Overbooking as a Means to Manage Restaurant No-Shows and Cancellations: A Novel Model Extension

Abstract

Newsvendor models have been developed to determine the optimal overbooking level of hotel rooms to manage no-shows and cancellations. This research note extends the newsvendor model to a restaurant context by taking into account the “stretched capacity” of restaurants to determine the optimal overbooking levels for restaurant seats. Data were collected from a restaurant in Taiwan to illustrate the model. The percentage of no-shows per day in this restaurant ranged from 11% to 16%. Utilizing its stretched capacity, the restaurant can overbook up to five more seats than the estimated number of no-shows. The extended model will be most suitable for restaurants that largely depend on reservations (rather than walk-ins), such as luxury or fine-dining restaurants. Directions for future research on restaurant overbooking are provided.

Keywords

cancellations no-shows luxury newsvendor overbooking stretched capacity

Highlights

An extension of the newsvendor model has been developed to manage no-shows and cancellations in the restaurant industry.

This novel model extension is especially suitable for luxury and fine-dining restaurants that can create a stretched capacity.

The proposed model is not suitable for smaller, independent restaurants, where the pattern of no-shows is more uncertain.

Introduction

Restaurant table management is the process in which a host assigns dining room tables to customers, and it involves overseeing when and where to seat customers. Hwang (2008) investigated the impact of table assignment policies on customer waiting time and found that seating guests in front-to-back and out-in arrangements resulted in improved performance and led to higher seat turnover and revenues. The front-to-back policy has customers first being seated away from the back area of a dining room (i.e., away from the kitchen or restrooms); the out-in policy initially placed customers in the outer area of a dining room and then gradually seated them toward the center. These are strategies that restaurants can use to increase revenue and profit.

Some restaurant guests like to make reservations before arriving at a restaurant to ensure they will have a proper table for dining; however, not all guests with reservations will show up. Restaurants can adopt a reservation policy to reduce the opportunity costs of no-shows. Some reservation policies include (1) credit card guarantee: customers submit a credit card number with their reservation; (2) “short-shows” or having an accurate headcount of the party: when the number of guests that shows up is less than the number booked in the reservation, the restaurant can levy a charge; (3) table holding—when guests are late, the restaurant gives the table away to the next available guests; (4) entire party seating: the restaurant delays seating guests at a table until the whole party has arrived; and (5) maximum duration: putting time limits on dining experiences (Kimes, 2011). These reservation policies also allow restaurant operators to identify the most profitable mix of customers. For example, Alexandrov and Lariviere (2012) found that customers who spend a lot are also more likely to reserve tables. Accepting reservations is thus warranted if the gain in the average bill compensates for the resulting no-shows. Gregorash (2016) investigated this optimization characteristic of reservations in five fine-dining restaurants by comparing which type of restaurant patron—one who makes a reservation, or one who walks in—spends more. Average spending for those who made reservations was higher, so Gregorash (2016) recommended that fine-dining restaurants should save more space for parties making reservations.

One option to manage this optimization issue is for restaurants to use the overbooking method, a common practice of revenue management (RM), where the restaurants accept more table or seat reservations than their available capacity allows. Research shows that overbooking can increase the efficiency of capacity usage; however, it can also be an inconvenience to guests, which is a necessary topic for management to appraise and discuss.

Hwang and Wen (2009) researched overbooking and found that 44% of respondents considered overbooking to be unfair; however, providing compensation to overbooked guests was seen as reasonable and fair treatment. Therefore, they emphasized that it is essential for managers to know what guests perceive as equal compensation for being overbooked. Customer loyalty and positive word-of-mouth could be enhanced with the assurance of at least a perceived “fair” treatment in cases where overbooking inconveniences guests.

Overbooking is not a simple method to implement. It implies intense marketing intentions, especially considering that the practice of overbooking directly conflicts with the mission of satisfying guest demands. For many restaurants, overbooking happens all the time, such as in famous restaurants that require reservations during every period of the year. For example, to book a table in a Michelin three-star restaurant, one would need to place a reservation many months or even years in advance. Some guests may change their minds, cancel their reservations, or become no-shows after waiting such a long time for a date to dine; therefore, many restaurants of similar status practice preventive overbooking to maximize their profit. If management continues to focus on delivering high-quality customer service, overbooking can be a realistic option for restaurants that heavily rely on reservations.

Restaurant managers must consider how to properly balance overbooking for profit with customer satisfaction. Restaurant tables, like hotel rooms or airplane seats, are perishable inventory, meaning their capacity cannot be inventoried; however, as Heo et al. (2013) pointed out, the restaurant business has certain features that can help manage unknown fluctuations in demand. For example, the way tables and seats are arranged in a restaurant can be easily changed. Since this feature offers relative flexibility for the service capacity, it is essential to consider when implementing RM practices. Thompson (2010) reviewed research on restaurant revenue optimization and introduced two related issues that are neglected in this context: capacity management and customer experience. For capacity management, the literature has emphasized using a variety of table sizes to match the capacity and demand better (Kimes, 2004; Kimes & Thompson, 2004; Thompson, 2002, 2003; Vidotto et al., 2007). The literature on customer experience, which explores how a customer responds to different pricing strategies, draws attention to how the number of occupied seats affects the expense of restaurants and service recovery, as well as the customer’s sensitivity to different reservation strategies (Kimes & Robson, 2004; Kimes & Wirtz, 2002; McGuire & Kimes, 2006).

Since it is a rare practice for restaurants to send guests to other restaurants when overbooking happens, the special characteristics aforementioned should be carefully considered to ease the burden of no-shows. Looking at the feature of flexibility, restaurants can rearrange the tables to create a “stretched capacity” when necessary. For example, a cramped arrangement of tables could be created in one area, with a more relaxed setup in another. The service crew could ask customers whether they are willing to sit in the less spacious section or if they prefer to wait. In other words, there could exist two capacity constraints in a restaurant: one is the desired capacity and the other is the stretched capacity. The desired capacity is the capacity the restaurant is designed for, in which guests have comfortable space to enjoy meals. On the other hand, the stretched capacity refers to when the seat limit is higher than the desired capacity, while still manageable and under the maximum capacity. When an overbooking situation happens, restaurants would still be able to squeeze in a few more guests using the stretched capacity but could still ultimately face a situation where they have to turn away guests.

Restaurant managers benefit from having some flexibility when dealing with reservations since they can choose between desired and stretched capacity, but there are certain costs that must be evaluated. Suppose that the number of guests arriving exceeds the desired capacity. A manager must not only calculate the extra revenue that could be generated but also the associated costs, such as the inconvenience or discomfort to other guests, the added pressure on the staff, and the possible reputation loss for the restaurant. Once the stretched capacity has been used, these costs may increase as more guests arrive, since guests must be denied seating due to the lack of physical space. The additional cost of denying guests consists of an unsatisfied guest, an apology from the restaurant, and providing a discount for the guest’s next visit.

This research therefore aims to explore how the optimal level of overbooking in a restaurant can be determined when both desired and stretched capacities are in play. The article is organized as follows: The second section proposes a novel model for determining the overbooking level in a restaurant, the third section utilizes data collected from a hotel restaurant in Taiwan to explore the applicability of the proposed model, and finally, the fourth section discusses the directions for future research.

Proposing a Model for Overbooking in Restaurants

To develop an overbooking model for restaurants, we draw from the overbooking literature in hotel RM. Netessine and Shumsky (2002) developed a model of determining the overbooking level for a specific room type; however, their model did not consider different probability distributions of no-shows and did not include the variety of room types. Vajpai (2018) suggested a simple and straightforward method for overbooking that did not require complex mathematical analysis. This method determined the overbooking level by calculating the ratio of late cancellations or no-shows based on the historical data of a hotel’s operations, and then found its probability. This study assumed that the number of late cancellations or no-shows would fit Poisson distribution, as shown by Equation (1) and the cumulative probability function shown in Equation (2):

P (x) = \frac{e^{- λ} λ^{x}}{x!}

(1)

F (k - 1) = \sum_{x = 1}^{k - 1} P (x)

(2)

In these equations, x represents the number of late cancellations or no-shows, and λ represents the average number of late cancellations or no-shows. At the time of the study, the authors calculated the probability of successfully dealing with the number of overbookings (k). Since the probability of the total number of late cancellations or no-shows is equal to one, the probability of the number of late cancellations or no-shows being higher than the overbookings (k) is 1 − F(k − 1). In other words, 1 − F(k − 1) is the probability of successfully handling overbookings (k). Given the said method, a table of the probability of effectively handling overbooking for a hotel based on different λ can be established.

Neither of these models consider the marginal cost resulting from late cancellations or no-shows. The issue of such opportunity costs was examined by Tse and Poon (2017). They collected and analyzed data from a restaurant at Hotel ICON in Hong Kong for 2 years and summarized the percentage of cancellations for each day of the week under lunchtime and dinnertime, as shown in Table 1.

Table 1

The Percentage of Cancellations for Each Day of the Week

Mealtime	Monday	Tuesday	Wednesday	Thursday	Friday	Saturday	Sunday
Lunchtime, %	10.08	10.36	10.52	9.07	10.60	12.95	10.53
Dinnertime, %	10.63	10.84	10.73	9.61	11.53	11.38	10.88

The authors assumed that the number of cancellations fits binomial distribution. Hence the revenue function can be represented as follows:

\begin{array}{l} R (N) = \sum_{k = 0}^{M_{1}} (\begin{matrix} N \\ k \end{matrix}) {(1 - p)}^{k} p^{N - k} k r + \sum_{k = M_{1} + 1}^{M_{2}} (\begin{matrix} N \\ k \end{matrix}) {(1 - p)}^{k} p^{N - k} F_{1} (k) \\ + \sum_{k = M_{2} + 1}^{N} (\begin{matrix} N \\ k \end{matrix}) {(1 - p)}^{k} p^{N - k} (F_{1} (M_{2}) - F_{2} (k)) \end{array}

(3)

In this equation, N represents the limit of overbookings, p the probability of cancellations, M₁ the desired capacity, M₂ the stretched capacity, and r the average bill for each guest. Also, $F_{1} (x) = (x - \frac{1}{40} {(x - M_{1})}^{2}) r$ constitutes the decreased revenue per capita due to the increased number of guests when the number of arrival guests are over the desired capacity (i.e., the revenue per capita decreases 2.5% [1/40] when guests increase by one person), whereas $F_{2} (x) = (\frac{x - M_{2}}{2}) r$ shows the loss of sending guests away when the number of arrival guests exceeds the stretched capacity. Based on its revenue function, the upper limit for maximizing the optimal number of overbookings can be obtained by using the simulation method.

When taking into account the number of walk-ins (m), Equation (3) can be rewritten as follows:

\begin{array}{l} R (N) = \sum_{k = 0}^{M_{1} - m} (\begin{matrix} N \\ k \end{matrix}) {(1 - p)}^{k} p^{N - k} (k + m) r \\ + \sum_{k = M_{1} - m + 1}^{M_{2} - m} (\begin{matrix} N \\ k \end{matrix}) {(1 - p)}^{k} p^{N - k} {(k + m)}^{N - k} F_{1} (k + m) \\ + \sum_{k = M_{2} - m + 1}^{M_{2}} (\begin{matrix} N \\ k \end{matrix}) {(1 - p)}^{k} p^{N - k} F_{1} (M_{2}) \\ + \sum_{k = M_{2} + 1}^{N} (\begin{matrix} N \\ k \end{matrix}) {(1 - p)}^{k} p^{N - k} (F_{1} (M_{2}) - F_{2} (k)) \end{array}

(4)

The models proposed by Vajpai (2018) and Tse and Poon (2017) both assumed that the number of late cancellations and no-shows would fit a discrete probability distribution. Phumchusri and Maneesophon (2014) proposed a model with a cost function in which the number of late cancellations or no-shows fits a continuous probability distribution. This model is the “newsvendor model” that is used in operation research to determine the optimal order quantity when facing stochastic demand. If demand is less than or equal to the order quantity, sales will equal the quantity demand; if demand is greater than the initial stock, sales will equal the order quantity. The demand is assumed to have a probability distribution; and since demand is uncertain, the optimal order quantity is one that will minimize total expected costs.

Using the newsvendor model as a starting point, a novel model can be put together for restaurants. Let Q denote the number of overbookings and x represent the number of late cancellations or no-shows. The total cost (TC) when x > Q can be written as

T C_{x > Q} = \int_{Q}^{\infty} r (x - Q) f (x) d x

(5)

where f (x) is the probability density function of late cancellations or no-shows, and r represents the average guest check. Also, the cost function of x < Q can be formulated as

T C_{x \leq Q} = \int_{0}^{Q} C (Q - x) f (x) d x

(6)

where C is the compensation given to guests who were overbooked. Therefore, the total cost of using the strategy of overbooking while considering late cancellations or no-shows can be shown as follows:

T C = T C_{x > Q} + T C_{x \leq Q} = \int_{Q}^{\infty} r (x - Q) f (x) d x + \int_{0}^{Q} C (Q - x) f (x) d x

(7)

When taking the first-order partial derivative of (7) with respect to Q, the optimal overbooking level is

F (Q^{*}) = \frac{r}{C + r}

(8)

From Equation (8), one can observe the straightforward results: the optimal overbooking level (Q^*) is determined by the loss of guest check, (r), the monetary value of compensation (C), and the cumulative probability function of the late cancellations or no-shows.

Looking back, Tse and Poon (2017) noted that restaurants can obtain more flexible space through the rearrangement of tables to create two capacity limits: the desired capacity and the stretched capacity. Our study is based on this perspective and extended by Phumchusri and Maneesophon’s (2014) model for restaurants. With this in mind, we denote the gap between the stretched capacity and the desired capacity as Q_S. Thus, the cost associated with overbooking can be rewritten as

T C_{x > Q} = \int_{Q}^{\infty} r (x - Q) f (x) d x

(9)

T C_{x \leq Q} = \int_{0}^{Q_{s}} C_{s} (Q - x) f (x) d x + \int_{Q_{s}}^{Q} C (Q - x) f (x) d x

(10)

The right side of Equation (10) consists of two parts: The first part represents the cost when the overbooking level is higher than the number of late cancellations or no-shows but still not beyond the stretched capacity (Cs denotes the cost of having uncomfortable guests due to having a tighter seating arrangement when using the stretched capacity, inconvenience or discomfort to other guests, and the added pressure on the staff). The second part shows the cost when the overbooking level is higher than the number of late cancellations or no-shows and is over the stretched capacity (C denotes the cost of turning away guests and the possible reputation loss for the restaurant).

By summing up Equations (9) and (10), we can obtain the function for total cost:

\begin{array}{l} T C = T C_{x > Q} + T C_{x \leq Q} \\ = \int_{Q}^{\infty} r (x - Q) f (x) d x + \int_{0}^{Q_{s}} C_{s} (Q - x) f (x) d x \\ + \int_{Q_{s}}^{Q} C (Q - x) f (x) d x \end{array}

(11)

\begin{array}{l} = - r \int_{Q}^{\infty} F (x) d x - r Q + C_{S} Q F (Q_{S}) - C_{S} Q_{S} F (Q_{S}) \\ + C Q_{s} F (Q_{s}) - C Q F (Q_{s}) + C_{s} \int_{0}^{Q_{s}} F (x) + C \int_{Q_{s}}^{Q} F (x) d x \end{array}

(12)

When taking the first-order partial derivative of (12) with respect to Q, the first-order condition can be shown as follows:

\begin{array}{l} \frac{\partial T C}{\partial Q} = r \cdot F (Q) - r + C_{S} F (Q_{S}) - C \cdot F (Q_{s}) + C \cdot F (Q) = 0 \\ F (Q^{*}) = \frac{(C - C_{S}) \cdot F (Q_{S}) + r}{C + r} \end{array}

(13)

Thus, Equation (13) is the optimal solution of the modified model that can be used for restaurants when deciding the ideal overbooking numbers. We compare Equation (13) with Equation (8) and find that the overbooking number determined by (13) is greater than that of (8). This means that if a stretched capacity was not available, fewer people ( $F^{- 1} (\frac{r}{C + r})$ ) could be overbooked; however, with the novelty of this model’s stretched capacity, more people ( $F^{- 1} (\frac{(C - C_{S}) \cdot F (Q_{S}) + r}{C + r})$ ) can be overbooked, optimizing space and income.

An Illustration

To demonstrate our model, we collected and analyzed data from the restaurant of a tourist hotel in the northern part of Taiwan. The restaurant is an all-you-can-eat semibuffet restaurant, in which the price per person during lunchtime is 968 new Taiwan dollars (NTD, about 35 U.S. dollars) and 1,298 NTD (about 47 U.S. dollars) during dinnertime. Its table capacity is 78 seats, but the lounge–bar (with 50 seats) will be utilized when arrival guests exceed 78, resulting in a total number of available seats for this semibuffet of 128 seats. If guests arriving exceed the desired capacity of 128, the restaurant manager could add an extra 8 seats along the side of the window for a total stretched capacity of 136 seats.

Detailed data were collected from this restaurant from January 2018 to December 2018. This included the number of daily reservations, cancellations, and arrivals during lunchtime and dinnertime, offering a total of 730 records. Supplemental Figure 1 demonstrates the average number of reservations and cancellations per day at lunchtime and dinnertime each month. Looking at this, we can see that the average number of reservations each day (per person reserved for) increased from month to month, escalating from 20 guests in January to almost 60 guests in December. The average number of cancellations corresponded to the average number of reservations, growing from 2 guests in January to approximately 12 guests in December.

Table 2 shows the analysis from looking at each separate day of the week. The upper part of Table 2 demonstrates the average number of guests that seats were reserved for each day at lunchtime and dinnertime, respectively.

Table 2

The Average Guests of Reservations and Cancellations or No-Shows by Day of the Week

Reservations/Cancellations	Monday	Tuesday	Wednesday	Thursday	Friday	Saturday	Sunday
Number of reservations
Lunch	24.96	32.04	30.60	35.81	40.21	72.04	71.35
Dinner	19.29	22.55	22.88	29.81	42.21	43.35	29.60
Combined	22.12	27.29	26.74	32.81	41.21	57.69	50.47
Number of cancellations or no-shows
Lunch	3.77 (0.114)	4.30 (0.110)	3.94 (0.0823)	6.73 (0.128)	7.40 (0.116)	12.25 (0.154)	12.87 (0.145)
Dinner	3.23 (0.105)	5.45 (0.203)	4.98 (0.187)	6.83 (0.196)	6.13 (0.130)	7.13 (0.128)	5.15 (0.166)
Combined	3.50 (0.109)	4.88 (0.154)	4.46 (0.135)	6.78 (0.162)	6.77 (0.123)	9.69 (0.141)	9.01 (0.155)

Note: The ratio of cancellations to reservations made are in parentheses.

There is a significant difference among the average number of reservations for each individual day of the week (F = 20.906, p < .001), with Saturday having the highest average number of reservations. Furthermore, the average number of reservations at lunchtime versus dinnertime for the entire week also shows a significant difference (F = 35.818, p < .001), with the average number of reservations at lunchtime (43.82) being higher than at dinnertime (29.93).

Looking at cancellations and no-shows in the lower part of Table 2, the average number for lunchtime on the weekend (12.25~12.87) was significantly higher than that of weekdays (3.77~7.40). Friday (7.40) had the highest average number of cancellations for weekdays. For dinnertime, there was no significant difference between the averages for the weekend (5.15~7.13) and weekdays (3.23~7.13); however, Saturday (7.13) had the highest average cancellations or no-shows during dinnertime and Monday (3.23) had the least (be it during lunch or dinner).

The ratio of cancellations to reservations made (the decimals in parentheses) is presented in the lower part of Table 2. During lunch, the ratio for the weekend was approximately 5% higher than that of weekdays; however, there was no significant difference between weekdays and weekends during dinner. Looking at weekdays, Thursday had the highest rate of cancellations during lunch, which was about 13%. For dinner, Tuesday and Thursday had similar ratios, around 20%.

Table 3 shows the frequency of arrival guests exceeding the desired capacity of the restaurant (128) during lunch and dinner of each day of the week. There was a total of 80 incidents of being over the desired capacity: 58 at lunch, and 22 at dinner; 48 during the weekend, and 32 during the weekdays. Lunchtime during the weekend had the highest frequency, with two occurrences that even exceeded the stretched capacity (not explicitly shown on Table 3).

Table 3

The Number of Arrival Guests Exceeding the Capacity in Each Day of the Week

Meal	Monday	Tuesday	Wednesday	Thursday	Friday	Saturday	Sunday
Lunch	1	4	4	3	4	20	22
Dinner	2	2	1	4	7	4	2
Combined	3	6	5	7	11	24	24

For the guests who made reservations but could not be seated immediately, the restaurant provided a drink as compensation. Thus, each instance cost 144 to 210 NTD. When the number of arrival guests was greater than the stretched capacity, guests who were overbooked were give a discount coupon (20%) for their next visit to the restaurant (which only happened twice). The average cost of turning away a guest was 194 NTD at lunchtime and 260 NTD at dinnertime.

As for the revenue loss from cancellations or no-shows, the cost was equivalent to the average price charged for a guest at lunchtime or dinnertime (r = 968 NTD during lunch, or 1,298 NTD during dinner). According to Tse and Poon (2017), the number of late cancellations or no-shows (by number of guests) at lunch or dinner fits binomial distribution—an assumption that this study has also adopted. We used a normal distribution to approximate the binomial distribution, because a binomial distribution can be approximated by the normal distribution if $n p \geq 5 a n d, n (1 - p) \geq 5$ hold, where n represents sample size and p the probability of success. In our case, n stands for the number of total seats (128 seats), and p represents the probability of no-shows that ranges from 0.083 to 0.203 (as shown in Table 4). Therefore, $n p \geq 10 a n d n (1 - p) \geq 102$ satisfy the conditions.

Table 4

Cumulative Probability Among Various Numbers of No-Shows and Overbookings

Probability of no-shows	0.08	0.09	0.1	0.11	0.12	0.13	0.14	0.15	0.16	0.17	0.18	0.19	0.2	0.21
M	10.24	11.52	12.8	14.08	15.36	16.64	17.92	19.2	20.48	21.76	23.04	24.32	25.6	26.88
SD	3.069	3.238	3.394	3.540	3.677	3.805	3.926	4.040	4.148	4.250	4.347	4.438	4.525	4.608
F(Q_S)	0.2328	0.1385	0.0786	0.0429	0.0226	0.0116	0.0058	0.0028	0.0013	0.0006	0.0003	0.0001	0.0001	0.0000
F(Q) lunchtime	0.872	0.856	0.846	0.840	0.837	0.835	0.834	0.834	0.833	0.833	0.833	0.833	0.833	0.833
F(Q) dinnertime	0.869	0.853	0.843	0.837	0.834	0.832	0.831	0.830	0.830	0.830	0.830	0.830	0.830	0.830
Number of overbookings
1	0.001	0.001	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
2	0.004	0.002	0.001	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
3	0.009	0.004	0.002	0.001	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
4	0.021	0.010	0.005	0.002	0.001	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
5	0.044	0.022	0.011	0.005	0.002	0.001	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
6	0.084	0.044	0.023	0.011	0.005	0.003	0.001	0.001	0.000	0.000	0.000	0.000	0.000	0.000
7	0.146	0.081	0.044	0.023	0.011	0.006	0.003	0.001	0.001	0.000	0.000	0.000	0.000	0.000
8	0.233	0.138	0.079	0.043	0.023	0.012	0.006	0.003	0.001	0.001	0.000	0.000	0.000	0.000
9	0.343	0.218	0.131	0.076	0.042	0.022	0.012	0.006	0.003	0.001	0.001	0.000	0.000	0.000
10	0.469	0.319	0.205	0.125	0.072	0.040	0.022	0.011	0.006	0.003	0.001	0.001	0.000	0.000
11	0.598	0.436	0.298	0.192	0.118	0.069	0.039	0.021	0.011	0.006	0.003	0.001	0.001	0.000
12	0.717	0.559	0.407	0.278	0.180	0.111	0.066	0.037	0.020	0.011	0.006	0.003	0.001	0.001
13	0.816	0.676	0.523	0.380	0.260	0.169	0.105	0.062	0.036	0.020	0.010	0.005	0.003	0.001
14	0.890	0.778	0.638	0.491	0.356	0.244	0.159	0.099	0.059	0.034	0.019	0.010	0.005	0.003
15	0.940	0.859	0.742	0.603	0.461	0.333	0.228	0.149	0.093	0.056	0.032	0.018	0.010	0.005
16	0.970	0.917	0.827	0.706	0.569	0.433	0.312	0.214	0.140	0.088	0.053	0.030	0.017	0.009
17	0.986	0.955	0.892	0.795	0.672	0.538	0.407	0.293	0.201	0.131	0.082	0.050	0.029	0.016
18	0.994	0.977	0.937	0.866	0.764	0.640	0.508	0.383	0.275	0.188	0.123	0.077	0.047	0.027
19	0.998	0.990	0.966	0.918	0.839	0.732	0.608	0.480	0.361	0.258	0.176	0.115	0.072	0.044
20	0.999	0.996	0.983	0.953	0.897	0.811	0.702	0.578	0.454	0.339	0.242	0.165	0.108	0.068
21	1.000	0.998	0.992	0.975	0.937	0.874	0.784	0.672	0.550	0.429	0.319	0.227	0.155	0.101
22	1.000	0.999	0.997	0.987	0.965	0.921	0.851	0.756	0.643	0.523	0.405	0.301	0.213	0.145
23	1.000	1.000	0.999	0.994	0.981	0.953	0.902	0.827	0.728	0.615	0.496	0.383	0.283	0.200
24	1.000	1.000	1.000	0.997	0.991	0.973	0.939	0.883	0.802	0.701	0.587	0.471	0.362	0.266
25	1.000	1.000	1.000	0.999	0.996	0.986	0.964	0.924	0.862	0.777	0.674	0.561	0.447	0.342
26	1.000	1.000	1.000	1.000	0.998	0.993	0.980	0.954	0.908	0.841	0.752	0.647	0.535	0.424
27	1.000	1.000	1.000	1.000	0.999	0.997	0.990	0.973	0.942	0.891	0.819	0.727	0.621	0.510
28	1.000	1.000	1.000	1.000	1.000	0.999	0.995	0.985	0.965	0.929	0.873	0.796	0.702	0.596
29	1.000	1.000	1.000	1.000	1.000	0.999	0.998	0.992	0.980	0.956	0.915	0.854	0.774	0.677
30	1.000	1.000	1.000	1.000	1.000	1.000	0.999	0.996	0.989	0.974	0.945	0.900	0.835	0.751
31	1.000	1.000	1.000	1.000	1.000	1.000	1.000	0.998	0.994	0.985	0.966	0.934	0.884	0.814
32	1.000	1.000	1.000	1.000	1.000	1.000	1.000	0.999	0.997	0.992	0.980	0.958	0.921	0.867
33	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	0.999	0.996	0.989	0.975	0.949	0.908
34	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	0.999	0.998	0.994	0.985	0.968	0.939
35	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	0.999	0.997	0.992	0.981	0.961
36	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	0.999	0.996	0.989	0.976

Note: The boldfaced entries represent the number closest to yet larger than the F(Q) in lunch or dinner.

Based on Table 2 and related parameters (C = 194 or 260, r = 968 or 1,298), the cumulative probability distribution can be obtained, as shown in Table 4.

In this case, the cumulative probability corresponding to the stretched capacity, F(Q_s), can be obtained by using a normal distribution, with the means and the standard deviations shown in the second and third rows of Table 4. The fourth row in Table 4 reports values for the cumulative probability of the stretched capacity at lunchtime. Thus, when the probability of no-shows is 0.08 (the second column of Table 4), the cumulative probability of the optimal overbooking level at lunchtime can be calculated by Equation (12), as displayed in the fifth row of Table 4: $F (Q) = (194 * 0.233 + 968) / (194 + 968) = 0.872$ . Then, to choose the optimal overbooking level at lunchtime, the number closest to yet larger than this number in the section that charts the number of overbookings (while still staying in the second column) — 0.890—points to the answer. Therefore, the optimal overbooking level in this situation for lunch is 14. In the same way, the cumulative probability of the optimal overbooking level at dinnertime (shown in the sixth row of Table 4) indicates what the optimal overbooking level at dinnertime will be: in this case, within the second column, 14.

The overbooking levels are higher than the estimated no-shows (Table 5), calculated by rounding the product of the total seats (at stretched capacity) and the ratios reported in Table 2. The difference between the overbooking levels and no-shows is still not over the stretched capacity (eight additional seats). Moreover, the cost paid to any booked guests who are not immediately seated (a free drink) is relatively lower than what is lost due to no-shows.

Table 5

The Number of Estimated No-Shows and the Optimal Overbooking Level for Each Day of the Week

Meal	Monday	Tuesday	Wednesday	Thursday	Friday	Saturday	Sunday
Lunch
Estimated no-shows	15	15	11	17	15	20	19
Number of overbookings	18	18	14	21	19	24	22
Dinner
Estimated no-shows	14	27	24	26	17	17	22
Number of overbookings	17	30	29	30	21	21	26

Other restaurants will, of course, have different figures to work with and different costs to weigh. If manager can use historical data to calculate the probability of no-shows, then the number of overbookings can be found (as in Table 4). From this, a restaurant manager can decide how many overbookings to allow while still focusing on exceptional customer service.

Practical Implications and Future Directions

Unlike hotels and airlines, a typical restaurant cannot afford the risk of inconveniencing customers by charging for no shows and cancellations; the connection between a restaurant customer and a restaurateur is more relational than transactional. Thus, restaurateurs must have a greater tolerance for no shows and cancellations so that they may gain or keep guest loyalty, and the risk of losing a customer when cancellation policies are not chosen prudently is much greater. This raises the question of what the level of tolerance for no-shows and cancellations should be in different types of restaurants.

To explore this, Parsa et al. (2020) classified restaurants in the United States based on a continuum ranging from greatly utilitarian to greatly hedonic, depending on the nature of the dining experience. Their results suggested that the U.S. restaurant industry can be classified into four major classes: luxury, fine-dining, casual, and quick-service. Since the restaurants belonging to the luxury and fine-dining classes emphasize top-notch service and limited use of technology (Parsa et al., 2020), most patrons will book tables in advance, and walk-ins are relatively rare. Thus, customer no-shows will significantly affect these restaurants’ revenue, especially since, according to Gregorash (2016), restaurant customers who make reservations spend more money. Furthermore, no-shows and cancellations are more critical in restaurants that have a very high average guest check (e.g., luxury restaurants, Michelin starred restaurants).

Therefore, this study’s proposed model is especially suitable for luxury and fine-dining restaurants. It is also important that a restaurant can implement a stretched capacity when using this model. In practice, this approach allows restaurants to optimally overbook on busy days to account for no-shows, particularly when walk-in demand is limited or unable to replace no-show demand.

Due to the COVID-19 pandemic, many restaurants have reduced capacities. Thus, each no-show at restaurants that are still under COVID-19 restrictions represents a bigger slice of the potential revenue pie (Manning, 2020). Not only will the restaurant’s revenue be affected, but the front-of-house staff will also be affected: when large reservations become no-shows, managers may decide to send servers home resulting in a server’s income being only a fraction of what it should have been during that shift. In addition, if restaurants are not using all the ingredients they have on hand, they will not need to order more from suppliers; therefore, local farms and purveyors may also experience a ripple effect resulting from no-shows.

Our proposed model may also be more helpful on certain days or at certain times than others. In this study, we separated the data for weekdays and weekends. As shown in Table 2, there were more reservations on weekends (around 70% of the total reservations made in a given week). Thus, no-shows and cancellations on those days may be more critical from a RM perspective. Moreover, cancellations and no-shows were at a tolerable level of around 5% for Monday through Thursday. Since the pattern of reservations and no-shows is quite different on weekdays than on weekends, the data should continue to be separated in future work.

When dealing with the demands of guests in a time interval, researchers usually assume that demand follows Poisson distribution. Even in overbooking, Vajpai (2018) also assumed that the probability of no-shows would follow Poisson distribution with the pattern of no-shows being fairly certain. Furthermore, Sierag et al. (2017) presented some evidence on the inhomogeneous Poisson nature of the probability distribution function that demand follows. They pointed out that “demand is more uncertain for smaller than for larger hotels.” In the current study, the proposed model utilized a continuous probability density function, that is, normal distribution, to approximate the binomial distribution. The model also assumed that no-shows would follow the binomial distribution. This limitation suggests that the proposed model may not be as suitable for smaller, (independent) restaurants, where the pattern of no-shows is more uncertain.

To improve future studies, researchers could include a dollar value on no-shows and cancellations. The cost of cancellations (the product of the number of cancellations, the average party size, and the average guest-check amount) could be reported on a weekly, monthly, quarterly basis, making it easier to analyze and compare. Including such information would be more meaningful when discussing results and practical implications. While this study has proposed a model for large luxury and fine-dining restaurants to appropriately deal with no-shows by overbooking, researchers in the future may find a way to include a solution for a larger variety of restaurants who are under different constraints. This and further studies will help restaurant staff more greatly master the art of restaurant management.

Supplemental Material

sj-docx-1-jht-10.1177_10963480211064356 – Supplemental material for Overbooking as a Means to Manage Restaurant No-Shows and Cancellations: A Novel Model Extension

Supplemental material, sj-docx-1-jht-10.1177_10963480211064356 for Overbooking as a Means to Manage Restaurant No-Shows and Cancellations: A Novel Model Extension by C. I. Chiang in Journal of Hospitality & Tourism Research

Footnotes

Author’s Note:

This study was supported by the Ministry of Science and Technology, under grant MOST 108-2410-H-364-006.

ORCID iD

C. I. Chiang

Supplemental material

Supplemental material for this article is available online.

C. I. Chiang, PhD (cichiang@hcu.edu.tw), is an associate professor at the department of hospitality management, Hsuan Chuang University, Hsinchu city, Taiwan.

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