Please solve the following question with complete steps:

You are the program manager of a $250M program to build an orbital transfer vehicle for NASA. To minimize technical risk and control developmental costs, your proposal --which formed the basis for the fixed price contract -- employs an already developed propulsion system which is in production. Unfortunately, your engineers -- in their competitive zeal to win the contract -- were somewhat optimistic in assessing the real performance of the propulsion system. After the contract was signed, an independent testing company performed tests on the 48 propulsion systems of the same production line as to whether they met their own specifications. The results of these tests are summarized in Figure 1. Based on the strict terms of the contract, if the propulsion system fails to meet its specifications, the orbital transfer mission will fail to inject a crucial payload into the proper kinematic state, and your company will be penalized the entire value of the contract. (Assume that this is the entire penalty for this failure.) Unfortunately, performing new developments to improve the propulsion performance is not within the domain of your authority. However, even with a subpar propulsion, the mission could still be successfully attained if the weight of the entire vehicle could be reduced, thereby permitting the total system thrust-to-weight ratio to be almost satisfactory; with acceptable compromises in the terminal velocity vector. Additionally, the customer has had a strategic goal to reduce the "dead weight" of your vehicles for many years, and offers a set of reduced penalties if you would embark upon weight reduction programs for your vehicle's fuel storage and transfer systems and for the overall structure of the vehicle -- even if the propulsion systems does not meet specs. This set of reduced penalties for weight reduction of the total of these two systems is summarized in Figure 2. You lash out at your engineers, challenging them to develop weight reduction programs for the fuel and structure systems of the vehicle. They present you with programs A and B which are summarized in Figure 3, providing the cost of each program as well as the probability mass functions of the likelihoods of reducing the weights of the fuel and structure systems. True positives: 39 Total Weight Reduction Penalty True negatives: 31+ lbs $ 5M False positives: WAN 21-30 lbs 15M False negatives: 11-20 lbs 30M 0-10 Ibs 60M Figure 1: Test Results on whether propulsion Figure 2: Schedule of penalties as a function system meets specs. of the total weight reduced. Program Cost Probability Mass Functions of Weight Reductions, (1b) A $5M Fuel system weight reductions: 10 3 Probabilities of above: 0. 1 08 0.1 Structure weight reductions: 15 10 5 Probabilities of above 0.3 0.4 0.3 B $15M Fuel System weight reductions: 15 10 Probabilities of above 0.3 0.6 0.1 Structure weight reductions: 30 20 10 Probabilities of above: 0.4 0.5 0.1 Figure 3: Probability mass functions and costs for weight reduction programs A and B. (Fuel and structure weight reduction probabilities are statistically independent.) You have these decision alternatives: (a) Ignore the likelihood of subpar propulsion and just pray (b) Buy an insurance policy for $45M, (c) Implement a weight reduction program regardless of propulsion performance, and (d) Hire the test contractor to do a $4M test on the specific propulsion system you have in your vehicle. If the test goes OK, ignore weight reduction; if not, perform a weight reduction program. For the situation above, draw a decision diagram, compute all expected values and determine the "optimum" (risk neutral) decision path and its expected value. Would the decision change if you were very risk averse