Random price coefficients

The first approach adopts the classical Markowitz s MV model to handle randomness in the objective function coefficients of prices, in which the expected profit is maximized while an appended term representing the magnitude of operational risk due to variability or dispersion in price, as measured by variance, is minimized (Eppen, Martin, and Schrage, 1989). The model can be formulated as minimizing risk (i.e., variance) subject to a lower bound constraint on the target profit (i.e., the mean return). 

Consideration of the expected value of profit alone as the objective function, which is characteristic of the classical stochastic linear programs introduced by Dantzig (1955) and Beale (1955), is obviously inappropriate for moderate and high-risk decisions under uncertainty since most decision makers are risk averse in facing important decisions. The expected value objective ignores both the risk attribute of the decision maker and the distribution of the objective values. Hence, variance of each of the random price coefficients can be adopted as a viable risk measure of the objective function, which is the second major component of the MV approach adopted in Risk Model I. 

Since the above derivation does not explicitly evaluate variances of the random price coefficients as given by V(c A() and V(cyAt), we consider the following alternative definition for variance from Markowitz (1952) that yields ... 

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