Consider the following LP problem.
max z = 2x1 + 3x2 + x3
s.t. x1 + x2 + x3 · 40
2x1 + x2 ¡ x3 ¸ 10
¡x2 + x3 ¸ 10
x1; x2; x3 ¸ 0
It can be transformed into the standard form by introducing 3 slack variables x4, x5 and x6. max z = 2x1 + 3x2 + x3
s.t. x1 + x2 + x3 + x4 = 40
2x1 + x2 ¡ x3 ¡ x5 = 10
¡x2 + x3 ¡ x6 = 10
x1; x2; x3; x4; x5; x6 ¸ 0
There is no obvious initial basic feasible solution, and it is not even known whether there exists one. We can use Phase I method to ¯nd out. Consider the following LP problem derived from the original one by relaxing the second and third constraints and introducing a new objective function.

min x7 + x8, (or max w = ¡x7 ¡ x8)
s.t. x1 + x2 + x3 + x4 = 40
2x1 + x2 ¡ x3 ¡ x5 + x7 = 10
¡x2 + x3 ¡ x6 + x8 = 10
x1; x2; x3; x4; x5; x6; x7; x8 ¸ 0
This problem (Phase I) has an initial basic feasible solution with basic variables being x4, x7 and x8. If the minimum value of x7 + x8 is 0, then both x7 and x8 are 0. As the result, the optimal solution of the Phase I problem is an basic feasible solution of the original problem. If the minimum value of x7 +x8 is bigger than 0, then the original problem is not feasible. We construct tableaus to solve the Phase I problem. The objective value w should be written in terms of non-basic variables:

...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...

...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...

...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...

...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 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...

...Interim report
Contents
1. Abstract page 2
2. Introduction page 2
3. How the Process Currently Works page 3
4. TwoPhase Flow page 4- 6
5. Flow Patterns page 7- 8
6. Interfacial Mass and Energy Exchange page 9
7. TwoPhase Flow in Horizontal Pipes page 10
8. Pressure Drop page 11
9. Future Work page 11
10. Timetable page 12
11. References page 13
1. Abstract
The petro-chemical industry employs a wide range of engineers from many different disciplines to design, problem solve, test, simulate and build, among other tasks, to bring many petroleum/oil based and other chemically manufactured products and to the customers who require or desire them. One of the many problems engineers face in this process is the transport of materials in either a liquid or vapour/gas state. The most common and widely used means of transporting substances in these states is through pipes. Whether be pipes that serve households and business with gas and water to a pipe that acts as an outlet for steam release from a boiler piping is an important transport tool in the modern world, none more so than in the petro-chemical industry. The later of the previous examples will be the focus of this project, examining whether or not conservative estimates in the calculations involved are...

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