Linear Programming Discussion

Linear Programming Discussion

Linear programming is a powerful technique that helps us optimize a solution given objective function, Constraints, and non-negative constrains. One of the methods used to solve linear programming is the graphical method. The constraints and the objective function is plotted on a graph. The region satisfied by all the constrains I the graph is the feasible solution. In this problem, there are two decision variables; the number of regular model baseball gloves and the number of Catcher’s model baseball gloves to be produced. The optimized solution is obtained by inputting the decision variables into the objective function.

Objective function

Maximize Z: 5x₁ + 8X₂

Where X₁ is the number of regular model baseball gloves, and X₂ is the number of Catcher’s model baseball gloves. The objective function is to maximize the total profit given that the unit profit for the regular model and Catcher’s model baseball gloves is $8 and $5, respectively.

Constraints

Cutting and sewing: The total number of hours required for the production of the two baseball types in the Cutting and Sewing department should be less than or equal to 900.

X₁ + 1.5X₂ 900

Finishing: The total number of hours required for the production of the two baseball types in the Finishing department should be less than or equal to 300. 0.5X₁ + 1/3X₂ 300

Packaging and Shipping: The total number of hours required for the production of the two baseball types in the Packaging and Shipping department should be less than or equal to 100. 1/8X₁ + 1/4X₂ 100

Non-negative constraints: The company cannot produce cloves that are less than zero. A non-negative constraint makes the production of either glove to be greater than or equal to zero. X₁, X₂ 0

Integer constraints: The company cannot produce a fraction number of cloves. The decision variables should be set to integers.  X₁, X₂ = Integer.

Solution

From the graph above, Kelson should manufacture 500 units of regular model baseball gloves and 150 units of Catcher’s model baseball gloves. The maximum profit Kelson manufacturers can earn is calculated as follows:

Z =5x₁ + 8X₂

Z = $5 * 500 + $8 x 150

Z = $2,500 + $1,200 = $3,700

Cutting and sewing department will utilize 725 hours of production, and 175 hours is the slack time. The other two departments: finishing department and packaging and shipment department will utilize all the allocated hours of production. There is zero slack time for the last two departments.