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Vol:.(1234567890)
Journal of Revenue and Pricing Management (2020) 19:190–209
https://doi.org/10.1057/s41272-019-00210-9
RESEARCH ARTICLE
Optimal overbooking model forcar rental business withtwo levels
ofprices having stochastic joint booking andshow‑up levels
NaragainPhumchusri1· PhatsakornSangsukiam1· NannapatChariyasethapong1
Received: 19 August 2019 / Accepted: 14 September 2019 / Published online: 17 October 2019
© Springer Nature Limited 2019
Abstract
Overbooking is a technique in revenue management which offers products or services more than the amount available because
there is a possibility that some purchasers may later cancel their purchases. In car rental overbooking problem, overbooking
model is complicated because in car rental business, different types of car must be taken into account. This paper presents a
mathematical overbooking model for car rental business with two levels of prices in order to find the optimal overbooking
levels which minimize the total cost, consisting of opportunity cost, outsourcing cost, and upgrading cost. Booking requests
and show-up customers are joint random variables which follow some known joint distributions. Sensitivity analysis is
performed to examine the effects of parameters in the overbooking model on the optimal overbooking levels. Due to the
complication of the overbooking model, several simplified models are presented as alternative methods for estimating the
solutions of the overbooking problem. The results show that the total cost difference between the optimal model and the
proposed regression models is in the range of about 3.2–14.09%. The expected total cost when the overbooking policy is
implemented is also compared to the total cost when there is no overbooking policy to explore how overbooking decision
is significant in minimizing total cost.
Keywords Overbooking· Revenue management· Car rental· Opportunity cost· Outsourcing cost· Upgrading cost
Introduction
At present, tourism industry is considered one of the most
important economic drivers of Thailand. Tourism plays a
significant part in generating revenues for the country, and
it also supports other business sectors. As the competition
between low-cost airlines becomes higher and airline tick-
ets’ prices decline, traveling behavior of tourists starts to
undergo major changes. This results in the growth of car
rental business because the tourists tend to travel by air-
lines and then rent cars from the service providers instead
of driving their own cars. Table1 shows the overview of
car rental business in Thailand in 2017 and 2018. It can be
seen that the growth of market value is expected to be as
high as 6–8%.
Revenue management is the practice of controlling cus-
tomer demand through the use of dynamic pricing and
capacity management to enhance profitability (El Haddad
etal. 2008). The main objective of revenue management is
to sell the right products or services to the right customers at
the right time (Cross 1997). Nowadays, revenue management
technique is a widely utilized decision tool for businesses
dealing with perishable products such as airlines industry,
hotels, hospitals, and car rental industry [see McGill and
van Ryzin (1999), Kimes and Wirtz (2003), Koide and Ishii
(2005), Chevalier etal. (2015), Fouad etal. (2014), Phum-
chusri and Maneesophon (2014), Sierag etal. (2015), and
Cetin etal. (2016)]
Cars in the car rental business are classified as perish-
able products because once the day begins, the leftover cars
which are not booked for the day will be “spoiled” hence
losing their opportunity to generate revenues. This is caused
by the uncertainties in the system, in terms of the number of
no-show customers and cancelations.
Although car rental revenue management and revenue
management in other industries, such as hotel and health-
care, can be done in similar manner, there are two distinct
* Naragain Phumchusri
naragain.p@chula.ac.th
1 Department ofIndustrial Engineering, Faculty
ofEngineering, Chulalongkorn University, Bangkok,
Thailand
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191
Optimal overbooking model forcar rental business withtwo levels ofprices having stochastic…
characteristics which separate car rental revenue manage-
ment from others. First, cars in car rental shops can be
rented from an outsource. This does not have an effect on
the customer’s satisfaction whereas products and services
in other industries can rarely be outsourced. For example,
hotel managers cannot provide customers with an outsourced
hotel room as the quality of the rooms are different. Second,
most car services provide more than one category of cars,
e.g., low-priced and high-priced cars. These issues add more
complexity to the car rental revenue management in the situ-
ation where there are excess customers who book low-priced
cars and the manager needs to decide whether to turn down
those customers’ booking requests or let them be upgraded
and use the leftover high-priced cars instead.
Overbooking is one of the key techniques in revenue man-
agement, as well as in car rental revenue management. The
concept is that providers and suppliers offer a greater num-
ber of products and services than the amount available so as
to decrease the loss of opportunity to generate revenues from
the situation where there are some purchasers or customers
who do not show up for the products or services. This con-
cept is proven to be beneficial for air cargo industry, yielding
40% decrease in the total cost (Wang and Kao 2008). When
applying the overbooking concept with car rental business,
there are two main things to be taken into consideration,
i.e., number of booking requests and number of cancelations
which are the uncertainties in the system. The objective of
overbooking is to minimize opportunity cost and outsourc-
ing cost. Opportunity cost is the cost incurred from reject-
ing booking requests of the customers. When the number of
booking requests exceeds the overbooking level, some book-
ing requests will be rejected, and the company loses oppor-
tunity to sell, called the opportunity cost. Outsourcing cost
is the cost incurred from using cars from an outsource. This
happens when the number of show-up customers exceeds the
supply. Finding the optimal overbooking level which bal-
ances these two possibilities and minimizes the total cost is
very important in generating more profit for the car rental
business. Thus, the objective of this paper is to develop an
overbooking model for car rental business with two levels
of prices to find the optimal overbooking level for each price
which minimize the total cost.
The main distinction of this paper compared to existing
papers is that (1) the considered opportunity cost only occurs
when the booking requests were rejected previously; (2)
booking and show-up requests are both uncertain with joint
distribution to capture what happens in reality; (3) upgrading
cost is considered when it is possible to offer high-priced
cars in the situation of low-priced cars shortage. This paper
is organized as follows. In the next section, we summarize
related literature review in overbooking study. “Overbook-
ing Model” section presents a mathematical model for two
classes of cars situations. Then, in “Computational Experi-
ments” section, we perform sensitivity analysis on the effects
of selected model parameters on the optimal solution. Sim-
plified methods of estimating overbooking levels are pre-
sented in “Sensitivity Analysis” section, and we finally con-
clude the important points obtained from this research in
“Model Simplification” section.
Literature review
Overbooking has become a very popular strategy in revenue
management. It has been comprehensively applied in various
industries such as airlines and hotel industries. However,
there are few researches in the area of car rental revenue
management compared to other industries. This section
briefly reviews some research papers from the area of car
rental revenue management.
The early study of revenue management in car rental
industry involved yield management as a decision tool for
managers to solve pricing problems and management deci-
sion (Carroll and Grimes 1995; Smith etal. 1992). Li and
Pang (2017) described the characteristics of car rental indus-
try for revenue management to be successful: car service is
considered perishable as the capacity is lost after the service
is opened for the day, it can be offered with different types
and price rates depending on the customer request and there
is an uncertainty in customer bookings. Li and Pang (2017)
also described the distinct attribute of the car rental industry
which is that the rental cars capacity is flexible and dynamic
whereas the products and services in other industries such
as airline seats and hotel rooms are fixed. For example, a
car rental service provider can use cars from other provid-
ers nearby to sufficiently service its customers for the day
and return them the day after. A few researches study fleet
planning in car rental industry revenue management in
Table 1 Overview of car rental
business in Thailand in 2017
and 2018 (Thairentacar.com
2019)
Business sector 2017 2018
Market value (in mil-
lion baht)
Growth rate
(%)
Market value (in mil-
lion baht)
Growth rate
(%)
Long-term car rental 29,300 7 31,200–31,800 7–9
Short-term car rental 13,200 13 13,800–14,100 5–7
Overall car rental 42,500 9 45,000–45,900 6–8
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
192 N.Phumchusri et al.
order to increase the short-term efficiency and reduce total
cost. Pachon etal. (2006) studied fleet management which
deals with a three-stage vehicle transfer planning including
pooling, transferring and servicing. Li and Tao (2010) pro-
posed a policy in a single-trip fleet planning for car rental
industry under the assumption of dynamic booking requests
and unlimited capacity. You and Hsieh (2014) also did a
research on fleet planning and transfer policy for car rental
problems. It was not until only recently that the overbook-
ing concept is applied in car rental revenue management.
Lazov (2017) developed a mathematical model simulating
customer requests and show-ups as a birth–death process.
The objective of this study is to find the optimal fleet size
which gives maximum profit. Oliveira etal. (2017) described
the overbooking problem in car rental industry dealing with
two types of car. This research applied fleet management to
minimize the car transferring between two service providers.
Overbooking has been widely used in other industries.
Rothstein (1985) developed an early study on overbooking
problem in the airline industry. He explored how overbook-
ing is developed and the importance of operations research.
Feng etal. (2004) studied an overbooking technique using
piecewise linear approach. They claimed that more traffic
with no downside could be permitted using this approach
for admissions. The study by LaGanga and Lawrence (2007)
presented a model to remove no-shows to increase the pro-
ductivity of a clinic. The results showed that even with an
inconsistent duration a patient takes, it was still profitable for
the clinic to apply this model. Their conducted experiments
showed that overbooking technique was highly beneficial
in the case where there was a high ratio of no-shows. Simi-
lar researches of overbooking technique in the healthcare
industry include El-Sharo etal. (2015), Muthuraman and
Lawley (2008), Salemi Parizi and Ghate (2016). For air-
lines, Suzuki (2006) found that almost 15% of bookings in
average resulted in no-shows without prior notification to
the airlines. The overbooking policy helped increase about
40% of profitability to airlines compared to the no overbook-
ing alternative. Wang and Kao (2008) studied the concept
of air cargo overbooking under uncertain capacity applying
the fuzzy knowledge system. Wannakrairot and Phumchusri
(2016) studied how overbooking concept could be applied
to two-dimensional air cargo. The overbooking model pre-
sented in this research included two-dimensional character-
istics for air cargo: volume and weight of a booking request.
The density, show-up rate, and booking requests are random
variables with known distributions.
Although some researches presented some ideas about car
rental overbooking models, they did not yet explore a model
with two types of products or services and did not consider
the joint distribution between two important uncertainties,
i.e., booking and show-up levels. In this paper, we develop
an overbooking model to find optimal overbooking level for
two car types (low and high-priced) for car rental business
that minimize total cost. This model also presents a consid-
eration of upgrading cost and provides managerial insights
for future applications.
Overbooking model
Overbooking problem
This section explains how the overbooking model for per-
ishable products with two levels of price is formulated.
The rental car capacities are fixed for both low-priced and
high-priced cars and the concepts of bookings are as fol-
lows. First, customers request for bookings to be serviced by
either low-priced or high-priced cars. Then, the number of
booking requests and the overbooking level are compared.
The booking requests which exceed the pre-determined
overbooking level are rejected. Next, there could be some
booking requests which do not show up for the service. On
the day of the service, the customers who show up for the
car are called the show-up booking requests or show-ups.
Finally, the show-up booking requests are compared with
the available capacities for both types of car. If the number
of show-up booking requests are more than the capacities,
outsourcing is needed in order to follow the contract created
with customers.
If the number of show-up booking requests are less than
the available capacities, the leftover capacities are consid-
ered spoiled only if there are rejections of customer book-
ing requests. This is what makes the cost calculation in
this research unique and different from other researches
because there can be a situation where the number of book-
ing requests are less than the capacities but no rejection has
occurred before. The leftover capacities in this circumstance
have nothing to do with the overbooking decision. Therefore,
the opportunity cost calculation in this research only occurs
when the number of booking requests exceed the overbook-
ing level and the number of show-up booking requests are
less than the available capacity.
To elaborate, customer rejection is used as a criterion in
determining whether there will be an opportunity cost or
not because the overbooking level influences the number
of customers whose bookings are rejected. If the car ser-
vice provider sets the overbooking level too low, the chance
of customer rejections will be high considering the same
amount of customer demand, so the car service provider
loses its opportunity to generate profit from these potential
customers. On the contrary, if the car service provider sets
the overbooking level higher, the chance of customer rejec-
tions will decrease resulting in the gain of profit. However,
setting the overbooking level too high does not give the
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193
Optimal overbooking model forcar rental business withtwo levels ofprices having stochastic…
optimal profit. The reason for this will be explained in the
following sections.
This paper presents the overbooking model for car rental
business with two levels of price which is much more com-
plicated than the model for one level of price. This is because
the model presents a new component of the total cost: the
upgrading cost, which happens in the circumstance where the
number of show-up booking requests is more than the avail-
able capacity for low-priced cars and there are some available
capacities for high-priced cars. Managers can consider having
these low-priced booking requests upgraded, rather than out-
sourcing. The upgrading cost per unit of service can be any
parameter; however, this paper assumes it to be the difference
between the opportunity cost per unit of high-priced and low-
priced cars for simplicity. In summary, the objective of the
overbooking model in this research is to find the overbooking
levels that minimize the total cost (the sum of opportunity
cost, outsourcing cost, and upgrading cost).
Before proceeding to the explanation of the overbooking
model, the model parameters, random variables, and deci-
sion variables in the model are clarified in Tables2, 3, 4.
Assumptions
Random variable distribution
In this research, the number of booking requests and that
of show-up customers are random variables. Since show-up
request depends on booking request and it cannot exceed the
booking request, these two variables have a joint distribu-
tion. In practice, the car rental service providers can find the
appropriate joint distribution function of these two variables
from their historical data to effectively use the result from
the overbooking model. The joint distributions are defined
as fL[bL,sL] for low-priced cars and fH[bH,sH] for high-priced
cars, respectively (Table5).
Opportunity cost
Opportunity cost is a profit loss cost. In reality, it can be con-
sidered as the difference between selling price and cost of
owning a car. In this research, we consider opportunity cost
only when there are customer rejections. These rejections
happen when there are a greater number of booking requests
than the overbooking level. That is, the overbooking level
has an effect on the number of customers who are rejected.
In the case where there is at least one customer rejection
and the number of show-up booking requests is less than the
available capacity, the quantity of the cars which produces
opportunity costs are equal to the minimum of the number of
customers rejected and the number of leftover cars.
Outsourcing
Outsourcing cost can be considered as the difference
between outsourcing price and cost of owning a car. If the
number of show-ups is greater than the available capacity,
the car rental service provider must use the cars from an
outsource. These cars must have to be similar in quality with
the cars of the provider itself for the purpose of customer
satisfaction, but with a higher cost to the car rental service
provider. Otherwise, the reputation of the provider may be
ruined, resulting in further loss in profit. The number of
Table 2 Parameter notations
Parameter Meaning
oL
Outsourcing cost per unit of low-priced car
oH
Outsourcing cost per unit of high-priced car
aL
Opportunity cost per unit of low-priced car
aH
Opportunity cost per unit of high-priced car
cL
Supply capacity of low-priced cars
cH
Supply capacity of high-priced cars
Table 3 Random variable notations
Random Variable Meaning
BL
Customer booking requests of low-priced cars
BH
Customer booking requests of high-priced cars
SL
Show-up booking requests of low-priced cars
SH
Show-up booking requests of high-priced cars
Table 4 Decision variable
notations Decision variable Meaning
QL
Overbook-
ing level
of low-
priced
cars
QH
Overbook-
ing level
of high-
priced
cars
Table 5 Other variables notations
Variable Meaning
COL
Outsourcing cost of low-priced cars
COH
Outsourcing cost of high-priced cars
CSL
Opportunity cost of low-priced cars
CSH
Opportunity cost of high-priced cars
CP
Upgrading cost
TC
Total cost
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
194 N.Phumchusri et al.
potential outsourced cars depends on the contract or the
agreement between two car rental service providers. In this
research, the number of allowed outsourced cars can exceed
the overbooking level for both types of cars.
Upgrading
In this paper, the upgrading cost per unit is assumed to be
the difference between opportunity cost per unit of high-
priced and low-priced cars. The explanation for this is that
when there is an upgrade, a high-priced car will be sold at
the price of a low-priced car. By offering the high-priced
car at the lower price, the car rental service provider loses
the opportunity to use the high-priced car to generate its
high-priced revenue.
Costs
As explained in the previous sections, overbooking model
cost is composed of opportunity cost, outsourcing cost, and
upgrading cost. Other costs which are not caused by the
overbooking decision, e.g., management costs, are negligi-
ble. Considering, situations for both random variables, there
are four cases in total.
Case 1
sL
<
cL
, and
sH
<
cH
In this case, the number of
show-up booking requests is less than the available capaci-
ties resulting in leftover cars for both low-priced and high-
priced cars. The costs occurring are opportunity costs for
both types of car.
Opportunity cost of low-priced cars occurring in case one
(CU1.1)
is as follows:
Opportunity cost of high-priced cars occurring in case
one
(CU1.2)
is as follows:
(1)
CU
1.1 =aL
c
H
∫
0
b
H
∫
0
maxb
L
∫
QL
−b
L
+c
L
+Q
L
∫
0
bL−QLfLbL,sLdsLdbL
fHbH,sHdsHdb
H
+aL
maxbH
∫
cH
cH
∫
0
maxbL
∫
QL
−bL+cL+QL
∫
0
bL−QLfLbL,sLdsLdbL
fHbH,sHdsHdbH
+aL
cH
∫
0
bH
∫
0
maxbL
∫
QL
cL
∫
−bL+cL+QL
cL−sLfLbL,sLdsLdbL
fHbH,sHdsHdbH
+aL
maxbH
∫
cH
cH
∫
0
maxbL
∫
QL
cL
∫
−bL+cL+QL
cL−sL
fL
bL,sL
dsLdbL
fH
bH,sH
dsHdbH
(2)
CU
1.2 =aH
maxb
H
∫
QH
−b
H
+c
H
+Q
H
∫
0
c
L
∫
0
b
L
∫
0
bH−QHfLbL,sLdsLdbL
fHbH,sHdsHdbH
+aH
maxbH
∫
QH
−bH+cH+QH
∫
0
maxbL
∫
cL
cL
∫
0
bH−QHfLbL,sLdsLdbL
fHbH,sHdsHdb
H
+aH
maxbH
∫
QH
cH
∫
−bH+cH+QH
cL
∫
0
bL
∫
0
cH−sHfLbL,sLdsLdbL
fHbH,sHdsHdbH
+aH
maxbH
∫
QH
cH
∫
−bH+cH+QH
maxbL
∫
cL
cL
∫
0
cH−sH
fL
bL,sL
dsLdbL
fH
bH,sH
dsHdbH
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
195
Optimal overbooking model forcar rental business withtwo levels ofprices having stochastic…
Case 2
sL
<
cL
and
sH
>
cH
In this case, the number of show-up booking requests
for low-priced cars is less than the available low-priced car
capacity resulting in leftover low-priced cars, while the
number of show-up booking requests for high-priced cars
exceeds the high-priced car capacity resulting in the short-
age of high-priced cars. The cost occurring are opportunity
cost for low-priced car and outsourcing cost for high-priced
cars.
Opportunity cost of low-priced cars occurring in case two
(CU2.1)
is as follows:
Outsourcing cost of high-priced cars occurring in case
two
(CO2.2)
is as follows:
Case 3
sL
>
cL
and
sH
<
cH
In this case, the number of show-
up booking requests for low-priced cars is greater than the
available low-priced car capacity resulting in the shortage
of low-priced cars, while the number of show-up book-
ing requests for high-priced cars is less than the available
high-priced car capacity resulting in leftover high-priced
cars. Because of the shortage of low-priced cars, the cost
(3)
CU
2.1 =aL
Q
H
∫
cH
b
H
∫
cH
maxb
L
∫
QL
−b
L
+c
L
+Q
L
∫
0
bL−QLfLbL,sLdsLdbL
fHbH,sHdsHdbH
+aL
maxbH
∫
QH
QH
∫
cH
maxbL
∫
QL
−bL+cL+QL
∫
0
bL−QLfLbL,sLdsLdbL
fHbH,sHdsHdb
H
+aL
QH
∫
cH
bH
∫
cH
maxbL
∫
QL
cL
∫
−bL+cL+QL
cL−sLfLbL,sLdsLdbL
fHbH,sHdsHdbH
+aL
maxbH
∫
QH
QH
∫
cH
maxbL
∫
QL
cL
∫
−bL+cL+QL
cL−sL
fL
bL,sL
dsLdbL
fH
bH,sH
dsHdbH
(4)
CO
2.2 =oH
Q
H
∫
cH
b
H
∫
CH
c
L
∫
0
b
L
∫
0
sH−cHfLbL,sLdsLdbL
fHbH,sHdsHdbH
+oH
maxbH
∫
QH
QH
∫
cH
cL
∫
0
bL
∫
0
sH−cHfLbL,sLdsLdbL
fHbH,sHdsHdbH
+oH
QH
∫
cH
bH
∫
cH
maxbL
∫
cL
cL
∫
0
sH−cHfLbL,sLdsLdbL
fHbH,sHdsHdbH
+oH
maxbH
∫
QH
QH
∫
cH
maxbL
∫
cL
cL
∫
0
sH−cH
fL
bL,sL
dsLdbL
fH
bH,sH
dsHdb
H
occurring with low-priced cars is the outsourcing cost.
Whereas in high-priced cars, the cost which occurs is the
opportunity cost. As stated in the previous section, when
the situation of case three happens, the car rental service
provider will allow excess customers who book low-priced
car service to use the service of the remaining high-priced
cars instead. Therefore, in this case the cost calculation
will be more complicated than the previous cases and can
be divided into two subcases depending on the number of
excess low-priced customers and the number of high-priced
cars available.
Subcase 3.1 The number of excess low-priced customers
is less than the number of high-priced cars available.
Upgrading cost occurring in case 3.1
(CP3.1a)
is as
follows;
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
196 N.Phumchusri et al.
(5)
CP
3.1a=(aH−aL)
cH
∫
cH+cL−QL
bH
∫
cH+cL−QL
cH−sH+cL
∫
cL
bL
∫
cL
sL−cLfLbL,sLdsLdbL
fHbH,sHdsHdbH
+(aH−aL)
cH
∫
cH+cL−QL
bH
∫
cH+cL−QL
QL
∫
cH−sH+cL
cH−sH+CL
∫
cL
sL−cLfLbL,sLdsLdbL
fHbH,sHdsHdb
H
+(aH−aL)
cH+cL−QL
∫
cH+cL−maxbL
bH
∫
cH+cL−maxbL
QL
∫
CL
bL
∫
CL
sL−cLfLbL,sLdsLdbL
fHbH,sHdsHdbH
+(aH−aL)
cH
∫
cH+cL−QL
cH+cL−QL
∫
cH+cL−maxbL
QL
∫
cL
bL
∫
cL
sL−cLfLbL,sLdsLdbL
fHbH,sHdsHdbH
+(aH−aL)
cH+cL−maxbL
∫
minbH
bH
∫
minsH
QL
∫
cL
bL
∫
cL
sL−cLfLbL,sLdsLdbL
fHbH,sHdsHdbH
+(aH−aL)
cH
∫
cH+cL−maxbL
cH+cL−maxbL.
∫
minsH
QL
∫
cL
bL
∫
cL
sL−cLfLbL,sLdsLdbL
fHbH,sHdsHdbH
+(aH−aL)
cH
∫
cH+cL−QL
bH
∫
cH+cL−QL
cH−sH+CL
∫
cL
bL
∫
cL
sL−cLfLbL,sLdsLdbL
fHbH,sHdsHdbH
+(aH−aL)
cH
∫
cH+cL−QL
bH
∫
cH+cL−QL
QL
∫
cH−sH+CL
cH−sH+cL
∫
cL
sL−cLfLbL,sLdsLdbL
fHbH,sHdsHdb
H
+(aH−aL)
cH+cL−QL
∫
cH+cL−maxbL
bH
∫
cH+cL−maxbL
maxbL
∫
QL
QL
∫
CL
sL−cLfLbL,sLdsLdbL
fHbH,sHdsHdbH
+(aH−aL)
cH
∫
cH+cL−QL
cH+cL−QL
∫
cH+cL−maxbL
maxbL
∫
QL
QL
∫
cL
sL−cLfLbL,sLdsLdbL
fHbH,sHdsHdbH
+(aH−aL)
cH+cL−maxbL
∫
minbH
bH
∫
minsH
maxbL
∫
QL
QL
∫
cL
sL−cLfLbL,sLdsLdbL
fHbH,sHdsHdbH
+(aH−aL)
cH
∫
cH+cL−maxbL
cH+cL−maxbL
∫
minsH
maxbL
∫
QL
QL
∫
cL
sL−cLfLbL,sLdsLdbL
fHbH,sHdsHdbH
+(aH−aL)
QH
∫
cH
cH
∫
cH+cL−QL
cH−sH+cL
∫
cL
bL
∫
cL
sL−cLfLbL,sLdsLdbL
fHbH,sHdsHdbH
+(aH−aL)
QH
∫
cH
cH
∫
cH+cL−QL
maxbL
∫
cH−sH+cL
cH−sH+cL
∫
cL
sL−cLfLbL,sLdsLdbL
fHbH,sHdsHdbH
+(aH−aL)
QH
∫
cH
cH+cL−QL
∫
minsH
QL
∫
cL
bL
∫
cL
sL−cLfLbL,sLdsLdbL
fHbH,sHdsHdbH
+(aH−aL)
QH
∫
cH
cH+cL−QL
∫
minsH
maxbL
∫
QL
QL
∫
cL
sL−cLfLbL,sLdsLdbL
fHbH,sHdsHdbH
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197
Optimal overbooking model forcar rental business withtwo levels ofprices having stochastic…
The opportunity cost for high-priced cars from this sub-
case can be further divided into two subcases depending on
the number of rejected customers and the remaining high-
priced cars after the event of upgrading.
Opportunity cost for high-priced cars where the number
of rejected customers is more than the remaining high-priced
cars
(CU3.2a)
is as follows:
(6)
CU
3.2a=aH
maxb
H
∫
QH
c
H
∫
cH+cL−QL
maxb
L
∫
QL
c
H
−s
H
+c
L
∫
cL
cH−sH−sL+cLfLbL,sLdsLdbL
fHbH,sHdsHdbH
+aH
maxbH
∫
QH
cH
∫
cH−bH−QH
maxbL
∫
QL
cH−sH+cL
∫
cL
cH−sH−sL+cLfLbL,sLdsLdbL
fHbH,sHdsHdbH
+aH
maxbH
∫
QH
cH−bH+QH
∫
cH+cL−QL
maxbL
∫
QL
cH−sH+cL
∫
cH−sH+cL−bH+QH
cH−sH−sL+cLfLbL,sLdsLdbL
fHbH,sHdsHdbH
+aH
maxbH
∫
QH
cH
∫
cH+cL−QL
maxbL
∫
QL
cH−sH+cL
∫
cH−sH+cL−bH+QH
cH−sH−sL+cLfLbL,sLdsLdbL
fHbH,sHdsHdbH
+aH
maxbH
∫
QH
cH+cL−QL
∫
cH−bH+QH
maxbL
∫
QL
QL
∫
cL
cH−sH−sL+cLfLbL,sLdsLdbL
fHbH,sHdsHdbH
+aH
maxbH
∫
QH
cH−bH+QH
∫
cH+cL−bH+QH−QL
maxbL
∫
QL
QL
∫
cH−sH+cL−bH+QH
cH−sH−sL+cLfLbL,sLdsLdbL
fHbH,sHdsHdb
H
+aH
maxbH
∫
QH
cH−bH+QL
∫
cH+cL−bH+QH−QL
maxbL
∫
QL
QL
∫
cH−sH+cL−bH+QH
cH−sH−sL+cLfLbL,sLdsLdbL
fHbH,sHdsHdb
H
+aH
maxbH
∫
QH
cH−bH+QH
∫
cH+cL−maxbL
maxbL
∫
QL
QL
∫
cH−sH+cL−bH+QH
cH−sH−sL+cLfLbL,sLdsLdbL
fHbH,sHdsHdbH
+aH
maxbH
∫
QH
cH+cL−QL
∫
cH+cL−maxbL
maxbL
∫
QL
QL
∫
cH−sH+cL−bH+QH
cH−sH−sL+cLfLbL,sLdsLdbL
fHbH,sHdsHdbH
+aH
maxbH
∫
QH
cH+cL−maxbL
∫
cH−bH+QH
maxbL
∫
QL
QL
∫
cL
cH−sH−sL+cLfLbL,sLdsLdbL
fHbH,sHdsHdbH
+aH
maxbH
∫
Q
H
cH−bH+QH
∫
c
H
+c
L
−b
H
+Q
H
−Q
L
maxbL
∫
Q
L
QL
∫
c
H
−s
H
+c
L
−b
H
+Q
H
cH−sH−sL+cL
fL
bL,sL
dsLdbL
fH
bH,sH
dsHdb
H
Opportunity cost for high-priced cars where the number
of rejected customers is less than the remaining high-priced
cars
(CU3.3a)
is as follows:
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198 N.Phumchusri et al.
(7)
CU
3.3a=aH
maxb
H
∫
QH
c
H
∫
cH+cL−QL
maxb
L
∫
QL
c
H
−s
H
+c
L
−b
H
+Q
H
∫
cL
bH−QHfLbL,sLdsLdbL
fHbH,sHdsHdbH
+aH
maxbH
∫
QH
cH
∫
cH−bH+QH
maxbL
∫
QL
cH−sH+cL−bH+QH
∫
cL
bH−QHfLbL,sLdsLdbL
fHbH,sHdsHdbH
+aH
maxbH
∫
QH
cH+bH−QH
∫
cH+cL−bH+QH−QL
maxbL
∫
QL
cH−sH+cL−bH+QH
∫
cL
bH−QHfLbL,sLdsLdbL
fHbH,sHdsHdb
H
+aH
maxbH
∫
QH
cH+cL−QL
∫
cH+cL−bH+QH−QL
maxbL
∫
QL
cH−sH+cL−bH+QH
∫
cL
bH−QHfLbL,sLdsLdbL
fHbH,sHdsHdb
H
+aH
maxbH
∫
QH
cH−bH+QH
∫
cH+cL−maxL
maxbL
∫
QL
cH−sH+cL−bH+QH
∫
cL
bH−QHfLbL,sLdsLdbL
fHbH,sHdsHdbH
+aH
maxbH
∫
QH
cH+cL−QL
∫
cH+cL−maxbL
maxbL
∫
QL
cH−sH+cL−bH+QH
∫
cL
bH−QHfLbL,sLdsLdbL
fHbH,sHdsHdbH
+aH
maxbH
∫
QH
cH+cL−maxbL
∫
cH−bH+QH
maxbL
∫
QL
QL
∫
cL
bH−QHfLbL,sLdsLdbL
fHbH,sHdsHdbH
+aH
maxbH
∫
QH
cH+cL−bH+QH−QL
∫
cH+cL−maxbL
maxbL
∫
QL
QL
∫
cL
bH−QHfLbL,sLdsLdbL
fHbH,sHdsHdbH
+aH
maxbH
∫
QH
cH−bH+QH
∫
cH+cL−bH+QH−QL
maxbL
∫
QL
cH−sH+cL−bH+QH
∫
cL
bH−QHfLbL,sLdsLdbL
fHbH,sHdsHdb
H
+aH
maxbH
∫
Q
H
cH+cL−bH+QH−maxbL
∫
c
H
+c
L
−b
H
+Q
H
−Q
L
maxbL
∫
Q
L
QL
∫
c
L
bH−QH
fL
bL,sL
dsLdbL
fH
bH,sH
dsHdbH
Subcase 3.2 The number of excess low-priced customers
is greater than the number of high-priced cars available.
In this subcase, the remaining high-priced cars cannot
support all the excess low-priced customers. The maximum
number of customers allowed to be upgraded is equal to
the number of remaining high-priced cars. The rest of the
customers who have not been serviced yet need to use the
cars from an outsource. Hence, it produces upgrading cost
and outsourcing cost.
Upgrading cost occurring in case 3.2
(CP3.1b)
is as
follows:
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199
Optimal overbooking model forcar rental business withtwo levels ofprices having stochastic…
Outsourcing cost for high-priced cars where the number
of rejected customers is greater than the remaining high-
priced cars
(CO3.2b)
is as follows:
Case 4
sL
>
cL
and
sH
>
cH
In this case, the number of
show-up booking requests is greater than the available
(8)
CP
3.1b=(aH−aL)
c
H
∫
cH+cL−QL
b
H
∫
cH+cL−QL
Q
L
∫
cH−sH+cL
b
L
∫
cH−sH+cL
cH−sHfLbL,sLdsLdbL
()fHbH,sHdsHdb
H
+(aH−aL)
cH
∫
cH+cL−QL
bH
∫
cH+cL−QL
maxbL
∫
QL
QL
∫
cH−sH+cL
cH−sHfLbL,sLdsLdbL
fHbH,sHdsHdbH
+(aH−aL)
maxbH
∫
cH
cH
∫
cH+cL−QL
QL
∫
cH−sH+cL
bL
∫
cH−sH+cL
cH−sHfLbL,sLdsLdbL
fHbH,sHdsHdbH
+(aH−aL)
maxbH
∫
c
H
cH
∫
cH
+
cL
−
QL
maxbL
∫
QL
QL
∫
c
H
−s
H
+c
L
cH−sH
fL
bL,sL
dsLdbL
fH
bH,sH
dsHdbH
(9)
CO
3.2b=oL
c
H
∫
cH+cL−QL
b
H
∫
cH+cL−QL
Q
L
∫
cH−sH+cL
b
L
∫
cH−sH+cL
sL−cL−cH+sHfLbL,sLdsLdbL
fHbH,sHdsHdb
H
+oL
cH
∫
cH+cL−QL
bH
∫
cH+cL−QL
maxbL
∫
QL
QL
∫
cH−sH+cL
sL−cL−cH+sHfLbL,sLdsLdbL
fHbH,sHdsHdbH
+oL
QH
∫
cH
cH
∫
cH+cL−QL
QL
∫
cH−sH+cL
bL
∫
cH−sH+cL
sL−cL−cH+sHfLbL,sLdsLdbL
fHbH,sHdsHdbH
+oL
QH
∫
c
H
cH
∫
c
H
+c
L
−Q
L
maxbL
∫
Q
L
QL
∫
c
H
−s
H
+c
L
sL−cL−cH+sH
fL
bL,sL
dsLdbL
fH
bH,sH
dsHdbH
capacities resulting in the shortages of both low-priced and
high-priced cars. The costs occurring is outsourcing cost for
both types of cars (Fig.1).
Outsourcing cost of low-priced cars occurring in this case
(CO4.1)
is as follows:
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
200 N.Phumchusri et al.
Outsourcing cost of high-priced cars occurring in this
case
(CO4.2)
s as follows:
The diagram showing the costs in each case is shown in
Fig.2
(10)
CO
4.1 =oL
Q
H
∫
cH
b
H
∫
cH
Q
L
∫
cL
b
L
∫
cL
sL−cLfLbL,sLdsLdbL
fHbH,sHdsHdbH
+oL
maxbH
∫
QH
QH
∫
cH
QL
∫
cL
bL
∫
cL
sL−cLfLbL,sLdsLdbL
fHbH,sHdsHdbH
+oL
QH
∫
cH
bH
∫
cH
maxbL
∫
QL
QL
∫
cL
sL−cLfLbL,sLdsLdbL
fHbH,sHdsHdbH
+oL
maxbH
∫
Q
H
QH
∫
c
H
maxbL
∫
Q
L
QL
∫
c
L
sL−cL
fL
bL,sL
dsLdbL
fH
bH,sH
dsHdb
H
(11)
CO
4.2 =oH
Q
H
∫
cH
b
H
∫
cH
Q
L
∫
cL
b
L
∫
cL
sH−cHfLbL,sLdsLdbL
fHbH,sHdsHdbH
+oH
maxbH
∫
QH
QH
∫
cH
QL
∫
cL
bL
∫
cL
sH−cHfLbL,sLdsLdbL
fHbH,sHdsHdbH
+oH
QH
∫
cH
bH
∫
cH
maxbL
∫
QL
QL
∫
cL
sH−cHfLbL,sLdsLdbL
fHbH,sHdsHdbH
+oH
maxbH
∫
Q
H
QH
∫
c
H
maxbL
∫
Q
L
QL
∫
c
L
sH−cH
fL
bL,sL
dsLdbL
fH
bH,sH
dsHdb
H
The total cost (TC) is composed of the sum of the costs
from all four cases. The total cost equation is as follows:
Computational experiments
The formulation of the total cost function was explained in
the previous section. In this section, we present a method to
obtain the optimal overbooking levels from the overbooking
model and example of distribution functions of booking and
(12)
TC
=
CU
1.1 +
CU
1.2 +
CU
2.1 +
CO
2.2
+CP3.1a+CU3.2a+CU3.3a+CP3.1
b
+CO
3.2b
+CO
4.1
+CO
4.2
Fig. 1 Bookings without customer rejections (top figure) and with
customer rejections (bottom figure)
show-up random variables. It is important to note that the
model is composed of many cost function terms based on the
conditions, and the model itself has a high degree of com-
plexity. The model is not continuous and cannot guarantee
the convexity property. As a result, the optimal overbooking
levels cannot be obtained by mathematical proof.
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201
Optimal overbooking model forcar rental business withtwo levels ofprices having stochastic…
Fig. 2 Diagram of costs occurring to the car rental service provider in each case
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
202 N.Phumchusri et al.
Joint distribution function forbooking andshow‑up
variables
Note that the model presented in the previous section can be
used for any joint distribution. To perform experiments for
insights, two functions are used in this paper.
Uniform distribution
Uniform distribution is a symmetric continuous probabil-
ity distribution such that for each member of the family, all
intervals of the same length on the distribution’s support
are equally probable. When the number of booking requests
and that of show-up requests are correlated, the relation-
ship between these two numbers can be established. Figure3
presents an example of uniform shape for the considered
random variables.
To find the joint distribution function for uniform distri-
bution, the concept of total probability is applied. The joint
distribution is defined as:
(13)
f[b,s]=
2
(maxb−minb)(max s−min s);
0
<
b< maxb, ;0 < s< maxs,s≤b
Fig. 3 Joint uniform distribution function of booking requests and
show-ups
Fig. 4 Joint triangular distribution function of booking requests and
show-ups
Table 6 Ranges of the parameters used in the computational experi-
ments
Parameter Value
oL
800, 1000, …, 1600
oH
1600, 1800, …, 2400
aL
100, 300, …, 900
aH
900, 1100, …, 1700
cL
28, 29, …, 32
cH
18, 19, …, 22
mins
, min
b
0
maxs
,
maxb
40
Fig. 5 Example output of the optimal overbooking level calculation
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203
Optimal overbooking model forcar rental business withtwo levels ofprices having stochastic…
Triangular distribution
Another distribution considered in the experiment is the
Triangular distribution, which is a continuous probability
distribution with a probability function shaped like a trian-
gle. It is defined by three values: the minimum value, the
maximum value, and the peak value. This research focuses
on the distribution where the peak value of the probability
function is the same as the maximum value. This is because
in high season where there is higher demand for the service
than usual, there is a great chance that the booking requests
are as high as the maximum bookings allowed. With this
specification, the overbooking model becomes more prag-
matic. Figure4 presents an example of triangular shape for
the considered random variables.
To find the joint distribution function for triangular distri-
bution, the total probability concept is applied. The sum of
the probability of all events in the triangular prism must be
equal to one, and the joint triangular distribution is defined
in Eq.14.
(14)
Slope
=Δ
f
Δs=
maxf
−
0
maxb−0=
f
−
0
b−0
f[b,s]=maxf∗s
maxs
=6s
maxs
2
maxb
;0 <b<maxb,0<s<maxs,s≤
b
There are several steps required to obtain the optimal
overbooking levels. First, the ranges of the parameters in the
model, i.e., opportunity cost per unit, outsourcing cost per
unit and capacities of low-priced and high-priced cars, need
to be defined. Then, the distribution of booking requests
and show-ups are specified. This section explores the effects
of the parameters on the overbooking levels when the ran-
dom variables are uniformly and triangular distributed. It
is imperative to note that in practice, booking requests and
show-ups can follow any variable distribution depending on
the nature of the demand. Next, the computer process begins
once the parameters and variables are defined in Table6. It
should be pointed out that the maximum allowable show-ups
and booking requests (maxs and maxb) arbitrarily chosen for
this experiment are with respect to the available capacity.
A change in these parameters is expected to have a conse-
quential effect on the optimal overbooking level. Then, each
of the predefined overbooking level is used to calculate the
expected total cost, which is composed of opportunity cost,
outsourcing cost, and upgrading cost based on the conditions
22
24
26
28
30
32
34
36
38
1000 1200 1400 1600
Overbooking le
vel
o
L
22
24
26
28
30
32
34
36
38
1600 1800 2000 2200 2400
Overbooking le
vel
o
H
QH
QL
QH
QL
Fig. 6 The optimal overbooking levels as oL and oH increase
20
22
24
26
28
30
32
34
36
500700 900
Overbooking
level
a
L
20
22
24
26
28
30
32
34
36
38
40
42
1100 1300 1500 1700
Overbooking
level
a
H
QH
QL
QH
QL
Fig. 7 The optimal overbooking levels as aL and aH increase
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
204 N.Phumchusri et al.
for each of the combination of the values of parameters. The
plausible values of the overbooking level are defined to be
within the range of the available capacity and the maximum
allowable booking request. The program then computes the
actual expected total cost for each overbooking level com-
bination and returns the combination of overbooking levels
which gives the least value of total cost. The output example
of the model calculation is displayed in Fig.5
Sensitivity analysis
This section displays computational experiments for the car
rental overbooking model presented in the previous section.
The objective is to study the effects of the model parameters
on the overbooking decision.
Figure6 shows the optimal overbooking levels for low-
priced cars and high-priced cars at different values of out-
sourcing costs per unit (oL and oH). It can be seen that as
oL increases, the optimal overbooking level for low-priced
cars decreases. This is due to the fact that as oL increases, it
costs more if the show-up booking requests need to be out-
sourced. As a result, the optimal overbooking level decreases
in order to reduce the risk of the bookings being outsourced.
However, the optimal overbooking level for high-priced cars
is not affected by the change of oL because the bookings
for low-priced and high-priced cars are independent of each
other. The same explanation goes for the change in the solu-
tion of high-priced cars and oH.
Figure7 shows the optimal overbooking levels at different
values of opportunity costs (aL and aH). As demonstrated
in Fig.7, the optimal overbooking level for low-priced cars
increases as aL increases. This is because as aL increases, it
costs more to have a spoilage of low-priced car capacity. The
same explanation goes for the relationship between the opti-
mal overbooking level of high-priced cars and aH. Subse-
quently, the optimal overbooking level increases to increase
the utilization of available cars. However, it is important
to pinpoint that there is a relationship between the optimal
overbooking level of high-priced cars and aL, and vice versa.
As aL increases, QH decreases. An increase in aL means the
upgrading cost per unit (aH – aL) decreases. Hence, it is more
desirable to have the low-priced booking requests upgraded
to use the high-priced cars. The optimal overbooking level
for high-priced cars decreases so as to lessen the chance of
the company having to outsource high-priced cars. In con-
trast, an increase in aH results in the decrease in QL. An
increase in aH means the upgrading cost per unit increases.
Hence, it is not desirable to provide an upgrade. The optimal
overbooking level of low-priced cars decreases to minimize
the chance of having excess low-priced booking requests,
minimizing the total costs.
Figure8 shows the optimal overbooking levels at differ-
ent values of available car capacities (cL and cH). For both
types of car, the optimal overbooking level increases as the
18
20
22
24
26
28
30
32
34
36
38
40
42
28 29 30 31 32
Overbooking level
c
L
20
22
24
26
28
30
32
34
36
38
40
42
18 19 20 21 22
Overbooking level
cH
QH
QL
QH
QL
Fig. 8 The optimal overbooking requests as cL and cH increase
Table 7 Sensitivity analysis of the overbooking decision and model
parameters
Parameter Effect on QLEffect on QH
oLNegative –
oHPositive Negative
aLPositive Negative
aHNegative Positive
cLPositive Negative
cHPositive Positive
Table 8 The r-squared scores and measures of accuracy of multiple
robust regression
R-squared score Mean
squared
error
Mean absolute percent-
age error (MAPE) (%)
QLuni
0.8315 2.61 3.89
QHuni
0.7585 2.10 4.34
QLtri
0.7294 7.61 6.61
QHtri
0.8720 2.02 4.78
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
205
Optimal overbooking model forcar rental business withtwo levels ofprices having stochastic…
capacity increases. This is to decrease the chance of capacity
spoilage, generating maximum profit. Besides, the relation-
ships between the optimal overbooking level for high-priced
cars and cL, and vice versa also exist. As observed in Fig.8,
an increase in cL results in a slight decrease of QH. As cL
and QL increases, the chance that there will be an excess of
low-priced customers also increases. This requires a higher
vacancy of high-priced cars hence the lower the high-priced
overbooking level. This is to lower the low-priced outsourc-
ing cost. On the other hand, an increase in cH results in an
increase of QL. As cH increases, there are more available
high-priced cars capacity and more room to support surplus
low-priced show-ups. Hence, we can increase QL to maxi-
mize the revenue generated.
In summary, the relationships between the optimal over-
booking levels for low-priced cars and high-priced cars and
the model parameters are shown in Table7.
Model simplication
In order to effectively utilize this overbooking model in
practice, simplified methods for estimating the optimal over-
booking level are proposed owing to the complexity of the
full model. These alternative methods are used to estimate
the overbooking without having to run the full model via
the program. After observing the effects of the parameters
on the optimal overbooking levels, we gather all insights
they give to build new models which are able to give proper
estimate of the answers.
This paper presents three methods for estimating the opti-
mal overbooking level from model parameters as independ-
ent variables. They are (1) robust multiple linear regression,
(2) nonparametric regression, and (3) naïve method.
Robust multiple linear regression
Robust multiple regression is one of many regression meth-
ods which is very efficient in predicting the value of the
response. This method is an alternative to ordinary least
squares (OLS) regression as it is less sensitive to high lev-
erage outliers in the response variable and to the assumption
of normality in the residuals of the response variable.
M-estimation with Huber weighting is used for estimating
the parameters of the robust multiple regression equation.
M-estimation defines a weight function such that the esti-
mating equation becomes
n
i=1
w
i
y
i
−x�b
x�
i
=
0
(Li 1985).
The equation is solved using Iteratively Reweighted Least
Squares (IRLS). In Huber weighting, observations with
small residuals get a weight of 1, and the larger the residual,
the smaller the weight. This is defined by the weight func-
tion of the residual w(e)
By means of this method, the issues of the complication
and tediousness of the model are solved. The results from the
computational experiments section were used as input data
for the regression model with optimal overbooking levels
(QL and QH) as response variables and the model parameters
are predictor variables. After running the regression model,
the robust multiple regression equations for estimating the
optimal overbooking levels are shown in Eqs.15 and 16 for
uniformly distributed booking requests and show-ups and in
Eqs.17 and 18 for triangular-distributed booking requests
and show-ups, respectively.
Table8 shows the r-squared scores and measures of accu-
racy of the estimated optimal overbooking level for the mul-
tiple robust regression model. The r-squared scores obtained
from Eqs.15 through 18 can be interpreted as the percentage
of the variance in the response variable, the optimal over-
booking level, which can be explained by the variance of the
predictor variables, the model parameters in this case. As
displayed in Table8, it can be seen that all of the r-squared
score values are above 0.7 or 70% which are considered to be
relatively high values. The errors results are also satisfactory
since the MAPEs are only around 3.89–6.61%.
w
(e)=
{
1for
|
e
|≤
k
k
|
e
|
for
|
e
|
>
k
(15)
Q
Luni =
16.325
−
0.000213
oL+
0.000299
oH+
0.0119
a
L
−0.003178
aH
+0.526261
cL
+0.097051
cH
(16)
Q
Huni =
10.342
+
0.000013o
L−
0.001770o
H−
0.00558aL
+0.004076a
H
−0.139406c
L
+1.028254c
H
(17)
Q
Ltri =−
31.371
−
0.000913o
L+
0.00062o
H+
0.009142aL
−0.00265a
H
+1.896874c
L
+0.493086c
H
(18)
Q
Htri =
14.605
+
0.0001o
L−
0.0022o
H−
0.0042aL
+0.0044
aH
−0.5083
cL
+1.2827
cH
Table 9 The r-squared scores and measures of accuracy of nonpara-
metric regression
R-squared score Mean
squared
error
Mean absolute percent-
age error (MAPE) (%)
QLuni
0.8246 2.72 3.94
QHuni
0.7442 2.18 4.26
QLtri
0.7034 7.75 6.52
QHtri
0.7780 2.05 4.77
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
206 N.Phumchusri et al.
Nonparametric regression
Another alternative method for estimating the optimal over-
booking levels is nonparametric regression. Nonparametric
regression is a regression analysis in which the predictors
and response does not take a predetermined form of distribu-
tion. Hence, it does not consider the assumption of the nor-
mality of the residuals of the response variable. In this paper,
Nadaraya–Watson kernel regression estimation is used as
the parameter estimation technique. This method estimates
m as a locally weighted average, using a kernel as a weight-
ing function. The Nadaraya–Watson estimator is
m
H(x)=
∑n
i=1KH
(
x−xi
)
yi
∑
n
j=1
KH(x−xj) where
KH
is a kernel with bandwidth
h. The denominator is a weighting term with sum 1
(Nadaraya 1964).
Similar to the robust multiple linear regression model,
the results from the computational experiments section were
used as input data for the nonparametric regression models
with optimal overbooking levels (QL and QH) as response
variables and the model parameters as predictor variables.
Then, the r-squared score and measures of accuracy were
compared.
As shown in Table9, it can be seen that the r-squared
scores of all responses are greater than 0.7 or 70% which
are considered as relatively high values. Furthermore, it is
important to note that the values of mean squared error and
mean absolute percentage error are very close to those of
robust linear regression. Therefore, these two regression
methods perform similarly in term of error performance.
Naïve model
Naive method presented in this paper is used to predict the
optimal overbooking levels which does not require computer
processing. This method of estimation is built on the simple
fact that the optimal overbooking level will always be between
the available capacity and the maximum booking requests
allowed. The reason for this is intuitive. If the overbooking
Table 10 Randomly generated combinations of model parameters
used in the experiment
oL
oH
aL
aH
cL
cH
928 2128 470 1682 32 22
850 1613 240 1131 28 21
1119 2339 389 1321 30 19
1599 1750 406 913 28 19
1051 1935 889 1148 28 22
1346 2227 173 1520 29 21
1030 1835 627 1014 32 21
1077 1619 372 1170 30 22
1127 1774 249 1068 28 21
1594 2262 464 1104 32 20
835 2143 198 1261 31 22
1587 1956 660 1149 29 19
1500 1853 750 1568 29 18
1351 2053 477 1419 32 21
1306 1960 761 1086 31 22
1456 1921 345 1685 32 21
1266 2221 788 1590 29 18
1577 2276 588 1101 30 21
1397 2032 431 1276 31 19
1011 2104 372 1622 31 18
1480 2243 578 1212 29 20
1447 2039 224 1117 29 22
1263 2226 728 1593 32 21
833 1922 796 933 29 22
869 2145 136 1606 31 21
Fig. 9 Total cost comparison for uniform distributed booking requests and show-ups
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
207
Optimal overbooking model forcar rental business withtwo levels ofprices having stochastic…
level falls below the capacity, the show-up booking requests
will always be less than the available capacity, increasing the
total opportunity cost. Also, the number of booking requests
cannot exceed maxb which is the maximum number of book-
ings a car service provider can accept each day. The parameters
used in this method include outsourcing cost per unit, oppor-
tunity cost per unit, capacity and maximum booking requests
allowed. If the opportunity cost per unit and outsourcing cost
per unit were the same, the optimal overbooking level would
always equal to the average of the available capacity and maxb.
However, in reality, outsourcing costs more than loss of sales
for both levels of car price. Thus, the ratio between opportunity
cost per unit and outsourcing cost per unit affects the optimal
overbooking level. If the ratio is close to zero, which means
the opportunity cost per unit is significantly lower than the
outsourcing cost per unit, the optimal overbooking level should
be lower to prevent the chance of outsourcing. On the other
hand, if the ratio is close to one meaning that the opportunity
cost per unit is almost the same as outsourcing cost per unit,
the optimal overbooking level should be close to the average
of the available capacity and maxb. By applying this idea, the
naïve model is expressed in Eq.19
In the next section of this paper, we present the compari-
son of the results obtained from these simplified models with
(19)
Q
naive =
a
o(maxb
−
c
2)
+
c
Table 11 Average total cost
difference from the optimal total
cost for uniformly distributed
booking requests and show-ups
Method Difference (%) p value for two
sample t test
Conclusion
Naïve method 50.67 < 0.0005 Significantly different
Robust multiple regression 3.53 0.749 Not significantly different
Nonparametric regression 3.20 0.772 Not significantly different
Fig. 10 Total cost comparison for triangular-distributed booking requests and show-ups
Table 12 Average total cost
differences between the optimal
total costs for triangular-
distributed booking requests and
show-ups
Method Difference (%) p value for two
sample t test
Conclusion
Naïve method 18.34 0.104 Not significantly different
Robust multiple regression 14.09 0.205 Not significantly different
Nonparametric regression 12.88 0.246 Not significantly different
Table 13 Average total cost differences between overbooking policy
and non-overbooking policy
Distribution Total cost with
overbooking
policy (Baht)
Total cost without
overbooking
policy (Baht)
Difference (%)
Uniform 800 1844 56.33
Triangular 1291 2236 42.68
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
208 N.Phumchusri et al.
the optimal response and the comparison of the performance
between these models as well as their insights.
Model comparison
This section explains in detail the procedure of comparing
the performance of each of the simplified models to the opti-
mal overbooking levels using the expected total cost. First,
several model parameter combinations are randomly gener-
ated. It is important to note that the values of every gener-
ated parameter are within the experimental ranges. Then, for
each combination of the model parameters in Table10, the
optimal overbooking levels are estimated from the simplified
models. We then use these pairs of estimated overbooking
levels to find the expected total cost occurred. The total cost
generated by the estimation methods is, in turn, compared
with the expected total cost obtained from the optimal over-
booking levels to see how much the costs occurred from
simplified model as compared to the optimal cost. If they
are not much different, it means the estimation methods are
satisfactory.
The expected total cost comparison of different simplified
models for uniform distributed booking requests and show-
ups is shown in Fig.9. It can be clearly seen that for every
combination of parameters, the robust multiple regression
and nonparametric regression models perform well, giving
the expected total costs consistently close to the expected
optimal total costs, while the total costs generated from the
naïve model are quite distant from the optimal total costs.
The average total cost difference from optimal total cost for
uniformly distributed booking requests and show-ups are
summarized in Table11.
The expected total cost comparison for triangular-distrib-
uted booking requests and show-ups follows the same pro-
cedure as the comparison for those of uniform distributed.
The result is shown in Fig.10. From Fig.10, we observe that
the robust multiple regression and the nonparametric regres-
sion models perform arguably worse compared with the uni-
formly distributed booking requests. However, the expected
total costs from naïve model are closer to the optimal total
costs. The average total cost differences between optimal
total costs for triangular-distributed booking requests and
show-ups are summarized in Table12.
From the experiments and statistical hypothesis tests,
we can conclude that the simplified regression models,
i.e., robust multiple regression and nonparametric regres-
sion models, can be appropriately used to estimate the opti-
mal overbooking levels, QL and QH, as the expected total
costs generated from these two models are not statistically
significantly different from the optimal total costs in both
types of random variable distributions. However, the naïve
model performs well only with triangular-distributed book-
ing requests and show-ups. A possible reason that the naïve
model performs badly with uniform distribution is that the
model generally underestimates the optimal overbooking
levels as the estimates obtained from the model are slightly
below the average of the available capacity and maximum
allowable booking requests in all cases whereas the actual
optimal overbooking levels of uniform distribution are gen-
erally higher. The reason for this is that in reality, overbook-
ing technique is used when the number of show-up custom-
ers is less than the available capacity; that is, it is used when
the probability of customers not showing up for the service
is high. Hence, for a distribution which has high expected
values or high ratio of show-ups per booking requests, higher
overbooking level is not needed. In summary, the naïve
model performs well with booking requests and show-ups
following a distribution with higher expected values.
Furthermore, by employing overbooking concept as
opposed to simply accepting booking requests equal to the
number of available capacities, the car rental service provid-
ers can reduce 42–56% of the overall total cost occurred.
This is shown in Table13. This result emphasizes the impor-
tance of the overbooking concept in the business dealing
with perishable products with more than one type.
Conclusion
Overbooking is one of the most important strategies in rev-
enue management. The key requirement for this strategy is
the formulation of an appropriate model which determines
the optimal overbooking levels so as to minimize the total
cost. This research presented a mathematical overbooking
model dealing with two levels of perishable products. The
main contribution of the model presented in this paper is
that unlike other researches, customer rejection is used as
a criterion for the occurrence of opportunity cost and the
model considers the joint distribution between two impor-
tant uncertainties, i.e., booking requests and show-ups. Also,
sensitivity analysis is performed to observe the effects of the
parameters in the model (outsourcing cost per unit, oppor-
tunity cost per unit, and available capacity) on the optimal
overbooking levels. The results give useful information in
predicting the optimal overbooking levels. In some cases, the
results might be intuitive. In others, they showed meaning-
ful trend for the overbooking levels especially in the situa-
tion where a change in the parameter of one level of price
was used to predict the optimal overbooking level of the
other level of price. As the full model is very complex, three
simplified models are proposed: (1) robust multiple regres-
sion model, (2) nonparametric regression model, and (3)
naïve model. The proposed regression models usually have
a satisfying performance in estimating the optimal over-
booking levels, predicting them accurately that the differ-
ence in expected total cost is negligible and not statistically
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
209
Optimal overbooking model forcar rental business withtwo levels ofprices having stochastic…
significant (3.2–14.09%); while the naïve model performs
well in only some cases. The performance of the naïve
model is acceptable when used with triangular-distributed
booking and show-up requests which has relatively high
expected values. However, the naïve method is still a helpful
method in predicting the optimal overbooking levels when
there is the absence of some variables or when the regression
models are not available. The final results showed that a car
rental service provider can save great cost when employing
the overbooking concept.
Many interesting aspects can be extended from this paper.
First, other joint distribution functions can be considered
for more generalization. Second, other estimation methods
can be explored if they can provide more accurate results.
Third, other independent variables, i.e., minimum and maxi-
mum allowable booking requests and show-ups (mins, maxs,
minb, maxb), can be included in the estimation of the over-
booking level. Finally, more types of cars can be studied to
apply in cases with more price levels.
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Publisher’s Note Springer Nature remains neutral with regard to
jurisdictional claims in published maps and institutional affiliations.
Naragain Phumchusri is an Associate Professor in the Department of
Industrial Engineering, Faculty of Engineering, Chulalongkorn Uni-
versity, Bangkok, Thailand. She received her master’s and doctoral
degrees in Industrial Engineering from The H. Milton Stewart School
of Industrial and Systems Engineering, Georgia Institute of Technol-
ogy, Georgia, USA in 2010. Her research interests include Operation
Research, Revenue Management, Applied Statistics, Stochastic Opti-
mization, and Data Analytics with Machine Learning.
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