Chapter 12, Problem 3P

Amy Lloyd is interested in leasing a new Honda 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:

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 this 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 costs for the three lease options. For example, she figures that the Hepburn Honda lease will cost her 36($299) + $0.15(36,000 – 36,000) = $10,764 if she drives 12,000 miles per year, 36($299) + $0.15(45,000 – 36,000) = $12,114 if she drives 15,000 miles per year, or 36($299) + $0.15(54.000 – 36,000) = $13,464 if she drives 18,000 miles per year.

1. a. What is the decision, and what is the chance event?
2. b. Construct a payoff table for Amy’s problem.
3. c. If Amy has no idea which of the three mileage assumptions is most appropriate, what is the recommended decision (leasing option) using the optimistic, conservative, and minimax regret approaches?
4. 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 Amy choose using the expected value approach?
5. e. Develop a risk profile for the decision selected in part (d). What is the most likely cost, and what is its probability?
6. f. 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. What decision should Amy make using the expected value approach?

To determine

Find the decision and the chance event faced by Amy.

Explanation of Solution

Here, the decision is based on selecting the best lease option. Thus the least option has three alternatives that are Hepburn Honda, Midtown Motors and Hopkins Automotive.

Here, the chance event is the number of miles Amy will drive.

To determine

Find the payoff table for Amy’s problem.

Answer to Problem 3P

Thus, the payoff table for Amy’s problem is given as:

	Actual Miles Driven Annually
Decision	12000	15000	18000
Hepburn Honda	$10,764	$12,114	$13,464
Midtown Motors	$11,160	$11,160	$12,960
Hopkins Automotive	$11,700	$11,700	$11,700

Explanation of Solution

The payoff for any combination of alternative and the chance event is the sum of the total monthly charges and total additional mileage cost that is given as follows:

For the Hepburn Honda lease option:

$\begin{array}{l}36000\text{ (}12000miles\times 3yrs):\text{ 36(\$299)+\$ 0}\text{.15}\times \text{(36000-36000)=\$10,764}\text{.}\\ 45000\text{ (}15000miles\times 3yrs):\text{ 36(\$299)+\$ 0}\text{.15}\times \text{(45000-36000)=\$12,114}\text{.}\\ 54000\text{ (}18000miles\times 3yrs):\text{ 36(\$299)+\$ 0}\text{.15}\times \text{(54000-36000)=\$13,464}\text{.}\end{array}$

For the Midtown Motors lease option:

$\begin{array}{l}36000\text{ (}12000miles\times 3yrs):\text{ 36(\$299)+\$ 0}\text{.20}\times \text{(36000-45000)=\$11,160}\text{.}\\ 45000\text{ (}15000miles\times 3yrs):\text{ 36(\$299)+\$ 0}\text{.20}\times \text{(45000-45000)=\$11,160}\text{.}\\ 54000\text{ (}18000miles\times 3yrs):\text{ 36(\$299)+\$ 0}\text{.20}\times \text{(54000-45000)=\$12,960}\text{.}\end{array}$

For the Hopkins Automotive lease option:

$\begin{array}{l}36000\text{ (}12000miles\times 3yrs):\text{ 36(\$299)+\$ 0}\text{.15}\times \text{(36000-54000)=\$11,700}\text{.}\\ 45000\text{ (}15000miles\times 3yrs):\text{ 36(\$299)+\$ 0}\text{.15}\times \text{(45000-54000)=\$11,700}\text{.}\\ 54000\text{ (}18000miles\times 3yrs):\text{ 36(\$299)+\$ 0}\text{.15}\times \text{(54000-54000)=\$11,700}\text{.}\end{array}$

Thus, the payoff table for Amy’s problem is given as:

	Actual Miles Driven Annually
Decision	12000	15000	18000
Hepburn Honda	$10,764	$12,114	$13,464
Midtown Motors	$11,160	$11,160	$12,960
Hopkins Automotive	$11,700	$11,700	$11,700

To determine

Find the decision using the optimistic, conservative and minimax regret approaches.

Answer to Problem 3P

The Hepburn Honda lease option gives the optimistic approach because it has the smallest minimum profit.

The Hopkins Automotive lease option gives the conservative approach because it has the smallest maximum profit.

The minimax regret is the Hopkins Automotive lease option because it minimizes the maximum regret.

Explanation of Solution

By using the decision tree in Part (a), the maximum and minimum profit based on the decisions Hepburn Honda, Midtown Motors and Hopkins Automotive.

Decision	Maximum Profit	Minimum Profit
Hepburn Honda	$10,764	$13,464
Midtown Motors	$11,160	$12,960
Hopkins Automotive	$11,700	$11,700

Optimistic approach:

The optimistic approach evaluates each decision alternative in terms of best payoff that can occur.

The Hepburn Honda lease option gives the optimistic approach because it has the smallest minimum profit (from the above table).

Conservative approach:

The conservative approach evaluates each decision alternative in terms of worst payoff that can occur.

The Hopkins Automotive lease option gives the conservative approach because it has the smallest maximum profit (from the above table).

Minimax Regret approach:

The minimax regret approach is the difference between the payoff associated with a particular alternative and payoff associated with the most decision that would yield the most desirable payoff for a given state of nature.

Regret or opportunity loss table:

Decision	12000	15000	18000	Maximum Regret
Hepburn Honda	$0	$954	$1,764	$1,764
Midtown Motors	$396	$0	$1,260	$1,260
Hopkins Automotive	$936	$540	$0	$936

The maximum regret for the decision Hepburn Honda is $1,764, Midtown Motors is $1,260 and Hopkins Automotive is $936.

Here, the Hopkins Automotive lease option has been selected because it minimizes the maximum regret.

To determine

Find the expected value approach for the probabilities of 0.5, 0.4 and 0.1.

Answer to Problem 3P

The expected value approach results in Midtown Motors lease option.

Explanation of Solution

The formula to find expected value (EV) of decision alternative d_i is given as follows:

$\text{EV(}{d}_i)={\displaystyle \sum _{j=1}^{N}P(s_j)V_{ij}}$

Here, the expected value for the payoffs associated with each of Amy’s three alternatives:

$\begin{array}{c}\text{EV(Hepburn Honda) =0}\text{.5(\$10,764)+0}\text{.4(\$12,114)+0}\text{.1(\$13,464)}\\ \text{ =\$11,574}\text{.}\\ \text{EV(Midtown Motors) =0}\text{.5(\$11,160)+0}\text{.4(\$11,160)+0}\text{.1(\$11,160)}\\ \text{ =\$11,340}\text{.}\\ \text{EV(Hopkins Automotive) =0}\text{.5(\$11,700)+0}\text{.4(\$11,700)+0}\text{.1(\$11,700)}\\ \text{ =\$11,700}\text{.}\end{array}$

The expected value approach results in Midtown Motors lease option because it has the minimum expected value of the three alternatives.

To determine

Find the most likely cost and its probability using risk analysis.

Answer to Problem 3P

The risk analysis for midtown motors, the most likely cost is $11,160 with the probability of 0.9.

Explanation of Solution

From the data in part (d).

The risk profile for the decision to lease from Midtown Motors is:

From the above risk analysis for midtown motors, the most likely cost is $11,160 with the probability of 0.9.

Here, there are only two unique costs on this graph because for the decision alternative (midtown motors) have two unique payoffs. The payoffs are associated with the three chance outcomes- the payoff (cost) associated with midtown motors lease is same for the two of the chance outcomes.

To determine

Find the expected value approach for the probabilities of 0.3, 0.4 and 0.3.

Answer to Problem 3P

The expected value approach results in either Midtown Motors lease option or Hopkins Automotive lease option.

Explanation of Solution

The formula to find expected value (EV) of decision alternative d_i is given as follows:

$\text{EV(}{d}_i)={\displaystyle \sum _{j=1}^{N}P(s_j)V_{ij}}$

Here, the expected value for the payoffs associated with each of Amy’s three alternatives:

$\begin{array}{c}\text{EV(Hepburn Honda) =0}\text{.3(\$10,764)+0}\text{.4(\$12,114)+0}\text{.3(\$13,464)}\\ \text{ =\$12,114}\text{.}\\ \text{EV(Midtown Motors) =0}\text{.3(\$11,160)+0}\text{.4(\$11,160)+0}\text{.3(\$11,160)}\\ \text{ =\$11,700}\text{.}\\ \text{EV(Hopkins Automotive) =0}\text{.3(\$11,700)+0}\text{.4(\$11,700)+0}\text{.3(\$11,700)}\\ \text{ =\$11,700}\text{.}\end{array}$

The expected value approach results in either Midtown Motors lease option or Hopkins Automotive lease option because both have the same minimum expected value of the three alternatives.