1. The manufacturer, WanTo3 produces products that have a probability to be defective.

These products are divided into batches of 200. Past experience reveals that some (batches) are of good quality (i.e. p=0.05) and some are of bad quality (i.e. p=0.25). In addition, 80% of the batches produced are of good quality and 20% of the batches are of bad quality. These products are then used in an assembly, and before the final assembly leaves the factory, their consistency is finally determined. At a gross average cost of 15 per item, the manufacturer may either screen each item in a batch and replace defective products, or use the items directly without screening. Ultimately, if the latter action is taken, the rework cost is 150 per defective product. For these data, the costs per batch can be calculated as follows:

p = 0.05 p = 0.25

Screen 3000 3000

Do not Screen 1500 7500

Because screening requires inspectors and equipment scheduling, it is important to make the decision to screen or not to screen 2 days before the proposed screening takes place.

However, manufacturers can take one item from a batch and submit it to a laboratory, and it is possible to report the test results (defective or non-defective) before making the screen/no-screen decision. The tested object is returned to its batch following the laboratory examination. The cost of this initial inspection is 175. Also note that the probability that a random sample item is defective is

0.8 * 0.05 + 0.2 * 0.25 = 0.09,

and the probability that an item in a lot is of good quality given a randomly sampled item is defective is 0.444 and the probability that an item in a lot is of good quality given a randomly sampled item is not defective is 0.835. The manufacture wants to minimize his or her cost.

2. Jose is interested in leasing a new Saab and has contracted three automobile dealers for pricing information. Each dealer offered Jose a closed-end 36-month lease with no down payment at the time signing. Each lease includes a monthly charge and a millage allowance. Additional miles receive a surcharge on a per-mile basis. The monthly lease cost, the mileage allowance, and the cost for additional miles as follows:

Dealer Monthly Cost Mileage Allowance Cost per Additional Mile

Forno Saab 299 36,000 0.15

Midtown Motors 310 45,000 0.20

Automotive 325 45,000 0.15

Jose decided to choose the lease option that will minimize her total 36-month cost. The difficulty is that Jose is not sure how many miles he will drive over the next three years.

For purposes of this decision he believes it is reasonable to assume that he will drive 12,000 miles per year, 15,000 miles per year, or 18,000 miles per year. With this assumption Jose estimated his total costs for the three lease options. For example, he figures that the Forno Saab lease will cost her 10,764 if he drives 12,000 miles per year,

12,114 if he drives 15,000 miles per year, or 13,464 if he drives 18,000 miles per year.

a. What is the decision, and what is the chance event?

b. Construct a payoff table for Jose's problem.

c. If Jose has no idea which of the three mileage assumptions is most appropriate, what is the recommended decision (leasing option) using the optimistic, conservative, laplace and minimax regret approaches?

d. Suppose that the probabilities that Amy drives 12,000, 15,000, and 18,000 miles per year are 0.5, 0.4, and 0.1, respectively. What option should Jose choose using the expected value approach?

e. Suppose that after further consideration Jose concludes that the probabilities that he will drive 12,000, 15,000, and 18,000 miles per year are 0.3, 0.4, and 0.3, respectively.

What decision should Jose make using the expected value approach?