Questions tagged [linear-programming]

For questions related to problems that optimize (i.e., minimize or maximize) a linear objective subject to linear constraints.

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1answer
30 views

CPLEX log always the same after hours (“Gap” and “Best Integer” always blanks)

I'm using the Java API of CPLEX 12.6.1 (with license) to solve a MILP (Mixed-Integer Linear Programming) maximization problem. The point is that, after 21 hours, CPLEX has consumed 32 GB of RAM and ...
3
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1answer
56 views

Need help with an appointment scheduling problem

I am currently stuck on writing a linear programming model to describe the process of appointment scheduling for an Oncological Center. I wanted to share it with you guys and see if anyone here could ...
1
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1answer
37 views

Employee allocation based on ranking: Mathematical Model

Suppose I have three employees and I have to assign three employees based on their ranks. If an employee has rank 1 that means he is best. Say, I have the following table ...
3
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0answers
61 views

Simplified risk game: writing a pratical Minimax objective for mixed integer programming

Problem To ensure fairness of the game, I am writing a bot that plays against itself. I have trouble rewriting a minimax objective to a pratical maximization in mixed integer programming. The amount ...
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1answer
69 views

MODM to create balanced groups of students to maximise diversity

I am very new to LP, Goal Seek, Decision Models etc and I have a multiple-objective decision making problem that may or may not require a mix of techniques although I am trying to solve it purely ...
4
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1answer
68 views

An efficient Integer programming model for the minimum spanning tree problem?

Let $T=(V, E')$ be a spanning tree of a graph $G=(V, E)$. Rather than verifying for any subset of vertices $S\subseteq V$ that $|E'(S)|=|S|-1$, is there an efficient way to satisfy the spanning tree ...
2
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1answer
91 views

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 ...
5
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1answer
239 views

What is a good way to penalise LP relaxation?

I have a binary integer program. It is of a large size and the solver is taking longer time. I am thinking of relaxing the binary integer variable and making it a continuous variable. How can I ...
2
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0answers
49 views

How is the dual revised simplex method equivalent to running the RSM on the dual problem?

I've seen the claim (in the title) several places, but can't quite understand why it's true. From what I understand so far, the revised simplex method solves an LP in standard computational form, $$\...
2
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2answers
86 views

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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1answer
81 views

Objective function with exponential coefficients

I have a linear programming problem, with $n$ variables and $a\leq x_{i} \leq b$ for each variable $x_{i}$, where the objective function is $\min \sum\limits_{i=1}^{n}{2^{i} x_{i}}$ Is it true that, ...
10
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1answer
592 views

Linear Optimization Library for C++ with GPU Support

Does anyone know any linear optimization libraries for C++ supporting GPUs for parallelization? If multiple, which do you recommend? The GPU support is important to me since I am dealing with large ...
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1answer
89 views

Multiple If else constraints in Mixed integer programming

How to formulate the following as constraints in MILP? a[0][0] = y, if x[0]= 0, a[0][0] = 0, if x[0] != 0, . . . . a[i][j] = b[i][j-1] + y, if x[j]=i, a[i][j] = a[i][j-1], if x[j] != i, ... . . . b[0]...
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0answers
44 views

Specializing Iterations of Dantzig-Wolfe Decomposition with an Oracle

This arises from an engineering problem I am working on. Let $\mathbf{c}_i,\mathbf{a}_i\in \mathbb{R}^{d}$ be a given set (collection) of vectors where $i\in\{1,\dots,n\}$. Define the bounded ...
4
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2answers
151 views

Asymmetric time-constrained capacitated vehicle routing problem

I am trying to add some more constraints to the flow-based ADVRP model in Almoustafa et al. (2013)1 (pp.4). The mentioned model caps the travel distance, while I cap the travel time. Let $U$ represent ...
3
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1answer
107 views

Extreme points of a simple polyhedron

Consider the polyhedron given by the set of inequalities \begin{align} \mathbf{b}^T\mathbf{x} ~&\leq~ c \\ \mathbf{e}^T\mathbf{x} - 1 ~&\leq~0 \\ \mathbf{x}~&\geq~0 \end{align} where $\...
4
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2answers
135 views

Any Solution for $k$-means with minimum and maximum cluster size constraint?

I am looking for an efficient approach to $k$-means clustering with minimum cluster size constraints. The clusters are non overlapping, so, one point can belong to only one cluster. $N$ be the number ...
2
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1answer
64 views

How to linearize inequalities having max or min?

I'm modeling an LP problem in which I have to maximize an objective function. Two of the constraints are the following, where $k_i$ are constants and $x_i$ decision variables (continuous). Could ...
5
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4answers
417 views

How to visit a subset of network nodes in a single trip?

I have a connected network where I want to visit a set of destinations which may require visiting intermediate nodes as well because there may be no direct edge between source and destination nodes. I ...
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2answers
92 views

How to deal with a decision variable in the objective function that depends on if-else conditions involving other decision variables?

I'm modeling an optimization problem in which a decision variable $x_1$ in the objective function depends on if-else conditions involving decision variables $x_2$ and $x_3$, as the following equation, ...
4
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1answer
305 views

Is there any automatic way to spot contradictory constraints in linear programming?

Let's have the following trivial linear program: \begin{align}\max&\quad z=20A+30B\\\text{s.t.}&\quad A\le60\\&\quad B\le50\\&\quad A+2B\ge220\\&\quad A,B\ge0\end{align} It's easy ...
3
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1answer
54 views

Name for subclass of ILP without any inequality constraints (including constraints on x)

In "Myths and Counterexamples of Mathematical Programming" myth "IP Myth 21" says: The problem of finding $x\in \mathbb{Z}$ such that $Ax=b$, where $A\in\mathbb{Z}^{m\times n}$ ...
1
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0answers
58 views

LP instead of IP formulation of assignment problem

In the example files of GLPK, the assignment problem is written as a linear program. I don't understand why this isn't an integer programming problem. The problem formulation: ...
3
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3answers
103 views

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 ...
1
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1answer
128 views

How to linearize the product of a binary and a continuous variable? [duplicate]

Suppose we have a binary variable $b \in \{0, 1\}$ and a continuous (possibly negative) variable $y \in \mathbb{R}$. How can we linearize the product $b \cdot y$?
4
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2answers
150 views

Scheduling optimisation constraint on consecutive shifts & consecutive night shifts (`python`)

I am trying to write a program to schedule a team of 8 individuals into shifts. I want to know how to model that every individual must get at least one night shift break, and must not work two ...
2
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3answers
122 views

Linearizing a Max Function in the constraint - not working

I have a minimization function which is in its simplest form looks like below. I am including the index of the variables. ...
1
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1answer
247 views

How to mathematically formulate the optimization problem?

I have a system with $S$ service points. There are also $U$ users in the system. We have $$U>S>G$$ One group can have maximum $M$ service points, but there is no restrictions on the number of ...
0
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1answer
85 views

Mixed Integer Programming - How to model the dependency of two variables in an objective function

I have two variables $a$ and $b$, in which $a$ is the amount of goods and $b$ is the amount of boxes of the given sizes. So $b$ (box size + number) is dependent on a (goods quantity). If $a$ is ...
3
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0answers
71 views

How to linearize a max min objective function?

Let us suppose that I have a $\max \min$ objective function that only depends on one set of variables: $\underset{x}\max \underset{y}\min dy$ Associated with the linear set of constraints and right ...
1
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1answer
74 views

Job Shop Scheduling Problem: jobs are scheduled on the same machine at the same time

I want to solve a job shop scheduling problem. I got $n$ Jobs that have to be scheduled on $k$ Machines. A Job $i$ has 2 or 3 Tasks $j$, and there is a known sequence of the Tasks of a Job. One ...
3
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0answers
102 views

0 1 solution of linear programming problem with only equality constraints

I have a linear programming problem $LP$ where all the variables $x_{i}$ take value in $\left[0, 1\right]$ (that is $0\leq x_{i} \leq 1$). All the constraints are as follow: $a_{1}+a_{2}+a_{3}=1$ that ...
0
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0answers
37 views

linear equation with conditions in Java

I have the following problem: I want to determine a linear equation for a data table with $x$ and $y$ value. Point 1 should be the point where the first time $y > 0$ and point 2 should be the point ...
1
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1answer
91 views

complexity order of the interior point method

I was wondering why the complexity order of the interior point method is O()^3 or O()^3.5? Much appreciate your time and consideration.
0
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1answer
79 views

How to linearise this nonlinear constraint?

I have a constraint in the form $\sum_{n=1}^{N}x_{m,n}\omega_{m,n}\ge (t_u-1)\beta_u, \forall u, u=1,2,\cdots, U$ where $x_{m,n}$ is binary variable $t_u$ and $\beta_u$ are continuous optimization ...
1
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1answer
89 views

How to transform this problem with logarithmic objective function into an approximated convex optimization problem?

I have an objective function as follows $\underset{x_{m,n}}{\max}\hspace{1mm}\hspace{1mm}\sum_{m=1}^{M}\log_2\left(\frac{\sum_{n=1}^{N}(1-x_{m,n})\omega_{m,n}+z}{\sum_{n=1}^{N}x_{m,n}\omega_{m,n}}\...
0
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0answers
40 views

Linear equation from pairs of values with conditions in Gurobi (Java)

How can I create a linear equation in Gurobi (Java) from values (x-y value pairs) that also has the following properties: $\forall x \leq 0 \Longrightarrow y = 0$ The linear line/equation should have ...
2
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1answer
62 views

Constraining flow into a node and out of a node using Min Cost Flow input

Let's say I have a graph $G=(V,E)$ where each vertex has an edge going both into and out of it (i.e. cyclic, akin to the kidney donor pair problem). Now suppose we have a capacity $u_{ij} = 1 \forall(...
3
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1answer
120 views

Min Cost Flow with lower bound reduction to MCF algorithm

We define the Min Cost Flow Problem with Lower Bounds (MCFPLB) as a generalization of the usual Min Cost Flow. The input consists of: a directed graph $G=(V,E)$ capacities $u_{ij} \geq 0$ for each ...
2
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1answer
112 views

Can I solve the separation problem efficiently, when I have access to an optimization oracle?

Assume I have given a convex feasible set $X$ and I have an oracle that can optimize some linear objective function $c$ over $X$. Assume that I have given a point $r$. I want to solve the separation ...
10
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2answers
630 views

Do LP solvers convert LPs to standard form?

To solve a linear program (LP) using the simplex method one first needs to bring the LP to standard form. This requires replacing every equality constraint with two inequalities and replacing every ...
1
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0answers
45 views

Unifying constraint matrices in sparse situations

$\DeclareMathOperator\Set{Set}$ Let $Set=\{x\in\mathbb Z^{n}:\exists y\in\mathbb Z^m\text{ satisfying } A[x,y]'\leq b\}$ where $A$ has $r=km$ rows and $k=O(1)$. I am trying to write $$ Set=\{x\in\...
3
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0answers
99 views

Two binding constraints - Linear Programming

I'm having some troubles to continue solving my system, I'm used to solve such systems but with "one" binding constraint, if someone could give me some helpful hints so I can solve it I will ...
4
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3answers
223 views

Clustering points based on a distance matrix

Although I asked this question on stackoverflow to possibly reach a broader audience, I wonder your inputs about this problem. Without giving much research into this, I thought p-center problem ($x_{...
2
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2answers
301 views

If-then constraint with continuous variables

I was usually using if-then constraints with integer variables but ended up using continuous variables and got confused. I have variables $x_{ij}\in\mathbb{R}_{\geq 0}$, and would like to force the ...
3
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1answer
220 views

Algorithms for sparse linear systems

I've long wondered this, but what is the algorithm(s) implemented in modern linear equation solvers for sparse systems? The obvious answer I can think of is Gauss-Jordan with a bunch of tricks to make ...
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0answers
44 views

To estimate new sales from history

A fruit supplier sells 3 types of fruits. The company has 3 salespersons. Here are the sales quantity of each person for each fruit. The total sales figure is available. (this is all the available ...
3
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0answers
34 views

PuLP Python: How to linearize an inequality involving an integer variable

I am working on a Copper payables problem where the objective function is to maximise the sum of copper payable over a time period, T. The total amount of payable tonnes i.e. what the customer will ...
1
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1answer
78 views

How to dynamically set variable names in Pyomo?

I am looking to set variables in my Pyomo model by using a loop, so that they can be created automatically. However, each variable also contains bounds. I was hoping that it can loop through a ...
4
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2answers
272 views

Two-Objective Optimization in CPLEX

Until now, I used CPLEX to solve single-objective optimization problems only, but now I need to solve a two-objective mixed-integer linear optimization problem and I noticed that CPLEX 12.6.9 (unlike ...

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