# Questions tagged [mixed-integer-programming]

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

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### What is the definition for an integer-friendly constraint?

I have seen some papers claiming that their proposed model is integer-friendly. I would like to get more information about what type of constraints we can call integer-friendly. Probably, it can be ...
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### 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 ...
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### Ways to improve lower bounds while solving MIPs

What are the ways to improve lower bounds while solving a minimization problem (MILP)?
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### Non-constant expressions cannot be multiplied

I am trying to solve a valued n-queens problem, in which queens in black squares worth double of those in white squares. I solved it in AMPL just fine, but I would like to try that in Python using a ...
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### Open source MILP solver for quick “good enough” solution

I have a problem that I have already posted elsewhere in OR.stack, but the question is focused around a large binary MILP (about 1 million decision variables). Ultimately, I am more time constrained ...
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### Modeling the sum of binary variables

Suppose $x_{1},x_{2}, \cdots ,x_{n}$ are binaries. I would like to model the following: IF $x_{1} + x_{2}+ \cdots +x_{n} \ge 2$ THEN $x_{1} + x_{2} = 2$ IF $x_{1} + x_{2}+ \cdots +x_{n} \ge 3$ ...
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### Using big M values for a constraint

I want to enforce $x_{i,j}=x_{k,j}\implies z_i \neq z_k$ where $k = i-1$ so I used \begin{align}z_k + 1 - (x_{i,j} - x_{k,j})) \leq z_i \leq z_k - 1 - (x_{i,j} - x_{k,j})\quad\text{for each $j$}\end{...
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### Enforcing discontinuous range for a variable

I want to enforce the following ranges for a set of variables, where the ranges for each variable are discontinuous. For example, I have two sets of ranges for each variable, \begin{align}\sf LB_1 &...
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### Does this problem fall into any common problem definition....Knapsack maybe?

I am struggling to find a representative problem formulation for this optimization challenge. I have implemented a MILP in Matlab, but the run time is taking more then a day. My goal is to see if it ...
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### "Out of the box" Mixed Integer Programming Heuristics

I currently have a complex Mixed Integer Program (a sort of vehicle routing problem variant, with multiple vehicle types and without assigned pickup -> delivery routes, among other complications) ...
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### What are the most useful evaluation metrics when comparing the performance of different model formulations

Suppose you are writing a paper about a certain new problem class. You have certain problem instances of different size (real-world as well as random) given. You developed different integer ...
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### Is the iteration-limited Simplex dual solution of a MIP node useful?

Idea Sometimes I encounter problems where Simplex spends many iterations for final convergence to the optimal objective value. Let's suppose, this happens when solving branch and bound-tree nodes as ...
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### How can I represent null or dashes in a cost matrix or incidence matrix in CPLEX?

In the image below, the cost matrix of customers and supplier has several dashes which indicates the impossibility of certain suppliers with certain customers. How can I represent these dashes in ...
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### Do Benders cuts exclude current solutions?

I am wondering if optimality cuts in Benders algorithm exclude the possibility to have the same solutions and as a result, have the same optimality cut? I don't know why it is not possible to have the ...
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### Mocking up conditional statements in LP

I would like to know how if condition statements in linear programming can be reformulated using indicator constraints, and hence solved as a mixed integer linear program. Specifically: 1. Is it ...
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### Where can I find resources to learn mathematical modelling for real life operation research problems like combinatorial optimization?

I find it hard to form math models for real life operations research problems, how can I learn this? Any books, tutorials available?
The following relationship is part of my optimization model ($W_k$ denotes a binary variable) $$W_{1} \le \cdots \le W_{k-1} \le W_{k} \le W_{k+1} \le \cdots \le W_{k^{\max}}$$ My question is, do ...