Universal Mines Inc. operates three mines in West Virginia. The ore from each mine is separated into two grades before it is shipped. The daily production capacities of the three mines, as well as their daily operating costs, are as follows: c/day Mine High-grade Ore, Low-grade Ore, Operating Cost Tons/day Tons/day in $/day 20,000 22,000 18,000 Mine 11 Mine III Universal has committed itself to deliver 54 tons of high-grade order and 65 tons of low-grade ore by the end of the week. Universal can run its mines seven days a week if required. Determine the number of days each mine should be operated during the upcoming week if Universal Mines is to fulfill its commitment at the minimum total cost. Round the answers to two decimal places.
a. The number of days Mine I should operate = _____days
b. The number of days Mine Il should operate = _days
c. The number of days Mine III should operate = _days
d. The total cost of the operation for next week = $ ____

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jepessoa jepessoa · Answer 1 · 2020-10-11T16:20:30+02:00

Answer:

this is a cost minimization problem, but it is missing some numbers, so I looked for similar questions (see attached PDF):

minimization equation = 20x₁ + 22x₂ + 18x₃ (costs per ton)

where:

x₁ = mine I

x₂ = mine II

x₃ = mine III

the constraints are:

4x₁ + 6x₂ + x₃ ≥ 54 (high grade ore)

4x₁ + 4x₂ + 6x₃ ≥ 65 (low grade ore)

x₁, x₂, x₃ ≤ 7 (only 7 days per week)

using solver, the optimal solution is

2x₁, 7x₂, and 5x₃

a. The number of days Mine I should operate = 2 days

b. The number of days Mine Il should operate = 7 days

c. The number of days Mine III should operate = 5 days

d. The total cost of the operation for next week = $284,000