Assortment Optmzaton under the Pared Combnatoral Logt Model Heng Zhang, Paat Rusmevchentong Marshall School of Busness, Unversty of Southern Calforna, Los Angeles, CA 90089 hengz@usc.edu, rusmevc@marshall.usc.edu Huseyn Topaloglu School of Operatons Research and Informaton Engneerng, Cornell Tech, New York, NY 10011 topaloglu@ore.cornell.edu June 19, 017 We consder uncapactated and capactated assortment problems under the pared combnatoral logt model, where the goal s to fnd a set of products to maxmze the expected revenue obtaned from each customer. In the uncapactated settng, we can offer any set of products, whereas n the capactated settng, there s a lmt on the number of products that we can offer. We establsh that even the uncapactated assortment problem s strongly NP-hard. To develop an approxmaton framework for our assortment problems, we transform the assortment problem nto an equvalent problem of fndng the fxed pont of a functon, but computng the value of ths functon at any pont requres solvng a nonlnear nteger program. Usng a sutable lnear programmng relaxaton of the nonlnear nteger program and randomzed roundng, we obtan a 0.6-approxmaton algorthm for the uncapactated assortment problem. Usng randomzed roundng on a semdefnte programmng relaxaton, we obtan an mproved, but a more complcated, 0.79-approxmaton algorthm. Fnally, usng teratve varable fxng and coupled randomzed roundng, we obtan a 0.5-approxmaton algorthm for the capactated assortment problem. Our computatonal experments demonstrate that our approxmaton algorthms, on average, yeld expected revenues that are wthn 3.6% of a tractable upper bound on the optmal expected revenues. Key words : Customer choce modelng, par combnatoral logt model, assortment optmzaton. 1. Introducton Tradtonal revenue management models commonly assume that each customer arrves nto the system wth the ntenton to purchase a partcular product. If ths product s avalable for purchase, then the customer purchases t; otherwse, the customer leaves the system wthout a purchase. In realty, however, customers observe the set of avalable alternatves and make a choce among the avalable alternatves. Under such a customer choce process, the demand for a partcular product depends on the avalablty of other products. In ths case, dscrete choce models provde a useful representaton of demand snce dscrete choce models capture the demand for each product as a functon of the entre set of products n the offer set. A growng body of lterature ndcates that 1
Zhang, Rusmevchentong, and Topaloglu: Assortment Optmzaton under the PCL Model capturng the choce process of customers usng dscrete choce models can sgnfcantly mprove the qualty of operatonal decsons; see, for example, Tallur and van Ryzn (004), Gallego et al. (004) and Vulcano et al. (010). Whle more sophstcated choce models yeld a more accurate representaton of the customer choce process, the assortment and other operatonal problems under more sophstcated choce models become more challengng. So, t s useful to dentfy sophstcated choce models, where the correspondng operatonal problems reman tractable. In ths paper, we study assortment problems under the pared combnatoral logt (PCL) model. In our problem settng, there s a fxed revenue assocated wth each product. Customers choose among the offered products accordng to the PCL model. The goal s to fnd an offer set that maxmzes the expected revenue obtaned from each customer. We consder both the uncapactated verson, where we can offer any subset of products, as well as the capactated verson, where there s a lmt on the number of products that we can offer. We show that even the uncapactated assortment problem s strongly NP-hard. We gve a general framework for constructng approxmaton algorthms for the assortment problem. We use ths framework to develop approxmaton algorthms for the uncapactated and capactated assortment problems. Our computatonal results demonstrate that our approxmaton algorthms perform qute well, yeldng solutons wth no larger than 3.6% optmalty gaps on average. The PCL model s a rather sophstcated choce model, allowng to capture ntrcate customer choce behavor. Ths model s known to be compatble wth the random utlty maxmzaton prncple, where each customer assocates random utltes wth the alternatves. The utltes are sampled from a certan dstrbuton. The customer knows the utltes and chooses the alternatve that provdes the largest utlty. Other commonly used choce models, such as the multnomal logt and nested logt models, are also compatble wth the random utlty maxmzaton prncple. An mportant feature of the PCL model s that t allows for a general correlaton structure among the utltes. In contrast, the multnomal logt model assumes that the utltes are ndependent. Under the nested logt model, the alternatves are grouped nto nests. The utltes of the alternatves n dfferent nests are ndependent and there s a sngle parameter that governs the correlaton between the utltes of the alternatves n the same nest. By allowng a general correlaton structure among the utltes, the PCL model captures the stuaton where the preference of a customer for one product offers nsght nto ther nclnaton towards other products. Furthermore, the PCL model subsumes the multnomal logt and nested logt models as specal cases. There s work n the lterature showng that the PCL model can provde advantages when capturng the demand process; see, for example, Prashker and Bekhor (1998), Koppleman and Wen (000) and Chen et al. (003). Although the PCL model can
Zhang, Rusmevchentong, and Topaloglu: Assortment Optmzaton under the PCL Model 3 provde advantages n capturng the demand process, there s lttle research on understandng the complexty of makng operatonal decsons under the PCL model and provdng effcent algorthms for makng such decsons. Our work n ths paper s drected towards fllng ths gap. Thus, we beleve that the approxmaton algorthms that we provde for assortment optmzaton under the PCL model wll enhance the practcal appeal of the PCL model, whose effectveness for capturng the demand process has already been demonstrated n the lterature. Man Contrbutons: We make four man contrbutons. Frst, we study the complexty of the assortment problem under the PCL model. Second, we gve a general framework that can be used to develop approxmaton algorthms under the PCL model. Thrd, we use ths framework to gve approxmaton algorthms for the uncapactated assortment problem under the PCL model. Fourth, we gve an approxmaton algorthm for the capactated problem, also by usng our approxmaton framework. Although we use the same general framework for the uncapactated and capactated problems, the detals of the approxmaton algorthms are qute dfferent. Consderng the computatonal complexty of the problem, we show that the uncapactated assortment problem under the PCL model s strongly NP-hard. Ths result s n contrast wth the assortment problem under the closely-related multnomal logt and nested logt models. In partcular, as dscussed shortly n our lterature revew, there exst polynomal-tme algorthms to solve even the capactated assortment problem under the multnomal logt and nested logt models. Motvated by the computatonal complexty result, we focus on approxmaton algorthms. We gve a general framework for constructng approxmaton algorthms for the assortment problem. In ths approxmaton framework, we show that the assortment problem s equvalent to fndng the fxed pont of a one-dmensonal functon f : R R, whose evaluaton at a certan pont requres solvng a nonlnear nteger program. To obtan an α-approxmaton algorthm, we follow three steps. Frst, we desgn an upper bound f R : R R to the functon f( ), possbly through lnear programmng (LP) or semdefnte programmng (SDP) relaxatons of the nonlnear nteger program assocated wth computng the value of f( ) at a certan pont. Second, we compute the fxed pont ẑ of f R ( ). Thrd, we develop an α-approxmaton algorthm for the nonlnear nteger program that computes the value of f( ) at ẑ. It turns out that an α-approxmate soluton to ths nonlnear nteger program s an α-approxmate soluton to the assortment problem. So, we compute the fxed pont ẑ of f R ( ) and buld an approxmate soluton for computng f( ) at ẑ. For the uncapactated assortment problem, we apply the framework above by constructng f R ( ) through an LP relaxaton of the nonlnear nteger program that computes f( ). We show that f we compute f R ( ) by solvng an LP, then we can compute the fxed pont of f R ( ) also by solvng an LP. After we compute the fxed pont ẑ of f R ( ), we get a soluton to the LP that computes
4 Zhang, Rusmevchentong, and Topaloglu: Assortment Optmzaton under the PCL Model f R ( ) at ẑ. We use randomzed roundng on ths soluton to get a 0.6-approxmate soluton to the nonlnear nteger program that computes f( ) at ẑ. Lastly, to obtan a determnstc algorthm, we de-randomze ths approach usng the standard method of condtonal expectatons; see Wllamson and Shmoys (011). Our framework can allow for a broad class of approxmaton algorthms. For example, f we construct f R ( ) by usng an SDP relaxaton of the nonlnear nteger program, then we can use the sphercal roundng method of Goemans and Wllamson (1995) to obtan a 0.79-approxmaton algorthm for the uncapactated assortment problem. Ths approxmaton algorthm requres solvng an SDP. Although we can theoretcally solve an SDP n polynomal tme, solvng large-scale nstances n practce can be computatonally dffcult. Therefore, the SDP relaxaton s arguably less appealng than the LP relaxaton from a practcal perspectve. We apply our framework to the capactated assortment problem as well. Here, we explot the structural propertes of the extreme ponts of the LP relaxaton and use an teratve varable fxng method, along wth coupled randomzed roundng, to develop a 0.5-approxmaton algorthm. In ths algorthm, f there are n products that can be offered to the customers, then we solve at most n successve LP relaxatons, fxng the value of one decson varable after solvng each LP relaxaton. Once we solve these LP relaxatons, we perform randomzed roundng on the bass of the soluton of the last LP relaxaton to obtan a soluton to the assortment problem. Usng the method of condtonal expectatons, we can de-randomze ths soluton to obtan a determnstc algorthm wth the same performance guarantee. Our computatonal experments demonstrate that our approxmaton algorthms for both the uncapactated and capactated versons perform qute well. Over a large collecton of test problems, the average optmalty gap of the solutons provded by these algorthms s no larger than 3.6%. Lterature Revew: Our broader obectve n ths paper s to enhance the practcal appeal of the PCL model n operatons management. As noted earler, there s lttle work on makng operatonal decsons under the PCL model. The only example we are aware of s L and Webster (015), where the authors consder prcng problems under the PCL model. In ther problem, the goal s to choose the prces for the products to maxmze the expected revenue obtaned from each customer. The authors gve suffcent condtons for the prce senstvtes of the products to ensure that the prcng problem can be solved effcently. We are not aware of other work that studes assortment or prcng problems under the PCL model. Despte lmted work on solvng operatonal problems under the PCL model, there s consderable work, especally n the transportaton lterature, on usng the PCL model to capture route choces. Koppleman and Wen (000) and Chen et al. (003) demonstrate that the general correlaton structure of the utltes under the PCL model s useful n modelng route choces, because dfferent routes overlap wth each other to varyng
Zhang, Rusmevchentong, and Topaloglu: Assortment Optmzaton under the PCL Model 5 extents through shared street segments, creatng complex correlatons between the utltes provded by dfferent routes. Prashker and Bekhor (1998) demonstrate that the PCL model can provde mprovements over the multnomal logt model n predctng route choces. Chen et al. (014) and Karoonsoontawong and Ln (015) study varous traffc equlbrum problems under the PCL model and dscuss the benefts from usng ths choce model. The work n route choce lterature clearly demonstrates that the PCL model provdes mprovements over the multnomal logt and nested logt models n predctng user choces, especally when the utltes provded by dfferent alternatves exhbt complex correlaton structures. There s sgnfcant work on assortment and prcng problems under the multnomal logt and nested logt models. In the multnomal logt model, the utltes of the products are ndependent of each other. In the nested logt model, the products are grouped nto dsont nests. Assocated wth each nest, there s a dssmlarty parameter characterzng the correlaton between the utltes of the products n the same nest, but the utltes of products n dfferent nests are ndependent of each other. In the PCL model, there exsts one nest for each par of products, so the nests are overlappng. Assocated wth each nest, there s a dssmlarty parameter characterzng the correlaton between the utltes of each par of products. Therefore, when compared wth the multnomal logt and nested logt models, we can use the PCL model to specfy a sgnfcantly more general correlaton structure between the utltes of the products. As dscussed earler n ths secton, the general correlaton structure under the PCL model allows better capturng the demand n numerous applcatons comng from the route choce doman. Tran (00) provdes a thorough dscusson of the multnomal logt, nested logt and PCL models. Tallur and van Ryzn (004) and Gallego et al. (004) study assortment problems under the multnomal logt model. The authors show that t s optmal to offer a nested-by-revenue assortment, whch ncludes a certan number of products wth the largest revenues. In ths case, the optmal assortment can be found effcently by computng the expected revenue of each nested-by-revenue assortment. Rusmevchentong et al. (010) gve an effcent algorthm for the capactated assortment problem under the multnomal logt model. Davs et al. (013) show that the assortment problem under the multnomal logt model wth varous constrants can be formulated as an LP. Hanson and Martn (1996), Song and Xue (007), and Dong et al. (009) study prcng problems under the multnomal logt model. Assumng that the prce senstvtes of the products are the same, the authors show that the expected revenue s concave when expressed as a functon of the purchase probabltes of the products. Zhang et al. (016) generalze ths result by showng that the expected revenue s concave n the purchase probabltes under any generalzed extreme value model, whch s a general class of choce models ncludng the multnomal logt,
6 Zhang, Rusmevchentong, and Topaloglu: Assortment Optmzaton under the PCL Model nested logt and PCL models. Zhang and Lu (013) show that even f the products have dfferent prce senstvtes, the prcng problem under the multnomal logt model can be formulated as a convex program, but ths result does not generalze to other choce models. Davs et al. (014) develop an effcent algorthm for the assortment problem under the nested logt model. Ther algorthm requres solvng an LP whose sze grows polynomally wth the number products and nests. Gallego and Topaloglu (014) study the assortment problem under the nested logt model when a capacty constrant lmts the number of products offered n each nest, whereas Feldman and Topaloglu (015) explore the same problem when a capacty constrant lmts the total number of products offered n all nests. L and Huh (011) nvestgate the prcng problem under the nested logt model when the products n the same nest share the same prce senstvtes and show that the problem can be formulated as a convex program. Gallego and Wang (014) analyze the same problem when the products have dfferent prce senstvtes. They show that the prcng problem can be cast as a search over a sngle dmenson. Rayfeld et al. (015) consder the prcng problem under the nested logt model wth arbtrary prce senstvtes and gve an algorthm to compute prces wth a performance guarantee. L and Huh (015) and L et al. (015) study prcng and assortment problems under the mult-level nested logt model, where the products are herarchcally organzed nto nests and subnests. Vulcano et al. (010) and Da et al. (014) use the multnomal logt and nested logt models n arlne applcatons. Organzaton: In Secton, we formulate both the uncapactated and capactated assortment problems and show that even the uncapactated assortment problem s strongly NP-hard. In Secton 3, we show that we can transform the assortment problem nto the problem of fndng the fxed pont of a functon f : R R. Computng the value of f( ) at any pont requres solvng a nonlnear nteger program. Thus, we approxmate f( ) by another functon f R : R R obtaned through an LP relaxaton of the nonlnear nteger program. We show how to fnd the fxed pont ẑ of f R ( ). We gve a suffcent condton for a certan subset of products to provde a performance guarantee for our assortment problem. The suffcent condton states that f the value of a sutable set functon computed at a subset of products exceeds α f R (ẑ), then the subset of products n queston provdes an α-approxmate soluton for the assortment problem. In Secton 4, we consder the uncapactated assortment problem and show how to obtan a subset of products satsfyng the suffcent condton wth α = 0.6. Also, we dscuss how we can obtan a 0.79-approxmaton algorthm by buldng on an SDP relaxaton. In Secton 5, we consder the capactated assortment problem and show how to obtan a subset of products satsfyng the suffcent condton wth α = 0.5. In Secton 6, we gve our computatonal experments. In Secton 7, we conclude.
Zhang, Rusmevchentong, and Topaloglu: Assortment Optmzaton under the PCL Model 7. Problem Formulaton and Complexty The set of products s ndexed by N = {1,..., n}. The revenue of product s r 0. We use the vector x = (x 1,..., x n ) {0, 1} n to capture the subset of products that we offer to the customers, where x = 1 f and only f we offer product. We refer to the vector x smply as the assortment or the subset of products that we offer. Throughout the paper, we denote the vectors and matrces n bold font. We denote the collecton of nests by M = {(, ) N : }. For each nest (, ) M, we let γ [0, 1] be the dssmlarty parameter of the nest. For each product, we let v the preference weght of product. Under the PCL model, we can vew the choce process of a customer as takng place n two stages. Frst, the customer ether decdes to make a purchase n one of the nests or leaves the system wthout makng any purchase. In partcular, lettng V (x) = v 1/γ x + v 1/γ x and usng v 0 > 0 to denote the preference weght of the no-purchase opton, f we offer the subset of products x, then a customer decdes to make a purchase n nest (, ) wth probablty V (x) γ /(v 0 + (k,l) M V kl(x) γ kl ). Second, f the customer decdes to make a purchase n nest (, ), then she chooses product wth probablty v 1/γ product wth probablty v 1/γ be x / V (x), whereas she chooses x / V (x). Note that f a customer decdes to make a purchase n nest (, ), then she must choose one of the products n ths nest. If we offer the subset of products x and a customer has already decded to make a purchase n nest (, ), then the expected revenue that we obtan from the customer s R (x) = r v 1/γ x + r v 1/γ V (x) We use π(x) to denote the expected revenue that we obtan from a customer when we offer the subset of products x. In ths case, we have π(x) = V (x) γ v 0 + V R (k,l) M kl(x) γ (x) = kl x. V (x) γ R (x) v 0 + V. (x) γ Throughout the paper, we consder both uncapactated and capactated assortment problems. In the uncapactated assortment problem, we can offer any subset of products to the customers. In the capactated assortment problem, on the other hand, we have an upper bound on the cardnalty of the subset of products that we can offer to the customers. To capture both the uncapactated and capactated assortment problems succnctly, for some c Z +, we use F = {x {0, 1} n : N x c} to denote the feasble subsets of products that we can offer to the customers. Snce there are n products, the constrant N x c s not bndng when we have c n. Therefore, we mmedately obtan the uncapactated assortment problem by choosng a value of c that s no smaller n, whereas we obtan the capactated assortment problem wth other values
8 Zhang, Rusmevchentong, and Topaloglu: Assortment Optmzaton under the PCL Model of c. In the assortment problem, our goal s to fnd a feasble subset of products to offer to the customers that maxmzes the expected revenue obtaned from each customer, correspondng to the combnatoral optmzaton problem z = max x F π(x) = max x F { V (x) γ R (x) v 0 + V (x) γ }. (Assortment) In the next theorem, we show that even the uncapactated verson of the Assortment problem s strongly NP-hard, ndcatng that t s unlkely that there exsts an effcent algorthm to solve the Assortment problem. We gve the proof of ths result n Appendx A. Theorem.1 (Computatonal Complexty) The Assortment problem s strongly NP-hard, even when we have F = {0, 1} n so that the problem s uncapactated. The proof of Theorem.1 uses a reducton from the max-cut problem, whch s a well-known NP-hard problem; see Garey and Johnson (1979). Motvated by ths complexty result, we focus on developng approxmaton algorthms for the Assortment problem. For some α [0, 1], an α-approxmaton algorthm s a polynomal-tme algorthm that, for every problem nstance, computes an assortment ˆx F, whose expected revenue s at least α tmes the optmal expected revenue; that s, notng that the optmal expected revenue s z, π(ˆx) α z. Closng ths secton, we note that our formulaton of the PCL model s slghtly dfferent from the one that often appears n the lterature. In the exstng lterature, the collecton of nests s often {(, ) N : < }, whereas n our formulaton, the collecton of nests s {(, ) N : }. If we let γ = γ for all (, ) N wth > and double the preference weght of the no-purchase opton n our formulaton, then t s easy to check that the two formulatons of the PCL model are consstent. Our formulaton of the PCL model wll sgnfcantly reduce the notatonal burden. 3. A General Framework for Approxmaton Algorthms In ths secton, we provde a general framework that s useful for developng approxmaton algorthms for the Assortment problem. Our framework s applcable to both the uncapactated and capactated assortment problems smultaneously. 3.1. Connecton to a Fxed Pont Problem The startng pont for our approxmaton framework s to relate the Assortment problem to the problem of computng the fxed pont of a one-dmensonal functon. Computng the value of ths functon at any pont requres solvng a nonlnear nteger program, but t turns out we can obtan
Zhang, Rusmevchentong, and Topaloglu: Assortment Optmzaton under the PCL Model 9 an approxmate soluton to the Assortment problem by usng a tractable approxmaton to the one-dmensonal functon. We defne the functon f : R R as { } f(z) = max x F V (x) γ (R (x) z). (Functon Evaluaton) Snce x = 0 R n s a feasble soluton to the Functon Evaluaton problem on the rght sde above and provdes an obectve value of zero, we have f(z) 0 for all z R. Furthermore, f(z) s decreasng and v 0 z s strctly ncreasng n z. Thus, there exsts a unque ẑ 0 satsfyng f(ẑ) = v 0 ẑ. Note that the value of ẑ that satsfes f(ẑ) = v 0 ẑ s the fxed pont of the functon f( )/v 0. In the next lemma, we show that ths value of ẑ corresponds to the optmal obectve value of the Assortment problem and we can use ths value of ẑ n the Functon Evaluaton problem to obtan an optmal soluton to the Assortment problem. We do not gve a proof for Lemma 3.1 snce ths lemma follows as a corollary to a more general result that we shortly gve n Theorem 3.. Lemma 3.1 (Optmal Assortment from Fxed Pont) Let ẑ 0 satsfy f(ẑ) = v 0 ẑ and ˆx be an optmal soluton to the Functon Evaluaton problem when we solve ths problem wth z = ẑ. Then, we have π(ˆx) = ẑ = z ; so, ẑ s the optmal obectve value of the Assortment problem and ˆx s an optmal soluton to the Assortment problem. Thus, f we fnd the fxed pont ẑ of f( )/v 0 and solve the Functon Evaluaton problem wth z = ẑ, then an optmal soluton to the Functon Evaluaton problem s an optmal soluton to the Assortment problem. Fndng the fxed pont of f( )/v 0 s dffcult snce the Functon Evaluaton problem s a nonlnear nteger program. Instead, assume that we have a tractable upper bound f R ( ) on f( ) so that f R (z) f(z) for all z R. We construct f R ( ) n such a way that f R (z) s decreasng n z and f R (z) 0 for all z R, so there exsts a unque ẑ 0 satsfyng f R (ẑ) = v 0 ẑ. In the next theorem, we show that ths value of ẑ upper bounds the optmal obectve value of the Assortment problem and we can use ths value of ẑ to obtan an approxmate soluton. Theorem 3. (Approxmaton Framework) Assume that f R ( ) satsfes f R (z) f(z) for all z R. Let ẑ 0 satsfy f R (ẑ) = v 0 ẑ and ˆx F be such that V (ˆx) γ (R (ˆx) ẑ) α f R (ẑ) (Suffcent Condton) for some α [0, 1]. Then, we have π(ˆx) α ẑ α z ; so, ẑ upper bounds the optmal obectve value of the Assortment problem and ˆx s an α-approxmate soluton to the Assortment problem.
10 Zhang, Rusmevchentong, and Topaloglu: Assortment Optmzaton under the PCL Model Proof: Notng that v 0 ẑ = f R (ẑ), we have α v 0 ẑ = α f R (ẑ) V (ˆx) γ (R (ˆx) ẑ) V (ˆx) γ (R (ˆx) α ẑ), where the frst nequalty uses the Suffcent Condton. Thus, we have α v 0 ẑ V (ˆx) γ (R (ˆx) α ẑ), n whch case, solvng for ẑ n the last nequalty, we get α ẑ V (ˆx) γ R (ˆx)/(v 0 + V (ˆx) γ ) = π(ˆx). Next, we show that ẑ z. We let x be an optmal soluton to the Assortment problem. Snce x s a feasble but not necessarly an optmal soluton to the Functon Evaluaton problem when ths problem s solved wth z = ẑ, we have f(ẑ) V (x ) γ (R (x ) ẑ). Notng that v 0 ẑ = f R (ẑ) f(ẑ), we obtan v 0 ẑ V (x ) γ (R (x ) ẑ). Solvng for ẑ n ths nequalty, we get ẑ V (x ) γ R (x )/(v 0 + V (x ) γ ) = π(x ) = z. Lemma 3.1 follows as a corollary to Theorem 3.. In partcular, as n Lemma 3.1, assume that ẑ 0 satsfes f(ẑ) = v 0 ẑ and ˆx s an optmal soluton to the Functon Evaluaton problem when we solve ths problem wth z = ẑ. In ths case, we have V (ˆx) γ (R (ˆx) ẑ) = f(ẑ), whch mples that ẑ and ˆx satsfes the Suffcent Condton wth f R ( ) = f( ) and α = 1. Therefore, by Theorem 3., we obtan π(ˆx) ẑ z. Snce ˆx s a feasble but not necessarly an optmal soluton to the Assortment problem, we also have z π(ˆx), whch yelds z π(ˆx) ẑ z. Thus, all of the nequaltes n the last chan of nequaltes hold as equaltes, n whch case, Lemma 3.1 follows. By Theorem 3., to obtan an α-approxmate soluton to the Assortment problem, we can execute the followng three steps. A General Approxmaton Framework Step 1: Construct an upper bound f R ( ) on f( ) such that f R (z) f(z) for all z R. Step : Fnd the fxed pont ẑ of f R ( )/v 0 ; that s, fnd the value of ẑ such that f R (ẑ) = v 0 ẑ. Step 3: Fnd an assortment ˆx F such that V (ˆx) γ (R (ˆx) ẑ) α f R (ẑ). In Secton 3., we show how to construct an upper bound f R ( ) on f( ) by usng an LP relaxaton of the Functon Evaluaton problem. In Secton 3.3, we show how to compute the fxed pont of f R ( )/v 0 by solvng an LP. Wth these results, we can execute Steps 1 and n our approxmaton framework. In Secton 4, we use randomzed roundng on an LP relaxaton of the Functon Evaluaton problem to construct an assortment ˆx that satsfes the Suffcent Condton wth α = 0.6, yeldng a 0.6-approxmaton algorthm. Also, we show that we can use an SDP relaxaton to satsfy the Suffcent Condton wth α = 0.79. In Secton 5, we tackle the capactated assortment problem and use an teratve varable fxng method to construct an assortment that satsfes the Suffcent Condton wth α = 0.5, yeldng a 0.5-approxmaton algorthm.
Zhang, Rusmevchentong, and Topaloglu: Assortment Optmzaton under the PCL Model 11 3.. Constructng an Upper Bound We construct an upper bound f R ( ) of f( ) by usng an LP relaxaton of the Functon Evaluaton problem. Notng the defnton of V (x) and R (x), we have V (x) γ (R (x) z) = (v 1/γ x + v 1/γ x ) (r γ z) v 1/γ v 1/γ x + (r z) v 1/γ x x + v 1/γ x. We let ρ (z) be the expresson on the rght sde above when x = 1 and x = 1 and θ (z) be the expresson on the rght sde above when x = 1 and x = 0. In other words, we have ρ (z) = (v 1/γ + v 1/γ ) (r γ z) v 1/γ v 1/γ + (r z) v 1/γ + v 1/γ and θ (z) = v (r z). There are only four possble values of (x, x ). In ths case, lettng µ (z) = ρ (z) θ (z) θ (z) for notatonal brevty, we can express V (x) γ (R (x) z) succnctly as V (x) γ (R (x) z) = ρ (z) x x + θ (z) x (1 x ) + θ (1 x ) x = µ (z) x x + θ x + θ x. Wrtng ts obectve functon as (µ (z) x x + θ x + θ x ), the Functon Evaluaton problem s equvalent to f(z) = max (µ (z) x x + θ (z) x + θ (z) x ) : x c, N x {0, 1} N. To construct an upper bound f R ( ) on f( ), we use a standard approach to lnearze the term x x n the obectve functon above. Defne the decson varable y {0, 1} wth the nterpretaton that y = x x. To ensure that y takes the value x x, we mpose the constrants y x + x 1, y x and y x. If x = 0 or x = 0, then the constrants y x and y x ensure that y = 0. If x = 1 and x = 1, then the constrant y x + x 1 ensures that y = 1. We defne our upper bound f R ( ) on f( ) by usng the LP relaxaton f R (z) = max (µ (z) y + θ (z) x + θ (z) x ) (Upper Bound) s.t. y x + x 1 (, ) M y x, y x (, ) M x c N 0 x 1 N, y 0 (, ) M. Snce the Upper Bound problem s an LP relaxaton of the Functon Evaluaton problem, we have f R (z) f(z) for all z R. Furthermore, settng x = 0 for all N and y = 0 for all (, ) M
1 Zhang, Rusmevchentong, and Topaloglu: Assortment Optmzaton under the PCL Model provdes a feasble soluton to the Upper Bound problem. Therefore, we have f R (z) 0 for all z R. It s not mmedately clear that f R (z) s decreasng n z snce t s not mmedately clear that the obectve functon coeffcent µ (z) n the Upper Bound problem s decreasng n z. In the next lemma, we show that f R (z) s ndeed decreasng n z. In ths case, snce f R (z) s decreasng n z and f R (z) 0 for all z R, there exsts a unque ẑ 0 satsfyng f R (ẑ) = v 0 ẑ. Lemma 3.3 (Monotoncty of Upper Bound) The optmal obectve value of the Upper Bound problem s decreasng n z. Proof: Consder z + z and let (x, y ) wth y = {y : (, ) M} be an optmal soluton to the Upper Bound problem when we solve ths problem wth z = z +. Snce µ (z) = ρ (z) θ (z) θ (z) by the defnton of µ (z), we obtan f R (z + ) = (µ (z + ) y + θ (z + ) x + θ (z + ) x ) = (ρ (z + ) y + θ (z + ) (x y) + θ (z + ) (x y)) (ρ (z ) y + θ (z ) (x y ) + θ (z ) (x y )) = (µ (z ) y + θ (z ) x + θ (z ) x ) f R (z ), where the frst nequalty s by the fact that ρ (z) and θ (z) are decreasng n z, along wth the fact that y x and y x, whereas the second nequalty s by the fact that (x, y ) s a feasble but not necessarly an optmal soluton to the Upper Bound problem wth z = z. One useful property of the Upper Bound problem s that there exsts an optmal soluton to ths problem where the decson varable x takes a nonzero value only when r z and the decson varable y takes a nonzero value only when r z and r z. Thus, we need to keep the decson varable x only when r z and we need to keep the decson varable y only when r z and r z. Ths property allows us to sgnfcantly smplfy the Upper Bound problem. In partcular, let N(z) = { N : r z} and M(z) = {(, ) N(z) : }. In Lemma B.1 n Appendx B, we show that there exsts an optmal soluton x = {x : N} and y = {y : (, ) N} to the Upper Bound problem where we have x = 0 for all N(z) and y = 0 for all (, ) M(z). The proof of ths result follows by showng that f x = {x : N} and y = {y : (, ) N} s a feasble soluton to the Upper Bound problem, then we can set x = 0 for all N(z) and y = 0 for all (, ) M(z) whle makng sure that the soluton (x, y) remans feasble to the Upper Bound problem and we do not degrade the obectve value provded by ths soluton. In ths case, lettng 1( ) be the ndcator
Zhang, Rusmevchentong, and Topaloglu: Assortment Optmzaton under the PCL Model 13 functon and droppng the decson varable x for all N(z) and the decson varable y for all (, ) M(z), we wrte the obectve functon of the Upper Bound problem equvalently as 1( N(z), N(z)) (µ (z) y + θ (z) x + θ (z) x ) + 1( N(z), N(z)) θ (z) x + 1( N(z), N(z)) θ (z) x. For the last two sums, we have 1( N(z), N(z)) θ (z) x = N \ N(z) θ N(z) (z) x and 1( N(z), N(z)) θ (z) x = N \ N(z) θ N(z) (z) x. Thus, the obectve functon of the Upper Bound problem takes the form (µ (z) (z) y + θ (z) x + θ (z) x ) + N \ N(z) θ N(z) (z) x. A smple lemma, gven as Lemma B. n Appendx B, shows that µ (z) 0 for all (, ) M(z). So, the decson varable y takes ts smallest possble value n the Upper Bound problem, whch mples that the constrants y x and y x are redundant. In ths case, the Upper Bound problem s equvalent to the problem f R (z) = max (µ (z) y + θ (z) x + θ (z) x ) + N \ N(z) θ (z) x (z) N(z) s.t. y x + x 1 (, ) M(z), (Compact Upper Bound) x c N(z) 0 x 1 N(z), y 0 (, ) M(z). We wll use the Upper Bound problem to fnd the fxed pont of f R ( )/v 0. We wll use the Compact Upper Bound problem above to fnd an assortment ˆx satsfyng the Suffcent Condton. 3.3. Computng the Fxed Pont To compute the fxed pont of f R ( )/v 0, we use the dual of the Upper Bound problem. For each (, ) M, let α, β, and γ be the dual varables of the constrants y x + x 1, y x and y x, respectvely. For each N, let λ be the dual varable of the constrant x 1. Let δ be the dual varable of the constrant x N c. The dual of the Upper Bound problem s f R (z) = mn c δ + N λ + α (Dual) s.t. α + β + γ µ (z) (, ) M δ + λ + 1( ) (α + α β γ ) (n 1) θ (z) N N α 0, β 0, γ 0 (, ) M, λ 0 N, δ 0. In the Dual problem above, the decson varables are α = {α : (, ) M}, β = {β : (, ) M}, γ = {γ : (, ) M}, λ = {λ : N} and δ. We treat z as fxed. In the next theorem, we show
14 Zhang, Rusmevchentong, and Topaloglu: Assortment Optmzaton under the PCL Model that f we treat z as a decson varable and add one constrant to the Dual problem that nvolves the decson varable z, then we can recover the fxed pont of f R ( )/v 0. Theorem 3.4 (Fxed Pont Computaton by Usng an LP) Let ( ˆα, ˆβ, ˆγ, ˆλ, ˆδ, ẑ) be an optmal soluton to the problem mn c δ + N λ + α s.t. α + β + γ µ (z) (, ) M δ + λ + 1( ) (α + α β γ ) (n 1) θ (z) N N c δ + N λ + α = v 0 z α 0, β 0, γ 0 (, ) M, λ 0 N, δ 0, z s free. (Fxed Pont) Then, we have f R (ẑ) = v 0 ẑ; so, ẑ s the fxed pont of f R ( )/v 0. Proof: Let z be the fxed pont of f R /v 0 so that f R ( z) = v 0 z. We wll show that ẑ = z. Let (ᾱ, β, γ, λ, δ) be an optmal soluton to the Dual problem when we solve ths problem wth z = z. Thus, we have c δ + λ N + ᾱ = f R ( z) = v 0 z, whch mples that the soluton (ᾱ, β, γ, λ, δ, z) satsfes the last constrant n the Fxed Pont problem n the theorem. Furthermore, snce the soluton (ᾱ, β, γ, λ, δ) s feasble to the Dual problem, t satsfes the frst two constrants n the Fxed Pont problem as well. Thus, (ᾱ, β, γ, λ, δ, z) s a feasble but not necessarly an optmal soluton to the the Fxed Pont problem, whch mples that v 0 z = f R ( z) = c δ + λ + ᾱ c ˆδ + ˆλ + N N ˆα = v 0 ẑ, where the last equalty uses the fact that ( ˆα, ˆβ, ˆγ, ˆλ, ˆδ, ẑ) satsfes the last constrant n the Fxed Pont problem. The chan of nequaltes above mples that z ẑ. Next, we show that z ẑ. Note that ( ˆα, ˆβ, ˆγ, ˆλ, ˆδ) s a feasble soluton to the Dual problem wth z = ẑ, so that the obectve value of the Dual problem at the soluton ( ˆα, ˆβ, ˆγ, ˆλ, ˆδ) s no smaller than ts optmal obectve value. Therefore, we have f R (ẑ) c ˆδ + ˆλ N + ˆα = v 0 ẑ, where the equalty uses the fact that ( ˆα, ˆβ, ˆγ, ˆλ, ˆδ, ẑ) satsfes the last constrant n the Fxed Pont problem. Snce f R ( ) s a decreasng functon, havng f R ( z) = v 0 z and f R (ẑ) v 0 ẑ mples that z ẑ. Snce µ (z) and θ (z) are lnear functons of z, the Fxed Pont problem s an LP. Thus, we can compute the fxed pont of f R ( )/v 0 by solvng an LP.
Zhang, Rusmevchentong, and Topaloglu: Assortment Optmzaton under the PCL Model 15 3.4. Extreme Ponts The approxmaton algorthms that we gve for the uncapactated and capactated assortment problems use the propertes of the extreme ponts of the polyhedron defned by the set of feasble solutons to the Compact Upper Bound problem. For the capactated assortment problem, we also work wth the extreme ponts of ths polyhedron after fxng the values of some of the decson varables. As a functon of c, let P c denote the polyhedron defned by the set of feasble solutons to the Compact Upper Bound problem. Furthermore, for any subset H N(z), let P c (H) denote the polyhedron defned by the set of feasble solutons to the Compact Upper Bound problem, after fxng the values of the decson varables {x : H} at 1. That s, we have { P c = (x, y) [0, 1] N(z) R M(z) + : y x + x 1 (, ) M(z), { P c (H) = P c (x, y) [0, 1] N(z) R M(z) + : x = 1 }. H N(z) x c In the next lemma, we consder the uncapactated assortment problem. We show that all of the decson varables {x : N(z)} take a value n {0, 1, 1} n any extreme pont of P c(h). We gve the proof of ths result n Appendx C. Lemma 3.5 (Extreme Ponts n Uncapactated Problem) Assume that c > n. For any H N(z), let (ˆx, ŷ) be any extreme pont of P c (H). Then, we have ˆx { 0, 1, 1} for all N(z). } In the uncapactated assortment problem, the decson varables {x : N(z)} take one of the values n {0, 1, 1} n any extreme pont of P c(h), but the same result does not hold n the capactated assortment problem, as shown n the next example. Example 3.6 (No Half-Integralty n Capactated Problem) Consder the case where we have N(z) = 7, c = 3 and H =. Let ˆx = (,,,,,, 3 ) and ŷ = 0 R M(z) 5 5 5 5 5 5 5 +. Note that we have ˆx = c and ˆx + ˆx 1 0 = ŷ for all (, ) M(z), whch mples that (ˆx, ŷ) P c (H). We clam that (ˆx, ŷ) s an extreme pont of P c (H). Assume on the contrary that there exst (ˆx + ɛ, ŷ + δ) P c (H) and (ˆx ɛ, ŷ δ) P c (H) wth (ɛ, δ) nonzero so that we have (ˆx, ŷ) = 1 (ˆx + ɛ, ŷ + δ) + 1 (ˆx ɛ, ŷ δ). Snce we have ŷ = 0, ŷ + δ R M(z) + and ŷ δ R M(z) +, t must be the case that δ = 0. Notng that (ˆx + ɛ, ŷ + δ) P c (H) and (ˆx ɛ, ŷ δ) P c (H), for each {1,..., 6}, the constrant y 7 x + x 7 1 yelds 0 = ŷ 7 + δ 7 ˆx + ɛ + ˆx 7 + ɛ 7 1 and 0 = ŷ 7 δ 7 ˆx ɛ + ˆx 7 ɛ 7 1. In the case, snce ˆx + ˆx 7 = 1 by the defnton of ˆx, the nequaltes above mply that ɛ 7 = ɛ for all {1,..., 6}. Also, the constrant N(z) x c yelds 7 =1 ˆx + 7 =1 ɛ c and
16 Zhang, Rusmevchentong, and Topaloglu: Assortment Optmzaton under the PCL Model 7 ˆx =1 7 ɛ =1 c, n whch case, notng that 7 ˆx =1 = 3 = c by the defnton of ˆx, we obtan 7 ɛ =1 = 0. Combnng the last equalty wth the fact that ɛ = ɛ 7 for all {1,..., 6}, t follows that ɛ = 0 for all {1,..., 7}. So, (ɛ, δ) s the zero vector, whch s a contradcton. In the next lemma, we consder the capactated assortment problem. We show that f none of the decson varables {x : N(z)} take a fractonal value larger than 1 n an extreme pont of P c (H), then all of these varables take a value n {0, 1, 1}. The proof s n Appendx C. Lemma 3.7 (Extreme Ponts n Capactated Problem) Assume that c n. For any H N(z), let (ˆx, ŷ) be any extreme pont of P c (H). If there s no product N(z) such that 1 < ˆx < 1, then we have ˆx {0, 1, 1} for all N(z). We use the lemma above when developng an teratve varable fxng method for the capactated assortment problem. Note that Example 3.6 does not contradct Lemma 3.7, snce 1 < ˆx 7 < 1 n the extreme pont (ˆx, ŷ) consdered n ths example. 4. Applyng the Approxmaton Framework to the Uncapactated Problem In Sectons 3. and 3.3, we construct an upper bound f R ( ) on f( ) by usng an LP relaxaton of the Functon Evaluaton problem and fnd the fxed pont of f R ( )/v 0 by usng the Fxed Pont problem. Ths dscusson allows us to execute Steps 1 and n our approxmaton framework. In ths secton, we focus on Step 3 n our approxmaton framework. In partcular, consderng the uncapactated assortment problem, we gve a tractable approach to fnd a subset of products ˆx that satsfes V (ˆx) γ (R (ˆx) ẑ) 0.6 f R (ẑ). In ths case, Theorem 3. mples that ˆx s a 0.6-approxmate soluton to the uncapactated assortment problem. Let ẑ satsfy f R (ẑ) = v 0 ẑ. Snce the value of ẑ s fxed, to smplfy our notaton, we exclude the reference to ẑ n ths secton. In partcular, we let µ = µ (ẑ), θ = θ (ẑ), ρ = ρ (ẑ), f R = f R (ẑ), ˆN = N(ẑ) and ˆM = M(ẑ). Fnally, snce we consder the uncapactated assortment problem, we omt the constrant N(z) x c. Therefore, we wrte the Compact Upper Bound problem as f R = max s.t. (µ y + θ x + θ x ) + N \ ˆN θ x ˆN (,) ˆM y x + x 1 (, ) ˆM 0 x 1 ˆN, y 0 (, ) ˆM. Our goal s to fnd ˆx that satsfes V (ˆx) γ (R (ˆx) ẑ) 0.6 f R, where f R s the optmal obectve value of the problem above. We use randomzed roundng for ths purpose. Let (x, y ) be
Zhang, Rusmevchentong, and Topaloglu: Assortment Optmzaton under the PCL Model 17 an optmal soluton to the problem above. We defne a random subset of products ˆX = { ˆX : N} by ndependently roundng each coordnate of x as follows. For each ˆN, we set { 1 wth probablty x ˆX = 0 wth probablty 1 x. For each N \ ˆN, we set ˆX = 0. Note that the subset of products ˆX s a random varable wth E{ ˆX } = x for all ˆN. In the next theorem, we gve the man result of ths secton. Theorem 4.1 (0.6-Approxmaton) Let approach. Then, we have { E ˆX be obtaned by usng the randomzed roundng } V ( ˆX) γ (R ( ˆX) ẑ) 0.6 f R. Proof: Here, we wll show that E{ V ( ˆX) γ (R ( ˆX) ẑ)} 0.5 f R and brefly dscuss how to refne the analyss to get the approxmaton guarantee of 0.6. The detals of the refned analyss are n Appendx D. As dscussed at the begnnng of Secton 3., we have V ( ˆX) γ (R ( ˆX) ẑ) = µ ˆX ˆX + θ ˆX + θ ˆX. So, snce { ˆX : N} are ndependent and E{ ˆX } = x, we have µ x x + θ x + θ x f ˆN, ˆN, E{V ( ˆX) γ θ x (R ( ˆX) f ẑ)} = ˆN, / ˆN θ x f / ˆN, ˆN 0 f / ˆN, / ˆN. Lettng [a] + = max{a, 0} and consderng the four cases above through the ndcator functon, we can wrte,) M E{V ( ˆX) γ (R ( ˆX) ẑ)} equvalently as E{V ( ˆX) γ (R ( ˆX) ẑ)} = = 1( ˆN, ˆN) (µ x x + θ x + θ x ) + 1( ˆN, ˆN) θ x + ˆM(µ x x + θ x + θ x ) + N \ ˆN (,) ˆN = 1( ˆN, ˆN) θ x θ x ˆM(µ [x + x 1] + + θ x + θ x ) + N \ ˆN (,) ˆN + µ (x x [x + x 1] + ) (,) ˆM = f R + (,) ˆM µ (x x [x + x 1] + ) f R + 1 4 (,) ˆM where the second equalty s by the fact that 1( ˆN, ˆN) θ x = N \ ˆN ˆN θ x = 1( ˆN, ˆN) θ x and the fourth equalty follows because µ 0 for all (, ) ˆM θ x µ,
18 Zhang, Rusmevchentong, and Topaloglu: Assortment Optmzaton under the PCL Model by Lemma B. so that the decson varable y takes ts smallest possble value n an optmal soluton to the Compact Upper Bound problem, whch mples that y = [x + x 1] +. The fnal nequalty follows from the fact that 0 ab [a + b 1] + 1/4 for any a, b [0, 1]. We gve a lower bound on f R. Let ˆx = 1 for all ˆN and ŷ = 0 for all (, ) ˆM. In ths case, (ˆx, ŷ) s a feasble soluton to the LP that computes f R at the begnnng of ths secton, whch mples that the obectve value under (ˆx, ŷ) provdes a lower bound on f R. So, we lower bound f R as f R (,) ˆM θ + θ + N \ ˆN ˆN θ (,) ˆM θ + θ (,) ˆM θ + θ ρ = (,) ˆM where the second nequalty holds snce θ 0 for all ˆN, the thrd nequalty holds snce ρ 0 for all (, ) ˆM and the equalty follows from the defnton of µ. Usng the lower bound above on f R n the earler chan of nequaltes, we have E{V ( ˆX) γ (R ( ˆX) ẑ)} f R + 1 4 (,) ˆM µ 1 f R, whch s the desred result. Next, we brefly dscuss how to refne the analyss to mprove the approxmaton guarantee to 0.6. The refned analyss s lengthy and we defer the detals of the refned analyss to Appendx D. The dscusson above uses a lower bound on f R that s based on a feasble soluton (ˆx, ŷ) wth ˆx = 1 for all ˆN and ŷ = 0 for all (, ) ˆM. Ths lower bound may not be tght. The feasble regon n the LP that computes f R at the begnnng of ths secton s gven by the polyhedron P c wth c > n, where P c s as defned n Secton 3.4. By Lemma 3.5, f c > n, then any extreme pont (ˆx, ŷ) of P c satsfes ˆx {0, 1, 1} for all ˆN. In Appendx D, we enumerate over a relatvely small collecton of solutons (ˆx, ŷ) P c, where we have ˆx {0, 1, 1} for all ˆN and ŷ = [x +x 1] + for all (, ) ˆM. We pck the best one of these solutons to obtan a tghter lower bound on f R. In ths case, we can show that E{V ( ˆX) γ (R ( ˆX) ẑ)} 0.6 f R. The subset of products ˆX s a random varable, but n Theorem 3., we need a determnstc subset of products ˆx that satsfes the Suffcent Condton. Snce we manpulate fractons n the proof of Theorem 3., the subset ˆX does not necessarly delver the desred performance guarantee n expectaton. We use the standard method of condtonal expectatons to construct a determnstc subset of products ˆx by buldng on the subset ˆX; see Wllamson and Shmoys (011). Method of Condtonal Expectatons In the method of condtonal expectatons, we nductvely construct a subset of products x (k) = (ˆx 1,..., ˆx k, ˆX k+1,..., ˆX n ) for all k N, where the frst k products n ths subset are determnstc and the last n k products are random varables. These subsets of products are constructed n such a way that they satsfy E{V (x (k) ) γ (R (x (k) ) ẑ)} 0.6 f R for all k N. In ths case, the subset of products x (n) corresponds to a determnstc subset of products and t satsfes µ,
Zhang, Rusmevchentong, and Topaloglu: Assortment Optmzaton under the PCL Model V (x (n) ) γ (R (x (n) ) ẑ) 0.6 f R, as desred. To nductvely construct the subsets of products x (k) = (ˆx 1,..., ˆx k, ˆX k+1,..., ˆX n ) for all k N, we start wth x (0) = ˆX. By Theorem 4.1, we have E{V (x (0) ) γ (R (x (0) ) ẑ)} 0.6 f R. Assumng that we have a subset of products x (k) that satsfes E{V (x (k) ) γ (R (x (k) ) ẑ)} 0.6 f R, we show how to construct a subset of products x (k+1) that satsfes E{V (x (k+1) ) γ (R (x (k+1) ) ẑ)} 0.6 f R. By the nducton assumpton, we have 0.6 f R E{V (x (k) ) γ (R (x (k) ) ẑ)}. Condtonng on ˆX k+1, we wrte the last nequalty as 0.6 f R P{ ˆX k+1 = 1} E{V (x (k) ) γ (R (x (k) ) ẑ) ˆX k+1 = 1} + P{ ˆX k+1 = 0} E{V (x (k) ) γ (R (x (k) ) ẑ) ˆX k+1 = 0}. We defne the two subsets of products as x (k) = (ˆx 1,..., ˆx k, 1, ˆX k+,..., ˆX n ) and as x (k) = (ˆx 1,..., ˆx k, 0, ˆX k+,..., ˆX n ). By the defnton of x (k), gven that ˆXk+1 = 1, we have x (k) = x (k). Gven that ˆX k+1 = 0, we have x (k) = x (k). So, we wrte the nequalty above as 0.6 f R P{ ˆX k+1 = 1} E{V ( x (k) ) γ (R ( x (k) ) ẑ)} + P{ ˆX k+1 = 0} E{V ( x (k) ) γ (R ( x (k) ) ẑ)} { max E{V ( x (k) ) γ (R ( x (k) ) ẑ)}, } E{V ( x (k) ) γ (R ( x (k) ) ẑ)}. Thus, ether E{V ( x (k) ) γ (R ( x (k) ) ẑ)} or E{V ( x (k) ) γ (R ( x (k) ) ẑ)} s at least 0.6 f R, ndcatng that we can use x (k) or x (k) as x (k+1). In both x (k) and x (k), the frst k + 1 products are determnstc and the last n k 1 products are random varables, as desred. We can also use an SDP relaxaton of the Functon Evaluaton problem to construct a subset of products ˆX that satsfes E{V ( ˆX) γ (R ( ˆX) ẑ)} 0.79 f R, but ths approach comes at the expense of solvng an SDP. We defer the detals to Appendx E. 5. Applyng the Approxmaton Framework to the Capactated Problem In ths secton, we consder Step 3 n our approxmaton framework for the capactated assortment problem. Lettng ẑ satsfy f R (ẑ) = v 0 ẑ, we focus on fndng a subset of products ˆx such that V (ˆx) γ (R (ˆx) ẑ) 0.5 f R (ẑ) and N ˆx c. In ths case, by Theorem 3., the subset of products ˆx s a 0.5-approxmate soluton. As n the prevous secton, snce the value of ẑ s fxed, to smplfy our notaton, we drop the argument ẑ n the Compact Upper Bound problem. In partcular, we let µ = µ (ẑ), θ = θ (ẑ), ρ = ρ (ẑ), f R = f R (ẑ), 19 ˆN = N(ẑ) and ˆM = M(ẑ). Furthermore, we observe that the set of feasble solutons n the Compact Upper Bound problem corresponds to the polyhedron P c gven n Secton 3.4. Therefore, notng that the optmal
0 Zhang, Rusmevchentong, and Topaloglu: Assortment Optmzaton under the PCL Model obectve value of the Compact Upper Bound problem s f R, we can equvalently wrte the Compact Upper Bound problem as { f R = max (µ y + θ x + θ x ) + N \ ˆN θ x : (x, y) P c }. (,) ˆM ˆN As ndcated by Example 3.6, t seems to be dffcult to predct the values of the varables {x : ˆN} n an extreme pont (x, y) of P c. Thus, we do not see a smple way of drectly roundng an optmal soluton to the LP above. Instead, we teratvely solve a modfed verson of the LP above, where we fx some of the decson varables {x : ˆN} at 1. Our goal s to obtan a soluton, where the decson varables {x : ˆN} ultmately all take values n {0, 1, 1} and the obectve value provded by ths soluton for the LP above s not too far from the optmal obectve value f R. In partcular, we consder the followng teratve varable fxng algorthm. Iteratve Varable Fxng Step 1: Set the teraton counter to k = 1 and ntalze the set of fxed varables H k =. Step : Let (x k, y k ) be an optmal soluton to the problem { f k = max (µ y + θ x + θ x ) + N \ ˆN } θ x : (x, y) P c (H k ), (Varable Fxng) ˆN (,) ˆM where the polyhedron P c (H) for H ˆN s as defned n Secton 3.4. Step 3: If there exsts some k ˆN such that 1 < xk k < 1, then set H k+1 = H k { k }, ncrease k by one and go to Step. Otherwse, stop. By the defnton of P c (H), the decson varables {x : H k } are fxed at 1 n the Varable Fxng problem n Step of the teratve varable fxng algorthm. Wthout loss of generalty, we assume that the optmal soluton (x k, y k ) to the Varable Fxng problem n Step s an extreme pont of P c (H k ). In Step 3 of the teratve varable fxng algorthm, f x k ( 1, 1) for all ˆN, then we stop. Lemma 3.7 mples that f x k ( 1, 1) for all ˆN, then x k {0, 1, 1} for all ˆN. Therefore, f the teratve varable fxng algorthm stops at teraton k wth the soluton (x k, y k ), then we have x k {0, 1, 1} for all ˆN. We use (x, y ) to denote the soluton to the Varable Fxng problem at the last teraton of the teratve varable fxng algorthm. We observe that ncludng each product ˆN n a subset wth probablty x may not provde a feasble soluton. In partcular, snce we have x {0, 1, 1} and ˆN x c, there can be as many as c decson varables such that x > 0. Includng each product ˆN n a subset wth probablty x may yeld a subset wth as many as c products. Instead, we use the followng coupled randomzed roundng approach to obtan a subset of products that satsfes the capacty constrant.