Mining scenario A mining company owns two mines, each of which produces three grades (high, medium, and low) of ore. The company has a contract to supply a smelting company with at least 12 tons of highgrade ore, at least 8 tons of medium-grade ore, and at least 24 tons of low-grade ore. Each hour of operation, mine 1 produces 6 tons of high- grade ore, 2 tons of medium-grade ore, and 4 tons of low-grade ore. Each hour of operation, mine 2 produces 2 tons of high-grade ore, 2 tons of medium-grade ore, and 12 tons of low-grade ore. It costs $200 per hour to operate mine 1 and $160 per hour to operate mine 2. How many hours should each mine be operated so as to meet the contractual obligations at the lowest total operating cost

the lowest operating cost is achieved when mine 1 is operated for 0 hours and mine 2 is operated for 2 hours.

Explanation:

Mining hours of mine 1: M1

mining hours of mine 2: M2

Objective function: Minimize operating cost < 200M1 + 160 M2

Explicit constraints:

6M1 + 2M2 ≥ 12

2M1+ 2M2≥ 8

4M1+ 12M2 ≥ 24

Implicit constraints:

M1> 0

M2>0

See the attachment for feasible region

The coordinates of feasible region are (0,0), (2,0),(0,2),(1.5,1.5)

To maximize cost substitute each of these coordinates in objective function

for(0,0)

cost=0 (neglect this cost since M1>0 and M2 >0)

for(2,0)

cost= 320

for (0,2)

cost= 400

for (1.5,1.5)

cost= 540

the lowest operating cost is achieved when mine 1 is operated for 0 hours and mine 2 is operated for 2 hours

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