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 can totally minimize it but I'm unsure of how to set up my equations. Thank you!
Um... pretty sure this was a problem for one of my classes. Not supposed to be on the internet. Is there anyway to request to have a post removed?
There seems to be a problem here. I am reluctant to help with this linear programming problem under the circumstances.
Why don't you use Excel for this?
To set up the problem using the tableau method, you need to define your decision variables, formulate the objective function, and specify the constraints.
1. Decision Variables:
Let's define:
- X1: The number of tons of coal shipped from mine M1 to purchaser P1
- X2: The number of tons of coal shipped from mine M1 to purchaser P2
- X3: The number of tons of coal shipped from mine M1 to purchaser P3
- X4: The number of tons of coal shipped from mine M2 to purchaser P1
- X5: The number of tons of coal shipped from mine M2 to purchaser P2
- X6: The number of tons of coal shipped from mine M2 to purchaser P3
2. Objective Function:
The objective is to minimize the total cost. To calculate the total cost, we need to multiply the amount shipped by the corresponding transportation cost and sum them up. Therefore, the objective function can be expressed as:
Minimize: 50X1 + 45X2 + 50X3 + 45X4 + 50X5 + 55X6
3. Constraints:
The constraints ensure that the requirements of each purchaser are satisfied and that the amount shipped from each mine is non-negative.
For P1 (Steel Mill 1):
X1 + X4 ≥ 1000 (Requires at least 1000 tons)
For P2 (Steel Mill 2):
X2 + X5 ≥ 1500 (Requires at least 1500 tons)
For P3 (Power Generating Station):
X3 + X6 ≥ 1200 (Requires at least 1200 tons)
Finally, all variables need to be non-negative:
X1, X2, X3, X4, X5, X6 ≥ 0
This set of equations represents the linear programming problem. Now, you can use a suitable LP solver or perform the simplex method to find the optimal solution.