June 2020 International Journal of Innovative Technology and Creative Engineering (ISSN:2045-8711) by Editor IJITCE

INTERNATIONAL JOURNAL OF INNOVATIVE TECHNOLOGY AND CREATIVE ENGINEERING (ISSN:2045-8711) VOL.10 NO 6 JUNE 2020

UK: Managing Editor International Journal of Innovative Technology and Creative Engineering 1a park lane, Cranford London TW59WA UK

USA: Editor International Journal of Innovative Technology and Creative Engineering Dr. Arumugam Department of Chemistry University of Georgia GA-30602, USA.

India: Editor International Journal of Innovative Technology & Creative Engineering 36/4 12th Avenue, 1st cross St, Vaigai Colony Ashok Nagar Chennai , India 600083 Email: editor@ijitce.co.uk

www.ijitce.co.uk

INTERNATIONAL JOURNAL OF INNOVATIVE TECHNOLOGY AND CREATIVE ENGINEERING (ISSN:2045-8711) VOL.10 NO 6 JUNE 2020

IJITCE PUBLICATION

International Journal of Innovative Technology & Creative Engineering Vol.10 No.6 June 2020

www.ijitce.co.uk

INTERNATIONAL JOURNAL OF INNOVATIVE TECHNOLOGY AND CREATIVE ENGINEERING (ISSN:2045-8711) VOL.10 NO 6 JUNE 2020

Dear Researcher, Greetings! Articles in this issue discusses about NUMERICAL COMPUTATION OF INTUITIONISTIC FUZZY ASSIGNMENT PROBLEM WITH SIMILARITY MEASURES. 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 6 JUNE 2020

Editorial Members Dr. Chee Kyun Ng Ph.D Department of Computer and Communication Systems, Faculty of Engineering,Universiti Putra Malaysia,UPMSerdang, 43400 Selangor,Malaysia. Dr. Simon SEE Ph.D Chief Technologist and Technical Director at Oracle Corporation, Associate Professor (Adjunct) at Nanyang Technological University Professor (Adjunct) at ShangaiJiaotong University, 27 West Coast Rise #08-12,Singapore 127470 Dr. sc.agr. Horst Juergen SCHWARTZ Ph.D, Humboldt-University of Berlin,Faculty of Agriculture and Horticulture,Asternplatz 2a, D-12203 Berlin,Germany Dr. Marco L. BianchiniPh.D Italian National Research Council; IBAF-CNR,Via Salaria km 29.300, 00015 MonterotondoScalo (RM),Italy Dr. NijadKabbara Ph.D Marine Research Centre / Remote Sensing Centre/ National Council for Scientific Research, P. O. Box: 189 Jounieh,Lebanon Dr. Aaron Solomon Ph.D Department of Computer Science, National Chi Nan University,No. 303, University Road,Puli Town, Nantou County 54561,Taiwan Dr. Arthanariee. A. M M.Sc.,M.Phil.,M.S.,Ph.D Director - Bharathidasan School of Computer Applications, Ellispettai, Erode, Tamil Nadu,India Dr. Takaharu KAMEOKA, Ph.D Professor, Laboratory of Food, Environmental & Cultural Informatics Division of Sustainable Resource Sciences, Graduate School of Bioresources,Mie University, 1577 Kurimamachiya-cho, Tsu, Mie, 514-8507, Japan Dr. M. Sivakumar M.C.A.,ITIL.,PRINCE2.,ISTQB.,OCP.,ICP. Ph.D. Project Manager - Software,Applied Materials,1a park lane,cranford,UK Dr. Bulent AcmaPh.D Anadolu University, Department of Economics,Unit of Southeastern Anatolia Project(GAP),26470 Eskisehir,TURKEY Dr. Selvanathan Arumugam Ph.D Research Scientist, Department of Chemistry, University of Georgia, GA-30602,USA. Dr. S.Prasath Ph.D Assistant Professor, Department of Computer Science, Nandha Arts & Science College, Erode , Tamil Nadu, India

INTERNATIONAL JOURNAL OF INNOVATIVE TECHNOLOGY AND CREATIVE ENGINEERING (ISSN:2045-8711) VOL.10 NO 6 JUNE 2020

Review Board Members Mr. Rajaram Venkataraman Chief Executive Officer, Vel Tech TBI || Convener, FICCI TN State Technology Panel || Founder, Navya Insights || President, SPIN Chennai 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, Australia Dr. Zhiming Yang MD., Ph. D. Department of Radiation Oncology and Molecular Radiation Science,1550 Orleans Street Rm 441, Baltimore MD, 21231,USA Dr. Jifeng Wang Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign Urbana, Illinois, 61801, USA Dr. Giuseppe Baldacchini ENEA - Frascati Research Center, Via Enrico Fermi 45 - P.O. Box 65,00044 Frascati, Roma, ITALY. Dr. MutamedTurkiNayefKhatib Assistant Professor of Telecommunication Engineering,Head of Telecommunication Engineering Department,Palestine Technical University (Kadoorie), TulKarm, PALESTINE. Dr.P.UmaMaheswari Prof &Head,Depaartment of CSE/IT, INFO Institute of Engineering,Coimbatore. Dr. T. Christopher, Ph.D., Assistant Professor &Head,Department of Computer Science,Government Arts College(Autonomous),Udumalpet, India. Dr. T. DEVI Ph.D. Engg. (Warwick, UK), Head,Department of Computer Applications,Bharathiar University,Coimbatore-641 046, India. 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 Visiting Scholar at INSEAD,INSEAD Social Innovation Centre,Boulevard de Constance,77305 Fontainebleau - France Y. BenalYurtlu Assist. Prof. OndokuzMayis University 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 SchoolRuaItapeva, 474 (8° andar),01332-000, São Paulo (SP), Brazil Dr. Lucy M. Brown, Ph.D. Texas State University,601 University Drive,School of Journalism and Mass Communication,OM330B,San Marcos, TX 78666 JavadRobati Crop Production Departement,University of Maragheh,Golshahr,Maragheh,Iran VineshSukumar (PhD, MBA) Product Engineering Segment Manager, Imaging Products, Aptina Imaging Inc. Dr. Binod Kumar PhD(CS), M.Phil.(CS), MIAENG,MIEEE HOD & Associate Professor, IT Dept, Medi-Caps Inst. of Science & Tech.(MIST),Indore, India Dr. S. B. Warkad Associate Professor, Department of Electrical Engineering, Priyadarshini College of Engineering, Nagpur, India

INTERNATIONAL JOURNAL OF INNOVATIVE TECHNOLOGY AND CREATIVE ENGINEERING (ISSN:2045-8711) VOL.10 NO 6 JUNE 2020 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 6 JUNE 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 Muhammad Javed Centre for Next Generation Localisation, School of Computing, Dublin City University, Dublin 9, Ireland

INTERNATIONAL JOURNAL OF INNOVATIVE TECHNOLOGY AND CREATIVE ENGINEERING (ISSN:2045-8711) VOL.10 NO 6 JUNE 2020 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 6 JUNE 2020

Contents NUMERICAL COMPUTATION OF INTUITIONISTIC FUZZY ASSIGNMENT PROBLEM WITH SIMILARITY MEASURES ……………………..…...[ 813]

INTERNATIONAL JOURNAL OF INNOVATIVE TECHNOLOGY AND CREATIVE ENGINEERING (ISSN:2045-8711) VOL.10 NO.6 JUNE 2020

NUMERICAL COMPUTATION OF INTUITIONISTIC FUZZY ASSIGNMENT PROBLEM WITH SIMILARITY MEASURES C.Praveenkumar #1, V.Pachamuthu +2, A.Sathishkumar *3 #,+ Department

of Mathematics, Bharathidasan College of Arts and Science, Erode, Tamilnadu, India 1praveenmukunth@gmail.com, 2pachamuthu1729@gmail.com

*Department

of Mathematics, Nandha Arts & Science College, Erode, Tamilnadu, India 3sathishattssa@gmail.com

Abstract- In this paper, we focus on the solution of intuitionistic fuzzy assignment(IFA) problem, Similarity measures model also considered into account. Implement the consideration model to IFA problem. In this paper determining the IFA composite relative degree of similarity đ?&#x2019;&#x2026;đ?&#x2019;&#x160;đ?&#x2019;&#x2039; . The concept of the score function has also been used to find the composite relative similarity degree model. The numerical example are illustrated to proposed method to obtain the optimal solution

Keywords: Intuitionistic fuzzy assignment problem, Intuitionistic Fuzzy number, Score function, Similarity measures of IFS. I. INTRODUCTION Assignment problem in operation research is used planetary in solving real world problems. An assignment problem is an main role in the assigning of persons to jobs or classes to rooms or operators to machines or drivers to trucks, trucks to routes or problems to research terms etc. Many researchers have worked on the concept of similarity measures of fuzzy sets. Chen M.S, investigate the similarity measures of fuzzy sets. Which are depends on the geometric model and set theoretic approach and matching function similarity measures of Intuitionistic fuzzy sets has been studied and developed by many authors chen M.S (1988) and Chen M.S (1955) introduced a matching function to calculate the degree of similarity between fuzzy sets. An Intuitionistic fuzzy set(IFS) A in X is defined as the following form, A = {x, ÎąA (x), Î˛A (x) >/xĎľX} where the functions ÎąA : X â&#x2020;&#x2019; [0,1] and Î˛A : X â&#x2020;&#x2019; [0,1]. Define the degree of membership and the degree of non-membership of element xĎľX respectively and for every xĎľX, 0 â&#x2030;¤ ÎąA (x) + Î˛A (x) â&#x2030;¤ 1 ; Î˛A (x) = 1 â&#x2C6;&#x2019; ÎąA (x). The value of, Ď&#x20AC;A (x) = 1 â&#x2C6;&#x2019; ÎąA (x) â&#x2C6;&#x2019; Î˛A (x) is called the

813

degree of non-determinacy or uncertainty of the element xĎľX to the intuitionistic fuzzy set A. II. PRELIMINARIES Similarity measures of intuitionistic fuzzy set: Let S:Ď&#x2020;(X)2 â&#x2020;&#x2019; [0,1], then the degree of similarity between Aâ&#x2C6;&#x2C6; Ď&#x2020;(X) and Bâ&#x2C6;&#x2C6; Ď&#x2020;(X) is defined as S(A,B) which satisfies the following conditions: 1. 0â&#x2030;¤ S(A, B) â&#x2030;¤1 2. S(A,B) = 1 iff A = B 3. S(A,B) = S(B,A) 4. S(A,C) â&#x2030;¤ S(A,B) iff S(A,C) â&#x2030;¤ S(B,C) if Aâ&#x2030;¤ B â&#x2030;¤ C Similarity measures based on matching function: Let A â&#x2C6;&#x2C6; Ď&#x2020;(X) and B â&#x2C6;&#x2C6; Ď&#x2020;(X), then the degree of similarity of A and B has been defined on matching function as, S(A, B) =

â&#x2C6;&#x2018;nj=1 ÎąA (xj )ÎąB (xj ) + Î˛A (xj )Î˛B (xj ) + Ď&#x20AC;A (xj )Ď&#x20AC;pi (xj ) 2

max {(ÎąA (xj ) + Î˛A (xj ) + Ď&#x20AC;A (xj ) ) , (ÎąB (xj ) + Î˛B (xj ) + Ď&#x20AC;B (xj ) )}

Considering the weight Wj of each element xj â&#x2C6;&#x2C6; X we get, S(A, B) =

â&#x2C6;&#x2018;nj=1 Wj [ÎąA (xj )ÎąB (xj ) + Î˛A (xj )Î˛B (xj ) + Ď&#x20AC;A (xj )Ď&#x20AC;pi (xj )] 2

max {Wj (ÎąA (xj ) + Î˛A (xj ) + Ď&#x20AC;A (xj ) ) , Wj (ÎąB (xj ) + Î˛B (xj ) + Ď&#x20AC;B (xj ) )}

III. MATHEMATICAL MODELS BASED ON SIMILARITY MEASURES Assume there are m persons and n jobs. Each job must be done by exactly one person to one job. MODEL I: The crisp assignment problem, the cost Cij is usually deterministic but real life situations it cannot be practical to know the exact values of these costs. In such an uncertain situations, instead of exact value of

www.ijitce.co.uk

INTERNATIONAL JOURNAL OF INNOVATIVE TECHNOLOGY AND CREATIVE ENGINEERING (ISSN:2045-8711) VOL.10 NO.6 JUNE 2020 costs, we apply the form of composite relative degree (αij + βij − 1) xij ≤ 0 (dij ) of similarity to ideal solution, we can replace Cij by αij xij ≥ βij xij dij in the crisp assignment problem we get, βij xij ≥ 0 m

Max Z = ∑ ∑ dij xij _____________________________(1) i=1 j=1

Subject to, m

∑ xij = 1

j = 1,2 … … … … … … . m

i=1 m

∑ xij = 1

i = 1,2 … … … … … … . m

i=1

xij = 0 or 1 i, j = 1,2 … … … … … . m MODEL II: The cost cij represent uncertain data like interval number,trapezoidal or triangular fuzzy or intuitionistic fuzzy number. In this chapter the cost has been considered to be IFN denoted by c̃ij which involves the positive and the negative evidence for thr membership of an element in a set. The cost of person i doing the job j is considered as an intuitionistic fuzzy number c̃ij = {(αij , βij )} , i,j = 1,2………m. Here αij represents the degree of acceptance and βij represent the degree of rejection of the cost of doing jth job by the ith person. The problem is to find the composite relative degree dij to ideal solution denote ith person to jth job with the costs cij in the form of IFN. Now we replace c̃ij by c̃ij = {(αij , βij )} then the crisp assignment problem becomes, m Max Z = ∑m i=1 ∑j=1{(αij , βij )}x ij _____________________(2) The objective is to maximize acceptance degree αij and minimize the rejection degree βij . The objective function (2) can be written as, m

SOLUTION PROCEDURE FOR FINDING OPTIMAL SOLUTION BY USING SIMILARITY MEASURES : For an IFAP, Consider P = {p1 , p1 … … … . pm } be a set of alternatives for a row or column in the cost matrix and consider c be an attribute describing the selection alternative. The characteristics of the alternative pi are represented by IFS as, pi = {(c, αpi (c), βpi (c))} , i =1,2…………m Where αpi (c) indicates the degree that the alternative pi satisfies the attribute c and βpi (c) indicates the degree that the alternative pi does not satisfy the attribute c and αpi ϵ [0,1] and βpi ϵ [0,1] , αpi (c) + βpi (c) ≤ 1. Step 1 : Let πpi (c) = 1 − αpi (c) − βpi (c) for all i = 1,2……… m find the positive ideal solution and negative ideal solution depends on IFN is defined as follows: p+ = {αp+ (c), βp+ (c)}____________________ (9) p− = {c, αp− (c), βp− (c)}__________________ (10) Where, αp+ (c) = maxi {αpi (c)} , βp+ (c) = αp− (c) = βp− (c) =

min i

{βpi (c)}_________ (11) min , i {αpi (c)} max i {βpi (c)}__________(12)

Step 2 : Next we calculate the degree of similarity of the positive ideals intuitionistic fuzzy set p+ and the alternative pi and negative ideal intitionistic fuzzy set p− and the alternative pi by using the similarity measures of IFS’s as follows : s(p+ , pi ) αp+ (c)αpi (c) + βp+ (c)βpi (c) + πp+ (c)πpi (c) ___ (13) max{(αp+ (c)2 + βp+ (c)2 + πp+ (c)2 ), (αpi (c)2 + βpi (c)2 + πpi (c)2 )} s(p− , pi ) αp− (c)αpi (c) + βp− (c)βpi (c) + πp− (c)πpi (c) = ___ (14) (c) max{(αp− 2 + βp− (c)2 + πp− (c)2 ), (αpi (c)2 + βpi (c)2 + πpi (c)2 )} =

Max Z1 = ∑ ∑ αij xij i=1 j=1 m m

Max Z2 = ∑ ∑ βij xij i=1 j=1

Hence the intuitionistic fuzzy assignment problem becomes, m Max Z1 = ∑m i=1 ∑j=1 αij x ij _______________(3)

For all i = 1,2……..m and j = 1,2……… m Step 3: Next we calculate the relative similarity measures di based on (13),(14) corresponding to the alternatives Ai as follows : di =

m Max Z2 = ∑m i=1 ∑j=1 βij x ij _______________(4)

Subject to, (αij + βij − 1) xij ≤ 0_____________________(5) αij xij ≥ βij xij ___________________(6) βij xij ≥ 0______________________(7) MODEL III : The above multi objective model for the intuitionistic fuzzy assignment problem can be written as a single objective function is as follows : m Max Z = ∑m i=1 ∑j=1(αij − βij )x ij ____________8)

s(p+ , pi ) , i = 1,2 … … . . m _________(15) − i ) + s(p , pi )

s(p+ , p

Step 4: continue step 1 to step 3 for remaining columns of the cost matrix and we find the relative similarity measure di corresponding to the alternative pi for these columns. That is, the job with respect to the persons. Step 5: Next we have to the form the matrix R1 where R1 = [rij ]m×m where rij is the relative similarity measures representing the jth person prefer to the ith job.

Subject to,

814

www.ijitce.co.uk

INTERNATIONAL JOURNAL OF INNOVATIVE TECHNOLOGY AND CREATIVE ENGINEERING (ISSN:2045-8711) VOL.10 NO.6 JUNE 2020 Step 6: we find the relative similarity measure dj (0.1)(0.3) + (0.7)(0.4) + (0.2)(0.3) corresponding to the alternative pi for these rows.The = 2 + 0.72 + 0.22 ) , (0.32 + 0.42 + 0.32 )} max{(0.1 person with respect to the jobs and form the matrix R 2 0.37 0.37 where R 2 = [sij ]m×m where sij is the relative similarity = = max{0.54,0.34} 0.54 measures representing the ith job is suitable for the jth − s(p , p1 )=0.685 person. s(p+ , p1 ) Step 7: Next we form the composite matrix [r11 ] = + s(p , p1 ) + s(p− , p1 ) com( R1 R 2 ) = (rij sij )m×m = (dij )m×m . This matrix 0.714 0.714 represent the composite relative degree of similarity of = = 0.714 + 0.685 1.399 preference or suitability to ith job to the jth person or jth = 0.51 person is chosen the ith job. s(p+ , p2 ) Step 8: The composite matrix considered as a initial αp+ (c)αp2 (c) + βp+ (c)βp2 (c) + πp+ (c)πp2 (c) table of assignment problem in the maximization form = max{(αp+ (c)2 + βp+ (c)2 + πp+ (c)2 ), (αp2 (c)2 + βp2 (c)2 + πp2 (c)2 )} (model 3) and using the Hungarian method to find the optimal assignment and the composite relative degree of (0.6)(0.1) + (0.2)(0.7) + (0.3)(0.2) similarity are maximizes. = max{(0.62 + 0.22 + 0.32 ) , (0.12 + 0.72 + 0.22 )} 0.26 0.26 NUMERICAL COMPUTATION: = = max{0.49,0.54} 0.54 The given IFAP with row representing three s(p+ , p2 )= 0.481 machines M1,M2,M3 and columns representing the s(p− , p2 ) three jobs J1,J2,J3 . The cost matrix are TIFN. To find αp− (c)αp2 (c) + βp− (c)βp2 (c) + πp− (c)πp2 (c) = the optimal assignment of jobs to machines. max{(αp− (c)2 + βp− (c)2 + πp− (c)2 ), (αp2 (c)2 + βp2 (c)2 + πp2 (c)2 )} jobs J1 J2 J3 Machines (0.1)(0.1) + (0.7)(0.7) + (0.2)(0.2) M1 (0.3,0.4) (0.5,0.1) (0.4,0.1) = max{(0.12 + 0.72 + 0.22 ) , (0.12 + 0.72 + 0.22 )} M2 (0.1,0,7) (0.7,0.1) (0.5,0.3) 0.54 0.54 M3 (0.6,0.2) (0.2,0.4) (0.3,0.2) = = max{0.54,0.54} 0.54 Solution : − s(p , p2 )= 1 Find the values of s(p+ , Mj ), s(p− , Mj ) and values of s(p+ , p2 ) [r21 ] = dj in R1 for machines with respect to jobs (column + s(p , p2 ) + s(p− , p2 ) wise).In first column we find, 0.481 = πpi (c) = 1 − αpi (c) − βpi (c) 0.481 + 1 0.481 i.e, πp1 (c) = 1 − αp1 (c) − βp1 (c) = 1 − 0.3 − 0.4 = 0.3 = 1.481 πp2 (c) = 1 − αp2 (c) − βp2 (c) = 1 − 0.1 − 0.7 = 0.2 = 0.325 π (c) = 1 − α (c) − β (c) = 1 − 0.6 − 0.2 = 0.2 p3

Here αp+ (c) = αp− (c) =

π (c) = 0.3 p+

and

s(p+ , p3 )

πp− (c) = 0.2

max min i {βpi (c)} = 0.2 i {αpi (c)} =0.6,βp+ (c) = min max i {αpi (c)} = 0.1, βp− (c) = i {βpi (c)} = 0.7

s(p+ , p1 )

αp+ (c)αp3 (c) + βp+ (c)βp3 (c) + πp+ (c)πp3 (c) max{(αp+ (c)2 + βp+ (c)2 + πp+ (c)2 ), (αp3 (c)2 + βp3 (c)2 + πp3 (c)2 )}

(0.6)(0.6) + (0.2)(0.2) + (0.3)(0.2) max{(0.62 + 0.22 + 0.32 ) , (0.62 + 0.22 + 0.22 )} = 2 2 2 2 2 2 (c) (c) (c) (c) (c) (c) + + + max{(αp + βp + πp ), (αp1 + β p1 + π p1 )} 0.46 = max{0.49,0.44} 0.46 (0.6)(0.3) + (0.2)(0.4) + (0.3)(0.3) = = 2 2 2 2 2 2 0.49 max{(0.6 + 0.2 + 0.3 ) , (0.3 + 0.4 + 0.3 )} s(p+ , p3 )= 0.939 0.35 0.35 = = s(p− , p3 ) max{0.49,0.34} 0.49 αp− (c)αp3 (c) + βp− (c)βp3 (c) + πp− (c)πp3 (c) s(p+ , p1 )= 0.714 = =

αp+ (c)αp1 (c) + βp+ (c)βp1 (c) + πp+ (c)πp1 (c)

max{(αp− (c)2 + βp− (c)2 + πp− (c)2 ), (αp3 (c)2 + βp3 (c)2 + πp3 (c)2 )}

s(p− , p1 ) =

αp− (c)αp1 (c) + βp− (c)βp1 (c) + πp− (c)πp1 (c) max{(αp− (c)2 + βp− (c)2 + πp− (c)2 ), (αp1 (c)2 + βp1 (c)2 + πp1 (c)2 )}

815

(0.1)(0.6) + (0.7)(0.2) + (0.2)(0.2) max{(0.12 + 0.72 + 0.22 ) , (0.62 + 0.22 + 0.22 )}

www.ijitce.co.uk

INTERNATIONAL JOURNAL OF INNOVATIVE TECHNOLOGY AND CREATIVE ENGINEERING (ISSN:2045-8711) VOL.10 NO.6 JUNE 2020 0.24 0.58 0.58 = = = max{0.54,0.44} max{0.66,0.54} 0.66 0.24 = 0.879 = s(p− , p2 ) 0.54 − αp− (c)αp2 (c) + βp− (c)βp2 (c) + πp− (c)πp2 (c) s(p , p3 ) = 0.444 = s(p+ , p3 ) max{(αp− (c)2 + βp− (c)2 + πp− (c)2 ), (αp2 (c)2 + βp2 (c)2 + πp2 (c)2 )} [r31 ] = + − s(p , p3 ) + s(p , p3 ) 0.939 (0.2)(0.7) + (0.4)(0.1) + (0.2)(0.2) = = 0.939 + 0.444 max{(0.22 + 0.42 + 0.22 ) , (0.72 + 0.12 + 0.22 )} 0.939 0.22 0.22 = = 0.679 = = 1.383 max{0.24,0.54} 0.54 In second column we find, = 0.407 πpi (c) = 1 − αpi (c) − βpi (c) s(p+ , p2 ) [r ] = 22 i.e, πp1 (c) = 1 − αp1 (c) − βp1 (c) = 1 − 0.5 − 0.1 = 0.4 s(p+ , p2 ) + s(p− , p2 ) πp2 (c) = 1 − αp2 (c) − βp2 (c) = 1 − 0.7 − 0.1 = 0.2 0.879 0.879 = = = 0.684 πp3 (c) = 1 − αp3 (c) − βp3 (c) = 1 − 0.2 − 0.4 = 0.4 0.879 + 0.407 1.286 + s(p , p3 ) Here πp+ (c) = 0.4 and πp− (c) = 0.2 αp+ (c) =

max i

{αpi (c)} =0.7

, βp+ (c) =

min i

{βpi (c)}

= 0.1 αp− (c) =

min i

{αpi (c)} = 0.2

αp+ (c)αp3 (c) + βp+ (c)βp3 (c) + πp+ (c)πp3 (c)

max{(αp+ (c)2 + βp+ (c)2 + πp+ (c)2 ), (αp3 (c)2 + βp3 (c)2 + πp3 (c)2 )}

βp− (c) =

(0.7)(0.2) + (0.1)(0.4) + (0.4)(0.4) max{(0.72 + 0.12 + 0.42 ) , (0.22 + 0.42 + 0.42 )} s(p+ , p1 ) 0.52 0.52 = = αp+ (c)αp1 (c) + βp+ (c)βp1 (c) + πp+ (c)πp1 (c) max{0.66,0.36} 0.66 = max{(αp+ (c)2 + βp+ (c)2 + πp+ (c)2 ), (αp1 (c)2 + βp1 (c)2 + πp1 (c)2 )}= 0.515 max i

{βpi (c)} = 0.4

s(p− , p3 )

(0.7)(0.5) + (0.1)(0.1) + (0.4)(0.4) max{(0.72 + 0.12 + 0.42 ) , (0.52 + 0.12 + 0.42 )} 0.52 = max{0.66,0.42} =

αp− (c)αp3 (c) + βp− (c)βp3 (c) + πp− (c)πp3 (c) max{(αp− (c)2 + βp− (c)2 + πp− (c)2 ), (αp3 (c)2 + βp3 (c)2 + πp3 (c)2 )}

(0.2)(0.2) + (0.4)(0.4) + (0.2)(0.4) max{(0.22 + 0.42 + 0.22 ) , (0.22 + 0.42 + 0.42 )} = = 0.788 0.66 0.28 0.24 s(p− , p1 ) = = max{0.24,0.36} 0.36 − − − (c)α (c) (c)β (c) (c)π (c) αp + βp + πp p1 p1 p1 = 2 2 2 2 2 2 = 0.778 max{(αp− (c) + βp− (c) + πp− (c) ), (αp1 (c) + βp1 (c) + πp1 (c) )} s(p+ , p3 ) [r32 ] = s(p+ , p3 ) + s(p− , p3 ) (0.2)(0.5) + (0.4)(0.1) + (0.2)(0.4) = 0.515 0.515 max{(0.22 + 0.42 + 0.22 ) , (0.52 + 0.12 + 0.42 )} = = 0.515 + 0.778 1.286 0.22 0.22 = = = 0.398 max{0.24,0.42} 0.42 In third column we find, = 0.524 πpi (c) = 1 − αpi (c) − βpi (c) s(p+ , p1 ) [r12 ] = i.e, πp1 (c) = 1 − αp1 (c) − βp1 (c) = 1 − 0.4 − 0.1 = 0.5 s(p+ , p1 ) + s(p− , p1 ) πp2 (c) = 1 − αp2 (c) − βp2 (c) = 1 − 0.5 − 0.3 = 0.2 0.788 0.788 = = πp3 (c) = 1 − αp3 (c) − βp3 (c) = 1 − 0.3 − 0.2 = 0.5 0.788 + 0.524 1.312 = 0.601 Here πp+ (c) = 0.5 and πp− (c) = 0.2 =

0.52

s(p+ , p2 ) =

αp+ (c) =

αp+ (c)αp2 (c) + βp+ (c)βp2 (c) + πp+ (c)πp2 (c)

max{(αp+ (c)2 + βp+ (c)2 + πp+ (c)2 ), (αp2 (c)2 + βp2 (c)2 + πp2 (c)2 )}αp− (c) = s(p+ , p1 )

(0.7)(0.7) + (0.1)(0.1) + (0.4)(0.2) max{(0.72 + 0.12 + 0.42 ) , (0.72 + 0.12 + 0.22 )}

816

max min i {βpi (c)} = 0.1 i {αpi (c)} =0.5,βp+ (c) = min max i {αpi (c)} = 0.3, βp− (c) = i {βpi (c)} = 0.3

αp+ (c)αp1 (c) + βp+ (c)βp1 (c) + πp+ (c)πp1 (c) max{(αp+ (c)2 + βp+ (c)2 + πp+ (c)2 ), (αp1 (c)2 + βp1 (c)2 + πp1 (c)2 )}

www.ijitce.co.uk

INTERNATIONAL JOURNAL OF INNOVATIVE TECHNOLOGY AND CREATIVE ENGINEERING (ISSN:2045-8711) VOL.10 NO.6 JUNE 2020 s(p− , p3 )

(0.5)(0.4) + (0.1)(0.1) + (0.5)(0.5) max{(0.52 + 0.12 + 0.52 ) , (0.42 + 0.12 + 0.52 )} 0.46 0.46 = = max{0.51,0.42} 0.51 = 0.902

αp− (c)αp3 (c) + βp− (c)βp3 (c) + πp− (c)πp3 (c) max{(αp− (c)2 + βp− (c)2 + πp− (c)2 ), (αp3 (c)2 + βp3 (c)2 + πp3 (c)2 )}

(0.3)(0.3) + (0.3)(0.2) + (0.2)(0.5) max{(0.32 + 0.32 + 0.22 ) , (0.32 + 0.22 + 0.52 )} 0.25 0.25 s(p− , p1 ) = = αp− (c)αp1 (c) + βp− (c)βp1 (c) + πp− (c)πp1 (c) max{0.22,0.38} 0.38 = max{(αp− (c)2 + βp− (c)2 + πp− (c)2 ), (αp1 (c)2 + βp1 (c)2 + πp1 (c)2 )}= 0.658 =

s(p+ , p3 ) s(p+ , p3 ) + s(p− , p3 ) 0.824 = 0.824 + 0.658 0.824 = 1.482 = 0.556

(0.3)(0.4) + (0.3)(0.1) + (0.2)(0.5) max{(0.32 + 0.32 + 0.22 ) , (0.42 + 0.12 + 0.52 )} 0.25 0.25 = = max{0.22,0.42} 0.42 = 0.595 s(p+ , p1 ) [r13 ] = s(p+ , p1 ) + s(p− , p1 ) 0.902 0.902 = = 0.902 + 0.595 1.497 =0.603 =

[r33 ] =

s(p+ , p2 ) =

αp+ (c)αp2 (c) + βp+ (c)βp2 (c) + πp+ (c)πp2 (c) max{(αp+ (c)2 + βp+ (c)2 + πp+ (c)2 ), (αp2 (c)2 + βp2 (c)2 + πp2 (c)2 )}

(0.5)(0.5) + (0.1)(0.3) + (0.5)(0.2) max{(0.52 + 0.12 + 0.52 ) , (0.52 + 0.32 + 0.22 )} 0.38 0.38 = = max{0.51,0.38} 0.51 = 0.745 =

s(p− , p2 ) =

αp− (c)αp2 (c) + βp− (c)βp2 (c) + πp− (c)πp2 (c) max{(αp− (c)2 + βp− (c)2 + πp− (c)2 ), (αp2 (c)2 + βp2 (c)2 + πp2 (c)2 )}

(0.3)(0.5) + (0.3)(0.3) + (0.2)(0.2) max{(0.32 + 0.32 + 0.22 ) , (0.52 + 0.32 + 0.22 )} 0.28 0.28 = = max{0.22,0.38} 0.38 = 0.737 s(p+ , p2 ) [r32 ] = s(p+ , p2 ) + s(p− , p2 ) 0.745 0.745 = = 0.745 + 0.737 1.482 = 0.503 =

Values of s(p+ , MJ ) for machines with respect to the jobs (column wise) s(p+ , Mj )

0.714

0.788

0.902

0.481

0.879

0.745

0.939

0.515

0.824

Values of s(p− , MJ ) for machines with respect to the jobs (column wise) s(p− , Mj )

0.685

0.524

0.595

0.407

0.737

0.444

0.778

0.658

s(p+ , p3 ) =

αp+ (c)αp3 (c) + βp+ (c)βp3 (c) + πp+ (c)πp3 (c) max{(αp+ (c)2 + βp+ (c)2 + πp+ (c)2 ), (αp3 (c)2 + βp3 (c)2 + πp3 (c)2 )}

Matrix R1 containing the values of dj for machines with respect to the jobs (column wise)

(0.5)(0.3) + (0.1)(0.2) + (0.5)(0.5) = max{(0.52 + 0.12 + 0.52 ) , (0.32 + 0.22 + 0.52 )} 0.46 0.46 = = max{0.51,0.38} 0.51 = 0.824

dj or rij

817

www.ijitce.co.uk

INTERNATIONAL JOURNAL OF INNOVATIVE TECHNOLOGY AND CREATIVE ENGINEERING (ISSN:2045-8711) VOL.10 NO.6 JUNE 2020 M1 M2

0.51 0.325

0.601 0.684

s(p− , p2 )

0.603

αp− (c)αp2 (c) + βp− (c)βp2 (c) + πp− (c)πp2 (c) max{(αp− (c)2 + βp− (c)2 + πp− (c)2 ), (αp2 (c)2 + βp2 (c)2 + πp2 (c)2 )}

0.503 (0.3)(0.5) + (0.4)(0.1) + (0.3)(0.4) max{(0.32 + 0.42 + 0.32 ) , (0.52 + 0.12 + 0.42 )} 0.31 0.31 = = max{0.34,0.42} 0.42 = 0.738 s(p+ , p2 ) [s12 ] = s(p+ , p2 ) + s(p− , p2 ) 0.902 0.902 = = 0.902 + 0.738 1.64 = 0.55 =

0.679

0.398

0.556

Find the values of s(p+ , Ji ), s(p− , Ji ) and values of di in R 2 for jobs with respect to machines (row wise). In first row we find, πpi (c) = 1 − αpi (c) − βpi (c) i.e, πp1 (c) = 1 − αp1 (c) − βp1 (c) = 1 − 0.3 − 0.4 = 0.3

πp2 (c) = 1 − αp2 (c) − βp2 (c) = 1 − 0.5 − 0.1 = 0.4 πp3 (c) = 1 − αp3 (c) − βp3 (c) = 1 − 0.4 − 0.1 = 0.5 Here πp+ (c) = 0.5 and πp− (c) = 0.3 αp+ (c) =

max min i {βpi (c)} = 0.1 i {αpi (c)} =0.5, βp+ (c) = min max i {αpi (c)} = 0.3, βp− (c) = i {βpi (c)} = 0.4

s(p+ , p3 ) =

αp+ (c)αp3 (c) + βp+ (c)βp3 (c) + πp+ (c)πp3 (c) max{(αp+ (c)2 + βp+ (c)2 + πp+ (c)2 ), (αp3 (c)2 + βp3 (c)2 + πp3 (c)2 )}

(0.5)(0.4) + (0.1)(0.1) + (0.5)(0.5) 2 + 0.12 + 0.52 ) , (0.42 + 0.12 + 0.52 )} max{(0.5 s(p+ , p1 ) 0.46 0.46 αp+ (c)αp1 (c) + βp+ (c)βp1 (c) + πp+ (c)πp1 (c) = = = max{(αp+ (c)2 + βp+ (c)2 + πp+ (c)2 ), (αp1 (c)2 + βp1 (c)2 + πp1 (c)2 )} max{0.51,0.42} 0.51 = 0.902 αp− (c) =

s(p− , p3 )

(0.5)(0.3) + (0.1)(0.4) + (0.5)(0.3) = max{(0.52 + 0.12 + 0.52 ) , (0.32 + 0.42 + 0.32 )} 0.34 0.34 = = max{0.51,0.34} 0.51 = 0.667

αp− (c)αp3 (c) + βp− (c)βp3 (c) + πp− (c)πp3 (c) max{(αp− (c)2 + βp− (c)2 + πp− (c)2 ), (αp3 (c)2 + βp3 (c)2 + πp3 (c)2 )}

(0.3)(0.4) + (0.4)(0.1) + (0.3)(0.5) max{(0.32 + 0.42 + 0.32 ) , (0.42 + 0.12 + 0.52 )} s(p− , p1 ) 0.26 0.26 αp− (c)αp1 (c) + βp− (c)βp1 (c) + πp− (c)πp1 (c) = = = max{(αp− (c)2 + βp− (c)2 + πp− (c)2 ), (αp1 (c)2 + βp1 (c)2 + πp1 (c)2 )} max{0.34,0.42} 0.42 = 0.619 s(p+ , p3 ) (0.3)(0.3) + (0.4)(0.4) + (0.3)(0.3) [s13 ] = = s(p+ , p3 ) + s(p− , p3 ) max{(0.32 + 0.42 + 0.32 ) , (0.32 + 0.42 + 0.32 )} 0.902 0.902 0.34 0.34 = = = = 0.902 + 0.619 1.521 max{0.34,0.34} 0.34 =0.593 =1 In second row we find, s(p+ , p1 ) πpi (c) = 1 − αpi (c) − βpi (c) [s11 ] = s(p+ , p1 ) + s(p− , p1 ) (c) i.e, π = 1 − αp1 (c) − βp1 (c) = 1 − 0.1 − 0.7 = 0.2 p1 0.667 0.667 = = = 0.4 πp2 (c) = 1 − αp2 (c) − βp2 (c) = 1 − 0.7 − 0.1 = 0.2 0.667 + 1 1.667 + s(p , p2 ) πp3 (c) = 1 − αp3 (c) − βp3 (c) = 1 − 0.5 − 0.3 = 0.2 αp+ (c)αp2 (c) + βp+ (c)βp2 (c) + πp+ (c)πp2 (c) Here πp+ (c) = 0.2 and πp− (c) = 0.2 = =

max{(αp+ (c)2 + βp+ (c)2 + πp+ (c)2 ), (αp2 (c)2 + βp2 (c)2 + πp2 (c)2 )}

αp+ (c) =

max i

{αpi (c)} =0.7

, βp+ (c) =

min i

{βpi (c)}

= 0.1

(0.5)(0.5) + (0.1)(0.1) + (0.5)(0.4) = max{(0.52 + 0.12 + 0.52 ) , (0.52 + 0.12 + 0.42 )} 0.46 0.46 = = max{0.51,0.42} 0.51 = 0.902

αp− (c) = max i

min i

{αpi (c)} = 0.1

βp− (c) =

{βpi (c)} = 0.7

s(p+ , p1 ) =

818

αp+ (c)αp1 (c) + βp+ (c)βp1 (c) + πp+ (c)πp1 (c) max{(αp+ (c)2 + βp+ (c)2 + πp+ (c)2 ), (αp1 (c)2 + βp1 (c)2 + πp1 (c)2 )}

www.ijitce.co.uk

INTERNATIONAL JOURNAL OF INNOVATIVE TECHNOLOGY AND CREATIVE ENGINEERING (ISSN:2045-8711) VOL.10 NO.6 JUNE 2020 s(p− , p3 )

(0.7)(0.1) + (0.1)(0.7) + (0.2)(0.2) max{(0.72 + 0.12 + 0.22 ) , (0.12 + 0.72 + 0.22 )} 0.18 0.18 = = max{0.54,0.54} 0.54 = 0.333

αp− (c)αp3 (c) + βp− (c)βp3 (c) + πp− (c)πp3 (c) max{(αp− (c)2 + βp− (c)2 + πp− (c)2 ), (αp3 (c)2 + βp3 (c)2 + πp3 (c)2 )}

(0.1)(0.5) + (0.7)(0.3) + (0.2)(0.2) max{(0.12 + 0.72 + 0.22 ) , (0.52 + 0.32 + 0.22 )} 0.3 0.3 s(p− , p1 ) = = αp− (c)αp1 (c) + βp− (c)βp1 (c) + πp− (c)πp1 (c) max{0.54,0.38} 0.54 = max{(αp− (c)2 + βp− (c)2 + πp− (c)2 ), (αp1 (c)2 + βp1 (c)2 + πp1 (c)2 )}= 0.556 s(p+ , p3 ) [s23 ] = (0.1)(0.1) + (0.7)(0.7) + (0.2)(0.2) s(p+ , p3 ) + s(p− , p3 ) = 2 2 2 2 2 2 0.778 0.778 max{(0.1 + 0.7 + 0.2 ) , (0.1 + 0.7 + 0.2 )} = = 0.54 0.54 0.778 + 0.556 1.334 = = =0.583 max{0.54,0.54} 0.54 In third row we find, =1 πpi (c) = 1 − αpi (c) − βpi (c) s(p+ , p1 ) [s21 ] = + − i.e, πp1 (c) = 1 − αp1 (c) − βp1 (c) = 1 − 0.6 − 0.2 = 0.2 s(p , p1 ) + s(p , p1 ) 0.333 0.333 πp2 (c) = 1 − αp2 (c) − βp2 (c) = 1 − 0.2 − 0.4 = 0.4 = = 0.333 + 1 1.333 πp3 (c) = 1 − αp3 (c) − βp3 (c) = 1 − 0.3 − 0.2 = 0.5 = 0.250 Here πp+ (c) = 0.5 and πp− (c) = 0.2 + =

s(p , p2 ) =

αp+ (c) =

αp+ (c)αp2 (c) + βp+ (c)βp2 (c) + πp+ (c)πp2 (c)

max{(αp+ (c)2 + βp+ (c)2 + πp+ (c)2 ), (αp2 (c)2 + βp2 (c)2 + πp2 (c)2 )}αp− (c) = s(p+ , p1 )

(0.7)(0.7) + (0.1)(0.1) + (0.2)(0.2) max{(0.72 + 0.12 + 0.22 ) , (0.72 + 0.12 + 0.22 )} 0.54 0.54 = = max{0.54,0.54} 0.54 =1

max i {αpi (c)} =0.6, min i {αpi (c)} = 0.2,

βp+ (c) = βp− (c) =

min i {βpi (c)} = 0.2 max i {βpi (c)} = 0.4

αp+ (c)αp1 (c) + βp+ (c)βp1 (c) + πp+ (c)πp1 (c) max{(αp+ (c)2 + βp+ (c)2 + πp+ (c)2 ), (αp1 (c)2 + βp1 (c)2 + πp1 (c)2 )}

(0.6)(0.6) + (0.2)(0.2) + (0.5)(0.4) max{(0.62 + 0.22 + 0.52 ) , (0.62 + 0.22 + 0.42 )} − 0.6 0.6 s(p , p2 ) = = αp− (c)αp2 (c) + βp− (c)βp2 (c) + πp− (c)πp2 (c) max{0.65,0.56} 0.65 = max{(αp− (c)2 + βp− (c)2 + πp− (c)2 ), (αp2 (c)2 + βp2 (c)2 + πp2 (c)2 )}= 0.923 =

s(p− , p1 )

(0.1)(0.7) + (0.7)(0.1) + (0.2)(0.2) max{(0.12 + 0.72 + 0.22 ) , (0.72 + 0.12 + 0.22 )} 0.18 0.18 = = max{0.54,0.54} 0.54 = 0.333 s(p+ , p2 ) [s22 ] = s(p+ , p2 ) + s(p− , p2 ) 1 1 = = 1 + 0.333 1.333 = 0.750

αp− (c)αp1 (c) + βp− (c)βp1 (c) + πp− (c)πp1 (c) max{(αp− (c)2 + βp− (c)2 + πp− (c)2 ), (αp1 (c)2 + βp1 (c)2 + πp1 (c)2 )}

(0.2)(0.6) + (0.4)(0.2) + (0.2)(0.4) max{(0.22 + 0.42 + 0.22 ) , (0.62 + 0.22 + 0.42 )} 0.28 0.28 = = max{0.24,0.56} 0.56 = 0.5 s(p+ , p1 ) [s31 ] = s(p+ , p1 ) + s(p− , p1 ) s(p+ , p3 ) 0.923 0.923 = = αp+ (c)αp3 (c) + βp+ (c)βp3 (c) + πp+ (c)πp3 (c) 0.923 + 0.5 1.423 = max{(αp+ (c)2 + βp+ (c)2 + πp+ (c)2 ), (αp3 (c)2 + βp3 (c)2 + πp3 (c)2 )} = 0.649 =

s(p+ , p2 )

(0.1)(0.5) + (0.7)(0.3) + (0.2)(0.2) max{(0.12 + 0.72 + 0.22 ) , (0.52 + 0.32 + 0.22 )} 0.42 0.42 = = max{0.54,0.38} 0.54 = 0.778

αp+ (c)αp2 (c) + βp+ (c)βp2 (c) + πp+ (c)πp2 (c) max{(α (c)2 + βp+ (c)2 + πp+ (c)2 ), (αp2 (c)2 + βp2 (c)2 + πp2 (c)2 )} p+

819

(0.6)(0.2) + (0.2)(0.4) + (0.5)(0.4) max{(0.62 + 0.22 + 0.52 ) , (0.22 + 0.42 + 0.42 )}

www.ijitce.co.uk

INTERNATIONAL JOURNAL OF INNOVATIVE TECHNOLOGY AND CREATIVE ENGINEERING (ISSN:2045-8711) VOL.10 NO.6 JUNE 2020 0.4 0.4 M2 0.333 1 0.778 = = max{0.65,0.36} 0.65 = 0.615 M3 0.923 0.615 0.723 s(p− , p2 ) =

αp− (c)αp2 (c) + βp− (c)βp2 (c) + πp− (c)πp2 (c) max{(αp− (c)2 + βp− (c)2 + πp− (c)2 ), (αp2 (c)2 + βp2 (c)2 + πp2 (c)2 )}

(0.3)(0.5) + (0.4)(0.1) + (0.3)(0.4) = max{(0.32 + 0.42 + 0.32 ) , (0.52 + 0.12 + 0.42 )} 0.28 0.28 = = = 0.778 max{0.65,0.36} 0.65 s(p+ , p2 ) [s32 ] = s(p+ , p2 ) + s(p− , p2 ) 0.615 = 0.615 + 0.778 0.615 = 1.393 = 0.441

s(p− , ji )

0.738

0.619

0.333

0.556

0.5

0.778

0.64

Matrix R2 containing the values of di for jobs with respect to the machines (row wise)

s(p+ , p3 ) =

Values of s(p− , ji ) for jobs with respect to machines (row wise)

αp+ (c)αp3 (c) + βp+ (c)βp3 (c) + πp+ (c)πp3 (c) max{(αp+ (c)2 + βp+ (c)2 + πp+ (c)2 ), (αp3 (c)2 + βp3 (c)2 + πp3 (c)2 )}

(0.6)(0.3) + (0.2)(0.2) + (0.5)(0.5) max{(0.62 + 0.22 + 0.52 ) , (0.32 + 0.22 + 0.52 )} 0.47 = max{0.65,0.38} 0.47 = 0.65 = 0.723

dij orsij

0.4

0.55

0.593

0.250

0.750

0.583

0.649

0.441

0.53

Next find the composite matrix representing the composite relative degree of similarity of jobs and machines. i.e., com( R1 R 2 ) = (rij sij )m×m = (dij )m×m

s(p− , p3 ) =

αp− (c)αp3 (c) + βp− (c)βp3 (c) + πp− (c)πp3 (c) max{(αp− (c)2 + βp− (c)2 + πp− (c)2 ), (αp3 (c)2 + βp3 (c)2 + πp3 (c)2 )}

(0.2)(0.3) + (0.4)(0.4) + (0.2)(0.5) max{(0.22 + 0.42 + 0.22 ) , (0.32 + 0.42 + 0.52 )} 0.32 = max{0.24,0.5} 0.32 = 0.5 = 0.64 s(p+ , p3 ) [s33 ] = s(p+ , p3 ) + s(p− , p3 ) 0.723 = 0.723 + 0.64 0.723 = = 0.530 1.363 =

0.667

0.902

0.204

0.331

0.358

0.124

0.513

0.293

0.441

0.176

0.295

Hence the optimal assignment is, M1 → J3 ; M2 → J2 ; M3 → J1 Hence the optimal solution is Z = 0.358 + 0.513 + 0.441 = 1.312 CONCLUSION In this paper, Mathematical models of similarity measures are considered. Solution procedure of Intuitionistic fuzzy Assignment (IFA) problem with similarity measures are also discussed. Numerical example also illustrated to the proposed method. The result of the proposed method give the effective optimal solution to the solving IFA problem.

Values of s(p+ , ji ) for jobs with respect to machines (row wise) s(p+ , ji )

(rij sij )m×m

J3 0.902

820

www.ijitce.co.uk

INTERNATIONAL JOURNAL OF INNOVATIVE TECHNOLOGY AND CREATIVE ENGINEERING (ISSN:2045-8711) VOL.10 NO.6 JUNE 2020 REFERENCES [1]

K. Atanassov, Intuitionistic fuzzy sets. Fuzzy Sets and Systems, 20(1986), 87-96. [2] K. Atanassov, More on intuitionistic fuzzy sets, Fuzzy Sets and Systems, 33(1989), 37-46. [3] S. Chanas and D. Kuchta, A concept of the optimal solution of the transportation problem with fuzzy cost coefficients, Fuzzy Sets and Systems, 82 (1996) 299–305. [4] S. Chanas and D. Kuchta, Fuzzy integer transportation problem, Fuzzy Sets and Systems,98 (1998), 291–298. [5] M. S. Chen, On a fuzzy assignment problem, Tamkang J., 22 (1985), 407–411. [6] S. M. Chen and P. H. Hsiao, A comparison of similarity measures of fuzzy values., Fuzzy Sets and Systems, 72(1995), 79-89. [7] Deng Yong, Shi Wenkang, Du Feng, Liu Qi, A new similarity measure of generalized fuzzy numbers and its application to pattern recognition, Pattern Recognition Letters, Volume 25, Issue 8, [8] Li Dengfeng and Cheng Chuntian, New similarity measures to Intuitionistic Fuzzy Sets and Applications to Pattern Recognitions, Pattern Recognition Letters, 23(2002), 221-225 [9] Huang Long-sheng and Zhang Li-pu, Solution method for Fuzzy Assignment problem with Restiction of Qualification, Proceedings of the Sixth International Conference on Intelligent Systems Design and Applications( ISDA’06 ), 2006. [10] Sathi Mukherjee and Kajla Basu, Solving Intuitionistic Fuzzy Assignment Problem by using Similarity Measures and Score Functions, Int. J. Pure Appl. Sci. Technol., 2(1) (2011), pp. 1-18 [11] X. Wang, Fuzzy optimal assignment problem, Fuzzy Math., 3(1987), 101–108. [12] Z. S Xu and R. R. Yager, Some geometric aggregation operators based on intuitionistic fuzzy sets, International Journal of General Systems, 35(2006), 417-433.

821

www.ijitce.co.uk

INTERNATIONAL JOURNAL OF INNOVATIVE TECHNOLOGY AND CREATIVE ENGINEERING (ISSN:2045-8711) VOL.10 NO.6 JUNE 2020

822