# Questions tagged [integer-programming]

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

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### Are there any good models for min-max vehicle routing problem?

I am trying to model a min-max VRP problem with multiple delivery vehicles and I have come up with a model using branch and cut but I do not think it is strong enough as it takes lot of time to ...
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### Linear functions in Lenstra's algorithm

I had asked this question at MathOverflow and was pointed here. I'm working on implementing Lenstra's algorithm. At the bottom of p.5 (at "construct $n+1$ linear functions"), he says to ...
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### Min-cost flow with per-edge flow conservation

I am trying to solve a linear program that is identical to a min-cost flow problem, except for a difference in the flow-conservation constraint. Instead of the summed outgoing flow equaling the summed ...
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### 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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### Linear objective function with non-linear constraints

I would like to choose a set of $\beta_j$s that maximizes a simple linear objective function of the type $$\underset{\beta_j}{\operatorname{max}}\sum_{j=1}^{J}X_j\beta_j \\$$ subject to the ...
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### Gurobi stuck at start - 16GB RAM exhausted

I'm solving a large instance using Gurobi 7.0, 64-bit version on machine with 16GB of RAM. Gurobi is stuck at the start and all the memory is exhausted. Here is the output: ...
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### How is Big M calculated?

Because of excessive pollution on the Momiss River, the state of Momiss is going to build pollution control stations. Three sites (1, 2, and 3) are under consideration. Momiss is interested in ...
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### How to choose between high number of binary variables or fewer number of integer (not only 0 and 1) variables in a IP formulation?

When I have to write the formulation of an IP, I usually have the choice between writing $i\times j$ binary variables with two indices such as $x_{i,j}$ or, writing $j$ integer variables $x_i$. Is ...
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### Integrality gap in bilevel binary linear programming problem

I have a bilevel max-min optimization problem over binary variables, with constraints expressed using linear inequalities. The inner (minimization) problem is  \begin{alignat}2 \min\limits_x&\...
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### Automatic detection of SOS variables and constraints

We've been working on a new feature for Octeract Engine, namely to automatically extract SOS structure from a model and then exploit it. While the literature is quite rich on what to do with SOS once ...
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### Linear constraint formulation (OR-statement)

I have the decision variable $X_{iz}$ And I have two parameters $T_i\in\{0,1\}$ and $IT_z\in\{0,1,2\}$. I can only assign $i$ to $z$ if the following holds: for $T_i=0$, $IT_z$ needs to be $0$ or $2$...
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### 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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### Integer programming example clarification

There is an explanation in my book for an integer programming example, which goes like this: A company is considering manufacturing three types of autos: compact, midsize, and large. The resources ...
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### MIP: If integer variable $>0$ it should be equal to other integer variables $>0$

I have an MIP problem where $n$ different types of cars are delivering packages. Sometimes multiple types of cars are required to go to a single location. For example if car $1$ makes two deliveries ...
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### A heuristic approach to solve a MILP problem?

I have the following optimization problem which is a MILP. I can solve it with a MILP solver. This one I posted here Is there a heuristic approach to the MILP problem? Since I have an additional but ...