To formulate a linear programming model for this problem, we need to define the decision variables, objective function, and constraints.
1. Decision Variables:
Let's denote the floor space allocated to each department as follows:
- x1: Men's clothing floor space (in ft2)
- x2: Women's clothing floor space (in ft2)
- x3: Children's clothing floor space (in ft2)
- x4: Toys floor space (in ft2)
- x5: Housewares floor space (in ft2)
- x6: Electronics floor space (in ft2)
- x7: Auto supplies floor space (in ft2)
2. Objective Function:
The objective is to maximize the total profit contribution. The profit contribution from each department is given per square foot. So, the objective function can be written as:
Maximize Z = 4.25x1 + 5.10x2 + 4.50x3 + 5.20x4 + 4.10x5 + 4.90x6 + 3.80x7
3. Constraints:
We need to consider the following constraints:
- Each department must have at least 15,000 ft2 of floor space:
x1 ≥ 15,000
x2 ≥ 15,000
x3 ≥ 15,000
x4 ≥ 15,000
x5 ≥ 15,000
x6 ≥ 15,000
x7 ≥ 15,000
- No department can have more than 20% of the total retail floor space:
x1 + x2 + x3 + x4 + x5 + x6 + x7 ≤ 0.2 * (x1 + x2 + x3 + x4 + x5 + x6 + x7)
- Inventory space constraint (10% of total retail floor space for toys, electronics, and auto supplies):
0.1 * (x4 + x6 + x7) = x4 + x6 + x7 - (x4 + x6 + x7)
- Total floor space should not exceed 140,000 ft2:
x1 + x2 + x3 + x4 + x5 + x6 + x7 ≤ 140,000
All the variables are non-negative.
This formulation represents a linear programming model that can be used to determine the optimal floor space allocation in order to maximize profit contribution for Wal-Mart's new store in Rock Springs, Maryland.