New compac formulaons for choce nework revenue managemen Sum Kunnumkal Kalyan Tallur Augus 13, 2015 Absrac The choce nework revenue managemen model ncorporaes cusomer purchase behavor as a funcon of he offered producs, and s he approprae model for arlne and hoel nework revenue managemen, dynamc sales of bundles, and dynamc assormen opmzaon. The opmzaon problem s a sochasc dynamc program and s nracable, and a lnear program approxmaon called choce deermnsc lnear program (CDLP ) s usually used o generae dynamc conrols. Recenly a compac lnear programmng formulaon of CDLP for he mulnomal log (MNL) model of cusomer choce has been proposed. In hs paper we oban a beer approxmaon o he dynamc program han CDLP whle reanng he appealng properes of a compac lnear programmng represenaon. Our formulaon s based on he affne relaxaon of he dynamc program. We frs show ha he affne relaxaon s NP-complee even for a sngle-segmen MNL model. Neverheless, by analyzng he affne relaxaon we derve new lnear programs ha approxmae he dynamc programmng value funcon provably beer, beween he CDLP value and he affne relaxaon, and ofen comng close o he laer n our numercal expermens. We gve exensons o he case wh mulple cusomer segmens where choce by each segmen s accordng o he MNL model. Fnally we perform exensve numercal comparsons on he varous mehods o evaluae her performance. 1 Inroducon Revenue managemen (RM) nvolves conrollng he avalably of producs o cusomers who arrve over me o purchase hem. RM models ncorporang cusomer choce behavor have receved much aenon n recen years as he purchasng decson depends on he assormen of producs made avalable for sale. Therefore, decsons on wha producs o make avalable for sale (he offer se) have o consder resource avalables as well as esmaes of cusomer purchase probables as a funcon of he offer se. In Nework Revenue Managemen (NRM), he producs consume mulple resources, and s relevan for he arlne, adversng, hoel and car renal ndusres. In he canoncal arlne example, producs correspond o nerares whle resources correspond o sea capaces on he flgh legs; for hoels, he producs correspond o mul-ngh says whle resources are hoel rooms. The NRM Indan School of Busness, Hyderabad, 500032, Inda, emal: sum kunnumkal@sb.edu ICREA and Unversa Pompeu Fabra, Ramon Tras Fargas 25-27, 08005 Barcelona, Span, emal: kalyan.allur@upf.edu 1
problem can be formulaed as a sochasc dynamc program. However, solvng he opmaly equaons and compung he value funcons become nracable even for moderaely szed problems. We refer he reader o Tallur and van Ryzn [19] for background on NRM. Consderng he nracably of he NRM dynamc program, Gallego, Iyengar, Phllps, and Dubey [6] and Lu and van Ryzn [11] proposed a lnear-programmng approxmaon called he CDLP (Choce Deermnsc Lnear Program). The opmal objecve funcon value of hs lnear program gves an upper bound on he value funcon. Upper bounds are useful boh for dervng conrols from hem, as well as o assess he sub-opmaly of polces. CDLP however, has a drawback: he number of columns are exponenal n he number of producs, and so has o be solved usng column generaon. Lu and van Ryzn [11] show ha he CDLP column generaon procedure s racable for he mulnomal log (MNL) choce model wh mulple cusomer segmens when he cusomers consderaon ses do no overlap. More recenly, Gallego, Ralff, and Shebalov [7] show ha CDLP has a compac lnear programmng formulaon under he MNL model wh dsjon consderaon ses. On he oher hand, f he consderaon ses overlap, CDLP s nracable even for he MNL model, as shown by Bron, Méndez-Díaz, and Vulcano [3], Rusmevchenong, Shmoys, Tong, and Topaloglu [16]. Relaed o hs body of research, Zhang and Adelman [23] propose an affne relaxaon o he NRM dynamc program and show ha obans a gher upper bound han CDLP. Ths paper bulds on hese advances and makes he followng research conrbuons: 1. We show ha he affne relaxaon of Zhang and Adelman [23] s NP-hard even for he snglesegmen MNL model, arguably one of he smples possble choce models. Ths movaes soluon mehods ha ghen he CDLP bound and reman racable a leas for he snglesegmen MNL model. 2. We propose new, compac lnear programmng formulaons ha gve a gher bound on he dynamc program value funcon han CDLP, mprovng upon he work of Gallego e al. [7]. Compac represenaons are aracve from an mplemenaon perspecve snce elmnaes he need for cusomzed codng n he form of consran-separaon or column-generaon echnques. To our knowledge, hese are he frs racable approxmaon mehods ha are provably gher han CDLP. 3. We show how our deas can be exended o he mxure of mulnomal logs (MMNL) model ha can approxmae any random uly choce model arbrarly closely; McFadden and Tran [12]. 4. We propose conrol polces based on he new formulaons and es her performance hrough an exensve numercal sudy. We show ha our mehods can yeld noceable benefs boh n erms of gher bounds and mproved revenue performance. Our compuaonal expermens reveal ha he proposed mehods srke a good balance beween he mproved qualy of he bound versus he ncrease n soluon me. An neresng observaon s ha he revenue mprovemens from our mehods ypcally exceed he correspondng mprovemens n he upper bounds. Our mehods obans sharper value funcon approxmaons owards he end of he sellng horzon when capacy s relavely scarce. I urns ou ha he ably o make mproved decsons n such capacy consraned scenaros has a sgnfcan payoff. The remander of he paper s organzed as follows: In 2 we revew he leraure and n 3 we descrbe he choce NRM model, he noaon, and he basc dynamc program. In 4 we descrbe 2
he CDLP and he affne relaxaon of he NRM dynamc program. Nex, n 5 we show ha he affne relaxaon s NP-hard even for he sngle-segmen MNL model. We descrbe our frs racable approxmaon mehod n 6 and buld on o oban racable, gher approxmaons n 7. 8 dscusses exensons o varans of he MNL model ncludng he MMNL model. 9 conans our compuaonal sudy usng he new formulaons. 2 Leraure revew As menoned earler, our paper advances he lne of research on choce NRM naed n Gallego e al. [6], Lu and van Ryzn [11] and Zhang and Adelman [23]. These papers use he well-known MNL choce model (see Ben-Akva and Lerman [2]) whch s aracve from an esmaon and opmzaon sandpon, and has been wdely used n ransporaon modelng, operaons and markeng. For nsance, van Ryzn and Mahajan [21] and Topaloglu [20] use he model n real assormen plannng, and Feldman, Lu, Topaloglu, and Zya [5] n an applcaon n healhcare. Da, Dng, Kleyweg, Wang, and Zhang [4] descrbe a choce RM projec a a major arlne where hey fnd ha opmzaon wh he MNL model gves superor polces o more complcaed models. Under he MNL choce model, Lu and van Ryzn [11] show ha column generaon for CDLP can be effcenly carred ou f he consderaon ses of he segmens are dsjon (consderaon ses arse hs way: he populaon s made of segmens, and each segmen s neresed n only a subse of producs, s consderaon se; for nsance a segmen may be neresed n an nerary and s consderaon se are he flghs on hs nerary). Recenly Gallego e al. [7] gve an equvalen, compac lnear programmng formulaon of CDLP for he MNL choce model whch elmnaes he need for column generaon. On he oher hand, f he consderaon ses overlap, CDLP s nracable even for he MNL model as shown n Bron e al. [3] and Rusmevchenong e al. [16]. There are a number of oher approaches o oban upper bounds on he NRM value funcon. Zhang and Adelman [23] and Messner and Srauss [13] use he lnear programmng approach o approxmae dynamc programmng wh Zhang and Adelman [23] usng affne approxmaons, and Messner and Srauss [13] pecewse-lnear approxmaons. Tallur [18] proposes a segmen-based concave program and Messner, Srauss, and Tallur [14] show how o furher srenghen hs formulaon by addng equales called produc-cus. There are wo mporan dmensons o assess he dfferen approxmaon mehods. One s he qualy of he upper bound and he oher s compuaonal racably. On he qualy dmenson, he approaches proposed by Zhang and Adelman [23] and Messner and Srauss [13] are provably gher han CDLP. However, n hs paper, we show ha he affne relaxaon (AF ) of Zhang and Adelman [23] urns ou o be nracable even for he MNL model wh a sngle segmen. Snce pecewse-lnear value funcon approxmaons nclude affne funcons as a specal case, hs mples a smlar hardness resul for he approach proposed by Messner and Srauss [13]. 3 Problem formulaon A produc s a specfcaon of a prce and he se of resources ha consumes. For example, a produc would be an nerary-fare class combnaon for an arlne nework, where an nerary s a combnaon of flgh legs; n a hoel nework, a produc would be a mul-ngh say for a parcular room ype a a ceran prce pon. 3
Tme s dscree and he sales horzon s assumed o conss of τ nervals, ndexed by. The sales horzon begns a me = 1 and ends a = τ; all he resources persh nsananeously a me τ + 1. We make he sandard assumpon ha he me nervals are fne enough so ha he probably of more han one cusomer arrvng n any sngle me perod s neglgble. We le I denoe he se of resources and J he se of producs. We ndex resources by and producs by j. We le f j denoe he revenue assocaed wh produc j and use I j I o denoe he se of resources used by produc j. We le 1 [ ] denoe he ndcaor funcon, 1 f rue and 0 f false and 1 [Ij ] denoe he vecor of resources used by produc j, wh a 1 n he h poson f I j and a 0 oherwse. We use J J o denoe he se of producs ha use resource. In each perod he frm offers a subse S of s producs for sale, called he offer se. We wre I S whenever here s a j S wh I j ; ha s, here s a leas one produc n he offer se S ha uses resource. We use superscrps on vecors o ndex he vecors (for example, he resource capacy vecor assocaed wh me perod would be r ) and subscrps o ndcae componens (for example, he capacy on resource n me perod would be r ). Therefore, r1 = [r 1 ] represens he nal capacy on he resources and r = [r ] denoes he remanng capacy on he resources a he begnnng of me perod. The remanng capacy r akes values n he se R = {0,..., r 1} and R = R represens he sae space a each me. 3.1 Demand model We have mulple cusomer segmens, each wh dsnc purchase behavor. We le L denoe he se of cusomer segmens. In each perod a cusomer from segmen l L arrves wh probably λ l so ha λ = l λ l s he oal arrval rae. Noe ha condoned on a cusomer arrval, λ l /λ s he probably ha he cusomer belongs o segmen l. Cusomer segmen l has a consderaon se C l J of producs ha consders for purchase. We assume hs consderaon se s known o he frm (by a prevous process of esmaon and analyss). The choce probables of a segmen-l cusomer are no affeced by producs no n s consderaon se. Gven an offer se S, an arrvng cusomer n segmen l purchases a produc j n he se S l = C l S or leaves whou makng a purchase. The no-purchase opon s ndexed by 0 and s always presen for he cusomer. Whn each segmen, choce s accordng o he MNL model. The MNL model assocaes a preference wegh wh each alernave ncludng he no-purchase alernave. We le wj l denoe he preference wegh assocaed wh a segmen-l cusomer for produc j. Whou loss of generaly, by suably normalzng he weghs, we se he no-purchase wegh w0 l o be 1. The probably ha a segmen-l cusomer purchases produc j when S s he offer se s Pj(S) l = wl j 1 [j S l ] 1 + k S l wk l. (1) The probably ha he cusomer does no purchase anyhng s P0(S) l = 1/(1 + k S l wk l ). We noe ha he preference weghs are npus o our model; esmang hem s ousde he scope of he paper. We refer he reader o Ben-Akva and Lerman [2] for furher background on hs popular choce model. Gven a cusomer arrval, and an offer se S, he probably ha he frm sells j S s gven by P j (S) = λ l l λ P j l(s) and makes no sale wh probably P 0(S) = 1 j S P j(s). The expeced 4
sales for produc j s herefore λp j (S) = l λ lpj l(s), whle 1 λ + λp 0(S) = 1 j S λp j(s) s he probably of no sales n a me perod. Gven an offer se S, Q l (S) = j J Pj l (S) denoes he expeced capacy consumed on resource condonal on a segmen-l cusomer arrval and Q (S) = λ l l λ Ql (S) denoes he expeced capacy consumed on resource condonal on a cusomer arrval. Noe ha λq (S) = l λ lq l (S) gves he expeced capacy consumed on resource n a me perod. The revenue funcons can be wren as R l (S) = j S l f j Pj l(s l) and R(S) = j S f jp j (S). In wha follows f we are consderng a sngle-segmen MNL model, we drop he subscrp l from he probables. We assume ha he arrval raes and choce probables are saonary. Ths s for brevy of noaon and all of our resuls go hrough wh non-saonary arrval raes and choce probables. 3.2 Choce dynamc program The dynamc program (DP) o deermne opmal conrols s as follows. Le V (r ) denoe he maxmum expeced revenue o go, gven remanng capacy r a he begnnng of perod. Then V (r ) mus sasfy he Bellman equaon V (r ) = max λp S S(r ) j (S) [ ( f j + V +1 r ( 1 [Ij])] + [λp0 (S) + 1 λ] V +1 r ), (2) where j S S(r) = { j 1 [ Ij ] r } represens he se of producs ha can be offered gven he capacy vecor r. The boundary condons are V τ+1 (r) = V (0) = 0 for all r and for all, where 0 s a vecor of all zeroes. V DP = V 1 (r 1 ) denoes he opmal expeced oal revenue over he sales horzon, gven he nal capacy vecor r 1. 3.3 Lnear programmng formulaon of he dynamc program The value funcons can, alernavely, be obaned by solvng a lnear program (LP). The lnear programmng formulaon of (2) has a decson varable for each sae vecor n each perod V (r) and s as follows: V DP LP = mn V (DP LP ) s.. V 1 (r 1 ) (3) V (r) j λp j (S) [ f j + V +1 ( r 1[Ij ]) V+1 (r) ] + V +1 (r) r R, S S(r),. Boh dynamc program (2) and lnear program DP LP are compuaonally nracable, bu lnear program DP LP urns ou o be useful n developng value funcon approxmaon mehods, as shown n Zhang and Adelman [23]. 5
4 Approxmaons and upper bounds In he followng, we oulne he wo approxmaons suded n hs paper. We frs descrbe he choce deermnsc lnear program and hen oulne he affne relaxaon mehod. 4.1 Choce deermnsc lnear program (CDLP ) The choce deermnsc lnear program (CDLP ) proposed n Gallego e al. [6] and Lu and van Ryzn [11] s a cerany-equvalence approxmaon o (2). We wre CDLP as he followng LP: V CDLP = max h (CDLP ) s.. λr(s)h S, S λq (S)h S,k r 1, (4) k=1 S h S, = 1 (5) S h S, 0 S,. The decson varable h S, can be nerpreed as he frequency wh whch se S (ncludng he empy se) s offered a me perod. The frs se of consrans ensure ha he oal expeced capacy consumed on resource up unl me perod does no exceed he avalable capacy. Noe ha snce h S, 0, consrans (4) are redundan excep for he las me perod. Sll, hs expanded formulaon s useful when we compare CDLP wh oher approxmaon mehods. The second se of consrans saes ha he sum of he frequences adds up o 1. The dual of CDLP urns ou o be useful n our analyss. Assocang dual varables γ = {γ,, } wh consrans (4) and β = {β } wh consrans (5), he dual of CDLP s V dcdlp = mn β,γ (dcdlp ) s.. β + γ, r 1 β + ( τ ) γ,k λq (S) λr(s), S (6) k= γ, 0,. Lu and van Ryzn [11] show ha he opmal objecve funcon value of CDLP, V CDLP s an upper bound on V DP LP. Besdes gvng an upper bound on he value funcon, CDLP can also be used o consruc dfferen heursc conrol polces. Le ˆγ = {ˆγ,, } denoe he opmal values of he dual varables assocaed wh consrans (4), we nerpre ˆγ, as gvng he value of an addonal un of capacy on resource from me perod o + 1. Wh hs nerpreaon, τ s= ˆγ,s gves he margnal value of capacy on resource a me perod. Zhang and Adelman [23] approxmae he value funcon as ˆV (r) = ( τ ) ˆγ,s r (7) s= 6
and f r s he vecor of remanng resource capaces a me, solve he problem [ max λp S S(r ) j (S) f j + ˆV ( +1 r 1 [Ij ]) ] + [λp 0 (S) + 1 λ] ˆV ( +1 r ), (8) j S and use he polcy of offerng he se ha acheves he maxmum n he above opmzaon problem. The number of decson varables n CDLP s exponenal n he number of producs and so has o be solved usng column generaon. The racably of column generaon depends on he underlyng choce model. Lu and van Ryzn [11] show ha he column generaon procedure can be effcenly carred ou when choce s accordng o he MNL model and he consderaon ses of he dfferen segmens do no overlap. Tha s, we have C l C m = for segmens l and m. Under he same se of assumpons, Gallego e al. [7] furher show ha CDLP has he followng equvalen, compac formulaon V SBLP = max x (SBLP ) s.. λ l f j x l j, l j C l l j J C l λ l x l j, r 1, (9) x l 0, + j C l x l j, = 1 l, x l j, w l j x l 0, 0 x l 0,, x l j, 0 l, j,. l, j C l, In he above sales-based lnear program (SBLP ), he decson varables x l j, can be nerpreed as he sales rae for produc j a me. Noe ha SBLP s a compac formulaon snce he number of consrans and decson varables s polynomal n he number of producs and resources. On he oher hand, f he consderaon ses overlap, Bron e al. [3] and Rusmevchenong e al. [16] show ha he CDLP column generaon s NP-complee even under he MNL choce model. 4.2 Affne relaxaon The second approxmaon mehod we consder s he affne relaxaon, where he value funcon s approxmaed as V (r) = θ + V,r. Noe ha V, can be nerpreed as he margnal value of capacy on resource a me. Subsung hs value funcon approxmaon no he formulaon DP LP we ge he affne relaxaon LP V AF = mn θ,v θ 1 + V,1 r 1 (AF ) s.. θ + V, r j λp j (S) f j V,+1 + θ +1 + Ij V,+1 r r R, S S(r), θ 0, V, 0, wh he boundary condons θ τ+1 = 0, V,τ+1 = 0. Zhang and Adelman [23] show ha he opmal objecve funcon value V AF s an upper bound on he value funcon and ha here exss an opmal soluon (ˆθ, ˆV ) of AF ha sasfes ˆV, ˆV,+1 0 for all and. 7
Whle he number of decson varables n AF s manageable, he number of consrans s exponenal boh n he number of producs as well as he number of resources. Vossen and Zhang [22] use Danzg-Wolfe decomposon o derve a reduced, equvalen formulaon of AF, where he number of consrans s exponenal only n he number of producs. We gve an alernave, smpler proof of he reducon below. The analyss we presen also urns ou o be useful n he developmen of our racable soluon mehods laer. We make a change of varables β = θ θ +1, and γ, = V, V,+1 and wre AF equvalenly as mn β,γ s.. β + β + γ, r 1 γ, r + j γ, 0,, λp j (S) Ij τ k=+1 γ,k f j 0 r R, S S(r), (10) where we use he fac ha V, = τ k= γ,k and so τ k= γ,k can be nerpreed as he margnal value of capacy on resource a me. Noe ha he nonnegavy consran on γ, s whou loss of generaly, snce here exss an opmal soluon o AF ha sasfes V, V,+1 0. Now, consrans (10) can be wren as mn r R,S S(r) β + γ, r + λp j (S) j Ij τ k=+1 γ,k f j 0 (11) for all. Snce γ, 0, he coeffcen of r n mnmzaon problem (11) s nonnegave, and we can assume r {0, 1} n he mnmzaon (as larger values of r would be redundan n S S(r) and would only ncrease he objecve value). Moreover, snce γ, 0, for any se S, we have r = 0 for I S. On he oher hand, feasbly requres we have r = 1 for I S. Therefore, (11) can be wren as mn S β + 1 [ IS ]γ, + j λp j (S) Ij τ k=+1 γ,k f j 0. and we can wre AF equvalenly as V RAF = mn β + β,γ (RAF ) s.. γ, r 1 β + 1 [ IS ]γ, + γ, 0,. [( τ ) ] γ,k λq (S) λr(s), S (12) k=+1 Noce ha he number of consrans n he reduced formulaon RAF s an order of magnude smaller han AF. Takng he dual of RAF by assocang dual varables h S, wh consrans (12), 8
we ge V draf = max h (draf ) s.. λr(s)h S, S S ( 1 ) λq (S)h S,k + 1 [ IS ]h S, r 1, k=1 h S, = 1 S h S, 0 S,. The above argumens mply ha Proposon 1. (Vossen and Zhang [22]) V AF = V RAF = V draf. We close hs secon wh wo remarks. Frs, n addon o gvng an upper bound on he opmal expeced oal revenue, he affne relaxaon can also be used o consruc heursc conrol polces. Leng ( ˆβ, ˆγ), wh ˆβ = { ˆβ } and ˆγ = {ˆγ,, }, denoe an opmal soluon o RAF, we use τ s= ˆγ,k o approxmae he margnal value of capacy on resource a me. We approxmae V (r) usng (7) and solve problem (8) usng hs value funcon approxmaon o decde on he se of producs o be offered a me perod. Second, Zhang and Adelman [23] show ha he upper bound obaned by AF s gher han CDLP. In ha sense, AF s a beer approxmaon han CDLP. A he same me, s mporan o undersand he compuaonal effor requred by AF o oban a gher bound. We explore hs queson n he followng secon. 5 Tracably of he affne relaxaon for MNL wh a sngle segmen In hs secon, we focus on he racably of he affne relaxaon for he sngle-segmen MNL model. We resrc our aenon o he sngle-segmen MNL snce s one of he few cases where CDLP s racable. We show ha he affne relaxaon s NP-complee even for hs smple choce model. Snce we resrc aenon o he sngle-segmen MNL model, we drop he segmen superscrp l for ease of noaon. So we wre he preference weghs as w j, and he choce probables, expeced resource consumpons and expeced revenues as 1 [j S]w j P j (S) = 1 + k S w k Q (S) = j J S w j 1 + j S w j R(S) = j S f jw j 1 + j S w. j Snce RAF has an exponenal number of consrans, we have o generae on he fly consrans (12) volaed by a soluon. Followng he resul of Gröschel, Lovász, and Schrjver [8] polynomalsolvably of a lnear program s equvalen o polynomal-me generaon of volaed consrans. Subsung he MNL choce probables, expeced resource consumpons and expeced revenues no consran (12), we oban β + γ S, + [( τ ) j J γ,k λ S w ] j 1 + j S j S w λ f jw j j 1 + j S w j k=+1 9
where γ S, = 1 [ IS ]γ,. Mulplyng boh sdes by he posve quany 1+ j S w j and smplfyng, consran (12) of RAF can be equvalenly wren as β γ S, 1 + w j ζ j, (β, γ), (13) j S j S where ζ j, (β, γ) = w j β + λ Ij τ k=+1 γ,k f j. Snce he consran has o be sasfed for every S and, we have β Π AF (β, γ) for all, where Π AF (β, γ) = max S γ S, 1 + w j ζ j, (β, γ) (14) j S j S and he affne relaxaon consran (12) can be equvalenly wren as β Π AF (β, γ). (15) Generang consrans on he fly nvolves checkng, gven a se of values (β, γ), f consran (13) s sasfed for all S. If no, we add he volaed consran o he LP. In oher words, he RAF separaon problem a me nvolves solvng opmzaon problem (14) and deermnng f β Π AF se (β, γ). If β Π AF (β, γ), hen consran (13) s sasfed for all S a me. Oherwse, he Ŝ whch aans he maxmum n problem (14) volaes he consran, and we add he consran for se Ŝ o he LP. Therefore, solvng problem (14) n an effcen manner s key o separang consrans (13) effcenly. Proposon 2 below saes ha he affne relaxaon separaon problem for MNL wh a sngle segmen, as gven n (13) s NP-complee. Proposon 2. The followng problem s NP-complee: Inpu: w j 0, 1 λ 0, f j 0, and values β and γ, 0. Queson: Is here a se S ha volaes (13)? Proof Our reducon s from he NP-complee maxmum edge bclque problem (Peeers [15]). We sae frs he defnons and noaon n he problem. The problem s defned on an undreced, bpare graph G = (V 1 V 2, E), wh V 2 = m 2. A (k 1, k 2 )-bclque s a complee bpare subgraph of G,.e., a subgraph conssng of a par (X, Y ) of verex subses X V 1 and Y V 2, X = k 1 > 1, Y = k 2 > 1, such ha here exss an edge (x, y) E, x X, y Y. Noe ha he number of edges n he bclque s k 1 k 2. Maxmum edge bclque problem (MBP) Inpu: A bpare graph G = (V 1 V 2, E) and a posve neger p. Queson: Does G conan a bclque wh a leas p edges. 10
Consder he complemen bpare graph Ḡ of G defned on he same verex se as G, where here s an edge e = (u, v) n graph Ḡ f and only f here s no edge beween u and v n G. Defne a cover C S V 2 of a subse S V 1 n he complemen graph Ḡ, as C S = {v V 2 e = (u, v) Ḡ, u S}. By defnon f C S s a cover of some subse S, means here s no edge from any u S o any v V 2 \C S n he graph Ḡ. Hence, as G s a complemen of Ḡ, here s an edge from every u S o every v V \C(S) n G, hus represenng a bclque beween S and V \C(S) n he graph G. Now we se up he reducon for he separaon for (13). In equaon (13), for each u V 1, we (p+1) assocae a produc j wh f j = m 2 p and w j = m 2. For each v V 2, we assocae a resource wh weghs γ, = 1 p and γ,k = 0, k >. The resource consumpons of he producs j are defned from he graph Ḡ: j conans all he such ha here s an edge beween he assocaed nodes n Ḡ. We le λ = 1, β = m 2. We now clam ha G has a (k 1, k 2 )-bclque wh k 1 k 2 > p f and only f here s a se S ha volaes he nequaly (13) for hs nsance. or, Wh he above values, S V 1, wh S = k 1, C(S) = m 2 k 2 volaes (13) f and only f (p+1) j S p (m 2 ) 2 m 2 (1 + j S m < 1 2) p C(S) m 2 (p ( + 1)m 2k ) 1 < p + k 1 1 m 2 (m 2 k 2 ) p or mulplyng boh sdes by he posve number p( 1 m 2 + k 1 ), ( ) ( ) 1 1 m 2 p + k 1 (p + 1)m 2 k 1 < (m 2 k 2 ) + k 1 m 2 m 2 or, The erm 0 < (m 2 k 2 ) m 2 < 1 mples, f and only f p < (m 2 k 2 ) m 2 + k 2 k 1. p < k 2 k 1. Therefore, even hough he affne relaxaon ghens he CDLP bound, comes a a sgnfcan cos. Ths movaes he soluon mehod ha we propose n he followng secon, whch ghens he CDLP bound whle remanng racable. 6 Weak affne relaxaon In hs secon we propose our frs racable approxmaon mehod ha ghens he CDLP bound. We also show ha our approxmaon mehod can, n fac, be formulaed as a compac LP. In our 11
nal developmen, we resrc aenon o he sngle-segmen MNL choce model. We emphasze ha hs s only for clary of exposon. In 8 we show how he deas can be readly exended o more realsc varans of he MNL model ha consder mulple cusomer segmens. Moreover, all of he es problems n our compuaonal expermens nvolve mulple cusomer segmens wh and whou overlappng consderaon ses. 6.1 Prelmnares All of our approxmaon mehods nvolve solvng an opmzaon problem of he form mn β,γ β + γ,r 1 subjec o he consrans β Π (β, γ), where Π (, ) s a scalar funcon of β = {β } and γ = {γ,, }. The followng observaon s useful n comparng he upper bounds obaned by he dfferen approxmaon mehods. Lemma 1. Le V I = mn β,γ β + γ, r 1 (I) s.. β Π I (β, γ) γ, 0,, and le V II = mn β,γ (II) s.. β + γ, r 1 β Π II (β, γ) γ, 0,. If Π I (β, γ) Π II (β, γ) for all, hen V I V II. Proof The proof follows by nong ha a feasble soluon o problem (II) s also feasble o problem (I) and boh opmzaon problems have he same objecve funcon. 6.2 CDLP vs. AF for sngle-segmen MNL We begn by comparng he CDLP and AF separaon problems for he sngle-segmen MNL model. For hs choce model, he CDLP consrans can be separaed effcenly, whle he AF separaon problem s nracable. Comparng he CDLP and AF separaon problems helps us denfy he dffcul erm n he affne relaxaon. Replacng hs dffcul erm n he AF separaon problem wh somehng more racable yelds our approxmaon mehod. Usng he sngle-segmen MNL formulas for he expeced resource consumpons and expeced revenues, he CDLP dual consran (6) can be wren as β w j β + λ τ γ,k f j, S j S Ij k= 12
whch looks smlar o he rgh-hand-sde of (13) excep ha he nner summaon over k runs from nsead of + 1. To make he comparson wh AF easer, we rewre he above consran as β Π CDLP (β, γ) (16) where Π CDLP (β, γ) = max S λ j S w j γ, ζ j, (β, γ). (17) Ij j S Snce 0 λ 1, and γ S, = 1 [ I S ]γ, I j γ, 0 for all j S, we have γ S, 1 + w j λ w j γ,. j S j S Ij Therefore Π AF (β, γ) Π CDLP (β, γ) and by Lemma 1, V AF V CDLP, whch gves an alernave proof of he AF bound beng gher han he CDLP bound. More mporanly, he comparson hns a how we can oban racable relaxaons ha are gher han CDLP. 6.3 A racable approxmaon We are now ready o descrbe our frs racable approxmaon mehod, whch we refer o as weak affne relaxaon (war). The dffcul erm n (14) s he γ S, (1 + j S w j), and CDLP s racable as replaces hs by λ j S w j( I j γ, ). We nsead replace he γ S, (1 + j S w j) erm n (14) wh γ S, + j S w j( I j γ, ) and solve he lnear program V war = mn β,γ (war) s.. β + γ, r 1 β Π war (β, γ) (18) γ, 0,, where Π war = max S γ S, j S w j γ, ζ j, (β, γ). (19) Ij j S Proposon 3 below shows ha war obans an upper bound on he value funcon ha s weaker han AF bu sronger han CDLP. Kunnumkal and Tallur [9] show ha also gves a gher upper bound han by workng wh a connuous relaxaon of Π AF (β, γ). Proposon 3. V AF V war V CDLP. Proof The proof follows by nong ha γ S, 1 + w j γ S, + w j γ, λ w j γ,. j S j S Ij j S Ij 13
Therefore Π AF (β, γ) Π war (β, γ) Π CDLP (β, γ) and he resul now follows from Lemma 1. In he remander of hs secon, we show ha he weak affne relaxaon upper bound, V war, can be obaned n a racable manner; moreover we show ha he weak affne relaxaon LP can, n fac, be reformulaed as a compac lnear program where he number of varables and consrans s polynomal n he number of producs and resources. Observe ha solvng problem (19) n an effcen manner s key o separang he weak affne relaxaon consrans effcenly. Therefore, we focus on solvng opmzaon problem (19). Inroducng decson varables q, and u j,, respecvely, o ndcae f resource and produc j are open a me, problem (19) can be formulaed as he neger program Π war (β, γ) = max q,u γ, q, j ζ j, (β, γ) + w j γ, u j, (20) Ij s.. u j, q, 0 I j, j (21) q, 1 (22) u j, 0, neger j. (23) Noe ha he frs consran ensures ha a produc s open only f all he resources uses are open. Now, observe ha he consran marx of he above neger program has exacly one +1 and one 1 coeffcen n each row, and hence s oally unmodular. So we can gnore he neger resrcon and solve (20) (23) exacly as a lnear program. In fac, problem (20) (23) can also be solved combnaorally as a flow problem: he dual of he LP can be ransformed o be a nework flow problem on a bpare graph wh one se of nodes represenng producs and he oher sde resources and edges represenng produc-resource ncdence, and flow from a source o a snk node, each conneced o he produc and resource nodes respecvely; fas algorhms of Ahuja, Orln, Sen, and Tarjan [1] can hen be used o solve he problem n me O( I E + mn( I 3, I 2 E )) where I s he number of resources and E s he number of edges n hs graph. Therefore, problem (20) (23) can be solved effcenly and separang he war consrans s racable. We nex show ha war can be formulaed as a compac LP elmnang he need for generang consrans on he fly. Snce he separaon problem can be solved as a LP where all he fxed values (β, γ) appear n he objecve funcon only, we can fold no he orgnal LP as follows: Frs ake he dual of (20) (23) wh dual varables π,j, correspondng o (21), and ψ, o (22): Π war (β, γ) = mn π,ψ s.. ψ, π,j, ζ j, (β, γ) + w j γ, I j Ij j J π,j, + ψ, = γ, π,j,, ψ, 0, j J. j 14
Then use he second consran n he above LP o elmnae he varable ψ, o wre he dual as Π war (β, γ) = mn π,j, γ, π j J s.. π,j, ζ j, (β, γ) + w j γ, j (24) I j Ij π,j, γ, (25) j J π,j, 0, j J. Now we fold n he above LP formulaon of Π war (β, γ) no consrans (18) and wre war equvalenly as V war = mn β,γ,π s.. β + β (24), (25) γ, r 1 π,j, γ, j J γ,, π,j, 0, j J,. The sze of he above LP s polynomal n he number of resources and producs. Hence, no only s war sronger han CDLP, s also racable and has a compac formulaon. Noce ha hs formulaon would have been hard o derve and jusfy whou he lne of reasonng sarng from AF. as The dual of he above LP gves more nsgh no he weak affne relaxaon. We ge he dual LP V war = max x,ρ λf j x j, j 1 (dwar) s.. x 0, + λx j,s + x j, ρ, r 1 s=1 j J j J x 0, + j x j, = 1, x j, w j x 0, + ρ, 0, j J, x 0,, x j,, ρ, 0, j,. If we nerpre x j, as he sales rae for produc j a me and x 0, ρ, as he resource level nopurchase rae a me, hen we can vew war as a refnemen of SBLP of Gallego e al. [7], where he sales raes a each me perod are modulaed by he expeced remanng resource capaces. 15
7 Tgher, racable relaxaons The weak affne relaxaon s based on solang he dffcul erm n he affne relaxaon and replacng wh a smpler, more racable erm. In hs secon, we buld on hs dea and propose wo racable approxmaon mehods ha furher ghen he war bound. We agan resrc aenon o he sngle-segmen MNL model o reduce noaonal overhead. In 8, we descrbe exensons o mul-segmen varans of he MNL model. 7.1 Weak affne relaxaon + (war + ) We descrbe a smple way o ghen he war bound, whle reanng he compac formulaon. Assocang decson varables q, and u j,, respecvely, o ndcae f resource and produc j are open, he AF separaon problem (14) can be wren as Π AF (β, γ) = max q,u γ, q, 1 + j s.. (21), (22), (23). w j u j, j ζ j, (β, γ)u j, Now war replaces he produc erm q, u j, for j / J n he frs summaon wh 0 and snce q, u j, 0, we have Π AF (β, γ) Π war (β, γ). Nong ha q, u j, q, + u j, 1, we propose replacng he rgh hand sde of consrans (15) wh Π war+ (β, γ) = max γ, q, γ, w j χ,j, ζ j, (β, γ) + w j γ, u j, q,u j Ij s.. (21), (22) The followng lemma s mmedae. Lemma 2. Π AF (β, γ) Π war+ j / χ,j, q, + u j, 1 u j,, χ,j, 0 j, / I j., j / (β, γ) Π war (β, γ). Therefore, we replace he rgh hand sde of consrans (15) wh Π war+ (β, γ) and solve he LP V war+ = mn β + γ, r 1 β,γ (war + ) s.. β Π war+ (β, γ) (26) γ, 0,. We refer o hs mehod as weak affne relaxaon + (war + ). Lemma 2 ogeher wh Lemma 1 mples ha V AF V war+ V war. Therefore, war + furher ghens he war bound. Noe however ha he war + separaon problem can have as many as I J addonal consrans compared o war. Sll, he war + separaon problem nvolves solvng a lnear program and hence s racable. Moreover, s possble o oban a compac formulaon of war + by followng he seps n 6.3; we om he deals. 16
7.2 A herarchcal famly of relaxaons In hs secon we show how o consruc a herarchcal famly of relaxaons ha a he hghes level (level-n, he number of producs) gves us he affne relaxaon. Naurally, because of he NPhardness of solvng he affne relaxaon, we canno expec racably, and so we concenrae on small levels. The level-1 relaxaon already urns ou o be a gher relaxaon han war. Whle he level-1 relaxaon separaon problem can be solved n a racable manner, a poenal drawback s ha, unlke war and war +, canno be folded no he orgnal problem o yeld a compac formulaon. For smplcy we descrbe he level-1 formulaon and remark on how exends o a herarchy of relaxaons. In he level-1 relaxaon, whch we refer o as herarchcal affne relaxaon (har), we replace he γ S, (1 + j S w j) erm n (14) wh γ S, + ( j S w j)(max j S I γ j,) and solve he LP V har = mn β + γ, r 1 β,γ (har) s.. β Π har (β, γ) (27) γ, 0,, where Π har We have he followng lemma. Lemma 3. Π AF = max S γ S, w j max j S j S (β, γ) Π har (β, γ) Π war (β, γ). I j γ, ζ j, (β, γ). j S Proof By defnon, we have I j I S for all j S. Therefore, γ S, = I S γ, I j γ, for all j S and so γ S, max j S I j γ,. The proof now follows by nong ha γ S, (1 + ) j S w j ( ) ( γ S, + j S w ) j max j S I γ j, γ S, + ( ) j S w j I γ j,. (28) Lemma 3 ogeher wh Lemma 1 mples ha V AF V har V war. Therefore, har obans a gher bound han war. Nex, we show ha har separaon problem (28) can be solved n a racable manner. Assocang bnary decson varables q, and u j,, respecvely, o ndcae f resource and produc j are open, problem (28) can be wren as Π har (β, γ) = max q,u s.. (21) (23). γ, q, j w j u j, max j γ, u j, j I j ζ j, (β, γ)u j, Alhough he above opmzaon problem has a nonlnear objecve funcon, we can solve hrough a sequence of lnear programs n he followng manner. We fx a produc ĵ as he one achevng he 17
maxmum value of max j γ, u j,. Snce ĵ acheves he maxmum value, we mus have uĵ, = 1 and u j, = 0 for j wh I j γ, > γ Iĵ,. Leng Jĵ ˆ { = j I j γ, > γ Iĵ, }, we solve he followng lnear neger program for produc ĵ: Π har,ĵ (β, γ) = max q,u s.. (21), (22) uĵ, = 1 u j, = 0 γ, q, j j ˆ Jĵ u j, 0 neger ζ j, (β, γ) + w j j J \ ˆ Jĵ. γ, u j, Iĵ Snce he consran marx s oally unmodular, we can solve he above lnear neger program equvalenly as a lnear program. So we solve he lnear program for each produc ĵ J and oban Π har (β, γ) = maxĵ J Π har,ĵ (β, γ). Snce problem (28) can be solved n a racable manner, separang he har consrans s racable, and har can be solved n polynomal me by he ellpsod mehod. However, unlke war +, har does no seem o have a compac lnear programmng formulaon. Ths s because he se Jĵ ˆ depends on he values of he γ s n a nonlnear fashon and he dualy argumen n 6.3 ha we used o fold he separaon problem back no he orgnal LP does no hold. On he oher hand, an appealng feaure of har s ha s separaon problem has fewer number of decson varables and consrans han war +. Remark: One can ge furher relaxaons by consderng pars of elemens j, j for a level-2 relaxaon (or rples for level-3, and so on) such ha we fnd he offer se S ha maxmzes 1 + w j max. j S γ, {j,j S} I {j,j } In hs way, we can conrol he degree of approxmaon o he affne relaxaon. numercal resuls o fxng a sngle elemen j. We lm our 8 Exensons In hs secon we descrbe how o exend he weak affne relaxaon of 6 o varans of he MNL model (he developmen for war + and har s smlar). In 8.1 we consder he MNL choce model wh mulple cusomer segmens and dsjon consderaon ses. In 8.2 we consder he case where he consderaon ses of he dfferen segmens may overlap. Ths model, also referred o as he mxure of mulnomal logs (MMNL), s a rch choce model ha can approxmae any random uly choce model arbrarly closely; McFadden and Tran [12]. I s also possble o exend he weak affne relaxaon dea o he general aracon model of Gallego e al. [7] n a ransparen manner. Kunnumkal and Tallur [9] show how he same deas can be exended o he nesed-log choce model. 18
8.1 Mulple segmens wh dsjon consderaon ses Now we consder he case where he oal demand s comprsed of demand from mulple cusomer segmens. The consderaon ses of he dfferen segmens are dsjon and so we have C l C m = for segmens l and m. We noe ha he case of dsjon consderaon ses for he segmens s one of he few known cases where he CDLP formulaon s racable. We descrbe below how war can be exended o ghen he CDLP bound n a racable manner. The key dea s o look a he AF separaon problem for each cusomer segmen, whch agan urns ou o be nracable. We apply he deas from he sngle-segmen case o ge a racable relaxaon. Le I l = { I j C l and j J } and L = {l L I l }. We can nerpre I l as he se of resources ha are used by segmen l and L as he se of segmens ha use resource. Now consder he separaon problem for AF. Usng λq (S) = l λ lq l (S l) and λr(s) = l λ lr l (S l ), where S l = S C l, consran (12) can be wren as β + 1 [ IS ]γ, + [( τ ] γ,k) λ l Q l (S) λ l R l (S). (29) k=+1 l l We frs spl hs consran no l separae consrans, one for each segmen, by nroducng varables β l,. The consran for segmen l a me s ha β l, + 1 [ ISl ] γ λ, l + [( τ ) ] γ,k λ l Q l I l l L λ (S l ) λ l R l (S l ) (30) l k=+1 for each S l = S C l. The proof of Proposon 4 below shows ha he segmen level consrans (30) mply (29) and ha we oban a looser upper bound by separang over (30) nsead of (29). We observe ha he segmen level consrans (30) have he same form as consrans (12) n he sngle-segmen case, and are herefore hard o separae. So we use he same relaxaon as we dd for he sngle-segmen case o oban a racable separaon problem a he segmen level: Π swar l, (β, γ) = max q,u I l λ l γ, q, l L λ l j C l w l j s.. (21) (23). β l, + λ l Ij τ k=+1 γ,k f j + Ij γ, l L λ l u j, We replace consran (30) wh β l, Π swar l, (β, γ) o oban a segmen-based weak affne relaxaon (swar): V swar = mn β,γ s.. β l, + l γ, r 1 β l, Π swar l, (β, γ) l, γ, 0,. Moreover, by followng he same seps as for he sngle-segmen case, s possble o show ha 19
swar can be formulaed as he compac LP V swar = mn β l, + γ, r 1 γ,β,π l (swar) s.. β l, λ π,j, l γ, l, I l j J,j C l l L λ l π,j, λ lj w lj j f j ( τ ) γ γ,k +, I j Ij k=+1 l L λ l λ π,j, l γ, 0, l L, j J,j C l l L λ l γ,, π,j, 0,, j J,, β l j, λ lj j, where l j denoes he segmen o whch produc j belongs. swar can be vewed as an exenson of war o he MNL model wh mulple segmens and dsjon consderaon ses. Noe ha swar s agan racable as s a compac LP. Proposon 4 below shows ha also obans an upper bound on he value funcon ha s gher han CDLP. Proposon 4. V AF V swar V CDLP. Proof Usng he MNL choce probably (1) and rearrangng erms, he swar consran β l, Π swar l, (β, γ) can be equvalenly wren as [ β l, λ l R l (S l ) ] τ Q l (S l )γ,k 1 [ ISl ] γ λ, l P I l k=+1 I l l L λ j(s l l ) + P0(S l l ) (31) l j J for all S l C l. Consder now wo nermedae problems: V = mn β l, + γ, r 1 β,γ l s.. (30) l, S l C l, γ, 0,, and V = mn β l, + γ, r 1 β,γ l [ s.. β l, λ l R l (S l ) ] τ Q l (S l )γ,k I l k= γ, 0,. l, S l C l, (32) We can nerpre he frs problem as a segmen based relaxaon of AF, whle he second problem can be vewed as a segmen based relaxaon of CDLP. We nex show ha V AF V V swar V = V CDLP, whch complees he proof of he proposon. 20
() V V swar V. Snce he objecve funcons of all he problems are he same, we only need o compare he correspondng consrans. Snce j J P l j (S l) + P l 0(S l ) 1, follows ha consran (31) mples consran (30) and we have V V swar. On he oher hand, he rgh hand sde of consran (32) can be wren as [ λ l R l (S l ) ] τ Q l (S l )γ,k λ l Q l (S l )γ,. I l k=+1 I l Now noe ha λ l Q l (S l )γ, = λ l 1 [ ISl ] Ql (S l )γ, = λ l 1 [ ISl ] Pj(S l l ) γ, j J λ l l L λ l 1 [ ISl ] Pj(S l l ) γ, j J λ l l L λ l 1 [ ISl ] Pj(S l l ) + P0(S l l ) γ, j J where he frs equaly holds snce f 1 [ ISl ] = 0, hen Ql (S l) = 0 and he frs nequaly holds snce l L λ l 1. Therefore consran (32) mples consran (31) and we have V swar V. () V AF V. Suppose ha ( ˆβ, ˆγ) sasfes consrans (30). We show ha sasfes consrans (29) as well. Fx a se S and le S l = S C l. Addng up consrans (30) for all he segmens ˆβ l, [ {λ l R l (S l ) ] τ Q l (S l )ˆγ,k } 1 ]ˆγ λ [ ISl, l l l I l k=+1 I l l L λ l [ = λ R(S) ] τ Q (S)ˆγ,k λ ˆγ, 1 [ ISl ] l k=+1 l L l L λ l [ λ R(S) ] τ Q (S)ˆγ,k λ ˆγ, 1 [ IS ] l k=+1 l L l L λ l [ = λ R(S) ] τ Q (S)ˆγ,k ˆγ, 1 [ IS ], k=+1 where he frs equaly uses he fac ha Q l (S l) = 0 for l / L and hence λq (S) = l λ lq l (S l) = l L λ l Q l (S l). The second nequaly holds snce 1 [ ISl ] 1 [ I S ]. Leng β = { β = ˆβ l l, }, follows ha ( β, ˆγ) sasfes consrans (29). Therefore V AF β + ˆγ, = V. Messner e al. [14] prove he followng ha we nclude for compleeness. () V = V CDLP. (Messner e al. [14]) 21
Consrans (6) n dcdlp are equvalen o { [ β = max λ R(S) ]} τ Q (S)γ,k S k= { [ = max λ l R l (S C l ) ]} τ Q l (S C l )γ,k S l I l k= = [ max {λ l R l (S l ) ]} τ Q l (S l )γ,k S l l I l k= where he las nequaly uses he fac ha he consderaon ses are dsjon. Therefore, he dcdlp consran s equvalen o he consrans β = l β l, and [ {λ l β l, = max S l R l (S l ) I l ]} τ Q l (S l )γ,k, k= whch s exacly consran (32). As we show n he nex secon, s possble o exend he swar formulaon o he MNL model wh mulple segmens when he consderaon ses overlap. The dual of swar, whch we gve below, urns ou o be useful for hs purpose. V dswar = max λ l f j x l j, x,ρ l j C l x (dswar) s.. λ l l 1 0, + x l j J j,s + C l x l j, l L l L λ l s=1 j J C l l L λ l ] x l 0, + j C l x l j, = 1 l, ρ l l L λ l r 1, (33) x l j, wj l x l 0, + ρ l, 0 l,, j J C l, (34) x l 0,, x l j,, ρ l, 0 l,, j J C l,. 8.2 Mulple segmens wh overlappng consderaon ses When he segmen consderaon ses overlap, he CDLP formulaon s dffcul o solve, even for MNL wh jus wo segmens. So one would magne ha s dffcul o fnd a racable bound gher han CDLP n hs case. One sraegy, pursued n Messner e al. [14] s o formulae he problem by segmens and hen add a se of conssency condons called produc-cu equales (PC-equales). These equales apply o any general dscree-choce model and appear o be que powerful n numercal expermens, ofen brngng he soluon close o CDLP value. Srauss and 22
Tallur [17] subsequenly show ha when he consderaon se srucure has a ceran ree srucure, he cus n fac acheve he CDLP value. Tallur [18] shows how o specalze he PC-equales o he MNL choce model. In hs secon we descrbe how he PC-equales, specalzed for MNL, can be added o dswar o ghen he formulaon. We begn wh a bref descrpon of he PC-equales: Messner e al. [14] allow dfferen ses o be offered o dfferen segmens. However, o ensure conssency, hey requre ha for any produc j C l C m, he lengh of me s offered o segmen l mus be he equal o he lengh of me s offered o segmen m. Ths leads o a se of conssency consrans whch hey erm as PC-equales. Tallur [18] uses choce probables (1) o specalze he PC-equales o he MNL model as: x l j, w l j = {S (C l C m ) j S} y l,m S l, m, j C l C m (35) y l,m S,j yl,m S l, m, S C l C m, j C l \ C m (36) w jy l l,m T,j + (1 + W T l )y l,m T = {T (C l C m) T S} j C l \C m wj m y m,l T,k + (1 + W T m )ym,l T l, m, S C l C m, (37) {T (C m C l ) T S} j C m\c l where WS l = j S wl j and we have new varables of he form yl,m S defned for all pars of segmens l, m and for all S C l C m ; see Tallur [18]. If he overlap n he consderaon ses of he dfferen segmens s no oo large, hen he number of PC-equales s manageable. Tallur [18] shows ha addng PC-equales (35)-(37) o he sales-based lnear program (SBLP ) of Gallego e al. [7] furher ghens he SBLP bound. We are also able o ghen he dswar bound by dong he same hng. Moreover, comparng dswar wh SBLP, s easy o see ha a feasble soluon o dswar s also feasble o SBLP. Therefore, dswar s gher han SBLP. I follows ha dswar augmened wh he PC-equales, connues o be gher han SBLP wh he same PC-equales. So n concluson, when segmen consderaon ses overlap, we also have Proposon 5. The objecve funcon value of dswar wh (35 37) added, s less han or equal o he objecve funcon value of SBLP wh (35 37) added. In closng, we noe ha dswar augmened wh he PC-equales s no guaraneed o be gher han CDLP. We numercally compare he performance of dswar wh CDLP n our compuaonal expermens ha we presen nex. 9 Compuaonal expermens In hs secon, we compare he upper bounds and he revenue performance of he polces obaned by he dfferen benchmark soluon mehods. We es he performance of our benchmark soluon mehods on a hub-and-spoke nework, wh a sngle hub servng mulple spokes. Whle comparng he revenue performance of he benchmark mehods, we dvde he bookng perod no fve equal nervals. A he begnnng of each nerval, we re-solve he benchmark soluon mehods o ge fresh esmaes for he margnal value of capacy on he resources. All of he benchmark mehods gve 23
a soluon of he form ( ˆβ, ˆγ) wh τ s= ˆγ,s beng an esmae for he margnal value of capacy on resource a me. We use hese margnal values o consruc a value funcon approxmaon ˆV (r) = ( τ s= ˆγ,s)r and solve problem (8) o decde on he offer se. We connue o use hs decson rule unl he begnnng of he nex nerval where we re-solve he benchmark soluon mehods. In all of our es problems, we have mulple cusomer segmens and choce whn each segmen s governed by he MNL model. We begn by descrbng he dfferen benchmark soluon mehods and he expermenal seup. 9.1 Benchmark soluon mehods Choce deermnsc lnear program (CDLP ) Ths s he soluon mehod descrbed n 4.1. Weak affne relaxaon (war) Ths s he verson of weak affne relaxaon ha apples o mulple segmens and descrbed n 8 (swar). Weak affne relaxaon + (war + ) Ths s he verson of war + ha apples o mulple segmens. As menoned, s possble o exend he war + formulaon descrbed n 7.1 o he seng wh mulple segmens by followng he seps n 8. Herarchcal affne relaxaon (har) Ths s he verson of he level-1 herarchcal affne relaxaon ha apples o mulple segmens. As menoned, s possble o exend he herarchcal affne relaxaon mehod descrbed n 7.2 o he seng wh mulple segmens by followng he seps n 8. Snce har does no adm a compac formulaon, we solve har by generang consrans on he fly and sop when we are whn 1% of opmaly. Affne relaxaon (AF ) Ths s he soluon mehod descrbed n 4.2. We work wh he reduced formulaon RAF of Vossen and Zhang [22]. Whle he number of decson varables n RAF s manageable, has a large number of consrans. We solve RAF by generang consrans on he fly (usng neger programmng) and sop when we are whn 1% of opmaly. 9.2 Hub-and-spoke nework We consder a hub-and-spoke nework wh a sngle hub ha serves N spokes. Half of he spokes have wo flghs o he hub, whle he remanng half have wo flghs from he hub so ha he oal number of flghs s 2N. Fgure 1 shows he srucure of he nework wh N = 8. We noe ha he flgh legs correspond o he resources n our NRM formulaon. The oal number of fare-producs s 2N(N +2). There are 4N fare producs connecng spoke-ohub and hub-o-spoke orgn-desnaon pars, of whch half are hgh fare-producs and he remanng half are low-fare producs. The hgh fare-produc s 50% more expensve han he correspondng low fare-produc. The remanng 4N 2 fare-producs connec spoke-o-spoke orgn-desnaon pars. Half of he 4N 2 fare-producs are hgh fare-producs and he res are low fare-producs, wh he hgh fare-produc beng 50% more expensve han he correspondng low fare-produc. Each orgn-desnaon par s assocaed wh a cusomer segmen and each segmen s only neresed n he fare-producs connecng ha orgn-desnaon par. Therefore, he consderaon ses are dsjon. Whn each segmen choce s governed by he MNL model. We sample he preference weghs of he fare-producs from a posson dsrbuon wh a mean of 100 and se he no-purchase preference wegh o be 0.5 j C l wj l. So he probably ha a cusomer does no purchase anyhng when all he producs n he consderaon se are offered s around 33%. 24
We measure he ghness of he leg capaces usng he nomnal load facor, whch s defned n he followng manner. Leng Ŝ = argmax S R(S) denoe he opmal se of producs offered a me perod when here s ample capacy on all flgh legs, we defne he nomnal load facor α = λ Q (Ŝ), r1 where λ denoes he arrval rae a me perod. Inally, we assume saonary arrvals and se he arrval rae o be 0.9 n each me perod. We have τ = 200 n all of our es problems. We label our es problems by (N, α) where N {4, 6, 8} and α {0.8, 1.0, 1.2, 1.6}, whch gves us 24 es problems n oal. Table 1 gves he upper bounds obaned by he benchmark soluon mehods. The frs column n he able gves he problem characerscs. The second o sxh columns, respecvely, gve he upper bounds obaned by CDLP, war, war +, har, and AF. The las four columns gve he percenage gap beween he upper bounds obaned by CDLP and war, CDLP and war +, CDLP and har, and CDLP and AF, respecvely. AF generaes he ghes upper bound and CDLP he weakes, wh he remanng upper bounds sandwched n beween. The average percenage gap beween war and CDLP s 1.59%, alhough we observe nsances where he gap s as hgh as 2.73%. The percenage gap beween war and CDLP seems o ncrease wh he nomnal load facor and he number of spokes n he nework. war + and har ghen he war bound and oban bounds ha are on average 1.81% and 1.63% gher han CDLP. AF obans bounds ha are on average 2.16% gher han CDLP. Table 2 gves he CPU seconds requred by he dfferen soluon mehods for dfferen numbers of spokes n he nework and dfferen numbers of me perods n he bookng horzon. All of our compuaonal expermens are carred ou on a Penum Core 2 Duo deskop wh 3-GHz CPU and 4-GB RAM. We use CPLEX 11.2 o solve all lnear programs. Snce we have dsjon consderaon ses, CDLP has a compac sales-based formulaon, SBLP, whch can be solved n a maer of seconds. The soluon mes of he oher mehods are generally n mnues. war ypcally runs faser han AF and he savngs can be sgnfcan especally for relavely large neworks. In lgh of he hardness resul n Proposon 2, we only expec he savngs n run mes o ncrease wh he problem sze. war + and har have addonal compuaonal overheads assocaed wh hem and can ake longer han war. If we compare he mprovemen n he upper bound relave o wha AF acheves over CDLP wh he correspondng ncrease n soluon me (relave o he overhead ncurred by AF over CDLP ), we fnd ha war closes around 70% of he gap beween he AF and CDLP bounds by ncurrng a compuaonal overhead of around 15% of ha of AF. In conras, war + closes around 80% of he gap by ncurrng a 45% compuaonal overhead, whle har closes around 75% of he gap by ncurrng roughly a 22% compuaonal overhead. Overall, war seems o acheve a good balance beween he qualy of he soluon and he compuaonal effor. Table 3 gves he expeced revenues obaned by he dfferen benchmark mehods. The columns have a smlar nerpreaon as n Table 1 excep ha hey gve he expeced oal revenues. We evaluae he revenue performance by smulaon and use common random numbers n our smulaons. In he las four columns, we use o ndcae ha he correspondng benchmark mehod generaes hgher revenues han CDLP a he 95% level, an f he dfference n he revenue performance of he benchmark mehod and CDLP s no sgnfcan a he 95% level and a f he benchmark mehod generaes lower revenues han CDLP a he 95% level. war on average generaes revenues ha are 2.28% hgher han CDLP, alhough we observe nsances where he gap s as hgh as 7%. war +, har and AF, respecvely, generae revenues ha are on average 3.02%, 2.94% and 2.34%, 25