# Questions tagged [mixed-integer-programming]

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

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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 ...
1 vote
28 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 ...
35 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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62 views

### 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
38 views

### 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 ...
• 900
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 ...
• 8,960
1 vote
79 views

### 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 ...
• 317
1 vote
56 views

### 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?
• 153
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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 ...
• 1
107 views

### 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 ...
24 views

### 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: $$Minimize \; cx + \frac{1}{\lvert S \rvert} \sum_{s \in S} \theta_s$$ ...
• 317
40 views

### 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, ...
• 33
81 views

### 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 ...
• 2,377
104 views

### 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
86 views

### 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 ...
• 33
74 views

### 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
78 views

### How to deal with mixed linear integral optimization problem with extra nonconvex but differentiable constraints?

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 ...
36 views

### Add second "constraint" to model a binary variable

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,...
128 views

### Is there any way to use Lazy constraints in Pyomo?

I would like to know if there is any way to implement a Lazy constraint in the Pyomo package. (As far as I know, the only way to implement such a constraint is by ...
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### Minimize Expenses For Workers

My goal is to minimize the labor expenses. Say we have 3 types of workers: $x_1$ = Permanent Driver, rate = 693.875/day $x_2$ = Reliever Drivers rate = 435/day $x_3$ = Crews rate = 400/day There are 6 ...
1 vote
86 views

### How to model $\max\limits_{x\in X} \min\limits_{y\in Y} \max\limits_{z\in Z} f(z)$ as single MILP

I have the following optimization problem: \begin{align*} \max\limits_{x\in X} &\min\limits_{y\in Y} \max\limits_{z\in Z} & f(z) \\ &\text{such that} & (x, y, z)\in P \end{align*} ...
66 views

### How to deal with performance bottlenecks in Stochastic Vehicle Routing Problem with Benders' decomposition?

I've been working on solving a stochastic vehicle routing problem using Benders' decomposition with CPLEX in C++. Initially, my implementation struggled with larger instances, but I've made ...
• 317
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### Improving the lower bound

Good afternoon. I have a very difficult to a MIP model. I have tried several different strategies to reduce the gap. I am using Gurobi and in this case, I already have an incumbent solution. I've set &...
51 views

### multiple shifts per day constraint for a workforce scheduling problem

This is an extention to the workforce scheduling problem discussed in this thread Workforce scheduling problem. Adding a new post to discuss an important new requirement in this model. So the new ...
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58 views

### How to linearize this L0 norm of a vector?

I have an QP optimization problem. $\bf x$ is the binary optimizaion variable of size $12\times 1$. One of the constraints is non-linear/non-convex. The constraint is L0 constraint. The constraint I ...
• 2,377
1 vote