Disentangling capacity control from price optimization

Abstract

Standard revenue management (RM) methods typically work with an integrated forecast of demand and customer choice at booking class level and then maximize revenue by optimally controlling class availability. We show that the structure of the RM problem can be decomposed into two sub-problems: capacity control and price optimization for incoming customer requests. For capacity control in practice often bid prices, the dual variables to the capacity constraints, are used. Price optimization on the other hand needs forecasts of passengers’ willingness to pay. We propose to use two separate and tailored forecast models for both optimization tasks and discuss the advantages of this forecast separation. Furthermore, for the task of bid price calculation, we present a robust forecast model that is independent of booking classes and the actual control mechanism. It depends on historical booking data and bid prices only, but does not require historical fares and availabilities, which are often not available in good quality.

Appendix

Deterministic network optimization

With the notation from Network RM section, the deterministic network optimization problem is given by

$$\begin{aligned} \mathop {\hbox{max} }\limits_{x \in X} \mathop \sum \limits_{k = 1}^{K} \mathop \sum \limits_{{j \in J_{k} }}^{{}} \mathop \sum \limits_{t = 0}^{T} f_{j} d_{j} \left( {t,x_{k} } \right) \hfill \\ {\text{subject to}}\mathop \sum \limits_{k = 1}^{K} \mathop \sum \limits_{{j \in J_{k} }}^{{}} \mathop \sum \limits_{t = 0}^{T} a_{i,k} d_{j} \left( {t,x_{k} } \right) \le C_{i} \quad \forall i = 1, \ldots ,m \hfill \\ \end{aligned}$$

(10)

In other words, we maximize total revenue while under the constrained that the expected number of booking for each resources does not exceed capacity. Different itineraries are only coupled through the capacity constraints, and the problem naturally lends itself to a dual decomposition approach: Let the row vector $\pi = \left( {\pi_{1} , \ldots ,\pi_{m} } \right) \in {\mathbb{R}}^{m}$ be the dual variables or Lagrange multipliers associated to the capacity constraints, i.e., (deterministic) bid prices. The Lagrangian of Eq. (10) is given by

$$\begin{aligned} L\left( {x,\pi } \right) & = \mathop \sum \limits_{k = 1}^{K} \mathop \sum \limits_{{j \in J_{k} }}^{{}} \mathop \sum \limits_{t = 0}^{T} f_{j} d_{j} \left( {t,x_{k} } \right) + \mathop \sum \limits_{i = 1}^{m} \pi_{i} \left[ {C_{i} - \mathop \sum \limits_{k = 1}^{K} \mathop \sum \limits_{{j \in J_{k} }}^{{}} \mathop \sum \limits_{t = 0}^{T} a_{i,k} d_{j} \left( {t,x_{k} } \right)} \right] \\ & = \mathop \sum \limits_{k = 1}^{K} \mathop \sum \limits_{{j \in J_{k} }}^{{}} \mathop \sum \limits_{t = 0}^{T} \left[ {f_{j} - \mathop \sum \limits_{i = 1}^{m} \pi_{i} a_{i,k} } \right]d_{j} \left( {t,x_{k} } \right) + \mathop \sum \limits_{i = 0}^{m} \pi_{i} C_{i}, \\ \end{aligned}$$

where in the second line the triple-sum is independent of capacity, while the last sum is independent of demand and the controls.

The Lagrange dual function is

$$\begin{aligned} g\left( \pi \right) & = \mathop {\hbox{max} }\limits_{x \in X} \left\{ {L(x,\pi )} \right\} \\ & = \mathop \sum \limits_{k = 1}^{K} \mathop {\hbox{max} }\limits_{{x_{k} \in X_{k} }} \left\{ {\mathop \sum \limits_{{j \in J_{k} }}^{{}} \mathop \sum \limits_{t = 0}^{T} \left[ {f_{j} - \mathop \sum \limits_{i = 1}^{m} \pi_{i} a_{i,k} } \right]d_{j} \left( {t,x_{k} } \right)} \right\} + \pi C \\ & = \mathop \sum \limits_{k = 1}^{K} H_{k} (\mu_{k} ) + \pi C, \\ \end{aligned}$$

where $\mu_{k} = \mathop \sum \limits_{i = 0}^{m} \pi_{i} a_{i,k}$ is again the bid price for each itinerary k and

$$H_{k} \left( \mu \right) = \mathop {\hbox{max} }\limits_{{x_{k} \in X_{k} }} \left\{ {\mathop \sum \limits_{{j \in J_{k} }}^{{}} \mathop \sum \limits_{t = 0}^{T} \left( {f_{j} - \mu } \right)d_{j} \left( {t,x_{k} } \right)} \right\}$$

is the expected total network contribution function of itinerary k over the course of the booking horizon given itinerary bid price μ.

Optimal dual variables π for the original problem Eq. (10) can be computed by solving the Lagrange dual problem

$$\mathop {\hbox{min} }\limits_{{}} \left\{ {g\left( \pi \right)| \pi \ge 0} \right\}.$$

This problem is non-linear but always convex by construction and hence can be solved efficiently. Given forecasts $\hat{h}_{k} \left( {\tau ,\mu } \right)$ for the network contribution functions as described in Network RM section, one can therefore compute optimal deterministic displacement costs by solving

$$\begin{aligned} &\mathop {\hbox{min} }\limits_{{\pi \in {\mathbb{R}}^{m} }} \mathop \sum \limits_{k = 1}^{K} \mathop \sum \limits_{\tau = 0}^{365} \hat{h}_{k} \left( {\tau ,\mathop \sum \limits_{i = 1}^{m} \pi_{i} a_{i,k} } \right) + \mathop \sum \limits_{i = 1}^{m} \pi_{i} C_{i} \hfill \\ &{\text{subject to}}\,\pi_{i} \ge 0\quad \forall i = 1, \ldots ,m \hfill \\ \end{aligned}$$

(11)