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NSGA - II: A multi-objective optimization algorithm

version (154 KB) by Aravind Seshadri
A function for multi-objective optimization using evolutionary algorithms


Updated 19 Jul 2009

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NSGA-II is a very famous multi-objective optimization algorithm. I submitted an example previously and wanted to make this submission useful to others by creating it as a function. Even though this function is very specific to benchmark problems, with a little bit more modification this can be adopted for any multi-objective optimization.

The function is nsga_2(pop,gen). The input arguments for the function are population size and number of generations. For customization purposes the user is free to modify the objective function (function of several decision variables) by modifying an m file (evaluate_objective.m). Couple of sample objective functions is already described in the file. The user also has the freedom to define the decision space.

For more information on NSGA-II visit Kanpur Genetic Algorithm Laboratory at

One of the main applications of multi-objective optimization that I am currently working on is tuning PID controllers using MOEA. I am hoping to share that work with everyone soon.

Update (January 27, 2009): I am unable to support user's request to modify this program to incorporate constraints in the optimization program since I have no time to delve into this field. Hence effective today (January 27, 2009) I release this program under GPLv3. This means that anyone and everyone can modify this code as and how they wish. Enjoy! But do remember to contribute the code back to the community.

Effective July 17, 2009 this code is re-licensed under BSD license to comply with Mathworks policy on submissions to MATLAB central.

Note: I no longer have the resources to maintain this code.

Cite As

Aravind Seshadri (2021). NSGA - II: A multi-objective optimization algorithm (, MATLAB Central File Exchange. Retrieved .

Comments and Ratings (118)

Murad Ali

hi, i need NSGA-II-TS with Tabu search code in matlab.


djamel eddine

Hi, can I ask a question ?!!!
if I want to change the crossover probability (pc) and the mutation probability (pm), how can I do it !!!
Thank you

tomsio john

it was so useful ,but i can't modify the objective function。

kanako machii

Hi. Can I ask a question?
If I want to maximize a objective function, can this code do that?
Only minimize?
Pease help.

zaki dahia

Hi everyone i am new to optimization. I want to optimize tow objective functions using NSGA - II I couldn't understand how can i implemented my problem because i am not understanding the code objective functions is
Tp=140 ;Tf= 600;
can anyone guide me how to implemented my evaluate_objective .
my email:

mohit bajaj

I am getting the following error:
Error using save
Unable to write file solution.txt: permission denied.
Please help

Mahtab Munawar

Best ever code.
Easy to handle.
Thanks a lot.

Anis Younes

Hi everyone i have a project Concerning Simulation under ANSYS and Multi-objective Optimization of a Wing Profile by NSGA II Genetic Algorithm.'' The part of Ansys is easy but the Optimization of a Wing Profile by NSGA II Genetic Algorithm can you help me
My email :

Lei Jing

sanchit goel

hi i am new to optimization. I want to optimize gear parameters but don`t know how to use constraints in this code and also my each variable have different range. Can anyone help me i am in desperate need of help? Plz help

Rui Guo


Karthick Kanagaraj

muthu pandi s

can anyone explain me in details about how to obtain the objective equation from the ANN toolbox


tilfani oussama

I would like to perform this algorithm with the following objective functions, can someone provide me a help for:
Sigma is a known (n,n) matrix
Max R= ∑wici + ∑wiβiRm
Min S= w’Sigmaw
Subject to ∑wi=1 and each wi is smaller than one in absolute value
And ci, βi and Rm are known
We would like to solve for w.

Iuga Mihai

Naveed Ahmed

SKS Labs

SKS Labs

Thanks for this very useful submission. However. there seems to be some issue in the determination of crowding distance.

Not all the solutions with the minimum and maximum objective function value are assigned the value of infinity as crowding distance. The crowding distance is dependent on the ordering of the solutions which should not be the case.

Check with [5 2 6; 4 2 7; 6 2 5; 5 8 1] and [6 2 5; 4 2 7; 5 2 6; 5 8 1] as the objective function values in the function non_domination_sort_mod. Though the points are identical (only with row swapping), the value of crowding distance is different.

Jian Yang

It seems that polynomial mutation exist some problem, i.e., in the "function f=genetic_operator", Line 142, where the "child_3(j) = child_3(j) + delta(j)" should be "child_3(j) = child_3(j) +(u_limit(j)-l_limit(j)) delta(j)" . In addtion, the function can not handle the constraints. But thanks for your sharing.

Lahoucine Ouhsaine

Hi, I opt to use this algorithm to optimise my system design. However, i fin a problem in the "non_dominated_sorting_mod.m" where my input variables (i have 5 variables) in the vector x are not all consistent to put it in the same matrix x, so how can you guide me please to fix this problem? Thank you!

Javier Gutierrez

Can we use parallel computing toolbox to apply this algorithm?


Mingxiao Liang

i found some mistake in the "function f = genetic_operator", where the "child_3(j) = child_3(j) + delta(j)" should be "child_3(j) = child_3(j) +(u_limit(j)-l_limit(j)) delta(j)" .


Hello; thank you for this great work.
I was wondering too with V, how if my decision variable are : routing, chosen vehicle and facility to be build. So, should I fill it with 3?
Then, I have a problem about "Subscripted assignment dimension mismatch." at "z(previous_index:current_index,:) = y;" in non_dominated_sort. Would you like to help me with this?

My research is about green vehicle routing problem (multiple objective for minimizing cost and emission), is anyone here can give me any references or teach me how to put the model of NSGA II into routing problem?
Thank you very much.


Thank you for sharing this great work.
I need your help to make the code account for integer variables.
Thank you


thank you.but it does not work well to DTLZ1.

Yishuai Xu

After reading the whole code, this is a great code for implementing NSGA II algorithm through Matlab.
But, just as user mer ve mentioned on 9 Mar 2012, there was a wrong code in genetic_operator.m file.

According to the definition of polynomial mutation:
child_3(j) = child_3(j) + delta(j); line
should be replaced by
child_3(j) = child_3(j) + (u_limit(j)-l_limit(j))*delta(j);

please regards guys.

anyway, thanks for the great work :)

liwei kou

Jonathan Roy

Great code, I use it a lot in past years. Now I get an prob that I cant solve.

-46 parameter to optimize
-2 objectives, "a" and b

"a" is evaluate by the 23 first parameter of my optimisation.
"b" is evaluate by the 23 last parameter + the result of the 23 first parameter

so "a" is a part of the "b" solution... and with std use of NSGA2 if "a" is good but not "b" parameter of "a" will not stop to change...

Any idea for that prob??

Ravindar Reddy

how to run the code??2016b

Tamiru Paulos

thank so much you for this

Tamiru Paulos

thank you for this

Swapna Prajoth

Can anyone help me in modifying the code when the decision variable is an array

Muhammad Ansab Ali

I am new to optimization, can anyone guide me how to use NSGA-II by using a simple mathematical function.
please email me on

Aryan Goyal


wang kai

thank you very much

mohamed fahmy

i need to max. or min. the objective to find the best solution, can anyone help me

Wen-Ling Lin

Can I use this for a single objective optimization problem? If so, Does anyone know what changes I would need to make to make it be a single optimization? I skimmed through the documentation in particular the objective_description_function.m and it seems that an error will be thrown if the number of objectives is less than 2, however, I read online that the ngsa2 algorithm can be modified to handle single objective optimizations. I'm a little confused about where to start as this is quite new to me. Thank you very much!



Having problem in running ZDT-4 test function. This algorithm is not giving the desired results and converging to local pareto optimal


Gu Micheal

It's very useful, thanks!

Qinyuan Xiong

I was wondering that V-the number of the decision variables. If the decision variable is entered as in the binary form or in the real value? and what is the meaning of number? is it like the digit of the decision variable or the real value of the decision variable?
Many thanks


I need it to complete some researches, and I want to konw how to download?

This algorithm was also featured in my PhD thesis on Applied Multiobjective Optimization Heuristics for Logistic Routing (Spanish, with Summary and conclusions in English):

Marwan Al abas

I need it to complete some researches

anil krishna

its required for me


kevin sun

When I employ this code in ZDT1 and ZDT3, I found that this code did not work well and cannot find the best true Pareto front. Does any one know the reason?

So, apparently jMetal already includes some version of NSGA3 based on this C code:

Oh BTW, please stop using roman numerals in acronyms for MH algorithms, as it becomes confusing. Better to use arabic numerals.


How can I learn to analyze the solution file? Since, I want to know what is the best result?


I want to know how to download?

lin jianhua

How to download?

Carlos Oliveira

Hello to all,
I want to maximize an objective and minimize other? How can I make this code? I'm having some difficulties.


Can anyone give me an example of how to do a maximization problem?


agree with 'mer ve', mutation operator requires a correction

Thank you very much for the code!

varun soni

How to use it for solving multiobjective constrainted problems??


Thanks, the code has been very useful for my research on the implementation of a multi objective Harmony Seach technique


mer ve

I think in genetic_operator.m,
child_3(j) = child_3(j) + delta(j); line
should be replaced by
child_3(j) = child_3(j) + (u_limit(j)-l_limit(j))*delta(j);
according to the attached file...

mer ve

how can we add a rule like, final chromosomes(the solutions) should be divisable by 100? It would be so nice of you if you can help. Thanks in advance...


Does anyone know how to solve the problem "Error: File: ndsort.m Line: 259 Column: 10"?
It would be nice if you could help me. Thanks in advance

aftab Ahmad


when I run the program one error "Error: File: ndsort.m Line: 259 Column: 10" appeared. The error place is "[~,ix]". I hope you can help me.

xiaoping wu

It's great.But I want to know how to use it for constrained problems.I need the help.


it's nice

Anisuzzaman Shuvo


Wafa Ben Yahia

pour travailler avec l'implémentation en langage C de Mr Deb, tu dois écrire le suivant dans l'espace réservé à la fonction objective:

f1 = xreal[0];
g = 0.0;
for (i=1; i<10; i++)
g += xreal[i]*xreal[i] - 10.0*cos(4.0*PI*xreal[i]);
g += 91.0;
h = 1.0 - sqrt(f1/g);
f2 = g*h;
f[0] = f1;
f[1] = f2;

si tu veux autre chose, tu peux me contacter :)


Did anyone try running problem ZDT4 described in paper "AFastandElitstMulti-Objetive Geneti Algorithm:NSGA-II"?

It doesn't give the correct or even close to Deb's original implementation in c language. I did this:

f(1) = x(1);
sum = 0;
for (i = 2:V)
sum = sum + x(i)^2-10*roundn(cos(4*pi*x(i)),-4);
g_x = 1+10*(V-1) + sum;

f(2) = g_x*(1-sqrt(x(1)/g_x));

xu kuiwen

Thank you very much

Aravind Krishnan S


Can you please advise me on how to use the code. As to which file to run? When I ran the objective_description_function, I could input the number of objectives, the number of decision variables and also the range of decision variables. But the code finished running just to display "ans=2". How do I rectify this problem? Thank you.


Gundian Martin

Thank you for the code
Please help me that how use it for constrained problems.

Thanks in advance

Wafa Ben Yahia

Hi, thanks, it's a good work
I want to ask "Ram Natamo" if c-implementation by Deb's group is faster than Matlab?
In my problem i have many constraints, so how to imput it.
thanks for answer :)


HI Arvind.
Really great work. Thank you for the code.Please help me that how use it for constrained problems.

In other way, I think the code could have more consistency in variables declaration.

yousaf zia

Thanks arvind.......
Great work is the knowledge which gives the benefit to other..........i would like to be same as you in this regard.........thanking you

liheng liu

great job! It can work well on my project! just a little modification to suit my constraint problem! thank you!

Tuan Pham

I had solved my problem with this NSGA-II codes but it does cost computer resources. However as I know latest Matlab versions have "gamultiobj" dealing with multiobjective optimization with genetic algorithm. I haven't worked with gamultiobj so I don't know what will be different between these two.

Tuan Pham

hailiang shen

The author seems to be used to c programming language. If implemented with more vector operation, the performance (speed) should be better.

Thnks for this. Where do i call the input arguments for the function popsize and number of arguments in the main file?

lin deng

the code can work well, and it can be used for constrained problems by little modification, thank u for the author!

Zhang hua yong

It can not work !

Zhang Tieliang

thank you for the code.Please help me that how use it for constrained problems. If not, would you know other matlab code that can do these contraints?

Javed Dhillon

ali mohammadpour

Mauro Kito

The code works well with all the test problems. Indeed, C code implementation will have faster performance in computation

Ram Natamo

After my previous comments I did some head to head testing with this implementation and original c-implementation by Deb's group (downloadable from, and this implementation is converging really slowly (in terms of objective function evaluations)! That's a pity, I could have used Matlab implementation... :(
My recommendation is that you should use this version only if you dont care a bit about performance.

Ram Natamo

Further, it seems that performance of this implementation is not even close to that of Deb's original implementation, with regard to generations.
Probably there is something wrong with this code?

Ram Natamo

To me it seems there is some room for improvement. In file genetic_operator.m line
was_cossover = 0;
should probably be
was_crossover = 0;

Even worse, in the same file, child_3 is of different size than V before call:
child_3(:,V + 1: M + V) = evaluate_objective(child_3, M, V);
This must be an error?!

elty sarvia

This is a very helpful set of files! It seems like this implementation does not handle constraints though. Deb's NSGA-II paper mentions a scheme for handling constraints (i.e. not just bound constraints on the decision variables, but "constraint functions"). If this is implemented in this version, could you point me towards whereI acn specify the constraints? If not, would you know of a matlab version that implements these contraints? Thanks!

Slim Bechikh

Hi Arvind, I am Slim Bechikh a tunisian student, i thank you for your efforts to develop such program but there is a small mistake in your code:
function f = replace_chromosome(intermediate_chromosome,
must be
function f = replace_chromosome(intermediate_chromosome,

I tried your program on MATLAB 7.0 and it did not run and after correcting this mistake it has run with succes.

Luis Felipe Giraldo

segaf husein

Joe Leo

hadj smail

Seth Soman

nima tavakkoli

thanx alot,it`s brilliant!

Fenggy Chen


Istadi Istadi

water bruce

thx, dear friend, you renew my life.

little cat


shahab shamshirband

HI dear friends
i need a source code for tsp based on quadratic problems

tong hunter

this toolbox is very useful to me .thanks
but,I haven't found there is some bugs

Mohammad Hemati

thank you for this report,but these codes don't use for decition variables space,please guide me ,How I get the variables value (x1,x2,.....).Also How use it for constrained problems.
Very thanks

Varun Aggarwal

this code is buggy, use at own risk. sbx is wrong, selection is not done the right way and others...

Piotr Wozniak

Generally a pleasure to work with, has been very useful to my work. The one minor thing I'd recommend would be ensure consistency in variable declarations in the m-files.

Great work!

moh hemati


Mukiwa Zimbani

This is a very helpful set of files!
It seems like this implementation does not handle constraints though. Deb's NSGA-II paper mentions a scheme for handling constraints (i.e. not just bound constraints on the decision variables, but "constraint functions"). If this is implemented in this version, could you point me towards whereI acn specify the constraints? If not, would you know of a matlab version that implements these contraints?


hamid ansari


xunkai wei

A good one, yet if you can use a mex interface, it will be more attractive!

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