Question

The J. R. Ryland Computer Company is considering a plant expansion that will enable the company to begin production of a new computer product. The company's president must determine whether to make the expansion a medium- or large-scale project. The demand for the new product involves an uncertainty, which for planning purposes may be low demand, medium demand, or high demand. The probability estimates for the demands are 0.20, 0.10, and 0.70, respectively. Letting x and y indicate the annual profit in thousands of dollars, the firm's planners developed profit forecasts for the medium- and large-scale expansion projects:

Medium-Scale Large-Scale

                   Expansion Profits Expansion Profits

x f(x)                                                      y                      f(y)

Low          50               0.20                                                    0                       0.20

Demand Medium 150 0.10 100                   0.10

High 200             0.70                                                   300                   0.70

A. Compute the expected value for the profit associated with the two expansion alternatives. Which decision is preferred for the objective of maximizing the expected profit?

B. Compute the variance for the profit associated with the two expansion alternatives. Which decision is preferred for the objective of minimizing the risk or uncertainty?

Answer 1

(A) We know that mean for a probability mass function is given as

For medium scale

For large scale

Therefore, expected value for large scale is greater than expected value for medium scale. So, large scale is preferred for the objective of maximizing the expected profit.

(B) Formula for variance for probability mass function is given as

where x are the given values with probability f(x) and mu is the expected value

Variance calculation for medium scale

Variance calculation for large scale

So, it is clear that the variance for medium scale is lower than large scale. Therefore, medium scale is preferred for the objective of minimizing the risk or uncertainty