Big-M method
The Big-M method of handling instances with artificial variables is the “common sense approach”. Essentially, the notion is to make the artificial variables, through their coefficients in the objective function, so costly or unprofitable that any feasible solution to the real problem would be preferred....unless the original instance possessed no feasible solutions at all. But this means that we need to assign, in the objective function, coefficients to the artificial variables that are either very small (maximization problem) or very large (minimization problem); whatever this value, let us call it Big M. In fact, this notion is an old trick in optimization in general; we simply associate a penalty value with variables that we do not want to be part of an ultimate solution (unless such an outcome Is unavoidable). Indeed, the penalty is so costly that unless any of the respective variables' inclusion is warranted algorithmically, such variables will never be part of any feasible solution.

This method removes artificial variables from the basis. Here, we assign large undesirable (unacceptable penalty) coefficients to artificial variables from the objective function point of view. If the objective function (Z) is to be minimized, then a very large positive price (penalty, M) is assigned to each artificial variable and if Z is to be minimized, then a very large negative price is to be assigned. The penalty will be designated by +M for minimization problem and by –M for a maximization problem and also M>0.

Example: Minimize Z= 600X1+500X2
subject to constraints,
2X1+ X2 >or= 80
X1+2X2 >or= 60 and X1,X2 >or= 0
Step1: Convert the LP problem into a system of linear equations. We do this by rewriting the constraint inequalities as equations by subtracting new “surplus & artificial variables" and assigning them zero & +M coefficientsrespectively in the objective function as shown below. So the Objective Function would be:...

...The SimplexMethod: Learning Team A
Mike Smith, Todd Jones
Math212/Introduction to Finite Mathematics
February 1, 2011
The SimplexMethod: Learning Team A
Sam’s Hairbows and Accessories is a small company preparing for the next scheduled craft fair. The owners, Sam and Todd, both have full-time jobs in addition to owning the company so they are only able to spend a combined total of 80 hours labor to prepare for the fair in four...

...The simplexmethod is used regularly on applied problem involving thousands of variables and problem constrains (Barnett, Ziegler, & Byleen, 2011). This was the method chosen when Wintel Technologies needed to figure out the best way to utilize time and schedule for their field engineers. With a busy schedule and being needed on several different areas in the United States, the simplexmethod was brought in to decide where...

...SimplexMethod Paper
SimplexMethod Paper
Many people may be wondering exactly what the simplexmethod is. The simplexmethod definition is a method for solving linear programming problems. According to Barnett, Byleen, and Karl (2011) the simplexmethod is used routinely on applied problems involving thousands of variables and problem...

...unboundedness and degeneracy. 5. Use the simplex tables to conduct sensitivity analysis. 6. Construct the dual problem from the primal problem.
Linear Programming: The SimplexMethod
LEARNING OBJECTIVES
After completing this chapter, students will be able to: 1. Convert LP constraints to equalities with slack, surplus, and artificial variables. 2. Set up and solve LP problems with simplex tableaus. 3. Interpret the meaning of...

...of its third chapter.
Key words: linear programming; simplexmethod.
I. Fundamentals and scope
Based on a prototype example, Linear Programming is presented, as well as the simplexmethod of resolution. This method was first presented by G. B. Dantzig in 1947 [MacTutor, 2007]. The text is based on the book by Hillier and Lieberman [2005], and begins with segments of the third chapter of the book.
II. Explanation...

...and SimplexMethods of Linear Programming
The graphical method is the more popular method to use because they are easy to use and understand. Working with only a few variables at a time they allow operations managers to compare projected demand to existing capacity. The graphical method is a trial and error approach that can be easily done by a manager or even a clerical staff. Since it is trial and error though, it does...

...The simplexmethod is a general mathematical solution technique for solving linear programming
problems. In the simplexmethod, the model is put into the form of a table,
and then a number of mathematical steps are performed on the table. These mathematical
steps in effect replicate the process in graphical analysis of moving from one extreme
point on the solution boundary to another. However, unlike the graphical...

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