Semi-Markov information model for revenue management and dynamic pricing

Abstract

In traditional airline yield management, when a customer requests a discount fare, the airline must decide whether to sell a seat at the requested discount or to hold the seat in hopes that a customer will arrive later who will pay more. In contrast to that, in dynamic pricing models of revenue management, when faced with a request for a seat the airline quotes a price that may or may not be accepted by that customer. In each approach different type of information is available to the seller and, consequently, there is usually a difference between optimal policies and their expected revenues. On the other hand many structural properties of optimal policies are shared. We provide a framework that includes these two types of models by introducing an information variable into the state description of the decision problem.

Appendix

The arguments utilized in this section generally proceed by conditioning on batch size and utilizing the independence of batch sizes and fares. To assist in this approach, a version of ${\user1{\mathcal{F}}}$ is defined conditioned on $\it\Phi _{\beta } = \beta$:

$$ {\user1{\mathcal{F}}}_{\beta } v{\left( {\eta ,x,t} \right)}: = {\mathop {\max }\limits_{u \in D{\left( {\eta ,x,t} \right)}} }{\left\{ {{\sum\limits_{k = 1}^K {{\left[ {r{\left( {\eta ,x,t;\rho _{k} ,\beta ;u} \right)} + v{\left( {\eta - z{\left( {\eta ,x,t;\rho _{k} ,\beta ;u} \right)},x,t} \right)}} \right]}p_{k} {\left( {x,t} \right)}} }} \right\}} $$

(A.1)

where $p_{k} {\left( {x,t} \right)} = \Pr {\left[ {{\it{\Phi}} _{\rho } = \rho _{k} \left| {{\it{\Phi}} _{\beta } } \right. = \beta ,x,t} \right]} = \Pr {\left[ {{\it{\Phi}} _{\rho } = \rho _{k} \left| {x,t} \right.} \right]}.$

The last equality follows from the independence of Φ _ρ and Φ _β given (x,t). Note that u ∈ D(η,x,t) can be replaced by u ∈ D(η∨β,x,t) since there is no benefit from offering more seats than the customer wants. Also define $\overline{p} _{k} {\left( {x,t} \right)}: = \Pr {\left( {{\it{\Phi}} _{\rho } \geq \rho _{k} \left| {x,t} \right.} \right)}.$Recall that a function $v \in {\user1{\mathcal{B}}}$ is discretely concave if $\Delta v{\left( {\eta ,x,t} \right)}: = v{\left( {\eta ,x,t} \right)} - v{\left( {n - 1,x,t} \right)}$ is decreasing in η = 1,2,... for each information x and time t.

Lemma A.1

Assume that the revenue function has affine form and r(η,x,t;0) is discretely concave in inventory for each state (x,t). For each (x,t), booking limits $\eta ^{ \star }_{k} {\left( {x,t} \right)}$ can be determined so that the maximizing allocation in ${\user1{\mathcal{F}}}_{\beta }$ is $u^{ \star }_{K} {\left( {x,t} \right)} = {\left[ {\eta - \eta ^{ \star }_{K} {\left( {x,t} \right)}} \right]}^{ + }$ and for k < K, $u^{ \star }_{k} {\left( {x,t} \right)} = {\left[ {\eta \wedge \eta ^{ \star }_{{k + 1}} {\left( {x,t} \right)} - \eta ^{ \star }_{k} {\left( {x,t} \right)}} \right]}^{ + }.$

Proof

Fix (x,t) and suppose that v(η,x,t) is discretely concave in η. For convenience we introduce some high fare ρ ₀ such that $\Pr {\left( {{\it{\Phi}} _{\rho } \geq \rho _{0} \left| {x,t} \right.} \right)} = 0$ i.e. offering a seat at ρ ₀ is the same as not offering it for sale. Without loss of generality we assume that r(η,x,t) is zero, since we can always add it to v and obtain a discretely concave function.

Let u be an allocation of seats given state (η,x,t). We start with an expression for the expected revenue for allocation u (the one under maximization sign in Eq. (A.1)) and then transform it to a more convenient form.

$${\sum\limits_{k = 1}^K {{\left[ {r{\left( {\eta ,t,x;\rho _{k} ,\beta ;u} \right)} + v{\left( {\eta - z{\left( {\eta ,x,t;\rho _{k} ,\beta ;u} \right)},x,t} \right)}} \right]}p_{k} {\left( {x,t} \right)} = v{\left( {\eta ,x,t} \right)} + {\sum\limits_{k = 1}^K {{\sum\limits_{m = 1}^{u_{k} } {{\left[ {\rho _{k} - \Delta v{\left( {\eta + 1 - {\left[ {{\sum\limits_{j = k + 1}^K {u_{j} + m} }} \right]},x,t} \right)}} \right]} \cdot \Pr {\left( {\Phi _{\rho } \geqslant \rho _{k} \left| {x,t} \right.} \right)} \cdot 1{\left( {\beta \geqslant {\sum\limits_{j = k + 1}^K {u_{j} } } + m} \right)} = } }} }} }v{\left( {\eta ,x,t} \right)}$$

(A.2)

Thus when written out fully the double sum in Eq. (A.2) consists of at most η terms. The j-th term (1 ≤ j ≤ η) is of the form

$${\left[ {\rho ^{j} - \Delta v{\left( {\eta - j + 1,x,t} \right)}} \right]}\Pr {\left( {\Phi _{\rho } \geqslant \rho ^{j} \left| {x,t} \right.} \right)} \cdot 1{\left( {\beta \geqslant j} \right)}$$

for some fare ${\left[ {\rho ^{j} - \Delta v{\left( {\eta - j + 1,x,t} \right)}} \right]}\Pr {\left( {\Phi _{\rho } \geqslant \rho ^{j} \left| {x,t} \right.} \right)} \cdot 1{\left( {\beta \geqslant j} \right)}$In order to find u that maximizes the sum we will show that we can maximize each term separately by finding a maximizer $\rho ^{\eta }_{ \star }$ and then by showing that the maximizers ρ ^j increase as the index j increases, which will be in agreement with the customer buying seats starting from the lowest priced.

Formally, for each inventory level η let

$$\rho ^{\eta }_{ \star } : = \arg \kern-17pt\max _{{\rho_{j} \in {\left\{ {\rho _{0} ,\rho _{1} ,...,\rho _{K} } \right\}}}} {\left\{ {{\left[ {\rho _{j} - \Delta v{\left( {\eta ,x,t} \right)}} \right]}\Pr {\left( {\Phi _{\rho } \geq \rho _{j} } \right)}} \right\}} = \arg \kern-17pt\max _{{\rho_{j} \in {\left\{ {\rho _{0} ,\rho _{1} ,...,\rho _{K} } \right\}}}} {\left\{ {{\left[ {\rho _{j} - \Delta v{\left( {\eta ,x,t} \right)}} \right]}\overline{p} _{j} {\left( {x,t} \right)}} \right\}},$$

(A.3)

with an appropriate convention in case of ties (e.g. “pick the smallest maximizer”). It follows that $\rho ^{\eta }_{ \star } \geq \Delta v{\left( {\eta ,x,t} \right)}$ since otherwise the expresssion would be negative, while rejecting the request (ρ = ρ ₀) yields 0 since the probability involved is 0.

We now turn our attention to finding ways to determine $\rho ^{\eta }_{ \star } .$It is instructive to start by comparing two prices, say ρ _j and ρ _i with ρ _j ≥ ρ _i , when requests are just for one seat. The question we want to answer is this: When is it better to offer ρ _j than ρ _i ? If v(η,x,t) represents future revenue the question becomes: When is

$${\left[ {\rho _{j} + v{\left( {\eta - 1,x,t} \right)}} \right]}\overline{p} _{j} {\left( {x,t} \right)} + {\left[ {1 - \overline{p} _{j} {\left( {x,t} \right)}} \right]}v{\left( {\eta ,x,t} \right)} \geq {\left[ {\rho _{i} + v{\left( {\eta - 1,x,t} \right)}} \right]}\overline{p} _{i} {\left( {x,t} \right)} + {\left[ {1 - \overline{p} _{i} {\left( {x,t} \right)}} \right]}v{\left( {\eta ,x,t} \right)}?$$

After some algebra this can again be restated as

$${\left[ {\rho _{j} - {\it{\Delta}} v{\left( {\eta ,x,t} \right)}} \right]}\overline{p} _{j} {\left( {x,t} \right)} \geq {\left[ {\rho _{i} - {\it{\Delta}} v{\left( {\eta ,x,t} \right)}} \right]}\overline{p} _{i} {\left( {x,t} \right)}$$

(A.4)

and then as

$${{\rho _j \overline p _j \left( {x,t} \right) - \rho _i \overline p _i \left( {x,t} \right)} \over {\overline p _j \left( {x,t} \right) - \overline p _i \left( {x,t} \right)}}\geq{\it{\Delta}} v\left( {\eta ,x,t} \right).$$

(A.5)

Thus ρ _j is better than ρ _i if and only if Eq. (A.5) holds for a given (η,x,t). The $\rho ^{\eta }_{ \star }$ so defined have a number of desired properties. They are clearly independent of β. Furthermore they are decreasing in η because v is discretely concave by assumption. We show next that they are determined by suitably defined booking limits $\eta ^{ \star }_{k} {\left( {x,t} \right)}$.

Define first

$$r_{k,i} \left( {x,t} \right): = {{\rho _k \overline p _k \left( {x,t} \right) - \rho _i \overline p _i \left( {x,t} \right)} \over {\overline p _k \left( {x,t} \right) - \overline p _i \left( {x,t} \right)}}.$$

and then for each 1 ≤ k ≤ K define $\overline{\eta } _{k} : = \min {\left\{ {\eta :r_{{k,i}} {\left( {x,t} \right)} \geq {\it{\Delta}} v{\left( {\eta ,x,t} \right)}} \right\}}$, for i < k} − 1 if the set in question is nonempty and put $\overline{\eta } _{k} {\left( {x,t} \right)} = c$ otherwise. By discrete concavity of v, it is better to offer a single seat at ρ _k rather than at any other ρ _i ≥ ρ _k , if and only if $\eta > \overline{\eta } _{k} {\left( {x,t} \right)}.$From Eq. (A.5) follows that ρ _k is better than ρ _i ρ _k if and only if $\eta \leq \overline{\eta } _{i} .$Thus ρ _k is optimal if and only if $\eta > \overline{\eta } _{k}$ but at the same time $\eta \leq \overline{\eta } _{i}$for i > k. So if it happens that for some j > k, $\overline{\eta } _{j} \leq \overline{\eta } _{k}$ then it will never be optimal to offer seats at ρ _k . Therefore we set $\eta ^{ \star }_{k} = \eta _{k}$ unless the above is true in which case we set $\eta ^{ \star }_{k} = c.$It now follows that the optimal number of seats to be allocated at ρ _k is $u^{ \star } = {\left[ {\eta \wedge \eta ^{ \star }_{{k + 1}} - \eta ^{ \star }_{k} } \right]}^{ + }.$

Note that it is OK to allocate more than β since revenue from the excess allocation is nil. From construction also follows that some of the booking limits could be equal to c, the latter implies that no seats are allocated at the corresponding price.

We now show that ${\user1{\mathcal{F}}}_{\beta }$ preserves discrete concavity, i.e. that $\Delta {\user1{\mathcal{F}}}_{\beta } v{\left( {\eta ,x,t} \right)}: = {\user1{\mathcal{F}}}_{\beta } v{\left( {\eta ,x,t} \right)} - {\user1{\mathcal{F}}}_{\beta } v{\left( {\eta - 1,x,t} \right)}$ is decreasing in η.

Lemma A.2

Assume that the revenue function has affine form and r(η,x,t;0) is discretely concave in inventory for each state (x,t). Then ${\user1{\mathcal{F}}}_{\beta } v$ is discretely concave whenever $v \in {\user1{\mathcal{B}}}$ is.

Proof

$$\Delta {\user1{\mathcal{F}}}_{\beta } {\left( {\eta ,x,t} \right)} = \Delta v{\left( {\eta ,x,t} \right)} + {\left( {\rho ^{\eta }_{ \star } - \Delta v{\left( {\eta ,x,t} \right)}} \right)}\Pr {\left( {\Phi _{\rho } \geqslant \rho ^{\eta }_{ \star } \left| {x,t} \right.} \right)} - {\left( {\rho ^{{\eta - \beta }}_{ \star } - \Delta v{\left( {\eta - \beta ,x,t} \right)}} \right)}\Pr {\left( {\Phi _{\rho } \geqslant \rho ^{{\eta - \beta }}_{ \star } \left| {x,t} \right.} \right)}.$$

(A.6)

By the assumed discrete concavity of v we obtain that

$$\Delta v{\left( {\eta ,x,t} \right)} + {\left( {\rho ^{\eta }_{ \star } - \Delta v{\left( {\eta ,x,t} \right)}} \right)}\Pr {\left( {\Phi _{\rho } \geqslant \rho ^{\eta }_{ \star } } \right)} = \Delta v{\left( {\eta ,x,t} \right)}{\left( {1 - \Pr {\left( {\Phi _{\rho } \geqslant \rho ^{\eta }_{ \star } } \right)}} \right)} + \rho ^{\eta }_{ \star } \Pr {\left( {\Phi _{\rho } \geqslant \rho ^{\eta }_{ \star } } \right)} \leqslant \Delta v{\left( {\eta - 1,x,t} \right)}{\left( {1 - \Pr {\left( {\Phi _{\rho } \geqslant \rho ^{\eta }_{ \star } } \right)}} \right)} + \rho ^{\eta }_{ \star } \Pr {\left( {\Phi _{\rho } \geqslant \rho ^{\eta }_{ \star } } \right)} \leqslant \Delta v{\left( {\eta - 1,x,t} \right)} + {\left( {\rho ^{{\eta - 1}}_{ \star } - \Delta v{\left( {\eta ,x,t} \right)}} \right)}\Pr {\left( {\Phi _{\rho } \geqslant \rho ^{{\eta - 1}}_{ \star } } \right)},$$

(A.7)

with the last inequality valid because $\rho ^{{\eta - 1}}_{ \star }$ maximizes Eq. (A.3) for η−1. Same arguments but in different order also work to show that

$$ - {\left( {\rho ^{{\eta - \beta }}_{ \star } - \Delta v{\left( {\eta - \beta ,x,t} \right)}} \right)}\Pr {\left( {\Phi _{\rho } \geqslant \rho ^{{\eta - \beta }}_{ \star } } \right)} \leqslant - {\left( {\rho ^{{\eta - 1 - \beta }}_{ \star } - \Delta v{\left( {\eta - \beta ,x,t} \right)}} \right)}\Pr {\left( {\Phi _{\rho } \geqslant \rho ^{{\eta - 1 - \beta }}_{ \star } } \right)} = - \rho ^{{\eta - 1 - \beta }}_{ \star } \Pr {\left( {\Phi _{\rho } \geqslant \rho ^{{\eta - 1 - \beta }}_{ \star } } \right)} + \Delta v{\left( {\eta - \beta ,x,t} \right)}\Pr {\left( {\Phi _{\rho } \geqslant \rho ^{{\eta - 1 - \beta }}_{ \star } } \right)} \leqslant - \rho ^{{\eta - 1 - \beta }}_{ \star } \Pr {\left( {\Phi _{\rho } \geqslant \rho ^{{\eta - 1 - \beta }}_{ \star } } \right)} + \Delta v{\left( {\eta - 1 - \beta ,x,t} \right)}\Pr {\left( {\Phi _{\rho } \geqslant \rho ^{{\eta - 1 - \beta }}_{ \star } } \right)} = - {\left( {\rho ^{{\eta - 1 - \beta }}_{ \star } - \Delta v{\left( {\eta - 1 - \beta ,x,t} \right)}} \right)}\Pr {\left( {\Phi _{\rho } \geqslant \rho ^{{\eta - 1 - \beta }}_{ \star } } \right)}.$$

(A.8)

Combining Eqs. (A.7) and (A.8) shows that $\Delta {\user1{\mathcal{F}}}_{\beta } v{\left( {\eta ,x,t} \right)} \leqslant \Delta {\user1{\mathcal{F}}}_{\beta } v{\left( {\eta - 1,x,t} \right)}$, i.e. that ${\user1{\mathcal{F}}}_{\beta }$ preserves discrete concavity.

Lemma A.3

Under assumptions of Lemma A.1 the operator ${\user1{\mathcal{F}}}$ preserves discrete concavity. For each (η,x,t) and each fare class k, the optimal allocation can be determined from the booking limits $\eta ^{ \star }_{k} {\left( {x,t} \right)}:u^{ \star }_{k} {\left( {x,t} \right)} = {\left[ {\eta \wedge \eta ^{ \star }_{{k + 1}} - \eta ^{ \star }_{k} } \right]}^{ + }$

Proof

Fix (x,t). By Lemma A.1 the allocation $u^{ \star } = {\left( {u^{ \star }_{K} ,...,u^{ \star }_{1} } \right)},u^{ \star }_{k} = {\left[ {\eta \wedge \eta ^{ \star }_{{k + 1}} - \eta ^{ \star }_{k} } \right]}^{ + }$ maximizes (A.1) for each β and consequently

$${\user1{ \mathcal{F}}}v{\left( {\eta ,x,t} \right)} = {\mathop {\max }\limits_{u \in D{\left( {\eta ,x,t} \right)}} }E{\left\{ {{\sum\limits_{i = 1}^K {{\left[ {r{\left( {\eta ,x,t;\rho _{i} ,\Phi _{\beta } ;u} \right)} + v{\left( {\eta - z{\left( {\eta ,x,t;\rho _{i} ,\Phi _{\beta } ;u} \right)},x,t} \right)}} \right]}p_{i} {\left( {x,t} \right)}} }} \right\}} \leqslant E{\mathop {\max }\limits_{u \in D{\left( {\eta ,x,t} \right)}} }{\left\{ {{\sum\limits_{i = 1}^K {{\left[ {r{\left( {\eta ,x,t;\rho _{i} ,\Phi _{\beta } ;u} \right)} + v{\left( {\eta - z{\left( {\eta ,x,t;\rho _{i} ,\Phi _{\beta } ;u} \right)},x,t} \right)}} \right]}p_{i} {\left( {x,t} \right)}} }} \right\}} = E{\mathop {\max }\limits_{u \in D{\left( {\eta ,x,t} \right)}} }{\left\{ {{\sum\limits_{i = 1}^K {{\left[ {r{\left( {\eta ,x,t;\rho _{i} ,\Phi _{\beta } ;u^{ \star } } \right)} + v{\left( {\eta - z{\left( {\eta ,x,t;\rho _{i} ,\Phi _{\beta } ;u^{ \star } } \right)},x,t} \right)}} \right]}p_{i} {\left( {x,t} \right)}} }} \right\}} \leqslant {\user1{\mathcal{F}}}v{\left( {\eta ,x,t} \right)},$$

(A.9)

where the expectation is with respect to the distribution of ${\it{\Phi}} _{\beta }$ conditioned on (x, t). The last inequality is by definition of ${\user1{\mathcal{F}}}$So the same $u^{ \star }$ is the optimal allocation for the original maximization problem. Again by Lemma A.1, ${\user1{\mathcal{F}}}_{\beta }$ is discretely concave for each β and so is the expected value.

From the proof of the Lemma A.1 it follows that an optimal allocation is determined by ρ’s maximizing ${\left[ {\rho - {\it{\Delta}} v{\left( {\eta ,x,t} \right)}} \right]}\Pr {\left( {{\it{\Phi}} _{\rho } \geq \rho \left| {x,t} \right.} \right)}$ for each η. When request size ${\it{\Phi}} _{\beta }$ is stochastically idependent from the customer reservation price ${\it{\Phi}} _{\rho }$ given current information and time of arrival (x, t), those can be determined as if the requests were for single seats.

Theorem 6.2

If the revenue function is discretely concave with affine form, then the optimal value function $v^{ \star } {\left( {\eta ,x,t} \right)}$ and the corresponding auxiliary function $v^{0} = {\user1{\mathcal{T}}}v^{ \star }$ are each discretely concave. Moreover, there is an optimal booking limit policy.

Proof

We note that the operator ${\user1{\mathcal{T}}}$ preserves discrete concavity since it is linear and transforms nonnegative functions into the same. Under assumptions of the theorem, the operator ${\user1{\mathcal{F}}}$, by Lemma A.3, preserves discrete concavity and therefore the composed operator ${\user1{\mathcal{F}\mathcal{T}}}$ does the same.

The function which is 0 for all states is discretely concave, so ${\left( {{\user1{\mathcal{F}\mathcal{T}}}} \right)}0$ is also discretely concave by the preceding lemma. Under our assumption, the successive approximations are valid so that $v^{ \star } = \lim _{{n \to \infty }} {\left( {{\user1{\mathcal{F}\mathcal{T}}}} \right)}^{n} 0$Hence, $v^{ \star }$ is discretely concave. Since ${\user1{\mathcal{T}}}$ preserves concavity, $v^{0} = {\user1{\mathcal{T}}}v^{ \star }$ is also discretely concave.

We will now turn to fleshing out an algorithm for determining $\rho ^{\eta }_{ \star }$ by comparing ${\it{\Delta}} v^{0} {\left( {\eta ,x,t} \right)} + {\it{\Delta}} r{\left( {\eta ,x,t;0} \right)}$ to ratios r _k,i.

Recall that we assumed that ρ ₁ > ... > ρ _K and therefore $\Pr {\left( {\Phi _{\rho } \geq \rho _{k} \left| {x,t} \right.} \right)}$ increases as k increases. From Eq. (A.5) follows that for any two fares ρ _k , ρ _j such that ρ _j > ρ _k , ρ _k will never be preferred to ρ _j if $\rho _{k} \overline{p} _{k} {\left( {x,t} \right)} < \rho _{j} \overline{p} _{j} {\left( {x,t} \right)},$i.e. if the expected revenue using ρ _k is smaller than for ρ _j . Therefore, we need to keep only all those fares from the set of all fares ${\user1{\mathcal{R}}}$ for which that expected revenue is monotone: ${\user1{\mathcal{R}}}^{\prime } : = {\left\{ {\rho \in {\user1{\mathcal{R}}}:\rho _{k} \overline{p} _{k} {\left( {x,t} \right)} \leqslant \rho _{j} \overline{p} _{j} {\left( {x,t} \right)},{\text{if}}\,\rho _{k} \leqslant \rho _{j} } \right\}}$

Further, from the same proof it also follows that for a fare ρ _k such that

$${\mathop {\min }\limits_{j < k} }\kern3ptr_{{k,j}} {\left( {x,t} \right)} < {\mathop {\max }\limits_{j > k} }r_{{j,k}} {\left( {x,t} \right)}$$

(A.10)

it will never be optimal to allocate at ρ _k so it can be dropped from consideration since then Eq. (A.10) would contradict Eq. (A.5).

So assume that we have dropped all such fares from ${\user1{\mathcal{R}}}$, the set of fares obtained above, to form a set of active fares ${\user1{\mathcal{R}}}_{A} {\left( {x,t} \right)}: = {\left\{ {\rho \in {\user1{\mathcal{R}}}^{\prime } :\min _{{j < k}} r_{{k,j}} {\left( {x,t} \right)} \geqslant \max _{{j > k}} r_{{j,k}} {\left( {x,t} \right)}} \right\}}$, and let us renumerate the fares in ${\user1{\mathcal{R}}}_{A} {\left( {x,t} \right)}$ from 1 to some K′. If we now form the ratios $r^{\prime }_{{k,k - 1}}{\left( {x,t} \right)}$ that correspond to two consecutive fares in ${\user1{\mathcal{R}}}_{A} {\left( {x,t} \right)}$ then $r^{\prime }_{{k,k - 1}} {\left( {x,t} \right)}$ have to be decreasing in k.

Now for a fare ρ _k in ${\user1{\mathcal{R}}}_{A} {\left( {x,t} \right)}$ the booking limit is determined by setting $\eta ^{ \star }_{k} {\left( {x,t} \right)}: = \min {\left\{ {\eta :r^{\prime }_{{k,k - 1}} {\left( {x,t} \right)} \geq {\it{\Delta}} v^{0} {\left( {\eta ,x,t} \right)} + {\it{\Delta}} r{\left( {\eta ,x,t;0} \right)}} \right\}} - 1$if the set in question is nonempty and put $\eta ^{ \star }_{k} {\left( {x,t} \right)} = c$ otherwise. For each fare ρ _k in ${\user1{\mathcal{R}}}\backslash {\user1{\mathcal{R}}}_{A} {\left( {x,t} \right)}{\text{set}}\,\eta ^{ \star }_{k} {\left( {x,t} \right)} = c$

The algorithm becomes relatively simple when the original ratios r _k,k−1 are ordered as one is then assured that $\eta ^{ \star }_{k}$’s are ordered. Chatwin (2000) assumes those ratios ordered as a starting point in his analysis, based on economic considerations. Feng and Xiao (2000), working without that assumption, introduce an equivalent concept of maximum concave envelope, which turns out to be supported by fares in ${\user1{\mathcal{R}}}_{A} {\left( {x,t} \right)}$They also describe an algorithm to determine it. The idea is further investigated in Talluri and van Ryzin (2004) in the context of general consumer choice model, where the term efficient frontier is used to denote what we called the set of active fares (or the maximum concave envelope in Feng and Xiao 2000, terminology).