1
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In this example, we want we do have an equal size circles and the radius of these circles are unknown.

2
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We want to find the biggest value of art that can have.

3
00:00:13,470 --> 00:00:22,680
And in this example, we are about to allocate an equal sized circle inside the circle and we want to

4
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find out and the maximum radius of these equally sized circles and the big circle has the radius equal

5
00:00:31,530 --> 00:00:32,090
to one.

6
00:00:32,700 --> 00:00:41,580
OK, and the condition for this specific optimization problem is that no two circles have any overlap

7
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with each other and they are all inside the big circle.

8
00:00:45,930 --> 00:00:52,810
So first, um, inequality is telling us they are not having any overlap.

9
00:00:53,070 --> 00:00:58,260
The second one is making sure that they are all inside the big circle.

10
00:00:58,320 --> 00:01:03,390
And that big circle is the center of that big circle is located on the point one on one.

11
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OK, for simplicity, let's have a look at the, uh, Python code.

12
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So first, we import every required package, then we define an abstract model.

13
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And number of circles is a parameter.

14
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It is mutable so it can be changed.

15
00:01:25,340 --> 00:01:35,220
Um, I is arranged, set, uh, started from one to end, which is the number of circles and J is the

16
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alias of I, R, X and Y are variables that are changing between a small number and two.

17
00:01:44,010 --> 00:01:52,200
And also uh we need to define um some uh rules for that.

18
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As we said, um, the first rule is making sure that no two circles have any overlap.

19
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The second one is making sure they are all inside the big circle.

20
00:02:05,120 --> 00:02:15,110
And the objective function is, let's say R-squared square or so, and we can maximize our or R-squared,

21
00:02:15,120 --> 00:02:23,480
so let's maximize are OK and also, um, instance, it will be model that create instance.

22
00:02:23,660 --> 00:02:32,870
So after D-line, all the constraints will be created and supplied with the input data and also the

23
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results is obtained here.

24
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Opt of instance.

25
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And the created instance will be solved using the software that we already specified.

26
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OK, and finally the results are plotted here.

27
00:02:49,930 --> 00:02:55,000
So I run the code for you and show you the results.

28
00:02:58,080 --> 00:03:05,040
OK, so here you can see that four, 20 circles and we can have these results.

29
00:03:05,490 --> 00:03:12,980
OK, and the location of these are small circles and as well as the radius of them is found.