Questions tagged [two-phase-simplex]

For questions about the two phase simplex method, which is an algorithm to solve a linear program which has no initial basic feasible solution.

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Two phase simplex when constraint is the same as minimize function

Consider the following linear program: Maximize $Z = x - y$ where $-x+y\ge 1$ and $x\ge 0, y\ge 0$ Add surplus and artificial variables to the constraint: $-x+y-s+a=1$ For phase 1 maximize $W = -...
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Two phase method in linear programming

Suppose following tableau came after one iteration in first phase of a two phase method problem, here $s_1$ is a surplus variable and $s_2$ is a slack variable $w$ is a artificial variable. I tried ...
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1answer
151 views

Two-phase simplex

The upper part of the image attached is the question and lower part is the solution. I did the exact same table as table4 in the picture, yet I put the ratio of [2] as 0.5/0.5=1 instead. Could someone ...
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280 views

Is simplex method weaker than other methods?

Given linear program: $$ \text{min } x_1 - x_2 + 2 x_3 $$ s.t.: $$ -3x_1 + x_2 + x_3 = 4 $$ $$ x_1 - x_2 + x_3 = 3 $$ $$ x_i \geq 0; i = \{1,2,3\} $$ solution by simplex method (with double pass) is ...
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Linear Program Transformations

I have a Linear Program with constrains of the form: $$a_{11}x_1+a_{12}x_2+\ldots\le 0$$ $$a_{21}x_1+a_{22}x_2+\ldots\le 0$$ $$a_{31}x_1+a_{32}x_2+\ldots\le 0$$ My problem is that if I try to ...
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I want to know what is the relation between network flow min cost problem and max flow problem with simplex method linear programming,

I want to know what is the relation between network flow min cost problem and max flow problem with simplex method linear programming, such as primal dual and complementary slackness and how can i ...
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1answer
38 views

Simplex Algorithm, determining Two Phase is required and choice of artificial variables

Given the following system : \begin{align*} \text{minimise } z = &2x_1 &+ 3x_2 &+ 3x_3 &+ x_4 &- 2x_5& \\ \end{align*} Subject to \begin{align*} &...
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Simplex Algorithm: Why does Phase I deliver a BFS immediately?

When trying to solve a Linear Program (I call it primary problem below) of the form min $c^{T}x$ s.t. $ $ $Ax = b, x \geq 0, b \geq 0 $ The two phase simplex algorithm suggests to starts with the ...
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354 views

Minimize using simplex method with mixed constraints

Using the simplex method, minimize $z= -2x_1+12x_2-3x_3-5x_4-x_5$ with $x_i \geq 0$ and subject to constraints: $x_1+4x_3-5x_4+2x_5 \leq 4$ $4x_2-x_4 \leq 1$ $5x_1-6x_2-3x_4 \leq 6$ $...
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Simplex Algorithm (Two-Phase) - Finding pivot entries consistent with the simplex method

I'm working through the examples in my professor's class notes, but I can't figure out how to solve this problem: Example: In the following standard tableau, mark by * the choices for the pivot ...
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1answer
148 views

List the six possible bases of the LP P and find their corresponding solutions

Consider the following LP P: max z = 22x_1 - 12x_2 subject to: 8x_1 + 4x_2 \le 15 2x_1 + 6x_2 \le 7 x_1, x_2 \ge 0 ...
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Converting a primal feasible problem to dual LP problem, does it implies a feasible solution?

I'm new in this Stack exchange,a friend of mine and I are using the lindo, to solve this LP problem: $$ \begin{align*} \max z= 18x_{11}+18x_{12}+20x_{21}+20x_{22}-6x_{41}-10x_{51}-4x_{61}-4x_{42}-...
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1answer
179 views

General question about new objective function W using the simplex method

In regards to the two-phase simplex method; When creating a new objective function that consists the sum of the constraint(s) with artificial variables, I am told that if the Min value of (wmin) w is >...
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A doubt regarding Vogel's Approximation Method

In VAM, we usually start with the row/column with maximum of differences (and we select the row/column accordingly). But, if I don't select the 'absolute maximum' and go on to start the process with ...
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Linear programming/seeing feasibility and unboudedness

Consider the dual linear programming problem and its simplex dually feasible table: $$\begin{array}{|c|c|c|c|c|c|c|c|}\hline -4& 0 & 1&5&16&0&4&0 \\ \hline -12& 0 &...
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Proof verification for simplex method problems

I was studying simplex method in LPP from "Introduction to Linear Optimization by Bertsimas and Tsitsiklis", and came across this problem: Consider the simplex method applied to a standard form ...
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116 views

calculating the extreme rays of a polyhedra

How could I go about the problem finding the extreme rays of a polyhedral defined by constraints $x_1-x_2 \geq -2$ $x_1+x_2 \geq 1$ $x_1,x_2 \geq 0$ I know for certain that given a max or min ...
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In Simplex Method, can I move from a basic solution $x^{*}$ to the other $y^{*}$ by pivoting zero cost in Tableau?

Suppose that $B_1$ is the bases for $x^{*}$ and I want to go to the second solution $y^{*}$ with bases $B_2$, how to go to the second optimal solution through pivoting? My attempt was trying to prove ...
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1answer
222 views

Operations Research: Simplex Table Unbound Optimal Solution

The problem statement, all variables and given/known data Hi I am looking at this quesiton attached part c). Relevant equations The attempt at a solution I can by my notes which tell me that if I ...
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70 views

Complexity of the first phase method

(first post on math SE) I am looking for a formal upper-bound on the number of iterations in the first phase, in terms of the number of artificial variables (A.V). I have some intuition that it ...
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1answer
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Show that it as no feasible solution.

An optimization problem: $$\text{ maximize } z=8x+6y$$ $$\text{ such that: } x-y ≤ 0.6 \text{ and } x-y≥2$$ Show that it as no feasible solution using SIMPLEX METHOD. It seems very logical that ...
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87 views

Maximising a Linear Programming Problem

Maximize $w=2x+3y+6z$ subject to $2x+y+z \le 5$ $3y+2z \le 6$ $x,y,z \ge 0$ Is the optimal solution unique? Justify your answer. I tried to solve it by Simplex method. In the second iteration, ...
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How to Adequately Implement Phase I of Two-Phase Simplex Algorithm on a Computer with Floating Point Error

I'm currently trying to write some code that implements Phase I of the two-phase Simplex Algorithm described here: http://www.statslab.cam.ac.uk/~ff271/teaching/opt/notes/notes8.pdf In order to test ...
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1answer
129 views

Simplex method state after first phase

I'm implementing a simplex method solver for a standard problem $$ \begin{aligned} \operatorname{minimize} \qquad&c^T x\\ \operatorname{subjected to} \qquad&Ax = b\\ &x \geq 0\\ \end{...
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260 views

If we start with a feasible tableau in simplex method, are we basically generating a different feasible point in every pivot step?

This is a true and false question. The actual question reads: "In solving a linear program by the simplex method, starting with a feasible tableau, a different feasible point is generated after every ...
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1answer
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Simplex method and basic solutions

I have put this into the form $0.5x_1 + 0.25x_2 + x_3=6$ $-x_1 - 3x_2 + x_4=-2$ $x_1 + x_2 = 10$ Is this correct? If so, how do I find a basic solution so that I can begin the simplex algorithm?
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linear program-Simplex method-Dual problem

At an exercise I am asked to solve a linear program using the simplex method(in Matlab).Then I have to formulate the dual of this problem and read off an optimal solution of the dual problem from the ...
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How to solve this question by the two-phase simplex algorithm?

A clever but ethically corrupted mathematics student used to sell assignment solutions to her lazy fellow students. The student, however, learned that she can make much more money by charging the ...