suggest a method/algorithm to solve Combinatorial optimization

I have three different set of variables

SC=[a,b,c,d,e,f]; % each option a, b, etc are actually temperature dependent and can be evaluated using empirical correaltions

AM=[p;q;r]; % each option p,q, etc are actually temp dependent and can be evaluated using empirical correaltions

HF=[m,n,x,y,z]; % each option m,n, etc are actually temp dependent and can be evaluated using empirical correaltions

, need to choose a best combination by picking one option from each set such that the resulting combination maximizes the efficiency

efficiency is the evaluated by putting value of these variables in a mathematical model.

Which opimization technique/algorithm can solve such problem.

Please suggest.

Thanks a lot.

[Bruno Luong]

There are 6*3*5 = 90 combinations, can't you just evaluate them all then pick the best?

Answer 1

[Ameer Hamza]

I don't think there is a function in MATLAB that operate on a list of optimization variables. The closest possible option is to use the optimizer to generate the index of these variables. Only ga() and surrogateopt() will work for nonlinear integer optimization.

For example, consider this

SC = [1 2 3 4 5 6];
AM = [1 2 3];
HF = [5 4 3 2 1];
    
lb = [1 1 1];
ub = [6 3 5]; % upper bound for indexes

sol = ga(@(x) objFun(x, SC, AM, HF), 3, [], [], [], [], lb, ub, [], 1:3)

SC_sol = SC(sol(1))
AM_sol = AM(sol(2))
HF_sol = HF(sol(3))

function y = objFun(ind, SC, AM, HF)
    y = myModel(SC(ind(1)), AM(ind(2)), HF(ind(3)));
end

function y = myModel(sc, am, hf)
    y = sc + am + hf;
end

However, I guess surrogateopt() is more appropriate here, since it is explicitly defined for "time-consuing objective function"

SC = [1 2 3 4 5 6];
AM = [1 2 3];
HF = [5 4 3 2 1];
    
lb = [1 1 1];
ub = [6 3 5]; % upper bound for indexes

opts = optimoptions('surrogateopt', 'PlotFcn', '');
sol = surrogateopt(@(x) objFun(x, SC, AM, HF),lb, ub, 1:3, opts)

SC_sol = SC(sol(1))
AM_sol = AM(sol(2))
HF_sol = HF(sol(3))

function y = objFun(ind, SC, AM, HF)
    y = myModel(SC(ind(1)), AM(ind(2)), HF(ind(3)));
end

function y = myModel(sc, am, hf)
    y = sc + am + hf;
end

[ANUBHAV Goel]

Thanks a lot for your help.

I would like to add one more concern to the problem and know your recommendation with that.

Here, objective function (efficiency) is not directly dependent on the mentioned variables. choice of these variables influence the calculation of intermediate terms required to evaluate the efficiency and thus indirectly influence the objective function.

would it be fine to use surrogateopt() for this case.

[ANUBHAV Goel]

Thanks a lot.

I have tried the GA for my problem and it worked.

[Ameer Hamza]

"Here, objective function (efficiency) is not directly dependent on the mentioned variables. choice of these variables influence the calculation of intermediate terms required to evaluate the efficiency and thus indirectly influence the objective function."

Yes, because if you study the optimization problem in my answer, you can see that the algorithm controls the indexes, which are even less relevant to the objective function compared to your variables. I would even say that using an optimizer like this is no better than running an exhaustive search using all the possible values. I don't think there is a direct link between indexes and the objective function value, which an optimizer can exploit. But if such a link exist, these optimizers will exploit it.

[ANUBHAV Goel]

Thanks a lot for sharing knowledge and your time..