A Lagrangian relaxation approach for network inventory control of stochastic revenue management with perishable commodities

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1 Journal of the Operational Research Society (2006), Operational Research Society Ltd. All rights reserved /06 $ A Lagrangian relaxation approach for network inventory control of stochastic revenue management with perishable commodities HJiang University of Cambridge, Cambridge, UK Airline seat inventory control is the allocation of seats in the same cabin to different fare classes such that the total revenue is maximized. Seat allocation can be modelled as dynamic stochastic programs, which are computationally intractable in network settings. Deterministic and probabilistic mathematical programming models are therefore used to approximate dynamic stochastic programs. The probabilistic model, which is the focus of this paper, has a nonlinear objective function, which makes the solution of large-scale practical instances with off-the-shelf solvers prohibitively time consuming. In this paper, we propose a Lagrangian relaxation (LR) method for solving the probabilistic model by exploring the fact that LR problems are decomposable. We show that the solutions of the LR problems admit a simple analytical expression which can be resolved directly. Both the booking limit policy and the bid-price policy can be implemented using this method. Numerical simulations demonstrate the effectiveness of the proposed method. Journal of the Operational Research Society advance online publication, 13 December 2006 doi: /palgrave.jors Keywords: allocation; transport; revenue management; seat inventory control; mathematical programming; Lagrangian relaxation Introduction Seat inventory control is an important research problem in revenue management (Belobaba, 1989; Williamson, 1992; Brumelle and McGill, 1993; Bertsimas and Popescu, 2003; Chen and Homem-de-mello, 2004; Talluri and van Ryzin, 2004). Its purpose is to decide whether to accept or reject a booking request during the booking horizon. On the one hand, it is costly for the company to accept too many lower-value bookings if no seats are available to higher-value customers who make bookings closer to the time when products perish. On the other hand, revenue is also lost if the company rejects too many lower-value bookings and leaves products unsold when they perish. Dynamic (and stochastic) programming (DP) models can be used for optimal seat inventory control (Talluri and van Ryzin, 2004). Unfortunately, these DP models are computationally intractable even for small sized networks. In the academic literature and in practice, DP models are therefore approximated by various mathematical programming models. Two well-known approximations are the deterministic linear program (DLP) and the probabilistic nonlinear program (PNLP) (Glover et al, 1982; Williamson, 1992; Talluri and Correspondence: H Jiang, Judge Business School, University of Cambridge, Trumpington Street, Cambridge CB2 1AG, UK. h.jiang@jbs.cam.ac.uk van Ryzin, 2004). Based on these approximate models, several inventory control (or booking) policies such as booking limits, bid-price controls and virtual nesting control can be implemented (Talluri and van Ryzin, 1999a, 2004; Williamson, 1992). Several numerical studies have been carried out to determine the effectiveness and efficiency of the DLP and the PNLP in conjunction with different booking policies (Williamson, 1992; Talluri and van Ryzin, 1999a,b; de Boer et al, 2002; Chen and Homem-de-mello, 2004; Higle and Sen, 2004). There is no doubt that the DLP is much easier to solve than the PNLP. An interesting observation is that the nested booking limit policy based on the DLP frequently produces higher revenues than the same policy based on the PNLP (de Boer et al, 2002; Williamson, 1992). This seems to contradict the intuition that on average, results from the DLP should be no better than those from the PNLP since the former is a simplified version of the latter. This phenomenon is explained in de Boer et al (2002) as a result of the nested booking process, which is not taken into account in either the DLP or the PNLP. Indeed, this interpretation is reinforced by the numerical results presented in Ciancimino et al (1999), where the PNLP does perform better than the DLP when the booking process is not nested. This shows that it is advantageous to use the PNLP rather than the DLP in some applications such as railway revenue management.

2 2 Journal of the Operational Research Society As has been mentioned, it is generally more difficult to solve the PNLP than the DLP because the former involves a nonlinear objective function. Sometimes it may not be computationally acceptable in terms of CPU time to solve largescale PNLP examples in practice. To remedy this problem, Talluri and van Ryzin (1999b) propose a randomized linear programming approach which approximates Lagrangian multipliers of the PNLP by taking the average of the Lagrangian multipliers obtained from solving a number of sampling DLPs of the PNLP. Ciancimino et al (1999) describe a nonlinear algorithm for the solution of the PNLP which is computationally more efficient than generic nonlinear programming algorithms. The numerical results in Ciancimino et al (1999) show that higher revenues can be generated from the PNLP than the DLP when a non-nested booking limit policy is used. It is thus desirable to develop further some efficient algorithms for solving the PNLP by exploring its special structure. Lagrangian relaxation (LR) tends to be most useful when a problem can be characterized as easy, except for a few complicating constraints (Magee and Glover, 1996; Lemarechal, 2001). The optimal solution of the original optimization problem is recovered under certain conditions by solving the socalled Lagrangian dual (LD). Typically, the LD is piecewise smooth and can be solved using subgradient algorithms or other nonsmooth optimization methods. In this paper, we shall propose a LR method as an alternative for solving the PNLP. Our method will provide both partitioned booking limits and Lagrangian multipliers for the PNLP which allow us to implement both the booking limit policy and the bid-price policy for accepting or rejecting booking requests, whereas the method proposed in Talluri and van Ryzin (1999b) can only approximate Lagrangian multipliers. Compared with Ciancimino et al (1999), our method is particularly suitable for seat inventory control with large-sized networks because the resulting LR problem can be decomposed into many one-dimensional optimization problems with simple bounds. It turns out that the solutions of the LR problems admit a simple analytic expression which can be resolved directly. This simple but powerful result is the main contribution of this paper. The paper is organized as follows. In the next section, we shall introduce the DLP and the PNLP models for the seat inventory allocation problem, and then discuss some basic properties of the PNLP. In the subsequent section, we define the LR problem of the PNLP and its LD. We also present analytical solutions of the LR problem and the equivalence between the PNLP and the LD problem. A standard subgradient optimization method for solving the LD is explained in Section Subgradient Optimization. Numerical results are reported in the penultimate section. We make concluding remarks in the final section. Seat inventory control We consider seat inventory control over a network with m types of resources which are used to form n products. For airlines, resources are flight seats on all m flight legs and products are origin destination tickets of various fare classes. Let i and j be indices of resources and products, respectively. Let A =[a ij ] be the incidence matrix, where a ij = 1ifand only if product j uses resource i. Assume C i is the remaining capacity of resource i, r j the unit revenue of product j,andd j the stochastic demand for product j over the remaining booking horizon. Let x j denote the decision variable, representing the amount of resources allocated to product j. The objective of seat inventory control is to maximize the expected total revenue by optimally allocating resources to products. This results in the following PNLP max f PNLP := j Ax C x j, j r j E[min{x j, d j }] where E[] denotes the expectation operator and 0. = 0 is often assumed in the literature. However, it is quite useful to allow positive values for, in particular when we attempt to re-optimize the same PNLP model during the booking period (Ciancimino et al, 1999). Note that a further integrality condition stating that x j is an integer could be added to the above formulation, but this addition makes the PNLP a mixed integer nonlinear program which is computationally very hard to solve. All approximation models presented in Chapter 3 of Talluri and van Ryzin (2004) use continuous values for capacity allocation. The above model is an approximation of the optimal model which can be formulated as a dynamic stochastic program (Talluri and van Ryzin, 2004), which is computationally intractable, due to the curse of dimensionality associated with such models. To our knowledge, unlike dynamic programs for seat inventory control in single-resource settings, no dynamic stochastic program arising from seat inventory control in network settings has been solved optimally. It is worth noting that in the PNLP, we assume that the stochastic demand d j for product j is independent of that for other products. In reality, demands for different products are correlated. Some customers may buy up, or buy down or switch to different itineraries depending on product availability. Some recent research on buy-up models and choice modelling can be found from Talluri and van Ryzin (2004). We next give a simple example of the PNLP as well as other models to be introduced in the rest of this paper. Figure 1 is a small hub-and-spoke network consisting of four nodes A, B, C and D and three legs AB, BC and BD. Assume there are only five itineraries on one traffic direction: AB, BC, BD, ABC and ABD, and for each itinerary there are exactly two classes: high fare and low fare. In total, there are 10 products over the network. The capacities of all legs and the fares and average demands of all products are shown in Tables 1 and 2, respectively.

3 H Jiang Lagrangian relaxation approach for network inventory control 3 A B C Table 2 Product fares and average demands High Low Fare Average demand Fare Average demand AB BC BD ABC ABD Figure 1 A simple airline hub-and-spoke network. Table 1 Leg capacities Leg AB BC BD Capacity The PNLP for this example is max 240 min(x 1AB, d 1AB )df 1AB min(x 1BC, d 1BC )df 1BC min(x 1BD, d 1BD )df 1BD min(x 1ABC, d 1ABC )df 1ABC min(x 1ABD, d 1ABD )df 1ABD min(x 2AB, d 2AB )df 2AB + 80 min(x 2BC, d 2BC )df 2BC + 80 min(x 2BD, d 2BD )df 2BD min(x 2ABC, d 2ABC )df 2ABC min(x 2ABD, d 2ABD )df 2ABD x 1AB + x 1ABC + x 1ABD + x 2AB + x 2ABC + x 2ABD 200 x 1BC + x 1ABC + x 2BC + x 2ABC 120 x 1BD + x 1ABD + x 2BD + x 2ABD 100 x 1AB, x 1BC, x 1BD, x 1ABC, x 1ABD, x 2AB, x 2BC, x 2BD, x 2ABC, x 2ABD 0 where F j represents the cumulative probability distribution of the demand for product j. In the PNLP, we add an upper bound to x j : u j x j for j, where u j = max{c i : a ij = 1}. This constraint is obviously implied by Ax C and is hence redundant. However, this cut D will be useful for the LR problem to be introduced in the next section. See the remark after Proposition 1. After adding this cut, the PNLP becomes: max r j E[min{x j, d j }] j Ax C x j u j, j A further simplification of the PNLP is the following DLP, which replaces the stochastic random demand variable d j by its expectation d j for al: max f DLP := j r j x j Ax C x j d j, j In particular, the concrete DLP for our simple example is max 240x 1AB + 200x 1BC + 200x 1BD + 400x 1ABC + 400x 1ABD + 100x 2AB + 80x 2BC + 80x 2BD + 150x 2ABC + 150x 2ABD x 1AB +x 1ABC +x 1ABD +x 2AB +x 2ABC +x 2ABD 200 x 1BC + x 1ABC + x 2BC + x 2ABC 120 x 1BD + x 1ABD + x 2BD + x 2ABD 100 x 1AB 30, x 1BC 20, x 1BD 10, x 1ABC 5, x 1ABD 5 x 2AB 100, x 2BC 60, x 2BD 50, x 2ABC 35, x 2ABD 20, x 1AB, x 1BC, x 1BD, x 1ABC, x 1ABD, x 2AB, x 2BC, x 2BD, x 2ABC, x 2ABD 0 Both booking limits x j and the dual prices or Lagrangian multipliers can be obtained from solving either the PNLP or the DLP. In the literature as well as in practice (Talluri and van Ryzin, 2004), these solutions are used to form several booking policies to be discussed below: partitioned booking limits and bid price control among others. Partitioned booking limits in the network setting are an extension of partitioned booking limits in the single-resource setting (Talluri and van Ryzin, 2004). In the network setting, a fixed amount of capacity of each resource is allocated to every product offered. The demand for each product has access only to its allocated capacity and no other product may use this

4 4 Journal of the Operational Research Society capacity. In both the DLP and the PNLP, these booking limits are set to be the respective optimal solutions x. In the network setting, a bid-price control policy sets a threshold price or bid price for each resource in the network. Roughly this bid price is an estimate of the marginal cost of consuming the next incremental unit of the resource s capacity. When a booking request for a product arrives, the revenue from the request is compared to the sum of the bid prices of all the resources required by the product. If the revenue exceeds the sum of the bid prices, the request is accepted, provided all the resources associated with the requested product are still available; if not, the request is rejected. In the DLP, the optimal dual variables associated with the constraints Ax C are used as bid prices. If the optimal solution is not degenerate (ie, active constraints of Ax C are linearly independent at the optimal solution), then the firstorder derivative of the value function (The value function is the objective of the optimization problem as a function of some parameters; in our case, the parameter is the remaining capacity C) of the deterministic model exists and is given by the unique vector of the optimal dual prices. If the optimal solution is degenerate, then there are multiple optimal dual price vectors, each of which is only a subgradient of the value function. See Talluri and van Ryzin (2004) for further discussion on this point. Similarly, the bid prices for the PNLP are the Lagrangian multipliers at its optimal solution. Another popular booking policy, virtual nesting control, is based on the fact that the nested booking limit policy is optimal in a single-resource setting. In a single-resource setting, all products are ordered according to revenue values. In the nested booking policy, any higher-value product can always access seats allocated to lower-value products, but not vice versa. An extension of this policy to a network setting is not obvious. Firstly, the ordering of fare classes is no longer straightforward. Secondly, the multiple capacities involved make it difficult to specify protection levels or booking limits for product combinations which use different resources in the network. Nevertheless, the virtual nesting control can be formed using the so-called indexing technique and virtual classes. This booking policy is discussed in Talluri and van Ryzin (2004) and will not be used in this paper. We conclude this section by presenting some analytical properties of the PNLP, which are well known in the literature, see Ciancimino et al (1999), de Boer et al (2002). With these properties, we are able to employ LR and to ensure equivalence between the LD and the PNLP. Lemma 1 (1) The objective function of the PNLP is concave. (2) The objective function of the PNLP is continuously differentiable if the demand random variable d j is continuous for al. (3) Suppose D is a continuous random variable. Let G(x) = E[min(x, D)]. Then the first-order derivative of G at x is G (x) = 1 F(x) where F is the cumulative distribution function of the random variable D. We remark that in the above lemma we assume that the demand is a continuous random variable. In real-world applications, passenger demands should be modelled by discrete random variables. In the revenue management literature and in practice, approximations using continuous random variables are a common strategy to simplify both analysis and computational processes; see page 96 of Talluri and van Ryzin (2004). If the probability distribution of the demand is not known, there are two ongoing research directions: a combination of sampling and simulation and robust optimization techniques. When the probability distribution is discrete, the LR method proposed in this paper is not valid any more, and the PNLP is equivalent to a large-sized linear program, for which the column generation approach is an alternative. Lagrangian relaxation Unlike the DLP, the solution of the PNLP requires nonlinear algorithms. Existing general nonlinear programming codes could not exploit the specific structure of the PNLP and are likely to be computationally inefficient. Talluri and van Ryzin (1999b) propose a randomized linear programming approach for approximating Lagrangian multipliers or bid prices. Ciancimino et al (1999) describe a penalty Lagrangian algorithm for the optimal solution, and thus the booking limit of the PNLP. In this section, we propose a LR approach for finding both booking limits and bid prices of the PNLP. Lagrangian relaxation Let μ i 0 be the Lagrangian multipliers for the constraints of the PNLP. We obtain a Lagrangian relaxation (LR) of the PNLP by dualizing the constraint Ax C: (LR) max E[r j min(x j, d j )] μ i a ij x j C i j i j x j u j, j which decomposes into a sum of one-dimensional optimization problems. For each product j, LR j is defined by ( ) (LR j ) max E[r j min(x j, d j )] μ i a ij x j x j u j. The objective function for the LR j is equivalent to r j E[min(x j, d j )] B j x j where B j = i μ ia ij and B j is non-negative as μ i 0 for all i. From Lemma 1, the objective function of (LR) as well as (LR j ) is concave, and continuously differentiable if the demand random variable d j is continuous for al. The i

5 H Jiang Lagrangian relaxation approach for network inventory control 5 following lemma shows that the optimal solution of (LR) can be found analytically. See Appendix for a proof. Proposition 1 Suppose the demand random variable d j is continuous for al. Then there exists an optimal solution to (LR). Let x be such an optimal solution. We have (1) If r j B j, then x j = min{max{,v j }, u j } with v j satisfying r j (1 F j (v j )) = B j, where F j is the cumulative distribution function of the demand d j for product j. (2) If r j < B j, then x j =. Remark The above proposition shows that it is necessary to include the redundant upper bound u j for x j in the PNLP. Without this upper bound, the existence of a feasible solution to (LR) is not guaranteed when B j = 0 for some j. Evenif a feasible solution exists for (LR), it does not seem desirable to have a feasible solution so that x j > u j for some j, which clearly violates the constraint Ax C of the PNLP. For our simple example, the LR for product 1AB is the following simple one-dimensional optimization problem: max 240 min(x 1AB, d 1AB ) df 1AB μ AB x 1AB l 1AB x 1AB u 1AB. Lagrangian dual The optimal objective function value of (LR) with any multipliers μ 0 provides an upper bound to the optimal objective function value of the PNLP. But which multipliers provide the best upper bound and how can they be found? This question is answered by solving the LD. Let L(x, μ) be the objective function of (LR) and q(μ) the optimal objective function value of (LR). The Lagrangian dual (LD) is the following optimization problem: min q(μ) μ 0. Remarks Since the objective function of the PNLP is concave and its constraints are linear, it can be shown (Magee and Glover, 1996; Lemarechal, 2001) that (1) The LD is a convex optimization problem with simple bounds; (2) For any μ 0, the objective function value q(μ) of the LR provides an upper bound to the optimal objective function value of the LD; and (3) solving the LD is equivalent to solving the PNLP. Subgradient optimization We remarked at the end of the last section that solving the PNLP is equivalent to solving the LD, which is a nonsmooth convex optimization problem with simple bounds. In this section, we present a standard subgradient method for solving the LD (Magee and Glover, 1996). Subgradient method (Input) A lower bound q for the LD; see Remark (I). A stopping criterion ε > 0. An initial value μ 0 0; see Remark (II). (Initialization) θ 0 = 2 (Subgradient iterations) for n := 0, 1,..., do Solve (LR) when μ = μ n.letx μ n be the optimal solution of (LR). Calculate a subgradient of q(μ) at μ n : γ n ; see Remark (III). Choose a step length: t n > 0; see Remark (IV). Update μ : μ n+1 = max{0, μ n + t n γ n }. Termination check: If μ n+1 μ n < ε, then stop. See remarks (V) and (VI). Update θ: If the minimum objective function value for (LR) in the last K iterations is not smaller than the minimum objective function value for (LR) prior to the last K iterations, then θ n+1 = θ n /2; otherwise, θ n+1 = θ n. Update q : q = q(μ n ) if x μ n is feasible to the PNLP; see Remark (I). Update iterations: n := n + 1. A number of points need to be clarified about the above subgradient method. We start with a result related to lower and upper bounds for the PNLP, which is required for determining the initial lower bound q in the subgradient method. See Appendix for a short proof. Lemma 2 Let x be an optimal solution to the DLP. Then f DLP (x ) and f PNLP (x ) are upper and lower bounds, respectively, for the LD (or the PNLP equivalently). Remarks (I) Let x be an optimal solution of the DLP. By Lemma 2, f PNLP (x ) can be used as an initial value for q in the subgradient method. If x μ n is feasible to the PNLP, then f PNLP (x μ n ) provides a lower bound for the PNLP. If q < f PNLP (x μ n ), then we can replace q by f PNLP (x μ n ). (II) The DLP often gives a good upper bound to the PNLP. Hence, its optimal Lagrangian multiplier should give a good estimate for the optimal Lagrangian multiplier, that is, a good choice for μ 0. (III) For any μ 0, C Ax μ is a subgradient of q(μ). (IV) A popular choice of the step length t n is t A n = θ n(q(μ n ) q )/ Ax μ n C 2. Other choices are also available in the literature (Magee and Glover, 1996). It is known that the subgradient

6 6 Journal of the Operational Research Society algorithm converges when the stepsize satisfies the following conditions: t B n = σ n Ax μ n C 2, σ n 0, σ n =+, see Lemarechal (2001). However, the method using this step length converges very slowly and it can often only offer an approximate solution to the LD. (V) If x μ n is feasible to the PNLP and if the complementarity condition μn T (Axμ n C) = 0 is satisfied, then x μ n is an optimal solution to the LD (and the PNLP). This can be proved as follows. On the one hand, f PNLP (x μ n ) is an upper bound to the PNLP by Lemma 2 because f PNLP (x μ n )= L(x μ n, μn ). On the other hand, f PNLP (x μ n ) is also a lower bound to the PNLP since x μ n is feasible to the PNLP. (VI) Note that the statement If x μ n is feasible to the PNLP and if the complementarity condition μn T (Axμ n C) = 0 is satisfied is equivalent to the statement q(μ n ) q, where q is the current best lower bound of the LD. Therefore, q(μ n ) q can be used as a termination criterion of the algorithm. Sometimes the subgradient method converges slowly and can be terminated using the criterion μ n+1 μ n < ε or when the number of iterations exceeds a pre-specified number. Numerical results In this section, we demonstrate effectiveness of the proposed method through simulation. The subgradient method presented in the previous section is implemented in Matlab Version 6.5 (MATLAB ) on a Windows XP machine. The DLP model is also implemented in Matlab for comparison and the DLP is solved using linprog. Asthe booking limits and Lagrangian multipliers can be obtained from solving both the PNLP and the DLP, both the booking limits policy and the bid-price policy are used in the booking process. Hence, we have implemented four booking schemes: the booking limit policy based on the DLP, the booking limit policy based on the PNLP, the bid-price policy based on the DLP, and the bid-price policy based on the PNLP. In order to take into account new information, including updated demand forecasts and remaining capacities, and to improve revenue by adjusting seat allocation adequately, both the PNLP and the DLP are resolved over the booking horizon. In our experiments, the booking horizon is divided into several booking periods and both the DLP and the PNLP models are resolved at the beginning of each booking period. We follow the procedure described in Talluri and van Ryzin (1999b) for our simulation experiments. For each test example with a particular booking scheme, we simulate the booking process 1000 times. In each simulation, booking requests are randomly generated in two steps. In Step 1, the number of requests for each product is randomly generated according n to a truncated normal distribution (left truncated at zero) with the given expected demand. In Step 2, booking arrival times for each product are randomly generated according to the prespecified booking curve. When a booking request arrives, it is either accepted or rejected based on either the booking limits in the booking limit policy or the bid price in the bid-price policy. The total revenue is calculated for all accepted bookings for each booking scheme in each simulation run. Finally, the average total revenue from 1000 simulation runs as well as its 95% confidence interval is calculated for each booking scheme. For a fair comparison, the same booking arrivals are used for all four booking schemes in each simulation run. Our first two airline test problems T1 and T2 are from Higle and Sen (2004). In T1, the airline network consists of seven cities and seven legs with eight itineraries. For each itinerary, there are two fare classes which result in 16 products in total. Bookings start from 180 days prior to the flight departure time. Demand forecasts for all products are updated on day 180, 120, 60, 30, 14 prior to the departure time. Those five days are used to divide the booking horizon into five booking periods. A fixed percentage of demand is assumed in each booking period for each product. More high fare customers are assumed to arrive in later booking periods while low fare customers are assumed to arrive in earlier booking periods. These percentages form a booking curve for each product. The expected demand for each product and the capacity on each leg and other detailed information can be found in Higle and Sen (2004). Test problem T2 is similar to T1 but with a different network. In particular, T2 is a hub spoke network with one hub, 20 spoke cities and 20 legs, and it has two fares and 50 itineraries (and hence 100 products). Two more test problems T3 and T4 are randomly generated in a way similar to test examples S1 and S2 used in Ciancimino et al (1999) where chain networks in railway systems are considered. In T3, the number of legs and the number of products are 5 and 15, respectively. In T4, those numbers are 10 and 55, respectively. The following characteristics are the same for both T3 and T4. Bookings are open 60 days prior to the train departure time. The 60-day booking horizon is divided into 15 periods which are separated by day 46, 31, 24, 17, 10, 9, 8, 7, 6, 5, 4, 3, 2, and 1 prior to the train departure time. For each product, the same percentage (that is 1/15) of demand is assumed in each booking period. The average revenues, as well as their 95% confidence intervals, generated by four different booking methods for four test problems are shown in Table 3. It can be seen that with a non-nested booking limit policy, the PNLP outperforms the DLP for all four test problems. This confirms the numerical results of Ciancimino et al (1999). However, using a bid-price booking policy, the DLP dominates the PNLP for all four test problems. In our admittedly limited simulations, on average, the booking limit policy based on the PNLP and the bid-price policy based on the DLP appeared to outperform the other two methods. Between those two better methods, neither

7 H Jiang Lagrangian relaxation approach for network inventory control 7 Table 3 Average revenues and 95% confidence intervals for various booking policies DLP PNLP Upper bound Problem Booking limit Bid price Booking limit Bid price T1 292, , , , ,250.0 (290,958.3, 293,530.2) (270,382.1, 273,804.2) (292,606.7, 295,383.5) (269,751.9, 272,928.1) T2 1,381, ,387, ,417, ,323, ,573,375.0 (1,378,196.1, 1,383,868.7) (1,383,964.1, 1,391,277.2) (1,413,960.4, 1,420,937.4) (1,318,083.9, 1,328,794.7) T3 307, , , , , (306,454.9, 309,130.8) (309,727.6, 312,349.1) (307,156.1, 309,717.3) (307,974.6, 310,504.8) T4 407, , , , ,061.0 (407,346.0, 407,947.5) (412,335.1, 412,841.5) (410,138.4, 410,668.0) (411,643.0, 412,097.9) Table 4 Average CPU time in seconds for solving the PNLP Problem Booking limits Bid prices T T T T completely dominates the other. It would be nice to have the optimal expected value available for each of the test problems. However, this is not possible because with modern computer power there is no way to solve the stochastic dynamic program of seat inventory control over even a very small network. For example, the capacity management problem has to be solved based on some approximation approaches for a single-leg network with multiple but substitutable flights; see Zhang and Cooper (2005). One way to roughly measure the quality of the solutions of the DLP and the PNLP is to provide an upper bound for the optimal total expected revenue for each test problem. It is known (Talluri and van Ryzin, 2004) that the optimal objective function value of the DLP is an upper bound to the optimal total expected revenue for a network seat inventory control problem. However, it is not known how good this upper bound is, though asymptotically this upper bound is tight when both product demand and resource capacity are scaled up at the same speed (Talluri and van Ryzin, 2004). Nevertheless, we provide this upper bound for each test problem in Table 3. The CPU times in seconds for solving the LD (or the PNLP equivalently) are shown in Table 4. It is not surprising to note that the CPU times for the methods based on the booking limit policy and the bid-price policy are comparable on average. It is noticeable that the CPU time increases as the network size increases. We expect that the CPU time for solving the PNLP can be reduced significantly if the algorithm is implemented in other programming languages such as C++. We remark that due to its slow convergence, the subgradient method terminates before an optimal solution is obtained for the LD and hence sometimes we only obtain approximations of the optimal solution and the optimal Lagrangian dual of the LD. We have also solved the PNLP using an available algorithm called fmincon in Matlab. Our experience is that on average, it takes considerably more CPU time than our LR method. More seriously, the code did not terminate after two hours of computer elapsed time for one simulation run of either T1 or T2 even if values of tolerance for several stopping criteria in fmincon were set to be much larger than their default values. Conclusions It is known that the PNLP is an approximation of the intractable dynamic program for seat inventory control. In this paper we have proposed a LR approach for solving the PNLP by exploiting its separable structure and its concavity. The solution of the LR subproblem admits a simple analytical expression. Numerical results have confirmed that the PNLP is useful in some revenue management applications where non-nested booking policies are employed in practice, and demonstrated that the subgradient method is a simple way of solving medium to large-scale problem instances and may well outperform conventional off-the-shelf solvers. It is well known that considerable progress has been made in solving the LD in the recent past; see Guignard (2003). A possible research topic is how we can extend the techniques developed in Barahona and Anbil (2000), and du Merle et al (1998) for solving the PNLP, which is stochastic and nonlinear. Acknowledgement I am grateful to Danny Ralph and Stefan Scholtes for their valuable discussions and comments, and to referees for their constructive comments, which have helped to improve the presentation of this paper. I am also thankful to Giovanna Miglionico for pointing out data inconsistency of some test problems in an early version of the paper. Appendix Proof of Proposition 1 Let F j and f j be the cumulative probability distribution and the probability density function of the demand for product j, respectively. Let H j (x j ) be the objective function of (LR j ).

8 8 Journal of the Operational Research Society Then H j ( ) = r j E[min(, d j )] B j = r j [ + u j u j Df j (D) dd + f j (D) dd ] f j (D) dd B j H j (u j ) = r j E[min(u j, d j )] B j u j = r j [ + u j u j Df j (D) dd + Df j (D) dd ] u j f j (D) dd B j u j By Lemma 1, H j is continuously differentiable and the firstorder derivative of H j at x j is H j (x j) = r j (1 F j (x j )) B j Obviously, H j (x j) = 0 has a solution if and only if r j B j. (1) If r j B j 0, then the following equation has a solution B j r j = 1 F j (v j ) (A1) as B j 0. Let v j be such a solution. It follows from the concavity of H j that the only candidates for the optimal solution of (LR j ) are, u j and v j. If v j u j,thenv j is a unique stationary point and hence a unique optimal solution of (LR j ). If >v j,then1 F j (v j ) 1 F j ( ) and [ u j H j (u j ) H j ( ) = r j (D ) f j (D) dd + u j ] (u j ) f j (D) dd B j (u j ) r j (u j ) f j (D) dd B j (u j ) r j (u j )(1 F j ( )) B j (u j ) r j (u j )(1 F j (v j )) B j (u j ) = 0 where the last equality follows from (A1). In this case, is an optimal solution. Similarly, it is easy to prove that u j is an optimal solution if u j <v j. (2) If r j B j < 0, then [ u j H j (u j ) H j ( ) = r j (D ) f j (D) dd + u j ] (u j ) f j (D) dd B j (u j ) r j (u j ) f j (D) dd B j (u j ) B j (u j ) B j (u j ) B j (u j ) = 0 f j (D) dd B j (u j ) which implies that H j (u j ) H j ( ). Since H j is continuously differentiable and H j (x) has no stationary point, the optimal solution must be attained at either and u j.theresult follows. Proof of Lemma 2 If x is an optimal solution to the DLP, then x is feasible to the PNLP. Hence f PNLP (x ) provides a lower bound to the LD. The fact that f DLP (x ) provides an upper bound for the PNLP is well known in the literature (Talluri and van Ryzin, 2004) and follows Jensen s inequality. References Barahona F and Anbil R (2000). The volume algorithm: Producing primal solutions with a subgradient method. Math Program 87: Belobaba P (1989). Application of a probabilistic decision model to airline seat inventory control. Opns Res 37: Bertsimas D and Popescu I (2003). Revenue management in a dynamic network environment. Transport Sci 37: Brumelle S and McGill J (1993). Airline seat allocation with multiple nested fare classes. Opns Res 41: Chen L and Homem-de-mello T (2004). Multi-stage stochastic programming models for airline revenue management. Technical Report, Northwestern University. Ciancimino A, Inzerillo G, Lucidi S and Palagi L (1999). A mathematical programming approach for the solution of the railway yield management problem. Transport Sci 33: de Boer S, Freling R and Piersma N (2002). Mathematical programming for network revenue management revisited. Eur J Opl Res 37: du Merle O, Goffin J and Vial J (1998). On improvements to the analystic center cutting plane method. Comput Optim Appl 11: Glover F, Glover R, Lorenzo J and McMillan C (1982). The passenger mix problem in the scheduled airlines. Interfaces 12: Guignard M (2003). Lagrangean relaxation. TOP 11: Higle J and Sen S (2004). A stochastic programming model for network resource utilization in presence of multi-class demand uncertainty. In: Ziemba W and Wallace S (eds). Applications of Stochastic Programming, SIAM Series on Optimization, chapter 16.

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