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Possibility Fuzzy Soft Expert Set Maruah Bashir School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor DE, Malaysia aabosefe@yahoo.com Abdul Razak Salleh School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor DE, Malaysia aras@ukm.my Abstract— In this paper we introduce the concept of possibility fuzzy soft expert set .We also define its basic operations, namely complement, union, intersection, AND and OR, and study some of their properties. Finally, we give an application of this theory in solving a decision making problem. Keywords—Fuzzy soft expert set, possibility fuzzy soft set, possibility fuzzy soft expert set 1. Introduction In 1999 Molodtsov [1] initiated a novel concept of soft set theory as a new mathematical tool for dealing with uncertainties . After Molodtsov’s work, some different operations and application of soft sets were studied by Chen et al. [2] and Maji et al. [3,4]. Furthermore Maji et al. [5] presented the definition of fuzzy soft set as a generalization of Molodtsov’s soft set. Roy and Maji [6] gave an application of this concept in decision making problem. In 2010 Çağman et al. introduced the concept of fuzzy parameterized fuzzy soft sets and their operations [7]. Alkhazaleh et al. [8] generalized the concept of fuzzy soft set to possibility fuzzy soft set and they gave some applications of this concept in decision making and medical diagnosis. They also introduced the concept of fuzzy parameterized interval-valued fuzzy soft set [9], where the mapping in which the approximate functions are defined from fuzzy parameters set to the interval-valued fuzzy subsets of universal set, and gave an application of this concept in decision making. Alkhazaleh and Salleh [10] introduced the concept of soft expert sets where the user can know the opinion of all experts in one model and gave an application of this concept in decision making problem. Salleh et al. [11] introduced the concept of multi paramatriezed soft set and studied its properties and basic operations. Alkhazaleh and Salleh [12] introduced the concept of fuzzy soft expert sets and gave an application of this concept in a decision making problem. In this paper we introduce the concept of possibility fuzzy soft expert set which is a combination of possibility fuzzy soft set and fuzzy soft expert set. We also define its basic operations namely complement, union, intersection, AND and OR. Finally,we give an application of this concept in adecision making problem. 2. Preliminaries In this section we recall some definitions and properties regarding soft expert set , fuzzy soft expert set and possibility fuzzy soft set. Alkhazaleh and Salleh [10] defined soft expert set and in [12] Alkhazaleh and Salleh defined a fuzzy soft expert set in the following way. Let Ube a universe,E a set of parameters, and X a set of experts (agents). Let Obe a set of opinions, Z EXO and A Z. Definition 2.1. [12] A pair , F Ais called a soft expert set over ,U where F is a mapping : F APU and PU denotes the power set of .U Definition 2.2. [12] A pair , F Ais called a fuzzy soft expert set over ,U where F is a mapping: U F AI,and U Idenotes all fuzzy subsets of .U Definition 2.3. [12] The complement of a fuzzy soft expert set , F A is denoted by , c F Aand is defined by , c F A= , Ί c F A where :Ί c F ApU is a mapping given by Ί,Ί, c F cF A where c is a fuzzy complement. Definition 2.4. [12] The union of two fuzzy soft expert sets , F Aand ,GB over ,U denoted by ,, F AGB , is a fuzzy soft expert set , H C where CAB and ,C , , , , FifAB HG ifBA sFGifAB where is an -norm. ss Definition 2.5. [12] The intersection of two fuzzy soft expert sets , F Aand ,GB over ,U denoted by ,,, F AGB is a soft expert set , H Cwhere andCAB ,C Open Journal of Applied Sciences Supplement：2012 world Congress on Engineering and Technology 208 Copyright © 2012 SciRes. , , , , FifAB HG ifBA tFGifA B where is a -norm. tt The following definitions are due to Alkhazaleh et al. [8]. Definition 2.6. Let 12 ,,..., n Uxx x be the universal set of elements and 12 ,,..., m Eee ebe the universal set of parameters. The pair (,)UE will be called a soft universe. Let :U F EIand be a fuzzy subset of ,E i.e. :U EI , where U I is the collection of all fuzzy subsets of U. Let :UU F EII be a function defined as follows: ()(), () F e Fexex , x U . Then F is called a possibility fuzzy soft set (PFSS in short) over the soft universe (,).UE For each parameter , (), ii ii eFeFe xex indicates not only the degree of belongingness of the elements of U in (), i F ebut also the degree of possibility of belongingness of the elements of U in (), i F e which is represented by() i e . So we can write () i F e as follows: 1 1 1 ( ),()(),...,,()() ()( )()(). n ii in iin xx Fee xe x Fe xFe x Sometime we write F as ,. F E If A Ewe can also have a PFSS ,. F A Definition 2.7. Union of two PFSSs F and ,G denoted by , F G is a PFSS :UU H EII defined by (), , H eHexex eE such that ()(), () H esFeGe and ()(), ()es ee where s is an s-norm. Definition 2.8. Intersection of two PFSSs F and ,G denoted by , F G is a PFSS :UU H EII defined by () , H e Hexex ,eE such that ()(), () H etFeGe and ()(), ()et ee where t is a t-norm. 3. Possibility Fuzzy Soft Expert Sets In this section we generalise the concept of fuzzy soft expert sets as introduced by Alkhazaleh and Salleh. [12]. In our generalisation of fuzzy soft expert set, a possibility of each element in the universe is attached with the parameterization of fuzzy sets while defining a fuzzy soft expert set. Definition 3.1. Let 12 ,,...,n Uuuu be the universal set of elements, 12 ,,..., m Eee ebe the universal set of parameters, X be a set of experts, and 1agree, 0disagreeO a set of opinions. Let Z EXOand . A Z Let :U F ZIand be a fuzzy subset of , Z i.e. :U Z I , where U I is the collection of all fuzzy subsets of U. Let :UU F ZII be a function defined as follows: ()(), () F z Fzuzu ,.uU Then F is called a possibility fuzzy soft expert set (PFSES in short) over the soft universe (, ).UZ For each , (), ii ii zFzFzu zu indicates the degree of belongingness of the elements of U in (), i F zand also the degree of possibility of such belongingness which is represented by() i z . So we can write () i F z as follows: 1 1 1 ( ),( )(),...,,( )( ) ()()()() . n ii in iin uu Fzz uz u Fz uFzu Sometime we write , F Z as . F If A Zwe can also have a PFSES ,. F A Definition 3.2. Let , F A and ,GB be two PFSESs over (,)UZ . , F A is said to be a possibility fuzzy soft expert subset (PFSE subset) of ,,GB and we write ,,, F AGB if A Band , A i. is a fuzzy subset of , ii. is a fuzzy subset of .FG Definition 3.3. A possibility agree - fuzzy soft expert set 1 , F A over U is a possibility fuzzy soft expert subset of , F A defined as follows : 1 ,,, where 1.FAFE X Definition 3.4. A possibility disagree - fuzzy soft expert set 0 , F A over Uis a possibility fuzzy soft expert subset of , F A defined as follows : 0 ,,, where 0.FAFEX Copyright © 2012 SciRes. 2 Definition 3.5. Let , F A be a PFSES over (,)UZ . Then the complement of , F A , denoted by ,c F A is defined by ,, , c F AcFcA ┐┐ ┐ where c is a fuzzy soft expert complement and cis a fuzzy complement. Definition 3.6. Union of two PFSESs , F A and , over ,GB U denoted by ,,, F AGB is a PFSES , v H Cwhere , isCAB defined by ()(),(), s C and () , H FG C where s is an s-norm and is a fuzzy soft expert union. Definition 3.7. Intersection of two PFSESs , F A and , over ,GB U denoted by ,,, F AGB is a PFSES , v H Cwhere , isCAB defined by ()(),(), et eeeC and () , H eFeGeeC where t is a t-norm and is a fuzzy soft expert intersection. 4. An application of possibility fuzzy soft expert set Ahkhazaleh and Salleh [12] applied the theory of fuzzy soft expert sets to solve a decision making problem. In this section, we present an application of PFSES in a decision making problem by generalizing Ahkhazaleh and Salleh’s algorithm to be compatible with our work. The problem we consider is as below. Suppose that electricity board wants to make an electric generator using waterfalls. Suppose there are three different locations and they want to take the opinion of some experts concerning these locations. Let 123 {,, }Uuuu be a set of locations, 1234 {,, , }Eeeee a set of decision parameters where 1, 2,3,4 i eidenotes the decision “Ebb and tide,” “wind power,” “the power of the regression” and “solar energy” respectively, and let , X qmbe a set of experts (two members). Suppose that the electricity board wants to choose one such location depending on the parameters. After a serious discussion the committee constructs the following possibility fuzzy soft expert set: 3 12 1 3 12 2 1 4 ,,,1,,0.3,,0.2,,0.3, 0.4 0.2 0.5 ,,1,,0.3,,0.4,,0.3, 0.1 0.20.7 ,,1,,0 0.9 u uu FZ eq u uu eq u eq 3 2 .1 ,,0.4,,0.5, 0.6 0.1 u u 3 12 1 3 12 2 1 3 ,,1,,0.1,,0,,0.2, 0.80.3 0.3 ,,1,,0.1,,0.5,,0.1, 0.1 00.2 ,,1,0. u uu em u uu em u em 3 2 3 12 1 12 2 ,0 ,,0 ,,0.1, 10.30.1 ,,0,,0.2,,0.3,,0.1, 0.1 0.6 0.2 ,,0,,0.5,,0.5, 0.3 0.6 u u u uu eq uu eq 3 3 12 3 3 12 1 ,0.1 , 0.5 ,,0 ,,0.2,,0.3,,0.1, 0.2 0.4 0.5 ,,0,,0.3,,0.5,,0.3, 0.3 0.50.4 u u uu eq u uu em 3 12 3 3 12 4 ,,0,,0.3,,0.7,,0.3, 0.2 0.40.1 ,,0,,0.1,,0.1,,0.1. 0.2 0.3 0.4 u uu em u uu em The following algorithm may be followed by the electricity board to determine the best location. 1. Input the PFSES ,. F Z 2. Find the highest numerical grade for the agree-PFSES and disagree-PFSES. 3. Compute the score of each such locations by taking the sum of the products of these numerical grades with the corresponding possibility , for the agree-PFSES which is denoted by j A and disagree-PFSES denoted by j D. 4. Find . j jj s AD 5. Find ,mfor which max . mj s s Then m s is the highest score. If mhas more than one value, then any one of them could be chosen by the electricity board using its option. Now we use this algorithm to find the best choice for the electricity board to make an electric generator. 210 Copyright © 2012 SciRes. TABLE I. GRADE FOR AGREE PFSES H i u Highest numerical grade i 1,eq 3 u 0.5 0.3 2,eq 3 u 0.7 0.3 4,eq 1 u 0.9 0.1 1,em1 u 0.8 0.1 2,em 3 u 0.2 0.1 3,em 2 u 0.3 0 1 score 0.90.10.80.10.17u 2 score 0.300u 3 score 0.50.30.70.30.20.10.38u TABLE II. GRADE FOR DIS-AGREE PFSES H i u Highest numerical grade i 1,eq 2 u 0.6 0.3 2,eq 2 u 0.6 0.5 3,eq 3 u 0.5 0.1 1,em2 u 0.5 0.5 3,em 2 u 0.4 0.7 4,em 3 u 0.4 0.1 1 score 0u 2 score 0.60.30.60.50.50.50.40.71.01u 3 score 0.50.10.40.10.09u For abbreviation suppose that the score of each numerical grade for agree-PFSES, j A the score of each numerical grade for dis-agree-PFSES. j D TABLE III. j jj s AD j A j D j jj s AD 10.17score u 10score u 0.17 20score u 21.01score u -1.01 30.38score u 30.09score u 0.29 Then max3j s s , so the electricity board will select the location with the highest score. Hence, they will choose location 3.u 5. Acknowledgement The authors would like to acknowledge the financial support received from Universiti Kebangsaan Malaysia under the research grant UKM-DLP-2011-038. REFERENCES [1]. D. Molodtsov, Soft set theory-first results, Computers and Mathematics with Applications, vol. 37, no. 4-5, pp. 19 – 31, 1999. [2] D. Chen, E. C. C. Tsang, D. S. Yeung and X. 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Hassan, Possibility Fuzzy Soft Set, Advances in Decision Sciences, 2011(2011) 18 pages. [9] S. Alkhazaleh, A. R. Salleh and N. Hassan, Fuzzy parameterized interval- valued fuzzy soft set, Applied Mathematical Sciences, 5(67): 3335-3346, 2011. [10] S. Alkhazaleh and A. R. Salleh, Soft expert sets, Advances in Decision Sciences, 2011(2011) 15 pages. [11] A. R. Salleh, S. Alkhazaleh, N. Hassan and A. G. Ahmad, multiparameterized soft set, Journal of Mathematics and Statistics, 8(1): 92-97, 2012. [12] S. Alkhazaleh , 2012. Soft sets and fuzzy soft sets: Some generalizations. PhD thesis, Universiti Kebangsaan Malaysia. Copyright © 2012 SciRes. |