THE PENN-OHIO BUILDING COMPANY BUILDS AND SELLS TWO TYPES OF HOUSES:"CLEVELAND CASTLES" AND "PITTSBUTGH PALACES". CONTRACTS CALL FOR USING TOTALS OF AT LEAST 20,000 FT OF FRAMING LUMBER AND 8,000 CUBIC FT OF CONCRETE; AND AT LEAST $9,000 FOR PROMOTION AND ADVERTISING. EACH CLEVELAND CASTLE REQUIRES 4,000 FT OF FRAMING LUMBER, 1,000 CUBIC FEET OF CONCRETE AND $1,000 PROMOTION AND ADVERTISING. EACH PITTSBURGH PALACE REQUIRES 4,000 FEET OF FRAMING LUMBER, 2,000 CUBIC FEET OF CONCRETE AND $3,000 PROMOTION AND ADVERTISING. ALL COSTS FOR EACH CASTLE COMES TO $4,000 AND ON EACH PALACE COME TO $6,000. WHAT IS THE MINIMUM TOTAL COST ACHIEVEABLE AND HOW WOULD I FIGURE THIS OUT STEP BY STE?
To determine the minimum total cost achievable for the houses built by The Penn-Ohio Building Company, you need to formulate a linear programming problem. Here are the steps to solve it:
1. Define the decision variables:
Let x represent the number of Cleveland Castles to be built.
Let y represent the number of Pittsburgh Palaces to be built.
2. Establish the objective function:
The objective is to minimize the total cost. The cost of each Cleveland Castle is $4,000, and the cost of each Pittsburgh Palace is $6,000. Therefore, the objective function can be stated as:
Total Cost = 4,000x + 6,000y
3. Set up the constraints:
- Framing Lumber Constraint: The total amount of framing lumber should be at least 20,000 ft. Each Cleveland Castle requires 4,000 ft of framing lumber, and each Pittsburgh Palace requires 4,000 ft as well. The constraint can be written as:
4,000x + 4,000y ≥ 20,000
- Concrete Constraint: The total amount of concrete should be at least 8,000 cubic ft. Each Cleveland Castle requires 1,000 cubic ft of concrete, and each Pittsburgh Palace requires 2,000 cubic ft. The constraint can be written as:
1,000x + 2,000y ≥ 8,000
- Promotion and Advertising Constraint: The total cost for promotion and advertising should be at least $9,000. Each Cleveland Castle requires $1,000 for promotion and advertising, and each Pittsburgh Palace requires $3,000. The constraint can be written as:
1,000x + 3,000y ≥ 9,000
- Non-negativity Constraint: The number of houses cannot be negative.
x ≥ 0
y ≥ 0
4. Solve the linear programming problem:
Use a linear programming solver or graphical method to find the feasible region and the optimal solution. The optimal solution will provide the values of x and y that minimize the total cost.
5. Interpret the solution:
Once you obtain the optimal solution, substitute the values of x and y into the objective function to find the minimum total cost achievable.
By following these steps, you can determine the minimum total cost achievable for building houses and understand the logic behind solving it.