Re: Linear Programing
Thanks for the response.
As far as I am aware the answer should be generated by both graphical and equation solving means.
My son has now managed to complete the question but is unsure as to how successful he has been at getting the right answer. His attempt at the questions are detailed below. Any advice on the correctness of his answers and or help with where he has gone wrong (if he has) would be most appreciated.
Corner Points
A (0, 100)
B (20, 50)
C (40, 20)
D (80, 0)
How many hours should each mine operate, and which type of ore is
overproduced if the running costs of mine 1 are:
i) the same as mine 2?
So if mine 1 costs the same to run as mine 2, hypothetically it could cost $20 an
hour to run both mines
A (0, 100) = 20(0) + 20(100) = $2000
B (20, 50) = 20(20) + 20(50) = $1400
C (40, 20) = 20(40) + 20(20) = $1200 - Min cost to run
D (80, 0) = 20(80) + 20(0) = $1600
If mine 1 cost the same to run as mine two, it would be cheapest to operate mine 1 40 hours a week and mine two for only 20 hours. While operating the mines at these times only 400 low-grade ore would be over produced.
ii) 50% higher than mine 2
If mine 1 costs 50% higher than mine 2, hypothetically it would then cost $30 an hour to run mine 1 and only $20 to run mine 2
A (0, 100) = 30(0) + 20(100) = $2000
B (20, 50) = 30(20) + 20(50) = $1600 - Min cost to run
C (40, 20) = 30(40) + 20(20) = $1600 - Min cost to run
D (80, 0) = 30(80) + 20(0) = $2400
If mine 1 cost 50% more to run than mine two, there would be two options to run the mines, the first option would require mine 1 to run for 20 hours and mine 2 to run for 50 hours while the second option would require mine 1 to run for 40 hours and mine 2 to run for 20 hours, both costing the same amount to keep running in this case. In this case running the mines for that period of time for option 1 would produce 400 extra high-grade while option 2 would only over produce 400 low-grade ore.
iii) twice that of mine 2
If mine 1 costs twice that of mine 2, hypothetically it would then cost $40 an hour to run mine 1 and only $20 to run mine 2
A (0, 100) = 40(0) + 20(100) = $2000
B (20, 50) = 40(20) + 20(50) = $1800 - Min cost to run
C (40, 20) = 40(40) + 20(20) = $2000
D (80, 0) = 40(80) + 20(0) = $3200
If mine 1 costs twice more to run than mine two, for the cheapest option to run the mines meeting minimum criteria it would require mine 1 to run for 20 hours and mine 2 to run for 50 hours. 400 high-grade ore would be over produced in the process.
b) Also show that if:
i) The running costs per hour of mine 1 exceeds 2.5 times the running costs of mine 2, minimum running costs per week are achieved by operating mine 2 only. How many hours per week would it operate?
If mine 1 is 2.5 times the cost to run as mine 2, hypothetically it would then cost $50 an hour to run mine 1 and only $20 to run mine 2
A (0, 100) = 50(0) + 20(100) = $2000 - Min cost to run
B (20, 50) = 50(20) + 20(50) = $2000 - Min cost to run
C (40, 20) = 50(40) + 20(20) = $2400
D (80, 0) = 50(80) + 20(0) = $4000
As the results show, you could obtain a minimum cost if you only operated mine 2 during that week if mine 1 cost 2.5 times the amount of mine 2 per hour. If this was the case you would have to run mine 2 for a total of 100 hours a week in order to achieve the contracted amount of ore. As such you would over produce 1200 high-grade ore and 400 medium-grade ore.
ii) The running costs of Mine 2 exceed twice the running cost per hour of Mine 1,
minimum running costs per week are achieved by operating Mine 1 only. How many hours per week would it operate?
If mine 2 is 2 times the cost to run of mine 1, hypothetically it would then cost $20 an hour to run mine 1 and $40 to run mine 2
A (0, 100) = 20(0) + 40(100) = $4000
B (20, 50) = 20(20) + 40(50) = $2400
C (40, 20) = 20(40) + 40(20) = $1600 - Min cost to run
D (80, 0) = 20(80) + 40(0) = $1600 - Min cost to run
The results indicate that you could achieve minimum costs by running only mine 1 for 80 hours a week in order to achieve the contracted amount of ore. From running mine 1 for this period of time you would over produce 800 medium-grade ore and 2000 low grade ore.
