CORC Technical Report TR Managing Correlated Callable Products on a Network

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1 CORC Technical Report TR Managing Correlated Callable Products on a Network G. Gallego G. Iyengar June 16, 2005 Abstract We formulate and solve the problem of managing correlated callable products on a network. A callable product gives the seller the option to recall a sold unit and replace it with an alternative from a pre-specified set this set can include substitute products or a cash compensation. We model demand by a generalized multinomial logit model that allows the constants determining the arrival rates to themselves be functions of subsets of offered products. A subset of products is said to be correlated if they influence each other s rate constant. We also introduce a new class of products called optional products that give the buyer the option to select an alternative from a pre-specified set. We show that the seller can significantly increase revenue by offering these optional products at a modest premium and then hedging the sales using a combination of flexible and callable products. 1 Introduction Gallego, Kou and Phillips [11] define a callable product as a product that gives the seller the option of recalling a sold unit at a later date by paying the buyer a pre-specified recall price. This is in contrast to the more traditional of a specific product in which a unit that is sold can never be recalled. Gallego et al introduce callable products in the context of airline revenue management as a hedge against the uncertain high-fare demand: when the high-fare demand exceeds available capacity, the callable units are recalled to accommodate this demand. They show that by carefully managing the recall price the airline can mitigate some of loss associated with spoilage, and, 1

2 thereby, increase revenue. Callable products can potentially enhance revenue via the following two mechanisms: (a) Improved capacity utilization: Since the seller can recall unit of callable products after the uncertainty in the high-fare demand is largely resolved, offering callables allows the seller to free up scarce capacity for the high-fare demand on an as-needed basis. (b) Demand Induction: Since buyers of callable products are uncertain about whether their units will be recalled, these products are likely to be viewed as inferior products, and can be offered at a lower price to attract additional customers without entirely cannibalizing the full-fare demand. It should be clear that unless the callable products are priced correctly they could cannibalize higher paying specific-product customers and actually reduce revenue. While the concept callable products is interesting for any industry selling fixed, perishable capacity to a population that is heterogeneous in its willingness to pay, for consistency and ease of understanding this paper is couched in terms of passenger airlines. Our main contributions in this paper are as follows. (a) We formulate and solve the problem of managing callable products on a network for both the independent demand model (see Section 2) and a choice-driven demand model (see Section 3). We broaden the definition of callable products by allowing the seller to replace the recalled unit by one unit of an alternative specific product from pre-specified list of alternatives. We show that the optimal solution of the stochastic control problem is closely approximated by the solution of a fluid relaxation. The fluid problem corresponding to the choice-driven demand model is a linear program (LP) with exponentially many columns; however, we show that column generation solves this LP efficiently. The techniques used to establish these results are a minor modification of those in [9]. (b) Suppose the firm offers a subset S of products. Then according to the multinomial logit (MNL) model the demand d i (S) for a product i S is given by d i (S) = g i (S)/(1 + j S g j) where g, g are given constant non-negative vectors independent of the set S being offered. In Section 4 we develop a generalized MNL model that allows the parameters g, g determining the arrival rates to themselves be functions of subsets of offered products, i.e. the parameters (g i, g i ) 2

3 corresponding to a given product i changes whether or not another product j is being offered. This model is motivated by the observation that the parameters corresponding to a specific product may change depending on whether the firm is simultaneously offering the callable version of the same product. We call a subset of products correlated if they influence each other s rate constant, and call the demand model an MNL model with correlation. Our main result is that for this new choice model the column generation subproblem is still efficiently solvable; consequently, the fluid LP remains efficiently solvable. Although we motivate this new choice model in the context of callable products, the model is able of express more general correlation structures and is of independent interest. (c) In Section 5 we introduce optional products that give the buyer the option to select a product from a pre-specified set of alternatives, possibly after paying a penalty. We show that the seller can significantly increase revenue by selling optional products at a modest premium and then hedging this sale by selling an appropriate number of flexible and callable products. The organization of this paper is as follows. Sections 2 and 3 are devoted to managing callable products on a network. Section 4 introduces the multinomial logit model with correlation. Section 5 discusses optional products. Section 6 reports the results of our numerical experiments and Section 7 has some concluding remarks. 2 Network model with independent demand and callable products We consider a network consisting of m resources and n specific products that consume one or more resources. The initial capacity is given by the vector c R m +. This capacity is available for consumption over the sales horizon [0, T ]. Any unused capacity at time T is worthless. Let A = [a li ] R m n denote the incidence matrix of the specific products, i.e. a li = 1 if the i-th specific product consumes resource l and 0 otherwise. (The analysis in this paper extends to the setting where a product may consume multiple units of a resource.) The revenue vector of the specific products is given by p R n +. In addition, products k = 1,..., f ( n), can be sold with a call option. Such a product will be called a callable product as opposed to a specific product. Let q R f + denote the revenue vector corresponding to the callable products. The seller guarantees that the customer who purchases 3

4 the k-th callable product will be assigned to some alternative in set {k} F k (k F k ), where F k consists of products that are considered substitutes. We let n k denote the cardinality of the set F k. Let B k denote the incidence matrix of the alternatives in the set F k. We allow the possibility that an alternative in the set F k does not consume any resource (i.e. the corresponding column is 0); such a column could represent either a cash payment or the cost of satisfying the customer by using a competing supplier. The seller incurs a penalty r kl each time the k-th callable product is called and reassigned to the specific product l F k. Let r k R n k P f k=1 n k denote the vector r k = (r kl ) l Fk and r R+ denote the vector r = (r 1,..., r f ). Two extreme cases of special interest are as follows. (i) Pure flexible products: In this case, r 0, and q < p f, where p f denotes the first f components of the p, i.e. the callable products are sold at a discount but the seller does not incur any penalties for reassigning customers. These products are similar to the flexible products discussed in [10]. In [10, 9] a flexible product is a menu of specific products, i.e. a buyer purchases the menu and not an individual product from this menu. In the model presented in this paper, the buyer purchases a flexible version of a given product (i.e. she has the expectation of consuming a given product) but gives the seller the right to reallocate. This difference manifests itself when the demand is driven by a choice model. (ii) Pure callable products: Here, r > 0, and q = p f, i.e. the seller only incurs penalties if the products are called. For ease of notation, the products labeled 1,..., n are the specific products and the products labeled n+1,..., n+f are the callable products. Then the revenue vector p R n+f is given by p = (p, q ). We consider a continuous time model. Requests arrive according to independent Poisson processes with the arrival rate vector given by λ R n+f +. While it is not conceptually difficult to allow the rate vector λ to be time-dependent, for ease of exposition we assume that the rates are constant. (We consider more general arrival processes in Section 2.2 and Section 2.3.) The firm has the option of accepting or rejecting the request of an arriving customer; however, it must ensure that every admitted request for a specific product is satisfied and every request for a callable product is assigned to some specific product in the set of allowed alternatives, i.e. once admitted the firm 4

5 cannot renege on the requests. Given the Poisson assumption, the network can be described by the pair of vectors (s, y) Z m+f +, where s Z m + denotes the residual capacity and y Z f + denotes the vector of accepted requests for callable products. The state (s, y) is said to be feasible if, and only if, the requests y can be satisfied by the residual capacity s. Let A denote the set of feasible states. P f k=1 Proposition 1 The state (s, y) A if, and only if, there exists a vector z Z n k + satisfying A f y + (B A f U)z c, y + Uz 0, where A f denotes the submatrix of A obtained by taking the first f columns, B = and 1 k denotes k-dimensional vector of all 1 s. Proof 1 n U = 0 1 n , n f (1) (B 1 B 2... B f ), Suppose (1) has a solution z. Partition this vector z into sub-vectors z i Z n k +, k = 1,..., f, such that z = (z 1,..., z f ). For each k = 1,..., f, assign y k 1 T k z k to the specific product k and split up the remaining allocation 1 T k z k among the alternatives in F k according to the vector z k. This assignment clearly satisfies the capacity constraints. The converse is easy to establish and is left to the reader. 2.1 Dynamic programming formulation We assume that the firm is risk-neutral and maximizes the total expected revenue. Let V (s, y, t) denote the maximum achievable revenue starting from the state (s, y) at time t. Then V (s, y, t) is characterized by the Bellman recursion V (s, y, t) t = n λ i max {p i + V (s A i, y, t) V t (s, y, t), 0} i=1 + f λ n+k max {q k + V (s, y + 1 k, t) V (s, y, t), 0}, (2) k=1 5

6 and boundary conditions: for all (s, y) A and t [0, T ], V (s, y, t) = ; and for (s, y) A, V (s, y, T ) = max r z, subject to A f y + (B A f U)z s, y + Uz 0, y, z 0. Note that in the case where (s, y) A the termination revenue is given by the minimum cost of accommodating the accepted the callable products by possibly calling some the requests and reassigning them. The goal of the recursion is to compute V (c) = V (c, 0, 0). Even for networks with modest size (i.e. small n and m) and moderate capacity vectors c, the size of the state space A is very large; consequently, the complexity of computing the solution of the Bellman recursion (2) is prohibitive. In practice, therefore, one has to resort to heuristics to approximate V (s, y, t). In the next section, we construct an asymptotically optimal policy by solving an LP. 2.2 Deterministic control problem In this section we develop a deterministic approximation for the stochastic revenue optimization problem where the requests arrive according to independent point processes. Note that we allow a more general stochastic structure in this section. corresponding stochastic control problem. Let V (c) denote the optimal value of the Note that for the assumed stochastic structure even characterizing V (c) is impossible! Bellman recursion (2) characterizes V (c) only when the arrival processes are Poisson. The first step in the approximation is to approximate the stochastic demand by the expected value of the demand. For i = 1,..., n + f, let d i (T ) denote the random number of requests for product i that arrive over the sales horizon [0, T ] and let d i (T ) = E[d i (T )]. Since the demand is deterministic, the decision of the firm reduces to computing the booking limit x = (x (s), x (c) ) Z n+f +, where x(s) and x (c) denote, respectively, the booking limit for specific and callable products. The second step in the approximation is to relax the integrality constraints on x. Then Proposition 1 6

7 implies that the firm s decision problem reduces to the following LP. V D (c) = max p x (s) + q x (c) r z subject to Ax (s) + A f x (c) + (B A f U)z c, x (c) + Uz 0, x d(t ), x, z 0. Define à = A A f 0 fn I f, B = B A f U U, c = c 0 f1 (3) where 0 st denotes a (s t)-dimensional matrix of all zeros, and I f denotes f-dimensional identity matrix. Then the above LP can be simplified as follows. V D (c) = max p x r z subject to Ãx + Bz c, x d(t ), x, z 0. (4) We will call (4) the Deterministic Network Allocation Problem with Callables (DNAPC). The formulation in (4) is a generalization of the linear programming formulation for the network revenue management problem studied by Williamson [21], Phillips [16], and Talluri and van Ryzin [18], among others. For future reference, we would like to emphasize that the columns of the matrix à R (m+f) (n+f) are given by (A i à i =, 0) i = 1,..., n, (5) (A i, e i ) i = n + 1,..., f, where e k denotes the k-th column of f-dimensional Identity matrix. 2.3 Asymptotic optimality of the deterministic control problem We show that all the standard results for LP approximation of the revenue management problem extends to DNAPC with simple modifications. We refine the asymptotic optimality result by establishing a lower bound on the rate of convergence. 7

8 Proposition 2 The deterministic value function V D (c) is an upper bound on the stochastic value function V (c). Proof Let x ρ denote the random number of requests admitted by any feasible control policy ρ for the stochastic problem by time T. Then feasibility implies that Ãx ρ + Bz c, P k for some z Z n k +. Taking expectations, it follows that the vector (E[x ρ ], E[z]) is feasible for DNAPC. Consequently, it follows that the expected reward R ρ of the policy ρ satisfies R ρ = E [ p x ρ r z ] = p E[x ρ ] r E[z] V D (c). Since the policy ρ was arbitrary, the result follows. Next, we prove a lower bound on V (c) in terms of the deterministic value function V D (c). For i = 1,..., n + f, let χ i (T ) = Var(di (T )) d i (T ) denote the coefficients of variation of the number of requests for the i-th product. Let χ(t ) = max 1 i n+f {χ i (T )} denote the maximum coefficient of variation. Lemma 1 V (c) Proof ( ) χ 2 3 (T ) V D (c) for all T such that χ 2 (T ) 1 2. Let (x, z ) denote the optimal solution of the DNAPC (4). Since all callable products k with x n+k = 0 can be dropped from further consideration, we will assume that x n+k Define the control policy ρ as follows: (a) Admit at most (1 ɛ) x requests. products that are accepted by the policy ρ. > 0 for all k. Let x ρ Z n+f + denote the number of request for each (b) Assign a ( x ) n+k 1 k z k x -fraction of the requests for the k-th callable product to the k-th specific n+k product and a ( zkl ) x -fraction to specific product l Fk. n+k Since x ρ (1 ɛ)x, it follows that the policy ρ is feasible. The expected revenue R ρ is given by [ n+f R ρ = E p i min{d i (T ), (1 ɛ)x i } + i=1 f k=1 8 ] ( q k r k z ) k x min{d n+k (T ), (1 ɛ)x n+k }. n+k

9 From Marshall s inequality, it follows that (see [9] for details) Consequently, ( E[min{d i, (1 ɛ)x i }] (1 ɛ) 1 χ2 (T ) ) ɛ 2 x i, i = 1,..., n + f, ( R ρ (1 ɛ) 1 χ2 (T ) ) ( ɛ 2 ( p x r z ) = (1 ɛ) 1 χ2 (T ) ) ( ɛ 2 V D (c) 1 ɛ χ2 (T ) ) ɛ 2 V D (c). Optimizing the lower bound over ɛ yields ɛ = ( 2χ 2 (T ) ) 1 3. Thus, for all such T such that ɛ < 1, i.e. 2χ 2 (T ) < 1, V (c) R ρ (1 ( )χ 2 3 (T ) )V D (c). The result follows by explicitly computing the constants. Corollary 1 Fix ζ > 1 and consider the scaled stochastic problem in initial capacity ζc and time horizon ζt. Then, for all ζ > 0 such that χ 2 (ζt ) 1 2, we have ) V (ζc) ζv D (c) (1 1.89χ 2 3 (ζt ). Proof Let (x, z ) denote the optimal solution of the DNAPC (4). Then it is easy to check that ζ(x, z ) is optimal for the NLPF corresponding to the stochastic problem with initial capacity ζc and time horizon ζt. Consequently the result follows from Lemma 1. When requests for the specific and callable products arrive according to independent Poisson processes, the maximum coefficient of variation χ(t ) = 1 λmin T, where λ min = min 1 i n+f {λ i }. Corollary 1 implies the following convergence rate ( V (ζc) ζv D (c) (ζλ min T ) 1 3 Lemma 1 and Corollary 1 are new. Taken together they refine the previously known asymptotic optimality results by providing a lower bound on the rate of convergence. ). 9

10 2.4 Bid-price heuristic for networks with flexible products We can use the dual of DNAPC to derive a bid-price heuristic for the network problem with callables. The dual of (4) is given by min c θ + d(t ) µ subject to A j θ + µ j p j, j = 1,..., n, A k θ + µ n+k η k q k, k = 1,..., f, (B k 1 k A k ) θ + η k 1 k r k, k = 1,..., f, θ, µ, η 0, (6) where r k denotes the subvector of r corresponding to the k-th callable product, k = 1,..., f. Note that in constructing this dual we have substituted for the Ã, B, p and c in terms of A, B, p, q, and c. Since we can remove the specific and callable products with zero demand, in our analysis we will assume that the demand vector d(t ) > 0. Let (x, z ) and (θ, µ, η ) denote a pair of strictly complementary optimal solutions for the pair of primal-dual LPs (4)-(6). First consider the specific products, i.e. products labeled 1,..., n. Since µ j max{p j A j θ, 0} and d j (T ) > 0, it follows that µ j = max{p j A j θ, 0}. Thus, for all j = 1,..., n, p j > e j A θ implies that µ j > 0, and, therefore, by complementary slackness, we have that x j = d j (T ). From the dual constraints corresponding the callable products, i.e. products labeled n + 1,..., n + f, we obtain Thus, µ n+k > 0 implies x n+k = γ k. that µ n+k = max{q k A k θ + ηk, 0}. (7) Since µ n+k is an increasing function of η k and d n+k > 0, the third set of dual constraints imply η k = max { max {1 j n k } { (Ak B kj ) θ r kj }, 0 }, k = 1,..., f, (8) where B kj denotes the j-th column in the matrix B k. For k = 1,..., f, define the set F k as follows: F k = { j F k : η k = (A k B kj ) θ r kj }. (9) This analysis leads to the bid-price heuristic displayed in Figure 1. In the deterministic case, the bid prices given by A jθ would be constant over time. In the situation where demand is uncertain, these bid prices would be updated as bookings are received 10

11 (a) Fix θ R m +. For each k = 1,..., f, compute ηk, F k and µ n+k using, respectively, (8), (9) and (7). (b) If p j > A jθ, accept a request for the j-th specific product; otherwise reject. (c) If µ n+k > 0, accept a request for the k-th specific product; otherwise reject. If F k (i.e. ηk > 0), assign a accepted request to any alternative in this set, otherwise assign the request to the k-th specific product. Figure 1: Bid price heuristic for accepting callable products and the expectation of future demands changes. The bid price heuristic has been shown to be an effective approach to the network management problem faced by airlines managing specific products across a complex network. Talluri and van Ryzin [18] have shown that it is asymptotically optimal as the number of seats and time periods increases. We believe that a similar asymptotic optimality property is likely to hold for the bid-price heuristic for combined flexible and specific products. The final result in this section compares the optimal revenues generated by callable and flexible products. For details on flexible products [10] and managing flexible products on a network see [9]. Lemma 2 Suppose n specific products and f flexible products are sold on a network. Suppose the set of alternatives for the k-th flexible product is given by F k {k}, where F k denotes the set of alternatives available for the k-th callable product. Suppose the revenue q k from the k-th flexible product satisfies q k p k max {1 j nk }{r kj }, where r k denotes the reassignment cost vector for the k callable. Then the revenue from callable products will be at least as large as the revenue from flexible products. Proof The proof is straightforward and left to the reader. 3 A Customer Choice Model The model in the previous section assumes that the arrival process of the products is exogenously defined. In this section, we consider a model where the rates of the products are generated endogenously. 11

12 At any t [0, T ] the requests for the products arrive according to independent Poisson processes with a rate vector λ(s) R n+f + that depends on the set S of products being offered. As before, the state of the network is given by the pair of vectors (s, y) Z m+f +, where s Z m + denotes the residual capacity and y Z f + denote the vector of accepted requests for callable products. The firm s control in this model is the choice of the set of products S to offer at any given time t [0, T ]. Unlike in the previous section, the firm must accept all requests for the set of products S open at t [0, T ]. 3.1 Dynamic programming formulation Let V (s, y, t) denote the maximum achievable revenue starting from the state (s, y) at time t. Then V (s, y, t) is characterized by the Bellman recursion V (s, y, t) t = max S { n λ i (S) ( p i + V (s A i, y, t) V t (s, y, t) ) i=1 + f λ n+k (S) ( q k + V (s, y + 1 k, t) V (s, y, t) )}, (10) k=1 and the boundary conditions: for all (s, y) A and t [0, T ], V (s, y, t) = ; and for (s, y) A, V (s, y, T ) = max r z, subject to A f y + (B A f U)z s, y + Uz 0, y, z 0. The goal of the recursion is to compute V (c) = V (c, 0, 0). Note that the only control in (10) is the choice of the set of products S; once the set is chosen the firm must accept all arriving requests. For most practical networks, the complexity of computing V (c) is prohibitive, and one has to resort to heuristics. In the next section, we construct an asymptotically optimal policy by solving a deterministic control problem. As in Section 2, we establish that this policy is asymptotically optimal even in the case when the requests arrive according to a stationary point process. Note that in the latter case it is very hard to even formulate the optimal control problem. 12

13 3.2 Deterministic control problem with customer choice In this section we construct a deterministic approximation for the stochastic control problem. We show that this deterministic approximation is valid as long as the requests for products arrive according to a independent stationary point processes with intensity λ(s) R n+f + when the subset S is offered at time t [0, T ]. The deterministic approximation is obtained by assuming that the requests arrive according to a deterministic rate. Thus, the decision problem reduces to choosing a collection S l, l = 1,..., L, of subsets of products to offer and corresponding time intervals t(s l ), l = 1..., L, over which to offer these sets in order to maximize revenue, i.e. the problem faced by the firm reduces to the following LP The goal of the network manager is to choose a collection S l N, l = 1,..., L, of subsets of products to offer and corresponding time intervals t(s l ), l = 1..., L, over which to offer these sets in order to maximize revenue. Let λ (s) (S) = (λ 1 (S),..., λ n (S)) and λ (f) (S) = (λ n+1,..., λ n+f (S)). Then, the optimization problem faced by the network manager is given by: V D (c) = max ( S p λ (s) (S)t(S) + q λ (f) (S))t(S) r z ( subject to Aλ (s) (S) + A f λ (f) (S) ) t(s) + (B A f U)z c, S S λ(f) (S)t(S) + Uz 0, S t(s) T, Using Ã, B, c defined in (3), the above LP can be simplified as follows t(s) 0, S, z 0. V D (c) = max S p λ(s)t(s) r z subject to S Ãλ(S)t(S) + Bz c, S t(s) T, t(s) 0, S, z 0. (11) The decision variables in (11) are the times {t(s) : S {1,..., n + f}} and the composition variables z, i.e., the number of variables are 2 n + f k=1 n k 1. Since there are only m + f + 1 constraints in the problem, at most m + f + 1 of the exponentially many variables in (11) can take 13

14 a positive value. Moreover, suppose the k-th callable product is offered for any period but is never reassigned then the row corresponding to k in the constraint S γ(s)t(s) Uz = 0 is effectively redundant and can be dropped without affecting the solution. On the other hand, if part of the demand for the k-th callable product is reassigned then at least one component of z k is strictly positive. Thus, we have the following result. Lemma 3 There exists an optimal solution of the LP (11) with t(s) > 0 for at most m + 1 subsets S. Since the number of variables that will be positive at the LP optimal solution is relatively modest, namely m + 1, a column generation ought be an efficient algorithm for solving (11). However, as we shall see, the effectiveness of column generation depends upon the consumer choice model underlying the calculations of γ(s) and λ(s). 3.3 Column generation algorithm The dual of (11) is given by min u c + βt subject to u Ãλ(S) + β p λ(s), S N, u B r, u, β 0, (12) where u R m+f + and β is a scalar. This linear program has an exponentially many constraints one for every S {1,..., n + f}. Let G (0) be a collection of subsets of N. Consider the following restricted LP where we consider only the sets in G (0). max S G (0) p λ(s)t(s) r z subject to S G (0) Ãλ(S)t(S) + Bz c, S G (0) t(s) T, t(s) 0, S G (0), z 0. (13) Notice that we have kept all the z variables in the restricted LP. 14

15 1. Select an initial collection G (0) of subsets of N. Set σ Solve the restricted primal LP (13) corresponding to G (σ). Let (t (σ), z (σ) ) denote the optimal solution of the restricted primal and let (u (σ), β (σ) ) denote the corresponding optimal dual vector. 3. Using the explicit construction in Lemma 4 compute the minimum set S (σ) corresponding to (u (σ), β (σ) ). Let denote the length of set S (σ). l(s (σ) ) = (u(σ) ) Ãλ(S (σ) ) +β p λ(s (σ), ) 4. If l(s (σ) ) 1, the solution of the current restricted primal problem is optimal for the full primal LP (11). Stop. 5. Else, set σ σ + 1, G (σ) G (σ 1) S (σ 1). Return to step 2. Figure 2: Column generation algorithm Suppose the restricted LP (13) is solved to optimality. Let (t (0), z (0) ) and (u (0), β (0) ) denote the primal and dual optimal vectors respectively. Since all the z variables are present in the restricted primal, the dual vector (u (0), β (0) ) satisfies u B r. Thus, (u (0), β (0) ) will be feasible for (12) provided Suppose (u (0), γ (0) ) satisfies (14). min S N { (u (0) ) Ã + β (0) } p 1. (14) λ(s) Then, by strong duality, we have that the primal variables (t (0), z (0) ) are optimal for the full primal LP (11). On the other hand, if (u (0), β (0) ) does not satisfy the constraint, we augment the collection G (0) by adding the set that achieves the minimum in (14) and repeat the process. This is called a column generation method because in every iteration a new column is added to the primal LP. For column generation to be successful we must be able to efficiently solve (14) for all values of the dual variables (u, β). This is not possible for arbitrary assignments of the demand rates λ(s). However, in reality, we can restrict ourselves to consideration of those values of λ(s) that are consistent with rational underlying consumer choice models. Specifically, at any time, we can 15

16 conceive of an arrival stream of potential consumers confronted with the current set of specific and callable products and their corresponding prices. Each customer will either choose not to purchase or will purchase one of the available products, depending upon her preferences. Thus, we can think of λ(s) as being determined as the outcome of a consumer choice model. In particular, we consider consumer choice models defined as follows. Definition 1 The demand vectors λ(s) R n+f are defined as follows λ j (S) = g j 1+ P k S g k, j S, 0, otherwise, (15) where g, g R n+f +. Note that both independent demands and the Multinomial Logit (MNL) model as well as other attraction models are included in the class of customer choice model defined in Definition 1 1. Consider the following nested choice model. In the first stage, the customer decides the specific through a model like δ k (S) = a k /(a(s)+1), k S. Then, if specific k is chosen, the customer decides whether or not he wants to keep it specific (not grant the option to turn it into a flexible call specific). This may be done via θ k = m k /(m k + 1), resulting in λ k (S) = θ k δ(s), and γ k (S) = (1 θ k )δ(s). Clearly, this model belongs to the class defined in Definition 1. Lemma 4 Let u R m +, β 0, and suppose that the demand vector λ(s) is given by (15). For j = 1,..., n + f, define w j = p j g j, and x j = (g j u à j + β g j )/w j. Let {π j : j = 1,..., n} denote the values {x j : j = 1,..., n} arranged in increasing order. Let k denote the largest index k such that where S k = {j : x j π k }. Then min S N j S k w j (x j π k ) + β 0, { u ( Aλ(S) + A f γ(s) ) v } { γ(s) + β u ( Aλ(S k ) + A f γ(s k ) ) v γ(s k ) + β p λ(s) + q = γ(s) p λ(s k ) + q γ(s k ) }, i.e. S k is a minimal set. 1 Ben-Akiva and Lerman [3] show how the Multinomial Logit model implies the choice probabilities given in Definition 1. Talluri and van Ryzin [19] apply the Multinomial Logit Model to a single-leg case. Anderson, Palma and Thisse [1] provide an overview of attraction models including the Multinomial Logit. 16

17 Remark 1 The minimal set Sk is the network analog of the efficient sets introduced in [19]. Proof The proof of this result closely parallels the proof of Lemma 2 in [9]. The crucial step in the proof is to rewrite the optimization problem (14) as the feasiblility problem u Ãλ(S)β ζ ( p λ(s) ), and minimize over ζ. For fixed ζ, the above inequality can be rewritten as w j (x j ζ) + β 0, j S Thus, for fixed ζ, the minimal set S ζ = {j : x j ζ}. The result follows by observing that S ζ same for all ζ [π j, π j+1 ), j = 1,..., n 1. is the 3.4 Asymptotic optimality of the deterministic control In this section we compare the value V (c) achievable in the stochastic problem to the value V D (c) achievable in the deterministic control problem. Proposition 3 The deterministic value function V D (c) is an upper bound for the stochastic value function V (c). Proof The proof of this result is straightforward and is, therefore, left for the reader to reconstruct. Next, we prove a lower bound for the stochastic value function V (c). Let d i (S, t) denote the random Var(di (S,t)) number of requests for the i-th product. Let χ i (S, t) = E[d i (S,t)], i = 1,..., n + f, denote the coefficients of variation of the number of requests for the various products when the set S is offered for t units of time; and let χ(s, t) = max 1 i n+f {χ i (S, t)}. Suppose χ(s, t) is a monotonically decreasing function of t for all S. Lemma 5 Let ({t(sl )} l=1,...,l, z ) denote the optimal solution of the LP (11). Suppose χ(t ) = ( max 1 l L {χ(sl, t l )} < 1 n+f. Then V D (c) V (c) V D (c) ( (n + f)χ 2 (t ) ) ) 1 3. Proof Define a control policy ρ as follows: 17

18 (a) Offer the sets S1,..., S L in sequence. (b) Offer the set S l for a random time { τ l = min t(sl ), inf { t : d(sl, t) λ(s l )t l }}. (c) The policy ρ assigns a ( P L l=1 λ n+k(sl )t (Sl ) P ) 1 k z k L -fraction of the total number of requests for l=1 λ n+k(sl )t (Sl ) the k-th callable product to the k-th specific product and a ( zkl ) P L -fraction to the l=1 λ n+k(sl )t (Sl ) specific product l F k. The choice of the time τ l and the reassignment policy ensures that the policy ρ is feasible. Moreover, the expected revenue R ρ satisfies R ρ = p ( L l=1 ) E[d(S l, τ l )] f ( 1 k z k L )E[d l=1 λ n+k(sl )t (Sl ) n+k (S l, τ l )]. k=1 The result now follows by combining techniques from the proof of Lemma 1 in this paper and Lemma 3 in [9]. The following Corollary establishes an asymptotic optimality result. Corollary 2 Suppose χ(t ) = max 1 l L {χ(sl, t l )}. Fix ζ > 1 and consider the scaled stochastic problem in initial capacity ζc and time horizon ζt. Then, for all ζ > (n + f)χ 2 (t ), the stochastic ( value function V (ζc) ζv D (c) ( (n + f)χ 2 (ζt ) ) ) 1 3. When the arrival processes are Poisson, the maximum coefficient of variation satisfies χ 2 (t ) = 1/ min {i S l,l=0,...,l}{λ i (Sl )t l }. Thus, Corollary 2 implies that ( V (ζc) ζv D (c) The proof of this result is identical to that of Corollary 1. ( (n + f)χ 2 (t ) ζ ) 1 3 ) 4 Multinomial Logit Model with Correlated Products The choice model proposed in the previous section treats callable and specific versions of the same product as two independent products in much the same manner as it treats two distinct specific products. This is a potentially serious flaw since it is clear that there is interaction between the 18

19 callable and specific versions of a given product, e.g. it is plausible that the arrival rate to the specific product might be significantly different whether or not the callable version is offered. One possible fix to this problem is to offer the callable version whenever a specific product is offered. This is, however, very restrictive and may reduce the firms profits. In this section we develop a new demand model that is able to express this interaction without restricting the set of options available to the seller. Recall that the first n products are the specific products and the products labeled n+1,..., n+f are, respectively, the callable versions of the first f products. Suppose the arrival rate λ i (S) is still given by the expression λ i (S) = g i (S) 1 + i S g i(s). However, now the parameters (g i, g i ) are themselves function of the set S being offered. In particular, for all i = 1,..., f, (g i (S), g i (S)) = (gi 0, g0 i ), i S, n + i S, (gi 1, g1 i ), i S, n + i S, 0, otherwise, and (g n+i (S), g n+i (S)) = (gn+i 0, g0 n+i ), i S, n + i S, (gn+i 1, g1 n+i ), j S, n + i S, 0, otherwise, i.e. the product pair (i, n+i) are correlated in the sense that they affect each other s rate constants. For products i = f + 1,..., n, the parameters (g i (S), g i (S)) = (g i, g i ) independent of the set S. Since one enumerates all sets when computing the Bellman recursion, the new demand model can be accommodated without a significant increase in computational effort. However, the column generation step fails since the weights (g i (S), g i (S)) depend on the set S being offered. In this section we propose an alternative view of this choice model that allows one to solve the column generation step efficiently. Generalizing the scenario discussed above, we define a Multinomial Logit Model with Correlated Products (MNLC) as follows. (a) The product set N = d l=1 N l, where n l = N l, l = 1,..., d. 19

20 (b) The products in each of the sets N l are correlated in the sense that the weight g i of each product i N l depends on which other products from the set are being offered, i.e. g i is a function g i (χ l ) where χ l {0, 1} n l being offered from the set N l. denotes indicator vector for the set of products that are (c) Suppose a subset S N of products are offered by the firm. Let χ S = (χ S 1,..., χs d ) denote the indicator vector of the set S, where χ S l, l = 1,..., d, denotes the sub-vector corresponding to the set N l. Then the arrival rate for product i N l is given by λ i (S) = g i (χ S l ) 1 + (16) d j=1 k N j g k (χ S j ). The optimization problem that we have to solve in the column generation step is given by ( ) u i N λ i(s)ãi + β min S N i N p iλ i (S). (17) When the rates {λ i (S)} are given by the usual MNL model we are able to solve this problem because the objective is a ratio of linear functions of the indicator vector χ S of the set S [9]. This is, however, not the case when the rates are given by (16). Thus, the previous analysis does not extend to the MNLC model. In the rest of this section, we show that the MNLC model can be linearized by introducing fictitious products and extra linear constraints. Consider the following alternative choice model. (a) For each set of products N l, create 2 n l fictitious products. For each new product (l, j), j = 0,..., 2 n l 1 define the weight, capacity and revenue rate as follows. h lj = i N l g i (bin(j)), Â lj = P i N l g i (bin(j))ãi h lj, h lj = i N l g i (bin(j)), ˆp lj = P i N l g i (bin(j)) p i h lj, (18) where bin(j) {0, 1} n i denotes the binary representation of j {0,..., 2 n i 1}. Let Ñl = {(l, j) : j = 0,..., 2 n l 1}. Thus, the new set of products is given by Ñ = d l=1ñi. (b) Let S Ñ denote a subset of products that are currently offered by the firm. Let χ S denote the indicator vector of the set S. (Note that for notational convenience we have labeled each 20

21 component of the indicator vector by a pair of indices). Then the arrival rate for the individual products (l, j) are given by λ lj ( S) = h lj χ S (l,j) 1 + (i,k) h ik χ S (i,k). (19) Thus, in the expanded space of pseudo-products, the MNLC arrival rates are are given by the ratio of two affine functions of the indicator vector χ S. (c) The firm is only allowed to offer sets S Ñ that satisfy S Ñi = 1, for all i = 1,..., n. All we have done in the alternative model is co-alesce all the offered correlated products into one psuedo-product. Consequently, we have to insist that seller can only offer one pseudo-product from each set Ñi. Since the number of new products is exponential in the size of the set N i, this method is practical only when n i are sufficiently small. When applied to the case of callable products, each n i = 2; therefore, the number of products increase by a factor of 2 n i = 4. Note that we co-alesce the products they loose their individual existence, i.e. we are unable to recover the per-product arrival rate described in (16). However, the objective function in (17) can be simplified to an expression only involving the new rates as follows. ( ) u s N λ s(s)ãs + β s N p sλ s (S) = = = u ( P d P l=1 s N g l s(χ S l )Ãs) 1+ P d l=1 P d P l=1 + β Ps N g l s(χ S l ), s N l g s(χ S l ) ps 1+ P d P l=1 s N g i s(χ S l ) u ( P lj h ljχ S (l,j) Â lj ) 1+ P h + β lj lj χ S (l,j) P, lj h lj p lj χ S (l,j) 1+ P h lj lj χ S (l,j) ( ) u (l,j) S λ lj( S)Âlj + β (l,j) S ˆp ljλ lj ( S), (20) where S {(l, j) : 1 l d, 0 j 2 n i 1} with (l, j) S if, and only if, χ S l = bin(j). The above reformulation uses pseudo-products to linearize the expression. In order to use this reformulation one has to restrict the firm to offering only one product from each set of pseudo-products Ñi. Let (t (0), z (0) ) denote the optimal solution of the restricted primal problem (13) and let (u, β) R m+f+1 + denote the corresponding optimal dual solution. Then (t (0), z (0) ) is an optimal solution of 21

22 the full primal problem (11) if, and only if, u ( s N λ s(s)ãs ) + β s N p sλ s (S), S, ( ) u ij S λ ij( S)Âij + β ij S ˆp ijλ ij ( S), S : S Ñi = 1, i. ij ((u  ij ˆp ij )h ij + β h ij )χ S (i,j) + β 0, S : S Ñi = 1, i. d i=1 min {0 j 2 n i 1}{(u  ij ˆp ij )h ij + β h ij } + β 0. Suppose the final inequality is violated, i.e. (u, β) is not feasible for the full dual, or equivalently, (t (0), z (0) ) is not optimal for the full primal. For each i = 1,..., d, let s i = argmin {0 j 2 n i 1} {(u A ij + β p ij ) g ij }. Define the set S N by setting χ S i = bin(j i ), i = 1,..., d. Then, ( ) u λ s ( S)A s + β < p s λ s ( S), s N s N i.e. S yields a cut in the dual, or equivalently, a column in the primal. Note that unlike in Section 3, the new column S does not correspond to the minimum set, or equivalently, the corresponding cut in the dual in not the deepest cut. A column generation algorithm that does not use the minimum set is also guaranteed to converge; however, such an algorithm might be slow in practice. We detail an algorithm for computing the minimum set in Appendix A. We conclude this section with the following result relating callable and specific products. Lemma 6 Suppose the first f( n) products can be offered either as specific products or as pure callable products, i.e. the customer pays the full price up front and is compensated for reassignment. Suppose the MNL rates for the product i (1 i f) is given by χ S i specific rate callable rate (0, 0) 0 0 (0, 1) 0 (g 0 n+i, g0 n+i ) (1, 0) (g 0 i, g0 n+i ) 0 (1, 1) (g 1 i, g1 i ) (g1 n+i, g1 n+i ) Suppose gi 1 + g1 n+i > g0 i and g i 1 + g1 n+i < g0 i, i.e. combined arrival rate of when both the specific and callable versions of a product i are offered exceeds the arrival rate when only the specific version 22

23 is offered. Then one can restrict oneself to S such that χ S i sacrificing optimality in LP (11). (1, 0) for all i = 1,..., f, without Proof In order to prove this result we will establish that the firm can be restricted to χ S i {(0, 0), (0, 1), (1, 1)} at any column generation step. We linearize the firm s problem by introducing 4 new fictitious products for each of the first f products. The characteristics of these new fictitious products are as follows. (i, j) rate h ij rate h ij consumption Ãij revenue p ij (i, 0) (i, 1) g 0 n+i g 0 n+i (i, 2) g 0 i g 0 i à n+i à i p i p i (i, 3) g 1 i + g1 n+i g 1 i + g1 n+i gi 1 à gi 1 i + g1 i à +g1 n+i gi 1 n+i p i +g1 n+i We want to argue that for any column generation step and for any i = 1,..., f, one can restrict the range of j to j = 0, 1, 3, without any loss of optimality. Suppose this is not the case, i.e. there exist (u, β) such that argmin {j=0,...,3} {(u  ij ˆp ij )h ij + β h ij } = 2 and j = 2 is the unique optimum. Since g i0 = 0 it follows that (u  i2 ˆp i2 )h i2 + β h i2 = (u à i p i )g 0 i + β g 0 i < 0. (21) The term corresponding to j = 3 satisfies (u  i3 ˆp i3 )h i3 + β h i3 = ((u à i p i )g 1 i + β g 1 i ) + ((u à n+i p i )g 1 n+i + β g 1 n+i) (22) Recall that (A i à i =, 0) i = 1,..., n, (A i, e i ) i = n + 1,..., f, where e k denotes the k-th column of (f f)-dimensional Identity matrix. Since u 0, it follows that for all i = 1,..., f, we have u à i u à n+i. Thus, (22) implies that (u  i3 ˆp i3 )h i3 + β h i3 ((u à i p i )(g 1 i + g1 n+i ) + β( g1 i + g1 n+i ), < ((u à i p i )g 0 i + β g 0 i, = (u  i2 ˆp i2 )h i2 + β h i2, 23

24 where the last equality follows from (21). Thus, we have a contradiction. Consequently, j = 2 cannot be the unique minimum. In concluding this section we would like to comment that the MNLC demand model has many potential applications beyond that of modeling correlated callable products. 5 Products involving buyer choice Callable and flexible products [10, 9, 11] the seller has all the flexibility and the buyer implicitly agrees to accept all seller decisions. In this section we propose optional products that allow buyer choice. In Section 6 we show that the seller is able to significantly increase revenue by selling optional products at a modest premium and then hedging the sales by selling flexible and callable products. The firm offers w optional products. Let p o R w + denote the corresponding revenue vector. Buyers of the j-th optional product purchase one unit of the specific product ι j and an option to switch from ι j to an alternative l Ω j {1,..., n}\{ι j } after paying a penalty r o jl. Let r 0 R P j Ω j denote the vector r o = (r o jl ) l Ω j,j=1,...,w. Note that the seller must accommodate all buyer switches. The scenario described here occurs by default. A business traveler might have to reschedule a flight because of altered plans. Presently, she is at the mercy of the airlines to get reassigned to another flight. An optional product will give such travelers a guaranteed seat while allowing the airline to extract a premium for providing this flexibility. The crucial step in analyzing the network with optional products is to characterize the set of feasible sales. Suppose firm sells ω Z w + units of optional products. Since customers purchasing the j-th optional product have the option to select any specific product from the set Ω j, their decisions can be characterized a vector v j Z Ω j + that satisfies l v jl ω j with the interpretation that ω j l v jl purchasers consume the product ι j and v jl customers switch to the alternative l Ω j. Let { P V = v : v = (v 1,..., v w) j Z Ω } j +, 1 Ω j v j ω j, j = 1,..., w. Then the network capacity consumed by the optional products belongs to the set {A o ω + (B o A o U o )v : v V} 24

25 where A o R m w denotes the sub-matrix of A corresponding to the specific product associated with each optional product, B o R m P j Ω j is the matrix obtained by collecting together columns corresponding to the alternatives Ω j, and U o is the matrix that sums the appropriate components of the vector v. Note that the exact capacity consumed by the optional products is unknown to the seller. The seller attempts to hedge the sale of optional products by selling callable products. Suppose y Z f + units of callable products are sold. Then the feasible reassignments of the callable products is given by vectors z that belong to the set H = { z : z = (z 1,..., z f ), z k Z n k +, 1 n k z k y k, k = 1,..., f }, and the corresponding capacity consumption is given by A f y + (B f A f U f )z. We assume that the seller chooses the reassignment of the callable products after observing the requests of the purchasers of optional products. Thus, we have that the sale of x R n + units of specific products, y R f + units of callable products, and ω Z w + units of optional products is feasible if, and only if, v V z H : Ax + A o ω + A f y + (B o A o U o )v + (B f A f U f )z c. (23) Note the asymmetry in the role of the buyer and the seller the seller plays second in this game and is able to choose the reassignment vector z after observing the vector v. Constraints of the form (23) are called adjustably robust constraints [4]. It is, in general, NP-hard to characterize the set of points that are feasible for adjustable robust constraint (23). In our problem the polytope V is the Cartesian product of w simplices and, therefore, its extreme points are explicitly known. We can leverage this fact to show that the set (23) is, in fact, a polytope. Lemma 7 Let {ˆv j : j = 1,..., J} denote the set of extreme points of the polytope V. Then (x, y, ω) satisfy (23) if, and only if, there exist vectors {ẑ j H : j = 1,..., J} such that Ax + A o ω + A f y + (B o A o U o )ˆv j + (B f A f U f )ẑ j c, j = 1,..., J. (24) Remark 2 Since the polytope V is the Cartesian product of simplices, it follows that the number of extreme points of V are given by w j=1 (1 + Ω j ). Proof Since ˆv j V for all j = 1,..., J, it is clear that (23) implies (24). establishing the reverse implication. Thus, our task reduces to 25

26 Suppose (24) holds, i.e. there exists vectors {ẑ j : j = 1,..., J} that hedge the extreme points. Pick any v V. Then it follows that v = J i=1 η j ˆv j for some η satisfying η j 0, J j=1 η j = 1. Let z = J j=1 η jẑ j. Then Ax + A o ω + A f y + (B o A o U o )v + (B f A f U f )z J ( ) = η j Ax + Ao ω + A f y + (B o A o U o )ˆv j + (B f A f U f )ẑ j, j=1 ( J ) η j c = c. j=1 Thus, we have established that (24) implies (23). Lemma 7 implies that the set of feasible sales (x, y, ω) are descried by linear inequalities. The next step is to characterize the analog of (4) for when the demand for specific, optional and callable products is given by an independent demand model. Let (d x, d y, d ω ) denote the expected demand for specific, callable and optional products, respectively. Given this information the seller wants to select booking limits (x, y, ω) for the three categories of products. Unlike in the scenario modeled by the LP (4) where the seller made all the choice, here the buyer also has flexibility; consequently there is uncertainty in the seller s revenue. In this paper, we take the robust optimization approach and estimate revenue by the minimum value that is guaranteed to the seller over all possible buyer decisions, i.e. the revenue is the optimal value of the LP max γ subject to p x + (p o ) ω + q y + (r o ) ˆv j r z j γ, j = 1,..., J, Ax + A o ω + A f y + (B o A o U o )ˆv j + (B f A f U f ) j c, j = 1,..., J, x, y, ω 0, z j H, j = 1,..., J. (25) where {ˆv j : j = 1,..., J} denotes the extreme points of the polytope V, and the decision vectors are (x, y, ω) R n+2f, γ R, and z j H R P k n k, j = 1,..., J. The first constraint in (25) ensures that the revenue is at least γ across all possible buyer decisions. The LP (25) is a clearly a pessimistic estimate of the revenue; one cannot, however, improve over this estimate without making some stochastic assumptions about the buyers decisions. 26

27 Departure Arrival Origin Destination Time Time Capacity Fare 1. LGA ORD $ LGA ORD LGA ORD ORD SFO ORD SFO ORD SFO JFK SFO JFK SFO LGA STL LGA STL JFK STL STL SFO STL SFO Table 1: Flight Subnetwork 6 Numerical experiments In this section, we report the results of the numerical experiments that we conducted to explore the benefits from offering callable and optional products. 6.1 Callables and correlated callables In this section, we numerically investigate the performance of callable products as a function of the up-front discount, i.e. p q, and the re-allocation cost r. We also study the effect of the correlation between specific and callable versions of the same product. For our numerical experiments we considered a subset of American Airlines flights from JFK, LGA, STL, ORD to SFO. The details of the flights in our model are shown in Table 1. There were n = 26 specific products consisting of all the direct flights shown in Table 1 and all connections from NYC to SFO satisfying time constraints. We assumed that the first f = 20 of these products were also offered as callable products. The set of alternatives F k for each callable product was set 27