Questions tagged [mixed-integer-programming]

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

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3
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2answers
185 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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4answers
436 views

What expresses the efficiency of an algorithm when solving MILPs

What is the appropriate measure to indicate the efficiency of an algorithm\model that solves a MILP through B&B? Is it the number of nodes examined to reach the optimum? the number of iterations ...
3
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2answers
140 views

Should I use Special Ordered Sets in MIP model

I investigating various aspects of MIP and I can see that my current modelling langaunge support Special Ordered Sets(https://python-mip.readthedocs.io/en/latest/sos.html) but looking at the papers ...
4
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1answer
124 views

Using LR-based method to solve mixed integer programming

When we use Lagrangian relaxation-based methods to solve mixed integer programming, does the convergence of multipliers to the optimum as well as convergence of primal variables to the optimum happen ...
3
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1answer
83 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 ...
2
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1answer
99 views

Defining Solution Space in MILP / LP using If Then Statements

I have the following statements for an MILP: Variables: $c$ (can be $1$ or $0$); $\alpha_j$ (real numbers with $0\le\alpha_j\le1$). I have a linear inequality system for $\alpha_j$: $\sum_jv_j\...
4
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2answers
131 views

Best practices regarding using “aggregated variables” in MIP

I have a variable that can be defined as a sum of other variables. Should I create this variable and add a constraint that they are sum of the other variables or should use the minimum number of ...
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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1answer
324 views

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, ...
3
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2answers
275 views

Calculated CPU time of C++ is different from actual time to solve MILP model via Gurobi

I am solving a MILP model in C++ using Gurobi 7.5.2. More specifically, I am determining the CPU time of my C++ program via the following commands: clock_t start_time; double elapsed_time; start_time=...
2
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2answers
101 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 ...
3
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1answer
237 views

Why does the getTime() function from cplex concert returns wrong value?

I am running a MILP formulation (implemented in C++) with the Cplex Concert Technology 12.10, and I am trying to get the total elapsed time. So till the moment, I have tried three approaches: Be ...
1
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0answers
82 views

Linearization of constraints with square root [closed]

I am trying to solve an optimization programming model involving a non-linear constraint with a square root. It follows (in a simplified form): $X_i\ge\sqrt{A_i/B_i}$ where $X_i,A_i,B_i$ are positive ...
5
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1answer
115 views

How to check whether two formulations are equivalent?

I am given two formulations, that is, two integer programs $ (IP1)\quad \min \{c^tx \mid Ax\geq a, x\in Z^n\} $ and $ (IP2)\quad \min \{d^ty \mid By\geq b, y\in Z^m\} $ and I wish to check whether ...
1
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1answer
631 views

How to describe the traveling salesman problem with an integer programming model?

I'm trying to describe the travelling salesman problem as an integer programming model. I'm interested in the asymmetric version of the problem. The problem can be summarized as finding the optimal ...
5
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2answers
237 views

Shortest path problem with underlying continuous variables

I recently got interested in the following variation of the shortest path problem. I've looked in the literature for days but I couldn't find any paper studying this problem. I'd like to ask if you ...
1
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2answers
123 views

Modeling the product of two variables

Suppose we have two continuous nonnegative variables $X_{1}$ and $X_{2}$ both bounded by the number $M$ from above. I would like to model the following: If $X_{1} > 0$ then $X_{2} = 0$ If $X_{2} &...
1
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0answers
56 views

Modeling the multiplication of two binary decision variables in undirected graph in python

In an undirected graph, I'm trying to model a constraint that forcing the optimizer to set an edge $(u,v)$ between two nodes to only exist (= $1$) if the two nodes have been selected to be $1$. The ...
1
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0answers
21 views

Implement geometric constraint using DOCplex

currently I'm working on a wind farm layout optimization problem. I found an appropriate model in literature (Fischetti et al.) and now I'm trying to reproduce it using the Python API of cplex with ...
3
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1answer
97 views

What are the flow based formulations?

What are the flow-based formulations? For what optimization problems are they applied, and in which form? Which are the specificities of such a formulation? Also, the same question for the time staged ...
1
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1answer
78 views

How to model this chain of logical implication II

I would like to seek some advice on modeling the following (chain of) logical implication: For instance $\omega_{xz}$ might indicate precedence, i.e., $x$, $z$ being the nodes $x$ and $z$, ...
0
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2answers
206 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 ...
1
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1answer
60 views

How to model this chain of logical implication

I would like to seek some advice on modeling the following (chain of) logical implication: For instance $\omega_{xy}$ might indicate precedence, i.e., $x$, $y$ being the nodes $x$ and $y$, ...
6
votes
2answers
383 views

Branch and Price Algorithm

Can branch and price be a good solution approach for a routing problem with min-max objective function? For example, minimizing the max length of any vehicle route in a VRP. In the literature, I haven'...
2
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1answer
160 views

If-Then-Else modeling in MILP using the Big M method

I have trouble finding a solution to the following problem. I have a decision variable $x$. If the value of $x$ is between 0 and a constant $A$, then the binary variable $y_1$ must be equal to 1. If $...
3
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1answer
55 views

Python modeling for ILP Minimum Dominating Set (MDS)

I'm writing a code for solving the MDS problem, the problem is: \begin{align}\min&\quad\sum_{v\in V}y_v\\\text{s.t.}&\quad y_v+\sum_{(u,v)\in E}y_u\ge1\quad\forall v\in V\\&\quad y_v\in\{0,...
22
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2answers
650 views

How much can we expect to increase the speed of mixed integer programming in the next 10 years?

Mixed-integer programming is a super powerful tool for operations researchers to solve many difficult problems. As described by Bixby[1] there has been an overall improvement in the performance of a ...
1
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1answer
75 views

Looking for an example of a heuristic implementation in GAMS

I am new to GAMS and the documentation is not helping me to make fast progress. I am looking for an example implementation similar to Relax and Fix heuristic where in several iterations subsets of ...
7
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1answer
209 views

No, Gurobi, I really do want this variable to be binary

When I mark a variable in a Gurobi MIP model as binary, sometimes Gurobi gives me a solution where that variable has a fractional value other than 0 or 1. How do I constraint a variable to be honest-...
7
votes
1answer
132 views

MILP formulation for minimum set Vertex cover problem

I’m sorry to bother you with this simple question. I would like to model a simple model of the minimum cover vertex set problem. I believe that the original problem is such as $$ \min \quad \sum_{v\in ...
6
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1answer
240 views

Multi-period linear dynamic programming with differing in-period dependencies and changes

I’m not sure if I’m wording this right but in a nutshell, my problem is: I’m modelling potential actions a boat owner can do to their boat. Let’s say he wants to know over the 50 year lifespan of the ...
6
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1answer
234 views

What is the performance improvement when using semi-continuous variables instead of binary + continuous variable pair?

I have a MILP model that solves a master production schedule including capacity decisions. In the model I have a production quantity that should either be 0 or at least the amount that can be produced ...
3
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2answers
105 views

Relaxation and complexity of two formulations

I have two different MILP formulations for the same scheduling problem with the same complexity but with different running times. Why it is recommended to compare the relaxed versions of each ...
3
votes
1answer
189 views

How do I interpret the CPLEX Optimization Studio MIP gap output?

I'm having difficulties understanding my FlowControl output compared to what the Engine Log shows me. My output from the FlowControl into the Scripting Log (yellow marks) is ...
0
votes
1answer
172 views

How to form math model to solve this problem using cplex

Hiring company has the following requirement within a year; week 1 to 5: 20 week 6 to 20: 40 week 21 to 40: 35 week 41 to 60: 55 week 61 to 80:75 week 81 to 100: 60 Training time for an ...
2
votes
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 ...
2
votes
1answer
71 views

MIP for assigning tasks with prerequisite tasks

I have a modified assignment problem for which I'm having difficulty formulating the constraints mathematically. I have a set of workers and a set of tasks which should be completed in the minimum ...
6
votes
1answer
271 views

Can GLPK be used to solve an optimal team selection problem?

My Problem I am quite new to optimisation, so any advice is appreciated. I am currently trying to solve a problem as follows: Given a pool of people, we want to create n teams such to find the optimal ...
4
votes
2answers
120 views

Benders subproblem with product of continuous and discrete variables

I am trying to solve the following problem. The decisions in the problem are $x, y, v, $ and $W$, where $x, y$ are binary and $v, W$ are continuous variables. \begin{equation}\label{eq:3} \begin{...
6
votes
1answer
168 views

Does the weighted sum approach find all pareto-optimal solutions in MILP

I use the weighted sum approach for a multiobjective optimization problem that is formulated as a MILP. This means that the objective function is linear. I read quite often that the weighted sum ...
1
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1answer
76 views

A discrete location problem

I have a discrete optimization problem in which there are some predetermined nodes in the set A. a vehicle must visit them as the traveling salesman problem. there are some other nodes in the set B ...
3
votes
1answer
172 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
76 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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0answers
96 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 ...
1
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2answers
128 views

Formulate a problem as Mixed Linear Programming problem

I need to formulate the following problem as a Mixed Integer Linear Programming problem A farmer needs to establish a 17-year business plan where he will decide when to sell or buy a new truck. The ...
2
votes
2answers
207 views

Does it make sense to use a MILP solver for a scheduling problem but without the obligation to schedule all the tasks?

I know that MILP solvers are bad with scheduling problems. However, if we are allowed to keep unscheduled some tasks (i.e a solution with 0 scheduled tasks is a feasible solution but we add the ...
0
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0answers
36 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 ...
7
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0answers
77 views

Building the Scheurman's Model II constraints for a multi period linear program

Scheurman's paper discusses Model I and model II Formulation to solve harvesting and scheduling problems. It is a specific implementation to solve multi period linear programs. Both models are also ...
2
votes
2answers
332 views

How can I identify the reason that makes a MILP model hard for solvers such as CPLEX?

I'm solving a MILP model whose native lower bound (via linear relaxation) is very poor. We could provide a lower bound by providing a given value (derived based on the problem itself). I know that ...
6
votes
4answers
741 views

Can this be formulated as one inequality

I have two binary variables $x_1$ and $x_2$ and a non-negative continuous variable $y$. In addition, I have the following two parameters $u>q>0$. I would like to formulate the following ...

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