Linear programming maximization

[eldrito]

Who knows solve this problem of linear programming of maximization, explaining me all the steps to reach the solution.

total cost to produce the products a,b,c: 100

cost to produce the product a (x)= ? quantity to produce of product a = 5
cost to produce the product b (y)= ? quantity to produce of product b = 1.39
cost to produce the product c (z)= ? quantity to produce of product c = 1.77 I have three production lines, each is equipped with a different technology that does not allow me to produce the different products a, b, c in a single production
line, so I necessarily use all three production lines, creating a product mix.

These are production functions related to each production line

line 1) ax + cy

line 2) (0.5by + 0.5y) + (0.5cz + o.5z)

line 3) bz

what are the production costs (x,y,z) that must support for each type of product (a,b,c) to be produced on the three production lines 1),2),3) to obtain ever
the highest possible gain.
Considering that the costs of each type of product will be equal to its sale price (example: (x) is the cost to produce the product "a" will be equal to its sale
price,The same goes for the other two products "b" and "c").
Consider also that the total cost must not exceed 100.

 

[Aaron Bex]

We can solve this problem using linear programming. We have 3 decision variables (x, y, z) and the objective is to maximize our gain.
The constraints are:
1. x + y + z <= 100 (the total cost should not exceed 100)
2. ax + cy <= 5 (quantity of product a should be 5)
3. 0.5by + 0.5y + 0.5cz + 0.5z <= 1.39 (quantity of product b should be 1.39)
4. bz <= 1.77 (quantity of product c should be 1.77)

We can solve this problem using the simplex method.
First, we will convert the objective and the constraints into a canonical form.

Maximize: P = x + y + z
subject to:
1. x + y + z <= 100
2. ax + cy <= 5
3. 0.5by + 0.5y + 0.5cz + 0.5z <= 1.39
4. bz <= 1.77

Now, we will add slack variables (s1, s2, s3, s4) to convert the above inequality constraints into equality constraints.

Maximize: P = x + y + z
subject to:
1. x + y + z + s1 = 100