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.