Approximation Methods for Pricing Problems under the Nested Logit Model with Price Bounds

Transcription

1 Approxmaton Methods for Prcng Problems under the Nested Logt Model wth Prce Bounds W. Zachary Rayfeld School of Operatons Research and Informaton Engneerng, Cornell Unversty, Ithaca, New York 14853, USA Paat Rusmevchentong Marshall School of Busness, Unversty of Southern Calforna, Los Angeles, Calforna 90089, USA Huseyn Topaloglu School of Operatons Research and Informaton Engneerng, Cornell Unversty, Ithaca, New York 14853, USA September 5, 2013 Abstract We consder two varants of a prcng problem under the nested logt model. In the frst varant, the set of products offered to customers s fxed and we want to determne the prces of the products. In the second varant, we jontly determne the set of offered products and ther correspondng prces. In both varants, the prce of each product has to be chosen wthn gven upper and lower bounds specfc to the product, each customer chooses among the offered products accordng to the nested logt model and the objectve s to maxmze the expected revenue from each customer. We gve approxmaton methods for both varants. For any ρ > 0, our approxmaton methods obtan a soluton wth an expected revenue devatng from the optmal expected revenue by no more than a factor of 1 + ρ. To obtan such a soluton, our approxmaton methods solve a lnear program whose sze grows at rate 1/ρ. In addton to our approxmaton methods, we develop a lnear program that we can use to obtan an upper bound on the optmal expected revenue. In our computatonal experments, we compare the expected revenues from the solutons obtaned by our approxmaton methods wth the upper bounds on the optmal expected revenues and show that we can obtan hgh qualty solutons qute fast.

2 1 Introducton When faced wth product varety, most customers make ther purchase decsons by comparng the offered products through attrbutes such as prce, rchness of features and durablty. In ths type of a stuaton, the demand for a certan product s determned not only by ts own attrbutes but also by the attrbutes of other products, creatng nteractons among the demands for dfferent products. Dscrete choce models are partcularly sutable to study such demand nteractons, as they model the demand for a certan product as a functon of the attrbutes of all products offered to customers. However, optmzaton models that try to fnd the rght set of products to offer or the rght prces to charge may quckly become ntractable when one works wth complex dscrete choce models and tres to ncorporate operatonal constrants. In ths paper, we consder prcng problems where the nteractons between the demands for the dfferent products are captured through the nested logt model and there are bounds on the prces that can be charged for the products. We consder two problem varants. In the frst varant, the set of products offered to customers s fxed and we want to determne the prces for these products. In the second varant, we jontly determne the products that should be offered to customers and ther correspondng prces. Once the products to be offered and ther prces are determned, customers choose among the offered products accordng to the nested logt model. In both varants, the objectve s to maxmze the expected revenue obtaned from each customer. We gve approxmaton methods for both varants of the problem. In partcular, for any ρ > 0, our approxmaton methods obtan a soluton wth an expected revenue devatng from the optmal by at most a factor of 1 + ρ. To obtan ths soluton, the approxmaton methods solve lnear programs whose szes grow lnearly wth 1/ log(1 + ρ). Notng that 1/ log(1 + ρ) grows at the same rate as 1/ρ for small values of ρ, the computatonal work for our approxmaton methods grows polynomally wth the approxmaton factor. Our approxmaton methods gve a performance guarantee over all problem nstances, but we also develop a lnear program that we can use to quckly obtan an upper bound on the optmal expected revenue for an ndvdual problem nstance. In our computatonal experments, we compare the expected revenues from the solutons obtaned by our approxmaton methods wth the upper bounds on the optmal expected revenues and demonstrate that our approxmaton methods can quckly obtan solutons whose expected revenues dffer from the optmal by less than a percent. Thus, our approxmaton methods have favorable theoretcal performance guarantees and they are useful to obtan hgh qualty solutons n practce. Man Results and Contrbutons. The frst problem varant we consder s a prcng problem where customers choose accordng to the nested logt model and there are bounds on the prces of the offered products. For the frst varant, assumng that there are m nests n the nested logt model and each nest ncludes n products to offer, we show that for any ρ > 0, we can solve a lnear program wth O(m) decson varables and O(mn + mn log(nσ)/ log(1 + ρ)) constrants to obtan a set of prces wth an expected revenue devatng from the optmal expected revenue by at most a factor of 1+ρ. In ths result, σ depends on the devaton between the upper and lower prce bounds of the 2

3 products. The second problem varant we consder s a jont assortment offerng and prcng problem, where we need to choose the products to offer and ther correspondng prces. For ths varant, we establsh a useful property for the optmal subsets of products to offer. In partcular, orderng the products accordng to ther prce upper bounds, we show that t s optmal to offer a certan number of products wth the largest prce upper bounds. Usng ths result, we show that for any ρ > 0, we can solve a lnear program wth O(m) decson varables and O(mn 2 + mn 2 log(nσ)/ log(1 + ρ)) constrants to fnd a set of products to offer and ther correspondng prces such that the expected revenue obtaned by ths soluton devates from the optmal expected revenue by at most a factor of 1 + ρ. Comparng our results for the two varants, we observe that the extra computatonal burden of jontly fndng a set of products to offer and prcng the offered products bols down to ncreasng the number of constrants n the lnear program by a factor of n. Prcng under the nested logt model has recently receved attenton, startng wth the work of L and Huh (2011) and Gallego and Wang (2011). L and Huh (2011) consder prcng problems wthout upper or lower bound constrants on the prces. Assumng that the products n the same nest share the same prce senstvty parameter and the so called dssmlarty parameters of the nested logt model are less than one, they cleanly show that the prcng problem can be reduced to the problem of maxmzng a scalar functon. Ths scalar functon turns out to be unmodal so that maxmzng t s tractable. Gallego and Wang (2011) also study prcng problems under the nested logt model wthout prce bounds, but they allow the products n the same nest to have dfferent prce senstvtes and the dssmlarty parameters of the nested logt model to take on arbtrary values. Surprsngly, ther elegant argument shows that the optmal prces can stll be found by maxmzng a scalar functon, but ths scalar functon s not unmodal n general and evaluatng ths scalar functon at any pont requres solvng a separate hgh dmensonal optmzaton problem nvolvng mplctly defned functons. Our paper flls a number of gaps n ths area. The earler work shows that the problem of fndng the optmal prces can be reduced to maxmzng a scalar functon, but ths functon s not unmodal and maxmzng t can be ntractable for two reasons. Frst, a natural approach to maxmzng ths scalar functon s to evaluate t at a fnte number of grd ponts and pck the best soluton, but t s not clear how to place these grd ponts to obtan a performance guarantee. Second, gven that computng the scalar functon at any pont requres solvng a nontrval optmzaton problem, t s computatonally prohbtve to smply follow a brute force approach and use a large number of grd ponts. Thus, whle the earler work shows how to reduce the prcng problem to a problem of maxmzng a scalar functon, as far as we can see, t does not yet yeld a computatonally vable and theoretcally sound algorthm to compute near optmal prces n general. Our work provdes practcal algorthms that delver a desred performance guarantee of 1 + ρ for any ρ > 0. To obtan our approxmaton methods, we transform the prcng problem nto a knapsack problem wth a separable and concave objectve functon, whch ultmately allows us to use dfferent arguments from L and Huh (2011) and Gallego and Wang (2011). Besde provdng computatonally vable algorthms to fnd prces wth a certan performance guarantee, a unque feature of our work s that t allows mposng bounds on the prces that can be 3

4 chosen by the decson maker. Such prce bounds do not appear n the earler prcng work under the nested logt model and there are a number of theoretcal and practcal reasons for studyng such bounds. On the theoretcal sde, f we mpose prce bounds, then even n the smplest case when the prce senstvtes of all products are equal to each other, the scalar functons n the works of L and Huh (2011) and Gallego and Wang (2011) are no longer unmodal. In such cases, we emphasze that the lack of unmodalty s purely due to the presence of the prce bounds, as the work of L and Huh (2011) shows that the scalar functons that they work are ndeed unmodal when the prce senstvtes of the products are equal to each other. Thus, prce bounds can sgnfcantly complcate the structural propertes of the prcng problem. Furthermore, nave approaches for satsfyng prce bound constrants may yeld poor results. For example, a frst cut approach for dealng wth prce bounds s to use the work of L and Huh (2011) or Gallego and Wang (2011) to fnd the optmal prces for the products under the assumpton that there are no prce bounds. If these unconstraned prces are outsde the prce bound constrants, then we can round them up or down to ther correspondng lower or upper bounds. Ths nave approach does not perform well and we can come up wth problem nstances where ths nave approach can result n revenue losses of over 20%, when compared wth approaches that explctly ncorporate prce bounds. There are also practcal reasons for studyng prce bounds. Customers may have expectatons for sensble prce ranges and t s useful to ncorporate these prce ranges explctly nto the prcng model. Furthermore, lack of data may prevent us from fttng an accurate choce model to capture customer choces, n whch case, we can gude the model by lmtng the range of possble prces through prce bounds. When we solve the prcng model wthout prce bounds, we essentally rely on the choce model to fnd a set of reasonable prces for the products, but dependng on the parameters of the choce model, the prces may not come out to be practcal. Thus, ncorporatng prce bounds nto the prcng problem s a nontrval task from a theoretcal perspectve and t has mportant practcal mplcatons. It s also worth mentonng that f there are no prce bounds, then fndng the rght set of products to offer s not an ssue as Gallego and Wang (2011) show that t s always optmal to offer all products at some fnte prce level. Ths result does not hold n the presence of prce bounds and our second varant, whch jontly determnes the set of products to offer and ther correspondng prces, becomes partcularly useful. Our approxmaton methods allow us to obtan prces wth a certan performance guarantee. In addton to these approxmaton methods, we gve a smple approach to compute an upper bound on the optmal expected revenue. Ths upper bound s obtaned by solvng a lnear program and we can progressvely refne the upper bound by ncreasng the number of constrants n the lnear program. By comparng the expected revenue from the soluton obtaned by our approxmaton methods wth the upper bound on the optmal expected revenue, we can bound the optmalty gap of the solutons obtaned by our approxmaton methods for each ndvdual problem nstance. Admttedly, our approxmaton methods provde a performance guarantee of 1 + ρ for a gven ρ > 0, but ths s the worst case performance guarantee over all problem nstances and t turns out that we can use the lnear program to obtan a tghter performance guarantee for an 4

5 ndvdual problem nstance. The lnear program we use to obtan an upper bound on the optmal expected revenue can be useful even f we do not work wth our approxmaton methods to obtan a good soluton to the prcng problem. In partcular, we can use an arbtrary heurstc or an approxmaton method to obtan a set of prces and check the gap between the expected revenue obtaned by chargng these prces and the upper bound on the optmal expected revenue. If the gap turns out to be small, then there s no need to look for better prces. Related Lterature. There s a long hstory on buldng dscrete choce models to capture customer preferences. Some of these models are based on axoms descrbng a sensble behavor of customer choce, as n the basc attracton model of Luce (1959). On the other hand, some others use a utlty maxmzaton prncple, where an arrvng customer assocates random utltes wth the products and chooses the product provdng the largest utlty. Such a utlty based approach s followed by McFadden (1974), resultng n the celebrated multnomal logt model. The nested logt model, whch plays a central role n ths paper, goes back to the work of Wllams (1977). Extensons for the nested logt model are provded by McFadden (1980) and Borsch-Supan (1990). An mportant feature of the nested logt model s that t avods the ndependence of rrelevant alternatves property suffered by the multnomal logt model. The dscusson of ths property can be found n Ben-Akva and Lerman (1994). There s a body of work on assortment optmzaton problems under varous dscrete choce models. In the assortment optmzaton settng, the prces of the products are fxed and we choose the set of products to offer gven that customers choose among the offered products accordng to a partcular choce model. Tallur and van Ryzn (2004) study assortment problems when customers choose under the multnomal logt model and show that the optmal assortment ncludes a certan number of products wth the largest revenues. As a result, the optmal assortment can effcently be found by checkng the performance of every assortment that ncludes a certan number of products wth the largest revenues. Rusmevchentong et al. (2010) consder the same problem wth a constrant on the number of products n the offered assortment and show that the problem can be solved n a tractable fashon. Wang (2012a) extends ths work to more general versons of the multnomal logt model. In Bront et al. (2009), Mendez-Daz et al. (2010) and Rusmevchentong et al. (2013), there are multple types of customers, each choosng accordng to the multnomal logt model wth dfferent parameters. The authors show that the assortment problem becomes NP-hard n weak and strong sense, propose approxmaton methods and study nteger programmng formulatons. Jagabathula et al. (2011) work on how to obtan good assortments wth only lmted computatons of the expected revenue from dfferent assortments. The work mentoned so far n ths paragraph uses the multnomal logt model, but there are extensons to the nested logt model. Rusmevchentong et al. (2009) develop an approxmaton scheme for assortment problems when customers choose under the nested logt model and there s a shelf space constrant for the offered assortment. Davs et al. (2011) study the same problem wthout the shelf space constrant and gve a tractable method to obtan the optmal assortment under the nested logt model. Gallego and Topaloglu (2012) show that t s tractable to obtan the optmal assortment when customers 5

6 choose accordng to the nested logt model and there s a cardnalty constrant on the number of products offered n each nest. They extend ther result to the stuaton where each product can be offered at a fnte number of prce levels and one needs to jontly choose the assortment of products to offer and ther correspondng prce levels. Ther approach does not work when the set of products to be offered s fxed and not under the control of the decson maker. Prcng problems wthn the context of dfferent dscrete choce models s also an actve research area. Under the multnomal logt model, Hanson and Martn (1996) note that the expected revenue functon s not concave n prces. However, Song and Xue (2007) and Dong et al. (2009) make progress by formulatng the problem n terms of market shares, as ths formulaton yelds a concave expected revenue functon. L and Huh (2011) extend the concavty result to the nested logt model by assumng that the prce senstvtes of the products are constant wthn each nest and the dssmlarty parameters are less than one. Gallego and Wang (2011) relax both of the assumptons n L and Huh (2011) and extend the analyss to more general forms of the nested logt model. Wang (2012b) consders a jont assortment and prce optmzaton problem to choose the offered products and ther prces. The author mposes cardnalty constrants on the offered assortment, but the customer choces are captured by usng the multnomal logt model, whch s more restrctve than the nested logt model. Recently, there s nterest n modelng large scale revenue management problems by ncorporatng the fact that customers make a choce dependng on the assortment of avalable tnerary products and ther prces. The man approach n these models s to formulate determnstc approxmatons under the assumpton that customer arrvals and choces are determnstc. Such determnstc approxmatons have a large number of decson varables and they are usually solved by usng column generaton. The assortment and prcng problems descrbed n ths and the paragraph above become nstrumental when solvng the column generaton subproblems. Determnstc approxmatons for large scale revenue management can be found n Gallego et al. (2004), Lu and van Ryzn (2008), Kunnumkal and Topaloglu (2008), Zhang and Adelman (2009), Zhang and Lu (2011) and Messner et al. (2012). Organzaton. In Secton 2, we formulate the frst varant of the problem, where the set of products to be offered s fxed and we choose the prces for these products. In Secton 3, we show that ths problem can be vsualzed as fndng the fxed pont of a scalar functon. In Secton 4, we develop an approxmaton framework by usng the fxed pont representaton and computng a scalar functon at a fnte number of grd ponts. In Secton 5, we show how to construct an approprate grd wth a performance guarantee and gve our approxmaton method. In Secton 6, we extend the work n the earler sectons to the second varant of the problem, where we jontly choose the products to offer and ther correspondng prces. In Secton 7, we show how to obtan an upper bound on the optmal expected revenue and gve computatonal experments to compare the performance of our approxmaton methods wth the upper bounds on the optmal expected revenues. In Secton 8, we conclude. In Appendces A and B, we gve the proofs that are omtted n the paper. In Appendx C, we gve a glossary to collect the crucal peces of notaton used throughout the paper. 6

7 2 Problem Formulaton In ths secton, we descrbe the nested logt model and formulate the prcng problem. There are m nests ndexed by M = {1,..., m}. Dependng on the applcaton settng, nests may correspond to dfferent retal stores, dfferent product categores or dfferent sales channels. In each nest there are n products and we ndex the products by N = {1,..., n}. We use p j to denote the prce of product j n nest. The prce of product j n nest has to satsfy the prce bound constrant p j [l j, u j ], for the upper and lower bound parameters l j, u j [0, ). We use w j to denote the preference weght of product j n nest. Under the nested logt model, f we choose the prce of product j n nest as p j, then the preference weght of ths product s w j = exp(α j β j p j ), where α j (, ) and β j [0, ) are parameters capturng the effect of the prce on the preference weght. Snce there s a one to one correspondence between the prce and preference weght of a product, throughout the paper, we assume that we choose the preference weght of a product, n whch case, there s a prce correspondng to the chosen preference weght. In partcular, f we choose the preference weght of product j n nest as w j, then the correspondng prce of ths product s p j = (α j log w j )/β j, whch s obtaned by settng w j = exp(α j β j p j ) and solvng for p j. For brevty, we let κ j = α j /β j and η j = 1/β j and wrte the relatonshp between prce and preference weght as p j = κ j η j log w j. Notng the upper and lower bound constrant on prces, the preference weght of product j n nest has to satsfy the constrant w j [L j, U j ] wth L j = exp(α j β j u j ) and U j = exp(α j β j l j ). We use w = (w 1,..., w n ) to denote the vector of preference weghts of the products n nest. Under the nested logt model, f we choose the preference weghts of the products n nest as w and a customer decdes to make a purchase n ths nest, then ths customer purchases product j n nest wth probablty w j / k N w k. Thus, f we choose the preference weghts of the products n nest as w and a customer decdes to make a purchase n ths nest, then we obtan an expected revenue of R (w ) = j N w j k N w k (κ j η j log w j ) = j N w j (κ j η j log w j ) j N w, j where the term w j / k N w k on the left sde above s the probablty that a customer purchases product j n nest gven ths customer decdes to make a purchase n ths nest, whereas the term κ j η j log w j captures the revenue assocated wth product j n nest. Each nest has a parameter γ (0, 1], characterzng the degree of dssmlarty between the products n ths nest. In ths case, f we choose the preference weghts of the products n all nests as (w 1,..., w m ), then a customer decdes to make a purchase n nest wth probablty ( Q (w 1,..., w m ) = 1 + l M j N w ) γ j ( j N w lj Dependng on the nterpretaton of a nest as a retal store, a product category or a sales channel, the expresson above computes the probablty that a customer chooses a partcular retal store, product category or sales channel as a functon of the preference weghts of all products. Wth 7 ) γl.

8 probablty 1 M Q (w 1,..., w m ), a customer leaves wthout makng a purchase. McFadden (1984) demonstrates that the choce probabltes above can be derved from a utlty maxmzaton prncple, where a customer assocates a random utlty wth each product and purchases the product that provdes the largest utlty. Thus, f we choose the preference weghts as (w 1,..., w m ) over all nests, then we obtan an expected revenue of Π(w 1,..., w m ) = M Q (w 1,..., w m ) R (w ) = 1 + M 1 ( j N w j ( ) γ ) γ w j M j N j N w j (κ j η j log w j ) j N w, (1) j where the second equalty s by the defntons of R (w ) and Q (w 1,..., w m ). Our goal s to choose the preference weghts to maxmze the expected revenue, yeldng the problem Z = max { Π(w 1,..., w m ) : w [L, U ] M }, (2) where we use L and U to respectvely denote the vectors (L 1,..., L n ) and (U 1,..., U n ) and nterpret the constrant w [L, U ] componentwse as w j [L j, U j ] for all j N. We close ths secton wth a remark on our formulaton of the nested logt model. In our formulaton of the nested logt model, f a customer chooses a partcular nest, then ths customer must purchase one of the products offered n ths nest. We can extend our model to allow the possblty that a customer may leave wthout purchasng anythng even after choosng a partcular nest. To make ths extenson, we use w 0 to denote the preference weght of the no purchase opton n nest, n whch case, f a customer decdes to make a purchase n nest, then ths customer leaves the nest wthout purchasng anythng wth probablty w 0 /(w 0 + j N w j). The preference weght w 0 s a constant, not dependng on the prces of any of the products. It turns out that our results contnue to hold when we allow customers to leave a nest wthout purchasng anythng. We come back to ths extenson at approprate places n the paper. 3 Fxed Pont Representaton In ths secton, we show that problem (2) can alternatvely be represented as the problem of computng the fxed pont of an approprately defned scalar functon. Ths alternatve fxed pont representaton allows us to work wth a sngle decson varable for each nest, rather than n decson varables w = (w 1,..., w n ) and t becomes crucal when developng our approxmaton methods. To that end, assume that we compute the value of z that satsfes z = { ( ) γ j N max w w j (κ j η j log w j ) ( ) γz j w [L,U ] M j N j N w w j }. (3) j j N Vewng the rght sde of (3) as a functon of z, fndng the value of z satsfyng (3) s equvalent to computng the fxed pont of ths scalar functon. There always exsts such a unque value of z 8

9 snce the left sde above s strctly ncreasng and the rght sde above s decreasng n z. Lettng ẑ be the value of z satsfyng (3), we clam that ẑ s the optmal objectve value of problem (2). To see ths clam, note that f (w1,..., w m) s an optmal soluton to problem (2), then we have ẑ { ( ) w γ j N w j (κ j η j log wj ) } ( γẑ j wj), M j N j N w j j N where we use the fact that ẑ s the value of z satsfyng (3) and w s a feasble but not necessarly an optmal soluton to the maxmzaton problem on the rght sde of (3) when we solve ths problem wth z = ẑ. In the nequalty above, f we collect all terms that nvolve ẑ on the left sde of the nequalty, solve for ẑ and use the defnton of Π(w 1,..., w m ) n (1), then t follows that ẑ Π(w 1,..., w m) = Z. On the other hand, f we let ŵ be an optmal soluton to the maxmzaton problem on the rght sde of (3) when we solve ths problem wth z = ẑ, then we observe that (ŵ 1,..., ŵ m ) s a feasble soluton to problem (2). Furthermore, snce ẑ s the value of z that satsfes (3), the defnton of ŵ mples that ẑ = { ( ) γ j N ŵ ŵj (κ j η j log ŵ j ) ( j M j N ŵj j N j N } ) γẑ ŵ j. (4) If we solve for ẑ n the equalty above and use the defnton of Π(w 1,..., w m ) n (1) once more, then we get ẑ = Π(ŵ 1,..., ŵ m ) Z, where the last nequalty uses the fact that (ŵ 1,..., ŵ m ) s a feasble but not necessarly an optmal soluton to problem (2). So, we obtan ẑ = Z, establshng the clam. Thus, we can obtan the optmal objectve value of problem (2) by fndng the value of z that satsfes (3). Furthermore, f we use ẑ to denote such a value of z and ŵ to denote an optmal soluton to the maxmzaton problem on the rght sde of (3) when ths problem s solved wth z = ẑ, then the dscusson n ths paragraph establshes that (ŵ 1,..., ŵ m ) s an optmal soluton to problem (2). Snce the left and rght sdes of (3) are respectvely ncreasng and decreasng n z, we can fnd the value of z satsfyng (3) by usng bsecton search. However, one drawback of usng bsecton search s that we need to solve the maxmzaton problem on the rght sde of (3) for each value of z vsted durng the course of the search. Ths maxmzaton problem nvolves a hgh dmensonal objectve functon. Also, t s not dffcult to generate counterexamples to show that ths objectve s not necessarly concave. To get around the necessty of dealng wth hgh dmensonal and nonconcave objectve functons, we gve an alternatve approach for fndng the value of z satsfyng (3). We defne g (y ) as the optmal objectve value of the nonlnear knapsack problem { } g (y ) = max w j y, w j [L j, U j ] j N. (5) j N w j (κ j η j log w j ) : j N We make a number of observatons regardng problem (5). We can verfy that the objectve functon of ths problem s concave. Also, f we do not have the frst constrant n the problem above, then by usng the frst order condton for the objectve functon of ths problem, we can check that the 9

10 optmal value of the decson varable w j s gven by mn{max{exp(κ j /η j 1), L j }, U j } for all j N. Thus, lettng Ū = j N mn{max{exp(κ j/η j 1), L j }, U j }, f we have y > Ū, then the frst constrant n problem (5) s not tght at the optmal soluton. On the other hand, lettng L = j N L j, f we have y < L, then problem (5) s nfeasble. Fnally, f we have y [ L, Ū], then t follows that the frst constrant n problem (5) s always tght at the optmal soluton. Thus, ntutvely speakng, the nterestng values for y take values n the nterval [ L, Ū]. In ths case, notng that problem (5) fnds the maxmum value of the numerator of the fracton n (3) whle keepng the denomnator of ths fracton below y, nstead of fndng the value of z satsfyng (3), we propose fndng the value of z that satsfes { z = M max y [ L,Ū] y γ g (y ) y y γ z }. (6) The value of z satsfyng (6) s unque snce the left sde above s strctly ncreasng and the rght sde above s decreasng n z. The maxmzaton problem on the rght sde above nvolves a scalar decson varable and the computaton of g (y ) requres solvng a convex optmzaton problem. In the next proposton, we show that (6) can be used to fnd the value of z satsfyng (3). Proposton 1 (Fxed Pont Representaton). The value of z that satsfes (3) and (6) are the same, correspondng to the optmal objectve value of problem (2). Proof. The value of z that satsfes (3) or (6) has to be postve. Otherwse, the left sdes of these expressons evaluate to a negatve number, but the rght sdes evaluate to a postve number. In ths case, comparng (3) and (6), f we can show that { ( ) γ j N max w w j (κ j η j log w j ) j w [L,U ] j N w j j N ( j N w j ) γz }= max y [ L,Ū] { y γ g (y ) y y γ z for any z > 0, then the value of z that satsfes (3) and (6) are the same. The equalty above can be establshed by showng that we can use the optmal soluton to one of the problems above to construct a feasble soluton to the other. We defer the detals to Appendx A. The proposton above provdes a temptng approach for solvng problem (2). In partcular, we can fnd the value of z that satsfes (6) by usng bsecton search. We observe that the maxmzaton problem on the rght sde of (6) nvolves a scalar decson varable and the computaton of g ( ) requres solvng a convex optmzaton problem. Thus, the optmzaton problems that we solve durng the course of the bsecton search may be tractable. We use ẑ to denote the value of z that satsfes (6) and ŷ to denote an optmal soluton to the maxmzaton problem on the rght sde of (6) when we solve ths problem wth z = ẑ. In ths case, we can solve problem (5) wth y = ŷ to obtan an optmal soluton ŵ. Once we solve problem (5) wth y = ŷ for all M, t follows that (ŵ 1,..., ŵ m ) s an optmal soluton to problem (2). It s possble to check that all of the dscusson n ths secton holds when we use the no purchase preference weght w 0 to allow a customer to leave nest wthout makng a purchase. To make ths extenson, all we need to do s to replace j N w j n (3) and (5) wth w 0 + j N w j. 10 }

11 Fgure 1: The functon y γ 1 1 (g 1(y 1 )/y 1 ) y γ 1 1 z as a functon of y 1. 4 Approxmaton Framework As mentoned at the end of the prevous secton, the maxmzaton problem on the rght sde of (6) nvolves a scalar decson varable and t s temptng to try to solve problem (2) by fndng the value of z satsfyng (6). Unfortunately, t turns out that the objectve functon of ths maxmzaton problem s not unmodal and t can be ntractable to solve the maxmzaton problem on the rght sde of (6). To gve an example where the objectve functon of the maxmzaton problem on the rght sde of (6) s not unmodal, consder a case wth a sngle nest and seven products. The problem parameters are gven by γ 1 = 0.4, (α 11,..., α 17 ) = (2.1, 1.0, 1.7, 1.4, 1.0, 12.0, 13.0), (β 11,..., β 17 ) = (0.07, 0.07, 0.07, 0.07, 0.07, 0.07, 0.07), (l 11,..., l 17 ) = (30, 30, 30, 30, 30, 251, 330) and (u 11,..., u 17 ) = (200, 200, 200, 200, 200, 368, 383). For ths problem nstance, Fgure 1 plots the objectve functon of the maxmzaton problem on the rght sde of (6) as a functon of y 1, fxng z at and shows that ths objectve functon s not necessarly unmodal. We note that the value of z that we use n ths fgure s sensble as the optmal objectve value of problem (2) s close to for ths problem nstance. So, we do not have unmodalty even wth sensble values of z. Interestngly, Gallego and Wang (2011) consder the case where there are no lower or upper bounds on the prces. The authors show that f the dssmlarty parameters of the nests satsfy γ 1 mn j N β j / max j N β j for all M, then the objectve functon of the maxmzaton problem on the rght sde of (6) s always unmodal. In the example above, we ndeed have γ 1 mn j N β j / max j N β j for all M, ndcatng that ths example satsfes the condton n Gallego and Wang (2011). However, due to the presence of the lower and upper bounds on the prces, we lose the unmodalty property. The objectve functon of the maxmzaton problem on the rght sde of (6) s not necessarly unmodal, but snce ths objectve functon s scalar, a possble strategy s to construct a grd over the nterval [ L, Ū] and check the values of the objectve functon only at the grd ponts. To 11

12 pursue ths lne of thought, we use {ỹ t : t = 1,..., T } to denote a collecton of grd ponts such that ỹ t ỹt+1 for all t = 1,..., T 1. Furthermore, the collecton of grd ponts should satsfy ỹ 1 = L and ỹ T = Ū to make sure that the grd ponts cover the nterval [ L, Ū]. In ths case, nstead of consderng all values of y over the nterval [ L, Ū] as we do n (6), we can focus only on the grd ponts and fnd the value of z that satsfes z = M max y {ỹ t : t = 1,...,T } { y γ g (y ) y y γ z }. (7) The mportant queston s that what propertes the grd should possess so that the soluton obtaned by lmtng our attenton only to the grd ponts has a quantfable performance guarantee. In the next theorem, we show that f the optmal objectve value g (y ) of the knapsack problem n (5) does not change too much at the successve grd ponts, then we can buld on the value of z satsfyng (7) to construct a soluton to problem (2) wth a certan performance guarantee. Theorem 2 (Requrements for a Good Grd). For some ρ 0, assume that the collecton of grd ponts {ỹ t : t = 1,..., T } satsfy g (ỹ t+1 ) (1 + ρ) g (ỹ t) for all t = 1,..., T 1, M. If ẑ denotes the value of z that satsfes (7) and Z denotes the optmal objectve value of problem (2), then we have (1 + ρ) ẑ Z. Proof. To get a contradcton, assume that (1 + ρ) ẑ < Z. For all M, we let y be an optmal soluton to the maxmzaton problem on the rght sde of (6) when ths problem s solved wth z = Z. Furthermore, we let t {1,..., T 1} be such that y [ỹt ρ Z > ẑ { (ỹt ) γ g (ỹ t ) ỹ t ( } ỹ t ) γ ẑ { ρ M M t, ỹt +1 (ỹt ) γ g (y ) ỹ t ]. We have ( ỹ t ) γ ẑ where the second nequalty follows from the fact that ẑ corresponds to the value of z that satsfes (7) and ỹ t s a feasble but not necessarly an optmal soluton to the maxmzaton problem on the rght sde of (7) when ths problem s solved wth z = ẑ. To see that the thrd nequalty holds, we observe that g ( ) s ncreasng, n whch case, snce y [ỹt, ỹt +1 (1+ρ) g (ỹ t ). In ths case, notng that γ 1 and ỹ t the chan of nequaltes above as { M ρ (ỹt ) γ g (y ) ỹ t ( ỹ t ) γ ẑ } M { ρ ], we obtan g (y ) g (ỹ t +1 ) y so that (ỹt )1 γ (y )1 γ, we contnue ( y ) γ g (y ) y ρ M ( y ) γ ẑ } } { (y ) γ g (y ) y ( } y ) γ Z, where the second nequalty uses the assumpton that (1+ρ) ẑ < Z. By usng the last two dsplayed chans of nequaltes and notng the defnton of y, t follows that Z > { (y ) γ g (y ) y ( } y ) γ Z = { max M y M [ L,Ū] y γ g (y ) y y γ Z }., 12

13 By Proposton 1, Z corresponds to the value of z that satsfes (6), but the last chan of nequaltes above shows that Z does not satsfy (6), whch s a contradcton. When we work wth grd ponts that satsfy the assumpton of Theorem 2, ths theorem allows us to obtan a (1 + ρ)-approxmate soluton to problem (2) n the followng fashon. We fnd the value of z that satsfes (7) and use ẑ to denote ths value. We let ŷ be an optmal soluton to the maxmzaton problem on the rght sde of (7) when ths problem s solved wth z = ẑ. For all M, we solve problem (5) wth y = ŷ and use ŵ to denote an optmal soluton to ths problem. In ths case, t s possble to show that the soluton (ŵ 1,..., ŵ m ) provdes an expected revenue that devates from the optmal expected revenue by at most a factor of 1 + ρ, satsfyng (1 + ρ) Π(ŵ 1,..., ŵ m ) Z. To see ths result, we note that snce ẑ s the value of z that satsfes (7) and ŷ s an optmal soluton to the maxmzaton problem on the rght sde of (7) when ths problem s solved wth z = ẑ, we have ẑ = M { ŷ γ g (ŷ ) ŷ ŷ γ ẑ }. (8) Also, snce ŷ [ L, Ū], the dscusson rght after the formulaton of problem (5) shows that the frst constrant n ths problem must be tght at the optmal soluton when ths problem s solved wth y = ŷ. Therefore, notng that ŵ s an optmal soluton to problem (5) when we solve ths problem wth y = ŷ, we obtan ŷ = j N ŵj and g (ŷ ) = j N ŵj (κ j η j log ŵ j ) for all M. Replacng ŷ and g (ŷ ) n (8) by ther equvalents gven by the last two equaltes, we observe that ẑ and (ŵ 1,..., ŵ m ) satsfy the equalty n (4). So, f we collect all terms that nvolve ẑ on the left sde of (4), solve for ẑ and use the defnton of Π(w 1,..., w m ), then we get ẑ = Π(ŵ 1,..., ŵ m ). When the grd ponts satsfy the assumpton of Theorem 2, we also have (1 + ρ) ẑ Z. So, we obtan (1 + ρ) Π(ŵ 1,..., ŵ m ) Z, showng that the expected revenue from the soluton (ŵ 1,..., ŵ m ) devates from the optmal by at most a factor of 1 + ρ. The precedng dscusson, along wth Theorem 2, gves a framework for obtanng approxmate solutons to problem (2) wth a performance guarantee. The crucal pont s that the collecton of grd ponts {ỹ t : t = 1,..., T } has to satsfy the assumpton of Theorem 2. Also, the number of grd ponts n ths collecton should be reasonably small to be able to solve the maxmzaton problem on the rght sde of (7) quckly. In the next secton, we show that t s ndeed possble to construct a reasonably small collecton of grd ponts that satsfes the assumpton of Theorem 2. Before dong so, however, we make a bref remark on how to fnd the value of z that satsfes (7). Thus far, we propose bsecton search as a possble method to obtan ths value of z. One shortcomng of bsecton search s that t may not termnate n fnte tme. To get around the fact that bsecton search may not termnate n fnte tme, we demonstrate that t s possble to obtan the value of z satsfyng (7) by solvng a lnear program. To formulate the lnear program, we note that the left sde of the equalty n (7) s ncreasng n z, whereas the rght sde s decreasng. Therefore, the value of z that satsfes (7) corresponds 13

14 to the smallest value of z such that the left sde of the equalty n (7) s stll greater than or equal to the rght sde. Ths observaton mmedately mples that fndng the value of z satsfyng (7) s equvalent to solvng the problem { mn z : z M max y {ỹ t : t = 1,...,T } { y γ g (y ) y y γ z If we defne the addtonal decson varables (x 1,..., x m ) so that x represents the optmal objectve value of the maxmzaton problem n the th term of the sum on the rght sde of the constrant above, then the problem above can be wrtten as { mn z : z x, x y γ g (y ) y γ y M }} z y {ỹ t : t = 1,..., T }, M. }, (9) where the decson varables are z and (x 1,..., x m ). The problem above s a lnear program wth 1 + m decson varables and 1 + M T constrants. So, as long as the number of grd ponts s not too large, we can solve a tractable lnear program to obtan the value of z satsfyng (7). 5 Grd Constructon In ths secton, our goal s to show how we can construct a reasonably small collecton of grd ponts {ỹ t : t = 1,..., T } that satsfes the assumpton of Theorem 2. By notng the dscusson that follows Theorem 2 n the prevous secton, such a collecton of grd ponts allows us to obtan a soluton to problem (2) wth a gven approxmaton guarantee. To construct the collecton of grd ponts, we begn by gvng a number of fundamental propertes of the knapsack problem n (5). After we gve these propertes, we proceed to showng how we can buld on these propertes to construct the collecton of grd ponts. 5.1 Propertes of Knapsack Problems The frst property that we have for problem (5) s that the optmal values of the decson varables n ths problem are monotoncally ncreasng n y as long as y [ L, Ū]. To see ths property, we assocate the Lagrange multpler λ wth the frst constrant n problem (5) and wrte the Lagrangan as L (w, λ ) = j N w j (κ j η j log w j λ ) + λ y, whch s a concave functon of w. Maxmzng the Lagrangan L (w, λ ) subject to the constrants that w j [L j, U j ] for all j N, the optmal soluton to problem (5) can be obtaned by settng { { ( κj w j = mn max exp 1 λ ) } }, L j, U j η j η j for all j N. We observe that the expresson on the rght sde above s decreasng n λ, showng that the optmal value of the decson varable w j s decreasng n the optmal value of the Lagrange multpler. On the other hand, snce we have y [ L, Ū], by the dscusson that 14 (10)

15 follows the formulaton of problem (5), the frst constrant n ths problem must be tght at the optmal soluton. Therefore, notng (10), the optmal value of the Lagrange multpler λ satsfes the equalty j N mn{max{exp(κ j/η j 1 λ /η j ), L j }, U j } = y. The expresson on the left sde of ths equalty s decreasng n λ, whch mples that the optmal value of the Lagrange multpler s decreasng n the rght sde of the frst constrant n problem (5). To sum up, f we use λ (y ) to denote the optmal value of the Lagrange multpler for the frst constrant n problem (5) as a functon of the rght sde of ths constrant, then λ (y ) satsfes { { ( κj mn max exp 1 λ (y ) } } ), L j, U j = y. (11) η j η j j N Furthermore, λ (y ) s decreasng n y. Snce the optmal value of the decson varable w j n problem (5) s decreasng n the optmal value of the Lagrange multpler and λ (y ) s decreasng n y, t follows that the optmal value of the decson varable w j n problem (5) s ncreasng n y, as desred. Therefore, we can let ζ j and ξ j be such that ( κj exp 1 λ (ζ ) ( j) κj = L j and exp 1 λ (ξ ) j) η j η j η j η j = U j, (12) n whch case, (10) mples that f y = ζ j, then we have w j = L j n the optmal soluton to problem (5), whereas f y = ξ j, then we have w j = U j. Also, snce the optmal value of the decson varable w j s ncreasng n y, the optmal value of the decson varable w j n problem (5) satsfes w j = L j for all y ζ j, whereas w j = U j for all y ξ j. In ths way, ζ j and ξ j correspond to the two threshold values of the rght sde of the frst constrant n problem (5) such that f y ζ j, then the optmal value of the decson varable w j s always L j, whereas f y ξ j, then the optmal value of the decson varable w j s always U j. We note that there may not exst a value of ζ j or ξ j satsfyng (12). If ths s the case, then we set ζ j = or ξ j =. Buldng on the dscusson above, we obtan the next lemma. Lemma 3 (Intervals). For any j N, there exsts an nterval [ζ j, ξ j ] such that the optmal value of the decson varable w j n problem (5) satsfes w j = L j when we have y ζ j, whereas w j = U j when we have y ξ j. Furthermore, f y [ζ j, ξ j ], then we can drop the constrant w j [L j, U j ] n problem (5) wthout changng the optmal soluton to ths problem. Proof. We let ζ j and ξ j be as defned n (12), n whch case, the frst part follows from the dscusson rght before the lemma. To show the second part, we let w be the optmal soluton to problem (5) and λ (y ) be the correspondng Lagrange multpler for the frst constrant. Snce y [ζ j, ξ j ] and λ (y ) s decreasng n y, (12) mples that exp(κ j /η j 1 λ (y )/η j ) L j and exp(κ j /η j 1 λ (y )/η j ) U j. By the last two nequaltes and (10), the optmal value of the decson varable w j n problem (5) s wj = mn{max{exp(κ j/η j 1 λ (y )/η j ), L j }, U j } = exp(κ j /η j 1 λ (y )/η j ). Also, the last two nequaltes mply that the max and mn operators for product j can be dropped from the sum n (11) wthout dsturbng the equalty, showng that 15

16 λ (y ) s stll the optmal value of the Lagrange multpler for the frst constrant n problem (5) when we drop the constrant w j [L j, U j ]. In ths case, we let ŵ be the optmal soluton to problem (5) when we drop the constrant w j [L j, U j ], together wth the correspondng Lagrange multpler λ (y ) for the frst constrant. When we drop the constrant w j [L j, U j ], settng L j = and U j = n (10) mples that the optmal value of the decson varable w j s gven by ŵ j = exp(κ j /η j 1 λ (y )/η j ). Thus, t follows that w j = ŵ j, as desred. The second property that we have for problem (5) s that we can partton the extended real lne [, ] nto a number of ntervals {[ν k, νk+1 ] : k = 1,..., K } such that f we solve problem (5) for any y [ν k, νk+1 ], then we can mmedately fx the values of some of the decson varables at ther upper or lower bounds and not mpose the upper and lower bound constrants at all on the remanng decson varables. To see ths property, we note that f we plot the 2n ponts n the set {ζ j : j N} {ξ j : j N} on the extended real lne [, ], then they partton the extended real lne nto at most 2n + 1 ntervals. We denote these ntervals by {[ν k, νk+1 ] : k = 1,..., K } wth ν 1 = and ν K +1 =. Snce the ntervals {[ν k, νk+1 ] : k = 1,..., K } are obtaned by parttonng the real lne wth the ponts {ζ j : j N} {ξ j : j N}, t follows that for any k = 1,..., K and j N, we must have [ν k, νk+1 ] [ζ j, ξ j ], or [ν k, νk+1 ] [, ζ j ], or [ν k, νk+1 ] [ξ j, ]. In ths case, we defne the sets of products L k, U k and F k as L k = {j N : [ν k, ν k+1 ] [, ζ j ]} U k = {j N : [ν k, ν k+1 ] [ξ j, ]} F k = {j N : [ν k, ν k+1 ] [ζ j, ξ j ]}. Consder problem (5) wth a value of y satsfyng y [ν k, νk+1 ] for some k = 1,..., K. If product j s n the set L k, then we have [νk, νk+1 ] [, ζ j ]. Snce y [ν k, νk+1 ], we obtan y ζ j, n whch case, Lemma 3 mples that the optmal value of the decson varable w j n problem (5) s L j. By followng the same reasonng, f product j s n the set U k, then the optmal value of the decson varable w j n problem (5) s U j. Fnally, f product j s n the set F k, then we have [ν k, νk+1 ] [ζ j, ξ j ], but snce y [ν k, νk+1 ], we obtan y [ζ j, ξ j ], n whch case, by Lemma 3, we can drop the constrant w j [L j, U j ] n problem (5) wthout changng the optmal soluton. Therefore, whenever we solve problem (5) wth a value of y [ν k, νk+1 ], we can fx the values of the decson varables n the sets L k and U k respectvely at ther lower and upper bounds and not mpose the upper and lower bound constrants on the decson varables n the set F k. The observatons n ths paragraph yeld the next lemma. Lemma 4 (Partton). There exst ntervals {[ν k, νk+1 ] : k = 1,..., K } parttonng [ L, Ū] such that for any y [ν k, νk+1 ], the optmal soluton to problem (5) can be obtaned by solvng { } max w j y, w j = L j j L k, w j = U j j U k (13) j N w j (κ j η j log w j ) : j N for some subsets of products L k, U k N that depend on the nterval k contanng y but not on the specfc value of y. Furthermore, we have K = O(n). 16

17 Proof. Constructng the ntervals {[ν k, νk+1 ] : k = 1,..., K } as defned n the dscusson rght before the lemma, the frst part follows by ths dscusson, as long as we take the ntersecton of each one of these ntervals wth [ L, Ū]. To see that the second part holds, snce the ponts {ζ j : j N} {ξ j : j N} partton the extended real lne nto at most 2n + 1 ntervals and these ntervals correspond to {[ν k, νk+1 ] : k = 1,..., K }, K s at most 2n + 1 = O(n). Lemma 4 becomes useful when constructng a collecton of grd ponts that satsfes the assumpton of Theorem 2. We focus on ths task n the next secton. 5.2 Propertes of Grd Ponts In ths secton, we turn our attenton to constructng a collecton of grd ponts {ỹ t : t = 1,..., T } that satsfes the assumpton of Theorem 2. To that end, we choose a fxed value of ρ > 0 and consder the grd ponts that are obtaned by = L j + U j + (1 + ρ) q (14) Ỹ kq j L k j U k for k = 1,..., K and q =..., 1, 0, 1,.... In the expresson above, L k, U k and K are such that they satsfy Lemma 4. In problem (5), once we fx the decson varables n L k at ther lower bounds and the decson varables n U k at ther upper bounds, the sum of the remanng decson varables s at least j N\(L k U k) L j and at most j N\(L k U k) U j. Therefore, we choose the possble values for q n (14) such that the smallest value of (1+ρ) q does not stay above j N\(L k U k ) L j and the largest value of (1 + ρ) q does not stay below j N\(L k U k ) U j. If L k U k = N, then usng a sngle value of q = suffces. Otherwse, usng and to denote the round down and round up functons, we can choose the smallest value of q as q L = log(mn j N L j )/ log(1 + ρ) and the largest value of q as q U = log(n max j N U j )/ log(1 + ρ). In ths case, lettng σ = max j N U j / mn j N L j, we have q U q L = O(log(nσ )/ log(1 + ρ)). To construct a collecton of grd ponts that satsfes the assumpton of Theorem 2, we augment the set of ponts {Ỹ kq : k = 1,..., K, q = q L,..., qu } defned above wth the set of ponts {ν k : k = 1,..., K + 1}, where the last set of ponts are obtaned from the set of ntervals {[ν k, νk+1 ] : k = 1,..., K } gven n Lemma 4. We obtan our set of grd ponts {ỹ t : t = 1,..., T } by orderng the ponts n {Ỹ kq : k = 1,..., K, q = q L,..., qu } {νk : k = 1,..., K + 1} n ncreasng order and droppng the ones that are not ncluded n the nterval [ L, Ū]. Also, we add the two ponts L and Ū nto the collecton of grd ponts to ensure that the smallest and the largest one of the grd ponts {ỹ t : t = 1,..., T } are respectvely equal to L and Ū. Thus, the collecton of grd ponts constructed n ths fashon satsfes ỹ t ỹt+1 for all t = 1,..., T 1, ỹ 1 = L and ỹ T = Ū. Snce K = O(n) and q U q L = O(log(n σ )/ log(1 + ρ)), the number of grd ponts n the collecton s T = O(n + n log(n σ )/ log(1 + ρ)). There are two useful propertes for the grd ponts {ỹ t : t = 1,..., T } constructed by usng the approach above. The frst property s that f ỹ t 17 and ỹt+1 are two consecutve grd ponts, then

18 they satsfy ỹ t, ỹt+1 [ν k, νk+1 ] for some k = 1,..., K. To see ths property, f ths property does for some k = 1,..., K wth one of the two nequaltes not hold, then we have ỹ t νk ỹ t+1 holdng as a strct nequalty. If ths chan of nequaltes holds, then snce ν k we get a contradcton to the fact that ỹ t and ỹt+1 frst property. The second property s that f ỹ t s a grd pont tself, are two consecutve grd ponts, establshng the and ỹt+1 are two consecutve grd ponts satsfyng ỹ t, ỹt+1 [ν k, νk+1 ] for some k = 1,..., K, then we have Ỹ kq ỹ t ỹ t+1 Ỹ k,q+1 for some q = q L,..., qu 1. The dea behnd the second property s smlar to the one used for the frst property. In partcular, f the second property does not hold, then ether we have ỹ t Ỹ kq ỹ t+1 for some q = q L,..., qu 1 or we have ỹ t Ỹ k,q+1 ỹ t+1 for some q = q L,..., qu 1 wth one of the last four nequaltes holdng as a strct nequalty. If ether one of the last two chans of nequaltes holds, then snce Ỹ kq and Ỹ k,q+1 are grd ponts themselves, we get a contradcton to the fact that ỹ t are two consecutve grd ponts, establshng the second property. In and ỹt+1 the next theorem, we use these propertes along wth Lemma 4 to show that the collecton of grd ponts {ỹ t : t = 1,..., T } satsfes the assumpton of Theorem 2. Theorem 5 (Grd Constructon). Assume that the collecton of grd ponts {ỹ t : t = 1,..., T } are obtaned by orderng the ponts n {Ỹ kq : k = 1,..., K, q = q L,..., qu } {νk : k = 1,..., K + 1} n ncreasng order. In ths case, we have g (ỹ t+1 ) (1 + ρ) g (ỹ t ) for all t = 1,..., T 1. Proof. If ỹ t and ỹ t+1 are two consecutve grd ponts, then the frst property rght before the statement of the theorem mples that there exsts k = 1,..., K such that ỹ t, ỹt+1 [ν k, νk+1 ], n whch case, by the second property, t follows that there exsts q = q L,..., qu 1 such that Ỹ kq ỹ t ỹt+1 Ỹ k,q+1. Subtractng j L k L j + j U k U j from the last chan of nequaltes and notng the defnton of Ỹ kq n (14), we obtan (1 + ρ) q ỹ t L j U j ỹ t+1 L j U j (1 + ρ) q+1. j L k j U k j L k j U k Usng the chan of nequaltes above, t follows that ỹ t+1 j L k L j j U k U j (1 + ρ) q+1 (1 + ρ) (ỹ t j L k L j j U k U j). Snce ỹ t, ỹt+1 [ν k, νk+1 ], Lemma 4 mples that the optmal soluton to problem (5) wth y = ỹ t or y = ỹ t+1 can be obtaned by solvng problem (13) respectvely wth y = ỹ t or y = ỹ t+1. We let w be the optmal soluton to problem (13) when we solve ths problem wth y = ỹ t+1. Note that wj = L j for all j L k and wj = U j for all j U k. We defne the soluton ŵ as ŵ j = wj /(1 + ρ) for all j N \ (Lk U k), ŵ j = L j for all j L k and ŵ j = U j for all j U k. In ths case, we have { ŵ j = 1 wj 1 ỹ t+1 L j } U j ỹ t L j U j, 1 + ρ 1 + ρ j N\(L k U k ) j N\(L k U k ) where the frst nequalty s by the fact that w s a feasble soluton to problem (13) when we solve ths problem wth y = ỹ t+1 and the second nequalty follows from the fact that ỹ t+1 j L k L j j U k U j (1 + ρ) (ỹ t j L k L j j U k U j), whch s shown at the begnnng of 18 j L k j U k j L k j U k