Optimizing Production with Linear Programming in Economics: A Math Question

[Izekid]

This is a economic math question see if you can help me out here.

I've got 3 variables A,B and C and I shall get the production to be optimal I have there variables

This is the forumla = Z = 800A+ 550B +400C
Under these condition it shall work

14A+10B+8C=<4500
12A+7C=<1000
12A+18B+10C=<480
A,B,C>0

With these numbers I shall get a number for each variable that should be the best production way.
It's called in mathematics, Linear Programming

http://www-unix.mcs.anl.gov/otc/Guide/faq/linear-programming-faq.html

I have tested many thing please help me if you can

 

[HallsofIvy]

First convert the inequalities into equations:

14A+ 10B+ 8C= 4500
12A+ 7C = 1000
12A+ 18B+ 10C= 480

Those equations define planes that form the boundary of the "feasible area". Solve for values of A, B, C to get the vertices of that area.
The evaluate Z at each of those vertices. The optimal value will be at one of the vertices.

 

[Architeuthis Dux]

I can provide some guidance on how to approach this problem using linear programming. Linear programming is a mathematical technique used to find the optimal solution to a problem with multiple variables and constraints. In this case, we have three variables (A, B, and C) and three constraints that must be satisfied in order to optimize production.

The first step in solving this problem is to define the objective function, which in this case is the formula Z = 800A + 550B + 400C. This represents the total profit or production value that we want to maximize. Next, we need to identify the constraints, which are the inequalities that limit the values of A, B, and C. These constraints are given as:

14A + 10B + 8C ≤ 4500
12A + 7C ≤ 1000
12A + 18B + 10C ≤ 480

The last constraint states that all variables must be greater than zero. This is a common requirement in linear programming problems.

Once we have defined the objective function and constraints, we can use a computer program or software to solve the problem and find the optimal values for A, B, and C. There are also manual methods for solving linear programming problems, such as the Simplex Method, but these can be time-consuming and prone to errors.

It's important to note that linear programming is a powerful tool, but it relies on certain assumptions and simplifications. For example, it assumes that the relationships between the variables and constraints are linear and that the problem has a unique optimal solution. In some cases, these assumptions may not hold true and other techniques may be needed to find the best solution.

In conclusion, linear programming can be a useful tool for optimizing production in economics, but it's important to carefully define the problem and understand its limitations. If you are struggling with this specific problem, I recommend seeking help from a math tutor or using a specialized software for linear programming.