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: Department A B C
Typesetting 0 2 3
Camera 3 1 3
Pressroom 3 6 2
Bindery 5 4 4

Assuming these units of work are produced using regular time, the contribution to profit R2 000f for each unit of Class A work, R3 000 for each unit of Class b work and R1 000 for each unit of Class C work. The firm currently has the following regular-time capacity (in hours) available in each department for the next time period.

Department Regular-time
Typesetting 40
Camera 60
Pressroom 200
Bindery 160

In addition to this regular time, the firm could utilise an overtime shift in typesetting which would make available an addition 35 hours in that department. The premium for this overtime (I e incremental costs in addition to regular time) would be R40/hour

Since the firm wants to find the optimal work 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 maximise its contribution to profit and overhead, we can formulate the above situation as a linear programming model as follows

Decision variables

XAR = Number of units of Class A work produced on regular time
XBR = Number of units of Class B work produced on regular time
XCR = Number of units of Class C work produced on regular time
XBO = Number of units of Class B work produced on overtime typesetting
XCO = Number of units of Class C work produced on overtime typesetting

Required:
a. Formulate LP model
b. What is the optimal production mix?
c. Is there any unused production capacity?
d. If the printing firm has a chance to sell a new type of work that requires no hours of typesetting, 2 hours of camera time, 2 hours in the pressroom and 1 hour of bindery time, what profit is required to make it attractive to print this type of work? Give details on all the constraints.
e. Is it advisable to use sensitivity analysis? Why