Arnold L. Sweet

SOLUTIONS FOR SOME DIFFUSION PROCESSES WITH TWO BARRIERS

in various applications of stochastic processes to problems of engineering interest. These applications frequently involve the presence of barriers. Although mathematical methods for solving Kolmogorovs forward equation for the above processes have previously been discussed ([1], [2]), many solutions for problems with two barriers do not seem to be available in the literature. Instead, one finds solutions for unrestricted processes or simulation used in place of analytical solutions in various applications ([3], [4], [5]). In this paper, solutions of Kolmogorovs forward equations in the presence of constant absorbing and/or reflecting barriers are obtained by means of separation of variables. This enables one to obtain expressions for the probability density functions for first passage times when absorbing barriers are present. The solution for the O.U. process is used to obtain a result of Breimans [6] concerning first passage times. 2. The Wiener process with drift

On the hazard rate of the lognormal distribution

It is argued that plots of the hazard rate for the lognormal random variable which have appeared in some recent literature are incorrect and/or misleading; the hazard rate always begins at zero, rises to a maximum, then decreases very slowly to zero. An equation for the location of the maximum of the hazard rate is derived. The maximum lies in a finite interval for all positive values of the standard deviation of the associated normal random variable. Approximations that can be used to compute the hazard rate for parameter values outside of the usual range in the tables associated with the normal (Gaussian) random variable are presented. >

Parameter Estimation Methods for Discrete Exponential Learning Curves

Abstract A discrete exponential model has been proposed for use as a learning curve. This model describes different learning behavior than the classical or a modified version of the power-form model and it provides some practical advantages. Tests for selection between the models are given. Methods of parameter estimation for the exponential model based on least-squared errors, maximum likelihood and mean likelihood criteria are shown for individual cycle time data. Estimation methods are also given for data on the number of completed cycles during a fixed time interval. Monte Carlo simulation experiments were conducted to test the past description and future forecast quality of this model using these estimation methods. The classical power-form model was also included in these experiments. Estimation precision is described for both models. Other features of the exponential model are shown in contrast to the power-form models.

Adaptive Smoothing for Forecasting Seasonal Series

Abstract Seasonal series can be forecast by using Browns adaptive forecasting method. The main advantage of this approach is that explicit expressions for the variance of the forecast error are derived without the use of numerical matrix inversion. This allows the forecaster to devise means to obtain a signal warning of possible failure of the forecasting model. Explicit expressions are also given for the smoothing vector, and the coefficients of the model can be updated without the use of either matrix inversion or multiplication, thus making the computation of the forecasts simple.

Determination of column-buckling criteria using vibratory data

A mathematical model is presented which describes a nondestructive testing procedure for determining buckling criteria for structures. The procedure requires identification of the structures support boundary conditions using vibration data. Column-buckling experiments are presented which validate the model. The results illustrate the feasibility of using such models to predict the buckling load for structures whose support boundary conditions are not known in advance of service.

Evaluating tolerances and process capability when using truncated probability density functions

An approach is presented on the evaluation of tolerances for the fit between a bore and a shaft, based on the construction of a statistical model of the fit, using truncated probability density functions. The model yields relationships between the probabilities that the bore, shaft and fit meet their specifications, and thus relates the parameters that define the probability density functions. Equations are presented in sufficient detail so that numerical values of capability indices, operating characteristics and Taguchi loss functions can be computed using currently available software. As an example, truncated normal distributions are used to illustrate the capabilities of the model.

Statistical design for the location of planes and circles when using a probe

Abstract A statistical approach to the problem of establishing the location of planes and circles by use of a touch-trigger probe mounted on a machining centre is presented. Equations for determining the minimum number of measurements to make and their location in order to locate a part within a specified confidence interval are given. An explicit solution to the parameter estimation problem for the circle is derived, and its properties are compared with solutions from two other methods. It is shown that one of the methods has an estimator whose accuracy does not improve as the number of measurements made is increased because the probe can move only along cartesian axes when it is mounted on a machining centre.

A simulation study of energy consumption by elevators in tall buildings

Abstract The problem of estimating the energy requirements of elevators in tall buildings is shown to be dependent on the integration of functions of random variables. In addition, the period of integration is also random. Hence, a discrete-continuous simulation language, GASP IV, is employed as a method of solving the equations which describe the model. The model is applied to estimating the energy requirements of a building which contains both apartments and commercial facilities. Amongst the topics studied are the effects of elevator speeds and counterweight values on energy consumption, and also the amount of energy available for regeneration. Although the prediction of energy consumption is not considered to be an Industrial Engineering function, Industrial Engineering concepts and tools are used in modelling the problem.

Identification of a model for predicting elastic buckling

Abstract By means of stochastic and deterministic models, we present a system-identification theory for predicting buckling loads of conservative elastic systems from vibration data, thus creating a procedure for non-destructive testing. By means of some illustrative examples (vibrating columns under axial loading) we discuss the problems of existence, uniqueness and sensitivity of the data obtained. We also present a non-linear problem which is associated with the Tsien analogy for the buckling of thin shells.

An Investigation of the Measurements Systems Analysis “Analytic Method” for Attribute Gages

ABSTRACT Three automotive corporations have developed and sanctioned the recently revised reference manual entitled Measurement Systems Analysis. This “standard” contains a procedure, called the “analytic method,” whose purpose is to estimate the gage bias and gage repeatability of an attribute gage. An improved estimation procedure for this standard is presented. The improved estimation procedure yields more accurate estimates than those obtained using the procedures currently presented in the standard. In addition, the improved procedure allows more flexibility in data collection than the current test protocol. A simulation study that evaluates the estimation procedure of the current standard and compares it with the improved estimation procedure is presented. Errors that are contained in the present standard are also noted.