R U C O R R E S E A R C H R E P O R APPROXIMAIONS FOR AND CONVEXIY OF PROBABILISICALLY CONSRAINED PROBLEMS WIH RANDOM RIGH-HAND SIDES M.A. Lejeune a A. PREKOPA b RRR 7-005, JUNE 005 RUCOR Ruges Cene fo Opeaons Reseach Ruges Unvesy 640 Baholomew Road Pscaaway, New Jesey 08854-8003 elephone: 73-445-3804 elefax: 73-445-547 Emal: @uco.uges.edu hp://uco.uges.edu/~ a eppe School of Busnes, Canege Mellon Unvesy, 5000 Fobes Avenue, Psbugh, PA 53; mlejeune@andew.cmu.edu b RUCOR - Ruges Unvesy Cene fo Opeaons Reseach, Pscaaway, NJ, USA; pekopa@uco.uges.edu
RUCOR RESEARCH REPOR RRR 7-005, JUNE 005 APPROXIMAIONS FOR AND CONVEXIY OF PROBABILISICALLY CONSRAINED PROBLEMS WIH RANDOM RIGH-HAND SIDES M.A. Lejeune A. Pékopa Absac. We consde pobablscally consaned poblems, n whch he mulvaae andom vaables ae locaed n he gh-hand sdes. he objecve funcon s lnea, and s opmzaon s subjec o a se of lnea consans as well as a jon pobablsc consan enfocng ha he jon fulfllmen of a sysem of lnea nequales wh andom gh-hand sde vaables be above a pescbed pobably level p. o deal wh such complex poblems, we descbe a solvng mehod based on he p- effcency concep fo dsceely dsbued andom vaables, and also popose some alenave fomulaons applcable o boh dscee and connuous pobably dsbuons, and nvolvng he subsuon of he jon pobablsc consan by a se of ndvdual consans, he Boole s nequaly, he bnomal momen boundng scheme, and Slepan s nequaly, especvely. he common advanage of hese fomulaons s ha hey nvolve he compuaon of jon pobablsc consans of lowe dmenson han hs of he jon pobablsc consan ncluded n he ognal fomulaon. We analyze he compuaonal acably, and evaluae he consanng powe elyng on hee daases, n whch andom vaables have a nomal dsbuon. We hen pove ha he funcon E[ ε xε x > 0] enfocng a elably level d s concave excep fo vey lage (small) values of x (d ). We sudy he elaonshp beween he sevce levels p and d defned by he funcons and Px ( ε ) p and E[ ε xε x > 0] d, and use he SABIL poblem o povde a powe managemen nepeaon of he esuls.
RRR 7-005 PAGE Objecves he wo followng pogams mn c x so. x= ε, and mn c x so. x ε, () whee A n an [m x n]-dmensonal max, n an [ x n]-dmensonal max, c and x ae n- dmensonal vecos, b and ε ae m- and -dmensonal vecos, especvely, ae he undelyng deemnsc pogams fo sac sochasc pogammng models, whose objecve funcon s lnea, and s subjec, n addon o a se of lnea and non-negavy consans, o pobablsc consans, he andom vaables of whch beng n he gh-hand sdes of he consans. Denong by F( z) = P( ε z) he known pobably dsbuon of he mulvaae andom vaable ε, he sochasc pogammng poblem assocaed wh () s he pobablscally consaned poblem expessed as follows: mn c x so. P( x ε ) p ha can alenavely be wen (Pékopa, 988, Sen, 99) as mn c x so. x ε 0 F( z) p, n whch p s a pescbed pobably level, usually close o. Pogammng unde pobablsc consans was noduced by Chanes e al. [958], who consdeed a se ( =,,) of ndvdual pobablsc consans mposed on each pacula andom even: mn c x so. P( x ε) p, =,..., he use of mulple ndvdual pobablsc consans =,..., s somewha elavely easy o handle. he obusness of such fomulaons s howeve quesonable; ndvdual pobablsc consans ae only appopae n case of a sysem composed of dffeen componens ha do no
PAGE RRR 7-005 affec each ohe. Howeve, n mos suaons, pobablsc consans, aken ndependenly, do no povde an accuae epesenaon of he consdeed sysem. Subsequenly, Mlle and Wagne [965] poposed a fomulaon fo jon pobablsc consans: mn c x so. P( x ε, =,..., ) p () wh he escon ha each of he andom vaables ae ndependen of each anohe: ρ, j= 0,, j, j. he pogam () can alenavely be ewen as: mn c x so. P( x ε) p By akng he logahm of boh sdes of he consan P ( x ε) p, = we oban he pogam below mn c x = so. ln F( x ) ln p, (4) = snce he elemens, =,, of he veco ε ae ndependen andom vaables wh pobably dsbuon F. he geneal case, ha does no eque he andom gh-hand sde vaables be ndependen of each anohe,.e., ρ, j 0, j, was noduced by Pékopa [970, 973]. he fomulaon s smla o () bu does no allow he ansfomaon of () no (3), no (4). he solvng of he geneal case poposed by Pékopa s vey challengng, and nvolves he complex ask of dealng wh possbly non-convex, non-lnea pogams: hey conan non-lnea consans enalng he compuaon of jon pobables fo mulvaae andom vaables. Fs, mulvaae pobably dsbuons ae vey dffcul o compue, o even o appoxmae. Asseng ha he handlng of nonlnea consans nvolvng mulvaae pobably dsbuons s moe dffcul ha of a nonlnea objecve funcon, Komaóm [986], fo example, poposed a dual mehod fo handlng pobablsc consaned poblems, n whch he dual poblems chaacezed by lnea consans and a concave objecve funcon s solved. (3)
RRR 7-005 PAGE 3 Second, mulvaae nomal dsbuons ae no always concave. Fo example, f he sandad unvaae nomal dsbuon s well known o be concave, he sandad bvaae nomal dsbuon Φ ( ε, ε; ρ) s concave [Pékopa, 970] n ε and ε f: p 0.5 and ρ 0 o ρ = ; p 0.355 and < ρ 0 ; p 0.638 egadless of he value of he coelaon coeffcen ρ. Moe geneally, he sandad n-vaae nomal dsbuon Φ ( z,..., z n ) s concave n he se { zz n, =,..., n} [Pékopa, 00]. Howeve, he concavy of ohe connuous and dscee dsbuons has ye o be suded n geae deal. In he nex secon, we focus on he pobablsc pogam (), n whch he objecve funcon s lnea and he mulvaae andom vaables ae locaed n he gh-hand sdes of he pobablsc consans. hs pogam enfoces ha he pobably of he jon fulfllmen of a sysem of lnea nequales wh andom gh-hand sdes be above a pescbed pobably level p. We pesen some alenave fomulaons fo he pogam (), analyze he p-effcency concep fo appoxmang he pobablsc consans of pogam () n whch he andom vaables ae dsceely dsbued, and descbe a dscee momen boundng scheme [Pékopa, 999]. he noducon of he boundng schemes no he sochasc poblems esul n he compuaon of he jon pobably dsbuon of seveal lowe dmensonal vecos nsead of hs of a sngle hghe dmensonal andom veco. In Secon 3, we evaluae he compuaonal acably and he consanng powe of some alenave fomulaons applcable o mulvaae andom vaables, and nvesgae he usefulness of he bnomal momen boundng schemes fo solvng pobablsc pogams. We epo he esuls on hee daa ses well known n he sochasc pogammng leaue. In Secon 4, we analyze he convexy of non-lnea consans enfocng a cean ype of elably level (Pékopa, 995), and nepe hem n he conex of powe managemen. Secon 5 conans some concludng emaks. Fomulaon In hs secon, we noduce alenave fomulaons fo he pobablsc pogam (6). he fs one, elyng on he p-effcency concep, s applcable o mulvaae andom havng a dscee pobably dsbuon, whle he ohe fomulaons can be appled o vaables egadless of he ype (dscee o nomal) of pobably dsbuon.. p-effcency appoxmaon fo dsceely dsbued andom vaables In he case of andom vaables dsceely dsbued, he pogam () can be ansfomed no a dsjuncve pogam usng he concep of p-effcency [Pékopa, 990]. Le p [0,], and F be s he pobably dsbuon funcon of a s-dmensonal nege andom vaable ε Z +, a pon v, s v R, s called a p-effcen pon of he pobably dsbuon funcon F f: Fv ( ) p, and hee s no v' v, v' v such ha F( v') p
PAGE 4 RRR 7-005 he nequaly fo veco mus be undesood coodnae-wse. I has been shown [Dencheva I e al., 000] ha fo any pobably dsbuon, he se of p-effcency pons, v,..., v, fo an nege andom veco s non-empy and fne. Consequenly, he pogam () can be subsued by he followng dsjuncve pogam: mn c x so. x v (5) p v Κ n whch I p s Κ = ( v + R + ), and he dsjuncve consan x v ha mus hold fo a leas one = eplaces he jon pobablsc consan P ( x ε, =,..., n) p. he p-effcen pons ae mul-dmensonal vecos ha mus be found po o he opmzaon of (5) wh espec o x. Fo small dmensonal poblems, he eases way s o enumeae all p-effcen pons v and o pocess he assocaed poblems. Howeve, fo lage poblems, he bue foce appoach ha consss of enumeang all p-effcen pons can be ovely me- and esouce-consumng; s han bee sued o convexfy he poblem (5) as follows: mn c x s.o x I = I = λ = λ v, (6) λ 0, =,..., I ha mposes x be geae han o equal o a convex combnaon of p-effcen pons. he opmal value of (6) s he lowe bound (fo a mnmzaon poblem) fo he opmal value of (5). Snce all p-effcen pons ae no know, he solvng of (6) s caed ou usng a column geneaon mehod. Fo connuous andom vaables, he cone geneaon mehod (Dencheva e al., 000, 00, 00, Beald and Ruszczyńsk, 00a, Pékopa, 003) s shown o be vey effcen.. Boundng scheme fo mulvaae andom vaables he appoxmaon schemes pesened n hs secon ae applcable egadless of he ype (.e., dscee o connuous) of he pobably dsbuon of he andom vaables. In he case of andom vaables havng a connuous pobably dsbuon, he compuaon of a jon pobably s exemely dffcul: he hghes jon nomal dsbuon fo whch a compuaon s povded [see, ne ala, Daley, 974, Szána, 988, Dezne, 994] s he vaae one.
RRR 7-005 PAGE 5.. Boole s lowe bound on he nesecon of evens An alenave fomulaon fo () can be obaned usng Boole s lowe bound elave o he nesecon of evens. Denong by F... j( z,..., zj) = P( ε z,..., ε j zj), < j he jon pobably assocaed wh he componens o j of he -dmensonal andom veco, by A he even ε z, =,...,, and seng k S = F ( z,..., z ), j = + k, k, k... k j = he followng nequaly follows fom he applcaon of Boole s lowe bound fo he nesecon of evens: P( A... A ) S ( ) (7) Usng (7) fo eplacng he jon pobablsc consan n (), he followng pogam esuls: mn c x = so. F( x ) ( ) p (8).. Fomulaon wh ndvdual pobablsc consans A fomulaon conanng a fne se of ndvdual pobablsc consans nsead of he jon pobablsc consan, and enfocng he equemens n () s gven below: mn c x so. x F ( p), =,..., ( p ) p (9) = n whch p,..., p ae andom vaables consaned o be a leas equal o p, whch s a fxed paamee, and F efes o he nvese pobably dsbuon of he andom vaable ε. he fomulaon (9) s vald, snce consans x F ( p), =,...,, and = ( p ) p
PAGE 6 RRR 7-005 mply ha x F ( p), =,...,, = P ( x ε, =,..., ) = P (( x< ε )... ( x< ε )) = = = P ( x< ε ) = ( P ( x< ε )) = ( p ) p..3 Bnomal momens boundng scheme he fomulaon pesened n hs secon s based on he bnomal momen poblem (Pékopa, 999, Dencheva e al., 00) fomulaed fo a fne numbe of evens A,..., A, defned n a specfed pobably space. We use shap lowe and uppe bounds fo he pobably of he followng funcons of evens: A... A, A... A. Denong by ε he numbe of evens ou of A,..., A, Pékopa [995] shows ha whee ε E = Sk, k =,..., k, k k j = S = P( A... A ), j = + k, k s he k h momen of he andom vaable ε, and S 0 =. he bnomal momen poblem s defned as he followng lnea pogammng poblem: mn(max) = 0 s. o v = Sk, k = 0,..., m = 0 k (0) v 0, = 0,..., whee f,..., 0 f ae some consans and m<. If, f =, f = 0, f <, () hen he opmal values of (0) povde lowe (uppe) bounds fo P ( A... A ). fv
RRR 7-005 PAGE 7 If, f, f = () 0, f = 0, hen he opmal values of (0) povde lowe (uppe) bounds fo P ( A... A ). Snce P( A... A) = P ( A... A), wh A beng he complemenay even of A, Pékopa [999] esablshes ha he shap lowe (uppe) bound fo P ( A... A ) usng he mnmzaon (maxmzaon) poblem (0) and he funcon () s he same as - he shap uppe (lowe) bound fo P ( A... A ) usng he maxmzaon (mnmzaon) poblem (0) and he funcon (). Usng he esuls above, we fomulae he followng lnea pogams mn(max) v so. v + v + v +... + v = 0 v + v + 3 v +... + v = F( x) 3 3 ( ) ( ) 3 m + ( m ) = v + v +... + v = F ( x, x), j = +... v m j j = ( ) m+ + vm+ +... + v = Fj( x,..., jx), j = + m m v0, v,..., v 0 whch opmal values povde lowe and uppe bounds fo F( z,..., z j). Usng hs boundng scheme on F( z,..., zj) and noducng n he fomulaon of () fo eplacng he jon pobablsc consans, we oban he wo followng appoxmaons fo (): mn c x+ αv so. Ax b v + v + v +... + v = 0 3 3 ( ) ( ) 3 m+ ( m ) + ( m) = = v + v + 3 v +... + v = F( x) v + v +... + v = F ( x, x), j = +... j j = m+ v + v +... + v = F ( x,..., x), j = + m m m j = v0, v,..., v 0 v p j, (3)
PAGE 8 RRR 7-005 and max - so. Ax b v + v + v +... + v = 0 v + v + 3 v +... + v = F( x) 3 3 ( ) ( ) 3 m+ ( m ) + ( m) = v + v +... + v = F ( x, x), j = +... c x+ αv j j = m+ v + v +... + v = F ( x,..., x), j= + m m m j = v, v,..., v 0 0 whee α s an abay non-negave numbe. v p j (4)..4 Bound usng Slepan s nequaly he Slepan nequaly can be saed as follows: f ε, =,..., has a sandad mulvaae nomal pobably dsbuon, and R and R ae wo possble coelaon maces whch especve elemens s j and s j ae such ha s s', j,, j =,...,, (5) j j hen can shown ha [Gupa, 965, Bawa, 973] P( ε x, =,..., ; R) P ( ε x, =,..., ; R') (6) fo a same value aken by he decson vaable x. Consequenly, when (5) holds, he solvng of () fo he coelaon max R povdes an uppe found on he objecve value of () fo he coelaon max R. Denong by z'* = c x'* ( z* = c x* ) he opmal value of () assocaed wh R (R), and by x * (x *) he opmal values of he decson vaables, he followng cung plane z* z'* (7) and dsjuncve cus * '* x x + δ M, =,...,, (8) δ, δ {0,}, =,..., = whee M s a lage posve numbe, and δ s a -dmensonal bnay veco, can be noduced n he pogam (). In pacula, f R s such ha s j 0, fo all,j, and R s such ha s j = 0, fo all j, hen (6) can be ewen as: P ( ε x, =,..., ; R) ln F( x: R') (9) = In he nex secon, we shall fuhe evaluae he especve compuaonal acably and consanng powe of he fomulaons pesened n hs secon.
RRR 7-005 PAGE 9 3 Compuaonal esuls he solvng of he sochasc pogam () conanng jon pobablsc consans wh dependen andom vaables s vey challengng; he complexy of he ask nceases wh he dmenson of he andom vaable. In he pecedng secon, we gve some alenave fomulaons fo (). In hs secon, we shall commen on he especve compuaonal acably. We noe ha he pogam (9) s convex. Indeed, f a andom vaable ε has a log-concave pobably densy funcon,.e., he nomal dsbuon s log-concave, hen he dsbuon of ε and s magnal dsbuons ae log-concave, whch mples ha he consan n (9) s convex. heefoe, F( x ) s concave fo x F ( p), =,...,. I has also been shown ha he log-concavy popey does no cay ove fo sums, whch means ha he consan F( x) ( ) p = n (8) s no guaaneed o be concave, whch n un mples ha he pogam (8) s no necessaly convex. he same emak apples fo he pogams (3) and (4). Moeove, he pogams (3) and (4) conan equaly consans nvolvng nonlnea funcons. In he emanng pa of hs secon, we shall evaluae he consanng powe of he fomulaon (), and of s appoxmaons (8), (9), (3) and (4). Compuaonal esuls ae deved fom he applcaon of he pogams lsed above o hee daa ses, fo whch dffeen paamee sengs ae consdeed. In each case handled, he andom vaables ae assumed o have a jon nomal dsbuon. 3.. Resevo Managemen he fs daa se s elaed o esevo managemen. he objecve s o desgn a esevo sysem, n whch esevos ae used o hedge agans he possbly of floodng ha may occu as a esul of andom seam of wae. he pobablsc consans mpose lms on he pobably ha he wae ses above esevo capaces. he capaces x j of he esevos j consdeed, ae he decson vaables, whch especve capaces ae lmed fom above by V j, and ae o be desgned n such a way ha hey ae able o ean he flood of wo dffeen wae souces comng n andom quanes ε j, pevenng fom connung o downseam locaons. he paamee c j efes o he cos pe un of capacy fo esevo j. he eade s efeed o Pékopa and Szána [978] and Pékopa [995] fo moe dealed explanaons. he assocae pogam fo a esevo sysem conanng wo esevos akes he followng fom: mn cx + cx x + x ε + ε so. P p x ε, 0 x V 0 x V when consdeng fomulaon ();
PAGE 0 RRR 7-005 mn cx + cx sof. ( x) + F( x + x) ( ) p 0 x V 0 x V when consdeng fomulaon (8); mn cx + cx sox. + x F ( p) x F ( p) (- p) + (- p) p 0 x V 0 x V when consdeng fomulaon (9); mn cx + cx s.o v0 + v + v = v + v = F( x) + F( x + x) v = F( x, x + x) 0 x V 0 x V v p when consdeng he fomulaon (3), wh α = 0. he andom vaables ε and ε ae nomally dsbued, ε N(,0.), ε N (, 0.), and he covaance ( cov[ ε, ε ] ) beween ε and ε s known. In he second pobablsc consan, he andom vaable ε s unvaae: (fo all he paamees sengs), whle, n he jon pobablsc consan, he andom vaable θ = ε + ε s bvaae wh expeced value: E[ θ] = E[ ε + ε] = E[ ε] + E[ ε ] = µ + µ, and vaance Va[ θ] = Va[ ε + ε] = E[( ε + ε µ µ ) ], = E[( ε µ ) + ( ε µ ) + ( ε µ )( ε µ )] = Va[ ε] + Va[ ε] + cov[ ε, ε] wh cov[ ε, ε ] = ρ σ σ,, whee ρ, denoes he coelaon level beween ε and ε and σ and σ ae gven, efeng o he sandad devaon of ε and ε.
RRR 7-005 PAGE able dsplays he dffeen poblem nsances sengs consdeed,.e., he dffeen values aken by he coeffcens of he objecve funcon, by he coelaon levels beween andom vaables, by he elably level p, and by he maxmal quany ha he esevos can conan, whle able epos he value of he objecve funcon and hese of he decson vaables fo he vaous poblem fomulaons dscussed n he pecedng secons. able : Paamee sengs Case # c c p ρ E[ θ] cov Va θ [ ] V V 0.9 0 3 0 0.05 0.8.5 0.9-0.8 3-0.06 0.08 0.8.5 3 0.9 0.8 3 0.06 0.08 0.8.5 4 0.9 0 3 0 0.05 0.8.5 5 0.9-0.8 3-0.06 0.08 0.8.5 6 0.9 0.8 3 0.06 0.08 0.8.5 7 0.99 0 3 0 0.05 5 8 0.99-0.8 3-0.06 0.08 5 9 0.99 0.8 3 0.06 0.08 5 0 0.99 0 3 0 0.05 0.8 3 0.99-0.8 3-0.06 0.08 0.8 3 0.99 0.8 3 0.06 0.08 0.8 3 I appeas ha he fou pogams (), (8), (9) and (3) esul n he same value of he objecve funcon n almos all he paamee sengs consdeed fo he esevo poblem. I uns ou ha appoxmaons of fom (8), (9) and (3) ae equally (vey) gh fo such poblems of modeae sze. Pogam () wh jon pobably consans able : Opmal soluons fo each paamee seng and fomulaon Pogam (8) wh Boole s nequaly Pogam (9) wh se of ndvdual consans Pogam (3) wh bnomal momen boundng scheme # z x x z x x z x x z x x 4.088 0.794.500 4.089 0.795.500 4.089 0.795.500 4.088 0.794.500 3.853 0.677.500 3.854 0.677.500 3.854 0.677.500 3.853 0.677.500 3 INFEASIBLE 4 5.789 0.8000.494 5.790 0.800.495 5.790 0.800.495 5.789 0.8000.494 5 5.585 0.8000.393 5.586 0.800.393 5.586 0.800.393 5.585 0.8000.393 6 INFEASIBLE 7 6.090.05.59 6.09.05.50 6.09.05.50 6.090.05.59 8 5.858 0.856.50 5.858 0.856.50 5.858 0.856.50 5.858 0.856.50 9 6.8.93.53 6.50.89.530 6.50.89.530 6.8.93.53 0 6.43 0.800.7 6.43 0.800.7 6.43 0.800.7 6.43 0.800.7 5.870 0.800.535 5.870 0.800.535 5.870 0.800.535 5.870 0.800.535 6.53 0.800.866 6.533 0.800.866 6.533 0.800.866 6.53 0.800.866
PAGE RRR 7-005 I appeas ha he pogams (3) and (4) esul n he same opmal value of he objecve funcon, and do no pesen majo dffeence n ems of compuaonal acably. he opmal value of he objecve funcon s oughly he same as hs of he pogams analyzed n able. Howeve, unlke pogams (8), (9) and (3) ha ae wdely applcable, pogams (3) and (4) can only be used f he andom vaables ae ndependen of each ohe. able 3: Opmal soluons wh (3) and (4) fo cases when andom vaables ae ndependen Pogam (3) Pogam (4) # z x x p p z x x p p 4.088 0.794.500 0.994 0.906 4.088 0.794.500 0.994 0.906 4 5.789 0.800.494 0.993 0.906 5.789 0.800.494 0.993 0.906 7 6.09.05.59 0.995 0.995 6.090.05.59 0.995 0.995 0 6.43 0.800.7.000 0.990 6.44 0.800.7.000 0.990 3.. Coffee blendng he second daa se s elaed o coffee blendng. he objecve funcon s lnea and s mnmzed subjec o a se of lnea consans elaed o he lmed avalably of he coffee ypes, he fulfllmen of qualy equemens, and a jon pobably consan ha mposes ha he demand fo coffee be sasfed wh pobably p, epesenng he elably level of he poducon sysem. he eade s efeed o Pékopa [995] and Szána [988] fo moe dealed explanaons. Denong by D he feasble se deemned by he lnea consans, and seng he pogam s fomulaed as: n ems of (); 8 8 8, ε = x, ε = x, and ε = x k k 3 k3 k= k= k= mn c x sox. D mn c x sox. D 8 xk ε k= 8 P xk ε p k= 8 xk 3 ε3 k=, k =,...,8 8 8 8 k k k3 k= k= k= n ems of (8); kl F( x ) + F( x ) + F( x ) (3 ) p x kl 0, k =,...,8
RRR 7-005 PAGE 3 n ems of (9); mn c x so. x D 8 k= 8 k= 8 k= 3 = x kl x F ( p ) k x F ( p ) k x F ( p ) k 3 3 3 ( p ) p 0, k =,...,8 mn c x so. x D v + v + v + v = 0 3 8 8 8 v + v + 3 v = F( x ) + F( x ) + F( x ) 3 k k k3 k= k= k= 8 8 8 8 8 8 v + 3 v = F( x, x ) + F( x, x ) + F( x, x ) 3 k k3 k k k k3 k= k= k= k= k= k= xkl 0, k =,...,8 v0, v, v 0 v3 p n ems of (3) wh α = 0 and when consdeng he fs wo bnomal momens. Assumng ha ξ N (3, 0.5) ξ N(40, 5), ξ 3 N(0,3) we can compue (able 4) he opmal objecve value and he values of he andom vaables p, p, p fo dffeen values of he elably level p fo he pogams (8) and (9). 3 p able 4: Opmal soluons fo pogams nvolvng Boole s nequaly and a se of ndvdual pobablsc consans Pogam (9) wh se of ndvdual consans p p p3 Objecve value Pogam (8) wh Boole s nequaly p p p3 Objecve value 0.9 0.99 0.94 0.983 988 0.99 0.94 0.983 988 0.95 0.996 0.963 0.99 390 0.996 0.963 0.99 390 0.99 0.999 0.993 0.998 5703 0.999 0.993 0.998 5703
PAGE 4 RRR 7-005 I can be seen ha pogams (8) and (9) especvely nvolvng Boole s nequaly and a se of ndvdual consans esul n he same opmal soluon. Fo pogams () and (3) nvolvng he jon pobablsc consan and he use of he bnomal momen boundng scheme, he coelaon levels beween andom vaables mus be specfed. We consde he hee followng coelaon maces R, R and R 3 : 0. 0. 0 0 0. 0. R = 0. 0.9, R = 0 0 and R3 = 0. 0.9 0. 0.9 0 0 0. 0.9 Resuls fo he pogams () and (3) ae gven n able 5. able 5: Opmal soluons fo he pogams nvolvng a jon pobablsc consan and p Coelaon level he bnomal momen boundng scheme Pogam () wh jon pobably consans Pogam (3) wh bnomal momen boundng scheme: MIN 0.9 R 564.0 575. Pogam (3) Pogam (4) 0.9 R 949.4 958. 958.3 958.4 0.9 R 3 96.6 970. 0.95 R 3603.6 3587.3 0.95 R 3866.6 3896.8 3896.8 3897. 0.95 R 3 3885. 390.3 0.99 R 5500.6 5493.7 0.99 R 570.0 5700.0 5700.0 570. 0.99 R 3 5680.6 570.5 Regadless of he elably level consdeed, can be seen ha he opmal objecve values obaned wh pogams () and (3) fo any of he hee coelaon maces ae lowe han he opmal soluon obaned wh pogams (8) and (9). I s logcal ha he opmal soluon of (3) be lowe han hese of pogams (8) and (9), snce (3) povdes a lowe bound on he objecve value, whle (8) and (9) ae appoxmaons of (), enfocng equemens ha a leas as demandng as hose of (). he low magnude of he gap beween he opmal soluons of (3) on he one sde and hose of (8) and (9) on he ohe sde ndcaes ha (8) and (9) ae gh appoxmaons of (3). Consdeng he coelaon maces R, R and R 3, can be seen ha (0) apples () () () s s, j,, j =,..., and s s (3), j,, j =,...,, j j heefoe allowng he elance on Slepan s nequaly P( ε x, =,..., ; R) P( ε x, =,..., ; R) and P( ε x, =,..., ; R) P( ε x, =,..., ; R3) and he noducon of he nequales (8). Fo any elably level enfoced, can be seen n able 5 ha he opmal values obaned when consdeng he coelaon max R ae lowe han obaned when consdeng coelaon maces R and R 3. he solvng of he sochasc pogam assocaed wh coelaon max R can moeove be solved wh he noducon of nequaly (9). Fnally, can also be seen ha (4) s a b less consanng han (3). j j
RRR 7-005 PAGE 5 3..3 Powe managemen: SABIL poblem he las poblem consdeed s he SABIL poblem (Pékopa e al., 976, 980) nvolvng he consucon of a plan fo he Hungaan eleccal enegy seco n he sevenes. I has a lnea objecve funcon mnmzng he pof funcon mulplyng by, whle sasfyng 06 deemnsc consans (manpowe balance, nvesmen feaues, foegn ade balance, balance of he sae budge, fnance, and eleccy demand sasfacon), as well as fou sochasc consans. he poblem s fomulaed by: mn c x soax. b, =,...,06 5x5 ε 6.67x6 ε () P p 0.8696x4 + x40 ε 3 0.9( x x x3 x4 ) 0.5x4 ε + + + 4 whee c x s gven by x 35 - x 36, x 35 and x 36 epesenng especvely he ncease n he wage bll and he enepse pof befoe axaon, and ε, =,..., 4 ae nomally dsbued andom vaables wh he followng means and vaances: µ = 4833, σ = 483 µ = 46, σ = 4. µ 3 = 6000, σ3 = 60 µ 4 = 4950, σ4 = 90 he jon dsbuon of he andom vaables s nomal, and he covaance max s gven by: 0.8 0.4 0.4 0.8 0. 0. C = 0.4 0. 0.9 0.4 0. 0.9 he fs wo componens of () esan he planned defc of foegn ade (n $US and oubles) o be below a cean level, whle he las wo componens expess he elaonshps beween he eleccal seco and he ohe secos of he Hungaan economy. he eade s efeed o Pékopa e al. [976 and 980] fo a moe dealed descpon of he model. Below, he solvng of he pogams assocaed wh fomulaons () conanng jon pobably consans, (8) based on Boole s nequaly, (9) conanng a se of ndvdual consans, and (3) based on he bnomal momen boundng scheme ae epoed and dscussed. wo elably levels (p = 0.9, 0.95) ae consdeed.
PAGE 6 RRR 7-005 Pogam () wh jon pobably consans able 6: Opmal soluons fo each fomulaon and p = 0.9 Pogam (8) wh Boole s nequaly Pogam (9) wh se of ndvdual consans Pogam (3) wh bnomal momen boundng scheme: MIN Pogam wh (4) bnomal momen boundng scheme: MAX x 643.37 6497.096 6497.683 6305.88 650.645 x 806.469 80.380 80.400 779.88 83.766 x 6 8.36 0.485 0.484 0.045 0 x 7.47.3..36.330 x 4 8400.000 8400.000 8400.000 8400.000 8400.000 x 5 89.749 897.938 897.940 889.79 889.60 x 6 4.676 5.09 5.09 4.643 4.65 x 40 97.45 93.08 93.030 93.93 93.885 x 35 93.76 93.780 93.780 93.77 93.78 x 36 4464.090 4463.8 4463.30 4465.968 4463.6 z - 4370.38-4369.349-4369.80-437.4-4369.444 In able 7 below, we epo he ndvdual elably levels assocaed wh he opmal soluon of (9) fo he SABIL poblem and wh an oveall sevce level p equal o 0.9. able 7 : Indvdual elably level wh p = 0.9 p 95.06% 99.96% 3 00.00% 4 94.97% Usng he bnomal momen boundng fomulaon (3), we oban a lowe bound on he objecve funcon equal o - 437.4 and an uppe bound equal o - 4369.444. I can be seen ha he bnomal boundng scheme povdes a vey naow boundng scheme, and ha he solvng of he poblems (), (8) and (9) povdes vey smla objecve values ha fall whn he bounds deemned by (3) and (4). A smla concluson can be obseved fo p se equal o 0.95. Usng he bnomal momen boundng fomulaon (3), we oban a lowe and uppe bound on he objecve funcon equal o - 4370.70 and - 4368.350, especvely. he opmal objecve value obaned wh he poblem fomulaons () and (8) s he same, equal o - 4368.56, and s also exemely close o he opmal objecve value obaned when p s equal o 0.9. he opmal ndvdual elably levels p, =,, 3, 4 n (8) ae equal o 97.9%, 99.8%, 00.00% and 97.9%, especvely. 4 Convexy analyss of consans enfocng elably level 4. Concavy of elably funcons In hs secon, we analyze he elably level enfocng ha he aveage measue E[ ε ε 0] d, =,..., () of volaons of he consan x ε be below a cean heshold d. Clealy, he nequaly () enfoces he aveage o be aken n he only cases when x ε s volaed.
RRR 7-005 PAGE 7 Below, we analyze he concavy of he funcon g () = E[ ε ε 0] (3) when he andom vaable ε s nomally dsbued. he funcon (3) can be ewen as ( x ) f( x ) dx -F() whee f(x) denoes he pobably densy funcon and F(x) denoes he cumulave pobably dsbuon of he andom vaable o, alenavely, as I can be seen ha ( F( x )) dx -F() ( ( )) ( ) ( ( )) F + f Fx dx g'( ) = ( F ( )) f() g'( ) = + g( ) < 0, F ( ) f() whee s he hazad ae and s deceasng. Assumng ha ε has a sandad nomal F ( ) dsbuon wh mean 0 and sandad devaon, and hus knowng ha / e f() = π, f '( ) = f( ) we oban he followng developmens fo he second devave of g(): d f() f() g"( ) = g( ) + g'( ) d F ( ) F ( ) f '( )( F( )) + f ( ) f( ) = g () + g'() ( F ( )) F ( ) f ()( F()) + f () f () f () = g () + g () + ( F ( )) F ( ) F ( ) fg () () f () g () f() f () g() = + + F ( ) ( F ( )) F ( ) ( F ( )) f ( ) g( ) f( ) = ( g( ) + ) ( F ()) F () f() f() g() = g() F ( ) F ( ) 0.
PAGE 8 RRR 7-005 o es he concavy of g(), we mus heefoe evaluae he sgn of f() g() ( g() )( F()) (4) he solvng of mn f() g() ( g() )( F()) (5) shows ha he expesson (4) s mnmzed fo = 7.58067 fo whch he objecve value of (5) s equal o -0.004798 s posve on ],7.3499[. he objecve value of (5) becomes negave beyond 7.3499, and s equal o -0.00045477 a hs pon. Fnally, we noe ha P( ε 7.3499) = 0.99999999999987, whch means ha g() (3) s concave excep fo exemely unlkely lage values of, and ha he funcon () wll also be concave excep fo exemely demandng elably levels d chaacezed by exemely low values fo d, equng exemely hgh values fo fo () o hold. Fgue : Funcon g() 0,50 0,45 0,40 d 0,35 0,30 0,5 0,0,0,5,0,5 3,0 hs also mples ha he nvese g - () of g() s convex fo mos values of. Fgue : Funcon g - () 3,0,5,0,5,0 0,0 0,5 0,30 0,35 0,40 0,45 0,50 d
RRR 7-005 PAGE 9 Consequenly, he sochasc poblem mn c x soe. [ ε x ε x > 0] d, =,...,, equvalen o s a non-lnea convex poblem. mn c x sox F d. ( ), =,..., 4.. Coespondence beween ypes of elably levels In hs secon, we sudy he coespondence beween he sevce level p enfoced by a pobablsc consan akng he fom P ( ε) p, whch s equvalen o F ( p) and he sevce level d enfoced by he consan (), alenavely wen as g ( p) o deemne whch value of d coesponds o vaous pobables (p = 0.9, 0.95, ) of no havng a sockou, we solve he followng non-lnea poblem: mn d ( F( x )) dx p so. p -F( ) d (6) d 0 whee p = F - (p ) s gven and d s he decson vaable. he coespondence beween he wo ypes of sevce levels obaned by solvng (6) fo dffeen values of p ae gven n able 8. able 8 : Coespondence beween p and d p 0.9 0.9 0.9 0.93 0.94 0.95 0.96 0.97 0.98 0.99 0.999 d 0.4734 0.4636 0.4533 0.443 0.4306 0.479 0.4037 0.3873 0.367 0.3389 0.769
PAGE 0 RRR 7-005 4.. Applcaon Afe havng shown n Secon 4. ha he enfocemen of elably levels of ype d wh nomally dsbued andom vaables esul n non-lnea convex poblems, we consde he SABIL poblem n whch smla elably levels ae pescbed. Moe pecsely, we wan o enfoce he elably level d 0 such ha mn c x soe. [ ε xε x> 0] d, =,..., ( d P( ε x > 0)) d0 (7) = d R, =,..., n whch d, =,, ae decson vaables. Subsung d, =,, by E[ ε x ε x > 0], =,...,, he consan becomes Snce and consan (8) can be ewen as = ( d P( ε x > 0)) d ( E[ ε x ε x > 0] P( ε x > 0)) d0. (8) = E[ ε x ε x > 0] = x 0 ( z x) f( z ) dz - Fx ( ) P( ε x > 0) = - F( x), x ( z x) f( z ) dz and s hus an negaed pobablsc consan (Klen Haneveld, 986), n whch = z x = [ ],,..., η + s he posve pa of he expesson z x, =,..., and can be nepeed, n he SABIL conex, as he amoun of unseved powe. I s hus clea ha poblem (7) esans he expeced amoun of unseved powe o be below a cean heshold d.,
RRR 7-005 PAGE In able 9, we epo he esuls fo a elably level d equal o 0.4734, 0.479 and 0.3389, whch especvely coespond o a pobably level p of no havng a sockou equal o 0.9, 0.95 and 0.99 (see able 8). able 9 : Relably level d fo he SABIL poblem p 0.9 0.95 0.99 d 0.473 0.48 0.339 ( z x) f( z) dz 0.58 0.39 0. x ( z x) f( z) dz 0.5 0.35 0. x ( z3 3x) f( z3) dz3 0 0 0 x 3 ( z4 4x) f( z4) dz4 0.63 0.44 0.6 x 4 Objecve value -4373.080-437.840-437.44 5 Concludng emaks We consde pobablscally consaned poblems, n whch he mulvaae andom vaables ae locaed n he gh-hand sdes. he jon pobablsc consan guaanees he jon fulfllmen of a sysem of lnea nequales wh andom gh-hand sde vaables o be above a pescbed pobably level p. We show he applcably of he p-effcency concep fo appoxmang jon pobablsc vaables wh dsceely dsbued andom vaables. We hen develop appoxmaon boundng schemes applcable o andom vaables wh ehe a dscee o a connuous pobably dsbuon. he boundng schemes nvolve he subsuon of he jon pobablsc consan by a se of ndvdual consans, he Boole s nequaly, he bnomal momen boundng scheme, and Slepan s nequaly, especvely. he advanage of such fomulaons s ha hey eque he compuaon of jon pobablsc consans of lowe dmenson han hs of he jon pobablsc consan ncluded n he ognal fomulaon. We analyze he compuaonal acably, and show he elave ghness. Fnally, we sudy a pacula elably level d, and show ha he funcon of fom E[ ε x ε x > 0] enfocng s concave excep fo vey lage (small) values of x (d ). We sudy he elaonshp beween he sevce levels p and d defned by he funcons Px ( ε) p and E[ ε xε x > 0] d, especvely. We llusae and nepe such a sevce level d n a powe managemen conex.
PAGE RRR 7-005 Refeences Bawa V. S. 973. On Chance Consaned Pogammng Poblems wh Jon Consans. Managemen Scence 0 (), 36-33. Beald P. and Ruszczyńsk A. 00a. he Pobablsc Se Coveng Poblem. Opeaons Reseach 50, 956-967. Beald P. and Ruszczyńsk A. 00b. A Banch and Bound Mehod fo Sochasc Inege Poblems unde Pobablsc Consans. Opmzaon Mehods and Sofwae 7 (3), 359-38. Boole G. 854. Laws of hough (Amecan Repn of 854 Edon). Dove, NY. Boole G. 868. Of Poposons Numecally Defne. In: ansacons of he Cambdge Phlosophcal Socey. Pa II. XI. Chanes A. and Coope W.W. 959. Chance-Consaned Pogammng. Managemen Scence 5, 903-93. Chanes A. and Coope W.W. 963. Deemnsc Equvalens fo Opmzng and Sasfcng unde Chance Consans. Opeaons Reseach, 8-39. Chanes A., Coope W.W. and Symonds G.H. 958. Cos Hozons and Ceany Equvalens: An Appoach o Sochasc Pogammng of Heang Ol. Managemen Scence 4, 35-63. Daley D.J. 974. Compuaon of B- and -Vaae Nomal Inegals. Appled Sascs 3 (3), 435-438. Dencheva D., Pékopa A. and Ruszczyńsk A. 000. Concavy and Effcen Pons of Dscee Dsbuons n Pobablsc Pogammng. Mahemacal Pogammng 89, 55-77. Dencheva D., Pékopa A. and Ruszczyńsk A. 00. On Convex Pobablsc Pogammng wh Dscee Dsbuons. Nonlnea Analyss 47, 997-009. Dencheva D., Pékopa A. and Ruszczyńsk A. 00. Bounds fo Pobablsc Inege Pogammng Models. Dscee Appled Mahemacs 4, 55-65. Dezne Z. 978. Compuaon of he Bvaae Nomal Inegal. Mahemacs of Compuaon 3 (4), 77-79. Dezne Z. 994. Compuaon of he vaae Nomal Inegal. Mahemacs of Compuaon 6, 89-94. Gassmann H.I. 000. Recangle Pobables of vaae Nomal Dsbuons. Unpublshed pepn avalable fom webse hp://www.mgm.dal.ca/sba/pofs/hgassmann/ Gupa S.S. 965. Pobably Inegals of Mulvaae Nomal and Mulvaae. Mahemacal Sascs 34, 79-88. Klen Haneveld W.K. 986. Dualy n Sochasc Lnea and Dynamc Pogammng. Lecue Noes n Economcs and Mahemacal Sysems 74. Spnge-Velag. New Yok, NY. Komáom É. 986. A Dual Mehod fo Pobablsc Consaned Poblems. Mahemacal Pogammng Sudy 8, 94-. Pékopa A. 970. On Pobablsc Consaned Pogammng. Poceedngs of he Pnceon Symposum on Mahemacal Pogammng. Pnceon Unvesy Pess, 3-38. Pékopa A. 973. Conbuons o he heoy of Sochasc Pogammng. Mahemacal Pogammng 4, 0-. Pékopa A. 988. Numecal Soluon of Pobablsc Consaned Pogammng Poblems. n
RRR 7-005 PAGE 3 Numecal echnques fo Sochasc Opmzaon. Eded by Emolev Y. and Wes R.J.-B. Spnge-Velag. Beln. 3-39. Pékopa A. 990. Dual Mehod fo a One-Sage Sochasc Pogammng wh Random hs Obeyng a Dscee Pobably Dsbuon. Zeschf of Opeaons Reseach 34, 44-46. Pékopa A. 995. Sochasc Pogammng. Kluwe. Dodech-Boson. Pékopa A. 999. he Use of Dscee Momen Bounds n Pobablsc Consaned Sochasc Pogammng Models. Annals of Opeaons Reseach 85, -38. Pékopa A. 00. On he Concavy of Mulvaae Pobably Dsbuon Funcons. Opeaons Reseach Lees 9, -4. Pékopa A. 003. Pobablsc Pogammng Models. Chape 5 n Sochasc Pogammng: Handbook n Opeaons Reseach and Managemen Scence 0. Eded by Ruszczyńsk A. and Shapo A. Elseve Scence Ld, 67-35. Pékopa A. and Szána. 978. Flood Conol Resevo Sysem Desgn Usng Sochasc Pogammng. Mahemacal Pogammng Sudy 9, 38-5. Pékopa A., Gancze S., Deák I. and Pay K. 975. he SABIL Sochasc Pogammng Model and s Expemenal Applcaon o he Eleccal Enegy Indusy of he Hungaan Economy. Alkalmazo ma Lapok, 3-. Pékopa A., Gancze S., Deák I. and Pay K. 980. he SABIL Sochasc Pogammng Model and s Expemenal Applcaon o he Eleccal Enegy Seco of he Hungaan Economy. In Sochasc Pogammng. Eded by Dempse M.A.H. Academc Pess. London, 369-385. Ruszczyńsk A. and Shapo A. 003. Sochasc Pogammng: Handbook n Opeaons Reseach and Managemen Scence 0. Elseve Scence Ld. Schage L. 997. Opmzaon Modelng wh LINDO, 5 h Edon. Duxbuy Pess. Pacfc Gove, CA. Sen S. 99. Relaxaons fo Pobablscally Consaned Pogams wh Dscee Random Vaables. Opeaons Reseach Lees, 8-86. Szána. 988. A Compue Code fo Soluon of Pobablsc-Consaned Sochasc Pogammng Poblems. In: Numecal echnques fo Sochasc Opmzaon. Eded by Emolev Y. and Wes R.J.-B. Spnge-Velag. New Yok, 9-35. Szána. 000. Impoved Bounds and Smulaon Pocedues on he Value of he Mulvaae Nomal Pobably Dsbuon Funcon. Annals of Opeaons Reseach 00, 85-0. Wolfam S. 003. he Mahemaca Book, 5 h Edon. Wolfam Meda Inc. Champagn, IL.