A chemist wants to mix two kinds of glycerins for the manufacture of a soap. The glycerin from supplier A cost $1 million per ton and the glycerin from supplier B costs $1.5 million per ton. The chemist is obligated to buy no more than twice as much glycerin from supplier B as she does from supplier A. Furthermore, the chemist must buy at least 7 tons but no more than 9 tons from supplier A. How much should the chemist buy from each supplier to minimize the cost of her purchase.
How much should the chemist buy from each supplier to minimize the cost of her purchase>>
buy the required 7 tons from A, zero from B. That minimizes the cost.
Wouldn't it be wonderful if problems asked what they really wanted?
How many different outcomes could there be when rolling three 6 sided dice ?
To find the optimal amount of glycerin to buy from each supplier, we need to minimize the cost of the purchase. Let's solve this problem step by step:
1. Let's define our variables:
- Let's call the amount of glycerin purchased from supplier A as x (in tons).
- Let's call the amount of glycerin purchased from supplier B as y (in tons).
2. Formulate the objective function:
- The cost of glycerin from supplier A is $1 million per ton, so the cost of glycerin from supplier A would be 1 million x.
- The cost of glycerin from supplier B is $1.5 million per ton, so the cost of glycerin from supplier B would be 1.5 million y.
- Our objective is to minimize the total cost, so our objective function would be:
Total Cost = 1 million x + 1.5 million y.
3. Define the constraints:
- The chemist must buy at least 7 tons but no more than 9 tons from supplier A, so the constraint would be:
7 ≤ x ≤ 9.
- The chemist must buy no more than twice as much glycerin from supplier B as from supplier A, so the constraint would be:
y ≤ 2x.
4. Combine the objective function and constraints:
- Our final optimization problem would be:
Minimize: 1 million x + 1.5 million y
Subject to: 7 ≤ x ≤ 9
y ≤ 2x.
5. Solve the optimization problem:
- We can solve this problem using various optimization techniques such as linear programming or graphical method.
- Using linear programming, we can graph the feasible region and find the point that results in the minimum cost.
Once we solve the optimization problem, we will get the optimal values of x (amount of glycerin from supplier A) and y (amount of glycerin from supplier B) that minimize the cost of the purchase.