A new approach for ranking trapezoidal vague numbers by using SAW method

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1 Science Road Publishing Corporation Trends in Advanced Science and Engineering ISSN: TASE 2() 57-64, 20 Journal homepage: A new approach or ranking trapezoidal vague numbers by using SAW method Tohid Komijani Department o Industrial Engineering, Iran University o Science and Technology t_komijani@ind.iust.ac.ir Ali Deheshvar Department o Industrial Engineering, Iran University o Science and Technology a_deheshvar@ind.iust.ac.ir Hesam Shams Department o Industrial Engineering, Shari University o Technology h_shams@ie.shari.edu Abstract Real lie problems are aected by various actors which are either unknown or too expensive to estimate thereore some aspects o the problems are not crisp. In this situation, the uncertainty theory is applied to deal with real world problems. Vague sets are one o the most important and the most requently applied tools or modeling problems in uncertain environments. Thereore, sometimes to make a decision it is necessary to compare two or more vague numbers. In spite o its application in decision making, no methods have been presented to compare two trapezoidal vague numbers. In this paper a two-sided decision making is used to rank vague numbers. In the proposed approach, Simple Additive Weighting (SAW) method is applied to rank trapezoidal vague numbers. At last, two numerical examples are illustrated to veriy the eiciency o the approach. Keywords: Numbers, Ranking Vague Numbers, SAW Method, Decision Making. Introduction Fuzzy sets or inexact data were irst proposed by Zadeh [] and many authors used the concept to ormulate uncertain conditions in real world problems and thus it has come to be used to overcome ambiguity in applicable problems [2], [3], [4], [5], [6] & [7]. Fuzzy sets are presented as vague sets by Gau [8], and Tan and Chen apply this type o uzzy sets in decision making under uncertainty [9]. We encounter vague numbers in many real world cases, thereore we need an approach to compare them and select the most desirable o them. Although these numbers are important and eicient, there is no approach or ranking vague sets. In this paper, we proposed a new approach based on the SAW method in order to compare and rank vague numbers. Two criteria, vagueness and location in this paper are considered in which less vagueness o a number and larger location, Science Road. All rights reserved 57

2 more important. The presences o decision makers are necessary to weigh all criteria in every step o problem solving. The paper is organized as ollows. A brie description o vague numbers, basic concepts, and saw method is considered in section 2. The proposed approach or ranking is presented in section 3. To illustrate the proposed approach, numerical examples are shown in section 4 and inally, conclusions are discussed in the inal section. 2. Basic Concepts 2.. Vague Sets A vague set is a set o objects, each o which has two values o grade o membership where these two values are called truth-membership unction and alse-membership unction. In such a set, truth-membership unction is shown by t ( ) A x and alse-membership unction is shown by A ( x ), or each element x in the vague set. The ollowing diagram depicts a vague set (Figure..). Figure.. A vague set It can be seen that a vague space has three major regions. Support region is a region where the evidences o existence o elements are kept there and existence o elements o the set is assured. Against region is a region where the evidences o reusal o elements o the sets are kept there and inexistence o elements o the set is assured. In hesitation region, we are under a shadow o vagueness and this because o lack o inormation. On the other hand membership unctions o all elements o the vague set are located there Decision Making by SAW Method SAW method is a type o multi-attribute value method. Value unction is obtained by a simple summation o scores in each criterion is multiplied by its weight. Decision maker should match the criteria by normalization. We obtain the rank value in the ollowing ormula ater normalization and scaling o the matrix. K v ( a ) w. v ( ( a )) with w 0 and w where w n k k k n k k k k k denotes the assigned weight or the k th criterion, K denotes numbers o criteria, n denotes numbers o decision matrix alternatives and v k ( k ( an )) is K 58

3 normalized value o k th criterion and n th alternative element. In this method we consider two normalizations: a) positive criteria In these criteria the greater values are more important. We normalize the criterion by dividing each element by the maximum element o that criterion. b) negative criteria In these criteria the lesser values are more important. We normalize the criterion by dividing the minimum element o that criterion by each element. 3. Ranking Numbers by Using SAW Method A simple trapezoidal vague number is depicted as ollows (Figure.2.): Figure. 2. A simple trapezoidal vague number This vague number is shown by six vagueness numbers and eight location numbers. A general orm o a trapezoidal vague number is considered as ollows: [(, 2, ab, e, cd, jh) A( a, b, j, c, d, h, e, )] where the selection approach o the criteria and description o the symbols are going as ollows. To rank such numbers by using SAW method, irst we survey them in two phases. Phase I is discussed about the vagueness and Phase II is discussed about the location. In vagueness phase, each trapezoidal number with vagueness is surveyed by six criteria and each trapezoidal number with location is surveyed by eight criteria and these criteria are considered as the main actors o every trapezoidal vague number. The purpose o these two phases is to choose a number in the best situation o vagueness and location. 3.. Vagueness Phase In this phase six criteria are considered. The decision maker speciies assigned weight o each criterion. Six criteria are: - Membership unction distance rom (we denote this by ). 2- Membership unction 2 distance rom membership unction (we denote this by 2 and less distance, less vagueness we have). 59

4 3- Let oot value (ab ) 4- Right oot value (e ) 5- Top middle value (cd segment) 6- Bottom middle value ( jh segment) 3.2. Location Phase Eight criteria are considered in this phase. Since these criteria are location criteria, greater value o them are more desirable. Eight criteria are: First location o let oot in the trapezoidal number (a ) Second location o let oot in the trapezoidal number (b ) Starting location o jh segment ( j ) Starting location o cd segment (c ) Ending location o jh segment ( h ) Ending location o cd segment (d ) First location o right oot in the trapezoidal number (e ) Second location o right oot in the trapezoidal number ( ) Note that according to commensurability in this phase there is no need to normalize Final Ranking Finally, ater the two phases by using SAW method, by considering the importance o vagueness and location rom decision maker perspective the inal rank o each trapezoidal vague number will be obtained. According to this rank or every numbers, we could compare them. Notice that in the inal decision making matrix, vagueness criteria are considered as negative criteria and location criteria are considered as positive criteria. Fig.3 is an example o intererences o some trapezoidal vague numbers. This shows that ranking o such numbers is not possible visually. Figure. 3. An example o intererences o some trapezoidal vague numbers Generally, ranking o trapezoidal vague numbers includes the ollowing steps: Step. Consider a table or vagueness phase and survey the six criteria in this table. Put the weight o criteria speciied by decision maker in the table and rank the matrix by SAW method. 60

5 Step 2. Consider a table or location phase and survey the eight criteria in this table. Put the weight o criteria speciied by decision maker in the table and rank the matrix by SAW method. Step 3. Put the obtained values rom vagueness and location matrices in new table. Note that vagueness criteria are considered as negative criteria and location criteria are considered as positive criteria. Decision maker speciies weights o vagueness and location. Ater normalizing the table, the inal ranking is obtained by using o SAW method. 4. Numerical Examples 4.. Example Consider the ollowing two trapezoidal numbers, A & B : [(, 3, 3.5, 4, 6, 6.5, 7, 9) A (0.4, 0.2, 2, 2, 3, 2)] [(2, 2.5, 4, 5, 6, 7, 7.5, 8) B (0.5, 0., 0.5, 0.5, 3, )] In this example the decision maker considered priorities same at phase & phase 2 thereore criteria are considered with equal values. But in the inal decision matrix, the weight o location criteria and the weight o vagueness criteria are considered 0.7 and 0.3 respectively. The data in table show the decision matrix or vagueness criteria. Table : vague table or two trapezoidal vague numbers Number A Number B Which its normalized orm is as ollows: Table 2: normalized vague table or two trapezoidal vague numbers Number A Number B The values obtained rom solution in this model are as ollow: U ( A ) 0.967, U ( B ) v Now, in table 3, ranking o location criteria by SAW method is shown. v 6

6 Vague Number A Vague Number B T. Komijani et al., Trends in Advanced Science and Engineering, 2(), 20 Table 3: location table or two trapezoidal vague numbers Criteri Criteri Criteri Criteri Criteri on 2 on 3 on 4 on 5 on 6 Criteri on Criteri on 7 Criteri on (As mentioned beore, no need to normalize this decision matrix) Ater solving the decision matrix we have: U ( A ) 5, U ( B ) 5.25 c And in the last step, the inal decision matrix (table 4) is obtained according to previous steps (note that in the matrix location criteria is considered positive and vagueness criteria is considered negative): Table 4: the inal decision matrix Vagueness Location Number A Number B weight Nature - + Ater normalizing, the inal values are as ollow: U ( A ) , U ( B ) Thereore, trapezoidal vague number A is preerred to trapezoidal vague number B Example 2 In this example, our trapezoidal vague numbers are considered: [(4, 5, 4.5, 6, 8, 9.5, 9, ) A (0.6, 0.,, 2, 5, 2)] [(6, 7, 6.5, 8,, 2, 3, 4) B (0.4, 0.,,, 5.5, 3)] [(, 2, 3, 3.5, 5.5, 6.5, 8, 8.5)C (0.2, 0.,, 0.5, 3.5, 2)] [(2, 3, 3.5, 3.5, 9, 9, 9.5, 0) D (0.5, 0.,, 0.5, 5.5, 5.5)] Again, in this example the decision maker considered priorities same at phase & phase 2 thereore criteria are considered with equal values. But in the inal decision matrix, the weight o location criteria and the weight o vagueness criteria are considered 0.8 and 0.2 respectively. In table 5 vagueness criteria decision matrix or trapezoidal vague numbers are shown. (Note that criterion 2 and criterion 3 are the same in the our numbers, we eliminate the related columns): c 62

7 Table 5: vague table or our trapezoidal vague numbers Number A Number B Number C number D By normalizing table 5 and solve it by SAW method, we have these results: U ( A ) 0.875, U ( B ) 0.68, U ( C ) 0.395, U ( D ) 0.77 v v v v Location criteria are shown in table 6. Table 6: location table or our trapezoidal vague numbers vague number A vague number B vague number C vague number D By solve it, we have the ollowing results: U ( A ) 7.25, U ( B ) , U ( C ) 4.75, U ( D ) c c c c Now with the previous results, we have the ollowing inal decision matrix: Table 7: the inal decision matrix Vagueness Location Number A Number B Number C Number D weight Nature - + Ater normalizing, the inal values are as ollow: U ( A ) , U ( B ) 0.962, U ( C ) 0.592, U ( D ) According to these values number B is selected as the best number and number D is selected as the worst number o these our trapezoidal vague numbers. 63

8 5. Conclusion Comparing o two trapezoidal vague numbers could apply in many problems in real world. Thereore we need a method to compare such numbers. In this paper a new approach based on SAW method is expanded or ranking them. In the proposed approach, by using speciied criteria or each trapezoidal vague numbers, the importance o vagueness and location is obtained and eventually by using the importance degree o vagueness and location, the rank o that number is determined. For urther study, applying MADM methods, such as AHP or ANP or ranking o these numbers and comparing them is suggested. Reerences: [] Zadeh, L.A.: Fuzzy sets. Inormation and Control 8 (965) [2] C.L.Hwang, K.Yoon. Multiple Attributes Decision Making Methods and Applications, Springer, Berlin Heidelberg, 98. [3] S,M,Chen. A new approach to handling multicriteria decision making problems, IEEE Trans. Systems Man and Cybernetic 8(988): [4] S.M. Chen, J.S.Ke, J.F.Chang. Techniques or handling multicriteria uzzy decision problems, Proc. 4th Internet Symp on Computer and inormation Sciences, Ceseme, Turkey, 989: [5] P.J.M. Laarhoven, W. Pedrycz, A uzzy extension o saaty s priority theory, Fuzzy Sets and System (983): [6] R.R.Yager. Fuzzy decision making including unequal objectives, Fuzzy Sets and Systems (978): [7] R.R. Yager. On ordered weighted averaging aggregation operators in multicriteria decision making, IEEE Trans. Systems Man and Cybernetics 8(988): [8]Gao, W.L., Danied, J.B.: Vague sets. IEEE Transactions on Systems, Man, and Cybernetics 23 (993) [9]Chen, S. M. and J. M. Tan. (994) Handling Multicriteria Fuzzy Decision making Problems Based on Vague set Theory, Fuzzy Sets and Systems 67,