MANE Engineering Decision Support Systems Problem 1 Reliability of a vehicle can be expressed as the probability of the vehicle failure at a certain time. Suppose you are planning for a trip to Houston this weekend and you worry about the reliability of your car. In considering this, you figured out that failure of your car can be caused by engine failure, transmission failure and chassis failure, while engine failure can be caused by electronic system failure. This is represented by the following diagram: Electronic system Failure Engine Failure Transmissio n Failure Chassis Failure Car Failure Assume F=Car Failure, E=Engine failure, T=Transmission Failure, C=Chassis Failure, ES=Electronic Failure, Use law of total probability to present the probability of car failure: P(F). ( for help, refer to nuclear regulator example in the lecture) Problem 2. 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 300 2500 Lottery Outcome 2 -150 -1250 Oscar 0 Mildred 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: Bond Fund Stock Fund CD .25 $1015.75 $4200 .6 $1042.50 $1125 1030 .15 $1030.25 $250 a. What is the best investment for Oscar? What is his expected utility? b. What is the best investment for Mildred? What is her expected utility? Problem 3 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 Medium quo 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 = $million) for cost across the service and a range from -1 to +2 for the (constructed) evaluation measure scale capability and from -1 to +1 for survivability. Note that preferences are monotonically increasing for capability and survivability ("more is better"), while preferences are monotonically decreasing for cost ("more is worse"). The scores for each of the three alternatives on these three evaluation measures are as follows: Table 2: Scores of Evaluation Measures for Alternatives Alternative Cost Capability Survivabilit ($M) y Status Quo 50 0 1 Twin 30-mm 150 1 0 guns 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: a. Compute the one-dimensional utility values for each alternative for each evaluation measure. b. Determine the swing weights for all three evaluation measures. c. Determine the utility values of the three alternatives, and find which alternative is most preferred