Question

A manufacturing company is considering expanding its production capacity to meet a growing demand for its product line of air fresheners. The alternatives are to build a new plant, expand the old plant, or do nothing. The marketing department estimates a 35% probability of a market upturn, a 40% probability of a stable market, and a 25% probability of a market downturn. Georgia Swain, the firm's capital appropriations analyst, estimates the following annual returns for these alternatives:

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​	Market Upturn	Stable Market	Market Downturn
​
Build new plant	$690,000	$(130,000)	$(150,000)
Expand old plant	490,000	(45,000)	(65,000)
Do nothing	50,000	0	(20,000)

​

a.   What should the company do?  Why?

b.   What returns will accrue to the company if your recommendation is followed?

c.   What is the Expected Value of  Perfect Information?

Answer 1

We know the following probabilities

The marketing department estimates

a 35% probability of a market upturn

P(Market upturn)=0.35

a 40% probability of a stable market,

P(Stable Market)=0.40

and a 25% probability of a market downturn

P(Market downturn)=0.25

The company have 3 alternatives

The expected value of Build new plant is

$\begin{align*} E(\text{build a new plant})&=P(\text{Market Upturn})\times (\text{annual return when upturn})+\\ &\qquad P(\text{Stable Market})\times (\text{annual return when Stable})+\\ &\qquad P(\text{Market Downturn})\times (\text{annual return when Downturn})\\ &=0.35\times 690000+0.4\times (-130000)+0.25\times (-150000)\\ &=152000 \end{align*}$

The expected value of Expand old plant is

$\begin{align*} E(\text{Expand old plant})&=P(\text{Market Upturn})\times (\text{annual return when upturn})+\\ &\qquad P(\text{Stable Market})\times (\text{annual return when Stable})+\\ &\qquad P(\text{Market Downturn})\times (\text{annual return when Downturn})\\ &=0.35\times 490000+0.4\times (-45000)+0.25\times (-65000)\\ &=137250 \end{align*}$

The expected value of Do nothing is

$\begin{align*} E(\text{Do nothing})&=P(\text{Market Upturn})\times (\text{annual return when upturn})+\\ &\qquad P(\text{Stable Market})\times (\text{annual return when Stable})+\\ &\qquad P(\text{Market Downturn})\times (\text{annual return when Downturn})\\ &=0.35\times 50000+0.4\times 0+0.25\times (-20000)\\ &=12500 \end{align*}$

a. What should the company do?  Why?

ans: The alternative "Build a new plant" has the highest expected value (expected annual return) of the 3 alternatives that the company has. Hence it should go for building a new plant ,in order to maximize the expected annual return.

b.   What returns will accrue to the company if your recommendation is followed?

ans: The annual returns that is expected to accrue to the company by building a new plant is $152,000

c.   What is the Expected Value of  Perfect Information?

If the company has perfect information about the market, it would have selected the alternative with the highest annual return for each market condition

The following table shows the alternative chosen

Market Upturn Stable Market Market Downturn Build new plant Expand old plant Do nothing $690,000 490,000 50,000 $(130,000) (45,000 0 $(150,000) (65,000) (20,000)

That is,

• if the company knows that it is going to be a Market Upturn then it would choose to build new plant at an annual return of $690,000.
• if the company knows that it is going to be a Stable Market then it would choose to do nothing at an annual return of $0
• if the company knows that it is going to be a Market Downturn then it would choose to do nothing at an annual return of -$20,000

The expected value with perfect information is

$\begin{align*} \text{EVwPI}&=P(\text{Market Upturn})\times (\text{annual return when upturn})+\\ &\qquad P(\text{Stable Market})\times (\text{annual return when Stable})+\\ &\qquad P(\text{Market Downturn})\times (\text{annual return when Downturn})\\ &=0.35\times 690000+0.4\times 0+0.25\times (-20000)\\ &=236500 \end{align*}$

The optimum action that was chosen in part a) was to Build a new plant at an expected value of 152000.

This is the Expected Value without Perfect Information

$\begin{align*} \text{EVwoPI}=152000 \end{align*}$

The expected value of perfect information is

$\begin{align*} \text{EVPI}=\text{EVwPI}-\text{EVwoPI}=236500-152000=84500 \end{align*}$

ans: the Expected Value of  Perfect Information is $84,500