# Operations Research:Payoff tables and decision tree

Problem #1 Auto Leasing

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

Dealer Monthly Cost Mileage Allowance Cost per Additional Mile

Forno Saab $299 36,000 $0.15

Midtown Motors $310 45,000 $0.20

Hopkins Automotive $325 54,000 $0.15

Amy decided to choose the lease option that will minimize her total 36-month cost. The difficulty is that Amy is not sure how many miles she will drive over the next three years. For purposes of the decision she believes it is reasonable to assume that she will drive 12,000 miles per year, 15,000 miles per year, or 18,000 miles per year. With this assumption Amy estimated her total cost for the three lease options. For example, she figures that the Forno Saab lease will cost her $10,764 if she drives 12,000 miles per year, $12,114 if she drives 15,000 miles per year, or $13,464 if she drives 18,000 miles per year.

a. Construct a complete payoff table for Amy's decision.

b. If Amy is uncertain as to which of the three mileage assumptions is most appropriate, determine the recommended decision (leasing option) using the optimistic, conservative, and minimax regret approaches.

c. 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. Determine the action Amy should choose using the expected value approach.

d. Suppose that after further consideration, Amy concludes that the probabilities that she will drive 12,000, 15,000, and 18,000 miles per year are 0.3, 0.4, and 0.3, respectively. Determine the action Amy should choose using the expected value approach.

Problem #2 Myrtle Air Express

Myrtle Air Express decided to offer direct service from Cleveland to Myrtle Beach. Management must decide between a full-price service using the company's new fleet of jet aircraft and a discount service using smaller capacity commuter planes. It is clear that the best choice depends on the market reaction to the service Myrtle Air offers. Management developed estimates of the contribution to profit for each type of service based upon two possible levels of demand for service to Myrtle Beach: strong and weak. The following table shows the estimated quarterly profits (in thousands of dollars).

Demand for Service

Service Strong Weak

Full price $960 -$490

Discount $670 $320

a. If nothing is known about the probabilities of the chance outcomes, determine the recommended decision using the optimistic, conservative, and minimax regret approaches.

b. Suppose that management of Myrtle Air Express believes that the probability of strong demand is 0.7 and the probability of weak demand is 0.3. Use the expected value approach to determine an optimal decision.

c. Suppose that the probability of strong demand is 0.8 and the probability of weak demand is 0.2. What is the optimal decision using the expected value approach?

d. Use sensitivity analysis to determine the range of demand probabilities for which each of the decision alternatives has the largest expected value.

Problem #3 Seneca Hill Winery

Seneca Hill Winery recently purchased land for the purpose of establishing a new vineyard. Management is considering two varieties of white grapes for the new vineyard: Chardonnay and Riesling. The Chardonnay grapes would be used to produce a dry Chardonnay wine, and the Riesling grapes would be used to produce a semi-dry Riesling wine. It takes approximately four years from the time of planting before new grapes can be harvested. This length of time creates a great deal of uncertainty concerning future demand and makes the decision concerning the type of grapes to plant difficult. Three possibilities are being considered: Chardonnay grapes only, Riesling grapes only, and both Chardonnay and Riesling grapes. Seneca management decided that for planning purposes it would be adequate to consider only two demand possibilities for each type of wine: strong or weak. With two possibilities for each type of wine it was necessary to assess four probabilities. With the help of some forecasts in industry publications, management made the following probability assessments.

Riesling Demand

Chardonnay Demand Weak Strong

Weak 0.05 0.50

Strong 0.25 0.20

Revenue projections show an annual contribution to profit of $20,000 if Seneca Hill only plants Chardonnay grapes and demand is weak for Chardonnay wine, and $70,000 if it only plants Chardonnay grapes and demand is strong for Chardonnay wine. If it only plants Riesling grapes, the annual profit projection is $25,000 if demand is weak for Riesling grapes and $45,000 if demand is strong for Riesling grapes. If Seneca plants both types of grapes, the annual profit projections are shown in the following table.

Riesling Demand

Chardonnay Demand Weak Strong

Weak $22,000 $40,000

Strong $26,000 $60,000

a. Develop a decision tree or a complete payoff table to represent the alternatives and outcomes for this problem.

b. Use the expected value approach to recommend which alternative Seneca Hill Winery should follow in order to maximize expected annual profit.

c. Suppose management is concerned about the probability assessments when demand for Chardonnay wine is strong. Some believe it is likely for Riesling demand also to be strong in this case. Suppose the probability of strong demand for Chardonnay and weak demand for Riesling is 0.05 and the probability of strong demand for Chardonnay and strong demand for Riesling is 0.40. How does this change the recommended decision? Assume that the probabilities when Chardonnay demand is weak are still 0.05 and 0.50.

d. Other members of the management team expect the Chardonnay market to become saturated at some point in the future, causing a fall in prices. Suppose that the annual profit projections fall to $50,000 when demand for Chardonnay is strong and Chardonnay grapes only are planted. Using the original probability assessment, determine how this change would affect the optimal decision.

Problem #4 Embassy Publishing

Embassy Publishing company received a six-chapter manuscript for a new college textbook. The editor of the college division is familiar with the manuscript and estimated a 0.65 probability that the textbook will be successful. If successful, a profit of $750,000 will be realized. If the company decides to publish the textbook and it is unsuccessful, a loss of $250,000 will occur.

Before making the decision to accept or reject the manuscript, the editor is considering sending the manuscript out for review. A review process provided either a favorable (F) or unfavorable (U) evaluation of the manuscript. Past experience with the review process suggests probabilities P(F) = 0.7 and P(U) = 0.3 apply. Let s1 = the textbook is successful, and s2 = the textbook is unsuccessful. The editor's initial probabilities of s1 and s2 will be revised based on whether the review is favorable or unfavorable. The revised probabilities are as follows:

P(s1|F) = 0.75 P(s1|U) = 0.417

P(s2|F) = 0.25 P(s2|U) = 0.583

a. Construct a decision tree or complete payoff table assuming that the company will first make the decision of whether to send the manuscript out for review and then make the decision to accept or reject the manuscript.

b. Use the expected value approach to recommend a decision strategy for the publishing company.

c. If the manuscript review costs $5000, what is your recommendation?

d. Determine the expected value of perfect information for this decision problem. What does this EVPI suggest for the company?

See attached file for full problem description.

#### Solution Summary

Word file contains step by step approach to solving these problems. It also contains payoff tables and decision trees.

Excel file contains detailed computations with formulas sheet.