Questions tagged [integer-programming]

For questions about mathematical optimization problems involving binary or general integer variables.

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Check VRP instance is feasibility

Beforehand, this is a very long thread, in case you want to know in advance, to see if this thread's interests match with yours, this thread concerns fast ways of determining whether a VRP instance is ...
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1answer
162 views

Scheduling minimization Integer Programming problem formulation

I am working with integer optimization. I have a problem with $t$ tasks and every task $i$ needs $w_i$ weeks to be completed and $p_{il}$ workers on a specific week $l$. There is a total time in weeks ...
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MIP: Do binary variables perform better that integers? [duplicate]

I have a model where investments can be done in blocks. Now I could model this with integer or binary variables. Does anybody know which one is the better choice in terms of computational performance?
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if $x = 0$ then $y \ne b$

I'm trying to model the following: if $x=0$ then $y \ne b$ $y$ is a positive integer number( $y\le U$) and $x$ is binary and $b$ is a constant.
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161 views

How to make the elements of the solution of gurobi belong to the elements of the specified list?

If I want to use the elements of the list as the range of the solution, like list1 = [10,20,50,60,30],and the elements of the solution must belong to the elements of the list The sample example as ...
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How can I set the solution of gurobi to be a multiple of 10 instead of all integers?

For example, the solution for gurobi has two solutions, as follows: [10,20,50,70] [55,79,30,80] I only want to output solutions that contain only multiples of 10. The sample example as follow: ...
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1answer
108 views

Is that Ok to exclude fixed components from an objective function?

Suppose we have the following objective function with one decision variable $x_i$ where $p_i$ is a fixed parameter for each $i$ and also, $a$ is a constant for the problem \begin{align} \label{eq} \...
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1answer
99 views

Capacitated Maximum Coverage Location Problem, Python and Gurobi

I am building a variant of the maximum coverage location model and want to limit the amount of points that each "facility" can cover. I am using Gurobi optimization . I have tried using the ...
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1answer
73 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 ...
4
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1answer
76 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 ...
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131 views

How to prove the following statement about convex hulls?

Consider $M$ finite sets of integer points $P_m$, $m=1,\ldots,M$. Let $$A = \left\{x_m\in\operatorname{conv}P_m, m=1,\dots,M, \sum_{m=1}^MN_mx_m=0\right\}$$ and $$B =\operatorname{conv}\left\{x_m\in ...
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Is there any academic reference which suggests/uses dual values as initialization of Lagrangian multipliers?

The Lagrangian relaxation approach is used to generate lower (upper) bounds for minimization (maximization) problems by moving some constraints to the objective function and multiplying them by "...
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133 views

What to do with cuts (constraints) when a fixation is contrary to a RHS in a ILP / LP relaxation?

I am trying to understand an algorithm in a paper by Crévits et al. (2012)1 (see algorithm 2, the cuts I'm referring to are from the reduced costs). It uses a series of successive cuts on a linear ...
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157 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 ...
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1answer
63 views

Finding the minimum of a group of timings

I would like to seek some modeling advice on the following: Say for instance I have 5 nodes representing workstations of the operation of 5 jobs, and that I have less than 5 vehicles. Say I have two ...
3
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1answer
128 views

Modeling that there is no feasible solution to a linear system in mixed integer programming

My question is about how to construct a mixed integer programming to model that there is no feasible solution to a given linear system. Specifically, given $x\in \mathbb{R}^{n}$ and $z\in \{0,1\}^{d}$,...
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Characterization for total dual integrality

A problem I study reduces to whether the polyhedron $P=\{\mathbf{x}\mid A\mathbf{x}=\mathbf{1}, \mathbf{x}\geq0\}$ is integral ($A$ is a matrix with coefficients in $\{0,1\}$). I know that the ...
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275 views

No-good cuts for general integer variables

Question: Suppose we have an integer program $\min\{c^\top{x}\mid{Ax\leq{b}},x\in\mathbb{Z}_+^n\}$, and suppose that $x^*$ is a feasible solution for this IP (or even that $x^*$ is an extreme point of ...
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MAX-CUT: are there any algorithms or codes for classical computers, that cater to this specific case?

I missed the opportunity to ask this on OR.SE by 24 days! I asked it at CS.SE on 6 May 2019 and OR.SE entered Private Beta on 30 May 2019. It's a problem about minimizing a sum of terms that are ...
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1answer
60 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 $A\in\mathbb{Z}^{m\times n}$ ...
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208 views

Faster implementation of “or” constraints in ILP

I have implemented a set of "or" constraints in my ILP using binary decision variables (as in this method). It works fine for smaller problems, but when I try to increase the number of ...
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399 views

Can a generic ILP solver find graph matchings as fast as a specialized algorithm?

Finding a maximum matching, or a maximum-weight matching, is a well-known problem, which has polynomial-time combinatorial algorithms. It can also be formulated as an integer linear program. In ...
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1answer
161 views

Efficient solver for multiway number partitioning

I am interested in the following problem. The input is a set of $n$ integers, and a fixed integer $k$. The required output is a partitioning of the integers into $k$ subsets, such that the smallest ...
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1answer
60 views

Constraints that set values to binary variables depending on other binaries

I am trying to write a mathematical problem that involves some conditions based on binary variables. More specifically, I have a set of three binary variables $d_1$, $d_2$, $d_3$ and depending on ...
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1answer
329 views

Indicator function in math programming

Let $x$ be an integer variable that takes the values $1$, $2$ or $3$. Let $y_1$ and $y_2$ be binary variables. I want to express the two following logical constraints: if $x=2$ then $y_1=1$ if $x=3$ ...
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1answer
59 views

How does the RCPSP's precedence constraint work?

In [1] the authors define the RCPSP (resource-constrained project scheduling problem) as follows: minimize $$ \sum_{t} t x_{n t} $$ subject to $$ \begin{array}{c} \sum_{t} x_{j t}=1, \quad j \in J, \\ ...
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2answers
77 views

IP model for k-rooted spanning forest

I am looking for an IP model for finding a $k$-rooted minimum spanning forest on an undirected graph $G$. Given a set of roots $R$ and a set of nodes $N$ $(R\cap N=\emptyset)$, I would find a forest ...
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1answer
145 views

Generating numbers that should add up to a fixed value while they follow a known distribution

Suppose a perishable item that is associated with a shelf life $m\in \mathcal{M} = \{1,\dots,M\}$. We have a periodic review system with stock level $S$, i.e., based on the inventory level of the item,...
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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 ...
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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 ...
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1answer
107 views

Assignment problem with variable tasks to be done

I'm dealing with a kind of assignment problem, in which I have a set of tasks $t$ to be executed by machines $w$, but these tasks depend on the variatns $v$ of components $m$ being selected, which is ...
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1answer
90 views

How can I linearise this nonlinear proportional relation constraint?

My optimisation problem has a constraint in the form \begin{equation} \begin{array}{*{35}{l}} \text{}\hspace{16.5mm}\text{ C4:} \hspace{2mm}\sum_{u=1}^U d_{u,1}L_{u}:\sum_{u=1}^U d_{u,2}L_{u}:\cdots:\...
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113 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 ...
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185 views

What underlies intlinprog in MATLAB?

When a paper says they used the intlinprog in MATLAB to solve an integer program, what system actually does the solving? I have seen documentation about Gurobi and MATLAB: does Gurobi always provide ...
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2answers
429 views

Modeling a constraint such that a set of binary decision variables do not equate to 1 simultaneously

I would like to seek some advice on modeling the following logical condition: I would like to ensure that a group of binary variables do not equate to 1 simultaneously, i.e., $\omega_{1}=1, \omega_{2}=...
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What's the name of a finite-capacity bin packing problem trying to minimize the weight of the heaviest bin?

I have a fixed number of bins which are themselves weightless. Each bin can hold only a fixed amount of weight. Not all bins have the same capacity. I also have a fixed number of objects each of which ...
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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\...
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1answer
110 views

Constraint programming and scheduling issues

I have a constraint problem that I need to resolve, but I did not how know to model the problem: I have 11 employees, I will name them from $a$ to $k$: $\{a,b,c,d,e,f,g,h,i,j,k\}$. I have a small ...
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1answer
181 views

Formulation of Assignment problem as integer programming

We need to maintain as quickly as possible a complex system. In particular, we need to replace six of its components $\{P_1,\ldots,P_6\}$. We have three 3D printers $\{M_1,M_2,M_3\}$ which we can use ...
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How can I find the optimal assignments for this MILP problem heuristically?

I have an assignment problem as follows $\begin{equation} \begin{array}{*{35}{l}} \underset{d_{u,c}}{\max}\hspace{1mm}\hspace{1mm}\sum_{u=1}^{U}\sum_{c=1}^{C}d_{u,c}\omega_{u,c}\\ \text{}\text{...
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Are there explainability approaches in optimization?

In the machine learning community there is the big topic of explainability, where you want to make the solution of ML models explainable or derive explainable models. This is also interesting for ...
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1k views

What are good reference books for introduction to operations research?

The reference books should cover the wide range of problem-solving techniques and methods.
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Bounding the size of the dual solution

Given an primal optimization with bounded feasible set: $\max \{cx: Ax \leq b\}$. The feasible region of the dual is $D = \{y:y^\top A = c^\top, y \geq 0\}$. If the primal feasbile region is a ...
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1answer
156 views

ILP Constraint to ensure exactly one constraint from a set of constraints is satisfied

Consider several Integer (0/1) ILP variables, i.e., Boolean variables, $x_i$'s. Consider an ILP constraint $x_1 + x_2 + x_3 \geq 1$ and another constraint $x_4 + x_5 + x_6 \geq 1$. I would like to ...
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415 views

How to maximize “contrast” between nodes on a graph?

I have an undirected graph such as the one shown below. I can make up to 3 choices about the color of each node. The edge weights are equal to the difference between the nodes, given by the "...
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1answer
86 views

Minimizing a quadratic binary nonconvex function by CPLEX

I am using CPLEX 12.8 to minimize a quadratic binary nonconvex function, according to quadratic function by CPLEX. In particular, my function is the following: $$ \sum_{i=1}^{m-1} \sum_{f=1}^{F} \sum_{...
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2answers
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How to improve relative mip GAP using CPLEX in a MIP

Supose that I have an integer feasible solution for a MIP and I provide this one for CPLEX. I have tested this situation in a problem and CPLEX have reported the following: ...
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1answer
135 views

Problems finding a feasible solution in a MIP

I am using CPLEX with Julia using the package JuMP to solve a MIP problem. In a small instance, I have tested my problem but, after 10 minutes, nothing happens. I have defined the following parameters ...
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1answer
120 views

Miller-Tucker-Zemlin subtour elimination constraints to obtain a minimum spanning tree

I need Miller-Tucker-Zemlin subtour elimination formulation for symmetric traveling salesman problem (STSP) to use to construct a minimum spanning tree. Ie, I need Miller-Tucker-Zemlin formulation ...
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457 views

How to establish constraint between variables with multiple indexes using CPLEX in Python

I am new in CPLEX and I am using docplex in Python to solve an ILP. I would like to translate the following constraint in docplex: $$\sum_{c}(X_p{_w}_{cj}+X_{p+1}{_{w'}}_{cj+1})\leqslant T_w{_{w'}}_{,...