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 product are given here:

Product Contribution Margin Raw Material $($kg$)$ Labour Hours for Production Labour Hours for Finishing & Packaging

P$1150101030$

P$2120101020$

P$3160202030$

P$4200302040$

P$5100102020$

Total Availability $-5050140$

The manager of the company insists that the products P$3$ 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 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 dual variables

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

$($f$)$ The marketing manager informs that the selling price of the product P$2$ has to be revised downward to Rs$.116.$ What would be the new optimal product mix and the profit at it$?$