Problem As a manager of a small business, you are considering to introduce a new product. The production requires a new machine. You figure out that you could buy it for $190,000, but the price could be in between $180,000 and $200,000. Because of the budget limitation you can only pay 60% of the machine price with your own saving. You will borrow the other 40% with an interest rate around 9% per year (but subject to change in between 8.5% and 10%). The demand of this product is predicted to be 15,000 per year and but could be in between 14800 and 15500. The unit price could be in between $2 and $3, and now you believe that $2.5 is a reasonable price right now. The raw material cost is estimated to be $0.9 but could be in between $0.5 and $1.2. The operation cost of the equipment is around $0.2 for one product but could be in between $0.1 and $0.25. The maintenance cost for this equipment is estimated to be $2000 per year but could be in between $1500 and $2300. Suppose you could always invest your cash in the money market that give a return at 8% per year for sure. Please do following analysis: Construct the Influence Diagram for this decision problem and identify the inputs and mathematic models that relate these inputs (decision variables) to the decision problem. Construct the Tornado Diagram Based on Tornado Diagram, identify two most sensitive variables, and do two-way sensitivity analysis based on these two variables. Illustrate your result in two-way sensitivity graph Can you decide whether to invest after these analyses? If not, what to do next? Problem Suppose you are asking two people, Oscar and Mildred, who gave the following responses to lottery questions: Given a 50-50 chance between the Lottery outcome #1 and Lottery outcome #2, Oscar stated that he found it equivalent to the dollar value in the third column, and Mildred stated that she found it equivalent to the dollar value in the last column. All amounts are in dollars. Lottery outcome1 Lottery Outcome 2 Oscar Mildred 300 -150 0 2500 -1250 0 You will solve a problem using the utility function for each person. Assume that each person has a risk-averse exponential utility function. Suppose that Oscar and Mildred are each offered a choice of investments for $1000: A CD paying 3% per year, a bond fund, and a stock fund. The bond and stock fund pay as shown after 1 year: Stock Market is: Up Same Down Probability: .25 .6 .15 Bond Fund $1015.75 $1042.50 $1030.25 Stock Fund $4200 $1125 $250 CD 1030 What is the best investment for Oscar? What is his expected utility? What is the best investment for Mildred? What is her expected utility? Problem The Royal Canadian Air Force is considering buying a new weapon system to use on its fighter aircraft for close-in air to air attack/defense after longer-range missiles have been expended. The existing Status Quo is the M61 20-mm gun, which are installed in the current aircraft but will need to be refurbished at a cost of $50 million. There are two additional alternatives: internally mounted twin 30 mm guns and a new high-tech laser system. There are three evaluation measures that the Air Force will use for this decision: cost of implementing across the fleet ($millions), capability, and survivability. The last two measures are constructive measures defined as follows: Table 1 Constructive Measures Score Capability Survivability -1 Less effective Low 0 Same as status quo Medium 1 More effective High 2 Very effective N/A (specific values are defined for these categories in measurable terms) For the following analysis, assume a range from $50M to $250M ($M = $million) for cost across the service and a range from -1 to +2 for the (constructed) evaluation measure scale capability and from -1 to +1 for survivability. Note that preferences are monotonically increasing for capability and survivability (“more is better”), while preferences are monotonically decreasing for cost (“more is worse”). The scores for each of the three alternatives on these three evaluation measures are as follows: Table 2: Scores of Evaluation Measures for Alternatives Alternative Cost ($M) Capability Survivability Status Quo 50 0 1 Twin 30-mm guns 150 1 0 Laser 250 2 -1 A multi-objective utility analysis is being done to evaluate these alternatives. The single dimensional utility function over cost is (normalized) exponential with an R value of $65M. The single dimensional utility function over capability is piecewise linear. The utility increment going from a capability score of -1 to a score of 0 is the same as the utility increment going from a capability score of 0 to a score of +1. The utility increment going from a capability score of +1 to a score of +2 is twice as great as the utility increment going from a capability score of 0 to a score of 1. The single dimensional utility function over survivability is linear from –1 to +1. The utility increment going from a capability score of -1 to a capability score of +2 is twice as great as the utility increment going from a cost per weapon system alternative of $250M to a cost of $50M, while the utility increment going from a survivability score of –1 to a survivability score of +1 is the same as the utility increment going from a capability of -1 to a capability score of +2. Questions: Compute the one-dimensional utility values for each alternative for each evaluation measure. Determine the swing weights for all three evaluation measures. Determine the utility values of the three alternatives, and find which alternative is most preferred.

Problem

As a manager of a small business, you are considering to introduce a new product. The production requires a new machine. You figure out that you could buy it for $190,000, but the price could be in between $180,000 and $200,000. Because of the budget limitation you can only pay 60% of the machine price with your own saving. You will borrow the other 40% with an interest rate around 9% per year (but subject to change in between 8.5% and 10%).

The demand of this product is predicted to be 15,000 per year and but could be in between 14800 and 15500. The unit price could be in between $2 and $3, and now you believe that $2.5 is a reasonable price right now. The raw material cost is estimated to be $0.9 but could be in between $0.5 and $1.2. The operation cost of the equipment is around $0.2 for one product but could be in between $0.1 and $0.25. The maintenance cost for this equipment is estimated to be $2000 per year but could be in between $1500 and $2300.

Suppose you could always invest your cash in the money market that give a return at 8% per year for sure.

Please do following analysis:

1. Construct the Influence Diagram for this decision problem and identify the inputs and mathematic models that relate these inputs (decision variables) to the decision problem.
2. Construct the Tornado Diagram
3. Based on Tornado Diagram, identify two most sensitive variables, and do two-way sensitivity analysis based on these two variables. Illustrate your result in two-way sensitivity graph
4. Can you decide whether to invest after these analyses? If not, what to do next?

Problem

Suppose you are asking two people, Oscar and Mildred, who gave the following responses to lottery questions: Given a 50-50 chance between the Lottery outcome #1 and Lottery outcome #2, Oscar stated that he found it equivalent to the dollar value in the third column, and Mildred stated that she found it equivalent to the dollar value in the last column. All amounts are in dollars.

You will solve a problem using the utility function for each person. Assume that each person has a risk-averse exponential utility function.

Suppose that Oscar and Mildred are each offered a choice of investments for $1000: A CD paying 3% per year, a bond fund, and a stock fund. The bond and stock fund pay as shown after 1 year:

1. What is the best investment for Oscar? What is his expected utility?
2. What is the best investment for Mildred? What is her expected utility?

Problem

The Royal Canadian Air Force is considering buying a new weapon system to use on its fighter aircraft for close-in air to air attack/defense after longer-range missiles have been expended. The existing Status Quo is the M61 20-mm gun, which are installed in the current aircraft but will need to be refurbished at a cost of $50 million. There are two additional alternatives: internally mounted twin 30 mm guns and a new high-tech laser system. There are three evaluation measures that the Air Force will use for this decision: cost of implementing across the fleet ($millions), capability, and survivability. The last two measures are constructive measures defined as follows:

Table 1 Constructive Measures

(specific values are defined for these categories in measurable terms)

For the following analysis, assume a range from $50M to $250M ($M = $million) for cost across the service and a range from -1 to +2 for the (constructed) evaluation measure scale capability and from -1 to +1 for survivability. Note that preferences are monotonically increasing for capability and survivability (“more is better”), while preferences are monotonically decreasing for cost (“more is worse”). The scores for each of the three alternatives on these three evaluation measures are as follows:

Table 2: Scores of Evaluation Measures for Alternatives

A multi-objective utility analysis is being done to evaluate these alternatives. The single dimensional utility function over cost is (normalized) exponential with an R value of $65M. The single dimensional utility function over capability is piecewise linear. The utility increment going from a capability score of -1 to a score of 0 is the same as the utility increment going from a capability score of 0 to a score of +1. The utility increment going from a capability score of +1 to a score of +2 is twice as great as the utility increment going from a capability score of 0 to a score of 1. The single dimensional utility function over survivability is linear from –1 to +1.

The utility increment going from a capability score of -1 to a capability score of +2 is twice as great as the utility increment going from a cost per weapon system alternative of $250M to a cost of $50M, while the utility increment going from a survivability score of –1 to a survivability score of +1 is the same as the utility increment going from a capability of -1 to a capability score of +2.

Questions:

1. Compute the one-dimensional utility values for each alternative for each evaluation measure.
2. Determine the swing weights for all three evaluation measures.
3. Determine the utility values of the three alternatives, and find which alternative is most preferred.