A real estate developer is planning to build an apartment building specifically for graduate students on a parcel of land adjacent to a major university. Four types of apartments can be included in the building: efficiencies, one-bedroom, two-bedroom, and three-bedroom units.

- Each efficiency requires 500 square feet.
- Each one-bedroom apartment requires 700 square feet.
- Each two-bedroom apartment requires 800 square feet.
- Each three-bedroom unit requires 1000 square feet.

The developer believes that the building should include no more than 15 one-bedroom units, 22 two-bedroom units, and 10 three-bedroom units. Local zoning ordinances do not allow the developer to build more than 40 units in this location and restrict the building to a maximum of 40,000 square feet. The developer has already agreed to lease 5 one-bedroom units and 8 two-bedroom units to a local rental agency that is a "silent partner" in this endeavor.

Market studies indicate that:
- Efficiencies can be rented for $350 per month.
- One-bedrooms for $450 per month.
- Two-bedrooms for $550 per month.
- Three-bedrooms for $750 per month.

How many rental units of each type should the developer include in the building plans in order to maximize the potential rental income from this building?

1. Formulate a linear programming model for this problem.
2. Create a spreadsheet model for this problem and solve it using Solver.

Solver will provide the optimal values for x1, x2, x3, x4, which represent the number of rental units of each type that should be included in the building plans to maximize the potential rental income.

Linear Programming Model:

Let x1 be the number of efficiencies,

x2 be the number of one-bedroom units,

x3 be the number of two-bedroom units,

x4 be the number of three-bedroom units.

Objective function: Maximize Z = 350x1 + 450x2 + 550x3 + 750x4

Subject to the following constraints:

Efficiency units: x1 >= 0

One-bedroom units: x2 <= 15

Two-bedroom units: x3 <= 22

Three-bedroom units: x4 <= 10

Total number of units: x1 + x2 + x3 + x4 <= 40

Total square footage: 50x1 + 700x2 + 800x3 + 1000x4 <= 40000

Spreadsheet Model using Solver:

Create a spreadsheet with the following columns:

A: Apartment Type

B: Number of Units

C: Square Footage

D: Rental Income

In the Apartment Type column (A), list the types of apartments: Efficiency, One-Bedroom, Two-Bedroom, Three-Bedroom.

In the Number of Units column (B), enter the variables x1, x2, x3, x4.

In the Square Footage column (C), enter the corresponding square footage values for each apartment type.

In the Rental Income column (D), calculate the rental income for each apartment type using the formula:

=Number of Units * Monthly Rent

Set up Solver by selecting the Objective cell (D), set it to Max, and select the changing cells (B2:B5) as the variable cells.

Set the constraints by adding the following constraints:

One-bedroom units <= 15

Two-bedroom units <= 22

Three-bedroom units <= 10

Total number of units <= 40

Total square footage <= 40000

Click Solve to find the optimal solution that maximizes the rental income.

Solver will provide the optimal values for x1, x2, x3, x4, which represent the number of rental units of each type that should be included in the building plans to maximize the potential rental income.

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