Chapter 5.S, Problem 4P

a)

Summary Introduction

To draw: A decision tree diagram.

Introduction:

Decision tree is one of the methods used in decision-making process. It would graphically represent the available alternatives and states of nature. It would also mention the payoffs and probabilities of the alternatives. It helps to choose the best alternative that would give the best result among the alternatives.

Answer to Problem 4P

After analyzing decision tree large facility need to build.

Explanation of Solution

Given information:

If small facility is build:

When demand is lower, then NPV is $400,000.

When demand is higher, then two alternatives are as follows:

First alternative: To maintain and NPV is $50,000

Second alternative: To expand and NPV is $450,000

If large facility is build:

Demand is high: NPV is $800,000

Demand is low: NPV is -$10,000

Decision tree diagram:

Explanation:

Here we see that there are (1) two alternative decision small facility to build or to large facility to build. First alternative is to build small facility and there is also two sub parts, first is low demand and second is high demand. In high demand there are (2) two alternative decisions which has also two parts; first has to be maintained and other is expand.

Decision is based on right to left in decisions. We can see that expand have higher NPV that other alternatives.

Calculate the decision alternative payoffs:

It is given that probability of low demand is 40%, and high demand is 60%.

Small facility to build:

If low demand:

$\begin{array}{l}{\text{Payoff = Portability×NPV}}_{\text{Demand low}}\\ =0.40\times \$400,000\\ =\$160,000\end{array}$

Therefore, the payoff of low demand is $160,000.

If high demand:

$\begin{array}{l}{\text{Payoff = Portability×NPV}}_{\text{Demand high }}\\ =0.60\times \$450,000\\ =\$270,000\end{array}$

Therefore, the payoff of high demand is $270,000.

Large facility to build:

If low demand:

$\begin{array}{l}{\text{Payoff = Portability×NPV}}_{\text{Demand low }}\\ =0.40\times -\$10,000\\ =-\$4,000\end{array}$

If high demand:

$\begin{array}{l}{\text{Payoff = Portability×NPV}}_{\text{Demand high }}\\ =0.60\times \$800,000\\ =\$480,000\end{array}$

Calculate the expected value of each alternative:

Small facility to build:

$\begin{array}{l}{\text{Expected Value}}_{\text{small}}{\text{ = Payoff}}_{\text{Demand Low}}{\text{+Payoff}}_{\text{Demand high }}\\ =\$160,000+\$270,000\\ =\$430,000\end{array}$

Large facility to build

$\begin{array}{l}{\text{Expected Value}}_{\text{Large}}{\text{ = Payoff}}_{\text{Demand Low}}{\text{+Payoff}}_{\text{Demand high }}\\ =\$4,000+\$480,000\\ =\$476,000\end{array}$

Here, we see that highest expected value is for larger facility to build so this alternative need to select and small facility alternative have double slash.

b)

Summary Introduction

To determine: To calculate expected value of perfect information.

Expected value of perfect information: It is the rate that a person is willing to pay to gain access to get perfect information. A common area which uses expected value of perfect information is the healthcare economy. This value tries to evaluate the expected cost of the uncertainty, which can be interpreted as the expected value of perfect information.

Answer to Problem 4P

The expected value of perfect information is $164,000.

Explanation of Solution

$\begin{array}{l}\text{EVPI}\text{=}\text{Expected value with perfect information –}\text{Maximum}\text{EMV}\\ \text{or}\\ \text{EVPI}\text{=}\text{Expected}\text{payoff}\text{under}\text{certainty}\text{–}\text{Expected}\text{payoff}\text{under}\text{risk}\end{array}$

Explanation

Calculation of expected value of perfect information:

Step 1: Calculate the expected value with perfection information or Expected payoff under certainty:

$\begin{array}{c}\text{EVwPI}={\displaystyle \sum \text{Max}\left(\text{Alternative}\text{payoff}\right)}\times \text{Probability}\text{state of nature }\\ =\left(400,000\times 0.40\right)+\left(800,000\times 0.60\right)\\ =\$640,000\end{array}$

Therefore, expected value with perfection information is $640,000.

Step 2: Calculation of the expected value of perfect information:

$\begin{array}{c}\text{EVPI}=\$640,000-\$476,000\\ =\$164,000\end{array}$

Hence, the expected value of perfect information is $164,000.

c)

Summary Introduction

To determine: The range over each alternative that are best in term of the value of P.

Introduction: Decision table to evaluate the range over each alternative. Decision table is formats or visual representations were data is expressed arranged, determined and calculated to make an effective decision making. A decision table is a tabular representation that is used to analyze decision alternatives and states of nature.

Explanation of Solution

Decision table is based on each alternative which are relative to P (low) and graph of low demand and high demand:

Alternative	High Demand	Low Demand
Build Small	$450,000	400,000
Build Large	$800,000	-$10,000.00

From the above graph we can obtain that optimal value of P (low) over each alternative are, for low value of P (low) here we decide to build large facilities because we are getting higher expected value. For high value of P (low) we decided to build small facility because we are getting higher expected value.

Calculate the exact value of range:

Equations:

Build Small: 450,000 – 50,000P (slope = 400,000 – 450,000)

Build Large:800,000 – 810,000P (slope = -10,000 – 800,000)

Find the intersection between the two lines:

$\begin{array}{c}450,000–50,000P=800,000–810,000P\\ -50,000P+810,000P=800,000–450,000\\ 760,000P=350,000\\ P=\frac{350,000}{760,000}\\ =0.4605\end{array}$

Optimal ranges:

Build Large: P (Low) = 0 to < .4605

Build Small: P (Low) > .4605 to 1.00