ISAHP 001, Berne, Swtzerlan, August -4, 001 WHY NOT USE THE ENTROPY METHOD FOR WEIGHT ESTIMATION? Masaak SHINOHARA, Chkako MIYAKE an Kekch Ohsawa Department of Mathematcal Informaton Engneerng College of Inustral Technology, Nhon Unversty 1--1 Izum-chou, Narashno, Chba 75-8575, Japan m7snoha@ct.nhon-u.ac.p Keywors: atve error, lnear scale, entropy metho, smulaton Summary: When a ecson maker eces each element of a parwse comparson matrx on the bass of lnear scale, the entropy metho s expecte to prouce a weght vector the closest to the true weght vector. On the contrary when a ecson maker eces on the bass of exponental scale, the egenvector metho an the geometrc mean metho are expecte to prouce weght vectors closer to the true weght vector. 1. Introucton Varous methos, such as the egenvector metho (EGV), the geometrc mean metho (GMM), an the entropy metho (ENT) [1], have been propose to estmate a weght vector from parwse comparson ata. In ths paper we wll show that when a ecson maker eces each element of a parwse comparson matrx A={ a } on the bass of lnear scale wth regular ntervals, among the egenvector metho, the geometrc mean metho, an the entropy metho, the entropy metho prouces a weght vector whch s statstcally the closest to the true weght vector.. Multplcatve Error an Atve Error Let A={ a } be a measure n n parwse comparson matrx, W={ w } be the consstent n n parwse comparson matrx, x 0 be a true weght vector, an E={ e } be an n n error matrx assocate wth A an W. In general, A can be expresse as a functon of W an E. A= f ( W, E ) (1) In ths paper we wll conser two types of error functons, multplcatve error an atve error. In case of multplcatve error wth error matrx M, f(w,m) s expresse by Eq.(), an n case of atve error wth error matrx D, f(w, D) s expresse by Eq.(3). f ( W, E ) = W * M () f ( W, D ) = W + D (3) Here, the operaton * n Eq.() ncates the matrx operaton for elementwse prouct, where =b c for D=B*C, an the operaton + n Eq.(3) s the ornary matrx aton, where D=B+C. =b + c for Proceengs 6 th ISAHP 001 Berne, Swtzerlan 431
3. Theorems concernng Scale Type an Error Type Snce we have ntrouce two types of errors n Sec., we wll gve two theorems whch relate an error type an a scale type. In the case of a lnear scale as n Fg.1, f (= x / x ), true ugment value of a ecson maker, s strbute unformly over the whole scale an the ecson maker fxes hs or her ugment value to a, whch s one of the nearest screte values, (= - w ), the fference from a to w a, strbutes unformly over [ -0.5, +0.5 ]. Therefore, followng Theorem 1 s obtane. [ Theorem 1 ] If w = x / x, true (or consstent) ugment value of a Decson Makng Unt on the 9-stage lnear scale for the (, )th element of a measure parwse comparson matrx A={ w }, s strbute unformly over [ 0.5, 9.5 ],, the (, )th element of the atve error matrx D={ }, s strbute unformly over [ -0.5, +0.5 ], where takes a screte value among 1,, 3, 4, 5, 6, 7, 8, an 9. a a Smlarly n the case of an exponental scale as n Fg., the same scusson hols f we take logarthms on the exponental scale, an followng Theorem s obtane. [ Theorem ] If = /, the true ( or consstent ) ugment value of a Decson Makng Unt on the 5-stage w x exponental scale for the (, )th element of a measure parwse comparson matrx A={ strbute log-unformly over [, ], m, the (, )th element of the multplcatve error matrx 0 M={ m }, s strbute log-unformly over [. 5, 0. 5 ], where a takes a screte value among 1,, x 4, 8, an 16. 0.5 4. 5 + a } s 4. Smulaton Experment In orer to compare the three weght estmaton methos, EGV, GMM, an ENT, n ts weght estmaton accuracy uner the two types of error structures, followng smulaton experment s carre out. 1. Assume a true weght vector x 0. ( n=4 an x 0 =( 1 3 4,,, ) T n our experment ). 10 10 10 10 w w. Make consstent parwse comparson matrx W={ } ( = x / x ) from the true weght vector x 0. 3. Make a set of perturbe parwse comparson matrces from the consstent matrx W by applyng atve or multplcatve error to each element of W. 4. Apply the three weght estmaton methos, EGV, GMN, an ENT, to each of perturbe matrces. Let,, an x be estmate weght vector by EGV, GMM, an ENT, respectvely. x1 x 3 5. For each sample from the perturbe set, calculate,, an, the stance from the true x 0 weght vector to each of estmate weght vector ( =1,, 3 ). 6. Average the stance 0 ( =1,, 3 ) over the perturbe set. 01 x 0 03 Proceengs 6 th ISAHP 001 Berne, Swtzerlan 43
5. Experment Result Tables 1 an show average stances 01, 0, an 03, for the case of atve errors strbute unformly over [ -0.5, +0.5 ] an the case of multplcatve errors strbute log-unformly over [ 1, ], respectvely. From Table 1 an other relate smulaton experment result, t s shown that uner the atve error the entropy metho prouces a weght vector statstcally the closest to the true weght vector (as epcte n Fg.3), an from Table an other relate smulaton experment result, t s shown that uner the multplcatve error both the egenvector metho an the geometrc mean metho prouce weght vectors whch are statstcally closer to the true weght vector (as epcte n Fg.4). Table 1. Average stances uner atve errors unformly strbute over [-0.5, 0.5] n=4 n=8 n=1 01 0.100940 0.060048 0.073549 0 0.101054 0.060453 0.07394 03 0.081136 0.038540 0.050749 Table. Average stances uner multplcatve errors log-unformly strbute over [ n=4 n=8 n=1 01 0.007990 0.00479 0.003394 0 0.007989 0.004791 0.003393 0 0.008179 0.005308 0.00407 1, ] 6. Concluson Together wth the smulaton experment result n Sec.5 an the two theorems n Sec.3, t s conclue that when a ecson maker eces hs or her -ugment on wely-use lnear scale bass, the entropy a metho s expecte to prouce a weght vector whch s the closest to the unknown true weght vector. To analyze how the error s strbute uner more general contons s one of the future research problems. References [1] Masaak Shnohara, Entropy AHP an ts comparson wth conventonal AHP s, The Ffth Internatonal Symposum on The Analytc Herarchy Process (ISAHP 99.), pp165-170 (1999.8). Proceengs 6 th ISAHP 001 Berne, Swtzerlan 433
True ugment w 0.5 9.5 a 1 3 4 e 5 6 7 8 9 Fg.1 True ugment w, measure scretze ugment a, an the error on the 9-stage lnear scale True ugment w 1 16 a 1 4 e 8 1 Fg. True ugment w, measure scretze ugment a, an the error on the 5-stage exponental scale m EGV GMM EGV GMM True Weght True Weght ENT Fg.3 Schematc agram for relatve poston of the four weght vectors uner atve error error ENT Fg.4 Schematc agram for relatve poston of the four weght vectors uner multplcatve Proceengs 6 th ISAHP 001 Berne, Swtzerlan 434
Proceengs 6 th ISAHP 001 Berne, Swtzerlan 435