Question

Two chemical plants, one at Multan and one at Mianwali, produce three types of fertilizer, low phosphorus (LP), medium phosphorus (MP), and high phosphorus (HP). At each plant, the fertilizer is produced in a single production run, so the three types are produced in fixed proportions. the Multan plant produces I ton of LP, 2 tons of Me, and 3 tons of FiP in a single operation, and it charges $600 for what is produced in one operation, whereas one operation of the Mianwali plant produces 1 ton of LP, 5 tons of M, and I ton of He, and it charges $ 1000 for what it produces in one operation. If a customer needs 100 tons of LP, 260 tons of MP, and 180 tons of HP, how many production runs should be ordered from each plant to minimize costs?

Answer 1

To determine the number of production runs from each plant that would minimize costs, we can set up a linear programming problem. Let's denote the number of production runs from the Multan plant as 'x' and the number of production runs from the Mianwali plant as 'y'.

Objective Function:
We want to minimize the total cost, so our objective function is the cost function. The total cost can be calculated as follows:

Cost = ($600 * x) + ($1000 * y)

Constraints:

1. For LP fertilizer:
The Multan plant produces 1 ton per run, and the Mianwali plant produces 1 ton per run. The total required is 100 tons, so the constraint is:
x + y = 100

1. For MP fertilizer:
The Multan plant produces 2 tons per run, and the Mianwali plant produces 5 tons per run. The total required is 260 tons, so the constraint is:
2x + 5y = 260

1. For HP fertilizer:
The Multan plant produces 3 tons per run, and the Mianwali plant produces 1 ton per run. The total required is 180 tons, so the constraint is:
3x + y = 180

Non-Negative Constraints:
Since the number of production runs cannot be negative, we have the non-negative constraints:
x ≥ 0
y ≥ 0

Now, we can solve this linear programming problem to find the optimal solution.

However, it seems there is an inconsistency in the given information. The LP fertilizer requirement has the constraint x + y = 100, which implies that the total number of production runs from both plants needs to be 100. But the requirements for MP and HP fertilizers suggest that the total number of production runs should be 260 and 180, respectively. These constraints cannot be simultaneously satisfied. Please double-check the information provided to ensure accuracy.