MOTOR CITY RENTAL Employees at Motor City are interested in determining the best
ID: 460389 • Letter: M
Question
MOTOR CITY RENTAL
Employees at Motor City are interested in determining the best fleet purchase assortment for the coming year. There are 4 types of vehicles that can be purchased: eco, standard, sports and SUV. All cars purchased are depreciated and paid off for over a two year period, after which they are sold in a secondary market. Table 1 below presents the wholesale purchase price and revenues (the second year includes the resale value) over the two-year period for each type of car:
Table 1 - Purchase Price and Net Revenue Summary
Type
Price ($)
1st yr. ($)
2nd yr. ($)
Eco
40,000
28,000
46,000
Standard
30,000
18,000
30,000
Sports
24,000
14,000
26,000
SUV
18,000
10,000
18,000
Motor City has established a budget of $10 million for funding the purchases this year. It can either pay the entire amount for the cars or make a down payment and pay the remaining balance over a two-year period. The loan company requires at least a 20% down payment and that at least 50% of the purchase price plus interest must be paid by the end of the first year. The loan company is currently charging 8%. Motor City uses a 10% discount rate for financial planning. Employees estimates that they can rent all of the cars that are purchased. However, in order to meet overall market demand they wish that each vehicle category represent at least 15 percent and no more than 50 percent of the total number of vehicles purchased.
1. Formulate a linear programming model for this problem. 2. Define the basic assumptions associated with this problem. 3. Determine the optimal fleet-mix to purchase including the payment schedule. 4. What is the impact on the mix if the financing charge is reduced to 6%?
5. What is the impact on the mix if they require at least 25% of each car type?
6. What is the maximum discount rate that supports paying for the cars over time?
Type
Price ($)
1st yr. ($)
2nd yr. ($)
Eco
40,000
28,000
46,000
Standard
30,000
18,000
30,000
Sports
24,000
14,000
26,000
SUV
18,000
10,000
18,000
Explanation / Answer
1) Linear Programming Model:
Minimize the cost:
40,000x1+30,000x2+24,000x3+18,000x4
Subject to the constraint
28,000x1+6,000x2<46,000
18,000x1+0x2<30,000
14,000x1+2000x2<26,000
10,000x1+0x2<18,000
x1+x2>0
Calculation for 2nd year:
Eco=46,000- 40,000=$6,000
Standard=30,000 - 30,000=$0
Sports=26,000 - 24,000=2,000
SUV=18,000- 18,000=$0
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3) Payment schedule:
Eco=40,000*20/100=$8,000(down payment)
First year Eco=40,000 -8,000=$32,000 *50/100=16,000+$1280(interest)(16,000*8/100=1280)=$17,280
Standard=30,000 *20/100=$6,000(down payment)
First year Eco=30,000 -6,000=$24,000 +$1,920(interest)(24,000*8/100=$1,920)=$25,920
Sports=24,000*20/100=$4,800
First year =24,000-$4,800=$19,200+$1536(interest)($19,200 *8/100=$1536)=$20,736
SUV=18,000 *20/100=$3,600
First year= 18,000 -$3600=$14,400+$1152(interest)($14,400*8/100=$1152)=$15,552
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