Problem 3 A car dealer is offering three two-year leasing options Plan Fixed Mon
ID: 416970 • Letter: P
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Problem 3 A car dealer is offering three two-year leasing options Plan Fixed Monthly Payment $200 $300 Additional cost per mile 0 095 per mile $0.061 for the first 6000, miles; $0.050 thereafter $0.000 for the first 6000 miles; S0.14 per mile thereafter II S170 Assume that a customer expects to drive between 15,000 and 35,000 miles during the next two years according to the following probability distribution: P(driving 15,000 miles) 0.1 P(driving 20,000 miles) = 0.2 Pi(driving 25,000 miles) 0.2 P(driving 30,000 miles)- 0.3 P(driving 35,000 miles) = 0.2 a. Construct a payoff matrix for this problem (keep in mind that the "payoffs here are costs, where less is better. So construct your payoff matrix in terms of negative rewards and use the rules normally)Explanation / Answer
a.
The payoff for event and alternative is calculated as follows:
Payoff = Fixed cost + Total miles cost till 6000 miles + Total miles cost over 6000 miles
Fixed cost = 2 years x 12 months x Fixed cost per month
Total miles cost upto 6000 miles = (Cost per miles upto 6000 miles) x 6000 miles
Total miles cost over 6000 miles = (Cost per miles over 6000 miles) x (miles traveled - 6000 miles)
For example if the individual selects plan II and drives upto 25,000 miles in 2 years, the payoff of total cost is given as follow:
Payoff = (2 x 12 x $300) + ($0.061 x 6000 miles) + ($0.050 x (25,000 – 6,000) miles) =
Payoff = 7200 + 366 + 950 = $8,516
Payoff matrix in terms of cost
Miles to drive
Plan
Fixed monthly Payment
Cost per mile upto 6000
Cost per mile over 6000
15000
20000
25000
30000
35000
I
200
0.095
0.095
6225
6700
7175
7650
8125
II
300
0.061
0.05
8016
8266
8516
8766
9016
III
170
0
0.14
5340
6040
6740
7440
8140
The Payoff in terms of Cost is as follows:
The payoff matrix in term of rewards is a shown below:
Miles to drive
Maximum Payoffs
Minimum Payoffs
Expected Payoffs
Probability
0.1
0.2
0.2
0.3
0.2
Plan
15000
20000
25000
30000
35000
I
-$6,225.00
-$6,700.00
-$7,175.00
-$7,650.00
-$8,125.00
-$6,225.00
-$8,125.00
-$7,317.50
II
-$8,016.00
-$8,266.00
-$8,516.00
-$8,766.00
-$9,016.00
-$8,016.00
-$9,016.00
-$8,591.00
III
-$5,340.00
-$6,040.00
-$6,740.00
-$7,440.00
-$8,140.00
-$5,340.00
-$8,140.00
-$6,950.00
Maximum Overall payoff
Maximum Overall payoff
Maximum Overall payoff
-$5,340.00
-$8,125.00
-$6,950.00
b. Optimistic or Maximax approach:
Under this approach decision maker will evaluate the decision alternatives, in terms of the best payoff that can occur or the alternative that provides the best of all possible maximum payoffs of each alternative selected. Under maximax principle, take the highest payoff under each alternative and then select the largest of those minimum payoffs. This criterion is also known as optimistic approach.
Maximum of Maximum payoffs from three plans = - $5,340 for plan III.
According to Maximax Approach: Plan III should be selected
C. Pessimistic or Maximin Approach:
Under this approach decision maker will evaluate the decision alternatives, in terms of the worst payoff that can occur or the alternative that provides the best of all possible minimum payoffs of each alternative selected. Under maximin principle, take the smallest payoff under each alternative and then select the largest of those minimum payoffs. This criterion is also known as conservative or pessimistic approach
Maximum of Minimum payoffs from three alternatives = - $8,125 for plan I
According to Maximin Approach: Plan I should be selected
D. Expected Value Decision rule:
Expected payoff for each alternative is calculated as sum of products of payoffs under each state of nature by the probability of the responsive state and sum the products.
Maximum expected value is -$6,950 for plan III
According to expected value decision select Plan III
Payoff matrix in terms of cost
Miles to drive
Plan
Fixed monthly Payment
Cost per mile upto 6000
Cost per mile over 6000
15000
20000
25000
30000
35000
I
200
0.095
0.095
6225
6700
7175
7650
8125
II
300
0.061
0.05
8016
8266
8516
8766
9016
III
170
0
0.14
5340
6040
6740
7440
8140
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