On Upper Bounds for Assortment Optimization under the Mixture of Multinomial Logit Models

Transcription

1 On Upper Bounds for Assortment Optimization under the Mixture of Mutinomia Logit Modes Sumit Kunnumka September 30, 2014 Abstract The assortment optimization probem under the mixture of mutinomia ogit modes is NPcompete and there are different approximation methods to obtain upper bounds on the optima expected revenue. In this paper, we anayticay compare the upper bounds obtained by the different approximation methods. We propose a new, tractabe approach to construct an upper bound on the optima expected revenue and show that it obtains the tightest bound among the existing tractabe approaches in the iterature to obtain upper bounds. Assortment optimization has important appications in retaiing and revenue management and has received much attention atey. In the assortment probem, we have a firm that is interested in maximizing revenues by seing products to customers, where each product has a revenue associated with it and customers choose among the offered products according to a given discrete choice mode. The goa therefore is to figure out the set of products, or the assortment, that maximizes the expected revenue obtained from a customer. Whie there are a arge number of discrete choice modes that can be used to describe customer choice behavior, the mutinomia ogit mode and its variants have been a popuar choice in the assortment optimization iterature. In this paper, we consider the assortment probem under a mixture of mutinomia ogit modes. In this mode, we have mutipe customers segments and an arriving customer beongs to a particuar segment with a given probabiity and chooses among the offered products according to the mutinomia ogit mode. The parameters of the mutinomia ogit mode are aowed to depend on the segment to which the customer beongs. The assortment optimization probem under the mixture of mutinomia ogit modes is NPcompete; see Bront, Mendez-Diaz and Vucano (2009) and Rusmevichientong, Shmoys, Tong and Topaogu (2013). On the other hand, McFadden and Train (2000) show that the mixture of mutinomia ogits is a rich choice mode that can approximate any random utiity choice mode arbitrariy cosey. So there has been considerabe interest in the assortment probem under the mixture of mutinomia ogit modes and there is a growing iterature that focuses on deveoping approximation methods that generate assortments with provabe performance guarantees; see for exampe Rusmevichientong et a. (2013) and Mitta and Schuz (2013). Indian Schoo of Business, Hyderabad, , India, emai: sumit kunnumka@isb.edu 1

2 Another interesting research direction is to obtain upper bounds on the optima assortment revenue, since upper bounds are usefu in getting a better hande on the revenue performance gaps of the candidate assortments obtained by the different approximation methods. Bront et a. (2009) formuate the assortment probem under the mixture of mutinomia ogit modes as a inear mixed integer program, the inear programming reaxation of which gives an upper bound on the optima expected revenue. Fedman and Topaogu (2013) propose a Lagrangian reaxation approach where they reax the constraints that the same assortment be offered to the different customer segments by associating Lagrange mutipiers with them. Whie the Lagrangian reaxation approach obtains an upper bound, soving it turns out to be intractabe. Therefore the authors propose a further approximation that is based on soving a continuous knapsack probem over a discrete set of grid points. They show that this approximation method is tractabe and continues to provide an upper bound on the optima expected revenue. One difficuty with the approximation method proposed by Fedman and Topaogu (2013) is that the quaity of the upper bound depends on the density of the grid, which makes it chaenging to estabish anaytica resuts. In this paper, we focus on soution methods to obtain upper bounds on the optima expected revenue for the assortment probem under the mixture of mutinomia ogit modes. We first provide a partia characterization of the optima assortment. In particuar, we show that an optima assortment incudes a certain revenue ordered subset of the products, and this subset can be obtained efficienty. This structure can be potentiay expoited to speed up the soution methods by reducing the size of the search space. We then anayticay compare the upper bounds obtained by the different soution methods in the iterature. We show that the Lagrangian reaxation approach obtains the tightest bound among the avaiabe approaches to obtain upper bounds. However, since the Lagrangian reaxation is NP-compete, we propose a new, aternative approach to obtain an upper bound on the optima expected revenue. Our approach buids on the ideas deveoped in Fedman and Topaogu (2013) and Martinezde-Abeniz and Kunnumka (2014). The Lagrangian reaxation approach of Fedman and Topaogu (2013) decomposes the assortment probem invoving mutipe customer segments into a number of singe segment probems where there is a fixed cost of introducing a product into the assortment. Martinez-de-Abeniz and Kunnumka (2014) study the assortment probem under the mutinomia ogit mode when there is a fixed cost of introducing a product into the assortment and propose tractabe soution methods. Whie our approach is a simpe adaptation of the ideas deveoped in the above mentioned papers, it yieds a soution method that is tractabe and provaby tighter than the other tractabe approaches in the iterature to obtain upper bounds on the optima expected revenue. In summary, we make the foowing contributions in this paper. 1) We provide a partia characterization of the soution to the assortment probem under the mixture of mutinomia ogit modes. 2) We anayticay compare the different methods proposed in the iterature to obtain upper bounds on the optima expected revenue. We show that the Lagrangian reaxation approach proposed by Fedman and Topaogu (2013) obtains the tightest upper bound. However, the Lagrangian reaxation cannot be soved efficienty. 3) We propose a new approach to obtain an upper bound that remains tractabe. The bound obtained by our approach is provaby tighter than the other tractabe soution methods in the iterature to obtain upper bounds. Moreover, in contrast to Fedman and Topaogu (2013), our soution method does not require a discrete set of grid points and thus reduces some of the subectivity invoved in choosing a discretization scheme. 4) Computationa experiments indicate that our approach can be beneficia both in terms of tighter bounds and faster run times. The rest of the paper is organized as foows. In Section 1, we describe the assortment probem under the mixture of mutinomia ogit modes. In Section 2, we describe the soution methods proposed in the iterature to obtain upper bounds on the optima expected revenue. We anayticay compare 2

3 the upper bounds obtained by the proposed methods in Section 3. In Section 4, we describe our soution approach and show that it obtains the tightest bound among the tractabe soution methods in the iterature to obtain upper bounds on the optima expected revenue. Section 5 presents our computationa experiments. 1 Probem Formuation We consider the assortment optimization probem under the mixture of mutinomia ogit modes. We have N products and the revenue associated with product {1,..., N} is p. We et x {0, 1} denote whether we offer product. We have L customer segments interested in purchasing a product from the offered assortment. The preference weight for customer segment {1,..., L} for product is v, whie the preference weight associated with a segment customer not purchasing anything is normaized to be 1. Within each segment, choice is governed by the mutinomia ogit mode and so the probabiity that a segment customer purchases product is v x /(1 + k v k x k). We et α denote the arriva rate of customer segment. The assortment probem is to decide which products to make avaiabe to an arriving customer in order to maximize the expected tota revenue. The optima assortment can be obtained by soving the probem Z OP T = max x x {0,1} α p v x v x + 1. (1) Soving assortment probem (1) and obtaining the optima expected revenue is intractabe; see Bront et a. (2009) and Rusmevichientong et a. (2013). Therefore, it is unikey that we can provide a compete characterization of the optima soution to probem (1). However, Lemma 1 beow provides a partia characterization of the structure of an optima soution. We begin with some preiminaries. Assume without oss of generaity that the products are indexed in order of decreasing revenues so that p 1 p 2... p N, and consider the assortment probem invoving customer segment aone max x x {0,1} p v x (2) v x + 1. It is known that revenue ordered assortments are optima for probem (2); see for exampe Liu and van Ryzin (2008). That is, there exists an optima soution ˆx = {ˆx } to probem (2) with ˆx = 1 for {1,..., ȷ } and ˆx = 0 for {ȷ + 1,..., N}, where ȷ {1,..., N}. We have the foowing emma. Lemma 1 Let ˆȷ = min {ȷ }. There exists an optima soution ˆx = {ˆx } to probem (1) with ˆx = 1 for {1,..., ˆȷ}. Proof. Suppose that the statement of the emma is fase. Let ˆx be an optima soution to probem (1) and et κ {1,..., ˆȷ} be the smaest index such that ˆx κ = 0. We et x = { x } be the same as ˆx except that x κ = 1. We show beow that the soution x generates as much revenue as ˆx, and is therefore aso optima. Let S = {1,..., N}\{κ} and T = {1,..., κ 1}. Fix a segment and note that S p v ˆx ( S v ˆx + 1 = T v ˆx ) + 1 T p v ˆx ( S v ˆx + 1 T v ˆx S\T v ˆx ) S\T p v ˆx S v ˆx + 1 S\T v ˆx. (3) 3

4 Since κ {1,..., ȷ} and ȷ ȷ, it foows that T {1,..., ȷ }, the optima revenue ordered assortment for segment. Lemma 2 in Liu and van Ryzin (2008) then impies that T p v ˆx T v ˆx +1 T p v ˆx +p κvκ T v ˆx +vκ+1. This T p v ˆx T v ˆx +1. together with Lemma 3.1 in Rusmevichientong and Topaogu (2012) impies that p κ On the other hand, since the products are revenue ordered, p κ p for a S\T and we have p κ S\T p v ˆx S\T v ˆx. The above statements together with (3) impy that S p v ˆx S v ˆx + 1 p κ. (4) Now consider the difference in the expected revenues from segment under x and ˆx. We have p v ˆx v ˆx = [ ] p v 1 ˆx + 1 S S v ˆx + vκ p κ vκ S v ˆx [ vκ S = S v ˆx + vκ p κ p v ˆx ] + 1 S v ˆx p v x v x + 1 S v ˆx + v κ + 1 where the first equaity uses the fact that x is the same as ˆx except that x κ = 1, whie ˆx κ = 0, and the inequaity foows from (4). It foows that α p v x v x +1 α p v ˆx v ˆx +1. Lemma 1 impies that an optima soution to probem (1) incudes the revenue ordered subset {1,..., ȷ}, and this subset can be determined efficienty. This may be usefu in speeding up approximation methods by reducing the size of the soution space. We emphasize that revenue ordered assortments are not optima for probem (1) in genera; see Rusmevichientong et a. (2013). 2 Upper Bounds In this section, we describe approximation methods that obtain upper bounds on the optima expected revenue. Upper bounds are usefu in assessing the revenue performance gaps of the assortments obtained by the different heuristic methods. 2.1 Linear Programming Reaxation Bront et a. (2009) formuate probem (1) as the mixed integer program Z OP T = max s.t α p v y (5) v y + w = 1 (6) y w, (7) y x, (8) w y 1 x, (9) y, w 0, x {0, 1}. (10) In the above optimization probem, the decision variabe x indicates if product is offered, whie the term v y can be interpreted as the probabiity that segment purchases product and w can be 4

5 interpreted as the probabiity that customer segment does not purchase anything. It is intractabe to sove the above mixed integer program, but its inear programming reaxation yieds an upper bound on the optima expected revenue. Letting Z LP denote the optima obective function vaue of the LP reaxation of probem (5)-(10), we have Z OP T Z LP. Z LR (λ) = max x x ϕ,x {0,1} 2.2 Lagrangian Reaxation A second upper bound is due to Fedman and Topaogu (2013) who consider reaxing the constraints that the same assortment be offered to the different customer segments by associating Lagrange mutipiers with them. Letting x {0, 1} denote the decision to offer product to segment, we can view probem (1) as requiring x = xϕ for a and, where xϕ {0, 1}. These constraints ensure that the decision to offer a product is uniform across the different customer segments. Fedman and Topaogu (2013) reax the constraints x = xϕ by associating Lagrange mutipiers λ with them. For a given set of Lagrange mutipiers λ = {λ, }, they consider soving probem p v x α v x + 1 λ x + x ϕ [ λ ]. (11) The authors show that for any set of Lagrange mutipiers λ, we have Z OP T Z LR (λ) and they find the set of Lagrange mutipiers that yied the tightest upper bound on the optima expected revenue by soving the probem Z LR = min Z LR (λ). λ However, the optimization probem for segment in (11) invoves finding the optima assortment when there are fixed costs associated with introducing the products, and this is known to be NP-compete in genera; see Kunnumka, Rusmevichientong and Topaogu (2009). An exception is when a the Lagrange mutipiers are set to zero. If λ = 0 for a and, then there are no costs associated with offering any of the products and the probem Z LR ( 0) can be soved efficienty; see Tauri (2014). On the other hand, if the Lagrange mutipiers are nonzero, then Z LR (λ) cannot be computed in a tractabe manner. 2.3 Lagrangian Reaxation over a discrete grid Fedman and Topaogu (2013) propose a tractabe approximation to Z LR (λ), which is based on writing the optimization probem for segment in (11) as max max α p v x θ x x [0,1], 1 + θ λ x. v x θ In the above representation, the inner maximization probem is a knapsack probem that depends on the parameter θ, where θ can be interpreted as the capacity of the knapsack and can take vaues in the interva [0, v ]. Instead of soving the optimization probem for a vaues of θ in its range, the authors focus on a set of discrete grid points Θ = {θ (g), g}, where 0 = θ (1) < θ (2) <... < θ (G) = v for a, and G is the number of grid points. For an interva [θ (g), θ (g+1) ], they sove the probem Z LR D,,g (λ, Θ) = max α p v x x 0 x 1, v x θ(g+1) 1 + θ (g) λ x. (12) 5

6 Fedman and Topaogu (2013) show that for any set of discrete grid points Z LR D (λ, Θ) = [ ] max g {1,...,G 1} ZLR D,,g (λ, Θ) + max x ϕ x ϕ {0,1} [ λ ] Z LR (λ), where the argument Θ emphasizes the dependence of the choice of grid points on the upper bound obtained. Note that probem (12) is a continuous knapsack probem and can be soved efficienty. Therefore, the upper bound Z LR D (λ, Θ) can be obtained in a tractabe manner. Furthermore, the authors show that the probem Z LR D (Θ) = min λ Z LR D (λ, Θ) (13) can be soved efficienty as a convex minimization probem. However, there is a trade-off between the quaity of the upper bound and the computationa efficiency, since a more refined set of grid points gives a tighter upper bound, but at the same time imposes a greater computationa burden. Therefore, the choice of the grid points becomes crucia to the performance of the method. Fedman and Topaogu (2013) propose a set of exponentia grid points and bound the performance gap between the set of exponentia grid points and an arbitrary set of grid points. 3 Comparison of upper bounds In this section, we compare the upper bounds obtained by the different approximation methods. We first compare the upper bound Z LR with Z LP. We begin by making the change of variabes z = v y so that probem (5)-(10) can be written as Z OP T = max α p z s.t z /v + k z k 1, z /v x z /v + k z k x, z 0, x {0, 1}, where the first and third constraints use the fact that w = 1 z. Introducing variabes x and xϕ and constraining them to be equa, the above optimization probem can be written as Z OP T = max α p z (14) s.t z /v + k z k 1, (15) z /v x, (16), z /v + k z k x, (17) z 0, x, xϕ x = xϕ, (18) {0, 1}. (19) Reaxing constraints (18) by associating Lagrange mutipiers λ = {λ, } with them, we get a reaxed probem Z(λ) = max α p z + λ [xϕ x ] (20) s.t z /v + k z k 1, (21) z /v x, (22) z /v + k z k x, (23) z 0, x, xϕ 6 {0, 1}. (24)

7 We have the foowing emma. Lemma 2 If {x, xϕ, z, } satisfies constraints (21)-(24), then we have z = v x /(1 + k v k x k ) for a and. Proof. Fix a customer segment and a product. The caim ceary hods if x = 0 since constraint (22) forces z to be zero. On the other hand, if x = 1, et J + = {k x k = 1}. Note that J +. Constraints (21) and (23) impy that zk /v k + zk = 1 (25) k J + for a k in J+. Considering equaity (25) for products and k J+, we have zk /v k = z /v for a k J+. Pugging this back into the constraint z /v + k J+ z k = 1, we get z = v /(1 + k J+ v k ) = v x /(1 + k v k x k ). Lemma 2 impies that probem (20)-(24) can be equivaenty written as max x x ϕ,x {0,1} α v p x 1+ k v k x k λ x + λ xϕ, k which is precisey the Lagrangian reaxation (11) proposed by Fedman and Topaogu (2013). So we have the foowing emma. Lemma 3 Z(λ) = Z LR (λ). Next, consider the LP reaxation of probem (14)-(19). Letting ˆλ = {ˆλ, } be an optima set of dua variabes associated with constraints (18), strong duaity impies that Z LP = max α p z + ˆλ [xϕ x ] (26) s.t z /v + k z k 1, (27) z /v x, (28) z /v + k z k x, (29) z 0, 0 x, xϕ 1. (30) Proposition 4 Z LR = min λ Z(λ) Z(ˆλ) Z LP. Proof. The equaity foows from Lemma 3, whie the first inequaity hods since ˆλ is a feasibe but not necessariy optima soution to the minimization probem and the ast inequaity hods since x and x ϕ are restricted to be binary in probem (20)-(24) whie they are aowed to take on fractiona vaues in probem (26)-(30). Proposition 4 shows that the Z LR bound is tighter than Z LP. It triviay foows that Z LR Z LR ( 0), whie Fedman and Topaogu (2013) show that Z LR Z LR D (Θ). Therefore the Z LR bound is the tightest upper bound among the avaiabe bounds. However, as mentioned, Z LR cannot be computed 7

8 in a tractabe manner, and Fedman and Topaogu (2013) propose the approximation Z LR D (Θ) that invoves soving a continuous knapsack probem over a set of discrete grid points. Whie the numerica study in Fedman and Topaogu (2013) indicates that their method performs quite we in practice and generates tight bounds, one difficuty with the discretization approach is that the quaity of the soution depends on how refined the grid is and it is not immediatey cear how to anayticay compare the upper bound with that obtained by the other approaches, such as the LP reaxation. 4 An aternative upper bound We propose an aternative approach to obtain an upper bound on the optima revenues. It is a simpe adaptation of the ideas deveoped in Fedman and Topaogu (2013) and Martinez-de-Abeniz and Kunnumka (2014), but our approach is tractabe and is provaby tighter than the existing tractabe approaches to obtain upper bounds. Moreover, our approach avoids the subectivity invoved in determining the density of the grid points. We begin by reaxing the constraints that the same assortment be offered to the different segments by associating Lagrange mutipiers λ = {λ, } with them and consider soving the probem p v x Z LR C (λ) = max x 0 x 1,xϕ {0,1} max α x 0 x 1 α v x + 1 λ x + x ϕ [ λ ]. (31) Note that the probem we sove is amost the same as Z LR (λ) except that we reax the integraity constraints on the x variabes. Again note that probem (31) separates by segment and the optimization probem for segment has the form p v x v x + 1 It can be shown that the above optimization probem can be soved efficienty, in fact, essentiay in cosed form; see Martinez-de-Abeniz and Kunnumka (2014). Therefore, given a set of Lagrange mutipiers, probem (31) can be soved efficienty. Since Z LR C (λ) Z LR (λ) Z OP T, we obtain an upper bound on the optima assortment revenue. We propose soving Z LR C = min λ Z LR C (λ) (32) to find the optima set of Lagrange mutipiers that yied the tightest upper bound. By foowing the same ine of reasoning as in Fedman and Topaogu (2013), it can be shown that the optima Lagrange mutipiers can be obtained efficienty. We now show that Z LR C obtains an upper bound that is uniformy tighter than the existing tractabe approaches in the iterature. Reca, that if a the Lagrange mutipiers are set to zero, then probem (11) can be soved in a tractabe manner. Proposition 5 beow shows that the resuting upper bound Z LR ( 0) is weaker than Z LR C, whie Propositions 6 and 7 beow show that the Z LR C bound is tighter than Z LP and Z LR D (Θ). λ x. Proposition 5 Z LR C Z LR ( 0). Proof. Setting a the Lagrange mutipiers to zero, we have Z LR C ( 0) = max α p v x v x + 1 = max x 0 x 1 8 α x x {0,1} p v x v x + 1 = ZLR ( 0)

9 where the second equaity hods since the obective function is quasiconvex; see for exampe, Lemma 1 in Liu and van Ryzin (2008). We have Z LR C = min λ Z LR C (λ) Z LR C ( 0) = Z LR ( 0). Proposition 6 Z LR C Z LP. Proof. Let ˆλ = {ˆλ, } be the optima set of dua variabes corresponding to constraints (18). Let ˆx = {ˆx, ˆxϕ, } be an optima soution to probem (31), when the Lagrange mutipiers are set as ˆλ. Letting ẑ = v ˆx /(1+ k v k ˆx k ), and ẑ = {ẑ, }, we show that (ˆx, ẑ) is a feasibe soution to probem (26)-(30). Note that k ẑ k = k v k ˆx k /(1 + k v k ˆx k ). So we have ẑ /v = ˆx /(1 + k v k ˆx k ) 1/(1 + k v k ˆx k ) = 1 k ẑ k and constraint (27) hods. Next, ẑ /v = ˆx /(1 + k v k ˆx k ) ˆx and so constraint (28) hods. Finay note that ẑk = vk ˆx k /(1 + vk ˆx k ) ˆx ( vk ˆx k )/(1 + vk ˆx k ) = ( ˆx 1 1/(1 + vk ˆx k )) = ˆx ẑ/v k k k k k k and constraint (29) aso hods. Therefore Z LR C (ˆλ) Z LP and the resut foows. Proposition 7 Z LR C Z LR D (Θ). Proof. We have Z LR C (λ) = { max θ max x 0 x 1, v x θ α p v x } 1+θ λ x + max x ϕ {0,1} xϕ [ λ ]. Now, for θ [θ (g), θ (g+1) ], max α p v x x 0 x 1, 1 + θ v x θ λ x Z LR D,,g (λ, Θ). Therefore, Z LR C (λ) Z LR D (λ, Θ) and the resut foows. 5 Computationa Experiments In this section we numericay test the quaity of the upper bound obtained by our proposed method. Fedman and Topaogu (2013) demonstrate that the upper bound obtained by the Lagrangian reaxation over a discrete grid tends to be significanty tighter than the inear programming reaxation bound and the bound obtained by setting a the Lagrange mutipiers to zero. So, in our numerica study, we focus on comparing the upper bound and the soution time of our proposed method with that of the Lagrangian reaxation over a discrete grid. We use LR-C to refer to the soution method described in Section 4 and LR-D(Θ) to the soution method described in Section 2.3. Foowing Fedman and Topaogu (2013), we use a set of exponentia grid points in LR-D(Θ). That is, we cover the interva [0, v ] using a set of grid points {θ(g) g 9

10 {1,..., G}}, where θ (g) = (1 + ρ) g 1 1 and ρ > 0. Note that the parameter ρ determines the density of the grid. As ρ decreases, the spacing between the adacent grid points decreases and we require a arger number of grid points to cover the interva [0, v ]. We consider three different grid densities in our computationa experiments by varying ρ over the set {0.1, 0.01, 0.001}. We note that Fedman and Topaogu (2013) use a vaue of ρ = in their computationa study. We sove probems (32) and (13) using subgradient search. In both cases, we initiaize a the Lagrange mutipiers to zero and run 200 iterations of the subgradient search agorithm with a step size of 20/(40 + t) at iteration t. Our computationa experiments are carried out in MATLAB on a Core i7 desktop with 3.4-GHz CPU and 16-GB RAM. In a of our test probems, we have 50 products and 25 customer segments. We generate our test probems by foowing an approach simiar to that described in Fedman and Topaogu (2013). We sampe the product revenues from the uniform distribution over [0, 2000]. We sampe β from the uniform distribution over [0, 1] and set the arriva rate of segment, α, as β / β. We set the preference weight for segment for product, v, as ν 1 ϕ ν ϕ. We sampe ν from the uniform distribution over [0, V ] and ϕ from the uniform distribution over [0, Φ], where V and Φ are parameters that we vary in our computationa experiments. Note that v /(1 + v ) = 1 ϕ and so the probabiity that a segment customer eaves without making a purchase when a the products are offered is ϕ. In our computationa experiments we vary V {5, 10, 20} and Φ {0.4, 0.6, 0.8}, which gives us a tota of nine parameter combinations. For each parameter combination, we generate 100 test probems by foowing the approach described above. For each test probem, we obtain the LR-C bound as we as the soution time. We obtain the corresponding numbers for LR-D(Θ) at different grid densities by varying ρ {0.1, 0.01, 0.001}. We use LR-C as a benchmark and for each of the three grid densities, we compute the percentage gap in the upper bound with respect to the upper bound obtained by LR-C, as we as the soution time reative to the soution time of LR-C. We report the averages across the 100 test probems. Tabe 1 summarizes the resuts of our computationa study. The first coumn in this tabe gives the parameter combinations. The second coumn gives the percentage gap in the upper bounds obtained by LR-D(Θ) with ρ = 0.1 and LR-C, averaged over the 100 probem instances, where as the third coumn gives the average of the ratio of the corresponding soution times. Coumns four and five have a simiar interpretation except that they compare LR-D(Θ) with ρ = 0.01 with LR-C. The ast two coumns, respectivey, compare the upper bounds and soution times of LR-D(Θ) with ρ = with LR-C. We observe that the upper bound obtained by LR-C is on average 9.553% tighter than that obtained by LR-D(Θ) with ρ = 0.1, 0.986% tighter than that obtained by LR-D(Θ) with ρ = 0.01 and 0.099% tighter than that obtained by LR-D(Θ) with ρ = The quaity of the upper bound obtained by LR-D(Θ) improves with the density of the grid, but this comes at the expense of an increase in the soution time. LR-C is abe to obtain bounds that are comparabe to LR-D(Θ) with ρ = but at a fraction of the computationa cost. The running time of LR-C is in between those of LR-D(Θ) with ρ = 0.1 and ρ = LR-C can be viewed as the imit of LR-D(Θ) as ρ tends to zero and the grid becomes infinitey dense. Proposition 3 in Fedman and Topaogu (2013) impies that no set of grid points, no matter how dense, can improve on the upper bounds obtained by LR-D(Θ) with ρ = 0.1, 0.01 and by more than 10%, 1% and 0.1%, respectivey. It is worthwhie noting that the gains from LR-C are quite cose to the theoretica imits. 10

11 Probem LR-D(Θ) ρ = 0.1 vs LR-C LR-D(Θ) ρ = 0.01 vs LR-C LR-D(Θ) ρ = vs LR-C (V, Φ) % Gap in Ratio of % Gap in Ratio of % Gap in Ratio of upper bound son. times upper bound son. times upper bound son. times (5, 0.4) (5, 0.6) (5, 0.8) (10, 0.4) (10 0.6) (10, 0.8) (20, 0.4) (20 0.6) (20, 0.8) Average Tabe 1: Comparison of the upper bounds and soution times of LR-D(Θ) with LR-C. References Bront, J. J. M., Mendez-Diaz, I. and Vucano, G. (2009), A coumn generation agorithm for choicebased network revenue management, Operations Research 57, Fedman, J. and Topaogu, H. (2013), Bounding expected revenues for assortment optimization under mixtures of mutinomia ogits, Technica report, Corne University, Schoo of OR&IE, Ithaca. Kunnumka, S., Rusmevichientong, P. and Topaogu, H. (2009), Assortment optimization under the mutinomia ogit mode with product costs, Technica report, Corne University, Schoo of OR&IE, Ithaca. Liu, Q. and van Ryzin, G. (2008), On the choice-based inear programming mode for network revenue management, M&SOM 10(2), Martinez-de-Abeniz, V. and Kunnumka, S. (2014), Tractabe modes and agorithms for assortment panning with product costs, Working paper, IESE Business Schoo, Barceona. McFadden, D. and Train, K. (2000), Mixed mn modes for discrete response, Journa of Appied Economics 15. Mitta, S. and Schuz, A. (2013), A genera framework for designing approximation schemes for combinatoria optimization probems with many obectives combined into one, Operations Research 61. Rusmevichientong, P., Shmoys, D., Tong, C. and Topaogu, H. (2013), Assortment optimization under the mutinomia ogit mode with random choice parameters, Production and Operations Management. (to appear). Rusmevichientong, P. and Topaogu, H. (2012), Robust assortment optimization under the mutinomia ogit choice mode, Operations Research 60(4). Tauri, K. (2014), New formuations for choice network revenue management, INFORMS Journa on Computing. 11