A manufacturing company introduces two product alternatives.

Full Answer Section

Once the states of nature have been identified, we can calculate the expected value of each product alternative for each state of nature. The expected value is calculated by multiplying the profit payoff by the probability of the state of nature occurring.

The following table shows the expected value of each product alternative for each state of nature:

To construct the decision tree, we start at the root node and draw a branch for each state of nature. The expected value of each product alternative is written at the end of each branch.

The following decision tree shows the expected value of each product alternative for each state of nature:

Decision Tree

Root Node: Product Alternative 1 (Expected value: 73) or Product Alternative 2 (Expected value: 80)

Up (0.35): Product Alternative 1 (Expected value: 100) or Product Alternative 2 (Expected value: 120)

Stable (0.35): Product Alternative 1 (Expected value: 60) or Product Alternative 2 (Expected value: 70)

Down (0.30): Product Alternative 1 (Expected value: 20) or Product Alternative 2 (Expected value: 30)

Recommendation

Based on the decision tree, the manufacturing company should select Product Alternative 2. This is because Product Alternative 2 has a higher expected value than Product Alternative 1 for all states of nature.

Expected Value of Perfect Information (EVPI)

The expected value of perfect information (EVPI) is the amount that a decision-maker would be willing to pay to obtain perfect information about the state of nature.

To calculate the EVPI, we first need to calculate the expected value with perfect information (EVPI). The expected value with perfect information is the expected value of the decision if the decision-maker knew the state of nature before making the decision.

The following table shows the expected value with perfect information for each state of nature:

To calculate the EVPI, we subtract the expected value without perfect information (73) from the expected value with perfect information for each state of nature and then weight the results by the probabilities of the states of nature occurring.

The following formula shows how to calculate the EVPI:

EVPI = Σ(EVPI_i - EV_i) * P(i)

where:

• EVPI_i is the expected value with perfect information for state of nature i
• EV_i is the expected value without perfect information for state of nature i
• P(i) is the probability of state of nature i occurring

Substituting the relevant values into the formula, we get the following:

EVPI = (120 - 73) * 0.35 + (70 - 73) * 0.35 + (30 - 73) * 0.30 = 4.2

Interpretation

The EVPI is 4.2. This means that the manufacturing company would be willing to pay up to 4.2 thousand dollars to obtain perfect information about the state of nature before making the decision.

Since the EVPI is relatively small, it suggests that the manufacturing company should not attempt to obtain a better estimate of the response. This is because the cost of obtaining perfect information is likely to outweigh the benefits.

Bayes' Theorem

Bayes' theorem is a mathematical formula that can be used to update the probabilities of events based on new information.

To use Bayes' theorem to compute the conditional probability of the demand being up, stable, or down, given each market research outcome, we need the following information:

• The probability

Sample Solution

Decision Tree

A decision tree is a graphical decision support tool that helps to choose the best course of action by mapping out all the possible outcomes of a decision.

To construct a decision tree for the given problem, we start by identifying the decision to be made, which is the choice of product alternative. The next step is to identify the states of nature, which are the possible outcomes of the decision, i.e., Up, Stable, and Down.