Not sure what the epsilon value is, or how it acts as a bound, but if you were to impose a penalty function process into the heuristic search, you are forced to deal with the defining (and defending) the amount of the penalty, and the potential of creating final solutions that are not feasible. The best solution would be the one with the highest objective function value, however that is defined, when all constraints are satisfied.

Pete Bettinger

The ε-constraint method is one of the classical methods that is used to handle multi-objective optimization problems (MOPs) by converting a MOP into single objective optimization problems (SOPs). This method depends on the epsilon value, which represents a boundary of the objective. For more details please check Section 3.2 in this book: Multi-Objective Optimization using Evolutionary Algorithms By Kalyanmoy Deb.

Dear Shaymah

This is the principle that SIMUS use, however, the criteria bounds are determined as a function of the values of the left hand side of the inequalities.

For each objective it delivers scores for the alternative that are optimum and then, Pareto efficient.

Once you get the complete set of values using all criteria as objectives, the partial scores for each alternative are added up and their result multiplied by a weight which is the he ratio between the number of criteria that each alternative satisfies and the total number of criteria

What Pete Bettinger says in his two paragraphs has a lot of sense, and this second paragraph is in some way how SIMUS works to determine the most important criteria, however, this is not a direct measure, but the result of using the marginal value of the criteria that define the selected objective and corresponding to the selected alternative.

Nolberto Munier

Thank you for your answer. What you have mentioned is called weighted sum method. Whereas, my question is about the ε-constraint method, which is completely uses different approach. Please check Section 3.1 in this book: Multi-Objective Optimization using Evolutionary Algorithms By Kalyanmoy Deb.

Shaymah Akram Yasear

You can perform a web search on "epsilon constrained method multi objective" which provides quite many hits on Google scholar - and see what others have done in the near past. It is a quite popular tool.

You may look at the useful links below:

Chapter Solving Hard Multiobjective Optimization Problems Using ε-Co...

https://dl.acm.org/citation.cfm?id=2610012

I hope it can help u!

Many thanks for the question and replies. My understanding is that epsilon constraint method is employed to convert an MOO problem into a single-objective optimization (SOO) problem and then use an efficient technique for solving SOO problem (very often, a deterministic method). If you are planning to use a multi-objective heuristic, then it can directly be used for solving the MOO problem without transformation into an SOO problem. This approach is the main focus of the book by Kalyanmoy Deb, which Shaymah Akram Yasear has cited in his replies.

By the way, we have two Excel-based programs for MOO (one is based on NSGA-II and another based on multi-objective differential evolution) free of cost. Both programs can handle inequality constraints (but not equality constraints) and progress-based termination criteria (besides maximum number of generations). Readers interested in either of these are welcome to contact me via email to [email protected].

Dear Shaynah

I am afraid that perhaps I did not explain myself very well.

In your original question you said: ' In the ε-constraint method, one objective will be used as the objective function, and the remaining objectives will be used as constraints using the epsilon value as the bound',

and what I said is that it is exactly what SIMUS does, meaning the bold sentence, but it does not use any epsilon value.

The weighted sum is a MCDM method, I know it.

What I said is that as a part of the SIMUS method it uses weighted sum.

Salah Mortada

Thank you for your reply. This paper like many other papers, does not discuss in details the ε-constraint method. The best references that I found are:

Nonlinear Multiobjective Optimization By Kaisa Miettinen

Multi-Objective Optimization using Evolutionary Algorithms By Kalyanmoy Deb

Based on these references, I have found way to handle the ε-constraint method with MHs.

Nolberto Munier

Many thanks for the clarification.

Gade Pandu Rangaiah

May I know, which constraints handling method you have used in the program?

Shaymah Akram Yasear

Our MS Excel program (e.g., EMOO) handles inequality constraints by the constrained-dominance approach of Deb et al. (2002). This approach does not require penalty coefficient/factor used in the penalty function method.

Gade Pandu Rangaiah Thank you