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It's examples, and we already know the radius of all circles.

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OK, and we do have circles and we want to place them inside the triangle at minimum surface without

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overlapping each other.

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So we have to write down some constraints to describe the problem.

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So as you can easily see, um, the area of the triangle is calculated by, um, X etch multiplied by

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Y up here, multiplied by half.

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It gives you the, um, surface of the area of the triangle.

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OK.

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And the first constraint can be easily understood because it is telling us that each pair of circles

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should not have any overlap with each other.

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And also, we do we can see some other constraints here.

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I will explain them to you.

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So first of all, we have to write down the.

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Equation that describes this red line.

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So this red line is passing through the origin and also this specific point up and up and this line

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is described by this equation.

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Hmm.

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And also, we have another line, the blue one.

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That blue line is passing through this point zero, and it's up and way up.

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So the relation between these parameters are given here.

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So this is the equation describing the blue line.

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And also we know that.

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We need to satisfy some constraint so the center of these circles should be, for example, on the right

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hand side of the red line and the left hand side of the blue one.

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This is one constraint.

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And the other one is that the distance between the center to the red one should be bigger than the radius

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of the circle and also the center to the blue.

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One should be bigger than the radius of the circle.

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OK, so these two constraints are given here.

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So these two are telling us, for example, we are below the red line and on the left hand side of the

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blue one, OK?

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And also these two will specify that and the distance between the center of the circle and the line

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is more than the radius of that specific circle.

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OK, so we have to write down the.

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A Python code to solve this optimization problem and the decision variables are, um, the, um, location

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of the centers and also, uh, the radius of them are already known for us.

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OK, so let's talk about the Python code.

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So first of all, we have to import everything that we need then, uh, define the sets I and J.

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Have you already know the meaning of them?

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Because I is representing the each circle.

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J is the alias of that R is a parameter and which is specified by us.

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It is telling us how much is the radius of each circle.

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And also we have to define some bounds for the variable X and Y, so it's between R and a big number

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and Y up an X each and X up are already unknown and the area is already the um, objective function

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for us.

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OK, so um, you can tell that OK, if I is less than J uh then it means that by just writing down this

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inequality, uh, we can make sure that, uh, any two circles are not overlapping each other and then

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we are making sure that we are below the red line.

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We are on the left hand side of the blue one in this one.

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And as you can see, these two equations are defined over I and also the distance between each center

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to the red or blue lines are more than the radius of each circle.

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OK, and also and this is another one that we already talked about it.

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And also, um, we should say that, OK, so x up and is less than X each.

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OK, this is an assumption.

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And and also the area is a specified here.

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OK, so these are the constraint and the objective function and also the uh input of the problem is

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um described here.

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So um, the radius of the circle should be specified in a dandified.

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OK, so I run the problem for you and see the results.

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OK, so the problem is, Ron, and you can see here that and the area of the circle is found, how much

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is it?

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How much is W and so on.

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OK, and you can also by using this command print results that sort the status and you can find out

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what's happening to the different variables like X, Y or what is the upper value, what is the lower

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value, what is the exact value of it and also is it fixed or not fixed and the X of X h y up and the

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area.

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And you can also have some information regarding the constraints and so on, on each equation's.

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OK, normally you don't need that kind of information, but if you want to have access to them, you

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can easily use this command to find out more about what's happening inside your model.

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Thank you very much.