Dear Iman,

sorry, I did not read your code too carefully - indeed, you only have two objective functions. Here, let me suggest that you follow Alireza's suggestion, to get an idea of the individual min and max for z1 and z2 separately. Then you only need to vary one parameter (w) in a small step width, to get the full 2d Pareto front. 

As far as I understand from your code snipped, you did not do weighted sum, but you solve three independent optimization problems instead. In order to have a real weighted sum method, you need to specify three weight parameters (say, lambda1, lambda2, lambda3), and define a surrogate objective function, such as z4 = lambda1*z1+lambda2*z2+lambda3*z3), and Minimize/maximize for z4. By varying the values for the three lambdas, you can get a point from the Pareto set (in three dimensions) in each run. So you need to run several times (perhaps, several million times), to obtain a full understanding of the Pareto set.

DEar prof Armin

actually objectives z1,and z2 are my two conflict objective and objective Z3 is composite objective from two objectives,i.e. z1,z2;that change with varying weights for example as I mentioned "z3=e=0.1*z1-0.9*z2" with numbers between 0 to 1 .but i do not know how to define MODEL and SOLVE statement in my code!

suppose all z1 to z3 are to be minimized. 

first find the max and min of each z_i separately. 

now optimize the min=sum(i,w_i (z_i-zimax)/(zimin-zimax) )

change wi :) 

Dear Iman,

sorry, I did not read your code too carefully - indeed, you only have two objective functions. Here, let me suggest that you follow Alireza's suggestion, to get an idea of the individual min and max for z1 and z2 separately. Then you only need to vary one parameter (w) in a small step width, to get the full 2d Pareto front. 

Another approach :

forget about z3,

first find the max and min of each z_i separately.

loop(counter,

change epsilon=max z2 to min z2 

min z1 

St: 

z2

Dear Alireza 

do I have to run this algorithm also for another objective

I mean:

loop(counter,

change epsilon=max z1 to min z1

min z2

St:

z1

No you only have two important objective functions (lets say independent) 

find the minimum value of z1 

z2 will definitely be equal to epsilon in each loop counter.

z3 can be easily calculated based on z1 and z2 .

Hi Dear Iman

you can use the multiobjectives methods for example weighted sum or e-constraint and etc.

in weighted sum method you can by changing the weight get pareto solution.

must be sum of weights equal 1.

you can use this reference : Marler, R. T. and J. S. Arora (2004). "Survey of multi-objective optimization methods for engineering." Structural and Multidisciplinary Optimization 26(6): 369-395.

Hi Iman,

Just I would like to remind some points...

Actually weighted sum method is successful in application on convex optimization problems. By the convex optimization, we mean the optimization of convex objective functions over convex sets. In multi objective optimization, if the convexity is not demonstrated, the other methods, e.g. epsilon constraint method, could be proposed to apply :)  

Dear Mehdi and Dear Shadan

Thank you for your response,but actually my problem is about GAMS coding that I have mentioned in my question. 

your welcome Iman.. Actually since you are dealing with a nonlinear model ( that is probable for non convexity) I wanted to remind the application of weighted sum method and attract your attention to verify firstly if it is the proper method for your model. But as you said the problem in obtaining the results can be originated from two reasons:

1- improper parameter setting

2- wrong formulation for multi objective problem.

I think may be it is better to solve your problem with each of the objective functions (solve three different single objective problem subject to your constraints) and try to find the parameter setting that offers feasible solution area for all the objectives. Once you define it, try to use the formula that Alireza has suggested for dealing with multi objective problem :)

http://www.gams.com/modlib/libhtml/epscm.htm

Hi Dear Iman

I think that you have problem in multi-objective methods for this reason I suggested to study above reference.

you must convert two objective functions into one objective function by using weights.

after convert، you code model with one objective function (W1*F1 + W2*F2).

Note that objective functions are normalized.

Dear Iman,

First of all, watch out as weighting method is not the optimal tool to use in your application. Then, try to use XPress MP or AIMMS instead of GAMS for your future career. Things like the one you're trying to do are a piece of cake in these applications.

Hi Soheil

 Can you please give some references about software you introduce here.

Thank you

http://www.or.unimore.it/corsi/MaterialeDidattico/Xpress/mosel_userguide.pdf

hi Mr. Rahimi.

I don't work with GAMS. I work with MATLAB for my programming. sorry I cant help you.

Dear Iman,

As I see, the variables used in the first objective have not been used in the second one. Therefore, the Pareto optimal solution will be stepwise. You can find  the Pareto optimal solution with the epsilon constrained method.

Thank you Dear Hossein

Dear Iman

I have developed a MDCM method based in Linear Programming that allows working with any number of objective functions, even contradictory, and with subjective and objective criteria in any mix.

Once the different projects or alternatives are specified as well as criteria for evaluation, the user selects amongst the criteria which of them are to be used as objective functions albeit he can use them all, which is recommended.

Once the decision-making matrix is built, with projects or alternatives in columns and criteria in rows, the method selects the first criterion and uses it as objective function by removing it from the criteria list. Applying the Simplex algorithm it finds, if exists, by using Solver - which is an Excel add-in - an optimal or Pareto efficient solution. The result is transferred to a matrix called ERM (Efficient Results Matrix).

The used restriction is returned to the criteria list and then the second criterion is chosen, used as an objective function, processed with Solver, and the solution transferred to a second line of the ERM. The used restriction is returned to the criteria list, and then the third criterion is picked up as objective function, results found and saved as a third row in the ERM.

This process is repeated as many times as criteria selected.

When all chosen criteria have been used as objective function, the ERM is a Pareto efficient matrix, and then it is normalized.

From here the ERM in analyzed in two ways: Vertically and horizontally.

In the vertical analysis, all the values for a project are added up and the result multiplied by a ‘Participation Factor’, that considers the number of times a project has been selected when all the criteria have been used as objective functions.

The final result in a row gives the ranking of projects, which can be in decimal or integer format, the higher the better, or in binary format, at analyst choice.

Then the normalized ERM is analyzed horizontally, computing the number of times a project outranks others. This produces a square matrix, called ‘Project Dominant Matrix’ or PDM, where dominant projects are on rows, while dominated projects are on columns.

Then, for each project, the method looks for the difference between number of times it dominates others and the number of times it is dominated by others. That is, we get another ranking in a column, with the highest value the better.

This way, two rankings are obtained for the same scenario, one from the ERM and the other for the PDM. Both rankings must coincide not in numerical values, but in their sequence.

That is, the model gets two solutions, by two different procedures, starting from the same data, and getting the same result. Naturally, this is not a Pareto efficient solution, but a compromise solution. It is a known fact than in multiobjective decision-making it is impossible to get an optimal solution, since for instance we cannot, at the same time, maximize a benefit, and also minimize a cost. However, it can be done with Simus.

One advantage of the system is that the process can be stopped using step-by-step procedure for each objective which allows for examining results and making modifications if necessary.  In addition, shadow prices are shown after each objective is processed, and at the end of the process. This allows for determining the actual ‘weigh’ of each criterion and from it to extract valuable information referred to each project.

The main advantage of the model is that, other that the interventions of the decision-maker to add, change, delete, etc, original data according to partial results, there is no need for weights and levels of acceptance between projects, which are sources of high subjectivity. There are neither ‘preferences’ from the analyst nor selection of ways to get the best value. That is, there is no subjectivity, other than the decision-maker’s actions mentioned above. Therefore, the decision-maker has a crucial role in the process.

There is efficient software used to process this model, written in Visual Basic, which can be sent free of charge and full operational to people requesting it. If interested, request it to [email protected].

A complete description of this method as well as descriptions and actual examples for other methods such as Electre, Promethee, AHP, Topsis, etc, can be found in

‘Strategy for Using Multicriteria Analysis en Decision Making’, by Nolberto Munier, Springer, 2011.

Please do not hesitate to contact me if you have any question or doubts about this methodology.

hi

your answer is attached

Dear Iman

You can use epsilon-constraint method or weighted sum approach in order to change the multi-objective model to single objective model then solve using GAMS optimization software.

Thanks a lot.

Sayyad

I agree with Sayyad Sir on use of weighted sum approach for conversion of multi-objective function to single objective model!

I run my model on Neos server using baron solver and every time I get this message

ERROR: Your job was terminated because it exceeded the maximum allotted time for a job.

How can I fix it ?

Dear all, I need to see a solved example of a multi-objective problem using the epsilon constraint method. can anyone help?

Here is the example written in GAMS. It is available in the latest released version of GAMS

https://www.gams.com/latest/psoptlib_ml/libhtml/psoptlib_ParetoOptimalFront.html

Converting multiple objectives into a single one can be done in a few alternative ways. Alireza and Sayyad have already pointed to some of them to get to a pareto optimal frontier. Two other methods are goal programming and compromise programming. The latter uses the concept of a "distance" from an ideal solution. The ideal solution is a collection of each individual objective optimal (say minimum cost and minimum loss) but it cannot be achieved simultaneously. A compromise solution finds the nearest optimal point on the pareto-optimal solution frontier. The form of the objective function is not too different from what Alireza described but is a quadratic (distance) function. One can of course change the weights on the objective function to explore the frontier. I had used it for capacity expansion planning problem and found it intuitive. Look for Mike Zeleny's text on Multi-criteria Decision Making that has details of both techniques (and a lot more but the names get fancier and less useful).