[8-8] (Automobile leasing 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?

Minimize cost = 420X₁ + 400X₂ + 370X₃

X₁ = รถที่มีสัญญาเช่า 3 เดือน

X₂ = รถที่มีสัญญาเช่า 4 เดือน

X₃ = รถที่มีสัญญาเช่า 5 เดือน

Constraints

          X₁ +X₂+ X₃ ≥ 2010

                      X₃ ≥ 0.5(X₁ +X₂+ X₃)

          X₁             ≥ 170

               X₂        ≥ 290+ X₁

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[8-8] (Automobile leasing problem)
				min
variable(Decision variable)	x1	x2	x3	objective function
units produced (จำนวนเช่ารถ)	170	460	1380	766000
cost per unit	420	400	370
constraints				LHS		RHS
c1	1	1	1	2010	>=	2010
c2	-0.5	-0.5	5	6585	>=	1005
c3	1			170	>=	170
c4	-1	1		290	>=	290

ดังนั้น บริษัท ควรมีการจัดการล่วงหน้า 6 เดือนโดยเช่ารถ

X₁ = รถที่มีสัญญาเช่า 3 เดือน = 170 คัน

X₂ = รถที่มีสัญญาเช่า 4 เดือน = 460 คัน

X₃ = รถที่มีสัญญาเช่า 5 เดือน = 1380 คัน