Hello,

In a single objective problem, one can have a minimization or a maximization program. In your case, the problem has two objectives: one regarding a minimization problem and other related to a maximization problem. Your problem has a multi objective nature.

The multi objective optimization leads to solutions for problems with this characteristic.

In your case, you can try: (i) plan a multi objective approach, like Pareto front or converting the problem in a single objective program, with weights; or (ii) change the problem, considering an objective function like a constraint.

Regards,

Bruno

I think you should define three objective functions. one for minimization,another for maximization and the third one would be the difference between first and second solution. Finally the third objective function should be minimized.

The second strategy would be the one that Dr. Prata said. But I think the first strategy is faster.

Best Regards,

Suppose as a seller point of view he/she wants to sell the prize at A (max func) and from buyer point of view he/she wants to buy that at B (min func). So, the fitness function must be (A-B) and which is minimization function.

You have to generate the equation for A and B using the parameters; there may be same parameters, same equations for A and B, but as the point of views are different, so the values must not be the same.

Thanks , Mahboobeh Parsapoor , Bruno Prata, Arash Khassetarash, Pritam Roy for your valuable answers.

What is the real-life meaning of your problem? The example does not help me much because I do not know what is meant by DG. On whose behalf is the optimization aimed? Electricity producer or consumer?  Why would someone maximize the prize? Maximizing the income may make more sense. However, what is the use of large income if the direct costs surpass it? I strongly favour maximizing the contribution, i.e. the difference between income and direct costs. If both facilities belong to a same owner, the overall effect should be optimised, i.e. the overall contribution.  If you want to make the two opposite parties equally happy, than I do not wonder if the result of this utopia can be nirvana – or mathematically, zero activity. In my experience, simpler models are much better accepted by the users, meaning that multicriteria optimization is more favoured by the scientist than users. From the above discussion I am mostly inclined to agree with Pritam Roy.

If A is the function you need to maximize and B is the one that need to be minimized, then the program MAX C=A-B should do the job.

You've already mentioned a GA algorithm to solve it, just let you know that there are different evolutionary algorithms that can be of interest.

For example the MOGA (multi-objective genetic algorithm) described in:

Murata, T.; Ishibuchi, H., "MOGA: multi-objective genetic algorithms," Evolutionary Computation, 1995., IEEE International Conference on , vol.1, no., pp.289,, Nov. 29 1995-Dec. 1 1995

doi: 10.1109/ICEC.1995.489161

keywords: {Genetic algorithms;Iterative methods;Pareto optimization},

URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=489161&isnumber=10439

I think you have to compromise between both. Therefore you can use Pareto function.

There are are lot of good answers above. It is quite hard to add anything useful. Defining a good fitness function is not easy and often it can take some time.  

With the problem at hand, the fitness function needs to take into consideration both aspects.  However, the lesser is the purchase cost, the profit is likely to increase. It would be worth to investigate the relationship between cost of buying and the selling price. Then you will find a good fitness function. 

Without too much knowlegde, about the problem. It appears you are looking at a price optimisation problem. 

I hope it helps.

Kind regards,

Patricia 

Thank you for the question and also for the answers by several researchers. I recommend multi-objective optimization (MOO), and one can perform this in Excel, Matlab etc. For example, we have one Excel-based MOO (EMOO) program available free of charge to interested researchers and practitioners. Please browse http://cheed.nus.edu.sg/stf/chegpr/Software.html for some details and/or contact me for the EMOO program.

Alternately, one can use epsilon-constraint method along with single objective optimization program for solving problems with 2 objectives; although this approach can be used for more than 3 objectives, it requires a lot more effort.

All the Best

(GP Rangaiah)

Dear  Vaiju Kalkhambkar,  if  you  have  more than one   objectie functions, you may  assign weigt  to  them  such that  more impotant objecte has  suiatable weight. Then you can use  linear  or  nonlinear program solver to solve  the problem.

Dear Vaiju Kalkhambkar

if I had understood your question, THE SAME OBJECTIVE FUNCTION has to be maximized and minimized at the same time. Multiobjectve optimization was not intended to deal with such situations. My opinion, you need to model the problem using game theory approaches instead.

Dear Sir,

Problem can be solved in MATLAB using Multiobjective Genetic Algorithm where objective functions value can be obtained in the form of pareto front. You can test goal programming for the multiobjective optimization which can be solved using MATLAB in the optimization tool box. Recently releases version of ther MATLAB are good in speed and computations.

I think your problem is little bit critical ...

Maximize and minimize at the same time...

according to me... First you decide priority, which is most important max. or min.... and according to that solve problem,

or other wise , try to choose a average value, means a value at which Max.(Fn) = Min(Fn).

Hi,

Your problem is to maximize for one player and minimize for another. If yes, Use either stackelberg game or Nash game theory and obtain most compromising solution. Here compromising means beneficial to both the parties. You can read some of my articles coauthored with Prof. Wee and Dr. Gede. 

So far as I know, the objective function of an optimization problem is written from the standpoint of one decision maker who wants to optimize it for his suitability. It is different from multi-objective case where more than one objectives (possibly conflicting) have to achieved at the same time. What I think is that in the case under consideration, the objective function may be written either in maximization or in maximization form depending upon the decision maker's choice or requirement.

Dear all,

Does the first researcher who had firstly highlighted that :

1. MOO was not a so good research path;

2. but Game Theory;

merit a vote?

I presume that

1. you have the same constraints

2. A single objective function

3. Two players outdoing each other.

Then I agree that game theory is suitable for this problem.

For MCDM you have multiobjectives to be considered whereby you need to prioritized or assign weights to them as pointed by previous commentators. Such is not the case that you are dealing with, from what you listed

I sympathize most comments. I think that the real issue is the problem formulation. If I come back to your example, you are trying to formulate two independent optimization problems in a single one. The DG owner wants a maximal price and for this purpose he has degrees of freedom he can play with. The customer on the other side would like to get minimum price. Does he have manipulated inputs? Are these inputs the same ?

Generally speaking if you have two different objectives functions, with the same degrees of freedom/manipulated inputs, then you should try MOO. If not, which is in my opinion your case here, then MOO is probably not appropriate and game theory is perhaps more suited.

The question does not clearly state the problem under consideration.

If you consider maximization and minimization of the same objective as two conflicting objectives of one decision maker in multiobjective optimization, the answer is trivial: all feasible points are Pareto optimal. So this can hardly be the problem under consideration.

With a game-theoretic approach one could interpret this as a two-player zero-sum game - the gain of the first (whom you call DG owner) is the loss of the other (company).

But in game theory both players have their own strategies (variables of the objective), influencing the outcome for the other. Is this the case in your problem? Exactly this question was raised in the previous answer by Gregory Francois.

Lets assume this is the case. I assume the strategy sets are not finite in your case. These problems possess a Nash euquilibrium under certain conditions including compactness of the strategy sets, continuity and pseudo-concavity of the payoff-function for all players. If the last requirement is not fulfilled, then there is no Nash equilibrium in pure strategies. Consider the example with x is chosen by the first player from [0,1], y by the second player also from [0,1] and the objective to be maximized by the first player and minimized by the second is (x-y)^2.

Another possible approach would be that of a bilevel program, where a leader sets part of the variables and a follower replies to these by solving a problem with another objective including additional variables.

If there are no separate variables for each player I cannot see the quality of a game or a bilevel program and I cannot understand the question. Furthermore I do not understand why it shout be solved with a stochastic search algorithm like  GA.

The first point to be cleared is: what is the real question in this problem?

in multi objective optimization problem for one global solution, you can use a deterministic approach like desirability function, or you can different solutions as many local optimization results, you can go through the artificial intelligent methods such as genetic algorithm, tabu search and others.    

These are solution methods - i.e. numerical methods - I think the Framework should be first discussed. This was indeed the meaning of my previous post, and I agree with Ralf Gollmer.  

I am really thankful to all experts for giving their valuable suggestion.  I have got many options to solve it. 

Evolutionary multiobjective optimization will help in this regard.

I will suggest to you to use the NSGA-II or SPEA2 algorithms. the one that you want to maximize, you should use a negative sign or the reciprocal because the nature of the algorithms always minimizes the objective functions. hopfully this will help you.

For all who suggest multiobjective optimization approaches here, a detailed explanation or a reference to an application of such approaches to optimizing the same objective with antagonist and opposite interests is appreciable.     

It is simple to explain why a multiobjective won't be of any use in the case of two objectives, one being the negative of the other (resp. maximizing one objective and at the same time minimizing it) -- provided that maximization and minization are carried out with respect to the same variables.

in the presence of multiple objectives there is one accepted optimality cncept - that of Pareto optimality (a.k.a. efficiency). A feasible point is called efficient, if there is NO feasible point beaing better with respect to one of the objectives while not being worse with respect to the others.

Now consider the same objective function being maximized as the first objective and minimized as the second one and an anrbitrary feasible point. If any other feasible point is better with respect to the first objective (i.e. maximization - the objective value is greater than in the first point) ist is clearly worse with respect to the second objective (minimization). Thus there is not feasible point being better in the sense of efficiency.

The same holds for weak efficiency.

The situation is different, if the question concers a bilevel problem, i.e. the vector of variables is divided into (x,y) (x being the variables chosen by the leader, y the variables chosen by the follower) and maximization is carried out with respect to x, minimization wrt. y. See e.g.  the survey

http://www.mathe.tu-freiberg.de/~dempe/Artikel/bilevel_survey.pdf

you have to assign different weight to the maximization and minimization part. these weights have to be decided by you depending on the value of each part

For the previous answer by Farhad Tari I would suggest to think about an equal positive weight for maximization and minimization. The weighted sum then is identical zero, so all feasible points are optimal - which is the full Pareto set :-))

There are few cases where it is that easy to describe the Pareto set in its entirety.

The question posed just makes no sense when being viewed as a multiobjective problem

Again I sympathize with Ralf's comment. If the costs functions are exactly the same, MOO does not make any sense. It's like saying I want to travel as fast and as slow as possible. Cost functions have to be conflicting to generate a compromise of interest but they cannot just be of opposite signs.

I think you can apply fractional Objective Function for example

Objective Function =  minimum cost/  maximum cost ;

I think you can apply the following

the original problem is Min A+Max B, you can assign to Max B by lamda , the objective becomes Min A+Lamda and add the relation Max B < lamda to the constraints 

The question has to be formulated better in order to provide a meaningful answer.

See my models MOORA and MULTIMOORA, never use weights but use dimensionless measurements

You need to provide more information. This type of function tends to be problem dependent.  I would suggests you look at the type of solutions. Then you should be able to find a mean to indicate when a solution is of good quality or lesser quality.  From this point you should be able to find out a way to indicate whether or not your objective has been me. 

I hope it helps.

Patricia

PSO codes in MATLAB

https://www.researchgate.net/publication/296636431_Codes_in_MATLAB_for_Particle_Swarm_Optimization?ev=prf_pub

Code Codes in MATLAB for Particle Swarm Optimization

https://www.researchgate.net/publication/297245624_Particle_Swarm_Optimization_Algorithm_and_its_Codes_in_MATLAB

Research Particle Swarm Optimization: Algorithm and its Codes in MATLAB

https://www.researchgate.net/publication/280244004_A_comparative_study_of_metaheuristic_optimization_approaches_for_directional_overcurrent_relays_coordination

Article A comparative study of metaheuristic optimization approaches...

https://www.researchgate.net/publication/297245624_Particle_Swarm_Optimization_Algorithm_and_its_Codes_in_MATLAB

Research Particle Swarm Optimization: Algorithm and its Codes in MATLAB

https://www.researchgate.net/publication/296636431_Codes_in_MATLAB_for_Particle_Swarm_Optimization

Code Codes in MATLAB for Particle Swarm Optimization

Mr. Alam, have you really thought about the question or did you just want to point to your heuristics?

Mr Alam, Please can you help me with this pso matlab code for optimizing the parameter of my antenna having the equation for finding the width and length of my antenna as w=c/2f(sqrt(2/4.4+1)) where my f =60e9 and c=3e8 and for the length is L=c/2f(sqrt(4.4))

Hello mr Ralf Gollmer please can you help me on how to optimize the width of my patch antenna or the resonance frequency of my [email protected] is my email addrss.

Very good answer of Ralf Gollmer 

This problem is multi-objective, convert the part one of the problem to maximum (say Minimum transfer to Maximum or vice versa ) such that two parts are minimum or maximum and use the whitening method or epslom constraint method for solving the multi-objective problem

The simple way is to consider only an objective function by using lagrange multiplicators.

Pls refer some papers in Venkata Rao papers , NIT, SURAT, Mechanical Engg Dept..

Thank you for the question and for all the responses. This question was posed several years ago. Indeed, applications similar to the question require multi-objective optimization (MOO). For those interested, we have two MS Excel based programs for MOO, available at no charge; one is based on GA (non-dominated sorting genetic algorithm) and another based on different evolution (integrated multi-objective differential evolution). Both have progress-based termination criterion besides the usual maximum number of generations. Researchers interested to use these programs can contact me at [email protected]. Please provide some details about yourself, organisation, planned use of programs etc. in your email.

The objective function evaluates the quality of each solution. It determines the way in which the search strategy is conducted.