A New Hendecagonal Fuzzy Number For Optimization Problems

International Journal of Trend in ScientificResearch and Development (IJTSRD)UGC Approved International Open Access JournalISSN No: 2456 - 6470 www.ijtsrd.com Volume - 1 Issue – 5A New Hendecagonal Fuzzy Number for Optimization ProblemsM. RevathiAssistant professor,Department of Mathematics, Tamilnadu College ofEngineering,Coimbatore,Tamilnadu, IndiaDr.M.ValliathalAssistant professor, Department of Mathematics,Chikkaiah Naicker College, Erode, Tamilnadu, IndiaR. SaravananAssistant professor,Department of Mathematics, Tamilnadu College ofEngineering,Coimbatore,Tamilnadu, IndiaDr.K.RathiAssistant Professor, Department of Mathematics,Velalar College of Engineering and Technology,Erode, TamilnaduABSTRACTA new fuzzy number called Hendecagonal fuzzynumber and its membership function is introduced,which is used to represent the uncertainty with elevenpoints. The fuzzy numbers with ten ordinates exists inliterature. The aim of this paper is to defineHendecagonal fuzzy number and its arithmeticoperations. Also a direct approach is proposed tosolve fuzzy assignment problem (FAP) and fuzzytravelling salesman (FTSP) in which the cost anddistance are represented by Hendecagonal fuzzynumbers. Numerical example shows the effectivenessof the proposed method and the Hendecagonal fuzzynumber.Keywords: Hendecagonal fuzzy number, Alpha cut,Fuzzy arithmetic, Fuzzy Assignment problem, Fuzzytransportation problem.I.INTRODUCTIONA fuzzy number is a quantity whose values areimprecise, rather than exact as in the case with singlevalued function. The generalization of real number isthe main concept of fuzzy number. In real worldapplications all the parameters may not be knownprecisely due to uncontrollable factors.L.A.Zadeh introduced fuzzy set theory in 1965.Different types of fuzzy sets [17] are defined in orderto clear the vagueness of the existing problems.D.Dubois and H.Prade has defined fuzzy number as afuzzy subset of real line [8]. In literature, many typeof fuzzy numbers like triangular fuzzy number,trapezoidal fuzzy number, pentagonal fuzzy number,hexagonal fuzzy number, heptagonal fuzzy number,octagonal fuzzy number, nanagonal fuzzy number,decagonal fuzzy number have been introduced withits membership function.Thesemembershipfunctions got many applications and many operationswere done using these fuzzy numbers [2], [12] [14],[15].In much decision analysis, the uncertainty existing ininput information is usually represented as fuzzynumbers [1],. S.H.Chen introduced maximization andminimization of fuzzy set, uncertainty andinformation [4]. The arithmetic operations , alpha cutand ranking function are already introduced forexisting fuzzy numbers by C.B.Chen andC.M.Klein,T.S.Liou and M.J.Wang [6],[11]. Whenthe vagueness arises in eleven different points it isdifficult to represent the fuzzy number. In this paper anew type of fuzzy number named as hendecagonalfuzzy number is defined with its membershipfunction. The arithmetic operations, alpha cut andranking procedure for hendecagonal fuzzy numbersare introduced to solve fuzzy assignment problem(FAP) and fuzzy travelling salesman problem(FTSP).In literature, many methods were proposed forfuzzy optimization with uncertain parameters[3],[5],[10],[13].Here uncertainty in assignment cost@ IJTSRD Available Online @ www.ijtsrd.com Volume – 1 Issue – 5 July – August 2017Page: 326

International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470and travelling distance are represented by new fuzzynumber named as Hendecagonal fuzzy number, whichare ranked using the ranking function introduced byR.R.Yager [16],[7]. Numerical examples show theeffectiveness of the proposed method and the newfuzzy number, It is simple and very easy tounderstand and can be applied in many real lifeproblems.II.PRELIMINARYDefinition 2.1: The membership grade corresponds tothe degree to which an element is compatible with theconcept represented by fuzzy set.Definition 2.2: Let X denote a universal set. Then thecharacteristic function which assigns certain values ora membership grade to the elements of this universalset within a specified range [0,1] is known asmembership function & the set thus defined is called afuzzy set.Definition 2.3: Let X denote a universal set. Then themembership function A by using a fuzzy set A isusually denoted as A : X I , where I [0,1]Definition 2.4: An -cut of a fuzzy set A is a crispestA that contains all the elements of the universal setX that have a membership grade in A greater or equalto specified value of ThusA x X , A ( x) ,0 1 Definition 2.5: A fuzzy set A is a convex fuzzy set ifand only if each of its cuts A is a convex set. Definition2.6: A fuzzy set A is a fuzzy number if(i) For all (0,1] the cut sets A is a convex set (ii) A is an upper semi continuous function.Definition 2.7: A triangular membership function isspecified by three parameters [a,b,c] as follows ( x a) /(b a), a x b ,x b (x:a,b,c) 1 (c x) /( c b), b x c , otherwise 0This function is determined by the choice of theparameter a, b, c where xij 0,1 Definition 2.8: A trapezoidal fuzzy number A (a, b, c, d ) is a fuzzy number with membershipfunction of the form ( x a ) /(b a ) , a x b 1, b x c ( x : a, b, c, d ) (d x) /( d c) , c x d 0, otherwise III.HENDECAGONAL FUZZY NUMBERSDefinition 3.1: The parametric form of HendecagonalFuzzyNumberisdefinedas forU P1 (r ), Q1 ( s), R1 (t ), S1 (u ), T1 (v), P2 (r ), Q2 ( s), R2 (t ), S 2 (u ), T2 (v) ,r [0,0.2] s [0.2,0.4] t [0.4,0.6] u [0.6,0.8] andv [0.8,1] where P1 (r ), Q1 (s), R1 (t ), S1 (u) and T1 (v) arebounded left continuous non decreasing functionsover [0,0.2], [0.2,0.4], [0.4,0.6], [0.6,0.8] and [0.8.1] ,P2 (r ), Q2 (s), R2 (t ), S 2 (u) and T2 (v) Are bounded leftcontinuousnonincreasingfunctionsover[0,0.2], [0.2,0.4], [0.4,0.6], [0.6,0.8] and [0.8.1] .Definition3.2:AfuzzynumberA (a1 , a 2 , a3 , a 4 , a5 , a6 , a7 , a8 , a9 , a10 , a11 ) is said tobe a Hendecagonal fuzzy number if its membershipfunction is given by 1 x a1 , a1 x a 2 5 a 2 a1 1 1 x a 2 , a 2 x a3 a a 5 5 2 3 x a3 2 1 a a , a3 x a 4 5 5 3 4 3 1 x a 4 , a 4 x a5 5 5 a5 a 4 x a5 4 1 , a5 x a 6 5 5 a 6 a5 1 x a6 , a6 x a7 U ( x ) 1 5 a7 a6 x a7 41 , a 7 x a8 5 5 a8 a 7 3 1 x a8 5 5 a a , a8 x a 98 9 2 1 x a 9 a a , a9 x a1055 9 10 a11 x 1 , a10 x a11 5 a11 a10 , Otherwise 0 Figure 1 shows the graphical representation ofHendecagonal fuzzy number.@ IJTSRD Available Online @ www.ijtsrd.com Volume – 1 Issue – 5 July – August 2017Page: 327

International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470Hendecagonal fuzzy 2(v)S1(u)Q2(s)P1(r)P2(r)0.0Figure 1: Graphical representation of Hendecagonal fuzzy numberIV.ARITHMETICOPERATIONSHENDECAGONAL FUZZY NUMBERSONIn this section, arithmetic operations between twoHendecagonal fuzzy numbers, defined on universalset of real numbers R, are presented.LetA (a1 , a 2 , a3 , a 4 , a5 , a6 , a7 , a8 , a9 , a10 , a11 )beB (b1 , b2 , b3 , b4 , b5 , b6 , b7 , b8 , b9 , b10 , b11 )hendecagonal fuzzy number thenandtwo(i) Addition of two hendecagonal fuzzy numbers A B (c1 , c 2 , c3 , c 4 , c5 , c6 , c7 , c8 , c9 , c10 , c11 ) (a1 b1 , a 2 b2 , a3 b3 , a 4 b4 , a 5 b5 , a 6 b6 ,a 7 b7 , a8 b8 , a9 b9 , a10 b10 , a11 b11 )(ii) Scalar multiplication of hendecagonal fuzzyThe ranking function r : F ( R) R where F(R) is aset of fuzzy number defined on set of real numbers,which maps each fuzzy number into the real line,where the natural order exists, i.e. (i ) A B iff r ( A) r ( B ) (ii ) A B iff r ( A) r ( B ) (iii ) A B iff r ( A) r ( B )LetA (a1 , a2 , a3 , a4 , a5 , a6 , a7 , a8 , a9 , a10 , a11 )andB (b1 , b2 , b3 , b4 , b5 , b6 , b7 , b8 , b9 , b10 , b11 )be two hendecagonal fuzzy numbers then a a2 a3 a4 a5 a6 a7 a8 a9 a10 a11r ( A) 111b1 b2 b3 b4 b5 b6 b7 b8 b9 b10 b11 r(B) 11andnumbers ( a , a , a , a , a , a , a , a , a , a , a ) if 0 A 1 2 3 4 5 6 7 8 9 10 11 ( a11, a10 , a9 , a8 , a7 , a6 , a5 , a4 , a3 , a2 , a1 ) if 0VI.(iii) Subtraction of two hendecagonal fuzzynumbersIn this section, mathematical formulation of fuzzyassignment problem is given and a direct approach isproposed to solve FAP and FTSP. The method isapplicable for all optimization problems. A B A B (a1 b11 , a 2 b10 , a 3 b9 , a 4 b8 , a 5 b7 , a 6 b6 ,a 7 b5 , a8 b4 , a 9 b3 , a10 b2 , a11 b1 )V.RANKING HENDECAGONAL FUZZY NUMBERSThe ranking method proposed in [4] is used to rankthe hendecagonal fuzzy numbers.FUZZY ASSIGNMENT PROBLEM ANDTRAVELLING SALESMAN PROBLEMFUZZYA. Formulation of Fuzzy Assignment Problem Let there be m Tasks and m Workers , C ij be the costof assigning ithWorker to the jthTask and theuncertainty in cost is here represented asHendecagonal fuzzy numbers. Let x ij be the decisionvariable define@ IJTSRD Available Online @ www.ijtsrd.com Volume – 1 Issue – 5 July – August 2017Page: 328

International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470 1 if the i th person is assigned to the jth jobxij 0 otherwiseThen the fuzzy assignmentmathematically stated as follows m m Minimize Z C ij x ijproblemVII.canbem xj 1m xi 1ijij 1 , i 1,2.m ; 1 , j 1,2.mB. Formulation of Fuzzy Travelling SalesmanProblemThe travelling salesman problem deals with findingshortest path in a n-city where each city is visitedexactly once. The travelling salesman problem issimilar to assignment problem that excludes subpaths. Specifically in an n-city situation define 1 , if city j is reached from city ixij 0 , otherwise Here d ij is the distance from city i to city j which isHendecagonal fuzzy number. Mathematically FTSPcan be stated asnn Minimize z d ij x ij, d ij for all i ji 1 j 1nSubject to xj 1n xi 1ijijIn this section numerical examples are given toillustrate the proposed method and it is shown that theproposed method offers an effective way for handlingFAP as well as FTSP.Example7.1:Amanufacturingcompanymanufactures a certain type of spare parts with threedifferent machines. The company official has toexecute three jobs with three machines. Theinformation about the cost of assignment is impreciseand here Hendecagonal Fuzzy numbers are used torepresent the cost. The fuzzy assignment problem isgiven in Table 1.i 1 j 1Subject toNUMERICAL EXAMPLESolution:Step 1: Calculate the ranking value of each fuzzy costis given in Table 2.Step 2: Encircle the fuzzy cost with least rankingvalue in each row and examine all the encircled fuzzycosts and identify the encircled fuzzy cost that isuniquely encircled in both row wise and column wise.Assign it and delete the corresponding row andcolumn. The resultant table is given in Table 3.Step 3:If the cost is not uniquely selected both rowwise and column wise then choose next minimum andproceed as in step 2.This process is continued until thefuzzy cost is uniquely selected row and column wise. 1 , i 1,2.n 1 , j 1,2.n , xij 0,1 Then the optimal assignment is J1 M1 , J 2 M 2 ,with the optimal assignment costJ3 M3(4,8,12,16,21,29,35,39,43,47,54) and its crisp value isr(C) 28JOBTable 1: Fuzzy Assignment Problem with Hendecagonal Fuzzy Cost 2)(4,6,7,9,10,11,18,23,24,26,27)@ IJTSRD Available Online @ www.ijtsrd.com Volume – 1 Issue – 5 July – August 2017Page: 329