THE NONLINEAR NATURE OF PREFERENCES, ITS IMPACT ON THE SENSITIVITY AND EFFECTIVENESS OF MULTIPLE CRITERIA ALTERNATIVES

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1 IJAHP Artile: Mu, Saaty/A Style Guide for Paper Proposals o Be Subitted to the International Syposiu of the Analyti Hierarhy Proess 2014, Washington D.C., U.S.A. HE NONLINEAR NAURE OF PREFERENCES, IS IMPAC ON HE SENSIIVIY AND EFFECIVENESS OF MULIPLE CRIERIA ALERNAIVES Rafael Sarkisyan Mosow State University of Railway Engineering E-ail: sarkisyanry@ail.ru Aleksandra Masalida Mosow State University of Railway Engineering asalido4ka@rabler.ru Elena Kobets Mosow State University of Railway Engineering elenavk2706@ail.ru ABSRAC he ehaniss of reating and proessing expert and/or statistial inforation, whih underlie the AHP/ANP, do not enable to take into aount the nonlinear nature of preferenes and their dependene upon intensiveness (or level) of easurable features and qualities of researhed and optiized systes. In the traditional ethods of ultiple riteria (ultiobetive) optiization based on the onept of preferenes and utilities, the nonlinear nature of preferenes as well as the loss of sensitivity and effetiveness of alternatives aused by it an be taken into aount, if to proeed fro the onave inreasing property of a orresponding evaluation funtion (preferene funtion). If to assue also that the onave inreasing preferene funtion 1 u : E E F 0 E is differentiable on the nonepty onvex set of estiates, u( u( u( ( f, f F0 then fro its peuliar differential inequality u( = u0 ( + u( f, f F0. one an dedue the deoposition shee u 0 ( where is the height of intersetion of the hyperplane H u = {( y, / y = u( + u( ( f, f F0 }, tangential to the surfae of u( f, u( the funtion, with the axle of the latter at the point and is the gradient of the funtion. he seond oponent of the deoposition, = u( f, is related to the relative sensitivity funtion t du( t σ ( li 1 u( f t 1 u( t dt u( = 1 f = with the orrelation Analyti Hierarhy Proess 1 Washington, D.C.

2 IJAHP Artile: Mu, Saaty/A Style Guide for Paper Proposals o Be Subitted to the International Syposiu of the Analyti Hierarhy Proess 2014, Washington D.C., U.S.A. = = 1 u( f f = σ ( u( ("sensitive" oponent u( v ( = f = σ ( u(, = 1,..., f u( ). Its ites,, in effet, haraterize the relative weight (iportane, return) of individual riteria and play the sae role in ω = ( ω,,..., ) 1 ω2 ω alternatives ranking as the eigenvetor oordinates do in the AHP/ANP shees, but now they depend on the state vetor. heir ratio, v ( / v ( = σ ( / σ ( = ( f / µ, i, 1,...,, i i i i = µ i, i, = 1,..., where are arginal rates of substitution between the riteria, allows to f f = 1/ µ = µ. i / i i find stable solutions that would satisfy the ondition he σ ( u ( = onst axiization of the funtions and on the surfae of f (λ) λ 1, generates a traetory of solutions, that satisfy both the axiu of σ ( the harateristis and, and a "onsistent" value of the riteria с с с θ ( = f Hf = (1 λ) = (1 λ) σ ( interation funtion, where λ 1 u( is the Lagrange ultiplier, H is the Hessian atrix of the funtion. Also the possibility of using the proposed analyti orrelations in the AHP/ANP traditional proedures is being disussed. As appliations a ultiple riteria proble of orporate resoures anageent and diagnosti essages proessing in transport systes are being onsidered. Keywords: preferenes, nonlinear nature, sensitivity, effetiveness of alternatives, optiization on surfae, traetory, stability. 1. Introdution Applied ultiple riteria probles and interative ethods of their solution serve as the ontext of the proposed study. raditionally, one of the iportant streas of sholarship in this field of knowledge was based on the onept of preferenes and utilities that show up on the set of estiates of alternatives by riteria. Experiene and intuition suggest that preferenes are of nonlinear nature whih leads to the loss of sensitivity and effetiveness of ausal relations. he neessity to desribe analytially these nonlinear effets and to take the into aount in hoie and deision aking proedures ated as a spur to the present study. Involving the theory of sensitivity allows to onstrut suitable analyti tools (easures of relative sensitivity and effetiveness for nonlinear ausal relations) ontributory to ranking solutions aording to their preferene. he goal is a ore adequate desription of preferenes, Analyti Hierarhy Proess 2 Washington, D.C.

3 IJAHP Artile: Mu, Saaty/A Style Guide for Paper Proposals o Be Subitted to the International Syposiu of the Analyti Hierarhy Proess 2014, Washington D.C., U.S.A. raise of auray and utility of sought-for solutions and of ations planned on their base. 2. Literature Review As far as we know, the idea of spatio-teporal dependene of preferenes was first stated in (Keeney R., & Raiffa H., 1976). In effet, the logi of onstruting effetive an ahine ethods and proedures for solving pratial ultiple riteria probles also proeeds fro an assuption of dependene of preferenes upon an (ahieved) level of riteria (Benayoun R., 1971; Hall A.D., 1962; Fandel G., & Wilhel J., 1976; Geoffrion, A.M., 1972). o develop the ideas of the authors (Geoffrion, A.M., 1972), in (Sarkisyan R., 1992) the possibility was substantiated of representing the relative iportane of riteria via the ratio of diretional derivatives of riterion funtions. he results of soe reent studies (DellaVigna S., 2009; Hands W., 2010.) also speak in favour of the dependene of preferenes upon the evaluated state and upon trends of its developent in tie. he approah ontained in the proposal and its analyti orrelations (easures of sensitivity and effetiveness, optiization on surfae) that haraterize the nonlinear preferenes are original results of applying the theory of sensitivity to odeling nonlinear effets whih show up during the foration of expert inforation about riteria and alternatives. 3. Obetives and Hypotheses he ain assuption is brought to the existene of the onave inreasing twie differentiable preferene funtion on the set of riteria values. Suh a funtion satisfies the differentiable inequality u( u( u( ( f, f F0, u( = u0( + u( allows a deoposition in the for = ( 1 σ ( ) u( + σ ( u(, f F0, σ ( = σ 1( σ ( where = u( f = v1 ( v ( and f is the easure of relative sensitivity, and is the easure of effetiveness (effetive return). he axiization of these harateristis on the preferene funtion level surfae allows to reah neessary ranking of riteria values. 4. Researh Design he interative ethods of ultiple riteria optiization are in harony with the spirit of the studies (Saaty.L., 1994; Saaty.L., 2001) aied at finding a reasonable oproise between rivaling riteria and requireents. In this sense the hierarhy <Goal, Criteria, Alternatives> is inherent to any proble of optiization on the base of ultiple riteria logi and serves, aording to. Saaty, as a ighty speulative odel for desribing and researhing oplex systes. 5. Preises In one of the earlier studies related to researhing the reliability of the oputerized syste ELEPERM-ME (Sieens) by eans of the AHP/ANP odels we ae to a onlusion that the idea of a "reasonable oproise" haronizes with that of "ideal proportions" (Geoffrion A., 1972). Suh proportions an be obtained by axiizing σ ( u ( = onst the funtions and on the surfae of. he ratios Analyti Hierarhy Proess 3 Washington, D.C.

4 IJAHP Artile: Mu, Saaty/A Style Guide for Paper Proposals o Be Subitted to the International Syposiu of the Analyti Hierarhy Proess 2014, Washington D.C., U.S.A. v ( / v ( = σ ( / σ ( = ( f / µ, i, 1,...,, i i i i = satisfy this quality of solutions. Optiization on the preferene funtion level surfae also satisfies Sion's priniple of satisfiing (Sion H., 1979), while the role of onsent index in AHP/ANP is played by a relation whih allows to evaluate the level of the riteria с с с θ ( = f Hf = (1 λ) = (1 λ) σ (. interation funtion his f ( λ), λ 1. relation is ust at all points of the optial traetory It eans that at every point of the optial traetory the level of the riteria interation funtion is σ (. defined by the axiu value of indiators and 6. Liitations It is assued that during the inrease of the riteria values the preferene funtion is aiing at its established value (i.e. it is "getting sated"). In this ase the easure of its relative sensitivity σ(f) is a onotonously dereasing funtion, while the "sensitive" oponent f) reahes its axiu on the level surfae of u(f) = onst and, therefore, the given devie an be suessfully applied. Aording to the Debreu theore, if the preferene relation on the set of riteria values is a oplete preorder and has properties of onotonousness and of ontinuity, then the preferene (or utility, value) funtion an be put in orrespondene with it; the property of differentiability is entered for onveniene of atheatial analysis. It is not expedient to apply the given approah unless it is assued that the preferene funtion is "getting sated". 7. Conlusion Our professional interest in the AHP/ANP tehnique and odels is onditioned only by issues and prospets of developent of interative ethods of applied ultiple riteria probles solution; the appliation aspet is to reate deision aking support systes for proeting, planning and anaging in tehnial and organizational systes. In the researh pratie related to applied ultiple riteria probles solution, the interative ethods and proedures play a leading role. he traditional view on the algorithization of the preferred solutions searh proess proeeds fro the onseutive revelation and desription of preferenes as a funtion of state and oveent diretion in the spae of estiates and alternatives. It is aepted that naely suh a ulti-step (ulti-stage) searh proess will allow to derease unertainty and non-oparableness and to reah a reasonable balaning (Saaty,L., 2001) between rivaling requireents of riteria. he preferenes are of nonlinear nature whih generates the loss of sensitivity and effetiveness of a orresponding ausal relation. he approah stated in the proposal allows to odel ore adequately these nonlinear effets and to take the into aount while ranking ultiple riteria estiates and orresponding alternative solutions. In our view, the relation of these fators to the AHP/ANP is obvious. 8. Key Referenes Geoffrion, A.M., Dyer J.S., & Fienberg A. (1972). An interative approah for ultiriterion optiization, with an appliation to the operation of an aadei departent. Manageent Siene, 19(4), part 1, Analyti Hierarhy Proess 4 Washington, D.C.

5 IJAHP Artile: Mu, Saaty/A Style Guide for Paper Proposals o Be Subitted to the International Syposiu of the Analyti Hierarhy Proess 2014, Washington D.C., U.S.A. Benayoun R., de Montgolfier J., ergny J., & Larithev O. (1971). Linear prograing with ultiple obetive funtions: SEP ethod. Matheatial Prograing, 1(3), Keeney R., & Raiffa H. (1976). Deisions with ultiple obetives: preferenes and value tradeoffs. New York: John Wiley & Sons, In. Sion H. (1979). Rational deision aking in business organization. Aerian Eonoi Review, 69(4), Hall A.D. (1962). A ethodology for systes engineering. Prineton: D. van Nostrand Cop., In. Fandel G., & Wilhel J. (1976). Zur Eintsheidng heorie bei ehrfaher Zielsetzung. Zeitshrift für Operation Researh, DellaVigna S. (2009). Psyhology and eonois: evidene fro the field. Journal of Eonoi Literature, 47(2), Hands W. (2010). Norative rational hoie theory: past, present and future. Fourth annual onferene on the history of reent eonoi (HISRECO), Cahan, Frane. Saaty.L. (1994). Fundaentals of deision aking and priority theory with the AHP, vol. 6. Pittsburgh, PA: RWS Publiations. Saaty,L. (2001). Deision aking with dependene and feedbak, he analyti network proess. PA: RWS Publiations. Sarkisyan R.(1992). Adaptive proedures for dialogue systes: Methods of diretional derivatives. Dotoral thesis, Yerevan Polytehni Institute. 9. Appliations he analytial apparatus stated in the proposal an be illustrated by the exaple of two appliations: a) Corporate resoures anageent during transport systes aintenane and repair planning. On the base of regarding the tehnologial relation between the planned work volue and osts of eonoi fators ade for that purpose as a у = F( q 1,..., q n ), prodution funtion a biriteria optiization proble is being forulated whih inludes iniization of the ost funtion r ( q ) = p q p q n n ) and axiization of the return funtion F F q) = q qn. q1 qn he prodution funtion, at that, is being regarded as a twie ontinuously differentiable onave inreasing funtion, thus allowing a F( q) = F0 ( q) + q) = (1 σ ( q)) F( q) + F( q) q, deoposition in the for As a suitable preferene funtion on the set of riteria values the approxiations α1 f1 α2 f2 u( = a f + (1/ 2) f Hf, a + Hf 0 u( = U 0(1 e )(1 e ) and are being onsidered, where the oordinates f 1 and f 2 represent orresponding riteria Analyti Hierarhy Proess 5 Washington, D.C.

6 IJAHP Artile: Mu, Saaty/A Style Guide for Paper Proposals o Be Subitted to the International Syposiu of the Analyti Hierarhy Proess 2014, Washington D.C., U.S.A. у = F( q 1,..., q n ) funtions. Optiization on the surfae is being ade using the Lagrange ethod. b) Diagnosti essages proessing for high speed vehile. As it is known, the proble of diagnostis an be onsidered fro the point of view of a proble of hoosing one of possible explanations on the base of available data. It is assued that an observed obet is exhaustively desribed by a set of paraeters x 1, x 2,, x n, whih ake up the desription vetor x = (x 1,., x n ) D, where D is the spae of desriptions, the set of various values of the vetor x. Also it is assued that to every state S i fro aong the set of states S = {S 1, S 2,, S } orresponds an area D i Ì D, i = 1,,. If these areas are already established, the proble of diagnostis an be brought to deision aking in favour of a state (or hypothesis) S i any tie its desription x falls on an area D i. As the solution riteria the following are being regarded: - axiization of the a posteriori probability funtion P (S i /x) =,, i = 1,, ; l( x Di ), - iniization of the loss funtion losses being aused by a false lassifiation. In ase of Eulidean desription spae, i. e. D E n, the riteria are interpreted in ters of orresponding probability density funtions on the subsets D i, i = 1,,., both appliations represent a hierarhial odel <Goal Criteria - In effet Alternatives>. In the first appliation we deal with a ontinued set of alternatives у = F( q 1,..., q n ) presented by the level surfae equation, while in the seond one the set of states S = {S1, S2,, S} subet to identifiation (or lassifiation) serves as a set of alternatives. Note that in both ases the preferenes depend upon the riteria values. In AHP/ANP odels the udgent atrix looks as A = (a i) ), a 11 = a 22 = 1, a 12 = 1/a 21. he orresponding priority estiates will be equal to: 1 = a 12 /(1 + a 12 ), 2 = 1/(1 + a 12 ), 1 / 2 = a 12, as follows fro the odel of nuerial udgents proessing A = ax, ax = 2. hese estiates of relative priorities are not aeptable for both (or siilar) appliations, despite the efforts of experts. Aording to the aterial of the proposal, their role an be played by the easures v i (f), i=1,,, ore preisely, by their optial values. Analyti Hierarhy Proess 6 Washington, D.C.