Expert: Josh - 7/8/2009

Question
Dear Sir

Below problem has given me to solve as assignment from my university.

Department    A    B   C
I             1    1   9
II            1    3   7
III           2    7   13

Please note, A,B,C are the products and the numeric values are the hours required for per unit of production.

Department I,II & III produces unit product per hour as mentioned above.

Now, Department I has 75 hours, II has 65 hours & III has 125 hours.

Question is, if profits per unit is $20 for A, $30 for B and $40 for C, then what is the maximum profit and the composition of the maximum-profit combination of outputs.

Please help, as there is a exam in my university and I don't have time.


Regards
Pervez  

Answer
Dear Pervez,

Your question is of the form

Maximize C=c1+c2+c3
 where c1=20x+30y+40z, c2=20u+30v+40w, c3=20r+30s+40t
subject to the inequality constraints
 x+y+9z<=75
 u+3v+7w<=65
 2r+7s+13t<=125.

Here, {x,u,r} denote the number of units of A produced by (i), (ii) and (iii), respectively. {y,v,s} and {z,w,t} are similarly defined for B and C.

This problem belongs to the class of linear programming.
Like convex optimization problems, there is no analytic formula for the solution; unlike least square problems.

There are quite a few fast, reliable and established methods for solving problems of this kind. The exact details are beyond the scope of my knowledge. They are generally formulated and solved using numerical packages. Perhaps you can ask your course instructor for some suggestions?

Regards,
Josh