A company produces two main products, M and N, which have unit profits of $180 and $140, respectively. Each unit of product must be processed on two assembly lines, where the required production times are as follows. Product M requires 6 hours in Line 1 and 2 hours in Line 2, while Product N requires 2 hours in Line 1 and 4 hours in Line 2. For a typical week, Line 1 has 30 hours available and Line 2 has 20 hours available for producing Products M and N. No more than 4 units of Product N can be sold per week.
a. Formulate the problem as an algebraic model to maximize the total profit. (Define decision variables – describe verbally, constraints, and the objective function – overall measure of performance.)
b. Solve the problem graphically. (Hint: Draw the feasible region of the LP problem and plot an objective function line first. Then identify the optimal point. Finally, you need solve the simultaneous equations of the optimal point to get the optimal solution.) How many units of each product should be produced in order to be optimal? What is the maximum profit?
Note: Use an Excel sheet to “draw” a coordinate system (grid); make sure consecutive (vertical and horizontal) lines are equally distanced. Insert lines for constraints as I do in the lecture (Insert – Shapes – select the line). Draw an objective function line. Point out the optimal point.
c. Identify the amount of unused resources, i.e. slack, at the optimal solution.
d. What would be the effect on the optimal solution if the production time on Line 1 were reduced to 15 hours? (Hint: In the graph, observe the change as the hours reduce from 30 to 15. Would the optimal solution change? If it does, what is the new optimal solution?)
e. Formulate the original problem as a linear programming model on the spreadsheet and solve it. In your spreadsheet model, follow the coloring practice and use range name and sumproduct function as in our examples.)
f. From e, change the unit profit for Product N by small amount each time. Find out the range of the unit profit such that the optimal solution (in e) stays unchanged.