A company has facilities for producing 5 products which require the same raw material and same type of production, finishing and packaging facilities. The unit contribution margin and the material and labour requirements for each of the products are given here:

The manager of the company insists that the products P3 and P 4 should be given top priority in the production, because they yield the maximum unit contribution.
Formulate this as a linear programming problem, obtain the optimal solution and state whether you agree with the manager's viewpoint. Further, attempt the following.

(a) Write the dual of this problem.

(b) Obtain the marginal profitability of (i) raw materials, (ii) the time in the production department, and (iii) the time in the finishing and packaging department.

(c) Over what range of values of the respective constraints the marginal profitabilities determined by you in

(b) earlier would be valid?

(d) Obtain the optimal values of the dual variables.

(e) Verify that the objective function values of the primal and the dual problems are identical.

(f) The marketing manager informs that the selling price of the product P2 has to be revised downward to Rs 116. What would be the new optimal product mix and the profit at it?

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Contribution Raw Material Labour Hours Product Margin (kg) Production Labour Hours Finishing & Packaging P 150 10 10 30 P2 120 10 P3 160 20 P4 160 30 P5 Total availability ('000) 100 20 50 228 20 20 10 20 10 20 20 10 80 140