John G. Wilson

Bayesian Group Replacement Policies

Much research has been performed in finding optimal group replacement policies for production systems consisting of parallel components, where the failure times of the components are independent identically distributed exponential random variables with a common parameter λ. This paper introduces a class of decision rules that utilizes the statistical information obtained during operation of the components. Two forms of statistical input are allowed. We assume that a prior distribution over the possible values of λ is available. It is not required that this prior distribution be in conjugate form. Statistical information that is provided by the actual failure times of the components is incorporated into the decision rule via the sufficient statistics for the problem. This results in group replacement policies that are intuitively attractive, easy to implement, and mathematically tractable.

Optimal m-failure policies with random repair time

The failure times of n machines are i.i.d. exponential random variables with parameter @l. This paper extends Assaf and Shanthikumars (1987) repair and replacement model to the case where repairtime is a nonnegative random variable. The behaviour of the optimal policy as a function of the cost parameters is investigated. The cost function is shown to be unimodal and an easily implemented algorithm for finding optimal policies is developed.

Optimal choice and assignment of the best m of n randomly arriving items

A total of n items arrive at random. The decision maker must either select or discard the current item. Ranks must be assigned to items as they are selected. The decision makers goal is to follow a procedure that maximises the probability of selecting the m best items and assigning them according to their rank order. For m=1 this is the classical secretary problem. Rose (1982) solved the m=2 case. Key mathematical properties for the general m out of n problem are developed: functional equations expressing the general problem in terms of lower dimensional problems and theorems regarding the structure of optimal strategies are provided. A key optimal stopping result for the general problem is provided. Using these results a procedure for solving the above problem for any given m and n is developed. Using this algorithm, explicit formulas--similar in form to those for the well known m=1 and m=2 cases--can be derived. As an example, explicit formulas for the previously unsolved m=3 finite secretary problem are provided.

Adaptive Replacement Policies for a System of Parallel Machines

Consider n machines or components operating in parallel. Fixed, replacement and downtime costs are associated with the system. This paper considers the case where n equals 2 or 3 and the machines have i.i.d. exponential failure times. However, unlike much of the literature, the parameter of the failure time distribution is unknown. Adaptive policies that incorporate both the cost structure and the statistical information gained while operating the machines are analysed.

Selecting And Implementing The Best Group Replacement Policy For A Non Markovian System

One of the most important procedures in the process of designing a new piece of equipment which ages over time is to set up the optimal preventive maintenance schedule. The schedule is usually based on laboratory experiments during the design period. Once the equipment starts producing goods on the manufacturing floor or in the computer lab, managers and engineers schedule preventive or, if necessary, corrective maintenance over time to keep the system operational. In general they will follow the recommended procedure obtained with the new piece of equipment.

On optimal search plans to detect a target moving randomly on the real line

A target, whose initial position is unknown, is performing a random walk on the integers. A searcher, starting at the origin, wants to follow a search plan for which E[[tau]k] is finite, where k >= 1 and [tau] is the time to capture. The searcher, who has a prior distribution over the targets initial position, can move only to adjacent positions, and cannot travel faster than the target. Necessary and sufficient conditions are given for the existence of search plans for which E[[tau]k] is finite and a minimum.

Adaptive time dynamic model for production volume prediction

Data from a continuously operating high volume lathe was collected over a period of a number of months. This machine would periodically fail and unplanned maintenance was often necessary. Failures were due to a myriad of causes. At the beginning of each month, management was required to estimate production output for the month. This paper shows how this goal might be facilitated via a sequence of Bayesian models analysing previous data. The validity of the procedure is justified by an empirical study.

A subjectivist approach to consecutive conflict

An easily applied approach is developed to provide one participant in a sequence of conflicts with an optimal strategy. A goal of this article is to demonstrate that it is mathematically feasible to incorporate a decision makers subjective distributions over the effects his actions will have on the outcomes of future conflicts. Unlike many other approaches, the model of this article does not restrict the beliefs that the participant is allowed to express. The participant, not the decision theorist, decides on what is relevent. Model assumptions required for updating rules, such as Bayesian updating, are not required unless they really are appropriate for the situation.

Approximating an Infinite Stage Search Problem with a Finite Horizon Model

A target is moving randomly on a lattice. A searcher, starting at the origin, wants to follow a search plan that has minimum expected cost. Very general cost functions are allowed. The cost function depends on the time to capture and the search procedure followed. At each stage, the searcher can move only to adjacent positions or remain at his current position. The targets motion may depend on the search procedure, thus allowing for the possibility that the target may take evasive action. It is shown that only a finite number of plans need to be considered in order to approximate the optimal cost. It is also shown that the leading elements of optimal finite horizon plans eventually become the leading elements of optimal infinite horizon plans. The case of search on the real line, where the cost is Oτk, where k ≥ 1 and τ is the time to capture, is considered in detail. The results of this paper also generalise the important planning horizon theorems of Bean and Smith Bean, J. C., Smith, R. L. 1984. Conditions for the existence of planning horizons. Math. Oper. Res.9 391--401..