Question

A steel company has two mills. Mill 1 costs $70,000 per day to operate, and it can produce 400 tons of high-grade steel,500 tons of medium-grade steel, and 450 tons of low-grade steel each day. Mill 2 costs $60,000 per day to operate, and it can produce 350 tons of high-grade steel, 600 tons of medium-grade steel, and 400 tons of low-grade steel each day. The company has orders totaling 100,000 tons of high-grade steel, 150,000 tons of medium-grade steel, and 125,000 tons of low-grade steel. How many days should the company run each mill to minimize its costs and still fill the orders?

1). (10’) Please set up the LP model for the above situation.

2). (20’) How many days should we run Mill 1 (M1) and how many days should we run Mill 2 (M2) at optimal?

3). (10’) Whether we have slack or surplus variable applicable in this question? If we do, what is the value of slack (or/and surplus) variable? What is the meaning behind it?

4). (10’) What is the cost at the optimal solution?

Answer 1

Let the no days of operation of M1 be x and the no of operations of M2 be y.
Aim -> Min cost = 70,000x+60,000y
Constraint for high grade steel (produce atleast 100,000 tons ofhigh grade steel):
400x+350y>=100,000 ( 8x+7y>=2000
Similarly for medium grade:
500x+600y>=150,000 ( 5x+6y>=1500
Similarly for low grade:
450x+400y>=125,000 ( 9x+8y>=2500
LP: Min cost = 70,000x+60,000y
Subject to:
8x+7y>=2000
5x+6y>=1500
9x+8y>=2500

(diag)
At A, cost = 18,750,000
At B, cost = 19,285,760
At C, cost = 21,000,000

Cost minimisation happens at point A
So, mill 1 doesnot run at all and mill 2 runs for 312.5 days

For high grade, we produce 350*312.5=109,375>100,000 => There is surplus
For medium grade, we produce 600*312.5=187,500>150,000 => Surplus
For low grade, we produce 400*312.5=125,000 => No surplus

Optimal cost is 18,750,000