1. Management of Sundown Rent-a-Car (see Problem 8-8) has decided that perhaps the cost during the six-month period is not the appropriate Cost to minimize because the agency may still be obligated to additional months on some leases after that time. For example, if Sundown had some cars delivered at the beginning of the sixth month, Sundown would still he obligated for two additional months on a three-month lease. Use LP to determine how many cars should be leased in each month on each type of lease to minimize the cost of leasing over the entire life of these leases.
Problem 8-8:
(Automobile leasing problem) Sundown Rent-a-Car, a large automobile rental agency operating in the Midwest, is preparing a leasing strategy for the next six months. Sundown leases cars from an automobile manufacturer and then rents them to the public on a daily basis. A forecast of the demand for Sundown's cars in the next six months follows:
Month March April May June July August
Demand 420 400 430 460 470 440
Cars may be leased from the manufacturer for either three, four, or five months. These are leased on the first day of the month and are returned on the last day of the month. Every six months the automobile manufacturer is notified by Sundown about the number of cars needed during the next six months. The automobile manufacturer has stipulated that at least 50% of the cars leased during a six-month period must be on the five-month lease. The cost per month on each of the three types of leases are $420 for the three-month lease, $400 for the four-month lease, and $370 for the five-month lease.
Currently, Sundown has 390 cars. The lease on 120 cars expires at the end of March. The lease on another 140 cars expires at the end of April, and the lease on the rest of these expires at the end of May.
Use LP to determine how many cars should be leased in each month on each type of lease to minimize the cost of leasing over the six-month period. How many cars are left at the end of August?
quiero resuelto
To solve this problem, we can use linear programming (LP) to determine how many cars should be leased in each month on each type of lease to minimize the cost of leasing over the entire six-month period. LP involves formulating an objective function and a set of constraints. The objective is to minimize the cost of leasing, and the constraints include the demand for cars in each month, the lease durations, and the manufacturer's requirement.
Here are the steps to approach this problem using LP:
1. Define the decision variables:
Let x_i_j represent the number of cars leased in month i on a lease duration of j (where i is the month index and j can be 3, 4, or 5).
2. Formulate the objective function:
The objective is to minimize the cost of leasing over the six-month period. The cost of leasing in each month can be calculated as the product of the lease duration and the monthly cost of each lease.
minimize Z = 420*x_1_3 + 400*x_1_4 + 370*x_1_5 +
420*x_2_3 + 400*x_2_4 + 370*x_2_5 +
...
420*x_6_3 + 400*x_6_4 + 370*x_6_5
3. Add constraints:
- Demand constraints: The number of cars leased in each month should meet or exceed the demand for cars in that month.
x_1_3 + x_1_4 + x_1_5 >= 420 (March demand)
x_2_3 + x_2_4 + x_2_5 >= 400 (April demand)
x_3_3 + x_3_4 + x_3_5 >= 430 (May demand)
x_4_3 + x_4_4 + x_4_5 >= 460 (June demand)
x_5_3 + x_5_4 + x_5_5 >= 470 (July demand)
x_6_3 + x_6_4 + x_6_5 >= 440 (August demand)
- Lease duration constraints: The sum of the cars leased in each month on each lease duration should be equal to the number of cars leased for that duration.
x_1_3 + x_2_3 + x_3_3 + x_4_3 + x_5_3 + x_6_3 = 390 (cars currently leased for 3 months)
x_1_4 + x_2_4 + x_3_4 + x_4_4 + x_5_4 + x_6_4 = 120 (March expiration)
x_1_5 + x_2_5 + x_3_5 + x_4_5 + x_5_5 + x_6_5 = 140 (April expiration) + (May expiration - 140)
- Manufacturer's requirement constraint: At least 50% of the cars leased during a six-month period must be on the five-month lease.
(x_1_5 + x_2_5 + x_3_5 + x_4_5 + x_5_5 + x_6_5) >= 0.5 * (x_1_3 + x_2_3 + x_3_3 + x_4_3 + x_5_3 + x_6_3 + x_1_4 + x_2_4 + x_3_4 + x_4_4 + x_5_4 + x_6_4 + x_1_5 + x_2_5 + x_3_5 + x_4_5 + x_5_5 + x_6_5)
4. Solve the LP problem:
Solve the LP problem using linear programming software or tools. The solution will provide the optimal values for the decision variables that minimize the cost of leasing over the six-month period.
Once you have the optimal values for the decision variables, you can determine the number of cars left at the end of August by subtracting the total number of cars leased from the initial number of cars (390).
Note: This solution assumes that the number of cars available for lease is not limited by the manufacturer or any other constraint.