# Questions tagged [mixed-integer-programming]

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

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### Linearization of two constraints: one with a conditional max and one with a sum with a variable as index

I have these two quite nasty constraints I have tried to linearize. I am trying to dynamically control if you are allowed to plan producing product p. You are allowed to do it if the product arrived (...
107 views

### How to reformulate the BigM constraints into its equivalent Convex-hull formulation?

I am trying to work on a scheduling problem based on its polyhedron reformulations. For that, I would like to reformulate a BigM model into its equivalent C-hull formulation. The transforming map is ...
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### Mixed integer optimization: what if X has to be a rounded multiple of a fixed value? [duplicate]

I am working on a Mosek project. The utility function is to maximize u^{t} x where x is a vector of N There is a tricky constraint: the first 3 items of X must be ...
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### What is the best open-source solver for large-scale MILP optimization?

Currentlly, I am using ortools with SCIP/CBC solvers for a large scale optimization problem, which turned out to be quite slow. The integer part of my problem is due to a piece-wise linear function in ...
46 views

### Cost Function in a Gas Market Model

I am trying to formulate the correct objective function for the following problem: Minimize the total cost in a gas market from the perspective of the consumer. There are storage from which you can ...
1 vote
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### MILP Model for Assigning Unique Number for Each Point

I am trying to solve a problem which is smilar to sudoku. I have a specific device location on map I need to gave them a unique integer numbers. However, as in the sudoku I can assign up to 9 and ...
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1 vote
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### How to avoid division by zero with a binary variable at the denominator in a network assignment problem?

I am currently working with some network assignment problem as a network engineer. My engineering application can be stated as follows: There are $M$ users that needs to be assign to $N$ servers. This ...
491 views

### Can I tell a MILP solver to prefer solutions with fewer fractions?

I have a linear program $$\max c^T x \text{ s.t. } A x\leq b, x\geq 0.$$ I would like a solution that, among all optimal solutions, has the largest number of integer variables. For example, if the ...
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### Limit the time in the root node

I have a hard problem where I fix some variables to solve it. However, the time elapsed in the root node is high. Is there a way to limit this time anf force the Gurobi start the ramification? I have ...
61 views

### Optimal way to formulate a piecewise linear function

I am working on LP problem whose objective function includes a piecewise linear function. I would like to figure out the optimal way to formulate the piecewise linear function in order to minimize the ...
43 views

### Total Unimodularity of constraint matrix

Given a directed graph $G=(V,E)$, I have the following integer program- \begin{align} \max & \sum_{(u,v) \in E} \sum_{s \in S} w_{uv} z_{uv,s} + \sum_{v \in V} \sum_{s \in S} b_{v,s} x_{v,s} \\ \...
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### Enforcing Order in a Linear Programming Question

I have an optimization model to fulfill the water requirements of a city's distribution network. The model includes water sources from rainfall collection, river extraction, reservoir storage, and ...
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1 vote
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### Multi-Commodity Flow with "group edges"

I'm currently working on a special variation of the Multi-Commodity flow problem. My goal is to solve this variation via column generation, because the graph can become very large. Description Given......
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### What are the "cutoff values" in Hybrid Branching?

In "Hybrid Branching" by Achterberg and Berthold, we have this line: the number of subproblems that could be pruned due to branching on this variable, called the cutoff values Can you ...
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1 vote
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### Tightening a specific constraint

I would like to know if there exists any way to reformulate the following constraint in which one can relax the binary variable $z_{j,m}$, and the solution still being an integer for that. The ...
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1 vote
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### How to achieve recording the change of objective or primal gap in solving MILP by SCIP?

I am a solver beginner, would like to ask you how to call the solver to achieve second-level recording, as documented in many articles, as shown below.
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### How to correctly use legacy callbacks in CPLEX with C++ to generate Benders' cuts?

I've been working on implementing Branch-and-Benders Cut using CPLEX version 20.1 with the C++ API, but I'm encountering some difficulties. I've followed the example provided by CPLEX, specifically ...
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1 vote
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### Does getting the second optimal solution in a general MILP require solving a MILP again?

This general question popped into my mind if finding all optimal solutions takes not much more time than finding just one optimal solution in a MILP why not gettting all of them in Gurobi?
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### Constraint formulation in MILP AGV Scheduling (using CPLEX)

I am currently working on a MILP formulation for optimizing AGV/AMR scheduling. The model aims to optimize task scheduling across multiple missions using a limited set of resources. Parameters such as ...
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### How do I convert Integer Programming to Constraint programming?

/********************************************* OPL 12.10.0.0 Model Author: Shahrukh Creation Date: 10-Apr-2024 at 11:55:50 PM The podrace tournament involves n competitors (where n is even) who all ...
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### Defining augmenting path for Traveling salesman problem

I am currently looking into solving TSP using Branch and Bound method. In the book by Wayne and Winston, they have solved each subproblem (i.e. tree nodes) using Hungarian algorithm. Using the lecture ...
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### How can I calculate the UB if I'm using Fischetti's Alternative Cut Generation for Benders'?

Assuming I have a stochastic MIP that I want to minimize using Benders' Decomposition. With the objective of the MP being: $$Minimize \; cx + \frac{1}{\lvert S \rvert} \sum_{s \in S} \theta_s$$ ...
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### Network formulation based side constraints for a workforce scheduling problem

For a workforce scheduling problem, I have formulated the "at most one shift per day" constraint as a network formulation based side constraint (with other constraints like demand coverage, ...
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### Is the linearization with first-order Taylor approximation correct?

I have a QP problem as $\min \hspace{2mm} x^TQx-c^Tx$ here $x$ in binary I want to transform it into a MILP by writing the objective function as $\min \hspace{2mm} z-c^Tx$ and then adding a constraint ...
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### Need Help for building two constraints in MIP model

Really need help for building two constraints with MIP model, I have written the required information and variable information on the diagram below. The way I think of it now is to use two ...
1 vote
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### How to track the first timestep at which a binary variable becomes 1 in an IP? [duplicate]

I have an MIP where I have a binary variable $y_t$ which is set to 1 or 0 and is indexed by time t. It can be set to 1 at multiple timestamps but it is never continuously 1 for more than single ...
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### How to constrain variables to fit a normal distribution

Given a time series $s_t$, I would like to define for each $t$ a perturbation $e_t$ such that the set of all $e_t$ "fits" (within some range) a normal distribution with mean $\mu=0$ and a ...
1 vote
I got a model of MILP but I want to add extra constraints. $$\min\{c^T x + d^T y:Ax+By=a,Cx+Dy \leq b,x,y\geq0,y \in \{0,1\}\}$$ More specificlly, I got extra original constraints with indicator ...
in my model I have the binary variable $f_{ij}$ which pushes a time-dependent $j$ integer variable $D_{ij}$ to zero if $f_{ij}$ takes the value 1 and keeps the integer number if $f_{ij}$ equals 0. Yes,...