# Questions tagged [mixed-integer-programming]

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

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### How to mathematically formulate the optimization problem?

I have a system with $S$ service points. There are also $U$ users in the system. We have $$U>S>G$$ One group can have maximum $M$ service points, but there is no restrictions on the number of ...
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### Clustering problem involving multidepots and customers requiring commodities located exclusively in an specific depot

I'm trying to solve a clustering problem that's similiar to a VRP Pickup and Delivery problem with multiple depots and customers. Each customer demands a commodity that is exclusively found on one ...
232 views

### Impose binary constraint on integer matrix with CVXPY

So I have the following matrix: \begin{equation} P_{i,j}= \begin{bmatrix} x_0 & x_1 & x_2 \\ y_0 & y_1 & y_2 \\ z_0 & z_1 & z_2 \end{bmatrix} \end{equation} where ...
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### Strange Result from GurobiPy

I am trying to understand why GurobiPy gives me a strange result for a simple linear programming model coded as below? Why is the optimality gap is 0%? Please let me know if you spot any errors in the ...
173 views

### Index of element in MILP vector decision variable that equals 1

Consider a decision variable in a MILP constrained: $$\sum_i p_i = 1$$ $$p_i\ \in \{0, 1\}$$ Obviously one element in $p$ is 1 and all others are 0. How can I set a decision variable to the index i of ...
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### Linearize x different of y in ILP

I am surprised I couldn't find an already written answer for my question in the internet. I want to linearize $x$ different of $y$ for two nonegative integer decision variables. I am not considering ...
169 views

### Oscillations with (online) mixed-integer optimization problem

I have the following mixed-integer optimization problem: \begin{aligned} \max_{x,y} \quad & \sum_i x_i - \|wx\|_2 \\ \text{s.t.} \quad & \sum_i x_i \leq A \\ \quad & x \leq x_{\max} y \\ ...
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### Mixed Integer Programming - How to model the dependency of two variables in an objective function

I have two variables $a$ and $b$, in which $a$ is the amount of goods and $b$ is the amount of boxes of the given sizes. So $b$ (box size + number) is dependent on a (goods quantity). If $a$ is ...
94 views

### Mixed Integer Programming - Model Formulation for A Resource Allocation Problem

There are a number of orders, which needs to be shipped. For each order, there may be 1 to 3 route options. The problem here is to find out the best allocation (combination) of orders among these ...
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### 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: ...
162 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 ...
147 views

### Formulating two non-negative variables without binary and/or big-M

There are two non-negative integer variables $q$ and $p$, where only one of them can take a positive value. To impose this relation, I write: \begin{align} q &\leq M(1 - y) \tag1 \\ p &\leq M(...
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### Are there strategies/rules/systematic approaches to derive alternative formulations for an optimization problem?

For a given optimization problem there are often many known math programming formulations. For example for the TSP here is a survey https://link.springer.com/chapter/10.1007/3-540-36626-1_5 of many ...
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### What are useful plots/statistics/metrics when analyzing the solution sensitivity in multi-objective optimization?

Consider an optimization problem with $n>3$ objectives. For handling this there exists often two approaches: a) some weighting of the objectives, b) fix an order of objectives and then optimize ...
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### MIP for resources to task assignments

I have few tasks t1, t2, t3, t4, t5 which need to be assigned to resources r1, r2, the tasks are located across multiple ...
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### k -MST problem based on Miller-Tucker-Zemlin subtour elimination constraints

Where can I find the formulation of the $k$-MST ($k$-minimum spanning tree) problem as (mixed) integer linear program based on Miller-Tucker-Zemlin subtour elimination constraints (MTZ)?
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### Free solver for MINP problems

I have a mixed-integer nonlinear programming (MINP) problem. Is there a free solver for such a problem?
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### Lagrangian Relaxation The Weak Lower Bound

I am applying Lagrangian Relaxation with Subgradient Optimization Method and trying to solve a MIP model. Before testing the large-scale instances, I wanted to see how it performs on small-size ones. ...
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### Working with large models

I am working with a variant of TSP, number of nodes that I need to test are in between 2500 to 3000 nodes, I am using docplex for modelling, I have a 8 gb Ram but it gets filled with only 400 nodes. ...
302 views

### If else condition to MILP

I have following problem: $c_i = 1$ if $X + \sum_j^N G_j = T$ else $c_i = 0$ Also there is another constraint which upper bounds equation $X + \sum_j^N G_j \le T$. $c_i$ is binary $X, T$ are ...
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### 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. ...
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### 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 ...
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### 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 ...