Linear Programming Model

[ardenarden]

I need help in formulating a linear programming model.

A company makes their product using 2 chemicals that provide nitrogen, phosphate, and potassium. A pound of ingredient 1 contributes 10 ounces of nitrogen and 6 ounces of phosphate, while a pound of ingredient 2 contributes 2 ounces of nitorgen, 6 ounces of phosphate, and 1 ounce of potassium. Ingredient 1 costs $3 per pound, and ingredient 2 costs $5 per pound. The company wants to know how many pounds of each chemical ingredient to put into a bag of ferilizer to meet the minimum requirements of 20 ounces of nitrogen, 36 ounces of phosphate, and 2 ounces of potassium while minimizing cost.

A. Formulate a linear programming model for this probem.
B. Solve this model b y using graphical analysis.

I would really appreciate any help given for this...

 

[Mrspi]

I need help in formulating a linear programming model.

A company makes their product using 2 chemicals that provide nitrogen, phosphate, and potassium. A pound of ingredient 1 contributes 10 ounces of nitrogen and 6 ounces of phosphate, while a pound of ingredient 2 contributes 2 ounces of nitorgen, 6 ounces of phosphate, and 1 ounce of potassium. Ingredient 1 costs $3 per pound, and ingredient 2 costs $5 per pound. The company wants to know how many pounds of each chemical ingredient to put into a bag of ferilizer to meet the minimum requirements of 20 ounces of nitrogen, 36 ounces of phosphate, and 2 ounces of potassium while minimizing cost.

A. Formulate a linear programming model for this probem.
B. Solve this model b y using graphical analysis.

I would really appreciate any help given for this...

What have you done for this so far?

Have you defined variables? For example, you might say

let x = pounds of ingredient 1
let y = pound of ingredient 2

Since each pound of ingredient 1 costs $3, and each pound of ingredient 2 costs $5, the total cost for x pounds of ingredient 1 and y pounds of ingredient 2 would be:

Cost = 3x + 5y

NOW...that cost is what you hope to minimize.

I'll help you get started on the "constraints," or the conditions that need to be met.

You're told this: A pound of ingredient 1 contributes 10 ounces of nitrogen and 6 ounces of phosphate, while a pound of ingredient 2 contributes 2 ounces of nitorgen, 6 ounces of phosphate, and 1 ounce of potassium.

A bag of fertilizer must have (according to the problem) at least 20 oz of nitrogen...so,

x pounds of ingredient 1 contains 10x ounces of nitrogen
y pounds of ingredient 2 contains 2y ounces of nitrogen

There must be at least 20 ounces of nitrogen, so

10x + 2y = 20

Ok...use the information about the other chemicals to write similar equations to give you some additional constraints.

Remember that you can't use a negative amount of either ingredient 1 or ingredient 2, so x > 0 and y > 0.

Graph all of the constraints on the same set of coordinate axes to find the "feasible region," or the region which contains all of the points which satisfy all of the constraints. The maximum and minimum values for the "objective function" (in your case, the COST) occur at vertices of the feasible region.