Dynamic pricing & capacity assignment problem with cancellation and mark-up policies in airlines

Asia Pacific Management Review xxx (2016) 1e7

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Dynamic pricing & capacity assignment problem with cancellation and mark-up policies in airlines* Moon Gil Yoon a, *, Hwi Young Lee b, Yoon Sook Song c a

Department of Business Administration, Korea Aerospace University, Gyunggi-do, South Korea Department of Aviation Management, Inha Technical College, Incheon, South Korea c Passenger Business Division, Korean Air, Seoul, South Korea b

a r t i c l e i n f o

a b s t r a c t

Article history: Received 12 August 2015 Accepted 19 August 2016 Available online xxx

Revenue management has started on the capacity control by booking classes for available seats, and has been recognized as a powerful tool to maximize the total revenue. Among various RM techniques, pricing and seat controls have played key roles in airlines. Since pricing and seat control problems are highly correlated in RM problem. These two decision problems need to be considered jointly. However, due to the complex interaction between them, a few researches focus on the joint pricing and seat control problem. In this paper we consider a comprehensive problem including a cancellation in booking processes and a mark-up policy in pricing strategy under uncertain demands. To manage the demand uncertainty efficiently, we applied a linear approximation technique and proposed a new approximation model, a simple mixed Integer Programming model. From the computational experiments with randomly generated data, we can find our model makes good performance for deciding pricing and seat controls simultaneously. © 2016 College of Management, National Cheng Kung University. Production and hosting by Elsevier Taiwan LLC. All rights reserved.

Keywords: Dynamic pricing Seat allocation Booking control Cancellation Mark-up policy

1. Introduction Revenue Management (RM) has received considerable attention both from practitioners and academics for many years. Among various RM techniques, pricing and seat controls have played key roles in airlines. Since airlines have offered competitive prices to customers for coping with severe competition, dynamic pricing, which is to adjust the price in a timely fashion for responding uncertain demand, has become a common strategy to maximize the total revenue. Also, they have tried to protect the high yield customers from the low ones to increase their revenue by seat control optimally (Talluri & van Ryzin, 2005).

* This work was supported by the National Research Foundation of Korea Grant funded by the Korean Government (NRF-2012S1A5A2A01021080). * Corresponding author. Fax: þ82 2 300 0225. E-mail addresses: [email protected] (M.G. Yoon), [email protected] (H.Y. Lee), [email protected] (Y.S. Song). Peer review under responsibility of College of Management, National Cheng Kung University.

Since Littlewood (1972) proposed the expected marginal revenue (EMR) rule for a single flight leg, a lot of studies have been carried on the seat allocation problem for a single flight (Belobaba, 1987, 1989; Brumelles & McGill, 1993; Curry, 1990; Glover, Glover, Lorenzo, & McMillan, 1982; McGill & van Ryzin, 1999), and for the entire network (Curry, 1990; Glover et al., 1982; Jiang, 2008; Song, Hong, Hwang, & Yoon, 2010; Talluri & van Ryzin, 1999). For pricing control to respond to a market change, many researches have been carried out to develop comprehensive models and algorithms. Gallego and van Ryzin (1994) dealt with the dynamic pricing problem for pricing decisions. They showed that the optimal price can be increased with decreasing inventory and approaching the departure time. Useful researches in dynamic pricing problem can be found in many literature (Bitran & Caldentey, 2003; Elmaghraby & Keskinocak, 2003; Feng & Gallego, 1995; Feng & Xiao, 2000; Sen, 2013; Zhao & Zheng, 2000). Since pricing and seat control problems are highly correlated in RM problem. These two decision problems need to be considered jointly. However, due to the complex interaction between them, a few researches focus on the joint pricing and seat control problem (McGill & van Ryzin, 1999). In fact, a recent survey finds that only

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Please cite this article in press as: Yoon, M. G., et al., Dynamic pricing & capacity assignment problem with cancellation and mark-up policies in airlines, Asia Pacific Management Review (2016), http://dx.doi.org/10.1016/j.apmrv.2016.08.004

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11% of 479 companies practicing revenue management in Europe and North America manage both price and capacity allocation decisions (Kocabiyikoglu, Popescu, & Stefanescu, 2011). In this paper, we focus on the simultaneous decision problem for pricing and capacity control to maximize revenue in airlines. Cote, Marcotte, and Savard (2003) presented the bi-level programming approach for solving a pricing and capacity control problem. Maglaras and Meissner (2006) considered the dynamic pricing and capacity control problem and showed that these two problems can be formulated as a single model. Feng and Xiao (2006) considered a pricing and capacity allocation problem under time and capacity constraints. Xiao, Chen, and Chen (2007) studied a semi-dynamic pricing and seat allocation problem in which with assumptions of stochastic customer arrival processes, they suggested an approximation method. However, they did not consider cancellation frequently encountered in the booking process, and the individual arrival rate for each price at each time is very difficult to estimate in reality. Mookherjee and Friesz (2008) considered a joint problem of pricing, resource allocation and overbooking over networks under competition. Recently, de Vericourt and Lobo (2009) developed the optimization model to solve prices and allocations in a dynamic setting under a multiplicative demand model. In this paper we consider a comprehensive problem including a cancellation in booking processes and a mark-up policy in pricing strategy under uncertain demands. The half of the bookings were resulted in cancellations or no-shows, and 15% of the flight seats would be unused, if bookings were only limited to the capacity (Smith, Leimkuhler, & Darrow, 1992). Furthermore, airlines have offered various discount prices with different refund policies for their customers (Vinod, 2008). For example, the full price tickets guarantee a full refund at any time. However, airlines may sell more than 50% discount tickets with no-refund restriction to attract price sensitive customers. Therefore, the booking cancellation has much attention in airline industry and the refund policy for cancellations becomes an important factor for seat control as well as pricing decision in airlines (Iliescu, Garrow, & Parker, 2008). Yoon, Lee, and Song (2012) studied the capacity allocation problem with cancelation and refund policy in airlines. This study is an extension of their work for combining pricing and capacity control decisions in a single framework. Recently, many low cost airlines have changed their price frequently depending on market demands to attract price sensitive passengers. However, most airlines do not allow the price goes down as the departure approaches. If the price switches to the lower ones, perceived unfairness of advanced purchased customers with higher prices will be increased and hence the airlines' longterm profit will be affected (Kimes & Noon, 2002; Talluri & van Ryzin, 2005). Also, customer demand becomes less price elastic as it gets closer to the departure time. Thus, airlines usually adopt the mark-up policy which is to raise the price continually to the departure time. The objective of our problem is to find a price and its booking limit at each decision time to maximize the total revenue over the whole planning period. We first develop a stochastic model for dynamic pricing and seat allocation under uncertain demand. To manage the demand uncertainty, we consider an approximation method which is widely adapted to solve the uncertainty with ease (Jiang, 2008; Mookherjee & Friesz, 2008; Szwarc, 1964; Gallego & van Ryzin, 1994). In section 2, we first describe the modeling processes for our stochastic problem. We will show ours can be formulated as a mixed Integer Programming model approximately. In section 3, to evaluate the performance of our model, we will test some computational experiments with randomly generated data. Some discussions and extensions of our research are given in the last section.

2. Dynamic pricing and seat allocation model with cancellation & mark-up policy In this paper, the whole planning period is divided into T discrete times. At each time, we can select a price from a predetermined price set. In our problem, we assume that the demand for each price is uncertain and independent with others. Let C be the seat capacity and rjt be the j-th price among alternative prices at time t. To formulate our problem, we define the following variables and notations. t g: a set of predetermined prices at time t, R t ¼ fr1t ; r2t ; ::; rM

Dt ¼ fDt1 ; Dt2 ; ::; DtM g : a set of demand for price rjt at time t, (Dtj is a random variable and the maximum demand at time t t

is Dj .) dtj : The cancellation rate and the refund rate for rjt at time t respectively t t ð0  qj ; dj  1Þ; t xj : the number of seats assigned for the demand for price rjt at time t. ztj : 0, 1 integer variable denoting the selection of price rjt at time t.

qtj ,

At time t, given a price rjt and the number of assigned seats xtj , the actual amount of sales is depend on the demand occurred at the time. If the demand Dtj is greater than xtj , the excess demand above the number of assigned seats xtj is rejected all. While if the demand is less than xtj , all requested demand is accepted. Then airlines can be obtained the actual sales by minðDtj ; xtj Þ: However, among the accepted demand, airlines may have some cancellations by various reasons, thereby they have to refund for the cancellation according to refund policy. Usually, airlines differentiate the refund policy for the different price. Considering the cancellation and refund rates qtj , dtj , the actual revenue can be represented as rjt ð1  dtj qtj Þ½minðDtj ; xtj Þ. Since the demand is a random variable, the expected revenue for price rjt at time t can be obtained by rjt ð1  dtj qtj ÞE½minðDtj ; xtj Þ. Noting the cancellation and the mark-up policy as a pricing strategy, our problem can be formulated as a probabilistic nonlinear programming (PNLP) model to maximize the total revenue.

ðP0 ÞMax: Zp ¼

T X M   h  i X rjt 1  dtj qtj E min Dtj ; xtj ;

(1)

t¼1 j¼1 T X M   X t 1  qj xtj  C;

(2)

t¼1 j¼1 t

xtj  Dj ztj ; M X

j ¼ 1; …; M; t ¼ 1; …; T;

ztj ¼ 1;

t ¼ 1; …T;

(3)

(4)

j¼1

rjt ztj 

M X

rltþ1 ztþ1 ; l

j ¼ 1; …; M; t ¼ 1; …; T  1;

(5)

l¼1

xtj  0; ztj 2f0; 1g;

j ¼ 1; …; M; t ¼ 1; …; T;

(6)

The objective function (1) describes the total expected revenue which has to be maximized. In general, airlines have made an overbooking to minimize the empty seat resulting from the cancellation, and the over-booking pad depends on the cancellation rate for each

Please cite this article in press as: Yoon, M. G., et al., Dynamic pricing & capacity assignment problem with cancellation and mark-up policies in airlines, Asia Pacific Management Review (2016), http://dx.doi.org/10.1016/j.apmrv.2016.08.004

M.G. Yoon et al. / Asia Pacific Management Review xxx (2016) 1e7

price. Noting that we focused on the cancellation, constraint (2) describes over-booking schemes related with the cancellation. In constraint (2), the cancellation rate should be incorporated into the number of assigned seats. Thereby, the total number of assigned PP t seats ð xj Þ may exceed the seat capacity (C). For each price rjt and t

ztj ¼ 1;

maximum demand. At each time, the selected price is only one among the predetermined price set. Constraint (4) denotes it. The mark-up policy can be represented as constraint (5). In constraint (5), when the price j is selected at time t, the prices for the remaining period have to be equal to or greater than that price. Without constraint (5), the price drop is permitted at any time during the planning period and it may result in a serious price unfairness problem. The integer conditions (6) represent the number of seats allocating for each price should have an integer value. Even though our problem is formulated as a mathematical model (P0), it contains random variables for denoting the uncertain demand. Since the demand Dtj is a random variable with a distribution function, the objective has a non-linear function and (P0) becomes very difficult to solve. Due to the problem complexity of (P0), it is useful to apply an approximation technique. When the deterministic demand is given for Dtj , (P0) can be represented as a simple deterministic integer programming (DIP) model being solved easy. The solution of DIP can provide some guideline for the pricing and the seat control decision, but it is far from the optimal solution. In this paper, we consider two approximation methods to solve (P0) efficiently: randomized integer programming (RIP) and linear approximation technique. RIP method comes from the concept of the randomized linear programming method for solving seat allocation problem in network RM problems proposed by Talluri and van Ryzin (1999). For each iteration in RIP, we generate the demand for price rjt at time t randomly. With the randomly generated demand, we apply DIP and obtain the price and the seat allocation at each time. After N iterations, we can get the price having the most frequency at each ~t time and the seat allocation for the price as simple average. Let D jk be the randomly generated demand at k iteration for price rjt at time t. In (P0), replacing the objective function (1) and the constraint (3) with (1)0 and (3)0 respectively, we can obtain k-th RIP model (RIPk): T X M   X ðRIPk ÞMax: Zp ¼ rjt 1  dtj qtj xtj ;

0

(1)

t¼1 j¼1

 t 1  qj xtj  C;

(2)

t¼1 j¼1

~ t zt ; xtj  D jk j

(3)0

j ¼ 1; …; M; t ¼ 1; …; T;

0 = α iKt j

t ¼ 1; …T;

rjt ztj 

M X

rltþ1 ztþ1 ; l

xtj  0; ztj 2f0; 1g;

(5)

j ¼ 1; …; M; t ¼ 1; …; T:

(6)

For applying the linear approximation technique, we will assume that the demand distribution of each price has a step function with several sub-intervals. The linear approximation technique was suggested by Szwarc (1964) for stochastic transportation problem. It was expanded by Tcha and Yoon (1985) for stochastic facility location problem, Song et al. (2010) and Yoon et al. (2012) for a seat allocation problem with stochastic demand. Let Kj be the number of sub-intervals for the demand of the j-th price. Let Dtjs and ptjs be the interval size and the probability at s-th subinterval for the demand of price rjt at time t respectively. Let atjs be P t t defined as follow: atjs ¼ Dj  sk¼1 Dtjk . Then we can see atj0 ¼ Dj and

atjKj ¼ 0 (Fig. 1). With the assumption of step functions for demand distributions, the demand Dtj has an uniform distribution and its probability ptjs

density function can be represented as ½atjs ; atjs1 .

Let's consider  xtj < atjs1 pected actual revenue for price rjt at time t

atjs

h 





ptj ¼ E rjt 1  qtj dtj Min Dtj ; xtj 

¼ rjt 1  qtj dtj

8 > Zxtj < > > > : at



Dtjs

in the sub-interval

and ptj denote the exin the interval ½atjs ; atjs1 :

i 

t

Zaj0

Dtj f Dtj dDtj þ

jKj

 n    o t t ¼ rjt 1  qj dj Z t1 xtj þ Z2t xtj





xtj f Dtj dDtj xtj

9 > > = > > ;

R t R t t ðxt Þ ¼ xj Dt f ðDt ÞdDt , Z t ðxt Þ ¼ aj0 Dt f ðDt ÞdDt . where, Zj1 j j j j2 j j j j atjK j xtj j t ðxt Þ and Z t ðxt Þ are non-linear functions in the interval Since Zj1 j j2 j ½atjs ; atjs1 , we replace them with linear functions approximately in the interval to reduce the problem complexity. Then ptj can be approximated as a following linear function in the interval ½atjs ; atjs1 :





ptj zHjst þ htjs xtj  atjs ;

p tj1

•• Δ jK j

j ¼ 1; …; M; t ¼ 1; …; T  1;

l¼1

p tjs t

(4)

j¼1

j

time t, constraint (3) represents the seat can be assigned only if the price rjt is selected, and the seat allocation cannot exceed the

T X M  X

M X

3

D jt = α tj 0

••• α

t js

Δ js t

α

t js −1

Δt j1

Fig. 1. Demand distribution for Dtj : step function.

Please cite this article in press as: Yoon, M. G., et al., Dynamic pricing & capacity assignment problem with cancellation and mark-up policies in airlines, Asia Pacific Management Review (2016), http://dx.doi.org/10.1016/j.apmrv.2016.08.004

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M.G. Yoon et al. / Asia Pacific Management Review xxx (2016) 1e7

0 t ¼ r t ð1  qt dt Þ@at where, Hjs j j j js

rjt ð1



qtj dtj Þ

ptjs 2

þ

Ps

t k¼1 pjk þ

1

PKi

ðatjk þatjk1 Þ t pjk A; htjs 2 k¼sþ1

¼

ðPÞMax: Zp ¼

!

Ps1

t k¼1 pjk

¼ ðxtj  atjs Þ; 0  xtjs  Dtjs . The expected actual revenue for price j

can be represented as

ptj

¼

htjs xtjs ;

(7)

t¼1 j¼1 s¼1

Consider a new variable xtjs defined in the interval [atjs , atjs1 ]: xtjs

Kj T X M X X

t Hjs

þ

htjs xtjs ,

0

xtjs



Dtjs .

Noting that

ptj

has convex piecewise linear functions over the whole interval ½atjKj ; atj0 , we can represent the total expected revenue over all the PKj t t t þ interval ½atjKj ; atj0 , as follows: HjK s¼1 hjs xjs . Consequently, j applying the linear approximation technique and using the new variables, (P0) can be transformed to a simple integer programming model (P).

Kj  T X M X X

 t 1  qj xtjs  C

(8)

t¼1 j¼1 s¼1

xtjs  Dtjs ;

Kj X

s ¼ 1; …; Kj ; j ¼ 1; …; M; t ¼ 1; ::T;

t

xtjs  Dj ztj ;

(9)

j ¼ 1; …; M; t ¼ 1; …; T;

(10)

s¼1

Table 1 Input demand for computational experiments.

Time

Price

Upper bounds

T¼1

100 95 90 85 70 100 95 90 85 70 100 95 90 85 70 100 95 90 85 70 100 95 90 85 70 100 95 90 85 70 100 95 90 85 70

6 15 20 25 30 9 20 25 30 45 9 20 25 30 45 12 22 30 45 50 14 25 30 45 50 6 15 20 25 30 9 20 25 30 45

T¼2

T¼3

T¼4

T¼5

T¼6

T¼7

8 17 22 27 32 11 21 26 31 46 11 21 26 31 46 14 24 32 47 51 16 26 32 47 52 8 17 22 27 32 11 21 26 31 46

10 18 23 28 33 12 22 27 32 47 12 22 27 32 47 15 25 34 48 52 17 27 34 49 53 10 18 23 28 33 12 22 27 32 47

12 19 24 29 34 13 23 28 33 48 13 23 28 33 48 16 26 35 49 53 18 28 35 50 54 12 19 24 29 34 13 23 28 33 48

14 20 25 30 35 14 24 29 34 49 14 24 29 34 49 17 27 36 50 54 19 29 36 51 55 14 20 25 30 35 14 24 29 34 49

15 21 26 31 36 15 25 30 35 50 15 25 30 35 50 18 28 37 51 55 20 30 37 52 56 15 21 26 31 36 15 25 30 35 50

16 22 27 32 37 16 26 31 36 51 16 26 31 36 51 18 29 38 52 56 21 31 38 53 57 16 22 27 32 37 16 26 31 36 51

17 23 28 33 38 17 27 32 37 52 17 27 32 37 52 20 30 39 53 57 22 32 39 54 58 17 23 28 33 38 17 27 32 37 52

18 24 29 34 39 18 28 33 38 53 18 28 33 38 53 21 31 40 54 58 23 33 40 55 59 18 24 29 34 39 18 28 33 38 53

20 25 30 35 40 19 29 34 39 54 19 29 34 39 54 22 32 41 55 59 24 34 41 56 60 20 25 30 35 40 19 29 34 39 54

22 27 32 37 42 20 30 35 40 55 20 30 35 40 55 24 33 42 56 60 25 35 42 57 61 22 27 32 37 42 20 30 35 40 55

24 29 34 39 44 22 32 37 42 57 22 32 37 42 57 26 34 44 58 62 26 36 43 58 62 24 29 34 39 44 22 32 37 42 57

Please cite this article in press as: Yoon, M. G., et al., Dynamic pricing & capacity assignment problem with cancellation and mark-up policies in airlines, Asia Pacific Management Review (2016), http://dx.doi.org/10.1016/j.apmrv.2016.08.004

M.G. Yoon et al. / Asia Pacific Management Review xxx (2016) 1e7

M X

ztj ¼ 1;

t ¼ 1; …T;

(11)

j¼1

rjt ztj 

M X

rltþ1 ztþ1 ; l

j ¼ 1; …; M; t ¼ 1; …; T  1;

5

Xk: the number of seat assigned at time k with corresponding k, price rj* Yt1: the number of already reserved seats up to time period t-1.

(12)

l¼1

3. Computational experiment

xtjs  0; and Integer; ztj 2f0; 1g s ¼ 1; …; Kj ; j ¼ 1; …; M; t ¼ 1; …T; (13) Since (P) is a simple mixed Integer Programming model, we can find the optimal solutions efficiently by using mathematical programming tools such as Lindo and CPLEX. The price and the number of seat assigned for that price at each time can be obtained from the t ) at each time can be optimal solutions. The optimal price (rj* defined as the price having ztj ¼ 1 in the optimal solution. The number of seats assigned can be considered as the protection level PKj t at each time period, and obtained by Xt ¼ s¼1 xj*s for the optimal t . Applying nesting structure, we can have the booking limit price rj*

at each time period. Let BL(t) be the booking limit at time t with k ; k ¼ 1; …; tg}. BL(t) can be represented corresponding price set {frj*

as:

BLðtÞ ¼

t X

Xk  Yt1 :

k¼1

Table 2 Cancellation rates and refund rates. T Cancellation rate (%) Refund rates (%) $100 $95 $90 $85 $70

1 10 90 85 80 70 60

2 8 90 85 80 70 60

3 5 90 85 80 70 60

4 2 90 85 80 70 60

5 1 90 85 80 70 60

6 1 90 85 80 70 60

7 1 90 85 80 70 60

(Departure time: T ¼ 8).

To test the performance of our model, we will try to find optimal solutions of (P) for sample data with randomly generated demands. We consider five types of prices ($100, $95, $90, $85, $70) and the planning period is divided into 7 discrete times. We assume T ¼ 1 is a starting time for seat reservation. The demand distribution of each price at each time is divided into 12 intervals and assumed a step function with the lower and the upper bounds. The demand distribution for each price at each time is given in Table 1. For example, for price $100 at T ¼ 1, the lower bound in the second interval is equal to 6 and the upper bound is equal to 8. The figure represents the probability that a demand will arrive in the interval is 7% for both prices. For cancellation, it is a common to pay a full refund for the high price. However, a partial refund may be given for the discount price. For the cancellation rate, considering the travel uncertainty, we assume different rates depending on the time of booking. For the refund rate, we assume also different rates depending on the price without respect to the time of booking. Table 2 shows the cancellation and the refund rates. Different levels of capacities that ranged from 110 to 290 are considered in the computational experiments. With the data in Tables 1 and 2, we apply RIP model and the model (P) to get the price and the seat allocation at each time by using CPLEX Optimization Studio 12.6.0 on personal computer (Pentium, 2.93 GHz). For all cases, the computation time with CPLEX is less than few seconds on a PC. To get the solution from RIP model, we generate 10 random samples from the demand distribution. From the optimal price for 10 samples, we obtain the price having the most frequent price at each time. Furthermore, the number of seat allocation at each time and the expected revenue can be obtained by simple average on that price. Tables 3 and 4 show the computational results for the model (P) and RIP model respectively. Noting that we consider the booking

Table 3 Computational results on the model (P). Capacity

110 130 150 170 190 210 230 250 270 290

Price/(booking limit/Protection level) T¼1

T¼2

T¼3

T¼4

T¼5

T¼6

T¼7

90 19/19 90 18/18 90 20/20 90 22/22 85 29/29 70 33/33 70 34/34 70 35/35 70 38/38 70 42/42

90 43/24 90 42/24 90 45/25 90 49/27 90 57/28 70 81/48 70 81/47 70 84/49 70 90/52 70 97/55

90 46/3 90 67/25 90 59/14 90 76/27 90 85/28 90 109/28 70 129/48 70 133/49 70 142/52 70 152/55

90 80/34 90 101/34 90 93/34 90 110/34 90 119/34 90 144/35 90 164/35 90 169/36 90 181/39 90 194/42

100 96/16 100 117/16 95 118/25 95 137/27 90 153/34 90 178/34 90 199/35 90 206/37 90 221/40 90 236/42

100 104/8 100 125/8 95 135/17 95 155/18 90 176/23 95 197/19 95 218/19 90 232/26 90 250/29 90 268/32

100 115/11 100 136/11 95 156/21 95 177/22 95 198/22 95 220/23 95 241/23 90 261/29 90 282/32 90 303/35

# Of assigned seats (expected revenue, $) 115 (11,649) 136 (14,385) 156 (16,161) 177 (17,930) 198 (19,352) 220 (20,720) 241 (22,499) 261 (23,340) 282 (23,900) 303 (24,051)

Please cite this article in press as: Yoon, M. G., et al., Dynamic pricing & capacity assignment problem with cancellation and mark-up policies in airlines, Asia Pacific Management Review (2016), http://dx.doi.org/10.1016/j.apmrv.2016.08.004

M.G. Yoon et al. / Asia Pacific Management Review xxx (2016) 1e7

30 Model P

25

RIP Model

20 $1,000

cancellation, the total number of seat assigned is greater than the capacity. For example, when Q ¼ 210, (P) gives us a price path {$70, $70, $90, $90, $90, $95, $95} and a set of booking limit {33, 81, 109, 144, 178, 197, 220}. That means the prices for the first two periods are $70 and we are assigned the seat available to sell up to 81. For the next three periods, we sell seats with the price $90 up to 178 including the number of seat sold in the first two periods. For the last two periods, we increase the price to $95 and sell seats up to 220 including the seat sold in the previous 5 periods. With the mark-up policy constraints, we can find the prices increase as the departure date approaches. For the same demand distribution, the prices decrease as the seat capacity Q increases. When the demand is relatively smaller than the seat capacity such as in off-peak seasons, airlines try to sell air tickets by offering various types of discount prices for increasing

Revenue

6

15 10 5 0 110

130

150

170 190 210 230 Seat Capacity Q

250

270

290

Fig. 3. Expected revenues for the model (P) and RIP model.

Table 4 Computational results on RIP model. Capacity

110 130 150 170 190 210 230 250 270 290

Price/(booking limit/Protection level) T¼1

T¼2

T¼3

T¼4

T¼5

T¼6

T¼7

100 15/15 95 20/20 95 20/20 95 23/23 90 27/27 85 33/33 85 35/35 70 38/38 70 39/39 70 37/37

100 30/15 95 44/24 95 44/24 95 50/27 90 58/31 90 65/22 85 74/39 85 75/37 70 93/54 70 88/51

100 45/15 95 69/25 95 68/24 95 77/27 90 89/31 90 96/31 90 107/33 85 112/37 85 132/39 70 139/51

100 63/18 100 86/17 95 96/28 95 108/31 95 118/29 90 134/38 90 147/40 85 165/53 85 187/55 85 191/52

100 84/21 100 106/20 95 126/30 95 141/33 95 149/31 90 171/37 90 187/40 90 204/39 90 228/41 85 244/53

100 99/15 100 121/15 100 141/15 100 159/18 95 172/23 95 193/22 95 211/24 90 232/28 95 253/25 90 271/27

100 114/15 100 136/15 100 156/15 100 177/18 95 198/26 95 219/26 95 240/29 95 260/28 95 282/29 90 302/31

# Of assigned seats (expected revenue, $) 114 (10,023) 136 (12,052) 156 (13,762) 177 (14,317) 198 (16,544) 219 (20,234) 240 (20,756) 260 (20,278) 283 (21,051) 302 (23,568)

Fig. 2. Examples of the price path and the booking limits for Q ¼ 110 and Q ¼ 210.

Please cite this article in press as: Yoon, M. G., et al., Dynamic pricing & capacity assignment problem with cancellation and mark-up policies in airlines, Asia Pacific Management Review (2016), http://dx.doi.org/10.1016/j.apmrv.2016.08.004

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their revenues. While, when the demand is relatively larger than the seat capacity such as in peak seasons, they offer only high price tickets for maximizing their revenues (Fig. 2). As shown in Tables 3 and 4, for both models, the total number of seats assigned is almost similar, but there is some difference on the prices and the number of assigned seat at each time. Since the computational results for RIP model are obtained from the average values for the results of randomly generated DIP models, Table 4 gives only feasible solutions for our complex problem. Also, the model (P) generates higher revenues than RIP model for all cases. Fig. 3 shows the expected revenues for the model (P) and RIP model. 4. Conclusion In this paper we dealt with the dynamic pricing and seat allocation problem with cancellation and mark-up policies under uncertain demand in airlines. To manage the demand uncertainty efficiently, we applied a linear approximation technique and proposed a new approximation model, a simple mixed Integer Programming model, as an alternative for solving the PNLP. With appropriate input parameters, we found that our model generates a price path and seat assignment at each time effectively by CPLEX program. Our model provides the information for a pricing and booking limit decision at each time for maximizing the expected revenue. Compared with RIP model, the model (P) provides more expected revenues for our problem. Referring the computational results, our model can be applied for providing a good guideline to make an efficient price decision and seat control for each time period. Even though we can have good results for small sample problems, it is challengeable to devise more efficient heuristic algorithm for solving large problems in real world practice. Furthermore, to enhance the applicability of our model in practice, it needs to develop more comprehensive models including diversions, noshows and passengers' behavioral factors in air travel markets. References Belobaba, P. P. (1987). Airline yield management: An overview of seat inventory control. Transportation Science, 21, 63e73. Belobaba, P. P. (1989). Application of a probabilistic decision model to airline seat inventory control. Operations Research, 37, 183e197. Bitran, G., & Caldentey, R. (2003). An overview of pricing models for revenue management. Manufacturing and Service Operations Management, 5(3), 203e229. Brumelles, S., & McGill, J. I. (1993). Airline Seat Allocation with multiple nested fare classes. Operation Research, 41(1), 127e137. Cote, J. P., Marcotte, P., & Savard, G. (2003). A bi-level modeling approach to pricing and fare optimisation in the airline industry. Journal of Revenue and Pricing Management, 2(1), 23e46.

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Curry, R. E. (1990). Optimal airline seat allocation with fare classes nested by origins and destinations. Transportation Science, 24, 193e204. Elmaghraby, W., & Keskinocak, P. (2003). Dynamic pricing in the presence of inventory considerations: Research overview, current practices and future directions. Management Science, 49, 1287e1309. Feng, Y., & Gallego, G. (1995). Optimal staring times for end-of-season sales and optimal stopping times for promotional fares. Management Science, 41, 1371e1391. Feng, Y., & Xiao, B. (2000). Optimal policies of yield management with multiple predetermined prices. Operations Research, 48, 332e343. Feng, Y., & Xiao, B. (2006). Integration of pricing and capacity allocation for perishable products. European Journal of Operational Research, 168(1), 17e34. Gallego, G., & van Ryzin, G. (1994). Optimal dynamic pricing of inventories with stochastic demand over finite horizons. Management Science, 40(8), 999e1020. Glover, F., Glover, R., Lorenzo, J., & McMillan, C. (1982). The passenger mix problem in the scheduled airlines. Interfaces, 12, 73e79. Iliescu, D. C., Garrow, L. A., & Parker, R. A. (2008). A hazard model of US airline passengers' refund and exchange behavior. Transportation Research B, 42(3), 229e242. Jiang, H. (2008). A Lagrangian relaxation approach for network inventory control of stochastic revenue management with perishable commodities. Journal of the Operational Research Society, 59(3), 372e380. Kimes, S. E., & Noon, B. M. (2002). Perceived fairness of yield management: An update. Cornell Hotel and Restaurant Administration Quarterly, 43(1), 28e29. Kocabiyikoglu, A., Popescu, I., & Stefanescu, C. (2011). Pricing and revenue management: The value of coordination. Fontainebleau: INSEAD (Working Paper). Littlewood, K. (1972). Forecasting and control of passenger bookings. In Proceedings of the twelfth annual AGIFORS symposium Nathanya, Israel. Maglaras, C., & Meissner, J. (2006). Dynamic pricing strategies for multiproduct revenue management problems. Manufacturing and Service Operations Management, 8(2), 136e148. McGill, J. I., & van Ryzin, G. (1999). Revenue management: Research overview and prospects. Transportation Science, 33(2), 233e256. Mookherjee, R., & Friesz, T. L. (2008). Pricing, allocation and overbooking in dynamic service network competition when demand is uncertain. Production and Operations Management, 17(4), 455e474. Sen, A. (2013). A comparison of fixed and dynamic pricing policies in revenue management. Omega, 41(3), 586e597. Smith, B. C., Leimkuhler, J. F., & Darrow, R. M. (1992). Yield management at American airlines. Interfaces, 22, 8e31. Song, Y. S., Hong, S. T., Hwang, M. S., & Yoon, M. G. (2010). MILP model for network revenue management in airlines. Journal of Business and Economic Research, 8(2), 155e162. Szwarc, W. (1964). The transportation problem with stochastic demand. Management Science, 11(1), 33e50. Talluri, K., & van Ryzin, G. (1999). A randomized linear programming method for computing network bid prices. Transportation Science, 33(2), 207e216. Talluri, K., & van Ryzin, G. (2005). The theory and practice of revenue management. Kluwer Academic Publishers. Tcha, D. W., & Yoon, M. G. (1985). A dual-based heuristic for the simple facility location problem with stochastic demand. IIE Transactions, 17(4), 364e369. de Vericourt, F., & Lobo, M. S. (2009). Resource and revenue management in nonprofit operations. Operations Research, 57(5), 1114e1128. Vinod, B. (2008). The continuing evolution: Customer-centric revenue management. Journal of Revenue and Pricing Management, 7(1), 27e39. Xiao, Y. B., Chen, J., & Chen, Y. (2007). On a semi-dynamic pricing and seat inventory allocation problem. OR Spectrum, 29, 85e103. Yoon, M. G., Lee, H. Y., & Song, Y. S. (2012). Linear approximation approach for a stochastic seat allocation problem with cancellation & refund policy in airlines. Journal of Air Transport Management, 23, 41e46. Zhao, W., & Zheng, Y. S. (2000). Optimal dynamic pricing for perishable assets with nonhomogeneous demand. Management Science, 46(3), 375e388.

Please cite this article in press as: Yoon, M. G., et al., Dynamic pricing & capacity assignment problem with cancellation and mark-up policies in airlines, Asia Pacific Management Review (2016), http://dx.doi.org/10.1016/j.apmrv.2016.08.004