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 rented 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.
Management of Sundown Rent-a-car 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 six-month, Sundown would still be 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.

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Ete a la vergaa

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, we can use Linear Programming (LP). LP is a mathematical optimization technique that helps find the optimal solution for a given set of constraints.

Let's define some variables:
- Let x3, x4, and x5 represent the number of cars leased for three, four, and five months, respectively, in a particular month.
- Let C3, C4, and C5 represent the costs per month for a three-month, four-month, and five-month lease, respectively.
- Let D represent the demand for cars in a particular month.

The objective is to minimize the total cost of leasing over the entire six-month period. The total cost can be calculated as:
Total Cost = (x3 * C3 + x4 * C4 + x5 * C5)

However, we have several constraints to consider:
1. Sundown Rent-a-Car must satisfy the demand for cars in each month:
x3 + x4 + x5 >= D

2. At least 50% of the cars rented during a six-month period must be on a five-month lease:
x5 >= 0.5 * (x3 + x4 + x5)

3. The number of leased cars must cover the demand and replace expired leases:
x3 >= D - 390 - 120
x4 >= D - 390 - 140
x5 >= D - 390

4. The number of leased cars cannot exceed the demand in each month:
x3 <= D
x4 <= D
x5 <= D

Using these variables and constraints, we can formulate an LP problem and solve it using software like Microsoft Excel Solver or specialized LP solvers.

1. Set up the LP model by defining the decision variables, objective function, and constraints.

2. Input the given data such as demand and lease costs into the LP model.

3. Define the objective function as minimizing the total cost, calculated as (x3 * C3 + x4 * C4 + x5 * C5).

4. Set up the constraints based on the demand, lease duration, and the requirements of at least 50% on a five-month lease.

5. Solve the LP problem using an LP solver or software.

6. Analyze the solution: The solver will provide the optimal values for x3, x4, and x5. These values represent the number of cars to be leased for three, four, and five months in each month.

By following these steps, you can use LP to determine the optimal number of cars to be leased in each month on each type of lease to minimize the cost of leasing over the entire six-month period.