A manufacturing company introduces two product alternatives.

Task 1 A manufacturing company introduces two product alternatives. The table below provides profit payoffs in thousands of dollars. The probabilities for the state of nature are P(Up) = 0.35, P(Stable) = 0.35, and P(Down) = 0.30. 1) Use a decision tree to recommend a decision with justification. [3 Marks] 2) Use the expected value of perfect information (EVPI) to determine whether the manufacturing company should attempt to obtain a better estimate of the response. Justify your answers. [3 Marks] 3) A test market study of the potential demand for the product is expected to report either a favourable (F) or unfavourable (U) condition. The relevant conditional probabilities are as follows: P(F|Up) = 0.5; P(F|Stable) = 0.3; P(F|Down) = 0.2 P(U|Up) = 0.2; P(U|Stable) = 0.3; P(U|Down) = 0.5 Use Bayes’ theorem to compute the conditional probability of the demand being up, stable, or down, given each market research outcome. Justify your answers. [4 Marks] Task 2 You are required to formulate a research problem in a field and use multiple linear regression technique to model the chosen problem. The potential fields are relevant to, but are not limited to, the following: • Logistics management • Supply chain management • Warehousing management • International trade • Transport management • Business policy and regulations Analyse the chosen problem through completing the following tasks: 1) Clearly state the formulated research problem and explain its background and significance. (Word limit: 100) [2 Marks] 2) Determine the potential dependent and independent variables (at least one categorical and two quantitative independent variables). Justify the choice of variables. (Word limit: 100) [3 Marks] 3) Collect real data for the chosen variables with the minimum sample size of 30 and perform the multiple linear regression model in Excel. [2 Marks] 4) Interpret the estimation result in terms of R^2, F-test, and individual t-tests. [3 Marks] Task 3 A soft drink manufacturing company has three factories—one in Orlando, one in Tampa, and one in Port St. Lucie—which supply soft drink bottles to three warehouses located in the city of Miami (W1, W2 and W3). The associated per-unit transportation cost table is provided below. The factory in Orlando has a manufacturing capacity of 15,000 units. The factory in Tampa has a manufacturing capacity of 18,000 units. The factory in Port St. Lucie has a manufacturing capacity of 8,000 units. The requirements of the warehouses are: 1) Determine how many of each company’s production should be shipped from its factory to each warehouse in order to minimise the total transportation cost? Justify your findings. [5 Marks] 2) Find an alternative optimal solution for this transportation problem? Justify your findings. [5 Marks]

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.

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:

State of Nature	Product Alternative 1	Product Alternative 2
Up	100	120
Stable	60	70
Down	20	30

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:

State of Nature	Product Alternative 1	Product Alternative 2
Up	100	120
Stable	60	70
Down	20	30

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

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