A chicken farm orders bags of feed from two farmers. Store X charges $20 per bag, and store Y charges $15 per bag. The farm must obtain at least 60 bags per week to care for the chickens properly. Store Y can provide a maximum of 40 bags per week, and the farm has committed to buy at least as many bags from store X as from store Y. The chicken farm wants to minimize the cost of ordering feed. Let x represent the number of bags of feed from store X and y represent the number of bags of feed from store Y.
To minimize the cost of ordering feed, we need to determine the optimum values of x and y.
Let's start by defining the constraints and objectives of this problem:
Constraints:
1. The farm must obtain at least 60 bags per week: x + y >= 60
2. Store Y can provide a maximum of 40 bags per week: y <= 40
3. The farm has committed to buy at least as many bags from store X as from store Y: x >= y
Objective:
Minimize the cost of ordering feed: Cost = 20x + 15y
To find the optimal values of x and y, we can use linear programming. Linear programming is a mathematical method used to optimize (maximize or minimize) a linear objective function subject to a set of linear equations or inequalities.
To set up the linear programming problem, we can use the above constraints and objective function:
Constraints:
1. x + y >= 60
2. y <= 40
3. x >= y
Objective: Minimize Cost = 20x + 15y
Now we can solve this linear programming problem using any solver or by graphical method.
The optimal values of x and y will give us the quantities of bags from store X and store Y respectively, that will minimize the cost of ordering feed while satisfying the given constraints.