Chapter 20, Problem 11CE

a.

To determine

Find the expected monetary value for each decision.

Answer to Problem 11CE

The expected monetary value for the decision neither is $0.

The expected monetary value for the decision product 1 only is $76.

The expected monetary value for the decision product 2 only is $67.5.

The expected monetary value for the decision both only is $129.

Explanation of Solution

From the given information, the probabilities for the state of nature are $P\left(S_1\right)=0.30$, $P\left(S_2\right)=0.50$ and $P\left(S_3\right)=0.20$.

Then, the expected monetary value for the decision neither is calculated as follows:

$\begin{array}{c}\text{Expected}\text{monetary}\text{value}\text{for}\text{decision}\text{neither}=P\left(S_1\right)V\left(A_1,S_1\right)+P\left(S_2\right)V\left(A_1,S_2\right)+P\left(S_3\right)V\left(A_1,S_3\right)\\ =\left(0.30\right)\left(0\right)+\left(0.50\right)\left(0\right)+\left(0.20\right)\left(0\right)\\ =0+0+0\\ =0\end{array}$

Thus, the expected monetary value for the decision neither is $0.

The expected monetary value for the decision product 1 only is calculated as follows:

$\begin{array}{c}\text{Expected}\text{monetary}\text{value}\text{for}\text{decision}\text{product}\text{1}\text{only}=P\left(S_1\right)V\left(A_2,S_1\right)+P\left(S_2\right)V\left(A_2,S_2\right)+P\left(S_3\right)V\left(A_2,S_3\right)\\ =\left(0.30\right)\left(125\right)+\left(0.50\right)\left(65\right)+\left(0.20\right)\left(30\right)\\ =37.5+32.5+6\\ =76\end{array}$

Thus, the expected monetary value for the decision product 1 only is $76.

The expected monetary value for the decision product 2 only is calculated as follows:

$\begin{array}{c}\text{Expected}\text{monetary}\text{value}\text{for}\text{decision}\text{product}2\text{only}=P\left(S_1\right)V\left(A_3,S_1\right)+P\left(S_2\right)V\left(A_3,S_2\right)+P\left(S_3\right)V\left(A_3,S_3\right)\\ =\left(0.30\right)\left(105\right)+\left(0.50\right)\left(60\right)+\left(0.20\right)\left(30\right)\\ =31.5+30+6\\ =67.5\end{array}$

Thus, the expected monetary value for the decision product 2 only is $67.5.

The expected monetary value for the decision both only is calculated as follows:

$\begin{array}{c}\text{Expected}\text{monetary}\text{value}\text{for}\text{decision}\text{both}=P\left(S_1\right)V\left(A_4,S_1\right)+P\left(S_2\right)V\left(A_4,S_2\right)+P\left(S_3\right)V\left(A_4,S_3\right)\\ =\left(0.30\right)\left(220\right)+\left(0.50\right)\left(110\right)+\left(0.20\right)\left(40\right)\\ =66+55+8\\ =129\end{array}$

Thus, the expected monetary value for the decision both only is $129.

b.

To determine

Give the recommended decision.

Answer to Problem 11CE

The decision neither would be recommended.

Explanation of Solution

The expected monetary value for the decision both is greater when compared to the other decisions. But, the risk of loss is minimum for the decision neither. Then, the recommended decision is neither.

Thus, the decision neither would be recommended.

c.

To determine

Construct an opportunity loss table.

Answer to Problem 11CE

An opportunity loss table is

Decision	Opportunity loss
	$S_1$	$S_2$	$S_3$
Neither	220	110	40
Product 1 only	95	45	10
Product 2 only	115	50	10
Both	0	0	0

Explanation of Solution

The opportunity loss table is obtained as follows:

Decision	Opportunity loss
	$S_1$	$S_2$	$S_3$
Neither	$220-0=220$	$110-0=110$	$40-0=40$
Product 1 only	$220-125=95$	$110-65=45$	$40-30=10$
Product 2 only	$220-105=115$	$110-60=50$	$40-30=10$
Both	$220-220=0$	$110-110=0$	$40-40=0$

From the given table if the decision is neither, then the value is 0 when the state of nature $S_1$. Now if the decision is Both, then the value is 220 when the state of nature $S_1$.

Then, the opportunity loss is obtained by taking the difference between $220 and $0 That is, $220-0=220$.

Thus, the opportunity loss for the decision is neither, given a state of nature is $S_1$, is $220.

From the given table if the decision is product 1 only, then the value is 125 when the state of nature $S_1$. Now if the decision is Both, then the value is 220 when the state of nature $S_1$.

Then, the opportunity loss is obtained by taking the difference between $220 and $125 That is, $220-125=95$.

Thus, the opportunity loss for the decision is product 1 only, given a state of nature is $S_1$, is $95.

From the given table if the decision is product 2 only, then the value is 105 when the state of nature $S_1$. Now if the decision is Both, then the value is 220 when the state of nature $S_1$.

Then, the opportunity loss is obtained by taking the difference between $220 and $105 That is, $220-105=115$.

Thus, the opportunity loss for the decision is product 2 only, given a state of nature is $S_1$, is $115.

From the given table if the decision is both, then the value is 220 when the state of nature $S_1$. Its opportunity loss is 0.

Thus, the opportunity loss for the decision is both, given a state of nature is $S_1$, is $0.

From the given table if the decision is neither, then the value is 0 when the state of nature $S_2$. Now if the decision is Both, then the value is 110 when the state of nature $S_2$.

Then, the opportunity loss is obtained by taking the difference between $110 and $0 That is, $110-0=110$.

Thus, the opportunity loss for the decision is neither, given a state of nature is $S_2$, is $110.

From the given table if the decision is product 1 only, then the value is 65 when the state of nature $S_2$. Now if the decision is Both, then the value is 110 when the state of nature $S_2$.

Then, the opportunity loss is obtained by taking the difference between $110 and $65 That is, $110-65=45$.

Thus, the opportunity loss for the decision is product 1 only, given a state of nature is $S_2$, is $45.

From the given table if the decision is product 2 only, then the value is 60 when the state of nature $S_2$. Now if the decision is Both, then the value is 110 when the state of nature $S_2$.

Then, the opportunity loss is obtained by taking the difference between $110 and $60 That is, $110-60=50$.

Thus, the opportunity loss for the decision is product 2 only, given a state of nature is $S_1$, is $50.

From the given table if the decision is both, then the value is 110 when the state of nature $S_2$. Its opportunity loss is 0.

Thus, the opportunity loss for the decision is both, given a state of nature is $S_2$, is $0.

From the given table if the decision is neither, then the value is 0 when the state of nature $S_3$. Now if the decision is Both, then the value is 40 when the state of nature $S_3$.

Then, the opportunity loss is obtained by taking the difference between $40 and $0 That is, $40-0=40$.

Thus, the opportunity loss for the decision is neither, given a state of nature is $S_3$, is $40.

From the given table if the decision is product 1 only, then the value is 30 when the state of nature $S_3$. Now if the decision is Both, then the value is 40 when the state of nature $S_3$.

Then, the opportunity loss is obtained by taking the difference between $40 and $30 That is, $40-30=10$.

Thus, the opportunity loss for the decision is product 1 only, given a state of nature is $S_3$, is $10.

From the given table if the decision is product 2 only, then the value is 30 when the state of nature $S_3$. Now if the decision is Both, then the value is 40 when the state of nature $S_3$.

Then, the opportunity loss is obtained by taking the difference between $40 and $30 That is, $40-30=10$.

Thus, the opportunity loss for the decision is product 2 only, given a state of nature is $S_3$, is $10.

From the given table if the decision is both, then the value is 40 when the state of nature $S_3$. Its opportunity loss is 0.

Thus, the opportunity loss for the decision is both, given a state of nature is $S_3$, is $0.

d.

To determine

Find the expected opportunity loss for each decision.

Answer to Problem 11CE

The expected opportunity loss for the decision neither is $129.

The expected opportunity loss for the decision product 1 only is $53.

The expected opportunity loss for the decision product 2 only is $61.5.

The expected opportunity loss for the decision both is $0.

Explanation of Solution

From the given information, $P\left(S_1\right)=0.30$, $P\left(S_2\right)=0.50$ and $P\left(S_3\right)=0.20$.

From the part c, the opportunity loss table is

Decision	Opportunity loss
	$S_1$	$S_2$	$S_3$
Neither	220	110	40
Product 1 only	95	45	10
Product 2 only	115	50	10
Both	0	0	0

Then, the expected opportunity loss for the decision neither is

$\begin{array}{c}\text{Expected}\text{opportunity}\text{loss}\text{for}\text{alternative }{A}_1=P\left(S_1\right)R\left(A_1,S_1\right)+P\left(S_2\right)R\left(A_1,S_2\right)+P\left(S_3\right)R\left(A_1,S_3\right)\\ =0.30\left(220\right)+0.50\left(110\right)+0.20\left(40\right)\\ =66+55+8\\ =129\end{array}$

Thus, the expected opportunity loss for the decision neither is $129.

The expected opportunity loss for the decision product 1 only is

$\begin{array}{c}\text{Expected}\text{opportunity}\text{loss}\text{for}\text{the}\text{decision}\text{product}\text{1}\text{only}=P\left(S_1\right)R\left(A_2,S_1\right)+P\left(S_2\right)R\left(A_2,S_2\right)\\ +P\left(S_3\right)R\left(A_2,S_3\right)\\ =0.30\left(95\right)+0.50\left(45\right)++0.20\left(10\right)\\ =28.5+22.5+2\\ =53\end{array}$

Thus, the expected opportunity loss for the decision product 1 only is $53.

The expected opportunity loss for the decision product 2 only is

$\begin{array}{c}\text{Expected}\text{opportunity}\text{loss}\text{for}\text{the}\text{decision}\text{product}\text{2}\text{only}=P\left(S_1\right)R\left(A_3,S_1\right)+P\left(S_2\right)R\left(A_3,S_2\right)\\ +P\left(S_3\right)R\left(A_3,S_3\right)\\ =0.30\left(115\right)+0.50\left(50\right)++0.20\left(10\right)\\ =34.5+25+2\\ =61.5\end{array}$

Thus, the expected opportunity loss for the decision product 2 only is $61.5.

The expected opportunity loss for the decision both is

$\begin{array}{c}\text{Expected}\text{opportunity}\text{loss}\text{for}\text{the}\text{decision}\text{both}=P\left(S_1\right)R\left(A_3,S_1\right)+P\left(S_2\right)R\left(A_3,S_2\right)\\ +P\left(S_3\right)R\left(A_3,S_3\right)\\ =0.30\left(0\right)+0.50\left(0\right)++0.20\left(0\right)\\ =0+0+0\\ =0\end{array}$

Thus, the expected opportunity loss for the decision both is $0.

e.

To determine

Find the expected value of perfect information.

Answer to Problem 11CE

The expected value of perfect information is –6.

Explanation of Solution

The expected value of perfect information is obtained by taking the difference between the expected value under conditions of certainty and expected value under conditions of uncertainty.

From the opportunity loss table in the part c, in the state of nature $S_1$ the opportunity loss is 0 for the decision both, in the state of nature $S_2$ the opportunity loss is 0 for the decision both and in the state of nature $S_3$ the opportunity loss is 0 for the decision both.

Thus, these are taken as the decisions.

The expected value under conditions of certainty is calculated as follows:

State of Nature	Decision	Payoff	Probability	Expected payoff
$S_1$	Both	200	0.30	$200\times 0.30=60$
$S_2$	Both	110	0.50	$110\times 0.50=55$
$S_3$	Both	40	0.20	$40\times 0.20=8$

Then, the expected value under conditions of certainty is obtained by adding 60, 55 and 8. That is, $60+55+8=123$.

Thus, the expected value under conditions of certainty is $123.

From the part d, the decision both is recommended, because its expected opportunity loss is less when compared to other decisions. From the part a, it’s expected payoff value is $129. This is the expected value under conditions of uncertainty.

The expected value of perfect information is

$\begin{array}{c}EVPI=123-129\\ =-6\end{array}$

Thus, the expected value of perfect information is –6.