Suppose you work for a manufacturer who has decided to alloc

Suppose you work for a manufacturer who has decided to allocate funds for the construction of up to ten new production facilities. There are three designs, type A, B and C, for the facilities which are equally efficient but produce a different number of each of three products, product x, y, and z. The type A facility is expected to produce 200 of x, 100 of y, and 50 of z per week. The type B facility is expected to produce 125 of x, 125 of y, and 100 of z per week. And the type C facility is expected to produce 175 of x, 75 of y, and 100 of z per week. The company wishes to produce x, y, and z in a ratio of 2:1:1 (i.e., we want an equal amount of y and z but twice as mu ch of x.), and achieving this ratio is most important. How many of each type of facility should you suggest be constructed? (You may use a computer or calculator to aid your computation.)

Solution

Let, x1, x2 and x3 be the number of respective A,B & C facilies constructed.

Thus, x1+ x2 +x3 <= 10

Since, A, B & C type facilies produces different no. of x, y & z products.

So for   x1, x2 and x3 be the number of respective A,B & C facilies,

no. of x product produced = 200x1 + 125x2 + 175x3

no. of y product produced = 100x1 + 125x2 + 75x3

and no. of z product produced = 50x1 + 100x2 + 100x3

As required, the no. of x, y & z produced to be in ratio 2:1:1, so following equation is given

2(200x1 + 125x2 + 175x3) = 100x1 + 125x2 + 75x3 = 50x1 + 100x2 + 100x3

By simplyfying the above, we get,

12x1 + 5x2 + 11x3 = 0

and 2x1 + x2 - x3 = 0

So the LP problem is as below:

Maximize : x1 + x2 + x3

Constraints: x1+ x2 +x3 <= 10

12x1 + 5x2 + 11x3 = 0

2x1 + x2 - x3 = 0

and x1, x2, x3 >= 0

Solving the above problem by using computer we get,

x1 = 2, x2 = 1 & x3 = 5

Thus, 2 numbers facility A, 1 number facility B and 5 number facility C to be constructed.