A Single Product Cycling Problem Under Brownian Motion Demand

This paper treats a continuous review, single product stochastic cycling problem with demand modelled as a Brownian motion process. A broad class of production policies is admitted: they may be nonstationary, non-Markovian, or, in fact, almost arbitrary. Control theory is used to show that, within this wide class of policies, a simple, stationary, two-number policy is optimal for the average cost minimization problem. This policy switches production on when it is currently off and net inventory reaches a low critical level, or switches it off when it is on and net inventory reaches a high critical level. Simple methods are developed for obtaining the optimal critical levels numerically. Examples are developed comparing the results with those given by Graves and Keilson for a different demand process having the same mean and variance per unit time.Brownian motion, single product, stochastic scheduling

Stochastic Dominance Tests for Decreasing Absolute Risk Aversion. I. Discrete Random Variables

Stochastic dominance (SD) theory is concerned with orderings of random variables by classes of utility functions characterized solely in terms of general properties. This paper discusses a type of stochastic dominance, called DSD, which is denned by the utility functions having decreasing absolute risk-aversion. Necessary and sufficient conditions for DSD are presented for discrete random variables which, after the possible addition of points of zero probability, are concentrated on finitely many equally-spaced points. The problem is cast as a nonlinear program, which is solved through an efficient dynamic programming routine. Examples are presented to illustrate the increased effectiveness of DSD relative to previous types of stochastic dominance.

Stochastic Dominance Tests for Decreasing Absolute Risk-Aversion II: General Random Variables

Necessary and sufficient conditions are given for stochastic dominance over the class of decreasing absolute risk-averse utility functions. The random variables being compared may be continuous as well as discrete but are assumed to be bounded from below, to have finite means, to have only finitely many mass points in finite intervals, and to have cumulative distribution functions which cross one another only finitely many times, or touch one another only finitely many times in finite intervals. The precise forms of the dominance tests depend upon the number of times the distribution functions cross. An example of the DSD test is presented, and a dynamic programming algorithm for carrying out the general test is given.