MULTICRITERIA EVALUATION AND OPTIMIZATION OF HIERARCHICAL SYSTEMS. Albert Voronin

Transcription

1 08 Iteratioal Joural "Iformatio Theories & Applicatios" Vol.5 / 2008 MULTICRITERIA EVALUATION AND OPTIMIZATION OF HIERARCHICAL SYSTEMS Albert Voroi Abstract. It is show that ay multicriteria problem ca be represeted by a hierarchical system. Separate properties of the obect are evaluated at the lower level of the system, usig a criteria vector, ad a compositio mechaism is used to evaluate the obect as a whole at the upper level. The paper proposes a method to solve complex multicriteria problems of evaluatio ad optimizatio. It is based o ested scalar covolutios of vectorvalued criteria ad allows simple structural ad parametrical sythesis of multicriteria hierarchical systems. Keywords: alterative choice, multicriteriality, hierarchical systems, compositio of criteria, ested scalar covolutios Itroductio I decisio-maig theory [,2], there are two differet approaches to evaluatig obects (alteratives subect to choice. Oe of them is to evaluate a obect as a whole ad to choose a alterative by comparig obects as gestalts (holistic images of obects without detailig their properties. The secod approach is to detail ad evaluate certai vectors of characteristics (properties of obects ad to mae a decisio after comparig these properties. The scheme of decisio maig ca be represeted by the formula [3] {{ χ}, Φ } χ*, where {χ} is a set of obects (alteratives; Φ is a choice fuctio, i.e., a rule that establishes preferability i the set of alteratives; ad χ * are chose alteratives (oe or more. The holistic approach implies choosig χ * usig the choice fuctio Φ. The vector approach requires decomposig (expadig the fuctio Φ ito a set (vector of some choice fuctios φ. By decompositio of the choice fuctio Φ is meat [] its equivalet represetatio by a certai set of other choice fuctios φ whose compositio is the iitial choice fuctio Φ. Both approaches have their advatages ad shortcomigs. Choice after comparig obects may substatially differ from choice after comparig the vector characteristics of obects [2]. This is because iformatio o vectors sometimes gives a isufficietly adequate descriptio of obects, eve with the most careful choice of characteristics of obects. Some portio of the iformatio o obects is lost whe the obects are described by a set of characteristics. Aother portio of the iformatio, which is ot directly cocered with the obects beig compared, is itroduced ito the model. The choice of a appropriate set of properties (characteristics of a obect is subective to a certai extet. Moreover, there is a assumptio [2] that huma thiig is ot has bee evolved to adapt for a atural (from a formal stadpoit chageover from prefereces o a set of obects to prefereces o a set of their characteristics. Nevertheless, moder decisio theory is iclied to usig the vector approach sice it is obective ad comprehesive ad allows employig formalized methods. The cocreteess ad clearess of a approach are also tae ito accout sice it is easier to collect idisputable facts ad reach a cosesus o a specific issue [4]. It is assumed that it is much easier for a decisio-maer to reveal a preferable alterative for a certai property of a obect. For example, i choosig the best desig of a aircraft, it is easier to compare desig A over a desig B i comfort, or reliability, or weight-liftig ability, tha to compare the whole desigs A ad B [3]. Selectig properties of alteratives is a decompositio that leads to a hierarchical structure of properties. The properties of the first hierarchical level ca be subdivided ito sets of ext specific properties, etc. The divisio depth is determied by tedig to reach the properties that are coveiet to be compared to each other. Ideed, i the example with a aircraft, it is easier to udge its comfort rather tha the aircraft as a whole; however, such a qualitative property is also ot always coveiet for compariso ad has to be decomposed for coveiet ad obective compariso of properties. Therefore, the property of comfort, i tur, is subected to hierarchical

2 Iteratioal Joural "Iformatio Theories & Applicatios" Vol.5 / decompositio ito levels: (a cabi oisiess, (b floor vibratios, (c seat spacig, etc. These characteristics ca be evaluated ad are obective. Properties for which obective umerical characteristics exist are called criteria. More strictly, quatitative characteristics of properties of a obect, whose umerical values are a quality measure of a obect of evaluatio with respect to the give property, are called criteria. Derivig a set of criteria is the fial result of hierarchical decompositio. The umber of levels depeds o the decompositio depth required. Complexity is that decompositio depth ca be differet for differet iitial properties, ad heterogeeous sets of criteria should be ormalized at each hierarchy level. Though attractive, the approach comparig idividual properties ivolves the serious problem of iverse passage to the required compariso alteratives i geeral. This problem assumes a compositio of criteria based o hierarchy levels, which is difficult, especially for a sigificat decompositio depth of properties. I a elemetary ad most popular case (two-level hierarchy, the compositio problem ca be solved traditioally, by derivig a sigle scalar covolutio of criteria. However, other approaches are required for three (ad more-level hierarchy. The aforesaid maes it reasoable to state that ay multicriteria problem ca be represeted as a hierarchical system, where at the lower level, the obect is evaluated i separate characteristics usig a vector of criteria, ad at the upper level, a compositio is used to evaluate the obect as a whole. The ey problem is compositio of criteria at hierarchy levels. Aalysis of the problem state The case where a multicriteria problem ca be represeted by a two-level hierarchical system is developed most thoroughly i decisio theory. The compositio problem is usually solved either usig a mai criterio (criteria costraits method or by usig a sigle scalar covolutio of a vector valued criterio [3]. The latter method is used more ofte, the umerical value of covolutio beig the quality evaluatio of the give obect (alterative as a whole. It is coveiet to use scalar covolutio i a traditioal form whe the umber of partial criteria is ot too great (usually, s 0. The each criterio plays a self-depeded role ad all of them are comparable i importace. However, there exist complex multicriteria problems with a large umber (say, several tes of partial criteria. The the value of each criterio separately has a wea effect o the solutio of the multicriteria problem. It is expediet to group them ito headigs (groups, clusters where scalar covolutios are cosidered as ew, higher-weight criteria. These aggregated criteria, i tur, are subected to scalar covolutio ad the are compared with higher weights durig the solutio of the multicriteria problem. Thus, as the dimesio of the criteria space icreases, the iitially two-level hierarchical system of criteria is trasformed ito multilevel oe ad requires a mechaism to compose criteria ito hierarchy levels. As a example, let us cosider alterate evaluatio of research proects i biological studies i space [5]. To evaluate the efficiecy of such proects, 28 partial criteria are used. Cosultatios with experts has allowed groupig these criteria ito four headigs (groups: (i geeral criteria, (ii scietific developmet criteria, (iii ecoomic criteria ad (iv social criteria (see Table. Table Criterio No. Proect quality criteria Poits (0-poit scale Geeral Criteria Compliace of the proect with the Space Program of Uraie Itegratio of the proect ito iteratioal programs of biological ivestigatios i space Probability that the approach will lead to the desired results Completeess of feasibility of the proect uder give coditios of space experimet 8.30 Scietific Developmet Criteria 5 Compliace of the ob structure ad ivestigatio methods with proect tass Furtherace of gatherig owledge o the ifluece of space flight factors o fudametal physiological processes 8.25

3 0 Iteratioal Joural "Iformatio Theories & Applicatios" Vol.5 / Furtherace of gatherig owledge o the adaptatio of biological obects for space flight coditios Novelty of ivestigatios Origiality ad iovatio of the purposes formulated Ifluece of ivestigatio o scietific cocepts ad methods i space biology ad medicie 9.75 Probability that ivestigatio will allow ew brea-through proects Refutig the paradigms available Share of worldwide ivestigatios i the proect Ehacig the prestige of Uraie i the world Iteratioal support of the proect Coverage i the scietific literature Usig the results i academic activity Popularizatio ad propagatio of owledge 7.75 Ecoomic Criteria 9 Probability of itroducig the techologies ito Uraiia ecoomy Adequate umber of specialists to brig the research to practical implemetatio Attractig ivestmets Reducig productio expeses Icreasig sales Adequacy of fiacig the tass plaed 9.33 Social Criteria 25 Icreasig the umber of worsites Icreasig the level of staff qualificatio Promotig developmet of small ad medium-scale busiess Ifluece o the activity of social ad youth orgaizatios 0.00 The right-had colum of the table cotais the results of expert evaluatio of the Biosorbet space research proect, which has bee icluded i the program of oboard experimets at the Iteratioal Space Statio. These data are the lower level of a three-level hierarchical system of criteria for evaluatig the proect as a whole. Totseo cosidered i [6] the geeral case of developig a decisio-maig support techology whe a alterative should be chose from a set of ihomogeeous alteratives for which it is impossible to formulate a uified set of quatitative evaluatio criteria. I this case, the problem ca be solved by methods based o hierarchical obective evaluatio of alteratives without criteria aalysis. Give qualitative properties, the problem of compositio ca be solved usig biary relatios, for example, by the hierarchy aalysis [7]. But the problem is facilitated substatially if quatitative (or reducible to them criteria that permit operatios i a ormalized criteria space are employed to evaluate alteratives. The theory of multicriteria evaluatio ad optimizatio is applicable to such problems. The preset paper addresses such a class of problems. Formulatio of the problem The state of a hierarchical system for a give alterative is defied by the followig parameters: I = {,2,..., } is a set of elemetary subsystems evaluated usig lower-level hierarchy criteria; { y i} i I, y = { y,..., y} are the estimates of elemetary subsystems based o scalar criteria ad a vectorvalued criterio of the lower hierarchy level. The efficiecy of each highest level depeds o the estimates accordig to the lowest-level hierarchy criteria. The additioal coditios that defie the hierarchical structure are as follows: J = {,2,..., m} is the set of hierarchical levels; { I } is the distributio of subsystems ito levels, I =,2,..., }; J {

4 Iteratioal Joural "Iformatio Theories & Applicatios" Vol.5 / 2008 {λ } are priority vectors. J It is required to fid a aalytical estimate ϕ * ad a qualitative efficiecy evaluatio of the hierarchical structure ad to choose the best alterative from available oes. Solutio techique To aalytically evaluate the hierarchical structures for efficiecy, we propose to apply the method of ested scalar covolutios [8]. A compositio is carried out by a ested doll ( matresha priciple: scalar covolutios of weighed compoets of vector criteria of the lowest level are compoets of vector criteria of the highest level. The scalar covolutio of criteria obtaied at the uppermost level automatically becomes the expressio for the efficiecy evaluatio for the whole hierarchical system. The algorithm of ested scalar covolutios ca be represeted as sequetial weighed scalar covolutios of vector criteria at each hierarchy level i view of priority vectors based o the compromise (trade-off scheme selected ( ( {(, ϕ ( } ( ϕ λ J, ϕ y, ad efficiecy evaluatio of the whole hierarchical system ca be expressed by determiig scalar covolutio of the upper hierarchy level: ( m ϕ * = ϕ. I choosig solutios, the umber of alteratives is a. Each alterative is characterized by a hierarchical structure. For a =, the problem posed is trasformed ito the evaluatio of the give hierarchical structure. If a >, the each structure is evaluated as a give oe ad the alterative whose hierarchical structure has bee evaluated best, is chose. Therefore. i case of discrete multicriteria optimizatio, the base problem is to evaluate a give hierarchical structure. However, this method ca oly be used if the umber of alteratives a is relatively small, whe simple eumeratio does ot ivolve sigificat computatioal difficulties. For large sets of alteratives, other optimizatio methods should be applied, for example those stated i [9]. Compromise scheme As a base trade-off scheme for the method of ested scalar covolutios, we propose to use the oliear scheme described i [0]. It was established that without loss of geerality, a premise for its applicatio is that all of the partial criteria are subect to miimizatio ad are bouded: yi Ai, A = { Ai }, i [, ], where A is the vector of costraits. The scalar covolutio or ϕ ( λ, y = λ [ A y ] i i i 0 = i[ y0i ], ϕ ( λ, y λ if quatitative criteria are ormalized by the formula y 0 =y/a, is a simple iformative model of utility fuctio of the decisio maer at the lower level of hierarchy for criteria beig miimized accordig to the cocept of oliear trade-off scheme. Qualitative (but reducible to quatitative criteria are usually determied by experts usig scale poits. A questioaire with partial criteria is give to experts. Criteria are associated with a cotiuous scale divided, for example, ito te itervals. Zero o the scale is idicative of o weight, 0 correspods to the maximum weight. A expert should estimate the relative ifluece of each partial criterio o the geeral estimate uder give

5 2 Iteratioal Joural "Iformatio Theories & Applicatios" Vol.5 / 2008 coditios ad to associate it with the correspodig poit o the scale, characterized by a umber f. It is admissible to select poits betwee umbers or to assig some criteria to oe poit o the scale. A aalysis of decisio-maig processes has show that evaluatig obects o a 0-poits scale, experts are guided by gradatios of a so-called fudametal scale represeted i a geeral form i Table 2 ad described i [7], where qualitative gradatios of properties of obects are associated with the correspodig quatitative estimates f. It is possible to say that i terms of fuzzy-set theory [], the fudametal scale appears as a uiversal membership fuctio for a passage from a umber to the correspodig qualitative gradatio ad bac. A passage from a liguistic variable (satisfactory quality, excellet quality, etc to correspodig quatitative estimates f accordig to poit scale (5.5, 7.0, i.e. passage from fuzzy qualitative gradatios to umbers ad bac, is carried out. Table 2 Quality category Uacceptable Low Satisfactory Good High Rages of fudametal scale for f Rages of ormalized iverse fudametal scale for y 0, ϕ Estimates f are determied accordig to a direct 0-poit fudametal scale for the criteria beig maximized. The techique applied i the paper for multicriteria evaluatio accordig to a oliear trade-off scheme is developed for ormalized miimized criteria y 0 whose estimates are obtaied from f by the formula [2] y 0 =- 0, f, y 0 [0;]. This is reflected i Table 2 by a iverse ormalized scale. This scale is used to measure ormalized scalar covolutios of criteria ϕ 0 as well. Allowace for priorities The simplex Γλ = { λ λ i 0, λ i = } ( is the domai of defiitio of priority coefficiets λ Γλ, where λ i =cost are formal parameters with double physical meaig. O the oe had, these are priority coefficiets that express the preferece of a decisiomaer accordig to certai criteria. O the other had, these are coefficiets of a iformative regressio model costructed based o the cocept of oliear trade-off scheme. The coefficiets λ ca be determied at each hierarchy level through optimizatio o a simplex usig the dual approach described i [0] or by the formula ( fi λ i = (, I, f ( where i efficiecy of the -th subsystem o the -th level; λ is the i-th compoet of the priority vector at the ( i -th hierarchy level i evaluatig the f i is the sigificace parameter of the i-th subsystem of the ( ( -th level for the -th subsystem of the -th level (determied by experts usig a 0-poit scale; ad is the umber of subsystems of the ( -th level that support the -th subsystem of the -th level. I the most simple ad popular case, a multicriteria problem without priorities is formulated ad solved, where the decisio-maer assumes that all of the sigificace parameters are idetical for all of the subsystems. I this case, a elemetary scalar covolutio uder a oliear trade-off scheme i a uified form [0] is used.

6 Iteratioal Joural "Iformatio Theories & Applicatios" Vol.5 / For a compositio of criteria o hierarchy levels, it is expediet to calculate all of the scalar covolutios from top to bottom based o the cocept of a oliear trade-off scheme. I this case, the efficiecy of the -th subsystem at the -th hierarchy level as a ormalized ested scalar covolutio, i view of priority coefficiets, ca be evaluated by the formula ( 0 i 0i ] ( ( ( ( ϕ = N λ [ ϕ, I ( where 0i hierarchy level for evaluatig the -th subsystem of the -th level; ad (, ϕ 0 y0, (2 ϕ is the estimate of the i-th compoet of the ormalized vector-valued criterio at the ( Normalizig coditios ( N is a ormalizig factor. Normalizatio of ested scalar covolutios at each hierarchy level is of importace for the theory preseted here. I [5,8], the possibility of calculatig ormalizig coditios based o the priciple of oit liability of criteria is cosidered. The value of the greatest (worst criterio is separated out i the set of ormalized criteria. It is agreed that if this criterio has attaied the worst value, the remaiig ormalized criteria are assiged the possibility of attaiig the same values, which costitute compoets of the ormalizig vector. Such a approach is simple but wors oly if the criteria are really equivalet. ( ( It is logical that if estimates with respect to all of the partial relative criteria ϕ, [, 0i i ] are idetical ad ( equal to ϕ0i ϕ0, the their ormalized scalar covolutio i formula (2 should express the same aalytical ad qualitative estimate accordig to the iverted ormalized fudametal scale: ( ( N ( ϕ =. 0 λi ϕ Sice by ormalizig coditios ( ( ( 0 ( λ, the expressio for the ormalizig factor becomes i = ( N = ϕ0 ( ϕ0. (3 ( ( Let us use formula (3 to perform calibratio calculatios of the ormalizig factor N ( ϕ 0i for the estimates ( ( ϕ 0i, i [, ]. Let us compose the measure of the total quadratic error that occurs sice the uow ( ( ( factor N is used rather tha exact values of the ormalizig factor at calibratio poits N ( ϕ : M = ( ( ( ( [ N N ( ϕ Usig the ecessary extremum coditio for the fuctio M ( = 0, N we get the ormalizig factor ad with allowace for (3, we get N ( ( ( N = ( N ( ϕ ( = ( ( ϕ ( 0i 2 0 i ]. ( 0i ( ( ϕ 0i. 0i -th

7 4 Iteratioal Joural "Iformatio Theories & Applicatios" Vol.5 / 2008 Coclusios The recurrece formula (2 allows us to evaluate qualitatively ad quatitatively the scalar covolutios of criteria, ormalized usig the iverted fudametal scale, with respect to all of the hierarchy levels up to the upper oe: ϕ 0* =ϕ (m 0. The solutio of the multicriteria evaluatio problem for the example represeted by Table has resulted i the followig. From the scalar covolutio of partial criteria of the lower level of hierarchy, goig uder the headig (2 Geeral Criteria, we obtaied a aggregated criterio of the secod hierarchy level ϕ 0 = 0,2. Similarly, we (2 (2 obtaied the value of the aggregated criterio ϕ 02 = 0, 43 for Scietific Developmet Criteria, ϕ 03 = 0, 022 (2 for Ecoomic Criteria, ad ϕ 04 = 0, 00 for Social Criteria. The scalar covolutio of the idicated aggregated criteria of the secod hierarchy level has allowed obtaiig the ormalized estimate for the whole Biosorbet space research proect as a aggregated criterio of the third (3 hierarchy level ϕ0 * = ϕ0 = 0,094. Compariso of the aalytical estimates to the coverted ormalized fudametal scale (Table 2 allows cocludig that all of the aggregated criteria are withi the limits of the High Quality gradatio. Note that i calculatios of this example, all of the criteria were assumed to be of idetical sigificace, i.e., a multicriteria problem without priorities was solved. Evaluatio of a give alterative ad selectio of the best oe pertai to the class of problems of structural sythesis. Problems of parametrical sythesis solved by the method of ested scalar covolutios are described i [2]. Thus, ay multicriteria problem ca be represeted as a hierarchical system; at its lower level partial properties of the obect are evaluated usig a vector of criteria, ad at the upper level, the obect is evaluated as a whole by meas of compositio. Bibliography. I.M. Maarov, T.M. Viogradsaya, A.A. Rubchisi, ad V.B. Soolov, The Theory of Choice ad Decisio Maig [i Russia], Naua, Moscow ( D.B. Yudi, Computatioal Methods i Decisio Theory [i Russia], Naua, Moscow ( V.A. Gubaov, V.V. Zaharov, ad A.N. Kovaleo, Itroductio to Systems Aalysis [i Russia], Izd. LGU, Leigrad ( O.I. Larichev, Problems of collective decisio-maig i small groups, i: Noliear Dyamics ad Cotrol (To the 70-th Aiversary of Acad. S.V. Yemel yaov [i Russia], Editorial URSS, Moscow (999, pp A.N. Voroi, Systems aalysis ad multicriteria evaluatio of space proects by expert methods, Probl. Upravl. Iform., No., 2-35 ( V.G. Totseo, Decisio Support Systems ad Methods [i Russia], Nauova Duma, Kiev ( T.L. Saaty, Multicriteria Decisio Maig: The Aalytical Hierarchy Process, McGraw-Hill, New Yor ( A.N. Voroi, Nested scalar covolutios of a vector-valued criterio, Probl. Upravl. Iform., No. 5, 0-2 ( A.N. Voroi, Adaptive approximatio models i optimizatio, Cyber. Syst. Aalysis, 30, No. 5, ( A.N. Voroi, Yu. K. Ziatdiov, ad A.I. Kozlov, Vector Optimizatio of Dyamic Systems [i Russia], Techia, Kiev, (999.. L.A. Zadeh, The Cocept of a Liguistic Variable ad its Applicatio to Approximate Reasoig [i Russia], Mir, Moscow, ( A.N. Voroi, Vector optimizatio of hierarchical structures, Probl. Upravl. Iform., No. 6, (2004. Authors Iformatio Albert N. Voroi Natioal aviatio uiversity, faculty of computer iformatio techologies, Dr.Sci.Eg., professor; 03058, Kiev-58, Kosmoavt Komarov aveue,, Uraie