Wal-Mart, a discount store chain, is planning to build a new store in
Rock Springs, Maryland. The parcel of land the company owns is large
enough to accommodate a store with 140,000 square feet of floor space.
Based on marketing and demographic surveys of the area and historical
data from its other stores, Wal-Mart estimates its annual profit
contribution per square foot for each of the store's departments to be
as shown in the following table.
Department Profit contribution per ft2
Men's clothing $4.25
Women's clothing $5.10
Children's clothing $4.50
Toys $5.20
Housewares $4.10
Electronics $4.90
Auto supplies $3.80
Each department must have at least 15,000 ft2 of floor space and no
department can have more than 20% of the total retail floor space. Men's
women's and children's clothing plus housewares keep all their stock on
the retail floor; however, toys, electronics, and auto supplies keep
some items (bicycles, televisions, tires, etc.) in inventory. Thus, 10%
of the total retail floor space devoted to these three departments must
be set aside outside the retail area for stocking inventory.
Formulate a linear programming model that can be used to determine
the floor space that should be devoted to each department in order to
maximize profit contribution.
Maximize: 4.25x + 5.10y + 4.50z + 5.20w + 4.10u + 4.90v + 3.80t
Subject to:
x + y + z ≥ 15000
x + y + z ≤ 140000
w + u + v ≤ 140000
w + u + v ≥ 15000
x + y + z ≤ 0.2(140000)
w + u + v ≤ 0.1(140000)
x, y, z, w, u, v, t ≥ 0
Let's denote the floor space to be allocated to each department as follows:
Let x1 = floor space for Men's clothing
Let x2 = floor space for Women's clothing
Let x3 = floor space for Children's clothing
Let x4 = floor space for Toys
Let x5 = floor space for Housewares
Let x6 = floor space for Electronics
Let x7 = floor space for Auto supplies
The objective is to maximize the total profit contribution, which can be represented as:
Maximize: 4.25x1 + 5.10x2 + 4.50x3 + 5.20x4 + 4.10x5 + 4.90x6 + 3.80x7
Now let's incorporate the constraints:
1) Each department must have at least 15,000 ft2 of floor space:
x1 ≥ 15000
x2 ≥ 15000
x3 ≥ 15000
x4 ≥ 15000
x5 ≥ 15000
x6 ≥ 15000
x7 ≥ 15000
2) No department can have more than 20% of the total retail floor space:
x1 + x2 + x3 + x4 + x5 + x6 + x7 ≤ 0.2(140000)
3) 10% of the total retail floor space devoted to Toys, Electronics, and Auto supplies must be set aside outside the retail area for stocking inventory:
(x4 + x6 + x7) - 0.1(140000) = 0
All variables should be non-negative.
The complete linear programming model is as follows:
Maximize: 4.25x1 + 5.10x2 + 4.50x3 + 5.20x4 + 4.10x5 + 4.90x6 + 3.80x7
Subject to:
x1 ≥ 15000
x2 ≥ 15000
x3 ≥ 15000
x4 ≥ 15000
x5 ≥ 15000
x6 ≥ 15000
x7 ≥ 15000
x1 + x2 + x3 + x4 + x5 + x6 + x7 ≤ 0.2(140000)
(x4 + x6 + x7) - 0.1(140000) = 0
x1, x2, x3, x4, x5, x6, x7 ≥ 0
This model will allow us to determine the floor space that should be devoted to each department in order to maximize the profit contribution.
To formulate a linear programming model to determine the floor space that should be devoted to each department in order to maximize profit contribution, we need to define the decision variables, objective function, and constraints.
Decision variables:
Let's denote the floor space devoted to each department as follows:
x1 = Men's clothing
x2 = Women's clothing
x3 = Children's clothing
x4 = Toys
x5 = Housewares
x6 = Electronics
x7 = Auto supplies
Objective function:
The objective is to maximize the profit contribution. The profit contribution per square foot for each department is given, so the objective function is:
Maximize: 4.25x1 + 5.10x2 + 4.50x3 + 5.20x4 + 4.10x5 + 4.90x6 + 3.80x7
Constraints:
1. Total floor space constraint: The total floor space of the store should not exceed 140,000 square feet.
x1 + x2 + x3 + x4 + x5 + x6 + x7 ≤ 140,000
2. Department-specific constraints:
a. Each department must have at least 15,000 square feet of floor space.
x1 ≥ 15,000
x2 ≥ 15,000
x3 ≥ 15,000
x4 ≥ 15,000
x5 ≥ 15,000
x6 ≥ 15,000
x7 ≥ 15,000
b. No department can have more than 20% of the total retail floor space.
x1 ≤ 0.2 * (x1 + x2 + x3 + x4 + x5 + x6 + x7)
x2 ≤ 0.2 * (x1 + x2 + x3 + x4 + x5 + x6 + x7)
x3 ≤ 0.2 * (x1 + x2 + x3 + x4 + x5 + x6 + x7)
x4 ≤ 0.2 * (x1 + x2 + x3 + x4 + x5 + x6 + x7)
x5 ≤ 0.2 * (x1 + x2 + x3 + x4 + x5 + x6 + x7)
x6 ≤ 0.2 * (x1 + x2 + x3 + x4 + x5 + x6 + x7)
x7 ≤ 0.2 * (x1 + x2 + x3 + x4 + x5 + x6 + x7)
c. Inventory space constraint (10% outside retail area for toys, electronics, and auto supplies):
0.10 * (x4 + x6 + x7) = (x4 + x6 + x7) - 0.10 * (x4 + x6 + x7)
All the decision variables and constraints need to be non-negative.
This linear programming model will help determine the floor space that should be devoted to each department in order to maximize the profit contribution for the new Wal-Mart store in Rock Springs, Maryland.