The Shop at Home Network sells various household goods during live TV broadcasts

Question

The Shop at Home Network sells various household goods during live TV broadcasts. The company owns several warehouses to hold many of the goods it sells but also leases extra warehouse space when needed. During the next five months the company expects that it will need to lease the following amounts of extra warehouse space:

Month 1 2 3 4 5
Sq. ft. needed 20,000 30,000 40,000 35,000 50,000

At the beginning of any month, the company can lease extra space for one or more months at the following costs:

Lease term (months) 1 2 3 4 5
Cost per sq. ft. leased $55 $95 $130 $155 $185

So, for instance, at the start of month 1 the company can lease as much space as it wants for 4 months at a cost of $155 per square foot. The company wants to know the least costly way of meeting its warehousing needs over the coming 5 months.

a. Formulate the problem as a linear programming model (i.e. define the variables, and write down the objective function and all constraints in algebraic form).

Explanation / Answer

Let xit be the amount of space leased at the beginning of month i for t months
Let yt be the amount of space leased in time period t


Objective is to minimize cost and is given by
xi1*55 + xi2*95 + xi3*130 + xi4*155 + xi4*185 sum(i) i in 1,2,3.4.5

subject to the following constraints

x11 <= 20000, x12<= 30000, x13<= 40000, x14 <= 35000, x15 <= 50000
x21 <= 30000, x22<= 40000, x23 <= 40000, x24 <= 50000
x31 <= 40000, x32<= 40000, x33 <= 50000
x41 <= 35000, x42 <= 50000
x51 <= 50000
y1>=20000, y2>=30000, y3>=40000, y4>=35000, y5>=50000
x1i>=y1, x2i >= y2, x3i >= y3, x4i >= y4, x5i >= y5 for all i = 1,2,3,4,5

xii >=0 for all i = 1,2,3,4,5

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