Lec 09: Fuzzy relation and fuzzy implications Fuzzy multi-valued logic Defuzzification

Hello everyone, my name is Dr. Santimoy Kundu I welcome you to this lecture of soft computing technique. This is the fifth lecture of module 2. So till now we have covered module 1 and in module 2 we have studied fuzzy relation and fuzzy implications, fuzzy multivalued logic and some of the defuzzification methods. Last class we have discussed defuzzification is a method for the conversion of the fuzzy set to the crisp set or crisp output. We have learned some of the defuzzification methods. These are lambda cut method, maxima methods and weighted average methods. Today we will try to understand the centroid methods. So there are Few types of centroid methods. First one is that center of gravity methods. Second one is that center of sum method. And the last one is that center of area method. So let's first we discuss the center of gravity method. So the output fuzzy sets. C is equal to let C1 union C2 union Cn. And here let C equal to these things. This is Cx mu Cx. So here the C represent like this. And so on. So here this is the straight line equation something like mu Cx. So here for continuous case x star equal to integration x into mu Cx dx divided by integration mu Cx dx. For discrete case x star equal to summation of xi mu ci x divided by mu ci. So this is the center of gravity method. So we will try to understand this concept by using one example. So here example is that suppose this set, fuzzy set represents c So here the points are 0, 2, 2.7, 3, 4, 5, 6. From 0 to 2 the value is 0.7. From 2 to 2.7 value is 0.7. From 0.7 to 3, this is 1. Then from 3 to 4, this is 1. Then from 4 to 6, it's become this straight line. So if you find the equation of the straight line, you will get this point is 0, 0. This point is 2, 0.7. So here y minus 0 equal to 0.7 minus 0 by 2 minus 0 x minus 0. So that is y equal to 0.35x. So thus the straight line if I write a name from 0 to O to A. Then A to B, then B to C, C to D, D to E. So this straight line is 0.35X, Y equal to 0.35X. This is Y equal to 0.7. This is Y equal to X minus 2. This is Y equal to 1. This is Y equal to minus 0.5X plus 3. So, Straight line. Now center of gravity formula that is x integration x mu Cx dx divided by integration mu Cx dx. So that is numerator by denominator. So here numerator n can be written as x into mu Cx dx. Here the denominator d can be written as integration mu Cx dx. So for numerator 0 to 2 that is 0.35x into x dx from 2 to 2.7 that is 0.7 into x dx. 2.7 to 3 it is x into x minus 2 that is x square minus 2x dx. From 3 to 4 that is 1 into x that is x dx. From 4 to 6 it is minus 0.5x plus 3. into x that is 0.5 x square minus of that plus 3x dx so here we'll get the answer is 10.98 that is the value of numerator now here denominator is 0 to 2 integration 0.35 x dx 2 to 7 0.7 dx 2.7 to 3 x minus 2 dx 3 to 4 only dx then 4 to 6 minus 0.5x plus 3 dx. So, her answer is, denominator is 3.445. So, by using the center of gravity, the value of diffusification is x star, that is 10.98 divided by 3.445, that is basically 3.187. I hope you have understood these things. the next one is the center of sum method so here also the output fuzzy set c equal to c1 union c2 and so on union cn so in this method the defuzzified value is x star which is equal to summation i equal to 1 to n xi into aci by summation of i equal to 1 to n aci where aci is the area of ci and xi is geometric center of aci. Suppose this is a trapezoid, so the area is A1 and geometric center is X1. So that is C1. This is mu C1x. So that is x. That is C2. So here it is mu C2. So area is A2. And this is x2. So here x star is simply x1 into A1 plus x2 into A2 by a1 plus a2 so this is the center of sum methods now try to understand same thing by so now see the example of center of sum methods so here we consider two trapezium one is this a1 so this trapezium is a1 and this trapezium is a2 so here you know the area is half of the sum of parallel sides a plus b into height. So the area of a1 is sum of the sides that is 8 minus 1, 7 plus 7 minus 3, 4 height is half, 0.5 so half, so 11 by 4 so that is 2.75 Now similarly area of A2, this is the trapezoid, height is 0.3, so the area of A2 is half, so that is 9 minus 3 is 6 plus 8 minus 4 that is 4 into height is 0.3, so it is 10 into 3 is 0.3 by 2, so it is 3 by 2, so 1.5. So here the x star is summation of i equal to 1 to n ai x i divided by summation i equal to 1 to n ai. So here the value of a1 is 2.75. So take the middle point that is 3 plus 7 by 2, 5. So that is a geometric center 5 into 2.75 plus here it is Again 8 plus 4 by 2 is 6. So geometric center is 6 into 1.5 divided by 2.75 plus 1.5. So that is 22.75 plus 4.25 is 5.35. So here you have seen we have considered the overlap. So overlapped area can be counted twice or more. So main thing here this overlapped area. So write it as a note. overlapped area may be counted twice or more. So that is the center of sum methods. Now try to understand the next example. This is the center of area method. You can check COA. So here thing is that overlap is not allowed. So each area can be counted exactly once. Not more than that. So diagram is same. So we need to divide it in separate areas so that overlap is not possible. So same this is the figure you can write 1. This figure you can write 2. This is the 3. This area is 4. This is 5. And this is 6. So total 6. Formula is same. X star is equal to summation of ACI into XI by summation of ACI. So here you need to calculate the area. So we divided these things into 6 areas. So first one is 1, 2, 3, 4, 5, 6. So area is 0.5. Here this area is 2. Here area is 0.05, 0.15, 0.15, area is 0.15. So the center is 2.33, it is 5, it is 7.166, it is 7.25, it is 7.75, it is 8.33. So, Ai into Xi is 1.665. It is 10. It is 0.3583. It is 1.0875. It is 1.1625. It is 1.621. It is 1.2499. So, it is Ai Xi into summation of Ai. So, 1.1665 plus 10 plus 0.3583 plus 1.0875 plus 1.1625 plus 1.2499 divided by 0.5 area plus 2 plus 0.5 plus 0.15 5.008 So this is the method is known as the center of area methods so now we have learned the three centroid method now try to solve some questions question is that many products such as tar petroleum jelly and petroleum are extracted from crude oil in a newly drilled oil well So, three sets of oil samples are taken and tested for their viscosity. The results are given in the form of three fuzzy sets like B1, B2 and B3. Given in figure as follows, find the most nearly representative viscosity value for all three oil samples. So, we need to find the defuzzified value of these things. So, here you can check it from 0 to 3. So 1.5 it is 1. Then from 1.5 it is 3. From 1 to 2 at 2 it is 1. So from 1 to 5 this is the 3. Value is 1. So you need to find the union of this. So combine these things are union. So from 1.5 this value is 1 then 1.2 this sets represent by this then from this point we will get these things and then from so we need to find all these intersect points so this point is 0 0 this point is 1 comma 5 1 so the equation of straight line we will get it y minus 0 equal to 1 minus 0, 1.5 minus 0, x minus 0. So y equal to, that is 3 by 3, so it is 3 by 2, so it is 2 by 3x. So y equal to 0.67x. Similarly, the straight line is 3, 0. So if you solve it, you will get this straight line as 2 minus 0.7x. So that is basically 2 minus 0.67x. Similarly, this straight line you find. So it is 1, 0. You will get it is 2, 1. So that is basically y minus 0 equal to 1 minus 0 that is 2 minus 1 x minus 1 so you will get it y equal to x minus 1. so this straight line is y equal to x minus 1 so this intersect point so 1 is that 2 so this point we need to find so this point is the intersection of the straight line which are 2y equal to 2 minus 2 by 3x and another is y equal to x minus 1. So if you solve it, you will get x minus 1 equal to 2 minus 2 minus 3 by x. So if you add it, you will get it, solve it, you will get it. 5 by 3x is equal to 3. So x equal to 9 by 5. So x equal to 1.8. So this point x is 1.8. Similarly you need to solve these things. So after solving all of this you can find that y equal to that is mu x equal to 0.67x from 0 to 1.5, from 1.5 to 1.8, it is 2 minus 0.67x. From 1.8 to 2, it is x minus 1. From 2 to 2.33, you will get 3 minus x. From 2.33 to 3, you will get 0.5x minus 0.5. From 3 to 5, you will get 2.5 minus 0.5x. So by using the center of gravity method, so that is you can write it numerator by denominator. So here numerator is equal to integrations 0 to 1.5. x into 0.67 x dx plus integration 1.5 to 1.8 x into 2 minus 0.67 x dx plus integration 1.8 to 2 you will get it x into x minus 1 dx plus 2 to 2.33, you will get it x into 3 minus x dx. Plus 2.33 to 3, you will get x into 0.5x minus 0.5 dx. And plus 3 to 5, you will get x into 3. You will get 2.5 minus 0.5x squared. into x dx. So, that is basically numerator. So, similarly you will get the denominator. So, for this case, answer is 2.5. So, defuzzified value. By the weighted average method, the formula is x star equal to mu cx into xi by summation of mu cxi. So, that is for symmetrical figure. So, in these questions, all figures are symmetric. So, you can check the symmetry figure. So, here the Value is 1.5 and mu is 1. Symmetric value x is 2, mu is 1, x is 3, mu is 1. So here the value is 1 into 1.5 plus 1 into 2 plus 1 into 3 by 1 plus 1 plus 1. So that is, answer is 2.25. Again if we apply these things for the center of sum methods. So you will get half that is 3 into 1 plus geometric center 1.5 plus half 2 into 1 geometric figure that. So here you will get 2.3. So that is this formula ACI into XI by summation of ACI. So by using this formula you will get this thing. So how to find the area? You know half base so base into height then geometric center that is 1.5 similarly you will get it 2.3. So now next example is that a rail company intends to lay a new rail line in a particular part of the country. The whole area through which the new line is passing must be purchased for the right of way consideration. It is surveyed in three stages and the data are collected for analysis. The surveyed data for the road are given by the sets of B1, B2 and B3 where the sets are defined of the universe of right of way width in meters. To purchase the land, it must have an assessment of the amount of land to be bought. The three surveys on right-of-way width are ambiguous. Three fuzzy sets B1, B2 and B3 represented the uncertainty and ease survey as the membership of the right-of-way width meters given as follows. So that is the first survey that is B1, Second is B2. Third is B3. So, we need to combine all the things. So, here the universe of discourse that is width x in meters. So, we need to solve the defuzzified value. So, here. similar way if we find the center of gravity method so you need to find the equation of each of straight lines so 0 to 1 0.3 x 1 to 3.6 0.3 then 3.6 to 4 it is x minus 0.3 by 2 from 4 to 5.5 0.5. Then it is from 5.5 to 6, x minus 5. 6 to 7, it is 1. 7 to 8, it is 8 minus x. So you know how we obtain these equations. So similarly, you find the center of gravity method. The value of defuzzified is these things. So if you solve it, you know the answer. So you will get it. It is like 4.9 meters. I hope you have understood the methods of centroid. Methods of centroid for defuzzified value. So this is the methods of center of gravity method by using integration Center of sums where we can allow overlapping Center of area the overlapping is not allowed. So we have learned three methods. So today's question is that Solve the previous questions the previous questions is solved by By the methods of center of gravity methods I mean we have solved it by the using of integrations. So you are requested to solve it by other methods. Other any three methods. Solve the same questions by other three methods. so you can solve it by center of some central area and the weighted average method so solve the same question by other method so these are the references for today's class thank you