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...

...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...

...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....

...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...

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