thanks for answer -- but i need to understand step 3 , plz can you explain it more?

how to apply j to new position although it is not included in j?

Your formulae are quite confusing. According to the second one (velocity update), Q is a position, but according to the first one it seems to be a fitness ... that does not contain any position.

Moreover, according to the third one, x is also a position ... that is not used in velocity update formula.

Remember:

1) fitness function = function of a position in the search space. And only of the position. Don't forget that the same fitness function could be used by some other algorithms, and most of them don't use any velocity.

2) velocity update = combination of the previous velocity and some positions. Usually

- the previous one of the particle

- the previous best one of this particle

- and the best known one by its neighbours.

The neighbours are either the whole swarm (global best approach), which is bad for multimodal functions, or one of the local best versions (the set ot neighbours is either fixed, variable or adaptive).

3) position update = old position + new velocity.

I suggest to have a look at a simple PSO code, like SPSO 2007 (Particle Swarm Central http://particleswarm.info), which uses a variable neighbourhood.

ok sir i understand your explanation. let me tell you about my code.

i have 2 files one for PSO and one for calculating my system fitness function. i have a conflict in linking them.

tell me if i am mistaken.

* when i run the file of fitness function i will get a matrix and also PSO file will return a matrix of particles position a nd velocity.

*when i decide best position and global position. will i take the best fitness and then take the corresponding best & global position in position matrix ?

this is the confusion, tell the right plz

Obviously you didn't read the source code of a simple PSO that I suggested, and not even the Wikipedia page about this method. If you read and understand them you will realize that your question in bold is nonsense.

i read a lot about it but i still have misunderstand about this point:

for example how to initialize position & velocity matrix and relate it to my finess function parameters.

Please read the source code and say precisely which part you don't understand.