Semiconductor wafer fabrication is very much error-prone. Assume that a wafer production facility received an order for a specially designed prototype wafer. The cost of producing each wafer is estimated to be $20,000. The customer agrees to pay $150,000 for 3 good wafers, $200,000 for 4 good wafers, and $250,000 for 5 good wafers. Other than 3, 4, or 5 good wafers, all other wafers must be destroyed (i.e. have no value). To obtain the contract, the wafer fab offers to pay the customer a penalty of $100,000 if at least three good wafers are not produced. Assume that each wafer is produced independently and the probability that a wafer is acceptable is 0.65.

This exercise requires setting up several tables and computing different measures (expected profits, probabilities etc.). This sort of calculation is easily done in a spreadsheet. It is important to note that your calculations should be general enough to hold when different inputs (revenues, probabilities etc.) are changed. For this purpose, always define all inputs as variables whose values can be changed.

Let Q be the number of wafers manufactured and X(Q) be the random variable corresponding to the number of good wafers. Construct a probability table for Q=3,4,,10, to compute, P(X(Q)3), the probability that there are at least 3 good wafers when Q wafers are manufactured. Note that this is a Binomial probability. You can use the BINOMDIST function in Excel to compute Binomial probabilities. Find the minimum value of Q that ensures that the probability of having at least 3 good wafers is greater than 0.95.

a.Construct a probability table for Q=1,2,,10, to compute, P(Q,x), the probability that there are exactly x good wafers when Q wafers are manufactured. Note that this is a Binomial probability. You can use the BINOMDIST function in Excel to compute Binomial probabilities.

b.Construct a profit table for Q=1,2,,10, to find, Z(Q,x), the profit obtained when Q wafers are manufactured and x of them are good.

c.Use the probability and profit tables to compute E[Z(Q)], the expected profit when Q wafers are manufactured. What is the optimal quantity to manufacture to maximize the expected profit?

d. How does the optimal quantity change when the probability of a wafer being acceptable changes in the range 0.5 to 0.95? How does the optimal order quantity change if the production cost per wafer changes in the range from $10,000 to $40,000?

e.The wafer company is risk-averse and would also like to avoid the risk of losing money. In order to compute the probability of losing money when Q wafers are manufactured, convert the profit table to a binary loss/gain table where L(Q,x), =0 if Z(Q,x) 0 and is 1 otherwise.

f.Using the binary loss/gain table. Compute the probability of losing money for Q=1,2,,10.

g.Now assume that consecutive wafer productions are not necessarily independent. Consider the following model: P(wafer i+1 is good | wafer i is good)=0.9 and P(wafer i+1 is bad| wafer i is good)=0.1. In addition, P(wafer i+1 is good | wafer i is bad)=0.6, P(wafer i+1 is bad | wafer i is bad)=0.4. Assume that the first wafer is good with probability 0.75 and find the expected profit when 4 wafers are scheduled.