A CONSISTENCY IMPROVING METHOD IN BINARY AHP

Jurnal f the Operatins Research Sciety f Japan Vl. 38, N., March 995 995 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 = ().max-n,)/(n -) where n is an rder f the cmparisn matrix and), max is its maximum eigen-value. 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

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 r-th 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 -2-3-4 f length 4, we are t have a cycle -2-4 r 2-3-4 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 (-2-3 and -4-3) 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.

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 i-th 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.

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 i-th rw f matrix S are all zer, pint i is nt invlved in any cycles. If j-th elements n i-th rw is n, pint i has n arcs n cycles incming frm pint j. Incidentally sum f each i-th rw cincides with the value f i-th 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 k-th rws and i, j and k-th 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 zer-rws r zer-clumns, the pint i, j and k frm n cycle (where a zer-rw 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.

Cnsistency Imprving Binary AHP 25 Fr Example 3, we have fllwing 4 submatrices, frming cycle (D-@-@ (~ S23 = ~), (r S24 = ~), with n cycle, 2 S34 = (~ ~), frming cycle (D-@-@, and S234 = (~ ~) with n cycle. As a result, we have fund tw cycles, (D-@-@ and (D--@ in Fig. 3. 4. 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 cycle-arc incidence matrix whse (i,j) element is "I" if i-th cycle in C includes j-th 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.

26 K. Nishizawa Fr Example 3 (which has Cl = (,,\ max -n)/(n - ) = (4.644-4)/(4- ) =.25) we have the cycle-arc incidence matrix shwn in Table. Table Cycle-arc incidence matrix fr Example 3 cycle~arc,2),3),4) (2, 3) (3,4) -2-3 -3-4 The crrespnding bipartite graph is shwn in Fig. 5. -2-3 ~ ~~:~~,4) -3-4 (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.25 -.4 =.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 six-university baseball league, Fall 99) First the Tky six-university 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.

Cnsistency Imprving Binary AHP 27 ill CID @ CID @ 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 i ~ @ @ '\! (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 cycle-arc incidence matrix in Table 2. Table 2 Cycle-arc incidence matrix fr Applicatin cyc!e"'arc (2.3)(2.4)(3.4) 2-3-4 The crrespnding bipartite graph is shwn in Fig. 8. Cpyright by ORSJ. Unauthrized reprductin f this article is prhibited.

28 K. Nishizawa ~ (2'3) 2-3-4 (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 teams @, CD and @ 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 Fig.. CD @ @ @ @ @ CD ---~) @ CD '" X @ x "- i ~ @ x x "- 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.

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 cycle-arc incidence matrix in Table 3. Table 3 Cycle-arc incidence matrix fr Applicatin 2 cyc(e"'-arc,2),3),4),5) (2,4) (3,4) (4, 5) -2-4 -3-4 -5-4 The crrespnding bipartite graph is shwn in Fig.. -2-4 -3-4 -5-4,2),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 (East-metrplis 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.

3 K. Nishizawa CD <ID @ @) @ @ CD ---~) <ID CD "- <ID x "- x i ~ @ X "- X X @ @ @) X X "- \: 7 X @ X X X "- @ x x x X " Fig. 2 Results f matches f Applicatin 3 @ @< Fig. 3 Graph f ApplicatIOn 3 The cmparisn matrix is shwn belw. () () () () /() /() n () () (5. ) [ /() () /() /() A == /() /() /() /() /() We btain Amax = 6.744 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.

Cnsistency Imprving Binary AHP 3 Thus we judge t.his case fur incnsistencies. We have the cycle-arc incidence matrix in Table 4. Table 4 Cycle-arc 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) 2-4-3 2-5-3 3-{j-4 4-!i-6 The crrespnding bipartite graph is shwn in Fig. 4. Lfhl 2-4-3 (2, 4) 2-5-3 2 (2. 5) (3, 4) (3, 5) 3-6-4 (3,6) 4-5-6 ::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 eigen-vectr,'..,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 eigen-vectr W, which are nrmalized with sum f elements equal t ( = 2). (6.3) W = [.258.79.5.5.44.7] Cpyright by ORSJ. Unauthrized reprductin f this article is prhibited.

32 K. Nishizawa First, using (6.2), fr cmparisn matrix A, (5.), we have (6.4)..279.45.45.386..37.48 [..] _.7.578.. e'j -.7.688.5. [.8.63.522.9.99.3.54.65.38.5.23.).93.35.477.596...38.63. 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)..376.273...449 [eij] =.24.326.592.927..4 [.6.375.542.92.528.2.246.48.58.352.6)..696.9...6.375..6.5 '.58.78..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..37.26.26..587).587...37.26.26 (6.6) [..] _.26...26.587.37 e'j -.26.587.26..37. ' [..26.37.587..26.37.26.587..26..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..456.34.327.66 2.429).839..5.38.233.22 (6.7) [..] _.32.52..33.64.585 e l ) -.485.64.233..38.364 ' [.99.33.38.64..486.78.268.4.57.947..A max = 6.782 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.

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), 234-28. [2] Takahashi, I: AHP Applied t Binary and Ternary Cmparisns, J. f O. R. Sciety f Japan, Vl. 33, (99), 99-26. [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 Asian-Pacific Operatinal Research Scieties within IFORS, (99), 325-33. [4] Busacker, R. G. and T. L. Saaty: Finite Graphs and Netwrks: An Intrductin with Applicatins (in Japanese), Baifu-kan, (98), 99-2. [5] Tne, K : Gemu kankaku ishi-kettei-hu : Intrductin t the AHP (in Japanese), Nikkagiren, (99), 38-4. Kazutm :~ishizawa Department f Mathematical Engineering, Cllege f Industrial Technlgy, Nihn University, -2-, Izumich, Narashin, Chiba, 275, Japan e-mail: nisizawa@su.cit.nihn-u.ac.jp Cpyright by ORSJ. Unauthrized reprductin f this article is prhibited.