An interactive procedure for multiple criteria decision tree

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1 An neracve roceure for ulle crera ecson ree Macej Nowa Unversy of Econocs, Kaowce Absrac. A lo of real-worl ecson robles are ynac, whch eans ha no a sngle, bu a seres of choces us be ae. Aonally, n serous robles, ulle crera an uncerany have o be consere. In he aer an neracve algorh for ulle crera ecson ree s roose. Varous yes of crera are aen no accoun, nclung exece value, cononal exece value an robably of success. The roceure consss of wo ses. rs, non-onae sraeges are enfe. Nex, he fnal soluon s selece usng neracve echnque. An exale s resene o show he alcably of he roceure. Keywors. ulle crera ecson ang, ynac ecson robles, ecson ang uner rs, ecson ree. Inroucon The ynacs characerze nuerous ecson robles. In real-worl he ecson rocess can rarely be ose n ers of a sngle choce. Ofen, a seres of nereenen ecsons us be ae a fferen eros of e n orer o acheve an overall goal. Moreover, as he fuure s unnown, he resuls of hese ecsons are usually unceran. A ecson ree s well-nown ool o oel an o evaluae such rocesses. In classcal verson roves a eho for enfyng a sraegy axzng exece rof or nzng exece loss. Thus, s alcaon s le o sngle crera robles n whch consequences are easure on carnal scale. Real-worl ecson robles, however, usually nvolve ulle crera. In aon, a leas soe of he are qualave n naure, an as a resul carnal scale canno be use. In hs aer, we analyze sequenal ulle crera ecson ang robles uner rs, whch can be characerze as follows:. The ecson rocess consss of T eros. A each ero, a ecson us be ae. Any ecson ae a ero eernes he characerscs of he roble a ero Rs s aen no accoun. I s assue ha saes of naure are efne for each ero an are oele by robablsc srbuons. 3. Mulle conflcng crera, boh quanave an qualave n naure, are consere. Anrzej M.J. Sulows (e.): Proceengs of KICSS'203,. Progress & Busness Publshers, Kraów 203

2 The a of hs wor s o roose a new roceure for he roble efne above. I cobnes a classcal ecson ree echnque an neracve aroach. The fnal soluon s enfe n wo ses. rs, non-onae soluons are enfe. Nex, neracve echnque s eloye o enfy he soluon sasfyng he ecson aer (DM). 2 Decson ree Decson ree s an effcen ool for oelng an solvng robablsc ulsage ecson-ang robles [3], [0], [8]. Through a grahcal reresenaon of he roble, even colex suaons can be clearly resene o he DMs. Two yes of noes are use n a ecson ree: ecson noes (reresene by squares) an chance noes (reresene by crcles). The ecson alernaves are escrbe by branches leavng ecson noes, whle saes of naure, whch are no conrolle by he ecson-aer, are reresene by branches leavng chance noes. The a of he analyss s o enfy oal sraegy secfyng ecsons ha us be ae n each ero. Decson rees are use n varous areas [-2], [7], [9], [6], [7]. However, as ecson ree sze ncreases roughly exonenally wh he nuber of varables [], can be use only for relavely sall-sze robles. In hs aer we conser only such robles. Decson rees are ycally use for sngle creron ecson robles. A ulle crera ecson ree was analyze by Haes e al. [8], who roose a eho for generang he se of effcen soluons. Loosa [3] cobne ecson ree wh wo carnal ehos: ullcave AHP an SMART n orer o aggregae ulensonal consequences. In [5] an [6] cononal exece value as a easure of he rs of rare evens was use. More recenly rn e al. [4] solve he ulcrera ecson ree roble whou generang he se of all effcen soluons. Ther aroach cobne avanages of ecooson wh he alcaon of ulcrera ecson a (MCDA) ehos a each ecson noe. Le us assue he followng noaon: T nuber of eros, D he se of ecson noes of ero (for =,, T), D T+ he se of ernal noes (for = T + ), { } D =, K,, K. n ( ) () n () nuber of ecson noes n ero, A = { a ( ), K, a( ), K. an ( )} he se of ecsons (alernaves) a noe a, = { e (, ), K, e j(, ), K, en ( )} e,, E he se of saes of naure eergng fro alernave a ( ). j(, ) robably, ha he sae of naure e j(, ) wll occur

3 or all =,, T, =,, n (), =,, n a (, ) he followng conon s fulflle: n (, ) j(, ) = (2) e j= Le Ω be ranson funcon efnng he relaon beween nexes of wo successve ecson noes an + of eros an + resecvely: (, ( ), j(, ( )) + = Ω (3) where ( ) saes for he nex of he ecson an j(, ( )) reresens he nex of he sae of naure. Our goal s o efne sraegy o secfy ecsons for he ecson noe of he frs ero an ecson noes ha can be acheve as a consequence of he ecsons ae n revous eros. Such a sraegy s coose of aral sraeges, whch secfy ecsons ae a he arcular noe of ero an ecsons ae n nex eros. We wll enoe aral sraegy by s l( ). As we assue ha exacly one ec- son noe s efne for he frs ero, so s l( ) enoes a sraegy for he whole ecson rocess. We assue here ha M objecves are consere. They can be efne for exale as follows: Maxze he rof, Maxze he robably of success, Maxze he evaluaon wh resec o a qualave creron, ec. In orer o analyze how goo s a arcular sraegy n relaon o he arcular objecve we us efne a creron. Exece value s he creron os ofen use for evaluang sraeges. When s use, folng-bac-an-averagng-ou roceure can be ale o solve he roble. I aes ossble o elnae nferor olces a nereae noes. Slar aroach can also be use n a ulle objecve envronen, when all objecves are evaluae by exece values [8]. In such case, no scalar values bu M- ensonal vecors are fole bac an average usng he ulobjecve rncle of oaly. However, as was shown by L [2], folng-bac-an-averagng-ou roceure can be use f an only f he objecve funcon s searable an onoonc. Whle exece value sasfes hs requreen, varous rs easures o no. Cononal exece value s an exale of such funcon. I efnes exece value of he oucoe gven ha he agnue of he oucoe aans a leas a gven hreshol β. In [6] a ecson ree roceure usng cononal exece value s roose. Ths aroach s use n hs wor. In aon, our suy aes also no accoun qualave crera. In such case we assue ha ornal scale s use o evaluae fnal resuls. As saes of naure for each ero are oele by robablsc srbuons, he evaluaon of a arcular sraegy s exresse by a robablsc srbuon of ornal values. In hs aer we wll use he robably of rano even o analyze how goo s a sraegy wh resec o a qualave creron. Thus, we wll use hree yes of easures for evaluang sraeges:

4 exece value: E X ; s l ( ), [ ] P( ; s l ), [ X ; s ] robably of even : ( ) cononal exece value: E l( ). Whle all easures can be use for a quanave objecve, only he secon wll be ale for a qualave one. Le s sar wh he exece value. By f() we enoe he value of he creron a ( s ) T + ernal noe. Le l( ) be he exece value for a aral sraegy s l( ). I can be calculae usng he followng recurren forula: ne (, ) j= ( sl ( )) = n e (, ) j= j(, ) f ( ) j(, ) ' + ( ) + sl ( ') f = T oherwse (4) where s secfe by he ranson funcon: (, ( ), j(, ( ) ) ' = (5) Ω Le us now assue ha he ecson aer s nerese n he robably ha a arcular rano even wll occur, an be he se of ernal noes corresonng o he occurrence of even. The robably ha even wll occur, assung ha a aral sraegy s l( ) s ale can be calculae usng he followng forula: ne (, ) T + j(, ) ρ( ) j= P ( sl ( )) = n e (, ) + + ( ) ( ) j, P sl j= ( ) T + where s efne by (5), an bnary varable ( ) ρ f = T oherwse s efne as follows: (6) ρ T + ( ) T + f = (7) 0 oherwse If cononal exece value s use as a creron, folng-bac-an-averagng-ou aroach canno be use recly. However, rohwen an Laber [6] showe ha n such case he roble can be convere no b-crera one. Le ( ) cononal exece value gven ha even occurs an sraegy be exresse as: be he s l () s l() s use. I can

5 ~ where ( s l (),) (7). Whle ( sl ( ) ) roceure, boh ( s ) ( s ) l() ( sl () ) ( s ) ~ = (8) P l( ), s he exece value of he varable X ρ, where ρ s efne by oes no sasfy conons for folng-bac-an-averagng-ou ( ) l( ) an s l ( ) P o. As a resul, boh us be reserve an use for calculang cononal exece value. The followng forula can be use ~ s, : for calculang ( ) l( ) ne (, ) T + j, f ( ') ρ ~ j= ( sl ( ), ) = n e (, ) ~ + + ( ) ( ) j, sl, j= ( ) ( ) ( ) f = T oherwse T + where (5) s use o enfy an (7) for calculang ( ) ρ. In he classcal ecson ree aroach we sar fro he las ero an enfy he bes alernave a each ecson noe. If sngle creron ecson ree s analyze an ~ cononal exece value s use as a creron, boh ( sl ( ),) an P ( sl ( )) us be calculae a each se. Nex, onae aral sraeges can be elnae. If he ecson-aer goal s o axze cononal exece value, hgh values ~ ( sl ( ),) an low values of P ( sl ( )) are referre. Conversely, when hs value s ~ nze, low values of ( sl ( ),) an hgh values of P ( sl ( )) have a reference. The sraeges non-onae wh resec o hese easures are reserve, an a he en, when he analyss reaches he frs ero, forula (8) s use for calculang cononal exece value an he oal sraegy. (9) 3 The roceure Before we sar he roceure, he se of crera us be efne. Deenng on he ecson-aer s references exece value, cononal exece value or robably easures can be use. We sar wh enfyng non-onae sraeges, whch are he ones ha are no onae by any oher. In orer o chec wheher sraegy s onaes sraegy s we us verfy wheher for all crera s s a leas as goo as s, an for a leas one creron s s beer. I s clear, ha aes no sense o analyze onae sraeges, as s ossble o rove he value of a leas one creron whou worsenng any oher. The se of effcen sraeges can be enfe by arwse coarsons. However, hs rocess can be accelerae by folng-bac-an-averagng-ou aroach. As was alreay noce, cononal exece value us be relace by wo crera

6 To slfy he escron of he roceure, we wll assue here, ha all crera (exece value, robably easures, cononal exece value) are axze. Then aral sraegy s l( ) onaes a aral sraegy s l ( ) f for each exece value creron he followng conon s sasfe: ( sl( )) ( sl ( )) (0) In he case of robably creron he conon s as follows: P ( sl( )) P ( sl ( )) () nally for cononal exece value creron he followng nequales us be fulflle: ~ ( sl( )) ~ ( sl ( )) an ( ) ( ( ) P sl P sl ( )) (2) Aonally a leas one conon shoul be sasfe as a src nequaly. Conons (0)-(2) us be use for eros T o 2. In he frs ero forula (8) can be use for calculang cononal exece value an consrans (2) can be relace by he followng conon: ( sl( ) ) ( sl ( ) ) (3) The roceure for enfyng non-onae sraeges consss of he followng ses:. Sar fro he las ero: = T; enfy aral effcen sraeges for all ecson noes of ero T. 2. Go o he revous ero: =. 3. or each ecson noe of ero, enfy sraeges sasfyng he necessary conon for effcency (ae no accoun effcen aral sraeges for all ecson noes of ero + ). 4. or each ecson noe of ero enfy sraeges sasfyng he suffcen conon for effcency coare sraeges arwsely usng forulas (0)-(2) n orer o elnae he ones ha are onae by any oher. 5. If > go o If cononal exece value s use for evaluang sraeges, use (8) o calculae s value an coare sraeges arwsely usng forulas (0)-() an (3) n orer o enfy non-onae sraeges. In ses 3 an 6 aral sraeges consse of he ecsons ae for noe an all cobnaons of effcen aral sraeges enfe for noes acheve fro are consere. The effcen aral sraeges enfe for he ecson noe of ero are he effcen sraeges for he whole ecson rocess

7 The se of non-onae sraeges can be large. The queson hen arses: wha eho can be use for enfyng he fnal soluon of he roble? Our roosal s o use a slfe verson of INSDECM roceure [4]. A each eraon he oency arx s generae an resene o he ecson aer. I consss of wo rows: he frs grous he wors (esssc), an he secon he bes (osc) values of crera aanable neenenly whn he se of effcen sraeges. The ecson aer s ase wheher esssc values are sasfacory. If he answer s yes, he/she s ase o ae a fnal choce. Oherwse, he ecson aer s ase o exress hs/her references efnng values, ha he crera shoul acheve, or a leas ncang he creron, for whch he esssc value shoul be rove. Le S (q) be he se of sraeges analyze n eraon q, an P (q) be he oency arx: where: ( q) g s he wors an ( q) ( q) ( q ) ( ) g L g L g q M P = ( ) ( ) ( ) q q q (4) g L g L g M ( q) g he bes value of -h creron aanable whn he se of sraeges consere n eraon q. In he frs eraon S () consss of all nononae sraeges. Each eraon of neracve roceure wors as follows:. Ienfy oency arx P (l). 2. As he ecson aer wheher he/she s sasfe wh esssc values. If he answer s yes, go o he fnal selecon hase. 3. As he ecson aer wheher he/she woul le o efne asraon levels for crera. If he answer s no go o (5). 4. As he ecson aer o secfy asraon levels ( l ) g ~ for =,, M. Ienfy ( q+) ( l+) S he se of sraeges sasfyng ecson aer s requreens. If S = nofy he ecson aer an go o (3), oherwse go o he nex eraon. 5. As he ecson aer o ncae he nex of he creron, for whch he esssc value s unsasfacory. Ienfy S he se of sraeges for whch he ( q+) ( l ) value of he -h creron excees he curren esssc value g. If a he en of he roceure ore han one sraegy s sll uner conseraon, he ecson-aer can ae a fnal choce secfyng he creron ha shoul be oze. 4 Nuercal exale The alcaon resene here escrbes he real roble ha was analyze by a coany rovng soluons for he ralway nusry. The roble concerns ecsons ae when he coany consere enerng a new are. I was ossble o

8 oerae as a general conracor or o cooerae wh a local coany. our objecves were consere: () o axze he robably of success, (2) o axze rof argn realze f he offer s accee, (3) o nze he cos of rearng a b f he offer s no accee, an (4) o axze he evaluaon escrbng he sraegc f. The eale escron of he roble can be foun n [5]. Here jus basc nforaon s rove. The ecson ree escrbng he roble s resene on fg.. We assue ha he rocess s successful f he coany eces o sub an offer an he roosal s accee. In our ecson ree, success s reresene by ernal noes h r (uer branches eanang fro fnal chance noes h r). The oose suaon s reresene by ernal noes h2 r2 (lower branches eanang for fnal chance noes h r reresenng he efea n ener) an by ernal noes ha are reache as a resul of ecsons o gve u he offer subsson (3A, 6C, 9B, 0B, 2A, 5B). Thus, o evaluae sraeges wh resec o he frs creron we us calculae robably ha he rocess wll reach any of success ernal noes. 6A h 6 6B A a 2 3 2A 2B 3A 3B c e 6C 7 7A 8 8A 9A 9 9B 0A 0 0B A j l n B b 2 2A 4A f 3 3A o 4 4 4A 4B g 5A q 5 5B 5 5A r g.. Decson ree of he roble As he coany analyzes searaely fnancal resuls for he success an efea, we use cononal exece value for evaluang sraeges wh resec o objecves (2) an (3). nally, each fnal sae s evaluae wh resec o he las creron usng 4 on scale, where 0 eans ha he coany s no successful n ener, he coany leens he rojec wh a local arner rovng ar of he equen, 2 he coany execues he rojec wh a local arner eloye for coleng a ar of nsallaon wor only, 3 he coany leens he rojec as a general conracor. Table escrbes robables of saes of naure, whle able 2 rofs, coss an qualave creron evaluaons for each ernal noe

9 Sae of naure Table. Probables of saes of naure Sae of naure Sae of naure Sae of naure Probably Probably Probably Probably a 0.7 c 0.6 e 0.4 g 0.3 a2 0.3 c2 0.4 e2 0.6 g2 0.7 b f 0.6 h r 0.6 b f2 0.4 h2 r2 0.4 nal ecson / sae of naure Table 2. Values of rof argn an sraegc f creron Prof / cos Sraegc f nal ecson / sae of naure Prof / cos Sraegc f 6A / h 634, A / 2-46, A / h2-46, B -34, B / 800,867 A / n 744, B / 2-34,500 0 A / n2-46, C -27, A -39, A / j 870,333 3A / o 694, A / j2-34, A / o2-46, A / 89, A / 750, A / 2-34, A / 2-46, A / l 760, A / q 70, A / l2-46, A / q2-46, B -39, B -39,67 0 3A -27, A / r 756, A / 694,67 3 5A / r2-34,22 0 The nuber of sraeges ha can be enfe n he ecson ree s 8. An exale of such sraegy ay be he followng: A/2A/6A/7A/3A. I eans ha ecson A shoul be ae a ecson noe. Then, f he frs sae of naure realzes a chance noe a, he ecson 2A shoul be ae a ecson noe 2. Afer ha eher ecson 6A or 7A shoul be ae eenng on he sae of naure ha wll realze a chance noe c. nally, f he secon sae of naure realzes a chance noe a, he ecson 3A shoul be ae a he ecson noe 2. Snce we use cononal exece values for evaluang sraeges wh resec o fnancal objecves (axzaon of rof argn an nzaon of he cos of rearng he b), each of he us be convere no wo crera n orer o use folng-bac-an-averagng-ou aroach. As o he qualave creron, he ecsonaer ece o analyze wo characerscs: he robably, ha he evaluaon wh resec o hs creron s exacly 3, an he robably, ha hs evaluaon s a leas 2. In orer o enfy non-onae sraeges we sar fro he las ero. A he ecson noe 6 hree ecsons are consere: 6A, 6B, 6C. Table 3 resens crera values for hese ecsons

10 Decson Table 3. Values of crera for ecsons analyze n noe 6. Objecve () Probably of success Objecve (2) Objecve (3) Objecve (4) ~ 3 ~ 3 P 3 3 P evaluaon equal o 3 evaluaon a leas 2 6A , B , C ax ax n n ax ax ax I can be noce ha he ecson 6A s onae by he ecson 6B, as s no beer for any creron, an for one s worse. The slar analyss s conuce for oher ecson noes of he las ero. Then he rocess goes o he secon ero an values of crera are average accorng o forulas resene n he revous secon. In ero he cononal exece values for objecves (2) an (3) are calculae. or exale, cononal exece rof argn assung ha he coany wns he ener for sraegy A/2A/6B/7A/3A s calculae as follows: ~ ( s, ) = 348,034 P ( s ) = 0,42 ( s ) l() l() l() ~ = P ( sl (), ) ( s ) ( ) l = 348,034 = 828,653 0,42 nally, forulas (0), () an (3) are use n orer o enfy non-onae sraeges, whch are he followng: s : A/2A/6B/7A/3A s 2 : A/2A/6B/7A/3B/0A/A s 3 : A/2A/6B/7A/3B/0B/A s 4 : A/2A/6C/7A/3A s 5 : A/2A/6C/7A/3B/0B/A s 6 : A/2B/8A/9A/3A s 7 : A/2B/8A/9A/3B/0A/A s 8 : A/2B/8A/9A/3B/0B/A s 9 : B/4B/4A/5A/5A Table 4. Evaluaons of non-onae sraeges Sraegy Cos of rearng a Evaluaon wh resec o sraegc f (robably) Probably Prof argn n b n case of of success case of success efea Equal o 3 A leas 2 s s s s s s s s s The evaluaons of non-onae sraeges are resene n able 4. An exale of he alog wh he ecson-aer o enfy he fnal soluon s as follows:

11 Ieraon :. The oency arx s enfe an resene o he ecson aer (able 5). Value Table 5. Poency arx resene o he ecson-aer n eraon Probably of success Prof argn n case of success Cos of rearng a b n case of Evaluaon wh resec o sraegc f (robably) efea Equal o 3 A leas 2 esssc osc The ecson-aer s no sasfe wh esssc values an eclares ha he s nerese only n sraeges wh robably of success no less han As 5 sraeges sasfy he ecson-aer s requreens, he roceure goes o eraon 2. The analyss s connue n he sae way. nally n eraon 3, he ecson aer eclares ha he s sasfe wh esssc values (able 6). Value Table 6. Poency arx resene o he ecson-aer n eraon 3 Probably of success Prof argn n case of success Cos of rearng a b n case of Evaluaon wh resec o sraegc f (robably) efea Equal o 3 A leas 2 esssc osc As sll 3 sraeges are uner conseraon, he s ase, whch creron shoul be oze frs. Snce he answer s rof argn, he sraegy s 8 s selece as a fnal soluon. 5 Conclusons Mulle objecves, ynacs an uncerany characerze any real-worl ecson robles Decson ree s an effcen ool o oel an solve such robles. In hs aer a roceure cobnng hs aroach wh neracve echnque was roose. The roceure can be use for robles wh u o oerae nuber of sraeges. In fuure wor he ecson aer s aue o rs wll be aen no accoun by alyng sochasc onance rules. Alcaons n varous areas, le rojec orfolo anageen, nnovaon anageen an suly chan anageen wll also be consere. Acnowlegens Ths research was suore by Polsh Mnsry of Scence an Hgher Eucaon n years as a research rojec no NN

12 Bblograhy. Chu, L., Gear, T.E.: An Alcaon an Case Hsory of a Dynac R&D Porfolo Selecon Moel. IEEE Trans. Eng. Manage. 26, 2-7 (979). 2. Corner, J.L., Krwoo, C.W.: Decson Analyss Alcaons n he Oeraons Research Leraure Oer. Res. 39, (99). 3. Covalu, Z., Olver, R.M.: Reresenaon an Soluon of Decson Probles Usng Sequenal Decson Dagras. Manage. Sc. 4, (995). 4. rn, A., Guoun, A., Marel, J.-M.: A General Decooson Aroach for Mulcrera Decson Trees. Eur. J. Oer. Res. 220, (202). 5. rohwen, H.I., Haes, Y.Y., Laber, J.H.: Rs of Exree Evens n Mulobjecve Decson Trees. Par 2: Rare Evens. Rs Anal. 20, (2000). 6. rohwen, H.I., Laber, J.H.: Rs of Exree Evens n Mulobjecve Decson Trees. Par : Severe Evens. Rs Anal. 20, 3-23 (2000). 7. Grano, D., Zuceran, D.: Oal Sequencng an Resource Allocaon n Research an Develoen Projecs. Manage. Sc. 37, (99). 8. Haes, Y., L, D., Tulsan, V.: Mulobjecve Decson Tree Meho. Rs Analyss 0, -29 (990). 9. Hess, S.W.: Swngng on he branch of a ree: rojec selecon alcaons. Inerfaces 23, 5-2 (993). 0. Keeney, R.L., Raffa, H.: Decsons wh Mulle Objecves: Preferences an Value Traeoffs. Wley, New Yor (976).. Krwoo, C.W.: An Algebrac Aroach o orulang an Solvng Large Moels for Sequenal Decsons Uner Uncerany. Manage. Sc. 39, (993). 2. L, D.: Mulle Objecves an Non-Searably n Sochasc Dynac Prograng. In. J. Sys. Sc. 2, Loosa,.A.: Mulcrera Decson Analyss n a Decson Tree. Eur. J. Oer. Res. 0, (997). 4. Nowa, M.: INSDECM An neracve roceure for scree sochasc ulcrera ecson ang robles. Eur. J. Oer. Res. 75, (2006). 5. Nowa, B., Nowa, M.: Mulcrera ecson ang n rojec lannng usng ecson ree an sulaon. In: Trzasal, T., Wachowcz, T. (es.) Mulle Crera Decson Mang '0-, Unversy of Econocs Press, Kaowce (20). 6. Sonebraer, J.S., Krwoo, C.W.: orulang an solvng sequenal ecson analyss oels wh connuous varables. IEEE Trans. Eng. Manage. 44, (997). 7. Thoas, H.: Decson analyss an sraegc anageen of research an eveloen: a coarson beween alcaons n elecroncs an ehcal haraceucals. R&D Manage. 5, 3-22 (985). 8. von Wnerfel, D., Ewars, W.: Decson Analyss an Behavoral Research. Cabrge Unversy Press, Cabrge (986)