Single Resource Revenue Management with Dependent Demands

Abstract

Revenue managers struggled for decades with the problem of finding optimal control mechanisms for fare class structures with dependent demands. In this context, a resource, such as seats on a plane, can be offered at different fares with potentially different restrictions and ancillary services, and the demand for those fares is interdependent. The question is what subset of the fares (or assortment of products) to offer for sale at any given time. Practitioners often use the term open, or open for sale, for a fare that is part of the offered assortment, and the term closed for fares that are not part of the offered assortment. For many years, practitioners preferred to model time implicitly by seeking extensions of Littlewood’s rule and EMSR type heuristics to the case of dependent demands. Finding the right way to extend Littlewood’s rule proved to be more difficult than anticipated. An alternative approach, favored by academics and gaining traction in industry, is to model time explicitly. In this chapter, we will explore both formulations but most of our attention is devoted to the more tractable model where time is treated explicitly.

Appendix

Proof of Proposition 6.2

We can linearize (6.5) by introducing a new variable, say y, such that y ≥ R _j − zΠ _j for all j ∈ K and z ≥ 0, which results in the linear program:

$$\displaystyle \begin{aligned} \begin{array}{rcl} \bar{V}(T,c) &\displaystyle = &\displaystyle \min_{z \geq 0}[\varLambda y+ cz],\\ \mbox{subject to} &\displaystyle &\displaystyle \varLambda y + \varLambda \varPi_j z \geq \varLambda R_j~~j \in K\\ &\displaystyle &\displaystyle z \geq 0, \end{array} \end{aligned} $$

where for convenience we have multiplied the constraints y + Π _j z ≥ R _j, j ∈ K by Λ > 0. The dual of this problem is given by

$$\displaystyle \begin{aligned} \begin{array}{rcl} \bar{V}(T,c) &\displaystyle = &\displaystyle \varLambda \max \sum_{j \in K} R_jt_j\\ {} \mbox{subject to} &\displaystyle &\displaystyle \varLambda \sum_{j \in K} \varPi_j t_j \leq c \\ {} &\displaystyle &\displaystyle \sum_{j \in K}t_j =1 \\ {} &\displaystyle &\displaystyle t_j \geq 0~~~\forall j \in K. \end{array} \end{aligned} $$

This linear program decides the proportion of time, t _j ∈ [0, 1], that each efficient set E _j is offered to maximize the revenue subject to the capacity constraint. Dividing the constraint by Λ and defining ρ = c∕Λ we see that $\bar {V}(T,c)/\varLambda = Q(\rho )$, or equivalently $\bar {V}(T,c) = \varLambda Q(c/\varLambda )$.