The 11th Asia Pacific Industrial Engineering and Management Systems Conference The 14th Asia Pacific Regional Meeting of International Foundation for Production Research Melaka, 7 – 10 December 2010
Dynamic Programming Model to Determine Overbooking Limits for Two Parallel Flights with Cancellations and No-Shows Ahmad Rusdiansyah1, Dira Mariana2, Hilman Pradhana3 Naning Aranti Wessiani4 Department of Industrial Engineering Institut Teknologi Sepuluh Nopember (ITS), Surabaya, Indonesia Emails:
[email protected] [email protected] [email protected] Email:
[email protected] Abstract - In this study we discuss an advanced problem in Airline Revenue Management (ARM). We develop a model to determine overbooking seat allocation for two single-leg-route parallel flights owned by the same airline. The model is an enhancement of the previous models in the literature on two parallel flights problem which none of them has explored the situation with overbooking, cancelations and no-shows.We attempt to fill this gap. To solve this problem, we have developed a dynamic programming algorithm. We also have conducted numerical experiments to show the behavior of this model. As managerial insight, our model can determine the optimal overbooking limits of both flights in terms of the total expected revenue. Keywords: Airline Revenue Management, Seat Allocation Control, Two Parallel Flights, Overbooking, No Shows, Cancellations, Seat Allocation Control.
1. INTRODUCTION Airline industry is one of service industries practicing revenue management. The service offered can be considered as perishable products. They have limited capacity as wel as limited time. The ticket will have no residual value if the flight has departed. The practice of revenue management in airline industry is called Airline Revenue Management (ARM). The objective of ARM is to maximize passenger revenue by selling the right seats to the right customer at the right time (Dunleavy, 2009). The ARM can be divided into two general areas; seat inventory control and dynamic pricing. This research will focus on the second area. We spesifically discuss the overbooking problem for two parallel flights owned by same airline. There exists some literature on seat inventory problem. Generally, the literature can be classified into two groups, static and dynamic models. The earlier research on static model was proposed by Littlewood (1972). This paper discussed seat allocation control for single flight with twofare classes. This model was then extended by Belobaba (1987) for the condition of multiple-fare classes. In the dynamic models, the booking control policy is not determined at the beginning of the booking period. The inventory seat allocation control should be dynamically
reviewed over the entire booking period in order to optimize the expected revenue. Subramanian et al. (1999) proposed a dynamic model for controlling seat allocation under disrete time condition time. This paper also considered overbooking situation. Feng and Xiao (2006) developed seat allocation model under continuous time. Both of them designed the models designed for single flight. Currently, in more advanced models, some papers discuss the situation where an airline may open more than one flight in a day for “busy” routes. The airline opens some flights in the same departure date with different closed time schedule. These joint flights are called parallel flights. The objective is to maximize the total revenue of the parallel flights. In the parallel flights, the airline should know the behavior of passengers in buying tickets. In the point of view passengers, the seat availability of parallel flights can be substituted each other. There are some papers discussed the seat allocation problem for parallel flights, for instances are Zhang and Cooper (2005) and Chen et al. (2010). Zhang and Cooper (2005) proposed model in dynamic seat allocation control considering passenger choice behavior while Chen et al (2009) proposed model of dynamic programming to optimize booking policy under seat allocation problem for two and multiple flights. In a different perspective, Xiao et al (2008) has developed a joint dynamic pricing model for
The 11th Asia Pacific Industrial Engineering and Management Systems Conference The 14th Asia Pacific Regional Meeting of International Foundation for Production Research Melaka, 7 – 10 December 2010 two parallel flights. However, none of these papers discussing the situation when overbooking problem arises. We attempt to fill this research gap. Thus, the objective of this research is to build a model to determine overbooking seat allocation with overbooking, cancelations and noshows for two single-leg-route parallel flights owned by the same airline. We enhance the model of Subramanian, et al (1999) on overbooking, cancellations, and no shows to meet the condition of two parallel flights. The objective is to maximize the total expected revenue of the two flights by determining the overbooking limits for each flight. In this model, we consider passenger choice behaviour of the two flights to group the type of passengers. In this case, we employ the model of Chen, et al (2010). The problem discussed can be described as follows. There are two parallel flights owned by same airline with single leg route flight. Both of flight have same departure day with short dispute different time. There are limited capacity in each flight, which Ck denotes the capacity of seats in flight k (k = 1, 2). Both flights offered m fare class with different class and service. Class 1 is corresponding to highest fare and class m to the lowest. Both of them serve the same route with different time departure. It means a passenger can choose which flight he/she is preferred. There are three type passengers based on their choice behavior. Passenger type 1 and 2 are passengers who want to buy only a ticket of flight 1 and 2 respectively. Type 3 passengers are flexible passengers. They are willing to take either of the two flights. The last category passenger is flexible because they are indifferent between two flights as long as they can book a seat at the fare class they want. During the booking request period, passengers who have already booked may cancel the reservation at any time before the departure time. If the passenger cancels their reservation, the airline will refund an amount of percentage of their ticket selling price. The amount of refund may be time and class dependent. Passenger who already booked can also no show at the time of departure. No show are also refunded by an amount which is different from the refund for cancellation. To anticipate cancelations and no-shows, the airline assumes implementing overbooking policy for both flights. Overbooking penalties are determined by overbooking penalties cost function. Amounts of overbooking penalties are higher than the highest fare class in each flight and different for two flights. The assumptions of our model are as follows. The cost of cancellation and refund for no-shows are same for all classes. It means that these costs are class-independent. Booking requests in each fare class arrives in random process and each customer requests only a single seat. The
probabilities of booking request were known by random generation.
2. MODEL 2.1 Notation
i m
: Capacity of flight k ;{ k = 1, 2} : Current number of reserved seats in flight k : Fare class ; {i = 1, 2,.., m} : Number of fare class ; {i = 1, 2,.., m} : Price of fare class i in flight 1; > >. . > : Price of fare class i in flight 2 > >⋯> : Probability of booking request of passenger type j on fare class i ;{ j = 1,2,3} : Probability of cancellation in flight k : Refund for cancellation in flight k : Refund for no show in flight k : Probability of no show by passenger in flight k : Overbooking limit of flight k : Overbooking penalty of flight k
2.2 Dynamic Programming Model We divide the booking period into small discrete time period, which no more than one event occured. Let N be number of total booking period for two flights, so the decision period or stage can be numbered in reverse chronological order, n = N, N1,…, 1, 0. Stage N is the opening of reservation and stage 0 means end of reservation or departure time. In each stage, only one of these events will occur: (1) booking request by type j passenger in class i, (2) cancellation by passenger currently holding reservation in flight k, (3) a null event or no booking request from the passenger. It is clear that in period n, we have: +∑
(
+
+
)+
( )+
( )=1
(1)
Let ( ) denote the probability of a cancellation given that the number of reserved seats in flight k equal xk. We assume that ( ) ; for k = 1,2 ; is non decreasing and concave function of xk. It means ( )= × . We assume that at the time of departure, passenger who reserved in flight k have probability no show . Let ( ) is number of people who show up for flight k, given that xk is number of reserved seat at time of departure, so
The 11th Asia Pacific Industrial Engineering and Management Systems Conference The 14th Asia Pacific Regional Meeting of International Foundation for Production Research Melaka, 7 – 10 December 2010 that − ( ) is the number of no shows. Each passenger in flight k have probability show up 1 − , it means ( ) has binomial distribution ( − ). If at the time of departure ( ) = , than we incur overbooking penalty ( ). We assume that ( ) is non-negative, convex, and non-decreasing in ≥ 0, with ( ) = 0 for ≤ . The objective function of the model is to maximize total expected net revenue over the horizon, starting from period N to 0, starting from state xk = 0, with no booked seat at the beginning of period N. This model considers overbooking, cancellation, and no show, so the variable state does not need to satisfy the constraint xk ≤ Ck. In this model, both flights offered seats in the same booking period. It means, booking request event can occur either for flight 1 or flight 2. However, it starts from the stage N with no booked seat, and at most one booking request can be accept in each stage, so the total number of booked seat in both flight for every stage are not more than N-n. Let x1 and x2 be the number of booked seats in flight 1 and 2 respectively, and let X is the total number of booked seats of two flights, so that = ( + ) and 0 ≤ ≤ − . This is the constraint of state in each stage n based on the assumption of discrete time setting. We also introduce overbooking maximum level in every flight, which called overbooking limit ( ) . It resulting an additional state constraint, 0 ≤ ≤ + . It can be optimal policy that rejects all request of seat in flight k if the current number of reserved seat, ( ), equal to + . Let ( , ) be the maximal expected revenue from period n to 0 when the confirmed number of booked seat in flight 1 and 2 are and respectively. This model employs backward dynamic programming, so that the value of ( , ) in stage N will be the total expected revenue over the horizon, start from period N to 0. Due to cancellations and no-shows, to calculate the net revenue we consider the expected cost of cancellations and no-shows. Let ( , ) denote the total expected loss of revenue over period n to 0 caused by cancellations and noshows. The period time horizon can be viewed into two period; the departure time period or n = 0 and decision time periods when n ≥ 1. At the departure time period (n = 0), we assume airline not accept any booking requests. There is
only no show event and overbooking penalty will occur when overestimated overbooking occured. The equation of this situation is given as follow: ( ,
) = [−
( ,
)=(
( ( ))] + [−
.
)+( .
.
.
( ( ))] … (2)
) … . … … … . (3)
The overbooking penalty is calculated in equation (2) and expected loss of no-shows is calculated in equation (3). For stage n ≥ 1, the dynamic programming equation is given as follows:
1, ∑
1,
( ,
)−[
)= {
( ,
)],
)−[
1) − [ =
1,
(
+
(
(
) +
)= +
)+∑
)−
+ 1,
( ,
+ 1,
+ 1) − [
)−
( ,
)} + ∑
( ,
)+
− 1,
+
( )
( ,
)
( )
0≤
≤
≤
−
+
( ,
+
;
)],
( ,
)+
( ,
(
)} +
( ,
+ 1) −
+
( ,
{ + ( ,
(
+
+
)} +
+
− 1) … … (4)
=
;0 ≤
+
≤
( )
+
(
− 1) … … … … … … . . (6)
with following constraints: 0≤
)],
)],
{ +
) − ( . ( )) − ( . ( )) … . . (5)
+
+
( ,
( ,
+ 1) −
(
1−∑ ( ,
(
( ,
( )
∑
( ,
−
;
+
The value function of equation (4) is the total net revenue of two flights in stage n to 0. The total expected loss of revenue over period n to 0 caused by cancellation and no-shows is showed by equation (6). Equation (5) is show the calculation of probability null event, . The output of this model is total expected revenue of two flights. We determine the number of overbooking for both flights by seeking for the combination of overbooking limit in both flights which give the highest total expected revenue. In the next section, we will show how to determine overbooking limit by using deterministic parameter in numerical experiments.
The 11th Asia Pacific Industrial Engineering and Management Systems Conference The 14th Asia Pacific Regional Meeting of International Foundation for Production Research Melaka, 7 – 10 December 2010
3. NUMERICAL EXPERIMENTS 3.1 Numerical experiment 1: overbooking limit
Determine
We used the parameters in Table 1 and 2 for the numerical experiment 1. The cancellation refunds, ck, are same for all class in each flight, and also the no-show, dk, are same for all class in each flight. Overbooking penalty is 300% of the highest fare class in each flight. Using the equation (2) until (6), the results of total expected revenue for all combinations of overbooking limit in both flights are shown in Table 3. Table 1: Parameters for numerical experiment 1 flight k
Parameter
1 50 0.1 0.9 2
Ck βk 1-βk m i
ri ($) 1 (high fare) 2 (low fare)
qk ck dk πk ($) N
2 50 0.1 0.9 2
Table 3 shows that the highest TER can be obtained when the airline determine 6 overbooking limit for flight 1 and also 6 overbooking limit for flight 2. For different parameters, the result of overbooking limit combination will be adjusted depends on the cancellation and no show level. The adjusted overbooking limit has significantly caused the value of TER. If an airline extremely sets the number of overbooking limit, the TER will decrease due to the cost of overbooking penalty caused by overestimated overbooking. Using the same parameters in Table 1 and 2, we set a set of different overbooking penalty level ranging from 120% to 300% from the highest fare class in both flight (r1). Table 4 shows the calculation result of TER in different overbooking penalty level and the optimal overbooking limit for each flight. Figure 1 shows the visualization of these results Table 4: TER in different values of overbooking penalty overbooking penalty
Ri ($)
100 95 60 55 0.001 0.001 50% x r2 50% x R2 20% x r2 20% x R2 300%r1 300%xr1 150 stage
v2
TER
6
8
8668.28
150%
7
7
8661.89
200%
6
6
8653.74
250%
6
6
8648.20
300%
6
6
8643.91
8675.00
Table 2: probability of booking requests
8670.00 8665.00
Probability booking request (Pji)
8660.00
i passenger type (j)
v1 120%
TER
8655.00 8650.00
1
2
1
0.1575
0.1575
8640.00
2 3
0.1575 0.135
0.1575 0.0675
8630.00
8645.00 8635.00 120%
150%
200%
250%
300%
%overbooking penalty X highest price (r1)
Table 3: TER in different number of overbooking limit
v1 0 1 2 3 4 5 6 7 8
0 8275.28 8318.84 8361.35 8402.75 8442.90 8481.75 8481.75 8501.71 8500.40
1 8313.82 8356.36 8397.79 8437.97 8476.86 8514.39 8530.78 8532.11 8530.41
Total Expected Revenue (TER) v2 2 3 4 5 8351.34 8387.78 8423.01 8423.01 8392.80 8428.02 8461.99 8494.62 8433.01 8466.97 8499.61 8530.81 8471.93 8504.57 8535.77 8565.49 8509.50 8540.70 8570.42 8598.54 8545.59 8575.31 8603.43 8629.85 8560.69 8589.05 8615.75 8640.73 8561.06 8588.49 8614.33 8638.52 8559.06 8586.28 8614.33 8636.21
6 8474.48 8510.89 8545.82 8579.20 8610.92 8640.89 8643.91 8635.06 8630.75
7 8477.08 8512.72 8546.91 8579.57 8610.65 8640.04 8638.89 8615.03 8603.74
8 8476.65 8512.07 8546.09 8578.64 8609.63 8638.98 8636.54 8609.42 8593.79
Figure 1: TER in different value of overbooking penalty The results have shown clearly that higher increasing level overbooking penalty will decrease the total expected revenue (TER) of both flights. Table 4 also shows that the increasing level of overbooking penalty influences the optimal decision of overbooking limit in both flights. Our model can be used to find the best combination of overbooking limits. In general, we may see that higher level overbooking penalty will decrease the overbooking limits. It is caused by higher overbooking penalty will increase the loss of potential revenue if the airline makes the overbooking limit too high. As managerial insight, using our proposed model, the airline can determine the level of overbooking penalty
The 11th Asia Pacific Industrial Engineering and Management Systems Conference The 14th Asia Pacific Regional Meeting of International Foundation for Production Research Melaka, 7 – 10 December 2010 as well as the overbooking limit for both flights.
3.2 Numerical experiment 2: Different values of cancellation probability (q1) Using the probability booking request in Table 2 and other parameter in the Table 5, we have run numerical experiments under different values of cancellation probability in flight 1, (q1). We determine probability cancellation in flight 1, (q1) ranging from 0.000 to 0.0025 and probability cancellation in flight 2, (q2) constant with value 0.001. Cancellation refunds, ck, and no-shows refund, dk, are same for all classes for each flight. Overbooking penalty is 200% of the highest fare class for each flight. Using using the equation (2) until (6), the results of total expected revenue with and without adjusted overbooking limit were shown in Table 6 and Figure 2. Table 5: Parameter for numerical experiment 2 flight k
Parameter
1 50 0.1 0.9 2
Ck βk 1-βk m ck dk πk ($) N
50% x r2 20% x r2 200%r1
2 50 0.1 0.9 2
50% x R2 20% x R2 200%xr1 150 stage
Figure 2: TER with different values of q1
3.3 Numerical experiment 3: Different values of no-shows probability (β1) In this experiment, we will elaborate the effect of different values of no-shows probability. We have conducted numerical experiments under different values of no-show probability in flight 1, (β1). We determine probability no-shows in flight 1, (β1) ranging from 0 to 0.25 and probability no-shows in flight 2, (β2) constant with value 0.1. Cancellation refunds, ck, and no-shows refund, dk, are same for all class in each flight. Using the equation (2) until (6), the result of total expected revenue with and without adjusted overbooking limit are shown in Table 7 and the graph presented in Figure 3. Table 7: TER with and without overbooking in different values of β1
Table 6: TER in different value of q1 q1
optimal overbooking pad
q2
v1
v2
TER
without overbooking
TER
v1
v2
0.0000
0.0010
7
17
9234.28
0
0
8490.36
0.0005
0.0010
6
15
9106.53
0
0
8461.06
0.0010
0.0010
6
6
8653.74
0
0
8275.28
0.0015
0.0010
5
6
7891.99
0
0
7794.56
0.0020
0.0010
5
6
7384.50
0
0
0.0025
0.0010
5
6
7053.80
0
0
The
results
showed
that
the
to
v1
v2
TER
8548.39 8607.32 8653.74 8676.37
0 0 0 0
0 0 0 0
8316.62 8295.05 8275.28 8257.29
7263.76
0.2
0.1
12
5
8656.09
0
0
8240.65
6916.03
0.25
0.1
16
5
8601.12
0
0
8225.03
cancellation
airline should apply overbooking policy in order to due
v2
without overbooking
6 7 6 6
without that of without overbooking. This implies that the revenue
v1
TER
3 3 6 9
TER with optimal overbooking limit has higher values than
total
optimal overbooking pad
0.1 0.1 0.1 0.1
cancellation probability the lower TER. We may see that
the
β2
0 0.05 0.1 0.15
probability influence the total expected revenue. The higher
maximize
β1
cancellations.
Specifically, the airline may determine the optimal overbooking limit based on the historical cancellation level.
The result showed that no-shows probability has influence significantly the TER of both flights. We may conclude also that the TER with optimal overbooking limit has higher value than without overbooking. This implies that in order to maximize total revenue, the airline management needs to determine the overbooking limit adjusted to historical no-show probability.
The 11th Asia Pacific Industrial Engineering and Management Systems Conference The 14th Asia Pacific Regional Meeting of International Foundation for Production Research Melaka, 7 – 10 December 2010
8800.00 8700.00 8600.00 8500.00 TER
8400.00 8300.00
no overbooking
8200.00
overbooking
8100.00 8000.00 7900.00 0
0.05
0.1
0.15
0.2
0.25
β1
Cancellation, And No Show. transportation science, vol. 33 no. 2 . Xiao, Y.B., Chen, J., Liu, X. L. (2008) Joint Dynamic Pricing For Two Paralel Flight Based On Passenger Choice Bahavior. System Engineering Theory & practice, vol. 28 pp. 46-55. Youyi Feng, Baichun Xiao. (2006) A Continuous Time Seat Control Model For Single Leg Flights With No Show And Optimal Overbooking Upper Bound. European Journal of Operation Research, 1298-1316.
Figure 3: TER with different value of β1 ACKNOWLEDGEMENT 4. CONCLUSION AND FUTURE RESULT In this paper, we have considered overbooking, cancellations, and no-shows for two paralell flights problem. The problem has not yet been discussed in any literature. We have extended the optimal seat determination model for two flights problem in literature to include the overbooking strategy. Our objective was to find the optimal overbooking limit for both flights. Our numerical experiments have showed that under cancellations and noshows condition, the overbooking should be applied. The Total expected Revenue (TER) were influenced by cancellations and no-shows probabilities. Using our proposed model we can determine the optimal overbooking limits of both flights thru examining the combination overbooking limits that give the highest TER. In this paper, we have not yet considered cancellations and no-shows depend to the class-fare and the case of flight switching The future research of this area may consider such cases.
REFERENCES Belobaba, P.P. (1987) Air Travel Demand And Airline Seat Inventory Management. Massachusset Institute of technology. Cambrige. MA: Ph.D Thesis. flight transportation laboratory. Chen, S., Gallego, G., Li, Michael Z.F., Lin, Bing. (2010) Optimal Seat Allocation For Two-Flight Problems With A Flexible Demand Segment. European Journal of Operation Research, 210, pp. 897-908. Dan Zhang, W.L. Cooper. (2005) Revenue Management For Paralel Flight With Customer Choice Behavior. Operation research, vol. 53 no.3 pp. 415-431. Dunleavy. H, Philiphs. G. (2009) The Future Of Airline Revenue Management. Journal or revenue management, vol. 0 pp. 1-8. Littlewood, K. (1972) Forecasting And Control Of Passenger Booking. AGIFORS Symposium proceeding. 103-105. Subramanian, J., Stidham, J.R., Lauthenbacher, C.J. (1999) Airline Yield Management With Overbooking,
We acknowledge the financial support of the research grant “Hibah Kompetensi 2010” received from the Ministry of National Education of Indonesia AUTHOR BIOGRAPHIES Ahmad Rusdiansyah is a senior lecturer at the Department of Industrial Engineering, Institut Teknologi Sepuluh November (ITS) Surabaya. He is currently the Principal Researcher of the Transportation and Distribution Logistics (TDLog) research group in the Logistics & Supply Chain Management (LSCM) Laboratory. Dira Mariana is a graduate student at the Department of Industrial Engineering, Institut Teknologi Sepuluh November (ITS) Surabaya. Her research area currently is Airline Revenue Management. She is actively as a student researcher at TDLog Research Group. Hilman Pradhana is an undergraduate student at the Department of Industrial Engineering, Institut Teknologi Sepuluh November (ITS) Surabaya. He is currently doing research in Airline Revenue Management. He is actively as a student researcher at TDLog Research Group. Naning Aranti Wessiani is a lecturer at the Department of Industrial Engineering, ITS. She is a member of Industrial System and Management Development (ISMD)Laboratory adjunct researcher of TDLog.
