An Effective Heuristic for Multistage Stochastic Linear Programming

Transcription

1 An Effective Heuristic for Multistage Stochastic Linear Programming C. Beltran-Royo L. F. Escudero J. F. Monge R. E. Rodriguez-Ravines May 28, 2013 Abstract The Multistage Stochastic Linear Programming (SLP) problem may become numerically intractable for huge instances, in which case one can solve an approximation as for example the well known Expected Value (EV) problem. In this paper we introduce a new approximation to the SLP problem that we call the Event Linear Programming (ELP) problem. Based in this problem we derive the ELP heuristic that produces a lower and an upper bound for the SLP problem. We have assessed the validity of the ELP heuristic by solving large scale instances of the network revenue management problem for the fight tickets selling (up to 98 millions of variables and almost 90 millions of constraints). The average CPLEX time for the EV, ELP and SLP approaches has been 89, 142 and 1816 seconds, respectively, for the instances of the testbed we have experimented with (CPLEX has failed to solve 19% of the SLP instances within a time limit of two hours). The average worst case optimality gap for the EV and ELP approaches, has been 3.63% and 0.45%, respectively. Thus, regarding the capacity to deal with uncertainty and the numerical tractability, the ELP approach lies between the SLP and the EV approaches. Keywords: Multistage stochastic programming, constraint aggregation, conditional expectation, scenario tree, revenue management. 1 Introduction The Multistage Stochastic Linear Programming (MSLP) problem corresponds to a linear programming problem with uncertainty in some of its parameters and with several decision stages. For notational simplicity, considering that all the problems here are multistage, we will drop the M and will use the shorter form SLP instead. The main feature of the SLP problem is that the uncertain parameters are revealed gradually over time and our decisions should be adapted to this process. The relevance, applications, properties, approaches and solution methods of this problem can be found in [7, 26, 46], among others. To address the SLP problem, one can use different approaches such as chance constraint optimization [42], robust optimization [2, 3] and scenario based optimization [7, 13], among others. In this paper we will focus in the last approach. Problem P SLP with a continuous stochastic process is, in general, numerically intractable. To overcome this difficulty one can approximate the original stochastic process by discretizing it. Of course, Statistics and Operations Research, Rey Juan Carlos University, Madrid, Spain. Statistics and Operations Research, Rey Juan Carlos University, Madrid, Spain. Universidad Miguel Hernandez de Elche, Spain. Bayes Forecast, Madrid, Spain. 1

2 the quality of the approximation will depend on the quality of the discretization [36]. The discretization process consists in approximating the original stochastic process by a stochastic process with a finite support which can be represented by a scenario tree. Thus, the first difficulty in scenario based optimization corresponds to built a representative and tractable scenario tree. See [8, 13, 18, 20, 25], among others. Once a representative scenario tree has been built one can write the so called deterministic equivalent model which corresponds to a large scale linear programming (LP) problem. The second difficulty corresponds to solve this LP problem. State of the art optimization software as for example CPLEX can be used to successfully solve SLP instances of moderate size. However, for many SLP instances one needs to use alternative methods which can be classified into exact and approximate ones. Exact methods can deal with millions of scenarios and are based on Lagrangian relaxation [17, 43], Benders decomposition [19, 23] and interior point methods [4, 9], among others. The performance of these optimization methods can be enhanced by using large computing systems (parallel computing, grid computing, etc.) as for example in [29, 35]. However, if the number of scenarios becomes too large, exact methods are impractical. In this case, either one solves the SLP problem approximately or one solves an approximation to the SLP problem. This is the case of schemes such as scenario aggregation [27], scenario sampling [48], stochastic dynamic programming [10, 37, 39], scenario refinement [8], and constraint aggregation [11], approximate dynamic programming [40], among others. However, even an approximated solution of the SLP problem by the Sample Average Approximation method requires an exponential number of sampled scenarios in order to attain a reasonable accuracy [45]. In this context it is useful to have some cost bound in order to assess the quality of the approximated solutions. Bounds based on Jensen or on Edmundson-Madansky inequalities can be found in [7]. In [49] Jensen s bound is improved by relaxing certain constraints and associating dual multipliers with them. One can also infer statistical bounds as in [44]. Another type of bounds in stochastic programming is obtained by aggregation of constraints, variables or stages: [28] constructs two discrete, stage-aggregated stochastic programs which provide upper and lower bounds on the optimal cost of SLP and are numerically tractable. The other way to obtain numerically tractable approximations to the SLP problem is by aggregating constraints and/or decisions. Such approximations are shown to provide bounds if the randomness appears exclusively either in the objective or in the right-hand-side [6, 50]. One of the most popular approximations to the SLP problem is the Expected Value (EV) problem. For the SLP problem with randomness appearing exclusively in the right-hand-side (rhs), it can be seen that the EV problem can be derived by applying the expectation operator to the constraints of the SLP problem. In this way all the SLP m-dimensional constraints are aggregated into one m-dimensional constraint. The scenario tree associated to the SLP problem, is reduced into a degenerate scenario tree (one scenario) associated to the EV problem. In order to perform a less radical aggregation, as an alternative to the expectation operator, we propose to apply the conditional expectation operator to the constraints of the SLP problem. We call the resulting model, the Event Linear Programming (ELP) problem. As we will see, in this new approach, the multistage scenario tree associated to the SLP problem, will be approximated by what we call the event spike associated to the ELP problem. As far as we now such approach has never been proposed in literature and it could be useful in the cases where the SLP problem were numerically intratable. Roughly speaking, as we have said, the ELP problem will be derived by applying the conditional expectation operator to the SLP constraints. In this paper we will concentrate on the SLP problem with randomness appearing exclusively in rhs of the constraints. In this case it can be seen that, both the EV problem and the ELP problem are relaxations of the SLP problem. We will try to answer the following questions: Is the ELP relaxation tighter than the EV relaxation? Is the ELP relaxation tractable? Is it possible to derive good SLP solutions by using the ELP relaxation? Which is the 2

3 performance of the ELP approach in the case of large scale instances? Thus, the objectives of this paper are to introduce the ELP problem, to study some of its theoretical properties, to compare it to the EV problem, to analyze the computational effort to solve large scale ELP instances and to to consider a scheme for deriving a (hopefully good) feasible solution for the SLP problem. With these objectives in mind, in Section 2, we describe the well known SLP problem. In Section 3, we introduce the ELP problem and study some of its theoretical properties. In Section 4 we describe the scenario tree and the event spike structures in the ELP context. In Section 5 we will see a scheme for obtaining feasible SLP solutions after solving the EV and ELP problems. In section 6 we present the computational results of comparing the EV, ELP and SLP solutions in a large testbed of instances of the network revenue management problem to study the effectiveness of the ELP approach. 2 The multistage LP problem with a stochastic right-hand-side Let us consider the following multistage deterministic LP problem that we name P DLP : min c x t x t (1) t T s. t. A 1 x 1 = b 1 (2) B tτ x τ + A t x t = b t t T + (3) x 0, (4) where we use the index set T = {1,..., T }, the reduced index set T + = T \ {1} and the scalar product c t x t of the vectors c t and x t, where c t is the vector of the objective function coefficients, A 1 and b 1 are the constraint matrix and the right-hand-side (rhs) related to stage t = 1. For all t T +, B tτ is the constraint matrix of the variable vector x τ related to stage τ < t, A t is the constraint matrix of the variable vector x t and b t is the rhs corresponding to stage t. In real-life instances, any of the parameters of P DLP may be stochastic. In order to introduce our new approach, we consider a simpler stochastic version of P DLP, where b t is the only random vector, which we denote by ξ t. Therefore {ξ t } t T is a multivariate stochastic process, that is, a sequence of random vectors ξ t = (ξ 1 t,..., ξ p t ). The sequential structure of the problem implies that the decision at stage t must be contingent to the random history ξ [t] := (ξ 1,..., ξ t ). This implies that decision x t is a function of ξ [t] and we will write x t (ξ [t] ). We name P SLP this stochastic rhs version of P DLP, which can be written as: min x z SLP (x) = t T E[c t x t(ξ [t] )] (5) s. t. A 1 x 1 = ξ 1 (6) B tτ x τ (ξ [τ] ) + A t x t (ξ [t] ) = ξ t t T + (7) x 0, (8) where ξ 1 is a deterministic vector. For the remaining of the paper we make the following assumptions: Assumption 1 The multivariate stochastic process {ξ t } t T has the following features: 3

4 1. Its first component, i.e., vector ξ 1, is deterministic. 2. It has a finite support. 3. It is stagewise independent, that is, the random vectors ξ t and ξ t are independent, for any t, t T. 4. It does not depend on the decision sequence {x t } t T. 3 Two approximations for the SLP problem 3.1 The expected value problem If problem P SLP becomes too difficult, one can use the popular multistage expected value (EV) problem, which lower bounds the SLP problem by using the expected values of the random process instead of the random process itself. In our context, the EV problem is as follows: min x z EV ( x) = t T c t x t (9) s. t. A 1 x 1 = ξ 1 (10) B tτ x τ + A t x t = ξ t t T + (11) x 0, (12) where ξ t = E[ξ t ]. See in [31] an approach for analyzing the quality of the EV solution. 3.2 The event linear programming problem If problem P SLP becomes too difficult, as a tighter alternative to the EV problem, we introduce the multistage event linear programming (ELP) problem: min ˆx, x z ELP (ˆx, x) = t T E[c t ˆx t(ξ t )] (13) s. t. A 1ˆx 1 = ξ 1 (14) B tτ x τ + A tˆx t (ξ t ) = ξ t t T + (15) x t = E[ˆx t (ξ t )] t T (16) ˆx 0, x 0. (17) The main difference with problem SLP corresponds to the dynamic constraint system (7) for t T +. In the ELP problem, this set of constraints is approximated at each stage t by considering not the past decisions x 1,..., x t 1 but its expectation x 1,..., x t 1 as it can be seen in (15). Furthermore, at each stage t, the SLP problem considers x t (ξ [t] ), that is, one decision at each so named scenario group, see e.g. [7] for the related definition in the multistage scenario tree. In contrast, at each stage t, the ELP problem only considers one decision per realization of ξ t, that is, ˆx t (ξ t ), as it can be seen in (15). Notice that, the SLP problem takes into account the random histories ξ [t], whereas the ELP problem 4

5 only takes into account the random vectors ξ t. Going further in the simplification, the EV problem just takes into account the expectations ξ t. In this way, the ELP and EV problems are approximations to the SLP problem that, obviously, reduce both the problem size and the stochastic accuracy. Thus, regarding tractability and level of stochastic information the ELP problem lies between the SLP and the EV ones. 3.3 Lower bounds By Assumption 1 the stochastic process {ξ t } t T is stagewise independent. It is well-known that for the stochastic problem with the uncertainty only in the rhs, the expected value problem gives a lower bound for the stochastic optimal cost [6]. In the context of this paper it means: z EV z SLP, (18) where zp stands for the optimal cost of a given problem P. In this section we will prove that the event problem gives a stronger bound, that is, it will be proved that z EV z ELP z SLP. (19) To prove this we need the following result on conditional expectation known as the law of total expectation [34]. Theorem 1 Let ξ 1,..., ξ n be a set of random vectors and r(ξ 1,..., ξ n ) = (r 1 (ξ 1,..., ξ n ),..., r m (ξ 1,..., ξ n )) (20) be any vectorial function of the random vectors. Then, for any j {1,..., n} we have that E[ E[ r(ξ 1,..., ξ n ) ξ j ] ] = E[ r(ξ 1,..., ξ n ) ]. (21) In Theorem 2 it will be proved that zelp z SLP by showing that the ELP problem is a relaxation of the SLP one. It is worth to mention that, the proof of Theorem 2 will also show the connection between SLP and ELP: roughly speaking the ELP problem can be derived by the constraint aggregation induced by the conditional expectation operator applied to the SLP constraints. Theorem 2 Let us consider problem P SLP with rhs defined by the multivariate stochastic process {ξ t } t T. If the stochastic process is stagewise independent then the ELP problem is a relaxation of the SLP one. Therefore z ELP z SLP. (22) Proof: Let us prove that the ELP problem can be derived by the constraint aggregation induced by the conditional expectation operator applied to the SLP constraints. The proof consists of five steps: First, constraints (7) can be aggregated by using the conditional expectation operator E[ ξ t ] : E[ B tτ x τ (ξ [τ] )) + A t x t (ξ [t] ) ξ t ] = E[ξ t ξ t ] t T + (23) B tτ E[ x τ (ξ [τ] ) ξ t ] + A t E[ x t (ξ [t] ) ξ t ] = ξ t (24) t 1 B tτ E[ x τ (ξ [τ] )] + A t E[ x t (ξ [t] ) ξ t ] = ξ t, (25) 5

6 where, to deduce the last equation we have used that by hypothesis the stochastic process {ξ t } t T is stagewise independent. Now, considering that x t (ξ [t] ) is a vectorial function of the random vectors ξ 1,..., ξ t, by Theorem 1 we can write E[ x t (ξ [t] )] = E[ E[ x t (ξ [t] ) ξ t ] ] t T. (26) By using this equality, equation (25) can be rewritten as B tτ E[ E[ x τ (ξ [τ] ) ξ τ ] ] + A t E[ x t (ξ [t] ) ξ t ] = ξ t t T +. (27) Second, constraint (6) is related to the first stage and then, being a deterministic one, it is equivalent to A 1 E[ x 1 (ξ [1] ) ξ 1 ] = ξ 1. (28) Third, constraint (8) can also be aggregated by using the conditional expectation operator: E[ x t (ξ [t] ) ξ t ] 0 (29) Fourth, by Theorem 1, the SLP objective function can be rewritten as z SLP (x) = t T E[c t x t(ξ [t] )] = t T c t E[x t(ξ [t] )] (30) = t T c t E[ E[ x t(ξ [t] ) ξ t ]] = t T E[ c t E[ x t(ξ [t] ) ξ t ]]. (31) Fifth, after applying the conditional expectation operator, it results the following aggregated problem min E[c x t E[ x t(ξ [t] ) ξ t ]] (32) t T s. t. A 1 E[ x 1 (ξ [1] ) ξ 1 ] = ξ 1 (33) B tτ E[ E[ x τ (ξ [τ] ) ξ τ ] ] + A t E[ x t (ξ [t] ) ξ t ] = ξ t t T + (34) E[ x t (ξ [t] ) ξ t ] 0 t T +. (35) Note that the constraint aggregation has induced an aggregation of variables. Let us call ˆx t (ξ t ) the aggregation of variables E[ x t (ξ [t] ) ξ t ], and so we can rewrite the previous problem as min ˆx E[ c t ˆx t(ξ t )] (36) t T s. t. A 1ˆx 1 (ξ 1 ) = ξ 1 (37) B tτ E[ ˆx τ (ξ τ ) ] + A tˆx t (ξ t ) = ξ t t T + (38) ˆx t (ξ t ) 0 t T +. (39) 6

7 or equivalently min ˆx, x E[ c t ˆx t(ξ t )] (40) t T s. t. A 1ˆx 1 = ξ 1 (41) B tτ x τ + A tˆx t (ξ t ) = ξ t t T + (42) x t = E[ ˆx t (ξ t ) ] t T (43) ˆx 0 x 0, (44) which is the ELP problem. In Theorem 3 it will be proved that zev z ELP by showing that the EV problem is a relaxation of the ELP one. The proof of Theorem 3 will also show the connection between ELP and EV: roughly speaking the EV problem can be derived by the constraint aggregation induced by the expectation operator applied to the ELP constraints. Theorem 3 The EV problem is a relaxation of the ELP problem. Therefore z EV z ELP. (45) Proof: Let us prove that the EV problem can be derived by the constraint aggregation induced by the expectation operator applied to the ELP constraints. The proof also consists of five steps: First, constraints (15) can be aggregated by using the expectation operator E[ ] for all t T + : where ξ t := E[ξ t ]. E[ B tτ x τ + A tˆx t (ξ t ) ] = E[ξ t ] t T + (46) t 1 B tτ x τ + A t E[ˆx t (ξ t ) ] = ξ t (47) Second, constraint (14) is related to the first stage and then, being a deterministic one, it is equivalent to A 1 E[ ˆx 1 (ξ 1 )] = ξ 1. (48) Third, constraint (16) can also be aggregated by using the expectation operator: E[ ˆx t (ξ t )] 0. (49) Fourth, the ELP objective function can be rewritten as z ELP (x) = t T = t T E[c t ˆx t(ξ t )] (50) c t E[ˆx t(ξ t )] (51) 7

8 Fifth, after applying the expectation operator, it results the following aggregated problem c t E[ ˆx t(ξ t )] (52) min ˆx, x t T s. t. A 1 E[ ˆx 1 (ξ 1 )] = ξ 1 (53) B tτ x τ + A t E[ ˆx t (ξ t )] = ξ t t T + (54) x t = E[ ˆx t (ξ t ) ] t T (55) E[ ˆx t (ξ t )] 0 t T +. (56) Note that the constraint aggregation has induced an aggregation of variables. Let us call x t the aggregation of variables E[ ˆx t (ξ t )], and so we can rewrite the previous problem as min c x t x t (57) which is the EV problem. t T s. t. A 1 x 1 = ξ 1 (58) B tτ x τ + A t x t = ξ t t T + (59) x 0. (60) Corollary 1 Let us consider problem P SLP with rhs defined by the multivariate stochastic process {ξ t } t T. If the stochastic process is stagewise independent then z EV z ELP z SLP. (61) 4 The multistage scenario tree and the event spike As its name implies, scenario based optimization relies on the scenario tree structure. Analogously, the event linear programming (ELP) problem is based in what we call the event spike structure, see Section 4.2. The objective of this section is to compare these two structures and to formulate the SLP and ELP problems accordingly. We find it convenient to list the following indexes, parameters and index sets that we will use throughout the paper: Indexes, parameters and index sets: t Index for stages, t T = {1,..., T } k Index for groups of nodes, k K t = {1,..., K t } t T l Index for nodes within the same group, l L t = {1,..., L t } t T T + Stands for {2,..., T } T Stands for {1,..., T 1} T K t L t Stands for T K t L t. t T 4.1 The multistage scenario tree By Assumption 1 the multivariate stochastic process {ξ t } t T has a finite support. This implies that each random vector ξ t has a finite support of cardinality, say L t, such that S ξt = { ξ t1,..., ξ tlt } t T. (62) 8

9 Let the probability of each realization ξ t be denoted π tl = P (ξ t = ξ ) tl tl T L t. (63) Given that {ξ t } t T has a finite support it can be represented by a multistage scenario tree, where each node at stage t T biunivoquely corresponds to a realization ξ [t] of the random history ξ [t] = (ξ 1,..., ξ t ). If the stochastic process is stagewise independent, then the multistage scenario tree is symmetric, i.e., at a given stage t, all the nodes have the same number of successors (L t ). In this case, the total number of scenarios is given by the product L 2 L 3 L T. Furthermore, if the number of successors is constant for all the nodes, then we have L T 1 scenarios. In practical applications, in order to have a tractable and representative multistage scenario tree, one usually considers a reduced set of scenarios, by using schemes such as the scenario reduction techniques [18], among others. Figure 1: Node labels. The scenario tree nodes are indexed by a triple tkl label (stage t, group k and leaf l). To compare the multistage scenario tree structure with the event spike structure, we find it convenient to label the nodes of the multistage scenario tree by using three indexes: tkl, where t is the stage, k is the group of nodes (nodes with the same ancestor) and l is the node in group k. For example, in Fig. 1 we observe that at stage 2 we have only one group of nodes (node 111 is the common ancestor) and at stage 3 we have two groups of nodes. At each stage t T the groups of nodes are labeled by k K t. At each group k of stage t the nodes are labeled by l L t. This notation is not the standard one but we use it since it will be useful to compare the ELP and SLP approaches. Note that K t L t is the number of scenario groups at stage t in the standard notation for symmetric multistage trees. On the other hand, the vectors and the parameters associated to each node are indexed accordingly. Associated to each node tkl of the multistage scenario tree there are the following elements: x tkl, decision vector. ξ tl, realization of the random vector ξ t. Note that in the multistage scenario tree of our consideration, ξ tkl = ξ tk l for any k, k K t and therefore we can drop the index k and write ξ tl. ξ [t], realization of the random history ξ [t], where ξ [t] = ( ξ 1,l(1),..., ξ t 1,l(t 1), ξ tl ) (64) 9

10 is the realization of ξ [t] that biunivoquely determines node tkl. p tkl, probability of reaching node tkl, that is, p tkl = P (ξ [t] = ξ ) [t] = π 1,l(1) π t 1,l(t 1) π tl. (65) For each stage t T, we have that kl K tl t p tkl = 1. The multistage scenario tree model Based on the multistage scenario tree, the deterministic equivalent model of the SLP problem can be written as follows: min z SLP (x) = p tkl c x t x tkl (66) tkl T K tl t s. t. A 1 x 111 = ξ 11 (67) B tτ x a t τ (tkl) + A t x tkl = ξ tl tkl T + K t L t (68) x 0, (69) where a(tkl) is the ancestor of node tkl, a 2 (tkl) = a(a(tkl)) is the second ancestor of node tkl and so on. Note that in (68), the rhs term ξ tl does not depend on the group of nodes k, i.e., we do not write ξ tkl. See in [7, 15, 26] some approaches for computing or estimating the value of the SLP solution based on decomposition schemes. 4.2 The event spike Figure 2: Event spike. The scenario tree approach considers the posible successions of events. In contrast, the event spike approach only considers the possible events. In order to introduce the event spike concept let us the following definition: 10

11 Definition 1 Event: Given a multivariate stochastic process {ξ t } t T, let us define event as each realization ξ tl of the random vector ξ t, for any t T. In the ELP approach, the multistage scenario tree of the SLP approach, is replaced by what we call the event spike, where we have the stem and one group of leaves per each future stage. On the one hand, the group of leaves at stage t corresponds to the events ξ tl. On the other hand, the stem corresponds to the sequence of expected decisions x 1,..., x T 1. (see Fig. 2). Therefore, each event spike has two types of nodes: nodes along the stem labeled by the index t T and leaf nodes labeled by the pair tl T L t. Note that the number of nodes along the stem is T 1 and the number of leaf nodes is L L T. If the number of leaf nodes is constant for all the stages, then the number of leaf nodes is (T 1)L, in contrast with the exponential number of nodes in the multistage scenario tree of our consideration. At each leaf node tl there are three elements: ˆx tl, decision vector. ξ tl, event tl. π tl, probability of event tl. For each stage t T, we have that l L t π tl = 1. Finally, the expected decision at stage t can be expressed x t = l L t π tl ˆx tl t T. (70) The event spike model Based on the event spike, the deterministic equivalent model of the ELP problem can be written as follows: min z ELP (ˆx, x) = π tl c t ˆx tl (71) ˆx, x tl T L t s. t. B tτ x τ + A tˆx tl = ξ tl, tl T + L t (72) x t = l L t π tl ˆx tl t T (73) ˆx 0, x 0. (74) 5 Algorithm for obtaining the expected result of using the expected value for multistage stochastic optimization The methodology for obtaining the Expected result of using the Expected Value (EEV) is very well established in the two-stage environment, see a good exposition in e.g. [7], but it is a difficult one for multistage problems, see [14, 30]. The main difficulty lies in the same concept of multistage EEV. Alternatively, we use the methodology introduced in [1] for obtaining EEV in a rolling horizon type of calculation. Basically, it is as follows: 1. Retain the EV solution for the first stage. 11

12 2. Once the first stage EV solution is fixed, stage t = 2 is considered, so that K t independent scenario subtrees remain. 3. The EV solutions are independently obtained for these K t scenario subtrees. 4. The first stage solution of the EV problem for each scenario subtree is retained and, then, fixed. 5. The procedure continues for the other stages in the scenario tree until stage T 1 is reached, where the two-stage (T 1 and T ) stochastic problem for each related node is solved and, then, their solution is obtained jointly with the solution of each (singleton) node in stage T. So, at the end of the process there is a solution for each scenario. The EEV is obtained by weighting the solution values for the scenarios as they have been calculated by the procedure. On the other hand, the Expected result of using the Event Linear Programming problem (EELP) could be obtained analogously, that is, in the previous methodology we should replace EV by ELP. Note: By construction, the EEV and EELP are upper bounds of the optimal SLP cost. 6 Computational experience 6.1 The network revenue management problem For computationally assessing the validity of the heuristic approach ELP that combines the ELP lower bound and the EELP upper bound, we have chosen as our pilot case the so named network revenue management problem for the flight tickets selling, taken from [32, 33]. The aim of revenue management in general consists of maximizing the revenue of selling limited quantities of a set of resources by means of demand management decisions. A resource in revenue management is usually a perishable product/service, such as seats on a single flight leg or hotel rooms for a given date. It is common in revenue management that multiple resources are sold in bundles. For instance, connecting flight legs are sold on a single ticket and hotel customers may stay multiple nights. In this case, the lack of availability of any resource will prevent sales of the bundle, which creates interdependence among these resources. Consequently, the demand management decisions of these resources must be coordinated. The following notation is used to describe the network revenue management problem: Sets: H, set of resources (with size H). I, set of bundles (with size I). J, set of fare classes (with size J). I h, set of bundles using resource h, for h H. IJ, stands for I J. Deterministic parameters: f ij, fare of bundle-class ij, for ij IJ. 12

13 C h, capacity on resource h, for h H. ub(p ij ij tkl ), upper bound on the protection level Ptkl (defined below) for ij IJ, tkl T K t L t. Uncertain parameters: ξ ij tl, demand for bundle-class ij in period t at node tkl, for tkl T + K t L t. Variables: b ij tkl, number of accepted bookings for bundle-class ij in period t at node tkl, for ij IJ, tkl T + K t L t. B ij tkl, cumulative number of accepted bookings of bundle-class ij along the path from the root to node tkl, for ij IJ, tkl T + K t L t. P ij a(tkl), protection level of bundle-class ij set at node a(tkl) for cumulative accepted bookings along the path from the root to node tkl, for ij IJ, tkl T + K t L t. (Notice in (77) that all the nodes with the same immediate ancestor share the same protection level.) SLP model The multistage stochastic lineal programming (SLP) model for the network revenue management problem with protection levels is as follows: max b,b,p p tkl tkl T + K tl t ij IJ f ij b ij tkl (75) s. t. B ij tkl = Bij a(tkl) + bij tkl tkl T + K t L t, ij IJ (76) B ij tkl P ij a(tkl) tkl T + K t L t, ij IJ (77) P ij T 1,kl Ch kl K T 1 L T 1, h H (78) ij I h J 0 b ij ij tkl ξ tl tkl T + K t L t, ij IJ (79) 0 P ij ij tkl ub(ptkl tkl T K t L t, ij IJ. (80) Constraints (76) define the booking balance equations and constraints. Notice that B ij 111 corresponds to the initial number of accepted booking, which is a parameter of the model for all ij IJ. Constraints (77) ensure that the cumulative number of accepted bookings along the path from the root to node tlk cannot exceed the protection level set at the ancestor node a(tlk). The protection levels across bundles and class fares are then bounded by the capacity on the resources in constraints (78). Constraints (79) reflect that the number of accepted bookings should be not greater than the demand. Constraints (80) bound the protection levels. Notice that the non-anticipativity constraints are satisfied by construction; see e.g. in [7, 26] good expositions of this important concept on stochastic optimization. We will refer to this model as SLP. 13

14 EV model The Expected Value (EV) model is as follows: max b, B, P t T + s. t. Bij t B ij t = ij I h J ij 0 b t ij 0 P t ij IJ f ij bij t (81) ij ij B t 1 + b t t T +, ij IJ (82) P ij t 1 t T +, ij IJ (83) P ij T 1 Ch h H (84) ξ ij t t T +, ij IJ (85) ij ub( P t ) t T, ij IJ. (86) ELP model The multistage Event Linear Programming (ELP) model is as follows: max ˆb, ˆB, B, P s. t. Bij π tl tl T + L t ij IJ tl = B ij ij tl P ij I h J B ij t 0 0 ij f ˆbij tl (87) ij ij B t 1 + ˆb tl tl T + L t, ij IJ (88) t 1 tl T + L t, ij IJ (89) P ij t 1 Ch h H (90) = l L t π tl Bij tl t T T +, ij IJ (91) ij ˆb tl ij P t ij ξ tl tl T + L t, ij IJ (92) ij ub( P t ) t T, ij IJ. (93) 6.2 Instance description In this numerical study, we consider a small network, a medium, and a large one. Next, we will introduce first the dimensions and then the structure of our test networks. Finally, we will discuss the fares and the demand model. Recall that, in this context, the dimensions of a network are given by its number of resources H, bundles I and fare classes J. So, they are as follows: SMALL network: H = 10, I = 18, J = 2; MEDIUM network: H = 100, I = 200, J = 2; and LARGE network: H = 500, I = 1000, J = 2. In terms of network structure, we follow the standard practice in the literature to use randomly generated networks with a hub-and-spoke structure [5, 12, 47], resembling networks seen in hub-based airlines. Among the resources, the first half are spoke-to-hub flight legs and the second half are hubto-spoke flight legs. The bundles are generated as follows. In the SMALL network, the first 10 bundles each use one of the 10 legs, and the remaining 8 each use two random legs, spoke1-to-hub and hub-to-spoke2. In the MEDIUM network, the first 100 bundles each use one of the 100 legs, the next 75 each use two random legs, spoke1-to-hub and hub-to-spoke2, and finally the last 25 each use four random legs spoke1-to-hub, hub-to-spoke2, spoke2-to-hub and 14

15 hub-to-spoke1. Similarly, in the LARGE network, the first 500 bundles each use one of the 500 legs, the next 375 each use two random legs, and finally the last 125 each use four random legs. See [16]. We now present the way the fares and the demands have been generated following the scheme presented in [16]. The class 1 fare of a single-resource bundle is generated from a normal distribution with mean 100 and standard deviation 40, truncated within the interval [20, 180]. The class 1 fare of a multi-resource bundle is the summation of the class 1 fares of the single-resource bundles associated with its resources. For all bundles, the fare of class 2 is 1.4 times that of class 1. As usual in the literature, demands are generated using a Poisson distribution. The demand for bundle-class ij in period t is generated from a Poisson distribution with mean µ ijt := β ijt µ, where h H Ch µ = η I J T. (94) Since all the parameters that define µ are constant, the higher η the higher the demand. Therefore, we refer to η as the load factor of demand, which has been set equal to 2 in this computational experience. With respect to β ijt, we have chosen them higher for single-resource bundles than for multi-resource ones. These parameters are also dynamic with respect to the time-to-service. For j = 1, 2, β ijt decreases when getting closer to the departure, eventually becoming zero. We use the optimization engine CPLEX v12.3, for solving the LP problems arising. Our experiments were conducted on a PC with a 2.33 GHz Intel Xeon dual core processor, 8.5 Gb of RAM and the operating system was LINUX Debian 4.0. Tables 1, 2 and 3 present for the SMALL, MEDIUM and LARGE networks the scenario tree structure and the dimensions of the three LP models subject of our computational comparison testing, namely, SLP (75)-(80), EV (81)-(86) and ELP (87)-(93) for the 21, 20 and 17 instances of our computational experience, respectively. The headings are as follows: case, instance name; T, the number of periods; L, number of immediate successor nodes (obviously, belonging to the next stage) from a node of any stage; N, number of nodes in the scenario tree; S, number of scenarios; n, number of variables; m, number of constraints nz, number of nonzero elements in the constraint matrix; dens, constraint matrix density in %. We can observe the high dimensions of the instances: up to 28.3 millions of variables and 25.8 millions of constraints for SMALL networks, up to 78.6 millions of variables and 71.5 millions of constraints for MEDIUM networks, and up to 98.3 millions of variables and 89.4 millions of constraints for LARGE networks. 6.3 Computational results Tables 4, 5 and 6 present for the SMALL, MEDIUM and LARGE networks the main computational results for the three LP models SLP, EV and ELP, respectively. The headings are as follows: Z SLP, Z EV and Z ELP, solution values of the original problem SLP and the models EV and ELP, respectively. Note: Z EV and Z ELP are upper bounds of Z SLP, such that Z EV Z ELP ; t SLP, t EV and t ELP, computing time (secs.) required for obtaining Z SLP, Z EV and Z ELP, respectively; Z EEV and Z EELP, solution values of the SLP feasible solutions obtained from the models EV and ELP, respectively, by using the EEV and EELP procedures described in Section 5; and t EEV and t EELP, computing time (secs.) required for obtaining Z EEV and Z EELP, respectively. Some gaps are reported as the relative 15

16 differences in % of several solution values, such as GAP 1 = (Z EV Z SLP ) /Z SLP % (95) GAP 2 = (Z SLP Z EEV ) /Z SLP % (96) GAP 3 = (Z EV Z EEV ) /Z EV % (97) GAP 4 = (Z ELP Z SLP ) /Z SLP % (98) GAP 5 = (Z SLP Z EELP ) /Z SLP % (99) GAP 6 = (Z ELP Z EELP ) /Z ELP % (100) Finally, mean gives the average values of the gaps and the computing times in the instances considered in each table. Our first observation is that the computing time for solving model SLP (i.e., solving the original problem by plain use of CPLEX) generally is high for non trivial instances and, even the original problem has not been solved for instances c18, c21, c33, c36, c38, c39, c41, c53, c55, c56 and c58 (i.e., a 19% of our testbed), due to we reaching the allowed computing time (2 hours in our experimentation). Additionally, we can observe that the computing time for obtaining the upper bounds Z EV and Z ELP (i.e., the solution values of the models EV and ELP, respectively) is very small, even it does not reach half a second. On the other side, the computing time requirements for obtaining the SLP feasible solutions whose value are Z EEV and Z EELP do not exceed and secs. for the SMALL network, respectively, and secs. for the MEDIUM network, respectively, and and secs. for the LARGE network, respectively. Although the computing time required for obtaining the solution value Z EELP is a bit higher than the time for obtaining Z EEV, the optimality gap of the former solution value is better than the optimality gap of the latter one. Thus for example, in Table 5 the mean of the EELP optimality gap is GAP 5 = 0.13% while the mean of the EEV optimality gap is GAP 2 = 1.48%. In any case, the gap is very small in both solution values. Finally, it is interesting to consider the quasi-optimality of the solutions values obtained by both approaches in the biggest instance (i.e., c56), being GAP 6 = 0.61% for the ELP-based approach and GAP 3 = 4.05% for the EV-based approach, being a representative comparison of the mean value of the three networks considered in or experiment, see last line of Tables 4, 5 and 6. 7 Conclusions In this paper we have considered the multistage stochastic linear programming (SLP) problem with a stochastic right-hand-side (rhs). We have introduced the multistage Event Linear Programming (ELP) problem whose objective is to approximate the SLP problem. The main difference is that, the former considers all the possible successions of events (scenario tree), whereas the ELP problem is bounded to only consider the events (event spike). From a theoretical point of view, we have proved that the ELP problem is a relaxation of the SLP problem, obtained by constraint aggregation. This aggregation is obtained by applying the conditional expectation operator to the SLP constraints. We have also proved that the EV problem is a relaxation of the ELP problem, obtained by constraint aggregation. This aggregation is obtained by applying the plain expectation operator to the ELP constraints. In this way, we have shown that the ELP relaxation is tighter than the EV relaxation. From a practical point of view, we have assessed the validity of the ELP approach by solving large scale instances of the network revenue management (NRM) problem for the fight tickets selling. We have used 58 instances whose SLP model ranges form one thousand of variables and constraints to over 98 millions of variables and almost 90 millions of constrains. For each NRM instance we have solved its 16

17 SLP model to obtain the optimal value Z SLP. We have also computed upper and lower bounds to Z SLP by means of the EV and ELP approaches. More specifically, by solving the EV and the ELP problems we have obtained the upper bounds Z EV and Z ELP, respectively. By computing the expected result of using the EV solution and the expected result of using the ELP solution, we have obtained the lower bounds Z EEV and Z EELP respectively. Finally, by combining these bounds, we have computed the worst case EV gap (Z EV Z EEV )/Z EV % and the worst case ELP gap (Z ELP Z EELP )/Z ELP %, which on average has been 3.63% and 0.45%, respectively (see Tables 4, 6 and 6). Therefore, in our computational experience the ELP approach has obtained SLP solutions which are almost optimal and significantly better than the EV solutions. Regarding the numerical tractability we have the following results. Within two hours of computing time CPLEX has failed to solve 19% of the SLP instances. On average, the computing time to solve the SLP instances has been 1816 seconds. In contrast, the average computing time to solve the EV and ELP instances has been 0.16 and 0.38 seconds, respectively. The average time to compute a suboptimal SLP solution by the EEV and the EELP approaches, has been 89 and 142 seconds, respectively (see Tables 4, 6 and 6). Therefore, in our computational experience, we have observed that even for large scale SLP instances the ELP heuristic is a tractable approximation. All in all, the main conclusion is that regarding the capacity to deal with uncertainty and the numerical tractability, the ELP approach lies between the SLP and the EV approaches. Finally, we would like to point out some limitations of the results here presented. The first limitation is that, although, in the computational experiment we have observed that the EEV solution is dominated by the EELP solution, we have not proved it theoretically. The second limitation is that we have only considered a stochastic rhs (stagewise independent) whereas the rest of the parameters have been considered deterministic. The third limitation is that this stochastic model is risk neutral. That is, it is based on the expected cost and it does not incorporate any risk averse measure as for example the conditional value at risk [24, 38, 41], or the first and second order stochastic dominance constraints [21, 22]. As a matter of future research, the authors are planning to improve the ELP approach regarding these aspects. Acknowledgments We are grateful to Jean-Philippe Vial and Alain Haurie (University of Geneva, Switzerland) for their comments and support at Logilab. We also thank the support of the grant S2009/esp-1594 (Riesgos CM) from Comunidad de Madrid (Spain) and the grant MTM C06-06 from the Spanish Ministry of Science and Innovation. 17

18 Table 1: Small network. Problem dimensions SLP EV ELP case T L N S n m nz dens n m nz dens n m nz dens c c c c c c c c c c c c c c c c c c c c c

19 Table 2: Medium network. Problem dimensions SLP EV ELP case T L N S n m nz dens n m nz dens n m nz dens c c c c c c c c c c c c c c c c c c c c

20 Table 3: Large network. Problem dimensions SLP EV ELP case T L N S n m nz dens n m nz dens n m nz dens c c c c c c c c c c c c c c c c c

21 Table 4: Small network. Solution values, gaps and computing time of SLP, EV and ELP SLP EV ELP case ZSLP tslp ZEV tev GAP 1 ZEEV teev GAP 2 GAP 3 ZELP telp GAP 4 ZEELP teelp GAP 5 GAP 6 c c c c c c c c c c c c c c c c c c c c c mean Note: The symbol means that the allowed computing time (7200 secs.) has been reached without obtaining the optimal solution. 21

22 Table 5: Medium network. Solution values, gaps and computing time of SLP, EV and ELP SLP EV ELP case ZSLP tslp ZEV tev GAP 1 ZEEV teev GAP 2 GAP 3 ZELP telp GAP 4 ZEELP teelp GAP 5 GAP 6 c c c c c c c c c c c c c c c c c c c c mean Note: The symbol means that the allowed computing time (7200 secs.) has been reached without obtaining the optimal solution. 22