A Linear Risk Model

A special structure optimization model is presented which includes many of the single variable risk problems that are encountered in operational problems. A risk function is assumed which is a piece-wise linear function of some random variable whose distribution is known; one seeks the value of the decision variable which minimizes expected risk. In this paper are presented the necessary and sufficient conditions for this optimization for random variables which are either continuously or discretely distributed. The important special case of a continuous risk function is discussed; multiple risk problems with a joint constraint are analyzed; and the change in policy for a small change in the distribution of the random variable is investigated. Examples illustrate the application of the model.