The Multiperiod Network Design Problem: Lagrangian-based Solution Approaches 2

Transcription

1 Konrd-Zuse-Zentrum für Informtionstechnik Berlin Tkustrße 7 D Berlin-Dhlem Germny ANASTASIOS GIOVANIDIS 1 JONAD PULAJ 1 The ultieriod Network Design Problem: Lgrngin-bsed Solution Aroches 2 1 Zuse Institute Berlin (ZIB), Discrete Otimiztion, Tkustr. 7 D-14195, Berlin-Dhlem, Germny, { giovnidis, ulj }@zib.de 2 This reserch hs been suorted by the Deutsche Forschungsgemeinschft DFG ZIB-Reort (July 2011)

2 The ultieriod Network Design Problem: Lgrngin-bsed Solution Aroches Anstsios Giovnidis, Jond Pulj 1 Abstrct We resent nd rove theorem which gives the otiml dul vector for which Lgrngin dul roblem in the Single Period Design Problem (SPDP) is mximized. Furthermore we give strightforwrd generliztion to the ulti-period Design Problem (PDP). Bsed on the otiml dul vlues derived we comute the solution of the Lgrngen relxtion nd comre it with the lgrngen relxtion nd otiml IP vlues. I. ODEL AND NOTATION We consider network reresented by strongly connected digrh G = (V, A) over finite time horizon T = {1,..., T }. We re given finite set C = {c 1,..., c } with = {1,..., }, where c 1 c 2... c nd Z +, m hold. For ech time eriod t T nd ech rc A we relte totl of integer design vribles = {,1,...,, }, ech with cost κ(t) m R +, m. Thus,m Z +, m is interreted s the number of ccities of size used on rc t given time eriod t. Also, for ech time eriod, set of commodities denoted by P should be routed throughout the network. Ech P is relted to fixed demnd er time eriod tht should be routed from single source node s() V to single destintion node t() V nd w.l.o.g we consider s() t(),. The flow of commodity over rc t given time eriod t is denoted by f, (t) R +. We mke use of the following dditionl nottion nd ssumtions: d (t) : P -dimensionl rel vector of the demnds t given time eriod t. It is ossible tht commodity hs zero entry t eriod t nd ositive one t t + 1, mening tht the demnd my er t lter time eriod. We lwys denote the set of commodities - including the ones with ossibly zero demnd - by P. It is imortnt to note tht the current model only considers non-decresing demnds, tht is d (t) d (t+1) := d (t+1) d (t), d (0) := 0, t T, P (1) U m : Uer bound for ech,m which stisfies the totl demnd of ll source destintion irs for the lst time eriod T, tht is U m := d T P r m (t) : Unit cost er ccity tye t given eriod t, i.e r m (t) := κ(t) m, nottion nd w.l.o.g we ssume the following ordering:, m (2) m. For simlicity of 0 < r (t)... r(t) 2 r (t) 1, t T (3)

3 2 r (t) : The difference in unit cost for c from eriod t to eriod t + 1, tht is r (t) := r(t) r(t+1), r(t +1) := 0, t = 1,..., T 1 (4) We ssume tht the costs of ech ccity tye re non-incresing over time, tht is m κ (t+1) m, t = 1,..., T 1, m (5) µ : A T dimensionl Lgrngin dul vector. Every entry of the vector is denoted by µ (t) for every A nd every t T. ρ : Rerrnged A T dimensionl Lgrngin dul vector. Ech entry of the vector is sum of µ (τ) from t to T, tht is T := µ (τ), t T, A (6) τ=t Z(ρ) (t) : The contribution of eriod t to the objective function of the Lgrngin roblem, tht is Z(ρ) (t) := ( ) κ (t),m x (t),m + f, (7) A m P A Asterisk suerscrits denote otiml vlues or vectors. The model under study considers ersistent routing. This mens tht if ortion of commodity s demnd is routed through secific edge set t time eriod t, then this will re true for future time eriods. In rticulr, re-routing is not llowed. A. Problem Sttement nd Lgrngin Relxtion We cn now rovide the multicommodity flow formultion of the ulti-period Design Problem (PDP) with incrementl flow nd ersistent routing: s.t t T A m δ + (v) P m,m, δ (v) t τ=1 f (τ),, R +, m,m Z +, f, (t) = [PDP] if v = s() if v = t() v, t, [FCONS] c t m τ=1 x(τ),m, t [CAPB],m U m, m,, t The theory of Lgrngin Relxtion nd its lictions in solving discrete otimiztion roblems cn be found in severl survey ers nd textbooks, such s [Geo74], [Sh79], [Fis04], [BW05], [Wol98]. To derive lower bound for the PDP, we follow the roch of relxing the CAPB constrints. For ech rc nd ech time eriod, the CAPB constrint is relted to dul vrible µ (t) 0, nd 0 s defined bove. After rerrnging terms we hve - for ny vector ρ - the consequently following Lgrngin roblem: (8)

4 3 Z(ρ) = s.t. ( t T δ + (v) A m,, R +, ( ) m δ (v),m Z +, f, (t) =,m + P A, if v = s() if v = t(),m U m ) v, t,, m,, t [L-PDP] [FCONS] The imiztion of the Lgrngin over rimry vribles decomoses into two subroblems for ech time eriod. The first is the Ccity Plnning Problem (CPP), which further decomoses into one subroblem for ech rc nd ech ccity tye: s.t ( m ),m,m Z +,,m U m,, m [CPP-t--m] The second subroblem is the unccitted multicommodity network flow roblem, which we stte in the following: s.t P A δ + (v),,, R + δ (v) f, (t) = if v = s() if v = t() v,, [UNCF-t] [FCONS-t] Furthermore, since the reing constrints do not coule the flow vribles for the individul commodities, for ech commodity P imum cost flow roblem should be solved. Altogether the solution from the imiztion of the Lgrngin is summrized in the following Theorem 1 Given Lgrngin dul vector µ, nd consequently vector ρ relted to the CAPB constrints, the riml vribles which imize the Lgrngin tke the following vlues: κ 0 (t) m > s.t δ + (v),m(ρ) =, δ (v) U m [0, U m ], = m m < = (9) (10) (11) t,, m (12) wheres the otiml flow for ech commodity P is the solution of shortest th roblem f, (t) A if v = s(), R + if v = t() v, t, t (13)

5 4 Since the bove holds for ny ρ, we re interested in the rticulr vector tht yields the tightest lower bound. This leds to the following Lgrngin dul roblem D(ρ): mx Z(ρ) [D-PDP] s.t 0, t Theorem 2 If Z D is the solution of D-PDP nd Z IP, Z LP the otiml solution of the PDP nd its solution considering liner relxtion of the integer instlltion vribles resectively, then it holds tht Z D = Z LP Z IP (14) The equlity on the left hnd side is due to the integrlity roerty [Geo74], which sttes tht the otiml vlue of the Lgrngin roblem does not lter when we dro the integrlity conditions on the vribles,m. The right hnd side inequlity results from the wek dulity theorem [BW05, Th.4.8]. Ner otiml Lgrngin dul vectors re in rctise obtined lgorithmiclly, nd there re severl well known methods tht re widely used. In the next section we find in closed form n otiml Lgrngin dul vector for the PDP. II. AIN RESULTS We will first give roof for n otiml Lgrngin dul vector for the Single Period Design Problem (SPDP), then generlize this result for the PDP. In order to recover n otiml Lgrngin dul vector for the SPDP we mke use of the second roosition, nd obtin the desired result s consequence of the liner rogrmg relxtion of the SPDP. A. Single Period Results Theorem 3 The otiml solution of the liner rogrmg relxtion of the SPDP equls x,m = { f, P c if m = where f, solves the UNCF with ρ = r, A., m (15) Proof: Consider ny fesible flow f,, of the liner relxtion of the SPDP. Then x,m = { P f, c if m =, m stisfies the CABP constrints with equlity for every rc. Thus the objective function equls κ x, A κ m x,m = m A = κ f, c A P = r f, (16) A P

6 5 Becuse of (3) we hve tht r is the smllest unit cost er ccity tye. If ny ortion of the flow for ny rticulr rc is stisfied by ccity tye other thn tht of size c, then it cn be relced by the rorite mmount of ccity tye c t smller cost in the objective function. Hence it is cler tht for ny given fesible flow, (16) is iml. It is cler tht the solution of the UNCF, ρ f, with ρ = r, A, is fesible flow for the liner rogrmg relxtion of SPDP. P A Then, r f, r f,. A P A P Theorem 4 An otiml vector µ for the Lgrngin dul of the SPDP equls r 1. Proof: We mke use of Theorem 2. By the left hnd side of (14), for ny otiml vector µ, we hve tht ( (κ m µ ) x,m + ) µ f, = r f, (x,m,f,) A m P A A P (κ m µ x,m ) x,m + µ f, f, = r f, P A A P A m Thus, in order to rove tht given vector µ is otiml for the Lgrngin dul of the SPND it is sufficient to check whether the eqution bove holds with equlity when µ is used in the left hnd side. Let µ = r 1. Then, κ m r = κ m κ 0 m (17) As consequence of Theorem 1, eqution (12) nd the bove inequlity (17), we hve tht (κ m r ) x,m = 0 (18) x,m All we hve left to rove is tht, f, A m P A r f, = A r f, (19) holds with equlity. But the left hnd side is the solution of the UNCF with ρ = r. Hence, due to Theorem 3, the equlity holds. B. ulti-period Result A nturl question for the multi eriod cse is the following: When is the otiml solution of the PDP the sum of eriodwise otiml solutions? Given our ssumtions on the monotoniclly decresing ccity to rice rtios it is cler tht when only one ccity tye is vilble, the otiml solution for the PDP will be the sum of the otiml solutions for ech eriod. When the otiml solution for ech eriod fulfills ech constrint with equlity then it is lso cler tht the solution of the PDP will be the sum of otiml solutions for ech eriod. To tret the more generl cse when the bove do not hold, it is imortnt to notice tht the L-PDP decomoses into time eriods. The substitution := T τ=t µ(τ), t T, A is intended to P

7 6 mke this esier to see. Thus for ech time eriod the design nd flow vribles deend only on ρ. Hence for ech time eriod t, we hve the following Lgrngin roblem: s.t. ( A m δ + (v), R +, ( ) m, δ (v),m Z +,, =,m + P A, ) if v = s() if v = t(),m U m v,, m, [L-PDP-t] [FCONS-t] (20) Theorem 5 An otiml vector µ for the D-PDP is equl entrywise to µ (t) of telescoing sums = r (t), t T nd A. = r (t), nd s result Proof: It is imortnt to note tht the theorem holds under the initil ssumtion (3) tht the costs of ech ccity tye re non-incresing over the given time horizon. This llows ech r (t) to be nonnegtive s required. Becuse of the decomosition of the L-PDP into time eriods which deend only on ρ, it is sufficient to consider the otiml vector for the Lgrngin dul of ech L-PDP-t. But this is simly the otiml vector for the Lgrngin dul of the SPDP for ech eriod t. Therefore by Theorem 4 we hve tht ρ (t) = r (t) 1. Theorem 6 The otiml solution of the liner rogrmg relxtion of the PDP is given by the solution of P T shortest th roblems. Proof: This is n immedite consequence of the revious theorems. since set of otiml Lgrngin duls hs been rovided in closed form, the CPP-t--m cn be solved by insection nd the UNCFs should be solved er eriod nd er commodity, given the otiml vector µ. III. COPUTATIONAL RESULTS Since we hve estblished tht the Lgrngin relxtion of the CAPB constrints yields the sme lower bound s the liner rogrmg relxtion of the PDP, it is nturl to investigte the qulity of this lower bound. Unfortuntely this lower bound cn be quite wek. The following exmle illustrtes our oint: Consider connected two node network with only one rc. The til of the rc indictes the source node, wheres the rrow indictes the sink. We hve c 1 = 100, κ 1 = 100 nd d 1 = 1. The objective function vlue of the liner rogrmg relxtion of the SPDP is 1 wheres the otiml integer solution vlue is 100. Next, we resent some comuttions for the network toology in the icture below. Prices re ket fixed, nd demnds nd ccity tyes re vried. As exected by the exmle given bove when the demnds re reltively smll comred to the ccity tyes the g between the liner relxtion nd the otiml is lrge. This cn be observed in the third row of the second tble.

8 7 Fig. 1. Exmle Network

9 8 Periods d1 d2 d3 c1 c2 c3 c t= t= t= t= t= t= t= t= t= t= t= t= t= t= t= Fig. 2. Tble I: 3 exmles of Demds, Ccities nd Prices er eriod for the bove network Time Periods LP Vlue LR Vlue Integer Vlue T= T= T= Fig. 3. Tble II: Comrison between the liner relxtion, lgrngen relxtion nd ctul integer solution