Mathematical Programming Models

The mathematical programming models that we provide in this section are aimed at finding the seat allocation that maximizes the total expected revenue of the network and satisfies the capacity constraints on the various flights. The models can therefore be used to approximate the value Vt(c) for a given t and c. This approximation can be used directly for the accept/deny decision. However, the models generally also provide the means to construct booking-limit and bid-price policies, which provide much more practical booking control policies. The main difference between the models that we present in this section is in the way they account for the stochastic nature of demand.

The first full network formulation of the network revenue management problem has been proposed by Glover et al. (1982). They formulate the problem as a minimum cost network flow problem with one set of arcs corresponding to the flights and another set corresponding to the route/fare-class combinations. The method is aimed at finding the flow on each arc in the network that maximizes revenue, without violating the capacity constraints on the flights and upper-bounds posed by the demand forecasts for the route/fare-class combinations. A drawback of the network flow formulation is that it cannot always discriminate between the routes chosen from an origin to a destination.

Therefore, this formulation only holds when passengers are path-indifferent. The advantage of the formulation is that it is easy to solve and can be re-optimized very fast.

A formulation of the problem that is able to distinguish between the different routes from an origin to a destination, is given by the integer programming model

underlying the network flow formulation. It is common practice to solve the LP relaxation of this model rather than the integer programming problem, since the integer programming problem is usually hard to solve when the number of decision variables and constraints is large and the LP relaxation provides a good approximation for it. The LP relaxation of the model is known as the deterministic linear programming (DLP) model and approximates the optimal expected revenue as follows:

= )

DLP(c

Vt max xr^T (2.10)

s.t. Ax≤ c ] [ 0≤x E≤ D ,

where E[D] denotes the expected demand for the various route/fare-class combinations and x gives the partitioning of the seat inventory. In this model, the demand is treated as if it takes on a known value, e.g. as if it is deterministic, and no information on the demand distributions is taken into account. Accordingly, the model produces the optimal seat allocation if the expected demand corresponds perfectly with the actual demand. The allocation of the seats obtained by the DLP model can be used as booking limits for an on-line policy. Chen et al. (1998) show that V_t^DLP(c) provides an upper-bound for the optimal expected revenue Vt(c). In sections 2.5.3 and 2.5.4 we discuss how to obtain nested booking limits and bid prices based on the mathematical programming models presented in this section.

The DLP model is a deterministic model and will never reserve more seats for a higher fare class than the airline expects to sell on average. In fact, it does not recognize the fact that Pr(D_j ≥k)≥Pr(D_j ≥k+1) at all. In order to determine whether reserving more seats for more profitable route/fare-class combinations can be rewarding, it is necessary to incorporate the stochastic nature of demand in the model. Wollmer (1986) develops a model which incorporates probabilistic demand into a network setting. It makes use of the same expected marginal revenue principles as Littlewood and Belobaba

do for the single flight case. Therefore the model is called the EMR model. It is

where Mj is an upper-bound to the number of seats allocated to route/fare-class combination j. A possible value for Mj is for example the smallest capacity remaining on any of the flights j uses. The decision variable z^k_j takes on the value 1 when k seats or more are allocated to the route/fare-class combination j and 0 otherwise. The coefficient of

k

zj in the objective function represents the expected marginal revenue of allocating an additional k^th seat to the route/fare-class combination. A drawback of the EMR model is clearly the large amount of decision variables, which makes the model impractical to use.

Also, the EMR model makes use of the full distribution function of the demand for each route/fare-class combination. Obtaining such a full distribution is often difficult and the accuracy with which this can be done is often dubious. Chen et al. (1998) show that

)

EMR(c

Vt provides a lower-bound for the optimal expected revenue Vt(c).

De Boer et al. (2002) introduce a variant to the EMR model for which they use a coarser demand discretization. They link the model to stochastic programming techniques and it is therefore known as the stochastic linear programming (SLP) model. It incorporates the stochastic nature of demand by discretizing it to a limited number of values: d¹_j <d_j²<...<d_j^Nj, where Nj is the number of discretization points for route/fare-class combination j. The SLP model is given by:

=

The decision variables z^k_j each accommodate for the part of the demand D_j that falls in the interval (d^k_j⁻¹,d^k_j]. Summing the decision variables z^k_j over all k, gives the total number of seats allocated to route/fare-class combination j. The EMR model is a special case of the SLP model that can be obtained by letting d¹_j = 1 and d^k_j −d^k_j⁻¹ = 1 for all k = 2, 3, ..., Mj. The SLP formulation is, however, more flexible because it allows a reduction of the number of decision variables by choosing a limited amount of demand scenarios. If only the expected demand is considered as a possible scenario, the SLP model reduces to the DLP model. In fact, the DLP and EMR models can be seen as the two extremes that can be obtained from the SLP model. The first is obtained by considering only one demand scenario, the latter by considering all possible scenarios. The fact that V_t^DLP(c) provides an upper-bound and V_t^EMR(c) provides a lower-bound for the optimal expected revenue Vt(c), supports the use of the SLP model which encompasses both models. De Boer et al.

(2002) show that the SLP model produces better results than the DLP model and that including more than 3 or 4 demand scenarios in the SLP model does not increase the performance significantly.

The mathematical programming models discussed in this section are capable of capturing the combinatorial aspects of the network revenue management problem. We discuss how to derive nested booking limits and bid prices from the models in the next sections.