Abstract. Pawel Tadeusz Kazibudzki Jan Dlugosz University in Czestochowa, Poland

Transcription

1 AN EXPONENTIAL AND LOGARITHMIC UTILITY APPROACH TO EIGENVECTOR METHOD IN THE ANALYTIC HIERARCHY PROCESS: THE IDEA VALIDATION WITH EXAMPLE LOGISTIC APPLICATION problems 85 Pawel Tadeusz Kazibudzki Ja Dlugosz Uiversity i Czestochowa, Polad poczta@gmail.com Abstract Thorough ivetory cotrol, plaig ad coordiatio, wise ad thoughtful selectio of warehousig ad meas of trasport are ecessary for efficiet decisios makig i logistics. I order to make optimal decisios, it is ecessary to deliberate may optios from the perspective of differet criteria. It was empirically verified that such problems are difficult to solve without methodological support as they exceed humas perceptio capabilities. However, there are some techiques which ca facilitate the process of decisio makig whe successfully applied. Oe of the most commoly used techique for structured problems solvig is the Aalytic Hierarchy Process. It is popular because its prescribed procedure supports sophisticated mathematical theory called the Eigevalue Method which is accurate ad uique. It does ot mea that there are ot other methods that ca support the Aalytic Hierarchy Process although their applicatio has pros ad cos. Two relatively ovel methods were proposed recetly ad their validatio studies are the essetial part of this research. I compariso with others, their simplicity ad compliace with the Eigevalue Method are their most crucial but ot exclusive advatages. At the ed of the research, the example logistic problem was solved with their applicatio, ad the results of all methods were compared. Keywords: Aalytic Hierarchy Process, eigevalue method, optimizatio models, decisio support techiques, selectio of trasportatio modes. Itroductio Obviously problem solvig activities pervade may aspects of huma life ad its levels. Additioally, due to the iterdepedet ature of most problems, they ca also ifluece ad be affected by other problems what costatly takes place i the real life at differet levels of huma activity. Geerally, problem solvig process is cosidered to be a multi-stage process ad although may models ad frameworks have bee proposed i a literature, oe of the most popular is Herbert Simo s (1960) categorizatio assumig its three phases: itelligece, desig, ad choice. The itelligece phase cocers the problem idetificatio. I this stage the problem is recogized ad iformatio are gathered i order to formulate a problem defiitio. Desig phase refers to the idetificatio of alterative solutios to the problem. I the choice phase, the solutio alteratives are selected ad implemeted. From the three differet phases of problem solvig metioed above, the act of decisio makig (i.e. choice) has received the most attetio i the academic literature. It is so, probably because we as the huma beigs, are uable to aalyze simultaeously may differet competig factors ad the sythesize them

2 problems 86 for the purpose of ratioal decisio. There is overwhelmig scietific evidece showig that the uaided huma brai is simply ot capable to do it. That is why oe eeds some decisio support tools which ca provide him/her with the required assistace. Psychologists have prove that the huma brai is limited i both its short term memory capacity ad its discrimiatio ability. It was scietifically verified that a huma beig will give iaccurate aswers whe forced to choose from a rage of twety alteratives, because the rage exceeds ma s badwidth of perceptio chael (Marti, 1973). It has bee demostrated that humas are ot capable of dealig accurately with more tha about seve thigs at a time. It is crucial to otice that results of umerous psychological experimets, icludig the well kow Miller s study (Miller, 1956) verified this otio. Example Techiques for Decisio Makig Facilitatio Advatage-Disadvatage Procedure to Decisio Aalysis This techique is perhaps the most simple ad fudametal of all the evaluatio methods. O the other had, it is also the approach that probably is used most frequetly what does ot ecessarily mea that it is used effectively. The process typically uses a list of all alteratives, which are the separately examied from the perspective of their stregths ad weakesses agaist certai criteria. Fially, it is chose the oe that best fits the problem objective. The choice is made either o the bases of greatest umber of advatages or largest et umber of pros over cos (Va Gudy, 1981). Obviously, the primary limitatio of the techique lies i the assumptio that all criteria must have the same weights what seems very ulikely ad costitutes the sigificat simplificatio of the whole problem. Weightig System Approach for Decisio Makig Support I order to avoid the mai drawback of the previous method, this commoly used procedure ivolves assigig weights to the differet evaluatio criteria. Sice ot all criteria are probably assiged the same weights, the weightig system approach provides a systematic method for assessig the stregths ad weakesses of each alterative. The geeral steps this method etails costitute: creatig a evaluatio criteria list, assigig weights to the criteria, ratig each alterative agaist the criteria ad selectig the alterative that best satisfies the criteria o the bases of total poits (Va Gudy, 1981). Obviously, at the ed of the process the questio arises, whether the attempt udertake i order to quatify value prefereces for both criteria ad alteratives was valid. Let us remember that ay measuremet system is oly as useful as the criteria applied to develop it ad the iformatio credibility, the ratigs were based o. Moreover, there are two basic kids of judgmets: absolute ad relative. However, it seems that humas ca make much better relative judgmets tha absolute oes. It is so because they have the better ability to discrimiate betwee the members of a pair tha compare oe thig agaist some recollectio from their log term memory (Saaty, 2000). The Aalytic Hierarchy Process It seems that makig pairwise comparisos is for huma beig as atural as biary coutig is for computers. It is so because humas have limited, so called badwidth of their chael for perceptio. As it was idicated (Marti, 1973, p. 334), we ca deal simultaeously oly with about seve uits of iformatio, ad this is our optimal performace as far as complex problem solvig is cocered. Fortuately, there are some ways to improve our efficiecy.

3 Pawel Tadeusz KAZIBUDZKI. A Expoetial ad Logarithmic Utility Approach to Eigevector Method i the Aalytic Hieararchy Process: The Idea Validatio with Example Logistic Applicatio The first is to eable a ma makig rather relative tha absolute judgmets. The secod is to orgaize tasks ito groups i order to make several judgmets i successio. All of this is just the reaso why the Aalytic Hierarchy Process (AHP) uses pairwise comparisos. The method was developed at the Wharto School of Busiess by Thomas Saaty (1980) ad sice its creatio has bee applied to umerous decisioal problems (Saaty, 2001). It has substatial theoretical ad empirical support icludig the studies of huma judgmetal processes by cogitive psychologists. It overcomes the weakesses of advatage-disadvatage procedure to decisio aalysis ad the weightig system techique by usig a hierarchical structure of the decisio problem, pairwise relative compariso of the elemets i the hierarchy, ad a series of idepedet judgmets. The AHP method reduces i this way possible iaccuracy of huma s judgmets ad provides a measure of their cosistecy. The first stage i AHP model buildig is to decompose the overall problem ito a hierarchy cosistig, miimally, of a goal, criteria, ad alteratives. Each elemet, or level, i the hierarchy ca be delieated further ito aother set of maageable compoets. For example, criteria ca be decomposed ito subcriteria. The hierarchical structurig cotiues dow to the most specific compoet of the problem, which usually ivolve specific alteratives or courses of actio available to the decisio maker. There are may possible hierarchical models that ca be applied to a wide variety of problems. The followig oes may be icluded as some of them (Dyer et al., 1991, p. 89): goal, criteria, alteratives; goal, criteria, subcriteria, alteratives; goal, criteria, subcriteria, scearios, alteratives; goal, actors, criteria, alteratives; goal, criteria, levels of itesities, may alteratives; The secod stage i this process is to establish priorities amog the elemets at each level of the hierarchy. The decisio maker evaluates (i a pairwise maer) the relative importace, preferece, or likelihood of each set of elemets with respect to elemets at the immediately higher level i the hierarchy. First pairwise comparisos of the relative preferece for the alteratives are made with respect to each of the lowest level (sub) criteria. Next, pairwise comparisos are made about the relative importace of the subcriteria with respect to each criterio, ad the for the relative importace of the top level criteria with respect to the goal. The process of makig pairwise comparisos uses ie-poit scale i order to evaluate prefereces for each pair of items. The scale provides both umerical ad correspodig verbal expressios for each judgmet ad is commoly available i the literature (Saaty, 2001, p. 26). Fially, the last stage ivolves sythesisig accordig to a well-prescribed procedure (stadard AHP aggregatio based o weightig ad addig) ad calculatio of so called icosistecy ratio that reflects the degree of judgmets icosistecy. Recapitulatig, that what makes AHP so differet ad effective from other similar methods is that (A) it eables makig judgmets i the verbal way, ad (B) it etails mathematical aalysis performed o the judgmets which provides relatively very accurate results ad the measure of their icosistecy. However, it is the mathematical aalysis i.e. the eigevalue method, which the AHP applies, that despite of its popularity, has as well its share of criticism. For istace some authors have oticed that Saaty s procedure does ot optimize ay objective fuctio, what etails it caot be iterpreted i statistical or optimizatio fashio. I particular, solutios obtaied with the applicatio of the eigevalue method i the AHP caot be compared to other oes received with a applicatio of differet commoly kow methods (Choo et al., 2004; Crawford et al., 1985). Besides, ulike may optimizatio models, the method applied by the AHP does ot allow a decisio maker for itroductio of additioal costraits that might be seemed ecessary accordig to his/her poit of view (Lam et al., 1995). Moreover, the method problems 87

4 problems 88 commoly applied to the AHP is supposed to operate oly with, so called, reciprocal pairwise comparisos matrices what etails a limited rage of applicatio. Aother drawback of the aalyzed procedure was revealed quite recetly (Baa e Costa, 2008), ad proved that the eigevalue method, commoly applied to the AHP, does ot satisfy a so called coditio of order preservatio. The Eigevector Method i the AHP I order to derive priorities from pairwise comparisos AHP uses a pricipal Eigevalue Method (EM) which has bee recetly a subject of more detailed aalysis (Kazibudzki, 2011a). Geerally speakig a problem of derivig priority weights from so called pairwise compariso matrix (PCM) deoted as A=[a ij ] x with elemets a ij =a i /a j is to estimate a priority vector (PV) T w = [ w1, w2, w3,, w ] o the bases of the matrix A which comprises a decisio maker (DM) pairwise comparisos judgmets cocerig the importace of a give biary set of alteratives. Commoly the priority weights w i, i=1,,, are chose to be positive ad ormalized to uity: wi = 1, ad the elemets a ij of the matrix A are the the DM judgmets about the priority i ratios w i /w j, i,j=1,,, where is the umber of all alteratives beig cosidered. I a perfect judgmet case the, oe has: A x w = x w [1] ad i this situatio the PV w ca be computed by solvig that so called eigevector equatio. It is so because the umber i the perfect case (matrix A is cosistet) is the pricipal eigevalue of A, i.e. the largest solutio of the characteristic equatio: det (A lambda x I) =0 [2] where I deotes idetity matrix of order. I this case it is also the oly ozero eigevalue. O the other had, whe the case is ot perfect (matrix A is icosistet) a estimate of the true PV is ormalized right eigevector associated with the largest eigevalue. Thus, i order to obtai the estimate oe eeds to solve geeral eigevector equatio: A x w = lambda max x w [3] where lambda max deotes the pricipal eigevalue which is ot smaller tha, is simple ad its existece is guarateed by Perro s Theorem (Saaty, 1994), while other lambdas are close to zero. If the elemets of a matrix A satisfy the coditio a ij =1/a ji for all i, j=1 the the matrix A is said to be reciprocal (RPCM). If its elemets satisfy the coditio a ik a kj =a ij for all i, j, k=1 ad the matrix is reciprocal, the it is called cosistet. Fially, the matrix A is said to be trasitive if the followig coditios hold: (A) if for ay l=1,,, a elemet a lj is ot less tha a elemet a lk the a a ij ik for i=1,,, ad (B) if for ay l=1,,, a elemet a jl is ot less tha a elemet a kl the a ji aki for i=1,,. Obviously, i the case of the reciprocal PCM the two coditios (A) ad (B) are equivalet. A Expoetial ad Logarithmic Utility Approach to EM i the AHP It has bee already deduced (Grzybowski, 2010) that istead of solvig the eigevalue equatio [3] oe may seek a vector w which best estimates the equatio [1]. So, i order to satisfy the equatio [1] as perfectly as possible it has bee recetly proposed (Kazibudzki, 2011a) to estimate the PV by solvig oe of the followig costraied optimizatio problems:

5 Pawel Tadeusz KAZIBUDZKI. A Expoetial ad Logarithmic Utility Approach to Eigevector Method i the Aalytic Hieararchy Process: The Idea Validatio with Example Logistic Applicatio problems 89 mi exp i= 1 j= 1 a w w ij j i 1 2 subject to: [4] i= 1 or respectively (Kazibudzki, 2011b): 2 wi = 1, wi 0, i, j = 1,, aij w j mi l [5] i= 1 j= 1 w i subject to: i= 1 wi = 1, wi > 0, i, j = 1,, Because the utility of both methods decreases expoetially or logarithmically, iversely proportioally to the icosistecy asced, it seems atural that they ca be commoly idetified, respectively as the Expoetial Utility Approach (EUA) ad the Logarithmic Utility Approach (LUA) to the problem of priorities weightig. The basic cocept of the above two methods is to miimize a distace betwee a ideal ukow PV givig cosistet PCM ad a PV resultig from a DM judgmets, typically ivolvig icosistet PCM. The quest for this solutio with the applicatio of commoly kow costraied optimizatio procedures results with the PV which best fits to the ideal oe deliverig cosistet PCM. I the ext sectio EM will be compared to the two approaches with the applicatio of computer simulatios. The Validatio Studies of EUA ad LUA to EM i the AHP I the simulatios with the help of sophisticated mathematical software the followig types of radom icosistet PCMs were geerated: reciprocal ad trasitive (RTPCM), oly reciprocal (RPCM), ad oly trasitive (TPCM). I order to compare results obtaied from EUA ad LUA with results give by EM, firstly te thousads of RTPCMs were geerated, i order to fid out if there is ay possibility of rak reversal pheomea betwee the methods, ad secodly oe thousad matrices from each of remaiig types (RPCM ad TPCM). For every matrix the PVs: w EUA, w LUA, ad w EM were calculated with the applicatio of EUA, LUA ad EM, respectively. Next were computed: Pearso correlatio coefficiet (PCC) betwee the priority vectors ad Spearma rak correlatio coefficiet (SRCC) betwee the priority raks. The cosidered umber of decisio alteratives was: 3, 5, 7, 9, 11 i the case of RTPCMs ad, for variety reasos: 4, 6, 8, 10 i the case of remaiig types of PCMs, i.e. RPCM ad TPCM. For the purpose of preferece ratios evaluatios, which etail the scope of elemets i PCMs, the stadard AHP scale was applied, i.e. itegers: 1,, 9 ad their iverses. Table 1, 2, ad 3 show the mea correlatio coefficiets betwee priority vectors ad priority raks geerated

6 problems 90 with the applicatio of the two ovel methods (EUA, LUA) ad EM, o the bases of the whole set of data i relatio to a give type of PCM ad umber of alteratives. Table 1. Simulatio results of comparative studies EUA ad LUA versus EM for RTPCMs. Number of alteratives LUA RTPCM EUA PCC SRCC PCC SRCC = , , =5 0, , , =7 0, , , =9 0, , , =11 0, , , Table 2. Simulatio results of comparative studies EUA ad LUA versus EM for RPCMs. Number of alteratives LUA RTPCM EUA PCC SRCC PCC SRCC =4 0, ,995 0, , =6 0, , , , =8 0, , , , =10 0, , , , Table 3. Simulatio results of comparative studies EUA ad LUA versus EM for TPCMs. Number of alteratives LUA RTPCM EUA PCC SRCC PCC SRCC =4 0, , , , =6 0, , , , =8 0, , , ,99563 =10 0, , , , As it ca be see from tables 1, 2 ad 3 the validatio studies of EUA ad LUA versus EM ca be successfully perceived. The very essetial result of them is the score of the RTPCMs simulatios for LUA which prove that this method fully complies with EM givig the same rak order i all te thousads cases of simulated matrices (SRCC=1) ad for every studied set of alteratives (=3, 5, 7, 9, 11). Amazigly close relatio betwee studied methods ca be also foud i the case of RPCMs ad TPCMs, however let us remember that i this case oe has to be coscious that EM is desiged oly for the applicatio with reciprocal matrices ad because of that i the case of matrices which are oreciprocal the EUA ad LUA have their advatage over EM, what may also certify that they have much broade rage of applicatios ad ca be adopted i cases where EM simply fails. Together with other importat fidigs cocerig

7 Pawel Tadeusz KAZIBUDZKI. A Expoetial ad Logarithmic Utility Approach to Eigevector Method i the Aalytic Hieararchy Process: The Idea Validatio with Example Logistic Applicatio EUA (Kazibudzki, 2011a) ad LUA (Kazibudzki, 2011b) it ca be cocluded, that there is at least oe valid method for derivig the priority vector from a pairwise compariso matrix, particularly whe the matrix is icosistet, which is equally satisfyig as eigevalue method ad sometimes it is eve better. This method is LUA ad it is so because it fully complies with EM, it ca be applied to both reciprocal ad oreciprocal matrices, it is computatioally simpler ad what is most importat it does ot violate coditio of order preservatio (Kazibudzki, 2011b). Obviously, aalyzed here approach eeds further studies ad aalysis to be stregthe. Further, both preseted ad aalyzed above methods will be used for the purpose of logistics applicatio. problems 91 Selectig Trasportatio Modes a Example of EUA ad LUA Applicatio It is a fact that physical distributio of goods ca represet a major cost category for may compaies. Due to poor plaig, lack of coordiatio ad lost or damage shipmets additioal costs must be uecessarily icurred. However, it is possible to avoid this burde through thorough ivetory cotrol, plaig ad coordiatio, wise ad thoughtful selectio of warehousig ad meas of trasport. Udoubtedly, durig trasportatio modes selectio, the cost costitutes the most importat criterio for decisio makers. However, there are also other criteria that ca affect a overall customer service which are recklessly eglected i the selectio process. I order to overcome this situatio, it is proposed to apply i such cases the AHP model preseted below which offers a meas of reviewig alteratives from differet perspectives o the bases of differet criteria. It seems worth to otice, that slight adaptatio of this model ca also assist i the evaluatio of the more geeral modes of trasportatio like for istace rail, air, water ad truckig, icludig formerly selected specific carriers withi each mode (Kotler, 1988, p. 747). The model is the simple oe. Uder the overall goal (the optimal mode of trasportatio selectio), there are four criteria: c1 (delivery time), c2 (availability), c3 (cost of shipmets), ad c4 (coformace with package size ad weight requiremets). Uder each criterio, there are four alteratives: a1 (carrier o.1), a2 (carrier o.2), a3 (carrier o.3), ad a4 (carrier o.4), which are the same for all the four criteria. The example judgmet matrices ad correspodig estimatio of PVs, as well as overall PV, obtaied with the applicatio of EM, LUA, ad EUA, respectively, are provided below: with respect to the GOAL: with respect to criterio c1:

8 problems 92 with respect to criterio c2: with respect to criterio c3: with respect to criterio c4: After sythesis, the followig overall rakig is obtaied: It is easy to otice that all three methods: the Eigevalue Method, Logarithmic Utility Approach ad Expoetial Utility Approach, provide the same overall alteratives raks, resultig: a4>a2>a3>a1. Obviously, it meas that i this hypothetic example carrier o. 4 should be the ultimate choice of a decisio maker. Coclusios Thorough ivetory cotrol, plaig ad coordiatio, wise ad thoughtful selectio of warehousig ad meas of trasport are ecessary for efficiet decisios makig i logistics. I order to make optimal decisios, it is ecessary to deliberate may optios from the perspective of differet criteria. It was empirically verified that such problems are difficult to solve without methodological support, as they exceed humas perceptio capabilities. However,

9 Pawel Tadeusz KAZIBUDZKI. A Expoetial ad Logarithmic Utility Approach to Eigevector Method i the Aalytic Hieararchy Process: The Idea Validatio with Example Logistic Applicatio oe has already got techiques which applicatio ca help. Oe of the most commoly used for structured problems solvig is the Aalytic Hierarchy Process. It is so popular because its prescribed procedure supports sophisticated mathematical theory called the Eigevalue Method which is accurate ad uique. However, two ew ad relatively ovel methods were proposed recetly. Their simplicity ad compliace with the Eigevalue Method are their most crucial but ot exclusive advatages which proved the validatio studies provided i this research. At the ed of the paper, the example logistic problem was solved with their applicatio, ad the results of all methods were compared as well. problems 93 Ackowledgemets I would like to express my gratitude to Adrzej Z. Grzybowski for his support durig the phase of article preparatio, ad especially for desigig the sophisticated software, without which validatio studies of the methods beig the research subject of this paper, would ot be possible. Refereces Baa e Costa, C. A., Vasick, J. C. (2008). A critical aalysis of the eigevalue method used to derive priorities i AHP. Europea Joural of Operatioal Research, Vol. 187, p Choo, E. U., Wedley, W. C. (2004). A commo framework for derivig preferece values from pairwise compariso matrices. Computers & Operatios Research, Vol. 31, p Crawford, G., Williams, C. A. (1985). A ote of the aalysis of subjective judgmet matrices. Joural of Mathematical Psychology, Vol. 29, p Dyer, R. F., Forma, E. H. (1991). A aalytic approach to marketig decisios. Pretice Hall, New Jersey. Grzybowski, A. Z. (2010). Goal programmig approach for derivig priority vectors some ew ideas. Scietific Research of the Istitute of Mathematics ad Computer Sciece, Vol. 1(9), p Kazibudzki, P. T. (2011a). A expoetial utility approach to eigevector method i the aalytic hierarchy process: a idea itro. Problems of Maagemet i the 21 st Cetury, Vol. 1, p Kazibudzki, P. T. (2011b). A sceario based aalysis of a logarithmic utility approach for derivig priority vectors i the aalytic hierarchy process. Scietific Research of the Istitute of Mathematics ad Computer Sciece, [submitted to Joural]. Kotler, P. (1988). Marketig Maagemet: Aalysis, Plaig, Implemetatio, ad Cotrol. 6 th Ed., Pretice Hall, Eglewood Cliffs, New York. Lam, K. F., Choo, E. U. (1995). Goal programmig i preferece decompositio. Joural of Operatioal Research Society, Vol. 46, p Marti, J. (1973). Desig of Ma-Computer Dialogues. Pretice Hall, Eglewood Cliffs, New York. Miller, G. A. (1956). The magical umber seve, plus or mius two: some limits o our capacity for iformatio processig, Psychological Review, Vol. 63, p Saaty, T. L. (2000). The Brai: Uravelig the Mystery of How it Works. RWS Publicatios, Pittsburgh, pp Saaty, T. L. (2001). Decisio Makig for Leaders. The Aalytical Hierarchy Process for Decisios i a Complex World. RWS Publicatios, Pittsburgh, pp Saaty, T. L. (1980). The Aalytic Hierarchy Process. Mcgraw Hill, New York.

10 problems 94 Saaty, T. L. (2001). Decisio Makig with Depedece ad Feedback. The Aalytic Network Process. RWS Publicatios, Pittsburgh. Saaty, T. L. (1994). Fudametals of the Aalytic Hierarchy Process. RWS Publicatios, Pittsburg. Simo, H. A. (1960). The New Sciece of Maagemet Decisio. Harper & Brothers, New York, pp Va Gudy, A. B. (1981). Techiques of Structured Problem Solvig. Litto Educatioal Publishig, Ic., New York, pp Advised by Adrzej Z. Grzybowski, Techical Uiversity of Czestochowa, Czestochowa, Polad Received: July 10, 2011 Accepted: September 25, 2011 Pawel Tadeusz Kazibudzki Dr., M.B.A., Assistat Professor, The Faculty of Social Scieces, Ja Dlugosz Uiversity i Czestochowa, Waszygtoa 4/8, Czestochowa, Polad. poczta@gmail.com Website: