Simplex Method Section 4 Maximization and Minimization with Problem Constraints Introduction to the Big M Method In this section, we will present a generalized version of the si l th d th t ill l b th i i ti dimplex method that will solve both maximization and minimization problems with any combination of ≤, ≥, = constraints 2 Example. Example: Simplex Method Solve the following problem by the simplex method: Max 12x1 + 18x2 + 10x3 s.t. 2x1 + 3x2 + 4x3 0 x2 - 1.5x3 0 x1, x2, x3 0 Example: Simplex Method Writing the Problem in Tableau Form We can avoid introducing artificial variables to the second and third constraints by multiplying each by -1.

Basic concepts and principles

Minimization Simplex Method Tutorial

Linear programming problem formulation

We will see in this section a practical solution worked example in a typical maximize problem. Sometimes it is hard to get to raise the linear programming, once done, we will use the methods studied in mathstools theory sections: Simplex, dual and two-phase methods.
We start with the statement of an optimization problem

We start understanding the problem. For this we construct the following tables
The first is the cost, or in this case, is a table of gains.

Material	Gains produced (monetary units /Tn)
Lignite	4
Anthracite	3

Material	Cutting hours (by Tn)	Washing time (by Tn)
Lignite	3	4
Anthracite	4	2

Attached to this table, we have the constraint of time available for each machine daily

Process	Hours available daily
Washing	8
Cutting	12

and finally, we have the objective produce at least 4 tons of coal daily

Exthended Theory

With these data, we have what we need to pose the problem of linear programming.
We proceed as follows:
Lets
x1=Tons of lignite produced
x2=Tons of Anthracite produced
We want to maximize the gains, ie, maximize the function 4x1 +3 x2, this will be our objective function.
now building restrictions
We start with the washing process, note that for every day we have
3x1+ 4x2 ≤ 12
Since for every day the number of hours available washing is 12.
Analogously we have for the cutting process
4x1+ 2x2 ≤ 8
Because 8 is the number of hours available for the cutting machine.
Finally, the last of the restrictions is our goal tons of production
x1 + x2 ≥ 4
Putting it together, we obtain the linear programming problem

Now, we can solve the linear programming problem using the simplex or the two phase method if necessary as we have seen in sections of theoryIn this case we use our famous calculator usarmos linear programming problems simplex method calculator.
We placed each of the steps, first introduce the problem in the program
Step 1:
Step 2:

Simplex Method Minimization Calculator

Step 3:
As can be seen, the output of method has gone unresolved optimal solution, this is because the restrictions are too strong, the feasible region is empty.
We will have to alter some (or more) of our restrictions for it, talk to the owner of the factory and send him an email, saying that it is not possible to produce 4 tons of coal daily and meet all these constraints, there are two options
1) Relax our objective function or
2) Relax constraints.
The chief replied that there was an error, and that the real goal is to produce 3 tons of coal (things that happens). We introduce this new information in our linear programming problem, that is, change the last of the restrictions
x1 + x2 ≤ 3
We introduce the problem again simplex method calculator and now yes, we get

Linear Programming Simplex Method Minimization

So, we talk again to the boss and say that the most we can produce with these restrictions is
And that is the solution of our linear programming problem (modified). Any Questions? run this problem here or send us a feedback.

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