Gerald J. Lieberman

Introduction to operations research

Overview of the operations research modelling approach introduction to linear programming solving linear programming problems - the Simplex Method the theory of the Simplex Method duality theory and sensitivity analysis other algorithms for linear programming the transportation and assignment problems network analysis, including Pert-CPM dynamic programming game theory integer programming non-linear programming Markov chains queueing theory the application of queueing theory inventory theory forecasting Markovian decision processes and applications decision analysis simulation.

Sampling Plans for Inspection by Variables

Abstract * This work was performed under the sponsorship of the office of Naval Research. The authors are indebted to Albert H. Bowker for invaluable aid.

On the Consecutive-k-of-n:F System

The linear (circular) k-of-n:F system has n linearly (circularly) ordered components. Each component either functions or fails. The system fails i.f.f.k consecutive components fail. This paper provides, in the i.i.d. case, recursive formulas and bounds for computing system reliability. It considers properties of system life distributions and, in the non i.i.d. case, questions of optimal system design.

An Exact Asymptotically Efficient Confidence Bound for Reliability in the Case of the Weibull Distribution

This paper presents a simple method for obtaining exact lower confidence bounds for reliabilities (tail probabilities) for items whose life times follow a Weibull distribution where both the “shape” and “scale” parameters are unknown. These confidence bounds are obtained both for the censored and non-censored cases and are asymptotically efficient. They are exact even for small sample sizes in that they attain the desired confidence level precisely. The case of an additional unknown “location” or “shift” parameter is also discussed in the large sample case. Tables are given of exact and asymptotic lower confidence bounds for the reliability for sample sizes of 10, 15, 20, 30, 50 and 100 for various censoring fractions.

On the optimal assignment of servers and a repairman

Abstract : Consider an N server queuing system in which service times of server i are exponentially distributed random variables with rate lambda sub i. Customers arrive in accordance with some arbitrary arrival process. If a customer arrives when all servers are busy, then he is lost to the system; otherwise, he is assigned to one of the free servers according to some policy. Once a customer is assigned to a server he remains in that status until service is completed. We show that the policy that always assigns an arrival to that free server whose service rate is largest (smallest) stochastically minimizes (maximizes) the number in the system. The result is then used to show that in an N component system in which the i superscript th power components up-time is exponential with rate lambda sub i and in which the repair times are exponential with rate mu, the policy of always repairing the failed components whose failure rate lambda is smallest stochastically maximizes the number of working components.

Use of Normal Probability Paper

Abstract Normal probability paper is so designed that the cumulative distribution function of a normally distributed chance variable appears as a straight line. It is a common practice to plot the observations of a sample on this paper to obtain a graphical check for normality or to obtain a graphical estimate of the mean and variance of the population. Textbooks, however, are not very specific about methods for plotting, for, although the ordered observations are plotted along the abscissa, some uncertainties about the corresponding ordinates are left unresolved. The purpose of this paper is to indicate, with a special example, that any graphical technique should depend to a large extent on the purpose for which the graph is drawn. In particular, it presents tables covering sample sizes up to 10, for selecting the ordinates on normal probability paper so as to obtain “optimum” graphical estimates of the mean ζ and the standard deviation σ of a normal distribution. The somewhat more complicated problem of...

A Stochastic Sequential Allocation Model

This paper considers the following model, described in terms of an investment problem. We have D units available for investment. During each of N time periods an opportunity to invest will occur with probability p. As soon as an opportunity presents itself, we must decide how much of our available resources to invest. If we invest y, then we obtain an expected profit P(y), where P is a nondecreasing continuous function. The amount y then becomes unavailable for future investment. The problem is to decide how much to invest at each opportunity so as to maximize total expected profit. When P(y) is a concave function, the structure of the optimal policy is obtained (§1). Bounds on the optimal value function and asymptotic results are presented in §2. A closed-form expression for the optimal value to invest is found in §3 for the special cases of P(y) = log y and P(y) = yα, for 0 < α < 1. §4 presents a continuous-time version of the model, i.e., we assume that opportunities occur in accordance with a Poisson ...

On the Use of Replacements to Extend System Life

We consider an extreme version of the replacement problem. A vital component of a system must be replaced before it fails, otherwise the system fails with no possibility of repairing the system. We assume that n spares are available and that the distribution of a component life is known; the objective is to schedule the replacements so that the expected life of the system is maximized. Our results range from an iterative formula for constructing the optimal schedule to more general mathematical properties of optimal schedules and expected times.

Confidence Intervals for Independent Exponential Series Systems

Abstract Suppose X1, X2, ···, Xn are independent identically distributed exponential random variables with parameter λ1. Let Y1, Y2, ···, Ym also be independent identically distributed exponential random variables with parameter λ2, and assume that Xs and Ys are independent. The problem is to estimate R(t) = e−(λ1+λ2)t. A procedure for determining on exact (1 − α) level lower confidence bound for R(t) is presented. let U = min(Σn t = 1Xi, Σm i = 1Yi), and K = {largest i ≤ n: Σj i − 1Xi ≤ U} + {Largest i ≤ m: Σj i = 1 Yi ≤ U}. Then given K = k, it is shown that U has a gamma distribution with parameters k and λ1 + λ2. Hence, a lower confidence bound for R(t) can be obtained. The suggested procedure is then compared with others presented in the literature.

Prediction Regions for Several Predictions from a Single Regression Line

When a linear relationship has been fitted by least squares, the methods for securing a prediction interval for the response at some fixed value of the independent variable are explained in many statistical text books. This paper describes the somewhat more complex problem of determining the joint prediction interval for the responses at each of K separate settings of the independent variables when all K predictions must be based upon the original fitted model.