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Abstract - Overbooking is a methodology in revenue
management to optimize important decision making for
perishable resources or services with uncertain demand.
Overbooking allows an incoming booking to be accepted in
exceedance of an available capacity because it is believed that
some booking will be cancelled later. It is a complicated and
risky decision since the decision maker needs to minimize
both outsourcing cost and opportunity-lost cost
simultaneously. When there are two classes of resources, it is
not necessary to always outsource the insufficient and low-
priced resources. Upgrading customers to high-priced
resources is possible. The objective of this research is to
develop overbooking models for (1) one class of resources and
(2) two classes of resources (i.e., high and low price) to
minimize total cost (i.e., opportunity cost, cost of upgrading
and outsource cost). The main contribution of this research is
that, unlike other existing literatures, the opportunity cost
considered is specifically identified in the situation where too
much booking request rejection of each type of resources is
present. Sensitivity analysis of our model is also shown for
managerial insights.
Keywords – Overbooking model, revenue management,
perishable resources
I. INTRODUCTION
Revenue management has played important roles in
effectively managing perishable resources or services such
as hotel rooms, airplane seats or car rentals. This group of
resources has limited time to sell and when a particular date
has passed, the resources for that date are perished. When
booking request is accepted, it is possible that the customer
will cancel it later as long as it is prior to an agreement
cancellation date. Since cancellation is uncertain, if
managers are allowed to accept booking requests only up
to capacity, it is possible to lose opportunity to sell when
some customers cancel the bookings. On the other hand, if
managers are allowed to accept booking requests that
exceed capacity, it is possible that the resources are not
enough to serve all show-up customers.
Overbooking is one of the techniques in revenue
management (RM) and is a concept that the company
allows more reservations by customers than the available
resources. Overbooking is a widely used revenue
management technique and one of the most effective tools
to manage limited resources where cancellation is possible
[1]. Overbooking has been used in a wide range of
industries, including airlines, hotels, clinics (see [2], [3],
[4], [5]) and car rental industries (see [6], [7]). For airlines,
Suzuki [8] found that 10-15% of booking requests resulted
in many no-shows without notifying airlines prior to the
flight date. Then, it has been found that this concept can
help increase profitability in airline businesses about 40%
when compared to the no overbooking alternative [9].
A research on overbooking problem in airline industry
is developed by Rothstein [10]. He explored how
overbooking is developed and why operations research is
vital. Smith et al. [11] did a study on yield management for
American Airlines, an airline company in the United
States. They found that the operation models developed by
American Airlines were divided into three parts. One of
those parts was overbooking, while the rest were discount
allocation and traffic management. An estimated benefit of
1.4 billion dollars was projected from this particular model.
Huang et al. [2] studied an overbooking technique using
piecewise linear approach. They claimed that by using this
approach for admissions, it could permit more traffic with
no downside. A research by LaGanga and Lawrence [3]
tried to eliminate no-shows to increase the productivity of
a clinic. Their results showed that even with inconsistent
and unpredictable duration that a patient takes, it still
benefited the clinic to use this model. Their conducted
experiments showed that overbooking was better applied
in the case as there was a high ratio of no-shows.
Muthuraman and Lawley [4] created a model with
overbooking concept for patients. It was used for the
purposes of making appointments. In the model’s schedule,
they included waiting time, overtime of the staff, as well as
the revenue resulting from the patients. Fouad et al. [5]
studied overbooking approaches in hotel industry by a
simulation-based approach. In this research, the processes
were executed with a Monte-Carlo simulation. They also
tried to prove their models’ effectiveness by implementing
this approach and model on an Egyptian hotel. Phumchusri
and Maneesophon [12] studied how overbooking can be
applied to hotel industry. They showed how to identify the
optimal overbooking level for hotel rooms and explored
how model parameters affect the optimal solutions.
Wannakrairot and Phumchusri [13] studied how
overbooking concept could be applied to air cargo industry.
Their overbooking model included two-dimensional
characteristics: booking request’s volume and weight. The
density, show-up rate, and booking requests are random
variables with known distributions.
In this paper, we develop a mathematical model to find
an optimal overbooking level for perishable resources
(such as car rentals). We also consider two classes of
resources situation where it is possible for customers to be
upgraded to high-class resources when low-class resources
are not available. This situation is complex because
overbooking decision of one class of resource affects the
other class. Another main contribution of this research is
that unlike existing literatures, we consider the opportunity
cost for left-over resources only when the booking requests
Optimal Overbooking Decision for Perishable Resources
with Jointly Stochastic Booking and Show-up Requests
S. Jongcheveevat, N. Phumchusri, A. Vilasdaechanont
Department of Industrial Engineering, Faculty of Engineering, Chulalongkorn University, Bangkok, Thailand
(naragain.p@chula.ac.th)
were rejected due to inappropriate overbooking levels.
Sensitivity analysis of the parameters is also conducted to
observe the effects of relevant parameters on an optimal
solution. Managerial insights are provided for future
applications.
This paper is organized as follows. In section 2, we
present mathematical models for one class and two classes
of resources situations, respectively. Then, in section 3, we
perform a sensitivity analysis on the effects of selected
model parameters on the optimal solution. In section 4, we
finally conclude the important points obtained from this
research.
II. OVERBOOKING MODEL
A. One Class of Resource
This section explains how the overbooking models are
formulated for one class of resource situation. The concepts
of booking are as follow. First, customers request a
booking to use the resources. If the total number of
accepted booking is lower than the overbooking level, the
booking request will be accepted. If not, that booking
requests will be rejected. In general, on the day of
committed booking, there will be a number of booking
requests that are absent. We called them no-shows. The rest
of booking requests are called show-up requests. Last, the
show-up booking requests and the available capacities are
compared. When the show-up booking requests exceed the
available capacities, the exceeded booking requests must
be outsourced with outsource cost since we must use other
companies for that committed resources. When the show-
up booking requests are below the available capacity, the
left-over capacity will be spoiled, considered as
opportunity cost. However, not all left-over capacity will
be considered as opportunity cost. The opportunity cost
only occurs when there is a booking request rejection. This
opportunity cost is considered as the revenue that the
company should have received if it had not rejected the
booking requests. To formulate this overbooking model,
there are important parameters and variables needed to be
clarified, as shown in Table 1 and Table 2
TABLE 1
PARAMETERS AND VARIABLES NOTATIONS FOR ONE CLASS
OF RESOURCES
TABLE 2
DECISION VARIABLE FOR ONE CLASS OF RESOURCES
Model Assumptions:
1. Both booking request and show-up request are
random variables with joint probability density
function f[b,s] since they are not independent.
Also, show-up request is always less than or equal
to booking request
2. Outsource cost is higher than opportunity cost
(o>a)
3. There is limited outsource capacity and it is
known.
Generally, overbooking model costs consist of
opportunity cost and outsource cost. Other costs that are
not caused by overbooking, e.g., management costs, are
negligible. As mentioned before, opportunity cost can be
considered as the revenue that the company should have
received, while outsource cost is the extra cost of paying
an outsourced company to get the resources for the show-
up customers when resource is overbooked. To specify the
total cost, consider the situation for booking request and
show-up random variables. There are two main cases, each
with two subcases as follow.
1. Outsource cost
Outsource cost occurs in two following cases:
1.1 The booking request (B) is between the capacity
(C) and the overbooking level (Q), where the show-
up request (S) is more than capacity (C): C<B<Q
and C<S≤B.
The outsource cost for this case is given by:
1.2 The booking request (B) is more than overbooking
level (Q), where the show-up request (S) is more
than capacity (C): B>Q and C<S≤Q. In this case
the company only accept the booking request up to
the overbooking level, but the show-up request is
more than capacity. So, outsourcing is required to
serve the exceeding customers.
The outsource cost for this case is given by:
2. Opportunity cost
Here, opportunity cost is considered only when there are
some booking request rejections and then there are some
left-over capacity. In this case, we have to compare
between the number of booking rejection and left-over
capacity. There are two subcases:
2.1 The booking rejection (B-Q) is less than left-over
capacity (C-S): B-Q < C-S. In this case, not all left-
over capacity cause opportunity cost. Only those
that the company has rejected before cause
opportunity cost and it is given by:
Notation
Meaning
C
Capacity
o
Outsource cost per unit
a
Opportunity cost per unit
f[b,s]
Joint probability density function of
booking request and show up request
B
Booking request (random variable)
S
Show-up (random variable)
CO
Outsource cost
CU
Opportunity cost
TC
Total cost
Notation
Meaning
Q
Overbooking level
2.2 The booking rejection (B-Q) is more than left-over
capacity (C-S): B-Q > C-S. In this case, all left-
over capacity cause opportunity cost and it is given
by:
Thus, the total cost from all cases can be written as:
The optimal overbooking decision (Q) is chosen to
minimize this total cost.
B. Two Classes of Resource
In this model, we consider two classes of resources
with different prices, i.e., high-class and low-class
resources. The question is what is the optimal overbooking
level for each class? This question is not simple because
the decision for each class is not independent from each
other. It is possible to upgrade customers who initially
submit a booking request for low-class resource to high-
class resource (without any additional charges), e.g., hotel
room upgrades or passenger seat upgrades. Most customers
are willing to accept resources with a higher price without
any additional charges when they intend to pay for low
prices.
The concepts of booking are as follow. First, customers
for each class request for booking to use the resources
independently. Then, the booking request for each class is
accepted if the total number of accepted booking does not
exceed the overbooking level for that class. The exceeded
booking requests will be rejected. Then, on the day of
committed booking, there will be some booking requests
that are absent. The rest of booking requests are called
show-up requests. The show-up booking requests and the
available capacity are compared. When the show-up high-
class requests exceed the available capacity, the exceeded
booking requests will be outsourced. For low class, if the
capacity is not sufficient, the company has to check for any
left-over high-class resources. If so, customers will be
upgraded (there will be some upgrade costs that the
company must absorb). If not, there will be outsource costs
from other contracted companies. Similar concepts for
opportunity cost in this situation, not all left-over capacity
should be considered as opportunity cost. It happens only
when they were previously rejected. Model parameters and
variables are shown in Table 3 and Table 4.
Model Assumptions
1. Both booking request and show-up request are
random variables with joint probability density
function fL[bL,sL] and fH[bH,sH] for low and high
class, respectively. Also, show-up request is less
than or equal to booking request
2. For each class of resource, outsource cost is
higher than opportunity cost (oL>aL and oH>aH)
3. The upgrade cost is approximately assumed to be
the difference between opportunity cost of each
class, e.g., aH - aL.
4. There is limited outsource capacity for each class
and those capacities are known.
TABLE 3
PARAMETERS AND VARIABLES NOTATIONS FOR TWO
CLASSES OF RESOURCES
TABLE 4
DECISION VARIABLES FOR TWO CLASSES OF RESOURCES
To specify the total cost, consider the situation for
show-up random variable for each class of resources. There
are four main cases as follow:
1. < and < (there are left-over
capacities for both classes)
2. < and > (there are left-over low-
class capacities and not enough high class)
3. > and < (there are left-over high-
class capacities and not enough low class)
4. > and > (there are not enough for
both low and high classes)
Note that for each case above, we have to compare
between the number of booking rejection and left-over
capacity. Also, we have to consider if it is possible to
upgrade low-class customers to high class. The cost
occurred for each case can be summarized as bellow:
Case 1 (opportunity cost for both classes)
Notation
Meaning
CL, CH
Capacity for low and high-class resources
oL ,oH
Outsource cost for low and high class
aL ,aH
Opportunity cost for low and high class
fL[bL,sL],
fH[bH,sH]
Joint probability density function of
booking request and show up request
for low and high class
BL ,BH
Booking request for low & high class (RV)
SL ,SH
Show-up for low & high class (RV)
CP
Upgrade cost
Notation
Meaning
QL ,QH
Overbooking level for low and high class
Case 2 (opportunity cost for low class and outsource cost
for high class)
Case 3 We have to consider how many low-class
customers can be upgraded to high class. There are two
subcases:
Case 3.1: < . So, all insufficient low
classes can be upgraded and there are left-over high class.
Thus, we have upgrade cost occurs for low class, while
opportunity cost occurs for high class.
∞
Case 3.2: > . So, only some insufficient
low classes can be upgraded and there are outsource costs.
Thus, we have outsource cost for low class as well as
upgrade cost.
Case 4: There are outsource costs for both classes
Total cost (TC) is equal to the sum of costs from all cases.
III. COMPUTATIONAL EXPERIMENT
This section demonstrates computational experiments
of the overbooking model for one class of resources,
presented in the former section. We perform a sensitivity
analysis on the effects of selected model parameters on the
optimal solution. Consider the following joint probability
function of B and S:
Interested model parameters are selected so that it covers
a wide range of situations (where C = 10, minb = 1, maxb
= 15, mins = 1, maxs = 15).
TABLE 5
INPUT PARAMETERS CONSIDERED IN THE EXPERIMENT
Fig. 1(a) shows the optimal overbooking level at
different outsource costs (o). It can be seen that the optimal
overbooking level decreases as the o increases. This is
because as o increases, it costs more if the show-up
booking requests are outsourced. As a result, the optimal
overbooking level decreases in order to reduce the risk of
the booking requests being outsourced. Fig. 1(b) shows the
optimal overbooking level at different opportunity cost (a).
It shows the optimal overbooking level increases as a
increases. This is because as a increases, it costs more if
the capacities are spoiled. Thus, the optimal overbooking
level increases in order to minimize the total costs.
Fig. 1. Optimal overbooking level at different outsource costs (a) and
opportunity costs (b)
IV. CONCLUSION
Overbooking is one of the important strategies in
revenue management. This strategy requires an appropriate
model to determine the optimal overbooking. This paper
presents mathematical models to find the optimal
overbooking level that can minimize total costs. Outsource,
opportunity and upgrade costs are considered. Unlike
existing literatures, this paper proposed that the
opportunity cost for left-over resources should be
considered only when the booking requests were rejected
due to inappropriate overbooking levels. Sensitivity
analysis of the parameters is also conducted to observe the
effects of relevant parameters on the optimal solution. For
future extension, it would be interesting to find out how
model parameters affect the solutions for two-class
resources, or how to estimate the solutions for a reasonable
total cost.
REFERENCES
[1] K. Talluri and G. Van Ryzin, “Revenue Management:
Research Overview and Prospects,” Transportation Science,
vol. 33, pp. 233-256, 1999
[2] F. Huang, C. Deccio, R. Ball, M. Clement and Q. Snell, “A
piecewise linear approach to overbooking,” High
Performance Switching and Routing, 2004
[3] L. LaGanga and S. Lawrence, “Clinic Overbooking to
Improve Patient Access and Increase Provider Productivity,”
Decision Sciences: A Journal of the Decisions Science
Institute, vol. 38, pp. 251-276, 2007
[4] K. Muthuraman and M. Lawley, “A stochastic overbooking
model for outpatient clinical scheduling with no-shows,” IIE
Transactions, vol. 40, pp. 820-837, 2010
[5] A. Fouad, A. Atiya, M. Saleh and A.M. Bayoumi, “A
simulation-based overbooking approach for hotel revenue
management,” Computer Engineering Conference
(ICENCO), 10th International, 2014
[6] I. Karaesmen and G. Van Ryzin, “Overbooking with
Substitutable Inventory Classes,” Operations Research, vol.
52, pp. 83-104, 2004
[7] D. Li and Z. Pang, “Dynamic booking control for car rental
revenue management: A decomposition approach,”
European Journal of Operational Research, vol. 256, pp.
850-867, 2017
[8] Y. Suzuki, “The net benefit of airline
Overbooking,” Transportation Research Part E: Logistics
and Transportation Review, vol. 42(1), pp. 1-19, 2006
[9] Y.-J. Wang and C.-S. Kao, “An application of a fuzzy
knowledge system for air cargo overbooking under uncertain
capacity,” Computers & Mathematics with Applications, vol.
56(10), pp. 2666-2675, 2008
[10] M. Rothstein, “OR Forum—OR and the Airline
Overbooking Problem,” Operations Research, vol. 33, pp.
237-248, 1985
[11] B. Smith, J. Leimkuhler and R. Darrow, “Yield Management
at American Airlines,” Interfaces, vol. 22, pp. 8-31, 1992
[12] N. Phumchusri and P. Maneesophon, “Optimal
overbooking decision for hotel rooms revenue
management,” Journal of Hospitality and Tourism
Technology, vol. 5, pp. 261-277, 2014
[13] A. Wannakrairot and N. Phumchusri, “Two-dimensional air
cargo overbooking models under stochastic booking request
level, show-up rate and booking request density,”
Computers & Industrial Engineering, vol. 100 (October), pp.
1-12, 2016
parameter
Min
Max
Gap
Sequence
o
1100
5000
100
1100,1200,…,5000
a
1000
5000
100
1000,1100,…,5000
