I just need help with the set up of this problem, that's where I'm getting lost.

A mining company operates two coal mines, M1 and M2. The company sells the coal to three purchasers: two steel mills P1 and P2, and a power generating station P3. The cost of transporting a ton of coal from each mine to each purchaser is given in the following table:

P1 P2 P3
M1 50 45 50
M2 45 50 55

Suppose purchasers P1, P2, and P3, require at least 1000 tons, 1500 tons, and 1200 tons respectively. How many tons should be shipped from each mine to purchaser to minimize total cost while satisfying the purchaser's requirements?

SO! I need help with the set up from here. We need to do it in a tableaux and I'm unsure of how to set up my equations. Thank you!

Setting equations is not necessary, you can do the following argument: you have that shipping 1 ton from M1 to P1 costs 50, while shipping 1 ton from M2 to P1 costs 45. So all the coal for P1 would be shipped from M2, as you pay less in this case than shipping it from M1. Then do the same argument for P2 and P3, (for them shipping from M1 is cheaper).

P1 needs 1000 ton, and shipping all the coal from M2 (the cheapest way) would cost you 45000,
P2 and P3 need 1500 and 1200 ton, and shipping all the coal from M1 cost 67500 and 60000 respectively.

So the total cost (the cheapest way possible) is 45000+67500+60000. Coal shipped will be 1000 ton from M2 and 1500+1200=2700 ton from M1.

If you really want an equation, you can set it as follows:

call (k=1,2, h=1,2,3) the cost of shipping 1 ton of coal from mine to consumer (you have their values in your table), and (k=1,2, h=1,2,3) the quantity of coal shipped from mine to consumer
(you have to determine them). Then impose:





and substituting you have


as you know , so the optimal choice is
. Similarly for other consumers.

A typo in the second part:

"and (k=1,2, h=1,2,3) the quantity of coal shipped"
instead of
"and (k=1,2, h=1,2,3) the quantity of coal shipped"