Untitled e ISSN 2582 5208 International Research Journal of Modernization in Engineering Technology and Science Volume 03/Issue 05/May 2021 Impact Factor 5 354 www irjmets com www irjmets com @Interna[.]
e-ISSN: 2582-5208 International Research Journal of Modernization in Engineering Technology and Science Volume:03/Issue:05/May-2021 Impact Factor- 5.354 www.irjmets.com INTUITIONISTIC TRISKAIDECAGONAL FUZZY NUMBER AND ITS APPLICATION IN ROBOTICS Son – Thanh Huynh1, Thanh – Van Tran1, Song – Duy Ngo1, Thoi – Nam Le 1Department of Technology, Dong Nai Technology University, Bien Hoa, Vietnam ABSTRACT Mathematics accomplish a remarkable part in robotic modelling, planning, and execution Robotics research has been enlarging aggressively and hallmarking a new industrial revolution Today, above one million robots are utilized worldwide and the number is expanding with time Today we have many robots with agility similar or over and above human intellect, physical potentiality, insight, and behavior Nowadays, Robots are playing a significant role in various aspects and disciplines in our daily lives It also plays a vital role in the field of medicine They are being used to perform major surgeries which reduces human effort This research idea aims in choosing the best robot with Intuitionistic Triskaidecagonal Fuzzy Number which enables us to an experience of a real life application I INTRODUCTION Robotics was introduced in the year 1920 by a Czech writer Karel Aoeapek Leonardo da Vinci introduced a programmable robot in the 18th century Charles Babbage and Ada Lovelace improvised the intelligence of robots in 19th century Their inventions and ideas have pushed us to explore more about robotics Robots are of various types and they are as follows: Mobile robots, Industrial robots, Field robots and Advance robots They imitate a lot like humans in all aspects They are designed in such a way that they seem to be caricatures of human beings In the making of robots, mathematics plays a significant role to deal with such complex system The complexities are being dealt with the concepts of mathematics which makes the robots to perform faster than human beings Being in a state of complex factors, fuzzy logic was introduced in the field of robotics It simplifies the complexities and make us more comprehensive to design intelligent robots It has also created an atmosphere for us to choose the best robots This paper has for major parts: the first part introduces the importance of robotics, the second part deals with introduction and definitions, the third part gives the algorithm and finally it ends with a numerical example on how to choose the best robot with Intuitionistic Triskaidecagonal Fuzzy Number DEFINITION 1.1 Intuitionistic Fuzzy Number: Let 𝒳 be a given set An Intuitionistic Fuzzy Set I in 𝒳 is given by I = {x, ψI (x), φI (x)/xϵ𝒳}, where ψI , φI : 𝒳 → [0,1], ψI (x) is the degree of belongingness of the element x in 𝒳 and φI (x) is the degree of nonbelongingness of x in 𝒳 and ≤ ψI (x) + φI (x) ≤ For each x ϵ 𝒳, πI (x) = − ψI (x) − φI (x) is the degree of hesitation www.irjmets.com @International Research Journal of Modernization in Engineering, Technology and Science [1565] e-ISSN: 2582-5208 International Research Journal of Modernization in Engineering Technology and Science Volume:03/Issue:05/May-2021 Impact Factor- 5.354 www.irjmets.com x − p1 ) 0.17 ( p2 − p1 x < p1 x − p2 ) p3 − p2 x − p3 0.33 + 0.17 ( ) p4 − p3 x − p4 0.5 + 0.17 ( ) p5 − p4 x − p5 ) 0.66 + 0.17 ( p6 − p5 x − p6 0.83 + 0.17 ( ) p7 − p6 x − p7 ) − 0.17 ( p8 − p7 x − p8 0.83 − 0.17 ( ) p9 − p8 x − p9 0.66 − 0.17 ( ) p10 − p9 x − p10 ) 0.5 − 0.17 ( p11 − p10 x − p11 0.33 − 0.17 ( ) p12 − p11 x − p11 0.17 ( ) p12 − p11 0.17 + 0.17 ( γT̃M (x) = { 1.2 Algorithm: q2 − x − 0.17 ( ) q2 − q1 q3 − x ) 0.83 − 0.17 ( q3 − q2 q4 − x 0.66 − 0.17 ( ) q4 − q3 q5 − x 0.5 − 0.17 ( ) q5 − q4 q6 − x ) 0.33 − 0.17 ( q6 − q4 q7 − x 0.17 ( ) q7 − q6 q8 − x (x) = γT̃ NM 017 + 0.17 ( ) q8 − q7 q9 − x 0.33 + 0.17 ( ) q9 − q8 q10 − x 0.5 + 0.17 ( ) q10 − q9 q11 − x ) 0.66 + 0.17 ( q11 − q10 q12 − x 0.83 + 0.17 ( ) q12 − q11 x − q13 0.17 ( ) q13 − q12 { p1 ≤ x ≤ p2 p2 ≤ x ≤ p3 p3 ≤ x ≤ p4 p4 ≤ x ≤ p5 p5 ≤ x ≤ p6 p6 ≤ x ≤ p7 p7 ≤ x ≤ p8 p8 ≤ x ≤ p9 p9 ≤ x ≤ p10 p10 ≤ x ≤ p11 p11 ≤ x ≤ p12 x > p13 q1 ≤ x ≤ q2 q2 ≤ x ≤ q3 q3 ≤ x ≤ q4 q4 ≤ x ≤ q5 q5 ≤ x ≤ q6 q6 ≤ x ≤ q7 q7 ≤ x ≤ q8 q8 ≤ x ≤ q9 q9 ≤ x ≤ q10 q10 ≤ x ≤ q11 q11 ≤ x ≤ q12 q12 ≤ x ≤ q13 x > q13 Step 1: Consider a multi – criteria decision making problem with p alternatives and q attributes Formulate a decision matrix using Intuitionistic Triskaidecagonal Fuzzy number, T = < t ij >p×q www.irjmets.com @International Research Journal of Modernization in Engineering, Technology and Science [1566] e-ISSN: 2582-5208 International Research Journal of Modernization in Engineering Technology and Science Volume:03/Issue:05/May-2021 Impact Factor- 5.354 www.irjmets.com [ < t11 > ⋮ < t p1 > ⋯ ⋱ ⋯ < t1q > ⋮ ] < t pq > Here, t ij (i = 1,2, … , p; j = 1,2, … , q) are Intuitionistic Triskaidecagonal Fuzzy number Step 2: Change Intuitionistic Triskaidecagonal Fuzzy number into crisp number using accuracy function ITFN(Mi ) = (a1 + b1 ) + (a + b2 ) + (a + b3 ) + ⋯ + (a11 + b11 ) + (a12 + b12 ) + (a13 + b13 ) 13 Step 3: Evaluate Xij = tij 1/2 p (∑j=1 tij ) ; i = 1,2, … q & j = 1,2, … p Step 4: Multiply the decision values obtained with the weight value taken Mij = wi × t ij Step 5: Obtain the positive absolute solution and negative absolute solution, ω+ & ω− ω+ = (M1+ , M2+ , … … Mn+ ), where Mi+ = {Max(Mij ), Min(Mij )} ω− = (M1− , M2− , … … Mn− ), where Mi− = {Min(Mij ), Max(Mij )} Step 6: Obtain the partition measures of each alternatives Pi+ Step 7: Obtain Ai = q = (∑ j=1 q Pi− = (∑ j=1 P− i − (P+ i +Pi ) (Mi+ 1/2 − Mij ) ) 1/2 (Mi− − Mij )2 ) Based on the values obtained, choose the highest values as the best one 1.3 Numerical Example: Consider given robots as T1 , T2 , T3 and T4 as alternatives with C1 , C2 , C3 , C4 , C5 , C6 , C7 , C8 and C9 as costs The values of the decision matrix are taken as Intuitionistic Triskaidecagonal Fuzzy number Obtain a solution of choosing the best robot using the proposed algorithm C1 C2 C3 C4 C5 C6 C7 C8 C9 R (5,5.2,5 4,5.6,5.8, 6.0,6.2,6 4,6.6,6.8, 7.0,7.2,7 4; 2,2.1,2.2, 2.3,2.4,2 5,2.6,2.7, 2.8,2.9,3 0,3.1,3.2 ) (5,5.2,5 4,5.6,5.8, 6.0,6.2,6 4,6.6,6.8, 7.0,7.2,7 4; 1.0,1.1,1 2,1.3,1.4, 1.5,1.6,1 7,1.8,1.9, 2.0,2.1,2 2) (5,5.2,5 4,5.6,5.8, 6.0,6.2,6 4,6.6,6.8, 7.0,7.2,7 4; 3,3.1,3.2, 3.3,3.4,3 5,3.6,3.7, 3.8,3.9,4 0,4.1,4.2 ) (5,5.2,5 4,5.6,5.8, 6.0,6.2,6 4,6.6,6.8, 7.0,7.2,7 4; 3,3.1,3.2, 3.3,3.4,3 5,3.6,3.7, 3.8,3.9,4 0,4.1,4.2 ) (5,5.2,5 4,5.6,5.8, 6.0,6.2,6 4,6.6,6.8, 7.0,7.2,7 4; 4,4.1,4.2, 4.3,4.5,4 6,4.7,4.8, 4.9,5.0,5 1,5.2,5.3 ) (5,5.2,5 4,5.6,5.8, 6.0,6.2,6 4,6.6,6.8, 7.0,7.2,7 4; 5,5.1,5.2, 5.3,5.4,5 5,5.6,5.7, 5.8,5.9,6 0,6.1,6.2 ) (5,5.2,5 4,5.6,5.8, 6.0,6.2,6 4,6.6,6.8, 7.0,7.2,7 4; 6,6.1,6.2, 6.3,6.4,6 5,6.6,6.7, 6.8,6.9,7 0,7.1,7.2 ) (5,5.2,5 4,5.6,5.8, 6.0,6.2,6 4,6.6,6.8, 7.0,7.2,7 4; 2.0,2.1,2 2,2.3,2.4, 2.5,2.6,2 7,2.8,2.9, 3.0,3.1,3 2) (5,5.2,5 4,5.6,5.8, 6.0,6.2,6 4,6.6,6.8, 7.0,7.2,7 4; 3,3.1,3.2, 3.3,3.4,3 5,3.6,3.7, 3.8,3.9,4 0,4.1,4.2 ) R (6,6.2,6 4,6.6,6.8, 7.0,7.2,7 4,7.6,7.8 0,8,8.2,8 4; 2,2.1,2.2, 2.3,2.4,2 5,2.6,2.7, 2.8,2.9,3 (6,6.2,6 4,6.6,6.8, 7.0,7.2,7 4,7.6,7.8 0,8,8.2,8 4; 3,3.1,3.2, 3.3,3.4,3 5,3.6,3.7, 3.8,3.9,4 (6,6.2,6 4,6.6,6.8, 7.0,7.2,7 4,7.6,7.8 0,8,8.2,8 4; 4,4.1,4.2, 4.3,4.5,4 6,4.7,4.8, 4.9,5.0,5 (6,6.2,6 4,6.6,6.8, 7.0,7.2,7 4,7.6,7.8 0,8,8.2,8 4; 4,4.1,4.2, 4.3,4.5,4 6,4.7,4.8, 4.9,5.0,5 (6,6.2,6 4,6.6,6.8, 7.0,7.2,7 4,7.6,7.8 0,8,8.2,8 4; 5,5.1,5.2, 5.3,5.4,5 5,5.6,5.7, 5.8,5.9,6 (6,6.2,6 4,6.6,6.8, 7.0,7.2,7 4,7.6,7.8 0,8,8.2,8 4; 5,5.1,5.2, 5.3,5.4,5 5,5.6,5.7, 5.8,5.9,6 (6,6.2,6 4,6.6,6.8, 7.0,7.2,7 4,7.6,7.8 0,8,8.2,8 4; 1.0,1.1,1 2,1.3,1.4, 1.5,1.6,1 7,1.8,1.9, (6,6.2,6 4,6.6,6.8, 7.0,7.2,7 4,7.6,7.8 0,8,8.2,8 4; 2,2.1,2.2, 2.3,2.4,2 5,2.6,2.7, 2.8,2.9,3 (6,6.2,6 4,6.6,6.8, 7.0,7.2,7 4,7.6,7.8 0,8,8.2,8 4; 3,3.1,3.2, 3.3,3.4,3 5,3.6,3.7, 3.8,3.9,4 www.irjmets.com @International Research Journal of Modernization in Engineering, Technology and Science [1567] e-ISSN: 2582-5208 International Research Journal of Modernization in Engineering Technology and Science Volume:03/Issue:05/May-2021 Impact Factor- 5.354 www.irjmets.com 0,3.1,3.2 ) 0,4.1,4.2 ) 1,5.2,5.3 ) 1,5.2,5.3 ) 0,6.1,6.2 ) 0,6.1,6.2 ) 2.0,2.1,2 2) 0,3.1,3.2 ) 0,4.1,4.2 ) R (8,8.2,8 4,8.6,8.8, 9.0,9.2,9 4,9.6,9.8, 10,10.2,1 0.4; 3,3.1,3.2, 3.3,3.4,3 5,3.6,3.7, 3.8,3.9,4 0,4.1,4.2 ) (8,8.2,8 4,8.6,8.8, 9.0,9.2,9 4,9.6,9.8, 10,10.2,1 0.4; 1.0,1.1,1 2,1.3,1.4, 1.5,1.6,1 7,1.8,1.9, 2.0,2.1,2 2) (8,8.2,8 4,8.6,8.8, 9.0,9.2,9 4,9.6,9.8, 10,10.2,1 0.4; 2,2.1,2.2, 2.3,2.4,2 5,2.6,2.7, 2.8,2.9,3 0,3.1,3.2 ) (8,8.2,8 4,8.6,8.8, 9.0,9.2,9 4,9.6,9.8, 10,10.2,1 0.4; 3,3.1,3.2, 3.3,3.4,3 5,3.6,3.7, 3.8,3.9,4 0,4.1,4.2 ) (8,8.2,8 4,8.6,8.8, 9.0,9.2,9 4,9.6,9.8, 10,10.2,1 0.4; 3,3.1,3.2, 3.3,3.4,3 5,3.6,3.7, 3.8,3.9,4 0,4.1,4.2 ) (8,8.2,8 4,8.6,8.8, 9.0,9.2,9 4,9.6,9.8, 10,10.2,1 0.4; 3,3.1,3.2, 3.3,3.4,3 5,3.6,3.7, 3.8,3.9,4 0,4.1,4.2 ) (8,8.2,8 4,8.6,8.8, 9.0,9.2,9 4,9.6,9.8, 10,10.2,1 0.4; 4,4.1,4.2, 4.3,4.5,4 6,4.7,4.8, 4.9,5.0,5 1,5.2,5.3 ) (8,8.2,8 4,8.6,8.8, 9.0,9.2,9 4,9.6,9.8, 10,10.2,1 0.4; 2,2.1,2.2, 2.3,2.4,2 5,2.6,2.7, 2.8,2.9,3 0,3.1,3.2 ) (8,8.2,8 4,8.6,8.8, 9.0,9.2,9 4,9.6,9.8, 10,10.2,1 0.4; 1.0,1.1,1 2,1.3,1.4, 1.5,1.6,1 7,1.8,1.9, 2.0,2.1,2 2) R (7,7.2,7 4,7.6,7.8, 8.0,8.2,8 4,8.6,8.8, 9.0,9.2,9 4; 2,2.1,2.2, 2.3,2.4,2 5,2.6,2.7, 2.8,2.9,3 0,3.1,3.2 ) (7,7.2,7 4,7.6,7.8, 8.0,8.2,8 4,8.6,8.8, 9.0,9.2,9 4; 2,2.1,2.2, 2.3,2.4,2 5,2.6,2.7, 2.8,2.9,3 0,3.1,3.2 ) (7,7.2,7 4,7.6,7.8, 8.0,8.2,8 4,8.6,8.8, 9.0,9.2,9 4; 3,3.1,3.2, 3.3,3.4,3 5,3.6,3.7, 3.8,3.9,4 0,4.1,4.2 ) (7,7.2,7 4,7.6,7.8, 8.0,8.2,8 4,8.6,8.8, 9.0,9.2,9 4; 3,3.1,3.2, 3.3,3.4,3 5,3.6,3.7, 3.8,3.9,4 0,4.1,4.2 ) (7,7.2,7 4,7.6,7.8, 8.0,8.2,8 4,8.6,8.8, 9.0,9.2,9 4; 5,5.1,5.2, 5.3,5.4,5 5,5.6,5.7, 5.8,5.9,6 0,6.1,6.2 ) (7,7.2,7 4,7.6,7.8, 8.0,8.2,8 4,8.6,8.8, 9.0,9.2,9 4; 4,4.1,4.2, 4.3,4.5,4 6,4.7,4.8, 4.9,5.0,5 1,5.2,5.3 ) (7,7.2,7 4,7.6,7.8, 8.0,8.2,8 4,8.6,8.8, 9.0,9.2,9 4; 4,4.1,4.2, 4.3,4.5,4 6,4.7,4.8, 4.9,5.0,5 1,5.2,5.3 ) (7,7.2,7 4,7.6,7.8, 8.0,8.2,8 4,8.6,8.8, 9.0,9.2,9 4; 3,3.1,3.2, 3.3,3.4,3 5,3.6,3.7, 3.8,3.9,4 0,4.1,4.2 ) (7,7.2,7 4,7.6,7.8, 8.0,8.2,8 4,8.6,8.8, 9.0,9.2,9 4; 2,2.1,2.2, 2.3,2.4,2 5,2.6,2.7, 2.8,2.9,3 0,3.1,3.2 ) Step 1: 3 3 2 6 6 6 5 4 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.34 0.39 0.27 0.29 0.22 0.08 0.00 0.34 0.23 0.43 0.30 0.27 0.29 0.22 0.21 0.64 0.43 0.32 0.53 0.64 0.69 0.63 0.77 0.74 0.52 0.62 0.68 0.53 0.47 0.48 0.52 0.36 0.48 0.41 0.43 0.50 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Step 2: Step 4: Positive Absolute Solution ω+ = {0.053,0.128,0.206,0.252,0.385,0.445,0.447,0.492,0.612} 0.00036 0.0026 0.0156 0.0182 0.0756 0.1576 0.1998 0.0501 0.1616 0.000081 0.0044 0.0156 0.0182 0.0756 0.1011 0.0222 0.1036 0 0 0 0.0064 0 0.0015 0.0039 0.002 0.0424 0.0252 0.0256 0.0222 0.0259 www.irjmets.com @International Research Journal of Modernization in Engineering, Technology and Science [1568] e-ISSN: 2582-5208 International Research Journal of Modernization in Engineering Technology and Science Volume:03/Issue:05/May-2021 Impact Factor- 5.354 www.irjmets.com Step 5: Negative Absolute Solution ω− = {0.034,0.061,0.081,0.117,0.110,0.048,0.00,0.269,0.209} 0.00028 0 0 0 0.00008 0 0 0.0062 0.1998 0.0056 0.0064 0.00036 0.0044 0.0156 0.01822 0.0756 0.1576 0.1346 0.0501 0.1616 0.00036 0.0011 0.0039 0.0081 0.0047 0.0566 0.0823 0.0056 0.058 Step 6: (a) Calculation of Pi+ q ∑ T1 T2 (b) Calculation of Pi− T3 T4 q ∑ 0.6815 0.8255 0.3408 0.5838 0.0064 0.0800 0.1487 0.3856 T2 T3 T4 Pi+ (Mi− − Mij )2 0.0003 0.0167 0.2181 0.4670 0.6181 0.7862 0.2207 0.4697 j=1 T1 Step 7: (Mi+ − Mij )2 j=1 Pi− Calculate the relative closeness to the ideal solution A+i = Pi− /(Pi+ + Pi− ) (Pi+ + Pi− ) T1 T2 T3 Robot T3 is the best T4 II A+i 0.84 0.02 1.05 0.44 0.87 0.91 0.86 0.55 Best CONCLUSION The problem so formed has been found with better solution using Intuitionistic Triskaidecagonal Fuzzy Number which reduces vagueness in a better way when compared with all other intuitionistic fuzzy numbers defined so far Through this numerical example, we come to know that the concept of IFSs can be applied to many real life problems which has more vagueness III REFERENCES [1] Deng Feng Li, “A ratio method of Triangular Intuitionistic Fuzzy Number & its Application to Multi criteria decision making methods, 2010, 1557 – 1570 [2] Shen L, Wang H, Feng X, “Ranking methods of Intuitionistic Fuzzy numbers in multi criteria decision making, 3rd International Conference of Information Management, Innovation Management and Engineering, 2010 [3] K Selvakumari, B Ummusalma, “Decision making problem in Pentagonal Intuitionistic Fuzzy Number using TOPSIS method, International Journal of Advance Engineering and Research Development”, Volume 4, Issue 7, July 2017 www.irjmets.com @International Research Journal of Modernization in Engineering, Technology and Science [1569] e-ISSN: 2582-5208 International Research Journal of Modernization in Engineering Technology and Science Volume:03/Issue:05/May-2021 Impact Factor- 5.354 www.irjmets.com [4] A Sahaya Sudha, K R Vijayalakshmi, “Arithmetic Operations of Hexagonal Intuitionistic Fuzzy Number using Extension Principle, International Journal of Mathematics and its Applications”, ISSN: 2347 – 1557, 219 – 225 [5] B Anushya, B Ramaand, L Sudha, “Transportation Problem using Intuitionistic Decagonal Fuzzy Number”, International Journal of Research and Analytical Reviews, Volume 6, Issue 1, Jan – March 2019 [6] Shiny Jose, Sunny Kuriakose, “Aggregation operator, Score function and accuracy function for multi criteria decision problems in intuitionistic fuzzy context”, Notes on Intuitionistic Fuzzy Sets, ISSN: 1310 – 5132, Vol 20, 2014 [7] A Rajkumar, D Helen, “ Tree Trigger Success of Door Bell using fuzzy number”, International Journal of Pure and Applied Mathematics, Volume 14, No 5, 2017, 71 – 77 [8] L Jeromia Anthvanet, A Rajkumar “Some properties of Intuitionistic Dodecagonal Fuzzy Number and its Application”, Advances in Mathematics: Scientific Journal, ISSN: 1857 – 8365, Special Issue on ICAM – 2020 www.irjmets.com @International Research Journal of Modernization in Engineering, Technology and Science [1570] ... 1,2, … , p; j = 1,2, … , q) are Intuitionistic Triskaidecagonal Fuzzy number Step 2: Change Intuitionistic Triskaidecagonal Fuzzy number into crisp number using accuracy function ITFN(Mi ) =... Triangular Intuitionistic Fuzzy Number & its Application to Multi criteria decision making methods, 2010, 1557 – 1570 [2] Shen L, Wang H, Feng X, “Ranking methods of Intuitionistic Fuzzy numbers in multi... with better solution using Intuitionistic Triskaidecagonal Fuzzy Number which reduces vagueness in a better way when compared with all other intuitionistic fuzzy numbers defined so far Through this