**In the variety of challenges encountered in restaurant business, the matter of choosing appropriate table sizes is often overlooked. By studying the choice of table sizes and their relative position directly in term of a generic layout and table mix problem, we find that it plays a key role in the eventual operation of bars and restaurants, enabling up to 29 % reduction of customer refusal in the chosen generic example. Hereby, it is key to install both some rows with larger tables and maintaining a sufficiently large pool of flexible smaller tables. As less distance in between tables is lost, rows of larger tables enable more seating capacity. Smaller tables on the other hand are less compact but can be combined flexibly by customers to fit the arriving party size.**

In a competitive
environment as the bar and restaurant business with other options just around
the corner or even next door it is of the utmost importance to maximize the efficiency
of the available resources. The restaurant business entails a varied set of
interacting operations ranging from inventory planning of highly perishable
goods to optimal staff occupation and staff allocation. In this sense, a
restaurant constitutes a highly vertical integrated mini factory. As these
subtasks are optimized integrally in practical settings, the table size problem
is seen as a subpart of the overall revenue management of the restaurant.
Several concrete problems such as dynamic pricing, seat allocation and
inventory management have received specific attention in literature. As opposed
to this, the problem of table sizes was focused on at best indirectly, as an integral
part of these other restaurant processes.

In search of results that are less
bound to specific cases, we study a model that explicitly focuses on the impact
of size and relative position of tables on customer service. Specifically, we
quantify the average customer refusal due to lack of available tables for a
generic example and a variety of system load values. Hereby, we avoid the
introduction of any parameter beyond a basic set capturing the customers’
arrivals and service process, similar to a classic queueing model. By doing so
we aim to obtain results that, if not quantitatively, at least qualitatively
can be applied to a broader set of other restaurants.

**Assumptions**

While the assumption
that a host or hostess seats the parties may hold true for many restaurants in
the United States, many European restaurants (especially the outdoor seating
area) and fast food restaurants all over the world very often use a seat
yourself policy. By introducing a wait to be seated policy in the model, the
lever of the possible performance improvement is shifted to the host that
decides which party is seated at which table at which point. Although this
additional control may result in performance improvement, it often yields
complicated seating policies that are hard to carry out in practice. Second,
the exact seating policy depends on the available table sizes, limiting the possibility
to extend the policies and results to other settings. Third, as these policies
often breach the FIFO rule, parties likely start complaining when they are
being overtaken by other customers. In contrast, our model assumes a seat
yourself policy, shifting the lever for performance improvement from the host
to the choice of table sizes. By doing so the determining factors for
performance improvement are in the hands of the restaurant manager rather than
the staff, resulting in more stable improvements which are easier to extend to
other settings.

Additionally we
assume parties that cannot be seated to leave immediately and search an empty
table at a neighboring restaurant. This corresponds to a setting where multiple
competing restaurants with a seat yourself policy are present in the same
neighborhood as for example is the case in a touristic city center. When
waiting customers are present both a model for the willingness to join the
queue (based on its length) and a patience model for prematurely leaving the
queue (based on the time already waiting) are necessary. As both processes are
highly dependent on the exact setting of a specific restaurant it would
complicate the necessary set of parameters even further. Moreover we expect
that a qualitative extension of our conclusions to settings with queues will be
possible without modelling this willingness to join or exit the queue. Settings
where the willingness of customers to wait is modelled in detail are thus
possible but non-vital. Nevertheless they provide an interesting path for
extending the current results in future work.

We assume customers
arrive in parties of two or four. As single customers, just as a party of two,
also occupy a table of two opposing seats, we can make an arrival of a party of
two represent both single and paired arrivals. Similarly, parties of four may
represent both arrivals of three and four persons. And with the addition of a
few chairs, parties of five and six often take place at a similar table and may
thus be modelled as parties of four. Parties consisting of more than six people
are not modelled explicitly. Although some more details are lost by introducing
these last two assumptions regarding parties larger than four, they are
acceptable as up to 90% of arriving parties are groups of four or less in real
life examples. For simplicity's sake, we limit the arrival process to two party
sizes (two and four), keeping the complexity of the model low without
sacrificing much on the accuracy of the eventual results. Similarly to the
queue length simplification, we expect our conclusions to make sense in
settings where the party sizes are modelled in more detail.

The arrivals are
modelled as a Poisson process, i.e. with exponentially distributed interarrival
times. Because of the memorylessness of the exponential distribution the
expected time until the next arrival is independent of the time expired since
the previous arrival. These completely independent arrivals correspond to the
case in which different parties do not influence each other's arrival times.
The service time, i.e. the time parties spend at the table is modelled as a
random variable with a lognormal distribution with a coefficient of variation
equal to 1/2, a common assumption in simulation studies of restaurants. In
contrast to a normal distribution a lognormal distribution avoids the negative
values in a natural way. Moreover the heavy-tailed lognormal distribution is a
good reflection of the tendency of some parties to linger their time at the
table.

**Towards 5 possible layouts**

We consider five
different layouts with varying number of rows, each row consisting of five
tables for two. The spacing in between two tables is assumed to be half of the
width of a table for two. The tables for two are assumed to be combinable, i.e.
if two adjacent tables are unoccupied at the same time they can be combined to
seat a party of four. Hereby, adjacent is defined as facing either an edge or a
corner of the table concerned. For a table for two completely enclosed by other
tables for two this gives a total of eight neighboring tables with which the
table is combinable. We assume a table can thus also be combined with adjacent
tables on adjacent rows if present and unoccupied. In our layout we also
introduce uncombinable tables of four, i.e. tables that have twice the width of
a table for two but that cannot be combined with other tables for two or four.
As larger tables require less spacing per surface unit, tables of four use the
available surface more efficiently. A row of five tables for two with a spacing
of half a table for two in between can indeed be replaced with three tables of
four with the same (absolute) spacing in between. As a result two additional
clients can be seated on a row with an equal length of seven table widths when
tables of four are used instead of tables for two. This is shown in the figure
below, which compares a single row of tables for two and four. Matching the
different table sizes to the party sizes like we do here is a natural choice
and is an often analyzed case in other related fields like batch-service
queueing systems, where matching the size of parties with the number of
customers that can be served at once leads to superior system performance.

The capacity
increase for tables of four however comes at a cost as the obtained capacity
can be used less flexibly. The big advantage of tables for two is their
combinability, allowing a perfect fit to the size of the arriving party. If a
party of four arrives, two tables for two are combined to a table of four. When
the group leaves, the table can be taken apart again to make two tables for
two. This is as opposed to a table of four which occasionally is underutilized
when a party of two is seated, since in practice, a party of two will not join
a table of four already taken by another party of two. Moreover the table of
four cannot be taken apart in two tables for two as this would result in a
violation of either the fixed spacing in between two tables or the total length
of a row. When a party of two arrives we assume they will prefer a table for
two. They will thus only occupy a table of four when they see all tables for
two occupied. This corresponds to a setting in which a waiter or other employee
can ask to switch tables if a couple unnecessarily occupies a table of four. As
both a row of tables for two and four have their particular advantage
(flexibility and capacity respectively) and disadvantage (more in between
spacing and underutilization respectively) it is typically better to combine
both in a single layout. This question lies at the heart of the layout and
table mix problem, that is investigated for a variety of parameter settings
below.

In our simulations
rows with (three) tables of four are introduced incrementally in the layout
from zero to all rows. The other rows are rows with (five) tables for two. For
each obtained layout we keep track of the number of lost parties of two and
four persons. The total capacity of the layout, the size of the arriving
parties and the average interarrival and service times are all connected by the
load ρ. As the system capacity increases with the number of rows with tables of
four, we have to take this into account when choosing the load of the system.
When for example the layout comprises five rows the capacity scales up from 50
to 60 seats as the number of rows of tables of four increases. To compare the
different table mixes of a layout on equal terms we therefore choose the
reference capacity (

*CAP*) as the capacity when no rows with tables of four are present in the layout. For the example of a layout with five rows the reference capacity is thus 50 seats. As we only consider two- and four-person parties, the size of arriving parties is characterized entirely by the relative share of four-person parties in the arrival mix, expressed in percent (_{0}*%*). This parameter is varied in different simulations to analyze the impact of a varying share in the arrival mix on the percentage of lost customers, our main performance measure. With the average interarrival time denoted by_{partyof4}*E[A]*, the average service time by*E[T]*, the load is given by:
By varying the load
in different simulations different arrival mixed can be compared. Note that as
a consequence of the variability in the arrival process and the service times
also for a load smaller than one, parties will sometimes not find an available
table and thus will be lost. The average service time

*E[T]*is normalized, i.e. set equal to one. The average interarrival time is thus changed in each simulation to match the chosen load, given the arrival mix and number of rows in the particular layout.

**Results**

In our simulation we
vary the total number of rows of tables from one to five. We thus evaluate five
different layouts each consisting of rows with five tables for two and rows
with three tables of four. In each layout the number of rows that have three
tables of four is changed from zero to the number of rows in the layout. The
other rows have five tables for two. The load is varied from 0.8 to 1.2 in
steps of 0.1, each step yielding five different values. The percentage of
customers that arrives as part of a party of four is varied from 0% to 100% in
steps of 10%. This amounts to 1100 combinations; for each 10

^{6}arrivals are simulated. The most important conclusions of these simulations are summarized below.
As we assume that a
table for two can be combined with all of its eight unoccupied neighboring
tables for two it is important to group the tables for two as much as possible.
In order to maximize the combination potential of the tables for two the rows
of tables for two are grouped and not alternated with rows with tables of four.
When the tables for two can only be combined with tables within the same row,
the different rows of tables for two do not have to be grouped. As a first
result, it is thus important to note that the dynamics of the combinable tables
are an essential boundary condition when determining the optimal layout and
should not be neglected. In what follows we keep the assumption that a table
for two can be combined with all of its unoccupied neighboring tables for two,
up to eight in number, and the rows with tables for two are thus grouped.

We first discuss the
results of a reference case with a total number of five rows, a load of 0.8 and
where 50% of customers arrive as part of a party of four. Note that this
corresponds with one in three (33%) parties representing parties of four, the
other two thirds accounting for parties of two. When no tables of four are used
(the classic layout) we have a capacity of 50 seats (CAP

_{0}) and on average 3.96 customers are lost per time unit, the time unit being the average service time. As rows with tables of four are introduced in this layout the capacity increases from 50 with zero rows of tables of four to 60 when all five rows have three tables of four. In the figure below the number of customers lost (or, equivalently, refused) per time unit is shown relative to the number of customers lost when no tables of four are present in the layout (3.96). The same figure shows the capacity relative to the capacity of 50 seats. The x-axis indicates the number of seats used for tables of four divided by the capacity. For example when two out of five rows have tables of four the total capacity equals 54 and approximately 44% of that capacity is provided by the rows with tables of four. It is clear that for each row of tables for two that is replaced with tables of four the capacity of the layout increases. From this figure we can also see that the minimal loss probability in this layout is reached when two out of five rows are replaced with tables of four which results in a layout with a total capacity of 54 seats consisting of six tables of four and 15 tables for two. With this optimal layout, shown in the next figure, the total number of customers lost per time unit decreases with 29% compared to the classic case when no tables of four are present. It is important to note that the points in the graphs are only connected as an illustration. The relation for both the capacity and loss probability when only a part of a row is replaced with tables of four is not linear as the capacity increase only occurs when a complete row is replaced with tables of four.
The graph below
shows results for almost the same setting, the only difference concerning the
arrival mix. The horizontal axis still shows the number of seats used for the
tables of four divided by the capacity. The vertical axis displays the number
of customers lost per time unit relative to the number of customers that are
lost when no tables of four are present (classic layout). Hereby, the latter
value varies under the influence of the arrival mix; all values are scaled to 100%.
Together with the reference case (33%), the graph displays cases in which 0%,
54% and 100% of parties is a party of four. For the case where all parties are
parties of two (0% case) it is clear that using tables of four is of no use.
Any table of four introduced is only used by parties of two and thus
underutilized. Although rows of tables of four increase the capacity of the
layout, this capacity increase cannot compensate for the loss of efficiency due
to partial use. If only parties of four arrive, it is clear that it is best to
replace all rows with tables for two by rows with tables of four. As no table
of four is ever underutilized no loss of efficiency comes along with it. In the
case where 54% of parties are parties of four, the layout with only tables for
two and the one with only tables of four perform equally well (or badly) in
terms of customer loss probability. In that case, any mix of rows with tables
for two and four will outperform the case where only a single type of table is
present, neatly demonstrating that in general a layout with a single type of
tables can be outperformed by an adapted mix of tables.

The graph below shows
the performance when we consider load values different from that of the
reference case. In the considered layout changing two rows to rows with tables
of four remains the optimal policy but the available performance improvement
decreases with an increased load. As the absolute number of lost customers is
higher for an increased load, the improvements in absolute terms are not
necessarily lower for higher loads.

In the tables below
the performance results for all parameters variations are summarized for a
fixed arrival mix with 33% of parties consisting of four persons. The first
table shows the number of rows that need to have tables of four to achieve the
lowest possible loss probability for the given setting. The second table shows
the optimal performance improvement achieved for this layout compared to the
case when only tables for two are present. From these tables it is clear that
for smaller layouts with fewer rows, fewer rows have to be equipped with tables
of four. For a layout with only one available row for example, it is not
advisable to use tables of four at all. In general however it is fair to say
that for most layouts it is advisable to change 25% to 50% of the available
rows to rows with tables of four. We are convinced that also for other
settings, where a more detailed modelling of the party size is appropriate or
where customers are willing to queue it is advantageous to have both combinable
tables for two and larger fixed size tables in the layout. Simulation results
out of the scope of this blogpost (not included) show that even when tables of
four do not result in a more efficient use of available space, an optimal mix
of the two types of tables still outperforms a layout where only a single type
of table is present by several percent, and this for a broad set of parameters.

**Conclusions**

In this blogpost we
define the layout and table mix problem and evaluate several design
alternatives for restaurants and bars by means of a discrete event simulator
and Monte Carlo simulation. Compared to the classic layout in which only
combinable smaller tables are used we propose to also use larger tables. As
less space in between the larger tables is lost, they enable a higher seating
capacity. As the available places of larger tables are used less effectively
when a small group takes a large table, a sufficiently large pool of flexible
smaller tables remains equally important to optimize the layout. This as
smaller tables are less compact but can be combined by customers to fit the
arriving party size. As we assume that a smaller table can be combined with all
of its neighboring smaller tables it is thereby important to group the smaller
tables as much as possible in order to maximize the possible combinations. By
means of a generic example with tables for two and four, we show that for most
layouts and parameter settings it is advisable to change 25% to 50% of the
available rows to rows with tables of four. By doing so customer loss can be
reduced by up to 29% when compared to the classic layout in which only tables
for two are present, showing the importance of the layout and table mix problem
in restaurant operations.

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