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INTERNATIONAL JOURNAL OF INNOVATIVE TECHNOLOGY AND CREATIVE ENGINEERING (ISSN:2045-8711) VOL.10 NO 4 APRIL 2020

INTERNATIONAL JOURNAL OF INNOVATIVE TECHNOLOGY AND CREATIVE ENGINEERING (ISSN:2045-8711) VOL.10 NO 4 APRIL 2020

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INTERNATIONAL JOURNAL OF INNOVATIVE TECHNOLOGY AND CREATIVE ENGINEERING (ISSN:2045-8711) VOL.10 NO 4 APRIL 2020

Dear Researcher, Greetings! Articles in this issue discusses about APPLICATIONS OF FUZZY GAME THEORY USING EPSILON FUZZY PAYOFF MATRIX. It has been an absolute pleasure to present you articles that you wish to read. We look forward many more new technologies in the next month.

Thanks, Editorial Team IJITCE

INTERNATIONAL JOURNAL OF INNOVATIVE TECHNOLOGY AND CREATIVE ENGINEERING (ISSN:2045-8711) VOL.10 NO 4 APRIL 2020

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INTERNATIONAL JOURNAL OF INNOVATIVE TECHNOLOGY AND CREATIVE ENGINEERING (ISSN:2045-8711) VOL.10 NO 4 APRIL 2020 Dr. S. B. Warkad Associate Professor, Department of Electrical Engineering, Priyadarshini College of Engineering, Nagpur, India Dr. doc. Ing. RostislavChoteborský, Ph.D. Katedramateriálu a strojírenskétechnologieTechnickáfakulta,Ceskázemedelskáuniverzita v Praze,Kamýcká 129, Praha 6, 165 21 Dr. Paul Koltun Senior Research ScientistLCA and Industrial Ecology Group,Metallic& Ceramic Materials,CSIRO Process Science & Engineering Private Bag 33, Clayton South MDC 3169,Gate 5 Normanby Rd., Clayton Vic. 3168 DR.ChutimaBoonthum-Denecke, Ph.D Department of Computer Science,Science& Technology Bldg.,HamptonUniversity,Hampton, VA 23688 Mr. Abhishek Taneja B.sc(Electronics),M.B.E,M.C.A.,M.Phil., Assistant Professor in the Department of Computer Science & Applications, at Dronacharya Institute of Management and Technology, Kurukshetra. (India). Dr. Ing. RostislavChotěborský,ph.d, Katedramateriálu a strojírenskétechnologie, Technickáfakulta,Českázemědělskáuniverzita v Praze,Kamýcká 129, Praha 6, 165 21 Dr. AmalaVijayaSelvi Rajan, B.sc,Ph.d, Faculty – Information Technology Dubai Women’s College – Higher Colleges of Technology,P.O. Box – 16062, Dubai, UAE

Naik Nitin AshokraoB.sc,M.Sc Lecturer in YeshwantMahavidyalayaNanded University Dr.A.Kathirvell, B.E, M.E, Ph.D,MISTE, MIACSIT, MENGG Professor - Department of Computer Science and Engineering,Tagore Engineering College, Chennai Dr. H. S. Fadewar B.sc,M.sc,M.Phil.,ph.d,PGDBM,B.Ed. Associate Professor - Sinhgad Institute of Management & Computer Application, Mumbai-BangloreWesternly Express Way Narhe, Pune - 41 Dr. David Batten Leader, Algal Pre-Feasibility Study,Transport Technologies and Sustainable Fuels,CSIRO Energy Transformed Flagship Private Bag 1,Aspendale, Vic. 3195,AUSTRALIA Dr R C Panda (MTech& PhD(IITM);Ex-Faculty (Curtin Univ Tech, Perth, Australia))Scientist CLRI (CSIR), Adyar, Chennai - 600 020,India Miss Jing He PH.D. Candidate of Georgia State University,1450 Willow Lake Dr. NE,Atlanta, GA, 30329 Jeremiah Neubert Assistant Professor,MechanicalEngineering,University of North Dakota Hui Shen Mechanical Engineering Dept,Ohio Northern Univ. Dr. Xiangfa Wu, Ph.D. Assistant Professor / Mechanical Engineering,NORTH DAKOTA STATE UNIVERSITY SeraphinChallyAbou Professor,Mechanical& Industrial Engineering Depart,MEHS Program, 235 Voss-Kovach Hall,1305 OrdeanCourt,Duluth, Minnesota 55812-3042 Dr. Qiang Cheng, Ph.D. Assistant Professor,Computer Science Department Southern Illinois University CarbondaleFaner Hall, Room 2140-Mail Code 45111000 Faner Drive, Carbondale, IL 62901 Dr. Carlos Barrios, PhD Assistant Professor of Architecture,School of Architecture and Planning,The Catholic University of America

INTERNATIONAL JOURNAL OF INNOVATIVE TECHNOLOGY AND CREATIVE ENGINEERING (ISSN:2045-8711) VOL.10 NO 4 APRIL 2020 Y. BenalYurtlu Assist. Prof. OndokuzMayis University Dr. Lucy M. Brown, Ph.D. Texas State University,601 University Drive,School of Journalism and Mass Communication,OM330B,San Marcos, TX 78666 Dr. Paul Koltun Senior Research ScientistLCA and Industrial Ecology Group,Metallic& Ceramic Materials CSIRO Process Science & Engineering Dr.Sumeer Gul Assistant Professor,Department of Library and Information Science,University of Kashmir,India Dr. ChutimaBoonthum-Denecke, Ph.D Department of Computer Science,Science& Technology Bldg., Rm 120,Hampton University,Hampton, VA 23688

Dr. Renato J. Orsato Professor at FGV-EAESP,Getulio Vargas Foundation,São Paulo Business School,RuaItapeva, 474 (8° andar)01332-000, São Paulo (SP), Brazil Dr. Wael M. G. Ibrahim Department Head-Electronics Engineering Technology Dept.School of Engineering Technology ECPI College of Technology 5501 Greenwich Road - Suite 100,Virginia Beach, VA 23462 Dr. Messaoud Jake Bahoura Associate Professor-Engineering Department and Center for Materials Research Norfolk State University,700 Park avenue,Norfolk, VA 23504 Dr. V. P. Eswaramurthy M.C.A., M.Phil., Ph.D., Assistant Professor of Computer Science, Government Arts College(Autonomous), Salem-636 007, India. Dr. P. Kamakkannan,M.C.A., Ph.D ., Assistant Professor of Computer Science, Government Arts College(Autonomous), Salem-636 007, India. Dr. V. Karthikeyani Ph.D., Assistant Professor of Computer Science, Government Arts College(Autonomous), Salem-636 008, India. Dr. K. Thangadurai Ph.D., Assistant Professor, Department of Computer Science, Government Arts College ( Autonomous ), Karur - 639 005,India. Dr. N. Maheswari Ph.D., Assistant Professor, Department of MCA, Faculty of Engineering and Technology, SRM University, Kattangulathur, Kanchipiram Dt - 603 203, India. Mr. Md. Musfique Anwar B.Sc(Engg.) Lecturer, Computer Science & Engineering Department, Jahangirnagar University, Savar, Dhaka, Bangladesh. Mrs. Smitha Ramachandran M.Sc(CS)., SAP Analyst, Akzonobel, Slough, United Kingdom. Dr. V. Vallimayil Ph.D., Director, Department of MCA, Vivekanandha Business School For Women, Elayampalayam, Tiruchengode - 637 205, India. Mr. M. Moorthi M.C.A., M.Phil., Assistant Professor, Department of computer Applications, Kongu Arts and Science College, India PremaSelvarajBsc,M.C.A,M.Phil Assistant Professor,Department of Computer Science,KSR College of Arts and Science, Tiruchengode Mr. G. Rajendran M.C.A., M.Phil., N.E.T., PGDBM., PGDBF., Assistant Professor, Department of Computer Science, Government Arts College, Salem, India. Dr. Pradeep H Pendse B.E.,M.M.S.,Ph.d Dean - IT,Welingkar Institute of Management Development and Research, Mumbai, India

INTERNATIONAL JOURNAL OF INNOVATIVE TECHNOLOGY AND CREATIVE ENGINEERING (ISSN:2045-8711) VOL.10 NO 4 APRIL 2020 Muhammad Javed Centre for Next Generation Localisation, School of Computing, Dublin City University, Dublin 9, Ireland Dr. G. GOBI Assistant Professor-Department of Physics,Government Arts College,Salem - 636 007 Dr.S.Senthilkumar Post Doctoral Research Fellow, (Mathematics and Computer Science & Applications),UniversitiSainsMalaysia,School of Mathematical Sciences, Pulau Pinang-11800,[PENANG],MALAYSIA. Manoj Sharma Associate Professor Deptt. of ECE, PrannathParnami Institute of Management & Technology, Hissar, Haryana, India RAMKUMAR JAGANATHAN Asst-Professor,Dept of Computer Science, V.L.B Janakiammal college of Arts & Science, Coimbatore,Tamilnadu, India Dr. S. B. Warkad Assoc. Professor, Priyadarshini College of Engineering, Nagpur, Maharashtra State, India Dr. Saurabh Pal Associate Professor, UNS Institute of Engg. & Tech., VBS Purvanchal University, Jaunpur, India Manimala Assistant Professor, Department of Applied Electronics and Instrumentation, St Joseph’s College of Engineering & Technology, Choondacherry Post, Kottayam Dt. Kerala -686579 Dr. Qazi S. M. Zia-ul-Haque Control Engineer Synchrotron-light for Experimental Sciences and Applications in the Middle East (SESAME),P. O. Box 7, Allan 19252, Jordan Dr. A. Subramani, M.C.A.,M.Phil.,Ph.D. Professor,Department of Computer Applications, K.S.R. College of Engineering, Tiruchengode - 637215 Dr. SeraphinChallyAbou Professor, Mechanical & Industrial Engineering Depart. MEHS Program, 235 Voss-Kovach Hall, 1305 Ordean Court Duluth, Minnesota 558123042 Dr. K. Kousalya Professor, Department of CSE,Kongu Engineering College,Perundurai-638 052 Dr. (Mrs.) R. Uma Rani Asso.Prof., Department of Computer Science, Sri Sarada College For Women, Salem-16, Tamil Nadu, India. MOHAMMAD YAZDANI-ASRAMI Electrical and Computer Engineering Department, Babol"Noshirvani" University of Technology, Iran. Dr. Kulasekharan, N, Ph.D Technical Lead - CFD,GE Appliances and Lighting, GE India,John F Welch Technology Center,Plot # 122, EPIP, Phase 2,Whitefield Road,Bangalore – 560066, India. Dr. Manjeet Bansal Dean (Post Graduate),Department of Civil Engineering,Punjab Technical University,GianiZail Singh Campus,Bathinda -151001 (Punjab),INDIA Dr. Oliver Jukić Vice Dean for education,Virovitica College,MatijeGupca 78,33000 Virovitica, Croatia Dr. Lori A. Wolff, Ph.D., J.D. Professor of Leadership and Counselor Education,The University of Mississippi,Department of Leadership and Counselor Education, 139 Guyton University, MS 38677

INTERNATIONAL JOURNAL OF INNOVATIVE TECHNOLOGY AND CREATIVE ENGINEERING (ISSN:2045-8711) VOL.10 NO 4 APRIL 2020

Contents APPLICATIONS OF FUZZY GAME THEORY USING EPSILON FUZZY PAYOFF MATRIX ………………………..…...[ 803]

INTERNATIONAL JOURNAL OF INNOVATIVE TECHNOLOGY AND CREATIVE ENGINEERING (ISSN:2045-8711) VOL.10 NO 4 APRIL 2020

APPLICATIONS OF FUZZY GAME THEORY USING EPSILON FUZZY PAYOFF MATRIX M.Suresh #1, A.Sathishkumar *2, S.Deepa **3 # Department

of Mathematics, Dr.R.A.N.M College of Arts and Science, Erode, Tamilnadu, India 1 msuresh201@gmail.com

*Department

of Mathematics, Nandha Arts & Science College, Erode, Tamilnadu, India 2

sathishattssa@gmail.com

**Department

of Mathematics, Bharathidasan College of Arts and Science, Erode, Tamilnadu, India 3sdeepamath@gmail.com

Abstract— The objective of this paper to construct the two-person zero sum game with Epsilon-delta fuzzy payoff matrix, Triangular epsilon fuzzy payoff matrix and its application by using max-min approximation, fuzzy optimum strategy is evolved.

Keywords: Fuzzy two person zero sum game, Epsilon fuzzy numbers and fuzzy optimal strategies. I. INTRODUCTION The set under consideration is said to be fuzzy set. It is denoted by đ?œ‡đ??´ (đ?‘Ľ) âˆś đ?‘‹ → [0,1] or A : đ?‘‹ → [0,1] . Each fuzzy set is completely and uniquely defined by one particular đ?‘šđ?‘’đ?‘šđ?‘?đ?‘’đ?‘&#x;đ?‘ â„Žđ?‘–đ?‘? đ?‘“đ?‘˘đ?‘›đ?‘?đ?‘Ąđ?‘–đ?‘œđ?‘›. Given an fuzzy set A defined on X. Then any number of Îą − cut can be associated with A is defined by đ??´đ?›ź = {đ?‘Ľ ∕ đ?œ‡đ??´ (đ?‘Ľ) ≼ đ?›ź}. Given an fuzzy set A defined on X. Then any number of đ?‘ đ?‘Ąđ?‘&#x;đ?‘œđ?‘›đ?‘” đ?›ź − đ?‘?đ?‘˘đ?‘Ą can be associated with A is defined by đ??´đ?›ź+ = {đ?‘Ľ ∕ đ?œ‡đ??´ (đ?‘Ľ) > đ?›ź }[8]. A convex and normalized fuzzy in R whose membership functions is piecewise continuous is called đ?‘“đ?‘˘đ?‘§đ?‘§đ?‘Ś đ?‘›đ?‘˘đ?‘šđ?‘?đ?‘’đ?‘&#x;. A fuzzy number X is a subset of real line R with membership function X : R → [0,1] such that a fuzzy number X is normal and fuzzy convex and đ?›ź-cut is closed for all đ?›ź ∈ [0,1]. Suppose a fuzzy number X is bounded and if the left hand curve and right hand curve are straight lines then the fuzzy number is called the đ?‘Ąđ?‘&#x;đ?‘–đ?‘Žđ?‘›đ?‘”đ?‘˘đ?‘™đ?‘Žđ?‘&#x; đ?‘“đ?‘˘đ?‘§đ?‘§đ?‘Ś đ?‘›đ?‘˘đ?‘šđ?‘?đ?‘’đ?‘&#x;[5]. If r is a real number then đ?œ€-fuzzy number is denoted by đ?‘&#x;đ?œ€ is also called đ?‘ đ?‘Śđ?‘šđ?‘šđ?‘’đ?‘Ąđ?‘&#x;đ?‘–đ?‘? đ?‘’đ?‘?đ?‘ đ?‘–đ?‘™đ?‘œđ?‘› đ?‘“đ?‘˘đ?‘§đ?‘§đ?‘Ś đ?‘›đ?‘˘đ?‘šđ?‘?đ?‘’đ?‘&#x; is the triangular fuzzy number for some đ?œ€ ∈ X ,( đ?œ€ > 803

0) is a fuzzy set đ?‘&#x;đ?œ€ âˆś đ?‘‹ → [0,1] is denoted by, đ?‘Ľâˆ’(đ?‘&#x;−đ?œ€) đ?‘–đ?‘“ đ?‘&#x; − đ?œ€ < đ?‘Ľ ≤ đ?‘&#x; đ?œ€ đ?‘Ľâˆ’(đ?‘&#x;+đ?œ€) đ?‘&#x;đ?œ€ (đ?‘Ľ) = { đ?‘–đ?‘“ đ?‘&#x; < đ?‘Ľ ≤ đ?‘&#x; + đ?œ€ . đ?œ€

0 đ?‘œđ?‘Ąâ„Žđ?‘’đ?‘&#x;đ?‘¤đ?‘–đ?‘ đ?‘’ A triangular fuzzy number X = (l, m, n ) in above notation is denoted by A = đ?‘šđ?‘šâˆ’đ?‘™,đ?‘›âˆ’đ?‘š . Also đ?‘&#x;đ?œ€ = (đ?‘&#x; − đ?œ€, đ?‘&#x;, đ?‘&#x; + đ?œ€) . A đ?œ€- đ?›ż Fuzzy number đ?‘&#x;đ?œ€,đ?›ż is ( đ?‘&#x; − đ?œ€(1 − đ?›ź), đ?‘&#x; + đ?›ż(1 − đ?›ź)), r ∈ R, đ?œ€, đ?›ż ∈X and đ?œ€, đ?›ż > 0, then âˆ? − đ?‘?đ?‘˘đ?‘Ą đ?‘œđ?‘“ đ?œ€ − đ?›ż đ?‘“đ?‘˘đ?‘§đ?‘§đ?‘Ś đ?‘›đ?‘˘đ?‘šđ?‘?đ?‘’đ?‘&#x; is denoted by, (đ?‘&#x;đ?œ€,đ?›ż )đ?›ź =[r đ?œ€(1 − đ?›ź), r +đ?›ż(1 − đ?›ź)], đ?›ź ∈ [0,1] The following evaluations are đ?‘Ąđ?‘&#x;đ?‘–đ?‘Žđ?‘›đ?‘”đ?‘˘đ?‘™đ?‘Žđ?‘&#x; ađ?‘?đ?‘?đ?‘&#x;đ?‘œđ?‘Ľđ?‘–đ?‘šđ?‘Žđ?‘Ąđ?‘–đ?‘œđ?‘›đ?‘ ođ?‘“ đ?‘šđ?‘Žđ?‘Ľ đ?‘Žđ?‘›đ?‘‘ đ?‘šđ?‘–đ?‘› đ?‘œđ?‘?đ?‘’đ?‘&#x;đ?‘Žđ?‘Ąđ?‘–đ?‘œđ?‘›đ?‘ of fuzzy numbers, (i)

If đ?‘&#x;đ?œ–1 ,đ?›ż1 and đ?‘ đ?œ–1 ,đ?›ż1 be any two epsilon-delta fuzzy numbers where r ≤ đ?‘ and r-đ?œ–1 ≤ s-đ?œ–2 , r+đ?›ż1 ≤ s+đ?›ż2 then we define ,

đ?‘&#x;đ?œ€1,đ?›ż â‹€đ?‘ đ?œ€2,đ?›ż = đ?‘&#x;đ?œ€1â‹ đ?œ€ ,đ?›ż â‹€đ?›ż and 1 2 2 1 2 đ?‘ đ?œ€2,đ?›ż = đ?‘ đ?œ€1∧đ?œ€ ,đ?›ż 2

(ii)

đ?‘&#x;đ?œ€1,đ?›ż ∨ 1

2 1 ∨đ?›ż2

If đ?‘&#x;đ?œ–1 ,đ?›ż1 and đ?‘ đ?œ–1 ,đ?›ż1 be any two epsilon-delta fuzzy numbers where r ≤ đ?‘ and s-đ?œ–2 < đ?‘&#x; − đ?œ–1 , r+đ?›ż1 ≤ s+đ?›ż2 then we define ,

đ?‘&#x;đ?œ€1,đ?›ż â‹€đ?‘ đ?œ€2,đ?›ż = đ?‘&#x;đ?œ€2−(đ?‘ −đ?‘&#x;),đ?›ż and 1 2 1 đ?‘ đ?œ€2,đ?›ż = đ?‘ đ?œ€1+(đ?‘ −đ?‘&#x;),đ?›ż 2

(iii)

đ?‘&#x;đ?œ€1,đ?›ż ∨ 1

2

If đ?‘&#x;đ?œ–1 ,đ?›ż1 and đ?‘ đ?œ–1 ,đ?›ż1 be any two epsilon-delta fuzzy numbers www.ijitce.co.uk

INTERNATIONAL JOURNAL OF INNOVATIVE TECHNOLOGY AND CREATIVE ENGINEERING (ISSN:2045-8711) VOL.10 NO 4 APRIL 2020 where r≤đ?‘ r-đ?œ–1 ≤ s-đ?œ–2 s+đ?›ż2 < r+đ?›ż1 then we define ,

and

đ?‘&#x;đ?œ€1,đ?›ż â‹€đ?‘ đ?œ€2,đ?›ż =đ?‘&#x;đ?œ€1,đ?›ż +(đ?‘ −đ?‘&#x;) and 1 2 2 đ?‘&#x;đ?œ€1,đ?›ż ∨ đ?‘ đ?œ€2,đ?›ż = đ?‘ đ?œ€2 ,đ?›ż1−(đ?‘ −đ?‘&#x;) v 1

(iv)

2

If đ?‘&#x;đ?œ–1 ,đ?›ż1 and đ?‘ đ?œ–1 ,đ?›ż1 be any two epsilon-delta fuzzy numbers where r ≤ đ?‘ and s-đ?œ–2 < đ?‘&#x; − đ?œ–1 , s+đ?›ż2 < đ?‘&#x; + đ?›ż1 then we define ,

đ?‘&#x;đ?œ€1,đ?›ż â‹€đ?‘ đ?œ€2,đ?›ż = đ?‘&#x;đ?œ€2−(đ?‘ −đ?‘&#x;),đ?›ż2+(đ?‘ −đ?‘&#x;) and 1 2 đ?‘ đ?œ€2,đ?›ż = đ?‘ đ?œ€1 +(đ?‘ −đ?‘&#x;),đ?›ż1−(đ?‘ −đ?‘&#x;)

đ?‘&#x;đ?œ€1,đ?›ż ∨ 1

2

Similarly, we get the results for the triangular epsilon fuzzy number by setting đ?œ– = đ?›ż . The sets of the possible feasible strategies of player-I are two fuzzy sets X and Y on R1 and R2 respectively. The two payoff functions P1 ( for player-I) and P2 ( for player-II) from R1Ă—R2→[0,1] are called đ?‘“đ?‘˘đ?‘§đ?‘§đ?‘Ś đ?‘Ąđ?‘¤đ?‘œ đ?‘?đ?‘’đ?‘&#x;đ?‘ đ?‘œđ?‘› đ?‘§đ?‘’đ?‘&#x;đ?‘œ đ?‘ đ?‘˘đ?‘š đ?‘”đ?‘Žđ?‘šđ?‘’[1]. An accentual measure of gratification of a person expressed in terms of epsilon fuzzy numbers is called the đ?‘’đ?‘?đ?‘ đ?‘–đ?‘™đ?‘œđ?‘› đ?‘?đ?‘Žđ?‘Śđ?‘œđ?‘“đ?‘“ đ?‘šđ?‘Žđ?‘Ąđ?‘&#x;đ?‘–đ?‘Ľ[2] . For player X has m activities and player Y has n activities. Then the đ?‘“đ?‘˘đ?‘§đ?‘§đ?‘Ś đ?‘?đ?‘Žđ?‘Śđ?‘œđ?‘“đ?‘“ đ?‘šđ?‘Žđ?‘Ąđ?‘&#x;đ?‘–đ?‘Ľ can be formed by some following rules. The player X’s fuzzy payoff matrix is,

Player Y

Strategies

1

2

‌..

j

‌..

n

đ?‘&#x;đ?œ€11 11 ,đ?œ€11

đ?‘&#x;đ?œ€12 12 ,đ?œ€12

‌..

1đ?‘—

1

đ?‘&#x;đ?œ€1đ?‘— ,đ?œ€1đ?‘—

‌..

đ?‘&#x;đ?œ€1đ?‘› 1đ?‘› ,đ?œ€1đ?‘›

đ?‘&#x;đ?œ€21 21 ,đ?œ€21

đ?‘&#x;đ?œ€22 22 ,đ?œ€22

‌..

đ?‘&#x;đ?œ€2đ?‘— ,đ?œ€2đ?‘—

2đ?‘—

‌..

đ?‘&#x;đ?œ€2đ?‘› 2đ?‘› ,đ?œ€2đ?‘›

â‹Ž â‹Ž

â‹Ž â‹Ž

â‹Ž â‹Ž

â‹Ž â‹Ž

â‹Ž â‹Ž

â‹Ž â‹Ž

đ?‘&#x;đ?œ€đ?‘–1 đ?‘–1 ,đ?œ€đ?‘–1

đ?‘&#x;đ?œ€đ?‘–2 đ?‘–2 ,đ?œ€đ?‘–2

‌..

đ?‘&#x;đ?œ€đ?‘–đ?‘— ,đ?œ€đ?‘–đ?‘—

‌..

đ?‘&#x;đ?œ€đ?‘–đ?‘› đ?‘–đ?‘› ,đ?œ€đ?‘–đ?‘›

â‹Ž â‹Ž

â‹Ž â‹Ž

‌..

â‹Ž â‹Ž

‌..

â‹Ž â‹Ž

đ?‘&#x;đ?œ€đ?‘š1 đ?‘š1 ,đ?œ€đ?‘š1

đ?‘&#x;đ?œ€đ?‘š2 đ?‘š2 ,đ?œ€đ?‘š2

‌..

đ?‘&#x;đ?œ€đ?‘šđ?‘— ,đ?œ€đ?‘šđ?‘—

đ?‘šđ?‘—

‌..

đ?‘&#x;đ?œ€đ?‘šđ?‘› đ?‘šđ?‘› ,đ?œ€đ?‘šđ?‘›

Player X

2 â‹Ž â‹Ž

đ?‘–đ?‘—

i

â‹Ž â‹Ž

m

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INTERNATIONAL JOURNAL OF INNOVATIVE TECHNOLOGY AND CREATIVE ENGINEERING (ISSN:2045-8711) VOL.10 NO 4 APRIL 2020 Let A and B are two different sugar factories in Namakkal district in Tamilnadu state. We consider near distance, maximum rate, more reliability and less deduction as constraints for obtaining optimal strategies.

The above payoff matrix we denoted by,

đ?‘‹đ??¸đ?‘ƒ,đ??¸đ?‘ƒ

đ?‘&#x;đ?œ€11 11 ,đ?œ€11 21 đ?‘&#x;đ?œ€21,đ?œ€21 = â‹Ž â‹Ž đ?‘š1 [ đ?‘&#x;đ?œ€đ?‘š1 ,đ?œ€đ?‘š1

đ?‘&#x;đ?œ€12 12 ,đ?œ€12 22 đ?‘&#x;đ?œ€22 ,đ?œ€22 â‹Ž â‹Ž đ?‘&#x;đ?œ€đ?‘š2 đ?‘š2 ,đ?œ€đ?‘š2

‌ ‌ đ?‘&#x;đ?œ€1đ?‘› 1đ?‘› ,đ?œ€1đ?‘› 2đ?‘› ‌ ‌ đ?‘&#x;đ?œ€2đ?‘› ,đ?œ€2đ?‘› â‹Ž â‹Ž â‹Ž â‹Ž ‌ ‌ đ?‘&#x;đ?œ€đ?‘šđ?‘› đ?‘šđ?‘› ,đ?œ€đ?‘šđ?‘› ]

Primary Data For The Season 2015-2016 :

��

Factory

Where,

A B

đ?‘&#x;11 đ?‘&#x; 21 [đ??´] = â‹Ž â‹Ž [ đ?‘&#x; đ?‘š1

đ?‘&#x;12 ‌ ‌ đ?‘&#x;1đ?‘› đ?‘&#x; 22 ‌ ‌ đ?‘&#x; 2đ?‘› , â‹Ž â‹Ž â‹Ž â‹Ž â‹Ž â‹Ž đ?‘&#x; đ?‘š2 ‌ ‌ đ?‘&#x; đ?‘šđ?‘› ]đ?‘šĂ—đ?‘› đ?œ€11 đ?œ€12 ‌ ‌ đ?œ€1đ?‘› đ?œ€21 đ?œ€22 ‌ ‌ đ?œ€2đ?‘› â‹Ž â‹Ž â‹Ž â‹Ž đ??¸đ?‘ƒ = [đ?œ€đ?‘–đ?‘— ] = â‹Ž â‹Ž â‹Ž â‹Ž [ đ?œ€đ?‘š1 đ?œ€đ?‘š2 ‌ ‌ đ?œ€đ?‘šđ?‘› ]đ?‘šĂ—đ?‘›

Rate in Rs. Rs.2500 Rs.2700

Reliability 0.7 0.8

Deduction inRs. Rs.150 Rs.108

Normalization of The Data :

đ?‘–đ?‘— (đ?‘&#x;đ?œ€đ?‘–đ?‘— ,đ?œ€đ?‘–đ?‘— )

and [3] put be the mĂ—n payoff matrix for a two person zero sum game. If đ?‘&#x; denotes the maximin value and đ?‘&#x; be the minimax value of the game then đ?‘&#x; ≼ đ?‘&#x; .

Factory

Distance

Reliability

A B

0.24 0.76

0.47 0.53

Rate in 0.48 0.52

Deduction 0.58 0.42

We define payoff matrix considering the following rules, To find diagonal elements: đ?‘ƒđ?‘–đ?‘– = đ??ľđ?‘– − đ??´đ?‘– , i = 1,2,3,4.

i,e, đ?‘–đ?‘— ⋠∧ 1≤đ?‘—≤đ?‘›[ 1≤đ?‘–≤đ?‘š{đ?‘&#x;đ?œ€đ?‘–đ?‘— ,đ?œ€đ?‘–đ?‘— }]

If

Distance (Km) 4 Km 13 Km

≼

đ?‘–đ?‘— ⋠∧ 1≤đ?‘—≤đ?‘›[ 1≤đ?‘–≤đ?‘š {đ?‘&#x;đ?œ€đ?‘–đ?‘— ,đ?œ€đ?‘–đ?‘— }] đ?‘–đ?‘— đ?‘&#x; = đ?‘&#x;đ?œ€đ?‘–đ?‘— ,đ?œ€đ?‘–đ?‘— for all

To find non diagonal elements: đ?‘ƒđ?‘–đ?‘— = đ??´đ?‘– Ă— đ??ľđ?‘— i≠j

đ?‘–đ?‘— ⋠∧ 1≤đ?‘–≤đ?‘š[ 1≤đ?‘—≤đ?‘›{đ?‘&#x;đ?œ€đ?‘–đ?‘— ,đ?œ€đ?‘–đ?‘— }]

=

đ?‘–đ?‘— ⋠∧ 1≤đ?‘–≤đ?‘š[ 1≤đ?‘—≤đ?‘›{đ?‘&#x;đ?œ€đ?‘–đ?‘— ,đ?œ€đ?‘–đ?‘— }]

i.e,

đ?‘&#x;= j=1,2,‌‌ n and i=1,2,‌‌ m then the đ?‘“đ?‘˘đ?‘§đ?‘§đ?‘Ś game has a đ?‘ đ?‘Žđ?‘‘đ?‘‘đ?‘™đ?‘’ đ?‘?đ?‘œđ?‘–đ?‘›đ?‘Ą at the cell (i,j). đ?‘–đ?‘— If fuzzy payoff đ?‘&#x;đ?œ€đ?‘–đ?‘— ,đ?œ€đ?‘–đ?‘— is a saddle point then the players have a fuzzy optimal strategies in a pure strategies that is, player-I have ith and player-II have jth fuzzy optimal đ?‘–đ?‘— strategies respectively. The fuzzy payoff đ?‘&#x;đ?œ€đ?‘–đ?‘— ,đ?œ€đ?‘–đ?‘— at the saddle point (i,j) is called the value of the game. A fuzzy game is said to be fair if the saddle point of the fuzzy đ?‘–đ?‘— game is zero. That is đ?‘&#x;đ?œ€đ?‘–đ?‘— ,đ?œ€đ?‘–đ?‘— = 0đ?œ€,đ?œ€ .

Calculations: đ?‘ƒ11 = đ??ľ1 − đ??´1 = 0.76-0.24= 0.52, đ?‘ƒ12 = đ??´1 Ă— đ??ľ2 = 0.24Ă— 0.53= 0.13. In this way, we can find the following payoff matrix,

II. PROBLEM DEFINITION: In our agriculture country like India there are direct and indirect opportunities in largest economic Factory scale. The present study has been made in NamakkalA district of Tamilnadu state. The primary data obtained from the decision maker is not deterministic. Hence it is necessary to process this non deterministic data by fuzzy set theory. Sugar industry needs to attract more sugarcane producers in order to optimize the benefit against various constraints such as near distance, maximum rate, more reliability, good recovery, less deduction, more availability of sugarcane in own zone etc. In present problem we develop payoff matrix by considering only four constraints.

Factory B

Near Distance More Reliability Maximum Rate Less Deduction

Near Distance

More Reliability

Maximum Rate

Less Deduction

0.52

0.13

0.12

0.10

0.36

-0.06

-0.24

-0.20

0.36

-0.25

-0.04

-0.20

0.44

0.31

0.30

-0.16

Development of Fuzzy Decision Making Model :

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INTERNATIONAL JOURNAL OF INNOVATIVE TECHNOLOGY AND CREATIVE ENGINEERING (ISSN:2045-8711) VOL.10 NO 4 APRIL 2020 Fuzzy payoff matrix : Factory B

Facto ry A

Near Distanc e More Reliabili ty Maximu m Rate Less Deducti on

Near Distan ce

More Reliabil ity

Maxim um Rate

Less Deducti on

5210,10

1310,10

1210,10

1010,10

3610,10

−610,10

−2410,10 −2010,10

3610,10

−2510,10

−410,10

−2010,10

4410,10

3110,10

3010,10

−1610,10

MAX-MIN Approximation: ROW MINIMUM: Row I: 5210,10 ⋀ 1310,10 ⋀ 1210,10 ⋀ 1010,10= 1010,10 ROW II: 3610,10 ⋀ −610,10 ⋀ −2410,10 ⋀−2010,10 = −2410,10 Row III:3610,10 ⋀ −2510,10 ⋀ −410,10 ⋀−2010,10 = −2510,10 Row IV: 4410,10 ⋀ 3110,10 ⋀ 3010,10 ⋀−1610,10 = −1610,10 Fuzzy Maxi-Min: 1010,10 ∨ −2410,10 ∨ −2510,10 ∨ −1610,10 = 1010,10 COLUMN MAXIMUM: Column I :5210,10 ∨ 3610,10 ∨ 3610,10 ∨ 4410,10 = 5210,10 Column II : 1310,10 ∨ −610,10 ∨ −2510,10 ∨ 3110,10 = 3110,10 Column III: 1210,10 ∨ −2410,10 ∨ −410,10 ∨ 3010,10 = 3010,10 Column IV:1010,10 ∨ −2010,10 ∨ −2010,10 ∨ −1610,10 = 1010,10

III. CONCLUSION Thus we consider a fuzzy two person zerosum game with epsilon fuzzy numbers and also this paper focuses the development of the applications of fuzzy game theory to industrial decision making. The information given by farmers in non-deterministic and is modeled in terms of fuzzy sets The 'fuzzy minimaxmaximin criterion' is used for obtaining best optimal strategy for sugar factory A and B. REFERENCES [1] M. D. Khedekar, S. N. Yadav, M. S. Bapat, and S. J. Aher, Two-Person Zero-Sum Game using MAX-MIN Approximation, International Journal on Recent and Innovation Trends in Computing and Communication, ISSN: 2321-8169, Volume: 4 Issue: 4. [2] S. J. Aher, S. N. Yadav, M. S. Bapat, and M. D. Khedekar, Linear Approximation Approach to Two-Person Zero-Sum Game Using Symmetric Epsilon Fuzzy Payoff Matrix, International Journal on Recent and Innovation Trends in Computing and Communication, ISSN: 2321-8169, Volume: 4 Issue: 4. [3] S. N. Yadav, M. S. Bapat, MAX-MIN Operations in Fuzzy Number, National Conference on Differential Equation2015, Shivaji University Kolhapur,29-30 January 2015. [4] Bellman.R and Zadeh L.A. (1970), Decision making in fuzzy environment, management science, 17(4). [5] S. N. Yadav, M. S. Bapat, Kamble .P.N., Triangular approximations of fuzzy numbers, International Journal of Statistika and Mathematika, ISSN: 2277-2790 EISSN:2249-8605, Volume 7, Issue 3, 2013. [6] Dubois. D., Prade. H., Fuzzy Sets and Systems: Theory and Applications, Academic Press, Boston, 1980. [7] Nurmi, H. (1976).on fuzzy games, Eur.Meet.Cybern.Syst.Res., 3rd,Vienna [8]

Zadeh L. A., Fuzzy sets, Information

Fuzzy Minimax: 5210,10 â‹€ 3110,10 â‹€ 3010,10 â‹€ 1010,10 = 1010,10 đ?‘–đ?‘—

đ?‘–đ?‘—

Thus, â‹€ ( â‹ đ?‘&#x;đ?œ€đ?‘–đ?‘— ,đ?›żđ?‘–đ?‘— ) = 1010,10 = â‹ ( â‹€ đ?‘&#x;đ?œ€đ?‘–đ?‘— ,đ?›żđ?‘–đ?‘— ) Thus, the fuzzy optimal strategy for factory A is I (Near distance) and the fuzzy optimal strategy for factory B is IV (Less deduction).

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INTERNATIONAL JOURNAL OF INNOVATIVE TECHNOLOGY AND CREATIVE ENGINEERING (ISSN:2045-8711) VOL.10 NO 4 APRIL 2020 [9] Control 8 (1965)338-353ases and in an orderly

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