Disjoint Programming in Computational Decision Analysis

Transcription

1 44444 Journal of Unceran Ssems Vol4, No, pp4-3, Onlne a: wwwusorgu Dson Programmng n Compuaonal Decson Analss aosong Dng,, Mas Danelson, Love Eenberg Inernaonal Busness School, Beng Foregn Sudes Unvers 89, Beng, People s Republc of Chna Deparmen of Compuer and Ssems Scences, Socholm Unvers and Roal Insue of Technolog Forum, Ksa, SE-64 4, Sweden Receved 8 December 8; Revsed Ma 9 Absrac Ths paper dscusses a seres of mprecse decson models and her correspondng compuaonal aspecs arsng n compuaonal decson analss The mprecse decson models rela he radonal pon esmaes no nervals and ncorporae varous pes of vague nformaon represened as lnear consrans from a decson-maer When he prncple of mamzng epeced ul s appled as he decson rule, he evaluaons of hese models become nonconve opmzaon problems and requre some global opmzaon sraeges Ths paper presens a class of global opmzaon algorhms for solvng such non-conve programs We ae advanage of polar cus and he dson srucural proper of he mprecse decson models o develop generalzed cung plane mehods ha are dfferen from he radonal class of branch and bound approaches World Academc Press, UK All rghs reserved Kewords: global opmzaon, dson programmng, cung plane, mprecse decson analss Inroducon Mos classcal decson analss approaches conss of a se of sraghforward decson rules appled o precse esmaes of weghs, probables, and/or ules no maer how unsure a decson-maer s of hs esmaes The requremen for numercall precse daa has been consdered unrealsc b an ncreasng number of researchers and decson-maers Technques allowng mprecson have been suggesed, among whch he nerval mehods have been wdel ulzed o srenghen oher decson models ha are bul upon he prncple of mamzng he epeced ul (PMEU), eg, [7, 8, 9, 3, 6, 7] When PMEU s appled, dson blnear programmng (DBLP) programs such as α or where pu αwu α or α are coeffcens, are obaned whle searchng for he bes alernaves f boh nerval probables or weghs and nerval ules are represened b varables When modelng mulcrera problems ha also have unceran evens, combnng hese wo nds of decson models gves rse o mulcrera probablsc models wh rlnear epressons such as α w p u, and elds compuaonall hard ( ) dson rlnear programmng (DTLP) programs ogeher wh consrans on weghs, probables, and ul values Moreover, for decson suaons ha possess dson probabl chan proper adopng PMEU ma even lead o dson mul-lnear programmng (DMLP) programs Alhough opmzaon echnques have been developed rapdl n recen ears, much of he area of opmzaon has been devoed o solvng large problems Neverheless, he parcular programs arsng n mprecse compuaonal decson analss requre approaches for solvng sequences of relaed smaller global opmzaon problems n neracve me because n mahemacal programmng, DBLP, DTLP and DMLP all ehb non-conve Therefore, some global opmzaon sraeges ha erae beween a global phase and a local phase n search of he global opmum are necessar Two man branches of deermnsc approaches for handlng such non-conve programs are cung plane mehods [,, 9,, ], and branch and bound mehods [,, 4, 4, 5, 8] Combned approaches have also been suggesed [3, ] Ths paper nends o dscuss a seres of mprecse decson models and her correspondng compuaonal aspecs A class of generalzed cung plane mehods s developed b ang advanage of polar cus and he dson srucural proper of he mprecse decson models The followng secon presens an mprecse decson framewor Correspondng auhor Emal: aosongdng@gmalcom ( Dng)

2 Journal of Unceran Ssems, Vol4, No, pp4-3, 5 ha can be readl ransformed no DBLP and he fundamenal echnque for solvng DBLP Thereafer, we eend he decson framewor no an mprecse probablsc mul-crera decson model, ransform no DTLP, and dscuss he correspondng global opmzaon algorhm n grea deal The fourh secon presens an mprecse mullevel probabl chan decson model ha can be ransformed no DMLP, and dscusses he global opmzaon algorhm eended from he hrd secon The fnal secon concludes hs paper and ndcaes furher research opcs Dson Blnear Programmng A Blnear Imprecse Decson Model In nerval decson analss, a decson-maer s ofen encouraged o be delberael mprecse n hs subecve udgemens Inpu senences n decson problems conssng of dscree alernaves wh evens and consequences, eg, nerval esmaes and qualave nformaon, are ranslaed no lnear consrans For nsance, he probabl (or ul of a consequence c beng beween he numbers a and b s epressed as p a, b ] (or u a, b ]) [ [ Relaons can be handled smlarl: a measure of c s greaer han a measure of c s epressed as p p (or analogousl u ) In hs wa each saemen s represened b one or more consrans u l Defnon : An nformaon frame represens a decson suaon wh m alernaves, each alernave havng m, =,, m consequences, as a srucure C, P, U = {{ c } }, P, U,,, m,,, m, where each denoes a m m consequence, P s a fne ls of lnear consrans over p -varables denong probables over he consequences c, and U s a fne ls of lnear consrans over u -varables denong ules over he consequences c l c l Fgure : A blnear mprecse decson model Suppose we have a decson suaon as shown n Fgure where D s a decson node, E and E are probabl nodes, represenng ndeermnsm, wh assocaed probabl dsrbuons, and he leaves are consequence nodes wh conve ses of assocaed value or ul funcons Ths decson suaon can be mapped no an nformaon frame C, P, U = {{ c } }, P U wh m =, m = and m 4 m m, Ths srucure s hen populaed wh user saemens represened as lnear nequales Snce a vecor n he polope can be consdered o represen a dsrbuon, a probabl base P can be nerpreed as consrans defnng he se of all possble probabl measures over he consequences Analogousl a ul base U consss of consrans defnng he se of all possble ul funcons over he consequences These wo bases ogeher wh a ree srucure consue he nformaon frame Defnon : Gven an nformaon frame {{ c } }, P U, he epeced ul of an alernave s m m, E ( A, where and are varables n P and U, respecvel; denoes he probabl of ) = p m u p u p occurrng gven ha alernave A s chosen; and u denoes he correspondng ul Usng precse numbers, evaluang he epeced ul of an alernave s raher sraghforward Neverheless, when numercall mprecse nformaon s nvolved, he epeced ul has o be evaluaed wh respec o P and U = A c

3 6 Dng, M Danelson and L Eenberg:Dson Programmng n Compuaonal Decson Analss B consderng all possble soluon vecors, a range of epeced ules for each alernave can be receved Once we have a well-defned nformaon frame and appl PMEU as he decson rule, we need o calculae he mnmal and mamal epeced ules for each alernave Frsl when mprecse nformaon represened as lnear consrans ess whn eher P or U, o evaluae he decson model s reduced o lnear programmng (LP) problems ha can be readl solved wh modern opmzaon echnques [8] Secondl as mprecse nformaon ess n boh P and U concernng ceran creron, he evaluaon of a decson model under hs creron based on PMEU resuls n a smplfed DBLP program n he sense ha onl conans cross produc erms such as pv bu no lnear erms For eample, f we need o evaluae he second 4 alernave n Fgure, we have o calculae ma/ mn = p u subec o dson lnear consrans concernng P and U, respecvel I should be noed ha for hs blnear mprecse decson model, we onl allow a decsonmaer o ssue hs mprecse saemens wh respec o each alernave such as p + p 3, bu eclude he cross lnear consrans among dfferen alernaves such as p + p 3 Then, hs decson model can be ransformed no DBLP ha s mahemacall saed as mn f (, = ( c + d )( c + d ) mn f (, = c + d + C () n s = { R n : A b, }, Y = { R : A b, } In (), A and represen he lnear consrans populaed n P and U, respecvel b A b Evaluaon of DBLP Local Opmzaon The mos mporan proper of DBLP s ha, even hough f (, ma no be quas-concave, here ess an ereme pon and an ereme pon Y such ha (, s an opmal soluon of DBLP [] Ths soluon proper and he srucure of DBLP self sugges an LP based vere followng algorhm ha converges o a Karush-Kuhn- Tucer pon [] Defnon 3: Consder P : mn f ( ) subec o S, where S s a compac polhedral se and f s non-conve A local sar mnmzer (LSM) of P s defned as a pon such ha f ( ) f ( ) for each N S (), where N S () denoes he se of ereme pons n S ha are adacen o For DBLP, an ereme pon s adacen o (, f and onl f s of he form eher (, or (, ), where N ( ) and N ( ) Y Defnon 4: An ereme pon (, s called a pseudo-global mnmzer (PGM) f f (, f (, for each B δ ( ) and for each, where Y B δ () s a δ neghborhood around Wh he above proper and defnon, we can have an LP based mounan clmbng procedure o oban a PGM of DBLP Algorhm : (a) Fnd a feasble ereme pon ~, where represens he reduced feasble regon n h eraon afer all cus have been added (b) () Solve: ( ~ mn{ f, Y} o eld an opmal ~ ; () Solve: mn{ (, ~ ) f } o eld an opmal ~ ; Se ~ ~ and repea sep (b) unl converges o an LSM (, (c) Suppose s non-degenerae and le ˆ N( ) be such ha f ˆ, ˆ) mn f ( ˆ, < mn f (, f (, ) ( = Y Y = Go o sep (b()) wh ~ ˆ (d) Termnae wh (, as a PGM Algorhm s relavel eas o mplemen and here are oher approaches concenrang on he locaon of dfferen pes of ereme pon [, 9] However, n hs paper, we wll no sep no he deals of oher local opmzaon mehods snce we focus on he ulzaon and generalzaon of such local algorhms n search of he global opmum The ncorporaon of oher approaches s supposed o be smlar

4 Journal of Unceran Ssems, Vol4, No, pp4-3, 7 Generaon of a Cung Plane Gven a PGM (, locaed b Algorhm, we emplo polar cus o cu off local vere soluons Assume s a non-degenerae ereme pon of ; le, N, be he n nonbasc varables a, where N s he nde se for he nonbasc varables Then has precsel n dsnc edges ncden o Each half lne ξ = { : = a λ, λ }, N conans eacl one such edge [5] Defnon 5: The generalzed reverse polar of Y for a gven scalar α s gven b Y ( α) = { : f (, α} for all Y Le (, be a PGM, le he ras ξ be defned as above, le α be he curren bes obecve value (CBOV) of f (,, and le λ be defned b λ = ma{ λ : f ( a λ, α for all Y } f ξ Y ( α) () As for he case when ξ Y ( α), we smpl se λ o raher han emplong he negave eenson of polar cus [9] because n ha case, he followng approach o compue λ canno be used an longer The nequal N / λ deermnes a vald cung plane Each λ can be compued b an approach on he bass of LP dual heor ha coss onl one LP eraon Consder () and (), n whch we need o oban ma{ λ : f ( a λ, α A b, } = ma{ λ : mn [ c ( a λ ) + d + ( a λ ) C] α A b, } Usng LP dual heor he foregong can be rewren as ma{ λ : ma [ c ( a λ ) + b u] α A u d + C ( a λ ), u } = ma{ λ : mn [ c a λ b u] c α C a λ + A u C + d, u } In mar form, hs epresson ass for he mamal λ n ma λ Hence, we can oban Global Opmzaon u u λ, u c a b λ c α s, u C a A u C + d λ when b solvng us one LP program ξ Y ( α ) A hs sage, we are able o presen a global opmzaon algorhm for solvng DBLP b usng Algorhm and polar cus Algorhm : (a) Le CBOV, ob = + ; le he nal bes feasble soluon {( ˆ, ˆ )} = φ Se = and = (b) If = φ, ermnae wh ob as he global mnmum and ( ˆ, ˆ ) as s correspondng global mnmzer (c) Fnd a PGM (, ) b usng Algorhm, and se he correspondng ob mn{, (, )}, ( ˆ, ˆ = ob f ) = arg mn{ ob, f (, )} (d) Compue λ s b solvng LP programs f, and se ξ Y ( α ) λ s o f Generae a polar cu ξ Y ( α) + + and defne = H ( ) (e) If here ess no λ such ha ξ Y ( α ), ermnae wh ob as he global mnmum and as s ( ˆ, ˆ ) correspondng global mnmzer (f) Se +, and reurn o (b) Accordng o he soppng rules, Algorhm elds an eac global mnmum for DBLP (3)

5 8 Dng, M Danelson and L Eenberg:Dson Programmng n Compuaonal Decson Analss Convergence Proof: Frs, noe ha Algorhm s fne so sep (c) n Algorhm elds eac soluons Consder he sequence of PGMs {(, )} generaed and le H ( ) be he cung plane ha elmnaes h In sep (d) of eraon, he algorhm s ermnaed as he consequence of nroducng polar cus, and he algorhm s ermnaed f + H ( ) = φ Oherwse, he cu ( + H ) s appled and a new PGM ( +, ) s found where + + H ( ) and + H ( ) I s possble for he process no o ermnae b an of he soppng rules n Algorhm An nfne sequence would hen be generaed, and we need o show ha he sequence { } has a lm pon such ha + lm H ( ) = φ Snce s a compac se, here ess a lm pon such ha for a gven ε and a posve neger ν, ε for nfnel man ν If + l l H ( ) φ for all ν, hen all subsequen PGMs (, ) generaed wll sasf he condon l H + ( ν ) for all l > ν From he defnon of a PGM, B ( ) and l B ( ) for some δ > Hence, l δ for all l > ν Ths conradcs he saemen ha s a lm pon δ Therefore, + lm H ( ) = φ and he cung plane algorhm s ermnaed 3 Dson Trlnear Programmng 3 A Probablsc Mul-Crera Decson Model The mprecse blnear decson model handles a decson suaon wh respec o onl one creron However, n man decson suaons s necessar o sud a decson problem from more han one perspecve usng mulple crera There are several mul-crera models, rangng from value funcon based mehods over goal and reference pon mehods o ouranng mehods For hs PMEU-based model, an addve mul-arbue ul heor (MAUT) approach s emploed, n whch he epeced ules of each alernave are weghed ogeher accordng o mporance weghs Consder n nformaon frames under n crera and suppose ha he mprecse saemens concernng mporance weghs are colleced n a se W wh w -varables Defnon 6: An eended nformaon frame represens a decson suaon wh m alernaves Each alernave s assessed under n crera,, n, and each creron has an nformaon frame I = {{ c} m } m, P U, as a δ, srucure W,{ I} where W s a fne ls of lnear consrans over w -varables denong weghs over he crera Snce he addve model s PMEU-based, evaluaon rules are based on an addve eenson of he epeced ul Defnon 7: Gven an eended nformaon frame W, { I },,, n, he weghed epeced ul of an n m alernave s G A ) = w ( p u ), where p and u are varables n P and U, respecvel; A ( = = c denoes he probabl of occurrng gven ha A s aen; and u denoes he correspondng ul valued under creron When mprecse nformaon ess n weghs, probables, and ules, he evaluaon of weghed epeced ul n an eended nformaon frame resuls n DTLP ha s mahemacall saed as: mn f (, = ( c + d )( c + d )( c z + d ) s = { R n Y = { R n n3 Z = { z R : A3 z b3, z } In (4), A and represen he lnear consrans populaed n P, U and W, respecvel To b, A b A3 z b3 receve a range of he weghed epeced ules for each alernave under n crera, we need o calculae boh mnmal and mamal values When he receved ranges for wo alernaves overlap, a decson-maer s suggesed o provde more nformaon for he ranges o become separaed, and hus ndcae a preference order : A b, } : A b, } z z p (4)

6 Journal of Unceran Ssems, Vol4, No, pp4-3, 9 3 Evaluaon of DTLP 3 Local Opmzaon For DTLP, because of he dson consran ses concernng P, U and W, respecvel he soluon proper of DBLP can be eended Theorem : If,Y and Z are nonemp and bounded, hen DTLP has an opmal soluon n whch (,, z,), and z are basc feasble soluons of,y and Z, respecvel Proof: Le, z be an opmal soluon Frs, consder he LP problem concernng, mn{ f (, }, and le be s opmal basc soluon Then we have f (, f (, Ne, consder he LP problem concernng mn{ f (, Y}, and le be s opmal basc soluon B he same argumen, we have f (,, f (, Fnall consder he LP problem concernng z, mn{ f (,, z Z }, and le z be s opmal basc soluon Then we have f (,, z ) f (,, Therefore,, and z are basc feasble soluons of,y and Z, respecvel Based on hs soluon proper and he nowledge n DBLP, we can have he followng local opmzaon algorhm o locae a PGM of DTLP Algorhm 3: (a) Fnd feasble ereme pons ~ and ~ Y (b) ()Solve: ( ~ ~ mn{ f,, z Z}, o eld an opmal ~ z ; () Solve: ~ ~ mn{ f (,, z ) }, o eld an opmal ~ ; (3)Solve: ( ~,, ~ mn{ f z ) Y}, o eld an opmal ~ ; Se ~ ~, ~ ~ and repea (b) unl converges o an LSM (, (c) Suppose s non-degenerae and le ˆ N( ) be such ha f ( ˆ, ˆ, = mn Y f ( ˆ, < f (, Go o (b()) wh ~ ˆ, ~ z z (d) Suppose s non-degenerae and le ˆ N( ) be such ha f ( ˆ, zˆ) = mn z Z f ( ˆ, < f (, Go o (b()) wh ~ ~ z zˆ (e) Termnae wh (, as a PGM 3 Global Opmzaon Gven a PGM (, of DTLP, we need o develop a generalzed cung plane mehod o oban he global opmum As a resul of he ndependenc beween,y and Z, we can develop a generalzed cung plane mehod solvng DTLP Algorhm 4: (a) Le CBOV, ob = +, le he nal bes feasble soluon {( ˆ, ˆ, zˆ )} = φ Se = and = (b) If = φ, ermnae wh ob as he global mnmum and he soluon ( ˆ, ˆ, zˆ ) as s correspondng global mnmzer (c) Fnd a PGM n b usng Algorhm 3, and se he correspondng ob mn{, (,, )},( ˆ, ˆ = ob f z, zˆ ) = arg mn{ ob, f (,, z )} (d) Compue λ concernng Y, and compue λ concernng Z for all N Y Z (e) If eher here ess no λ such ha ξ Y ( α ) or λ such ha ξ Z ( α), ermnae wh ob as he global mnmum and ( ˆ, ˆ, zˆ ) as s correspondng global mnmzer Y + + (f) Le λ = mn{ λ, λ } for all N, generae a polar cu, and le = H ( ) Y Z Z

7 Dng, M Danelson and L Eenberg:Dson Programmng n Compuaonal Decson Analss and (g) Se = +, and reurn o (b) The fundamenal dea n Algorhm 4 s ha a a local mnmzer n Z, respecvel and choose he smaller one as, we compue λ Y and λ concernng Z λ for each edge Ths wll guaranee he global opmal as wha we have done n DBLP Convergence of Algorhm 4 can be proved analogousl o he prevous wor Proposon: Gven an eended nformaon frame, ma G ( A and can be compued b DTLP ) mn G( A ) opmzaon Ths wa weghed epeced ul ranges can be obaned for each alernave Ths follows from he observaon ha he problem srucure of evaluang PMEU wh respec o an eended nformaon frame concdes wh he formulaon of (4) 4 Dson Mul-Lnear Programmng 4 A Mullevel Decson Model Wh approprae modfcaons o an nformaon frame, a well defned decson srucure as shown n Fgure can be obaned, n whch we have a sequence of chance nodes represenng unceran Neverheless, P s now broen no several dson pars, eg,, P and, wh respec o each level P P 3 Y Fgure : A mul-lnear decson model To calculae he epeced ul we have E( A ) = p( p( p3u + p3u) + p( p33u3 + p34u4)) + p( p3u5 + p4u6)) (5) = p p p3u + p p p3u + p p p33u3 + p p p34u4 + p p3u5 + p p4u6 As mprecse nformaon prevals, we ranslae no lnear consrans whn each level I should be noed ha currenl we onl allow consrans from he same level raher han dfferen levels For eample, he nerval saemen p 3 s consdered proper, whle he nerval saemen + p3 p + p3 3 s consdered mproper snce p and are from level and level, respecvel and hs wll desro he dson srucural p3 proper In oher words, mprecse saemens are supposed o be confned whn each level To evaluae he alernave b usng PMEU wh mprecse nformaon, we have o compue wo ereme values, e, mamum and mnmum, of (5) subec o dson lnear consrans n order o receve a range of epeced ul for he alernave Ths wll resul n a specal case of DMLP ha s nrnscall hard o solve If we ae, and, hen he mprecse mullevel decson = p = ( p,, pn ), =,, n n+ = ( u,, un ) u model (5) can be ransformed no he followng DMLP model as T = mn f (,, n+ ) = J n s = { R : A b, }, =,, n + (6) In (6), J denoes he nde se, and we can have a mos one decson varable from for each Tang (5) J as an eample, here ess a mos one decson varable from each level whn each produc erm Ths proper of he obecve funcon and he dson lnear consran ses, s, demonsrae he dson proper of DMLP, and herefore, s soluon proper s smlar o ha of DTLP

8 Journal of Unceran Ssems, Vol4, No, pp4-3, 4 Evaluaon of DMLP 4 Local Opmzaon Based on he soluon proper of DMLP and he nowledge n DBLP and DTLP, we can have he followng local h opmzaon algorhm o locae a PGM of DMLP, n whch denoes he feasble regon of n eraon; and N ( n+ ) denoes he se of ereme pons n ha are adacen o n Algorhm 5: ~ (a) Fnd feasble ereme pons, =,, n (b) () Solve: ( ~ ~ mn{ f,, n, n+ ) n+ n+ } o eld ~ n+ () Solve: ~ ~ mn{ f (,,, n+,) } o eld ~ (3) Solve: ( ~ ~,,,, ~ mn{ f 3 n+,) } o eld ~ n + n+ + n+ M (n+) Solve: ( ~ ~,,,, ~ mn{ f n n n+,) n n} o eld ~ n Se ~ ~, =,, n, and repea (b) unl he soluon converges o an LSM (,, n + ) (c) Suppose s non-degenerae, and for each n+, =,, n, le ˆn+ N( n+ ) be such ha f,,, ˆ,,,, ˆ ) mn f (,,,,,,, ˆ ) f (,, Go o (b()) wh ( + n n+ = + n n+ < n+ ~, =,,, +,, n and ~ ˆ (,, n as a PGM + (d) Termnae wh ) 4 Global Opmzaon Now we can develop he generalzed cung plane mehod for solvng DMLP b ang advanage of s dson proper As before, for he generalzed reverse polar, f ξ (α ), =,, n, N, we smpl se λ = oher han emplong he negave eenson of polar cus Algorhm 6: (a) Le CBOV, ob = +, le he nal bes feasble soluon {( ˆ,, ˆ )} = φ, and se n+ = (b) If = φ, ermnae wh as he global mnmum and as s correspondng global n+ ob ( ˆ,, ˆ n+ ) soluon (c) Fnd a PGM n b usng Algorhm 5, and se he correspondng n + ob mn{, (,, )},( ˆ,, ˆ = ob f n+ n+ ) = arg mn{ ob, f (,, n+ )} (d) Compue λ wh respec o for all N, =,, n (e) If here ess no λ such ha ξ ( ob ), =,, n, ermnae wh ob as he global mnmum and ˆ,, ˆ ( n+ ) as s correspondng global mnmzer + + (f) Le λ = mn{ } for all N, =,, n, generae a polar cu, and le = H ) λ ) n+ n+ ( n+ (g) Se = +, and reurn o (b) The basc dea n Algorhm 6 s smlar o hose eplaned n Algorhm 4 and s convergence can be proved analogousl o he prevous wor 5 Conclusons and Furher Research In hs paper, we have dscussed hree pes of mprecse decson models arsng n compuaonal decson analss, he correspondng soluon proper of DBLP, DTLP and DMLP, and her global opmzaon algorhms Beng based on onl LP operaons, he do no conan radonal nonlnear compuaonal elemens Even hough we have ganed some compuaonal eperence n Algorhm ; see [3, 4,, ], Algorhm 4 and Algorhm 6 sll need furher nvesgaons and modfcaons wh respec o her compuaonal performance benchmar agans he general-purpose desgned branch and bound mehods [4, 5]

9 Dng, M Danelson and L Eenberg:Dson Programmng n Compuaonal Decson Analss As for he opmzaon algorhms hemselves, here are several aspecs for furher developmen and mprovemen Frsl s possble o ncorporae ceran lower boundng echnques n branch and bound procedures o fahom he res of he feasble regon and acheve faser convergence [] For eample, we are able o oban an nerval such L U as for each varable Then usng he arhmec nervals [,, 4], he conve envelope of a rlnear erm z subec o bo consrans can be calculaed as Each blnear erm can be furher lower bounded b he mamum of wo lnear consrans, and we need o compue he mamum of 3 lnear consrans o oban an underesmae of z In ha case, boh algorhms ma be furher mproved o acheve fas convergence Secondl he generaon of a cung plane a a degenerae ereme pon as n Algorhms, 4 and 6 can be furher nvesgaed snce s a common suaon n compuaonal decson analss [5, 6] Fnall we ma ncorporae he negave eenson of polar cus when ξ Y ( α) n order o generae more effcen cung planes, and focus on he ereme pons belongng o he orgnal feasble se ha are canddaes for he global soluon raher han hose nduced b he generaed cung planes [9] 6 Acnowledgemens Ths wor s parall suppored b Naonal Naural Scence Foundaon of Chna under Gran #7773 and Proec References [] Al-Khaal, FA, Jonl consraned blnear programs and relaed problems: An overvew, Compuers & Mahemacs wh Applcaon, vol9, no, pp53 6, 99 [] Al-Khaal, FA, and JE Fal, Jonl consraned bconve programmng, Mahemacs of Operaons Research, vol8, pp73 86, 983 [3] Alare, S, C Aude, B Jaumard, and G Savard, Concav cus for dson blnear programmng, Mahemacal Programmng, vol9, no, pp , [4] Aude, C, P Hansen, B Jaumard, and G Savard, A smmercal lnear mamn approach o dson blnear programmng, Mahemacal Programmng, vol85, no3, pp573 59, 999 [5] Balas, E, Inersecon cus A new pe of cung planes for neger programmng, Operaons Research, vol9, pp9 39, 97 [6] Carvaal-Moreno, R, Mnmzaon of concave funcons subec o lnear consrans, Operaons Research Cener Repor, Unvers of Berele Calforna, 97 [7] Danelson, M, Generalzed evaluaon n decson analss, European Journal of Operaonal Research, vol6, no, pp44 449, 5 [8] Danelson, M, and L Eenberg, A framewor for analsng decsons under rs, European Journal of Operaonal Research, vol4, no3, pp , 998 [9] Danelson, M, and L Eenberg, Compung upper and lower bounds n nerval decson rees, European Journal of Operaonal Research, vol8, no, pp88 86, 7

10 Journal of Unceran Ssems, Vol4, No, pp4-3, 3 [] Dng,, and FA Al-Khaal, Accelerang convergence of cung plane algorhms for dson blnear programmng, Journal of Global Opmzaon, vol38, no3, pp4 436, 7 [] Gallo, G, and A ülücü, Blnear programmng: An eac algorhm, Mahemacal Programmng, vol, pp73 94, 977 [] Konno, H, A cung plane algorhm for solvng blnear programs, Mahemacal Programmng, vol, pp4 7, 976 [3] Ros Insua, D, and S French, A framewor for sensv analss n dscree mul-obecve decson mang, European Journal of Operaonal Research, vol54, pp76 9, 99 [4] Roo, HS, and NV Sahnds, Analss of bounds for mullnear funcons, Journal of Global Opmzaon, vol9 4, pp43 44, [5] Roo, HS, and NV Sahnds, Global opmzaon of mulplcave programs, Journal of Global Opmzaon, vol6 4, pp387 48, 3 [6] Sage, AP, and CC Whe, ARIADNE: A nowledge-based neracve ssem for plannng and decson suppor, IEEE Transacons on Ssems, Man and Cbernecs, vol4, pp35 47, 984 [7] Salo, A, and RP Hamalanen, Preference programmng hrough appromae rao comparsons, European Journal of Operaonal Research, vol8, no3, pp , 995 [8] Sheral HD, and A Alameddne, A new reformulaon lnearzaon algorhm for blnear programmng problems, Journal of Global Opmzaon, vol, pp379 4, 99 [9] Sheral, HD, and CM She A fnel convergen algorhm for blnear programmng problems usng polar cus and dsuncve face cus, Mahemacal Programmng, vol9, pp4 3, 98 [] Tu H, Concave programmng under lnear consrans, Sove Mah Dol, vol5, pp437 44, 964 [] Vash, H, and CM She A cung plane algorhm for he blnear programmng problem, Naval Research Logscs Quarerl vol4, pp83 94, 977

11 8 S Dora: Dfferen Tpes of Convergences for Random Varables Theorem 8 Le (Ω, d) be a merc space and le B be a paron of Ω For ever B B wh posve and fne Hausdorff ouer measure n s dmenson denoe b µ = P (A B) he resrcon o he Borel σ-feld of he upper condonal probabl defned as n Theorem Le L (B) be he class of all Borel measurable random varables on B Then he convergence n µ-dsrbuon of a sequence of random varables of L (B) o a random varable s equvalen o he ponwse convergence of epecaon funconals on all bounded and connuous funcon f ha s lm n fdµn = fdµ Proof: If n and are Borel-measurable random varables and H s a Borelan se hen he ses n (H) and (H) are also Borelan ses; moreover snce ever Hausdorff s-dmensonal ouer measure s counabl addve on he Borel σ-feld hen he (upper) condonal probables µ n and µ nduced respecvel b n and on (R,F ) are probabl measures Then convergence n µ-dsrbuon s equvalen o he ponwse convergence of epecaon funconals on all bounded and connuous funcon f 6 Conclusons Ths paper nvesgaes he relaons among dfferen pes of convergence for random varables when he are based on an upper probabl approach where condonal upper epecaons wh respec o Hausdorff ouer measures are used whenever we have o condon on a se wh probabl zero Upper (lower) condonal prevsons defned wh respec o Hausdorff ouer measures are proven o be he upper (lower) envelopes of all lnear eensons o he class of all random varables of he resrcon o he Borel-measurable random varables of he gven upper condonal prevsons I s proven ha he relaons among dfferen pes of convergences of random varables defned wh respec o upper condonal probabl defned b Hausdorff ouer measures are he same ha hold f convergences are defned wh respec o a probabl measure When he condonng even has fne Hausdorff ouer measure n s dmenson hese resuls are obaned because Hausdorff ouer measures are Borel regular ouer measures and so connuous from below and connuous from above on he Borel σ-feld In general f upper condonal probabl s defned as naural eenson of a coheren merel fnel addve probabl defned on a σ-feld we have ha µ-sochascall convergence does no mpl convergence n µ-dsrbuon snce n hs case he upper condonal probabl s no connuous from above References [] Bllngsle P, Probabl and Measure, Wle 986 [] Couso, I, S Mones, and P Gl, Sochasc convergence, unform negrabl and convergence n mean on fuzz measure spases, Fuzz Ses and Ssems, vol9, pp95 4, [3] de Cooman, G, MCM Troffaes, and E Mranda, n-monoone lower prevsons and lower negrals, Proceedngs of he Fourh Inernaonal Smposum on Imprecse Probabl: Theores and Applcaons, pp45 54, 5 [4] Dora, S, Probablsc ndependence wh respec o upper and lower condonal probables assgned b Hausdorff ouer and nner measures, Inernaonal Journal of Appromae Reasonng, vol46, pp67 635, 7 [5] Denneberg, D, Non-addve Measure and Inegral, Kluwer Academc Publshers, 994 [6] Falconer, KJ, The Geomer of Fracals Ses, Cambrdge Unvers Press, 986 [7] Rogers, CA, Hausdorff Measures, Cambrdge Unvers Press, 998 [8] Sedenfeld, T, M Schervsh, and JB Kadane, Improper regular condonal dsrbuons, The Annals of Probabl vol9, no4, pp6 64, [9] Walle P, Sascal Reasonng wh Imprecse Probables, Chapman and Hall, London, 99