A Constant-Factor Approximation Algorithm for Network Revenue Management

Transcription

1 A Constant-Factor Approxmaton Algorthm for Networ Revenue Management Yuhang Ma 1, Paat Rusmevchentong 2, Ma Sumda 1, Huseyn Topaloglu 1 1 School of Operatons Research and Informaton Engneerng, Cornell Tech, New Yor, NY 10044, 2 Marshall School of Busness, Unversty of Southern Calforna, Los Angeles, CA 90089, ym367@cornell.edu, rusmevc@marshall.usc.edu, ms3268@cornell.edu, ht88@cornell.edu Aprl 8, 2018 We provde a constant-factor approxmaton algorthm for networ revenue management problems. In our approxmaton algorthm, we construct an approxmate polcy usng value functon approxmatons that are expressed as lnear combnatons of bass functons. We use a bacward recurson to compute the coeffcents of the bass functons n the lnear combnatons. If each product uses at most L resources, then the total expected revenue obtaned by our approxmate polcy s at least 1/(1 + L of the optmal total expected revenue. In many networ revenue management settngs, although the number of resources and products can become large, the number of resources used by a product remans bounded. In ths case, our approxmate polcy provdes a constant-factor performance guarantee. To our nowledge, our approxmate polcy s the frst constant-factor approxmaton algorthm for networ revenue management problems. Our approach can ncorporate the customer choce behavor among the products and allows the products to use multple unts of a resource, whle stll mantanng the performance guarantee. In our computatonal experments, we demonstrate that our approxmate polcy performs qute well, provdng total expected revenues that are substantally better than ts theoretcal performance guarantee. 1. Introducton In networ revenue management problems, we manage the lmted capactes for a collecton of resources to satsfy the requests for dfferent products that arrve randomly over tme. Such problems fnd applcatons n a varety of settngs, ncludng arlnes, hosptalty, ralways, and cloud computng. In arlnes, for example, the resources are the flght legs and the products are the tnerares offered to customers that can consume capactes on multple flght legs. In hosptalty, on the other hand, the resources are the avalabltes of hotel rooms on each day and the products are the multple nght stays offered to customers that can consume capactes on multple days. The man tradeoff n networ revenue management problems nvolves eepng a balance between acceptng a product request that s currently n the system to generate some 1

2 2 Ma, Rusmevchentong, Sumda and Topaloglu: An Approxmaton Algorthm for Networ Revenue Management mmedate revenue and reservng the resource capactes for a potentally more proftable product request that can arrve n the future. Nevertheless, the ey dffculty n fndng the optmal course of acton arses from the fact that servng a request for a product consumes capactes of the dfferent resources used by the product. Thus, computng the optmal course of acton requres eepng trac of the remanng capactes for all the resources smultaneously, creatng the curse of dmensonalty as the number of resources ncreases. In ths paper, we provde a constant-factor approxmaton algorthm for networ revenue management problems. Our problem setup follows the standard networ revenue management lterature. We have access to resources wth lmted capactes that can be used to serve the requests for products arrvng randomly over a fnte sellng horzon. At each tme perod n the sellng horzon, a customer enters the system wth a request for a partcular product. If we accept the product request, then we generate a certan amount of revenue and consume capactes of the dfferent resources used by the product. The goal s to fnd a polcy to determne whch product requests to serve to maxmze the total expected revenue over the sellng horzon. The dynamc programmng formulaton for ths problem requres a hgh-dmensonal state varable that eeps trac of the remanng capactes of all the resources. Therefore, t s ntractable to compute the optmal polcy even when we have a relatvely small number of resources. Contrbutons: Lettng L be the maxmum number of resources used by a product, we gve an approxmate polcy that s guaranteed to obtan at least 1/(1 + L of the optmal total expected revenue. In many networ revenue management settngs, the number of resources and products can become large, but the number of resources used by a product remans bounded. In arlnes, for example, L corresponds to the maxmum number of flght legs ncluded n an tnerary, whch usually does not exceed two or three. When the number of resources used by a product s bounded, our approxmate polcy provdes a constant-factor performance guarantee. To our nowledge, our approxmate polcy s the frst constant-factor approxmaton algorthm for networ revenue management problems. Moreover, note that our performance guarantee s ndependent of the numbers of resources and products, and t does not nvolve any hdden constants that can potentally depend on other nput data. The dea behnd our approxmate polcy s to use value functon approxmatons that are expressed as lnear combnatons of bass functons. The coeffcents n the lnear combnatons are computed through a bacward recurson over the tme perods n the sellng horzon. The approach that we use to construct our approxmate polcy provdes flexblty on two mportant dmensons. Frst, our value functon approxmatons are a member of a relatvely broad class. In our value functon approxmatons, we have one bass functon for each product. The bass functon

3 Ma, Rusmevchentong, Sumda and Topaloglu: An Approxmaton Algorthm for Networ Revenue Management 3 assocated wth each product taes the value of zero when we do not have suffcent capactes to serve a request for the product. Therefore, we refer to our bass functons as avalablty-tracng bass functons. For any choce of avalablty-tracng bass functons, we can use our bacward recurson over the tme perods n the sellng horzon to compute coeffcents for the bass functons n the lnear combnatons. In our bacward recurson, we have a tunng parameter θ whose specfc allowable values are determned by the avalablty-tracng bass functons that we use. We prove that f we construct an approxmate polcy usng the value functons computed through our bacward recurson, then the approxmate polcy s guaranteed to obtan at least 1/(1 + θl of the optmal total expected revenue. Ths result holds for any choce of avalablty-tracng bass functons. The performance guarantee of 1/(1 + θl mproves as θ gets smaller. In our approach, the tunng parameter θ must be at least one, and there exst avalablty-tracng bass functons that permt choosng the smallest possble value of one for the tunng parameter θ, n whch case, we obtan the performance guarantee of 1/(1 + L n the prevous paragraph. Second, we start wth a networ revenue management setup where each customer enters the system to purchase a partcular product and each product uses at most one unt of a resource. Ths setup allows us to convey the ey deas wthout notatonal clutter, but we can extend our approach to more general setups. In partcular, we show how to extend our approach to a case n whch we offer a set of products to each arrvng customer, and the customer chooses among the offered products. Smlarly, we show how to extend our approach to a case n whch each product uses more than one unt of a resource, whch happens to be the case n arlnes, for example, when group reservatons are allowed. If the customers choose among the offered products, then the performance guarantee of our approxmate polcy s stll 1/(1 + L, whereas f a product consumes at most M unts of a partcular resource, then the performance guarantee of our approxmate polcy s 1/(1 + (2M 1 L. Lastly, our bacward recurson s smple, and does not requre solvng any nvolved optmzaton problems, but t can ncorporate the soluton to a lnear programmng approxmaton, whle retanng the performance guarantee of 1/(1 + L. Our computatonal experments demonstrate that the approxmate polcy that we provde performs substantally better than ts theoretcal performance guarantee, when compared wth a tractable upper bound on the optmal total expected revenue. We also compare our approxmate polcy wth the standard bd prce polcy from a lnear programmng approxmaton and demonstrate that our approxmate polcy performs notceably better. We emphasze that the attractve feature of our approxmate polcy s that t provdes a constant-factor performance guarantee, and t s the frst polcy n the lterature to possess ths feature. As we dscuss below n our lterature revew, there are heurstc polces to address varous shortcomngs of the standard bd

4 4 Ma, Rusmevchentong, Sumda and Topaloglu: An Approxmaton Algorthm for Networ Revenue Management prce polcy, but none of these polces have constant-factor approxmaton guarantees. The queston of whether we can further mprove the theoretcal or practcal performance of our approxmate polcy s certanly an nterestng research queston to pursue. Lterature Revew: One approach to construct polces n networ revenue management problems s based on bd prces, where we assocate a bd prce for each resource, measurng the value of a unt of capacty. In ths case, we are wllng to accept a product request f the revenue from the product request exceeds the total value of the resources consumed by the product. Smpson (1989 and Wllamson (1992 compute bd prces usng a lnear programmng approxmaton constructed under the assumpton that the product requests tae on ther expected values. They use the optmal values of the dual varables assocated wth certan capacty avalablty constrants to measure the value of a unt of capacty. Tallur and van Ryzn (1998 show that such a bd prce polcy s asymptotcally optmal, as the expected numbers of product requests and the capactes of the resources scale lnearly wth the same rate. Tallur and van Ryzn (1999 use sampled realzatons of the product requests n the lnear program to capture some nformaton about the dstrbutons of the product requests. Bertsmas and Popescu (2003 measure the value of a unt of capacty drectly usng the change n the optmal obectve value of the lnear program n response to a change n the rght sde of a capacty avalablty constrant. The value of a unt of capacty of a resource should depend on the tme left n the sellng horzon to utlze the resource, as well as the remanng capacty of the resource. Thus, bd prces should, n prncple, be tme and capacty dependent. There s wor on computng such bd prces. Adelman (2007 uses lnear value functon approxmatons. Cooper and de Mello (2007 decompose the dynamc programmng formulaton of the problem by pars of resources. Amaruchul et al. (2007 develop bd prce polces for cargo revenue management. Topaloglu (2009 computes upper bounds on the optmal total expected revenue, where he allows acceptng a product request partally by consumng capactes only on some of the resources used by the product. Kunnumal and Topaloglu (2010a develop an approxmaton strategy by relaxng the capacty constrants through Lagrange multplers. Zhang (2011 decomposes the dynamc programmng formulaton by the resources. Krshner and Neda (2015 use a second order code programmng approxmaton. The approaches dscussed n ths paragraph yeld ether tme or capacty dependent bd prces. Tong and Topaloglu (2013, Vossen and Zhang (2015a,b, and Kunnumal and Tallur (2016a show that some of these approaches are equvalent, although ther dervatons use seemngly unrelated paths. There s also wor on ncorporatng customer choce behavor, where customers choose among the offered products. Gallego et al. (2004 gve an analogue of the lnear programmng approxmaton under customer choce behavor. The number of decson varables n the lnear program ncreases

5 Ma, Rusmevchentong, Sumda and Topaloglu: An Approxmaton Algorthm for Networ Revenue Management 5 exponentally wth the number of products, so t s common to solve the lnear program usng column generaton. Zhang and Cooper (2005, 2009 gve bounds on the value functons when customers choose among the flght legs between the same orgn and destnaton. Lu and van Ryzn (2008 develop approxmatons by decomposng the dynamc programmng formulaton of the problem by the resources. Kunnumal and Topaloglu (2008 gve a more refned lnear programmng approxmaton to capture the customer arrval process more accurately. Zhang and Adelman (2009 construct lnear value functon approxmatons. Bront et al. (2009 analyze the complexty of the column generaton subproblem for the lnear programmng approxmaton when customers choose under a mxture of multnomal logt models. Mendez-Daz et al. (2010 gve vald cuts for the same subproblem. Kunnumal and Topaloglu (2010b heurstcally decompose the problem by the resources by allocatng the revenue from a product over the resources t uses. Messner and Strauss (2012 construct separable and pecewse-lnear approxmatons to the value functons. Messner et al. (2012 observe that f there are multple customer segments choosng accordng to dfferent choce models, then the lnear programmng approxmaton can be challengng to solve and they develop tractable relaxatons. Tallur (2014 tghten a smlar relaxaton usng sampled customer arrvals and vald cuts. Kunnumal and Tallur (2016b theoretcally compare the upper bounds on the optmal total expected revenue provded by the dfferent methods n the exstng lterature. Strauss and Tallur (2017 gve propertes for the sets of products consdered by dfferent customer segments that ensure that the lnear programmng approxmaton can be solved tractably. Lastly, van Ryzn and Vulcano (2008a,b use stochastc approxmaton to compute boong lmts, and Topaloglu (2008, and Chaneton and Vulcano (2011 extend ths wor to computng bd prces. None of the wor revewed thus far provdes constant-factor performance guarantees. A number of papers characterze the loss n the optmal total expected revenue for the polces derved from lnear programmng approxmatons. All of ths wor consders an asymptotc regme, where the total expected demands and the capactes of the resources ncrease lnearly wth the same rate. Cooper (2002 and Maglaras and Messner (2006 bound the loss by O(, whereas Jasn and Kumar (2012 bound the loss by a constant ndependent of, as long as we perodcally solve the lnear program over the sellng horzon. Jasn and Kumar (2013 show that the standard bd prce polcy has a loss of at least Ω(. We emphasze that these losses hold n an asymptotc regme as ncreases and they nvolve constants that are dependent on the nput data, ncludng the numbers of resources and products. Moreover, the losses are addtve. In contrast, our performance guarantee of 1/(1 + L s multplcatve, t holds wthout an asymptotc regme, and t does not depend on any nput data other than L. Lastly, the recent wor by Wang et al. (2016, Gallego et al. (2016, and Rusmevchentong et al. (2017 develops polces wth constant-factor performance guarantees for dynamc resource

6 6 Ma, Rusmevchentong, Sumda and Topaloglu: An Approxmaton Algorthm for Networ Revenue Management allocaton problems. In all of ths wor, however, each product request, f accepted, uses only one resource. In contrast, we consder here networ revenue management problems where each product request consumes a combnaton of resources, whch are sgnfcantly more challengng and rather nontrval. The networ revenue management settng requres buldng value functon approxmatons that consder the nteractons between the resources, and desgnng a recurson to update the coeffcents of the bass functons n the value functon approxmatons. Organzaton: In Secton 2, we gve a dynamc programmng formulaton for the networ revenue management problem. In Secton 3, we construct our approxmate polcy, provde a performance guarantee, and show that ths guarantee s tght. In Secton 4, we gve extensons to customer choce behavor and products consumng multple unts of a resource. We also dscuss how to leverage a lnear programmng approxmaton when constructng our approxmate polces. In Secton 5, we provde computatonal experments. In Secton 6, we conclude. 2. Problem Formulaton We have m resources ndexed by L = {1,..., m} and n products ndexed by J = {1,..., n}. The capacty of resource s C. If we accept a request for product, then we consume one unt of capacty of each resource n the set A L. We use L to denote the maxmum number of resources that can be used by a product, so L = max A. Acceptng a request for product generates a revenue of r. We have T tme perods n the sellng horzon ndexed by T = {1,..., T }. Each tme perod corresponds to a suffcently small nterval of tme so that there s at most one product request at each tme perod. We get a request for product at tme perod t wth probablty λ t. Wth the remanng probablty 1 λt, there s no request for a product. Our goal s to fnd a polcy to determne whch product request to accept at each tme perod to maxmze the total expected revenue over the sellng horzon, whle adherng to the capacty avalabltes of the resources. To capture the state of the system, we let x be the remanng capacty of resource at the begnnng of a generc tme perod. Therefore, we can use the vector x = (x 1,..., x m Z m + to capture the state of the resources. The set of possble states s Q = {x Z m + : x C L}. We can accept a request for product when we have at least one unt of capacty for each resource used by product. Therefore, lettng 1l { } be the ndcator functon, we can accept a request for product f and only f A 1l {x 1} = 1. We use V t (x to denote the maxmum total expected revenue over tme perods t, t + 1,..., T, gven that the system s n state x at tme perod t. Lettng e {0, 1} m denote the th unt vector

7 Ma, Rusmevchentong, Sumda and Topaloglu: An Approxmaton Algorthm for Networ Revenue Management 7 and defnng [a] + = max{a, 0}, we can fnd the optmal polcy by computng the optmal value functons {V t (x : x Q, t T } through the dynamc program V t (x = ( } λ t max {r + V (x t+1 A e, V t+1 (x A 1l {x 1} = V t+1 (x + λ t A 1l {x 1} ( + 1 λ t + ( λ t 1 ( [r V t+1 (x + V t+1 A 1l {x 1} V t+1 (x x A e ] +, (1 wth the boundary condton that V T +1 = 0. In the dynamc program above, f we have a request for product at tme perod t and we have capactes on all resources used by product, then we have a choce to accept or reect the request. If we accept, then we generate a revenue of r and the state of the resources at the next tme perod s x A e. If we reect, then we do not generate revenue and the state of the resources at the next tme perod remans at x. In addton, f there s no request at tme perod t or there s a request for some product, but we do not have capacty on some resource used by the product, then we do not accept a request, n whch case, the state of the resources at the next tme perod remans at x. The second equalty above follows smply by arrangng the terms. By the dynamc program above, gven that the state of the resources at tme perod t s x, f r V t+1 (x V t+1 (x A e, then t s optmal to accept a request for product as long as we have capacty on all resources used by product. Lettng C = (C 1,..., C m be the ntal resource capactes, the optmal total expected revenue s V 1 (C. The sze of the state space s Q = O ( C L, whch ncreases exponentally wth the number of resources, mang the computaton of the optmal value functons ntractable. 3. Approxmate Polcy We construct approxmatons to the optmal value functons usng a lnear combnaton of bass functons. Our approxmate polcy s guded by these value functon approxmatons. We use the followng outlne. In Secton 3.1, we descrbe our bass functons. In Secton 3.2, we show how to compute the coeffcent of each bass functon n the value functon approxmatons. In Secton 3.3, we show the performance guarantee for our approxmate polcy. In Secton 3.4, we show that ths performance guarantee s tght. 3.1 Requrements for Bass Functons We approxmate the optmal value functon V t usng the value functon approxmaton H t. To construct the value functon approxmaton H t, we use a lnear combnaton of bass functons,

8 8 Ma, Rusmevchentong, Sumda and Topaloglu: An Approxmaton Algorthm for Networ Revenue Management where, for each subset A L of resources that can be used by a product, we have a bass functon ϕ A : Q [0, 1]. In partcular, the value functon approxmaton H t s gven by H t (x = γ t ϕ A (x, (2 where B = {ϕ A : A L} s a prespecfed collecton of bass functons ndexed by subsets of resources and {γ t : J, t T } are adustable coeffcents. If no product uses the subset of resources A, then ϕ A s not needed n (2. We refer to our bass functons as avalablty-tracng bass functons because we wll mpose the condton that ϕ A (x taes the value of zero f the vector of resource capactes x does not provde enough avalablty to serve a request for a product usng the subset of resources A. In the next defnton, we fully specfy the condtons that we mpose on avalabltytracng bass functons. Defnton 3.1 (Avalablty-Tracng Bases The collecton B = {ϕ A : A L} s called a collecton of avalablty-tracng bass functons f t satsfes the followng condtons. (a Avalablty Tracng: For each subset A L and x Q, ϕ A (x = 0 whenever x = 0 for some A. That s, ϕ A (x 1l A {x 1}. (b Lmted Dependence: For each subset A L and x Q, the value of the bass functon ϕ A (x only depends on (x : A. That s, ϕ A (x = ϕ A (y whenever x = y for all A. (c Normalzaton: For each subset A L, we have ϕ A (C = 1. As dscussed rght before the defnton, the range of ϕ A s the nterval [0, 1], and the frst property ensures that ϕ A (x taes the value of zero f the resource capactes x are not suffcent to serve a request for a product usng the subset of resources A. The second property ensures that ϕ A (x s ndependent of the components of the resource capacty vector x that are not n A. The thrd property ensures that ϕ A (x s one when the resource capactes x are at ther largest possble x values. For example, the mnmum bass functon ϕ A (x = mn A C functon ϕ A (x = A satsfy the three propertes n the defnton above. x C and the polynomal bass Gven a collecton B = {ϕ A : A L} of avalablty-tracng bass functons, we defne the maxmum scaled ncremental contrbuton B from a unt of resource as B = max L,A L A collecton wth a smaller value of B max C ( ϕ A (x ϕ A (x e. x Q : x 1 wll result n an approxmate polcy wth a better performance guarantee, but B can never be smaller than one, as shown n the next lemma.

9 Ma, Rusmevchentong, Sumda and Topaloglu: An Approxmaton Algorthm for Networ Revenue Management 9 Lemma 3.2 If B = {ϕ A : A L} s a collecton of avalablty-tracng bass functons, then we must have B 1. Proof: Consder an arbtrary subset A L and a resource A. By parts (a and (c of Defnton 3.1, we have ϕ A (C C e = 0 and ϕ A (C = 1, so by a telescopng sum, we get [ ( ( C C 1 = ϕ A (C ϕ A (C C e = ϕ A C s e s +h e ϕ A C s e s +(h 1 e ] s s h=1 h=1 B C = B, where the nequalty follows from the defnton of B. As shown n the next two examples, the lower bound n Lemma 3.2 s tght. Example 3.3 (Mnmum Let ϕ A (x = mn A x C. It s smple to verfy that B = {ϕ A : A L} satsfes the three propertes n Defnton 3.1, so B 1 by Lemma 3.2. Moreover, 1 x { } { } xl xl 1l C f x C = mn l l A C l {l=} x ϕ A (x ϕ A (x e = mn mn = mn l l A x 1 x C l A C l l A C l C f C > mn l A l 0 f x 1 C x l C l x 1 C > mn l A x l C l. The quantty n all three cases on the rght sde above s no larger than 1/C, so that C (ϕ A (x ϕ A (x e 1, n whch case, we get B 1. Thus, we must have B = 1. Example 3.4 (Polynomal Let ϕ A (x = A x C. It s smple to verfy that B = {ϕ A : A L} satsfes the three propertes n Defnton 3.1, so B 1 by Lemma 3.2. Furthermore, ϕ A (x ϕ A (x e = x l C l l A x 1 C x l C l l A\{} = 1 C l A\{} x l C l 1 C, where the last nequalty follows because x l C l. Therefore, we have C (ϕ A (x ϕ A (x e 1, ndcatng that B 1. So, we get B = 1. In the next lemma, we show that we can construct avalablty-tracng bass functons from exstng ones by tang a composton wth a contnuously dfferentable functon. Lemma 3.5 (Constructng Bases Let B = {ϕ A : A L} be a collecton of avalablty-tracng bass functons such that ϕ A s componentwse nondecreasng for all A L. If f : [0, 1] [0, 1] s a dfferentable functon wth f(0 = 0, f(1 = 1 and max a [0,1] f (a <, then B = {f ϕ A : A L} s also a collecton of avalablty-tracng bass functons wth B ( max a [0,1] f (a B.

10 10 Ma, Rusmevchentong, Sumda and Topaloglu: An Approxmaton Algorthm for Networ Revenue Management Proof: It s smple to verfy that B satsfes the three propertes n Defnton 3.1. Moreover, by the mean value theorem, for each x Q wth x 1, there exsts y [0, 1] such that ( f(ϕ A (x f(ϕ A (x e = f (y (ϕ A (x ϕ A (x e max f (a a [0,1] (ϕ A (x ϕ A (x e, where the last nequalty follows from our assumpton that ϕ A s componentwse nondecreasng, so ϕ A (x ϕ A (x e 0. The nequalty above mples that B ( max a [0,1] f (a B. Snce f(0 = 0 and f(1 = 1, we must have max a [0,1] f (a 1 n the lemma. So, the lemma does not mply that B B. One can chec that f ϕ A (x s a lnear functon of x, then t does not satsfy the propertes n Defnton 3.1. Thus, lnear bass functons are not avalablty-tracng. 3.2 Approxmate Polcy and Performance Guarantee In ths secton, we gve an algorthm to compute the coeffcents {γ t : J, t T } n the value functon approxmatons n (2. Ths algorthm requres a recursve computaton over the tme perods n the sellng horzon. Once we construct our value functon approxmatons, the greedy polcy wth respect to the value functon approxmatons yelds our approxmate polcy. Then, we gve a performance guarantee for our approxmate polcy. To compute the coeffcents {γ t : J, t T } n the value functon approxmatons n (2, we use the algorthm below. Intalzaton: Let B = {ϕ A : A L} be any collecton of avalablty-tracng bass functons and θ 0 be a tunng parameter. Intalze γ T +1 = 0 for all J. Coeffcent Computaton: For each t = T, T 1,..., 1, use the coeffcents {γ t+1 compute {γ t : J } as γ t = γ t+1 : J } to [ + λ t r θ ] + 1 1l C { A } γ t+1. (3 A The above algorthm allows us to compute {γ t : J, t T }, whch specfes the approxmate value functons {H t : t T } gven n (2. Gven that the state of the resources at tme perod t s x, f r V t+1 (x V t+1 (x A e, then t s optmal to accept a request for product as long as we have capacty on all the resources used by product. In other words, f r V t+1 (x V t+1 (x A e, then t s optmal to accept a request for product as long as we have A 1l {x 1} = 1. We obtan our approxmate polcy by replacng V t+1 n the last nequalty wth H t+1, whch corresponds to the greedy polcy wth respect to the value functon approxmatons {H t : t T }. To formally state our approxmate polcy, we use u App,t (x = {u App,t (x : J } to denote the decson functon of the approxmate polcy at tme perod t. Gven that the state of the

11 Ma, Rusmevchentong, Sumda and Topaloglu: An Approxmaton Algorthm for Networ Revenue Management 11 resources at tme perod t s x, we have u App,t (x = 1 f we accept a request for product at tme perod t. Otherwse, we have u App,t (x = 0. Thus, u App,t (x s gven by 1l u App,t {x 1} f r H t+1 (x H (x t+1 A e, (x = A 0 otherwse. The next theorem gves a performance guarantee for our approxmate polcy as a functon of the tunng parameter θ n (3 and the maxmum number of resources L used by a product. (4 Theorem 3.6 (Performance If the tunng parameter θ satsfes θ B, then the total expected revenue obtaned by the approxmate polcy s at least 1/(1 + θl of the optmal. We gve the proof of Theorem 3.6 n the next secton. To obtan the best performance guarantee, we need to choose the tunng parameter θ as small as possble and the smallest possble value of θ n the theorem s B. By Lemma 3.2, we have B 1, but as shown n Examples 3.3 and 3.4, there are choces of bass functons under whch B = 1. Therefore, worng wth these bass functons, we can choose the tunng parameter θ as one and obtan an approxmate polcy whose total expected revenue s at least 1/(1 + L of the optmal. In many networ revenue management problems, arsng n settngs such as arlnes and hotels, each product uses only a small number of resources. Therefore, even though the number of resources can be large, as long as the number of resources used by a product s unformly bounded, Theorem 3.6 provdes a constant-factor approxmaton guarantee. Moreover, although we obtan the best performance guarantee by choosng θ at ts smallest possble value of B, our computatonal experments ndcate that ncreasng θ beyond B can mprove the total expected revenue of the approxmate polcy. We vew the tunng parameter θ as a nob that provdes flexblty n the mplementaton of our approxmate polcy. Lastly, as s the case for most constant-factor approxmaton algorthms, the performance guarantee n Theorem 3.6 s a worst-case guarantee. In our computatonal experments, we compare the total expected revenue obtaned by our approxmate polcy wth a computatonally tractable upper bound on the optmal total expected revenue, and demonstrate that the approxmate polcy performs substantally better than ts worst-case performance guarantee. We provde some nsght nto the computaton of the coeffcents {γ t : J, t T }. Lettng L = {1,..., m}, we consder the case wth m + 1 resources ndexed by {0} L and m products ndexed by L. The capacty of resource 0 s C 0 and the capactes of all other resources are one. Product uses resources 0 and. Thus, resource 0 s a common resource used by all products. For each L, resource has one unt of capacty, so we can serve at most one request

12 12 Ma, Rusmevchentong, Sumda and Topaloglu: An Approxmaton Algorthm for Networ Revenue Management for product, but t may be optmal to reect a request for product because servng such a request also consumes the common resource. For ths problem nstance, (3 reduces to [ ] γ t = γ t+1 + λ t r 1 + { } γ t+1 γ t+1 = λ t max r 1 γ t+1, γ t+1 + (1 λ t C 0 C γ t+1, 0 L where we choose the tunng parameter θ as one. Intutvely, γ t L captures the net expected proft contrbuton from product at tme perod t. Wth probablty λ t, we have a request for product at tme perod t. If we accept the request, then we generate a revenue of r, but we also consume a unt of the common resource, whch lmts our ablty to serve a request for another product n the future. The opportunty cost assocated wth the consumpton of the common resource appears as 1 C 0 L γt+1 n our algorthm, yeldng the term r 1 C 0 L γt+1 n the max operator. As 1 C 0 ncreases, C 0 L γt+1 decreases, reflectng the ntuton that f we have more of the common resource, then acceptng the request less severely lmts our ablty to serve a request for another product n the future. If we have a request for product at tme perod t, then we have the opton of reectng ths request, n whch case, the net expected proft contrbuton from product at tme perod t s the same as the net expected proft contrbuton at the next tme perod, yeldng the term γ t+1 n the max operator. Smlarly, wth probablty 1 λ t, we do not get a request for product at tme perod t, yeldng the term (1 λ t γ t+1 on the rght sde. 3.3 Proof of Theorem 3.6 The proof of Theorem 3.6 maes use of two lemmas. The next lemma bounds the opportunty cost of acceptng a request for product under our approxmate value functons. Lemma 3.7 (Bound on Opportunty Cost For a collecton of avalablty-tracng bass functons B = {ϕ A : A L}, let H(x = γ ϕ A (x, where the coeffcents {γ : J } satsfy γ 0 for all J. Then, for each J and x Q such that x A e 0, we have H(x H (x A 1 e B 1l C { A }γ. A Proof: We prove the result usng nducton on the cardnalty of A. Consder the base case where A = 1 so that we have A = {} for some L. In ths case, we get H(x H (x e = 1l { A }γ (ϕ A (x ϕ A (x e B C 1l { A }γ, where the equalty holds because ϕ A (x ϕ A (x e = 0 whenever A by the second property n Defnton 3.1 and the nequalty follows from the defnton of B. Thus, the base case holds.

13 Ma, Rusmevchentong, Sumda and Topaloglu: An Approxmaton Algorthm for Networ Revenue Management 13 Suppose that the result holds for any A s. Consder a case n whch A = s+1, so A = B {l} for some B L wth B = s and l L wth l B. Lettng y = x e B, we obtan ( H(x H (x A e = H(x H x e e l B ( ( ( = H(x H x e + H x e H B B 1 B 1l C { A }γ + H (y H (y e l B B 1 B 1l C { A }γ 1 = B 1l C { A }γ, A + B C l 1l {l A }γ x B e e l where the frst nequalty follows from the nducton assumpton, the second nequalty follows from the base case, and the last equalty s by the fact that A = B {l}. Thus, θ A 1 C 1l { A }γ n (3 s an upper bound on H(x H ( x A e for any θ B. In the next lemma, we bound the optmal total expected revenue usng {H t : t T }. Lemma 3.8 (Upper Bound on Optmal Total Expected Revenue If the value functon approxmatons {H t : t T } are constructed usng (3, then V 1 (C (1 + θl H 1 (C. Proof: Notng the dynamc program n (1, t s a well-nown result that the optmal total expected revenue s gven by the optmal obectve value of the lnear program mn Ṽ 1 (C s.t. Ṽ t (x Ṽ t+1 (x + λ t A 1l {x 1} [r Ṽ t+1 (x + Ṽ t+1 ( x A e ] + x Q, t T, where the decson varables are {Ṽ t (x : x Q, t T }; see Adelman (2007. In the lnear program above, we follow the conventon that Ṽ T +1 (x = 0 for all x Q. Snce we mnmze the obectve functon n the lnear program above, f {ν t (x : x Q, t T } s a feasble soluton to the lnear program above, then ν 1 (C s an upper bound on the optmal total expected revenue. Lettng {α t : t T } and {β t : L, t T } be defned as α t = γ t and β t = θ 1l C { A }γ, t we construct the soluton {ν t (x : x Q, t T } to the lnear program as ν t (x = α t + L βt x. We wll now show that {ν t (x : x Q, t T } s a feasble soluton to the lnear program above. It

14 14 Ma, Rusmevchentong, Sumda and Topaloglu: An Approxmaton Algorthm for Networ Revenue Management follows from (3 that γ t γ t+1, so we have β t β t+1. Moreover, notng that ν t+1 (x s lnear n x, we have ν t+1 (x ν t+1 (x A e = A β t+1. In ths case, evaluatng the rght sde of the constrant n the lnear program at the soluton {ν t (x : x Q, t T }, we get [r ν t+1 (x + ν t+1 (x A e ] + ν t+1 (x + λ t A 1l {x 1} = = = γ t+1 + β t+1 x + λ t L γ t+1 + β t+1 x + λ t L γ t+1 + L β t+1 x + γ t + L β t x = ν t (x, A 1l {x 1} A 1l {x 1} A 1l {x 1} [ [ r A β t+1 ] + r θ 1 1l C { A }γ t+1 A (γ t γ t+1 ] + where the second equalty uses the defnton of β t+1, the thrd equalty follows from (3, and the nequalty uses the fact that β t+1 β t along wth the observaton that γ t γ t+1 0, so ( (γ t γ t+1 γ t γ t+1. The chan of nequaltes above shows that the soluton A 1l {x 1} {ν t (x : x Q, t T } s feasble for the lnear program. Thus, ν 1 (C s an upper bound on the optmal total expected revenue, satsfyng ν 1 (C V 1 (C. In ths case, we have V 1 (C ν 1 (C = α 1 + β 1 C L = γ 1 + θ 1l { A }γ 1 L = γ 1 + θ γ 1 1l { A } L = γ 1 + θ γ 1 A (1 + θl γ 1 = (1 + θl H 1 (C, where the last nequalty follows because A L for all J and the last equalty holds snce we have ϕ A (C = 1 for all A L by the thrd property n Defnton 3.1, n whch case, t follows that γ1 = γ1 ϕ A (C = H 1 (C. To compute the total expected revenue obtaned by the approxmate polcy, we can use a dynamc programmng recurson smlar to (1. Let U t (x be the total expected revenue obtaned by the approxmate polcy over tme perods t, t + 1,..., T gven that the system s n state x at tme

15 Ma, Rusmevchentong, Sumda and Topaloglu: An Approxmaton Algorthm for Networ Revenue Management 15 perod t. Notng the decson functon for the approxmate polcy n (4, we can compute {U t : t T } through the recurson U t (x = λ t u App,t (x = U t+1 (x + [r + U t+1 (x A e ] λ t u App,t (x ( + 1 λ t + λ t (1 u App,t (x ] [r U t+1 (x + U (x t+1 A e U t+1 (x, (5 wth the boundary condton that U T +1 = 0. In the dynamc program above, we have a request for product at tme perod t wth probablty λ t, n whch case, f we have u App,t (x = 1 so that the approxmate polcy accepts ths request, then we obtan a revenue of r and the state of the resources at the next tme perod s x A e. If there s a request for product at tme perod t, but we have u App,t (x = 0, then the state of the resources at the next tme perod remans at x. Smlarly, wth probablty 1 λt, there s no request for a product at tme perod t, n whch case, the state of the resources at the next tme perod remans at x as well. The second equalty above follows smply by arrangng the terms. Usng {U t : t T } computed as above, the total expected revenue obtaned by the approxmate polcy s gven by U 1 (C. Observe that the coeffcents of U t+1 on the rght sde of the frst equalty n (5 are all nonnegatve. Therefore, f we replace U t+1 on the rght sde of the frst equalty n (5 wth a functon G t+1 satsfyng G t+1 U t+1, then the expresson on rght sde of the frst equalty becomes smaller. In ths case, snce the expresson on the rght sde of the second equalty n (5 s obtaned by arrangng the terms n expresson on the rght sde of the frst equalty, t follows that f we replace U t+1 on the rght sde of the second equalty n (5 wth a functon G t+1 satsfyng G t+1 U t+1, then the expresson on the rght sde of the second equalty n (5 becomes smaller as well. Ths observaton wll become useful when we gve the proof of Theorem 3.6 below. Proof of Theorem 3.6: In the proof, we wll show that the nequalty U t (x H t (x holds for all x Q and t T, where U t (x s gven by (5 and H t (x s the value functon approxmaton n (2 wth the coeffcents {γ t : J } computed through the recurson n (3. Then, applyng ths nequalty at the frst tme perod wth the ntal capactes of all of the resources, t follows that U 1 (C H 1 (C V 1 (C/(1 + θl, where the second nequalty follows from Lemma 3.8. Thus, the total expected revenue obtaned by the approxmate polcy s at least 1/(1 + θl of the optmal total expected revenue, whch s the desred result. We now use nducton over the tme perods n the sellng horzon to prove that U t (x H t (x holds for all x Q and t T. Consder the base case at tme perod T + 1. Snce U T +1 = H T +1 = 0, the base case holds. Suppose that the results hold at tme perod t + 1, so U t+1 (x H t+1 (x for

16 16 Ma, Rusmevchentong, Sumda and Topaloglu: An Approxmaton Algorthm for Networ Revenue Management all x Q. Snce H t+1 U t+1, replacng U t+1 on the rght sde of the second nequalty n (5 wth H t+1 and notng the dscusson rght before the proof, we have ] [r H t+1 (x + H (x t+1 A e U t (x H t+1 (x + λ t u App,t (x [r H t+1 (x + H t+1 (x A e ] + = H t+1 (x + λ t H t+1 (x + λ t H t+1 (x + λ t = H t+1 (x + H t+1 (x + A 1l {x 1} A 1l {x 1} A 1l {x 1} A 1l {x 1} [ [ ϕ A (x ( γ t γ t+1 r B A 1 C 1l { A }γ t+1 r θ 1 1l C { A }γ t+1 A (γ t γ t+1 = H t (x, where the frst equalty above follows because the defnton of the decson functon of the approxmate polcy n (4 mples that u App,t (x = 1 f and only f A 1l {x 1} = 1 and r H t+1 (x + H t+1 (x A e 0. The second nequalty follows from Lemma 3.7. The thrd nequalty s due to the fact that the tunng parameter θ s chosen such that θ B. The second equalty follows from the defnton of γ t n (3. The fourth nequalty follows from the frst property n Defnton 3.1, along wth the fact that γ t γ t+1 ] + ] + 0. The last equalty holds because we have H t+1 (x = γt+1 ϕ A (x by the defnton of the value functon approxmatons. By the chan of nequaltes above, we have U t (x H t (x, completng the nducton argument. 3.4 Tghtness of the Analyss We descrbe a problem nstance to demonstrate that the performance guarantee n Theorem 3.6 s tght. We consder a problem nstance wth a sngle resource. All products use ths resource. Thus, the maxmum number of resources L used by a product s one. Notng Examples 3.3 and 3.4, f we use the mnmum or polynomal bass functons, then we have B = 1, n whch case, by Theorem 3.6, we can choose the tunng parameter θ as one. Wth L = 1 and θ = 1, Theorem 3.6 mples that the total expected revenue obtaned by our approxmate polcy s at least 1/2 of the optmal. We gve a problem nstance where the rato between the total expected revenue from our approxmate polcy and the optmal total expected revenue s arbtrarly close to 1/2. There are T tme perods n the sellng horzon. We have a sngle resource. The capacty of the resource s gven by C 1 = T. We have two products ndexed by J = {1, 2}. The revenues assocated

17 Ma, Rusmevchentong, Sumda and Topaloglu: An Approxmaton Algorthm for Networ Revenue Management 17 wth the two products are r 1 = 1 T (1 1 T and r 2 = 1. At the frst T 1 tme perods, we have a request for product 1 at each tme perod wth probablty one. At the last tme perod T, we have a request for product 2 wth probablty one. That s, we have λ t 1 = { 1 f t < T 0 f t = T and λ t 2 = { 0 f t < T 1 f t = T. Snce the capacty of the resource s equal to the number of tme perods, we never run out of capacty by acceptng the requests. Thus, the optmal polcy accepts all requests, n whch case, the total expected revenue of the optmal polcy s OPT = (T 1 r 1 + r 2 = (T 12 T Consderng our approxmate polcy, snce we have a sngle resource and both products use ths resource, lettng x = (x 1 denote the remanng capacty of the resource, we have mn A x C = A x C = x 1 C 1 = x 1 T for all = 1, 2. Therefore, rrespectve of whether we use the mnmum or polynomal bass functons, the value functon approxmaton n (2 s gven by H t (x = {1,2} γt x 1 T, whch mples that H t (x H t (x e 1 = 1 T (γt 1 + γ t 2. In ths case, usng the fact that we never run out of capacty by acceptng the requests for the products, the decson functon of our approxmate polcy gven n (4 can be wrtten as u App,t (x = 1 f and only f r 1 T (γt γ t+1 2. Observe that the decson functon of our approxmate polcy n ths problem nstance depends on the sum γ t γ t+1 2, but not on the ndvdual values of γ t+1 1 and γ t+1 2. Next, we compute the total expected revenue obtaned by our approxmate polcy. Snce there s a sngle resource wth capacty C 1 = T and we choose θ = 1, the recurson n (3 taes the form γ t = γ t+1 + λ t [r 1 T {1,2} γt+1 ] + for all = 1, 2 and t T. In ths case, notng that the decson functon of our approxmate polcy depends on the sum γ t 1 + γ t 2, addng the last recurson over all = 1, 2 and lettng Γ t = γ t 1 + γ t 2, we wrte ths recurson as Γ t = Γ t+1 + {1,2} λ t [ r 1 T Γt+1 ] +, wth the boundary condton that Γ T +1 = 0. Snce λ T 2 = 1 and r 2 = 1, from the recurson above, we get Γ T = 1. Snce λ T 1 1 = 1, r 1 = 1 T (1 1 T and ΓT = 1, we get Γ T 1 = 1 + [ 1 T (1 1 T 1 T ]+ = 1. Snce λ T 2 1 = 1, r 1 = 1 T (1 1 T and ΓT 1 = 1, we get Γ T 2 = 1 + [ 1 T (1 1 T 1 T ]+ = 1. Contnung n a smlar fashon, t follows that Γ t = 1 for all t T. At each one of the tme perods t = 1,..., T 1, we have a request for product 1 wth probablty one. Snce r 1 = 1 (1 1 < 1 = 1 T T T T Γt+1 = 1 T (γt+1 1 +γ t+1 2, our approxmate polcy reects the requests for product 1 at the tme perods t = 1,..., T 1. At tme perod T, we have a request for product 2 wth probablty one. Snce r 2 = 1 0 = 1 T ΓT +1 = 1 T (γt γ T +1 2, our approxmate polcy accepts the request for product 2 at tme perod T. Thus, the total expected revenue from

18 18 Ma, Rusmevchentong, Sumda and Topaloglu: An Approxmaton Algorthm for Networ Revenue Management our approxmate polcy s APP = r 2 = 1. In ths case, the rato between the total expected revenue (T 12 from our approxmate polcy and the optmal total expected revenue s APP/OPT = 1/( + 1, T 2 whch becomes arbtrarly close to 1/2 as T becomes arbtrarly large. 4. Extensons We extend our approxmate polcy to cases n whch the customers choose among the offered products, and a product can use more than one unt of the capacty of a resource. We also dscuss leveragng a lnear programmng approxmaton to buld value functon approxmatons. 4.1 Customer Choce Behavor In the model n Secton 2, each customer enters the system wth a request for a partcular product. We decde whether to accept or reect the request for ths product. Here, we extend our model and performance guarantee to a case n whch we offer a subset of products to each arrvng customer, and the customer chooses among the offered products or decdes to leave wthout a purchase. Therefore, the customer does not arrve wth a request for a partcular product, and the product that the customer ends up choosng may depend on the subset of products that we offer. The notaton that we use closely follows the one ntroduced for the model n Secton 2. We only dscuss the addtonal notaton that we need. If we offer the subset S J of products to a customer arrvng at tme perod t, then the customer chooses product S wth probablty φ t (S. Naturally, we have φ t (S = 0 for all S. Note that the choces of the customers at dfferent tme perods may be governed by dfferent purchase probabltes. We refer to a subset of products that we offer to the customers as an assortment. We use F 2 J to denote the set of feasble assortments that we can offer to an arrvng customer. We mpose the followng mld assumpton on the choce probabltes and the set of feasble assortments that we can offer. Assumpton 4.1 (Substtutablty and Feasblty For all t T, S F, S, and S, we have φ t (S φ t (S {}. Moreover, f S F, then we have R F for all R S. The frst part of the assumpton ensures that f we ntroduce an addtonal product nto the assortment S, then the choce probablty of a product that s already n the assortment S does not ncrease. Ths property holds for all choce models that are based on the random utlty maxmzaton prncple. The second part of the assumpton ensures that f we remove products from a feasble assortment, then the assortment remans feasble. To formulate the problem as a dynamc program, we let V t (x be the optmal total expected revenue over tme perods t, t + 1,..., T gven

19 Ma, Rusmevchentong, Sumda and Topaloglu: An Approxmaton Algorthm for Networ Revenue Management 19 that the capactes of the resources at tme perod t s x. We can compute the optmal value functons {V t (x : x Q, t T } usng the dynamc program { ( ] V t (x = max φ t (S 1l {x 1} [r + V (x t+1 A e S F A ( + 1 φ t (S + ( φ t (S 1 } 1l {x 1} V t+1 (x A { ( } = V t+1 (x + max φ t (S (r V t+1 (x + V (x t+1 A e, (6 S F A 1l {x 1} wth the boundary condton that V T +1 = 0. In the dynamc program above, there s a customer arrval at each tme perod wth probablty one. If there s a strctly postve probablty of no customer arrval at tme perod t, then lettng λ t [0, 1 be the customer arrval probablty at tme perod t, all we need to do s to replace the choce probablty φ t (S n the dynamc program above wth the choce probablty φ t (S = λ t φ t (S. In the frst equalty n (6, f we offer the assortment S at tme perod t, then the arrvng customer chooses product wth probablty φ t (S. If we have suffcent resource capactes to serve product, so that A 1l {x 1} = 1, then the customer purchases product, n whch case, we generate a revenue of r and the state of the resources at the next tme perod s x A e. On the other hand, the arrvng customer does not choose any product and decdes to leave wthout a purchase wth probablty 1 φt (S, n whch case, the state of the resources at the next tme perod remans at x. Lastly, the arrvng customer chooses product wth probablty φ t (S, but f we do not have suffcent resource capactes to serve product, so that A 1l {x 1} = 0, then the customer leaves wthout a purchase, n whch case, the state of the resources at the next tme perod remans at x as well. In (6, we allow offerng a product for whch we lac suffcent resource capactes to serve. If the customer ends up choosng such a product, then the customer leaves wthout a purchase. Offerng a product for whch we lac suffcent resource capactes to serve may not sound realstc, but t s smple to argue that there exsts an optmal polcy that never offers such a product anyway. To establsh ths result, note that n the optmal soluton to the second maxmzaton problem n (6, f we attempt to offer products for whch we do not have enough resource capacty to serve, then we can drop all such products from the assortment, along wth each product such that r V t+1 (x + V t+1 (x A e < 0, n whch case, by Assumpton 4.1, the choce probabltes of all other remanng products n the assortment do not decrease, yeldng another assortment that provdes an obectve value that s as large as the orgnal one. As noted prevously, computng the optmal value functons {V t : t T } s ntractable. We use a value functon approxmaton of the form H t (x = γt ϕ A (x where B = {ϕ A : A L} s a

20 20 Ma, Rusmevchentong, Sumda and Topaloglu: An Approxmaton Algorthm for Networ Revenue Management collecton of avalablty-tracng bass functons. We compute the coeffcents {γ t : J, t T } n the value functon approxmatons usng a slght varaton of our earler algorthm. Intalzaton: Let B = {ϕ A : A L} be any collecton of avalablty-tracng bass functons and θ 0 be a tunng parameter. Intalze γ T +1 = 0 for all J. Coeffcent Computaton: For each t = T, T 1,..., 1, use the coeffcents {γ t+1 : J } to compute the assortment Ŝt F at tme perod t as { ( Ŝ t = arg max φ t (S r θ } 1 1l S F C { A }γ t+1. (7 A Then, use the coeffcents {γ t+1 {γ t : J } as : J } and the assortment Ŝt computed above to compute ( γ t = γ t+1 + φ t (Ŝt r θ 1 1l C { A }γ t+1. (8 A The algorthm above specfes the coeffcents {γ t : J, t T }, whch, n turn, specfy the approxmate value functons {H t : t T }. Gven that the state of the resources at tme perod t s x, we solve the maxmzaton problem on the rght sde of the second equalty n (6 to fnd the optmal assortment to offer. We construct our approxmate polcy by replacng V t+1 n ths problem wth H t+1. In ths case, gven that the state of the resources at tme perod t s x, our approxmate polcy offers the assortment { ( } S App,t (x = arg max φ t (S (r H t+1 (x + H (x t+1 A e. (9 S F A 1l {x 1} An optmal soluton to the problem above can be vewed as the decson functon of our approxmate polcy under customer choce behavor. Applyng the followng theorem, our approxmate polcy enoys the same performance guarantee as n Theorem 3.6. The proof of ths theorem uses a technque smlar to the one n Secton 3.3. We defer the proof to Appendx A. Theorem 4.2 (Performance under Choce If the choce probabltes and the feasble set of assortments satsfy Assumpton 4.1 and the tunng parameter θ satsfes θ B, then the total expected revenue obtaned by the approxmate polcy s at least 1/(1 + θl of the optmal. 4.2 Multple Unts of Capacty Consumpton In the model that we descrbed n Secton 2, each product consumes at most one unt of each resource. However, we can extend our approach to allow products to use multple unts of each

21 Ma, Rusmevchentong, Sumda and Topaloglu: An Approxmaton Algorthm for Networ Revenue Management 21 resource. Once agan, the notaton that we use closely follows the one ntroduced for the model n Secton 2. We only dscuss the addtonal notaton that we need. For each product and resource, we use a to denote the number of unts of resource used by product. In our earler model, we have a {0, 1} for all L, J. In ths secton, we consder the case where a can be any nonnegatve nteger. As before, A = { L : a 1} denotes the set of resources used by product, and L = max A denotes the maxmum number of resources used by a product. We use m = max a to denote the maxmum number of unts of resource that s used by any product. Wthout loss of generalty, we assume that the ntal capacty of each resource satsfes C m. Otherwse, there s a product that uses more unts than the ntal capacty of a resource, n whch case, we can drop such a product from consderaton. To fnd a polcy that maxmzes the total expected revenue over the sellng horzon, we can use a dynamc program that s smlar to the one n (1. All we need to do s to replace all occurrences of A 1l {x 1} wth A 1l {x a } and all occurrences of A e wth A a e n the dynamc program. We modfy our bass functons as follows. For each product J, let G : Q Q be a mappng such that for each x Q, G (x = ( x 1l {x a } : L. Thus, G (x leaves the th component of x unchanged when the value of the component exceeds the amount of resource consumed by product ; otherwse, G (x sets the component to zero. For a collecton of avalablty-tracng bass functons B = {ϕ A : A L}, we use the value functon approxmaton H t gven by H t (x = γ t ϕ A (G (x. (10 We compute the coeffcents {γ t : J, t T } n the value functon approxmaton above usng the followng algorthm. Intalzaton: Let B = {ϕ A : A L} be any collecton of avalablty-tracng bass functons and θ 0 be a tunng parameter. Intalze γ T +1 = 0 for all J. Coeffcent Computaton: For each t = T, T 1,..., 1, use the coeffcents {γ t+1 compute {γ t : J } as γ t = γ t+1 + λ t [ r θ A 2 m 1 C 1l { A } γ t+1 : J } to ] +. (11 To construct our approxmate polcy, we use the decson functon n (4 after replacng A 1l {x 1} wth A 1l {x a } and A e wth A a e. Ths decson functon provdes the decsons made by our approxmate polcy gven that the state of the resources at tme perod t s x. The followng theorem gves a performance guarantee for our approxmate polcy when a product can consume multple unts of a resource. The proof s n Appendx B.

22 22 Ma, Rusmevchentong, Sumda and Topaloglu: An Approxmaton Algorthm for Networ Revenue Management Theorem 4.3 (Performance under Multple Unts Consumpton Let M = max L, a be the maxmum number of unts of a resource used by a product. If the tunng parameter θ satsfes θ B, then the total expected revenue obtaned by the approxmate polcy s at least 1/(1 + θ(2m 1L of the optmal. The approach that we use to compute the coeffcents {γ t : J, t T } n (11 and the performance guarantee n Theorem 4.3 reduce to the approach n (3 and the performance guarantee n Theorem 3.6 when each product uses at most one unt of a resource because, n the latter case, we have m = 1 for all L, M = 1, and G (x = x for all x Q. 4.3 Leveragng a Lnear Programmng Approxmaton It s common to formulate lnear programmng approxmatons to networ revenue management problems under the assumpton that the arrvals of the customers tae on ther expected values. The optmal obectve values of such lnear programmng approxmatons can provde upper bounds on the optmal total expected revenues, whch become useful when assessng the performance of varous heurstcs. We can leverage an optmal soluton to a lnear programmng approxmaton to come up wth an approxmate polcy wth the same performance guarantee as n Theorem 3.6. We explan the dea usng the model gven n Secton 2, but we can ncorporate customer choce behavor as shown n Secton 4.1, and allow for products consumng multple unts of a resource as shown n Secton 4.2. For the model gven n Secton 2, the optmal total expected revenue s V 1 (C, where the value functons {V t : t T } are obtaned through the dynamc program n (1. To formulate the lnear program, we use the decson varables {z : J }, where z s the expected number of requests that we accept for product. Usng Λ = t T λt to denote the total expected number of requests for product over the whole sellng horzon, we consder the lnear program { } max r z : 1l { A }z C L, 0 z Λ J. (12 The obectve functon above accounts for the total expected revenue over the sellng horzon. The frst constrant ensures that the total expected capacty consumptons of the resources do not exceed the ntal capactes. The second constrant ensures that the expected numbers of requests that we serve for the products do not exceed the expected demands. It s well nown that the optmal obectve value of the lnear program n (12 provdes an upper bound on the optmal total expected revenue V 1 (C; see Bertsmas and Popescu (2003. For a collecton of avalablty-tracng bass functons B = {ϕ A : A L}, we approxmate the optmal value functons {V t : t T } usng value functon approxmatons {H t : t T } of

23 Ma, Rusmevchentong, Sumda and Topaloglu: An Approxmaton Algorthm for Networ Revenue Management 23 the form H t (x = γt ϕ A (x. We use the followng algorthm to compute the coeffcents {γ t : J, t T } n the value functon approxmatons. Intalzaton: Let B = {ϕ A : A L} be any collecton of avalablty-tracng bass functons, θ 0 be a tunng parameter, and {z : J } be an optmal soluton to the lnear program n (12. Intalze γ T +1 = 0 for all J. Coeffcent Computaton: For each t = T, T 1,..., 1, use the coeffcents {γ t+1 compute {γ t : J } as γ t = γ t+1 + z Λ λ t : J } to [ r θ ] + 1 1l C { A } γ t+1. (13 A Once we construct the value functon approxmatons {H t : t T } usng the algorthm above, we use the same decson functon n (4 n our approxmate polcy. The followng theorem gves a performance guarantee for our approxmate polcy. The proof s n Appendx C. Theorem 4.4 (Performance wth Lnear Programmng Approxmaton If the tunng parameter θ satsfes θ B, then the total expected revenue obtaned by the approxmate polcy s at least 1/(1 + θl of the optmal. Intutvely, f product s unlely to contrbute to the optmal total expected revenue, then we expect z to be close to zero. In ths case, notng (13, the coeffcents {γ t : t T } for ths product do not contrbute sgnfcantly to the value functon approxmaton. 5. Computatonal Experments In ths secton, we descrbe computatonal experments we conducted on a collecton of test problems to assess the numercal performance of our approxmate polcy. 5.1 Expermental Setup In our computatonal experments, we use the test problems n Topaloglu (2009. A number of other papers, ncludng Hu et al. (2013, Brown and Smth (2014, Vossen and Zhang (2015a,b, and Kunnumal and Tallur (2016a, adopted these test problems n ther computatonal experments as well. The test problems n Topaloglu (2009 orgnate from the arlne settng, where the resources correspond to the flght legs and the products correspond to tnerares n the arlne networ. In our test problems, we have a sngle hub and N spoes. There s a flght leg from each spoe to the hub and a flght leg from the hub to each spoe. Therefore, the number of flght legs

24 24 Ma, Rusmevchentong, Sumda and Topaloglu: An Approxmaton Algorthm for Networ Revenue Management s 2N. In Fgure 1, we show the structure of the arlne networ wth N = 6. We vary N n our computatonal experments. There are N orgn-destnaton pars that connect the hub to a spoe, N orgn-destnaton pars that connect a spoe to the hub, and N(N 1 orgn-destnaton pars that connect a spoe to another spoe. For each orgn-destnaton par, there are two tnerares, hgh-fare and low-fare. Therefore, the number of tnerares s 2 (2N + N(N 1. For a certan orgn-destnaton par, the revenue assocated wth the hgh-fare tnerary connectng ths orgndestnaton par s κ tmes the one assocated wth the correspondng low-fare tnerary. We vary κ n our computatonal experments as well. Recallng that λ t s the probablty that we have a request for product at tme perod t, f product corresponds to a hgh-fare tnerary, then λ t s an ncreasng functon of t, whereas f product corresponds to a low-fare tnerary, then λ t s a decreasng functon of t. Therefore, the requests for the hgh-fare tnerares tend to arrve later n the sellng horzon, whch maes t mportant to reserve the capactes for the hgh-fare tnerary requests by reectng the requests for the low-fare tnerares early n the sellng horzon. Notng that 1l t T { A } λ t gves the total expected demand for the capacty on flght leg, the ntal capacty of flght leg s set to be C = 1 1l α t T { A } λ t. Thus, larger values for α yeld tghter capactes. We vary α n our computatonal experments. Lettng N, κ, and α be as above, and recallng that T s the length of the sellng horzon, we vary T {200, 600}, N {4, 5, 6, 8}, κ {2, 3}, and α {1.0, 1.2, 1.6}, to get 48 test problems. 5.2 Benchmar Methods We compare two benchmar methods. The frst one s our approxmate polcy. The second one s based on the lnear programmng approxmaton n (12. Approxmate Polcy: Ths benchmar corresponds to our approxmate polcy wth the decson functon gven n (4. We refer to ths benchmar as AP, standng for approxmate polcy. In all our computatonal experments, we use the mnmum bass functons gven n Example 3.3. We repeated our computatonal experments wth polynomal bass functons gven n Example 3.4 as well, but the performance of AP under the polynomal bass functons was slghtly nferor. In our practcal mplementaton of AP, we mae two modfcatons. Frst, we dvde the sellng horzon nto fve equal segments and reconstruct our value functon approxmatons at the begnnng of each segment. In partcular, the begnnng of segment corresponds to tme perod ( 1 T + 1. If the remanng 5 capactes on the flght legs at the begnnng of segment are gven by the vector x, then we replace C n the recurson n (3 wth x and use ths recurson over tme perods T, T 1,..., ( 1 T to compute the coeffcents {γ t : J, t = ( 1 T + 1,..., T }. These coeffcents specfy the value 5 functon approxmatons that we use when mang the decsons over segment. When we reach

25 Ma, Rusmevchentong, Sumda and Topaloglu: An Approxmaton Algorthm for Networ Revenue Management 25 Fgure 1 Structure of the arlne networ wth N = 6 spoes. the begnnng of the next segment, we reconstruct our value functon approxmatons n a smlar fashon. Second, we calbrate the value for the tunng parameter θ at the begnnng of each segment. The values of the coeffcents {γ t : J, t T } n (3 depend on θ, whch, n turn, mples that the total expected revenue obtaned by AP also depends on θ. When reconstructng our value functon approxmatons at the begnnng of each segment, we search for the best tunng parameter over the nterval [1, 15] wth a precson of Gven that we use the tunng parameter θ when reconstructng our value functon approxmatons at the begnnng of segment, let U,θ (x be the total expected revenue obtaned by AP over tme perods ( 1 T + 1,..., T startng wth the 5 capactes x for the flght legs. Computng the total expected revenue U θ,t (x exactly s ntractable, but we estmate ths quantty usng smulaton. At the begnnng of segment, we choose the value of the tunng parameter θ as arg max{u,θ (x : θ {1, 1.01, 1.02,..., 15}}. We use ths value for the tunng parameter untl we reach the begnnng of the next segment. The performance guarantee for AP that we gve n Theorem 3.6 holds as long as θ B and t mproves when we use smaller values for the tunng parameter θ. Accordng to Example 3.3, we have B = 1 for the mnmum bass functons. So, we can choose the tunng parameter θ as one to obtan the best possble performance guarantee for AP, but our computatonal experments ndcated that although settng θ = 1 gves the best possble theoretcal performance guarantee, choosng a dfferent value for the tunng parameter may provde better practcal performance. Bd Prce Polcy: Ths benchmar s the standard bd prce polcy n the networ revenue management lterature; see Secton 3.3 n Tallur and van Ryzn (2005. We refer to ths benchmar as BP, standng for bd prce polcy. The man dea behnd BP s to use the optmal values of the dual varables assocated wth the frst constrant n the lnear program n (12 to estmate the value of a unt of capacty on each flght leg. In ths case, f the revenue from a certan tnerary exceeds the value of the capactes used by ths tnerary, then we accept the request for the tnerary. To be specfc, lettng {µ : L} be the optmal values of the dual varables assocated wth the