I think you can use global algorithm that observe all of equations

Well, first you should find an equation or a function in order to find the value of two variables

then you can use GA algorithm or PSO algorithm

please check the following links

https://de.mathworks.com/help/gads/ga.html?s_tid=srchtitle

https://de.mathworks.com/help/gads/particleswarm.html?searchHighlight=pso&s_tid=doc_srchtitle

Nicolas Dupin Thank you very much for the detailed answer. This seems to be very helpful. I will go through the provided information.

Mohamed Azab Thank you very much for your suggestion.

Nicolas Dupin In this attached research paper ''ε constraint combined with integer cut constraints (ICC)'' is used. Can I use this method in my case?

Dear Manish Khanra ,

In most of the cases people has worked on these algorithms (for example genetic, PSO ) and you wont find your work different. I suggest you to go with some algorithm which has been used very less for example Grasp algorithm or some hybrid of genetic or PSO or any other combination suitable for this type of problem.

With wishes,

Vivek

I think you can use global algorithm that observe all of equations

Well, first you should find an equation or a function in order to find the value of two variables

then you can use GA algorithm or PSO algorithm

please check the following links

https://de.mathworks.com/help/gads/ga.html?s_tid=srchtitle

https://de.mathworks.com/help/gads/particleswarm.html?searchHighlight=pso&s_tid=doc_srchtitle

Vivek Kumar Chouhan Thanks for your suggestion.Actually in my work, results are more important than the algorithms but still i am interested in PSO.Can you please share me if you have any related literature which can help me to deal with the kind of problem i mentioned above using PSO?

Thank you very much for your time.

Nicolas Dupin Thanks alot!

I am quite astonished that nearly everywhere, even for the simples problems, particle swarm or other stochastic search (meta-)heuristics are proposed, though the problems can be solved by exact algorithms.

This is by no means the only way or even the best way to solve problems!

Heuristics don't give any information on the quality of solutions found. The only problems where I accept resorting to them is optimization of black-box systems, where no properties of the resulting function and no (sub-)gradient information is available.

It is just something the users don't need to know any theory ;-)

To me it seems the main reason for proposing that stuff everywhere.

You have to decide two things:

- How should the unit commitment problem be modeled

- Which multi-objective technique to use. There are many, like epsilon-constraints, weighted sum (which dies not fit for a non-convex problem like we have here), weighted Tchebycheff norm.

There were several state-of.-the-art overview papers over the years.

Myself I prefer the reference point approach, derived from Zeleny's displaced ideal. Search for articles by A. Wierzbicki (there are e.g. IIASA working papers from the '80s).

In my opinion there is not 'the optimal' solution to a multiobjective problem. In its nature it is about finding a compromise between conflicting goals. I don't believe that there is a way of finding a user's preference function - otherwise there were no real multiobjective problem.

And the preference for one or another solution not only takes into account the values of the goal function like here generation costs and CO2 emission, but the way the units are utilized - looking also at stability against sudden demand changes and the number of startups.

So an interactive approach for generating different Pareto-optima (and of course there is no need for all of them!) for the evaluation by a decision-maker seems the most appropriate to me.

Unit commitment is a MIP or MINLP problem, depending on the way of representing the generation cost of the thermal plants.

We did it with real data and the engineers from the enterprise were convinced that the generation cost is nonlinear - but the measure data deviated only very slightly from a straight line. When plotting it on A4 paper one could not even see the deviation. So we modeled it as an affine funtion (linear + fixed costs) and could use a MIP solver like Gurobi, CPLEX or, if you don't have access to a commercial one, SCIP. But at the university Kassel you should have access to Gurobi, which is faster.

One other complication is modeling the min-up-/downtimes and the downtime dependent startup costs. Have a look at the page of my friend Miguel Carrion here at Researchgate for papers on models.

Best regards

Ralf

I have used the epsilon constraint method multiple times in research articles with MILP models. It works fine if there are only two objectives. With three or more it might become too time-consuming.

First of all, you should define which is your priority. 1. how you define the multiobjective function or the quality of solution. if you have a nonlinear multiobjective function, then you can go ahead with metaheuristic algorithms but if you are willing to live with a linear multiobjective function as in the paper you uploaded, then you can use an exact MILP solvers like @ralf mentioned. Personally for a two-objective problem, i like to compare the product of the ratio of the values to the global minimum i,e (fi/min(f))(gi/min(g)) where f and g are the objective functions.