Questions tagged [integer-programming]

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

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4
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2answers
258 views

Combinatorial Optimization using AMPL

I want to solve the following integer programming problem using AMPL. The problem is the following (It was already asked on mathstackexchange.com, but I need to know how to solve it using AMPL): Let $...
2
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0answers
143 views

Operation hours optimization for circular schedule

Here is my problem. A store has X = 15 electical devices with the ability to work non-stop, fully charged, up to 8 hours. Their battery charge lasts 2 hours and the operating hours of the store differ ...
5
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3answers
181 views

How can I optimize this integer programming constraint problem without running out of memory?

I am trying to run this constraint problem but the memory runs out, $$S_{i}$$ are 1975 Students that need to be assigned to one of 188 teacher assistants classes, each teacher assistant has to chose a ...
4
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1answer
240 views

Indicator function in math programming

I have the following doubt: Being $x$ an integer variable that takes the values $1$, $2$ or $3$. Being $y_1$ a binary variable. Being $y_2$ a binary variable. I want to express the two following ...
3
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1answer
142 views

Formulation of assignment problem as Linear Optimization

In the anti-tumor treatment with radiotherapy, it is possible to irradiate the tumoral mass from different positions with different intensities. For each of these possibilities, however, one has to ...
5
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1answer
75 views

Formulation of Machine allocation as an optimization problem

A manufacturing company from Eastern Finland will have to decide on the machines to use in order to produce 2800 units of a given item. Each machine has some given characteristics: a setup cost if it ...
12
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3answers
1k views

Allocating credit card points

I’m interested in the idea behind this in general, so I thought this would be the best place to post, though I have a practical and semi-urgent need of allocating the points on my credit card towards ...
4
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1answer
827 views

What are good reference books for introduction to operations research?

The reference books should cover the wide range of problem-solving techniques and methods.
3
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2answers
386 views

How to add logical OR constraint in OR-Tools?

Let's say nurses normally do 1 shift for d in all_days: for s in all_shifts: model.Add(sum(shifts[(n, d, s)] for n in all_nurses) == 1) But I want to ...
2
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4answers
78 views

Linearity of an optimization problem which comprises the product of variables with constant values from a non-linear function

In a mathematical integer optimization problem, if the objective function is represented as $\sum x_k \cdot M_k$, where $M_k$ is a non-linear function whose value is known and just plugged in to the ...
3
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1answer
94 views

What kind of scheduling problem is this?

I am trying to develop algorithm to solve following basic version of the problem. 1) I would like to know what this problem is called in literature so that I can look it up 2) What are efficient ...
7
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1answer
1k views

How to use the least number of colours to colour different routes of a bus route such that no two intersecting routes will have the same colour

I would like to know of a method in which if provided say 10 routes with details regarding which route intersects with which another route, we can use the least number of colours to colour the routes, ...
8
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3answers
1k views

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 ...
5
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1answer
96 views

Linearize a product of an integer variable (not just binary) and a continuous variable?

I have a constraint in my formulation that contains multiplication of an integer variable $y$ and a continuous variable $x$, which is $xy=q$ where $y$ is the number of units in which $q$ gets equally ...
4
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1answer
236 views

How to set these constraints in a linear optimization problem (PuLP)?

I am trying to implement an employee (nurse) scheduling problem and seek some advice on how to implement a specific constraint.The problem is as follows: There is a set of employees and days (both ...
2
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1answer
72 views

Inequality Constraint Linearization of a product of an integer and a binary variable

I have thought I had found the answer here: How to linearize the multiplication of an integer and a binary integer variable? But the answers to that questions didn't help me find a solution for my ...
5
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3answers
80 views

Requiring exactly $n_j$ slots for job $j$ (if scheduled)

Let $x_{j}(t)=1$ iff job $j$ is scheduled at time $t$. I want to say that if the job is scheduled at all, then it is scheduled at $n_j$ slots. I wrote this as: $$x_{j}(t)\sum_{s=1}^{T}x_{j}(s)=n_jx_{...
3
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1answer
161 views

Problem solving a linear program using Excel

The exercise is as follows: ACI has decided to put an order for golf shoes twice every year and expects to receive one shipment of $960$ pallets of shoes by the beginning of January and another ...
3
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2answers
272 views

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 ...
16
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7answers
2k views

Does there exist an aggregation of videos on optimization?

Is there a website or otherwise maintained list of talks regarding mathematical optimization? This would be a big help for the community it seems. I'm most interested in those relating to integer ...
4
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1answer
64 views

Inputting logical constraints into a binary programming model in Gurobi

I am very new to Gurobi and OR in general (I'm in my first class for it now), so apologies if this is a very obvious answer. For a project in that class, I am generating a flight schedule for a ...
4
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1answer
54 views

Find the number of idle intervals with weights

We have one job $i$ and one machine. Let $\mathbf{x}_i=[x_{i,1},x_{i,2},\ldots,x_{i,T}]$ be a binary vector where $x_{i,t}=1\iff$ job $i$ is scheduled at time $t$. Let $u$ be a positive number. I ...
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0answers
77 views

How to communicate number of integer combinations to a user

I'm working on a nifty little feature for our next release, i.e., to print the number of possible integer combinations left during branch and bound. This is really handy for the user because they ...
7
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1answer
768 views

How to find the idle intervals in integer programming?

I have a scheduling problem with one machine and one job. I defined a binary variable $z_t$ that is 1 iff the job is scheduled at time $t$ (the job can be served in multiple times that are not ...
10
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2answers
162 views

Use integer/quadratic programming to maximize consecutive zeros in a binary array

A binary array $t = [t_1, t_2, t_3, t_4, t_5]$ with each element a binary integer variable taking values 0 or 1. You can think this vector as slots with 1 representing the slot being taken and 0 ...
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2answers
1k views

How to formulate problems in the language of mathematical programming?

The question says it all. I am having difficulties formulating general problems (meaning no numbers just variables). When I read the solution, I understand but I can't figure how to formulate myself ...
4
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0answers
63 views

Continue on “Is there a known MILP to schedule routes after routes are made”

I have made some progress on my previous question (Is there a known MILP to schedule routes after routes are made). I have derived the sets of the problem, which are: 1) Itineraries of vehicle: $i \in ...
6
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1answer
122 views

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: ...
7
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1answer
171 views

Custom Nurse Rostering Problem

I've asked this question also on Math Stack Exchange. It's a custom nurse rostering problem: $N$ is a set of nurses; $S$ is the set of shift-type (morning, afternoon, night, rest) $n_\mathrm{...
5
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2answers
719 views

Integer programming problem

I have the following exercise: Stockco is considering four investments. Investment 1 will yield a net present value (NPV) of \$16,000; investment 2, an NPV of \$22,000; investment 3, an NPV of \$12,...
5
votes
1answer
285 views

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 ...
9
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1answer
108 views

Equivalence of formulations

I have a simple model such as: \begin{align}\max&\quad z=X_1+X_2+X_3+X_4\\\text{s.t.}&\quad y_1+y_2+y_3+y_4=2\\&\quad X_1 \leq y_1\\&\quad X_2 \leq y_1+y_2\\&\quad X_3 \leq y_2+...
13
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2answers
809 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 ...
12
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0answers
156 views

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&\...
8
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0answers
85 views

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 ...
23
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3answers
3k views

What is the difference between integer programming and constraint programming?

At first glance both approaches appear to be very similar. What are the major differences between integer programming and constraint programming?
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5answers
3k views

When to use indicator constraints versus big-M approaches in solving (mixed-)integer programs

Various optimization modeling languages and solvers allow for both indicator constraints (see for example here, here and here) and traditional binary variable and big-M approaches can be used to model ...
7
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2answers
144 views

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 $2$...
9
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0answers
70 views

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 ...
5
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1answer
185 views

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 ...
10
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1answer
99 views

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 ...
6
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2answers
172 views

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 ...
10
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1answer
406 views

TSP problem: traveller does not visit all nodes - Google OR-tools

Context: I am dealing with a kind of scheduling problem, in which I have a set of tasks and machines. All tasks must be assigned to machines (not necessary all of them). In addition to that, I must ...
8
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1answer
268 views

When using docplex.cp is it possible to get all feasible solutions?

I would like to solve an ILP and get all feasible solutions (even the worst one). How could I do that using docplex.cp? I've seen a similar question in: Using CPLEX "solution pool" to count ...
8
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1answer
501 views

How to linearize the multiplication of an integer and a binary integer variable?

I have the following constraints \begin{align}\sum_{i=1}^{N}{x_it_i}&= M\\\sum_{i=1}^{N}{t_i}&\le S\end{align} where $x_i\ge 0$ is an integer variable, $t_i\in\{0,1\}$ is a binary variable ...
15
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2answers
543 views

Divisibility constraints in integer programming

In the study of a certain pure mathematical problem (related to infinite-dimensional Lie algebras) I found myself in a situation where it would be very desirable to be able to solve an integer ...
12
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2answers
314 views

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 ...
15
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1answer
505 views

Duality in mixed integer linear programs

I know that the standard duality theory for the linear programming problem does not hold for mixed integer linear programming problems. I was wondering why an integer program does not have a dual ...
18
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1answer
240 views

Combinatorial Optimization: Metaheuristics, CP, IP — “versus” or “and”?

"Recently" someone asked on Twitter whether "people still use genetic algorithms for integer programs". The "majority answer", i.e., 1 out of 1, was: "Yes" . So, my follow-up question is: With all ...
11
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2answers
720 views

In an integer program, how can I “activate” a constraint only if a decision variable has a certain value?

Suppose we have the constraint $$a_1x_1 + \cdots + a_nx_n \gtreqless b,$$ where $a_i$ and $b$ are constants and $x_i$ are decision variables. Suppose also that we want the constraint to hold if $y=1$ (...