Consider a linear program to minimize c\'x subject to Ax b, x 0. Suppose that th

Question

Consider a linear program to minimize c'x subject to Ax b, x 0. Suppose that the components cj of the vector c are random variables distributed independently of each other and of the x-variables, and that the expected value of cj is cbarj, j = 1,,n. Show that the minimum expected cost is obtained by solving the problem to minimize cbar-tx subject to Ax b, x 0, where cbar = (cbar1,,cbarn)t. Suppose that a firm makes two products that consume a common resource, which is expressed as follows: 5x1 + 6x2 30, where xj is the amount of product j produced. The unit profit for product 1 is normally distributed with mean 4 and variance 2. The unit profit for product 2 is given by a chi2-distribution with 2 degrees of freedom. Assume that the random variables are independently distributed and that they are not dependent upon x1 and x2. Find the quantities of each product that must be produced to maximize expected profit. Will your answer differ if the variance for the first product were 4?

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