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 weeks. Sam’s offers five main product lines: basic bows, elaborate bows, bug clips, flower clips, and headbands. Sam’s want to calculate the mix of products they should bring to the fair to maximize their potential profit.
Sam’s believes it is important to give their customers a variety of products. They want every product to make up at least 10% of the total items offered for sale but no more than 30%. Sam’s also knows from past festivals, that headbands are their biggest seller and want at least 15% of their product mix to be headbands. To fill their booth, they want to take at least 400 items. The cost, selling price, and labor requirements for each product are listed in Table 1.
Table 1
| Cost to Make | Selling Price | Labor Required (in minutes) |
Basic Bow | 0.27 | 2.50 | 7 |
Elaborate Bow | 1.07 | 4.00 | 15 |
Bug Clip | 0.22 | 2.50 | 10 |
Flower Clip | 0.94 | 3.00 | 5 |
Headband | 0.82 | 4.00 | 20 |
This problem, as outlined above, is an...

...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 and when employees should be spending their time and the company’s money. The organization will do an analysis of the impact of the different compensation methods and benefit programs on employees and the organizations will discuss and gain an understanding of how these programs impact the organization. A comprehensive look at the compensation methods and benefit program is necessary to reveal any holes in the system. The company will then explore the many options available to it when deciding on a compensation package suite to both the employee and the organization.
An objective function is when you have one word that is the keyword. And that keyword is either minimized or maximized. You can do this to a name, colon, or a linear equation. “Although a particular linear program must have one objective function, a model may contain more than one objective declaration.” (Fourer, Gay, & Kernighan, 2003, pg. 134, Chapter 8). Constraints are a little...

...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 constraints. George B. Dantzig developed the simplexmethod in 1947. In this paper the topic of discussion includes how to solve a simplexmethod problem that a private artist creates paintings in a variety of sizes. Below describes the problems that the artist is facing.
Painting A: 8 X 10: requires 20 hours of labor, 1 hour to mat and frame
Painting B: 10 X 24: requires 60 hours of labor, 1 hour to mat and frame
Painting C: 24 X 48: requires 80 hours of labor, 2 hours to mat and frame
The artist is only able to spend 20 hours per week creating paintings.
The profits for each painting are: A: $400, B: $800, C: $1000
How many paintings should the artist create (and what sizes) within 1 year to maximize profits?
What would the artist’s maximum profits be?
Objective Function and Constraints
To find the objective function and constraints present in the situation described evaluation of the elements...

...MODULE
7
4. Recognize special cases such as infeasibility, 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 every number in a simplex tableau.
CHAPTER OUTLINE
M7.1 M7.2 M7.3 M7.4 M7.5 M7.6 M7.7 Introduction How to Set Up the Initial Simplex Solution Simplex Solution Procedures The Second Simplex Tableau Developing the Third Tableau Review of Procedures for Solving LP Maximization Problems Surplus and Artificial Variables M7.8 M7.9 M7.10 M7.11 M7.12 M7.13 Solving Minimization Problems Review of Procedures for Solving LP Minimization Problems Special Cases Sensitivity Analysis with the Simplex Tableau The Dual Karmarkar’s Algorithm
Summary • Glossary • Key Equation • Solved Problems • Self-Test • Discussion Questions and Problems • Bibliography
M7-1
M7-2
MODULE 7 • LINEAR PROGRAMMING: THE SIMPLEXMETHOD
M7.1
Introduction
In Chapter 7 we looked at examples of linear programming (LP) problems that contained two...

...course. It begins with segments 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 of the simplexmethod 3 Introduction to Linear Programming
(H&L 25)
The development of linear programming has been ranked among the most important scientific advances in the mid-20.th century, and we must agree with this assessment. Its impact since just 1950 has been extraordinary. Today it is a standard tool that has saved many thousands or millions of dollars for most companies or businesses of even moderate size in the various industrialized countries of the world; and its use in other sectors of society has been spreading rapidly.
3.1 Prototype example
(H&L 26)
Table 1 Data for the Wyndor Glass Co. problem
Plant A B C Profit per batch Production time per batch (h) Product 1 2 1 0 0 2 3 2 3 000 5 000 Production time available per week (h) 4 12 18
x1 = number of batches of product 1 produced per week x2 = number of batches of product 2 produced per week Z = total profit per week (in $1000) from producing these...

...Graphical 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 not necessarily generate the optimal plan. One downside of this method though is that it can only be used with two variables at the maximum. The graphical method is broken down into the following five steps:
1) Determine the demand in each period.
2) Determine the capacity for regular time, over time, and subcontracting each period.
3) Find labor costs, hiring and labor costs, and inventory holding costs.
4) Consider company policy that may apply to the workers or to stock levels
5) Develop alternative plans and examine their total costs.
When a company has a LP problem with more than two variables it turns to the simplexmethod. This method can handle any number of variables as well as for certain give the optimal solution. In the simplexmethod we examine corner points in a methodical fashion until we arrive at the best solution which is either the highest...

...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 graphicalmethod, in
which we could simply search through all the solution points to find the best one, the simplexmethod moves from one better solution to another until the best one is found, and
then it stops.
The manual solution of a linear programming model using the simplexmethod can
be a lengthy and tedious process.Years ago, manual application of the simplexmethod
was the only means for solving a linear programming problem. Now computer solution
is certainly preferred. However, knowledge of the simplexmethod can greatly enhance
one’s understanding of linear programming. Computer software programs like QM
for Windows or Excel spreadsheets provide solutions to linear programming problems,
but they do not convey an in-depth understanding of how those solutions are derived. To
a certain extent, graphical analysis provides an understanding of the solution process,
and...

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