Consistent inguistic fuzzy preference reation with muti-granuar uncertain inguistic information for soving decision making probems Siti mnah Binti Mohd Ridzuan, and Daud Mohamad Citation: IP Conference Proceedings 750, 0007 (06); doi: 0.06/.495456 View onine: https://doi.org/0.06/.495456 View Tabe of Contents: http://aip.scitation.org/toc/apc/750/ Pubished by the merican Institute of Physics rtices you may be interested in Consistent inguistic fuzzy preference reations method with ranking fuzzy numbers IP Conference Proceedings 65, 09 (05); 0.06/.490585
Consistent Linguistic Fuzzy Preference Reation with Muti- Granuar Uncertain Linguistic Information for Soving Decision Making Probems Siti mnah Binti Mohd Ridzuan a) and Daud Mohamad b) a,b Department of Mathematics, Facuty of Computer and Mathematica Sciences, Universiti Teknoogi MR, 40450, Shah am, Seangor, Maaysia. a) Corresponding author: amnahridzuan@gmai.com b) daud@tmsk.uitm.edu.my bstract. Muti-granuar information in decision making is an approach used to cater heterogeneity in the evauation process which normay presented using inguistic form. Sometimes the decision maker may use non homogeneous scaes in the evauation. This wi increase the compexity in computation. This paper proposes a hybrid decision making mode that combines the muti-granuar uncertain inguistic information and consistent inguistic preference reations (CLFPR) to cater this probem. The proposed method abe to evauate the aternatives or criteria based on the different opinions with different scaes using a minimum number of pairwise comparisons. The method uses severa interva representations of inguistic judgment as its muti-granuar representation to give decision makers fexibiity in the evauation. The evauation wi be aggregated into CLFPR vaues and before obtaining the fina resut. n iustrative exampe wi be given to show the effectiveness of the proposed method. Keywords: Muti-granuar uncertain inguistic information; Consistent inguistic fuzzy preference reations (CLFPR); Fuzzy decision making. PCS: 0.50.Le; 07.05.Mh INTRODUCTION Decision is made in our daiy ives, either as individua or in a group. Group decision making is an evauation process which invoves more than one evauator in obtaining the outcome or decision. It is natura that each evauator has his/her own opinion and preference in the evauation process due to different educationa background, cuture and experience []. There are severa approaches to manage this situation such as muti-granuar inguistic information and inear programming mode [], fusion method [], preference ranking organization method (PROMETHEE) [4], muti-granuar uncertain inguistic information [5] as to name a few. In this paper, we focus on muti-granuar uncertain inguistic information as a too to evauate the aternatives based on different opinion with different measurement scaes. In many situations, information invoved in making decision is vague and subjective. Fuzzy set theory has been appied effectivey to obtain a reaistic resut for modeing quaitative information [6]. It manages fuzziness effectivey and represents quaitative aspects as a inguistic variabe. The use of inguistic variabes aows experts to express their assessment in a more fexibe and reiabeway that cose to human intuition and perception. Uncertainty in evauation commony occurs when deaing with human perception in the evauation of decision making probems. Many methods have been introduced to cater this situation [7]. n effective approach is to consider a criteria invoved and compare them in pairwise in order to choose the best avaiabe aternatives. The consistent inguistic fuzzy preference reation (CLFPR) has been proven to be a desirabe approach in deaing with minimum number of pair-wise comparisons [8]. The CLFPR ony requires (n-) comparisons from the given n criteria as opposed to many other methods that wi require a vast number of comparisons when n increases. dvances in Industria and ppied Mathematics IP Conf. Proc. 750, 0007-0007-9; doi: 0.06/.495456 Pubished by IP Pubishing. 978-0-754-407-5/$0.00 0007-
In this paper, a group decision making mode with muti-granuar uncertain inguistic information based on CLFPR method is deveoped. The evauation is made in the form of interva inguistic judgment as its muti-granuar representation. This wi give an advantage to the decision makers in the evauation due to the fexibiity of the interva used where the decision makers can choose their desired interva according to their preference. The transformation of muti-granuar uncertain inguistic information into trapezoida fuzzy numbers is then made so that the evauation can be cacuated using CLFPR in order to obtain the preference ordering. This aggregation procedure is abe to hande different opinion of decision makers effectivey with heterogeneous interva scaes and the CLFPR wi reduce the pairwise cacuations but with consistent resuts. METHODOLOGY In 965, Zadeh introduced fuzzy set theory to measure the uncertainty of information. Fuzzy number is a subset of fuzzy set theory that is norma and convex [8] which can be defined in the form of a trianguar, trapezoida or any appropriate shapes of membership functions. Definition [8]: Let X be a universa set. The membership function by which a fuzzy set in is defined, is a mapping : X 0, where, 0 represents the interva of rea numbers from 0 to incusive. Definition [9]: If X is a coection of objects denoted genericay by x, then a fuzzy set in X is a set of ordered pairs {( x, ( x))}, x X where (x ) is caed the membership function or membership grade (aso degree of compatibiity or degree of truth) of x in. ~,, 4, is defined on the universa set of rea number, is said to be a generaized fuzzy number if its membership function has the foowing characteristics: Definition [0]: fuzzy set a, a a a w i. ~ ( x) : 0, is continuous. ii. 0 for a x (, a] [ a4, ) iii. ) is stricty increasing on [ a, a ] and stricty decreasing on [, a 4 ] iv. ~ ( x) a, a, where 0 w. ~ ( x) ~ ( x ~ ( x) w for a If w is equa to, then the generaized fuzzy number is essentiay a standard fuzzy number.. Muti-Granuar Uncertain Linguistic (MGUL) [5] In soving group decision making probems, decision makers may use uncertain inguistic terms, according to pre-estabished inguistic term set to express his preference on the aternative. Consequenty, the actua meaning of the uncertain inguistic term [ s a, sb ], the expert may not be abe to provide a precise judgment among mutipe inguistic terms ( sa, sa,..., sb, sb). This representation is desirabe as it gives fexibiity in the evauation and known as Muti-Granuity Uncertain Linguistic Information (MGUL). th Definition 4: Let S be the pre-estabished finite and totay ordered inguistic term set with odd cardinaities. n uncertain inguistic term is expressed by non-fixed interva [ s a, sb ], where sa, s S b. The terms s a and s b are the ower and upper imits of the given interva. The greater b a, the greater the imprecision of [ s a, sb ] wi be. In particuar, if a b, then [ s a, sb ] is reduced to a certain inguistic term s a or s b. The evauation in MGUL wi be transformed into fuzzy numbers form as foows: th Let S s, s, s,..., s } be the pre-estabished finite and totay ordered inguistic term set with odd { 0 T cardinaities and T is the cardinaity of S. If [ s a, sb ] is an uncertain inguistic term, s can be expressed as the foowing fuzzy number: a b a sb, s S, then [ s, ] 0007-
~ (,, a a b d d d d ) max,0,, min, ab ab ab ab T T T. () The fuzzy numbers wi be used to construct a decision matrix based on the method of Consistent Linguistic Fuzzy Preference Reation (CLFPR) []. B. Consistent Linguistic Fuzzy Preference Reations The Consistent Linguistic Fuzzy Preference Reation (CLFPR)[] wi be used in obtaining the fina outcome. It was introduced to dea with uncertainty or vagueness judgments in inguistic form. The method constructs fuzzy ~ preference reation matrices using fuzzy inguistic assessment variabes ~ L M R P ( pij ) ( pij, pij, p ij ) in the form of a trianguar fuzzy number. There are two propositions as given in [] that wi be utiized in the CLFPR process of obtaining the resut, which are as foows: Proposition : Supposed a set of aternative { a, a,..., an} which is associated with a reciproca mutipicative preference reation R ( r ij ) for r ij 9,9. Then the corresponding reciproca fuzzy preference reation P ( p ij ) (/ )( og 9 rij ) with pij 0, verify the additive reciproca, namey, the foowing statement is equivaent: L p ij p R ji i, j {,..., n}. M ij M ji p p i, j {,..., n}. R ij L ji p p i, j {,..., n}. ~ R ij Proposition : For a reciproca fuzzy inguistic preference reation ~ L M P ( pij ) ( pij, pij, p ) to be consistent, the foowing statements are equivaent. L L R pij p jk pki, i j k M M M pij p jk pki, i j k R R L pij p jk pki, i j k L L L R j i p i ( i ) p( i)( i)... p( j) j p ji, i j M M M M j i p i ( i ) p( i )( i )... p( j ) j p ji, i j R R R L j i p i ( i ) p( i)( i)... p( j) j p ji, i j Note that if the vaues are different, then we woud have obtained a matrix P ~ with the eements ~ pij which are not in the interva [ 0,], but in the interva [ c, c], being ( c 0). In order to obtain the fuzzy number form, transformation to the interva 0, is required using a transformation function f :[ c, [0,] which preserves reciprocity and additive consistency. The function f has the foowing properties[]: f ( c) 0. f ( c) 0. 0007-
L R f ( x ) f ( x ) x [ c,. M M f ( x ) f ( x ) x [ c,. R L f ( x ) f ( x ) x [ c,. L L R L L R L L R f ( x ) f ( y ) f ( z ), x, y, z [ c, such that x y z. M M M M M M M M M f ( x ) f ( y ) f ( z ), x, y, z [ c, such that x y z. R R L R R L R R L f ( x ) f ( y ) f ( z ), x, y, z [ c, such that x y z. Figure shows the framework of combination between muti-granuarity uncertain inguistic information and consistent inguistic fuzzy preference reations in deriving the proposed method. FIGURE. Framework of muti-granuarity uncertain inguistic information with consistent inguistic fuzzy preference reations. NUMERICL EXMPLE We iustrate the proposed method by ooking into a seection of textbooks. There are three choices of Cacuus textbooks to be used by mathematics undergraduate students at a university denoted by B, B and B. They are evauated based on five criteria, namey content ( C ), anguage ( C ), assessment ( C ) suitabiity to student ( C 4 ) and graphic and iustration ( C 5 ). The evauation process aso invoved three evauators E, E and E. The three different predetermined inguistic term sets for coective of criteria and books are represented as foows: S S : Equay important, S : Marginay more important, S : More important, S : Very more important, { 4 0007-4
5 S : Extremey more important} S { S : Equay important, S : Sighty more important, S : Marginay more important, S 4 : More important, S : Very more important, S : Extremey more important, S : Exceptionay more important} 5 6 S { S : Equay important, S : Very sighty more important, S : Sighty more important, S 4 : Marginay more important, S : More important, S : Very more important, S : Very very more important, : 5 9 Extremey more important, S : Exceptionay more important } 6 s an iustration, the pairwise comparisons of books with respect to content (C ) by the evauators E, E and E is given in Tabe. Each evauator provides pairwise evauation for the compement entries. For exampe, B refers to the evauation between B and B. TBLE. Evauations of Books with Respect to Content (C) From E, E and E. Decision Makers Book B B B E E E B - [ S, S ] - [ S, S B - - ] B - - - B - [ S 5, S6 ] - [ S 6, S7 B - - ] B - - - B - [ S 6, S6 ] - [ S, S B - - ] B - - - 7 7 S 8 eements in the decision matrix in Tabe are converted into the fuzzy number using the given transformation function f earier. The foowing cacuation shows an exampe of how to appy the equation () towards B from decision maker E in Tabe : () () () B max,0,, min, (5) (5) (5) 0.7,0.6,0.46 Tabe shows the evauation of three decision makers in terms of fuzzy numbers after the conversion process. 0007-5
TBLE. Pairwise comparison of a books with respect to content ( C ) from E, E and E in fuzzy numbers form. Decision Makers Book B B B E E E B - ( 0.7,0.6,0.46) - ( 0.7,0.6,0.46 B - - ) B - - - B - ( 0.60,0.67,0.87) - ( 0.7,0.80, B - - ) B - - - B - ( 0.58,0.6,0.68) - ( 0.6,0.,0.7 B - - ) B - - - Proposition and Proposition are appied in order to construct the decision matrix for each eement of fuzzy numbers in Tabe into 0,. The decision matrix based on CLFPR is constructed and this is depicted in Tabe. E E E TBLE. Decision Matrix Based on CLFPR. B B B B ( 0.50,0.50,0.50) ( 0.0,0.7,0.) ( 0.09,0.04,0.4) B ( 0.64,0.68,0.7) ( 0.50,0.50,0.50) ( 0.0,0.7,0.) B ( 0.78,0.86,0.96) ( 0.64,0.68,0.7) ( 0.50,0.50,0.50) B ( 0.50,0.50,0.50) ( 0.8,0.4,0.47) ( 0.,0.6,0.47) B ( 0.5,0.5,0.59) ( 0.50,0.50,0.50) ( 0.4,0.45,0.50) B ( 0.5,0.5,0.64) ( 0.50,0.50,0.55, ) ( 0.50,0.50,0.50) B ( 0.50,0.50,0.50) ( 0.8,0.40,0.4, ) ( 0.07,0.,0.9) B ( 0.57,0.57,0.6) ( 0.50,0.50,0.50) ( 0.0,0.4,0.7) B ( 0.77,0.8,0.87) ( 0.70,0.7,0.76) ( 0.50,0.50,0.50) For brevity, the aggregated evauation of a aternatives from decision makers with respect to a criteria is stipuated and is given in Tabe 4. 0007-6
TBLE 4. Coective fuzzy evauation of textbooks with respect to criteria from decision makers. Criteria B B B verage B ( 0.,0.7,0.47) ( 0.,0.8,0.9) ( 0.0,0.5,0.7) ( 0.,0.6,0.8) C B (0.7,0.4,0.59) ( 0.,0.7,0.47) ( 0.9,04,0.5) ( 0.0,0.4,0.47) B ( 0.44,0.5,0.75) ( 0.40,0.47,0.6) ( 0.,0.7,0.47) ( 0.9,0.45,0.6) B ( 0.,0.4,0.8) ( 0.09,0.,0.4) ( 0.00,0.04,0.6) ( 0.09,0.,0.5) C B ( 0.7,0.48,0.56) ( 0.7,0.,0.4) ( 0.09,0.,0.6) ( 0.6,0.0,0.6) B ( 0.,0.6,0.75) ( 0.,0.7,0.57) ( 0.7,0.,0.4) ( 0.,0.6,0.48) B ( 0.0,0.,0.44) ( 0.0,0.6,0.8) ( 0.00,0.07,0.0) ( 0.,0.9,0.) C B ( 0.9,0.46,0.66) ( 0.0,0.,0.44) ( 0.0,0.4,0.6) ( 0.0,0.4,0.49) B ( 0.4,0.5,0.79) ( 0.4,0.9,0.57) ( 0.0,0.,0.44) ( 0.6,0.4,0.60) B ( 0.8,0.,0.48) ( 0.8,0.,0.9) ( 0.,0.6,0.4) ( 0.9,0.,0.9) C 4 B ( 0.0,0.5,0.58) ( 0.8,0.,0.48) ( 0.0,0.5,0.40) ( 0.6,0.0,0.48) B ( 0.,0.8,0.68) ( 0.0,0.4,0.55) ( 0.8,0.,0.48) ( 0.0,0.5,0.56) B ( 0.0,0.,0.44) ( 0.,0.8,0.) ( 0.08,0.4,0.8) ( 0.7,0.,0.4) C 5 B ( 0.8,0.44,0.6) ( 0.0,0.,0.44) ( 0.5,0.0,0.4) ( 0.,0.6,0.49) B ( 0.9,0.46,0.70) ( 0.,0.5,0.50) ( 0.0,0.,0.44) ( 0.,0.8,0.55) Evauation of the five criteria is aso made to determine the criteria weight vector. Tabe 5 shows the evauation for weight vector of the five criteria by the decision maker E, E and E. TBLE 5. Evauation of weight vector on each criteria by E, E and E. C C C C 4 C 5 E [ 4 E [ 6,S 7 E [ 8, S 9 S, S ] [ S,S ] [ S,S ] [ S,S 5 ] [ S,S ] S 6,S ] S 8, S ] S ] [ S 4,S 5 ] [ S,S 4 ] [ S 6, S 7 ] [ 7 S ] [ S 5, S ] [ S 6 4, S ] 5 [ S 5, S ] [ 6 9 eements in Tabe 5 wi be transformed into fuzzy numbers. The evauation from each expert wi be aggregated using arithmetic mean. The weight vectors for the three books are given in Tabe 6. 0007-7
TBLE 6. Evauation of criteria weight in fuzzy numbers. C C C C 4 C 5 B ( 0.,0.6,0.8) ( 0.09,0.,0.5) ( 0.,0.9,0.) ( 0.9,0.,0.9) ( 0.7,0.,0.4) B ( 0.0,0.4,0.47) ( 0.6,0.0,0.6) ( 0.0,0.4,0.49) ( 0.6,0.0,0.48) ( 0.,0.6,0.49) B ( 0.9,0.45,0.6) ( 0.,0.6,0.48) ( 0.6,0.4,0.60) ( 0.0,0.5,0.56) ( 0.,0.8,0.55) The evauation of the books with respect to the criteria and the criteria weight is composed together as subscribed by CLFPR. Finay, the rank the books are obtained by appying a simpe defuzzification formua Wi L M ) ( pij pij p L ij [] depicted in Tabe 7. TBLE 7. The Ordering of the Three Books. Fuzzy Number Fina Score Order B (0.6, 0.,0.) 0. B (0.6,0.,0.46) 0.4 B (0.,0.7,0.56) 0.4 Hence, the ranking order of the textbooks is obtained where B is the preferred choice. CONCLUSION The objective of this study is to propose a method that can manage the evauation from a different opinion with different assessment measurements in uncertain information and reduce the cacuation when the number of variabes or parameters increases. new decision making mode based on the muti-granuar uncertain inguistic information and CLFPR is deveoped. Muti-granuarity uncertain inguistic information is a usefu too in managing evauation from a different opinion with different assessment measurement. This approach gives the decision makers fexibiity in the evauation process as the judgment can be made in term of interva of inguistics vaues. The advantage using CLFPR as compared to other pairwise comparison methods in the rating of aternatives is that it preserves the consistency (by using proposition and ). Moreover, maintaining the data in the form of fuzzy numbers is important to avoid oss of information. The proposed method is capabe to sove various decision making probems in many areas of appications with a simiar setup. CKNOWLEDGMENTS The authors woud ike to thank the Facuty of Computer & Mathematic Sciences, Universiti Teknoogi MR (UiTM), the Research Management Institute (RMI) of UiTM and the Minister of Education Maaysia for the financia support from Fundamenta Research Grant Scheme (FRGS) : 600-RMI/FRGS 5/ (/0). REFERENCES. E. Herrera-Viedma, L. Martinez, F. Mata and F. Chicana, Fuzzy Systems,, 644 658 (005).. Y. Jiang, Z. Fan and J. Ma, Info. Sciences: n Inter. Journa, 78, 098 09 (008).. F. Herrera, E. Herrera-Viedma, L. Mart and J. S. Pedro, Inte. Sens. Evauation, 7-9 (004). 4. N. Haouani and L. Martínez, Muti-granuar Linguistic Promethee Mode. IFS/EUSFLT 009, 8 (009). 5. Z. Fan and Y. Liu, Expert Syst. With ppications, 7, 4000 4008 (00). 6. F. Herrera and L. Martinez, Syst., Man and Cybernetics-Part B: Cybernetics,, 7 4 (00). 7. T. C. Wang and Y.H. Chen, Consistent Fuzzy Linguistic Preference Reations for Computer Integrated Manufactory Systems Seection. Proceedings of the 9th Joint Conference on Information Sciences, Kaohsiung,Taiwan, 5 (006). 8. G.J. Kir and T.. Foger, Fuzzy Sets, Uncertainty and Information, Engewood Ciffs: Prentice, Ha. 988. 9. H.J. Zimmermann, Fuzzy Set Theory, Boston: Kuwer, 996. 0. S. Banerjee, and T. Kumar Roy, Fuzzy System,, 6 44 (0). 0007-8
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