Estimating failure time distribution and its parameters based on intermediate data from a Wiener degradation model

Instead of measuring a Wiener degradation or performance process at predetermined time points to track degradation or performance of a product for estimating its lifetime, we propose to obtain the first‐passage times of the process over certain nonfailure thresholds. Based on only these intermediate data, we obtain the uniformly minimum variance unbiased estimator and uniformly most accurate confidence interval for the mean lifetime. For estimating the lifetime distribution function, we propose a modified maximum likelihood estimator and a new estimator and prove that, by increasing the sample size of the intermediate data, these estimators and the above‐mentioned estimator of the mean lifetime can achieve the same levels of accuracy as the estimators assuming one has failure times. Thus, our method of using only intermediate data is useful for highly reliable products when their failure times are difficult to obtain. Furthermore, we show that the proposed new estimator of the lifetime distribution function is more accurate than the standard and modified maximum likelihood estimators. We also obtain approximate confidence intervals for the lifetime distribution function and its percentiles. Finally, we use light‐emitting diodes as an example to illustrate our method and demonstrate how to validate the Wiener assumption during the testing. © 2008 Wiley Periodicals, Inc. Naval Research Logistics, 2008     

Nonrobustness of the normal approximation of lead‐time demand in a (Q,R) system

For computing an optimal (Q, R) or kindred inventory policy, the current literature provides mixed signals on whether or when it is safe to approximate a nonnormal lead‐time‐demand (“LTD”) distribution by a normal distribution. The first part of this paper examines this literature critically to justify why the issue warrants further investigations, while the second part presents reliable evidence showing that the system‐cost penalty for using the normal approximation can be quite serious even when the LTD‐distribution's coefficient of variation is quite low—contrary to the prevalent view of the literature. We also identify situations that will most likely lead to large system‐cost penalty. Our results indicate that, given today's technology, it is worthwhile to estimate an LTD‐distribution's shape more accurately and to compute optimal inventory policies using statistical distributions that more accurately reflect the LTD‐distributions' actual shapes. © 2003 Wiley Periodicals, Inc. Naval Research Logistics, 2003     

Lanchester-type models of warfare and optimal control

The optimization of the dynamics of combat (optimal distribution of fire over enemy target types) is studied through a sequence of idealized models by use of the mathematical theory of optimal control. The models are for combat over a period of time described by Lanchester-type equations with a choice of tactics available to one side and subject to change with time. The structure of optimal fire distribution policies is discussed with reference to the influence of combatant objectives, termination conditions of the conflict, type of attrition process, and variable attrition-rate coefficients. Implications for intelligence, command and control systems, and human decision making are pointed out. The use of such optimal control models for guiding extensions to differential games is discussed.     

An operator theory of parametric programming for the generalized transportation problem: I. Basic theory

This paper gives a new organization of the theoretical results of the Generalized Transportation Problem with capacity constraints. A graph-theoretic approach is utilized to define the basis as a one-forest consisting of one-trees (a tree with an extra edge). Algorithmic development of the pivot-step is presented by the representation of a two-tree (a tree with two extra edges). Constructive procedures and proofs leading to an efficient computer code are provided. The basic definition of an operator theory which leads to the discussion of various operators is also given. In later papers we will present additional results on the operator theory for the generalized transportation problem based on the results in the present paper.     

A computation procedure for the exact solution of location-allocation problems with rectangular distances

A method is presented to locate and allocate p new facilities in relation to n existing facilities. Each of the n existing facilities has a requirement flow which must be supplied by the new facilities. Rectangular distances are assumed to exist between all facilities. The algorithm proceeds in two stages. In the first stage a set of all possible optimal new facility locations is determined by a set reduction algorithm. The resultant problem is shown to be equivalent to finding the p-median of a weighted connected graph. In the second stage the optimal locations and allocations are obtained by using a technique for solving the p-median problem.     

On the treatment of force-level constraints in times-equential combat problems

The treatment of force-level constraints in time-sequential combat optimization problems is illustrated by further studying the fire-programming problem of Isbell and Marlow. By using the theory of state variable inequality constraints from modern optimal control theory, sharper results are obtained on necessary conditions of optimality for an optimal fire-distribution policy (in several cases justifying conjectures made in previous analysis). This leads to simplification of the determination of the domains of controllability for extremals leading to the various terminal states of combat. (Additionally, some new results for the determination of boundary conditions for the adjoint variables in optimal control problems with state variable inequality constraints have arisen from this work.) Some further extensions of previous analysis of the fire-programming problem are also given. These clarify some key points in the solution synthesis. Some important military principles for target selection and the valuation of combat resources are deduced from the solution. As a result of this work, more general time-sequential combat optimization problems can be handled, and a more systematic solution procedure is developed.     

An operator theory of parametric programming for the transportation problem-I

This paper investigates the effect on the optimum solution of a (capacitated) transportation problem when the data of the problem (the rim conditions-i. e., the warehouse supplies and market demands-the per unit transportation costs and the upper bounds) are continuously varied as a (linear) function of a single parameter. An operator theory is developed and algorithms provided for applying rim and cost operators that effect the transformation of optimum solution associated with changes in rim conditions and unit costs. Bound operators that effect changes in upper bounds are shown to be equivalent to rim operators. The discussion in this paper is limited to basis preserving operators for which the changes in the data are such that the optimum basis structures are preserved.     

An economic model for planning strategic mobility posture

This paper describes a linear programming model used to perform cost-effectiveness analysis to determine the requirements for military strategic mobility resources. Expenditures for three classes of economic alternatives are considered: purchase of transportation resources; prepositioning of material; and augmentation of port facilities. The mix of economic alternatives that maximizes the effectiveness of the resultant transportation system is determined for a given investment level. By repeating the analysis for several levels of investment, a cost-effectiveness tradeoff curve can be developed.     

Location of facilities with rectangular distances among point and area destinations

This article is concerned with the optimal location of any number (n) of facilities in relation to any number (m) of destinations on the Euclidean plane. The criterion to be satisfied is the minimization of total weighted distances where the distances are rectangular. The destinations may be either single points, lines or rectangular areas. A gradient reduction solution procedure is described which has the property that the direction of descent is determined by the geometrical properties of the problem.