AN INTERACTIVE APPROACH FOR MULTI-CRITERIA SORTING PROBLEMS

Transcription

1 AN INTERACTIVE APPROACH FOR MULTI-CRITERIA SORTING PROBLEMS A THESIS SUBMITTED TO THE GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES OF MIDDLE EAST TECHNICAL UNIVERSITY BY BURAK KESER IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE IN INDUSTRIAL ENGINEERING APRIL 2005

2 Approval of the Graduate School of Natural and Appled Scences. Prof. Dr. Canan Özgen Drector I certfy that ths thess satsfes all the requrements as a thess for the degree of Master of Scence. Prof. Dr. Çağlar Güven Head of the Department Ths s to certfy that we have read ths thess and that n our opnon t s fully adequate, n scope and qualty, as a thess for the degree of Master of Scence. Prof. Dr. Murat Kösalan Supervsor Examnng Commttee Members Prof. Dr. Nesm Erp (METU, IE) Assoc. Prof. Yasemn Sern (METU, IE) Prof. Dr. Erdal Erel (Blent U., MAN) Prof. Dr. Murat Kösalan (METU, IE) Assst. Prof. Esra Karasaal (METU, IE)

3 Plagarsm I hereby declare that all nformaton n ths document has been obtaned and presented n accordance wth academc rules and ethcal conduct. I also declare that, as requred by these rules and conduct, I have fully cted and referenced all materal and results that are not orgnal to ths wor. Name, Last Name : Bura KESER Sgnature :

4 ABSTRACT AN INTERACTIVE APPROACH FOR MULTI-CRITERIA SORTING PROBLEMS Keser, Bura M. Sc., Department of Industral Engneerng Supervsor: Prof. Dr. Murat Kösalan Aprl 2005 Ths study s concerned wth a sortng problem; the placement of alternatves nto preference classes n the exstence of multple crtera. An nteractve model s developed to address the problem, assumng that the decson maer has an underlyng utlty functon whch s lnear. A recent methodology, Even-Swaps, whch s based on value tradeoff s utlzed n the model for both mang an estmaton of the underlyng utlty functon and generatng possble domnance among the alternatves on whch t s performed. Convex combnatons, domnance relatons, weght space reducton, Even-Swaps and drect decson maer placements are utlzed to place alternatves n preference classes. The proposed algorthm s expermented wth randomly generated alternatve sets havng dfferent characterstcs. Keywords: multple crtera decson mang, sortng, even swaps v

5 ÖZ ÇOK KRİTER ALTINDA SIRALAMA PROBLEMLERİ İÇİN ETKİLEŞİMLİ BİR YAKLAŞIM Keser, Bura Yüse Lsans, Endüstr Mühendslğ Bölümü Tez Yönetcs: Prof. Dr. Murat Kösalan Nsan 2005 Bu çalışma, alternatflern ço rter altında terch sınıflarına yerleştrlmes le lgldr. Karar vercnn gzl fayda fonsyonunun doğrusal olduğu varsayılara, etleşml br yöntem gelştrlmştr. Değer ödünleşmeler üzerne urulu yen br metodoloj olan Eş-Taas yöntemnden, hem arar vercnn gzl fayda fonsyonunu tahmnleme hem de üzernde uygulandığı alternatfler arasında olası br basınlı lşs oluşturma üzere faydalanılmıştır. Alternatfler terch sınıflarına yerleştrme çn onves ombnasyonlar, basınlı lşs, ağırlı uzayı daraltılması, Eş-Taas ve arar vercnn doğrudan yerleştrmelernden faydalanılmıştır. Önerlen algortma, rastsal oluşturulmuş farlı araterde alternatf ümeleryle denenmştr. Anahtar Kelmeler: ço rterl arar verme, sıralama, eş-taas v

6 To my lovely famly v

7 ACKNOWLEDGMENTS I would le to than my thess supervsor Prof. Dr. Murat Kösalan for hs contnuous support, gudance and patence throughout my wor. I am also thanful to Onur Atuğ, Cen Güray, Bulut Aslan, Ümt Sönmez, Engn Ayüre, and Gzem Karslı. Fnally my dear famly; Elf, Remzye and Üzeyr Keser; ths wor would not be possble wthout ther support. v

8 TABLE OF CONTENTS PLAGIARISM... ABSTRACT...v ÖZ... v ACKNOWLEDGMENTS...v TABLE OF CONTENTS...v LIST OF FIGURES... x LIST OF TABLES...x 1. INTRODUCTION Problem Defnton Lterature Survey Even - Swaps Even - Swaps Example The Evoluton of the algorthm DEVELOPMENT OF THE MODEL Some Notaton and Assumptons Even Swaps Selectng the alternatves for the even swap Performng the Even-Swap Even-Swap on more than two crtera Estmatng the utlty functon usng the Even-Swap Alternatve Selecton Determnng Best and Worst Classes Convex Combnaton Chec Utlzng LPs for best and worst classes Fndng equvalent dummy ponts Decson maer placement THE ALGORITHM Summary of the Algorthm v

9 3.2 The Algorthm AUTOMATED APPROACH AND EPERIMENTATION Development of the Automaton and User Screens Expermentaton SUMMARY AND CONCLUSION REFERENCES APPENDI A Detaled flow of the algorthm...a-1 APPENDI B Expermentaton... B-1 x

10 LIST OF FIGURES Fgure 1. An overall flow of the Even-Swap framewor... 8 Fgure 2. Even-Swap example - graphcal representaton of the swap Fgure 3. Even-Swap on more than two crtera Fgure 4. Graphcal representaton of the example Fgure 5. Step 6 shown graphcally Fgure 6. Graphcal representaton of domnance and weght space reducton Fgure 7. Flow of the ntalzaton phase Fgure 8. Flow of the "placng alternatves phase" Fgure 9. Example - Alternatves graphcally represented Fgure 10. Example - Fnal status represented graphcally Fgure 11. Insertng number of alternatves to be consdered Fgure 12. Insertng the consstency ndex Fgure 13. Even Swap Screen Fgure 14. DM Placement Fgure 15. Consstency ndex not vald Fgure 16. Swap done n the wrong drecton Fgure 17. DM placed the alternatve to a wrong class Fgure 18. Averages wth dfferng alternatve set szes Fgure 19. Averages wth dfferng consstency ndexes Fgure 20. Results for 20 alternatves Fgure 21. Results for 50 alternatves Fgure 22. Results for 100 alternatves x

11 LIST OF TABLES Table 1. Real world applcatons of classfcaton and sortng problems... 2 Table 2. Even Swap Example - Consequence Table Table 3. Even-Swap Eample - Elmnated Alternatve by Domnance Table 4. Even-Swap Example - A Crteron and Alternatve Elmnated Table 5. Even-Swap Example - Elmnated Alternatve Table 6. Even-Swap Example - Fnal Table Table 7. Even-Swap example - alternatves Table 8. Even-Swap example - the swap Table 9. Decsons for dfferent values of decson varables Table 10. Example - Lst of alternatves and ther crtera values Table 11. Example - Current status of the alternatves (1) Table 12. Example - Current status of the alternatves (2) Table 13. Example - Fnal status Table 14. Example - Means of placements Table 15. Run codes and Run parameters Table 16. Run Results Summary x

12 CHAPTER 1 INTRODUCTION 1.1 Problem Defnton The problem consdered n ths study s a mult-crtera decson mang (MCDM), problem where the decson maer (DM) ntends to group a set of alternatves nto preference classes. The approach developed n ths study wll be more sutable for problems where the alternatve set s large, and the number of crtera to consder s small. The placement of alternatves nto predefned classes or groups s referred to as classfcaton or sortng problems dependng on whether the groups are nomnal or ordnal. The problem of sortng / classfcaton has numerous practcal applcatons, some of whch are lsted below (Zopounds, C., Doumpos, M., 2002) Medcne: performng medcal dagnoss through the classfcaton of patents nto dseases groups on the bass of some symptoms (Stefanows and Slowns, 1998; Tsumoto, 1998; Belacel, 2000; Mchalows et al., 2001). Pattern recognton: examnaton of the physcal characterstcs of objects or ndvduals and ther class-fcaton nto approprate classes (Rpley, 1996; Young and Fu, 1997; Neddu and Patrz, 2000). Letter recognton s one of the best examples n ths feld. Human resources management: assgnment of personnel nto approprate occupaton groups accordng to ther qualfcatons (Rulon et al., 1967; Gochet et al., 1997). 1

13 Producton systems management and techncal dagnoss: montorng the operaton of complex producton systems for fault dagnoss purposes (Nowc et al., 1992; Catelan and Fort, 2000; Shen et al., 2000). Maretng: customer satsfacton measurement, analyss of the characterstcs of dfferent groups of customers, development of maret penetraton strateges, etc. (Duta, 1995; Ssos et al., 1998). Envronmental and energy management, ecology: analyss and measurement of the envronmental mpacts of dfferent energy polces, nvestgaton of the effcency of energy polces at the country level (Daoula et al., 1999; Ross et al., 1999; Flnman et al., 2000). Fnancal management and economcs: busness falure predcton, credt rs assessment for frms and consumers, stoc evaluaton and classfcaton, country rs assessment, bond ratng, etc. (Altman et al., 1981; Slowns and Zopounds, 1995; Zopounds, 1998; Doumpos and Zopounds, 1998; Greco et al., 1998; Zopounds et al., 1999a,b). Below table (Zopounds, C., Doumpos, M., 2002) shows some real-world applcatons of classfcaton / sortng problems (Table 1). Some of these studes use real-world data for llustratve purposes n order to present the practcal applcablty of classfcaton and sortng theory n real-world data sets. Other studes use real-world data for performance evaluaton of selected methods orgnatng from the developed methods wth exstng methods, most commonly orgnatng from the feld of statstcs. Table 1. Real world applcatons of classfcaton and sortng problems Applcaton Area Studes Busness falure predcton Mahmood and Lawrence (1987), Gupta et al. (1990), Slowns and Zopounds (1995), Gehrlen and Wagner (1997b), Greco et al. (1998), Zopounds and Dmtras (1998), Zopounds and Doumpos (1999), Zopounds et al. (1999b), Konno and Kobayash (2000) 2

14 Applcaton Area Studes Credt cards assessment Lam et al. (1996), Zopounds et al. (1998) Country rs evaluaton Doumpos and Zopounds (2001a) Ecology Ross et al. (1999), Flnman et al. (2000) Educatonal admnstraton Choo and Wedley (1985), Lam et al. (1993) Energy plannng Daoula et al. (1999) Medcne Stefanows and Slowns (1998), Belacel (2000), Mchalows et al. (2001) Personnel management Gochet et al. (1997) Portfolo selecton and Zopounds et al. (1999a), Naayama and Kagau management (1998), Doumpos et al. (2000) R&D project evaluaton Jacquet-Lagreze (1995) Techncal dagnoss Nowc et al. (1992) Venture captal Stam (1990) nvestments Ths broad range of applcaton doman lead researchers to develop dfferent approaches for constructng sortng/classfcaton models. The approach developed n ths study s an nteractve method, whch utlzes Even-Swap method, weght space reducton technques and domnance relatons. The use of Even-Swap method n the study s slghtly dfferent from t s orgnal use, where the orgnal method ams to fnd the best alternatve n a gven alternatve set by nteractng wth the DM. 1.2 Lterature Survey Consderng a set of alternatves descrbed by a number of crtera; dfferent problems are consdered n the lterature. One, whch s more frequently addressed, s to dentfy the best alternatve or select a lmted set of the best alternatves (ths problem s also referred as the choce problem ). Another problem s to construct a ran orderng of the alternatves from best to the worst 3

15 ones (ths problem s also referred as the ranng problem ). One other problem s to classfy or sort the alternatves nto groups, where the groups may ether have a preference relaton or not (ths problem s also referred as the classfcaton/sortng problem ). The problem of dentfyng the most preferred alternatve among a number of alternatves where each alternatve s defned by several crtera s well studed n the lterature. The studes of Keeney and Raffa (1976) and Green and Srnvasan (1978) attempt to solve ths problem by fttng a utlty functon that explans the preferences of the decson maer (DM), and then fndng the alternatve that performs best accordng to the ftted utlty functon. Another approach for fndng most preferred alternatve had been nteractve approach. Interactve approaches typcally assume that the DM has an underlyng utlty functon. However, the exact form of the utlty functon s assumed to be unnown to both the DM and the analyst. The DM s expected to be consstent wth hs/her underlyng utlty functon whle expressng hs/her preferences. For the case where the underlyng utlty functon s assumed to be lnear, Zonts (1981) and Kösalan (1984) developed nteractve approaches. Several nteractve approaches have been developed for the quas-concave utlty functon case (Korhonen et al. 1984, Kösalan et al. 1984, Kösalan and Taner 1992, Malaoot 1989); Kösalan and Sagala (1995) also developed an approach for the general monotone utlty functon case. Korhonen (1998) developed a vsual nteractve approach that maes no assumpton on the underlyng utlty functon of the DM. Kösalan and Öden (1989) and Kösalan and Rz (2001) have also developed vsual nteractve approaches and utlzed graphcal ads n ther nteractve approaches. A more recent revew of the mult-crtera lterature s provded by Jacquet-Lagreze and Ssos (2001). The problem of constructng a ran orderng of the alternatves from has also been 4

16 a problem of nterest. Assumng that the attrbutes have been measured at least on an ordnal scale, Korhonen and Somaa (1981) attempt to fnd a complete ran orderng of alternatves. Malaoot (1989) uses quas-concave non-lnear multattrbute utlty functons to ran multple crtera alternatves, and shown that par comparson questons can be used to generate partal nformaton on the weghts. Another type of problem s the assgnment of alternatves nto predefned groups, whch s referred to as classfcaton or sortng problems. Whle both classfcaton and sortng refer to the assgnment of a set of alternatves nto predefned groups, they dffer wth respect to the way that the groups are defned. Classfcaton refers to the case where the groups are defned n a nomnal way. On the contrary, sortng refers to the case where the groups are defned n an ordnal way startng from those ncludng the most preferred alternatves to those ncludng the least preferred alternatves. Earler wor on classfcaton can be traced bac to Fsher (1936), whose wor was on the lnear dscrmnant analyss. Some other statstcal approaches was developed followng Fsher (Blss 1934, Berson 1944, McFadden 1974), whch was later on crtczed for ther statstcal assumptons (Altman et al. 1981). Recent research on developng classfcaton and sortng models s manly based on operatons research and artfcal ntellgence. Compared to other approaches, mult-crtera decson adng research (MCDA) does not focus solely on developng automatc procedures for analyzng an exstng data set n order to construct a classfcaton/sortng model. MCDA researchers also emphasze on the development of effcent preference modelng methodologes that wll enable the decson analyst to ncorporate the decson maer s preferences n the developed classfcaton/sortng model (Zopounds, C., Doumpos, M., 2002). Outranng relaton and utlty functon are the most wdely used crtera aggregaton models n MCDA lterature, whch are also employed for classfcaton and sortng purposes. The most wdely used sortng method based on outranng relatons s the ELECTRE TRI method (Yu 1992, Roy and 5

17 Bouyssou 1993). An alternatve approach for outranng relaton, the utlty theory framewor, s used n UTADIS for sortng purposes (Jacquet-Lagreze 1995, Zopounds and Doumpos 1999). Kösalan and Ulu (2001, 2003), developed an nteractve procedure for parttonng the alternatves nto preference classes regardng dfferent forms of utlty functons of the DM. Domnance, weght space reducton and drect DM placement technques are used to place alternatves. More recently sgnfcant research has been conducted on the use of the rough set approach as a methodology of preference modelng n mult-crtera decson problems (Greco et al. 1999, 2000). The rough approxmatons of decson classes nvolve domnance relaton, nstead of ndscernblty relaton consdered n the basc rough sets approach. They are bult of reference alternatves gven n the sortng example. Decson rules derved from these approxmatons consttute a preference model. Also, the domnance-based rough set approach s able to deal wth sortng problems nvolvng both crtera and regular attrbutes (whose domans are not preference ordered), (Greco et al., 2002), and mssng values n the evaluaton of reference alternatves (Greco et al., 1999, 2000b). The use of neural networs s another nterestng approach that can be used for preferental modelng purposes n mult-crtera classfcaton and sortng problems. Neural networs enable the modelng of hghly complex non-lnear behavors of decson-maers. Man dsadvantage of the neural networs s that, the results of a neural networs are dffcult to nterpret n terms of the gven nputs to the networ. The major advantage on the other hand s that, neural networs can be used to assess utlty functons, wthout posng any assumptons or restrctons on ther partcular structure or propertes. Arhcer and Wang (1993) showed that neural networs can provde an effcent mechansm for preference modelng n sortng problems. It s mportant to note that, the development of decson support systems that wll 6

18 enable decson-maers to tae advantage of the capabltes that the classfcaton and sortng approaches provde. Several mult-crtera decson support systems have been developed over the past decade mplementng MCDA classfcaton and sortng methods. The most characterstc are the RANGU system developed by Stam and Ungar (1995), the PREFDIS system of Zopounds and Doumpos (2000a), the ELECTRE TRI-Assstant system of Mousseau et al. (2000), the ROSE system of Pred et al. (1998) and the 4eMa system of Greco et al. (1999a) (Zopounds, C., Doumpos, M., 2002). 1.3 Even Swaps Even Swaps (Hammond et al. 1998, 1999) s a mult-crtera decson mang method based on value trade-offs whch are called even swaps. Performng sensble trade-offs s one of the most mportant and dffcult challenges n decson mang (Keeney and Raffa 1976; Keeney 2002). The even swaps method s developed n order to fll the gap of clear, easy-to-use and ratonal trade-off methodology. It provdes a practcal way of mang trade-offs among any set of objectves across a range of alternatves. It s a form of barterng that forces the decson maer to thn about the value of one objectve n terms of another. The even swap method does not argue that t provdes a mechansm whch maes complex decsons easy, but what t does provde s a relable mechansm for mang trades and consstent framewor n whch to mae them. In an even swap, the value of an alternatve n one attrbute s changed and ths change s compensated wth a preferentally equal value change n some other attrbute. The new alternatve wth these revsed values s equally preferred to the ntal one and thus t can be used nstead. The am of the method s to carry out even swaps that mae ether attrbutes rrelevant, n the sense that all the alternatves have equal values on ths attrbute, or alternatves domnated, n the sense that some other alternatve s at least as good as ths alternatve on every attrbute. Such attrbutes and alternatves can be elmnated, and the process contnues untl one alternatve,.e. the most preferred one, remans. 7

19 The man requrement for usng the Even Swaps method s to understand the dea of an even swap. The decson maer (DM) does not need to have a mathematcal bacground to use the method. Hammond et al. (1998, 1999) emphasze on the practcal aspects of the process, and let the DM to focus on the most mportant wor of decson mang: decdng the real value to hm/her. The general flow of the even swap framewor s provded n Fgure 1: Problem Intalzaton Elmnate Domnated Alternatves Mae an Even Swap Determne alternatves and crtera to perform even swap on them Elmnate Irrelevant Attrbutes More than one remanng alternatve? YES Determne the requred change Assess the requred change on the other crteron for compensaton Perform the swap NO Most preferred alternatve found Fgure 1. An overall flow of the Even-Swap framewor At the problem ntalzaton step, a consequence table s constructed n order to have a clear pcture of the alternatves and ther consequences for each crteron. The mportant thng when constructng consequence tables s to use consstent 8

20 terms for each crteron. Once the consequences table s constructed and crteron values for each alternatve s mapped, loo for opportuntes to elmnate one or more alternatves. If an alternatve A s better than alternatve B n some crtera and not worse than B n all other crtera, alternatve B can be elmnated from consderaton (alternatve B s domnated by alternatve A). Another ssue s to elmnate rrelevant crtera wth an obvous tenet; f every alternatve s rated equally on a gven crteron, that crteron can be gnored whle mang decsons. Now, the challenge s to choose the most proper even-swap to perform among numerous choces. Ths selecton shall be made consderng the nformaton that wll be provded after the swap. Even-swaps that can lead to a possble domnance or rrelevant crtera shall be preferred. Determnng the relatve value of dfferent crteron values s hard. The orgnators of the even-swap approach, Hammond, Keeney and Raffa, quoted some suggestons to mae sound trade-offs (Hammond et al. 1998, 1999). There are few reported applcatons of even-swaps n the lterature; where one of them s on strategy selecton n a rural enterprse (Kajanus et al. 2001) and another one on envronmental plannng (Gregory and Wellman 2001). Despte of the smplcty of the method, the lac of use may be due to nsuffcent computatonal help provded. Recently Mustajo and Hämälänen (2004) developed a decson support system for even swap approach, whch s supported by Preference Programmng. Preference programmng s a framewor for modelng ncomplete nformaton wthn mult-attrbute value theory. 1.4 Even Swap Example To llustrate the above dscussed methodology, a small example s gven below. The problem s about selectng a second-hand car among a number of alternatves. Alternatves are evaluated on three crtera (of course, there should be more, but three of them are selected for smplcty), these are: Age of the car (gven by model year) Mleage of the car (gven n lometers) 9

21 Prce of the car (n YTL.) Model year of the car s a hgher the better type crteron. But the last two, mleage and prce are lower the better. The alternatves are selected from a popular car sales web-ste from Turey. The alternatves and ther values on three crtera are gven n Table 2 (ths s the consequence table for the DM): Table 2. Even Swap Example - Consequence Table Toyota Corolla GL Peugeot 206 R Opel Corsa Swng Honda Cvc HB Ford Festa Model Mleage m m m m m Prce YTL YTL YTL YTL YTL Now, we wll loo for opportuntes to elmnate one or more alternatves by domnance. It can be observed that, Opel s domnated by both Ford and Toyota, so t can be elmnated for further consderaton, (see Table 3). Table 3. Even-Swap Eample - Elmnated Alternatve by Domnance Toyota Corolla GL Peugeot 206 R Opel Corsa Swng Honda Cvc HB Ford Festa Model Mleage m m m m m Prce YTL YTL YTL YTL YTL Now, we wll perform an even-swap on Toyota and as the DM How much wll you ncrease the prce, for an ncrease n the model from 1999 to 2000?. The DM says I wll ncrease the prce from YTL to YTL. The new consequence table s gven below (Table 4), the swapped values are hghlghted. 10

22 All alternatves score the same on Model crteron, so ths alternatve s now rrelevant and can be elmnated; furthermore, Toyota s now domnated by Ford, so t can be elmnated from the alternatve set, these are also shown n Table 4. Table 4. Even-Swap Example - A Crteron and Alternatve Elmnated Toyota Corolla GL Peugeot 206 R Honda Cvc HB Ford Festa Model Mleage m m m m Prce YTL YTL YTL YTL The consequence table s reduced to a much smaller form than the orgnal table. But, three alternatves are left, so we need more swaps and propose the DM another one on Ford : How much wll you ncrease the prce, for a decrease n the mleage from to 55000?. The DM says I wll ncrease the prce from YTL to YTL. Now, Peugeot s domnated by Ford and elmnated for further consderaton (Table 5). Table 5. Even-Swap Example - Elmnated Alternatve Peugeot 206 R Honda Cvc HB Ford Festa Mleage m m m Prce YTL YTL YTL Two alternatves are left, one more swap s requred, the followng queston s ased to DM, for performng an even-swap on Honda : How much wll you ncrease the prce, for a decrease n the mleage from to 55000?. The DM says I wll ncrease the prce from YTL to YTL. Fnally Honda s domnated by Ford, and elmnated. Ths reveals Ford to be the preferred alternatve for the DM (Table 6). 11

23 Table 6. Even-Swap Example - Fnal Table Honda Cvc HB Ford Festa Mleage m m Prce YTL YTL 1.5 The Evoluton of the algorthm Ths study s based on dfferent approaches developed n the feld of mult-crtera decson mang and sortng problems lterature; and proposes a new nteractve approach for mult-crtera sortng problems. The weght space reducton deas generated n Kösalan and Ulu (2001, 2003) are utlzed, but LPs used for weght space reducton are dfferent n order to be more effcent. Even-Swaps methodology (Hammond et al. 1998, 1999) s ncluded n the algorthm both for elctng nformaton from the DM and for placng the alternatves. Even-Swaps approach s orgnally proposed for selectng the best alternatve among a set of alternatves, however, n ths study t s used for sortng a set of alternatves to preference classes. The followng chapter dscusses the approaches developed n each step of the proposed algorthm. The thrd chapter presents the algorthm step by step, and llustrates a manual example. The fourth chapter ntroduces the developed automaton for the algorthm, and presents some results, whch are obtaned by usng the algorthm. The fnal chapter gves a summary of the study, presents some conclusons and proposes some possble future wor, whch can be performed as extensons to ths study. 12

24 CHAPTER 2 DEVELOPMENT OF THE MODEL Ths chapter dscusses the approach developed n each step of the algorthm. A detaled flow of the algorthm s gven n Appendx A. The proposed algorthm s two phased, the ntalzaton phase and then the selectng alternatves phase. At the ntalzaton phase an ntal estmaton of the DM s underlyng utlty functon s made, usng the nformaton ganed from an Even-Swap whch s proposed to DM by selectng two alternatves from the alternatve set. At the placng alternatves phase, ntally an alternatve s selected to be placed, among the set of unplaced alternatves. The algorthm tres to place the selected alternatve nto a preference class ether usng domnance relatons, convex combnatons or weght space reducton technques. If the alternatve cannot be placed, an Even-Swap s performed on the alternatve for swappng to a dummy alternatve, whch can be placed by convex combnatons. If the alternatve has not been placed yet, the DM s ased to place the alternatve among the range of possble preference classes. Followng sectons dscusses all the approaches developed n the algorthm. 2.1 Some Notaton and Assumptons Followng lst defnes some notaton that wll be used n ths study, th alternatve that s to be placed n preference classes s represented as = ( x, 1, x,2,..., x, j,..., x, p ) where x, j s the score of the th alternatve 13

25 n the th j crteron C s the th class that the alternatves can be placed n; C 1 beng the best class and Ct beng the worst, where there exst classes. Other notaton wll be ntroduced when they are defned n the flow of the algorthm. The model assumes that the DM has a lnear utlty functon. That s, the utlty of alternatve s gven by: = j U λ ) ( ) ju j ( x, j where U j ( x, j ) s the crteron score of on crteron j, and crteron j. λ j s the weght of It s assumed that the DM s consstent wth hs/her responses and can place alternatves consstently whenever s/he I ased. One other assumpton s that, at least one of the crtera s ordnal and contnuous. Ths assumpton s requred to enable performng the swap on that crteron. The model assumes that λ j s crtera weghts- are not nown, and tres to generate an estmated regon for λ j s usng DM s responses. A consstency ndex, α, s used when evaluatng the Even-Swap nformaton. Ths s for evaluatng the DM s swap response wthn a precson bound, the dscussons on ths α value wll be gven n the followng sectons. 2.2 Even Swaps At the ntalzaton phase of the approach, an Even-Swap s performed. The ntent of the Even-Swap s to have an dea about the DM s underlyng utlty functon. Ths early nformaton on the utlty functon provdes the nfrastructure for the latter steps. Followng sectons dscuss: the selecton of the alternatves that wll 14

26 be taen nto consderaton when performng the Even-Swap, performng the Even-Swap, and the case when there exsts more than two crtera for the alternatves Selectng the alternatves for the even swap Two alternatves wll be selected to perform the Even-Swap. Selectng the alternatves s crucal snce we want to mae the best use of the DM s response whle performng the swap. The orgnal Even-Swap methodology s utlzed for generatng possble domnance relatonshps among alternatves whch do not domnate each other. So, the selected alternatves shall not be domnatng each other. Another concern n alternatve selecton s the ease of the swap. If the crteron values of the alternatves are too far, t wll be hard for the DM to mae such a bg swap, and as the sze of the swap ncreases, the sze of the error may ncrease. The two alternatves are selected among the alternatves n such a way that. they do not domnate each other. they have the smallest Eucldean dstance The selected alternatves are presented to the DM Performng the Even-Swap The Even-Swap wll be performed wth the motvaton dscussed n secton 1.3. Followng steps wll be followed whle performng the swap: Tae one of the selected alternatves as the base (call t the base alternatve); the swap wll be performed on the other alternatve (swapped alternatve). Select one of the crtera (call t fxed crteron) and equate the value of the swapped alternatve on that crteron to that of the base alternatve. As the DM, how much he/she wants to swap on the other crteron, whch s not fxed, to compensate the change on the fxed crteron. 15

27 Even swap s llustrated on a two crtera example: Assume that we have two alternatves A and B havng the followng scores on two crtera (Table 7): Table 7. Even-Swap example - alternatves Alternatve A Alternatve B Crteron Crteron Let the DM mae a swap on alternatve A, and alternatve B s selected as the base alternatve. Then, we equate the frst crteron value of alternatve A to that of alternatve B, and as the DM how much he/she s wllng to decrease on crteron 2 value, to compensate the ncrease n crteron 1 (Table 8): Table 8. Even-Swap example - the swap Alternatve A Alternatve A swapped Crteron Crteron 2 78??? Say the DM s wllng to decrease the value of the second crteron from 78 to 65, to compensate the ncrease n the frst crteron from 45 to 60. So the alternatve A swapped becomes (60, 65). Intally no domnance relatons are apparent between alternatves A and B. However, now alternatve A swapped domnates alternatve B. So, Af B. Ths example s show graphcally n Fgure 2. 16

28 even-swap Alt. A crt Even-Swap Alt. A swapped 60 Alt. B crt 1 Fgure 2. Even-Swap example - graphcal representaton of the swap Even-Swap on more than two crtera As dscussed n secton 1.3. the Even-Swap method s orgnally proposed to be performed on two crtera. However, we use the approach for comparng two alternatves, and alternatves can have more than two crtera. For that reason, the method s expanded to enable the comparson possble for alternatves havng more than two crtera. Ths s done by performng consecutve swaps. The dea s llustrated on a four crtera example below: Illustraton: Let and j be the selected alternatves, havng the crteron values respectvely: = ( x, 1, x,2, x,3, x, 4 ) and j = ( x j, 1, x j,2, x j,3, x j, 4). Let be the fxed alternatve and the Even-Swap be performed on wll be used: Intally choose the frst crteron to be the fxed crteron 17 j. The followng steps

29 Equate the frst crteron value of j to that of Swap the second crteron value of j to compensate the change n the frst crteron, say the swapped value s s x j, 2 Equate the second crteron value of j, that s s x j 2,, to that of Swap the thrd crteron value of j to compensate the change n the second crteron, say the swapped value s s x j, 3 Contnue n the same manner The approach dscussed above s pctured n Fgure 3, where dashed squares represent the Even-Swaps, arrows representng the swaps. c 1 j, c 2 j, ' j are all equvalent alternatves to j, whch are generated after the Even-Swaps. In 1, the DM chooses the value x s j, 2 such that, n the frst two crtera, the DM c j s ndfferent between ( x j, 1, x j, 2 ) and ( x, 1, x s j, 2 ). Then the DM chooses x s j, 3 such that s/he s ndfferent between ( x j, 1, x j, 2, x j, 3) and ( x, 1, x, 2, x s j, 3 ). Fnally, the DM choose x s j, 4 such that = x, x, x, x ) and j ( j, 1 j,2 j,3 j, 4 ' = ( x, x, x, x s j, 4). It would be benefcal to leave the easer swap to the last j,1,2,3 step, snce all the changes on the other crtera wll be compensated by ths swap. 18

30 x, 1 x, 2 x, 3 x, 4 j x j, 1 x j, 2 x j, 3 x j, 4 c 1 s x j, 1 x j, 2 x j, 3 x j, 4 c 2 s x j, 1 x, 2 x j, 3 x j, 4 ' s x j, 1 x, 2 x, 3 x j, 4 Fgure 3. Even-Swap on more than two crtera Estmatng the utlty functon usng the Even-Swap An LP s constructed, n order to mae an estmaton of the DM s underlyng utlty functon, usng the nformaton obtaned from the swaps made n the prevous step. In fact the Even-Swap mples a drect rato relatonshp between the weghts of the crtera on whch t s performed. However, the developed LP evaluates ths drect relatonshp wthn a consstency nterval, dependng on the precson of the DM; the sze of the nterval can be changed. There are two constrants comng from each performed Even-Swap and one from the mpled preference relatonshp after the swaps. These constrants are elaborated below: Constrants obtaned from the swap: Assume that the DM performs the swap on alternatve and the Even-Swap wll be performed on crteron 1 and crteron 2, where both crtera are hgher the better type. The values of the alternatve on these par of crtera whch the swap s performed are x, 1 and x, 2. Agan assume that the DM maes a swap from x, 2 to 19

31 s x,2, to compensate the change for gong from, 1 x to x s, 1. Where, f x, 1 s greater than x s, 1, x, 2 shall be smaller than x s, 2 ; and f x, 1 s smaller than x s, 1, x shall be greater than x s, 2, that s the swap shall be performed n the reverse,2 drecton. By ths swap, the DM mples the followng rato between the weghts of crteron 1 and 2 (for smplcty x, j s used nstead of u j ( j x, )): λ2 = λ 1 x x s,1,2 x x,1 s,2 However, as mentoned above, the response of the DM s evaluated wthn a consstency nterval. Let α represent the consstency ndex for the DM s mpled rato on the crteron weghts. Then, the relatonshp becomes: x (1 α ) x s,1,2 x x,1 s,2 λ2 x (1 + α) λ x 1 s,1,2 x x,1 s,2 Ths relatonshp gves two constrants for the weght space: s x,1 x,1 λ1 ( 1+ α) λ2 0 s x,2 x,2 and s x,1 x,1 λ 2 λ1 (1 α) 0 s x,2 x,2 Constrants obtaned from the preference relaton on the ntal alternatves: After the swap a domnance relaton appears between the swapped alternatve and the base alternatve, snce all but one of the crtera values are equalzed. That unequal crteron value determnes the drecton of the domnance. Let s assume that, the crteron value of the swapped alternatve s hgher than that of the base alternatve. Then the swapped alternatve domnates the base alternatve. Snce the swapped alternatve s assumed to have the same utlty value wth ts orgnal state and all the ntermedate alternatves generated durng 20

32 the Even-Swap (when there exst more than two crtera), all these alternatves are preferred to the base alternatve. Then, correspondng constrants are added to constran the weght space. If the swapped alternatve s domnated by the base alternatve, all preference relatons are reversed. The preference relaton s represented wth f or p. For all mpled preference relaton the followng constrant s added: ( q r ) where q f r λ ε Wth the constrants generated from the performed Even-Swap, the followng ntal model s developed to defne the weght space for a two crtera example: max ε s. t. x λ1 (1 + α) x s x λ2 λ1 (1 α) x ε λ( q r ) λ j = 1 λ 0 ε 0 s,1,2 x x s,2,1,2,1 x x λ ε 2,1 s,2 ε q f r All the constrant are wrtten to be greater than a value of ε, whch s to be maxmzed, to force the LP to fnd a weght set whch s most dstant to the nearest bound. Solvng the LP, an estmated weght set of the crtera s obtaned, whch wll be used n the followng stages of the approach. Each tme when a new preference relaton s mpled by DM placement, a new constrant s added to the model, and the estmated crteron weghts are recalculated. 21

33 2.3 Alternatve Selecton The model utlzes some set of rules to select the alternatve that wll be treated n the algorthm for placng t n a preference class. Alternatve selecton s very crucal for the performance of the algorthm, snce a good selecton of alternatves wll reduce the DM s effort. To provde such a good selecton, the model uses the estmated utlty functon, obtaned from the ntally performed Even-Swap, and some set of rules. The rules for alternatve selecton are gven below: 1. If the bounds for all preference classes are not defned The bounds for the preference classes are determned usng the estmated utlty functon. For a preference class, among the prevously placed alternatves to that class, the one havng the hghest estmated utlty s the upper bound for that preference class, and the one havng the lowest estmated utlty s the lower bound. For the best class, f only one alternatve s placed prevously, than that defnes the lower bound, smlarly f only one alternatve s placed n the worst class than that defnes the upper bound for that class. If one of the followng condtons s satsfed, there s no way of determnng all the bounds for the problem.. No alternatves placed n the best class:. No alternatves placed n the worst class. Less than two alternatves placed n an ntermedate class 1. No alternatves placed n that ntermedate class 2. One alternatve s placed n that ntermedate class Intally number of alternatves n each class s calculated, consderng the number of all alternatves and the number of preference classes, and assumng there s approxmately equal number of alternatves n each class. Then the alternatves are sorted accordng to the estmated utlty functon. Ths sortng wll gve an estmated groupng of the alternatves. Fnally, followng decsons are made for the alternatve selecton, for the above stated cases:. Select the alternatve whch has the mnmum estmated 22

34 utlty value n the estmated best class (note that estmated best class term s used, snce no alternatves are actually placed n ths class yet, the selecton s based on the sortng made usng estmated utlty functon). Select the alternatve whch has the maxmum estmated utlty value n the estmated worst class. Agan the selecton wll be based on the estmated sortng of the alternatves a) Select the alternatve havng the maxmum estmated utlty value, among the estmated set of alternatves for that ntermedate class, ths alternatve s expected to form the upper bound for that class. b) Select the alternatve havng the mnmum estmated utlty value, among the estmated set of alternatves for that ntermedate class, ths alternatve s expected to form the lower bound for that class. (note that f there s only one alternatve actually placed n that class, that alternatve wll be formng the upper bound for that class snce the algorthm tres to place the alternatves havng hgher estmated utlty values frst) 2. If the bounds for all preference classes are defned The unplaced alternatves, whch are out of bounds defned by the estmated utlty values, have precedence when selectng the alternatve to place. So, once the bounds for all alternatves are defned, the algorthm searches for the exstence of alternatves whch are out of the estmated bounds.. If there exsts some alternatves that fall out of the defned estmated boundares, among those alternatves, select the one whch s most dstant from the closest boundary n 23

35 terms of estmated utlty. The reason for selectng the most dstant alternatve s to maxmze the beneft of the nformaton that wll be obtaned from the placement of that alternatve. The bound defned by estmated utltes s enlarged to the maxmum extent possble by ether placng that alternatve to the better class or the worse class.. If there are no alternatves fallng outsde the estmated boundares, select the alternatve whch s closest to the closest boundary. 2.4 Determnng Best and Worst Classes Along the executon of the algorthm, the best and worst possble classes that the selected alternatve can belong to, are utlzed. The ntent s to narrow down the range of possble classes that the alternatve may be placed, and whenever best possble class s the same as the worst possble class, the algorthm places the alternatve to that class. Intally, best class ndex for all alternatves s set to 1 and worst class ndex for all alternatves s set to t (number of classes). Then, as alternatves are placed to preference classes, gong over the prevously placed alternatves, alternatves that are domnated by the selected alternatve are searched. Among the domnated alternatves set, ones that have the smallest class ndex (lower the ndex, better the class), determnes the worst class that selected alternatve may belong to, and t wll be denoted by W. Smlar approach s followed for determnng the best class that the selected alternatve may belong to. Gong over the prevously placed alternatves, the alternatves that domnate the selected alternatve are searched. Among the domnatng alternatves set, ones that have the largest class ndex (hgher the ndex, worse the class), determnes the best class that the selected alternatve may belong to, and t s denoted by B. 24

36 The alternatve set s traced to dentfy domnance relatonshps. For each alternatve, set of domnatng alternatves and set of domnated alternatves are constructed. These sets are useful when the alternatve s placed n the 1 st class (best class) or t th class (worst class). If the alternatve s placed n the 1 st class, then all the alternatves domnatng that alternatve can be safely placed to the 1 st class. Smlarly f the alternatve s placed n the t th class, then all the alternatves domnated by that alternatve can be safely placed to the t th class. 2.5 Convex Combnaton Chec If the selected alternatve can be expressed as a convex combnaton of alternatves belongng to the same class, then the selected alternatve also belongs to that class. The followng LPs are used to decde whether the selected alternatve s a convex combnaton, where under consderaton. e 1, 2 [ ε ε,...,] e 2 = 2,..., 2 ε 2, where ε 1 and ε 2 are scalars. s the selected alternatve and C t s the class e are vectors such that = [ ε ε,...,] e 1 1,..., 1 ε1 and LP1 max ε s.t. Ct µ = 1 µ 0 1 µ e 1 0 LP2 max ε 2 s.t. µ = 1 µ 0 Ct µ e

37 Startng from the best class that the selected alternatve may belong to, the above LPs are solved. By loong at the values of ε 1 and ε 2 a decson s made about the class that the selected alternatve shall belong to. If ether ε 1 or ε 2 s zero, that means the selected alternatve can be expressed as a convex combnaton, so t belongs to C t. If both ε 1 and ε 2 are greater than zero, that means there exst two convex combnatons where the frst one domnates the selected alternatve, and the other s domnated by the selected alternatve; so the alternatve belongs to If ε 1 s greater than zero but ε 2 s smaller than zero, that means there exst some convex combnatons that domnates the selected alternatve, but no convex combnaton s domnated by the selected alternatve; so ths s the best class that the alternatve may belong to. If ε 2 s greater than zero but ε 1 s smaller than zero, that means there exst some convex combnatons that are domnated by the selected alternatve, but no convex combnaton domnates the selected alternatve; so ths s the worst class that the alternatve may belong to. If both ε 1 and ε 2 are smaller than zero, that means there are no convex combnatons ether domnatng or domnated by the selected alternatve. Ths result gves no nformaton on the possble class of the selected alternatve. All ths dscusson s summarzed n the below table. Table 9. Decsons for dfferent values of decson varables C t. ε 1 < 0 =0 > 0 B < 0 No nfo Ct = Ct ε 2 =0 Ct Ct Ct W > 0 = C t Ct Ct Gong over all classes that the selected alternatve may belong to, the alternatve s ether placed or ts best or worst possble class ndex s updated. 26

38 Fgure 4 provdes a two dmensonal graphcal example to the above dscusson. Suppose that, alternatves 1, 2 and 3 belong to the same class (say, class f), and other alternatves are evaluated wth above LPs: As t can be observed graphcally alternatve 4 can be represented as a convex combnaton of alternatves 1, 2 and 3. So, for alternatve 4, both ε > 0 and 0 1 ε > 2 and alternatve 4 also belongs to class f. For alternatve 5, ε > 1 0 and ε < 0 2, that means class f s the best possble class alternatve 5 can belong to. For alternatve 6, ε < 1 0 and ε > 0 2, that means class f s the worst possble class alternatve 6 can belong to. For alternatves 7 and 8, both ε < 1 0 and ε < 0 2, and no nformaton provded. For alternatves on the lnes boundng the shaded regon, ether ε = 1 0 and ε 2 = 0, that can be expressed as a convex combnaton. Alternatve 9 s such an alternatve, and t also belongs to class f crt t c crt 1 Fgure 4. Graphcal representaton of the example 27

39 2.6 Utlzng LPs for best and worst classes As mentoned above, the algorthm tres the narrow the range of possble classes that the selected alternatve can be placed n. Two smlar LPs are used to chec the possblty of the placement of the selected alternatve n preference classes wth respect to the defned weght space. The developed LPs are smlar to those developed by Kösalan and Ulu (2003), where each alternatve n a certan class s consdered separately. Here all alternatves n a class s consdered n one LP. The reasonng behnd ths s that, to decde whether the class s the worst/best class that the selected alternatve can belong to there should not exst any feasble weght set that maes the selected alternatve worse/better than all alternatves n that certan class. Startng wth the ntal utlty estmaton, weght space s defned. New constrants are added (weght space wll be reduced) n ether of the followng cases: whenever a preference relaton s mpled by an Even-Swap whenever an alternatve s placed to a class, and there are alternatves n worse classes whch that alternatve s not domnatng. Ths mples a preference relaton. If there exst some alternatves n better classes whch are not domnatng that alternatve, ths wll also mply a preference relaton. The followng LP s used for determnng the best class that the selected alternatve may belong to wth respect to the defned weght space where S represents the defned weght space. max ε s.t. λ ( λ S h ) ε h C t 28

40 The above LP s solved for each class C t startng from the worst possble class untl the worst possble class the selected alternatve may belong to. If ε < 0or the LP s nfeasble, t s concluded that there s no weght set n the defned weght space whch maes the selected alternatve,, better than all alternatves n that class under consderaton. So, the class s mared to be the best class that belong to, B. If the LP s feasble, consder the next class. may A Smlar approach s followed to determne the worst class. The followng LP s constructed for that purpose: max ε s.t. λ ( λ S h ) 0 h C The above LP s solved for each class C t startng from the best possble class untl the worst possble class the selected alternatve may belong to. If ε < 0or the LP s nfeasble, t s concluded that there s no weght set n the defned weght space whch maes the selected alternatve,, worse than all alternatves n that class under consderaton. So, the class s mared to be the worst class that may belong to, W. If the LP s feasble, consder the next best class. Whenever, B preference class. = the selected alternatve can safely be placed n that W 2.7 Fndng equvalent dummy ponts As mentoned n Secton 2.5 f the selected alternatve can be expressed as a convex combnaton of alternatves belongng to the same class, that selected alternatve also belongs to that class. The followng LP s constructed n order to 29

41 chec for a dummy pont that domnates a convex combnaton of alternatves of a class, and s domnated by another convex combnaton of the same class and equvalent to the selected alternatve n terms of estmated utlty value. Snce the dummy pont s both domnatng and domnated by two dfferent convex combnatons, t wll be a member of that class. max ε s. t. λ ( µ C t est µ = 1 β = 1 β µ 0; β 0; dum dum C t ) = 0 dum ε ε dum 0; ε 0 Fgure 5 shows an example of the case where ths step can be used. Here, the dummy alternatve can be expressed as a convex combnaton of alternatves 1, 2 and 3, whch belong to same class. And usng the estmated utlty functon, t can be sad that alternatve 5 and the dummy alternatve has the same utlty value. The DM s ased to perform an even-swap on alternatve 5, to come around the dummy alternatve. If the swapped pont can also be expressed as a convex combnaton t s expected because of the estmated utlty functon we can place alternatve 5 n the same class. 30

42 5 E-S 2 crt 2 1 Dummy 3 crt 1 Fgure 5. Step 6 shown graphcally Startng from the best class that the selected alternatve may belong to, the above LP s solved. If a feasble dummy pont can be found, an Even-Swap wll be performed on the selected alternatve, to generate a swapped alternatve around the dummy pont. Snce the dummy alternatve wll be a member of that class, the swapped alternatve s a close canddate to be a member. But before placng domnance relatons wth the convex combnatons wll be checed, snce the dummy pont s found usng the estmated weghts, so the swapped pont can be slghtly dfferent. 2.8 Decson maer placement If the selected alternatve cannot be placed to a preference class wth one of the above procedures, then the DM wll be ased to place the alternatve to a preference class between W and B. Let the DM place the alternatve to a preference class. If there exsts some 31

43 alternatve d belongng to a worse class but not domnated by wll be added to the weght space ndcatng there exsts some alternatve d, a constrant f, that s λ ( ) 0. If b belongng to a better class but not domnatng, a constrant wll be added to the weght space ndcatng λ ( b ) 0. (Kosalan, M.M., Ulu, C., 2003). b d f, that s Fgure 6, s a graphcal example of how domnance and weght space reducton s utlzed for alternatve placements (Kosalan, M.M., Ulu, C., 2003). Suppose, DM places domnated by to the worst class, the alternatves n the dashed rectangle are, and they also belong to the worst class. Assume that, t s nown by weght space reducton constrants that slope of the DM s underlyng utlty functon les between l 1 and l 2. Then, the alternatves whch are mared wth a *, wll also belong to same class by weght space reducton. crt 2 l 1 l 2 crt 1 Fgure 6. Graphcal representaton of domnance and weght space reducton 32

44 CHAPTER 3 THE ALGORITHM 3.1 Summary of the Algorthm The algorthm used for sortng can be dvded nto two phases: ntalzaton placng alternatves At the ntalzaton phase two alternatves are selected from the alternatve set and an Even-Swap s performed on them. Utlzng the nformaton comng from the Even-Swap, constrants are generated and an estmaton of the DM s utlty functon s made. An overall flow of the ntalzaton phase s gven n Fgure 7 (for the detaled flow, loo at Appendx A): Select two alternatves Perform an Even-Swap on them Generate constrants & defne weght space Estmate the utlty functon Fgure 7. Flow of the ntalzaton phase At the placng alternatves phase, frst the alternatve to be placed s selected among the unplaced alternatves. Then accordng to domnance relatons, best and worst classes for the alternatves are determned. The next step s to chec the selected alternatve for convex combnatons of the preference classes. Then utlzng generated LPs and the reduced weght space, possble range of classes that the selected alternatve may belong to s narrowed down. Then an equvalent dummy alternatve, whch belongs to a preference class, n terms of estmated utlty s searched and an even swap on the selected alternatve s performed to 33

45 come around the dummy alternatve. Whenever t s found that the possble best and worst classes are the same for the selected alternatve, durng the executon of the steps of the algorthm, the alternatve s placed to that class. If ths does not happen, DM s ased to place the alternatve to a class among the range of classes that the alternatve may belong to. An overall flow of the placng alternatves phase s gven n Fgure 8 (for the detaled flow, loo at Appendx A): Select an alternatve to place Determne best and worst classes by domnance Chec for convex combnatons Search for best class wth WSR Place usng domnance As the DM to place Fnd an equvalent dummy pont Search for worst class wth WSR Fgure 8. Flow of the "placng alternatves phase" 3.2 The Algorthm INITIALIZATION Step 0. Defne to be the set of all alternatves, and let there be n alternatves, defne each as. Then =,,..., }. { 1 2 n Defne the th j crteron value of the alternatve as x, j, and let there be p crtera, then = x, x,..., x,..., x ). Let (, 1,2, j, p B be the ndex of the best class that the worst class that belongs to, when t s placed. Let can belong to; and can belong to; C be the set of alternatves that belong to W be the ndex of C be the ndex of the class that preference classes where C 1 represents the best class and th class; and let there be t Ct represents the 34

46 worst. Go to next step. Step1. Select two alternatves from the alternatve set. Among the alternatve pars whch do not domnate each other, the closest one, n terms of Eucldean dstance s selected. Let the selected alternatves be, Go to next step. q and r. Step 2. As the DM to perform Even-Swap on the selected alternatves. Let the alternatve to be swapped be q. If crtera and l are consdered, equate the th crteron value of q to that of r, and as the DM how much s/he wants to swap on the th l crteron. Let the DM swap from r l x, to s x r, l. s q, xr, ( x q, x > s x >?( x r, ) r, l l ) c s s After ths swap, becomes = x,..., x q,, x q, l,..., x ). Perform all the q q ( q, 1 q, p swaps on all crtera. Fnally s s s s becomes s q = ( x q, 1,..., x q,, x q, l,..., x q, p ) q where all crteron values except the last are equal to that of Go to next step. r s. Step 3. One of s q or r domnates the other dependng on the value of ther last crteron. Let s q domnate r, wrte a preference constrant between r to constran the weght space: q f r q and 35

47 36 Wrte couple of constrants for each swap performed, usng prevously defned consstency ndex α : 0 ) (1,,,, + q s q l q l q s l x x x x λ α λ and 0 ) (1,,,, q s q l q l q s l x x x x α λ λ Step 4. Solve the followng LP, and fnd an estmate of the DM s utlty functon. ( ) ) (1 ) (1.. max,,,,,,,, = + ε λ λ ε λ ε α λ λ ε λ α λ ε j r q r q s l l s l s l l s l x x x x x x x x s t f Go to next phase. PLACING ALTERNATIVES Step 5. (ntalzng all alternatves) Intally equate all best possble class ndexes for all alternatves to best class, that s 1 = B ; and equate all worst possble class ndexes for all alternatves to worst class, that s t W =. Equate all class ndexes for all alternatves to zero, 0 = C, that means they do not belong to any class yet. Go to next step. For each swap performed

48 Step 6. (select one alternatve) Select an alternatve among those currently unplaced. Let the selected alternatve be. Go to next step. Step 7. (decde best and worst classes) Loo for the set of alternatves that domnate, let D{ } denote the set of alternatves domnatng. Loong at the class ndexes ( C ) of all alternatves n the domnatng set, assgn the hghest class ndex as the best possble class ndex of B ; = maxmum class ndex n D{ }. Loo for the set of alternatves that are domnated by, let D ' { } denote the set of alternatves that are domnated by C. Loong at the class ndexes ( ) of all alternatves n the domnatng set, assgn the lowest class ndex as the worst possble class ndex of W ' ; = mnmum class ndex n D { }. If W = y, then place to class y. Go to Step13 B = Go to next step. Step 8. (convex combnaton chec) Startng from class B to W solve the followng LP couple to chec whether can be expressed as a convex combnaton of alternatves that belong to same class. 37

49 LP1 (where = [ ε ε,...,] max ε s.t. Ct µ = 1 µ 0 1 µ e 1 1,..., 1 ε1 ) e 1 0 LP2 (where = [ ε ε,...,] max ε 2 s.t. µ = 1 µ 0 Ct e 2 2,..., 2 ε 2 ) µ e 2 0 If both ε 1 and ε 2 are postve for class y, then place to class y, go to Step 13. If ether ε 1 or ε 2 s equal to zero, agan place to class y, and go to Step 13. If ε > 1 0 and ε < 0 for class y, then equate best possble class ndex of 2 to y, B = y ; f ε 0 and ε 0 for class y, then equate worst possble class ndex of to y, 1 < 2 > W = y ; f both ε 0 and ε 0 for class y, ths gves no nformaton; go on wth the next class. 1 < 2 < If there are less than two prevously placed alternatves n a class under consderaton, sp that class and consder the next. When all classes are consdered go to next step. Step 9. (chec for best class wth WSR) 38

50 Startng from the worst class that can belong to, W for all possble classes, where S representng the weght space. LP3 max ε s.t. λ( λ S h ) ε h C, solve the followng LP If the problem s nfeasble or ε 0 n the optmal soluton, then class s the best class that then place can belong to, that s to class y, go to Step 13. If B B W =, termnate ths step. If = y,, go to Step 10. B W = If the problem s feasble and ε > 0 n the optmal soluton, go on wth the next B worst class, loop ths step tll the class just before the best possble class, + 1, s consdered. If there are no prevously placed alternatve n a class under consderaton, sp that class and consder the next. When all classes are consdered go to Step 10. Step 10. (chec for worst class wth WSR) Startng from the best class that B can belong to,, solve the followng LP for all possble classes, where S representng the weght space. LP4 max ε s.t. λ( h λ S ) ε h C If the problem s nfeasble or ε 0 n the optmal soluton, then class s the worst class that can belong to, that s W =, termnate ths step. If 39

51 10. W = y, then place to class y, go to Step 13. If B =, go to Step B W If the problem s feasble and ε > 0 n the optmal soluton, go on wth the next W best class, loop ths step tll the class just before the worst possble class, 1, s consdered. If there are no prevously placed alternatve n a class under consderaton, sp that class and consder the next. When all classes are consdered go to Step 11. Step 11. <fnd a dummy equvalent alternatve> Startng from the best possble class that LP. LP5 max ε s. t. λ ( C t est µ µ = 1 β = 1 Ct β µ 0; β 0; dum dum ) = 0 dum ε ε dum 0; ε 0 can belong to, solve the followng If a feasble dummy pont, on to compare wth to that of swapped pont, dum, can be found for class y, perform an Even-Swap dum. That s step-by-step equatng s crteron values dum s, and performng the swap. After the Even-Swap chec the s, f t can be expressed as convex combnaton of the alternatves of the class under consderaton, go to Step 8. 40

52 If no feasble soluton can be found, consder the next best class. Loop ths step tll all possble classes are consdered. If there are less than two prevously placed alternatves n a class under consderaton, sp that class and consder the next. When all classes are consdered and no feasble soluton obtaned, go to next step. Step 12. (as DM to place) As the DM to place to a preference class among the range of classes B to W. If the DM chooses class y, go to Step 13. Go to next step. Step 13. (assgnments, placement by domnance, update best & worst classes) Assgn to class y, that s C = y. If the alternatve s placed drectly by the decson maer: add a constrant to the weght space showng worse class and not domnated by f d, that s λ ( d ) 0, for all d belongng to a. Update the estmated utlty functon. If the alternatve s placed drectly by the decson maer: add a constrant to the weght space showng better class and not domnatng b f, that s λ ( b ) 0, for all b belongng to a. Update the estmated utlty functon. If y = 1, that s the best class, then all alternatves domnatng, D{ }, shall also belong to class 1; place those alternatves to class 1. Recursvely, the set of alternatves that are domnatng these alternatves shall also be n class 1, chec domnatng sets for all placed alternatves. If y = t, that s the worst class, then all alternatves that are domnated by, 41

53 D ' { }, shall also belong to class t; place those alternatves to class t. Recursvely, the set of alternatves that are domnated by these alternatves shall also be n class t, chec sets of domnated alternatves for all placed alternatves. If y s an ntermedate class, the best possble class ndex for the set of alternatves that are domnated by shall be at least (lower the better) equal to y; f there exst some alternatves havng best possble class ndex lower than y, n the domnated alternatves set, equate ther ndex to y, that s B = y. Agan, f y s an ntermedate class, the worst possble class ndex for the set of alternatves that are domnatng shall be at most (hgher the worse) equal to y; f there exst some alternatves havng best possble class ndex lower than y, n the domnatng alternatves set, equate ther ndex to y, that s W = y. If all the alternatves are placed, ext the algorthm and present the preference classes to the DM. If there are some unplaced alternatves left, go to Step 6. An Example The algorthm s llustrated on an example, wth 20 alternatves havng values on two crtera. Assume that more s better n both crtera. The DM tres to sort these 20 alternatves n three preference classes. The consstency ndex s selected to be, α = The alternatves are presented n Fgure 9, and the alternatve IDs and crteron values are tabulated n Table 10. Whle executng the algorthm for ths example, two settngs are made to smulate DM s responses: a lnear underlyng utlty functon of U ( ) = 0.7*, *, 2 s set, and DM s responses are declared accordngly. DM s boundares between preference classes are set le the followng: o If U ( ) > then place to class 1, C1 o If < U ( ) then place to class 2, C2 42

54 o If U ( ) then place to class 3, C crt crt Fgure 9. Example - Alternatves graphcally represented Table 10. Example - Lst of alternatves and ther crtera values Alt ID crt.1 crt.2 Alt ID crt.1 crt Step 1. Eucldean dstance between all alternatves are calculated, the closest alternatves whch are not domnatng each other are alternatves 2 and 7. These two alternatves are selected for ntal Even-Swap. 43

55 Step 2. An Even-Swap wll be performed on alternatve 2 (0.8292, ) and alternatve 7 (0.8454, ). Let the swap be performed on alternatve 2. For alternatve 2: Crt 1: Crt 2: ? Assume that asng the DM to mae the swap to compensate the change, the followng response s taen: the ncrease n the frst crteron from to s equal to a decrease n the second crteron from to Step 3. Then, alternatve ' 2 becomes (0.8454, ), and ths swapped alternatve s domnated by alternatve 7. Then t can be sad that 7 s preferred to 2, and followng constrant can be wrtten: f 7 2 Followng two constrants can be generated from the swap performed and usng the consstency ndexα : s x 2,1 x 2,1 λ1 ( ) λ2 0 s x 2,2 x 2,2 and whch gves.450λ λ s x 2,1 x 2,1 λ 2 λ1 (1 0.05) 0 whch gves λ s λ1 0 x 2,2 x 2,2 Step 4. The followng LP wll be solved to mae an estmaton of the DM s crtera weghts: 44

56 max ε s. t λ λ ε λ 0.407λ ε λ 2 ( ) 7 λ = 1 λ 0 ε ε for 7 f Intal estmated weghts are found to be: λ1 = and λ 2 = Step 5. B All best possble class ndexes are equalzed to 1 ( = 1 for all ), all worst W possble class ndexes are equalzed to 3 ( = 3 for all ) and all class ndexes C are equalzed to zero, snce no alternatves have been placed yet ( = 0 for all ). Step 6. Alternatve 4 s selected to be placed. Snce all the preference classes are ntally empty, none of the steps gve results tll Step 12; the algorthm steps forward to Step 12 (DM placement). Step 12. The DM s ased to place 4, and he places 4 to class 1. Step forward to Step 13. Step 13. C =1 4, and all alternatves domnatng 4 wll also be placed n class 1, these C C C C are alternatves 2, 7, 15, 20. Then, = = = 1. Select a new alternatve and contnue. Step forward to Step = 45

57 Step 6. Alternatve 13 s selected to be placed. Snce some preference classes are empty, none of the steps gve results tll Step 12; the algorthm steps forward to Step 12 (DM placement). Step 12. The DM s ased to place 13, and he places 13 to class 2. Step forward to Step 13. Step 13. C 13 = 2, update possble worst class ndexes for all unplaced alternatves W domnatng 13 to class 2, there s only one 10, 10 = 2. Update possble best class ndexes for all unplaced alternatves domnated by 13 to class 2, there are 10 alternatves domnated by 13, B B B B B B B B B B = = = = = = = = = = Snce 13 s not domnated by 4 but n a worse class, a preference relaton s mpled, 4 f 13, and the followng constrant s added to weght space: λ ( 4 13) 0 Solvng the LP for estmated utlty functon agan wth the new constrant, t s seen that the estmated weghts do not change. Select a new alternatve and contnue. Step forward to Step 6. Step 6. Alternatve 16 s selected to be placed. Snce some preference classes are empty, none of the steps gve results tll Step 12; the algorthm steps forward to Step 12 (DM placement). 46

58 Step 12. The DM s ased to place 16, and he places 16 to class 2. Step forward to Step 13. Step 13. C 16 = 2, update possble worst class ndexes for all unplaced alternatves domnatng 16 to class 2, there are two unplaced domnatng alternatves, W 11 = W 19 = 2. Update possble best class ndexes for all unplaced alternatves domnated by 16 to class 2, there are 2 alternatves domnated by 16, alternatves 5 and 17, but ther worst class ndex s already 2, no update requred. Snce 16 s not domnated by 4 but n a worse class, a preference relaton s mpled, 4 f 16, and the followng constrant s added to weght space: λ ( 4 16) 0 Solvng the LP for estmated utlty functon agan wth the new constrant, t s seen that the estmated weghts do not change. Select a new alternatve and contnue. Step forward to Step 6. Step 6. Alternatve 9 s selected to be placed. Snce some preference classes are empty none of the steps gve results tll Step 12; the algorthm steps forward to Step 12 (DM placement). Step 12. The DM s ased to place 9, and he places 9 to class 3. Step forward to Step

59 Step 13. C 9 = 3, and all alternatves domnated by 9 shall also be placed n class 3, C C these are alternatves 3 and 12. Then, = = Snce 9 s not domnated by 16 but n a worse class, a preference relaton s mpled, 16 f 9, and the followng constrant s added to weght space: λ ( ) Solvng the LP for estmated utlty functon agan wth the new constrant, t s seen that the estmated weghts do not change. Currently 10 alternatves are placed, and the algorthm status s shown n Table 11, hghlghted alternatves are placed: Table 11. Example - Current status of the alternatves (1) Alternatve ID class best worst Alternatve ID class best worst

60 classes Select a new alternatve and contnue. Step forward to Step 6. Step 6. Alternatve 8 s selected to be placed. Step 7. W and 3. B 8 =1 8 = Step 8. Startng from best class that 8 can belong to, that s class 1; convex combnaton chec wll be performed, by solvng LP1 and LP2. For the frst class LP1 gves ε > 0 1 and LP2 gves < B ε 2 0 ; ths mples 8 = 1. For the second class, both ε 1 < 0 and ε 0 ; no nformaton ganed. For the thrd class ε 0 and ε 0 ; 2 < W ths mples 3. 8 = Step forward to next step. 1 < 2 > Step 9. Startng from worst class that 8 can belong to, that s class 3; LP3 wll be solved. For both class 3 and 2 LP3 gves feasble and postve solutons; so, no updates to B 8. Step forward to next step. 49

61 Step 10. Startng from best class that 8 can belong to, that s class 1; LP4 wll be solved. For class 1, LP4 gves a feasble soluton, but ε < 0 ; so the worst class that 8 W can belong to s updated, 1. 8 = Now, B 8 = W 8 = 1, then place 8 to class 1. Go to Step 13. Step 13. C 8 =1, no updates needed for best and worst possble class ndexes of other alternatves due to domnance relatons. Select a new alternatve and contnue. Step forward to Step 6. Step 6. Alternatve 11 s selected to be placed. Step 7. W and 2. B = = Step 8. Startng from best class that 11 can belong to, that s class 1; convex combnaton chec wll be performed, by solvng LP1 and LP2. For the frst class LP1 gves ε 0 and LP2 gves ε 0 ; no nformaton ganed. For the second 1 < 2 < class, both ε < 1 0 and > 0 W ε 2 ; ths mples = Step 9. Startng from worst class that 11 can belong to, that s class 2; LP3 wll be solved. For both class 2 LP3 gves feasble soluton, butε < 0 ; so the best class B that 11 can belong to s updated, = 50

62 Now, B 11 = W 11 = 2, then place 11 to class 2. Go to Step 13. Step 13. C = 2 11, update possble best class ndexes for all unplaced alternatves domnated by 11 to class 2, there s only one unplaced alternatve domnated by B, alternatve 19, = Now, B 19 = W 19 = 2, then place 11 to class wll be placed to class 2. Go to Step 13 to place alternatve 19. Step 13. C 19 = 2, no updates requred for best and worst possble ndexes of classes. Select a new alternatve and contnue. Step forward to Step 6. Step 6. Alternatve 10 s selected to be placed. Step 7. W and 2. B 10 =1 10 = Step 8. Startng from best class that 10 can belong to, that s class 1; convex combnaton chec wll be performed, by solvng LP1 and LP2. For the frst class LP1 gves ε 0 and LP2 gves ε 0 ; no nformaton ganed. For the second 1 < 2 < class, ε < 1 0 and > 0 W ε 2 ; ths mples 10 = 2. Step forward to next step. Step 9. Startng from worst class that 10 can belong to, that s class 2; LP3 wll be 51

63 solved. Solvng for both class 2, LP3 gves feasble postve soluton, so no update to B 10 Go to Step 10. Step 10. Startng from best class that 10 can belong to, that s class 1; LP4 wll be solved. For class 1, LP4 gves a feasble soluton, but ε < 0 ; so the worst class that 10 W can belong to s updated, = Now, B 10 = W 10 = 1, then place 10 to class 1. Go to Step 13. Step 13. C 10 =1, no updates requred for best and worst possble ndexes of classes, and no preference relatons mpled. Select a new alternatve and contnue. Step forward to Step 6. Step 6. Alternatve 14 s selected to be placed. Step 7. B = 2 W 14 and = Step 8. Startng from best class that 14 can belong to, that s class 2; convex combnaton chec wll be performed, by solvng LP1 and LP2. For class 2 LP1 B gves ε 0 and LP2 gves ε 0 ; ths mples 2. For the second class, 1 > 2 < ε < 0 and > 0 W 1 ε 2 ; ths mples = Step forward to next step. 14 = Step 9. Startng from worst class that 14 can belong to, that s class 3; LP3 wll be 52

64 solved. For class 3 LP3 gves feasble soluton, butε < 0 ; so the best class that B can belong to s updated, = Now, B 14 = W 14 = 3, then place 14 to class 3. Go to Step 13. Step 13. C = 3 14, no updates requred for best and worst possble ndexes of classes. Select a new alternatve and contnue. Step forward to Step 6. Step 6. Alternatve 1 s selected to be placed. Step 7. B = 2 W 1 and 3. 1 = Step 8. Startng from best class that 1 can belong to, that s class 2; convex combnaton chec wll be performed, by solvng LP1 and LP2. For class 2 LP1 gves ε 1 > 0 B and LP2 gves ε 0 ; ths mples 2. For class 3, ε 0 and ε 0 ; ths W mples 3. 1 = 2 < Step forward to next step. 1 = 1 < 2 > Step 9. Startng from worst class that 1 can belong to, that s class 3; LP3 wll be solved. For class 3 LP3 gves feasble postve soluton, so no update on B 1. Step 10. Startng from best class that 1 can belong to, that s class 2; LP4 wll be solved. For class 1, LP4 gves a feasble soluton, but ε < 0 ; so the worst class that 1 W can belong to s updated, 2. Now, B 2, then place 1 to class 2. 1 = 1 = W 1 = 53

65 Go to Step 13. Step 13. C 1 = 2, no updates requred for best and worst possble ndexes of classes. Currently 16 alternatves are placed, and the algorthm status s shown n the below table, hghlghted alternatves are placed: 54

66 Table 12. Example - Current status of the alternatves (2) Alternatve ID class best worst Alternatve ID class best worst classes Remanng four alternatves are also placed by weght space reducton. 5 and 17 are placed to class 3, 6 and 18 are placed to class 2. The fnal placements are gven n the below table (Table 13), and shown graphcally n Fgure 10. Table 13. Example - Fnal status classes

67 crt Class Class crt Class 1 Fgure 10. Example - Fnal status represented graphcally Below table shows how the alternatves are placed, weght space reducton (WSR), domnance (DOM) or DM placement (DM). 9 placements are made by WSR, 7 placements are by DOM and 4 by DM placements. Loong at the alternatves that are placed by DM, t s observed that these are the alternatves whch consttute the boundary for the classes, ths proves the effectveness of selectng alternatves for placement technque; all other alternatves are placed ether by WSR or DOM. Only one even-swap s requred, whch was at the ntalzaton phase of the algorthm. 56

68 Table 14. Example - Means of placements WSR DOM DM Alternatve ID Class ID Alternatve ID Class ID Alternatve ID Class ID

69 CHAPTER 4 AUTOMATED APPROACH AND EPERIMENTATION An automaton of the proposed algorthm s developed for b-crtera problems n order to provde an nfrastructure for DMs to mplement the algorthm. The developed automaton s also used to test and nterpret the behavor of the algorthm to problems wth certan characterstcs. The extenson of the automaton to more than two crtera problems may result n longer run-tmes to execute and some complcatons especally dealng wth Even-Swaps. 4.1 Development of the Automaton and User Screens The automaton s developed usng Vsual Basc wth MS Excel, and utlzes Excel Solver for the LPs n the algorthm. The alternatves are read from a worsheet and model parameters - number of alternatves to be consdered and the consstency ndex - are taen nteractvely. The followng snapshots show nteracton screens; Fgure 11 for nsertng number of alternatves and Fgure 12 for nsertng consstency ndex. 58

70 Fgure 11. Insertng number of alternatves to be consdered Fgure 12. Insertng the consstency ndex Two other nteracton ponts wth the DM are: Performng the Even-Swap : Step 2 at the ntalzaton phase and Step 11 at the placng alternatves phase. Placng alternatves drectly : Step 12 at the placng alternatves phase. The frst one requres DM to perform even swaps. Whenever an Even-Swap s requred, the screen shown n Fgure 13 appears and ass the DM the decrease or ncrease n the value of one crteron to compensate the change n the other crteron. The crtera values for the both alternatves, and the change n the other crteron s presented to the DM, and s/he s expected to mae the swap. 59

71 Fgure 13. Even Swap Screen If none of the former steps can place the selected alternatve, the algorthm ass the DM to place the alternatve to a class n between ts possble best and worst classes. These best and worst possble class ndexes, and the crtera values of the alternatve s presented to the DM n the screen shown n Fgure 14, and s/he s expected to place the alternatve n one class. 60

72 Fgure 14. DM Placement Enterng correct nfo s crucal, so some checs are done n order to reject erroneous data entry, error messages appear when one of the followng cases occur: Consstency ndex not between 0 and 1. (Fgure 15) When swap s done n the wrong drecton. (Fgure 16) When DM tres to place the alternatve out of the presented best and worst possble class ndexes bound. (Fgure 17) The followng error messages appear, and the algorthm contnues when the error s corrected. Fgure 15. Consstency ndex not vald 61

73 Fgure 16. Swap done n the wrong drecton Fgure 17. DM placed the alternatve to a wrong class 4.2 Expermentaton The proposed algorthm s expermented wth the developed automaton, n order to analyze the behavor of the algorthm to alternatve sets wth dfferent characterstcs. Four dfferent parameters can be consdered when testng the algorthm: number of classes: ths parameter s fxed at 3 for all runs, the mplementaton may be extended to handle more classes, but currently t places alternatves to three preference classes number of alternatves: three dfferent szes of alternatve sets are used, sets wth 20, 50 and 100 alternatves are consdered weghts of the utlty functon: two dfferent weght sets are used, weght1 / weght2 rato of the frst one s 0.7/0.3 and the other one s 0.1/0.9. consstency ndex value, alpha: three dfferent consstency ndexes are used, these are; 0.05, 0.15 and The followng table summarzes the runs made, wth ther run codes and run parameters (Table 15): 62