Applying the fuzzy preference relation to the software selection

Transcription

1 Proceedings of the 007 WSEAS International Conference on Comuter Engineering and Alications, Gold Coast, Australia, January 17-19, Alying the fuzzy reference relation to the software selection TIEN-CHIN WANG 1, YING-HSIU CHEN (1) Deartment of Information Management I-Shou University 1, Section 1, Hsueh-Cheng Road, Ta-Hsu Hsiang, Kaohsiung 840 TAIWAN htt://wwwkingidvtw () Deartment of Information Engineering I-Shou University 1, Section 1, Hsueh-Cheng Road, Ta-Hsu Hsiang, Kaohsiung 840 TAIWAN Abstract: - Software selection is not a well-defined or structured decision roblem Lai et al [5] uses the Analytic Hierarchy Process (AHP) method to solve a Multimedia authoring systems (MAS) selection roblem However, the AHP method is difficult to obtain a convincing consistency index when the number of attributes or alternatives is increased Hence, this aer alies the fuzzy reference relation (Fuzzy PreRa) method to solve the roblems mentioned in the aer written by Lai et al [5] Alying the Fuzzy PreRa method to decision making is more efficient, as not only the ratio of the airwise comarison times of the riority weight for the six factors between Fuzzy PreRa and AHP is 5:15, but also the outcome obtained by Fuzzy PreRa almost coincides with the one that roduced by the AHP method Key-Words: - Fuzzy reference relation, Fuzzy PreRa, AHP, Multimedia authoring systems (MAS) 1 Introduction Software selection is not a well-defined or structured decision roblem Owing to the comlexity of the roduct and rofusion of alternatives, a systematic rocess of selection can be formidable and exensive Hence, Lai et al [5] uses the AHP method to solve a MAS (Multimedia authoring systems) selection roblem Analytic Hierarchy Process (AHP) is an imortant material analysis method which was roosed by Saaty [8,9] in 1977 The strength of AHP is that it organizes tangible and intangible factors in a systematic way, and rovides a structured yet relatively simle solution to decision making roblems [11] U to resent, the AHP method has been alied in many different domains, including risk assessment [1], enterrise resource lanning assessment [16], roect management [1] and so forth However, additional to the fact that the AHP method must erform very comlicated airwise comarisons amongst elements (attributes or alternatives), and it is also difficult to obtain a convincing consistency index with an increasing number of attributes or alternatives Hence, there are imroved techniques such as Fuzzy AHP [4], referenced AHP [10], extended Fuzzy AHP [15], modified AHP [13], random AHP [6], chainwise aired comarisons [7] and DS/AHP [] which have been roosed in rofessional ournals and conferences involving various discilines Herrera-Viedma et al [3] develoed a new method, the fuzzy reference relation (Fuzzy PreRa), which focuses on avoiding the inconsistent solutions in the decision making rocesses The obect of this aer is alying the Fuzzy PreRa method to solve the roblems of in the aer written by Lai et al [5] The aer is organized as follows: section outlines the basis and concet of the fuzzy reference relation (Fuzzy PreRa) method, section 3 contains two arts, 31 gives a brief review of the method of Lai et al [5] for software selection, and 3 emloys the examle from [5] to illustrate the selection of a multimedia authoring system of Fuzzy PreRa Finally, discussion and some concluding remarks are resented in section 4 and 5, resectively Fuzzy Preference Relation Perfect consistency is difficult to obtain in ractice, esecially when measuring references on a set with a large number of alternatives The Fuzzy PreRa [3]

2 Proceedings of the 007 WSEAS International Conference on Comuter Engineering and Alications, Gold Coast, Australia, January 17-19, method focuses on avoiding the inconsistent solutions in the decision making rocesses The concet of this method is that if there are n attributes X = { x1,, xn, n }, then we can obtain the airwise reference relation data { 1, 3,, n 1n}, from n 1 comaring and constructing a consistent recirocal fuzzy reference relations P This method follows the one of traditional AHP method characteristics, which is reference relation satisfied transitivity roerty Herrera-Viedma et al [3] had roof that for a recirocal additive fuzzy reference relation P= ( i ), the following statements are equivalent: 3 i + k + ki = i< <k (1) i + 1 ii ( + 1) + ( i+ 1)( i+ ) + + ( i) + i= i< () According to (1) (), therefore, we can deduce that i+ 1 i = i( i+ 1) ( i+ 1)( i+ ) ( 1) (3) and according to the additive recirocal + = 1 i, {1,, n } (4) i i i The indicates the ratio of the reference intensity of alternative x i to that of x If i = 1/ imlies there is no difference between x i and x ( xi ~ x ), i > 1/ imlies xi is referred to x ( xi > x ), i = 1 indicates that x i is absolutely referred to x The stes are described in the following: 1 Comute the set of reference values B as B= { i, i< i { 1, 3,, n 1n}} (5) Find P P= { 1, 3,, n 1n} B (6) {1 1,1 3,,1 n 1n} B 3 The consistent fuzzy reference relation P is obtained as P = f( P) such that f :[ a,1 + a] [0,1] x + a f( x) = (7) 1 + a If there is a set of alternatives X = { x1,, x n }, which is associated with a recirocal multilicative reference relation A= ( a i ), and ai [1/ 9,9], then we can use equation (8) to find the corresonding recirocal additive fuzzy reference relation P= ( i ), and i [0,1] = ga ( ) = 1/ (1+ log a) (8) i i 9 i Thus, we rofessed that equation (8) is a transformation function g, and using the function g a recirocal multilicative reference relation matrix can be transformed into a kind of reference relation After the Fuzzy PreRa method was roosed, Wang and Chen [14] adated it with the linguistic concet, which uses variables as fuzzy linguistic assessments, and roosed it as the fuzzy linguistic reference relations method 3 Methodology 31 Review of MSA selection with AHP by Lai et al [5] The evaluation and selection of a MAS are imortant arts of multimedia develoment roects In their aer, Software selection: a case study of the alication of the analytical hierarchical rocess to the selection of a multimedia authoring system, Lai et al [5] alied AHP method in software selection Decision makers udge the imortance of one alternative over another can be made subectively and converted to a numerical value using a scale of 1-9 where 1 denotes equal imortance and 9 denotes the highest degree of favoritism There are six imortant factors used to measures the erformance of MSA roduct: develoment interfaces (F1); grahic suort (F); multi-media data suort (F3); data file suort (F4); cost effectiveness (F5) and vendor suort (F6) The reference relation matrix for airwise comarison of criteria is shown in Table 1 (which reresents the evaluation of member 1 of the grou) [5] Table 1 Preference relation matrix for airwise comarison of criteria (AHP) F1 1 1/3 1/4 1/3 5 7 F 3 1 1/3 1/4 5 6 F F / F5 1/5 1/5 1/7 1/6 1 5 F6 1/7 1/6 1/8 1/8 1/5 1 After obtaining the airwise comarisons, a normalized matrix was develoed in Table by

3 Proceedings of the 007 WSEAS International Conference on Comuter Engineering and Alications, Gold Coast, Australia, January 17-19, dividing each element by the sum of its resective columns The row entries in the last two columns of the normalized matrix table comrised the sum of six elements in the row and the average of those row elements (rincial vector), resectively Table Normalized matrix of Table 1 F F F F F F Thus, the relative weights for each attribute, after calculation, are found to be: F1=0108, F=01606, F3=03596, F4=0786, F5=00548 and F6=0056 Given by the mathematical exression shown below: F3>F4>F>F1>F5>F6 3 Fuzzy PreRa for MSA selection This study uses fuzzy reference relation method re-comutation with the six evaluation criteria in [5] First, this study alies Eq (8) to transform the original data a1, a3, a34, a45, a56, which transforms the recirocal multilicative reference relation with an interval scale [1/ 9,9] into a recirocal additive fuzzy reference relation with an interval scale [0,1] from a set of n 1 reference values { 1, 3,, n 1n} In this case, there are only five comarisons required for six evaluation criteria, which are as follows: 1 = (1 + log 91/ 3) / = 05 3 = (1 + log 91/3) / = = (1 + log9 ) / = = (1 + log9 6) / = = (1 + log 5) / = Then, using Eq (1) ~ (4), we obtain the entire reference relation matrix, which is shown in Table 3 For clarity, examles for 1, 31, 4 and 5 are shown below: 1 = 1 1 = 1 05 = = = = = = = 0593 = = Table 3 Preference relation matrix for airwise comarison of criteria (Fuzzy PreRa) F F F F F F We make note that, the rimary values in Table 3 are not in the interval [0,1], but in an interval[ a, 1 + a], being a > 0, we need to transform the obtained values using Eq (7) a transformation function which reserves recirocity and additive consistency, that is a function f :[ a,1 + a] [0,1] According to Eq (7), a = 04317, therefore, the entire reference relation matrix can be transformed as shown in Table 4 and the normalized matrix is shown in Table 5 Table 4 After transformation matrix F F F F F F Table 5 Normalized matrix of Table 4 F F F F F F Thus, the relative weights for each attribute, after calculation, are found to be: F1=01469, F=01968, F3=0466, F4=015, F5=01338 and F6=00608 Given by the mathematical exression shown below: F3>F4>F>F1>F5>F6 4 Discussion This study alies the fuzzy reference relation (Fuzzy PreRa) to solve the roblems in Lai s aer

4 Proceedings of the 007 WSEAS International Conference on Comuter Engineering and Alications, Gold Coast, Australia, January 17-19, [5] The analyzed outcome obtained by Fuzzy PreRa almost coincides with that roduced by the AHP method, shown in Table 6 Notably, the ratio of the airwise comarison times of the riority weight for the six influential factors between Fuzzy PreRa and AHP is 5:15 Table 6 Comaring AHP and Fuzzy PreRa comute result AHP ranking Fuzzy PreRa ranking F F F F F F Conclusion Although the AHP method has been widely alied in many research domains, it still has some roblems that need to be solved Therefore, owing to the fact that the AHP method must erform very comlicated airwise comarison amongst elements (attributes or alternatives), and since it is difficult to obtain a convincing consistency index with an increasing number of attributes or alternatives Then, Herrera-Viedma et al [3] develoed the fuzzy reference relation (Fuzzy PreRa) method to imrove above roblem If we have n attributes, using the AHP method we need to count the airwise comarison times of nn ( 1)/ to be able to obtain a result However, using the fuzzy PreRa (fuzzy reference relation) method we only need to count the airwise comarison times of n 1 In this case, there are six criteria, using AHP we need to comare the criteria 15 times, but using fuzzy PreRa we only need to comare them 5 times Regarding the airwise comaring times, fuzzy PreRa requires less comarisons than AHP by 10 times Fuzzy PreRa is clearly faster to execute and more efficient than AHP Alying Fuzzy PreRa method, it is ossible to assure better consistency of the fuzzy reference relations rovided by the decision makers, and in such a way as to avoid the inconsistent solutions in the decision making rocesses References: [1] Al-Harbi, Kamal M Al-Subhi, Alication of the AHP in roect management, International Journal of Proect Management, Vol19, 001, 19-7 [] Beynon, M, DS/AHP method: A mathematical analysis, including an understanding of uncertainty, Euroean Journal of Oerational Research, Vol140, No1, 00, [3] Herrera-Viedma, E, Herrera, F, Chiclana, F and Luque, M, Some issues on consistency of fuzzy reference relations, Euroean Journal of Oerational Research, Vol154, No1, 004, [4] Laarhoven, PJM and Pedrycz, W, A fuzzy extension of Saaty's riority theory, Fuzzy Sets and Systems, Vol11, No3, 1983, 9-41 [5] Lai, Vincent S, Trueblood, Robert P and Wong, Bo K, Software selection: a case study of the alication of the analytical hierarchical rocess to the selection of a multimedia authoring system, Information and Management, Vol 36, No4, 1999, 1-3 [6] Liovetsky, S and Tishler, A, Interval estimation of riorities in the AHP, Euroean Journal of Oerational Research, Vol114, No1, 1999, [7] Ra, JW, Chainwise aired comarisons, Decision Sciences, Vol30, No, 1999, [8] Saaty, TL, Decision Making for Leader: The Analytic Hierarchy Process for Decisions in a Comlex World, RWS Publications, Pittsburgh, PA, 1988 [9] Saaty, TL, The Analytic Hierarchy Process, McGraw-Hill, New York, 1980 [10] Schoner, B and Wedley, WC, Ambiguous criteria weights in AHP: Consequences and solutions, Decision Sciences, Vol0, No3, 1989, [11] Skibniewski, MJ and Chao, L, Evaluation of advanced construction technology with AHP method, Journal of Construction Engineering and Management, Vol118, No3, 199, [1] Tsai, Ming-Chih and Su, Chin-Hui, Political risk assessment of five East Asian orts the viewoints of global carriers, Marine Policy, Vol9, No4, 005, [13] Tung, SL and Tang, SL, A comarison of the Saaty s AHP and modified AHP for right and left eigenvector inconsistency, Euroean Journal of Oerational Research, Vol106, No1, 1998, [14] Wang, TC and Chen, YH, A new method on decision-making using fuzzy linguistic assessment variables and fuzzy reference relations, The 9th World Multi-Conference on Systemics, Cybernetics and Informatics

5 Proceedings of the 007 WSEAS International Conference on Comuter Engineering and Alications, Gold Coast, Australia, January 17-19, (WMSCI 005), July 10-13, Orlando, Florida, USA (005) [15] Weck, M, Klocke, F, Schell, H and Rueauver, E, Evaluating alternative roduction cycles using the extended fuzzy AHP method, Euroean Journal of Oerational Research, Vol 100, No, 1997, [16] Wei, Chun-Chin, Chien, Chen-Fu and Wang, Mao-Jiun J, An AHP-based aroach to ERP system selection, International Journal of Production Economics, Vol 96, No1, 005, 47-6