Questions tagged [integer-programming]

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

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66 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 ...
2
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1answer
47 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 ...
4
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1answer
88 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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1answer
57 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, \\ ...
2
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2answers
68 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 ...
3
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0answers
70 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
140 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,...
3
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1answer
97 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 $z\in \{0,1\}^{d}$,...
3
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0answers
100 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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1answer
103 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 ...
1
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1answer
86 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:\...
2
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1answer
110 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 ...
5
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1answer
167 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 ...
4
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2answers
420 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}=...
7
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4answers
1k views

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 ...
1
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0answers
44 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\...
4
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3answers
148 views

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{...
5
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0answers
80 views

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 ...
5
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1answer
147 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 ...
9
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3answers
364 views

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 ...
2
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1answer
80 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_{...
3
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2answers
69 views

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: ...
3
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1answer
125 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 ...
0
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1answer
99 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 ...
2
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1answer
52 views

Has the concept of TU other application than proving convex hull characterizations?

If a matrix is totally unimodular (TU), then we know that $\text{\{}x| Ax\leq b \text{\}}$ is integral for all integral $b$'s. This is often used for convex hull proofs, but does the concept of TU has ...
3
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2answers
179 views

Modeling in integer programming vs modeling in constraint programming

I have some experience with linear and integer programming modeling (I read Model Building In Mathematical Programming by Williams). Now I am trying to learn how to model with constraint programming. ...
4
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1answer
91 views

Contiguous service area constraint

Background: I have a set of ZIP codes (e.g., all of the state of Wisconsin), and am trying to figure out an optimization-based approach to identify a subset of these ZIP codes for a logistics service ...
9
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2answers
407 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 "...
3
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1answer
78 views

In integer programming what's the difference between using lower upper bound constraints and using a big M constraints?

I've noticed that for integer programming models with binary variables some use upper bound constraints and others use big M constraints in order to have two mutually exclusive choices. I have trouble ...
3
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1answer
52 views

Strict inclusion for facility location formula and aggregate facility location formula

I am trying to prove that $P_{FL} \subset P_{AFL}$ where \begin{align}P_{FL}&=\left\{({\bf x},{\bf y})\,\,\middle\vert\,\,\forall i,j:\sum_{j=1}^nx_{ij}=1,x_{ij}\le y_j,0\le x_{ij},y_j\le1\right\}\...
3
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0answers
61 views

Optimal Seat Allocation Problem

I have to do an operations research assignment based on optimal seat allocation. The problem goes something like this. There are 5 rooms in an office each with a separate seating capacity. We now have ...
2
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2answers
98 views

Constraint to handle the machine-configuration's change between initial position and its first occurrence in the process

I am working with a kind of a reconfigurable process planning, meaning that the same machine can have different configurations and perform multiple operations. Each machine has an initial ...
1
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1answer
51 views

Priority Constraint

Suppose I have the following set of binary variables: $X_i$: $I$ ranges from {1,..,4} Highest priority among the three variables $X$ , $Y$ and $Z$ $Y_j$: $J$ ranges from {1,..,3} $Z_k$: $K$ ranges ...
8
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4answers
310 views

Understanding integer programming solvers

I would like to verify if I understand the nature or workings of integer programming solvers. My understanding is that for integer programming problems like the knapsack problem or the traveling ...
3
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2answers
158 views

How to get an extreme ray of an LP from Gurobi

I am working on a problem of form \begin{equation} \begin{array}{l @{\quad} l} \mathrm{max}_{x, u} & p^{\top} u \\ \text{st.} & A u + a x \leq 0 \\ & x \in \{0, 1\...
0
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2answers
204 views

How can I formulate this specific if-then constraint?

IF $\sum\limits_d X_{i,d}\ge6$ THEN $Y_i = 1$ (strictly) AND IF $\sum\limits_d X_{i,d}<6$ THEN $Y_i = 0$ (strictly) $X$ and $Y$ are binary variables. What I'm actually trying to do is to charge the ...
4
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2answers
208 views

How to solve this convex problem heuristically?

I have the following problem $$\max_{X_{i,j},i\in N_{U},j\in N_{B}}\sum_{i=1}^{N_U}\sum_{j=1}^{N_B}R_{i,j}X_{i,j}$$ $$\text{subject to}$$ $$a_{\min}\le\sum_{j=1}^{N_B}X_{i,j}\le a_{\max}, \forall i$$ $...
3
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0answers
50 views

Flexible Job Shop with Preemption

I'm trying to solve a flexible job shop problem variant that has precedence constraints on jobs along with a few other issues. We have a MIP formulation and also a simulated annealing algorithm to ...
1
vote
1answer
106 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 ...
2
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1answer
311 views

Divisibility constraint in Integer programming

I have a simple question regarding the divisibility in integer programming suppose the objective function is $\text{max}\quad x_1 + x_2$ where the constraint is that the sum of $x_1$ and $x_2$ are ...
2
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1answer
78 views

confusing results of two models with different complexity

i have two models that address the same problem. the first one is : the second one is: for different instances for the same size (n=30) i found the following results ( the first column on the left ...
5
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1answer
215 views

How to determine the size of a model?

I want to know about the number of variables and constraints of this formulation (exp: $o(n)$ variables and constraints or $o(n^2)$ ....). Is the number of variables $\mathcal O(n^3)$ because we have ...
6
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1answer
137 views

Literature on “simcity-like” problems

As it will become apparent, my field is not operation-research and so this question will sound very naive. I am sorry for that. I have a set of "buildings" that I want to place on a small 2d ...
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1answer
105 views

Any references to the ROADEF 2020 Challenge?

The problem description of the challenge is given here. Does anyone has some references to similar problem. I would like to participate but I don't know where to start.
2
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0answers
71 views

Indicator function for integer variable with inequality constraint

I have $n$ integer variables $\vec{x}$ with the following integer programming problem. $$ COST = \sum^{n-1}_{i = 0} a_i x_i + \sum^{n-1}_{j=0} b_j I(x_j > 0) $$ Here, $a_i, b_j \in \mathbb{R}_+$ ...
2
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1answer
39 views

Finding bounds on a data sensitivity scenario ILP problem

This is a follow up to a problem I posted here: Modelling a data-sensitivity scenario as an ILP problem As a recap, I was interested in finding the minimum number of cells that need to be suppressed ...
3
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1answer
166 views

Issue in solving a large scale MIQP problem

I am solving a large scale MIQP optimisation problem at each step of a model predictive control problem. The problem description is as below. \begin{align} \min_{u} \quad (x_{k}&-x_\text{ref})^{T}...
2
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0answers
95 views

Condition for an integer program and its linear relaxation to have the same value

Let $A$ be a $(0,1)$-matrix where no row or column is a zero vector, and consider the following optimization programs \begin{align}(1):\min&\quad y\cdot1\\\text{s.t.}&\quad yA\ge w\\&\quad ...
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0answers
34 views

Interger programming using gray encoding

Could anyone suggest me a tool or library which takes an integer programming problem written in DOCPLEX or CVXPY as input and outputs the equivalent problem using Gray binary encoding? I am happy to ...
3
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2answers
673 views

What are the solvers that give a feasible solution within a given time?

What are the solvers that take the maximal computation time as a parameter and gives the best found feasible solution within this time.