A General Attraction Model and an Efficient Formulation for the Network Revenue Management Problem *** Working Paper ***

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1 A General Attraction Model and an Efficient Formulation for the Network Revenue Management Problem *** Working Paper *** Guillermo Gallego Richard Ratliff Sergey Shebalov updated June 30, 2011 Abstract This paper addresses two concerns with the state of the art in network revenue management with dependent demands. The first concern is that the basic attraction model (BAM), of which the multinomial model (MNL) is a special case, may overestimate recapture in certain cases. The second concern is that the choice based deterministic linear program (CBLP), currently in use to derive heuristics for the stochastic network revenue management (SNRM) problem, has an exponential number of variables. We introduce a generalized attraction model (GAM) that has both the BAM and the independent demand model (IDM) as special cases. We also provide an axiomatic justification for the GAM and an E-M to estimate its parameters. As a choice model, the GAM should be of interest to those seeking a model that is not as optimistic as the BAM nor as pessimistic as the IDM in estimating recaptured demand. Our second contribution is a new formulation called the Sales Based Linear Program (SBLP) for the GAM. This formulation avoids the exponential number of variables in the CBLP and is essentially the same size as the formulation for the IDM. The SBLP should be of interest to revenue managers (even if their preferred choice model is the BAM) as it dramatically reduces the number of variables. Together these two contributions move forward the state of the art for network revenue management and allow for a wide range of effects including: partial demand dependencies, multiple time periods, inventory sharing across cabins, and competitive effects. In addition, the formulation yields new insights into the assortment problem that arises when capacities are infinite. 1 Introduction One of the leading areas of research in revenue management (RM) has been incorporating demand dependencies into forecasting and optimization models. Developing effective models 1

3 demand dependencies. The MNL is a special case of the basic attraction model (BAM) that states that the demand for a product is the ratio of the direct attractiveness of the product divided by the sum of the direct attractiveness of the available products (including a no purchase alternative). This means that closing a high demand product will result in spill, and a significant portion of that spill may be recaptured by other available products. In practice, however, we have seen situations where the BAM overestimates the relative attractiveness of alternative products thus leading to underestimation of spill to the competitor and nopurchase alternatives. In a sense, one could also say that the BAM tends to be too optimistic about recapture. Our goal in demand modeling is to develop a more general attraction model (GAM) that is less pessimistic than the IDM but not as optimistic as the BAM. Indeed, under the GAM the choice probabilities depend both on the products offered and the products not offered; this approach provides more flexibility in terms of handling product-level spill. Moreover, the GAM includes both the IDM and the BAM as special cases. The GAM is justified in this paper by slightly modifying one of the axioms that gives rise to the BAM. It can also be justified by studying the BAM under competition allowing for different mean utilities for the same product at different locations but assuming that customers have the same idiosyncratic noise associated with their valuations at different locations. The second concern addressed in this paper is that the linear programming formulation for the BAM, known as the Choice Based Linear Program (CBLP), has an exponential number of variables [12]. This is because the model strives to find the amount of time each subset of products is offered, and each subset requires a variable. The number of products itself is usually very large; for example, in airlines, it is the number of origin-destination fares (ODFs), and the number of subsets is two raised to the number of ODFs for each market segment. This forces the use of column generation and makes resolving frequently or solving for randomly generated demands impractical. Our goal here is to reformulate the Choice Based Linear Program (CBLP) to avoid using an exponential number of variables. For this purpose we propose an alternative formulation called the sales based linear program (SBLP), which we show is equivalent to the CBLP not only under the BAM but also under the GAM. Due to a unique combination of demand balancing and scaling constraints, the new formulation has a linear number of variables and constraints. The SBLP allows for direct solution without the need for large-scale column generation techniques. This has the benefit of enabling more efficient heuristics for the SNRM problem and providing interesting insights into the assortment problem (a special case that arises when capacities are infinite). The remainder of this paper is organized as follows. In 2 we present the GAM as well as a justification of the model from first principles and then present an E-M to estimate its parameters. In 3 we present the stochastic, choice based, network revenue management problem. The choice based linear program is reviewed in section 4. The new, sales based linear program is presented in 5 as well as examples that illustrate the use of the formulation. The infinite capacity case gives rise to the assortment problem and is studied in 6. An upgrade model is presented in 7, sales of flexible products in 8, and our conclusions and directions for future research are in 9. 3

5 This idea can be used to revisit the red-bus, blue-bus paradox, see Ben-Akiva and Lerman [4], where a person has a choice between driving a car and taking either a red or blue bus (it is implicitly assumed that both buses have ample capacity and depart at the same time). The paradox emerges because the probability of selecting to drive the car decreases when the second bus is added. Let v 1 represent the direct attractiveness of driving a car and let v 2,w 2 represent the direct and switching attractiveness of the two buses. If w 2 = v 2 then the probabilities are unchanged if one of the buses is removed (as one intuitively expects). The model with w 2 = 0 represents the case where there is no second bus. To formally define the GAM we assume that in addition to the direct attraction values v k,k N = {1,...,n}, there are switching attraction values w k [0,v k ],k N such that for any subset S N v j π j (S) = j S, (2) v 0 + w(s )+v(s) where S = {j N : j / S} and for any R N, w(r) = j R w j. For the GAM, π 0 (S) = v 0+w(S ) is no-purchase probability. The case w v 0 +w(s )+v(s) k =0,k N recovers the BAM, while the case w k = v k,k N recovers the independent demand model (IDM). As for the BAM it is possible to normalize the parameters so that v 0 = 1. We will later describe how to estimate the parameters of the GAM from data. The parsimonious formulation w j = θv j j N for θ [0, 1] can serve to test the competitiveness of the market, e.g., H 0 : θ =0orH 0 : θ =1 against obvious alternatives to determine whether one is better off deviating from either the BAM or the IDM. There is an alternative, perhaps simpler, way of presenting the GAM by using the following transformation: ṽ 0 = v 0 + w(n) andṽ k = v k w k,k N. For S N, letṽ(s) = j S ṽj. With this notation the GAM becomes: v j π j (S) = j S and π 0 (S) =1 π S (S). (3) ṽ 0 +ṽ(s) where π S (S) = j S π j(s). For S T we will use the notation π S+ (T )= j S + π j (T ). We now revisit the Luce Axioms, see [19], that give birth to the BAM and then suggest a generalization of one of the axioms to show that the GAM satisfies the resulting set of axioms. The original Luce axioms are: Axiom 1: If π i ({i, j}) (0, 1) for all i, j T, then for any R S +, S T π R (T )=π R (S)π S+ (T ). Axiom 2: If π i ({i, j}) =0forsomei, j T, then for any S T such that i S π S (T )=π S {i} (T {i}). The most celebrated consequence is that π i (S) satisfies the Luce Axioms if and only if π i (S) follows the basic attraction model (1). Axiom 1 holds trivially for R = since π (S) =0for all S N. For R S the formula can be written as π R (T ) π R (S) = π S + (T )=1 π T S (T ), 5

6 so the right hand side is independent of R, and it is the complement of the probability of selecting T S when T is offered. We want to generalize Axiom 1 so that the ratio remains independent of R for any non-empty R S T but in such a way that we have more control over how much of the relative demand is lost by adding alternatives T S to S. Indeed, empirical evidence suggests that the BAM allows too much of this demand to be lost, and the flip side of the coin is that too much of the demand π T S (T ) will find its way to R S when T S is removed from T. This suggests the following generalization of Axiom 1. Axiom 1 : If π i ({i, j}) (0, 1) for all i, j T, then for any R S T π R (T ) π R (S) =1 π T S(T )+ θ j π j (T ) for some set of values θ j [0, 1],j N. j T S Notice that the special case θ j =0forallj recovers Axiom 1 and therefore results in the BAM. On the other hand, the case θ j =1forallj makes the right hand side equal to one, so π R (S) is independent of S for all S containing R, corresponding to the IDM. We will call the set of Axioms 1 and 2 the Generalized Luce Axioms (GLA). The next result states that a choice model satisfies the GLA if and only if it is a GAM. Theorem 1 A choice model π i (S) satisfies the GLA if and only if it is of the GAM form. A proof of this theorem may be found in the Appendix. The GAM can also be justified from the multinomial logit under competition. A detailed justification of this construction is provided in Appendix. 2.1 Limitation and Heuristic Uses of the GAM One limitation of the GAM as we propose it is that, in practice, the utility associated with the products at different locations may depend on a set of covariates that may be customer or product specific. This can be dealt with either by assuming a latent class model on the population of customers or by assuming a customer specific random set of coefficients to capture the impact of the covariates. Another limitation is that we are implicitly modeling customers who visit the store first. This can be fixed by dividing the customers among market segments and having another GAM model for customers who visit the competitor s store first. The issues also arise when using the BAM. In fact, often a single BAM is used to try to capture the choice probabilities of all customers whether they visit the store first or not. This stretches the limits of the BAM and of the GAM, as this should really be handled by adding market segments. However the GAM can cope much better with this situation as the following example illustrates. Example 1: Suppose there are two stores (denoted l), and each market store attracts half of the population. We will assume that there are two products (denoted k) and that the GAM parameters v lk and w lk for the two stores are as given in Table 1: 6

7 v 10 v 11 v 12 v 20 v 21 v 22 w 11 w 12 w 21 w Table 1: Attraction Values (v, w) With this type of information we can use the GAM to compute choice probabilities π j (S, R) that a customer will purchase product j S from Store 1 when Store 1 offers set S, and Store 2offerssetR. Suppose Store 1 has observed selection probabilities for different offer sets S conditional on Store 2 offering R = {1, 2} as provided in Table 2: S % 0% 0% {1} 82.1% 17.9% 0% {2} 82.8% 0.0% 17.2% {1, 2} 66.7% 16.7% 16.7% Table 2: Observed π j (S, R) forr = {1, 2} To estimate these choice probabilities using a single market GAM we seek to find v k,w k,k = 1, 2 with v 0 = 1 to minimize the sum of squared errors of the predicted probabilities. This results in the GAM model: v 0 =1,v 1 = v 2 =0.25, w 1 =0.20 and w 2 =0.15. This model perfectly recovers the probabilities in Table 2. When limited to the BAM, the largest error is 1.97% in estimating π 0 ({1, 2}). A similar situation arises when demand comes from a mixture of the BAM and the IDM. In this case, the two extreme models (BAM and IDM) fit the overall data very poorly, but the GAM can do a reasonable job of approximating the choice probabilities. 2.2 GAM Estimation and Examples It is possible to estimate the parameters of the GAM by refining and extending the Expectation Maximization method developed by Vulcano, van Ryzin and Ratilff (VvRR) [31] for thebam. The details of the estimation procedure are given in the Appendix. Here we present three examples to illustrate how the procedure works when w =0,w = θv and w v. These examples are based on Example 1 in VvRR. The examples involve five products and 15 periods where the set N = {1, 2, 3, 4, 5} wasofferedinfourperiods,set{2, 3, 4, 5} was offered in two periods, while sets {3, 4, 5}, {4, 5} and {5} were each offered in three periods. Notice that product 5 is always offered making it impossible to estimate w 5. On the other hand, estimates of w 5 would not be needed unless the firm planned to offer other products in the absence of product 5. 7

8 The parameters of Example 1 of VvRR are λ = 50, v 0 =1,andv =(1.0, 0.7, 0.40, 0.20, 0.05), resulting in share s = v(n)/(1 + v(n)) =.71. To construct a GAM-version of the original VvRR problem, we assume the true parameter for the parsimonious GAM is θ =0.2, and the true parameters for the non-parsimonious GAM are w =(0.25, 0.35, 0.05, 0.05). In their paper, VvRR present the estimation results for a specific realization of sales z t,t=1,...,t =15 corresponding to the 15 offer sets S t,t =1,...,15. We instead generate 500 simulated instances of sales, each over 15 periods, and then compute our estimates for each of the 500 simulations. We report the mean and standard deviation of λ, v and either θ or w over the 500 simulations. We feel these summary statistics are more indicative of the ability of the method to estimate the parameters. We report results when s is assumed to be known, as in [31], and some results for the case when s is unknown 2. Estimates of the model parameters tend to be quite accurate when s is known and the estimation of λ and v become more accurate when w 0. Theresultsfors unknown tend to be accurate when w = 0 but severely biased when w 0. The bias is positive for v and w and negative for λ. Since it is possible to have biased estimates of the parameters but accurate estimates of demands, we measure how close the estimated demands ˆv i ˆλˆπ i (S) =ˆλ 1+ˆv(S)+ŵ(S ) i S, S N are from the true expected demands λπ i (S), i S, S N and also how close the estimated demands ˆλˆp i (S t ) i S t,t =1,...,T are from sales z ti,i S t,t =1,...,T. These measures are reported, respectively, as the Model Mean Squared Error (M-MSE ) and the Data Mean Squared Error (D-MSE). Surprisingly the results with unknown s, although strongly biased in terms of the model parameters, often produce M-MSEs and D-MSEs that are similar to the corresponding values when s is known. Nonetheless, due to the strong bias, we suggest the fitting a GAM with unknown s is as an area for future research. Example 2 (BAM) In the original VvRR example, the true value of w is 0. Assuming s is known, our estimates of λ and v based on 500 simulations had means ˆλ = and ˆv =(0.9875, , 04003, , ) with respective bias 0.53% and ( 1.25%, 0.44%, 0.07%, 0.20%, 1.80%). The standard deviation among the 500 estimates was for λ and (0.1001, , , , ) for v. The D-MSE and the M-MSE were respectively and We also tried our method when s is unknown and obtained λ = , v = (1.2117, , , , ) with respective bias 2.518% and (21.77%, 16.90%, 12.13%, 45.50%, 7.00%). The standard deviation among the 500 estimates was for λ and (0.7581, , , , ) for v. The mean share was s = and the standard deviation of the share was Finally, the D-MSE and the M-MSE were respectively and Example 3 (Parsimonious GAM) For this example v isthesameasabovebutw = θv with θ =0.20. Assuming s is known, our estimates of λ, v and θ based on 500 simulations had mean ˆλ = , ˆv =(0.9988, , , , ) and ˆθ = with respective bias 0.25%, ( 0.12%, 0.31%, 0.68%, 0.50%, 1.45%), and 19.2%. The standard deviations among the 500 estimates was for λ, (0.1110, , , , ) for v and for 2 The authors are grateful to Anran Li for coding the EM procedure and conducting the experiments. 8

9 θ. Finally, the D-MSE and the M-MSE were respectively and We also tried our method when s is unknown and obtained severely biased estimates of the parameters so they are not reported. However, the D-MSE and M-MSE with s unknown were respectively and Example 4 (GAM) We tried the same example but with w =(0.25, 0.35, 0.15, 0.05, 0.05). Assuming s is known, our estimates of λ, v and w based on 500 simulations had mean ˆλ = , ˆv =(0.9935, , , , ) and ŵ =(0.3213, , , , ), with respective bias.38%, ( 0.65%, 0.01%, 0.90%, 1.15%, 1.20%) and (28.50%, 17.69%, 23.86%, %, 79.77%). The standard deviation among the 500 estimates was for λ, (0.1025, , , , ) for v and (0.2819, , , , ) for w. Finally, the D-MSE and the M-MSE were respectively and We also tried our method when s is unknown and obtained hugely biased results so they are not reported. However, the D-MSE and M-MSE with s unknown were and with a surprisingly good fit to expected demands. The results from these experiments suggest that the EM method devised for the GAM improves the estimates of λ and v when s is known and w 0. Theresultsfors unknown result in parameter estimates that are severely biased when w 0. However, the fit to the data and to the model is almost as good as the case when s is known. This can be best appreciated by reporting the largest absolute relative error in estimating demands λπ i (S) i S, S N for a randomly generated sales instance of each of the three examples. For the GAM examples, the largest relative error was only 2% higher for s unknown than for s known. The BAM has been universally used over the last few years in revenue management (RM) as an alternative to the pessimistic IDM that ignores upsell and recapture. Our experience, however, reveals that the BAM can suffer from being overly optimistic. As the GAM allows greater flexibility and has both the BAM and the IDM as special cases, we feel that it is a more appropriate model for network revenue management. In sections 3, 4 and 5, applications of the GAM to the network revenue management problem will be discussed in detail. 3 Stochastic Revenue Management over a Network We will assume that there are L customer market segments and that customers in a particular market segment are interested in a specific set of itineraries and fares 3 from a set of origins to one or more destinations. For example, a market segment could be morning flights from New York to San Francisco that are less than \$500, or all flights to the Mexican Caribbean. Notice that there may be multiple fares with different restrictions associated with each itinerary. At any time the set of origin-destination-fares (ODFs) offered for sale to market segment l is a subset S l N l,l L= {1,...,L} where N l is the consideration set for market segment l L with associated fares p lk,k N l. If the consideration sets are non-overlapping over the market segments then there is no need to add further restrictions to the offered sets. This is also true when consideration sets overlap if the capacity provider is free to independently decide what 3 The airfare value used in the RM optimization may be adjusted to account for other associated revenues (e.g. baggage fees) or costs (e.g. meals or in-flight wireless access charges). 9

10 fares to offer in each market. As an example, different fares are offered for round-trips from the US to Asia depending on whether the customer buys the fare in the US or in Asia. If the provider cannot make independent decisions by market segment, then he needs to post asingleoffersets so that the projection subsets S l = S N l for all l are offered for sale. Customers in market segment l select from S l and the no-purchase alternative. We will abuse notation slightly and write π lk (S l ) to denote the probability of selecting k S l+ from market segment l. For convenience we define π lk (S l )=0ifk/ S l+. The state of the system will be denoted by (t, x) wheret is the time-to-go and x is an m-dimensional vector representing remaining inventories. The initial state is (T,c). For ease of exposition we will assume that customers arrive to market segment l as a Poisson process with time homogeneous rate λ l. We will denote by V (t, x) the maximum expected revenues that can be obtained from state (t, x). The Hamilton-Jacobi-Bellman (HJB) equation is given by V (t, x) t = l L λ l max π lk (S l )(p lk Δ lk V (t, x)) (4) S l N l k S l where Δ lk V (t, x) =V (t, x) V (t, x A lk ), where A lk is the vector of resources consumed by k N l. The boundary conditions are V (0,x)=0andV (t, 0) = 0. Talluri and van Ryzin [28] developed the first stochastic, choice-based, dependent demand model for the single resource case in discrete time. The first choice-based, network formulation is due to Gallego, Iyengar, Phillips and Dubey [12]. 4 The Choice Based Linear Program An upper bound V (T,c) on the value function V (T,c) can be obtained by solving a deterministic linear program. The justification for the bound can be obtained by assuming a perfect foresight model, arguing that the objective is concave on the realized demands and then using Jensen s inequality. Approximate dynamic programming with affine functions can also be used to justify the same bound. We now introduce additional notation to write the linear program with value function V (T,c) V (T,c). Let π l (S l )bethen l = N l -dimensional vector with components π lk (S l ),k N l. Clearly π l (S l ) has zeros for all k/ S l. The vector π l (S l ) gives us the probability of sale for each fare in N l per customer when we offer set S l.letr l (S l )=p l π l(s l )= n l k=1 p lkπ lk (S l )be the revenue rate per customer associated with offering set S l N l where p l =(p lk ) k Nl is the fare vector for segment l L. Notice that π l ( ) =0 R n l and r l ( ) =0 R. Let A l π l (S l ) be the consumption rate associated with offering set S l N l where A l is an identity matrix with columns A lj associated with ODF j N l. 10

11 The choice based deterministic linear program (CBLP) for this model can be written as V (T,c)=max l L λ lt S l N l r l (S l )α l (S l ) (5) subject to l L λ lt S l N l A l π l (S l )α l (S l ) c S l N l α l (S l ) = 1 l L α l (S l ) 0 S l N l l L. The decision variables are α l (S l ),S l N l,l Lwhich represent the proportion of time that set S l N l is used. The case of non-homogeneous arrival rates can be handled by replacing λ l T by T λ 0 lsds. There are l L 2n l decision variables. The CBLP is due to Gallego, et al. [12]. They showed that V (T,c) V (T,c) and proposed a column generation algorithm for a class of attraction models. This formulation has been used and analyzed by other researchers including Bront, Mendez-Diaz and Vulcano [5], Kunnumkal and Topaloglu [17], [18], Liu and van Ryzin [20] and Zhang and Adelman [34]. 5 The Sales Based Linear Program for the GAM In this section we will present a linear program that is equivalent to the CBLP but is much smaller in size (essentially the same size as the well known IDM) and easier to solve as it avoids the need for column generation. The new formulation is called the sales based linear program (SBLP) because the decision variables are sales quantities. The number of variables in the formulation is l L (n l +1), where x lk,k =0, 1...,n l,l Lrepresents the sales of product k in market segment l. LetD l = λ l T,l L(or T λ 0 lsds if the arrival rates are time varying). With this notation the formulation of the SBLP is given by: R(T,c)= max subject to capacity : balance : l L l L k N l p lk x lk k N l A lk x lk c l L ṽ l0 v l0 x l0 + ṽ lk k N l v lk x lk = D l l L x lk v lk scale : x l0 v l0 0 k N l l L non-negativity: x lk 0 k N l l L (6) Theorem 2 If for each market segment π lj (S l ) is of the form (2) then V (T,c)=R(T,c). We demonstrate the equivalence of the CBLP and SBLP formulations by showing that the SBLP is a relaxation of the CBLP and then proving that the dual of the SBLP is a relaxation of the dual of the CBLP. Full details of this proof may be found in the Appendix. 11

12 The intuition behind the SBLP formulation is as follows. Our decision variables represent the amount of inventory to allocate for sales across ODFs for different market segments. The objective is to maximize revenues. Three types of constraints are required. The capacity constraints limit the sum of the allocations. The balance constraints ensure that the sale allocations are in certain hyperplanes, so when some service classes are closed for sale, the demand for the remaining open service classes (including the null alternative) must be modified to offset the amount of the closed demands. The scale constraints operate on the principle that demands are rescaled depending on the availability of other items in the choice set. The decision variables x l0 play an important role in the SBLP. Since the null alternative has positive attractiveness and is always available, the dependent demand for any service class is upper bounded based on its size relative to the null alternative. The scale constraints imply that x lk v lk x l0 /v l0. Therefore, the scale constraints favor a large value of x l0. On the other hand, the balance constraint can be written as k N ṽlkx lk /v lk D l ṽ l0 x l0 /v l0,favoringa small value of x l0. The LP solves for this tradeoff taking into account the objective function and the capacity constraint. We will have more to say about x l0 in the absence of capacity constraints in Section 6. The combination of the balance and scale constraints ensures that GAM properties are represented and that the correct ratios are maintained between the sizes of each of the open alternatives. The special case of the SBLP with ṽ lk = v lk for all k N l,l Lreduces to the BAM. For the BAM the demand balance constraint simplifies to x l0 + k N l x lk = D l. For the GAM, the balance constraint can alternatively be written as x l0 + k N l x lk + k N l w lk (x l0 /v l0 x lk /v lk )=D l. Since the third term of this equation is non-negative on account of the scale constraints, it follows that for the GAM x l0 + k N l x lk D l. For the independent demand case ṽ lk = 0 for all k N l and ṽ l0 = v l0 + k N l v lk, so the balance constraint reduces to x l0 =(v l0 /ṽ l0 )D l, and the scale constraint becomes x lk (v lk /ṽ l0 )D l,k N l, so clearly x l0 + k N l x lk D l. A major advantage of the SBLP formulation is that it neatly avoids the usual non-linearities associated with calculating dependent demands under the general attraction model. The SBLP allows us to determine which flight service classes should be open versus closed for sale. Based on the specific pattern of open versus closed classes, the demands will vary (due to sameflight upsell and cross-flight recapture effects). The compact formulation of the SBLP makes practical the use of randomized linear programming (RLP) to obtain tighter upper bounds on revenues and robust bid-price controls by repeatedly solving the SBLP for simulated demands and averaging the objective function and the dual variables of the capacity constraints. See Talluri and van Ryzin [29] for more on the RLP for the independent demand model as well as Talluri [27] who proposes a concave programming method for general choice models. Most modern RM optimization models for handling dependent demands require special transformations of fare inputs and/or pre-calculation of maximum demands (conditional on closure of lower valued classes). Weatherford et al. [32] discusses several of these approaches (e.g. DAVN-MR and choice-based EMSR); however, the required fare and demand transformations complicate both the optimization and real-time control processing. An important practical benefit of the SBLP formulation is that it does not require any special transformations of the original inputs; the special structure of the demand, scale and balance constraints 12

13 enforces automatic rescaling of the dependent demands and correctly models the associated revenues (inclusive of upsell and recapture effects) in the LP. Another point worth noting is that the CBLP and SBLP formulations remain equivalent even if the market segments have products that overlap. The caveat is that both formulations assume in this case that each market segment will be managed independently. This works only if the capacity provider has the ability to offer a product for a market segment without offering it to other market segments. On the other hand, if a fare cannot be offered in one market without offering to other markets then both the CBLP and the SBLP formulations are upper bounds that can be tightened by adding additional constraints. This approach was used by Bront et al. [5] for the CBLP. They use column generation assuming that each market segment follows a BAM and show that the column generation step is NP-hard. We remark that the CBLP and therefore the SBLP can be extended to multi-stage problems since the inventory dynamics are linear. This allows for different choice models of the GAM family to be used in different periods. The different GAMs may reflect time heterogeneity and the products that planners expect competitors to be offering. Thus, the probabilities may be of the form π j (S, R i )fort T i if R i is the set of products offered by competitors over thetimeintervalt i. 5.1 Feasibility, Sub-optimality and Numerical Examples In this section we present two results that relate the solutions to the extreme cases w =0and w = v to the case 0 w v. The proofs of the propositions are in the Appendix. The reader may wonder whether the optimistic solution that assumes w j =0forallj N is feasible when w j (0,v j ) j N. The following proposition confirms and generalizes this idea. Proposition 1 Suppose that α l (S l ),S l N l,l Lis a solution to the CBLP for the GAM for fixed v lk,k N l+,l Land w lk [0,v lk ],k N l,l L. Then α l (S l ),S l N l,l Lis a feasible, but suboptimal, solution for all GAM models with w lk (w lk,v lk ],k N l,l L. An easy consequence of this proposition is that if α l (S l ),l Lis an optimal solution for the CBLP under the BAM, then it is also a feasible, but suboptimal, solution for all GAMs with w k (0,v k ]. As mentioned in the introduction, the IDM underestimates demand recapture. As such optimal solutions for the SBLP under the IDM are feasible but suboptimal solutions for the GAM. The following proposition generalizes this notion. Proposition 2 Suppose that x lk,k N l,l Lis an optimal solution to the SBLP for the GAM for fixed v lk,k N l+ and w lk [0,v lk ],k N l,l L. Then x lk,k N l+,l Lis a feasible solution for all GAM models with w lk [0,w lk],k N l,l Lwith the same objective function value. 13

14 An immediate corollary is that if x lk,k N l,l Lis an optimal solution to the SBLP under the independent demand model then it is also a feasible solution for the GAM with w k [0,v k ] with the same revenue as the IDM. Example 5: The following example problem was originally reported in Liu and van Ryzin [20]. There are three flights AB, BC and AC. All customers depart from A with destinations to either B or C. The relevant itineraries are therefore AB, ABC and AC. For each of these itineraries there are two fares (high and low) and three market segments. Market segment 1 are customers who travel to B. Market segment 2 are customers who travel to C but only at a high fare. Market segment 3 are customers who travel to C but only at the low fare. The six fares as well as v lj,j N l and v l0 for l =1, 2, 3 are given in Table 3. Assume that market sizes are λ 1 =6,λ 2 =9,λ 3 = 15 and that capacity c =(10, 5, 5) for flights AB, BC and AC Products AC H ABC H AB H AC L ABC L AB L Null Fares \$1,200 \$800 \$600 \$800 \$500 \$300 (AB, v 1j ) (AC high,v 2j ) (AC low,v 3j ) Table 3: Market Segment and Attraction Values v In this example we will consider the parsimonious model w lj = θv lj for all j N l,l {1, 2, 3} for some global parameter θ [0, 1]. We are interested in seeing how robust are the solutions to the two extreme cases θ =0andθ = 1 to situations where θ (0, 1). The extreme case θ = 0 corresponds to the BAM. The solution to the CBLP for the BAM is given by α({1, 2, 3}) = 60%, α({1, 2, 3, 5}) =23.3% and α({1, 2, 3, 4, 5}) =16.7% with α(s) = 0 for all other subsets of products. Equivalently, the solution spends 18, 7 and 5 units of time, respectively, offering sets {1, 2, 3}, {1, 2, 3, 5} and {1, 2, 3, 4, 5}. The solution results in \$11, in revenues. 4 By Proposition 1, the solution α({1, 2, 3}) = 60%, α({1, 2, 3, 5}) =23.3%, α({1, 2, 3, 4, 5}) =16.7% is feasible for all θ (0, 1] with a lower objective function value. The corresponding SBLP solution is shown in Table 4 and provides the same revenue outcome as the CBLP. The case θ = 1 corresponds to the IDM. The solution to the SBLP for the IDM and the corresponding revenue is given in Table 5. By Proposition 2, the solution to the IDM is a feasible solution for all θ [0, 1) with the same objective value function. Figure 1 shows how the two extreme solutions (CBLP for θ =0)and(SBLPforθ =1) perform compared to the optimal solution for the GAM for θ (0, 1). A couple of observations are in order. First, the BAM solution does well only for small values of θ, sayθ 20% and performspoorlyfor largevalues of θ. This result is intuitive because the modified problem with θ treats the competitor alternatives much differently than the (uncorrected) BAM assumes; in 4 This result is a slight improvement on the one reported in Liu and van Ryzin [20] for which we calculate a profit of \$11, based on an allocation of 16.35, 2.48, 10.3 and 0.87 units of time, respectively, on the sets {1, 2, 3}, {1, 2, 3, 4}, {1, 2, 3, 5}, {1, 2, 3, 4, 5}. 14

15 Products AC H ABC H AB H AC L ABC L AB L x l0 Profit (AB, x 1j ) \$2, (AC high,x 2j ) \$7, (AC low,x 3j ) \$1, Total \$11, Table 4: SBLP Allocation Solution for BAM with (θ =0) Products AC H ABC H AB H AC L ABC L AB L x l0 Profit (AB, x 1j ) \$2, (AC high,x 2j ) \$7, (AC low,x 3j ) \$1, Total \$11, Table 5: SBLP Allocation Solution for IDM with (θ =1) practice, ad-hoc increases to the size of the null alternative would be required in the BAM to avoid overstating expected recapture rates. Left uncorrected (which is what these results are based on), the BAM solution gives up about 10% of the optimal profits at the extreme value of θ = 1. Second, the solution for the IDM gives up significant profits only for small values of θ. In particular, the IDM solution loses over 4% of the optimal profits for θ = 0 but less than 0.25% of profits for θ 40%. Finally, the optimal value function for the GAM is only marginally better than the maximum of the (GAM-estimated) value functions for the BAM and the IDM. Indeed, a heuristic that uses the BAM for θ 20% and the IDM for θ>20% works almost as well as the GAM. Please note that these results apply to the special case where all items in the choice set have had their attractiveness downgraded by the same relative amount. The more typical application of the GAM in practice is that at most a few of the items in the choiceset would be modified (because they are especially competitive relative to the rest of the set), so the revenue differences between the BAM and GAM results wouldn t be quite as large. Nonetheless, the example does highlight the asymmetry of revenue performance associated with overestimating versus underestimating the attractiveness of competitor offerings. A related finding was reported by Hartmans [14] of Scandinavian Airlines; his simulation studies showed clear positive revenue potential of including sellup effects, but assuming strong sellup in cases where it doesn t really exist led to revenue loss. Example 6: We continue using the running example in Liu and van Ryzin but we now assume that demand is governed by the specific GAM presented in Table 6. The model reflects positive w values for lower fares. The optimal revenue under this model is \$11,225.00, and the optimal allocations are given in Table 7. We know from Proposition 1 that the CBLP solution for the BAM is feasible for this GAM. The resulting revenue is \$11, The solution to the IDM for the SBLP is also feasible for the GAM and results in revenue \$11, It is clear that in this case the BAM performs better than the IDM solution, and both are slightly worse than the GAM. 15

16 \$11,600 \$11,500 \$11,400 Legend gam bam idm \$11,300 \$11,200 \$11,100 \$11,000 Revenue \$10,900 \$10,800 \$10,700 \$10,600 \$10,500 \$10,400 \$10,300 \$10,200 \$10,100 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% Theta Figure 1: GAM, BAM and IDM Revenue for Different θ Values 5.2 Non-Linear Objective Functions Recently, Topaloglu, Birbil, Frenk, and Noyan [30] extended the results in this paper to deal with non-linear objective functions for the BAM. They do this by showing how to construct a primal solution to the CBLP from the SBLP. Here we present how to construct a primal solution to the CBLP for the GAM to allow also for non-linear objective functions 5 : Suppose that x lk is an optimal solution to the SBLP. For each market segment, sort the items in decreasing order of x lk /v lk,sothat x l0 /v l0 x l1 /v l1... x lnl /v lnl where n l is the cardinality of N l. Then form the sets S l0 = and S lk = {l1,...,lk} for k =1,...,n l.fork =0, 1,...,n l set ( xlk α l (S lk )= x ) l,k+1 ṽ(slk ), v lk v l,k+1 D l where for convenience we define x l,nl +1/v l,nl +1 =0. Setα l (S l ) = 0 for all S l N l not in the collection S l0,s l1,...,s lnl. It is then easy to verify that α l (S l ) 0, that S l N l α l (S l )=1 and that S l N l α l (S l )π lk (S l )=x lk. Applying this to the CBLP we can immediately see that all constraints are satisfied and the objective function coincides with that of the SBLP. Notice that for each market segment the set of offered sets S lk,k =0, 1...,n l are nested 5 The authors are indebted to Huseyin Topalogulu for suggestions this result. 16

17 Products AC H ABC H AB H AC L ABC L AB L Null Fares \$1,200 \$800 \$600 \$800 \$500 \$300 (AB, v 1j,w 1j ) (5, 0) (8, 1) 2 (AC high,v 2j,w 2j ) (10, 2) (5, 1) 5 (AC low,v 3j,w 3j ) (5, 1) (10, 0) 10 Table 6: Market Segment and Attraction Values (v, w) Products AC H ABC H AB H AC L ABC L AB L x l0 Profit Market 1 (AB) \$2, Market 2 (AC high ) \$7, Market 3 (AC low ) \$1, Total \$11, Table 7: SBLP Allocation Solution with GAM (w >0) S l0 S l1... S lnl and that set S lk is offered if and only if x lk /v lk >x l,k+1 /v l,k+1.inmany situations the sequence looks like: x l0 /v l0 = x l1 /v l1 =...= x lk /v lk >x l,k+1 /v l,k+1 > 0=x l,k+2 /v l,k+2 =...= x lnl /v lnl. In this case only the set S lk and S l,k+1 are offered for positive amounts of time. In this case fare k +1 is the marginal fare for market segment l. Unlike the BAM and the IDM the sorting order is not necessarily in decreasing order of fares. This is because fares that face more competition (w lj close to v lj ) are more likely to be offered even if the corresponding fare p lj is low. We will have more to say about sort ordering when we discuss the Assortment Problem in the next section. 6 The Assortment Problem As a corollary to Theorem 2, the infinite capacity formulations for the SBLP and CBLP are equivalent. With infinite capacity the problem separates by market segment so it is enough to study a single market segment. To facilitate this we will drop the subscript l both in the formulation and in the analysis. The resulting problem can be interpreted as the that of finding the assortment of products S N and the corresponding sales x k,k N to maximize profits from a market segment of size D that makes decision based on a GAM choice model. Under this interpretation the p k values are gross unit profits, net of unit costs. 17

18 R(T) = max subject to balance : scale : k N p kx k ṽ 0 v 0 x 0 + ṽ k k N v k x k = D x k v k x 0 v 0 0 k N (7) non-negativity: x k 0 k N + The problem is similar to a bin-packing problem with constraints linking the x k s to x 0. An analysis of the problem reveals that the allocations should be in descending order of the ratio of p k to ṽ k /v k. This ratio can be expressed as p k v k /ṽ k. For the BAM the allocation is in decreasing order of price. Including lower priced products improves sales, but it cannibalizes demands and dilutes revenue from higher priced products. As such, the allocation of demand to products needs to balance the potential for more sales against demand cannibalization. The sorting of products for the GAM depends not only on price but on the ratio v k /ṽ k,soa low priced product with a high competitive factor θ k = w k /v k may have higher ranking than a higher priced product with a low competitive factor. To explain how to obtain an efficient allocation of demand over the different products we will assume that the products are sorted in decreasing order of p k v k /ṽ k.letx j = {1,...,j} and consider the allocation x k = v k x 0 /v 0 for all k X j and x k =0fork/ X j. Solving the demand constraint for x 0 we obtain x 0 = v 0 D, (8) ṽ 0 +ṽ(x j ) x k = π k (X j )D k X j, (9) and revenue R j = k S j p k π k (X j )D. (10) Now consider the problem of finding the revenue R j+1 for set X j+1 = X j {j +1}. It turns out that R j+1 can be written as a convex combination of R j and p j+1 Dv j+1 /ṽ j+1.this implies that R j+1 >R j if and only if R j <p j+1 Dv j+1 /ṽ j+1. Thus the consecutive set X j with highest profit has index j =max{j 1:p j k X j p k π k (X j )(1 θ j )}. (11) The following proposition whose proof can be found in the Appendix establishes the optimality 6 of X j. Theorem 3 The assortment X j with j given by equation (11) with x k, k X j given by (9) is optimal. 6 H. Topaloglu at Cornell independently found the same sorting for the GAM. Personal communication. 18

19 Notice that j in equation (11) is independent of D. This means that products 1,...,j are offered for all demand levels D>0 and products j +1,...,nare closed regardless of the value of D. Products j + 1,...,n are inefficient in the sense they do not increase sales enough to compensate for the demand they cannibalize. Table 4 illustrates precisely this fact as product 5 priced at \$105 is inefficient because p 5 /(1 θ 5 ) <R 4. Notice that in this example it was better to keep product 1 at \$100 fare with a competitive factor θ 1 =0.9 than the higher product 5 at \$105 with competitive factor θ 5 = 0. The intuition behind this is that closing product 5 is more likely to result in recapture than closing fare 1 (whose demand will most likely be lost to competition or to the no purchase alternative). k p k v k θ k x k p k x k \$ \$ \$ \$ \$ \$ \$ \$ \$ \$0.00 Total \$ Table 8: Example Assortment Problem If the problem consists of multiple market segments and each market segment is allowed to post its own available product set X lj then the problem reduces to sorting the products for market segment l in decreasing order of p lk /(1 θ lk ), where θ lk = w lk /v lk, and then computing j(l) using (11) for each l. The resulting sets X lj(l),l Lare optimal. If, however, a single assortment set S is to be offered to all market segments then the problem is NP-hard, see Rusmevichientong et al. [23], even if there are only two market segments and the choice model is the BAM. However, the problem for the BAM is somewhat easier to analyze since the preference ordering for all the market segments is p 1 >p 2 >... > p n whereas the preference ordering for the GAM may be different for different market segments. Assuming the order p 1 >p 2 >... > p n, [23] provide a two market segment example where it is optimal to offer X 1 = {1} to market segment one, to offer X 2 = {1, 2} to market segment two and to offer X = {1, 3} to a composite of 50% of each of the two market segments. Given the NP-hardness of the assortment problem with overlapping sets, the research activity has concentrated on bounds, heuristics and performance warranties. We refer the reader to the working papers by Rusmevichientong, Shen, and Shmoys [24] and Rusmevichientong and Topaloglu [25]. If prices are also decision variables, with choices reflecting price sensitivities, then the problem involves finding the offer sets and prices for each market segment. We refer the reader to Schön [26] who allows for price discrimination, e.g., to offer different prices and offer sets to different market segments. The problem of offering a single offer set and a single vector of prices for all the market segments is again NP-hard. 19

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