Questions tagged [mixed-integer-programming]

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

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8
votes
1answer
306 views

How to resolve this issue in multi-objective optimization?

I have the following multiobjective optimization problem. The objectives are non-conflicting. The Optimization Problem: $$\underset{\large{a^{(l)}_{c,u},f^{(l)}_{c,u},z_{l,t},l\in\mathcal{L}}}{\max}\...
12
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2answers
172 views

Linearisation techniques for MINLPs

I am wondering what kinds of linearisations people do for MINLPs outside my field of expertise. I work in global optimisation, so by "linearisation" we would typically mean one of the following: ...
7
votes
2answers
639 views

Finding an Objective Function for Assigning Employees to Sequence Dates

I am using a mixed-integer-program to schedule employees to projects. These projects can have a time window to get completed from a few weeks to a few months. At the moment I am working in a ...
9
votes
1answer
104 views

How to model 24 hours demand into a daily shift schedule?

I am working on a weekly staff scheduling optimization problem with 24/7 demand. The binary decision variable is: $X_{\text{staff},\,\text{day},\,\text{shift}}$ whether to assign the staff $s$ to day ...
12
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3answers
809 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 ...
15
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3answers
1k views

How does the search space affect the speed of an ILP solver?

Let us suppose we have an optimization problem which we have modeled as an ILP. Suppose we solve this problem using some set of constraints which restricts the search space. Let us suppose we model ...
11
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2answers
191 views

Is deciding the presence of mixed-integer points in the relative interior of a polyhedron in NP?

Given $P = \{x\in\mathbb R^n: Ax \leq b\}$, I want to decide if $(\mathbb Z^\ell \times \mathbb R^{n-\ell}) \cap \operatorname{relint}(P)$ is non-empty. Is this problem in NP? One idea is to check ...
9
votes
1answer
192 views

Profit Maximization vs Cost Minimization for Employee Scheduling

I wanted to write two objective functions for an employee scheduling problem (MIP) until it occurred to me, that one objective function may be redundant. Is there a difference between the cost ...
6
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5answers
2k views

Algorithms vs LP or MIP

Is there a way of writing an algorithm with if-, while-statements to find an optimal solution without using linear-programming (LP)/MIP? If so, what would the benefits be against the LP/MIP? Is it ...
14
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2answers
974 views

State-of-the-art algorithms for solving linear programs

Průša and Werner (2019) show that the general linear programming problem reduces in nearly linear time to the LP relaxations of many classical NP-hard problems (assuming sparse encoding of instances)....
16
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4answers
895 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 ...
10
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5answers
498 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 ...
21
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5answers
317 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 ...
5
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2answers
281 views

Bridge the gap between theory and practice in Integer Programming

I've finished Wolsey's book on Integer programming. It's a theoretic book. I aim to learn how the ideas presented in the book can be applied to solve real-world non-academic problems. I am looking ...
25
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6answers
2k views

How to compare two different formulations of a problem?

I somewhat know how to compare two MILP formulations of a problem that both use the same set of decision variables (as in the classical MTZ vs DFJ formulations of the TSP). I was wondering how two ...
12
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2answers
347 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 ...
13
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4answers
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 ...
12
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1answer
347 views

Mixed-Integer Linear Programming (Capacity Planning)

I'm currently developing a small capacity planning problem and right now I am struggling with the "activation" of a subset. Needless to say I am not an expert in this kind of things. I have a set of $...
9
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1answer
162 views

Static stochastic knapsack problem: unbounded version

In the static stochastic knapsack problem (SSKP) the weights $w_i$ of the items are distributed according to a probability distribution. Each item $i \in I$ can be selected at most once. So, ...
26
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1answer
362 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 ...
13
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7answers
868 views

What are the examples (applications) of the MIPs in which the objective function has nonzero coefficients for only continuous variables?

I'm specifically looking for real applications of the following form of MIP: $$\max\,Cx$$ subject to: \begin{align}Ax +By &= D\\Ax &= E\\By &= F\\ x &\ge 0\\ y &\in \mathbb{...
11
votes
2answers
212 views

Assignment Problem with Decreasing Costs

Problem: I have $i$ jobs that I can assign to $j$ workers. Each job has a cost. Each worker can perform up to an arbitrary max number of jobs. However, there is a cost efficiency for each job that is ...
6
votes
1answer
2k views

How to linearize min function as a constraint?

I'm trying to solve an optimization problem including following constraint, and I need to linearize it in a maximization nonlinear programming model. Please help me to reformulate it with mixed ...
10
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2answers
155 views

Are there examples of spatially explicit MIP problems?

Disclosure: I am an MSc student in economics, but not an expert by any means in OR. I am trying to model a spatial MIP problem of an invasive species similar to this academic paper, however, my ...
14
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1answer
575 views

Estimation of the size of Branch-and-Bound trees using ML

A short background: A paper [1] published in 2006 intends to show that the time needed to solve mixed-integer programming problems by branch and bound can be roughly predicted early in the solution ...
12
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1answer
537 views

Complexity of verifying optimality in (mixed) integer programming

I looked around for a while, but I couldn't find a precise answer to the following question. If I'm given a candidate solution for a (mixed) integer (convex) program, what's the complexity of ...
17
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1answer
752 views

Warm start CPLEX using google or-tools

I have been trying to use the SetHint python API in google or-tools to warm start MIPs and solve it using CPLEX. It looks like my hints are accepted by the SetHint function but I am not sure whether ...
13
votes
2answers
318 views

Querying attributes of LP relaxation at MIP-optimality in Gurobi

Is there a way to configure Gurobi to allow the LP relaxation associated with the optimal solution leaf of a MIP branch-and-bound tree to be queried for shadow prices & other general LP properties-...
9
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1answer
149 views

Structural Optimization

Currently, I am working on a problem in which I need to use MILP to model equilibrium equations in a lightweight structure. Although this is an application based question, I wondered if there is a ...
8
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1answer
656 views

How to set a maximum time to improve a solution with Pyomo and CBC

In relation to using CBC via Pyomo: I was wondering if anyone knew if it was possible to set a maximum allowed time to improve a solution, rather than a maximum total time. This way, if you've been ...
11
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4answers
414 views

Good resources for solving techniques (Metaheuristics, MILP, CP etc)

I want some resources (tutorials, online courses, lecture notes, articles, books, etc.) to learn the different techniques to solve OR problems (metaheuristics, CP, MILP, etc). It would be better if ...
2
votes
1answer
115 views

Reduction of Unnecessary Parameters and Variables in an MIP

Let's observe an example constraint: $\sum \limits^E_{e\ \in \ A_a \ \cap \ B_b \ \cap \ C_c} x_{e,a,b,c} \geq n_{a,b,c} \; \; \; \forall a \in A,b \in B,c \in C$ with $e \in E$ an element and $A_a$ ...
5
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2answers
234 views

Formulation of a constraint in a MIP for an element in different Sets

I have an element e $\in E$ with $E$ the set containing all elements e and $e \in Y_i$ with $Y_i \subseteq E$. Each set $Y_i$ has different attributes. $G_j$ is a set of sets and the following holds: $...
19
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4answers
3k views

How to evaluate the performance of open source solver?

I am looking for a reliable open source solver to solve LP and MILP (with a few thousand variables). How can I evaluate the performance of a given solver for a particular use case?
12
votes
2answers
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 ...
8
votes
2answers
843 views

CPLEX exceeds time limit issue

I am solving a MILP model using CPLEX 12.8.0, but CPLEX exceeds the time limit on some test instances. More specifically, I set the time limit for 30 minutes using the cplex.setParam(IloCplex::TiLim, ...
4
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0answers
95 views

Conditional constraint formulation [duplicate]

How can I create constraints to make sure $x=1$ if $k\geq 0$ and $x=0$ if $k<0$, where $x\in \{0,1\}$ and $k\in \mathbb{R}$? Here is my attempt: \begin{equation}\label{cons:1} \begin{aligned} ...
4
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0answers
63 views

If and then constraint for a special case [duplicate]

I have the following constraint: $$f\geq C_1\left(d-z-E_1-E_2\right)+C_2\left(E_1+E_2\right).$$ Here $f, d,z\in \mathbb{R}_{\geq 0}$ and $y\in \{0,1\}$ are variables and $C_1$, $C_2$, $E_1$, and $E_2$...
27
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4answers
2k views

“Best practices” for formulating MIPs

Often there are many alternatives ways for formulating a MIP. For example: The model contains inequality constraints that must hold with equality in an optimal solution. The model contains ...
8
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1answer
681 views

Read an LP/MPS file in the PuLP

I was trying to read a MPS file by using the PuLP package in the Python, but I can't find any related documents on it. Does ...
11
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1answer
154 views

Heuristics for mixed integer linear and nonlinear programs

What are some primal heuristics that mixed-integer linear and nonlinear program solvers use to quickly obtain a reasonably good feasible solution?
12
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0answers
111 views

Has the expressibility of 'non-integrality testing' as extension to MILP been studied before?

It turns out that extending MILP with any of the constraints $y=\lfloor x\rfloor$, $y=\lceil x\rceil$, $0 < x$, or $x\notin \mathbb{Z}$ is 'equally hard'. (see my answer here, and below) ...
9
votes
4answers
1k views

Solving MIP in Java with free software

I would like to solve mixed integer programs in Java. I need an approach that relies purely on software that is free (not only for academic use, but also for people outside academia). What would be ...
11
votes
3answers
213 views

Efficiency of solving LP relaxation

I'm building a mixed-integer programming model, and the solver is experiencing a very long run time. So I tried to solve the LP relaxation to the MIP, and I get a similarly long solve time, which ...
11
votes
3answers
212 views

Applicability of Lagrange Multipliers in the analysis of large-scale MILPs?

Qualitatively, in my experience in the solving of large scale MILPs, it is common that binary variables corresponding to "edge possibility" components are frequently chosen. Intuitively, these seem ...
5
votes
1answer
134 views

What is the impact of making flow fractional rather than integer?

When creating a network flow formulation you can set up sinks with integer flow requirement $\ge 1$. This yields solutions with the total amount of flow along an edge. I have also seen this as a ...
3
votes
0answers
52 views

Cellular Automata and MIP Resource Repository? [closed]

Apologies if this is not an appropriate question for the SE, however, considering that it is in beta testing it might be worthwhile. Note that "Cellular Automata" is not yet a tag, so perhaps there is ...
13
votes
2answers
393 views

Application of complex numbers in Linear Programming?

The theory surrounding Linear Programming is based on variables, bounds and coefficients that take on values in $\mathbb R$, the set of real numbers. I have long wondered whether there might be ...
9
votes
1answer
384 views

Advantages of IBM CPLEX Studio over CPLEX in MATLAB?

This question comes from the perspective of an MSc student new to both OR and programming OR problems in a MIP framework. My research has led me down this path of needing to program an optimization ...
16
votes
4answers
577 views

Relationship between Benders’ decomposition and Dantzig-Wolfe decomposition

It’s often said that “Benders’ decomposition is Dantzig-Wolfe applied to the dual”. How can this statement be made precise? I know that in Dantzig-Wolfe, cuts are added in one-to-one correspondence ...

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