Questions tagged [simplex]
For questions related to the simplex method for linear programming (LP), which solves LPs optimally by moving iteratively from in corner point of the feasible region to a better one.
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Are there two types of 'slack variables'?
In simplex algorithm, in order to handle inequation constraints, we need to convert them into equations by introducing so-called 'slack variables', like
$$
\mathbf{a}^T\mathbf{x} + b \leq 0\quad\...
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1
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Short cuts when using the simplex algorithm
I haven't really kept myself up to date with research on the simplex algorithm for several years. I have taught linear programming, but it has not focused on the cutting edge of things.
I remember ...
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Computing simplex tableu for a given basis
I found the following problem in my book.
I know that I can compute the simplex tableau , let's call it S for a basis X_b=(x_1, x2, x_5)^T as ...
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3
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Any good write up for Dual simplex with boxed variables?
I was wondering if there is any write up that shows how to perform dual simplex with boxed variables where l <= x <= u and preferably with a small example (...
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Help me reproduce this tableau from the 'Integer Programming' book
From the Integer Programming book by Conforti et al, I've sniped the image below. At the bottom of this image there is the remnants of a tableau, presumably from several iterations of the simplex ...
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2
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Simplex method for multiple objectives
I am a user of Google OR Tools, which can interface with many LP & MIP solvers, plus it's own SAT based constraint programming solver.
My question, in the context of OR-Tools, is: how should I ...
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3
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Question regarding primal Simplex method
Given the following degenerate
optimization problem
\begin{align}\min&\quad c^Tx\\\text{s.t.}&\quad Ax=b,\\&\quad x\ge 0\end{align}
Using primal simplex algorithm (either revised or ...
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0
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64
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How to incorporate artificial variables into the revised simplex method
Seems like we can convert >= constraints to <=, either by multiplying by -1 or using variable substitution. But for ...
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1
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Finding a dual feasible basis for use with the dual simplex algorithm
I have learnt that the dual simplex method requires reduced costs to be non-negative or else it cannot be used.
I wanted to know what could be done to find a dual feasible basis and came across this ...
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1
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Inconsistent teachings on how to choose a non basic variable to enter the basis (primal simplex)
During the primal simplex algorithm, a non-basic variable must be chosen to enter the basis.
Many resources on the subject choose a variable based solely on its coefficient in the row of the tableau ...
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Column generation: decreasing value of restricted master problem
I am using column generation to solve a minimization problem.
At a given iteration, my subproblem finds a column with reduced cost $-1$, and in the following restricted master problem, this new column ...
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0
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Simplex Method Manual pivoting in GLPK for a Warm Start
My question is kind of related to this 2015 post at Stackoverflow.
About Simplex method tableau pivoting in Linear Programming -- this is proving to be difficult in GLPK.. I am using Linux, GLPK 5.0 ...
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1
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SoPlex basis file format
I'm running SoPlex from the command line to solve some linear programming problems. I'd like to get the basis of the solution, so I use the --writebas flag. However,...
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Phase I of the simplex method and Farkas certificates
Phase I of the simplex method solves an auxiliary optimization problem to determine an initial basic feasible solution, or concludes that no such exists. Is there a way to use the solution of this ...
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Upper bound on number of pivots to escape a degenerate point
Is it always possible to escape a degenerate point by a single pivot, or is it possible that several pivots are required?
In other words is it possible to get away from a degenerate point by a single ...
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How to make unconstrained variables non-negative (as in excel solver) in AMPL?
This is a sequencing problem.
I've got this variables
...
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4
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Where is the original Dantzig Simplex 1947 paper?
I see from here and many other sources that Dantzig invented the Simplex method in 1947. After much searching, I found that the earliest publication is this in 1956. Does anyone know where the ...
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Why are several of the decision variables zero at the corner point of a polytope?
I have the following equational Linear Program: \begin{align}\max&\quad c^T x\\\text{s.t.}&\quad Ax=b\\&\quad x\ge0\end{align}
The matrix $A$ is $m\times n$, where $m\le n$, $c\in \mathbb{...
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Dual instability, degeneracy, tailing off effect - Which are the causes and which are the effects?
Dual instability, degeneracy, and the tailing off effect are often mentioned together in papers. However, I cannot seem to find a clear explanation on which of these cause the other and vice versa? ...
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Why is the tailing off effect only a problem in column generation?
Why is the tailing off effect only a problem in column generation? If all columns were pregenerated, and one used the simplex method, wouldn't one see the tailing off effect?
Is it simply not an issue ...
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3
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Can the primal Simplex Method walk all optima in linear time?
It's typical that the Simplex Method implementation exits once it finds an optimum value. However, if I want to find all optima that exist at extreme points (not those that exist along a face), is it ...
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Why is it called the "Simplex" Algorithm/Method?
I have been trying to learn more about the Simplex Algorithm/Method. In particular, I am interested in knowing why this algorithm is called the "Simplex Algorithm".
For instance, when ...
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Interpretation of Reduced Costs
I am looking for an answer to a question I can't quite get behind. (continuation of Linear Programming: Integer and non-integer decision variables)
I am given the following mathematical optimization ...
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1
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Linear Programming: Integer and non-integer decision variables
I am looking for an answer to a question I can't quite get behind.
I am given the following mathematical optimization problem: \begin{align}\min&\quad\sum_{t\in T}s_t\cdot z_t+h_t\cdot i_t+p_t\...
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Best way to add dummy to transportation problem? Zero cost will be always chosen first?
I know that an unbalanced transportation problem could be made a balanced transportation problem by adding a dummy node which equals the difference between demand and supply.
In literature, dummy ...
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2
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Transformations that leave the linear program unchanged
A typical linear program is written as $$L_0:\min_{x \geq 0; A^\top x \leq b}c^\top x.$$ Here, $x \in \mathbb{R}^n$, $c \in \mathbb{R}^n$, $A \in \mathbb{R}^{m \times n}$, and $b \in \mathbb{R}^m$.
...
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3
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warmstarting simplex algorithm- how much can problems differ from each other?
I'm working on an implementation of the simplex algorithm.
I want to solve problems in real time every 30 minutes. They could be interpreted as a classic transportation problem.
I couldn't really say ...
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3
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How Close Is Linear Programming Class to What Solvers Actually Implement for Pivot Algorithms
As part of a final project for my linear programming course, I have been asked to discuss implementations of pivot algorithms, including which combinations of the ideas we have talked about in class ...
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1
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Simplex (GLPK) doesn't find a feasible solution on this simple assignment problem, but there is an obvious one
Problem
Assign 11 projects to 11 students, based on their preference.
For this example, each students chooses only one project, for simplicity shake (as shown below). Student 1 one chooses project 1, ...
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0
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Provide basic solution to CLP
I'm using Pyomo to formulate an LP with approx 500,000 constraints and 200,000 decision variables. The LP is solved using CLP. Some instances fail to return even a feasible solution after many ...
6
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2
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Simplex algorithm and extreme points
For this question my short-hand is LP = linear program, BFS = basic feasible solution, SEF = standard equality form.
Since the Simplex algorithm iterates from extreme point to extreme point (...
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Simplex Multiplier
I am reading through a book which provides an example of a linear program given by \begin{align}\min&\quad-24y_{1}-28y_{2}\\\text{s.t.}&\quad6y_{1}+10y_{2} \leq 2400\\&\quad8y_{1}+5y_{2} \...
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Simplex - Network flow problem : Arc from 1 to P with infinite capacity
The Network - Maximum flow problem below aims to find the maximum flow using simplex method :
With the LP as follow :
LP : \begin{Bmatrix}
Z(Max) = \sum_{i=1}^{m} fi \\
Af =0
\end{Bmatrix}
...
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2
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Linear optimization problem with user-defined cost function
I have a linear optimization problem for which I am looking for a suitable optimization solution that can fulfill my requirements. Here is an explanation of the optimization problem:
There are a ...
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0
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Proof that the leaving variable cannot be selected as the entering one in the next round
Using the Dantzig's pivoting rule, can it be proven that the leaving variable of one round cannot be selected as the entering variable in the next round?
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Does anyone have the criss cross algorithm programming code to solve linear programming problems?
I have a project that requires programming code for the simplex algorithm and criss-cross algorithm. The purpose of this project is to compare the two methods. I've tried to find it, but the ...
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Specific usecase of two-phase simplex algorithm
The problem below aims to find to most optimal way to transport the fuel :
A company Er must transport a type of fuel from its two refineries Ra and Rb to its two points of sale PV1 and PV2. The ...
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0
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On solving the Restricted Master Problem in Column Generation technique
I am working on developing a column generation (CG) based optimization framework for a large-scale airline crew pairing problem (a set-covering problem). First, I generate an initial feasible solution ...
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1
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Manually indicate initial basis for coin-or lp solver CLP
I have a set partitioning formulation with each constraint being an equality constraint to meet the given demand (right-hand side of the constraint).
For each constraint, I have a slack and a surplus ...
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2
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Is the iteration-limited Simplex dual solution of a MIP node useful?
Idea
Sometimes I encounter problems where Simplex spends many iterations for final convergence to the optimal objective value. Let's suppose, this happens when solving branch and bound-tree nodes as ...
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Solving a minimization problem using a Simplex method
There is a method of solving a minimization problem using the simplex method where you just need to multiply the objective function by -ve sign and then solve it using the simplex method. All you need ...
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Derivations for two formulae for obtaining optimal dual variable values from the optimal primal tableau
We're being taught Industrial Engineering and Operations Research for the first time this semester. Referring to the book by Hamdy A. Taha, I noticed the mention of two formulae for swiftly obtaining ...
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In Linear programming, how to determine if shadow price does not change linearly?
As the title says, in linear programming, is there a way to determine that the shadow price does not change linearly for a resource?
I understand one way is to simulate but is there a way to tell ...
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All optimal solutions
I have a following problem:
If I have some function $aX+bY+cZ+mD+nF$ and I want to maximize it and have some constraints, how can I find ALL solutions for this maximum value of the function?
To sum ...
5
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1
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Preemptive Goal programming by fixing nonbasic variables with non-zero reduced costs
I have been using the method of fixing nonbasic variables with non-zero reduced costs to do preemptive goal programming. It works for the most part. But I have recently noticed in a certain instance ...
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Negative reduced cost for basic variable
I am observing something unusual : after solving a linear program, some basic variables have negative reduced costs (instead of $0$) :
...
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Demonstrating a given solution is basic for a MCFP?
Given a minimum-cost flow problem, how could I go about demonstrating that a specific solution (as sets of basic and non-basic (!) arcs that build a rooted spanning tree for a given graph) is basic ...
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1
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Introducing a big M variable in given equations
While I do understand the general workings of the Big-M-method I am struggling with the following sample exercise, in which the Big-M-method has to be used to find a first feasible solution:
\begin{...
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How to access neighboring extreme points to an optimal extreme point of an LP?
Suppose that I have access to an optimal non-degenerate extreme point of an LP. I need to find some $\epsilon$-optimal extreme points. That is, a point $x$ where $c'x \le z^{*} + \epsilon$.
One way ...
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Doubt on finding simplex's initial canonical tableau (II Phase)
Good day.
Given the following notation for an initial canonical tableau for a linear program in standard form:
$$
T_1 =
\begin{bmatrix}
I & B^{-1}N & \bar{x}_{B} \\
0^\intercal & \hat{x}...
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