# [Excel spreadsheet available at http://web.mit.edu/15.053/www/Exer3.2.xls] A commercial printing…

[Excel spreadsheet available at http://web.mit.edu/15.053/www/Exer3.2.xls] A commercial printing firm is trying to determine the best mix of printing jobs it should seek, given its current capacity constraints in its four capital-intensive departments: typesetting, camera, pressroom, and bindery. It has classified its commercial work into three classes: A, B, and C, each requiring different amounts of time in the four major departments. The production requirements in hours per unit of product are as follows:

Assuming these units of work are produced using regular time, the contribution to overhead and profit is \$200 for each unit of Class A work, \$300 for each unit of Class B work, and \$100 for each unit of Class C work. The firm currently has the following regular-time capacity available in each department for the next time period: typesetting, 40 hours; camera, 60 hours; pressroom, 200 hours; bindery, 160 hours. In addition to this regular time, the firm could utilize an overtime shift in typesetting, which would make available an additional 35 hours in that department. The premium for this overtime (i.e., incremental costs in addition to regular time) would be \$4/hour. Since the firm wants to find the optimal job mix for its equipment, management assumes it can sell all it produces. However, to satisfy long-established customers, management decides to produce at least 10 units of each class of work in each time period.

Assuming that the firm wants to maximize its contribution to profit and overhead, we can formulate the above situation as a linear program, as follows:

a) What is the optimal production mix?

b) Is there any unused production capacity?

c) Is this a unique optimum? Why?

d) Why is the shadow price of regular typesetting different from the shadow price of overtime typesetting?

e) If the printing firm has a chance to sell a new type of work that requires 0 hours of typesetting, 2 hours of camera, 2 hours of pressroom, and 1 hour of bindery, what contribution is required to make it attractive?

f) Suppose that both the regular and overtime typesetting capacity are reduced by 4 hours. How does the solution change? (Hint: Does the basis change in this situation?)

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