelihood estimates of the three parameters. In addition, this computer program serves to illustrate a computational process for solving nonlinear systems of equations.

Simultaneously estimating the three parameters of the generalized gamma distribution

Abstract This paper is concerned with maximizing the log likelihood function of the the generalized gamma distribution that was introduced in 1962 by E. W. Stacy. This task is facilitated by using an equation obtained from differentiating with respect to one of the three parameters the log likelihood function and from equating to zero the result of the differentiation; then this pararmeter is expressed in terms of the other two parameters. A Lanczos approximation is used to calculate the gamma function, which is a part of the density function. Some important special cases of this distribution are the exponential distribution, the Weibull distribution, and the gamma distribution. Included in the present paper are two complete programs in BASIC and their outputs for two numerical examples from the literature.

Simultaneously estimating with ease the three parameters of the generalized gamma distribution

Abstract This paper is concerned with maximizing the log likelihood function of the generalized gamma distribution that was introduced in 1962 by E. W. Stacy. This task is greatly facilitated by using two equations obtained from differentiating with respect to the parameters the log likelihood function and from equating to zero the results of the differentiation. One then obtains the key result that one parameter is expressed in terms of another parameter, not in terms of the other two parameters. Included in the present paper are three complete programs in BASIC and their outputs for three numerical examples from the literature. These outputs suggest that these computer programs can be useful as models to find with ease the maximum likelihood estimates of the three parameters of the generalized gamma distribution of Stacy.

A note on reliability allocation under a preventive maintenance schedule

Abstract A previously published computer program is used as a model in this paper to deal with a reliability and replacement problem from a published article. The problem of interest is to determine the reliabilities and the replacements simultaneously. The minimization of the objective function used in this problem corresponds to the maximization of the mission reliability. Presented here are solutions slightly better than the solution given in the original article. Modelled after a published computer program, the complete computer program and the first part of its output are included in this paper. This BASIC computer program as is can be run on IBM-compatible personal computers with or without compilers.