# Questions tagged [modeling]

For questions related to the process of converting a real-world problem into a mathematical model. Can include questions related to linearization, logical constraints, tightness of formulations, and so on.

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### How to determine the correct level of detail when modelling?

"All models are wrong, but some are useful" - George E. P. Box I usually work on what one could call operational problems. There I usually do not have too much trouble figuring out the level of ...
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### Modeling floor function exactly

Suppose we want to enforce a constraint $$y=\lfloor{x}\rfloor$$ where $x$ is some continuous variable. One option is to use $$x-1\leq{y}\leq{x},\quad y\in\mathbb{Z},$$ which fails on the edge case ...
364 views

### The rationale to improve MTZ?

Currently I need to solve a quite specific problem involving symmetric TSP as a sub-problem (i.e., a Hamiltonian cycle is a necessary condition for optimizing some problem-specific variables that ...
1k views

### Examples of machine learning applied to operations research?

Can someone give me a few examples, if they exist, of problems in operations research that could be solved using machine learning. I am aware that machine learning examples are data-driven and do not ...
432 views

### How to determine if a given problem seems to be a good fit to be solved using combinatorial Benders decomposition

Combinatorial Benders decomposition is a mathematical programming technique consisting into dividing a problem into a master problem and a sub problem. The master problem is solved to optimality (or ...
3k views

### Combinatorial problem in my daughter’s class

In Denmark, a rather substantial amount of work and effort has gone into reducing bullying in the Danish public schools. Many initiatives, which purposes are to strengthen the unity and solidarity in ...
2k views

### Validation and verification of mathematical models

Within the subject of simulation I have found some literature on validation and verification (e.g. Sargent's paper). My question is, what techniques do you use to validate and verify your mathematical ...
318 views

### Tightness of an LP relaxation without using objective function

How can we measure the tightness of a linear programming relaxation for a mixed integer linear program without using the objective value? I would like to get a measure in terms of the feasible set and ...
600 views

### How can I best handle symmetries in my MIP?

When dealing with mixed-integer-programs with many symmetric solutions it can take very long until the branch-and-bound-tree search is finished because symmetric optimal solutions cannot be pruned. ...
272 views

### Mathematically creating the 'perfect' permutation for reservations in a hostel

I am working at a hostel which uses a reservation system for each room and the beds in the room (e.g. $14$ beds in one room, bed numbers $1-14$.) When we get bookings for multiple people, we assign ...
2k views

### Infeasibility in mathematical optimization models

Sometimes, when solving mathematical optimization models (especially MIPs), they may be infeasible. Is there any comprehensive method to deal with the infeasibility conditions? (especially in complex ...
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### Presolve is cutting down a lot of binary variables. Should I rethink my formulation?

I built my model on Python and am passing it to Gurobi to solve the problem. The presolve phase of Gurobi cuts down ~80% of the integer/binary variables and I am wondering if I should rethink my ...
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### As an Operations Research professional, how is your time divided when working on an optimization project?

When working on an optimization project, what is the typical time division (in percentage) between the various tasks that you have to work on: Problem understanding/definition (figuring out what is ...
896 views

### Best model for precedence constraints within scheduling problem

Suppose I'm modeling a problem where I want to compute the start time bucket for some jobs. All time buckets have equal duration. There are some additional constraints involved but I also have to ...
974 views

### Bin Packing with Relational Penalization

There are $N$ bins with equal capacity $C$. In addition, there are $N$ objects $x_1, x_2, \dots, x_N$ that need to be accommodated using the least amount of bins. Each object $x_i$ has a volume ...
2k views

### Is there a SQL/English like language that lets you define formulations given some data?

It would be very useful for beginning and non technical users to be able to define models in a way that was natural for them. Further this could perhaps assist generating some kind of generic ...
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### How to formulate: each pair of elements in $A$ has one common unit in $B$

We have two sets, $A$ and $B$. Some elements of $A$ must be connected to some elements of $B$, but no element of a given set is connected to another element of the same set. (Think of a bipartite ...
240 views

### Connectivity of two nodes in an arbitrary undirected graph

Is there an efficient way to model the connectivity of two nodes in an arbitrary undirected graph? I would like to have a binary variable representing this connectivity: 1 if there exists a path ...
841 views

### 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 ...
730 views

### On using correct notation in research papers

This question is about using correct notation while writing a research paper. Say I have a directed graph $G$, partitioned into $T$ layers. Denote the set of nodes in layer $t$ by $V_t$. Suppose, I ...
372 views

### Column generation for TSP

For teaching purposes, I would like to solve the Travelling Salesman Problem with a column generation approach. In the academic literature, an approach is proposed (for example here), where columns ...
204 views

### Symmetry-breaking ILP constraints for square binary matrix

Setup I have a binary $N \times N$ matrix. The objective is to minimize the number of ones in the matrix, subject to various constraints. This leads to symmetries by rotating 90 degrees and/or ...
889 views

### Where can I find documentation on good practices for efficient formulations of a problem?

I am sort of new to mathematical optimization and have to build some fairly complicated models for my thesis. I was wondering where I could find literature to help me develop more efficient versions ...
817 views

### What is a “hard problem” in the context of Mixed-integer programming?

As a practical (real-world problems) point of view, it's important we could solve optimization problems as quickly as possible (for instance, to release a daily schedule). Maybe a problem with many ...
506 views

### Expressing a chain of boolean ORs using ILP

How to express a chain of OR operations in an ILP in which each expression is a less than or equal constraint and the left hand side variable in all inequalities is always the same? All the variables ...
342 views

### How to handle real-world (soft) constraints in an optimization problem?

Cross-posted at Stats.SE (aka Cross Validated) I am working on a problem which involves optimizing for minimum power consumption in a large compressor network interconnected through pipelines (think ...
1k views

### Number of aircraft to operate in an airline company

Suppose that an airline company has X planes, in general the companies keep a number of these aircraft (say 80% of X) in reserve in case of hazards. My question is how the airline companies compute ...
386 views

### Debugging cplex model

I implemented a cplex model and I am convinced that the model should allow a better solution on a specific instance. However, when I impose the variable values of the solution onto the model, it ...
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### Expressing an implication as ILP where each implication term comprises a chain of boolean ORs

Consider an implication of the form $A \implies B$ where both $A, B$ comprises a chain of Boolean OR variables. For example, $(a_1 \lor a_2 \lor a_3) \implies (b_1 \lor b_2 \lor b_3)$. How can this ...
348 views

### Is there a way to proportionalize fixed costs in a MILP?

So assume we have a MILP (e.g. inventory or capacity planning) and the objective is to minimize total costs (inventory costs, set-up costs, backorder costs, production costs etc.). The production of a ...
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### How do we call this problem in literature and how to model it?

I have a set of sources and a set of sinks. Each source $s$ can produce a set of different products $P_s$. The transportation inter-sources and inter-sinks are allowed. The sources do not ...
116 views

### Pricing of blends/mixtures across multiple timesteps

I have a simple blending problem, where each final product is a blend or mixture of several raw materials, and want to calculate the price per unit of weight for each of the products. So for a given ...
617 views

### Comparison of Algebraic modelling languages and general programming languages

Some optimization software/frameworks (commercial or open-source) such as AMPL, GAMS, Cplex, ... have a specific Algebraic modelling language. Some of them have another type of programming that uses ...
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### Expressing a chain of boolean ORs using ILP involving different variables

How can I express a chain of OR operations in an ILP, given that each operand is an inequality between two binary variables? I have asked a similar question here: Chain of Boolean ORs. In that ...
194 views

### partitioning hub assignment models

When solving large-scale hub assignment models (1000+ candidate hubs and 1000+ demand nodes), it is possible that parts of a cost matrix are not connected to one another. A typical workflow would be: ...
311 views

### Minimize number of pieces required to cover distances, with overlap

The specific optimization problem I'm trying to solve is this: Find the minimum integer number of $2$m pieces required to cover $2$ or $4$ distances of length $D$ given that adjacent pieces must have ...
323 views

### Decoding a Deep Neural Network as an Analytical Expression for Optimization Purpose

This post is not really about a specific question but rather a topic I am curious about to know more. We know that when it comes to integrate machine/statistical learning with optimization for the ...
229 views

### How to fit a Beta distribution to three estimates from an “expert”?

I'm modeling a process time, $X$, for a simulation study and have an "expert" estimate of the minimum, $\hat a$, the most likely (mode), $\hat m$, and the maximum, $\hat b$. I'd prefer to avoid the ...
271 views

### Modeling an assignment/scheduling problem to minimize total wait

I have an assignment type problem in which there is a set of students, $S$, and a set of training classes $T$. Each training has a fixed start and end day and can accommodate at most 1 student. In ...
501 views

### Dealing with non-overlapping constraints

Let us consider the following problem: Let $T$ be a set of tasks. Each task $t \in T$ has a duration $d_t$ and a target start time $s_t$. No two tasks can be executed in parallel. The objective is to ...
418 views

### How to linearize a constraint with a maximum of binary variables times some coefficient in the right-hand-side

I have the following constraint that I'd like to linearize: $P$ is a given set $b_p \in \{0,1\} , \forall p \in P$ a binary variable associated with each element of $P$ $c_p \in \mathbb{R}^+$, a ...
704 views

### Running a linear programming model to maximize binned predictions

I have a dataframe like: ...
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### I've formulated my optimization model; now what?

I've formulated my linear/nonlinear/integer/mixed-integer optimization problem in algebraic form (possibly with the help of the folks on this site). Now what? How do I solve it?
You have a ticket allowing you to visit up to $n$ of $M$ carnival booths offering games of chance. At each booth you have probability $p_{i}$ of winning a reward with average value $r_{i}$. Each booth ...