A company has three production plants each of which produces

A company has three production plants, each of which produces three different models of a particular product. The daily capacities (in thousands of units) of the three plants are as follows. The total demand for Model 1 is 300,000 units, for Model 2 is 172.000 units, and for Model 3 is 249.5(H) units. Moreover, the daily operating cost for Plant I is $55,000. for Plant 2 is $60,000. and for Plant 3 is $60,000. How many days should each plant be operated in order to fill the total demand, and keep the operating cost at a minimum?

Solution

Let plant 1 be operated for x days, 2 for y days and 3 for z days

Then Operating cost C= 55000x+60000y+60000z

Our objective is to minimise C

Constraints are: The demand

The constraints can be represented as

8x+6y+12z>=300000 - I

4x+6y+4z>=172000 - II

8x+3y+8z>=249500 - III

I - III gives 3y+4z >= 50500

2(II)-III gives 9y >= 94500

y>= 10500

Substitute in 3y+4z >=50500

4z >= 40000 : z >=10000

Substitute in I

8x+6y+12z>=300000

8x + 63000+120000>=300000

8x >= 117000

x>=14625

Thus x = 14625

y = 10500

z = 10000

will meet the full demand at the same time keep the cost at minimum as plants willbe utilized fully.

Min cost C = 55000(14625) +10500(60000)+10000(60000)

= 807375000+630000000+600000000

= 2034375000