Wayne Winston

Optimal Control of Discrete and Continuous Time Maintenance Systems with Variable Service Rates

A maintenance system consists of o finite number of machines and a single-server repair facility that may be operated at several rates. Machines are subject to failure and machines that fail are sent to the repair facility. Under the assumption that costs depend on the repair rate and lost production, we derive conditions that ensure that, for a discrete time maintenance system, the optimal repair rate is a non-increasing function of the number of machines in good condition. By considering a continuous time maintenance system as a limit of a sequence of discrete time maintenance systems, we derive analogous conditions that ensure that the optimal repair rate for the continuous time maintenance system is a non-increasing function of the number of machines in good condition.

Constant Absolute Risk Aversion, Bankruptcy, and Wealth-Dependent Decisions

The existence of insurance, futures markets, and stock markets indicates the importance of risk aversion in economics. In spite of this, there are few studies which attempt to estimate the degree of an agent s risk aversion (see Friend and Blume 1975).1 Any study which seeks to quantify risk aversion, regardless of the setting, must necessarily resort to indirect measures. In this article We construct a model in which it is rational for an agent with constant absolute risk aversion to select the more risky of two investments if and only if his wealth is small. This paradoxical behavior is induced by the presence of discounting and a bankruptcy constraint, and it bodes ill for the empirical resolution of the controversial assumption of risk-averse agents. *This research was partially supported by the National Science Foundation through grant SOC-7808985. 1. This gap is especially significant given the continued controversy regarding the nature of the objective function when agents must produce in an uncertain environment. Some (see Sandmo 1971) have assumed that firms maximize expected utility where utility is a strictly concave function of profits. The model based on the strict concavity assumption gives rise to implications that are quite different from those flowing from expected profit maximization. For example, fixed costs affect output decisions. One objection to this model asserts that firms with strictly concave (or strictly convex) utility functions will not survive when confronted with risk-neutral competitors. In the long run, the average profits of the risk-averse firms will be smaller than those of the risk-neutral firm. Moreover, there will be a tendency for them to disappear or be acquired by risk neutral entrepreneurs (see, e.g., Gould 1976). Furthermore, the demise of the risk-averse entrepreneur will be beneficial for stockholders who will want each firm in their diversified portfolio to maximize expected profits. On the other hand, if there are perfect markets, then Fisher separation obtains and the firms goal is profit maximization which, empirically, is indistinguishable from a linear utility function. (The empirical work on efficient markets starts with the assumption of perfect capital markets.) This problem continues to be the subject of theoretical inquiry (see Radner 1974).

Assignment of Customers to Servers in a Heterogeneous Queuing System with Switching

We consider a discrete time queuing system composed of a set of servers in parallel in which both customers and servers may be of several types. Customers in service may be switched from one server to another at the beginning of each period. We consider dynamic assignment rules and derive conditions that ensure that it is optimal (in a very strong sense) always to assign customers with longer service times to faster servers.

A Controlled Birth and Death Process Model of Optimal Product Pricing Under Stochastically Changing Demand

We consider the problem of product pricing when the firms market share is changing stochastically according to a birth and death process. The current market share together with the price prevailing determine the current rate of profit made as well as the birth and death rates. The optimal pricing policy must balance the immediate advantage of setting a high price in terms of increased current profit against the disadvantage in terms of a possible erosion of the future market share. We formulate a continuous-time Markov decision model and analyse it using a recent technique developed by Lippman [6] for optimization of exponential queueing systems. The optimal pri6ing policy is characterized as having a sort of monotonicity property. We also analyse the dependence of the optimal policy on the problem parameters and indicate further extensions of the model. BIRTH AND DEATH PROCESS; QUEUE CONTROL; OPTIMAL PRICING STRATEGY; DYNAMIC PROGRAMMING

Stochastic Control of Competition Through Prices

We assume that the price of a product set by a firm affects its immediate profit rate as well as the probabilistic rate of arrival of new firms into the industry. Therefore, the firms optimal dynamic pricing strategy must balance the increased current profits from setting a high price against the expected dilution of future profits due to additional competition. We provide a continuous-time Markov decision model and characterize the structure of the optimal control strategy and its sensitivity to the problem parameters. We also indicate the relationship of our problem to the queue control literature.

A Zero-Sum Stochastic Game Model of Duopoly

We consider a discrete time zero-sum stochastic game model of duopoly and give a partial characterization of each firms optimal pricing strategy. An extension to a continuous time model is also discussed.

Eli Lilly and Company Uses Integer Programming to Form Volunteer Teams in Impoverished Countries

Each year, Eli Lilly and Company Lilly offers its worldwide employees the opportunity to participate in paid volunteer teams serving communities in impoverished countries. The company’s Connecting Hearts Abroad service program gives employees a unique opportunity to take part in service trips aimed at improving global health. Lilly annually offers about 23 trips, enabling employees to serve some of the world’s most resource-constrained regions where people lack basic resources or access to healthcare. A selection committee at Lilly manually forms volunteer teams from a large pool of applicants. Unfortunately, the manual selection process is time consuming and often fails to meet employee preference or adequately represent some applicant groups. This paper describes how we developed a mathematical programming model to improve Lilly’s process of volunteer selection. We incorporated the model into a decision support tool that assigns applicants to volunteer assignments and maximizes the chosen volunteers’ preferences under constraints that help ensure fair team compositions. Running the model against the prior year’s applicant data pool took less than two minutes to configure teams such that all volunteers received their first-choice assignment. The automated decision support system also provides a more consistent method of configuring teams that appears fair to the applicants.

ON THE USE OF CLIENT-DRIVEN PROJECTS IN THE MATHEMATICS CLASSROOM

ABSTRACT In this article, we discuss the use of client-driven projects – projects that are posed by business, government, and non-profit organizations and based upon real problems facing the organization. Although client-driven projects have long been used in business and engineering education, their use in the mathematics classroom is rare. Client-driven projects represent authentic connections between standard mathematics content and the world outside the classroom, but their use as teaching tools also raises many curricular and pedagogical issues. In limited evaluation, client-driven projects seem to have a positive impact on students attitudes and motivation.

Optimal pricing of a product with periodic enhancements

Abstract Consider a company whose customers may purchase annual (or periodic) updates of the product (the book of the year for an encyclopedia, for example). We determine how the profitability of the update affects the price of the companys main product (the encyclopedia). Our major result is that the products price should be more than the ‘myopic price’ if and only if the profitability of the update is less than the profitability per potential customer. We also show that an increase in the profitability of the update decreases the price of the product, and we derive several other sensitivity results.

The effect of uncertainty and instability on dynamic economic models with an application to cournot oligopoly

Abstract Consider a dynamic discrete-time economic model in which a state-dependent payoff is earned during each period. We give conditions which ensure that a mean preserving increase in the riskiness of the models parameters increases (or decreases) the expected payoff earned during n periods.