Internatonal Conference 20th EURO Mn Conference Contnuous Optmaton and Knowledge-Based Technologes (EurOPT-2008) May 20 23, 2008, Nernga, LITHUANIA ISBN 978-9955-28-283-9 L. Saalausas, G.W. Weber and E. K. Zavadsas (Eds.): EUROPT-2008 Selected papers. Vlnus, 2008, pp. 223 228 Insttute of Mathematcs and Informatcs, 2008 Vlnus Gedmnas Techncal Unversty, 2008 TRADE-OFF ANALYSIS TOOL FOR INTERACTIVE NONLINEAR MULTIOBJECTIVE OPTIMIZATION Petr Eselnen, Kasa Mettnen 2 Helsn School of Economcs, P.O. Box 20, FI-000 Helsn, Fnland 2 Dept. of Mathematcal Informaton Technology P.O. Box 35 (Agora), FI-4004 Unversty of Jyväsylä, Fnland E-mal: petr.eselnen@hse.f; 2 asa.mettnen@jyu.f Abstract: In nteractve methods, a decson maer (DM) drects the search for the most preferred Pareto optmal soluton wth hs/her preferences. We propose a tool that can be used to support the DM. Wth ths tool, the DM can convenently learn about local trade-offs and judge whether they are worthwhle. The tool s based on an dea where the DM s able to vary a selected Pareto optmal objectve vector. The vared vector s treated as a reference pont whch s then projected to the tangent hyperplane of the Pareto optmal set at the selected Pareto optmal soluton. Ths nformaton can be used to reflect what nd of Pareto optmal solutons and trade-offs are avalable n a local neghborhood of the selected soluton. Ths tool s especally useful when trade-off analyss must be carred out fast and wthout ncreasng computaton worload. Keywords: multobjectve optmaton, nteractve methods, reference pont method, trade-off analyss.. Introducton When solvng multobjectve optmaton problems, we must optme several conflctng objectve functons smultaneously. Because of the conflctng nature of the objectves, we can dentfy compromses, so-called Pareto optmal solutons, where we cannot mprove any objectve wthout mparng at least one of the others. The purpose of multobjectve optmaton methods s to offer support and ways to fnd the best compromse soluton. In ths, a decson maer (DM) and hs/her preference nformaton play an mportant role. By a DM we mean a person who s an expert n the doman of the problem consdered and who typcally s responsble for the fnal soluton. Multobjectve optmaton methods can be classfed n many ways (see e.g. (Mettnen, 999)). A wdely used class of methods s nteractve methods where the DM teratvely drects the soluton procedure by ndcatng hs/her preferences related to shown soluton canddates. The method then utles the gven preference nformaton and tres to produce new soluton canddates whch are more satsfyng for the DM. The teraton contnues untl the DM s satsfed or les to stop the soluton process. The ey feature of nteractve methods s that durng the soluton process the DM s able to learn about the underlyng problem as well as hs/her own preferences. So far, many nteractve methods have been proposed n the lterature (see e.g. (Mettnen, 999)). They dffer from each other, for example, by the type of preference nformaton utled. It s mportant that the DM s able to specfy the nd of nformaton (s)he s ased to and that the concepts used are famlar to hm/her. Reference pont based nteractve methods are popular because a reference pont has a natural meanng for the DM. It conssts of desrable objectve functon values, so-called aspraton levels (for each objectve). After the DM has specfed the reference pont, the feasble Pareto optmal soluton s found that best corresponds to t. If the DM s not satsfed, (s)he can specfy another reference pont. However, these methods are sometmes crtced for the fact that they do not provde support for the DM how to change the reference pont n order to get more preferred solutons. In ths paper, we propose a trade-off analyss tool that can offer the DM a way to analye soluton canddates. Ths tool s best suted for local analyss of solutons where the DM s nterested n studyng whether t s worthwhle to search for a better soluton n the neghborhood of some soluton. Ths study s strongly motvated by the experences gven by several DMs who have used the NIMBUS method (see e.g. (Mettnen, 999)). NIMBUS s a classfcaton based method and t s shown n (Mettnen and Mäelä, 2006) how t s closely related to reference pont based methods. The deas proposed here are 223
P. Eselnen, K. Mettnen drected to users of both classfcaton and reference pont based methods. The motvaton here s that real DMs n certan cases mss addtonal local trade-off nformaton so that they could get to now how values of objectves are changng, n other words, n whch drectons to drect the soluton process so that they could avod tral-and-error, that s, specfy some preference nformaton so that more preferred solutons wll be generated. Especally, n the case where the underlyng problem s computatonally demandng, t does not necessarly mae sense to compute addtonal solutons around some soluton just to grasp an dea of what nd of trade-offs there are avalable. The trade-off analyss tool to be presented n ths paper s a bt smlar to the automatc trade-off concept n the nteractve STOM method (Naayama and Sawarag, 984) where the DM classfes objectves at each teraton to three classes (mproved, relaxed, or accepted as they are). In STOM, a central dea s that the DM only gves desrable amounts of change for the objectves to be mproved and objectve trade-off nformaton s used to estmate how much relaxaton s needed n the others. The am s to decrease the cogntve burden set on the DM by asng less preference nformaton. The dfference to our approach s that we am at provdng the DM support for specfyng the classfcaton, that s, whch objectves to mprove or to relax. In our approach, the DM s assumed to determne desres of how to change objectve values to get a more preferred soluton. However, nstead of usng ths preference nformaton to generate new Pareto optmal solutons, we use trade-off analyss to see whether desred mprovements are possble. In other words, we use a lnear approxmaton of the soluton set to reflect local trade-offs near some soluton of a nonlnear problem. Because trade-off analyss s computatonally nexpensve, we can save n computatonal cost. Furthermore, t may tae tme to generate the new soluton. Wth our approach, the DM can get confrmaton whether t really maes sense to go to the drecton specfed and wat for the new soluton to be produced. Examples of other nteractve methods that utle objectve trade-off nformaton nclude ISWT (Chanong and Hames, 977), SPOT (Saawa, 982), GRIST (Yang, 999), IMOOP (Tappeta and Renaud, 999). In the frst three methods, objectve trade-offs are an ntegral part of the method whereas n the last one, trade-offs are offered for the DM n a sense of trade-off analyss. Here we understand trade-off analyss as addtonal supportng nformaton whch s generated to ad the DM but t s not necessarly drectly used n the method. In what follows, n Secton 2 we brefly present concepts and notaton. The trade-off analyss tool s presented n Secton 3. Fnally, we dscuss the potental of the tool and conclude n Secton 4. 2. Multobjectve problem and nteractve methods We consder nonlnear multobjectve optmaton problems n the form mnme { f ( ),..., f ( x)} () 224 x subject to x S n wth conflctng dfferentable objectve functons f : R R whch are mnmed subject to decson n n vector x belongng to a feasble set S R. For each x S we can use a mappng f : R R to T form a feasble objectve vector f ( x) = ( f( x),..., f ( x)) R where R s called an objectve space. The mage f ( S ) R of the feasble set s called a set of feasble objectve vectors. For the DM, potentally nterestng solutons of problem () can by dentfed by usng the concept of Pareto optmalty. A decson vector x S and the correspondng objectve vector f ( x ) = are Pareto optmal f there exsts no other decson vector x S such that f ( x) f ( x ) for all =,..., and at least one of the nequaltes s strct. We denote the set of Pareto optmal decson vectors by E S. We use a term Pareto surface for the set f ( E R ). A decson vector x S s wealy Pareto optmal f there does not exst another decson vector x S such that f ( x) < f ( x ) for all =,...,. A general nteractve multobjectve optmaton method can be outlned as follows:. Fnd an ntal Pareto optmal soluton; 2. Interact wth the DM; 3. Obtan a Pareto optmal soluton. If the new soluton or some of the prevous solutons s acceptable to the DM, stop; Otherwse, go to step 2. The man advantage of ths nd of nteractve methods s that the DM s allowed to gude the soluton procedure to
TRADE-OFF ANALYSIS TOOL FOR INTERACTIVE NONLINEAR MULTIOBJECTIVE OPTIMIZATION areas of the feasble set where the most nterestng Pareto optmal solutons are located. Here we concentrate on reference pont and classfcaton based methods. To be more specfc, we assume that an achevement scalarng problem (Werbc, 982) s used n steps and 3 to produce (wealy) Pareto optmal solutons for problem (). In other words, we solve the problem mnme α subject to w f ( x ) ) α, for =,..., (2) 225 ( x S, α R. For brevty of presentaton, we dscuss a scalarng formulaton that generates wealy Pareto optmal solutons. Wea Pareto optmalty can be avoded by consderng an augmented verson (see e.g. (Mettnen, 999)). In step 2 of an nteractve procedure, the DM sets aspraton levels to ndcate desrable levels of the objectve functons f, for =,...,. These values can be used to form a reference pont R. The scalng coeffcents w > 0 can be used to determne how the gven reference pont s projected to hgh low the set of Pareto optmal solutons. One possble scalng s w = /( ), where hgh and approxmate (computed or gven by the DM) hghest and lowest values for the objectve functon the Pareto optmal set, respectvely. In what follows, when we say that mplctly refer to soluton ( x, α ). 3. Interactve trade-off analyss tool low are f n x s an optmal soluton of (2) we In nteractve multobjectve optmaton methods, the DM consders Pareto optmal solutons n the set E. Because of the defnton of Pareto optmalty, movng from some Pareto optmal soluton to another one always necesstates some trade-off n objectve functon values. When usng an nteractve method, the DM mght be nterested n nowng what nd of trade-off taes place f some partcular Pareto optmal soluton s altered. From the practcal pont of vew, t s not always necessarly effcent or purposeful to carry out accurate computatons to reflect what s happenng n the set of Pareto optmal solutons. In such a case an approxmaton of Pareto surface can be used to study nterconnectons between the objectves. Let us assume that a Pareto optmal soluton x for problem () s produced by solvng problem (2) and the DM wants to study the trade-offs n a local neghborhood of = f ( x ) n order to determne n whch drecton to move from ths soluton. When avalable, a tangent hyperplane P for the Pareto surface f (E) at can be used to reflect trade-off nformaton. In other words, we produce a lnear ap- proxmaton for the nonlnear Pareto surface. In what follows, we refer to P wth a term trade-off plane. The trade-off nterpretaton of Karush-Kuhn-Tucer (KKT) multplers related to a soluton of problem (2) can be used to obtan ths trade-off plane (Yano and Saawa, 987, 990). ( Theorem 3. (Partal trade-off usng KKT-multplers) Let x be a soluton of (2) for some w and, wth optmal KKT-multplers λ related to constrants w f ( x ) ) α, for =,...,. Let us as- sume that. second-order KKT suffcency condtons are satsfed at x, 2. at x, gradents of actve constrants are lnearly ndependent and 3. all actve constrants have strctly postve KKT-multplers at x. Then partal trade-off nvolvng objectves f and f at x s f ( x ) / f = λ w / λw, for = 2,...,. Partal trade-off nformaton can be nterpreted as a lnear approxmaton for a relatve change n the value of f when the value of f ( = 2,..., ) s changng by one unt and at the same tme all the other objectves f j ( j = 2,...,, j ) reman at ther current levels. Ths nformaton can be presented n the form of partal trade-off vectors t j ( x ) R, where the st component s λ j w j / λw, j th component s, and the other components ( j = 2,...,, j ) are ero. We can drectly conclude that vector
P. Eselnen, K. Mettnen T n = ( λ w, w,..., w ) λ2 2 λ s orthogonal to ( x ) t j, for all j = 2,...,. In other words, we can express a normal vector n an explct form whenever we are able to compute partal trade-off vectors (see e.g. (Yang, 999) and (Yang and L, 2002)). The normal vector n can be used to charactere objectve vectors P R n the form n T ( ) = 0. Because P s gvng only a lnear approxmaton for the Pareto surface n a nonlnear case, we must pont out that most lely P s not Pareto optmal, and t mght be even nfeasble. However, we can use P to roughly reflect what nd of Pareto optmal objectve vectors are avalable n a local neghborhood of. In ths paper, we restrct our consderaton to Pareto optmal solutons of problem () where t s possble to compute the trade-off plane P usng KKT-multplers (see, e.g., (Ku et al., 997) how the assumptons n Theorem 3. can be verfed). Based on ths nformaton, the DM can judge whether more preferred solutons could be located n the neghborhood of the Pareto optmal soluton, n other words, n whch drectons to loo for better solutons by executng steps of the actual nteractve procedure (wthout approxmaton). Let us assume that the DM wants to study the trade-off behavor n a local neghborhood of a Pareto optmal. The DM s assumed to determne desres of how to change objectve values to get a more preferred soluton. However, nstead of usng ths preference nformaton to generate new Pareto optmal solutons, we use trade-off analyss to see whether desred mprovements are possble. Because trade-off analyss s computatonally nexpensve, we can save n computatonal cost. Furthermore, t may tae tme to generate the new soluton. Wth the trade-off tool, the DM can get confrmaton whether t really maes sense to go to the drecton specfed. A vector d R (expressed by the DM) s used to ndcate desred changes n the objectve functon values at. Ths determnes a reference pont = ( + d) whch can be used n a lnear achevement scalaraton problem mnme α subject to w ( ( + d )) α, for =,..., (3) n T ( ) = 0 R, α R In other words, the Pareto surface s temporarly replaced by a lnear approxmaton at (see Fg. ) and the soluton closest to the reference pont s found on ths trade-off plane and shown to the DM. Ths gves some rough nformaton for the DM about the feasblty of the desres expressed n the reference pont. It s preferable that the varaton d s set n such a way that t obeys the concept of Pareto optmalty, that s, f value of some objectve s mproved then the value of at least one objectve should be degraded. Problem (3) has a unque soluton except f the normal vector n s parallel to some of the coordnate axs n the objectve space R. For our purposes, t s enough to use nformaton that the projecton drecton n achevement scalarng problem (2) s determned by a scalng vector w R (.e. we do not have to pay specal attenton to wea Pareto optmalty). Thus, nstead of solvng the lnear problem (3), we can drectly compute where the projecton drecton vector ( + d + tw) and trade-off plane n T T ( ) = 0 are ntersectng. Ths means solvng for t n equaton n (( + d + tw) ) = 0 whch ~ T T we can wrte n the form t = n d / n w (we do not need to consder case n T w= 0 because w > 0, n 0, and n 0 ). Then, the approxmated Pareto optmal objectve vector obtaned usng the preference nformaton gven by the DM s ~ = ( + d) + ~ t w = + ~ t w. In Fg., we demonstrate the trade-off tool wth a smple problem of two objectves, where the DM consders soluton and s nterested n mprovng f 2 value by allowng f to mpar as ndcated by the reference pont = ( + d). Accordng to the trade-off analyss, the approxmated soluton reflectng s ~ (located on the trade-off plane). It s actually rather close to whch s the actual Pareto opt- 226
TRADE-OFF ANALYSIS TOOL FOR INTERACTIVE NONLINEAR MULTIOBJECTIVE OPTIMIZATION mal objectve vector correspondng to f problem (2) had been solved. From ~, the DM gets fast some understandng about the feasblty of the desres specfed wthout solvng (2) at all. In a general nonlnear case, too large d may lead to a poor approxmaton. In practce, the DM can be nformed about an approprate maxmal varaton (see Secton 4). Fg. 2 gves an example from a practcal pont of vew. Now the problem consdered has three objectves. The DM has selected for trade-off analyss a Pareto optmal soluton for whch objectve func- ton values are vsualed usng a bar chart. The arrows ndcate a reference pont determned by the DM. The crcles ndcate the approxmated Pareto optmal objectve vector obtaned usng trade-off analyss, that s, plane P. Now by alterng the DM can get an dea of what nd of trade-offs are avalable n the local neghborhood of the selected Pareto optmal soluton. Fg.. Approxmaton of the Pareto surface Fg. 2. Varyng the selected soluton 4. Dscusson and conclusons The defnte strength of the method proposed s that the lnear approxmaton of the Pareto surface can be produced as a sde-product when a Pareto optmal soluton s computed. Furthermore, n the tradeoff analyss, all computatons are fast and can be performed n real-tme. Because of ths, the DM s able to capture, wth a relatvely small effort, an dea about the trade-off rates and local behavor of the Pareto surface around the selected Pareto optmal soluton. The reference pont determned n trade-off analyss can be used drectly to produce the correspondng actual Pareto optmal soluton. Therefore, ths tool can be used to buld up confdence for the DM whle s(he) s settng the next reference pont n a reference pont based nteractve method. However, the DM s not forced to set preferences accordng to nformaton obtaned n the analyss. Ths method can be seen as an addtonal decson supportng tool that can be used n an nteractve framewor when needed. The beneft of ths approach s that the trade-off analyss can be made dynamcally through vsualaton, le presented n Fg. 2, nstead of consderng just statc numercal trade-off values. On the other hand, one potental drawbac related to ths method s the qute demandng assumptons that must be satsfed when solvng problem (2). For nstance, the second order KKT suffcency condtons assume that the functons related to the problem are twce contnuously dfferentable. Ths may be dffcult to guarantee n the case of practcal problems. Furthermore, lnear ndependence of actve constrants may also be problematc f we have, for nstance, more objectves than varables (however, ths happens very rarely n practcal applcatons). Because we are dealng wth an approxmaton n the objectve space, the connecton to the decson space s temporarly lost. Ths may be a drawbac especally n desgn problems where, n addton to the objectve functon values, the correspondng decson vector mght be needed to produce decson supportng vsualatons. As ponted out, the assumptons of Theorem 3. can be verfed at any soluton selected for trade-off analyss but from the computatonal pont of vew, ths may be worthwhle only f the second order nformaton s avalable n an analytc form. Furthermore, we must emphase that the solver used to solve problem (2) must be able to produce KKT-multplers. These aspects should be ept n mnd when the trade-off analyss tool s utled n practce as a part of some nteractve method. Topcs for further research contan a study of effcent qualty and error measures that could be convenently related to the approxmaton used for trade-off analyss. The am s that qualty measures developed should have some concrete meanng for the DM. For nstance, t must be studed how all Pareto 227
P. Eselnen, K. Mettnen optmal solutons produced n an nteractve procedure, at the moment when the trade-off analyss taes place, could be used to produce qualty measures. In addton, an approxmaton error between the tradeoff plane and the Pareto surface s also an mportant topc. Furthermore, the proposed tool must be tested wth real problems and decson maers. References Chanong, V. and Hames, Y. Y. 977. The Interactve Surrogate Worth Trade-off (ISWT) Method for Multobjectve Decson Mang, n Zonts, S. (Ed.). Multple Crtera Problem Solvng. Sprnger-Verlag, 42 67. Ku, H.; Tanno, T. and Tanaa, M. 997. Trade-off Analyss for Vector Optmaton Problems va Scalaraton, Journal of Informaton & Optmaton Scences 8: 75 87. Mettnen, K. 999. Nonlnear Multobjectve Optmaton. Kluwer, Boston. Mettnen, K. and Mäelä, M. M. 2006. Synchronous Approach n Interactve Multobjectve Optmaton, European Journal of Operatonal Research 70(3): 909 922. Naayama, H. and Sawarag, Y. 984. Satsfcng Trade-off Method for Multobjectve Programmng, Interactve Decson Analyss, n Grauer, M. and Werbc, A. P. (Ed.). Proceedngs of an Internatonal Worshop on Interactve Decson Analyss and Interpretatve Computer Intellgence. Sprnger-Verlag, New Yor, 3 22. Saawa, M. 982. Interactve Multobjectve Decson Mang by the Sequental Proxy Optmaton Technque: SPOT, European Journal of Operatonal Research 9: 386 396. Tappeta, R. V. and Renaud, J. E. 999. Interactve Multobjectve Optmaton Procedure, AIAA Journal 37: 88 889. Werbc, A. A. 982. Mathematcal Bass for Satsfcng Decson Mang, Mathematcal Modellng 3(25): 39 405. Yang, J.-B. 999. Gradent Projecton and Local Regon Search for Multobjectve Optmsaton, European Journal of Operatonal Research 2: 432 459. Yang, J.-B. and L, D. 2002. Normal Vector Identfcaton and Interactve Tradeoff Analyss Usng Mnmax Formulaton n Multobejctve Optmaton, IEEE Transactons on Systems, Man, and Cybernetcs Part A: Systems and Humans 32(3): 305 39. Yano, H. and Saawa, M. 987. Trade-off Rates n the Weghted Tchebycheff Norm Method, Large Scale Systems, 3: 67 77. Yano, H. and Saawa, M. 990. Trade-off Rates n the Hyperplane Method for Multobjectve Optmaton Problems, European Journal of Operatonal Research 44: 05 8. 228