$\backslash$table$[[$PRODUCT$,\backslash$table$[[$CONTRIBUTION$],[$MARGIN$]],\backslash$table$[[$RAW$],[$MATERIAL$],[($KG$)]],\backslash$table$[[$LABOR HOURS$],[$PRODUCTION$]],\backslash$table$[[$LABOR HOURS$],[],[$PRODUCTION$]]],[$P$1,150,10,10,30],[$P$2,120,10,20,20],[$P$3,160,20,10,20],[$P$4,160,30,10,20],[$P$5,100,20,20,10],[,$ TOTAL,,$50,80,140],[$AVAILABLE$,,,,],[,(000),,,,]]$A company has facilitated for producing $5$ products which require the same raw material and

the same type of production, finishing and packaging facilities. The unit contribution margin

and the material and labor requirements for each of the products are given here.

PRODUCT CONTRIBUTION

MARGIN

RAW

MATERIAL

$($KG$)$

LABOR HOURS

PRODUCTION

LABOR HOURS

FINISHING &

PRODUCTION

P$1150101030$

P$2120102020$

P$3160201020$

P$4160301020$

P$5100202010$

TOTAL

AVAILABLE

$(000)$

$5080140$

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 contribution.

a$)$ Formulate the above problem as a linear programing model

b$)$ Obtain optimal solution to the problem and state whether you agree with the managers

viewpoint $($use the simplex method$)$

c$)$ Write the dual of this problem

d$)$ 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

e$)$ Over what range of values of the respective constraints the marginal profitability

determined by you in $($b$)$ earlier would be valid?

f$)$ Verify that the objective function values of the primal and the dual problems are identical.

g$)$ The marketing manager informs that the selling price of the product P$2$ has to be revised

downward to birr $116.$ What will be the new optimal product mix and the profit at it$?$

I need an accurate answer and well explained one