Equivalence of Piecewise-Linear Approximation and Lagrangian Relaxation for Network Revenue Management

Transcription

1 Equvalence of Pecewse-Lnear Approxmaton and Lagrangan Relaxaton for Network Revenue Management Sumt Kunnumkal Kalyan Tallur Ths verson: November 2012 (June 2012) Barcelona GSE Workng Paper Seres Workng Paper nº 608

2 Equvalence of pecewse-lnear approxmaton and Lagrangan relaxaton for network revenue management Sumt Kunnumkal Kalyan Tallur November 5, 2012 Abstract The network revenue management (RM) problem arses n arlne, hotel, meda, and other ndustres where the sale products use multple resources. It can be formulated as a stochastc dynamc program, but the dynamc program s computatonally ntractable because of an exponentally large state space, and a number of heurstcs have been proposed to approxmate ts value functon. Notable amongst these both for ther revenue performance, as well as ther theoretcally sound bass are approxmate dynamc programmng methods that approxmate the value functon by bass functons (both affne functons as well as pecewse-lnear functons have been proposed for network RM) and decomposton methods that relax the constrants of the dynamc program to solve smpler dynamc programs (such as the Lagrangan relaxaton methods). In ths paper we show that these two seemngly dstnct approaches concde for the network RM dynamc program,.e., the pecewse-lnear approxmaton method and the Lagrangan relaxaton method are one and the same. Key words. network revenue management, lnear programmng, approxmate dynamc programmng, Lagrangan relaxaton methods. Revenue management s the control of the sale of a lmted quantty of a resource (hotel rooms for a nght, arlne seats, advertsng slots etc.) to a heterogenous populaton wth dfferent valuatons for a unt of the resource. The resource s pershable, and for smplcty sake, we assume that t pershes at a fxed pont of tme n the future. Sale s onlne, so the frm has to decde what products to offer at a gven prce for each product, so as not to sell too much at too low a prce early and run out of capacty, or, reject too many low-valuaton customers and end up wth excess unsold nventory. In ndustres such as hotels, arlnes and meda, the products consume bundles of dfferent resources (mult-nght stays, mult-leg tnerares) and the decson to accept or reject a partcular product at a certan prce depends on the future demands and revenues for all the resources used by the product and, ndrectly, on all the resources n the network. Network revenue management (network RM) s control based on the demands for the entre network. Chapter 3 of Tallur and van Ryzn [13] contans all the necessary background on network RM. The network RM problem can be formulated as a stochastc dynamc program, but computng the value functon becomes ntractable due to the hgh dmensonalty of the state space. As a result, Indan School of Busness, Hyderabad, , Inda, emal: sumt kunnumkal@sb.edu ICREA and Unverstat Pompeu Fabra, Ramon Tras Fargas 25-27, Barcelona, Span, emal: kalyan.tallur@upf.edu 1

3 researchers have focussed on developng approxmaton methods. Notable amongst these both for ther revenue performance, as well as ther theoretcally sound bass are approxmate dynamc programmng methods that approxmate the value functon by bass functons (both affne functons as well as pecewse-lnear functons have been proposed for network RM), and decomposton methods that relax the constrants of the dynamc program to solve smpler dynamc programs (such as the Lagrangan methods). In ths paper we show that these two seemngly dstnct approaches concde for the network RM dynamc program. Specfcally, we show that the pecewse-lnear approxmaton method and the Lagrangan relaxaton method are one and the same, usng a novel mnmal number of bndng constrants argument. As a by-product, we derve some auxlary results of ndependent nterest: () we gve a polynomaltme separaton procedure for the pecewse-lnear approxmaton lnear program, () we show that the optmal soluton of the pecewse-lnear approxmaton satsfes monotoncty condtons smlar to that of a sngle-resource dynamc program, and () sketch an extenson to a model of network RM wth overbookng. The rest of the paper s organzed as follows. In 1 we gve a bref survey of the relevant lterature. In 2 we frst formulate the network RM problem as a dynamc program. We then descrbe the approxmate dynamc programmng approach wth pecewse-lnear bass functons and the Lagrangan relaxaton approach. In 3 we gve the man body of proofs showng the two approaches are equvalent, along wth a smple calculus-based ntuton behnd our result. In 4 we dscuss a choce-model based network RM problem where the equvalence between the two approaches does not hold. Ths hghlghts the mportance of formulaton and customzaton of the number of Lagrange multplers to the problem at hand. In 5 we gve a small set of numercal results comparng the soluton values and runnng tmes of the two approaches. In the appendx we gve the proofs and also a sketch an extenson to an overbookng model of network RM. 1 Relevant lterature Approxmate dynamc programmng s the name gven for methods that replace the value functon of a (dffcult) dynamc program (DP) wth bass functons and solve the smplfed problem as an approxmaton. In ths stream of lterature, the lnear programmng approach conssts of formulatng the dynamc program as a lnear program wth state-dependent varables representng the value functons and then replacng them by partcular functonal forms to fnd the best approxmaton wthn that class of functons. In the network RM context, ths approach was frst nvestgated by Adelman [1] who uses affne functons; Zhang and Adelman [17] extend ths to the choce model of network RM. A natural extenson s to use pecewse-lnear functons nstead of affne as they are very flexble and ndeed, for the sngle-resource dynamc program, optmal. In ths ven, Faras and van Roy [4] propose a pecewse-lnear approxmaton wth concavty constrants. The resultng lnear program has an exponental number of constrants and cannot be solved easly. So they gve a constrant samplng heurstc to solve t. Messner and Strauss [11] extend the pecewse-lnear approxmaton approach to the choce model of network RM usng aggregaton over states to reduce the number of varables. Another stream of lterature revolves around the Lagrangan relaxaton approach to dynamc programmng. Here the dea s to relax certan constrants n the dynamc program by assocatng Lagrange multplers wth them so that the problem decomposes nto smpler problems. For network RM, Topaloglu [15] and Kunnumkal and Topaloglu [10] take ths approach. Computatonal results from Topaloglu [15] ndcate that Lagrangan relaxaton wth product-specfc Lagrange multplers 2

4 gves consstent and clearly superor revenues compared to the other methods, ncludng the affne approxmaton of Adelman [1]; the pecewse-lnear approxmaton was not ncluded, perhaps because t was not known how to solve t exactly. How do these seemngly dfferent approaches compare wth each other? In a recent paper, Tong and Topaloglu [14] establsh the equvalence between the affne approxmaton of Adelman [1] and the Lagrangan relaxaton of Kunnumkal and Topaloglu [10]. Vossen and Zhang [16] gve an alternate proof based on Dantzg-Wolfe decomposton. In ths paper we show that the pecewselnear approxmaton, wthout the explct concavty constrants of [4], and the product and tmespecfc Lagrangan relaxaton of Topaloglu [15] concde; that s, they represent the same lnear program. Note that both Adelman [1] and Faras and van Roy [4] formulate the affne and pecewselnear approxmatons but leave the queston of ther tractablty open. Indeed the soluton methods that they propose (nteger-programmng, constrant samplng) do not guarantee tractablty n theory or scale well n practce. So t s of consderable theoretcal and practcal nterest to show that these strong approxmatons are also tractable. Adelman and Mersereau [2] gve an equvalence result smlar to ours for a class of nfnte-horzon dscounted restless bandt problems. Infnte-horzon dscounted DP problems have a very dfferent flavor than fnte-horzon DPs and our technques are qute dfferent from thers. We beleve the result n ths paper s the frst such for a non-trval fnte-horzon stochastc DP, and the man mplcaton s that strong tractable approxmatons for dffcult stochastc fnte-horzon DPs are ndeed possble. The technques and the ntuton developed n ths paper can potentally be appled to derve such results; however t appears that even when such a result s true, some amount of customzaton and problem-specfc arguments are requred to prove t. For example, for a related problem, namely choce-based network RM, the rght formulaton and number of Lagrange multplers to use appear crtcal. We dscuss ths further n 4 of ths paper; see also [9]. 2 Network revenue management We consder a network RM problem wth a set of I = {1,..., m} resources (for example flght legs on an arlne network), J = {1,..., n} products (for example tnerary-fare combnatons) that use the resources n I at the end of τ tme perods (bookng horzon), wth tme beng ndexed from 1 to τ. We assume that each product uses at most one unt of each resource. 2.1 Notaton Throughout, we ndex resources by, products by j, and tme perods by t. We smplfy the notaton sgnfcantly by makng ths consstent; for nstance f j uses resource, we represent t as j, and all j that use resource by {j j }. We use ths notaton nstead of the somewhat more cumbersome, albet a bt more precse, opton of defnng I j I as the set of resources used by product j and J J as the set of products that use resource and then wrtng I j and j J. We use 1 [ ] as the ndcator functon, 1 f true and 0 f false. Bookng requests for products come n over tme and we let p jt denote the probablty that we get a request for product j at tme perod t. Ths s the so-called ndependent demands model n the revenue management lterature. We make the standard assumpton that the tme perods are small enough so that we get a request for at most one product n each tme perod. Throughout we assume that p jt > 0 for all j. Note that ths s wthout loss of generalty because f p jt = 0 3

5 for some product j, then we can smply dscard that product and optmze over a smaller number of products. We also assume that j p jt = 1 for all tme perods t. Ths s also wthout loss of generalty because we can add a dummy product wth neglgble revenue on each resource. We let f j denote the revenue assocated wth product j. Gven a request for product j, the arlne has to decde onlne whether to accept or reject the request. An accepted request generates revenue and consumes capacty on the resources used by the product; a rejected request does not generate any revenue and smply leaves the system. Throughout, we use boldface for vectors. We represent capacty vectors by r. We use superscrpts on vectors to ndex the vectors (for example, the resource capacty vector assocated wth tme perod t would be r t ) and subscrpts to ndcate components (for example, the capacty on resource n tme perod t would be r t ). We let r 1 = [r 1] represent the ntal capacty on the resources and rt = [r t ] denote the remanng capacty on resource at tme perod t. The remanng capacty r t takes values n the set R = {0,..., r 1} and R = R represents the state space. We represent control vectors by u. The control vectors could be at the level of the network or at the level of the ndvdual resources. So, u t = [u t j ] represents a vector of controls at the network level. We let u t j {0, 1} ndcate the acceptance decson 1 f we accept product j at tme t and 0 otherwse. On the other hand, u t = [u t j ] represents a control vector assocated wth resource at tme t. In ths case u t j {0, 1} represents the acceptance decson on resource 1 f we accept product j on resource at tme t and 0 otherwse. 2.2 Dynamc Program The network RM problem can be formulated as a DP. Let U(r) = {u {0, 1} n u j r j, j}, be the set of acceptable products when the state s r. The value functons V t ( ) can be obtaned through the optmalty equatons V t (r) = max p jt u j [f j + V t+1 (r e ) V t+1 (r)] + V t+1 (r), (1) u U(r) j j where e s a vector wth a 1 n the th poston and 0 elsewhere, and the boundary condton s V τ+1 ( ) = 0. Notng that r 1 represents the ntal capacty on the resources, V 1 (r 1 ) gves the optmal expected total revenue over the bookng horzon. The value functons can, alternatvely, be obtaned by solvng the lnear program wth an exponental number of decson varables {V t (r) t, r R} and an exponental number of constrants: (DP LP ) mn V s.t V 1 (r 1 ) V t (r) j p jt u j [f j + V t+1 (r j e ) V t+1 (r)] + V t+1 (r) V τ+1 ( ) = 0. t, r R, u U(r) 4

6 Both the recursve equatons (1) as well as the lnear program (DP LP ) are ntractable. followng sectons, we descrbe two approxmaton methods. In the 2.3 Pecewse-lnear approxmaton for Network RM We approxmate the value functons 1 of (1) by V t (r) v t (r ), r R. Substtutng ths approxmaton nto (DP LP ), we obtan the lnear program v 1(r 1) (P L) V P L = mn v s.t v t(r ) j p jt u j [f j + j {v,t+1 (r 1) v,t+1 (r )}] (2) + v,t+1 (r ) t, r R, u U(r) v,τ+1 ( ) = 0, v t ( 1) =, t,, where the decson varables are {v t (r ) t,, r R }. The number of decson varables n (P L) s r1 τ whch s manageable. However, snce (P L) has an exponental number of constrants of type (2), we need to use a separaton algorthm to generate constrants on the fly to solve (P L) (Grötschel, Lovász, and Schrjver [5]). In 3, we show that the separaton can be carred out effcently for (P L). A natural queston s to understand the structural propertes of an optmal soluton to (P L). Ths s useful snce mposng structure can often sgnfcantly speed up the soluton tme. Lemma 1 below shows that an optmal soluton to (P L) satsfes certan monotoncty propertes. In partcular, f we nterpret v t (r ) as the value of havng r unts of resource at tme perod t, then v t (r ) v t (r 1) s the margnal value of capacty on resource at tme perod t. Part () of Lemma 1 shows that the margnal value of capacty s decreasng n t keepng r constant, whle part () shows that the margnal value of capacty s decreasng n r for a gven t. These propertes are qute natural and turn out to be useful for a couple of reasons. Frst, Lemma 1 mples that the optmal objectve functon value of (P L) s not affected by addng constrants of the form v t (r ) v t (r 1) v t (r + 1) v t (r ) to the lnear program. So the lnear program proposed by Faras and van Roy [4], whch explctly mposes the monotoncty constrants, s n fact equvalent to (P L). More mportantly, the decreasng margnal value property turns out to be crucal n showng the equvalence between the pecewse-lnear approxmaton and the Lagrangan relaxaton approaches. Lemma 1. There exsts an optmal soluton {ˆv t (r ) t,, r R } to (P L) such that () ˆv t (r ) ˆv t (r 1) ˆv,t+1 (r ) ˆv,t+1 (r 1) for all t, and r R, () ˆv t (r ) ˆv t (r 1) ˆv t (r + 1) ˆv t (r ) for all t,, and r R, where we defne ˆv t (r 1 + 1) = ˆv t (r 1 ) for all t and. Proof. Appendx. 1 Adelman [1] uses the affne relaxaton V t (r) θ t + r v t but we do not need the offset term θ t for pecewselnear approxmatons as we can use the transformaton v t (r ) = θ t/m + v t (r ); see also Adelman and Mersereau [2]. 5

7 2.4 Lagrangan Relaxaton Topaloglu [15] proposes a Lagrangan relaxaton approach that decomposes the network RM problem nto a number of sngle-resource problems by decouplng the acceptance decsons for a product over the resources that t uses va product and tme-specfc Lagrange multplers. Let {λ jt t, j, j} denote a set of Lagrange multplers and U (r ) = {u {0, 1} n u j r, j } denote the set of feasble controls on resource when ts capacty s r. We solve the optmalty equaton ϑ λ t(r ) = max u U (r ) p jt u j[λ jt + ϑ λ,t+1(r 1) ϑ λ,t+1(r )] + ϑ λ,t+1(r ) for resource, wth the boundary condton ϑ λ,τ+1 ( ) = 0. j If we defne V λ t (r) = τ p jt [f j λ jt ] + + j j t =t ϑ λ t(r ) (3) t s possble to show that Vt λ (r) s an upper bound on V t (r), where we use [x] + = max{0, x}. The Lagrangan relaxaton approach fnds the tghtest upper bound on the optmal expected revenue by solvng V LR = mn V1 λ (r 1 ). λ Tallur [12] shows that the optmal Lagrange multplers satsfy j λ jt = f j for all j and t. Proposton 1. There exsts {ˆλ jt t, j, j} arg mn λ V1 λ (r 1 ) that satsfy ˆλ jt 0 and ˆλ j jt = f j for all j and t. Proof. Appendx. Proposton 1 mples that we can fnd the optmal Lagrange multplers by solvng V LR = mn {λ ϑ λ 1(r 1 j λjt=fj,λjt 0, t,j, j} ). Usng Proposton 1, we can nterpret the Lagrange multpler λ jt as the porton of the revenue assocated wth product j that we allocate to resource at tme perod t. Wth ths understandng ϑ λ 1 (r1 ) s the value functon of a sngle-resource RM problem wth revenues {λ jt j, t} on resource. Therefore, we can also obtan the optmal Lagrange multplers through the lnear programmng formulaton of the sngle-resource RM dynamc program, wth a set of lnkng constrants 6

8 (5) as below: (LR) V LR = mn λ,ν s.t. ν 1 (r 1 ) ν t (r ) p jt u j[λ jt + ν,t+1 (r 1) ν,t+1 (r )] j +ν,t+1 (r ) t,, r R, u U (r ) (4) λ jt = f j t, j (5) j λ jt 0 t, j, j; ν,τ+1 ( ) = 0. The lnear programmng formulaton (LR) turns out to be useful when comparng the Lagrangan relaxaton approach wth the pecewse-lnear approxmaton. 3 Equvalence of the pecewse-lnear approxmaton and the Lagrangan relaxaton approaches In ths secton we show that the pecewse-lnear approxmaton and the Lagrangan relaxaton approaches are equvalent, n that they yeld the same upper bound on the value functon. Ths also shows that the Lagrangan relaxaton approach yelds the tghtest separable, pecewse-lnear upper bound to the value functon of the network RM dynamc program. Proposton 2. V P L = V LR. It s easy to see that V P L V LR snce (LR) gves a separable approxmaton that s an upper bound, whle (P L) gves the tghtest separable approxmaton that s an upper bound; we gve a formal proof n 3.2. So the dffcult part s the other drecton: In the Lagrangan problem, we solve each of the resources ndependently and a product mght be accepted on one resource and rejected on another, and there s no reason to beleve that the Lagrange multplers co-ordnate perfectly ndeed there are few known dynamc programs where they do. For the network RM problem, Proposton 4 below shows that there exsts a set of Lagrange multplers that perfectly coordnate the acceptance decsons across the resources. In 3.1 we frst set up the separaton problem for (P L), a smpler alternatve to solvng (P L) drectly. We then show that constrants (2) n (P L) can be separated by solvng a lnear program. We use ths result to prove Proposton 2 n 3.2. We descrbe a polynomal-tme separaton algorthm for (P L) n Separaton for (P L) Snce (P L) has an exponental number of constrants of type (2), we use a separaton algorthm to solve (P L). Equvalence of effcent separaton and solvablty of a lnear program s due to the well-known work of Grötschel et al. [5]. The dea s to start wth a lnear program that has a small subset of constrants (2) and solve t to obtan V = { v t (r ) t,, r R }. We then check for volated constrants by solvng the 7

9 followng separaton problem: Prove that V satsfes all the constrants (2), and f not, fnd a volated constrant. Throughout we assume that V satsfes v t (r ) v t (r 1) v t (r + 1) v t (r ) for all t, and r R. Ths s wthout loss of generalty, snce by Lemma 1, we can add these constrants to (P L) wthout affectng ts optmal objectve functon value. Let t (r ) = v,t+1 (r ) v t (r ) and ψ t (r ) = v t (r ) v t (r 1) for r R, so that the separaton problem for (P L) for perod t can be wrtten as Φ t ( V) = max p jt u j [f j ψ,t+1 (r )] + t (r ). (6) r R,u U(r) j j Note that ψ t ( ) s just the margnal value of capacty on resource at tme perod t. The separaton problem for a set of values V s resolved by obtanng the value of Φ t ( V) and checkng f for any t, Φ t ( V) > 0. By Lemma 1, ψ t (r ) s nonncreasng n r. By defnton, v t (r 1 + 1) = v t(r 1 ). Snce ψ t(r 1 ) = v t (r 1 ) v t(r 1 1) v t(r 1 + 1) v t(r 1 ) = 0, we also have ψ t(r ) 0 for all r R. We show that problem (6) can be solved effcently as a lnear program. Ths result s useful for two reasons. Frst, t helps us n establshng the equvalence between the pecewse lnear approxmaton and the Lagrangan relaxaton approaches. Second, t shows that separaton can be effcently carred out for (P L). We begn by descrbng a relaxaton of problem (6) that decomposes t nto a number of sngle resource problems. For tme perod t, we splt the revenue of product j, f j, among the resources that t consumes usng varables λ jt. We nterpret λ jt as a Lagrange multpler and t represents the revenue allocated to resource j at tme t. As a result, the Lagrange multplers satsfy j λ jt = f j and λ jt 0 for all j, for all j. Gven such a set of Lagrange multplers, we solve the problem Π λ t( V) = max r R,u U (r ) p jt u j[λ jt ψ,t+1 (r )] + t (r ) (7) j for each resource. The followng lemma states that Πλ t ( V) s an upper bound on Φ t ( V). Lemma 2. If {λ jt t, j, j} satsfy j λ jt = f j and λ jt 0 for all t, j and j, then Φ t ( V) Πλ t ( V). Proof. If ( r = [r ], u U(r) ) s optmal for problem (6), then u U (r ) and consequently (r, u) s feasble for problem (7). We next show that the upper bound s tght. That s, lettng Π t ( V) = mn {λ Π λ t( V) (8) j λjt=fj j;λjt 0 j, j} we have the followng proposton. Proposton 3. Φ t ( V) = Π t ( V). Before we gve a formal proof, we provde some ntuton as to why the result holds. The equvalence of Φ t ( V) and Π t ( V) turns out to be the key result n showng the equvalence between the pecewse-lnear and the Lagrangan relaxaton approaches. It s qute possble that ths type of result apples to other problems (say, to some classes of loosely-coupled dynamc programs, n the framework of Hawkns [6], wth a threshold-type or ndex polcy), and we hope the heurstc argument and the ntuton we gve below wll help n such cases. 8

10 3.1.1 Intuton behnd Proposton 3 Lemma 2 shows that Φ t ( V) Π t ( V). So we only gve a heurstc argument for why Φ t ( V) Π t ( V). Consder problem (6). Notng that ψ,t+1 (0) =, an optmal soluton wll have u j = 1 only f the dfference f j j ψ,t+1(r ) > 0. Therefore, we can wrte u j [f j j ψ,t+1(r )] n the objectve functon as [f j j ψ,t+1(r )] +. Next, recall that ψ,t+1 ( ) s a decreasng functon of r. Assumng t to be nvertble (say t s strctly decreasng), we can wrte problem (6) wth ψ,t+1 s as the decson varables nstead of the r s. Therefore, we can wrte problem (6) as Φ t ( V) = max p jt [f j ψ,t+1 ] + + t (ψ,t+1 ). (9) ψ j j The above problem s not dfferentable. However, by smoothng the [ ] + operator and assumng that t ( ) s dfferentable, we can solve a dfferentable problem whch s arbtrarly close to problem (9). So we can assume that an optmzer of the above maxmzaton problem { ˆψ,t+1 } satsfes the frst order condton j p jt 1 [fj k j ˆψ k,t+1 >0] + t( ˆψ,t+1 ) = 0 (10) for all, where t ( ) denotes the dervatve of t(ψ,t+1 ) wth respect to ψ,t+1. We emphasze that the above arguments are heurstc; our goal here s to only gve ntuton. We use the optmal soluton { ˆψ,t+1 } descrbed above to construct a set of Lagrange multplers n the followng manner. Let ˆψ,t+1 ˆλ jt = f j k j ˆψ j, j, k,t+1 and note that they are feasble to problem (8). We have ˆλ jt ˆψ,t+1 = [f j k j ˆψ,t+1 ˆψ k,t+1 ] k j ˆψ. (11) k,t+1 Snce the rato on the rght hand sde s postve, ths mples 1 [ˆλjt ˆψ,t+1>0] = 1 [f j k j ˆψ k,t+1 >0]. Snce, { ˆψ,t+1 } satsfes (10), t also satsfes j p jt1 [ˆλjt ˆψ,t+1 >0] + t ( ˆψ,t+1 ) = 0, whch s the frst order condton assocated wth an optmzer of max ψ j p jt[ˆλ jt ψ,t+1 ] + + t (ψ,t+1 ). That s, { ˆψ,t+1 } s an optmzer of max ψ j p jt[ˆλ jt ψ,t+1 ] + + t (ψ,t+1 ). So problem (7) can be wrtten as Πˆλ t ( V) = max p jt [ˆλ jt ψ,t+1 ] + + t (ψ,t+1 ) = p jt [ˆλ jt ˆψ,t+1 ] + + t ( ˆψ,t+1 ), (12) ψ j j where the last equalty follows from above observatons. Puttng everythng together, we have Π t ( V) Πˆλ t ( V) = { p jt [ˆλ jt ˆψ,t+1 ] + + t ( ˆψ },t+1 ) j = p jt [ˆλ jt ˆψ,t+1 ] + + t ( ˆψ,t+1 ) = Φ t ( V), j j 9

11 where the frst nequalty holds snce {ˆλ jt j, j} s feasble for problem (8) and the last equalty uses (11) and the fact that { ˆψ,t+1 } s optmal for (9). Note also that the Lagrange multplers {ˆλ jt j, j} coordnate the decsons for each product across the dfferent resources: product j s accepted on resource j only f ˆλ jt ˆψ,t+1 > 0. By (11), ether ˆλ jt ˆψ,t+1 > 0 for all j or ˆλ jt ˆψ,t+1 0 for all j. That s, we ether accept the product on all the resources t consumes or reject the product on all the resources t consumes. We once agan emphasze that the above arguments are heurstc; we gve a formal proof n the followng secton that s qute dstnct from the above reasonng. Nevertheless, t s nterestng to note the man condtons for the heurstc argument to work. 1. A threshold type optmal control once we decompose the problem, so the soluton can be reconstructed from the Lagrange problem. In ths partcular case, note that we have u j = 1 only f the revenue f j exceeds j ψ,t+1(r ). 2. A monotone decreasng functon (ψ) that allowed us to change the separaton problem from optmzaton over the r s to optmzaton over the ψ s and use the frst-order condton. 3. An adequate number of Lagrange multplers that allowed us to splt the revenue functon. In ths partcular case, we requre a Lagrange multpler λ jt for each resource and product j at tme t. We dscuss ths pont further n Proof of Proposton 3 We begn wth some prelmnary results. Frst, we show that problem (7) can be wrtten as the followng lnear program (SepLR ) Π λ t( V) = mn w,z s.t. w t w t j z jtr + t (r) r R z jtr p jt [λ jt ψ,t+1 (r)] z jtr 0 j, r R. j, r R Lemma 3. The lnear program (SepLR ) s equvalent to (7). Proof. Appendx. 10

12 We can, therefore, formulate problem (8) as the lnear program Π t ( V) = mn w t λ,w,z (SepLR) s.t. w t z jtr + t (r), r R (13) j z jtr p jt [λ jt ψ,t+1 (r)], j, r R (14) λ jt = f j j (15) j λ jt 0, j (16) z jtr 0, j, r R. (17) Wth a slght abuse of notaton, we let (λ, w, z) = ( {λ jt v j, j}, {w t }, {z jtr, j, r R } ) denote a feasble soluton to (SepLR). Let ξ t (r) = w t [ j z jtr + t (r)] denote the slack n constrant (13), and B (λ, w, z) = {r R ξ t (r) = 0} denote the set of bndng constrants of type (13) and B c (λ, w, z) denote ts complement. Note that f (ˆλ, ŵ, ẑ) s an optmal soluton, then B (ˆλ, ŵ, ẑ) s nonempty, snce for each resource, there exsts some r R such that constrant (13) s satsfed as an equalty. The followng proposton s a key result. It says that there exsts a set of optmal Lagrange multplers that perfectly coordnate the acceptance decsons for each product across all the resources. That s, even though we solve the sngle resource problems n a decentralzed fashon, the Lagrange multplers are such that we ether accept the product on all the resources or reject t on all the resources. We say that (ˆλ, ŵ, ẑ) s an optmal soluton to (SepLR) wth a mnmal set of bndng constrants I B (ˆλ, ŵ, ẑ), f there s no other optmal soluton (ˆλ, ŵ, ẑ ) whch has a set of bndng constrants that s a strct subset of the bndng constrants of (ˆλ, ŵ, ẑ); that s, B (ˆλ, ŵ, ẑ). I B (ˆλ, ŵ, ẑ ) I Proposton 4. There exsts an optmal soluton (ˆλ, ŵ, ẑ) to (SepLR) wth a mnmal set of bndng constrants and {ˆr ˆr B (ˆλ, ŵ, ẑ), } such that for each j, we ether have ˆλ jt ψ,t+1 (ˆr ) for all j or ˆλ jt ψ,t+1 (ˆr ) for all j. Proof. Appendx. By Lemma 2, Φ t ( V) Π t ( V). We show below that Φ t ( V) Π t ( V), whch completes the proof. Let (ˆλ, ŵ, ẑ) and {ˆr } be as n Proposton 4; so we have Π t ( V) = ŵ t = ẑ,j,t,ˆr + t (ˆr ) = p jt [ˆλ jt ψ,t+1 (ˆr )] + + t (ˆr ), j j j where the second equalty holds snce ˆr B (ˆλ, ŵ, ẑ) for all. The last equalty holds snce f ẑ,j,t,ˆr > p jt [ˆλ jt ψ,t+1 (ˆr )] +, then we can decrease ẑ,j,t,ˆr by a small postve number contradctng ether the optmalty of (ˆλ, ŵ, ẑ) or the fact that (ˆλ, ŵ, ẑ) s an optmal soluton wth a mnmal set of bndng constrants amongst all optmal solutons. Let J 1 = {j ˆλ jt ψ,t+1 (ˆr ) j} and 11

13 J 2 = J \J 1 where J = {1,..., n}. By Proposton 4, every product j J 2 satsfes ˆλ jt ψ,t+1 (ˆr ) for all j. Therefore, Π t ( V) = p jt [ˆλ jt ψ,t+1 (ˆr )] + t (ˆr ) j J 1 j = p jt û j [ˆλ jt ψ,t+1 (ˆr )] + t (ˆr ) j j = p jt û j [f j ψ,t+1 (ˆr )] + t (ˆr ) j j Φ t ( V) where we defne û j = 1 for j J 1 and û j = 0 for j J 2. Note that the last equalty follows from constrant (15). The last nequalty holds snce r = [ˆr ], u = [û j ] s feasble to problem (6) by the followng argument: we trvally have û j ˆr for all j J 2 and j. On the other hand, snce ˆλ jt s fnte and ψ,t+1 (0) =, we have ˆr 1 for all j J 1 and j. It follows that û j ˆr for all j J 1 and j; so the 0-1 controls û = {û j } satsfy the constrant that û j = 0 f ˆr = 0 for any j. 3.2 Proof of Proposton 2 We frst show that V P L V LR. Consder a feasble soluton ( {ˆλ jt t, j, j}, {ˆν t (r ) t,, r R } ) to (LR). For a gven t, r = [r ] and u U(r), note that u U (r ) for all. Summng up constrants (4) for r and u U (r ) for all, ˆν t (r ) p jt u j [ˆλ jt + ˆν,t+1 (r 1) ˆν,t+1 (r )] + ˆν,t+1 (r ) j = j p jt u j [f j + j ˆν,t+1 (r 1) ˆν,t+1 (r )] + ˆν,t+1 (r ) where the equalty holds snce j ˆλ jt = f j. So {ˆν t (r ) t,, r R } s a feasble soluton to (P L) wth the same objectve functon value and we have V P L V LR. To show the reverse nequalty, let V P L = mn v v 1(r 1 )+ t Π t(v), where V = {v t (r ) t,, r R }. We have V P L = mn {v Φ t(v) 0} v 1 (r 1 ) = mn {v Π t(v) 0} v 1 (r 1 ) mn {v Π t(v) 0} v 1 (r 1 ) + t Π t (V) V P L, where the frst equalty follows from (6) whle the second one follows from Proposton 3. The frst nequalty follows snce Π t (V) s constraned to be nonpostve, whle the last equalty uses the fact that V P L does not have the constrants Π t (V) 0. Usng (7) and the fact that Π λ t (V) appears n the objectve functon of a mnmzaton problem, 12

14 we have V P L = mn π t + λ,π,v t s.t. π t j v 1 (r 1 ) p jt u j[λ jt + v,t+1 (r 1) v,t+1 (r )] + v,t+1 (r ) v t (r ) λ jt = f j j t, j t,, r R, u U (r ) λ jt 0 t, j, j; v,τ+1 ( ) = 0. Lettng π t = θ t θ,t+1 wth θ,τ+1 = 0, the above objectve functon becomes θ 1 + v 1 (r 1 ) whle the frst set of constrants become θ t + v t (r ) j p jtu j [λ jt + v,t+1 (r 1) v,t+1 (r )] + θ,t+1 + v,t+1 (r ). Fnally, lettng ν t (r ) = θ t + v t (r ), we have V P L s.t. = mn λ,ν ν 1 (r 1 ) ν t (r ) j λ jt = f j j p jt u j[λ jt + ν,t+1 (r 1) ν,t+1 (r )] + ν,t+1 (r ) t, j t,, r R, u U (r ) λ jt 0 t, j, j; ν,τ+1 ( ) = 0. The above lnear program s exactly (LR), the lnear programmng formulaton of the Lagrangan relaxaton. So V LR = V P L V P L. Therefore V LR = V P L as we argued the other, easer, drecton earler. 3.3 Polynomal-tme separaton for (P L) The separaton for (P L) can be done by solvng the compact lnear program (SepLR) for a gven set of V varables. If ts optmal objectve functon value Π t ( V) 0 for all t then V s feasble n (P L). If Π t ( V) > 0 for some t, then we fnd a state-acton par (r R, u U(r)) that volates constrant (2) n the followng manner. 13

15 Separaton Algorthm Step 1: Let (ˆλ (0), ŵ (0), ẑ (0) ) be an optmal soluton to (SepLR). Set k = 0. Step 2: Let {ˆr (k) } be as defned n Proposton 4. (k) If, for all j, ˆλ jt ψ,t+1(ˆr (k) (k) ) for all j or ˆλ jt ψ,t+1(ˆr (k) ) for all j, set u j = 1 for all j J 1 and u j = 0 for all j J 2, where J 1 and J 2 are as defned n Proposton 3. Set r = {ˆr (k) } and u = {u j j} and stop. (k) Else, pck a product j such that for j, we have ˆλ jt < ψ,t+1(ˆr (k) ), whle for l j, we (k) have ˆλ ljt > ψ l,t+1(ˆr (k) l ). Let ( λ (k), w (k), z (k) ) be as descrbed n Proposton 4. Step 3: Set ˆλ (k+1) = λ (k), ŵ (k+1) = w (k) and ẑ (k+1) = z (k). Set k = k + 1 and go to Step 2. By Proposton 4, (ˆλ (k), ŵ (k), ẑ (k) ) s an optmal soluton to (SepLR) for all k. Proposton 4 also mples that (ˆλ (k+1), ŵ (k+1), ẑ (k+1) ) has strctly fewer number of bndng constrants of type (13) than (ˆλ (k), ŵ (k), ẑ (k) ). Snce the number of bndng constrants of type (13) n any optmal soluton s at least m and at most r1, Separaton Algorthm termnates n polynomal tme. Fnally, by Proposton 3, u U(r). 4 Network RM wth customer choce falure of the equvalence We consder the network RM problem wth customer choce behavor. In contrast to the ndependent demands settng descrbed earler, customers do not come n wth the ntenton of purchasng a fxed product; rather ther purchasng decson s nfluenced by the set of products that are made avalable for sale. Ths problem appears to be an order of magntude more dffcult to approxmate than the ndependent-class network RM problem. Indeed, n Kunnumkal and Tallur [8] we show that even the affne approxmaton of the dynamc program under the smplest possble choce model, a snglesegment multnomal-logt model, s NP-complete. In choce-based RM, a customer chooses product j wth probablty p j (S), when S s the set of products offered. Note that p j (S) = 0 f j / S and 1 j p j(s) s the probablty that the customer does not choose any of the offered products. Lettng Q(r) = {j 1 [j ] r } denote the set of products that can be offered gven the resource capactes r, the value functons V t ( ) can be obtaned through the optmalty equatons V t (r) = max p j (S)[f j + V t+1 (r e ) V t+1 (r)] + V t+1 (r), S Q(r) j j and the boundary condton s V τ+1 ( ) = 0. 14

16 The value functons can alternatvely be obtaned by solvng the lnear program (CDP LP ) mn V t ( ) s.t V 1 (r 1 ) V t (r) j p j (S)[f j + V t+1 (r j e ) V t+1 (r)] + V t+1 (r) V τ+1 ( ) = 0. t, r R, S Q(r) Computng the value functons ether through the optmalty equatons or the lnear program s ntractable. In the followng sectons, we descrbe extensons of the pecewse-lnear and Lagrangan relaxaton approaches to choce-based RM. 4.1 Pecewse-lnear approxmaton The lnear program from usng a separable pecewse-lnear approxmaton to the value functon V t (r) v t(r ) for all r R s (CP L) V CP L = mn v s.t v 1(r 1 ) v t(r ) j p j (S) [ f j + j {v,t+1 (r 1) v,t+1 (r )} ] (18) + v,t+1 (r ) t, r R, S Q(r) v,τ+1 ( ) = 0. Messner and Strauss [11] propose the pecewse-lnear approxmaton for choce-based network RM. In order to make the formulaton more tractable, they use an aggregaton over the state varables to reduce to number of varables. 4.2 Lagrangan Relaxaton A natural extenson of the Lagrangan relaxaton approach to choce-based RM s to use product and tme-specfc Lagrange multplers to decompose the network problem nto a number of sngle resource problems. We show that the Lagrangan relaxaton approach usng product and tmespecfc multplers turns out to be weaker than the pecewse-lnear approxmaton. The number of Lagrange multplers that we use appears to be crucal to get the equvalence between the Lagrangan relaxaton and the pecewse-lnear approxmaton Lagrangan relaxaton usng product-specfc multplers A natural extenson of the Lagrangan relaxaton approach to the choce-based network RM problem s to use product and tme-specfc Lagrange multplers {λ jt t, j, j} to decompose the network problem nto a number of sngle resource problems. Lettng Q (r ) = {j 1 [j ] r }, we solve the 15

17 optmalty equaton ϑ λ t(r ) = max p j (S)[λ jt + ϑ λ,t+1(r 1) ϑ λ,t+1(r )] + ϑ λ,t+1(r ) S Q (r ) j for resource. It s possble to show that V λ t (r) = ϑλ t (r ) s an upper bound on V t (r). We fnd the tghtest upper bound on the optmal expected revenue by solvng V CLRp = mn {λ λ λ V1 (r 1 ). jt 0, j λjt=fj j,t} In contrast to the ndependent demands settng, the example below llustrates that we can have V CP L < V CLRp Example where V CP L < V CLRp Consder a network RM problem wth two products, two resources and a sngle tme perod n the bookng horzon. The frst product uses only the frst resource, whle the second product uses only the second resource, and we have a sngle unt of capacty on each resource. Note that n the arlne context, ths example corresponds to a parallel flghts network. The revenues assocated wth the products are f 1 = 10 and f 2 = 1. The choce probabltes are gven n Table 1. In the Lagrangan relaxaton, snce we have Lagrange multplers only for j, we have only two multplers λ 1,1,1 and λ 2,2,1. Moreover, the constrant j λ jt = f j mples that all feasble Lagrange multplers satsfy λ 1,1,1 = f 1 and λ 2,2,1 = f 2. Notng that there s only a sngle tme perod n the bookng horzon and that Q (1) = J for = 1, 2, ϑ λ 11(1) = max S J p 1(S)f 1 = 5 and so that V CLRp = 65/11. ϑ λ 21(1) = max S J p 2(S)f 2 = 10/11 Lettng S 1 = {1}, S 2 = {2} and S 3 = {1, 2}, the lnear program assocated wth the pecewselnear approxmaton s V CP L = mn v v 11 (1) + v 21 (1) s.t (CP L) v 11 (1) + v 21 (1) max{p 1 (S 1 )f 1, p 2 (S 2 )f 2, p 1 (S 3 )f 1 + p 2 (S 3 )f 2, 0} v 11 (1) + v 21 (0) max{p 1 (S 1 )f 1, 0} v 11 (0) + v 21 (1) max{p 2 (S 2 )f 2, 0} v 11 (0) + v 21 (0) 0. Note that the frst, second, thrd and fourth constrants correspond to the states vectors [1, 1], [1, 0], [0, 1] and [0, 0], respectvely. It s easy to verfy that an optmal soluton to (CP L) s v 11 (1) = 5, v 11 (0) = 10/11, v 21 (1) = 0, v 21 (0) = 0 and we have V CP L = 5 < V CLRp. Therefore, the Lagrangan relaxaton approach wth product and tme-specfc Lagrange multplers s weaker than the pecewse-lnear approxmaton for choce-based network RM. 16

18 S p 1 (S) p 2 (S) {1} 1/2 0 {2} 0 10/11 {1, 2} 1/12 10/12 Table 1: Choce probabltes for the example where V CP L < V CLRp. 4.3 Lagrangan relaxaton usng offer set-specfc multplers Next we consder a Lagrangan relaxaton approach wth an expanded set of multplers. We ntroduce some notaton frst. For a resource and offer set S, we wrte S f there exsts a product j S wth j. We nterpret S n a smlar fashon. Lettng R(S) = j p j (S)f j (19) be the revenue assocated wth offer set S, we let λ St be the porton allocated to resource S and λ ϕst = R(S) S λ St denote the dfference between the revenue assocated wth the offer set S and the allocatons across the resources. Gven a set of Lagrange multplers {λ ϕst, λ St t, S, S}, we solve the optmalty equaton ϑ λ t(r ) = max 1 [S ]λ St + p j (S)[ϑ λ,t+1(r 1) ϑ λ,t+1(r )] + ϑ λ,t+1(r ) S Q (r ) j for resource, wth the boundary condton that ϑ λ,τ+1 ( ) = 0. Lettng ϑ λ ϕt = max S 2 J λ ϕst + ϑ λ ϕ,t+1, where 2 J denotes the collecton of all subsets of J and ϑ λ ϕ,τ+1 = 0, t s possble to show that Vt λ (r) = ϑλ t (r ) + ϑ λ ϕt s an upper bound on V t(r). We fnd the tghtest upper bound on the optmal expected revenue by solvng (say as a lnear program) V CLRo = mn {λ λ ϕst + V S λ 1 λ (r 1 ). St=R(S) S,t} In Kunnumkal and Tallur [9] we show the followng usng a more elaborate verson of the mnmalbndng-constrants argument employed here: Proposton 5. ([9]) V CP L = V CLRo. 4.4 Number of Lagrange multplers Compared to the ndependent demands case consdered prevously, the dfference n the choce settng s that the revenue assocated wth an offer set S, R(S), s much more complcated than before and s not naturally gven as a sum of expected revenues from each product. Ths s what precludes equvalence wth product-specfc multplers. In fact, requrng S λ St = R(S) can be overly restrctve also and we do not necessarly have V CP L = V CLRo, f we mpose ths constrant on the Lagrange multplers. We llustrate wth an 17

19 example. Consder a network revenue management problem wth two products, two resources and a sngle tme perod n the bookng horzon. The frst product uses only the frst resource, whle the second product uses only the second resource, and we have a sngle unt of capacty on each resource. Note that n the arlne context, ths example corresponds to a parallel flghts network. S p 1 (S) p 2 (S) {1} 50/99 0 {2} 0 51/101 {1, 2} 1/2 1/2 Table 2: Choce probabltes for the example V CP L < mn λ S λ St=R(S), S,t V λ 1 (r 1 ). The revenues assocated wth the products are f 1 = 99 and f 2 = 101. The choce probabltes are gven n Table 2. Lettng S 1 = {1}, S 2 = {2} and S 3 = {1, 2}, we have R(S 1 ) = 50, R(S 2 ) = 51 and R(S 3 ) = 100. If we mpose the constrant S λ St = R(S) on the Lagrange multplers, then we have λ 1,S1,1 = 50, λ 2,S2,1 = 51 and λ 1,S3,1 + λ 2,S3,1 = 100 for all feasble Lagrange multplers. We have ϑ λ 11(1) = max{50, λ 1,S3,1} and ϑ λ 21(1) = max{51, λ 2,S3,1}. It can be verfed that mn {λ λ 1,S1,1=50,λ 2,S2,1=51,λ 1,S3,1+λ 2,S3,1=100} ϑλ 11(1) + ϑ λ 21(1) = 101 > V CP L = 100. The optmal soluton can be reached by the Lagrangan f we ntroduce a multpler λ ϕst for each S: Set λ ϕ,s1,t = λ ϕ,s2,t = λ ϕ,s3,t = 1 n the above problem, and the Lagrangan relaxaton V CLRo acheves the value 100. Therefore, the choce as well as the number of Lagrangan multplers seems crtcal and specfc to each problem. 5 Numercal results As the separaton problem for (P L) s solvable n polynomal tme, a plausble soluton procedure s by lnear programmng, generatng the constrants on the fly. In ths secton, we nvestgate how (P L) compares (LR). By our theoretcal result both should gve the same objectve functon value (as the numercal results ndeed show), so our man nterest s n comparng soluton tmes. We consder a hub and spoke network wth a sngle hub that serves N spokes. There s one flght from the hub to each spoke and one flght from each spoke to the hub. The total number of flghts s 2N. Note that the flght legs correspond to the resources n our network RM formulaton. We have hgh fare-product and low fare-product connectng each orgn-destnaton par. Consequently, there are 2N(N + 1) fare-products n total. In all of our test problems, the hgh fare-product connectng an orgn-destnaton par s twce as expensve as the correspondng low fare-product. We measure the tghtness of the flght leg capactes by α = t r1 j p jt where the numerator measures the total expected demand over the flght legs and the denomnator gves the sum of the capactes of the flght legs. We ndex the test problems usng (τ, N, α), where τ s the number of perods n the bookng horzon and N and α are as defned above. We use τ {25, 50, 100}, N {2, 3, 4} and α {1.0, 1.2, 1.6} so that we get a total of 27 test problems. We note that our test problems are adapted from those n Topaloglu [15]., 18

20 We use constrant generaton to solve (P L). By Lemma 1 we can add constrants of the form v t (r ) v t (r 1) v t (r + 1) v t (r ) for all t, and r R to (P L) wthout affectng ts optmal objectve functon value. So, whle solvng (P L), we start wth a lnear program that only has the above mentoned constrants and the nonnegatvty constrants. We add constrants of type (2) on the fly by solvng the separaton problem descrbed n 3.1. By Proposton 3 and Lemma 3, we can solve the separaton problem as a lnear program. We add volated constrants to (P L) and stop when we are wthn 1% of optmalty. We use constrant generaton to solve (LR) as well. It can be verfed that there exsts an optmal soluton {ˆν t (r ) t,, r R } to (LR) that satsfes ˆν t (r ) ˆν t (r 1) ˆν t (r + 1) ˆν t (r ) for all t, and r R. Whle solvng (LR), we start wth a lnear program that only has the above mentoned constrants n addton to constrants (5) and the nonnegatvty constrants. We add constrants ( of type (4) on the fly by solvng the followng separaton problem. Gven a soluton {ˆλjt t, j, j}, {ˆν t (r ) t,, r R } ) to the restrcted lnear program, we check for each and t f max p jt u j[ˆλ jt + ˆν,t+1 (r 1) ˆν,t+1 (r )] + ˆν,t+1 (r ) ˆν t (r ) r R,u U (r ) j s greater than zero. Note that the separaton problem for (LR) s easy to solve snce for r R \{0}, the maxmum s attaned by settng u j = 1 [ˆλ jt+ˆν,t+1(r 1) ˆν,t+1(r )>0] for j. On the other hand, f r = 0, the only feasble soluton s u j = 0 for all j. We add volated constrants to (LR) and stop when we are wthn 1% of optmalty. Table 3 gves the objectve functon values of (P L) and (LR) and the CPU seconds when they are solved to wthn 1% of optmalty. We solve the test problems usng CPLEX 11.2 on a Pentum Core 2 Duo PC wth 3 GHz CPU and 4 GB RAM. The frst column gves the characterstcs of the test problem n terms of (τ, N, α). The second column gves the objectve functon value of (P L), whle the thrd column gves the CPU seconds requred by (P L). The fourth and ffth columns do the same thng, but for (LR). Comparng the second and fourth columns, we see that the objectve functon values of (P L) and (LR) are very close; the dfferences are wthn 1%. On the other hand, the soluton tmes for (P L) on the test problems wth a relatvely large number of spokes and tme perods can be orders of magntude greater than (LR). We note that t s possble to solve both (P L) and (LR) more effcently; see [4] and [15]. Our goal here s to smply compare the objectve functon values and soluton tmes of comparable mplementatons of both methods. However, our results are n broadly n lne wth the computatonal studes reported [4] and [15]. 6 Conclusons We make the followng research contrbutons n ths paper: (1) We show that the approxmate dynamc programmng approach wth pecewse-lnear bass functons (Faras and van Roy [4]) and the Lagrangan relaxaton approach (Topaloglu [15]) are n fact equvalent. Ths result shows that there mght be surprsng connectons between the approxmate dynamc programmng approach and the Lagrangan relaxaton approach for complcated dynamc programs, and one can beneft from unfyng forces as t were. (2) We show that the separaton problem for the pecewse-lnear approxmaton s solvable n polynomal-tme. (3) We show that there exsts a separable concave approxmaton that yelds the tghtest upper bound among all separable pecewse-lnear approxmatons to the value functon. Ths mples that the Lagrangan relaxaton approach obtans the tghtest upper bound among all separable pecewse-lnear approxmatons that are upper bounds on the value functon. 19

21 Problem (P L) (LR) (τ, N, α) V P L CPU V LR CPU (25, 2, 1.0) (25, 2, 1.2) (25, 2, 1.6) (25, 3, 1.0) (25, 3, 1.2) (25, 3, 1.6) (25, 4, 1.0) 1, ,188 1 (25, 4, 1.2) 1, ,048 1 (25, 4, 1.6) (50, 2, 1.0) 1, ,306 1 (50, 2, 1.2) 1, ,117 1 (50, 2, 1.6) (50, 3, 1.0) 2, ,038 2 (50, 3, 1.2) 1, ,845 2 (50, 3, 1.6) 1, ,500 1 (50, 4, 1.0) 2,496 1,556 2,497 6 (50, 4, 1.2) 2, ,263 4 (50, 4, 1.6) 1, ,856 3 (100, 2, 1.0) 3,652 2,149 3, (100, 2, 1.2) 3,242 1,409 3, (100, 2, 1.6) 2, ,603 8 (100, 3, 1.0) 5,529 17,821 5, (100, 3, 1.2) 4,967 9,314 4, (100, 3, 1.6) 4,131 4,000 4, (100, 4, 1.0) 6, ,297 6, (100, 4, 1.2) 6,141 51,708 6, (100, 4, 1.6) 4,910 12,250 4, Table 3: Comparson of the upper bounds and soluton tmes of (P L) and (LR) both solved to 1% of optmalty by lnear programmng. 20

22 Our proof technque uses a novel mnmal-bndng-constrants argument. We gve heurstc condtons for the mnmal-bndng-constrants technque to work. The checklst s based on a smple calculus-based argument whch provdes ntuton and may be useful when applyng a smlar lne of reasonng to other classes of loosely-coupled dynamc programs, n the framework of Hawkns [6]. Indeed, we sketch n the appendx how the equvalence result extends to network RM wth overbookng under some assumptons on the dened-boardng cost functon. We show that the result does not drectly extend to choce-based network RM and dscuss why our nsght s that the number of Lagrange multplers plays a crtcal role n the equvalence, requrng problem-specfc customzaton. As to computatonal mpact, we solved the pecewse-lnear approxmaton usng the lnearprogrammng based separaton we descrbe n ths paper, but our results suggest that solvng the Lagrangan relaxaton s stll faster. Ths s n lne wth the computatonal studes reported n Faras and van Roy [4], and not too surprsng as solvablty by separaton s based on the ellpsod algorthm that s well known to be slow n practce. Tong and Topaloglu [14] make a smlar observaton for the affne relaxaton of Adelman [1]; they also fnd that the Lagrangan relaxaton approach descrbed n Kunnumkal and Topaloglu [10] s more effcent. Improvng the effcency of the separaton, say by a faster, combnatoral, algorthm, would be an nterestng area for future research. 21

23 References [1] Adelman, D Dynamc bd-prces n revenue management. Operatons Research 55(4) [2] Adelman, D, A. J. Mersereau Relaxatons of weakly coupled stochastc dynamc programs. Operatons Research 55(3) [3] Erdely, A., H. Topaloglu A dynamc programmng decomposton method for makng overbookng decsons over an arlne network. INFORMS Journal on Computng. [4] Faras, V. F., B. van Roy An approxmate dynamc programmng approach to network revenue management. Tech. rep., Sloan School of Management, MIT, Massachussetts, MA. [5] Grötschel, M., L. Lovász, A. Schrjver Geometrc Algorthms and Combnatoral Optmzaton, vol. 2. Sprnger. [6] Hawkns, J A Lagrangan decomposton approach to weakly coupled dynamc optmzaton problems and ts applcatons. Phd dssertaton, Massachusetts Insttute of Technology, Cambrdge, MA. [7] Karaesmen, I., G. J. van Ryzn Coordnatng overbookng and capacty control decsons on a network. Tech. rep., Graduate School of Busness, Columba Unversty, New York, NY. Workng paper. [8] Kunnumkal, S., K. T. Tallur A new compact lnear program formulaton for choce network revenue management. Tech. rep., Unverstat Pompeu Fabra. [9] Kunnumkal, S., K. T. Tallur Pecewse-lnear approxmatons for choce network revenue management (n preparaton). Tech. rep., Unverstat Pompeu Fabra. [10] Kunnumkal, S., H. Topaloglu Computng tme-dependent bd prces n network revenue management problems. Transportaton Scence [11] Messner, J., A. K. Strauss Network revenue management wth nventory-senstve bd prces and customer choce. European Journal of Operatonal Research 216(2) [12] Tallur, K. T On bounds for network revenue management. Tech. Rep. WP-1066, UPF. [13] Tallur, K. T., G. J. van Ryzn The Theory and Practce of Revenue Management. Kluwer, New York, NY. [14] Tong, C., H. Topaloglu On approxmate lnear programmng approach for network revenue management problems. Tech. rep., ORIE, Cornell Unversty. [15] Topaloglu, H Usng Lagrangan relaxaton to compute capacty-dependent bd prces n network revenue management. Operatons Research [16] Vossen, T. W. M., D Zhang A dynamc dsaggregaton approach to approxmate lnear programs for network revenue management. Tech. rep., Leeds School of Busness, Unversty of Colorado at Boulder, Boulder, CO. [17] Zhang, D., D. Adelman An approxmate dynamc programmng approach to network revenue management wth customer choce. Transportaton Scence 43(3)