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:

March	April	May	June	July	August
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 is $420 for the three-month lease

The cost per month is $400 for the four-month lease

The cost per month is $370 for the five-month lease.

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,

The lease on another 130 cars 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?

Let X_ij = number of new leases in month ifor j-months, i = 1, . . . , 6; j = 3, 4, 5

Minimize cost = 1260X₁₃ + 1260X₂₃ + 1260X₃₃ + 1260X₄₃ + 840X₅₃ + 420X₆₃ + 1600X₁₄ + 1600X₂₄ + 1600X₃₄ + 1200X₄₄ + 800X₅₄ + 400X₆₄+ 1850X₁₅ + 1850X₂₅ + 1480X₃₅ + 1110X₄₅ + 740X₅₅ + 370X₆₅

Note: X₁₃ is a designation for the 1 month of cars leased for 3 months; X₂₃is 2nd month and so on. The $1260 = total cost for 3 month lease(420*3).

subject to: X₁₃ + X₁₄ + X₁₅ 420 390

X₁₃ + X₁₄ + X₁₅ + X₂₃ + X₂₄ + X₂₅ 400 270

X₁₃ + X₁₄ + X₁₅ + X₂₃ + X₂₄ + X₂₅ + X₃₃ + X₃₄ + X₃₅ 430 130

X₁₄ + X₁₅ + X₂₃ + X₂₄ + X₂₅ + X₃₃ + X₃₄ + X₃₅ + X₄₃+ X₄₄ + X₄₅ 460

X₁₅ + X₂₄ + X₂₅ + X₃₃ + X₃₄ + X₃₅ + X₄₃ + X₄₄ + X₄₅+ X₅₃ + X₅₄ + X₅₅ 470

X₂₅ + X₃₄ + X₃₅ + X₄₃ + X₄₄ + X₄₅ + X₅₃ + X₅₄ + X₅₅+ X₆₃ + X₆₄ + X₆₅ 440

X₁₅ + X₂₅ + X₃₅ + X₄₅ + X₅₅ + X₆₅ 0.50(X₁₃ + X₁₄+ X₁₅ + X₂₃ + X₂₄ + X₂₅ + X₃₃ + X₃₄ + X₃₅ + X₄₃ + X₄₄+ X₄₅ + X₅₃ + X₅₄ + X₅₅ + X₆₃ + X₆₄ + X₆₅)

All variables 0

Solving this on the computer results in the following solution:

X₁₅ = 30 5-month leases in March

X₂₅ = 100 5-month leases in April

X₃₅ = 170 5-month leases in May

X₄₅ = 160 5-month leases in June

X₅₅ = 10 5-month leases in July

All other variables equal 0.

Total cost = $677,100.