Question

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

Answer 1

Answer:

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