Proactive Walking and Overbooking With Class Dependent Walk-Out Costs

Abstract

The increasing uncertainty in travel has resulted in elevated cancelation and no-show rates across many aspects of travel, elevating the importance of overbooking practices. Overbooking helps address travel uncertainty by accepting reservations beyond available rooms but may result in walk or re-accommodation costs if all (most) of these reservations materialize. Walk costs are not homogeneous across all customer types, with costs potentially different for loyal (branded, direct) versus non/less-loyal (third-party-intermediated) guests. We formulate an optimal overbooking model with class-dependent walk-out costs for a hotel with two classes of reservations—loyal members with higher walk-out costs, and nonmembers with lower walk-out costs, but with each class at the same room rate. We embed a dynamic walk-out model, one where guests may be proactively walked, that is, walked while rooms still available, into an overbooking model. The joint model determines optimal walk-out decisions to minimize expected walk-out costs while also determining optimal overbooking levels. We investigate how class-dependent no-show rates and walk-out costs impact optimal walk-out decisions and optimal overbooking levels. We find that changes in the no-show rates for a customer class only impact the overbooking levels of the related class whereas changes in class-specific walk-out costs impact all customer class overbooking levels. We offer managerial insight into a proactive and strategic walk-out policy for the lodging industry, aiming to achieve optimal overbooking levels.

Keywords

overbooking revenue management hotel,walk out loyalty program

Introduction

Overbooking (OB) is one of the earliest revenue management tools as it was essential during the early days of regulated air travel where prices were regulated, and consumers had considerable flexibility in when they used their ticketed reservations (Anderson & Xie, 2010). OB is the practice of service firms (airlines, rental cars, hotels, etc.) accepting reservations beyond capacity, anticipating that some of those reservations may not materialize as guests may cancel or no-show. As outlined by Lefever (1988), OB is the artful effort of maximizing profit as it balances the costs of having surplus guests that need to be re-accommodated at other hotels or bumped to later flights with the forgone revenue (during a high demand period) of having empty rooms (seats) that in the absence of cancelations would otherwise be occupied. The revenue loss of an empty room is easy to estimate, whereas the cost of a re-accommodated guest is more subjective and may be heterogeneous across customers, for example, the costs of walking a platinum member of a branded loyalty program may be much higher than that of a less brand-loyal customer who books through an online travel agent. In this article, we formulate an OB model to calculate the optimal OB levels for two types of customers (loyal members and non-members) with the help of the marginal analysis and the dynamic programming for the walk-out model. Our model is the first formalized approach to help firms decide when they should proactively walk less firm/brand-loyal guests even though some rooms are available.

OB is due to the existence of uncertainty of customers showing up for an event, such as taking a flight and checking in a hotel room. The ease of online reservations has dramatically elevated the need for OB as Accor’s D-Edge reported that almost 40% of all online hotel reservations are canceled before arrival.¹ OB has been a subject of academic literature for more than 60 years. Beckmann (1958) calculated the optimal limits of overselling problems with consideration of gamma distributions of cancelation and no-shows. Subsequent papers for example, Lyle (1970) looked to generalize the cancelation and no-show distributional assumptions. Rothstein (1968) developed the first dynamic programming models for OB in airline industries. Later, Subramanian et al. (1999) developed a dynamic programming model for jointly solving seat allocation and OB levels. Toh and Dekay (2002) first discussed factors that might be involved in executing an OB model for a hotel such as early departure and stay-over. Ivanov (2007) discovered dynamic OB limits based on whether hotel bookings are guaranteed or not. Moreover, Ivanov (2015) explored optimal OB limits with both upgrades and downgrades involved between three room types. Phumchusri and Maneesophon (2014) proposed optimal OB decisions based on the marginal cost involved with walk-out costs and vacant rooms from no-shows. Jongcheveevat et al. (2018) calculated optimal OB limits with joint stochastic bookings and show-up requests and room upgrades allowed. Chun and Ovchinnikov (2019) was the first paper to discuss optimal OB decisions for different distribution channels such as hotel-direct channels and intermediated (OTA) channels. Phumchusri and Maneesophon (2014), Ivanov (2015), and Chun and Ovchinnikov (2019) all studied the OB problems with the help of marginal analysis. While many of these models, especially those with integrated seat allocation, allow for different revenue rates (airline fares) by customer class they all assume homogeneous walk costs and as a result provide no insight into which guests should be walked.

Hwang and Wen (2009) proposed the first paper to empirically study the customer reactions to hotel OB and how OB impacts customer relationship management. The paper indicated that hotels must spend money on compensation for walking out customers to make customers have a positive perception of fairness in the OB policy. Moreover, the customers perceived fairness to the OB and compensation policies has a strong impact on the word of mouth for the hotels. Noone and Lee (2011) found that the intent to return of customers who were walked cannot be easily changed even with overcompensation.

Most often optimal OB levels are calculated through some form of marginal analysis with the optimal OB level calculated by comparing the expected marginal revenue (MR) and marginal cost for an incremental overbooked customer. For example, a hotel with capacity C of homogeneous rooms and a room rate r, with walk cost w needs to decide on x the number of rooms to reserve. If guests have a no-show rate q, then the number of guests who show follows a binomial distribution Y(x) ~ Binomial (x, 1–q). As a result, the probability the (x−C)th incremental customer is walked is

P (x) = \sum_{i = 0}^{x - C - 1} (\begin{matrix} x \\ i \end{matrix}) q^{i} {(1 - q)}^{n - i}, x > C

(1)

The expected marginal walk-out cost (MC_x) is then MC_x = wP(x), with the corresponding MR_x = r(1−P(x)).

The optimal OB level, b, is then

b = \max {x : M C_{x} \leq M R_{x}}

(2)

In traditional OB settings where the walk cost of a guest is assumed constant, w, walking of guests occurs only once all rooms/seats are occupied. But in practice, especially for branded properties with numerous repeat customers (Lefever, 1988), hotels may proactively walk lower valued guests (i.e., those booking via OTAs), given the strategic and long-term impacts of loyal guests, if there are still higher value guests yet to arrive (and the hotel is oversold). Marriott is one of the few brands to explicitly state in writing its compensation policies if a reservation cannot be honored with its Ultimate Reservation Guarantee² indicating Marriott will pay for your accommodation at a nearby hotel plus provide $100 in additional compensation, with this compensation increasing to $200 and 140,000 points depending on your Bonvoy status. In the following, we address this operationally important aspect of OB—specifically the proactive walking of lower-valued guests. For a setting with two customer types of differentiated walk costs, we develop an optimal walk policy (i.e., the proactive walking of lower walk cost guests) as a function of remaining rooms and reservations on the books for each customer type. The result of optimal walk policy is the optimal walk cost as a function of reservations by customer type. This optimal walk cost is then used to determine the optimal OB policy using traditional marginal analysis. Together, these models present the first approach to setting optimal OB policies with differentiated walk costs.

Model Development

Model development is separated into two parts: the proactive walk model and the multi-class probabilistic OB model with the walk-out model integrated. The walk-out model is formulated as a dynamic programming model, whereas the OB model is an extension of the single-class marginal analysis.

Walk-Out Model

Our focus is on two customer classes: members with walk cost W_m and nonmembers with walk costs W_n such that W_m ≥ Wn > r. As members (those with the highest walk cost) would never be proactively walked, the model focuses on deciding if an arriving nonmember should be checked-in or walked. Let m and n (m, n ≥ 0) represent the reservations on hand for member and nonmember customers with no-show probabilities for members and nonmembers q_m and qn (q_m, q_n ≥ 0), respectively. Let U_c(m, n) represent the expected walk cost of having c remaining rooms with m and n remaining reservations. Accordingly, we define D_c(m, n) as the walk-out decision matrix for arriving nonmembers given c remaining rooms with m and n reservations outstanding. In the following formulation, we assume that no further reservations are accepted once customers start to arrive (i.e., no day of reservations or walk-ins), all guests stay one night with homogeneous revenue r, and guests arrive randomly proportionate to the number of outstanding reservations of each customer type.

Once all rooms are occupied (c = 0), no further walk-out decisions are required as any arriving guests will have to be re-accommodated elsewhere, as such:

\begin{array}{l} U_{0} (m, n) = (1 - q_{m}) * m * W_{m} + \\ (1 - q_{n}) * n * W_{n}, \\ (m, n \geq 0, c = 0) \end{array}

(3)

Similarly, if n or m = 0, no decisions are required as arriving guests are accepted until c = 0, with any remaining arrivals re-accommodated.

Arriving nonmembers are checked-in if U_c₋₁(m, n−1) ≤ U_c(m, n−1) + w_n. or proactively walked if U_c(m, n−1) + wn < U_c₋₁(m, n−1).

Accordingly, the objective is to minimize the expected walk-out costs of the system on the stay date where U_c is determined recursively by,

U_{c} (m, n) = \frac{m [\begin{array}{l} (1 - q_{m}) m i n {U_{c} (m - 1, n) + w_{m}, U_{c - 1} (m - 1, n)} + q_{m} U_{c} (m - 1, n)] \\ + n [(1 - q_{n}) \min {U_{c} (m, n - 1) + w_{n}, U_{c - 1} (m, n - 1)} + q_{n} U_{c} (m, n - 1)] \end{array}}{m + n}

(4)

where U_c(0, 0) = 0.

Similarly, a general function is formulated to denote the optimal walk-out decision for nonmembers over stages from c to 0. When c≠ 0, the optimal walk-out decision functions, D_c are

D_{c} (m, n) = {\begin{matrix} \begin{array}{l} 1, U_{c - 1} (m, n - 1) - U_{c} (m, n - 1) \leq \\ w_{n} (1 - q_{n}) and m, n > 0 \end{array} \\ 0, otherwise \end{matrix}

(5)

and the optimal walk-out decision function at stage 0, D₀, is determined by

D_{0} (m, n) = 0, m, n \geq 0

(6)

where a 1 indicates an arriving nonmember is accommodated and 0 walked.

Multiclass OB Model

Through the integration of the above walk-out model, the multi-class OB model determines optimal OB levels for classes m and n. Members and nonmembers each show up following separate binomial distributions with rates q_m and q_n, respectively. The probability that an incremental customer will be walked out based on m and n reservation on hand is calculated as a two-dimensional probability P(m, n). P(m, n) is expressed as:

\begin{array}{l} P (m, n) = \sum_{i = C + 1 - n}^{m} \sum_{j = C + 1 - i}^{n} (\begin{matrix} m \\ i \end{matrix}) (\begin{matrix} n \\ j \end{matrix}) q_{m}^{m - i} {(1 - q_{m})}^{i} \\ q_{n}^{n - j} {(1 - q_{n})}^{j}, m + n > C \end{array}

(7)

The walk-out model’s minimal expected walk-out costs are then translated to marginal costs to determine optimal OB levels. The marginal walk-out cost (MC_m) is calculated based on P_{(m, n)}, the member no-show probability q_m, and the expected walk-out costs from the walk-out model. MC_m is expressed as:

M C_{m} = \frac{U_{c = C} (m, n) - U_{c = C} (m - 1, n)}{1 - q_{m}} P (m, n)

(8)

with

M C_{n} = \frac{U_{c = C} (m, n) - U_{c = C} (m, n - 1)}{1 - q_{n}} P_{(m, n)}

(9)

The MR is calculated based on P_{(m, n)} and the hotel standard room price r and is expressed as:

M R = r (1 - P_{(m, n)})

(10)

The optimal booking limits of members and nonmembers are then simply b_m = max{m: MC_m ≤ MR} and bn = max{n: MC_n ≤ MR}.

Results

We run numerical tests for both the walk-out model and the multiclass OB model with a focus on optimal OB levels and proactive walk out of nonmembers as a function of relative walk costs and no-show rates. The walk-out model is focused on the optimal walking of customers with lower walk-out costs to avoid walking the customers with higher walk-out costs. As an illustration, Figure 1 below illustrates the maximum number of nonmember reservations accepted for a 20-room property (C = 20) with r = 100, q_m = q_n = 0.25, w_m = 300, w_n = 150, as a benchmark the figure illustrates results for homogeneous walk costs (i.e., w_m = w_n = 150) which would be comparable to existing approaches. In other words, exceeding the maximum number indicates that the nonmember customer has to be walked out. Figure 1 illustrates that after member reservations exceed 18, nonmembers are proactively walked once w_m > w_n.

Figure 1.

Proactive Walk-Out Decisions: Cost Implications.

Similar to Figure 1, Figure 2 illustrates changes in proactive walk-out decisions as a function of differences in no-show rates. As before with r = 100, q_m = q_n = 0.25, w_m = 300, w_n = 150 with the figure also showing q_m = 0.25 and q_n = 0.35. Figure 2 shows that proactive walking of nonmembers decreases as the ratio of nonmember/member no-show rate increases.

Figure 2.

Proactive Walk-Out Decisions: No-Show Rate Implications.

We study how the no-show rate impacts the optimal OB levels for members and nonmembers. We test four cases with adjustment only to the nonmember no-show rate with homogeneous walk-out costs, the four cases are:

$\begin{array}{l} C = 20, r = 100, w_{m} = w_{n} = 150, q_{m} = 0.35, \\ q_{n} = 0.05 \end{array}$

$\begin{array}{l} C = 20, r = 100, w_{m} = w_{n} = 150, q_{m} = 0.35, \\ q_{n} = 0.15 \end{array}$

$\begin{array}{l} C = 20, r = 100, w_{m} = w_{n} = 150, q_{m} = 0.35, \\ q_{n} = 0.25 \end{array}$

$\begin{array}{l} C = 20, r = 100, w_{m} = w_{n} = 150, q_{m} = 0.35, \\ q_{n} = 0.35 \end{array}$

From Figure 3, we observe that the nonmember no-show rate only impacts the nonmember optimal OB level. From the left plot, it indicates the member optimal OB levels as a function of nonmember reservations, with the right plot showing nonmember optimal OB levels as a function of member reservations. The smooth behavior of the right panel on Figure 3 shows typical nonmember OB level changes as a function of no-show rates, whereas the nonsmooth nature of the left panel indicates that costly member walks are dramatically impacted by no-show rates of other (less-costly) rate classes or customers.

Figure 3.

Optimal Overbooking Level Changes by No-show Rates.

We use four additional cases to illustrate the impacts of walk costs on optimal OB levels for members and nonmembers. We test cases by adjusting the walk-out cost for members:

$\begin{array}{l} C = 20, r = 100, q_{m} = q_{n} = 0.35, w_{m} = 300, \\ w_{n} = 150 \end{array}$

$C = 20, r = 100, q_{m} = q_{n} = 0.35, w_{m} = w_{n} = 225$

$\begin{array}{l} C = 20, r = 100, q_{m} = q_{n} = 0.35, w_{m} = 900, \\ w_{n} = 150 \end{array}$

$C = 20, r = 100, q_{m} = q_{n} = 0.35, w_{m} = w_{n} = 525$

Figure 4 illustrates that member walk costs not only impact the member optimal OB levels as shown from the left plot but also impact the nonmember optimal OB levels as shown in the right plot. The increase in the walk-out cost for members causes the decrease in both optimal OB levels for members and nonmembers. Moreover, we can observe that for Case 5 and 6, the series do not overlap exactly indicating that the optimal OB levels vary even though the sum of singular walk-out cost for members and nonmembers are at the same level.

Figure 4.

Optimal Overbooking Level Changes by Walk-Out Costs.

Discussion

Traditional OB models are predicated on not proactively walking guests but rather gambling or hoping that guests who are walked are no more costly to walk than those accommodated earlier. Clearly in today’s differentiated online world where guests’ firm loyalty may be strongly correlated with their booking behaviors (i.e., intermediated guests potentially less loyal to firm/brand direct bookers) OB models should account for this heterogeneity when deciding OB levels and more importantly how OB should be executed. Our model is the first to formally illustrate the need to proactively walk guests in the presence of heterogenous walk costs and the resulting impact of this proactive walking upon OB levels.

We connect the multi-class OB model with a proactive walk-out model to obtain optimal OB levels for member and nonmember customers given that firms optimally (proactively) walk lower walk cost customers. The models assist hotels in generating an optimal business performance if the proportion of overbooked member and nonmember customers is carefully calculated. The analysis allows the walk-out model to offer the optimal walk-out decisions based on remaining capacity and the number of reservations on hand for multiple customer types. Our numerical analysis illustrates those changes in any class’s no-show rates or walk-out costs impact optimal OB and walk-out decisions for all customer classes.

We realize the process of OB and deciding whom to walk is a very complicated process for many properties as rates, length of stay and no-show rates may be dramatically different by customer classes. Our stylized model is a first step at highlighting the importance of customer or class-specific walk costs upon optimal OB levels. The model illustrates the need for a formalized proactive walk policy—versus simply waiting till all rooms are occupied. The model presented uses a homogeneous rate by customer type but can be extended to use class-specific rates. Extensions to 3 or more customer classes, while possible, become more cumbersome given the dynamic programming implementation for optimal walk-out costs.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, or publication of this article.

ORCID iDs

Shuwen Liang

Chris K. Anderson

Notes

Author Biographies

Shuwen Liang is graduate student in Applied Statistics and Hotel Administration at Cornell University. Prior to Cornell Shuwen worked in revene management for 4 years with Marriott Interational in Hangzhou China.

Chris K. Anderson is a Professor at Cornell University’s Nolan School of Hotel Administration, SC Johnson College of Business. He teaches and conducts research in data analytics, pricing, distribution and revenue management.

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