Epsilon constraint with a bi-objective mathematical model

Answered

John Raphy Karippery

October 13, 2021 10:36 Edited

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Hello,
I looking for an example code for epsilon constrain in gurobi. I didn't find any code yet

the definition of e-constraîn is if you want to consider the first objective function as the main objective function, consider the second objective function as a constraint.

systematically very bound on constraints to generate Pareto frontier. 

I would like to use e-constrain to find Pareto front  and  illustrate in a graph(plot)

I don't know how to start please help me. 

it will be more helpful when you apply a generic MOVRP model as an example code.

\[ min \ \ {f_1(X),f_2(X)} \]

\[ f_1(X)  \ \sum x_{ijk}  \ c_{ijk}\]  #minimize cost

\[ f_2(X)  \ \sum x_{ijk}  \ t_{ijk}\]  #minimize time

The formulation uses a set X of binary variables x_ijk equal to 1 if vehicle k travels directly from i to j, and to 0 otherwise.

Constraints (1) guarantee that each customer will be visited exactly once

Constraints(2) each customer get served before the deadline

Constraints(3) each vehicle start from the depot and end in the depot.

Constraints(4) all vehicles have a capacity limit

This is a general mathematic model of MOVRP I need to find Pareto front by using e-constraint.  

Thank you

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• Official comment

  

  Simranjit Kaur

  • Gurobi Staff

  June 23, 2025 02:46 Edited

  This post is more than three years old. Some information may not be up to date. For current information, please check the Gurobi Documentation or Knowledge Base. If you need more help, please create a new post in the community forum. Or why not try our AI Gurobot?.
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  Matthias Miltenberger

  • Gurobi Staff

  October 13, 2021 09:43

  Hi John,

  Could you please reformulate your question? It is not clear what you are asking for.

  Please also review our article Posting to the Community Forum – Gurobi Support Portal

  Thanks,
  Matthias

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  Matthias Miltenberger

  • Gurobi Staff

  October 13, 2021 10:23 Edited

  There is no Gurobi example code for the \(\epsilon\)-constraint method. You would need to search the literature and implement this yourself. Wikipedia refers to this book:

  Kaisa Miettinen (1999). Nonlinear Multiobjective Optimization. Springer. ISBN 978-0-7923-8278-2. Retrieved 29 May 2012.

  I hope that helps.

  Cheers,
  Matthias

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  John Raphy Karippery

  October 13, 2021 15:04

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  hello Matthias,
  
  Is that possible to implement  ϵ-constraint by using Gurobi?

  Could you please implement any simple model code for me? for reference 

  https://www.researchgate.net/profile/Naveenbalaji-Gowthaman/post/How-to-use-the-epsilon-constraint-method/attachment/5a7ac1464cde266d58896040/AS%3A591325149822976%401517994310757/download/1509.08302.pdf

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