Aviation Data Analysis by Linear Programming in Airline Network Revenue Management

Abstract

Aviation data comprise, e.g., bookings and cancellations by consumers as well as no show situations before departure, aircraft-type assignments to flight legs and overbooking-decisions to avoid empty seats in airplanes. Here deterministic linear programming (DLP) is a widely used approach to process this kind of data in an area called airline network revenue management for which adaptions of a basic DLP-model to overbooking situations as well as the offering of flexible products are known. We combine these concepts in a model which simultaneously allows the incorporation of overbooking-decisions and the offering of specific as well as flexible products. Additionally, we further extend this integrated formulation to allow the treatment of different booking-classes and aircraft-type assignment considerations. We present characteristics of the new approach, which uses the overlapping science directions of data analysis and operations research, point out differences to already known results in airline network revenue management, describe an example which illustrates how the different aspects can be considered, and indicate the advantages of our model in view of various data settings.

A Appendix

In this appendix, we consider different DLP (deterministic linear programming)-model adaptions for airline network revenue management to support the understanding of our new approach that we have called “class-specific DLP$_t$-flex-over with aircraft-type assignment”. The notation corresponds to what has already been explained in Chap. 2. Based on the formulation of the starting DLP-model (A1) in which only the offering of specific products was described (Simpson 1989), the incorporation of overbooking-decisions (A2) has been rewritten (Bertsimas and Popescu 2003) as well as the extension by Gallego et al. (2004) to include a distinction between specific and flexible offerings (A3). We further state a DLP-model with simultaneous treatment of both aspects (A4) which was the starting point for our treatise on aviation data analysis by linear programming.

1.1 A1 The Basic DLP-Model

An early formulation of a DLP-model in network revenue management is as follows (see Simpson 1989):

$$\begin{aligned} max \sum _{i \in I} r_i \cdot x_i \end{aligned}$$

$$\begin{aligned} s.t.: \sum _{i \in I} a_{hi} \cdot x_i \le c_{h} \quad \forall h \in H \end{aligned}$$

(A1.1)

$$\begin{aligned} \left( {\varvec{DLP}}_{t}\right) \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad x_i \le \overline{D}_{it} \quad \forall i \in I \end{aligned}$$

(A1.2)

$$\begin{aligned} x_i \ge 0 \quad \forall i \in I \end{aligned}$$

(A1.3)

The objective function maximizes the revenue. The constraints (A1.1) secure that the sum of sold products does not exceed the capacities $c_h$. Constraints (A1.2) and (A1.3) stand for upper bounds $\overline{D}_{it}$ of expected demand data and the nonnegativity of the number of accepted products.

1.2 A2 DLP Adaption to Overbooking

An extension of DLP to overbooking is expressed by the following model (see Bertsimas and Popescu 2003):

$$\begin{aligned} max \sum _{i \in I} r_i \cdot x_i - \sum _{h \in H} d_{h} \cdot u_{h} \end{aligned}$$

$$\begin{aligned} s.t.: \sum _{i \in I} a_{hi} \cdot x_i \le z_{h} \quad \forall h \in H \end{aligned}$$

(A2.1)

$$\begin{aligned} x_i \le \overline{D}_{it} \quad \forall i \in I \end{aligned}$$

(A2.2)

(A2.3)

$$\begin{aligned} u_{h} \ge p_{h} \cdot z_{h} - c_{h} \quad \forall h \in H \end{aligned}$$

(A2.4)

$$\begin{aligned} z_{h} \ge c_{h} \quad \forall h \in H \end{aligned}$$

(A2.5)

$$\begin{aligned} u_{h} \ge 0 \quad \forall h \in H \end{aligned}$$

(A2.6)

In the objective function, additionally, the costs $d_h$ for denied services are subtracted. Constraints (A2.1) are similar as (A1.1) but, now, overbooking-limits $z_h$ for all resources $h \in H$ as right-hand side replace the capacities. While constraints (A2.2) and (A2.3) are unchanged compared to formulation (A1) conditions (A2.4)–(A2.6) delineate the overbooking-extension where $p_h$ denotes show-probabilities.

1.3 A3 DLP Adaption to Flexible Offerings

The extension to the case in which specific as well as flexible products can be offered (see Gallego et al. 2004) can be stated as follows:

$$\begin{aligned} max \sum _{i \in I} r_i \cdot x_i + \sum _{j \in J} f_j \cdot \sum _{m \in M_j} y_{jm} \end{aligned}$$

$$\begin{aligned} s.t.: \sum _{i \in I} a_{hi} \cdot x_i + \sum _{j \in J} \sum _{m \in M_j} a_{hm} \cdot y_{jm} \le c_{h} \quad \forall h \in H \end{aligned}$$

(A3.1)

$$\begin{aligned} x_i \le \overline{D}_{it}^{s} \quad \forall i \in I \end{aligned}$$

(A3.2)

(A3.3)

$$\begin{aligned} \sum _{m \in M_j} y_{jm} \le \overline{D}_{jt}^{f} \quad \forall j \in J \end{aligned}$$

(A3.4)

$$\begin{aligned} y_{{}_{jm}} \ge 0 \quad \forall j \in J, m \in M_j \end{aligned}$$

(A3.5)

Now, the objective function maximizes the revenue from specific and flexible products with $M_{j}\subseteq I$ as execution-mode set that describes the allocation of the flexible product j to a subset of specific products. Also, capacity-constraints (A3.1) are extended by a term for flexible products. While (A3.2) and (A3.3) are already known from the (A1) and (A2) model descriptions (with a distinction of the demand data terms ($\overline{D}_{it}^{s}$ for specific products and $\overline{D}_{jt}^{f}$ for flexible products)) the restrictions for flexible products have now to take into consideration the execution-mode sets $M_j$ in (A3.4) and (A3.5).

1.4 A4 DLP-Formulation of Integrated Specific/Flexible Offerings and Overbooking

A DLP-model in which both aspects (overbooking as well as specific and flexible products) are handled simultaneously is now easy to formulate and was the starting point for our treatise of aviation data analysis within airline network revenue management:

$$\begin{aligned} max \sum _{i \in I} r_i \cdot x_i + \sum _{j \in J} f_j \cdot \sum _{m \in M_j} y_{jm} - \sum _{h \in H} d_{h} \cdot u_{h} \end{aligned}$$

$$\begin{aligned} s.t.: \sum _{i \in I} a_{hi} \cdot x_i + \sum _{j \in J} \sum _{m \in M_j} a_{hm} \cdot y_{jm} \le z_{h} \quad \forall h \in H \end{aligned}$$

(A4.1)

$$\begin{aligned} x_i \le \overline{D}_{it}^{s} \quad \forall i \in I \end{aligned}$$

(A4.2)

(A4.3)

$$\begin{aligned} \sum _{m \in M_j} y_{jm} \le \overline{D}_{jt}^{f} \quad \forall j \in J \end{aligned}$$

(A4.4)

$$\begin{aligned} y_{{}_{jm}} \ge 0 \quad \forall j \in J, m \in M_j \end{aligned}$$

(A4.5)

$$\begin{aligned} u_{h} \ge p_{h} \cdot z_{h} - c_{h} \quad \forall h \in H \end{aligned}$$

(A4.6)

$$\begin{aligned} z_{h} \ge c_{h} \quad \forall h \in H \end{aligned}$$

(A4.7)

$$\begin{aligned} u_{h} \ge 0 \quad \forall h \in H \end{aligned}$$

(A4.8)

The objective function maximizes the revenue of both kinds of offered products from which the costs for rejected customers have to be subtracted. Constraints (A4.1) secure that the sum of allocated flexible and specific products does not exceed the overbooking-limits for all resources $h \in H$. Constraints (A4.2) and (A4.4) consider the expected demand data of specific and flexible products. (A4.3) and (A4.5) are nonnegativity restrictions. Conditions (A4.6)–(A4.8) stand for the overbooking-extension already formulated in (A2).

Strictly speaking, the formulation described in (A4) provides already a new approach in which overbooking and the incorporation of flexible products are integrated. Even more challenging is the additional consideration of booking-classes (which, e.g., allows to restrict flexible products to lower valued classes) and aircraft-type assignments (which, e.g., combines demand with physically available seat capacity) in the DLP-model handled in this paper (see Fig. 1.2).

Fig. 1.2

From DLP to class-specific DLP-flex-over with aircraft-type assignment