A CONSISTENCY IMPROVING METHOD IN BINARY AHP


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1 Jurnal f the Operatins Research Sciety f Japan Vl. 38, N., March The Operatins Research Sciety f Japan A CONSISTENCY IMPROVING METHOD IN BINARY AHP Kazutm Nishizawa Nihn UniVfTsity (Received March 23, 992; Final September 2, 994) Abstract In this paper, a cnsistency imprving f the cmparisn matrix in binary AHP (Analytic Hierarchy Prcess) is studied. As its byprducts, we prpse a methd t cunt the cycles flength 3 and t find the lcatins f the cycles in a cmplete directed graph. We apply ur prpsed cnsistency imprving methd t varius examples including three actual sprts games. Cmparing ur methd with rdinary imprving methd, we can shw the usefulness f ur methd.. Intrductin In the AHP (Analytic Hierarchy Prcess []), the case where an element aij(i # j) f a cmparisn matrix A takes ne f nly tw intensity scale f imprtance values [2], either B r / B( B > ), is called "binary AHP". In general, cnsistency f a cmparisn matrix is usually measured by cnsistency index called "Cl". If Cl <., a cmparisn matrix is cnsisten t. In this paper, a cnsistency imprving f a cmparisn matrix at cmplete infrmatin case in incnsistent binary AHP is studied. In binary case, we can represent A by a directed graph [2] and can evaluate the incnsistency by the number f directed cycles in the graph. It is cnsidered that misjudgments cause cycles in the graph. It is nted that even if we suggest incnsistency in a cmparisn matrix, decisin maker judges cllecting nes r nt. First we prpse a methd t calculate the number f cycles in a directed graph thrugh Therem in 2. Secnd in 3 we prpse a methd t lcate each cycle. Next, in 4 an algrithm t imprve the cnsistency mst effectively by crrecting sme dubtful judgments is prpsed. In 5, we apply ur methd t several sprts g;ames, which are typical examples f binary AHP. In 6, we cmpare ur methd with rdinary imprving methd. 2. Prpsed Criterin f Cnsistency In the AHP, cnsistency f a cmparisn matrix is usually measured by [] (2. ) Cl = ().maxn,)/(n ) where n is an rder f the cmparisn matrix and), max is its maximum eigenvalue. It is said that if the value f Cl is less than. the cmparisn matrix is cnsistent. The dissatisfadins f Cl that we felt are as fllws. () The justificatin fr value. f Cl is nt theretically clear. (2) In incmplete infrmatin cases, it is impssible t have the value f Cl [3]. Since we want t have cnsistency befre estimating unknwn cmparisns. Nw we are ging t prpse a new criterin instead f Cl. We prpse ur new criterin f cnsistency fr a binary case. This idea is based n graphs and netwrks. If aij = B then 2
2 22 K. Nishizawa we represent it by a directed arrw (i, j) r i t j in the crrespnding cmplete graph in which any tw pints are cnnected by an arrw. Accrding t netwrk thery [4], we intrduce a vertex matrix V whse (i,j) element Vij = () if aij = (/) and Vii = O. The rth pwer f the vertex matrix represents the relatin cnnecting r + pints. In the binary case the incnsistency f a cmparisn matrix is caused by directed cycles in the crrespnding graph. It is clear that if there are n cycles in the graph it attains the maximum cnsistency [2J. Thus we are able t measure the incnsistency by the number f directed cycles (simple cycles hereafter) in the graph. There may be cycles f varius lengths, but we have nly t cnsider the cycles f length 3. Because, if we have a cycle f length 4, we are t have a cycle 24 r f length 3 accrding t 2 t 4 r 4 t 2. Thus we can measure incnsistency by the number f cycles f length 3. Here examples f cmparisn matrices and their graphs [2J are shwn. (2.2) Example f ~) CD Fig. Graph f Example (2.3) Example 2 Example 3 (2.4) A = (I~O CD Fig. 2 Graph f Example 2 I~\ CD Fig. 3 Graph f Example 3 In Example we have n cycle, since it is cmpletely cnsistent. We have ne cycle f length 3 in Example 2 and tw cycles (23 and 43) f length 3 in Example 3, since the frmer has ne incnsistency and the latter has tw incnsistencies. In rder t find the number f cycles f length 3 we can use the vertex matrix crrespnds t the cmparisn matrix. In binary case we can easily cnstruct the vertex matrix V frm the cmparisn matrix A as mentined abve. Thus we have the fllwing vertex matrices fr Examples,2 and 3: (2.5) V = (~ ~ i) fr Example, (2.6) fr Example 2, n i (2.7) v ~ ~ n fe Example 3. Cpyright by ORSJ. Unauthrized reprductin f this article is prhibited.
3 Cnsistency Imprving Binary AHP 23 We can calculate the number f cycles f length 3 in a given cmplete directed graph, r the incnsistency f a binary cmparisn matrix, by the fllwing therem. Therem Given a cmparisn matrix A in the binary AHP, the trace (sum f diagnal elements) f the third pwer f the vertex matrix V crrespnding t A is three times the number f the cycles f length 3; that is, (2.8) tr(v 3 )/3 = the number f direct cycles f length 3, where tr(m) is the trace f a matrix M. Prf The ith diagnal element f V 3 is (V 3 )ii = L L ViaVabVbi where ViaVabVbi == if a=lb=l and nly if pints i, a, b, i frm a cycle f length 3 (where Vij is (i,j) element f V ). Thus n (V 3 )ii represents the number f cycles that g thrugh pint i. Then L(V 3 )ii(= tr(v 3 )) i=l represents the number f all cycles cunted triple, which states frmula (2.8). Thus ur prpsed criterin f cnsistency is judged by diagnal elements f the matrix V 3 If diagnal elements f V 3 are all, this shws the crrespnding graph is acyclic and we judge it cnsistent, but if nt, the graph is a cyclic and we judge it incnsistent. Fr Example, we have (2.9) ). We see that Fig. is an acyclic graph, since it is cnsistent. Fr Example 2, we have (2.) V 3 = L ( ~) and tr(v3)/3 =. We see that there is ne cycle in Fig. 2. Fr Example 3, we have (i (2.) V 3 = and tr(v 3 )/3 = 2. We see that the graph in Fig. 3 has tw cycles f length 3. Having fund the methd t calculate the number f cycles, we prceed t lcate each cycle and this serves t imprve cnsistency. 3. Lcating f Cycles (3. ) Infrmatin n lcatins f cycles is cmpletely included in the fllwing matrix S; where * peratin means elementwise multiplicatin f tw matrices, that is, fr A [aij], B = [bij] we have A * B = [aijbij] and VT means the transpsed matrix f V. n n Cpyright by ORSJ. Unauthrized reprductin f this article is prhibited.
4 24 K. Nishizawa Therem 2 Let Sij be (i, j) element f matrix S defined as abve then Sij is the number f cycles that includes arc (i,j) in the crrespnding graph. n Prf By definitin we have Sij = L ViaVajVji, where "a" runs n all pints in the graph. a=l If and nly if "a" satisfies the cnditin as shwn in Fig. 4, that is, (i,a,j,i) frms a cycle, ViaVajVji =. Thus Sij is the number f cycles including arc (i,j). Of curse, we have Sij = if Vji =, and Sii = O. '~la VaJ=l ~ VJI~. J Fig. 4 Cnditin f fnning a cycle Therefre if the elements n ith rw f matrix S are all zer, pint i is nt invlved in any cycles. If jth elements n ith rw is n, pint i has n arcs n cycles incming frm pint j. Incidentally sum f each ith rw cincides with the value f ith diagnal element f V 3 which represents the number f cycles passing thrugh pint i. Using the infrmatin included in matrix S we can find the cycles f length 3 in a given directed graph r a cmparisn matrix. We make submatrix Sijk cmpsed f elements in the i, j and kth rws and i, j and kth clumns f matrix S as fllws (i, j, k E {I, 2,..,n}, i < j < k): (3.2) If and nly if 3 x 3 submatrix Sijk has zerrws r zerclumns, the pint i, j and k frm n cycle (where a zerrw means a rw whse cmpnents are all zers). Fr Example, we have (3.3) Example 2, (3.4) and Example 3, (3.5) Cpyright by ORSJ. Unauthrized reprductin f this article is prhibited.
5 Cnsistency Imprving Binary AHP 25 Fr Example 3, we have fllwing 4 submatrices, frming cycle (~ S23 = ~), (r S24 = ~), with n cycle, 2 S34 = (~ ~), frming cycle and S234 = (~ ~) with n cycle. As a result, we have fund tw cycles, and in Fig A Cnsistency Imprving Methd Generally speaking, incnsistency in AHP, r the number f cycles in the graph crrespnding t the cmparisn matrix A, is caused by misjudgments. In ur binary case "judge" is the directin f arc in ur graph. We ften ascertain that reversing the directin f specific arc disslves many cycles. The directin crrespnding such an arc must be based n misjudgment. If we find (i,j) element with value k in matrix S, we can reduce the number f cycles by k by reversing the directin f arc (i,j). Rughly speaking, we can reduce the incnsistency by reversing the directins f arcs that have large values in matrix S. When there are many such candidates, it is nt s clear which arc's directin shuld be changed mst effectively. Here we shw an effective algrithm reducing as many cycles as pssible by reversing directins f minimum number f arcs. Algrithm fr extinguishing cycles Step : Enumerate all cycles f length 3 in the given graph (by the methd in 3), and dente the set f all cycles by C. Step 2 : Enumerate all arcs included in any cycle in C, and dente the set f these arcs by A. Step 3 : Cnstruct the cyclearc incidence matrix whse (i,j) element is "I" if ith cycle in C includes jth arc in A, and therwise "". Or equivalently cnstruct a bipartite graph (C, A; J) where J is the set f edges cnnecting e( E C) and a( E A) are cnnected by an edge in.j if and nly if cycle e includes arc a. Step 4 : Find the minimum cvering set M(S;; A) which cvers all elements f C, where if a cycle e includes an arc a we say "a cvers e". (There are varius algrithms t find the minimum cvering sets in graph thery.) Step 5 : Reverse the directins f all arcs in M, then all cycles in the given graph eliminate. We explain the algrithm thrugh an example. Cpyright by ORSJ. Unauthrized reprductin f this article is prhibited.
6 26 K. Nishizawa Fr Example 3 (which has Cl = (,,\ max n)/(n  ) = ( )/(4 ) =.25) we have the cyclearc incidence matrix shwn in Table. Table Cyclearc incidence matrix fr Example 3 cycle~arc,2),3),4) (2, 3) (3,4) The crrespnding bipartite graph is shwn in Fig ~ ~~:~~,4) 34 (2, 3) (3, 4) Fig. 5 Bipartite graph fr Example 3 We have minimum cvering set M = {(I, 3)} as underlined in Fig. 5. If we reverse the directin f arc (,3) t (3,) then all cycles in the graph in Fig. 3 are eliminated and we have Cl =.4 reducing the value f Cl by =.75. This example shws that ur algrithm is useful t suggest the decisin maker his misjudgments. He had thught that is better than CD, which resulted a wrse cnsistency,.25. If he changed his judgment, that is, if he thught CD is better than then it wuld reduce the value f Cl by.75. Thus we can insist that his first decisin was a misjudgment. In the fllwing sectin we treat mre detailed examples, sprts games that any tw teams have a match and ne f them wins (and the ther lses) with n tie. Let us call such a game "league game". The fact that team i wins (team j lses) crrespnds t judgment "i is better than j". Thus league game can be treated as ur binary AHP, and ur methd can be applied t league games. 5. Applicatins t Sprts Games In rder t cnfirm the validity f ur methd, we apply ur methd in AHP t decide ranking in league sprts games. In an incnsistent case, cycles f three teams that is triangular cntest ften ccur. It is cnsidered that "misjudgment" in rdinary AHP crrespnds t "accidental victry (r defeat)". Our main bject is t find ut and suggest these accidental results. Imprving cnsistency crrespnds t crrecting accidental results. In an incnsistent case the islated cycle f three teams that is a triangular cntest ften ccurs. Then these three teams are cnsidered t have almst equal ability. Thus we cnsider there are n accidental results. 5. Applicatin (Tky sixuniversity baseball league, Fall 99) First the Tky sixuniversity baseball league is cnsidered. Its results f matches and its graph are shwn in Fig. 6 and Fig. 7, respectively. In this case there is nly ne cycle in Fig. 7. Cpyright by ORSJ. Unauthrized reprductin f this article is prhibited.
7 Cnsistency Imprving Binary AHP 27 ill ill '" CID X '" X X X X X '" CID X X X X X X X X X '" Fig. 6 Results f matches f Applicatin ill ';» CID '\! (5)< Fig. 7 "Graph f Applicatin The cmparisn matrix is shwn belw. (5. ) A _ l/b  [ l/b l/b l/b I l/b l/b B l/b fill We btain A max = 6.39 and Cl =.78 where = 2. T find cycles, we have V and S as belw. (5.2) V= [!!l (5.3) s= [!!l Then we find nly ne cycle as fllws. (5.4) Thus we judge this case ne incnsistency. We have the cyclearc incidence matrix in Table 2. Table 2 Cyclearc incidence matrix fr Applicatin cyc!e"'arc (2.3)(2.4)(3.4) The crrespnding bipartite graph is shwn in Fig. 8. Cpyright by ORSJ. Unauthrized reprductin f this article is prhibited.
8 28 K. Nishizawa ~ (2'3) (2,4) (3, 4) Fig. 8 Bipartite graph fr Applicatin We have minimum cvering set M = {(2,3)} r {(2,4)} r {(3,4)}. Thus we can nt suggest incnsistency lcatin in cmparisn matrix A shwn (5.). It is just a triangular cntest, and we cnsider that CD have almst equal ability. 5.2 Applicatin 2 (Private tennis league) Its results f matches and its graph are shwn in Fig. 9 and Fig., respectively. In this case there are three cycles in @ CD CD '" x " i x x x x " x x " x x x x x ), Fig. 9 Results f matches f Applicatin 2 ~ Fig. raph f Applicatin 2 The cmparisn matrix is shwn belw.!l [ (5.5) A = l~o /() l() We btain). max = 6.64 and Cl =.28 where = 2. T find cycles we have V and S as belw. (5.6) V= (5.7) s= [I 3 Cpyright by ORSJ. Unauthrized reprductin f this article is prhibited.
9 Cnsistency Imprving Binary AHP 29 Then we find three cycles as fllws. (5.8) (5.9) (5.) Thus we judge this case three incnsistencies. We have the cyclearc incidence matrix in Table 3. Table 3 Cyclearc incidence matrix fr Applicatin 2 cyc(e"'arc,2),3),4),5) (2,4) (3,4) (4, 5) The crrespnding bipartite graph is shwn in Fig ,2),3) ~, 5) ~ (2,4) ~ (3,4)  (4,5) Fig. Bipartite graph fr Applicatin 2 We have M = {(,4)} as underlined in Fig.. Thus we suggest that incnsistency lcatin in cmparisn matrix A, (5.5), is a4. If we reverse the directin f arc (4,) t (,4) then all cycles in Fig. are eliminated and we have Cl =.54. We cnsider that team wn an accidental victry ver team CD. 5.3 Applicatin 3 (Eastmetrplis universities baseball league, Fall 99) Its results f matches and its graph are shwn in Fig. 2 and Fig. 3, respectively. In this case there are fur cycles in Fig. 3. Cpyright by ORSJ. Unauthrized reprductin f this article is prhibited.
10 3 K. Nishizawa CD ~) <ID CD " <ID x " x i X " @) X X " \: 7 X X X x x x X " Fig. 2 Results f matches f Fig. 3 Graph f ApplicatIOn 3 The cmparisn matrix is shwn belw. () () () () /() /() n () () (5. ) [ /() () /() /() A == /() /() /() /() /() We btain Amax = and Cl =.49 where = 2. T find cycles we have V and S as belw. V~ [! (5.2) S~ 2 (5.3) 2 [I Then we find fur cycles as fllws. 2 (5.4) S234 = (~ 2 (5.5) S235 = (~ 2 (5.6) S346 = (~ (5.7) S456 = (~ ~) ~) ~) ~)!l Cpyright by ORSJ. Unauthrized reprductin f this article is prhibited.
11 Cnsistency Imprving Binary AHP 3 Thus we judge t.his case fur incnsistencies. We have the cyclearc incidence matrix in Table 4. Table 4 Cyclearc incidence matrix fr Applicatin 3 cyc!e""arc (2,3) (2,4) (2,5) (3,4) (3,5) (3,6) (4,5) (4,6) (5,6) {j4 4!i6 The crrespnding bipartite graph is shwn in Fig. 4. Lfhl (2, 4) (2. 5) (3, 4) (3, 5) (3,6) ::s:: (4, 5) ~  (5.6) Fig. 4 Bipartite graph fr Applicatin 3 We have M = {(2, 3), (4, 6)} as underlined in Fig. 4. Thus we suggest that incnsistency lcatins in cmparisn matrix A, (5.), are a2:l and a46. If we reverse the directin f arc (3,2) t (2,3) and arc (6,4) t (4,6) then all cycles in Fig. 3 are eliminated and we have Cl =.54. In this case, we have three candidates (2,3), (3,4) and (4,6), include tw cycles in Table 4. Rewriting these candidates, f curse, all cycles are eliminated. In Fig. 4, we see that it is nt always necessary t rewriting all candidates. 6. Cmparisn f Our Imprving Methd with Ordinary One The rdinary r cnventinal cnsistency imprving methd [5], is (a) t find the maximum errr element, (b) t crrect it and (c) t calculate the new eigenvectr,'..,wn f the crrected cmparisn matrix. In this methd an errr calculatin is based n (6.). (6. ) [eij] = (aij  w;/oj) Hwever, estimating incnsistency, here, rela,tive errr (6.2) instead f (6.) was calculated. (6.2) Fr Applicatin 3, we have A max = 6.744, Cl =.49 and the fllwing eigenvectr W, which are nrmalized with sum f elements equal t ( = 2). (6.3) W = [ ] Cpyright by ORSJ. Unauthrized reprductin f this article is prhibited.
12 32 K. Nishizawa First, using (6.2), fr cmparisn matrix A, (5.), we have (6.4) [..] _ e'j [ ) The maximum element f (6.4) is e46 (as underlined). We crrect a46 frm lib t B (a64 frm B t lib) and estimate again, we have (6.5) [eij] = [ ) ' A max = 6.52 and Cl =.2. The maximum element f (6.5) is e23 (as underlined). We crrect a23 frm lib t B (a32 frm B t lib) and estimate again, we have ) (6.6) [..] _ e'j ' [ A max = 6.27 and Cl =.54. The maximum element f (6.6) is e6 (as underlined). We crrect a6 frm B t lib (a6 frm lib t B) and estimate again, we have ) (6.7) [..] _ e l ) ' [ A max = and Cl =.56, thus we have a wrse result than (6.6), since stp the iteratin and (6.6) can be cnsidered as a final result. This cincides with ur result mentined in 5.3. By rdinary methd they must have much larger calculatin. Further, accrding t ur methd, we can easily suggest t crrect a23 and a46 at the same time with very small calculatin. If cmparisn matrix size becmes larger and has a lt f cycles, ur methd must be very effective. 7. Cnclusin T measure incnsistency f cmparisn matrix in binary AHP, we prpsed a new criterin t cunt hw many cycles in its graph. Further, we develped a cnsistency imprving methd, by finding and remving cycles in the graph. Our algrithm eliminates cycles in a graph mst effectively. We have applied ur prpsed cnsistency imprving methd t varius examples including three actual sprts games. Cmparing ur methd with rdinary imprving methd, we can shw the usefulness f ur methd. Cpyright by ORSJ. Unauthrized reprductin f this article is prhibited.
13 Cnsistency Imprving Binary AHP 33 Acknwledgments The authr wishes t thank Dr. Iwar Takahashi, Prfessr f Nihn University, fr useful advice n this investigatin. The authr is grateful t the referees fr their valuable and helpful cmments. References [IJ Saaty, T. L. : A Scaling Methd fr Pririties Hierarchical Structures, J. f Mathematical Psychlgy, Vl. 5, (977), [2] Takahashi, I: AHP Applied t Binary and Ternary Cmparisns, J. f O. R. Sciety f Japan, Vl. 33, (99), [3] Takahashi, I and M. Fukuda: Cmparisns f AHP with ther methds in binary paired cmparisns, Prceedings f the Secnd Cnference f the Assciatin f AsianPacific Operatinal Research Scieties within IFORS, (99), [4] Busacker, R. G. and T. L. Saaty: Finite Graphs and Netwrks: An Intrductin with Applicatins (in Japanese), Baifukan, (98), [5] Tne, K : Gemu kankaku ishiketteihu : Intrductin t the AHP (in Japanese), Nikkagiren, (99), Kazutm :~ishizawa Department f Mathematical Engineering, Cllege f Industrial Technlgy, Nihn University, 2, Izumich, Narashin, Chiba, 275, Japan Cpyright by ORSJ. Unauthrized reprductin f this article is prhibited.
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