COMPARISON OF ANALYTIC HIERARCHY PROCESS AND SOME NEW OPTIMIZATION PROCEDURES FOR RATIO SCALING

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1 Please cte ths artcle as: Paweł Kazbudzk, Comparso of aalytc herarchy process ad some ew optmzato procedures for rato scalg, Scetfc Research of the Isttute of Mathematcs ad Computer Scece, 0, Volume 0, Issue, pages The webste: Scetfc Research of the Isttute of Mathematcs ad Computer Scece, (0) 0, 0-08 COMPARISON OF ANALYTIC HIERARCHY PROCESS AND SOME NEW OPTIMIZATION PROCEDURES FOR RATIO SCALING Paweł Kazbudzk The Faculty of Socal Sceces, Ja Długosz Uversty, Polad Abstract. Dervg true prorty vectors from tutve parwse comparso matrces costtutes a key part of the Aalytc Herarchy Process. The Egevalue Method, commoly appled the Aalytc Herarchy Process, s the most popular cocept the process of rato scalg. It s kow that the Egevalue Method captures trastvty matrces that are ot cosstet a uque way. However, there are other methods such as statstcal estmato techques ad methods based o costraed optmsato models that are equally terestg. Ths artcle compares two ovel methods for prorty vectors dervg, whch combe the egevalue cocept wth a costraed optmsato based approach. Evdece s provded that cotrary to the logarthmc least squares method, they cocde wth the Egevalue Method capturg the rato scale rak order heret cosstet parwse comparso judgmets. Itroducto Plety of methods desged for the purpose of prortes establshmet o the bass of tutve judgmets ca be foud lterature. Some of them are based o dfferet statstcal cocepts [-], whle others focus o costraed optmzato models [-8]. Obvously, every method proposed the lterature has ts ow pros ad cos debate ad thus oe ca fd supporters ad adversares for each of them. Comparatve studes of dfferet prortzato methods [9-5], as well as suggestos to bled varous prortzato techques for better true prorty vector estmates [6], ca be foud as well. It seems that most of the kow prortzato methods ca be umbered amog costraed optmzato oes [7]. However, there are also a few others, cludg the most popular Egevalue Method ad two recetly troduced oes, whch combe the egevalue approach wth a certa costraed optmzato procedure.. Costraed optmzato methods These methods ca be descrbed the followg maer. Let us presume that we have oly judgmets (estmates) of the relatve weghts of a set of actvtes.

2 0 P. Kazbudzk The we ca express them a parwse comparso matrx (PCM) deoted as A wth elemets a j = a /a j that ca be preseted as follows: a / a a / a A = / a Μ a / a a / a a / a / a Μ a / a a / a / / Μ a / a / a a / a / a Μ a / a () Let us also deote A(w) as the symbol of a matrx wth elemets w j = w /w j that ca be preseted as follows: w / w w / w w / w K w / w w / w w / w w / w w / w K A( w ) = w / w w / w w / w K w / w M M M M w / w w / w w / w K w / w () Now, f we would lke to recover the vector of weghts w = [ w, w, w,, w ] whose true relatve weghts of a set of actvtes ca be created from, as the case of the above matrx A(w), we ca apply a optmzato method whch seeks a vector w as a soluto to the followg mmzato problem: m D( A, A( w )) () subject to some assged costrats such as postve coeffcets ad the ormalzato codto. As the dstace fucto D measures a terval betwee matrces A ad A(w), varous ways of ts defto lead to dfferet prortzato cocepts. It seems that the most popular oe s called the logarthmc least squares method (LLSM), kow also as the geometrc mea method [, 5, 6, 5]. I ths method the objectve fucto measurg the dstace betwee A ad A(w) s gve by: m D( A, A(w) ) = (l a l w + l w ) (), I order to receve the estmate of the prorty vector, objectve fucto () eeds to be mmzed wth subjecto to the followg costrats: = w =, w > 0, =,, j j T

3 Comparso of aalytc herarchy process ad some ew optmzato procedures for rato scalg 0 The LLSM soluto also has the followg closed form ad s gve by the ormalzed products of the elemets each row: w / / = a j a j (5) =. The egevalue method There s a method that caot be recogzed as oe of those characterzed as costraed optmzato oes. Ths method s a fudametal part of the mathematcal theory for dervg rato scale prorty vectors (PV) from postve recprocal matrces wth etres set o the bass of parwse comparsos. The theory s called the Aalytc Herarchy Process (AHP) ad t uses the prcpal Egevalue Method () to derve prorty vectors [,, 8-0]. It ca be descrbed the followg maer. Let us presume that we kow the relatve weghts of a set of actvtes. The we ca express them a PCM lke A(w) whch was descrbed above. Now, f we would lke to recover the vector of weghts w whch the ratos A(w) ca be created from, we could take the matrx product of matrx A(w) = [w j ] x wth vector w order to receve: w/ w w/ w w/ w Μ w / w w/ w w/ w w/ w Μ w / w w/ w w / w w/ w Μ w / w w/ w w w w/ w w w w/ w w = w Μ Μ Μ w / w w w If we kow A(w), but ot w, we ca solve ths problem for w. Solvg for a ozero soluto for ths set of equatos s a very commo procedure ad s kow as a egevalue problem: (6) A ( w) w = λ w. (7) I order to fd the soluto to ths set of equatos, geeral, oe eeds to solve a th order equato for λ that, geeral, leads to uque values for λ, wth a assocated vector w for each of the values. However, the case of PCM based o prorty weghtg, matrx A(w) has a specal form, sce each row s a costat multple of the frst row. I ths case, matrx A(w) has oly oe ozero egevalue ad sce the sum of the egevalues of a postve matrx s equal to the sum of ts dagoal elemets, the oly ozero egevalue such case equals the sze of the matrx ad ca be deoted as λ max =. If the elemets of a matrx A(w) satsfy codto w j = /w j for all, j =,,, the matrx A(w) s sad to be recprocal. If ts elemets satsfy codto w k w kj = w j for all, j, k =,, ad the

4 0 P. Kazbudzk matrx s recprocal, the t s called cosstet. Fally, matrx A(w) s sad to be trastve f the followg codto holds: f elemet w j s ot less tha elemet w k the wj wk for =,,. It s obvous that real lfe durg prorty weghg we do ot have A(w) but oly ts estmate A cotag our tutve judgmets, more or less close to A(w) accordace to our sklls, experece, etc. I such a case, the cosstecy property obvously does ot hold ad the relato betwee elemets of A ad A(w) ca be expressed the followg form: a = e w (8) j j j where e j s a perturbato factor whch should be close to. It has bee show that for ay matrx, small perturbatos the etres mply smlar perturbatos the egevalues, that s why order to estmate true prorty vector w, oe eeds to solve the followg matrx equato: A w = λ max w (9) where λ max s the prcpal egevalue, t s ot smaller tha, ad other characterstc values are close to zero. The estmates of true prorty vector w ca be foud the by ormalzg the egevector correspodg to the largest egevalue equato (9) whch s smple ad ts exstece s guarateed by Perro s Theorem.. Least absolute- ad least squared devato approxmato method It has bee devsed [] that stead of solvg egevalue equato (9), oe may seek a vector w whch best estmates equato (7). I order to satsfy equato (7) as accurately as possble, two ew methods were recetly proposed commucato [], they were called: least absolute devato approxmato (deoted ) ad least squared devato approxmato (deoted ). I order to estmate PV from the, the followg goal programmg model was formulated: ( d ) + + d m (0) = subject to: + d d + a w = w j j where [ + w j =, w 0, d 0, d 0, =,..., d, d, d,, d ] = Aw w. T

5 Comparso of aalytc herarchy process ad some ew optmzato procedures for rato scalg 05 I order to estmate PV from the, the followg costraed optmzato model was formulated: subject to:. [ ] T where B ( A I ) ( A I ) m w T Bw () wj =, w 0, =,...,, = wth I beg a detty matrx of order.. A example scearo based aalyss I ths secto of the artcle, we provde the ad effcacy aalyss based o already publshed case studes. Some examples provded the lterature [] showed a sequetally small ad drastc dscrepacy betwee the results obtaed wth the applcato of the ad LLSM. We adopt here the AHP model preseted there order to aalyze f there s the same dscrepacy betwee the, ad. The frst two scearos are smple AHP models. For the overall goal, there are four crtera: c, c, c, ad c. For each crtero, there are four alteratves: a, a,, ad, whch are the same for all the four crtera. The judgmet matrces ad correspodg estmato of PVs obtaed wth the applcato of the,, ad, respectvely, are provded below. We start from scearo o. : wth respect to the GOAL: c c c c c / / / c / / c / c wth respect to crtero c ad c: a a a / / / a / / /

6 06 P. Kazbudzk a a a a / / / wth respect to crtero c ad c: / / / After sythess, we obta the followg overall rakg: a a We ote that all three methods cocde wth the alteratves raks, resultg a > > a >. Now, we aalyze scearo o. : c c c c c / / / c / / wth respect to the GOAL: c c / wth respect to crtero c ad c: a a a a a / / / a a a / / / / / / wth respect to crtero c ad c: / / /

7 Comparso of aalytc herarchy process ad some ew optmzato procedures for rato scalg 07 After sythess, we obta the followg overall rakg: a a We ote that all three methods aga cocde wth the alteratves raks, resultg : > > a > a. We resume ow wth the followg coclusos. Coclusos To summarze, there are other vald methods for dervg the prorty vector from a parwse comparso matrx, especally whe the matrx s cosstet, that are equally satsfyg as the egevalue method. As was preseted ths artcle, o the bass of a example scearo aalyss, there are at least two such methods: the least absolute devato approxmato ad least squared devato approxmato. What s more, the two latter methods, as optmzato based, allow the decso maker to troduce addtoal costrats reflectg some addtoal requremets coected wth the preferece modellg. Ackowledgmets The author would lke to thak Adrzej Z. Grzybowsk for the sprato of ths survey ad costructve commets durg the preparato of ths paper. Refereces [] Basak I., Comparso of statstcal procedures aalytc herarchy process usg a rakg test, Mathematcal Computato Modellg 998, 8, [] Crawford G., Wllams C.A., A ote o the aalyss of subjectve judgmet matrces, Joural of Mathematcal Psychology 985, 9, [] Lpovetsky S., Tshler, A., Iterval estmato of prortes the AHP, Europea Joural of Operatoal Research 997,, 5-6. [] Bryso N., A goal programmg method for geeratg prorty vectors, Joural of the Operatoal Research Socety 995, 6, [5] Cook W.D., Kress M., Dervg weghts from parwse comparso rato matrces: A axomatc approach, Europea Joural of Operatoal Research 988, 7, [6] Hashmoto A., A ote o dervg weghts from parwse comparso rato matrces, Europea Joural of Operatoal Research 99, 7, -9. [7] L C-C., A ehaced goal programmg method for geeratg prorty vectors, Joural of the Operatoal Research Socety 006, 57, 9-96.

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