This example is going to talk about the multi objective decision making, and we are about to find out
the Parata optimal front so far, the examples that we have analyzed with each other and are all referring
to a single objective.
So we had one objective function that we wanted to minimize it or maximize it and find out the optimal
decisions, variables.
And as you can see on this plot and suppose we want to maximize to different objective functions simultaneously.
And, um, there are some solutions found, um, as you can see on this graph.
So if I show you another solution on this space, the blue one.
The dark blue one.
So if I look at these these solutions, there are five solutions you can see here.
Now, um, if I want to compare these solutions together, I can say that these this solution is better
than the dark one because it's it is better than the dark one in terms of if or if one or two.
And it has a solution in every aspect.
OK, and also the white one, the white one is has has bigger or if one or two but the green one has
a bigger of two but smaller one.
So I can say that the solution can be dominated by this blue one and the white one.
So if there is any solution that can dominate this guy, then we have to eliminate that and other solution.
Consider this one.
Now, if I want to compare the red one and the orange one, which one is dominating one?
The red one.
The red one is dominating the orange one.
OK, so now I hope you understand the meaning of, uh, domination.
So if I ignore these two new solutions that I showed you and just look at these for these solutions
are not better than each other in every aspect.
So if I improve one objective function, I'm deteriorating the other one.
So the question is, how can I if I have a problem with two different objective functions, how can
I find out these optimal solution?
So but before I proceed to the coding and everything else, I want to ask you, why do we need multi
objective optimizations?
So can we combine the objective functions and then solve the augmented one in one shots, for example?
Let me give you another example.
So you want to build, um.
Production plant, and there are some costs associated with that construction.
There are some operating costs.
There are some investment costs.
So you can easily do the summation and then ask your decision making tool to find out the best decision
for you.
OK, um, but sometimes the objective functions cannot be added together.
They are from different dimensions, for example, one of them is environmental, for example, pollutions,
and the other one is cost.
Some people say that the environmental aspects can be translated into cars.
Sometimes it's true, sometimes it's not.
Or there are some other aspects, for example, the beauty of something and the cost of something,
you know, so these two cannot be added together because the dimensions are different or sometimes these
are costs.
But both are most of the objective functions are costs or benefits or whatever financial.
But and the size of each is completely different with the other one.
So the order of magnitude is not at the same level.
So optimizing one of them is not putting any effect on the other one, you know, and sometimes the
objective functions are the same.
For example, they are both cost, but two different people are going to pay them, so.
Adding them together and minimizing the total cost is not always beneficial for both.
OK, so we have to find out some set of solutions, some set of non dominated solutions and give them
to the decision maker and ask them to choose.
And there is another thing.
If we go with the weightism approach and add the different objective functions using different weights.
Then we will find out.
Uh, just one, um, objective function and, uh, one set of decision variables.
OK, and what happens if that weighting factors change?
So we have to run the optimization again, OK, but now we want to find out the set of solutions, as
you can see here, those that are connected with dashed lines and in a way that if you're waiting for
your change, you can change.
Your solution is not needed to run the program again.
And also, you can see that if you move toward any of those objective functions, what will happen to
the other one?
So if I am at the Orange Solution, if I want to go to the blue one, for example, improve it in terms
of two, I want to increase the have to this is how much I will lose in terms of one.
OK, so for simplicity, consider this very easy example.
There are two constraints, or if one is something or if two is something else.
And the range of variation for X1 and X2 are also a specified.
OK, so let's have a look at the Python code, I import the required packages.
Sometimes some of these packages are not really necessary, but there is no harm if you keep them right
there, because maybe you need them in there during your coding.
So am I.
Specify the bounds of X1 or next to you can see here they are both non-negative real's.
And also there are two objective functions.
Or if one or two, they are defined as variables.
And also equation one is something.
Equation two, I will explain it later on.
A question three is the other equation.
So what is equation two.
Equation two is and telling us that or if two is less than something, OK.
So in order to solve that problem, just let me show you something else on the slides and then come
back to the points.
So if if you look at this graph and if I want to, for example, maximize the F one, what will happen?
What is the obvious solution for that?
The obvious solution is here.
OK, but and every time you just maximize the fun, which is the capability of the promo package, you
can only maximize or minimize the one objective function, OK, at the time.
And if you want to maximize or F1, this is a solution that you will get.
OK, but in order to be able to not into not trapping here, we add another constraint for o f two I
can say or two should be and bigger than something.
Hmm.
Or smaller than something and then solve the problem.
So I change that limit for off to and then try to optimize the other one.
I could do it for the other objective function, the o f one and I could say OK, the minimum value
or F one or a maximum value of OIF one is something.
Now optimize the other um objective function.
But how can I write it down in a pile.
So this is the concern I was talking about.
So I will limit the off to less than something and I will change that limit.
OK iteratively and ask the panel to solve it.
So as you can see here, this is strange.
I do have more than one objective function that's fine in the definition phase, but when I want to
solve that, I have to deactivate one of them.
OK, so I put a limit for or of two and I deactivate objective two.
OK, pay full attention here.
And then I asked the Palomo to solve it.
So what objective is being optimized.
This is the active objective, the objective one which is o f one.
So it is solved in this way.
The maximum value of of one is found, OK?
And how much is the Absolon at the beginning?
It's a very big number.
See, so actually there is no limit happening here, but and next time I will deactivate objective one
and keep the objective to activate it.
Hmm.
And then resolve the problem again.
So look at the objective functions.
The first time, the objective function one is its maximum value, second time objective function two
is has its maximum value.
OK, so the range of that Parata optimal is found.
Hmm.
And the X1 annexure are found.
But how should I continue?
I have to do something about the debt limit.
I already specified that the Epsilon.
OK, so if I am talking about the oath to I should look at the valuation ratio of two.
It is changing between zero point two to nine point fifty eight.
So I can say and it can start from this quantity and go down up to here and then try to optimize the
other objective function.
OK, so look at here.
At each point, I write down a for loop, and at each iteration I am updating the value of the Absolon,
so at the first iteration it is of a minimum of one, OK?
And at the final one it becomes a maximum of F.
OK, so in this way, when I solve the problem and optimize it, I can easily find out the value for
the other objective function.
So if you can see here I am starting from zero point two and little by little I am increasing this Epsilon
C.
And try to optimize the other objective function.
OK.
And if you want to graphically show it, you can see it here.
OK, so, um, the number of points depends on the, um, how many steps you have to specify for your
programming.
OK.
And that's it.
Thank you very much.