On a piecewise-linear approximation for network revenue management

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1 On a pecewse-lnear approxmaton for network revenue management Sumt Kunnumkal Kalyan Tallur January 31, 2014 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. In ths paper we show that the pecewse-lnear approxmaton to the network RM dynamc program s polynomal-tme solvable; specfcally we show that the separaton problem can be solved a lnear program. Moreover, the resultng compact formulaton of the approxmate dynamc program turns out to be exactly equvalent to the Lagrangan relaxaton, an earler heurstc method proposed for the same problem. We perform numercal comparson of solvng the problem by generatng separatng cuts or as our compact lnear program. We dscuss extensons to versons of the network RM problem wth overbookng as well as the dffcultes of extendng t to the choce model of network revenue RM. 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) nvolves makng the acceptance decsons based on the product demands as well as the resource capactes for the entre network. Chapter 3 of Tallur and van Ryzn [18] 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 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

2 result, researchers have focused on developng approxmaton methods. In ths paper we show that the pecewse-lnear approxmaton to the network RM dynamc program s polynomal-tme solvable; specfcally we show that the separaton problem can be solved a lnear program. Moreover, the resultng compact formulaton of the approxmate dynamc program turns out to be exactly equvalent to the Lagrangan relaxaton, an earler heurstc method proposed for the same problem. We perform a numercal comparson of solvng the problem by generatng separatng cuts or as our compact lnear program. We dscuss extensons to versons of the network RM problem wth overbookng as well as the dffcultes of extendng t to the choce model of network revenue RM. 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 prevously proposed Lagrangan relaxaton approach. In 3 we gve the man body of proofs showng that the lnear programmng formulaton of the pecewse-lnear approxmaton s polynomal-tme separable. In 4 we dscuss a choce-model based network RM problem where we show 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 solvng the pecewse-lnear approxmaton by on-the-fly separaton or as a compact lnear program. 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. 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. 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 [20] and Kunnumkal and Topaloglu [11] take ths approach. Topaloglu [20] uses Lagrange multplers to relax the constrants that lnk the product acceptance decsons across the resources, whle Kunnumkal and Topaloglu [11] relax the resource capacty constrants usng Lagrange multplers. Computatonal results from Topaloglu [20] ndcate that Lagrangan relaxaton wth product-specfc Lagrange multplers 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. In a recent paper, Tong and Topaloglu [19] establsh the equvalence between the affne approxmaton 2

3 of Adelman [1] and the Lagrangan relaxaton of Kunnumkal and Topaloglu [11]. A by-product of the results of ths paper s that the product and tme-specfc Lagrangan relaxaton of Topaloglu [20] and the pecewse-lnear approxmaton represent the same lnear program. Network RM ncorporatng more realstc models of customer behavor, as customers choosng from an offer set, have recently become popular after the sngle-resource model proposed n Tallur and van Ryzn [17]; see Gallego, Iyengar, Phllps, and Dubey [6], Lu and van Ryzn [13], Kunnumkal and Topaloglu [12], Messner and Strauss [14], Bodea, Ferguson, and Garrow [3], Bront, Méndez- Díaz, and Vulcano [4], Méndez-Díaz, Bront, Vulcano, and Zabala [15] for subsequent extensons to network RM. 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. Kunnumkal and Tallur [9] show that even the affne approxmaton of the dynamc program under the smplest possble choce model, a sngle-segment multnomal-logt model, s NP-complete. 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 The underlyng network has m resources (usually ndexed by ) and n products (usually ndexed by j), and we refer to the set of all resources as I and the set of all products as J. A product j uses a subset of resources I j I, and ts sale brngs n revenue f j. A resource s used by a subset J J of products. A resource s sad to be n product j ( I j ) f j uses resource, and conversely we wrte j J. We use 1 [ ] as the ndcator functon, 1 f true and 0 f false. The resources used by product j are represented by the 0-1 ncdence vector 1 [Ij ], whch has a 1 n the th poston f I j and a 0 otherwse. Bookng requests for products come n over tme and we let p j,t 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 j,t > 0 for all j. Note that ths s wthout loss of generalty because f p j,t = 0 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 j,t = 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 frm 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. 3

4 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 and u t = [u t j ] represents a vector of controls at the network level at tme t. We let u t j {0, 1} ndcate the acceptance decson 1 f we accept product j 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, I 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 j,t u j [f j + V t+1 (r e ) V t+1 (r)] + V t+1 (r), (1) u U(r) I 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 V τ+1 ( ) = 0. p j,t u j [f j + V t+1 (r I j e ) V t+1 (r)] + V t+1 (r) t, r R, u U(r) 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. 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]. 4

5 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 j,t u j [f j + I 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 [7]). 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. Ths decreasng margnal value property turns out to be crucal n showng the polynomal-tme solvablty of the separaton problem for the pecewse-lnear approxmaton. 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. 2.4 Lagrangan Relaxaton Topaloglu [20] 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. The dea s to wrte the network RM problem n terms of resource-level controls and then relax the constrants that lnk ] represent a control vector assocated wth resource at tme t, wth u t j {0, 1} representng the acceptance decson on resource 1 f we accept product j on resource at tme t and 0 otherwse, the network RM problem requres that the resource-level decsons u t j be coordnated across all resources I j. That s, ether u t j = 0 for all I j or u t j = 1 for all I j. Topaloglu [20] relaxes the constrants that lnk the resource-level decsons by assocatng Lagrange multplers wth them. the resource-level decsons together. Lettng u t = [u t j 5

6 Let {λ,j,t t, j, I j } denote a set of Lagrange multplers and U (r ) = {u {0, 1} n u j r, j J } denote the set of feasble controls on resource when ts capacty s r. We solve the optmalty equaton ϑ λ,t(r ) = max p j,t u j[λ,j,t + ϑ λ,t+1(r 1) ϑ λ,t+1(r )] + ϑ λ,t+1(r ) u U (r ) j J for resource, wth the boundary condton ϑ λ,τ+1 ( ) = 0. We note that ϑλ,t ( ) can be nterpreted as the value functon assocated wth a sngle resource RM problem where {λ,j,t t, j J } are the revenues of the products. Collectng the sngle resource value functons together, we let τ Vt λ (r) = ϑ λ,t(r ). (3) t =t j p j,t [f j I j λ,j,t ] + + We can nterpret V λ t (r) as an approxmaton to V t (r) and ndeed t s possble to show that V λ t (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 [16] shows that the optmal Lagrange multplers satsfy I j λ,j,t = f j for all j and t. Proposton 1. There exsts {ˆλ,j,t t, j, I j } arg mn λ V1 λ (r 1 ) that satsfy ˆλ,j,t 0 and I ˆλ,j,t j = 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 ). I λ j,j,t=f j,λ,j,t 0, t,j, I j} Usng Proposton 1, we can nterpret the Lagrange multpler λ,j,t 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,j,t j 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 (5) as below: (LR) V LR = mn λ,ν s.t. ν,1 (r 1 ) ν,t (r ) j J p j,t u j[λ,j,t + ν,t+1 (r 1) ν,t+1 (r )] +ν,t+1 (r ) t,, r R, u U (r ) (4) I j λ,j,t = f j t, j (5) λ,j,t 0 t, j, I j ; ν,τ+1 ( ) = 0. 6

7 The lnear programmng formulaton (LR) turns out to be useful when comparng the Lagrangan relaxaton approach wth the pecewse-lnear approxmaton. 3 Polynomal-tme solvablty of the pecewse-lnear approxmaton In ths secton we show that the lnear programmng formulaton of the pecewse-lnear approxmaton s separable n polynomal tme. A byproduct of our result s that the Lagrangan relaxaton s equvalent to the pecewse-lnear approxmaton, n that they yeld the same upper bound on the value functon. 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 3.3. 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. As a result, we can use these Lagrange multplers to dsaggregate constrant (2) of (P L) nto I constrants, one for each resource. The dsaggregated resource level constrants essentally have the same form as constrants (4) of (LR), whch n turn can be used to show that V LR V P L. 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 descrbe a polynomal-tme separaton algorthm for (P L) n 3.2. We buld on these results and gve a formal proof of Proposton 2 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. [7]. 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 followng separaton problem: Prove that V satsfes all the constrants (2), and f not, fnd a volated constrant and add t to the lnear program. 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 r R,u U(r) p j,t u j [f j ψ,t+1 (r )] + j I j,t (r ). (6) 7

8 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,j,t, so that λ,j,t represents the revenue allocated to resource I j at tme t. Naturally, we have I j λ,j,t = f j and λ,j,t 0 for all I j and for all j. Followng Topaloglu [20], we can alternatvely nterpret {λ,j,t I j } as Lagrange multplers assocated wth the constrants that lnk the acceptance decsons for product j across the resources that t consumes. Gven such a set of Lagrange multplers, we solve the problem Π λ,t( V) = max p j,t u j[λ,j,t ψ,t+1 (r )] +,t (r ) (7) r R,u U (r ) j J for each resource. The followng lemma states that Πλ,t ( V) s an upper bound on Φ t ( V). Lemma 2. If {λ,j,t t, j, I j } satsfy I j λ,j,t = f j and λ,j,t 0 for all t, j and I 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) I λ j,j,t=f j j;λ,j,t 0 j, I 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 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 I j ψ,t+1 (r ) > 0. Therefore, we can wrte u j [f j I j ψ,t+1 (r )] n the objectve functon as [f j I 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 8

9 as the decson varables nstead of the r s. Therefore, we can wrte a relaxed contnuous verson of problem (6) as Φ t ( V) = max p j,t [f j ψ,t+1 ] + +,t (ψ,t+1 ). (9) ψ j I 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 J p j,t 1 [ f j k I j ˆψk,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,j,t = f j j, I j, k I ˆψk,t+1 j and note that they are feasble to problem (8). We have ˆλ,j,t ˆψ,t+1 = [f j k Ij ˆψ,t+1 ˆψ k,t+1 ]. (11) k I ˆψk,t+1 j Snce the rato on the rght hand sde s postve, ths mples 1 [ˆλ,j,t ˆψ,t+1 >0] = 1[ f j k I j ˆψk,t+1 >0]. Snce, { ˆψ,t+1 } satsfes (10), t also satsfes j J p j,t 1 [ˆλ,j,t ˆψ,t+1>0] +,t( ˆψ,t+1 ) = 0 (12) whch s the frst order condton assocated wth an optmzer of max ψ j J p j,t [ˆλ,j,t ψ,t+1 ] + +,t (ψ,t+1 ). Examnng (12) we see that the frst term s always negatve and decreasng so the slope of, has to be postve at ths pont. But the term j J p j,t [ˆλ,j,t ψ,t+1 ] + s decreasng n ψ, so the can cross t wth postve slope only once. That s, { ˆψ,t+1 } s an optmzer of max ψ j J p j,t [ˆλ,j,t ψ,t+1 ] + +,t (ψ,t+1 ). So a relaxed contnuous verson of problem (7) can be wrtten as Πˆλ,t ( V) = max p j,t [ˆλ,j,t ψ,t+1 ] + +,t (ψ,t+1 ) = p j,t [ˆλ,j,t ˆψ,t+1 ] + +,t ( ˆψ,t+1 ), (13) ψ j J j J where the last equalty follows from above observatons. Puttng everythng together, we have Π t ( V) Πˆλ,t ( V) = { p j,t [ˆλ,j,t ˆψ,t+1 ] + +,t ( ˆψ },t+1 ) j J = p j,t [ˆλ,j,t ˆψ,t+1 ] + +,t ( ˆψ,t+1 ) = Φ t ( V), j I j 9

10 where the frst nequalty holds snce {ˆλ,j,t j, I 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,j,t j,, j} coordnate the decsons for each product across the dfferent resources: product j s accepted on resource I j only f ˆλ,j,t ˆψ,t+1 > 0. By (11), ether ˆλ,j,t ˆψ,t+1 > 0 for all I j or ˆλ,j,t ˆψ,t+1 0 for all I j. That s, we ether accept the product on all the resources t consumes or reject the product on all the resources t consumes. Fxng the Lagrange multplers as ˆλ, we clam ths s an optmal soluton to (7) that happens at an nteger value of r. Note that (7) s composed of two pecewse-lnear functons, the functon that has breakponts only at nteger r and the frst part whch as r ncreases has ψ,t+1 (r ) decreasng, and hence [ˆλ,j,t ψ,t+1 (r )] + ncreasng wth ncreasng r. If we decrease ψ contnuously n j J p j,t [ˆλ,j,t ψ] +, we agan obtan a contnuous pecewse-lnear functon wth possble nonnteger breakponts that moreover has ncreasng slope after each new breakpont (.e. s convex between the nteger breakponts). So between two nteger r, as t s a sum of a lnear functon and a pece-wse lnear convex functon, the maxmzer to the one-dmensonal relaxed problem max ψ j J p j,t [ˆλ,j,t ψ,t+1 ] + +,t (ψ,t+1 ) should happen at an nteger r. We therefore have Φ t ( V) = Φ t ( V) = Π t ( V) f the optmal ψ 1,t+1 ( ˆψ) s an nteger. 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 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 z,j,t,r +,t (r) r R j J z,j,t,r p j,t [λ,j,t ψ,t+1 (r)] j J, r R z,j,t,r 0 j J, r R. Lemma 3. The lnear program (SepLR ) s equvalent to (7). Proof. Appendx. 10

11 We can, therefore, formulate problem (8) as the lnear program (SepLR) Π t ( V) = mn λ,w,z s.t. w,t w,t j J z,j,t,r +,t (r), r R (14) z,j,t,r p j,t [λ,j,t ψ,t+1 (r)], j J, r R (15) I j λ,j,t = f j j (16) λ,j,t 0, j J (17) z,j,t,r 0, j J, r R. (18) Wth a slght abuse of notaton, we let (λ, w, z) = ( {λ,j,t j, I j }, {w,t }, {z,j,t,r, j J, r R } ) denote a feasble soluton to (SepLR). Let ξ,t (r) = w,t [ j J z,j,t,r +,t (r)] denote the slack n constrant (14), and B,t (λ, w, z) = {r R ξ,t (r) = 0} denote the set of bndng constrants of type (14) and B,t c (λ, w, z) denote ts complement. Note that f (ˆλ, ŵ, ẑ) s an optmal soluton, then B,t (ˆλ, ŵ, ẑ) s nonempty, snce for each resource, there exsts some r R such that constrant (14) 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,t(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,t (ˆλ, ŵ, ẑ). I B,t (ˆλ, ŵ, ẑ ) I Proposton 4. There exsts an optmal soluton (ˆλ, ŵ, ẑ) to (SepLR) wth a mnmal set of bndng constrants and a vector ˆr wth ˆr B,t (ˆλ, ŵ, ẑ) for all, such that for each j, we ether have ˆλ,j,t ψ,t+1 (ˆr ) for all I j or ˆλ,j,t ψ,t+1 (ˆr ) for all I j. Proof. Appendx. We remark that Proposton 4 formalzes the ntuton that there exsts a set of optmal Lagrange multplers that perfectly coordnate the acceptance decsons for each product across the resources t consumes. Indeed, f for product j, we have ˆλ,j,t ψ,t+1 (ˆr ) for all I j, then f j = I ˆλ,j,t j I j ψ,t+1 (ˆr ). Therefore, 1 [ˆλ,j,t ψ,t+1(ˆr )>0] = 1[ f j ]. An mplcaton of ths, k I ψ k,t+1 (ˆr k )>0 j as the followng arguments show, s that the acceptance decsons for product j are exactly the same for all I j. By Lemma 2, Φ t ( V) Π t ( V). We show below that Φ t ( V) Π t ( V), whch completes the proof. 11

12 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 j,t [ˆλ,j,t ψ,t+1 (ˆr )] + +,t (ˆr ), j J j I j where the second equalty holds snce ˆr B,t (ˆλ, ŵ, ẑ) for all. The last equalty holds snce f ẑ,j,t,ˆr > p j,t [ˆλ,j,t 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,j,t ψ,t+1 (ˆr ) I j } and J 2 = J \J 1 where J = {1,..., n}. By Proposton 4, every product j J 2 satsfes ˆλ,j,t ψ,t+1 (ˆr ) for all I j. Therefore, Π t ( V) = p j,t [ˆλ,j,t ψ,t+1 (ˆr )] +,t (ˆr ) j J 1 I j = p j,t û j [ˆλ,j,t ψ,t+1 (ˆr )] +,t (ˆr ) j I j = j p j,t û j [f j I j ψ,t+1 (ˆr )] +,t (ˆr ) 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 (16). 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 I j. On the other hand, snce ˆλ,j,t s fnte and ψ,t+1 (0) =, we have ˆr 1 for all j J 1 and I j. It follows that û j ˆr for all j J 1 and I j ; so the 0-1 controls û = {û j } satsfy the constrant that û j = 0 f ˆr = 0 for any I j. 3.2 Polynomal-tme separaton for (P L) Proposton 3 mples that 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. Separaton Algorthm: If Π t ( V) 0 Then stop, Else: 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,j,t ψ,t+1(ˆr (k) (k) ) for all I j or ˆλ,j,t ψ,t+1(ˆr (k) ) for all I 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 I j, we have ˆλ,j,t < ψ,t+1(ˆr (k) ), whle for l I j, (k) we have ˆλ l,j,t > 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. 12

13 Proposton 5. The separaton for (P L) can be carred out n polynomal tme. Proof. The lnear program (SepLR) can be solved n polynomal tme. If Π t ( V) 0, then Separaton Algorthm termnates mmedately and we are done. Otherwse, 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 (14) than (ˆλ (k), ŵ (k), ẑ (k) ). Snce the number of bndng constrants of type (14) 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). Therefore, we have obtaned a state-acton par (r, u) that volates constrant (2) n polynomal tme. 3.3 Proof of Proposton 2 We frst show that V P L V LR. Consder a feasble soluton ( {ˆλ,j,t t, j, I 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 j,t u j [ˆλ,j,t + ˆν,t+1 (r 1) ˆν,t+1 (r )] + ˆν,t+1 (r ) j J = j p j,t u j [f j + I j ˆν,t+1 (r 1) ˆν,t+1 (r )] + ˆν,t+1 (r ) where the equalty holds snce I j ˆλ,j,t = 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,1 (r 1 ) = mn v,1 (r 1 ) mn v,1 (r 1 ) + } Π t (V) V P L, {v Φ t (V) 0} {v Π t (V) 0} {v Π t (V) 0} t 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, we have V P L = mn λ,π,v s.t. π,t + t v,1 (r 1 ) π,t j J p j,t u j[λ,j,t + v,t+1 (r 1) v,t+1 (r )] + v,t+1 (r ) v,t (r ) I j λ,j,t = f j t, j t,, r R, u U (r ) λ,j,t 0 t, j, I 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 J p j,t u,j [λ,j,t + v,t+1 (r 1) 13

14 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 J p j,t u j[λ,j,t + ν,t+1 (r 1) ν,t+1 (r )] + ν,t+1 (r ) I j λ,j,t = f j t, j t,, r R, u U (r ) λ,j,t 0 t, j, I 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. 4 Network RM wth customer choce mportance of number of Lagrangan multplers We consder the network RM problem wth customer choce behavor. 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 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) I j and the boundary condton s V τ+1 ( ) = 0. j 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 I 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. 14

15 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 + I j {v,t+1 (r 1) v,t+1 (r )} ] (19) + v,t+1 (r ) t, r R, S Q(r) v,τ+1 ( ) = 0. Messner and Strauss [14] 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 the choce-based network RM problem s to use product and tme-specfc Lagrange multplers {λ,j,t t, j, I j } to decompose the network problem nto a number of sngle resource problems. Lettng Q (r ) = {j 1 [j J] r }, we solve the optmalty equaton ϑ λ,t(r ) = max p j (S)[λ,j,t + ϑ λ,t+1(r 1) ϑ λ,t+1(r )] + ϑ λ,t+1(r ) S Q (r ) j 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 CLR = mn {λ λ,j,t 0, V 1 λ (r 1 ). I λ,j,t =f j j,t} j In contrast to the ndependent demands settng, the Lagrangan relaxaton turns out to be weaker than the pecewse-lnear approxmaton; we gve a smple example n the appendx whch llustrates that we can have V CP L < V CLR. Wth customer choce, the decsons to the offer the dfferent products get nterlnked through the choce probabltes, and the product and tme-specfc Lagrange multplers are not rch enough to ensure coordnaton of the offer sets across the resources. Kunnumkal and Tallur [10] propose a new Lagrangan relaxaton approach usng an expanded set of Lagrange multplers that acheves equvalence to the pecewse-lnear approxmaton. Therefore, the number and form of the Lagrange multplers appear to be crucal to get the equvalence between the Lagrangan relaxaton and the pecewse-lnear approxmaton. 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 15

16 (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 a 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 J p j,t 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 [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 ( {ˆλ,j,t t, j, I j }, {ˆν,t (r ) t,, r R } ) to the restrcted lnear program, we check for each, and t f max r R,u U (r ) j J p j,t u j[ˆλ,j,t + ˆν,t+1 (r 1) ˆν,t+1 (r )] + ˆν,t+1 (r ) ˆν,t (r ) 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,j,t +ˆν,t+1 (r 1) ˆν,t+1 (r )>0] for j J. On the other hand, f r = 0, the only feasble soluton s u j = 0 for all j J. We add volated constrants to (LR) and stop when we are wthn 1% of optmalty. Table 1 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 16

17 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 1: Comparson of the upper bounds and soluton tmes of (P L) and (LR) both solved to 1% of optmalty by lnear programmng. 17

18 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 [20]. Our goal here s to smply compare the objectve functon values and soluton tmes of comparable mplementatons of both methods. 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 s solvable n polynomaltme, (2) The resultng compact lnear program s equvalent to the lnear programmng verson of the Lagrangan relaxaton approach proposed n Topaloglu [20]. 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. (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. We sketch n the appendx how the polynomal-tme solvablty 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, 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, ether as a lnear program or by subgradent optmzaton, s stll faster. Tong and Topaloglu [19] make a smlar observaton for the affne relaxaton of Adelman [1]; they also fnd that the Lagrangan relaxaton approach descrbed n Kunnumkal and Topaloglu [11] s more effcent. Improvng the effcency of the separaton, say by a faster, combnatoral, algorthm, would be an nterestng area for future research. 18

19 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] Bodea, T., M. Ferguson, L. Garrow Choce-based revenue management: Data from a major hotel chan. Manufacturng and Servce Operatons Management [4] Bront, J. J. M., I. Méndez-Díaz, G. Vulcano A column generaton algorthm for chocebased network revenue management. Operatons Research 57(3) [5] Erdely, A., H. Topaloglu A dynamc programmng decomposton method for makng overbookng decsons over an arlne network. INFORMS Journal on Computng. [6] Gallego, G., G. Iyengar, R. Phllps, A. Dubey Managng flexble products on a network. Tech. Rep. TR , Dept of Industral Engneerng, Columba Unversty, NY, NY. [7] Grötschel, M., L. Lovász, A. Schrjver Geometrc Algorthms and Combnatoral Optmzaton, vol. 2. Sprnger. [8] 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. [9] Kunnumkal, S., K. T. Tallur A new compact lnear programmng formulaton for choce network revenue management. Tech. rep., Unverstat Pompeu Fabra. [10] Kunnumkal, S., K. T. Tallur Pecewse-lnear approxmatons for choce network revenue management (n preparaton). Tech. rep., Unverstat Pompeu Fabra. [11] Kunnumkal, S., H. Topaloglu Computng tme-dependent bd prces n network revenue management problems. Transportaton Scence [12] Kunnumkal, S., H. Topaloglu A new dynamc programmng decomposton method for the network revenue management problem wth customer choce behavor. Producton and Operatons Management 19(5) [13] Lu, Q., G. J. van Ryzn On the choce-based lnear programmng model for network revenue management. Manufacturng and Servce Operatons Management 10(2) [14] Messner, J., A. K. Strauss Network revenue management wth nventory-senstve bd prces and customer choce. European Journal of Operatonal Research 216(2) [15] Méndez-Díaz, I., J. Mranda Bront, G. Vulcano, P. Zabala A branch-and-cut algorthm for the latent-class logt assortment problem. Dscrete Appled Mathematcs ((forthcomng)). [16] Tallur, K. T On bounds for network revenue management. Tech. Rep. WP-1066, UPF. [17] Tallur, K. T., G. J. van Ryzn Revenue management under a general dscrete choce model of consumer behavor. Management Scence 50(1) [18] Tallur, K. T., G. J. van Ryzn The Theory and Practce of Revenue Management. Kluwer, New York, NY. 19

20 [19] Tong, C., H. Topaloglu On approxmate lnear programmng approach for network revenue management problems. INFORMS Journal of Computng (to appear). [20] Topaloglu, H Usng Lagrangan relaxaton to compute capacty-dependent bd prces n network revenue management. Operatons Research

21 Appendx Proof of Lemma 1: Our analyss s essentally an adaptaton of analogous structural results for the revenue management problem on a sngle resource (Tallur and van Ryzn [18]). We ntroduce some notaton to smplfy the expressons. Fxng a resource l, we let R l (r l ) = {x R x l = r l } be the set of capacty vectors where the capacty on resource l s fxed at r l. Gven a separable pecewse-lnear approxmaton V = {v,t (r ) t,, r R }, we let { ϵ l,t (r l, V) = mn v,t (r ) r R l (r l ),u U(r) j p j,t u j [ fj + I j [v,t+1 (r 1) v,t+1 (r )] ] } v,t+1 (r ) where the argument V emphaszes the dependence on the gven approxmaton. Note that f V s feasble to (P L), then ϵ,t (r, V) 0 for all t, and r R. We begn wth a prelmnary result. Lemma 4. There exsts an optmal soluton ˆV = {ˆv,t (r ) t,, r R } to (P L) such that for all t, and r R, we have ϵ,t (r, ˆV) = 0. Proof. Let V = {v,t (r ) t,, r R } be an optmal soluton to problem (P L). Let s be the largest tme ndex such that there exsts a resource l and r l R l wth ϵ l,s (r l, V) > 0. Snce V s feasble, ths means that ϵ,t (r, V) = 0 for all t > s, and r R. We consder decreasng v l,s (r l ) alone by ϵ l,s (r l, V) leavng all the other elements of V unchanged. That s, let ˆV = {ˆv,t (r ) t,, r R } where ˆv,t (x) = { v,t (x) ϵ,t (x, v) f = l, t = s, x = r l v,t (x) otherwse. (20) Note that snce ˆv,t (r ) v,t (r ) for all t, and r R, we have ˆv,1(r 1) v,1(r 1 ). Next, we show that ˆV s feasble. Snce ˆV dffers from V only n one element, we only have to check those constrants where ˆv l,s (r l ) appears. Observe that ˆv l,s (r l ) appears only n the constrants for tme perods s 1 and s. For tme perod s 1, we have [ p j,s 1 u j fj + ˆv,s (r 1) ] + [1 p j,s 1 u j ]ˆv,s (r ) j I j j J j p j,s 1 u j [ fj + I j v,s (r 1) ] + [1 j J p j,s 1 u j ]v,s (r ) = v,s 1 (r ) ˆv,s 1 (r ) for all r R and u U(r), where the frst nequalty follows snce ˆv,s (r ) v,s (r ) and j J p j,s 1 u j 1, the second nequalty follows from the feasblty of V and the equalty follows from (20). For tme perod s, ˆv l,s (r l ) appears only n constrants correspondng to r R l (r l ). 21

22 For r R l (r l ), we have ˆv,s (r ) = v,s (r ) ϵ l,s (r l, V) j p j,s u j [ fj + I j {v,s+1 (r 1) v,s+1 (r )} ] + v,s+1 (r ) = j p j,s u j [ fj + I j {ˆv,s+1 (r 1) ˆv,s+1 (r )} ] + ˆv,s+1 (r ) for all u U(r), where the nequalty follows from the defnton of ϵ l,s (r l, V) and the last equalty follows from (20). Therefore ˆV s feasble, whch mples that ϵ,t (r, ˆV) 0 for all t, and r R. Next, we note from (20) that ϵ,t (r, ˆV) = 0 for all t > s, and r R. For tme perod s, snce ˆv,s (r ) v,s (r ) and ˆv,s+1 (r ) = v,s+1 (r ), t follows that ϵ,s (r, ˆV) ϵ,s (r, V). Therefore, f ϵ,s (r, V) was zero, then ϵ,s (r, ˆV) s also zero. Moreover, ϵ l,s (r l, ˆV) = 0 < ϵ l,s (r l, V). To summarze, ˆV s an optmal soluton wth ϵ,t (r, ˆV) = 0 for all t > s, and r R and {ϵ,s (r, ˆV) ϵ,s (r, ˆV) > 0} < {ϵ,s (r, V) ϵ,s (r, V) > 0}. We repeat the above procedure fntely many tmes to obtan an optmal soluton ˆV wth ϵ,t (r, ˆV) = 0 for all t s, and r R. Repeatng the entre procedure for tme perods s 1,..., 1 completes the proof. We are ready to prove Lemma 1. By Lemma 4, we can pck an optmal soluton ˆV = {ˆv,t (r ) t,, r R } such that ϵ,t (r, ˆV) = 0 for all t, and r R. The proof proceeds by nducton on the tme perods. It s easy to see that the result holds for tme perod τ. Fx a resource l and assume that statements () and () of the lemma hold for all tme perods s > t. We show below that statements () and () hold for tme perod t as well. Snce ˆv,t ( 1) =, statement () holds trvally for r l = 0. For r l = 1, Lemma 4 mples that there exsts x R l (0) and u U(x) such that ˆv l,t (0) + l ˆv,t (x ) = j J l p j,t u j [f j + l 1 [ Ij ][ˆv,t+1 (x 1) ˆv,t+1 (x )] ] +ˆv l,t+1 (0) + l ˆv,t+1 (x ). (21) where 1 [ ] denotes the ndcator functon and we use the fact that snce x l = 0, u j = 0 for all j J l. Next, consder the capacty vector y wth y = x for l and y l = r l = 1. Snce x y, U(x) U(y) and t follows that u U(y). Snce ˆV s feasble, we have ˆv l,t (1) + l ˆv,t (x ) j J l p j,t u j [f j + l 1 [ Ij][ˆv,t+1 (x 1) ˆv,t+1 (x )] ] +ˆv l,t+1 (1) + l ˆv,t+1 (x ). (22) Subtractng (21) from (22), we have ˆv l,t (1) ˆv l,t (0) ˆv l,t+1 (1) ˆv l,t+1 (0). We next show that statement () holds for r l R l \{0, 1}. By Lemma 4, there exsts x R l (r l 1) 22