The Simplex Method: Learning Team A
Mike Smith, Todd Jones
Math212/Introduction to Finite Mathematics
February 1, 2011

The Simplex Method: 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

This problem, as outlined above, is an example of a linear programming problem. Linear programming is part of the Optimization Techniques field of Mathematics, used for resource allocation and organization. With linear programming problems, one takes the inequalities that exist within a given situation and deduces a best case scenario under those particular conditions (Stapel, 2009). One particularly effective method of solving linear programming problems is the Simplex Method. The Simplex Method was invented in 1947 by the Mathematician George B. Dantzig. One of the greatest achievements in...

...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...

...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...

...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...

...1
Topics on "Operational Research" Mar. 2007, IST
Linear Programming, an introduction
MIGUEL A. S. CASQUILHO IST, Universidade Técnica de Lisboa, Ave. Rovisco Pais, IST; 1049-001 Lisboa, Portugal
Linear Programming is presented at an introductory level, mainly from the book by Hillier and Lieberman [2005], abridged and adapted to suit the objectives of the “Operational Research” 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...

...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...

...Unit 1 Lesson 9 : The Big M Method
Learning outcomes • The Big M Method to solve a linear programming problem.
In the previous discussions of the Simplex algorithm I have seen that the method must start with a basic feasible solution. In my examples so far, I have looked at problems that, when put into standard LP form, conveniently have an all slack starting solution. An all slack solution is only a possibility when all of the constraints in the problem have or = constraints, a starting basic feasible solution may not be readily apparent. The Big M method is a version of the Simplex Algorithm that first finds a basic feasible solution by adding "artificial" variables to the problem. The objective function of the original LP must, of course, be modified to ensure that the artificial variables are all equal to 0 at the conclusion of the simplex algorithm. Steps 1. Modify the constraints so that the RHS of each constraint is nonnegative (This requires that each constraint with a negative RHS be multiplied by 1. Remember that if you multiply an inequality by any negative number, the direction of the inequality is reversed!). After modification, identify each constraint as a , or = constraint. 2. Convert each inequality constraint to standard form (If constraint i is a < constraint, we add a
slack variable si; and if constraint i is a > constraint, we subtract an excess...

...USING STEPPING STONE AND MODI METHODS TO SOLVE TRANSPORTATION PROBLEMS
BY
ABDUSSALAM MUHAMMAD MUSTAPHA
09/211306009
A SEMINAR PAPER PRESENTED TO THE DEPARTMENT OF MATHEMATICS, FACULTY OF SCIENCE, USMANU DANFODIYO UNIVERSITY, SOKOTO IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE AWARD OF THE DEGREE OF MASTER OF SCIENCE (MATHEMATICS)
SUPERVISORY TEAM:
MAJOR SUPERVISOR: DR. U. A. ALI
CO – SUPERVISOR I: DR. I. J. UWANTA
CO – SUPERVISOR II: DR. MU’AZU MUSA
DATE: 07TH NOVEMBER, 2012
TIME: 10:00 AM
VENUE: U.G. COMPUTER LABORATORY
Abstract
This research is posed with investigation of transportation problems of a bus transit company considering the importance derived from the transport sector of the economy to the leaders and the led. The research considered the balanced type of transportation problems and showed how the initial solutions can be obtained as well as the optimal solution which is of great importance. The Vogel’s approximation method (VAM) was used to determine the initial solution while the Stepping stone (SS) and the Modified Distribution (MODI) methods were used to test for optimality. The algorithms for obtaining the VAM, SS and MODI methods are presented and used to maximize the transportation problem of Katsina State Transport Authority for four busiest routes selected. The optimal solution obtained yielded an increase of about 1.4% of the daily income generated by the...

...
Managerial Decision Modeling w/ Spreadsheets, 3e (Balakrishnan/Render/Stair)
Chapter 2 Linear Programming Models: Graphical and Computer Methods
2.1 Chapter Questions
1) Consider the following linear programming model:
Max X12 + X2 + 3X3
Subject to:
X1 + X2 ≤ 3
X1 + X2 ≤ 1
X1, X2 ≥ 0
This problem violates which of the following assumptions?
A) certainty
B) proportionality
C) divisibility
D) linearity
E) integrality
Answer: D
Page Ref: 22
Topic: Developing a Linear Programming Model
Difficulty: Easy
2) Consider the following linear programming model:
Min 2X1 + 3X2
Subject to:
X1 + 2X2 ≤ 1
X2 ≤ 1
X1 ≥ 0, X2 ≤ 0
This problem violates which of the following assumptions?
A) additivity
B) divisibility
C) non-negativity
D) proportionality
E) linearity
Answer: C
Page Ref: 21
Topic: Developing a Linear Programming Model
Difficulty: Easy
3) A redundant constraint is eliminated from a linear programming model. What effect will this have on the optimal solution?
A) feasible region will decrease in size
B) feasible region will increase in size
C) a decrease in objective function value
D) an increase in objective function value
E) no change
Answer: E
Page Ref: 36
Topic: Special Situations in Solving Linear Programming Problems
Difficulty: Moderate
4) Consider the following linear programming model:
Max 2X1 + 3X2
Subject to:
X1 ≤ 2
X2 ≤ 3
X1 ≤ 1
X1, X2 ≥ 0
This linear programming model has:
A) alternate optimal solutions
B) unbounded...