Scenario tree reduction for multistage stochastic programs

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1 Compu Manag Sci (009) 6:7 33 DOI 0.007/s y ORIGINAL PAPER Scenario ree reducion for mulisage sochasic programs Holger Heisch Werner Römisch Published online: 0 December 008 Springer-Verlag 008 Absrac A framework for he reducion of scenario rees as inpus of (linear) mulisage sochasic programs is provided such ha opimal values and approximae soluion ses remain close o each oher. The argumen is based on upper bounds of he L r -disance and he filraion disance, and on quaniaive sabiliy resuls for mulisage sochasic programs. The imporan difference from scenario reducion in wo-sage models consiss in incorporaing he filraion disance. An algorihm is presened for selecing and removing nodes of a scenario ree such ha a prescribed error olerance is me. Some numerical experience is repored. Keywords Sochasic programming Scenario reducion Scenario ree Mulisage Inroducion Numerical mehods for solving applied sochasic programming models (in finance, producion, energy, ransporaion, ec.) mosly rely on approximaing he underlying probabiliy disribuion by a finiely discree probabiliy measure. This approximaion echnique reduces he original infinie-dimensional opimizaion problem o a finie-dimensional program. To avoid ha hese opimizaion problems are oo high-dimensional, a scenario reducion mehodology was suggesed in Dupačová e al. (003) and furher developed in Heisch and Römisch (003, 007). These scenario reducion mehods are based on quaniaive sabiliy resuls for sochasic programs (see he survey, Römisch 003, and he recen supplemen, Römisch and Wes 007, for wo-sage models wih random recourse) and on he use of disances of probabiliy disribuions relying on Monge Kanorovich mass ransporaion problems H. Heisch W. Römisch (B) Insiue of Mahemaics, Humbold-Universiy Berlin, 0099 Berlin, Germany romisch@mah.hu-berlin.de

2 8 H. Heisch, W. Römisch (Rachev and Rüschendorf 998). Alhough opimal scenario reducion problems are combinaorial opimizaion models of k-median ype, and, hence, NP-hard, he forward and backward heurisics suggesed in Dupačová e al. (003), Heisch and Römisch (003) and refined in Heisch and Römisch (007) provide encouraging resuls and are ofen used in pracical applicaions. The general idea was recenly exended in Henrion e al. (007, 008) o chance consrained and mixed-ineger wo-sage sochasic programming models. An imporan class of sochasic programs for pracical applicaions are models wih measurabiliy consrains, e.g., mulisage sochasic programs. Recenly, he sabiliy behavior of mulisage linear sochasic programs was sudied in Heisch e al. (006). Is main resul saes ha he disance of opimal values of original and approximae models can be bounded by he L r -disance (for some r ) and a so-called filraion disance of he underlying sochasic processes. The main compuaional approach for solving mulisage models consiss in approximaing he original sochasic process by a process having finiely many scenarios exhibiing ree srucure. Presenly, several approaches for he generaion of such scenario rees are available. Here, we refer o he survey (Dupačová e al. 000) and o he original papers (Casey and Sen 005; Heisch and Römisch 008; Hochreier and Pflug 007; Høyland e al. 003; Kuhn 005, 008; Pennanen 009). If such a scenario ree is available, i may again be of ineres o reduce i by deleing some of is nodes. Due o he sabiliy behavior of muli-sage models, i is argued in Heisch e al. (006, Example.7) ha scenario ree reducion in mulisage models should be based on L r -disances as well as on filraion disances. In his paper, we ake up he laer issue and develop a sound heoreical basis for scenario ree reducion in mulisage sochasic programming models. To do so, we review sabiliy resuls for he mulisage siuaion (in Sec. ) and derive new bounds for boh he L r -disance and he filraion disances beween a scenario ree and is reduced version (in Sec. 3). These bounds moivae algorihms for reducing scenario rees. In Sec. 4 we presen a specific algorihm based on recursive single node reducion. In Sec. 5 we repor on numerical experience of he ree reducion algorihm and show ha is oucomes srongly depends on he use of boh ypes of disances, namely, he L r -disance and he filraion disance. In paricular, he resuls indicae ha applying he scenario reducion echniques from Heisch and Römisch (003, 007) (i.e., mehods ha are only based on he L r -disance) o he mulisage siuaion is no appropriae. A review of sabiliy in mulisage sochasic programming Le ξ ={ξ } = T be a sochasic process defined on some probabiliy space (Ω, F, P) and wih ξ aking values in R d. I is assumed ha his process eners an opimizaion model and ha he (sochasic) decision x a maps from Ω o R m is nonanicipaive, i.e., depends only on ξ := (ξ,...,ξ ). The laer propery is equivalen o he measurabiliy consrain saing ha x is measurable wih respec o he σ -field F (ξ) F generaed by ξ. We assume ha ξ is deerminisic, i.e., ha F (ξ) ={,Ω}. Then he sochasic process ξ is accompanied by a filraion (F (ξ)) = T of σ -fields saisfying

3 Scenario ree reducion for mulisage sochasic programs 9 F (ξ) ={,Ω} F (ξ) F + (ξ) F T (ξ) F. We consider he linear mulisage sochasic programming model [ T min E b (ξ ), x ] x X, x is F (ξ)-measurable, =,...,T, =, () A,0 x + A, (ξ )x = h (ξ ), =,...,T where he subses X of R m are nonempy and polyhedral, he cos coefficiens b (ξ ) belong o R m, he righ-hand sides h (ξ ) are in R n, A,0 R n m are fixed recourse marices and A, (ξ ) R n m echnology marices, respecively. We assume ha coss b ( ), righ-hand sides h ( ) and echnology marices A, ( ) depend affinely on ξ covering he siuaion ha some of he componens of b and h, and of he elemens of A, are random. Noe ha he wo consrains x X and A,0 x + A, (ξ )x = h (ξ ) mean x (ω) X and A,0 x (ω) + A, (ξ (ω))x (ω) = h (ξ (ω)) for P-almos every ω Ω. In addiion o he poinwise consrain wih probabiliy, measurabiliy, filraion or informaion consrains appear in (). They are funcional and non-poinwise a leas if T > and F (ξ) F (ξ) F T (ξ) for some < < T. The presence of such qualiaively differen consrains consiues he origin of boh he heoreical and compuaional challenges of mulisage models. Nex we record resuls of he recen papers (Heisch e al. 006; Heisch and Römisch 008). We assume ha he sochasic inpu process ξ belongs o he Banach space L r (Ω, F, P; R s ) wih s := Tdand r. The mulisage model () is regarded as an opimizaion problem in he space L r (Ω, F, P; R m ) wih m = T = m and endowed wih he norm x r := ( T = where he number r is defined by E[ x r ]) r ( r < ) or x := max =,...,T ess sup x, r r, if only coss are random, r r, if only righ-hand sides are random, := r =, if only coss and righ-hand sides are random,, if all echnology marices are random and r T. () The choice of r and he definiion of r are moivaed by he knowledge on exising momens of he inpu process and by having he sochasic program well defined (in paricular, such ha b (ξ ), x is inegrable for every decision x and =,...,T ). Nex we need o inroduce some noaions. Le F denoe he obecive funcion defined on L r (Ω, F, P; R s ) L r (Ω, F, P; R m ) R by F(ξ, x) := E[ T = b (ξ ), x ],le

4 0 H. Heisch, W. Römisch X (x ; ξ ) := {x X A,0 x + A, (ξ )x = h (ξ )} denoe he h feasibiliy se for every =,...,T and { X (ξ) := x = (x, x,...,x T ) = T L r (Ω, F (ξ), P; R m ) } x X, x X (x ; ξ ) he se of feasible elemens of () wih inpu Ξ. Then he mulisage sochasic program () may be rewrien as min{f(ξ, x) : x X (ξ)}. (3) Furhermore, le v(ξ) denoe is opimal value and, for any α 0, S α (ξ) := {x X (ξ) : F(ξ, x) v(ξ) + α} and S(ξ) := S 0 (ξ) denoe he α-approximae soluion se and he soluion se of he sochasic program (3) wih inpu ξ, respecively. The following condiions are imposed on (3): (A) ξ L r (Ω, F, P; R s ), i.e., Ω ξ(ω) r dp(ω) <,forsomer. (A) There exiss a δ>0 such ha for any ξ L r (Ω, F, P; R s ) wih ξ ξ r δ, any =,...,T and any x X ( ξ ), x τ X τ (x τ ; ξ τ ), τ =,...,, here exiss an F ( ξ)-measurable x X (x ; ξ ) (relaively complee recourse locally around ξ). (A3) The opimal values v( ξ) of (3) wih inpu ξ are finie for all ξ in a neighborhood of ξ and he obecive funcion F is level-bounded locally uniformly a ξ, i.e., for some α>0 here exis a consan δ>0 and a bounded subse B of L (Ω, F, P; R m ) such ha S α ( ξ) is conained in B for all ξ L r (Ω, F, P; R s ) wih ξ ξ r δ. The following sabiliy resul saes ha mulisage models behave sable a some sochasic inpu process if boh is probabiliy disribuion and is filraion are approximaed simulaneously in erms of he L r -disance and of one of he filraion disances D f, (ξ, ξ) := D f, (ξ, ξ) := sup T E[x F (ξ)] E[x F ( ξ)] r, (4) x sup = x = T E[x F (ξ)] E[x F ( ξ)] r, (5) where F (ξ) and F ( ξ) denoe he σ -fields generaed by ξ and ξ, respecively, and E[ F (ξ)] and E[ F ( ξ)] he corresponding condiional expecaions.

5 Scenario ree reducion for mulisage sochasic programs Fig. Example scenario ree ξ wih I =6, I,i =3and I T = (, i) I,i = T = 4 Theorem Le (A), (A) and (A3) be saisfied and X be bounded. Then here exis posiive consans L and δ such ha he esimae v(ξ) v( ξ) L( ξ ξ r + D f, (ξ, ξ)) (6) holds for all random elemens ξ L r (Ω, F, P; R s ) wih ξ ξ r δ. Furhermore, if he soluion ses S(ξ) and S( ξ) are nonempy, here exis L > 0 and ε >0 such ha he esimae dl (S ε (ξ), S ε ( ξ)) L ε ( ξ ξ r + Df, (ξ, ξ)) (7) holds for any ε (0, ε). Here,dl denoes he Pompeiu Hausdorff disance of bounded subses of L r. The firs par of Theorem is essenially Heisch e al. (006, Theorem.), where compared o Heisch e al. (006), condiion (A3) allows o make use of he filraion disances D f, or Df, (cf. he discussion in Heisch and Römisch 008, Sec. 3). The second par of Theorem is proved in Heisch and Römisch (006). Finally, we menion ha Theorem remains valid if he expecaion E in he obecive of () is replaced by a muli-period polyhedral risk funcional saisfying a cerain uniform level boundedness propery (see Eichhorn and Römisch 008). Muli-period polyhedral risk funcionals and heir incorporaion ino muli-sage sochasic programming models are sudied in Eichhorn and Römisch (005). 3 Bounding he L r -minimal and filraion disance Le ξ = {ξ } = T be a sochasic process on he probabiliy space (Ω, F, P) having a finie number of scenarios ξ i wih probabiliies p i, i =,...,N, informof a scenario ree. Le I denoe he index se of realizaions of ξ.ifwesea i := (ξ,...,ξ ) ({(ξ i,...,ξi )}) for every i I, he sysem {A i } i I is a pariion of

6 H. Heisch, W. Römisch Fig. Example scenario ree ξ red wih I red =5, J, = and I red T =8 (, ) = T = 4 Ω and generaes he σ -field F (ξ). Wesep i := P(A i ), i I, =,...,T, and have, in paricular, ha I T ={,...,N} and p i T = p i for every i I T. Furhermore, we have ξ = i I ξ i l A i ( =,...,T ), where l A denoes he characerisic funcion of a subse A of Ω. The ree srucure of ξ implies ha I is a singleon and ha I I I I + I T ={,...,N} holds. Moreover, if I,i I + denoes he index se of successors o ξ i a +, he relaions p i = + (i I ) I,i p are valid for every =,...,T. Any node of he ree corresponds o a pair (, i) {,...,T } I (Fig. ). Now, le ξ red be a sochasic process on (Ω, F, P) ha we regard as reduced scenario ree obained from ξ. IfI red denoes he index se of realizaions of ξ red,he laer means I red = I and I red I ( =,...,T ), where a leas for some {,...,T } we have I red I.Leξ,red, IT red, denoe he scenarios of ξ red. Le us furher denoe by J := I \ I red he index se of all wihdrawn realizaions a ime and by E he se E :=(ξ red,...,ξ red ) ({(ξ,...,ξ )}) for every I red. The sysem {E } I red forms a pariion of Ω and generaes he

7 Scenario ree reducion for mulisage sochasic programs σ -field F (ξ red ). Moreover, for every I red,lej, I denoe he index se such ha ξ i = ξ, red = ξ holds for all i J,, i.e., he index se of all scenarios in I which have been idenified wih ξ during he reducion process (Fig. ). The index ses {J, } I red form a pariion of I and i holds E = π i J, A i, := P(E ) = i J, p i ( I red ). If he assumpions of Theorem are saisfied for some r < +, he disances of opimal values and ε-approximae soluion ses ge small if boh disances ξ ξ red r and D f, (ξ, ξ red ) (8) are small. Hence, if a olerance ε>0 is given, i is reasonable o require w ξ ξ red r + w D f, (ξ, ξ red ) ε, (9) where w i, i =,, denoe posiive weighing facors such ha w ξ ξ red r and w D f, (ξ, ξ red ) belong o (0, ]. The condiion (9) appears as a naural condiion for reducing he scenario ree ξ. A canonical choice for he facors w and w is obained by selecing i I T such ha he corresponding scenario ξ i represens he bes approximaion of ξ wih respec o he L r -disance. More precisely, if ξ denoes he corresponding deerminisic scenario process, i.e., ξ (ω) = ξ i, for all ω Ω, wehave ξ ξ r ξ ξ r, for all deerminisic processes ξ consising of only one given scenario, i.e., for all processes wih ξ (ω) = ξ i, for all ω Ω, where i I T. Then he weighing facors are defined by w = ξ ξ r and w = D f, (ξ, ξ ). (0) Nex we derive bounds for boh disances in (9). They are of he form ξ ξ red r = D f, (ξ, ξ red ) = ( T = T = [ ]) r E ξ ξ red r, () sup x E[x F (ξ red )] r, () x L (F (ξ)) x

8 4 H. Heisch, W. Römisch where L (F (ξ)) = L (Ω, F (ξ), P; R m ). The laer formula is a consequence of he ideniy D f, (ξ, ξ red ) = sup x = = sup x = = sup x x L (F (ξ)) T E[x F (ξ)] E[x F (ξ red )] r T E[x F (ξ)] E[E[x F (ξ)] F (ξ red )] r T x E[x F (ξ red )] r, = which is due o he inclusion F (ξ red ) F (ξ), and he fac ha he condiion x is equivalen o E[x F (ξ)] for every =,...,T. To derive explici expressions for (), we use he measurabiliy of x wih respec o F (ξ red ) and denoe he scenarios x by x i, i I, for every =,...,T.We obain for he condiional expeced values E[x E ]= E x P(dω) P(E ) = π i J, p i xi for every I red and =,...,T.For r < we ge from () D f, (ξ, ξ red ) = T sup = x r E x i l A i E[x E ]l E i I I red r and coninue D f, (ξ, ξ red ) = = T sup E x i l A i E[x E ] i J, = x T sup = x I red I red i J, p i xi I red π p k xk k J, i J, l Ai r r r r. (3)

9 Scenario ree reducion for mulisage sochasic programs 5 3. The L r -disance Now we sar o discuss he L r -disance ξ ξ red r beween he wo processes ξ and ξ red. According o our noaions we direcly obain from () T ξ ξ red r r = = = Ω T = I red ξ ξ red r = T = I red i J, A i ξ ξ red r p i ξ i ξ r. (4) i J, This means ha he L r -disance beween a given process and a reduced one depends on he probabiliies of all wihdrawn scenario componens and on heir disances o some of he remaining scenario componens. 3. The filraion disance Nex we derive an esimae for Df, (ξ, ξ red ) given by (3) in case of r < and x := max i I x i for every x L (F (ξ)). Proposiion Consider he l r -norms y := ( m s= in R m for every =,...,T. Then we have y,s r ) r Df, (ξ, ξ red ) max m r =,...,T T { max = I red ( ) } f,r p i : J J, i J r, (5) where he funcion f,r is defined by f,r (p) := p(π p)r + (π p)pr (π )r [0,π ], I red,=,...,t. Proof From (3) we obain D f, (ξ, ξ red ) = T sup = x I red i J, p i xi k J, pk π x k for every p r Le x i,s, s =,...,m, denoe he componens of x i R m for every i I and =,...,T. We may coninue r.

10 6 H. Heisch, W. Römisch D f, (ξ, ξ red ) = = T sup = x T sup = x I red s= m p i i J, s= I red π m xi,s i J, pi π k J, pk π xi,s k J, x k,s pk π r r x k,s r Hence, o esimae he filraion disance we have o solve maximum problems of he form max λ i y i r λ k y k : y R J, max y i i J, (6) i J k J where J is a given finie index se wih cardinaliy J and λ i > 0, i J, aregiven wih i J λ i =. Le y ( ), =,..., J, denoe he verices of he polyope Y := {y R J : max i J y i }. Any elemen y Y can be represened as convex combinaion of he verices, i.e., r. J y = α y ( ), = where α 0 and J α =. = Since he obecive funcion g(y) := i J λ i yi k J λ k y k r in (6) is convex, one obains J J g(y) = g α y ( ) α g(y ( ) ) = = max g(y( ) ). =,..., J Hence, he maximum in (6) is aained a some y Y wih yi {+, } for all i J. LeJ + J and J J denoe he index ses, where y is posiive and negaive, respecively. Furhermore, le λ + = λ i and λ = λ i. i J + i J Then we have λ + + λ =, and, we obain λ i yi r λ k yk = λ i ( λ + + λ ) r + λ i ( + λ + λ ) r i J k J i J + i J = r λ + ( λ + ) r + r ( λ + )(λ + ) r.

11 Scenario ree reducion for mulisage sochasic programs 7 If we solve he problem (6) for all s {,...,m }, I red λ i := pi we ge as final esimae for he filraion disance π Df, (ξ, ξ red ) max m r =,...,T M,r := max T = I red M,r r, where { p(π p) r + (π p)p r (π : J J,, p = ) r wih J := J, and Proposiion says ha he filraion disance beween he given process ξ and he reduced one ξ red only depends on he paricular pariion srucure of he scenarios of ξ and on he choice of represenaive scenarios of ξ red. The ypical cases r = and r = are now discussed in more deail. i J p i }. The filraion disance for r = and r = In case r = he esimae of he filraion disance is of he form D f, (ξ, ξ red ) M, := max max =,...,T m T = I red { p(π p) π M,, where J J,, p = i J p i } π (7) This allows he following inerpreaion. For any scenario cluser J, defined by he realizaion I red, a some sage, he conribuion M, o he oal filraion disance depends on he oal probabiliy π of he se J, as well as on he pariioning of he probabiliy weighs. Noe ha M, is always bounded from above by π. Example: Le J, ={k,...,k n } and p k + + p k n = π. (a) In case p k i M, = π = p k for all i, =,...,n we obain M if n is odd or even, respecively. (b) In case of one dominan probabiliy in he sense ha p k 0 [, ] we obain M, M, = π, = n n π and = λ π wih λ π = λ( λ). For example, if λ =.95 we have.

12 8 H. Heisch, W. Römisch Fig. 3 Obecive funcion f,r (p) o deermine M,r in cases r = andr = π π 4 0 π π Likewise, in case r = we obain from Proposiion D f, (ξ, ξ red ) M, := max max m =,...,T T = { p(π p) π I red M,, where J J,, p = i J p i } π 4. (8) Figure 3 shows a plo of he obecive in problem (7) and (8), respecively, for deermining M, and M,. 4 Scenario ree reducion in mulisage models In his secion we are going o describe a simple scenario reducion algorihm which is based on recursive single node reducion. For a given ree srucure of ξ, he crierion (9), he represenaion (4), and he esimae (5) sugges he following reducion sraegy of ξ given some olerance ε>0. Algorihm (Single node reducion) [Iniializaion] The reducion procedure is iniialized by saring from he iniial process, i.e., by seing I red := I for all =,...,T, I,i red := I,i for all =,...,T, i I, J, := { } for all =,...,T, I, q := p for all =,...,T, I, ε appr := 0, where ε appr denoes he approximaion error of he reducion process.

13 Scenario ree reducion for mulisage sochasic programs 9 [Node selecion] The node selecion aims a deermining an accepable pair of nodes. A pair (, ) and (τ, i) is called accepable whenever = τ, i =, he unique predecessors of boh nodes coincide, and, simulaneously, he approximaion error is small enough. To his end, we search for some (, ) {,...,T } I red such ha here exiss i I red wih i =, i, I,k red for some k I red and such ha he pair (, ) and (, i) of nodes saisfies he esimae ε sep ε ε appr, ε sep := w (q i where ) r ξ i ξ +w ( ) q i(q ) r + (q i)r q r q i + q. (9) If such a pair canno be found, go o he erminaion sep, oherwise coninue wih he following reducion sep. [Reducion] In his sep we perform he node reducion according o he selecion of accepable nodes before. We adus he relevan index ses and probabiliies of he scenario ree by enlarging he se J,, reducing he se I, updaing he successor informaion and changing he probabiliies. The new index ses and probabiliies are given by J, := J, J,i, := I \{i}, I, red := I, I,i, I red q := q + q i. Finally, he approximaion error is updaed by ε appr := ε appr + ε sep, and he ieraion is coninued by a new node selecion sep. [Terminaion] Whenever he erminaion sep is reached all relevan index ses giving he srucure of he reduced scenario ree process are sored as I red, I, red and J, (cf. Sec. 3). I remains o define he (node) probabiliies by π := q for all =,...,T, I red. The process ξ red is well-defined now by he given index ses and probabiliies. We conclude his secion wih some commens on he above algorihm. In fac, we obain for he probabiliies of he scenario ree process ξ red ha π =. i J, p i for all =,...,T, I red

14 30 H. Heisch, W. Römisch Moreover, he approximaion error beween he iniial scenario ree process and he reduced one can be bounded by ε appr. More precisely, i holds w ξ ξ red r + w D f, (ξ, ξ red ) ε appr, which is a direc consequence of (9) and he riangle inequaliy for boh he L r -disance and he filraion disance Df,. Noe ha he reducion algorihm also can be easily performed wih respec o only one disance by seing he weighing facors w = 0 and w = 0, respecively. If we define w = 0 in condiion (9) he erm conrolling he L r -disance disappears. On he oher hand, when using w = 0 he filraion erm disappears and, hence, he reducion is only performed wih respec o he L r -disance. Table Srucure of he scenario ree processes ξ red obained by he single node reducion algorihm saring from he inpu ree conaining nodes and erminaing wih rees conaining 50 and 00 nodes, respecively Nodes Reducion Nodes (per Sage) Scenarios w.r L r only L r and Df, Df, only L r only L r and Df, Df, only Fig. 4 Srucure of he rivariae iniial scenario ree ξ serving as inpu for he single node reducion algorihm

15 Scenario ree reducion for mulisage sochasic programs 3 Fig. 5 Illusraion of he reduced (rivariae) scenario rees ξ red. The rees are obained by he single node reducion algorihm unil 50 (above) and 00 (below) remaining nodes are reached. The reducion is carried ou wih respec o he L r -disance only (lef), boh he L r -andhedf, -disance (middle), and he Df, -disance only (righ) 5 Numerical experience Finally, we repor on some preliminary numerical experience for scenario ree reducion in mulisage sochasic programs. For esing he single node reducion algorihm of he previous secion, we consider a sochasic opimizaion model for elecriciy porfolios of a German municipal power company. The porfolio consiss of he own (hermal) elecriciy producion, he spo marke conracs, supply conracs and elecriciy fuures. For deails of he opimizaion model we refer o Eichhorn and Römisch (005). I akes ino accoun he sochasic naure of he inpu parameers for every hour of he underlying ime horizon, namely, he elecriciy demand, he hea demand, he EEX spo prices, and base and peak fuure prices (for each monh). Here, we focus on he inpu scenario ree process and assume ha i is obained by he scenario ree generaion mehod of Heisch and Römisch (008). Since he fuure prices are considered as fair prices and can be derived from he spo prices, he inpu scenarios correspond o a rivariae ime discree sochasic inpu process whose componens are elecriciy demand, hea demand, and (EEX) spo prices.

16 3 H. Heisch, W. Römisch For our purposes a generaed scenario ree process ξ is singled ou and reduced by he algorihm in Sec. 4 unil a prescribed number of nodes is reached. To sudy, in paricular, he impac of he filraion disance, scenario rees ξ red are compued by he single node reducion algorihm, where he reducion is done wih respec o he L r -disance and he Df, -disance separaely as well as wih respec o he sum of boh disances as advised by he sabiliy analysis of Sec.. Due o modeling reasons he inpu scenario ree exhibis a monhly branching srucure. For our numerical es we considered a ime horizon of 6 monhs which correspond, hence, o six sages of he sochasic program (Table ). In order o cope wih his monhly srucure, each componen of he scenarios (corresponding o elecriciy demand, hea demand or spo prices) is represened by 6 vecors, where each vecor conains he inpus of one monh in hourly discreizaion. The ree srucure of he inpu process is illusraed in Fig. 4. Figure 5 illusraes he resuls of he scenario ree reducion by applying he single node reducion algorihm unil 50 and 00 nodes remain, respecively. They show ha he filraion disance influences he srucure of he reduced scenario rees noiceably. The incorporaion of he filraion disance leads o a smaller number of remaining scenarios in boh cases. The opposie effec appears when using he L r -disance only. 6 Conclusions Summarizing our heoreical argumens and preliminary numerical experience indicaes ha he incorporaion of he filraion disance ino he reducion of scenario rees is indispensable. This implies, in paricular, ha deleing scenarios in inpu rees for muli-sage models according o he mehodology presened in Dupačová e al. (003) and Heisch and Römisch (003, 007) isno appropriae as he informaion (filraion) srucure is no aken ino accoun. The numerical resuls in Sec. 4 are obained by a simple sraighforward sraegy of reducing single nodes recursively. Bu, he esimaes (4) and (5)forheL r - and filraion disance offer furher poenial for algorihmic exensions. Acknowledgmens This work was suppored by he DFG Research Cener Maheon Mahemaics for key echnologies in Berlin and by he BMBF under he gran 03SF03E. References Casey M, Sen S (005) The scenario generaion algorihm for mulisage sochasic linear programming. Mah Oper Res 30:65 63 Dupačová J, Consigli G, Wallace SW (000) Scenarios for mulisage sochasic programs. Ann Oper Res 00:5 53 Dupačová J, Gröwe-Kuska N, Römisch W (003) Scenario reducion in sochasic programming: an approach using probabiliy merics. Mah Program 95:493 5 Eichhorn A, Römisch W (005) Polyhedral risk measures in sochasic programming. SIAM J Opim 6 :69 95 Eichhorn A, Römisch W (006) Mean-risk opimizaion models for elecriciy porfolio managemen. In: Proceedings of PMAPS 006 (Probabilisic Mehods Applied o Power Sysems), Sockholm, Sweden

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