research article dynamic pricing for airline revenue...

Research ArticleDynamic Pricing for Airline Revenue Management underPassenger Mental Accounting

Yusheng Hu, Jinlin Li, and Lun Ran

School of Management and Economics, Beijing Institute of Technology, Beijing 100081, China

Correspondence should be addressed to Yusheng Hu; [email protected]

Received 10 August 2014; Accepted 2 December 2014

Academic Editor: Fazal M. Mahomed

Copyright © 2015 Yusheng Hu et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Mental accounting is a far-reaching concept, which is often used to explain various kinds of irrational behaviors in human decisionmaking process.This paper investigates dynamic pricing problems for single-flight andmultiple flights settings, respectively, wherepassengers may be affected by mental accounting. We analyze dynamic pricing problems by means of the dynamic programmingmethod and obtain the optimal pricing strategies. Further, we analytically show that the passenger mental accounting depth has apositive effect on the flight’s expected revenue for the single flight and numerically illustrate that the passenger mental accountingdepth has a positive effect on the optimal prices for the multiple flights.

1. Introduction

In airline revenue management, revenue maximization couldbe attained by dynamic pricing and capacity control [1–6], inwhich dynamic pricing is regarded as the engine and coretechnique of airline revenue management and plays a veryimportant role in improving airline’s revenue. Obviously, therapid development of Internet has enabled airlines to collectenormous pieces of market information. Therefore, airlinescould timely change flight prices to increase their revenueaccording to the market demand and the remaining seat levelof flight.The success of dynamic pricing depends on whetherthe passengers’ response to price change can be anticipatedproperly.

Most classical dynamic pricing models suppose thatpassenger behavior is rational [7]; that is, a passengerchooses the flight as soon as the flight price is below his/hervaluation. However, in the practical condition, passengersusually exhibit some irrationality; for example, passengercouldmentally aggregate or segregate purchase systematicallyin view of factors such as time or some uncertain eventsbefore making valuations. Thus, it is important to examinepassengers’ decision processes and then set the correspond-ing flight price for the airline. Mental accounting first studiedby Thaler has a far-reaching impact on finance and behavior

economics [8], which is often used to interpret all kinds ofirrational behaviors in the decision making process. It refersto the tendency that people divide their assets into severalindependent accounts based on various subjective criteria,such as the source of the assets and the intention of everyaccount. In the light of mental accounting, different levels ofthe utility are allocated to every account by individuals, whichinfluences their spending decisions.

In recent years, as the customer behavior issue has beenintroduced into the operations management field, modelingcustomer behavior increasingly attracts extensive attention inthe area of dynamic pricing and revenuemanagement [9–12].

In the customer choice behavior model, dynamic pricingofmultiple products is studied extensively. Zhang andCooper[13] develop a Markov decision process formulation of adynamic pricing problemwithmultiple parallel flights, takinginto account customer choice among the flights. Akcay etal. [14] use stochastic dynamic programming method toapproach the joint dynamic pricing issue of multiple ver-tically differentiated products and characterize the optimalprices. Lin and Sibdari [15] establish a gamemodel to describethe dynamic pricing competition between companies thatsell substitutable products and prove the existence of theNash equilibrium prices. Suh and Aydin [16] investigate thedynamic pricing problem of two substitutable products based

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015, Article ID 836434, 8 pageshttp://dx.doi.org/10.1155/2015/836434

2 Mathematical Problems in Engineering

on multinomial logit (MNL) choice model, and the resultsindicate that, under the optimal pricing strategy, themarginalrevenue function increases with the remaining time anddecreases with the inventory of each product. Dong et al.[17] apply MNL choice model to study dynamic pricing andinventory control of substitutable products.

In the strategic customer model, dynamic pricing prob-lems are still under development. Aviv and Pazgal [18] focuson the dynamic pricing problem of seasonal goods in thepresence of strategic customer. Levin et al. [19, 20] studythe influence of strategic customer behavior on the dynamicpricing policies in the case of monopoly and oligopoly,respectively. Levina et al. [21] consider dynamic pricingbased on online learning and strategic customers. Chen andZhang [22] solve the dynamic targeted pricing problem withstrategic customers and show that the dynamic targetedpricing can get more profits for competing firms, if theyactively pursue customer recognition on the basis of customerpurchase records.

The concept of customer mental accounting has beenwidely studied in the operations management literature. Liaoand Chu [23] utilize the mental accounting theory to studyhow customers’ economic psychology associated with pur-chasing a new product is influenced when a consciousness ofthe possibility of online retail is aroused. The result indicatesthat customers’ consciousness of the retail value of alreadypossessed product can affect their decision to buy a newproduct. Erat and Bhaskaran [24] formulate a simple modelto investigate how mental accounting associated with a baseproduct influences a customer’s add-on purchase decision.Chen et al. [25] study the inventory order quantity problembased on the principle of mental accounting. In the existingliteratures related to mental accounting, researchers havepaid less attention to dynamic pricing problem.

In this paper, we investigate the multiperiod dynamicpricing problems for a single flight and multiple flights inthe case of a monopoly airline selling the ticket to passengerswho could be affected by mental accounting. We explicitlymodel the passenger dynamic choice process by MNL choicemodel, in which the probability of purchasing a ticket isspecified as a function of mental accounting depth. At eachdecision period, airline dynamically determines the flightprice in order to maximize the expected revenue over thebooking horizon. Given the passenger choice model, weformulate the dynamic pricing problems using the dynamicprogramming method for a single flight and multiple flightsin the presence of passenger mental accounting. Then weobtain the optimal pricing strategies. The optimal prices aresignificantly affected by the depth of mental accounting.Furthermore, we numerically demonstrate the positive effectof passenger mental accounting on airline’s dynamic pricingstrategies.

The model of passenger mental accounting in this papercomplements the related research on behavioral operations,and the aim of this paper is to establish a manageablemodeling framework that embodies mental accounting. Theremainder of this paper is organized as follows. In Section 2,we present a single-flightmodel formulation and characterizethe optimal price. In Section 3, we make further discussion

on the dynamic pricing problem of multiple flights followedby a numerical study presented in Section 4. Finally, wesummarize this paper in Section 5.

2. Dynamic Pricing for a Single Flight

In this section, we investigate a single-flight dynamic pricingproblem. The model can be regarded as a building blockof the multiple-flight case. Consider a single-leg flight withcapacity 𝐶. We divide the booking horizon into 𝑇 discretetime periods such that there is at most one passenger arrivalin each time period and suppose that each passenger booksnomore than one ticket.The time period is counted in reversechronological order. Therefore, the first time period is 𝑇 andthe last time period is 1. In each time period, passengersarrive independently, and there is one passenger arrival withprobability 𝜆. The airline’s goal is to maximize the expectedrevenue from the booking horizon subject to the capacityconstraint.

2.1. Dynamic Pricing Formulation. We first present a single-flight decision framework to capture passenger’s mentalaccounting. Consider a potential passenger deciding whetherto purchase a flight ticket or not to buy at all. Suppose thatpayment of purchasing the flight ticket is classified in mentalaccounting, such as travel mental accounting. Let 𝛾 (≥ 0) bethe trigger increment of mental accounting. When 𝛾 > 0,a passenger possesses this mental accounting; otherwise, apassenger does not possess this mental accounting. We couldinterpret 𝛾 as the depth of mental accounting, because itcaptures how strongly each passenger is influenced bymentalaccounting.

Assume that the utility from passenger’s purchasing is 𝑈and the utility fromnonpurchasing is𝑈

0. In detail,𝑈 depends

on the ticket price and passenger’s valuation for the ticket,and 𝑈

0depends on the passenger’s external option such as

the future purchase and the change of travel plans. Given𝑈 and 𝑈

0, the perfectly rational passenger will purchase the

flight ticket if and only if 𝑈 ≥ 𝑈0[2]. However, when mental

accounting is considered, the passenger will purchase theflight ticket if and only if

𝑈 + 𝛾 ≥ 𝑈0. (1)

We describe the passenger choice process by MNL choicemodel. The MNL choice model has a wide range of appli-cations in the marketing literature, since it is analyticallytractable, considerably accurate, and could be estimatedeasily by standard statistical methods [26]. The distinction ofapplication is that passenger mental accounting is capturedin our model. At the beginning of each period, airline sets itsflight price, and then a passenger decideswhether to purchaseone or not to purchase at all. When facing a price 𝑝

𝑡, for each

passenger, the utility from purchasing a flight ticket is equalto

𝑈𝑡= 𝑎 − 𝑝

𝑡+ 𝛾 + 𝜁, (2)

where 𝑎 is the quality index of the flight, which describes theservice level, brand image, and the popularity of the flight,

Mathematical Problems in Engineering 3

and 𝜁 is a Gumbel random variable with shift parameterzero and scale parameter one.The utility fromnonpurchasingis 𝑈0; without loss of generality, we suppose that 𝑈

0is

normalized to zero.Let 𝑞𝑡be the probability that a passenger chooses to

purchase the flight ticket at price 𝑝𝑡in each period 𝑡. Based

onMNL choice model, we can write the purchase probability𝑞𝑡as

𝑞𝑡=𝑒𝑎−𝑝𝑡+𝛾

1 + 𝑒𝑎−𝑝𝑡+𝛾; (3)

therefore, in each period 𝑡, the probability that a passengerdecides not to purchase any flight ticket is 𝑞

0𝑡= 1 − 𝑞

𝑡.

Given the remaining seat level 𝑥𝑡, let𝑉𝑡(𝑥𝑡) be the optimal

expected revenue from period 𝑡 to the last period. Then weformulate the single-flight dynamic pricing problem as in thefollowing Bellman equation:

𝑉𝑡(𝑥𝑡) = max𝑝∈𝑅+

{𝜆 [𝑞𝑡(𝑝𝑡+ 𝑉𝑡−1(𝑥𝑡− 1)) + 𝑞

0𝑡𝑉𝑡−1(𝑥𝑡)]

+ (1 − 𝜆)𝑉𝑡−1 (𝑥𝑡)}

= max𝑝∈𝑅+

{𝜆 [𝑞𝑡(𝑝𝑡+ 𝑉𝑡−1(𝑥𝑡− 1) − 𝑉

𝑡−1(𝑥𝑡))]

+ 𝑉𝑡−1(𝑥𝑡)}

(4)

with boundary conditions𝑉0(𝑥𝑡) = 0, for all𝑥

𝑡, and𝑉

𝑡(0) = 0,

for 𝑡 = 1, 2, . . . , 𝑇.LetΔ𝑉

𝑡−1(𝑥) = 𝑉

𝑡−1(𝑥)−𝑉

𝑡−1(𝑥−1) represent themarginal

expected revenue with remaining seat level 𝑥 in period 𝑡 − 1.Using this notation, we can rewrite (4) as follows:

𝑉𝑡(𝑥𝑡) = max𝑝∈𝑅+

{𝜆𝑞𝑡[𝑝𝑡− Δ𝑉𝑡−1(𝑥𝑡)] + 𝑉

𝑡−1(𝑥𝑡)} . (5)

2.2. Optimal Pricing Strategy. In the following section, wedescribe the optimal pricing strategy. Before obtaining theoptimal price and expected revenue, we first rewrite 𝑝

𝑡as a

function of the purchase probability 𝑞𝑡. From (3), we have

𝑝𝑡= 𝑎 + 𝛾 − ln (𝑞

𝑡) + ln (1 − 𝑞

𝑡) . (6)

Then we can reformulate the expected revenue function as

𝑉𝑡(𝑥𝑡)=max𝑝∈𝑅+

{𝜆𝑞𝑡[𝑎 + 𝛾 − ln (𝑞

𝑡) + ln (1 − 𝑞

𝑡) − Δ𝑉

𝑡−1(𝑥𝑡)]

+𝑉𝑡−1(𝑥𝑡)} .

(7)

Next, we define function

𝜑 (𝑥𝑡, 𝑝𝑡) = 𝜆𝑞

𝑡[𝑝𝑡− Δ𝑉𝑡−1(𝑥𝑡)]

= 𝜆𝑞𝑡[𝑎 + 𝛾 − ln (𝑞

𝑡) + ln (1 − 𝑞

𝑡) − Δ𝑉

𝑡−1(𝑥𝑡)] .

(8)

InTheorem 1, we can show that𝜑(𝑥𝑡, 𝑝𝑡) is a concave function

of 𝑞𝑡in the case of a single flight.

Theorem 1. Function 𝜑(𝑥𝑡, 𝑝𝑡) is concave in 𝑞

𝑡.

Proof. BecauseΔ𝑉𝑡−1(𝑥𝑡) is not influenced by 𝑞

𝑡, we view it as

a constant. Taking the first derivative of 𝜑(𝑥𝑡, 𝑝𝑡)with respect

to 𝑞𝑡yields

𝜕𝜑 (𝑥𝑡, 𝑝𝑡)

𝜕𝑞𝑡

= 𝜆 [𝑎 + 𝛾 − ln (𝑞𝑡) + ln (1 − 𝑞

𝑡) − Δ𝑉

𝑡−1(𝑥𝑡) −

1

1 − 𝑞𝑡

] .

(9)

In order to derive the concavity of 𝜑(𝑥𝑡, 𝑝𝑡), we take the

second derivative of 𝜑(𝑥𝑡, 𝑝𝑡) with respect to 𝑞

𝑡, which leads

to

𝜕2𝜑 (𝑥𝑡, 𝑝𝑡)

𝜕𝑞2

𝑡

= −𝜆1

𝑞𝑡(1 − 𝑞

𝑡)2< 0. (10)

It then follows that 𝜑(𝑥𝑡, 𝑝𝑡) is concave in 𝑞

𝑡.

FromTheorem 1, we can also conclude that the expectedrevenue function𝑉

𝑡(𝑥𝑡) is concave in 𝑞

𝑡. InTheorem 2,we can

derive the optimal price 𝑝∗𝑡(𝑥𝑡) to problem (5).

Theorem 2. For a given remaining seat level 𝑥𝑡in period 𝑡, the

optimal price is set to

𝑝∗

𝑡(𝑥𝑡) = Δ𝑉

𝑡−1(𝑥𝑡) + 𝜔 (𝑡, 𝑥

𝑡) , (11)

where 𝜔(𝑡, 𝑥𝑡) is the unique solution such that

[𝜔 (𝑡, 𝑥𝑡) − 1] 𝑒

𝜔(𝑡,𝑥𝑡)− 𝑒𝑎+𝛾−Δ𝑉

𝑡−1(𝑥𝑡)= 0. (12)

Furthermore, the optimal expected revenue is given by

𝑉∗

𝑡(𝑥𝑡) = 𝜆

𝑡

∑

ℎ=1

[𝜔 (ℎ, 𝑥𝑡) − 1] . (13)

Proof. From Theorem 1, we have known the concavity of𝜑(𝑥𝑡, 𝑝𝑡). In order to obtain the optimal price 𝑝∗

𝑡(𝑥𝑡), we

compute the first derivative of 𝜑(𝑥𝑡, 𝑝𝑡)with respect to 𝑞

𝑡.The

first-order condition yields

𝜕𝜑 (𝑥𝑡, 𝑝𝑡)

𝜕𝑞𝑡

= 𝜆 [𝑎 + 𝛾 − ln (𝑞𝑡) + ln (1 − 𝑞

𝑡) − Δ𝑉

𝑡−1(𝑥𝑡) −

1

1 − 𝑞𝑡

] = 0.

(14)

Defining 𝜔(𝑡, 𝑥𝑡) = 1/(1 − 𝑞

𝑡) and then substituting 𝜔(𝑡, 𝑥

𝑡)

into (14), we have

𝑞𝑡=

1

𝜔 (𝑡, 𝑥𝑡)𝑒𝑎+𝛾−Δ𝑉

𝑡−1(𝑥𝑡)−𝜔(𝑡,𝑥

𝑡). (15)

Since 𝑞𝑡= 1 − 1/𝜔(𝑡, 𝑥

𝑡), we can get

1 −1

𝜔 (𝑡, 𝑥𝑡)=

1

𝜔 (𝑡, 𝑥𝑡)𝑒𝑎+𝛾−Δ𝑉

𝑡−1(𝑥𝑡)−𝜔(𝑡,𝑥

𝑡). (16)


This means that 𝜔(𝑡, 𝑥𝑡) is one solution to

[𝜔 (𝑡, 𝑥𝑡) − 1] 𝑒


𝑡−1(𝑥𝑡)= 0. (17)

We then show that 𝜔(𝑡, 𝑥𝑡) is the unique solution to (17).

Let Φ(𝜉) = (𝜉 − 1)𝑒𝜉 − 𝑒𝑎+𝛾−Δ𝑉𝑡−1(𝑥𝑡); we immediately getΦ(1) < 0, Φ(+∞) = +∞. Since 𝜕Φ(𝜉)/𝜕𝜉 = 𝜉𝑒𝜉 > 0 for any𝜉 ∈ (1, +∞), it follows that Φ(𝜉) is monotonically increasingin 𝜉 for 𝜉 ∈ (1, +∞). This indicates that 𝜔(𝑡, 𝑥

𝑡) ∈ (1, +∞)

is the unique solution to (17). We then obtain the optimalsolution 𝑝∗

𝑡(𝑥𝑡) from 𝜔(𝑡, 𝑥

𝑡). Substituting 𝑞

𝑡= 1 − 1/𝜔(𝑡, 𝑥

𝑡)

into (6), we can derive 𝑝∗𝑡(𝑥𝑡) = 𝑎 + 𝛾 − ln[𝜔(𝑡, 𝑥

𝑡) − 1].

Rewriting the above expression, we have 𝜔(𝑡, 𝑥𝑡) − 1 =

𝑒𝑎+𝛾−𝑝

∗

𝑡(𝑥𝑡). From (17), we then get

𝑝∗

𝑡(𝑥𝑡) = Δ𝑉

𝑡−1(𝑥𝑡) + 𝜔 (𝑡, 𝑥

𝑡) . (18)

Substituting (18) into (5), we have

𝑉∗

𝑡(𝑥𝑡) = 𝜆 [𝜔 (𝑡, 𝑥

𝑡) − 1] + 𝑉

𝑡−1(𝑥𝑡) . (19)

Applying the boundary conditions, we can get

𝑉∗


𝑡

∑

ℎ=1

[𝜔 (ℎ, 𝑥𝑡) − 1] . (20)

Theorem 2 describes the evolution of optimal prices. Forconvenience of exposition, 𝜔(𝑡, 𝑥

𝑡) is defined as the revenue

margin of selling one seat in period 𝑡; namely, 𝜔(𝑡, 𝑥𝑡)

denotes the immediate contribution from selling one seat,and Δ𝑉

𝑡−1(𝑥𝑡) denotes the future value of one remaining seat

to the overall revenue. From (18), we can observe that theoptimal price 𝑝∗

𝑡(𝑥𝑡) depends on the interaction between

the immediate contribution and the future value of oneremaining seat. From (17), it is clear that an increase of𝜔(𝑡, 𝑥

𝑡)

leads to a decrease of Δ𝑉𝑡−1(𝑥𝑡), which leads to complicated

optimal price patterns.Theorem 3 provides the effect ofmental accounting depth

on flight’s expected revenue, considering no seat scarcity.

Theorem 3. Let 𝑥𝑡denote the remaining seat level in period 𝑡;

if 𝑥𝑡≥ 𝑡, then 𝑉

𝑡(𝑥𝑡) is nondecreasing with respect to mental

accounting depth 𝛾.

Proof. To prove this theorem, we need to show that, for all𝑖 ≤ 𝑡, Δ𝑉

𝑖(𝑥𝑡) = 0 given that 𝑥

𝑡> 𝑡.

First, let us consider 𝑡 = 1. Since Δ𝑉0(𝑥) = 0 for all 𝑥,

𝜔(1, 𝑥1) is determined by solving

[𝜔 (1, 𝑥1) − 1] 𝑒

𝜔(1,𝑥1)− 𝑒𝑎+𝛾= 0. (21)

This implies that 𝜔(1, 𝑥1) is independent of 𝑥

1. So all 𝑉

1(𝑥1)

are equal for any 𝑥1> 0. It follows that Δ𝑉

1(𝑥1) = 𝑉

1(𝑥1) −

𝑉1(𝑥1−1) = 0, if 𝑥

1> 1;Δ𝑉

1(1) = 𝑉

1(1)−0 = 𝑉

1(1), if 𝑥

1= 1.

Next, we consider 𝑡 = 2. Note that since Δ𝑉1(𝑥2) = 0 for

𝑥2≥ 2, 𝜔(2, 𝑥

2) for 𝑥

2≥ 2 is determined by solving

[𝜔 (2, 𝑥2) − 1] 𝑒

𝜔(2,𝑥2)− 𝑒𝑎+𝛾= 0. (22)

This means that all 𝜔(2, 𝑥2) are equal for any 𝑥

2≥ 2.

Therefore, if 𝑥2≥ 2, all 𝑉

2(𝑥2) values are the same and

Δ𝑉2(𝑥2) = 𝑉

2(𝑥2) − 𝑉2(𝑥2− 1) = 0 for any 𝑥

2> 2. But if

𝑥2= 2, Δ𝑉

2(2) = 𝑉

2(2) − 𝑉

2(1) = 0.

Similarly, we can deduce that all 𝑉𝑡(𝑥𝑡) are equal for any

𝑥𝑡≥ 𝑡, and Δ𝑉

𝑡(𝑥𝑡) = 0, if 𝑥

𝑡> 𝑡. It follows that for all 𝑖 ≤

𝑡Δ𝑉𝑖(𝑥𝑡) = 0 given that 𝑥

𝑡> 𝑡. From the above derivation

process, we can conclude that all𝜔(𝑖, 𝑥𝑡) are equal for all 𝑖 ≤ 𝑡,

if 𝑥𝑡≥ 𝑡. We then prove the theorem.Consider 𝑡 = 1. Since 𝑉

1(𝑥1) is independent of 𝛾, 𝑉

1(𝑥1)

is nondecreasing with respect to 𝛾.Consider 𝑡 ≥ 2 and 𝑥

𝑡> 𝑡. The corresponding 𝜔(𝑡, 𝑥

𝑡) is

determined by the equation

[𝜔 (𝑡, 𝑥𝑡) − 1] 𝑒


𝑡−1(𝑥𝑡)= 0. (23)

Let 𝑔1= [𝜔(𝑡, 𝑥

𝑡) − 1]𝑒

𝜔(𝑡,𝑥𝑡), 𝑔2= 𝑒𝑎𝑡+𝛾−Δ𝑉

𝑡−1(𝑥𝑡). Obviously,

𝑔1is monotonically increasing with respect to 𝜔(𝑡, 𝑥

𝑡), and

𝑔2is monotonically increasing with respect to 𝛾. Therefore,

𝜔(𝑡, 𝑥𝑡) is monotonically nondecreasing with respect to 𝛾.

Because all 𝜔(𝑖, 𝑥𝑡) are equal for any 𝑖 ≤ 𝑡, if 𝑥

𝑡≥ 𝑡, we can

easily verify that 𝑉𝑡(𝑥𝑡) is monotonically nondecreasing with

respect to 𝛾 from (20).

3. Dynamic Pricing for Multiple Flights

In this section, we deal with amultiple-flight dynamic pricingproblem. The airline’s goal is to maximize the expectedrevenue from the booking horizon by setting the appropriateprice for each flight in each period; in other words, inevery period 𝑡, for each current remaining seat level 𝑥

𝑡=

(𝑥1𝑡, 𝑥2𝑡, . . . , 𝑥

𝑚𝑡), the airline must determine the price vector

𝑝𝑡= (𝑝1𝑡, 𝑝2𝑡, . . . , 𝑝

𝑚𝑡).

3.1. Dynamic Programming Formulation. There are𝑚 substi-tutable flights in a single origin-destination pairwith capacity𝑐𝑖of flight 𝑖. The payments of different flights are classified

among different mental accounts. The booking horizon isdiscrete and is divided into𝑇 time periods such that there is atmost one passenger arrival in each time period, and supposethat each passenger books no more than one ticket, and thetime period runs backwards. Passengers arrive independentlyacross time periods. Let 𝜆 be the probability of a passengerarrival in each period.The utility that passenger would obtainfrom the purchase of flight 𝑗 is 𝑈

𝑗(𝑗 = 1, 2, . . . , 𝑚) and the

utility from nonpurchasing is 𝑈0; without loss of generality,

we suppose that 𝑈0is normalized to zero.

Passengers choose among the substitutable flights orchoose nothing; the probability that a passenger choosesflight 𝑗 is 𝑞

𝑗(𝑝𝑡); 𝑞0(𝑝𝑡) represents the probability that an

arriving passenger does not make any purchase. In period 𝑡,we define passenger’s utility from the purchase of flight 𝑗 atprice 𝑝

𝑗𝑡as

𝑈𝑗𝑡= 𝑎𝑗− 𝑝𝑗𝑡+ 𝑏𝑗𝛾𝑗+ 𝜁𝑗, (24)

where 𝑎𝑗represents the quality level of flight 𝑗, 𝑏

𝑗is the men-

tal accounting coefficient, which represents the satisfactiondegree that a passenger derives from choosing flight 𝑗 and


∑𝑚

𝑗=1𝑏𝑗= 1, and 𝜁

𝑗is a Gumbel random variable with shift

parameter zero and scale parameter one. Defining 𝑏𝑗𝛾𝑗= Λ𝑗,

based on MNL choice model, we conclude that a passengerwill choose flight 𝑗 with probability

𝑞𝑗𝑡=

𝑒𝑎𝑗−𝑝𝑗𝑡+Λ𝑗

1 + ∑𝑚

𝑘=1𝑒𝑎𝑘−𝑝𝑘𝑡+Λ 𝑘

, (25)

and the passenger will decide to purchase nothing withprobability 𝑞

0𝑡= 1/(1 + ∑

𝑚

𝑘=1𝑒𝑎𝑘−𝑝𝑘𝑡+Λ𝑘).

Let 𝑉𝑡(𝑥𝑡) be the optimal expected revenue from period

𝑡 to the last period. Then we can use the following dynamicprogramming to formulate the multiple-flight dynamic pric-ing problem:

𝑉𝑡(𝑥𝑡)

= max𝑝𝑡∈𝑅+

𝑚

{

{

{

𝜆[

[

𝑚

∑

𝑗=1

𝑞𝑗𝑡(𝑝𝑗𝑡+ 𝑉𝑡−1(𝑥𝑡− 𝜀𝑗)) + 𝑞

0𝑡𝑉𝑡−1(𝑥𝑡)]

]

+ (1 − 𝜆𝑡) 𝑉𝑡−1(𝑥𝑡)}

}

}


𝑚

{

{

{

𝜆[

[

𝑚

∑

𝑗=1

𝑞𝑗𝑡(𝑝𝑗𝑡+ 𝑉𝑡−1(𝑥𝑡− 𝜀𝑗) − 𝑉𝑡−1(𝑥𝑡))]

]

+ 𝑉𝑡−1(𝑥𝑡)}

}

}

,

(26)

with boundary conditions 𝑉0(𝑥𝑡) = 0, 𝑉

𝑡(0) = 0 for

𝑡 = 1, 2, . . . , 𝑇, where 𝜀𝑗denotes a 𝑛-vector with the 𝑗th

component 1 and zeros elsewhere.For the expected revenue function 𝑉

𝑡(𝑥𝑡), define the

marginal revenue of flight 𝑗 by

Δ𝑥𝑗

𝑉𝑡−1(𝑥𝑡) = 𝑉𝑡−1(𝑥𝑡) − 𝑉𝑡−1(𝑥𝑡− 𝜀𝑗)

for 𝑡 = 1, 2, . . . , 𝑇, 𝑗 = 1, 2, . . . , 𝑚.(27)

Using this notation, we can rewrite (26) as follows:

𝑉𝑡(𝑥𝑡) = max𝑝𝑡∈𝑅+

𝑚

{

{

{

𝜆[

[

𝑚

∑

𝑗=1

𝑞𝑗𝑡(𝑝𝑗𝑡− Δ𝑥𝑗

𝑉𝑡−1(𝑥𝑡))]

]

+ 𝑉𝑡−1(𝑥𝑡)}

}

}

.

(28)

3.2. Optimal Pricing Strategy. Wenext investigate the optimalpricing strategy. Before obtaining the optimal price, we firstrewrite the price of each flight as a function of passenger’s

choice probability. From 𝑞0𝑡/𝑞𝑗𝑡= 1/𝑒𝑎𝑗−𝑝𝑗𝑡+Λ𝑗 , we have 𝑝

𝑗𝑡=

𝑎𝑗+Λ𝑗− ln(𝑞

𝑗𝑡/𝑞0𝑡); substituting 𝑝

𝑗𝑡into (28), we can derive

𝑉𝑡(𝑥𝑡)


𝑚

{

{

{

𝜆[

[

𝑚

∑

𝑗=1

𝑞𝑗𝑡(𝑎𝑗+ Λ𝑗− ln𝑞𝑗𝑡

𝑞0𝑡

− Δ𝑥𝑗


]

+ 𝑉𝑡−1(𝑥𝑡)}

}

}

.

(29)

To facilitate description of structural properties, we define

𝜓 (𝑥𝑡, 𝑝𝑡) = 𝜆[

[

𝑚

∑

𝑗=1

𝑞𝑗𝑡(𝑝𝑗𝑡− Δ𝑥𝑗


]

= 𝜆[

[

𝑚

∑

𝑗=1

𝑞𝑗𝑡(𝑎𝑗+ Λ𝑗− ln𝑞𝑗𝑡

𝑞0𝑡

− Δ𝑥𝑗


]

,

(30)

which can be proved to be concave in 𝑞𝑡= (𝑞0𝑡, 𝑞1𝑡, 𝑞2𝑡, . . . ,

𝑞𝑚𝑡) in Theorem 4. Intuitively, 𝜓(𝑥

𝑡, 𝑝𝑡) is the expected

additional revenue realized in period 𝑡 by booking a singleunit of remaining seat level 𝑥

𝑡at price 𝑝

𝑡. We can interpret

𝜓(𝑥𝑡, 𝑝𝑡) as the marginal revenue of time at the remaining

seat level 𝑥𝑡in period 𝑡 when flight prices are set at 𝑝

𝑡.

Consequently, we set the optimal prices in order to maximize𝜓(𝑥𝑡, 𝑝𝑡). Therefore, exploring the structural properties of

𝜓(𝑥𝑡, 𝑝𝑡) is the key to find the optimal prices. We discuss the

structural properties of 𝜓(𝑥𝑡, 𝑝𝑡) as follows.

Theorem 4. Function 𝜓(𝑥𝑡, 𝑝𝑡) is concave in 𝑞

𝑡.

Proof. Taking the second derivatives of 𝜓(𝑥𝑡, 𝑝𝑡), we can

derive

𝜕2𝜓 (𝑥𝑡, 𝑝𝑡)

𝜕𝑞2

0𝑡

= −𝜆

𝑚

∑

𝑗=1

𝑞𝑗𝑡

(𝑞0𝑡)2,


𝜕𝑞0𝑡𝜕𝑞𝑘𝑡

=𝜕2𝜓 (𝑥𝑡)

𝜕𝑞𝑘𝑡𝜕𝑞0𝑡

=𝜆

𝑞0𝑡

,

for 𝑘 = 1, 2, . . . , 𝑚.


𝜕𝑞2

𝑘𝑡

= −𝜆

𝑞𝑘𝑡

, for 𝑘 = 1, 2, . . . , 𝑚,


𝜕𝑞𝑙𝑡𝜕𝑞𝑘𝑡

=𝜕2𝜓 (𝑥𝑡, 𝑝𝑡)

𝜕𝑞𝑘𝑡𝜕𝑞𝑙𝑡

= 0, for 𝑙 = 𝑘.

(31)


Then the Hessian matrix of 𝜓(𝑥𝑡, 𝑝𝑡) follows as

𝐻(𝜓 (𝑥𝑡, 𝑝𝑡))

=

((((((((

(

−𝜆

𝑚

∑

𝑗=1

𝑞𝑗𝑡

(𝑞0𝑡)2

𝜆

𝑞0𝑡

𝜆

𝑞0𝑡

⋅ ⋅ ⋅𝜆

𝑞0𝑡

𝜆

𝑞0𝑡

−𝜆

𝑞1𝑡

0 ⋅ ⋅ ⋅ 0

𝜆

𝑞0𝑡

0 −𝜆

𝑞2𝑡

⋅ ⋅ ⋅ 0

⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅

𝜆

𝑞0𝑡

0 0 ⋅ ⋅ ⋅ −𝜆

𝑞𝑚𝑡

))))))))

)

.

(32)

Next, we show that the Hessian matrix 𝐻(𝜓(𝑥𝑡, 𝑝𝑡)) is

negative semidefinite. Let 𝑍 = (𝑧0, 𝑧1, . . . , 𝑧

𝑚) = 0; then

𝑍𝑇𝐻(𝜓 (𝑥

𝑡, 𝑝𝑡)) 𝑍

= −𝜆[

[

𝑧2

0

𝑚

∑

𝑗=1

𝑞𝑗𝑡

(𝑞0𝑡)2+

𝑚

∑

𝑗=1

𝑧2

𝑗

𝑞𝑗𝑡

−2

𝑞0𝑡

𝑧0

𝑚

∑

𝑗=1

𝑧𝑗]

]

.

(33)

If 𝑧0∑𝑚

𝑗=1𝑧𝑗≤ 0, then 𝑍𝑇𝐻(𝜓(𝑥

𝑡, 𝑝𝑡))𝑍 ≤ 0. This implies

that 𝐻(𝜓(𝑥𝑡, 𝑝𝑡)) is negative semidefinite. Therefore, we let

𝑧0∑𝑚

𝑗=1𝑧𝑗> 0. If 𝑧

0> 0 and ∑𝑚

𝑗=1𝑧𝑗> 0, then

− 𝑍𝑇𝐻(𝜓 (𝑥

𝑡, 𝑝𝑡)) 𝑍

≥2𝜆𝑧0

𝑞0𝑡

[

[

√(

𝑚

∑

𝑗=1

𝑞𝑗𝑡)(

𝑚

∑

𝑗=1

𝑧2

𝑗

𝑞𝑗𝑡

) −

𝑚

∑

𝑗=1

𝑧𝑗]

]

.

(34)

Since

√(

𝑚

∑

𝑗=1

𝑞𝑗𝑡)(

𝑚

∑

𝑗=1

𝑧2

𝑗

𝑞𝑗𝑡

)

= √

𝑚

∑

𝑗=1

𝑧2

𝑗+ 2∑

𝑖 =𝑗

(𝑞𝑖𝑡

𝑞𝑗𝑡

𝑧2

𝑗+𝑞𝑗𝑡

𝑞𝑖𝑡

𝑧2

𝑖)

≥ √

𝑚

∑

𝑗=1

𝑧2

𝑗+ 2∑

𝑖 =𝑗

𝑧𝑖𝑧𝑗=

𝑚

∑

𝑗=1

𝑧𝑗,

(35)

it follows that −𝑍𝑇𝐻(𝜓(𝑥𝑡, 𝑝𝑡))𝑍 ≥ 0. This implies that

𝐻(𝜓(𝑥𝑡, 𝑝𝑡)) is negative semidefinite.

Similarly, we can conclude that 𝐻(𝜓(𝑥𝑡, 𝑝𝑡)) is negative

semidefinite, if 𝑧0< 0 and∑𝑚

𝑗=1𝑧𝑗< 0. It directly follows that

𝜓(𝑥𝑡, 𝑝𝑡) is concave in 𝑞

𝑡.

Here we show that 𝜓(𝑥𝑡, 𝑝𝑡) is concave in 𝑞

𝑡so that we

can investigate the property of optimal pricing strategy in theeasier way.We proceed to analyze the property of the optimalprices.

Theorem 5. For a given remaining seat level 𝑥𝑡in period 𝑡, the

optimal price of flight 𝑖 is set to

𝑝∗

𝑖𝑡(𝑥𝑡) = Δ

𝑥𝑖

𝑉𝑡−1(𝑥𝑡) + 𝜂 (𝑡, 𝑥

𝑡) , (36)

where 𝜂(𝑡, 𝑥𝑡) is the unique solution to [𝜂(𝑡, 𝑥

𝑡) − 1]𝑒

𝜂(𝑡,𝑥𝑡)−

∑𝑚

𝑗=1𝑒𝑎𝑗+Λ𝑗−Δ𝑥𝑗𝑉𝑡−1(𝑥𝑡)= 0.

Furthermore, the optimal expected revenue is given by

𝑉∗


𝑡

∑

ℎ=1

[𝜂 (ℎ, 𝑥𝑡) − 1] . (37)

Proof. FromTheorem 4,𝜓(𝑥𝑡, 𝑝𝑡) is concave in 𝑞

𝑡. In order to

obtain the optimal price 𝑝𝑗𝑡, we use the first-order condition

𝜕𝜓(𝑥𝑡, 𝑝𝑡)/𝜕𝑞𝑗𝑡= 0; that is,

𝜕𝜓 (𝑥𝑡, 𝑝𝑡)

𝜕𝑞𝑗𝑡

= 𝜆[𝑎𝑗+ Λ𝑗− ln (𝑞

𝑗𝑡) + ln(1 −

𝑚

∑

𝑘=1

𝑞𝑘𝑡)

−Δ𝑥𝑗

𝑉𝑡−1(𝑥𝑡) −

1

1 − ∑𝑚

𝑘=1𝑞𝑘𝑡

] = 0.

(38)

Substituting 𝜂(𝑡, 𝑥𝑡) = 1/(1 − ∑

𝑚

𝑘=1𝑞𝑘𝑡) into (38), we can get

𝑞𝑗𝑡=

1

𝜂 (𝑡, 𝑥𝑡)𝑒𝑎𝑗+Λ𝑗−𝜂(𝑡,𝑥

𝑡)−Δ𝑥𝑗𝑉𝑡−1(𝑥𝑡). (39)

Therefore, ∑𝑚𝑗=1𝑞𝑗𝑡= (1/𝜂(𝑡, 𝑥

𝑡)) ∑𝑚

𝑗=1𝑒𝑎𝑗+Λ𝑗−𝜂(𝑡,𝑥

𝑡)−Δ𝑥𝑗𝑉𝑡−1(𝑥𝑡).

It follows that

[𝜂 (𝑡, 𝑥𝑡) − 1] 𝑒


𝑚

∑

𝑗=1

𝑒𝑎𝑗+Λ𝑗−Δ𝑥𝑗𝑉𝑡−1(𝑥𝑡)= 0, (40)

due to ∑𝑚𝑗=1𝑞𝑗𝑡= 1 − 1/𝜂(𝑡, 𝑥

𝑡). Next, we show that 𝜂(𝑡, 𝑥

𝑡) is

the unique solution to (40).Let Ω(𝜉) = (𝜉 − 1)𝑒𝜉 − ∑𝑚

𝑗=1𝑒𝑎𝑗+Λ𝑗−Δ𝑥𝑗𝑉𝑡−1(𝑥𝑡); then we

can conclude that Ω(1) < 0 and Ω(+∞) = +∞. Since𝜕Ω(𝜉)/𝜕𝜉 = 𝜉𝑒

𝜉> 0 for any 𝜉 ∈ (1, +∞), it follows thatΩ(𝜉) is

monotonically increasing in 𝜉 for 𝜉 ∈ (1, +∞). This indicatesthat 𝜂(𝑡, 𝑥

𝑡) ∈ (1, +∞) is the unique solution to (40); that is,

∑𝑚

𝑘=1𝑞𝑘𝑡< 1. From (25) and (39), we can derive

𝑝∗

𝑖𝑡= 𝜂 (𝑡, 𝑥

𝑡) + Δ𝑥𝑖

𝑉𝑡−1(𝑥𝑡) , (41)

where 𝜂(𝑡, 𝑥𝑡) is the unique solution to [𝜂(𝑡, 𝑥

𝑡) − 1]𝑒


∑𝑚

𝑗=1𝑒𝑎𝑗+Λ𝑗−Δ𝑥𝑗𝑉𝑡−1(𝑥𝑡)= 0.

Substituting (41) into (28), we have

𝑉∗

𝑡(𝑥𝑡) = 𝜆 [𝜂 (𝑡, 𝑥

𝑡) − 1] + 𝑉

𝑡−1(𝑥𝑡) . (42)

Applying the boundary conditions, we can get

𝑉∗


𝑡

∑

ℎ=1

[𝜂 (ℎ, 𝑥𝑡) − 1] . (43)


T

P

Flight 1Flight 2

11.5

11

10.5

10

9.5

9

8.51 2 3 4 5 6 7 8 9 10

Figure 1: Optimal prices of both flights in time remaining 𝑡.

4. Numerical Examples

In this section, we numerically illustrate the effect of pas-senger mental accounting on the optimal pricing of flights.We examine sets of examples with two substitutable flights inwhich 𝑎

1= 10, 𝑎

2= 6, 𝜆 = 0.7, initial seat levels 𝑐

1= 4, 𝑐2= 6,

and corresponding mental accounting depths 𝛾1= 8 and

𝛾2= 6. The mental accounting coefficients that a passenger

derives from choosing flights 1 and 2 are 𝑏1= 0.4 and 𝑏

2= 0.6,

respectively.Figure 1 depicts the optimal prices for both flights with

respect to remaining time, at fixed seat levels 𝑐1= 4, 𝑐

2= 6.

In Figure 1, we observe time monotonicity, the optimal priceof flight 1 increases in remaining time 𝑡, and the optimalprice of flight 2 decreases in remaining time 𝑡. Moreover, theoptimal prices of both flights converge to the identical value.This means that the uniform pricing strategy begins to takeeffect when the remaining seat level is abundant relative topassenger demand.

Figure 2 illustrates the optimal prices of both flights as afunction of the remaining seat level of flight 1 in period 10,with the remaining seat level of flight 2 fixed as 𝑐

2= 6. The

optimal price of flight 1 is nonincreasing with respect to itsown remaining seat level, and the optimal price of flight 2 isnonmonotonic in the remaining seat level of flight 1.

Next, we examine how the mental accounting depthaffects the optimal prices of both flights. Specifically, we wantto depict the change in the optimal prices of both flightsover mental accounting depth when 𝑡 = 10, 𝑐

1= 4, and

𝑐2= 6. In this numerical simulation, we keep the depth of

mental accounting 2 constant; that is, 𝛾2= 6. The depth of

mental accounting 1 can be chosen from 1 to 10. Figure 3gives the optimal prices of both flights for each depth level ofmental accounting 1. From Figure 3, we observe that mentalaccounting depth significantly affects the optimal prices ofboth flights. Given the remaining seat level and remaining

P

1 2 3 4 5 6 7 8 9 10

X1 (the remaining seat level of flight 1)

14

13.5

13

12.5

12

11.5

11

10.5

10

9.5

9

Flight 1Flight 2

Figure 2: Optimal prices of both flights in the remaining seat levelof flight 1.

P

Flight 1Flight 2

1 2 3 4 5 6 7 8 9 10

12.5

12

11.5

11

10.5

10

9.5

9

8.5

8

𝛾1 (the depth of mental accounting 1)

Figure 3: Effect of mental accounting on the optimal prices.

time, the optimal prices of both flights are nondecreasingwiththe depth level of mental accounting 1.

5. Conclusion

In this paper, we study the dynamic pricing problems for asingle flight and multiple flights over finite booking horizonin the presence of mental accounting. We use Bellman equa-tion to obtain the optimal prices. Furthermore, we present thenumerical simulation to investigate how mental accountingdepth affects the optimal prices. According to the numericalexample, we derive that passenger’s mental accounting has apositive impact on the optimal prices.


In the future study, we can incorporate the batch demandinto dynamic pricing instead of unit purchase. Of course, itis also useful to incorporate competition into the dynamicpricing problem or consider demand learning in our model.

Conflict of Interests

The authors declare that there is no conflict of interests aboutthe named companies regarding the publication of this paper.

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (nos. 71172172 and 71272058), Spe-cialized Research Fund for Doctoral Program of HigherEducation of China (no. 20121101110054), Special Programfor International Science, and Technology Cooperation ofBeijing Institute of Technology (no. GZ2014215101).

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