GSU manufactures three types of board at its facility in Reston, VA. The products; sheetrock U_A, U_AB, and U_B, vary by composition, application, and durability. The Reston facility has the capacity to produce a maximum of 2,000 tons of these products each week. It costs $80 to produce a ton of U_A wallboard, $150 per ton of U_AB, and $50 per ton of U_B. The production process requires 1.0 hours of labor for a ton of U_A, 1.2 hours for a ton of U_AB, and 1.8 hours for a ton of U_B. The facility has 800 hours of normal production labor available each week and expects demand for 800 tons of U_A, 900 tons of U_AB, and 1,100 tons of U_B during the same period. GSU has established the following goals at this facility, in order of priority:
The company does not want to spend over $20,000 per week on production costs, excluding labor.
The company would like to limit overtime to not more than 100 hours per week.
The company wants to meet expected demand levels for all three products; however, it considers it twice as important to meet the demand for U_A as it is to meet the demand for U_AB, and twice as important to meet the demand for U_AB as it is to meet the demand for U_B.
The company considers it desirable to avoid producing under capacity, if possible.
To avoid complications from union agreements, the company wants to avoid any underutilization of labor.
a. Formulate a linear goal programming model that can be used to determine the number of tons of each product to manufacture each week that will best satisfy the stated goal priorities established by GSU.