The Terrell Company can manufacture three products: Alpha, Beta, and Zeta, which sell for $20, $25, and $40, respectively. The materials cost for one unit is $4 for Alpha, $12 for Beta, and $10 for Zeta. Labor costs per unit are $5 for Alpha, $7 for Beta, and $15 for Zeta. When total labor costs are $20,000, factory overhead amounts to $80,000. When total labor costs are $30,000, factory overhead is $85,000.

Factory output is constrained by the time available on two machines. The firm has only 10,000 hours of time available on its grinding machines and 8,000 hours available on the polishing machine. Alpha requires 2 hours of grinding per unit, 4 units of Beta can be ground per hour, while Zeta requires 30 minutes of grinding time per unit. Two units of Alpha can be polished per hour, each unit of Beta requires 1 hour of polishing, while Zeta requires 2 hours of polishing per unit.

REQUIRED

A. Calculate the contribution margin per unit per product and set up the target (contribution margin) function for this problem. (Hint: First use the high-low method to estimate the cost function for factory overhead costs.)

B. List the constraint formulas, including the hours of constraint.

C. Set up a spreadsheet for Excel Solver or another linear programming package, and solve for the optimal product mix.

D. Now suppose that Beta’s selling price increased by $1.26, and the contribution margin therefore increases by $1.26. Re-solve this problem for the optimal product mix. Did the product mix change? Explain the results.