The Shop at Home Network sells various household goods during live television 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 Square Feet Needed 20,000 30,000
40,00035,000 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 Cost per Sq. Ft. Leased$55$95 $130$155S S185 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. Similarly, at the start of month 3 it can lease any
amount of space for 2 months at a cost of S95 per square foot. The company wants to determine the least costly way of meeting its
w arehousing needs over the coming 5 months. a. Formulate an LP model for this problem. b. Create a spreadsheet model for this
problem and solve it using Solver c. What is the optimal solution? d. How much would it cost the company to meet its space needs if
in each month it leases for one month exactly the amount of space required for the month?
x
Let ij is the number of square feet leased by the company at the starting of ith month for the
period of j months.
The prices for month ‘i’ to month ‘j’ was given as
c11 $55; c12 $95; c13 $130; c14 $155; c15 $185;
The Objective is to minimize the total lease cost.
The amount of square feet leased by the company for one month
x11 , x21 , x31 , x41 , x51 with the
cost $55
The amount of square feet leased by the company for two months
x12 , x22 , x32 , x42 with the
cost $95
The amount of square feet leased by the company for three months
x13 , x23 , x33 with the cost
of $130
The amount of square feet leased by the company for four months
x14 , x24 with the cost of
$155
The amount of square feet leased by the company in last month
x15 with the cost of $185
The objective function is
Minimum z =
$55 x11 x21 x31 x41 x51 $95 x12 x22 x32 x42 $130 x13 x23 x33
$155 x14 x24 185 x15
The constraints are for first month requirement
x11 x12 x13 x14 x15 20,000
5
x
2
1j x21 x22 x23 x24 30,000
Second month requirement
5 4
x x
3
1j
2
2j x31 x32 x33 40,000
For third month
5 4 3
x1 j x2 j x3 j x41 x42 35,000
4 3 2
For fourth month
For last month
x15 x24 x33 x42 x51 50,000
xij 0
All the
LP Model:
Min z $55 x11 x21 x31 x41 x51 $95 x12 x22 x32 x42
$130 x13 x23 x33 $155 x14 x24 185 x15
Subject to constraints
x11 x12 x13 x14 x15 20,000
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 Square Feet Needed 20,000 30,000
40,00035,000 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 Cost per Sq. Ft. Leased$55$95 $130$155S S185 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. Similarly, at the start of month 3 it can lease any
amount of space for 2 months at a cost of S95 per square foot. The company wants to determine the least costly way of meeting its
w arehousing needs over the coming 5 months. a. Formulate an LP model for this problem. b. Create a spreadsheet model for this
problem and solve it using Solver c. What is the optimal solution? d. How much would it cost the company to meet its space needs if
in each month it leases for one month exactly the amount of space required for the month?
x
Let ij is the number of square feet leased by the company at the starting of ith month for the
period of j months.
The prices for month ‘i’ to month ‘j’ was given as
c11 $55; c12 $95; c13 $130; c14 $155; c15 $185;
The Objective is to minimize the total lease cost.
The amount of square feet leased by the company for one month
x11 , x21 , x31 , x41 , x51 with the
cost $55
The amount of square feet leased by the company for two months
x12 , x22 , x32 , x42 with the
cost $95
The amount of square feet leased by the company for three months
x13 , x23 , x33 with the cost
of $130
The amount of square feet leased by the company for four months
x14 , x24 with the cost of
$155
The amount of square feet leased by the company in last month
x15 with the cost of $185
The objective function is
Minimum z =
$55 x11 x21 x31 x41 x51 $95 x12 x22 x32 x42 $130 x13 x23 x33
$155 x14 x24 185 x15
The constraints are for first month requirement
x11 x12 x13 x14 x15 20,000
5
x
2
1j x21 x22 x23 x24 30,000
Second month requirement
5 4
x x
3
1j
2
2j x31 x32 x33 40,000
For third month
5 4 3
x1 j x2 j x3 j x41 x42 35,000
4 3 2
For fourth month
For last month
x15 x24 x33 x42 x51 50,000
xij 0
All the
LP Model:
Min z $55 x11 x21 x31 x41 x51 $95 x12 x22 x32 x42
$130 x13 x23 x33 $155 x14 x24 185 x15
Subject to constraints
x11 x12 x13 x14 x15 20,000
