Linear Programming Production Line

[masrad]

I have absolutely no ides where to go from here, I am horrible at this, If you could help me I would appreciate it, I want help doing it, not just answers.

Homework Statement

You are the owner of a manufacturing plant. We've been hired by Apple to produce iPhone and iPods. Apples pays us $50 for every iPhone and $30 for every iPod. The constraints are that I only have 480 minutes of skilled labor and 2,000 minutes of machine use.

The iPhones require 2 minutes of skilled labor and 6 minutes of machine use time.
The iPods require 1 minute of skilled labor, and 10 minutes of machine use time.

In TOTAL, I can only make 250 unitsP= Profit
A= iPhone
B= iPod

The Attempt at a Solution

a = 2(skilled) + 6(machine) + 50(profit)
b = 1(skilled) + 10(machine) + 30(profit)

 

[HallsofIvy]

I have absolutely no ides where to go from here, I am horrible at this, If you could help me I would appreciate it, I want help doing it, not just answers.

Homework Statement

You are the owner of a manufacturing plant. We've been hired by Apple to produce iPhone and iPods. Apples pays us $50 for every iPhone and $30 for every iPod. The constraints are that I only have 480 minutes of skilled labor and 2,000 minutes of machine use.

The iPhones require 2 minutes of skilled labor and 6 minutes of machine use time.
The iPods require 1 minute of skilled labor, and 10 minutes of machine use time.

In TOTAL, I can only make 250 units


P= Profit
A= iPhone
B= iPod

Much better to write full sentences in order to be clear and precise. I presume that what you mean is that you can make P dollars if you make A iphones and B ipods.

The Attempt at a Solution

a = 2(skilled) + 6(machine) + 50(profit)
b = 1(skilled) + 10(machine) + 30(profit)

Again, you haven't defined your terms. Are "a" and "b" the same as "A" and "B"? If so that makes no sense. Those numbers look like the numbers given in how many minutes are required to make an iPhone or iPod. But then those are NOT equations. It looks like you are multiplying "2" by "skilled" and that is meaningless. And you certainly cannot add "minutes" and "dollars".

Let A be the number of iPhones made and let B be the number of iPods made. How much money would you make? That is the "object function".

How many minutes of skilled labor would be required? That must be larger than or equal to 0 and less than or equal to 480.

How many minutes of machine time would be required? That must be larger than or equal to 0 and less than or equal to 2000.

The total number of units made, A+ B, must be greater than or equal to 0 and less than or equal to 250.

The two inequalities define your "feasible region". The basic theorem of Linear programing is that max or min of a linear object function occur at the vertices of a convex polygonal feasible region.

 

[Architeuthis Dux]

The objective function would be: P = 50A + 30B

The constraints would be:
2A + B ≤ 480 (skilled labor constraint)
6A + 10B ≤ 2000 (machine use constraint)
A + B ≤ 250 (total unit constraint)

To solve this linear programming problem, you can use the simplex method or a graphing calculator. The optimal solution would be to produce 200 iPhones and 50 iPods, resulting in a profit of $12,500. This solution satisfies all of the constraints and maximizes the profit.

In order to improve production efficiency or increase profit, you could also consider adjusting the production mix or investing in more skilled labor or machines. It is important to regularly analyze and optimize the production line to ensure maximum efficiency and profitability.