The J. R. Ryland Computer Company is considering a plant expansion that will ena
ID: 3181409 • Letter: T
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, 0.50, and 0.30, 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 0.50 100 0.50
High 200 0.30 300 0.30
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?
Explanation / Answer
Solution
Back-up Theory
If a discrete random variable, X, has pmf (probability mass function) p(x), then
Mean (average) of X = E(X) = sum{x.p(x)} summed over all possible values of x………..…. (1)
Mean of a function f(x) of variable X = E{f(X)} = sum{f(x).p(x)} summed over all possible values of x………………………………………………………………………………………(2)
In particular, E(X2) = sum{(x2).p(x)} summed over all possible values of x…………………..(3)
Variance of X = V(X) = E(X2) – { E(X)}2……………………………………………………..(4)
Now, to work out solution,
X = Expansion Profits for Medium-Scale and Y = Expansion Profits for Large-Scale
All computations are shown in the table below:
Variable
X
Y
Demand
50 (Low)
150 (Medium)
200 (High)
Total
0 (Low)
100 (Medium)
300 (High)
Total
Probability
0.2
0.5
0.3
1.0
0.2
0.5
0.3
1.0
x.p(x); y.p(y)
10
75
60
145
0
50
90
140
x2.p(x); y2.p(y)
500
11250
12000
23750
0
5000
27000
32000
Part (A)
Expected value for the profit associated with Medium-Scale expansion = E(X) = 145.
Expected value for the profit associated with Large-scale expansion = E(Y) = 140.
Since 145 > 140, medium scale expansion is preferred for the objective of maximizing the expected profit. ANSWER
Part (B)
Variance for the profit associated with the medium scale expansion
= V(X) = E(X2) – {E(X)}2 = 23750 - 1452 = 2725.
Variance for the profit associated with the large scale expansion
= V(Y) = E(Y2) – {E(Y)}2 = 32000 - 1402 = 12400.
Since 2725 < 12400, medium scale expansion is preferred for the objective of
minimizing the risk or uncertainty. ANSWER
Variable
X
Y
Demand
50 (Low)
150 (Medium)
200 (High)
Total
0 (Low)
100 (Medium)
300 (High)
Total
Probability
0.2
0.5
0.3
1.0
0.2
0.5
0.3
1.0
x.p(x); y.p(y)
10
75
60
145
0
50
90
140
x2.p(x); y2.p(y)
500
11250
12000
23750
0
5000
27000
32000
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