Online Companion to

Using Simulation to Help Manage the Pace of Play in Golf

MoonSoo Choi

Industrial Engineering and Operations Research, Columbia University,

New York, NY, USA {moonsoo.choi@columbia.edu}

Qi Fu

Credit Suisse, Eleven Madison Avenue,

New York, NY, USA {qf2112@columbia.edu}

Ward Whitt

Industrial Engineering and Operations Research, Columbia University

New York, NY, USA {ww2040@columbia.edu}

April 30, 2017

Abstract

(from the main paper) To help design and manage golf courses, Whitt (2015) developed anew stochastic queueing network model of a conventional 18-hole golf course and characterizedthe capacity of each hole. The primitives in that model are the random times required for eachgroup of golfers to play each stage on each hole, where a stage is an appropriate set of steps. Themodel is relatively high level, but the model still captures the important property that multiplegroups can play at the same time on the longer holes, subject to precedence constraints. Inthis paper, we develop a flexible and efficient simulation algorithm of that model and applyit to study important operational issues in golf. We substantiate the characterization of holecapacities and study: (i) the costs and benefits of a wave-up rule designed to increase the paceof play and (ii) the costs and benefits of alternative tee schedules, i.e., the intervals betweensuccessive groups starting play on the first hole. We estimate that the wave-up rule can increasethe daily number of groups that can play each day by about 13%, while a two-level tee schedulerule can increase the daily number of groups that can play each day by about 3%. We show thatthe costs of making the tee interval too short tend to be far greater than the costs of making ittoo long.

Keywords: pace of play in golf, queueing models of golf, using simulation to manage golf courses,

waving up in golf, alternative tee schedules on golf courses

1

1 Overview

This is an online companion to the main paper, Choi et al. (2017), which we will refer to that way.

We present additional in which we present material supporting and elaborating upon our findings

in the main paper.

The online companion is organized into four sections beyond this introduction section. Section

2 elaborates on our simulation algorithm. In particular, we discuss (i) the overall structure of

our simulation algorithm, (ii) methodology for calculating statistical estimates, and (iii) efficiency

of our algorithm design. Section 3 presents more data visualization figures like Figure 4 of the

main paper. These provide a holistic perspective of how various parameters–such as tee times, the

presence of the wave-up rule, and different course designs–affect the throughput levels of a golf

course, and how they effect the sojourn and departure times of each playing group.

Section 4 provides additional waiting time statistics to portray a clearer context of the impact

of making the course under-loaded. Section 5 shows the various throughput levels attained in golf

courses that are balanced but have different hole sequences, thereby showing that hole sequence

does not have a significant impact on the throughput as long as the courses are balanced.

Again, we use the notation — P3, P3WU , SP3, P4, and P5 — respectively, to denote the five

hole types considered, namely, par-3 hole (unscaled par-3 without the wave-up rule), par-3 hole

with wave-up, scaled par-3 hole, par-4 hole, and par-5 hole.

2 Simulation Algorithm: Design and Methodology

This section describes the simulation algorithm for the stochastic group play model reviewed in §2of the main paper. We expand on the brief overview given in §1.2 of the main paper.

The basic goal was to develop a simulation algorithm that is flexible and scalable in imple-

menting variety of golf course models, as well as variety of parameters associated with the golf

courses. Our simulation algorithm provides flexibility by allowing users to choose various param-

eters, including but not limited to (i) hole sequence, (ii) number of playing groups per day, (iii)

number of simulation replications, (iv) playing time parameters and distributions, (v) arrival time

distributions, and (vi) hole types. Our simulation algorithm also is scalabile, because the algorithm

can handle large numbers of playing groups and simulation replications of a complex golf course

model. In order to make our simulation efficient, our simulation model makes use of matrices, as

opposed to loops, to provider a faster means of tracking and calculating golf performance statistics

2

of multiple playing groups.

In addition, our simulation algorithm allows the simulation results to be statistically analyzed

in multiple ways. The model breaks down playing times and waiting times by each stage in each

hole for all playing groups; such breakdown structure allows us to analyze each playing group’s

performance on a hole-by-hole, stage-by-stage basis. Additionally, the algorithm provides a platform

for the simulation optimization algorithm presented in Section 4 of the main paper, which the tee

interval that maximizes the throughput in the golf course subject to constraints.

2.1 Algorithm Design and Methodology

We now provide an overview of the simulation algorithm. We first describe the data structure of the

algorithm, and then we specify the algorithm steps. A graphic overview of the simulation process

is shown in Figure 1.

2.1.1 Data Structure

We now describe the key variables and their data structure.

Here are the user-specified variables (with our common choice are in parentheses):

• n grp: number of groups playing in the course (102 groups)

– We normally set n g to be slightly larger than the actual number of groups to be analyzed.

The reason is because each playing group’s stage playing times depend on those of the

precedent and the subsequent groups. For example, the playing times of the 100th group

at the P3WU holes depend on those of the 99th and the 101st group. Therefore, even

though we target to analyze the performance of only the first 100 groups, we set n g to

be larger, e.g. 102.

• n rep: number of simulation replications (2000 replications)

• Playing time distribution parameters: ~m, a

– For our simulations, we used the triangular distribution to generate random playing

times.

3

– ~m is the vector of mean stage playing time parameters and is different for each hole type.

As outlined in §2.1 of the main paper, Par-3 and Par-4 holes have 3 stage, whereas P5

holes have 5 stages.

∗ ~m(3) =(mu1 P3,mu2 P3,mu3 P3): mean stage playing times for Par-3 stages

∗ ~m(4) =(mu1 P4,mu2 P4,mu3 P4): mean stage playing times for Par-4 stages

∗ ~m(5) =(mu1 P5,mu2 P5,mu3 P5,mu4 P5,mu5 P5): mean stage playing times for Par-

5 stages

– a is the variability parameter that indicates the spread in the triangular distribution.

– For each of the hole types above, we set the mean parameters to be (3.50, 2.00, 2.67),

(4.00, 2.00, 4.00) and (4.00, 2.00, 2.00, 1.33, 4.00), respectively. However, afterwards we

made minor adjustments to achieve a balanced course, as described in §2.2 of the main

paper.

∗ More detailed description on generating random playing times via triangular distri-

bution is available in the Section 2.2 of the main paper.

• Lost ball parameters: p and L

– p: Probability of a playing group losing a ball during the first shot for each hole

– L: The alternative longer playing time required for a group to play on the first stage of

a hole due to the lost ball

• intv array: an array that establishes tee interval schedule settings for the arrival times. This

array may contain multiple values; for example, with intv array= [5.0, 5.5, 6.0, 6.5, 7.0], the

simulation program will run 5 rounds of golf play simulation, each with different tee schedule

policy (and with n rep number of replications).

• Hole Parameters: seq, n h and n s

– seq: an 1 by n h array that records the sequence of holes in the golf course (this array

can be scaled to contain fewer than 18 holes in the course).

– n h: Number of holes (18 holes)

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– n s: Number of stages (60 stages). n s is calculated by adding the number of stages in

each of the n h holes. In particular,

n s= 3·(Number of Par-3 and Par-4 holes) + 5·(Number of Par-5 holes).

Please refer to Section 2.1 of the main paper for stage diagrams for Par-3, Par-4 and

Par-5 holes (3-stage diagram for Par-3 and Par-4 holes, and 5-stage diagram for Par-5

holes).

Here is the list of output variables in each replication of the simulation:

• A-matrix: an (n g) by 1 array that records each playing group’s arrival time at the golf course.

• P -matrix: an (n g) by (n s+2·n h) records each playing group’s stage clearing time after

finishing each stage.

As mentioned in the description for n s, there are 3 stages in Par-3 and Par-4 holes and 5

stages in Par-5 holes. We add 2 · 18 columns in the P -matrix to record (i) hole arrival times

of each group n (denoted by An) and (ii) beginning times of each group n (i.e. tee-off times,

denoted by Bn). Please refer to §2.1 in the main paper for more details on performance

measures. Since there are n h number of holes in the course (typically 18), we multiply these

two times by n h, or 18.

• wh,1 and wh,2: n g by 18 arrays that respectively record the first and second moments of

waiting times of each group before beginning the play at each hole. The waiting times are

calculated from subtracting An, the arrival time of group n at the hole, from Bn, the time

that group n begins playing in the hole.

• x1 and x2: n g by18∑i=1

ni arrays that respectively record the first and second moments of

playing times of each group. The playing times are calculated from subtracting Bn, the time

that group n begins playing in the hole, from Gn, the time that group n clears the green and

departs from the hole.

• w1total and w2

total: n g by 1 arrays that records the first and second moments of total waiting

times of each group in the entire golf course, averaged over n rep number of replications.

5

• x1total and x2total: n g by 1 arrays that records the first and second moments of total playing

times of each group in the entire golf course, averaged over n rep number of replications.

• TSO, TSO 2, ETSO: these arrays, each with a size of n g by 1, respectively denote the first

moment of the sojourn times of each playing group, the second moments, and the average

sojourn times for n g groups. Sojourn times are calculated by subtracting the first column

of the P -matrix (the arrival times of the groups at the first hole) from the last column of

the P -matrix (the clearing times of the groups at the last hole). TSO 2 is the square of

the first-moment array, ETSO is the average of the TSO arrays generated in n rep number of

replications.

– Similarly as above, sample variance of the sojourn times can be generated from sub-

tracting (TSO)2 from (TSO 2).

2.1.2 Calculating Statistical Estimates

In this section, we indicate how the algorithm calculates the statistical estimates of sample means

and variances for 2000 samples of waiting and playing times of each playing group. As mentioned

previously, 2000 is our common parameter choice for n rep, or the number of simulation replications,

because we found that it is large enough to provide good statistical precision. The samples of waiting

and playing times for each group come from running 2000 simulation replications.

Furthermore, we provide two-sided confidence intervals with 95% confidence over 2000 repli-

cations for waiting and playing times. We remark that, since the sample size is sufficiently large,

there is no need to calculate Student t-Statistic, even when the variance is unknown.

• Estimating wh,1 and wh,2:

– For variables wh,1 and wh,2 at hole h (∀h = 1, ...,n h) for group j (∀j = 1, ...,n g) over

n r i.i.d. replications, we provide the estimates as

wjh,1 ≡

n r∑

k=1

[wjh,1

]k/(n r), and

wjh,2 ≡

n r∑

k=1

[wjh,2

]k/(n r)

6

With the above estimates, the simulation calculates the difference wjh,2− (wj

h,1)2 to yield

variance s2w and subsequently calculates the square root of the value to yield standard

deviation sw, again for each group i at hole j.

– To compute half-width of the 95% confidence interval for the waiting times, we calculate

1.96sw/√n r. Because our default setting for n rep is 2000, the half-width would then

be 1.96sw/√2000.

• Estimating x1 and x2:

– Similarly as the estimates of x1 and x2, for variables x1 and x2 at hole h (∀h = 1, ..., 18)

for group j (∀j = 1, ...,n g) over n r i.i.d. replications, we provide the estimates as

x1 ≡n r∑

k=1

[x1(j, h)

]k/(n r), and

x2 ≡n r∑

k=1

[x2(j, h)

]k/(n r)

With the above estimates, the simulation calculates the difference x2 −(x1)2

to yield

variance s2x and subsequently calculates the square root of the value to yield standard

deviation sx, again for each group j at hole h.

– To compute half-width of the 95% confidence interval for the waiting times, we calculate

1.96sx/√n r. Because our default setting for n r is 2000, the half-width would then be

1.96sx/√2000.

• Estimating w1total and w2

total:

– The total waiting time for group j (∀j = 1, ...,n g) across all the holes h (∀h = 1, ..., 18)

in the course is calculated as18∑h=1

wjh,1 for each replication k. Similarly, w2

total is defined

as the sum of the second moments (i.e.18∑h=1

wjh,2). Subsequently, the first and second

moments of the total waiting time are estimated as

w1total =

[n r∑

k=1

(18∑

h=1

[wjh,1

]

k

)]/(n r), and

7

w2total

=

[n r∑

k=1

(18∑

h=1

[wjh,2

]

k

)]/(n r)

• Estimating x1total and x2total:

– The total playing time for group j (∀j = 1, ...,n g) across all the holes h (∀h = 1, ..., 18)

in the course is calculated as18∑j=1

[x1(j, h)

]kfor each replication k. Similarly, x2total is

defined as the sum of the second moments (i.e.18∑j=1

[x2(j, h)

]k). Subsequently, the first

and second moments of the total playing time are estimated as:

x1total =

[n r∑

k=1

(18∑

h=1

[x1(j, h)

]k

)]/(n r), and

x2total =

n r∑

k=1

18∑

j=1

[x2(j, h)

]k

/(n r)

2.1.3 Algorithm Steps

Below is a list of steps in the simulation algorithm. An overview of the steps listed below is also

presented in Figure 1.

(i) First, seq, or the hole sequence array, with a dimension of 1×n h (n h is the number of holes

in the course) is defined in the simulation algorithm. Again, our common parameter choice

for n h is 18.

(ii) Next, all relevant parameters—such as n g (the number of playing group), tee intervals,

playing/waiting time distributions—are declared and defined.

(iii) Once all the relevant parameters are defined, the algorithm initiates the first replication of

the simulation by generating A (array of arrival times of all the playing groups) according to

the tee schedule policy defined in the simulation model (with a dimension of N × 1, in which

n g denotes the number of playing groups). For example, setting an even deterministic tee

schedule at 5 minutes means that the groups would arrive at times 0, 5, 10, 15, etc.

8

(iv) Once the arrival time array is generated, the algorithm begins generating stage playing times

for each playing group at each of the stages in the first hole. For generating stage play-

ing times, the algorithm calls a different sub-function for each hole type (P3/SP3, P3WU ,

P4, and P5). Each sub-function entails the recursive mathematical representation of the

stochastic models of group play, which are outlined in Section 2 of the main paper.

(v) Subsequently, the main function of the simulation algorithm will supply the type of playing

time distribution and relevant parameter values to the appropriate sub-function based on the

hole type of the first hole. The sub-function will recursively compute and record hole arrival,

waiting, and playing times for each stage in the first hole, for all n g groups.

(vi) The playing time matrix, which has a dimension of n g by n1 (the number of playing stages

at hole 1), is returned to the main function of the algorithm.Then, the hole departure times

from the matrix (i.e. the last column) are copied to the arrival time array to begin simulations

for the second hole.

(vii) Subsequently, Step 4 to 6 is repeated for the next 17 holes until the last playing group departs

from the last hole in the sequence. The stage playing time arrays for 18 holes are augmented

to create “P -matrix”: a n g by18∑i=1

ni matrix that records the stage playing times of each

playing group.

(viii) The entire simulation, from steps 1 to 7, is then run for multiple independent and identically

distributed replications. The simulation model is currently set to run n sim (e.g. 2000)

replications, which we found is sufficient to yield a reasonably high level of statistical precision.

(ix) Finally, the golf simulator concludes with computation of various statistics, including means,

second moments, and variances of sojourn times, waiting times, and playing times. Please

refer to Sec 2.1.2 for more details on our methodology for calculating statistical estimates.

In regards to the steps above, we make one concluding remark. In the P -matrix, we calculate and

populate the playing times group-by-group—before doing so hole-by-hole. The reason we cannot

populate the matrix the other way (i.e. populating playing times for group 1 hole-by-hole for 18

holes—before doing so for group 2 and subsequent groups) is because of the wave-up rule for P3WU

holes. The wave-up rule stipulates that each group’s playing time depends not only on that of the

precedent group, but also on that of the subsequent group.

9

Generate an arrival matrix based on the

tee schedule

Define a golf course (1 x 18 array)

Define all the parameters, including: N

(number of groups), playing and waiting

time distributions, and tee schedule

policy

nN

A

Generate random playing times for the

first group

Repeat Steps 3 and 4 for the next 17

holes, while populating the P-matrix, or

the stage playing time recording matrix.

Repeat for the next 2000 replications

1

2

3

5

Recursively generate the playing times for

the subsequent groups, according to the

stochastic modelsof play with precedent

constraints outlined in Section 2.

Subsequently, calculate and record the

waiting times.

4

1

2 3

6 4

6

5

P-matrix (100 groups x Number of stage playing times)

4 5 4 3 4 4 3 4 4 5 4 3 4 4 4 5 5 4

2000 Replications

Figure 1: an overview of the data structure and steps of the simulation algorithm.

2.2 Advantages of Our Simulation Algorithm Design

Because of the complexity of golf course simulations–and because of the large number of repli-

cations (i.e. 2000) run during each round of simulation–we have made our simulation design as

efficient as possible. There are several aspects of our simulation design that help us achieve the goal.

2.2.1 Computation Scale of the Simulation Experiments

We now elaborate upon §1.2 of the main paper and provide a more detailed context of the scale

of running our simulation algorithm. Overall, the scale of our simulation experiment is fairly large

in terms of run length, and the large scale can be attributed to several factors. Primarily, the run

length of our simulation experiment can be determined by three factors: (i) the number of i.i.d.

replications, (ii) stage playing times generated per day of golf play, and (iii) various design factors

considered.

First, the number of i.i.d. replications (or, as denoted above, n r) plays a key role in determining

the run length of the simulation. Our common choice for the parameter n r) is 2000, which, we

believe, is sufficient to provide robust statistical precision for various variable estimates. Indeed,

10

the statistical precision is evidenced by the small values of half-widths displayed in the bottom row

of Table 3 in §3.1 of the main paper.

Second, we consider the number of stage playing times generated per day of golf play. As

stipulated above in the list of input variables for our simulation experiments, our common parameter

choice for n s, which denotes the number of stage playing times, is 60. For all course designs, we

have three stage playing times for 3 Par-3 and 12 Par-4 holes, and five stage playing times for

three Par-5 holes; thus, the total number of stage playing times sum up to 60. Since the common

parameter choice for the number of groups is set to be 102, each simulation replication, then, will

generate 6120 stage playing times.

Third, we also consider a number of various design factors. There are at least three additional

design factors when we run the simulation experiment: (i) the number of course designs, (ii) the

number of alternative tee intervals and (iii) the number of Par-3 hole types. As mentioned in

the introduction to §3 of the main paper, golf courses commonly have 12 identical Par-4 holes,

3 identical Par-3 holes and 3 identical Par-5 holes. However, each golf course has a different

combination of hole sequence; using the combinatoric formula, there is a total of 18!3!·12!·3! = 371, 280

different combinations available. Practically speaking, we do recognize that the possible number of

hole combinations in the real world is roughly 20, but the number can be easily larger.

We also consider the number of alternative tee intervals (also denoted as n t). For each course

design and for each Par-3 hole type, we obtain the simulation results for 30 different tee time

intervals. Lastly, as mentioned in the previous paragraph, we also consider the three different types

of Par-3 hole types. Considering these three design factors, we conclude that total number of design

factors to be considered, at the minimum, equals to 20 · 30 · 3 = 1800.

Furthermore, we note that the scale of our simulation experiments can become larger for two-

level tee intervals (see §5 of the main paper for more details), for which we consider various combi-

nations of multiple tee parameters. We considered 3 cut-offs (i.e. the group number at which the

tee interval increases), 3 types of Par-3 holes for 93 combinations of two-level tee intervals. We also

have run our simulation experiment for 5 different course designs. This gives us additional 4185

scenarios to consider in addition to the 1800 cases mentioned above.

Considering all of the factors mentioned above, we can calculate the total number of stage play-

ing times generated, which is 6120 ·5985 ·2000 ≈ 7.3 ·1010 (or approximately 73 billion stage playing

times). To portray a clearer picture of the required computational capacity to run our simulation

11

experiment, we note that the entire simulation experiment is estimated to take approximately 1.8

million seconds or approximately 500 full hours, based on our observations of time performance

measurements of the MATLAB software.

An important remark is that the above 500-hour requirement meets the bare minimum amount

of time required to conduct our research. The time requirement does not include other efforts in

designing our simulation experiment; we have run additional simulation runs to carefully choose

and test our parameter choices, evaluate robustness of our model, etc. We plan on extending our

simulation study to carefully study the impact of slow groups in the golf course.

2.2.2 Other Advantages of the Simulation

There also are other advantages of our simulation design. First, our simulation program is modu-

larized into individual functions. In our simulation algorithm, the main function is designed as the

main underlying framework for running different types of golf play scenarios. The random playing

times for each hole are generated not in the main function, but in one of the following sub-functions:

(i) P3 hole (also includes SP3 holes), (ii) P3WU hole, (iii) P4 hole, and (iv) P5 hole. Allocating

various parts of the simulation algorithm to different functions helps us to quickly identify certain

parts that need improvements.

Second, we have minimized the usage of loops by extensively using arrays. The clear advantage

of running simulations with arrays is that doing so helps the simulation program to run faster than

running loops does since dealing with arrays often involve fewer lines of codes.

Furthermore, we have divided the columns of time-stamp array P more granularly by different

playing stages rather than holes. Such array construction indeed provides a more detailed analysis

of playing and waiting times for each stage; but more importantly, it helps us to easily identify

bottleneck stages.

12

3 Data Visualization

A significant component of the simulation algorithm is creating data-visualization figures that help

reveal important insights about system performance.

One such data visualization figure was illustrated in Figure 3 of the main paper. We present

others now. These figures show how sojourn times and departure times vary with different tee

interval times and playing groups.

In §4.1 of the main paper, we formulated an optimization framework that aims to maximize

the throughput (i.e. the number of groups that can play per day) subject to (i) sojourn time

constraint and (ii) departure time constraint. The visualization figures in this section provide a

comprehensive view of sojourn time and departure time trends across a wide range of tee times.

We created the visualization figures to better understand how the throughput levels change as tee

interval changes, and to understand the behaviors of sojourn time convergence at each tee interval:

at which group sojourn times converge, and what value they converge to. Through depicting this

dynamic interaction behavior, the visualization figures in this section seek to visually depict the

tradeoff between sojourn times and departure times in the optimization framework in §4.1 of the

main paper.

13

5.0 5.5 6.0 6.5 7.0 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 8.0 8.1 8.2 8.3 8.4 8.5 9.0 9.5188.03 188.23 188.17 188.14 188.27 188.00 188.07 188.23 188.30 188.39 188.24 188.16 188.24 188.27 188.06 188.08 188.32 188.19 188.05 187.97 188.05 188.12

195.59 195.29 194.65 194.06 193.77 193.67 193.45 193.54 193.40 193.31 193.35 193.12 193.02 193.06 192.73 192.69 192.85 192.51 192.34 192.36 192.03 191.38

201.56 200.66 199.53 198.50 197.66 197.19 197.07 197.14 197.03 196.52 196.55 196.38 196.27 196.02 195.60 195.53 195.48 195.15 194.96 194.93 194.07 193.33

206.72 205.32 203.57 202.21 200.68 200.31 200.03 199.91 199.54 199.21 199.22 198.68 198.61 198.11 197.92 197.55 197.37 197.28 196.79 196.70 195.61 194.48

211.35 209.63 207.37 205.40 203.44 202.99 202.58 202.27 202.01 201.47 201.20 200.65 200.68 199.87 199.61 199.39 198.86 198.52 198.35 197.91 196.45 195.03

215.85 213.56 210.84 208.29 205.94 205.46 204.91 204.49 204.19 203.39 203.15 202.52 202.21 201.58 201.13 200.95 200.28 199.89 199.47 198.89 197.14 195.44

220.07 217.35 214.17 211.00 208.23 207.64 206.96 206.58 206.08 205.32 204.93 204.13 203.64 203.04 202.53 202.33 201.45 200.96 200.59 199.76 197.75 195.94

224.19 221.07 217.44 213.74 210.34 209.64 208.97 208.56 207.98 206.92 206.46 205.70 205.00 204.26 203.71 203.35 202.51 201.85 201.60 200.53 198.23 196.20

228.35 224.65 220.44 216.34 212.49 211.47 210.88 210.37 209.61 208.60 207.95 207.21 206.38 205.41 204.92 204.32 203.57 202.82 202.18 201.25 198.60 196.42

232.28 228.06 223.52 218.75 214.46 213.53 212.75 212.12 211.27 210.23 209.49 208.43 207.63 206.64 205.93 205.33 204.40 203.58 202.87 201.92 198.92 196.44

236.26 231.46 226.46 221.05 216.51 215.51 214.57 213.78 212.82 211.68 210.92 209.65 208.58 207.71 206.84 206.07 205.13 204.38 203.44 202.55 199.07 196.47

240.27 234.85 229.59 223.46 218.37 217.32 216.28 215.32 214.34 213.15 212.14 210.87 209.78 208.66 207.89 206.91 205.93 204.98 203.99 202.97 199.38 196.40

244.05 238.29 232.51 226.00 220.42 219.11 217.93 216.96 215.82 214.43 213.44 212.13 210.87 209.78 208.71 207.56 206.72 205.75 204.48 203.47 199.65 196.55

248.02 241.71 235.25 228.51 222.33 220.91 219.54 218.55 217.32 215.91 214.71 213.45 211.99 210.81 209.50 208.44 207.42 206.32 204.93 203.90 199.81 196.54

251.89 244.87 238.12 230.72 224.21 222.78 221.00 220.04 218.68 217.38 216.05 214.55 213.20 211.76 210.38 209.23 208.02 206.74 205.52 204.32 199.97 196.50

255.75 248.25 240.91 233.02 226.09 224.58 222.69 221.51 220.20 218.62 217.26 215.74 214.10 212.63 210.99 209.91 208.74 207.30 205.97 204.74 200.08 196.50

259.58 251.57 243.62 235.36 227.77 226.22 224.37 223.08 221.59 219.91 218.40 216.85 215.19 213.51 211.72 210.65 209.33 207.90 206.43 205.10 200.18 196.56

263.26 254.78 246.33 237.63 229.51 227.93 226.03 224.56 222.95 221.19 219.55 217.90 216.07 214.36 212.47 211.43 210.03 208.25 206.90 205.49 200.29 196.59

267.03 258.00 249.22 239.92 231.29 229.73 227.77 226.02 224.22 222.50 220.78 218.96 217.04 215.25 213.30 212.07 210.62 208.68 207.41 205.81 200.30 196.59

270.80 261.26 251.90 242.27 232.96 231.33 229.42 227.55 225.44 223.79 221.99 219.73 218.00 216.02 214.16 212.86 211.12 209.21 207.83 206.16 200.39 196.42

274.55 264.58 254.52 244.61 234.80 232.98 230.96 228.93 226.80 224.97 223.03 220.85 218.96 216.77 215.03 213.57 211.64 209.65 208.28 206.56 200.51 196.35

278.31 267.79 257.31 246.80 236.59 234.56 232.36 230.40 228.11 226.02 224.22 222.01 219.85 217.65 215.73 214.26 212.17 210.28 208.67 206.79 200.49 196.44

282.06 271.00 259.98 248.92 238.43 236.14 233.78 231.91 229.43 227.27 225.26 222.88 220.79 218.51 216.59 214.90 212.87 210.68 209.01 207.06 200.60 196.47

285.69 274.32 262.79 251.12 240.18 237.80 235.39 233.12 230.75 228.42 226.22 223.99 221.76 219.47 217.35 215.45 213.38 211.20 209.44 207.48 200.66 196.55

289.48 277.48 265.42 253.28 241.87 239.35 236.84 234.40 232.06 229.64 227.22 225.15 222.66 220.22 218.07 216.05 213.87 211.62 209.86 207.72 200.74 196.88

293.19 280.74 268.14 255.48 243.63 240.87 238.26 235.67 233.38 230.76 228.41 226.06 223.54 221.01 218.81 216.74 214.30 212.14 210.03 207.96 200.80 196.87

296.89 283.94 270.80 257.87 245.35 242.53 239.74 237.06 234.65 232.03 229.55 226.99 224.37 221.87 219.61 217.35 214.81 212.46 210.34 208.19 200.79 196.89

300.50 287.02 273.61 259.99 247.02 244.08 241.23 238.38 236.00 233.20 230.63 227.88 225.29 222.51 220.28 217.89 215.23 212.80 210.73 208.43 200.85 196.82

304.20 290.28 276.14 262.18 248.69 245.56 242.67 239.79 237.24 234.38 231.68 228.90 226.07 223.34 221.16 218.49 215.84 213.32 210.92 208.57 200.95 196.80

307.88 293.50 278.93 264.28 250.32 247.05 244.10 241.14 238.65 235.52 232.84 229.79 226.87 224.07 221.73 219.10 216.25 213.69 211.34 208.79 201.07 196.82

311.60 296.71 281.54 266.55 251.94 248.68 245.57 242.40 239.78 236.70 233.89 230.80 227.75 224.77 222.24 219.66 216.74 214.03 211.74 208.92 201.04 196.82

315.22 299.87 284.20 268.70 253.43 250.11 246.98 243.84 241.13 237.95 235.09 231.78 228.70 225.66 222.92 220.22 217.14 214.40 212.01 209.26 200.86 196.75

318.82 302.97 286.84 270.86 255.11 251.74 248.41 245.28 242.38 239.16 236.14 232.74 229.60 226.46 223.45 220.69 217.71 214.64 212.29 209.37 200.73 196.73

322.52 306.06 289.63 272.99 256.66 253.26 249.95 246.79 243.62 240.44 237.15 233.65 230.60 227.29 224.09 221.43 218.14 215.05 212.72 209.70 200.89 196.80

326.23 309.23 292.25 275.04 258.39 254.92 251.33 248.19 245.11 241.53 238.27 234.62 231.37 228.12 224.72 221.91 218.53 215.43 212.96 209.83 200.82 196.75

329.82 312.38 294.85 277.15 260.04 256.58 252.75 249.57 246.36 242.57 239.33 235.56 232.36 228.91 225.37 222.46 219.02 216.01 213.32 210.15 201.05 196.65

333.44 315.41 297.59 279.23 261.64 257.93 254.21 250.91 247.61 243.68 240.39 236.48 233.19 229.65 226.00 223.05 219.56 216.28 213.54 210.45 201.09 196.76

336.99 318.56 300.19 281.41 263.30 259.44 255.62 252.04 248.87 244.77 241.50 237.39 234.08 230.36 226.60 223.59 219.94 216.76 213.70 210.74 201.03 196.76

340.65 321.61 302.84 283.58 264.96 260.85 257.08 253.42 250.11 245.94 242.41 238.27 234.95 231.00 227.27 224.04 220.40 217.14 214.20 210.93 201.11 196.67

344.41 324.81 305.46 285.78 266.62 262.45 258.49 254.78 251.25 246.98 243.34 239.20 235.70 231.68 227.89 224.48 220.94 217.52 214.57 211.19 201.06 196.73

348.02 327.95 308.16 287.87 268.09 263.93 259.99 256.05 252.39 248.24 244.42 240.15 236.50 232.47 228.57 225.03 221.43 217.82 214.77 211.41 200.94 196.78

351.73 331.06 310.74 290.00 269.78 265.64 261.50 257.36 253.53 249.40 245.38 241.08 237.30 233.21 229.22 225.60 221.86 218.17 214.92 211.60 201.01 196.86

355.24 334.22 313.30 292.10 271.55 267.12 262.88 258.71 254.74 250.50 246.36 241.93 238.17 233.94 229.83 226.09 222.27 218.35 215.11 211.88 200.84 197.01

358.84 337.32 315.91 294.14 273.06 268.54 264.29 260.13 255.89 251.54 247.36 242.87 238.93 234.58 230.43 226.63 222.66 218.72 215.39 212.15 200.86 196.98

362.47 340.43 318.54 296.19 274.70 270.08 265.82 261.51 257.05 252.61 248.30 243.82 239.76 235.27 231.10 227.16 223.09 219.21 215.60 212.28 200.84 197.02

366.06 343.51 321.21 298.21 276.17 271.68 267.17 262.79 258.39 253.54 249.16 244.69 240.44 235.96 231.68 227.78 223.51 219.51 215.79 212.44 200.90 196.92

369.72 346.50 323.79 300.37 277.77 273.19 268.55 264.10 259.86 254.61 250.31 245.74 241.41 236.57 232.29 228.26 224.04 219.72 216.07 212.56 201.00 196.85

373.15 349.69 326.45 302.46 279.37 274.66 269.93 265.37 261.07 255.81 251.24 246.70 242.21 237.34 232.89 228.78 224.46 220.06 216.24 212.75 201.07 196.74

376.71 352.82 329.04 304.65 280.88 276.33 271.22 266.63 262.11 256.87 252.26 247.68 242.97 238.16 233.42 229.24 224.94 220.37 216.55 212.89 201.02 196.73

380.27 355.87 331.58 306.66 282.50 277.84 272.64 267.97 263.42 257.90 253.14 248.57 243.81 238.80 233.96 229.77 225.36 220.63 216.74 213.00 201.05 196.75

383.86 359.00 334.19 308.69 284.21 279.39 274.07 269.23 264.62 259.07 254.15 249.42 244.54 239.63 234.64 230.23 225.79 220.88 216.93 213.16 201.08 196.88

387.36 361.99 336.78 310.74 285.76 280.74 275.39 270.47 265.76 260.25 255.18 250.28 245.37 240.24 235.34 230.74 226.23 221.25 217.25 213.45 201.16 196.82

390.94 365.13 339.40 312.93 287.35 282.25 276.83 271.66 266.94 261.32 256.26 251.15 246.12 240.94 235.99 231.30 226.61 221.63 217.43 213.63 201.26 196.70

394.47 368.15 341.90 315.03 288.98 283.64 278.31 272.93 267.98 262.54 257.26 252.11 246.89 241.63 236.60 231.83 226.99 221.96 217.62 213.81 201.22 196.71

398.06 371.26 344.45 316.95 290.59 285.14 279.60 274.20 269.20 263.59 258.21 253.07 247.69 242.29 237.19 232.28 227.33 222.34 217.91 213.94 201.37 196.82

401.80 374.26 347.10 319.03 292.24 286.61 281.03 275.62 270.34 264.71 259.22 253.79 248.63 242.84 237.74 232.86 227.67 222.55 218.08 214.23 201.31 196.85

405.34 377.30 349.65 321.09 293.87 288.01 282.43 276.84 271.51 265.73 260.23 254.61 249.32 243.61 238.31 233.26 228.12 222.90 218.41 214.35 201.43 196.79

408.95 380.39 352.41 323.18 295.31 289.55 283.83 278.24 272.65 266.82 261.19 255.50 250.16 244.30 238.93 233.78 228.48 223.18 218.63 214.51 201.36 196.81

412.48 383.57 355.00 325.27 296.97 291.02 285.14 279.52 273.81 267.89 262.41 256.38 250.90 244.96 239.50 234.31 228.90 223.51 218.81 214.64 201.32 196.70

416.04 386.57 357.63 327.37 298.48 292.57 286.54 280.81 275.10 268.93 263.39 257.27 251.71 245.70 240.11 234.83 229.24 223.88 219.12 214.65 201.30 196.79

419.66 389.58 360.16 329.38 300.07 293.95 287.96 281.99 276.34 270.05 264.42 258.10 252.40 246.33 240.70 235.30 229.53 224.10 219.33 214.72 201.42 196.90

423.34 392.84 362.71 331.46 301.67 295.51 289.27 283.06 277.58 271.12 265.38 259.05 253.27 246.96 241.17 235.63 230.03 224.45 219.53 214.86 201.45 196.85

426.90 395.89 365.11 333.73 303.11 296.97 290.66 284.32 278.76 272.14 266.30 259.86 254.13 247.63 241.66 236.10 230.45 224.66 219.77 215.05 201.39 196.73

430.61 398.86 367.67 335.88 304.81 298.38 292.06 285.58 279.76 273.27 267.34 260.82 254.84 248.27 242.21 236.57 230.70 225.00 220.09 215.23 201.36 196.82

434.15 401.99 370.11 337.84 306.40 299.79 293.39 286.90 280.94 274.34 268.31 261.67 255.59 248.95 242.69 237.14 231.12 225.23 220.28 215.35 201.35 196.82

437.71 405.08 372.68 339.92 307.96 301.22 294.74 288.25 282.27 275.38 269.34 262.50 256.29 249.59 243.28 237.61 231.41 225.56 220.39 215.56 201.22 196.75

441.25 408.15 375.27 341.98 309.55 302.59 296.11 289.58 283.39 276.47 270.28 263.42 257.12 250.30 243.87 238.04 231.69 225.78 220.67 215.71 201.18 196.69

444.75 411.20 377.81 343.96 310.99 304.08 297.53 290.90 284.63 277.60 271.25 264.26 257.80 250.99 244.49 238.42 232.07 226.03 220.77 215.86 201.25 196.86

448.44 414.26 380.41 346.02 312.58 305.49 298.87 292.22 285.74 278.60 272.20 265.15 258.43 251.72 245.12 238.82 232.50 226.38 220.91 215.89 201.32 196.86

451.94 417.24 383.13 348.11 314.31 306.87 300.05 293.47 286.82 279.66 273.29 265.98 259.27 252.38 245.66 239.42 232.85 226.77 221.07 215.94 201.21 196.67

455.53 420.26 385.67 350.18 315.87 308.33 301.42 294.74 288.02 280.85 274.30 266.94 259.99 253.10 246.24 239.91 233.14 227.11 221.29 216.02 201.19 196.64

459.16 423.38 388.28 352.20 317.46 309.85 302.78 296.00 289.24 281.88 275.24 267.70 260.72 253.75 246.87 240.38 233.45 227.46 221.56 216.14 201.13 196.62

462.71 426.45 390.76 354.33 319.06 311.41 304.11 297.30 290.34 282.95 276.22 268.48 261.49 254.37 247.47 240.82 233.80 227.74 221.71 216.32 201.17 196.64

466.19 429.55 393.47 356.19 320.60 312.85 305.36 298.54 291.48 284.09 277.15 269.43 262.21 255.00 247.93 241.21 234.20 227.99 221.98 216.28 201.33 196.73

469.74 432.55 395.94 358.39 322.17 314.32 306.70 299.66 292.63 285.04 278.14 270.35 263.09 255.70 248.47 241.64 234.60 228.24 222.18 216.51 201.23 196.72

473.26 435.62 398.67 360.32 323.71 315.84 308.09 300.93 293.88 286.17 279.06 271.14 263.77 256.38 249.06 242.07 235.14 228.42 222.45 216.61 201.32 196.57

476.81 438.68 401.10 362.42 325.24 317.11 309.57 302.25 294.94 287.26 279.97 271.84 264.62 257.14 249.70 242.46 235.50 228.93 222.71 216.72 201.24 196.73

480.42 441.75 403.63 364.52 326.91 318.61 310.93 303.54 296.30 288.31 280.77 272.66 265.31 257.94 250.23 242.99 235.79 229.27 222.91 216.84 201.16 196.72

484.07 444.79 406.12 366.50 328.42 320.09 312.26 304.80 297.49 289.32 281.68 273.56 266.04 258.56 250.85 243.53 236.15 229.36 223.04 217.05 201.17 196.72

487.59 447.84 408.65 368.52 329.96 321.59 313.64 306.01 298.75 290.37 282.63 274.43 266.82 259.10 251.36 244.02 236.53 229.61 223.19 217.09 201.15 196.78

491.13 450.94 411.13 370.59 331.46 323.10 314.95 307.40 299.78 291.40 283.61 275.15 267.58 259.68 251.94 244.43 236.94 229.99 223.34 217.19 201.21 196.88

494.68 454.01 413.74 372.65 332.93 324.50 316.13 308.69 300.89 292.41 284.48 276.09 268.25 260.32 252.39 244.71 237.32 230.29 223.49 217.30 201.17 196.86

498.17 457.03 416.31 374.68 334.58 326.01 317.52 309.86 302.04 293.41 285.39 276.83 269.00 260.96 252.93 245.31 237.66 230.49 223.73 217.38 201.19 196.73

501.78 460.02 418.98 376.80 336.10 327.39 318.79 311.21 303.13 294.39 286.45 277.78 269.65 261.62 253.47 245.87 237.94 230.82 223.86 217.46 201.13 196.67

505.43 463.13 421.56 378.92 337.76 328.86 320.14 312.40 304.23 295.50 287.36 278.66 270.32 262.29 253.99 246.29 238.26 231.11 223.94 217.61 201.12 196.63

509.07 466.16 424.05 380.97 339.22 330.28 321.44 313.64 305.45 296.57 288.37 279.56 271.17 262.82 254.62 246.68 238.63 231.47 224.17 217.73 201.22 196.78

512.56 469.18 426.68 383.06 340.85 331.68 322.72 314.76 306.61 297.69 289.30 280.53 271.85 263.56 255.06 247.21 238.94 231.73 224.27 217.87 201.14 196.92

516.15 472.42 429.19 385.00 342.23 333.06 323.99 315.98 307.73 298.75 290.20 281.33 272.63 264.20 255.71 247.67 239.34 232.00 224.44 217.98 201.18 196.77

519.65 475.36 431.63 387.02 343.61 334.48 325.40 317.15 308.68 299.71 291.11 282.14 273.37 264.82 256.10 248.07 239.79 232.18 224.73 218.06 201.29 196.76

523.12 478.40 434.16 389.26 345.14 335.84 326.68 318.34 309.83 300.67 291.92 282.91 274.12 265.59 256.70 248.50 240.09 232.43 224.90 218.09 201.26 196.68

526.58 481.45 436.67 391.32 346.67 337.33 327.89 319.58 310.97 301.65 292.90 283.77 274.94 266.14 257.29 248.89 240.38 232.67 225.05 218.04 201.29 196.73

530.12 484.53 439.17 393.42 348.37 338.83 329.41 320.87 312.08 302.76 293.83 284.58 275.76 266.81 257.69 249.42 240.70 233.03 225.18 218.32 201.19 196.74

533.70 487.65 441.66 395.43 349.93 340.21 330.75 322.13 313.30 303.79 294.82 285.36 276.56 267.47 258.30 249.90 240.97 233.26 225.31 218.47 201.16 196.88

537.22 490.68 444.30 397.38 351.33 341.62 332.16 323.28 314.42 304.82 295.92 286.11 277.32 268.05 258.78 250.31 241.26 233.52 225.65 218.42 201.07 196.82

540.62 493.62 446.85 399.39 352.85 343.11 333.60 324.51 315.62 305.80 296.80 287.05 277.93 268.62 259.32 250.65 241.57 233.79 225.75 218.47 201.01 196.90

544.14 496.60 449.38 401.39 354.45 344.57 334.99 325.74 316.78 306.86 297.71 288.00 278.67 269.21 259.94 251.03 241.88 233.98 225.93 218.57 201.13 196.80

547.74 499.51 451.87 403.44 355.97 345.99 336.40 326.98 317.87 307.76 298.72 288.78 279.52 269.91 260.60 251.44 242.23 234.36 226.02 218.53 201.17 196.85

551.30 502.50 454.40 405.44 357.44 347.47 337.71 328.14 319.12 308.78 299.66 289.63 280.13 270.52 261.15 251.95 242.57 234.47 226.26 218.66 201.20 196.78

554.78 505.58 456.80 407.48 358.97 348.88 339.10 329.37 320.32 309.63 300.57 290.51 280.94 271.20 261.71 252.54 242.78 234.75 226.44 218.75 201.35 196.75

558.36 508.59 459.37 409.43 360.53 350.33 340.36 330.65 321.27 310.66 301.54 291.33 281.65 271.84 262.14 252.95 243.04 235.04 226.60 218.78 201.33 196.79

70

80

90

100

Groups / Tee

10

20

30

40

50

60

5.0 5.5 6.0 6.5 7.0 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 8.0 8.1 8.2 8.3 8.4 8.5 9.0 9.5188.03 188.23 188.17 188.14 188.27 188.00 188.07 188.23 188.30 188.39 188.24 188.16 188.24 188.27 188.06 188.08 188.32 188.19 188.05 187.97 188.05 188.12

200.59 200.79 200.65 200.56 200.77 200.77 200.65 200.84 200.80 200.81 200.95 200.82 200.82 200.96 200.73 200.79 201.05 200.81 200.74 200.86 201.03 200.88

211.56 211.66 211.53 211.50 211.66 211.39 211.47 211.74 211.83 211.52 211.75 211.78 211.87 211.82 211.60 211.73 211.88 211.75 211.76 211.93 212.07 212.33

221.72 221.82 221.57 221.71 221.68 221.61 221.63 221.81 221.74 221.71 222.02 221.78 222.01 221.81 221.92 221.85 221.97 222.18 221.99 222.20 222.61 222.98

231.35 231.63 231.37 231.40 231.44 231.39 231.38 231.47 231.61 231.47 231.60 231.45 231.88 231.47 231.61 231.79 231.66 231.72 231.95 231.91 232.45 233.03

240.85 241.06 240.84 240.79 240.94 240.96 240.91 240.99 241.19 240.89 241.15 241.02 241.21 241.08 241.13 241.45 241.28 241.39 241.47 241.39 242.14 242.94

250.07 250.35 250.17 250.00 250.23 250.24 250.16 250.38 250.48 250.32 250.53 250.33 250.44 250.44 250.53 250.93 250.65 250.76 250.99 250.76 251.75 252.94

259.19 259.57 259.44 259.24 259.34 259.34 259.37 259.66 259.78 259.42 259.66 259.60 259.60 259.56 259.71 260.05 259.91 259.95 260.40 260.03 261.23 262.70

268.35 268.65 268.44 268.34 268.49 268.27 268.48 268.77 268.81 268.60 268.75 268.81 268.78 268.61 268.92 269.12 269.17 269.22 269.38 269.25 270.60 272.42

277.28 277.56 277.52 277.25 277.46 277.43 277.55 277.82 277.87 277.73 277.89 277.73 277.83 277.74 277.93 278.23 278.20 278.28 278.47 278.42 279.92 281.94

286.26 286.46 286.46 286.05 286.51 286.51 286.57 286.78 286.82 286.68 286.92 286.65 286.58 286.71 286.84 287.07 287.13 287.38 287.44 287.55 289.07 291.47

295.27 295.35 295.59 294.96 295.37 295.42 295.48 295.62 295.74 295.65 295.74 295.57 295.58 295.56 295.89 296.01 296.13 296.28 296.39 296.47 298.38 300.90

304.05 304.29 304.51 304.00 304.42 304.31 304.33 304.56 304.62 304.43 304.64 304.53 304.47 304.58 304.71 304.76 305.12 305.35 305.28 305.47 307.65 310.55

313.02 313.21 313.25 313.01 313.33 313.21 313.14 313.45 313.52 313.41 313.51 313.55 313.39 313.51 313.50 313.74 314.02 314.22 314.13 314.40 316.81 320.04

321.89 321.87 322.12 321.72 322.21 322.18 321.80 322.24 322.28 322.38 322.45 322.35 322.40 322.36 322.38 322.63 322.82 322.94 323.12 323.32 325.97 329.50

330.75 330.75 330.91 330.52 331.09 331.08 330.69 331.01 331.20 331.12 331.26 331.24 331.10 331.13 330.99 331.41 331.74 331.80 331.97 332.24 335.08 339.00

339.58 339.57 339.62 339.36 339.77 339.82 339.57 339.88 339.99 339.91 340.00 340.05 339.99 339.91 339.72 340.25 340.53 340.70 340.83 341.10 344.18 348.56

348.26 348.28 348.33 348.13 348.51 348.63 348.43 348.66 348.75 348.69 348.75 348.80 348.67 348.66 348.47 349.13 349.43 349.35 349.70 349.99 353.29 358.09

357.03 357.00 357.22 356.92 357.29 357.53 357.37 357.42 357.42 357.50 357.58 357.56 357.44 357.45 357.30 357.87 358.22 358.08 358.61 358.81 362.30 367.59

365.80 365.76 365.90 365.77 365.96 366.23 366.22 366.25 366.04 366.29 366.39 366.03 366.20 366.12 366.16 366.76 366.92 366.91 367.43 367.66 371.39 376.92

374.55 374.58 374.52 374.61 374.80 374.98 374.96 374.93 374.80 374.97 375.03 374.85 374.96 374.77 375.03 375.57 375.64 375.65 376.28 376.56 380.51 386.35

383.31 383.29 383.31 383.30 383.59 383.66 383.56 383.70 383.51 383.52 383.82 383.71 383.65 383.55 383.73 384.36 384.37 384.58 385.07 385.29 389.49 395.94

392.06 392.00 391.98 391.92 392.43 392.34 392.18 392.51 392.23 392.27 392.46 392.28 392.39 392.31 392.59 393.10 393.27 393.28 393.81 394.06 398.60 405.47

400.69 400.82 400.79 400.62 401.18 401.10 400.99 401.02 400.95 400.92 401.02 401.09 401.16 401.17 401.35 401.75 401.98 402.10 402.64 402.98 407.66 415.05

409.48 409.48 409.42 409.28 409.87 409.75 409.64 409.60 409.66 409.64 409.62 409.95 409.86 409.82 410.07 410.45 410.67 410.82 411.46 411.72 416.74 424.88

418.19 418.24 418.14 417.98 418.63 418.37 418.26 418.17 418.38 418.26 418.41 418.56 418.54 418.51 418.81 419.24 419.30 419.64 420.03 420.46 425.80 434.37

426.89 426.94 426.80 426.87 427.35 427.13 426.94 426.86 427.05 427.03 427.15 427.19 427.17 427.27 427.61 427.95 428.01 428.26 428.74 429.19 434.79 443.89

435.50 435.52 435.61 435.49 436.02 435.78 435.63 435.48 435.80 435.70 435.83 435.78 435.89 435.81 436.28 436.59 436.63 436.90 437.53 437.93 443.85 453.32

444.20 444.28 444.14 444.18 444.69 444.36 444.27 444.19 444.44 444.38 444.48 444.50 444.47 444.54 445.16 445.29 445.44 445.72 446.12 446.57 452.95 462.80

452.88 453.00 452.93 452.78 453.32 452.95 452.90 452.84 453.25 453.02 453.24 453.09 453.07 453.17 453.73 454.00 454.05 454.39 454.94 455.29 462.07 472.32

461.60 461.71 461.54 461.55 461.94 461.68 461.57 461.40 461.78 461.70 461.89 461.80 461.75 461.77 462.24 462.66 462.74 463.03 463.74 463.92 471.04 481.82

470.22 470.37 470.20 470.20 470.43 470.21 470.18 470.14 470.53 470.45 470.69 470.48 470.50 470.56 470.92 471.32 471.34 471.70 472.41 472.76 479.86 491.25

478.82 478.97 478.84 478.86 479.11 478.94 478.81 478.88 479.18 479.16 479.34 479.14 479.20 479.26 479.45 479.89 480.11 480.24 481.09 481.37 488.73 500.73

487.52 487.56 487.63 487.49 487.66 487.56 487.55 487.69 487.82 487.94 487.95 487.75 488.00 487.99 488.09 488.73 488.74 488.95 489.92 490.20 497.89 510.30

496.23 496.23 496.25 496.04 496.39 496.32 496.13 496.39 496.71 496.53 496.67 496.42 496.57 496.72 496.72 497.31 497.33 497.63 498.56 498.83 506.82 519.75

504.82 504.88 504.85 504.65 505.04 505.08 504.75 505.07 505.36 505.07 505.33 505.06 505.36 505.41 505.37 505.96 506.02 506.51 507.32 507.65 516.05 529.15

513.44 513.41 513.59 513.23 513.64 513.53 513.41 513.71 514.01 513.68 513.99 513.68 513.99 514.05 514.00 514.65 514.76 515.08 515.94 516.45 525.09 538.76

521.99 522.06 522.19 521.91 522.30 522.14 522.02 522.14 522.67 522.27 522.70 522.29 522.68 522.66 522.60 523.29 523.34 523.86 524.50 525.24 534.03 548.26

530.65 530.61 530.84 530.58 530.96 530.65 530.68 530.82 531.31 530.94 531.21 530.87 531.35 531.20 531.27 531.84 532.00 532.54 533.40 533.93 543.11 557.67

539.41 539.31 539.46 539.28 539.62 539.35 539.29 539.48 539.85 539.48 539.74 539.50 539.90 539.78 539.89 540.38 540.74 541.22 542.17 542.69 552.06 567.23

548.02 547.95 548.16 547.87 548.09 547.93 547.99 548.05 548.39 548.24 548.42 548.15 548.50 548.47 548.57 549.03 549.43 549.82 550.77 551.41 560.94 576.78

556.73 556.56 556.74 556.50 556.78 556.74 556.70 556.66 556.93 556.90 556.98 556.78 557.10 557.11 557.22 557.70 558.06 558.47 559.32 560.10 570.01 586.36

565.24 565.22 565.30 565.10 565.55 565.32 565.28 565.31 565.54 565.50 565.56 565.33 565.77 565.74 565.83 566.29 566.67 566.95 567.91 568.88 578.84 596.01

573.84 573.82 573.91 573.64 574.06 573.84 573.89 574.03 574.09 574.04 574.16 573.97 574.33 574.28 574.43 574.93 575.26 575.62 576.59 577.65 587.86 605.48

582.47 582.43 582.54 582.19 582.70 582.48 582.62 582.71 582.65 582.61 582.70 582.62 582.96 582.87 583.10 583.56 583.89 584.41 585.20 586.28 596.84 615.02

591.06 591.01 591.21 590.71 591.17 591.18 591.17 591.29 591.39 591.04 591.16 591.19 591.44 591.46 591.68 592.28 592.51 593.01 593.79 594.94 605.90 624.42

599.72 599.50 599.79 599.37 599.77 599.79 599.75 599.90 600.26 599.61 599.91 599.94 600.21 599.97 600.29 600.86 601.24 601.52 602.47 603.56 615.00 633.85

608.15 608.19 608.45 607.96 608.37 608.36 608.33 608.47 608.87 608.31 608.44 608.60 608.81 608.64 608.89 609.48 609.86 610.16 611.04 612.25 624.07 643.24

616.71 616.82 617.04 616.65 616.88 617.13 616.82 617.03 617.31 616.87 617.06 617.28 617.37 617.36 617.42 618.04 618.54 618.77 619.75 620.89 633.02 652.73

625.27 625.37 625.58 625.16 625.50 625.74 625.44 625.67 626.02 625.40 625.54 625.87 626.01 625.90 625.96 626.67 627.16 627.33 628.34 629.50 642.05 662.25

633.86 634.00 634.19 633.69 634.21 634.39 634.07 634.23 634.62 634.07 634.15 634.42 634.54 634.63 634.64 635.23 635.79 635.88 636.93 638.16 651.08 671.88

642.36 642.49 642.78 642.24 642.76 642.84 642.59 642.77 643.16 642.75 642.78 642.98 643.17 643.14 643.34 643.84 644.43 644.55 645.65 646.95 660.16 681.32

650.94 651.13 651.40 650.93 651.35 651.45 651.23 651.26 651.74 651.32 651.46 651.55 651.72 651.74 651.99 652.50 653.01 653.23 654.23 655.63 669.26 690.70

659.47 659.65 659.90 659.53 659.98 659.94 659.91 659.83 660.18 660.04 660.06 660.21 660.29 660.33 660.60 661.13 661.59 661.86 662.82 664.31 678.22 700.21

668.06 668.26 668.45 667.95 668.59 668.54 668.40 668.40 668.80 668.59 668.61 668.87 668.89 668.89 669.19 669.68 670.13 670.54 671.51 672.94 687.37 709.82

676.80 676.76 677.10 676.53 677.24 677.11 677.03 677.12 677.34 677.21 677.22 677.29 677.63 677.34 677.74 678.36 678.67 679.05 680.08 681.73 696.31 719.35

685.34 685.30 685.65 685.09 685.87 685.61 685.63 685.64 685.91 685.73 685.83 685.81 686.12 686.01 686.31 686.86 687.32 687.70 688.81 690.35 705.43 728.79

693.95 693.89 694.41 693.68 694.31 694.25 694.23 694.34 694.45 694.32 694.39 694.40 694.76 694.60 694.93 695.48 695.88 696.28 697.43 699.01 714.36 738.31

702.48 702.57 703.00 702.27 702.97 702.82 702.74 702.92 703.01 702.89 703.21 702.98 703.30 703.16 703.50 704.11 704.50 704.91 706.01 707.64 723.32 747.70

711.04 711.07 711.63 710.87 711.48 711.47 711.34 711.51 711.70 711.43 711.79 711.57 711.91 711.80 712.11 712.73 713.04 713.58 714.72 716.15 732.30 757.29

719.66 719.58 720.16 719.38 720.07 719.95 719.96 719.99 720.34 720.05 720.42 720.10 720.40 720.33 720.70 721.30 721.53 722.10 723.33 724.72 741.42 766.90

728.34 728.34 728.71 727.96 728.67 728.61 728.47 728.36 728.98 728.62 728.98 728.75 729.07 728.86 729.17 729.73 730.23 730.75 731.93 733.36 750.45 776.35

736.90 736.89 737.11 736.73 737.11 737.17 737.06 736.92 737.56 737.14 737.50 737.26 737.73 737.43 737.66 738.30 738.85 739.26 740.57 742.05 759.39 785.73

745.61 745.36 745.67 745.38 745.81 745.68 745.66 745.48 745.96 745.77 746.14 745.92 746.24 745.97 746.21 746.87 747.30 747.90 749.29 750.73 768.36 795.32

754.15 753.99 754.11 753.84 754.40 754.19 754.19 754.10 754.54 754.34 754.71 754.47 754.79 754.55 754.69 755.54 755.92 756.43 757.88 759.35 777.35 804.82

762.71 762.58 762.68 762.42 762.96 762.72 762.74 762.75 763.27 762.88 763.34 763.00 763.29 763.09 763.28 764.11 764.41 765.06 766.39 768.06 786.22 814.25

771.25 771.15 771.27 770.98 771.55 771.19 771.31 771.38 771.79 771.47 771.88 771.62 771.92 771.70 771.87 772.64 772.89 773.58 775.07 776.71 795.18 823.69

779.75 779.70 779.81 779.46 779.99 779.78 779.93 780.00 780.43 780.10 780.45 780.16 780.40 780.29 780.49 781.12 781.47 782.13 783.57 785.36 804.25 833.36

788.44 788.26 788.41 788.02 788.58 788.29 788.47 788.62 788.94 788.60 789.00 788.75 788.83 788.92 789.12 789.62 790.10 790.78 792.11 793.89 813.32 842.86

796.94 796.74 797.13 796.61 797.31 796.77 796.85 797.17 797.42 797.16 797.69 797.28 797.47 797.48 797.66 798.32 798.65 799.47 800.67 802.44 822.21 852.17

805.53 805.26 805.67 805.18 805.87 805.33 805.42 805.74 806.02 805.85 806.30 805.94 805.99 806.10 806.24 806.91 807.14 808.11 809.29 811.02 831.19 861.64

814.16 813.88 814.28 813.70 814.46 813.95 813.98 814.30 814.64 814.38 814.84 814.40 814.52 814.65 814.87 815.48 815.65 816.76 817.96 819.64 840.13 871.12

822.71 822.45 822.76 822.33 823.06 822.61 822.51 822.90 823.14 822.95 823.42 822.88 823.09 823.17 823.47 824.02 824.20 825.34 826.51 828.32 849.17 880.64

831.19 831.05 831.47 830.69 831.60 831.15 830.96 831.44 831.68 831.59 831.95 831.53 831.61 831.70 831.93 832.51 832.80 833.89 835.18 836.78 858.33 890.23

839.74 839.55 839.94 839.39 840.17 839.72 839.50 839.86 840.23 840.04 840.54 840.15 840.29 840.30 840.47 841.04 841.40 842.44 843.78 845.51 867.23 899.72

848.26 848.12 848.67 847.82 848.71 848.34 848.09 848.43 848.88 848.67 849.06 848.64 848.77 848.88 849.06 849.57 850.14 850.92 852.45 854.11 876.32 909.07

856.81 856.68 857.10 856.42 857.24 856.71 856.77 857.05 857.34 857.26 857.57 857.04 857.42 857.54 857.70 858.06 858.70 859.73 861.11 862.72 885.24 918.73

865.42 865.25 865.63 865.02 865.91 865.31 865.33 865.64 866.10 865.81 865.97 865.56 865.91 866.24 866.23 866.69 867.19 868.37 869.71 871.34 894.16 928.22

874.07 873.79 874.12 873.50 874.42 873.89 873.86 874.20 874.69 874.32 874.48 874.16 874.44 874.76 874.85 875.33 875.75 876.76 878.24 880.05 903.17 937.72

882.59 882.34 882.65 882.02 882.96 882.49 882.44 882.71 883.35 882.87 883.03 882.73 883.02 883.20 883.36 883.92 884.33 885.31 886.79 888.59 912.15 947.28

891.13 890.94 891.13 890.59 891.46 891.10 890.95 891.40 891.78 891.40 891.61 891.15 891.58 891.68 891.94 892.43 892.94 893.99 895.34 897.19 921.21 956.88

899.68 899.51 899.74 899.15 899.93 899.60 899.33 899.99 900.29 899.91 900.08 899.79 900.05 900.22 900.39 900.81 901.52 902.59 903.89 905.80 930.17 966.36

908.17 908.03 908.31 907.68 908.58 908.21 907.92 908.46 908.84 908.41 908.59 908.23 908.60 908.76 908.93 909.51 910.06 911.09 912.53 914.38 939.19 975.73

916.78 916.52 916.98 916.30 917.10 916.69 916.39 917.11 917.33 916.89 917.25 916.88 917.05 917.32 917.47 918.17 918.54 919.72 921.06 922.96 948.13 985.17

925.43 925.13 925.56 924.92 925.76 925.26 924.94 925.60 925.83 925.50 925.76 925.46 925.52 925.89 925.99 926.69 927.06 928.31 929.54 931.61 957.12 994.63

934.07 933.66 934.05 933.47 934.22 933.78 933.44 934.14 934.45 934.07 934.37 934.06 934.17 934.32 934.62 935.18 935.63 936.97 938.17 940.23 966.22 1004.28

942.56 942.18 942.68 942.06 942.85 942.28 941.92 942.56 943.01 942.69 942.90 942.73 942.65 942.96 943.06 943.81 944.14 945.53 946.67 948.87 975.14 1013.92

951.15 950.92 951.19 950.50 951.23 950.76 950.39 951.08 951.53 951.25 951.40 951.23 951.23 951.50 951.71 952.37 952.74 954.10 955.24 957.48 984.18 1023.27

959.65 959.36 959.63 959.02 959.61 959.28 959.00 959.55 959.88 959.71 959.91 959.74 959.77 960.02 960.10 960.87 961.39 962.58 963.93 966.06 993.29 1032.76

968.12 967.90 968.16 967.76 968.14 967.74 967.48 968.04 968.43 968.17 968.32 968.21 968.32 968.69 968.70 969.40 969.89 971.13 972.50 974.59 1002.26 1042.18

976.58 976.45 976.67 976.32 976.67 976.33 975.89 976.58 976.97 976.65 976.90 976.77 976.94 977.14 977.29 977.89 978.38 979.67 981.05 983.04 1011.29 1051.73

985.12 985.03 985.17 984.92 985.37 984.93 984.61 985.17 985.48 985.26 985.43 985.28 985.56 985.71 985.69 986.52 986.90 988.33 989.58 991.82 1020.19 1061.24

993.70 993.65 993.66 993.43 993.93 993.41 993.15 993.73 994.10 993.79 994.02 993.76 994.16 994.27 994.30 995.10 995.37 996.86 998.11 1000.47 1029.16 1070.88

1002.22 1002.18 1002.30 1001.88 1002.33 1001.92 1001.76 1002.18 1002.62 1002.32 1002.72 1002.21 1002.72 1002.75 1002.78 1003.61 1003.86 1005.42 1006.85 1008.92 1038.07 1080.32

1010.62 1010.62 1010.85 1010.39 1010.85 1010.51 1010.40 1010.71 1011.22 1010.80 1011.20 1010.85 1011.13 1011.22 1011.32 1012.05 1012.37 1013.99 1015.35 1017.47 1047.01 1089.90

1019.14 1019.10 1019.38 1018.89 1019.45 1019.07 1018.99 1019.24 1019.78 1019.36 1019.71 1019.50 1019.67 1019.71 1019.94 1020.53 1020.88 1022.48 1023.93 1026.07 1056.13 1099.30

1027.74 1027.51 1027.87 1027.44 1027.97 1027.59 1027.60 1027.78 1028.27 1027.76 1028.32 1027.98 1028.32 1028.31 1028.60 1029.04 1029.43 1031.16 1032.42 1034.53 1065.17 1108.85

1036.30 1036.00 1036.40 1035.94 1036.44 1036.17 1036.11 1036.24 1036.92 1036.28 1036.86 1036.53 1036.73 1036.82 1037.15 1037.65 1037.97 1039.57 1041.06 1043.16 1074.20 1118.28

1044.78 1044.58 1044.80 1044.48 1044.97 1044.68 1044.70 1044.77 1045.52 1044.63 1045.37 1045.11 1045.34 1045.40 1045.71 1046.34 1046.38 1048.15 1049.64 1051.75 1083.35 1127.75

1053.36 1053.09 1053.37 1052.93 1053.53 1053.23 1053.16 1053.35 1053.87 1053.16 1053.94 1053.63 1053.85 1053.94 1054.14 1054.85 1054.84 1056.74 1058.20 1060.28 1092.33 1137.29

70

80

90

100

Groups / Tee

10

20

30

40

50

60

Figure 2: Sojourn times (above) and departure times (below) as a function of the group and thetee interval for a golf course with P3 holes (without the wave-up rule) in the base course.

14

3.1 Interpreting the Color Visualization

3.1.1 Case 1: P3 holes Without the Wave-up Rule

The first pair of figures in Figure 2 show average sojourn times (top) and departure times (bottom)

of each group (averaged over n r replications) as a function of the group number (y-axis) and the

fixed tee interval (x-axis) for the base case with unscaled P3 holes. Here, the base case refers to

the hole sequence 454—434—454—434—454—434, as discussed in §3 of the main paper. When

enlarged, these figures will show the actual numeric sojourn and departure times.

The top and the bottom figures are color-coded figures; the colors associated with the sojourn

and departure times for each group at each fixed tee interval will vary based on how large these

times are. For both top and bottom figures, darker color indicates a larger numerical value.

For the top figure, starting from the top (i.e. group 1) to the bottom (i.e. group 100), we see

7 different colors. These colors, from the lightest to the darkest, respectively indicate the sojourn

time ranges of [0, 230], (230, 240], (240, 250], (250, 260], (260, 270], (270, 280] and (280,∞) minutes.

In the top figure, the thick black lines indicate whether the sojourn-time constraint is violated;

sojourn times that appear below the black lines indicate that the constraint is violated.

For the bottom figure, we see 2 different colors. The lighter color indicates that a departure

time for a particular group and tee occurs before 840 minutes (14 hours) since the golf course begins

its operations, while the red color indicates that the departure time occurs after 840 minutes (thus

violates the departure-time constraint). In the bottom figure in (Figure 2), the number of groups

is bounded by the red region according to the departure-time constraint.

From Figure 2, we draw couple of insights. The top figure of Figure 2 shows that more groups

will satisfy the sojourn-time constraint as tee interval increases; when groups arrive less frequently,

they experience shorter sojourn times. Consistent with intuition, the average sojourn time for the

group increases as the group number increases, but decreases as the tee interval gets longer (and

the arrival rate decreases). The average departure time for the group also increases as the group

number increases, but increases as the tee interval gets longer (i.e. as the arrival rate decreases).

Furthermore, Figure 2 visually shows that the sojourn-time constraint tends to be much more

binding than the departure-time constraint. There is a rapid change in the constraint region for

the sojourn times at the top, but only a gradual change in the departure times below.

15

5.0 5.5 6.0 6.5 7.0 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 8.0 8.1 8.2 8.3 8.4 8.5 9.0 9.5190.18 190.34 190.55 190.25 190.22 190.25 190.44 190.46 190.25 190.08 190.40 190.21 190.31 190.46 190.12 189.19 189.43 189.37 188.81 190.17 189.96 189.78

198.89 198.63 198.25 197.58 197.01 197.13 196.99 196.91 196.45 196.48 196.58 196.27 196.35 196.07 195.81 194.58 194.66 194.55 194.05 195.75 194.98 194.64

205.79 204.91 203.98 202.88 201.98 202.11 201.57 201.65 200.98 200.87 200.82 200.40 200.50 200.16 199.97 198.53 198.09 197.89 197.45 199.22 198.07 197.25

211.84 210.39 208.68 207.09 205.80 206.03 205.39 205.39 204.69 204.57 204.20 203.72 203.63 203.48 202.98 201.40 200.85 200.38 199.86 201.99 200.33 198.95

216.99 215.17 212.91 210.65 209.25 209.16 208.57 208.33 207.74 207.35 207.07 206.49 206.28 205.92 205.27 203.68 202.88 202.45 201.72 203.85 202.10 200.30

221.64 219.31 216.50 213.98 211.99 211.70 210.89 210.70 210.17 209.60 209.26 208.76 208.37 207.90 207.25 205.47 204.54 204.16 203.19 205.42 203.38 201.27

225.80 223.08 219.81 216.89 214.42 214.05 213.27 212.71 212.23 211.38 211.00 210.51 210.09 209.58 208.87 206.88 205.94 205.30 204.49 206.60 204.17 202.07

229.71 226.68 222.93 219.38 216.32 216.02 215.09 214.50 214.01 213.05 212.64 211.94 211.55 210.93 210.01 208.12 207.19 206.38 205.59 207.50 204.75 202.30

233.40 229.77 225.76 221.73 218.16 217.68 216.78 215.99 215.47 214.34 214.00 213.19 212.73 211.95 211.12 209.23 208.13 207.21 206.40 208.15 205.22 202.65

237.09 232.84 228.32 223.74 219.69 219.10 218.28 217.35 216.77 215.56 215.02 214.17 213.55 212.82 212.04 209.97 208.95 207.90 206.96 208.73 205.62 202.85

240.41 235.64 230.67 225.57 221.25 220.42 219.64 218.59 217.83 216.69 215.86 215.14 214.29 213.61 212.62 210.57 209.50 208.56 207.63 209.16 205.92 202.80

243.59 238.41 232.83 227.31 222.65 221.69 220.75 219.68 218.86 217.78 216.68 215.92 215.08 214.21 213.34 211.10 209.97 208.94 208.02 209.48 206.12 202.92

246.67 240.92 234.98 228.92 223.87 222.65 221.75 220.56 219.67 218.62 217.36 216.50 215.65 214.62 213.87 211.56 210.37 209.40 208.40 209.85 206.42 202.96

249.68 243.46 236.99 230.52 224.99 223.62 222.72 221.42 220.45 219.33 217.97 217.20 216.25 215.12 214.25 211.93 210.81 209.74 208.66 210.04 206.67 202.93

252.65 245.92 238.88 231.82 225.98 224.59 223.52 222.23 221.19 219.96 218.51 217.73 216.71 215.45 214.62 212.29 211.16 209.91 208.82 210.15 206.68 203.15

255.51 248.23 240.71 233.26 226.91 225.52 224.25 223.00 221.79 220.51 219.03 218.09 217.12 215.81 214.97 212.58 211.29 210.16 208.99 210.23 206.60 203.19

258.28 250.51 242.53 234.55 227.82 226.33 224.98 223.60 222.26 221.03 219.49 218.43 217.42 216.08 215.35 212.80 211.52 210.22 209.23 210.22 206.50 203.19

261.04 252.84 244.28 235.84 228.75 227.00 225.57 224.21 222.75 221.58 219.86 218.77 217.64 216.37 215.53 212.94 211.61 210.38 209.34 210.26 206.58 203.19

263.81 255.04 246.00 237.11 229.60 227.68 226.15 224.67 223.26 222.06 220.31 219.14 217.97 216.63 215.77 213.05 211.71 210.59 209.26 210.29 206.64 203.19

266.47 257.17 247.74 238.30 230.30 228.35 226.82 225.18 223.69 222.37 220.63 219.42 218.23 216.77 215.90 213.23 211.84 210.79 209.49 210.36 206.64 203.20

269.10 259.30 249.37 239.56 230.93 228.98 227.31 225.63 224.06 222.68 220.91 219.78 218.40 216.80 216.02 213.40 211.95 210.86 209.60 210.39 206.59 203.16

271.73 261.43 250.97 240.65 231.55 229.57 227.70 226.06 224.41 222.96 221.21 219.91 218.52 216.98 216.05 213.40 211.93 210.94 209.57 210.43 206.43 202.98

274.21 263.53 252.59 241.88 232.16 230.11 228.20 226.52 224.69 223.28 221.43 220.11 218.76 217.19 216.07 213.47 212.16 210.98 209.61 210.42 206.31 203.04

276.74 265.55 254.12 242.95 232.73 230.56 228.58 226.88 224.95 223.57 221.65 220.18 218.86 217.38 216.11 213.48 212.22 210.98 209.65 210.48 206.40 203.19

279.17 267.60 255.58 243.98 233.16 231.00 229.03 227.21 225.16 223.83 221.91 220.41 219.04 217.57 216.27 213.58 212.24 211.09 209.60 210.48 206.36 203.15

281.64 269.57 257.09 244.98 233.80 231.40 229.44 227.52 225.43 223.98 222.15 220.48 219.10 217.60 216.36 213.66 212.28 211.05 209.62 210.46 206.50 203.13

284.10 271.50 258.58 245.96 234.32 231.93 229.81 227.79 225.67 224.18 222.26 220.63 219.12 217.62 216.42 213.78 212.25 211.16 209.66 210.45 206.50 203.10

286.43 273.37 259.99 246.94 234.71 232.28 230.17 228.10 225.97 224.43 222.39 220.75 219.17 217.61 216.31 213.80 212.32 211.06 209.66 210.49 206.38 202.99

288.79 275.21 261.43 247.83 235.18 232.66 230.52 228.41 226.22 224.58 222.63 220.77 219.24 217.71 216.30 213.89 212.30 211.10 209.75 210.39 206.29 202.82

291.22 277.10 262.84 248.68 235.60 233.09 230.83 228.62 226.39 224.62 222.69 220.85 219.15 217.85 216.45 213.91 212.24 211.05 209.62 210.44 206.40 202.74

293.54 278.97 264.24 249.57 236.03 233.48 231.06 228.96 226.51 224.76 222.84 220.87 219.21 217.81 216.58 213.98 212.28 211.00 209.70 210.40 206.48 202.88

295.97 280.79 265.53 250.39 236.49 233.85 231.31 229.15 226.65 224.72 222.90 220.86 219.25 217.91 216.57 214.00 212.20 210.98 209.73 210.38 206.55 203.04

298.31 282.60 266.81 251.29 236.92 234.15 231.69 229.36 226.92 224.75 223.04 220.84 219.19 217.99 216.57 214.02 212.24 211.04 209.86 210.46 206.57 203.07

300.67 284.45 268.13 252.10 237.31 234.45 231.98 229.57 227.04 224.90 223.11 221.02 219.09 218.10 216.63 213.99 212.25 211.11 209.88 210.43 206.56 203.17

302.93 286.23 269.45 252.91 237.72 234.79 232.24 229.74 227.23 225.01 223.15 221.08 219.21 218.13 216.59 213.98 212.40 211.15 209.82 210.56 206.41 203.21

305.27 287.96 270.74 253.72 238.04 235.03 232.47 229.97 227.32 225.11 223.17 221.07 219.15 218.06 216.66 213.97 212.40 211.04 209.81 210.66 206.46 203.21

307.57 289.71 272.03 254.55 238.38 235.33 232.65 230.11 227.40 225.24 223.12 221.15 219.09 218.05 216.76 214.03 212.36 211.04 209.72 210.73 206.39 203.20

309.80 291.52 273.24 255.31 238.74 235.62 232.85 230.21 227.48 225.28 223.19 221.11 219.20 217.99 216.61 213.97 212.35 211.01 209.65 210.69 206.44 203.12

312.06 293.27 274.51 256.06 239.08 235.85 233.00 230.30 227.57 225.40 223.26 221.17 219.20 217.92 216.51 214.08 212.25 210.97 209.67 210.75 206.37 203.12

314.37 295.05 275.76 256.85 239.39 236.07 233.19 230.42 227.62 225.50 223.29 221.16 219.25 217.90 216.56 214.06 212.18 211.04 209.67 210.70 206.39 203.13

316.58 296.76 277.06 257.58 239.70 236.30 233.32 230.49 227.68 225.51 223.30 221.09 219.26 217.83 216.53 214.03 212.10 211.08 209.76 210.63 206.33 203.11

318.85 298.45 278.30 258.24 239.92 236.56 233.54 230.62 227.80 225.46 223.34 221.19 219.34 217.90 216.53 214.05 212.20 211.04 209.69 210.57 206.33 203.02

321.03 300.13 279.52 259.05 240.14 236.85 233.64 230.78 227.81 225.59 223.39 221.20 219.36 217.92 216.54 213.96 212.29 211.09 209.62 210.57 206.41 202.98

323.26 301.90 280.75 259.76 240.51 237.11 233.82 230.91 227.92 225.61 223.43 221.20 219.37 217.93 216.56 213.97 212.34 211.06 209.56 210.71 206.55 203.23

325.48 303.58 282.05 260.46 240.83 237.35 233.93 231.03 227.97 225.65 223.47 221.29 219.44 217.92 216.60 213.87 212.37 211.08 209.49 210.76 206.54 203.22

327.78 305.28 283.20 261.17 241.12 237.61 234.14 231.09 228.08 225.67 223.45 221.27 219.43 217.91 216.63 213.84 212.31 211.12 209.53 210.79 206.61 203.41

329.97 306.96 284.34 261.86 241.40 237.83 234.28 231.25 228.09 225.71 223.54 221.31 219.49 217.97 216.61 213.75 212.31 211.21 209.65 210.73 206.65 203.34

332.19 308.66 285.54 262.57 241.69 238.02 234.41 231.34 228.10 225.77 223.53 221.29 219.63 217.87 216.68 213.73 212.37 211.25 209.78 210.74 206.57 203.26

334.30 310.28 286.71 263.24 241.96 238.19 234.62 231.44 228.22 225.74 223.57 221.22 219.69 217.81 216.70 213.68 212.44 211.26 209.75 210.70 206.55 203.31

336.40 311.97 287.89 263.91 242.19 238.36 234.76 231.49 228.30 225.67 223.59 221.22 219.69 217.78 216.65 213.68 212.39 211.19 209.72 210.62 206.65 203.25

338.53 313.61 289.04 264.57 242.33 238.47 234.79 231.51 228.30 225.63 223.55 221.24 219.72 217.76 216.48 213.66 212.37 211.18 209.68 210.63 206.61 203.04

340.71 315.19 290.18 265.23 242.44 238.55 234.90 231.59 228.33 225.68 223.58 221.28 219.73 217.76 216.52 213.76 212.37 211.14 209.61 210.64 206.55 202.91

342.82 316.89 291.35 265.90 242.67 238.71 234.97 231.67 228.39 225.62 223.61 221.23 219.74 217.77 216.46 213.73 212.39 211.26 209.66 210.54 206.60 202.96

345.02 318.53 292.39 266.50 242.84 238.83 235.07 231.67 228.44 225.69 223.59 221.40 219.76 217.75 216.42 213.71 212.42 211.30 209.65 210.57 206.62 202.93

347.24 320.11 293.50 267.07 243.04 238.90 235.13 231.78 228.52 225.74 223.62 221.36 219.77 217.80 216.39 213.70 212.41 211.22 209.76 210.64 206.49 203.00

349.30 321.78 294.64 267.74 243.18 239.08 235.21 231.78 228.47 225.69 223.59 221.43 219.89 217.87 216.45 213.72 212.36 211.17 209.73 210.61 206.41 203.17

351.40 323.41 295.72 268.41 243.45 239.21 235.32 231.81 228.47 225.74 223.52 221.44 219.87 217.85 216.43 213.84 212.36 211.24 209.89 210.61 206.40 203.10

353.50 325.04 296.80 268.99 243.70 239.35 235.37 231.78 228.44 225.65 223.51 221.39 219.79 217.84 216.38 213.95 212.41 211.29 209.86 210.62 206.42 203.12

355.51 326.61 297.95 269.53 243.86 239.47 235.42 231.85 228.51 225.61 223.57 221.36 219.73 217.74 216.34 213.93 212.35 211.27 209.90 210.71 206.43 203.16

357.62 328.21 299.07 270.13 244.01 239.58 235.52 231.87 228.43 225.57 223.50 221.34 219.68 217.74 216.34 213.90 212.38 211.29 210.05 210.71 206.35 203.37

359.72 329.79 300.12 270.82 244.18 239.73 235.57 231.87 228.59 225.64 223.55 221.33 219.62 217.76 216.39 213.84 212.39 211.14 209.97 210.57 206.32 203.28

361.82 331.37 301.14 271.42 244.30 239.91 235.56 231.89 228.67 225.65 223.60 221.32 219.66 217.74 216.43 213.84 212.46 211.03 209.76 210.55 206.32 203.13

363.87 332.84 302.28 271.99 244.54 239.92 235.62 231.97 228.64 225.63 223.55 221.31 219.59 217.75 216.41 213.89 212.32 210.95 209.74 210.56 206.46 202.99

365.95 334.38 303.36 272.55 244.71 240.01 235.68 231.97 228.67 225.60 223.45 221.34 219.60 217.82 216.47 213.80 212.34 210.91 209.82 210.41 206.42 202.98

368.01 335.99 304.42 273.19 244.88 240.15 235.75 232.00 228.68 225.66 223.38 221.24 219.68 217.87 216.34 213.84 212.35 210.98 209.88 210.49 206.35 203.06

370.11 337.57 305.59 273.86 245.05 240.21 235.73 232.20 228.66 225.71 223.35 221.25 219.62 217.80 216.35 213.88 212.34 211.02 209.69 210.43 206.52 203.08

372.15 339.15 306.65 274.47 245.19 240.29 235.83 232.22 228.61 225.72 223.29 221.25 219.71 217.94 216.33 213.88 212.34 211.07 209.69 210.40 206.54 203.07

374.22 340.71 307.71 274.99 245.35 240.42 235.81 232.26 228.65 225.79 223.26 221.30 219.63 217.93 216.31 213.87 212.31 211.03 209.59 210.53 206.53 203.15

376.28 342.28 308.85 275.56 245.40 240.47 235.87 232.31 228.67 225.91 223.26 221.33 219.61 217.94 216.34 213.92 212.23 210.98 209.66 210.56 206.48 203.22

378.31 343.85 309.88 276.11 245.58 240.58 235.90 232.30 228.73 225.85 223.33 221.34 219.53 217.89 216.35 213.90 212.19 210.96 209.60 210.74 206.55 203.16

380.31 345.37 311.01 276.75 245.67 240.66 235.95 232.29 228.78 225.94 223.36 221.33 219.53 217.88 216.40 213.89 212.28 210.95 209.71 210.64 206.63 203.14

382.36 346.97 312.03 277.32 245.74 240.72 235.97 232.25 228.81 225.83 223.40 221.30 219.52 217.85 216.48 213.88 212.26 211.06 209.74 210.69 206.55 203.18

384.47 348.48 313.07 277.88 245.85 240.85 236.04 232.26 228.80 225.86 223.44 221.39 219.51 217.87 216.48 213.87 212.32 211.07 209.76 210.60 206.41 203.01

386.48 350.00 314.13 278.40 245.96 241.00 236.08 232.20 228.82 225.80 223.49 221.43 219.48 217.93 216.54 213.86 212.30 211.07 209.82 210.67 206.25 203.03

388.55 351.57 315.19 278.92 246.07 241.04 236.10 232.21 228.89 225.80 223.50 221.42 219.49 217.87 216.51 213.98 212.32 211.08 209.87 210.66 206.19 203.03

390.58 353.04 316.23 279.43 246.20 241.05 236.00 232.23 228.84 225.75 223.50 221.38 219.50 217.86 216.52 213.92 212.31 211.04 209.79 210.79 206.17 203.09

392.59 354.60 317.28 279.92 246.33 241.09 236.00 232.19 228.78 225.75 223.58 221.28 219.52 217.78 216.58 213.82 212.27 211.12 209.83 210.76 206.25 203.12

394.64 356.05 318.33 280.45 246.44 241.19 235.99 232.17 228.81 225.72 223.59 221.30 219.49 217.71 216.55 213.79 212.34 211.16 209.89 210.88 206.29 203.16

396.64 357.58 319.31 280.92 246.63 241.27 235.98 232.18 228.77 225.71 223.55 221.22 219.43 217.83 216.45 213.76 212.36 211.04 209.83 210.81 206.40 203.12

398.62 359.16 320.36 281.51 246.77 241.26 235.97 232.21 228.77 225.80 223.52 221.24 219.41 217.84 216.49 213.61 212.35 211.01 209.89 210.71 206.51 203.02

400.72 360.63 321.37 281.98 246.81 241.35 236.03 232.23 228.79 225.80 223.51 221.19 219.38 217.87 216.54 213.70 212.37 211.06 209.81 210.69 206.46 202.99

402.70 362.23 322.37 282.56 246.96 241.40 236.03 232.18 228.81 225.80 223.58 221.17 219.39 217.92 216.65 213.71 212.38 211.13 209.84 210.64 206.31 202.99

404.73 363.73 323.37 283.03 247.05 241.44 236.05 232.19 228.81 225.78 223.53 221.19 219.40 217.89 216.61 213.81 212.35 211.12 209.64 210.70 206.25 203.13

406.70 365.19 324.38 283.49 247.17 241.48 236.07 232.28 228.85 225.73 223.50 221.28 219.44 217.97 216.67 213.71 212.45 211.12 209.69 210.72 206.44 203.16

408.66 366.68 325.38 284.08 247.34 241.43 236.11 232.26 228.90 225.75 223.54 221.31 219.45 217.93 216.65 213.69 212.43 211.19 209.71 210.56 206.59 203.29

410.63 368.21 326.38 284.62 247.38 241.53 236.13 232.26 228.87 225.83 223.51 221.35 219.45 217.88 216.68 213.63 212.41 211.14 209.70 210.64 206.64 203.30

412.66 369.67 327.40 285.11 247.50 241.57 236.08 232.29 228.95 225.81 223.54 221.32 219.44 217.77 216.57 213.63 212.41 211.17 209.71 210.66 206.47 203.25

414.60 371.08 328.38 285.62 247.55 241.68 236.11 232.33 228.95 225.85 223.51 221.29 219.48 217.75 216.55 213.66 212.38 211.33 209.69 210.63 206.57 203.27

416.62 372.58 329.36 286.09 247.60 241.69 236.10 232.36 228.99 225.85 223.50 221.28 219.45 217.68 216.58 213.60 212.34 211.31 209.71 210.63 206.54 203.30

418.63 374.01 330.26 286.62 247.67 241.76 236.21 232.32 229.01 225.87 223.54 221.34 219.48 217.58 216.56 213.72 212.32 211.29 209.64 210.53 206.50 203.20

420.66 375.49 331.29 287.07 247.81 241.84 236.28 232.34 228.90 225.86 223.53 221.34 219.62 217.59 216.63 213.74 212.54 211.17 209.71 210.62 206.55 203.20

422.61 376.97 332.23 287.53 247.95 241.81 236.28 232.37 228.94 225.82 223.53 221.30 219.56 217.69 216.52 213.77 212.54 211.14 209.69 210.80 206.63 203.19

424.55 378.45 333.27 288.06 248.01 241.84 236.27 232.35 228.89 225.85 223.59 221.39 219.53 217.77 216.52 213.91 212.54 211.22 209.74 210.88 206.51 203.38

426.58 379.96 334.25 288.61 248.08 241.87 236.40 232.34 228.86 225.75 223.62 221.38 219.58 217.79 216.49 213.91 212.45 211.36 209.77 210.90 206.44 203.20

428.56 381.43 335.23 289.10 248.15 241.97 236.35 232.36 228.93 225.80 223.50 221.45 219.61 217.78 216.37 213.99 212.40 211.27 209.80 210.87 206.49 203.31

430.52 382.90 336.23 289.56 248.15 241.97 236.39 232.40 228.97 225.82 223.60 221.38 219.51 217.83 216.41 213.86 212.41 211.25 209.89 210.78 206.45 203.20

432.49 384.36 337.24 290.04 248.23 241.99 236.39 232.39 229.01 225.80 223.53 221.39 219.53 217.90 216.32 213.89 212.38 211.23 209.74 210.85 206.39 203.07

434.47 385.89 338.24 290.45 248.34 241.99 236.46 232.38 228.99 225.84 223.52 221.38 219.48 217.95 216.33 213.82 212.38 211.21 209.81 210.82 206.50 203.12

436.46 387.40 339.20 290.92 248.33 242.00 236.56 232.46 229.00 225.90 223.50 221.37 219.36 218.00 216.31 213.83 212.35 211.23 209.84 210.67 206.49 203.08

438.31 388.86 340.20 291.33 248.47 242.10 236.56 232.34 228.96 225.93 223.41 221.33 219.38 217.97 216.28 213.93 212.39 211.27 209.83 210.63 206.26 203.16

60

70

80

90

100

Groups / Tee

10

20

30

40

50

5.0 5.5 6.0 6.5 7.0 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 8.0 8.1 8.2 8.3 8.4 8.5 9.0 9.5190.18 190.34 190.55 190.25 190.22 190.25 190.44 190.46 190.25 190.08 190.40 190.21 190.31 190.46 190.12 189.19 189.43 189.37 188.81 190.17 189.96 189.78

203.89 204.13 204.25 204.08 204.01 204.23 204.19 204.21 203.85 203.98 204.18 203.97 204.15 203.97 203.81 202.68 202.86 202.85 202.45 204.25 203.98 204.14

215.79 215.91 215.98 215.88 215.98 216.31 215.97 216.25 215.78 215.87 216.02 215.80 216.10 215.96 215.97 214.73 214.49 214.49 214.25 216.22 216.07 216.25

226.84 226.89 226.68 226.59 226.80 227.33 226.99 227.29 226.89 227.07 227.00 226.82 227.03 227.18 226.98 225.70 225.45 225.28 225.06 227.49 227.33 227.45

236.99 237.17 236.91 236.65 237.25 237.56 237.37 237.53 237.34 237.35 237.47 237.29 237.48 237.52 237.27 236.08 235.68 235.65 235.32 237.85 238.10 238.30

246.64 246.81 246.50 246.48 246.99 247.20 246.89 247.20 247.17 247.10 247.26 247.26 247.37 247.40 247.25 245.97 245.54 245.66 245.19 247.92 248.38 248.77

255.80 256.08 255.81 255.89 256.42 256.65 256.47 256.51 256.63 256.38 256.60 256.71 256.89 256.98 256.87 255.48 255.14 255.10 254.89 257.60 258.17 259.07

264.71 265.18 264.93 264.88 265.32 265.72 265.49 265.60 265.81 265.55 265.84 265.84 266.15 266.23 266.01 264.82 264.59 264.48 264.39 267.00 267.75 268.80

273.40 273.77 273.76 273.73 274.16 274.48 274.38 274.39 274.67 274.34 274.80 274.79 275.13 275.15 275.12 274.03 273.73 273.61 273.60 276.15 277.22 278.65

282.09 282.34 282.32 282.24 282.69 283.00 283.08 283.05 283.37 283.06 283.42 283.47 283.75 283.92 284.04 282.87 282.75 282.60 282.56 285.23 286.62 288.35

290.41 290.64 290.67 290.57 291.25 291.42 291.64 291.59 291.83 291.69 291.86 292.14 292.29 292.61 292.62 291.57 291.50 291.56 291.63 294.16 295.92 297.80

298.59 298.91 298.83 298.81 299.65 299.79 299.95 299.98 300.26 300.28 300.28 300.62 300.88 301.11 301.34 300.20 300.17 300.24 300.42 302.98 305.12 307.42

306.67 306.92 306.98 306.92 307.87 307.85 308.15 308.16 308.47 308.62 308.56 308.90 309.25 309.42 309.87 308.76 308.77 309.00 309.20 311.85 314.42 316.96

314.68 314.96 314.99 315.02 315.99 315.92 316.32 316.32 316.65 316.83 316.77 317.30 317.65 317.82 318.25 317.23 317.41 317.64 317.86 320.54 323.67 326.43

322.65 322.92 322.88 322.82 323.98 323.99 324.32 324.43 324.79 324.96 324.91 325.53 325.91 326.05 326.62 325.69 325.96 326.11 326.42 329.15 332.68 336.15

330.51 330.73 330.71 330.76 331.91 332.02 332.25 332.50 332.79 333.01 333.03 333.59 334.12 334.31 334.97 334.08 334.29 334.66 334.99 337.73 341.60 345.69

338.28 338.51 338.53 338.55 339.82 339.93 340.18 340.40 340.66 341.03 341.09 341.63 342.22 342.48 343.35 342.40 342.72 343.02 343.63 346.22 350.50 355.19

346.04 346.34 346.28 346.34 347.75 347.70 347.97 348.31 348.55 349.08 349.06 349.67 350.24 350.67 351.53 350.64 351.01 351.48 352.14 354.76 359.58 364.69

353.81 354.04 354.00 354.11 355.60 355.48 355.75 356.07 356.46 357.06 357.11 357.74 358.37 358.83 359.77 358.85 359.31 359.99 360.46 363.29 368.64 374.19

361.47 361.67 361.74 361.80 363.30 363.25 363.62 363.88 364.29 364.87 365.03 365.72 366.43 366.87 367.90 367.13 367.64 368.49 369.09 371.86 377.64 383.70

369.10 369.30 369.37 369.56 370.93 370.98 371.31 371.63 372.06 372.68 372.91 373.78 374.40 374.80 376.02 375.40 375.95 376.86 377.60 380.39 386.59 393.16

376.73 376.93 376.97 377.15 378.55 378.67 378.90 379.36 379.81 380.46 380.81 381.61 382.32 382.88 384.05 383.50 384.13 385.24 385.97 388.93 395.43 402.48

384.21 384.53 384.59 384.88 386.16 386.31 386.60 387.12 387.49 388.28 388.63 389.51 390.36 390.99 392.07 391.67 392.56 393.58 394.41 397.42 404.31 412.04

391.74 392.05 392.12 392.45 393.73 393.86 394.18 394.78 395.15 396.07 396.45 397.28 398.26 399.08 400.11 399.78 400.82 401.88 402.85 405.98 413.40 421.69

399.17 399.60 399.58 399.98 401.16 401.40 401.83 402.41 402.76 403.83 404.31 405.21 406.24 407.17 408.27 407.98 409.04 410.29 411.20 414.48 422.36 431.15

406.64 407.07 407.09 407.48 408.80 408.90 409.44 410.02 410.43 411.48 412.15 412.98 414.10 415.10 416.36 416.16 417.28 418.55 419.62 422.96 431.50 440.63

414.10 414.50 414.58 414.96 416.32 416.53 417.01 417.59 418.07 419.18 419.86 420.83 421.92 423.02 424.42 424.38 425.45 426.96 428.06 431.45 440.50 450.10

421.43 421.87 421.99 422.44 423.71 423.98 424.57 425.20 425.77 426.93 427.59 428.65 429.77 430.91 432.31 432.50 433.72 435.16 436.46 439.99 449.38 459.49

428.79 429.21 429.43 429.83 431.18 431.46 432.12 432.81 433.42 434.58 435.43 436.37 437.64 438.91 440.30 440.69 441.90 443.50 444.95 448.39 458.29 468.82

436.22 436.60 436.84 437.18 438.60 438.99 439.63 440.32 440.99 442.12 443.09 444.15 445.35 446.95 448.45 448.81 450.04 451.75 453.22 456.94 467.40 478.24

443.54 443.97 444.24 444.57 446.03 446.48 447.06 447.96 448.51 449.76 450.84 451.87 453.21 454.81 456.58 456.98 458.28 460.00 461.70 465.40 476.48 487.88

450.97 451.29 451.53 451.89 453.49 453.95 454.51 455.45 456.05 457.22 458.50 459.56 461.05 462.81 464.57 465.10 466.40 468.28 470.13 473.88 485.55 497.54

458.31 458.60 458.81 459.29 460.92 461.35 462.09 462.96 463.72 464.75 466.24 467.24 468.79 470.79 472.57 473.22 474.64 476.64 478.66 482.46 494.57 507.07

465.67 465.95 466.13 466.60 468.31 468.75 469.58 470.47 471.24 472.40 473.91 475.12 476.49 478.80 480.63 481.29 482.85 485.01 487.08 490.93 503.56 516.67

472.93 473.23 473.45 473.91 475.72 476.19 477.04 477.94 478.83 480.01 481.55 482.88 484.41 486.73 488.59 489.38 491.20 493.35 495.42 499.56 512.41 526.21

480.27 480.46 480.74 481.22 483.04 483.53 484.47 485.47 486.32 487.61 489.17 490.57 492.15 494.56 496.66 497.47 499.40 501.54 503.81 508.16 521.46 535.71

487.57 487.71 488.03 488.55 490.38 490.93 491.85 492.91 493.80 495.24 496.72 498.35 499.89 502.45 504.76 505.63 507.56 509.84 512.12 516.73 530.39 545.20

494.80 495.02 495.24 495.81 497.74 498.32 499.25 500.31 501.28 502.78 504.39 506.01 507.80 510.29 512.61 513.67 515.75 518.11 520.45 525.19 539.44 554.62

502.06 502.27 502.51 503.06 505.08 505.65 506.60 507.70 508.77 510.40 512.06 513.77 515.60 518.12 520.51 521.88 523.85 526.37 528.87 533.75 548.37 564.12

509.37 509.55 509.76 510.35 512.39 512.97 513.99 515.12 516.22 518.00 519.69 521.46 523.45 526.00 528.56 529.96 531.98 534.74 537.27 542.20 557.39 573.63

516.58 516.76 517.06 517.58 519.70 520.30 521.32 522.49 523.68 525.51 527.30 529.09 531.26 533.83 536.53 538.03 540.10 543.08 545.76 550.63 566.33 583.11

523.85 523.95 524.30 524.74 526.92 527.66 528.74 529.92 531.20 532.96 534.94 536.89 539.14 541.80 544.53 546.15 548.40 551.34 554.09 559.07 575.33 592.52

531.03 531.13 531.52 532.05 534.14 535.05 536.04 537.38 538.61 540.59 542.59 544.60 546.96 549.72 552.54 554.16 556.69 559.69 562.42 567.57 584.41 601.98

538.26 538.40 538.75 539.26 541.51 542.41 543.42 544.81 546.12 548.11 550.23 552.30 554.77 557.63 560.56 562.27 564.94 567.96 570.76 576.21 593.55 611.73

545.48 545.58 546.05 546.46 548.83 549.75 550.73 552.23 553.57 555.65 557.87 560.09 562.64 565.52 568.60 570.27 573.17 576.28 579.09 584.76 602.54 621.22

552.78 552.78 553.20 553.67 556.12 557.11 558.14 559.59 561.08 563.17 565.45 567.77 570.43 573.41 576.63 578.34 581.31 584.62 587.53 593.29 611.61 630.91

559.97 559.96 560.34 560.86 563.40 564.43 565.48 567.05 568.49 570.71 573.14 575.51 578.29 581.37 584.61 586.35 589.51 593.01 596.05 601.73 620.65 640.34

567.19 567.16 567.54 568.07 570.69 571.72 572.81 574.44 575.90 578.27 580.73 583.19 586.23 589.17 592.68 594.43 597.77 601.35 604.58 610.24 629.57 649.76

574.30 574.28 574.71 575.24 577.96 578.99 580.22 581.84 583.42 585.74 588.37 590.82 594.09 597.01 600.70 602.48 606.04 609.66 612.95 618.70 638.55 659.31

581.40 581.47 581.89 582.41 585.19 586.26 587.56 589.19 590.90 593.17 595.99 598.52 601.89 604.88 608.65 610.58 614.19 617.89 621.32 627.12 647.65 668.75

588.53 588.61 589.04 589.57 592.33 593.47 594.79 596.51 598.30 600.63 603.55 606.24 609.72 612.76 616.48 618.66 622.37 626.18 629.68 635.63 656.61 678.04

595.71 595.69 596.18 596.73 599.44 600.65 602.10 603.89 605.73 608.18 611.18 613.98 617.53 620.66 624.52 626.86 630.57 634.44 638.01 644.14 665.55 687.41

602.82 602.89 603.35 603.90 606.67 607.91 609.37 611.27 613.19 615.62 618.81 621.63 625.34 628.57 632.46 634.93 638.79 642.86 646.46 652.54 674.60 696.96

610.02 610.03 610.39 611.00 613.84 615.13 616.67 618.57 620.64 623.19 626.39 629.50 633.16 636.45 640.42 643.01 647.02 651.20 654.85 661.07 683.62 706.43

617.24 617.11 617.50 618.07 621.04 622.30 623.93 625.98 628.12 630.74 634.02 637.16 640.97 644.40 648.39 651.10 655.21 659.42 663.36 669.64 692.49 716.00

624.30 624.28 624.64 625.24 628.18 629.58 631.21 633.28 635.47 638.19 641.59 644.93 648.89 652.37 656.45 659.22 663.36 667.67 671.73 678.11 701.41 725.67

631.40 631.41 631.72 632.41 635.45 636.81 638.52 640.61 642.87 645.74 649.12 652.64 656.67 660.25 664.43 667.44 671.56 676.04 680.29 686.61 710.40 735.10

638.50 638.54 638.80 639.49 642.70 644.05 645.77 647.88 650.24 653.15 656.71 660.29 664.39 668.14 672.38 675.65 679.81 684.39 688.66 695.12 719.42 744.62

645.51 645.61 645.95 646.53 649.86 651.27 653.02 655.25 657.71 660.61 664.37 667.96 672.13 675.94 680.34 683.73 687.95 692.67 697.10 703.71 728.43 754.16

652.62 652.71 653.07 653.63 657.01 658.48 660.32 662.57 665.03 668.07 671.90 675.64 679.88 683.84 688.34 691.80 696.18 700.99 705.65 712.21 737.35 763.87

659.72 659.79 660.12 660.82 664.18 665.73 667.57 669.87 672.59 675.64 679.55 683.33 687.62 691.76 696.39 699.84 704.39 709.14 713.97 720.57 746.32 773.28

666.82 666.87 667.14 667.92 671.30 673.01 674.76 677.19 680.07 683.15 687.20 691.02 695.46 699.64 704.43 707.94 712.66 717.33 722.16 729.05 755.32 782.63

673.87 673.84 674.28 674.99 678.54 680.12 682.02 684.57 687.44 690.63 694.75 698.71 703.19 707.55 712.41 716.09 720.72 725.55 730.54 737.56 764.46 791.99

680.95 680.88 681.36 682.05 685.71 687.31 689.28 691.87 694.87 698.10 702.25 706.44 711.00 715.52 720.47 724.10 728.94 733.81 739.02 745.91 773.42 801.48

688.01 687.99 688.42 689.19 692.88 694.55 696.55 699.20 702.28 705.66 709.78 714.04 718.88 723.47 728.34 732.24 737.15 742.18 747.48 754.49 782.35 811.06

695.11 695.07 695.59 696.36 700.05 701.71 703.73 706.70 709.66 713.21 717.35 721.75 726.62 731.30 736.35 740.38 745.34 750.52 755.69 762.93 791.52 820.58

702.15 702.15 702.65 703.47 707.19 708.89 711.03 714.02 717.01 720.72 724.89 729.45 734.51 739.34 744.33 748.48 753.54 758.87 764.09 771.40 800.54 830.07

709.22 709.21 709.71 710.49 714.35 716.12 718.21 721.36 724.45 728.29 732.46 737.20 742.23 747.23 752.31 756.57 761.71 767.13 772.39 780.03 809.53 839.65

716.28 716.28 716.85 717.56 721.40 723.27 725.47 728.71 731.87 735.91 740.06 744.93 750.01 755.14 760.34 764.72 769.83 775.38 780.86 788.56 818.48 849.22

723.31 723.35 723.88 724.61 728.58 730.48 732.70 736.00 739.33 743.35 747.73 752.64 757.73 762.99 768.35 772.80 777.99 783.66 789.20 797.24 827.55 858.66

730.31 730.37 731.01 731.75 735.67 737.66 739.95 743.29 746.78 750.94 755.36 760.33 765.53 770.88 776.40 780.89 786.28 791.95 797.71 805.64 836.63 868.14

737.36 737.47 738.03 738.82 742.74 744.82 747.17 750.55 754.21 758.33 763.00 768.00 773.32 778.75 784.48 788.98 794.46 800.36 806.14 814.19 845.55 877.68

744.47 744.48 745.07 745.88 749.85 752.05 754.44 757.86 761.60 765.86 770.64 775.79 781.11 786.67 792.48 797.07 802.72 808.67 814.56 822.60 854.41 887.01

751.48 751.50 752.13 752.90 756.96 759.30 761.68 765.10 769.02 773.30 778.29 783.53 788.88 794.63 800.54 805.16 810.90 816.97 823.02 831.17 863.25 896.53

758.55 758.57 759.19 759.92 764.07 766.44 768.90 772.41 776.49 780.80 785.90 791.22 796.69 802.47 808.51 813.38 819.12 825.28 831.47 839.66 872.19 906.03

765.58 765.54 766.23 766.93 771.20 773.55 776.00 779.73 783.84 788.25 793.50 798.88 804.50 810.36 816.52 821.42 827.31 833.54 839.79 848.29 881.17 915.59

772.59 772.60 773.28 773.92 778.33 780.69 783.20 786.99 791.18 795.75 801.18 806.48 812.32 818.18 824.58 829.42 835.47 841.92 848.23 856.76 890.25 925.12

779.64 779.55 780.33 780.95 785.44 787.89 790.39 794.27 798.61 803.22 808.79 814.20 820.09 826.01 832.55 837.49 843.74 850.26 856.69 865.38 899.29 934.66

786.64 786.58 787.31 787.92 792.63 795.07 797.58 801.58 805.97 810.71 816.35 821.82 827.83 834.03 840.45 845.56 851.96 858.44 865.03 873.81 908.40 944.12

793.62 793.66 794.36 795.01 799.77 802.16 804.77 808.91 813.37 818.30 823.92 829.54 835.61 841.94 848.49 853.51 860.15 866.71 873.49 882.21 917.51 953.52

800.72 800.63 801.37 801.98 806.81 809.35 812.03 816.23 820.79 825.80 831.51 837.19 843.38 849.87 856.54 861.70 868.37 875.06 881.81 890.69 926.46 962.99

807.70 807.73 808.37 809.06 813.96 816.50 819.23 823.48 828.21 833.30 839.18 844.87 851.19 857.82 864.65 869.81 876.58 883.43 890.24 899.14 935.31 972.49

814.73 814.73 815.37 816.03 821.05 823.64 826.45 830.79 835.61 840.78 846.73 852.59 859.00 865.69 872.61 878.01 884.75 891.72 898.44 907.70 944.25 982.13

821.70 821.69 822.38 822.99 828.17 830.78 833.67 838.18 843.05 848.23 854.30 860.38 866.84 873.67 880.67 886.01 893.05 900.02 906.89 916.22 953.44 991.66

828.66 828.68 829.38 830.08 835.34 837.83 840.91 845.46 850.50 855.75 861.94 868.11 874.65 881.53 888.65 894.09 901.23 908.39 915.31 924.56 962.59 1001.29

835.63 835.71 836.38 837.12 842.38 845.03 848.13 852.76 857.87 863.33 869.51 875.85 882.45 889.38 896.68 902.13 909.41 916.64 923.70 933.14 971.64 1010.80

842.66 842.67 843.40 844.11 849.50 852.17 855.28 860.09 865.35 870.81 877.14 883.52 890.24 897.17 904.57 910.23 917.61 924.97 932.11 941.66 980.47 1020.25

849.60 849.58 850.38 851.12 856.55 859.38 862.51 867.43 872.75 878.35 884.71 891.19 898.08 905.05 912.55 918.36 925.78 933.43 940.49 950.13 989.57 1029.77

856.62 856.58 857.36 858.09 863.60 866.49 869.70 874.76 880.19 885.85 892.30 898.88 905.85 912.88 920.58 926.40 933.94 941.71 948.91 958.63 998.54 1039.30

863.63 863.51 864.26 865.12 870.67 873.66 877.01 882.02 887.61 893.37 899.94 906.64 913.68 920.68 928.56 934.62 942.12 949.99 957.24 967.03 1007.50 1048.70

870.66 870.49 871.29 872.07 877.81 880.84 884.28 889.34 894.90 900.86 907.53 914.34 921.62 928.59 936.63 942.74 950.54 958.17 965.71 975.62 1016.55 1058.20

877.61 877.47 878.23 879.03 884.95 887.91 891.48 896.67 902.34 908.32 915.13 922.00 929.36 936.59 944.52 950.87 958.74 966.44 974.09 984.30 1025.63 1067.69

884.55 884.45 885.27 886.06 892.01 895.04 898.67 903.95 909.69 915.85 922.79 929.79 937.13 944.57 952.52 959.11 966.94 974.82 982.54 992.88 1034.51 1077.38

891.58 891.46 892.25 893.11 899.08 902.17 906.00 911.24 917.06 923.25 930.42 937.48 944.98 952.49 960.49 967.21 975.05 983.26 990.97 1001.40 1043.44 1086.70

898.56 898.43 899.23 900.10 906.15 909.37 913.15 918.56 924.53 930.80 937.90 945.25 952.81 960.38 968.37 975.39 983.20 991.47 999.40 1009.87 1052.49 1096.31

905.52 905.40 906.23 907.06 913.15 916.47 920.39 925.90 931.97 938.32 945.60 952.88 960.51 968.33 976.41 983.36 991.41 999.75 1007.89 1018.28 1061.45 1105.70

912.49 912.36 913.24 914.04 920.23 923.59 927.59 933.19 939.41 945.80 953.13 960.59 968.33 976.30 984.32 991.49 999.58 1008.03 1016.14 1026.85 1070.39 1115.07

919.47 919.39 920.24 920.95 927.34 930.69 934.86 940.48 946.79 953.34 960.72 968.28 976.08 984.25 992.33 999.52 1007.78 1016.31 1024.61 1035.32 1079.50 1124.62

926.46 926.40 927.20 927.92 934.33 937.80 942.16 947.86 954.20 960.90 968.30 975.97 983.76 992.20 1000.31 1007.63 1015.95 1024.63 1033.04 1043.67 1088.49 1134.08

933.31 933.36 934.20 934.83 941.47 945.00 949.36 955.04 961.56 968.43 975.81 983.63 991.58 1000.07 1008.28 1015.83 1024.19 1032.97 1041.43 1052.13 1097.26 1143.66

60

70

80

90

100

Groups / Tee

10

20

30

40

50

Figure 3: Sojourn times (above) and departure times (below) as a function of the group and thetee interval for a golf course, but now with P3WU holes in the base course. Note that the colored-regions (i.e. constraint regions) are much smaller in the sojourn time and departure time figureswhen P3WU holes are used.

16

3.1.2 Case 2: P3WU holes

The second pair of figures in Figure 3 show average sojourn times (top) and departure times

(bottom) of each group (averaged over n r replications) as a function of the group number (y-axis)

and the fixed tee interval (x-axis) for the base case, but with P3WU holes. Again, the base case

refers to the hole sequence 454—434—454—434—454—434, as discussed in §3 of the main paper.

Similarly to Figure 2, the top and the bottom figures in Figure 3 are color-coded. The top

figure shows 7 different colors to indicate different ranges of sojourn times; these colors, from the

lightest to the darkest, respectively indicate the sojourn time ranges of [0, 230], (230, 240], (240, 250],

(250, 260], (260, 270], (270, 280] and (280,∞) minutes. Sojourn times that appear below the black

lines indicate that the constraint is violated. In the bottom figure, the ivory color indicates that

the departure time occurs before 840 minutes, and the red color indicates that the departure time

occurs after 840 minutes. The red color indicates that the departure-time constraint is violated.

When compared to Figure 2, Figure 3 shows the advantage of the wave-up rule. First, the top

figure shows that the sojourn times of the groups across various tee intervals are much lower than

those in Figure 2; the rate of change in the constraint region of Figure 3 is much more rapid than

the rate in Figure 2. For instance, when the tee interval is 7.2, while only 27 groups in Figure

2 satisfy the sojourn-time constraint, all of the 100 groups in Figure 3 satisfy the sojourn-time

constraint.

For the departure time figure in the bottom, Figure 3 shows that the departure times decrease

as the tee interval gets longer. The decreasing trend in departure times across tee intervals is more

noticeable in Figure 3 than in Figure 2.

Let us further compare the two figures. In the top figure of Figure 3, we see that, when the tee

interval is equal to or greater than 7.2 minutes, the sojourn-time constraint is no longer binding,

but the departure-time constraint is. When the departure time figure (bottom) is juxtaposed with

the sojourn time figure (top), we see that the maximum throughput of 84 occurs when the tee

interval is 7.2. On the other hand, the maximum throughput in Figure 2 is attained at a larger tee

interval around 8.2 minutes, where the maximum throughput is 74. Indeed, we see the advantage

of the P3WU holes over P3 holes. Please see Table 5 in §4.1 of the main paper for more details on

the throughput. Further notice that, in Figure 3, when the tee interval is greater than 7.2, sojourn

times converge as group numbers increase, whereas the sojourn times converge in Figure 2 when

the tee is 8.4 or greater. Convergence in sojourn times, indeed, are strong indicators that the golf

17

course is substantially under-loaded.

On the other hand, when examined closely, Figure 3 also shows the disadvantage of the wave-up

rule. When the tee interval is 9.0 minutes or higher, sojourn and departure times of the playing

groups in Figure 3 are higher than those in Figure 2. This trend may be attributed to the complexity

of the P3WU hole, as mentioned in §4.1 of the main paper.

18

Groups / Tee 7.0 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 8.0 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 9.0 9.5188.05 188.13 188.05 188.30 188.01 188.20 187.97 188.35 188.05 188.28 188.06 188.14 188.17 188.18 188.11 188.30 188.21 188.20 187.99 188.12 188.24 187.96

193.85 193.98 193.83 193.89 193.84 193.81 193.78 193.93 193.72 194.00 193.82 193.75 193.98 194.00 193.70 193.89 193.91 193.97 193.84 193.85 193.83 193.82

197.85 197.92 197.75 197.87 197.91 197.69 197.70 197.77 197.67 198.01 197.85 197.91 197.88 197.99 197.60 198.00 197.74 197.86 197.76 197.86 197.76 197.80

201.10 201.10 200.87 201.19 201.17 200.95 200.81 201.09 201.08 201.27 201.09 201.10 201.16 201.29 201.07 201.16 201.04 201.19 201.00 201.01 201.10 201.00

203.93 203.99 203.79 203.88 204.10 203.90 203.57 203.97 203.80 204.02 204.00 203.94 204.00 204.08 204.06 203.89 203.82 203.89 203.95 203.96 203.92 203.85

206.61 206.53 206.48 206.54 206.59 206.51 206.34 206.61 206.37 206.65 206.49 206.51 206.60 206.76 206.64 206.55 206.39 206.43 206.56 206.48 206.46 206.50

209.02 209.01 208.76 208.96 209.00 208.96 208.73 209.01 208.80 209.13 208.90 208.91 208.90 209.16 209.07 209.00 209.02 208.75 209.04 208.93 208.96 208.87

211.24 211.34 211.05 211.30 211.26 211.22 211.06 211.32 211.08 211.47 211.19 211.19 211.20 211.54 211.44 211.30 211.33 211.05 211.32 211.24 211.28 211.17

213.58 213.45 213.37 213.43 213.34 213.46 213.16 213.41 213.26 213.57 213.46 213.38 213.49 213.82 213.66 213.48 213.58 213.25 213.47 213.49 213.41 213.26

215.71 215.68 215.53 215.63 215.58 215.50 215.39 215.60 215.42 215.76 215.51 215.40 215.52 215.94 215.80 215.55 215.62 215.38 215.60 215.68 215.49 215.31

217.81 217.70 217.54 217.81 217.62 217.53 217.36 217.68 217.44 217.80 217.55 217.52 217.62 218.10 217.82 217.64 217.68 217.46 217.61 217.66 217.55 217.38

219.65 219.60 219.59 219.73 219.65 219.58 219.38 219.67 219.46 219.87 219.52 219.60 219.61 220.09 219.73 219.68 219.71 219.53 219.61 219.69 219.66 219.35

221.74 221.53 221.58 221.67 221.67 221.62 221.45 221.55 221.41 221.74 221.51 221.46 221.56 221.98 221.73 221.71 221.73 221.53 221.60 221.78 221.62 221.34

223.66 223.51 223.56 223.63 223.59 223.57 223.39 223.54 223.37 223.65 223.47 223.49 223.41 223.96 223.66 223.62 223.53 223.53 223.50 223.68 223.57 223.28

225.51 225.42 225.48 225.47 225.51 225.44 225.35 225.42 225.33 225.54 225.39 225.29 225.32 225.69 225.60 225.50 225.47 225.38 225.44 225.61 225.56 225.28

227.37 227.33 227.35 227.29 227.43 227.32 227.20 227.29 227.10 227.47 227.18 227.27 227.17 227.54 227.39 227.37 227.34 227.23 227.32 227.48 227.42 227.17

229.28 229.13 229.23 229.13 229.18 229.12 229.08 229.20 228.93 229.23 228.96 229.05 228.99 229.38 229.22 229.15 229.23 229.05 229.23 229.40 229.23 229.01

231.16 231.00 231.10 230.91 231.11 230.99 230.86 230.96 230.82 231.16 230.84 231.02 230.83 231.24 231.10 230.97 231.12 230.92 230.99 231.15 231.04 230.82

232.83 232.72 232.95 232.67 232.83 232.80 232.74 232.80 232.62 232.97 232.64 232.80 232.58 232.99 232.84 232.75 232.95 232.66 232.87 232.94 232.90 232.66

234.65 234.51 234.77 234.49 234.64 234.63 234.48 234.63 234.43 234.76 234.39 234.67 234.34 234.83 234.61 234.40 234.77 234.47 234.64 234.81 234.74 234.56

236.39 236.21 236.41 236.04 236.04 235.90 235.66 235.67 235.42 235.59 235.22 235.37 234.99 235.28 235.00 234.72 234.88 234.49 234.67 234.66 234.48 233.77

238.02 237.87 238.00 237.53 237.49 237.17 236.73 236.85 236.31 236.42 235.96 236.09 235.54 235.79 235.34 235.01 235.12 234.58 234.57 234.59 234.22 232.95

239.76 239.58 239.48 238.95 238.83 238.54 237.84 237.94 237.33 237.32 236.70 236.80 236.18 236.19 235.70 235.28 235.28 234.70 234.53 234.43 234.00 232.25

241.56 241.23 241.07 240.41 240.18 239.77 238.96 238.90 238.20 238.16 237.56 237.42 236.70 236.62 236.04 235.54 235.39 234.81 234.56 234.27 233.78 231.47

243.32 242.86 242.57 241.79 241.41 240.88 240.05 239.89 239.20 238.97 238.35 238.00 237.23 237.07 236.48 235.75 235.47 234.83 234.42 234.07 233.55 230.73

245.09 244.57 244.11 243.23 242.75 242.17 241.10 240.85 240.11 239.87 239.06 238.66 237.73 237.45 236.71 236.02 235.69 234.81 234.27 233.92 233.25 229.93

246.80 246.10 245.62 244.57 244.10 243.42 242.31 241.89 240.99 240.65 239.72 239.17 238.26 237.86 237.11 236.17 235.82 234.80 234.21 233.71 232.92 229.26

248.52 247.74 247.07 246.07 245.43 244.59 243.33 242.95 241.95 241.54 240.39 239.86 238.77 238.28 237.39 236.46 235.85 234.81 234.09 233.56 232.79 228.41

250.25 249.39 248.63 247.47 246.73 245.74 244.46 243.93 242.94 242.28 241.11 240.37 239.24 238.61 237.73 236.58 236.04 234.73 234.01 233.30 232.49 227.45

251.89 251.03 249.99 248.92 248.06 246.87 245.58 244.91 243.87 243.11 241.82 240.98 239.70 239.03 238.00 236.82 236.23 234.68 233.82 233.17 232.11 226.69

253.53 252.56 251.49 250.32 249.38 248.05 246.71 245.88 244.77 243.94 242.59 241.58 240.15 239.42 238.25 236.98 236.27 234.57 233.72 232.95 231.86 225.87

255.27 254.13 252.88 251.70 250.63 249.23 247.79 246.82 245.49 244.73 243.27 242.13 240.72 239.71 238.57 237.22 236.30 234.59 233.62 232.74 231.59 225.10

256.91 255.70 254.39 253.20 251.88 250.46 248.87 247.77 246.38 245.46 243.98 242.77 241.19 240.15 238.81 237.47 236.42 234.64 233.46 232.50 231.23 224.22

258.66 257.25 255.85 254.56 253.25 251.68 249.88 248.73 247.35 246.31 244.56 243.38 241.60 240.62 239.05 237.72 236.45 234.53 233.31 232.15 230.86 223.44

260.33 258.79 257.22 255.94 254.47 252.79 250.96 249.62 248.16 247.02 245.27 243.95 242.10 240.97 239.34 237.83 236.50 234.44 233.22 231.85 230.54 222.67

262.00 260.30 258.77 257.29 255.72 253.94 252.12 250.63 249.02 247.74 245.98 244.58 242.53 241.33 239.70 237.80 236.57 234.36 233.12 231.57 230.18 221.82

263.63 261.77 260.29 258.71 257.02 255.13 253.10 251.67 249.94 248.56 246.55 245.15 243.01 241.63 239.92 237.92 236.61 234.32 233.00 231.45 229.88 220.93

265.29 263.35 261.80 259.94 258.25 256.29 254.09 252.61 250.75 249.27 247.19 245.64 243.53 241.97 240.19 238.17 236.63 234.22 232.84 231.21 229.46 220.00

266.91 264.93 263.14 261.30 259.50 257.41 255.14 253.56 251.63 249.98 247.90 246.24 243.95 242.33 240.46 238.29 236.62 234.22 232.62 230.91 229.13 219.08

268.57 266.57 264.63 262.54 260.76 258.52 256.13 254.50 252.38 250.75 248.56 246.82 244.39 242.64 240.69 238.40 236.66 234.16 232.41 230.58 228.87 218.27

270.22 268.17 266.03 263.75 261.95 259.61 257.16 255.47 253.27 251.51 249.20 247.43 244.88 242.93 240.94 238.59 236.70 234.10 232.16 230.35 228.43 217.53

271.81 269.70 267.46 265.07 263.16 260.80 258.14 256.41 254.03 252.27 249.85 247.90 245.26 243.31 241.18 238.65 236.71 234.09 232.00 230.06 228.02 216.61

273.38 271.21 268.86 266.44 264.46 261.92 259.18 257.24 254.83 252.97 250.42 248.42 245.73 243.58 241.49 238.75 236.73 233.95 231.77 229.77 227.68 215.77

275.03 272.72 270.38 267.78 265.53 262.96 260.22 258.18 255.66 253.72 251.04 248.85 246.18 243.89 241.72 238.86 236.74 233.84 231.55 229.58 227.25 214.96

276.65 274.17 271.84 268.97 266.79 264.09 261.30 259.14 256.45 254.39 251.60 249.40 246.60 244.28 241.93 238.92 236.72 233.77 231.47 229.34 226.86 214.15

278.26 275.72 273.28 270.29 268.10 265.22 262.35 260.11 257.26 255.13 252.13 249.93 247.08 244.62 242.16 238.99 236.74 233.69 231.28 229.14 226.54 213.26

279.84 277.26 274.71 271.60 269.25 266.25 263.29 260.96 258.17 255.85 252.81 250.38 247.53 245.01 242.43 239.11 236.73 233.61 231.12 228.79 226.18 212.45

281.48 278.77 276.15 272.99 270.43 267.41 264.23 261.92 259.04 256.51 253.37 250.91 247.88 245.29 242.66 239.27 236.77 233.54 231.02 228.48 225.80 211.67

283.10 280.23 277.47 274.17 271.64 268.51 265.33 262.80 259.69 257.24 253.98 251.51 248.28 245.59 242.86 239.31 236.72 233.55 230.92 228.15 225.49 210.98

284.74 281.78 278.90 275.50 272.83 269.65 266.32 263.74 260.59 257.97 254.66 251.98 248.74 245.87 243.07 239.53 236.67 233.41 230.67 227.82 224.98 210.17

286.41 283.34 280.29 276.82 273.97 270.67 267.32 264.69 261.46 258.79 255.24 252.40 249.15 246.24 243.18 239.58 236.63 233.36 230.49 227.63 224.59 209.41

287.93 284.70 281.66 278.18 275.25 271.78 268.30 265.58 262.31 259.49 255.81 252.84 249.47 246.49 243.35 239.68 236.65 233.32 230.41 227.33 224.27 208.71

289.50 286.24 283.13 279.42 276.57 272.85 269.23 266.45 263.15 260.14 256.46 253.34 249.94 246.73 243.57 239.88 236.63 233.21 230.24 227.11 224.00 207.99

291.16 287.85 284.52 280.70 277.75 273.95 270.24 267.36 263.95 260.91 257.05 253.82 250.40 246.98 243.78 239.84 236.63 233.12 230.04 226.72 223.55 207.29

292.72 289.26 285.91 282.01 278.94 275.04 271.20 268.31 264.72 261.58 257.68 254.34 250.74 247.28 244.04 239.95 236.68 233.01 229.86 226.47 223.17 206.58

294.31 290.72 287.34 283.21 280.17 276.19 272.19 269.21 265.51 262.19 258.29 254.84 251.08 247.59 244.25 240.04 236.69 232.90 229.71 226.06 222.68 205.99

295.97 292.22 288.70 284.42 281.38 277.35 273.15 270.12 266.39 262.85 258.82 255.32 251.53 247.96 244.46 240.23 236.62 232.81 229.56 225.86 222.28 205.42

297.51 293.77 290.08 285.70 282.49 278.45 274.14 271.00 267.21 263.54 259.50 255.73 251.99 248.34 244.64 240.27 236.67 232.70 229.31 225.60 221.89 204.78

299.05 295.26 291.47 286.97 283.70 279.50 275.20 271.79 267.95 264.11 260.13 256.19 252.41 248.67 244.90 240.36 236.68 232.69 229.06 225.21 221.47 204.18

300.63 296.75 292.97 288.30 284.85 280.59 276.06 272.69 268.77 264.73 260.63 256.71 252.75 248.90 245.15 240.51 236.61 232.54 228.89 224.86 221.08 203.78

302.22 298.26 294.23 289.58 286.05 281.68 277.10 273.61 269.59 265.50 261.18 257.12 253.17 249.28 245.28 240.52 236.59 232.51 228.71 224.53 220.66 203.31

303.81 299.71 295.68 290.93 287.17 282.79 278.14 274.53 270.37 266.20 261.66 257.65 253.48 249.53 245.45 240.63 236.51 232.31 228.49 224.23 220.28 202.86

305.44 301.25 297.05 292.18 288.37 283.70 279.09 275.45 271.25 266.90 262.30 258.11 253.81 249.82 245.65 240.76 236.50 232.15 228.39 223.93 219.94 202.38

307.06 302.76 298.38 293.48 289.56 284.82 280.00 276.32 272.00 267.62 262.85 258.51 254.18 250.08 245.80 240.82 236.45 232.04 228.17 223.67 219.53 202.09

308.46 304.11 299.80 294.71 290.75 285.92 280.98 277.18 272.80 268.22 263.50 259.05 254.60 250.32 245.90 240.87 236.43 232.00 227.96 223.35 219.15 201.90

310.00 305.51 301.15 296.05 291.88 287.03 281.97 278.03 273.53 268.86 264.06 259.53 254.97 250.77 246.07 240.98 236.32 231.94 227.81 223.00 218.75 201.57

311.51 307.01 302.61 297.30 293.00 288.08 282.82 278.98 274.30 269.50 264.62 260.04 255.33 251.01 246.23 241.07 236.31 231.77 227.58 222.66 218.37 201.37

313.13 308.53 303.99 298.55 294.13 289.21 283.89 279.82 275.06 270.25 265.16 260.46 255.76 251.25 246.40 241.16 236.27 231.59 227.41 222.38 217.97 201.10

314.67 310.09 305.33 299.83 295.45 290.26 284.88 280.66 275.89 270.96 265.75 260.97 256.13 251.52 246.62 241.28 236.20 231.46 227.19 222.13 217.57 200.99

316.27 311.50 306.70 301.00 296.54 291.26 285.90 281.55 276.67 271.56 266.36 261.47 256.50 251.81 246.76 241.35 236.19 231.31 226.89 221.80 217.22 200.75

317.85 312.85 308.02 302.24 297.69 292.39 286.94 282.38 277.39 272.30 266.92 262.00 256.82 252.00 246.84 241.42 236.11 231.22 226.72 221.52 216.85 200.69

319.46 314.28 309.37 303.56 298.88 293.43 287.94 283.23 278.14 272.98 267.52 262.45 257.21 252.27 246.99 241.49 236.15 231.13 226.52 221.24 216.45 200.44

320.96 315.77 310.75 304.87 300.04 294.55 288.92 284.06 278.90 273.57 268.05 262.85 257.56 252.53 247.20 241.62 236.21 231.06 226.19 220.96 216.11 200.28

322.58 317.27 312.09 306.16 301.21 295.61 289.97 284.84 279.59 274.33 268.58 263.34 258.01 252.84 247.33 241.69 236.24 230.92 225.98 220.65 215.79 200.22

324.19 318.74 313.42 307.38 302.40 296.71 290.90 285.65 280.35 274.94 269.15 263.86 258.37 253.20 247.51 241.77 236.18 230.70 225.75 220.38 215.42 200.10

325.76 320.27 314.79 308.58 303.53 297.82 291.98 286.60 281.12 275.63 269.71 264.33 258.71 253.42 247.65 241.77 236.13 230.59 225.48 220.09 215.13 200.00

327.24 321.62 316.03 309.82 304.78 298.94 292.93 287.50 281.92 276.21 270.33 264.85 259.09 253.59 247.79 241.84 236.12 230.46 225.26 219.82 214.75 199.85

328.77 323.06 317.39 311.07 305.87 299.96 293.87 288.36 282.60 276.88 270.88 265.29 259.43 253.98 247.93 241.89 236.17 230.28 225.08 219.55 214.35 199.72

330.43 324.53 318.70 312.30 306.98 300.97 294.96 289.17 283.37 277.48 271.40 265.83 259.89 254.11 248.06 242.04 236.02 230.16 224.87 219.27 214.02 199.69

331.90 325.99 320.02 313.62 308.17 302.09 295.88 290.02 284.04 278.13 271.87 266.34 260.26 254.37 248.15 242.11 235.94 230.07 224.59 218.99 213.73 199.63

333.53 327.49 321.34 314.93 309.42 303.16 296.86 290.95 284.77 278.73 272.33 266.80 260.59 254.60 248.30 242.08 235.88 229.92 224.33 218.75 213.46 199.51

335.05 328.95 322.61 316.11 310.61 304.21 297.77 291.84 285.53 279.39 272.87 267.24 260.95 254.84 248.47 242.06 235.92 229.74 224.16 218.36 213.12 199.57

336.56 330.40 323.99 317.38 311.78 305.32 298.69 292.69 286.33 280.05 273.39 267.68 261.36 255.16 248.70 242.16 235.83 229.54 223.89 218.11 212.87 199.54

338.07 331.80 325.33 318.69 312.95 306.36 299.68 293.55 287.10 280.68 273.90 268.20 261.75 255.39 248.88 242.16 235.82 229.42 223.68 217.76 212.58 199.49

339.68 333.24 326.59 319.87 314.05 307.36 300.63 294.39 287.83 281.26 274.50 268.59 262.08 255.57 248.99 242.12 235.75 229.32 223.46 217.51 212.36 199.42

341.12 334.76 328.10 321.17 315.26 308.38 301.51 295.21 288.68 282.03 275.03 269.07 262.35 255.73 249.16 242.14 235.74 229.17 223.28 217.24 212.10 199.42

342.77 336.14 329.35 322.35 316.44 309.46 302.55 296.02 289.40 282.60 275.56 269.51 262.70 256.01 249.25 242.24 235.69 229.08 223.06 216.95 211.84 199.40

344.31 337.64 330.70 323.61 317.56 310.53 303.44 296.81 289.99 283.27 276.09 269.88 263.12 256.28 249.35 242.29 235.65 228.91 222.84 216.68 211.50 199.27

345.87 339.01 332.03 324.91 318.66 311.70 304.37 297.69 290.81 283.96 276.62 270.36 263.47 256.60 249.55 242.32 235.70 228.73 222.61 216.47 211.25 199.28

347.40 340.44 333.47 326.13 319.79 312.71 305.28 298.51 291.50 284.57 277.17 270.81 263.91 256.84 249.70 242.40 235.70 228.54 222.44 216.09 211.01 199.34

348.94 341.86 334.72 327.36 320.90 313.77 306.29 299.32 292.22 285.28 277.65 271.23 264.22 257.03 249.81 242.43 235.63 228.43 222.16 215.79 210.74 199.32

350.53 343.33 336.00 328.59 322.01 314.82 307.24 300.16 292.98 285.82 278.21 271.59 264.57 257.27 250.05 242.41 235.53 228.22 221.94 215.52 210.53 199.27

351.98 344.88 337.36 329.81 323.13 315.94 308.23 300.97 293.69 286.46 278.69 272.05 264.96 257.58 250.21 242.51 235.47 228.12 221.70 215.29 210.29 199.18

353.54 346.28 338.65 331.14 324.24 316.98 309.11 301.80 294.46 287.20 279.24 272.45 265.23 257.82 250.39 242.58 235.40 228.01 221.50 215.01 210.06 199.17

355.10 347.67 340.00 332.33 325.42 317.97 310.08 302.64 295.10 287.80 279.81 273.03 265.66 258.05 250.51 242.59 235.36 227.86 221.34 214.78 209.86 199.13

356.64 349.06 341.44 333.56 326.61 318.88 310.92 303.42 295.82 288.56 280.35 273.45 265.95 258.31 250.64 242.67 235.34 227.69 221.09 214.65 209.68 199.20

358.20 350.49 342.70 334.84 327.66 319.96 311.88 304.29 296.58 289.13 280.85 273.83 266.28 258.57 250.71 242.79 235.21 227.62 220.75 214.41 209.46 199.19

359.68 352.07 344.06 336.02 328.76 321.07 312.94 305.10 297.30 289.74 281.37 274.33 266.56 258.83 250.81 242.68 235.22 227.48 220.60 214.19 209.28 199.26

361.28 353.52 345.52 337.22 329.99 322.18 313.87 305.88 298.05 290.44 281.84 274.76 266.91 259.06 251.02 242.74 235.10 227.26 220.49 213.97 208.99 199.31

362.76 354.81 346.75 338.41 331.11 323.26 314.80 306.71 298.81 291.01 282.40 275.26 267.27 259.31 251.14 242.73 235.02 227.15 220.26 213.69 208.76 199.32

70

80

90

100

10

20

30

40

50

60

Groups / Tee 7.0 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 8.0 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 9.0 9.5188.05 188.13 188.05 188.30 188.01 188.20 187.97 188.35 188.05 188.28 188.06 188.14 188.17 188.18 188.11 188.30 188.21 188.20 187.99 188.12 188.24 187.96

200.85 201.08 201.03 201.19 201.24 201.31 201.38 201.63 201.52 201.90 201.82 201.85 202.18 202.30 202.10 202.39 202.51 202.67 202.64 202.75 202.83 203.32

211.85 212.12 212.15 212.47 212.71 212.69 212.90 213.17 213.27 213.81 213.85 214.11 214.28 214.59 214.40 215.00 214.94 215.26 215.36 215.66 215.76 216.80

222.10 222.40 222.47 223.09 223.37 223.45 223.61 224.19 224.48 224.97 225.09 225.40 225.76 226.19 226.27 226.66 226.84 227.29 227.40 227.71 228.10 229.50

231.93 232.39 232.59 233.08 233.70 233.90 233.97 234.77 235.00 235.62 236.00 236.34 236.80 237.28 237.66 237.89 238.22 238.69 239.15 239.56 239.92 241.85

241.61 242.03 242.48 243.04 243.59 244.01 244.34 245.11 245.37 246.15 246.49 247.01 247.60 248.26 248.64 249.05 249.39 249.93 250.56 250.98 251.46 254.00

251.02 251.61 251.96 252.76 253.40 253.96 254.33 255.21 255.60 256.53 256.90 257.51 258.10 258.96 259.47 260.00 260.62 260.95 261.84 262.33 262.96 265.87

260.24 261.04 261.45 262.40 263.06 263.72 264.26 265.22 265.68 266.77 267.19 267.89 268.60 269.64 270.24 270.80 271.53 271.95 272.92 273.54 274.28 277.67

269.58 270.25 270.97 271.83 272.54 273.46 273.96 275.01 275.66 276.77 277.46 278.18 279.09 280.22 280.86 281.48 282.38 282.85 283.87 284.69 285.41 289.26

278.71 279.58 280.33 281.33 282.18 283.00 283.79 284.90 285.62 286.86 287.51 288.30 289.32 290.64 291.40 292.05 293.02 293.68 294.80 295.78 296.49 300.81

287.81 288.70 289.54 290.81 291.62 292.53 293.36 294.68 295.44 296.80 297.55 298.52 299.62 301.10 301.82 302.64 303.68 304.46 305.61 306.66 307.55 312.38

296.65 297.70 298.79 300.03 301.05 302.08 302.98 304.37 305.26 306.77 307.52 308.70 309.81 311.39 312.13 313.18 314.31 315.23 316.41 317.59 318.66 323.85

305.74 306.73 307.98 309.27 310.47 311.62 312.65 313.95 315.01 316.54 317.51 318.66 319.96 321.58 322.53 323.71 324.93 325.93 327.20 328.58 329.62 335.34

314.66 315.81 317.16 318.53 319.79 321.07 322.19 323.64 324.77 326.35 327.47 328.79 330.01 331.86 332.86 334.12 335.33 336.63 337.90 339.38 340.57 346.78

323.51 324.82 326.28 327.67 329.11 330.44 331.75 333.22 334.53 336.14 337.39 338.69 340.12 341.89 343.20 344.50 345.87 347.18 348.64 350.21 351.56 358.28

332.37 333.83 335.35 336.79 338.43 339.82 341.20 342.79 344.10 345.97 347.18 348.77 350.17 352.04 353.39 354.87 356.34 357.73 359.32 360.98 362.42 369.67

341.28 342.73 344.43 345.93 347.58 349.12 350.68 352.40 353.73 355.63 356.96 358.65 360.19 362.18 363.62 365.15 366.83 368.25 370.03 371.80 373.23 381.01

350.16 351.70 353.50 355.01 356.91 358.49 360.06 361.86 363.42 365.46 366.84 368.72 370.23 372.34 373.90 375.47 377.32 378.82 380.59 382.45 384.04 392.32

358.83 360.52 362.55 364.07 366.03 367.80 369.54 371.40 373.02 375.17 376.64 378.60 380.18 382.39 384.04 385.75 387.75 389.26 391.27 393.14 394.90 403.66

367.65 369.41 371.57 373.19 375.24 377.13 378.88 380.93 382.63 384.86 386.39 388.57 390.14 392.53 394.21 395.90 398.17 399.77 401.84 403.91 405.74 415.06

376.39 378.21 380.41 382.04 384.04 385.90 387.66 389.67 391.42 393.59 395.22 397.37 398.99 401.28 403.00 404.72 406.88 408.49 410.67 412.66 414.48 423.77

385.02 386.97 389.20 390.83 392.89 394.67 396.33 398.55 400.11 402.32 403.96 406.19 407.74 410.09 411.74 413.51 415.72 417.28 419.37 421.49 423.22 432.45

393.76 395.78 397.88 399.55 401.63 403.54 405.04 407.34 408.93 411.12 412.70 415.00 416.58 418.79 420.50 422.28 424.48 426.10 428.13 430.23 432.00 441.25

402.56 404.53 406.67 408.31 410.38 412.27 413.76 416.00 417.60 419.86 421.56 423.72 425.30 427.52 429.24 431.04 433.19 434.91 436.96 438.97 440.78 449.97

411.32 413.26 415.37 416.99 419.01 420.88 422.45 424.69 426.40 428.57 430.35 432.40 434.03 436.27 438.08 439.75 441.87 443.63 445.62 447.67 449.55 458.73

420.09 422.07 424.11 425.73 427.75 429.67 431.10 433.35 435.11 437.37 439.06 441.16 442.73 444.95 446.71 448.52 450.69 452.31 454.27 456.42 458.25 467.43

428.80 430.70 432.82 434.37 436.50 438.42 439.91 442.09 443.79 446.05 447.72 449.77 451.46 453.66 455.51 457.17 459.42 461.00 463.01 465.11 466.92 476.26

437.52 439.44 441.47 443.17 445.23 447.09 448.53 450.85 452.55 454.84 456.39 458.56 460.17 462.38 464.19 465.96 468.05 469.71 471.69 473.86 475.79 484.91

446.25 448.19 450.23 451.87 453.93 455.74 457.26 459.53 461.34 463.48 465.11 467.17 468.84 471.01 472.93 474.58 476.84 478.33 480.41 482.50 484.49 493.45

454.89 456.93 458.79 460.62 462.66 464.37 465.98 468.21 470.07 472.21 473.82 475.88 477.50 479.73 481.60 483.32 485.63 486.98 489.02 491.27 493.11 502.19

463.53 465.56 467.49 469.32 471.38 473.05 474.71 476.88 478.77 480.94 482.59 484.58 486.15 488.42 490.25 491.98 494.27 495.57 497.72 499.95 501.86 510.87

472.27 474.23 476.08 478.00 480.03 481.73 483.39 485.52 487.29 489.63 491.27 493.23 494.92 497.01 498.97 500.72 502.90 504.29 506.42 508.64 510.59 519.60

480.91 482.90 484.79 486.80 488.68 490.46 492.07 494.17 495.98 498.26 499.98 501.97 503.59 505.75 507.61 509.47 511.62 513.04 515.06 517.30 519.23 528.22

489.66 491.55 493.45 495.46 497.45 499.18 500.68 502.83 504.75 507.01 508.56 510.68 512.20 514.52 516.25 518.22 520.25 521.63 523.71 525.85 527.86 536.94

498.33 500.19 502.02 504.14 506.07 507.79 509.36 511.42 513.36 515.62 517.27 519.35 520.90 523.17 524.94 526.83 528.90 530.24 532.42 534.45 536.54 545.67

507.00 508.80 510.77 512.79 514.72 516.44 518.12 520.13 522.02 524.24 525.98 528.08 529.53 531.83 533.70 535.30 537.57 538.86 541.12 543.07 545.18 554.32

515.63 517.37 519.49 521.51 523.42 525.13 526.70 528.87 530.74 532.96 534.55 536.75 538.21 540.43 542.32 543.92 546.21 547.52 549.80 551.85 553.88 562.93

524.29 526.05 528.20 530.04 532.05 533.79 535.29 537.51 539.35 541.57 543.19 545.34 546.93 549.07 550.99 552.67 554.83 556.12 558.44 560.51 562.46 571.50

532.91 534.73 536.74 538.70 540.70 542.41 543.94 546.16 548.03 550.18 551.90 554.04 555.55 557.73 559.66 561.29 563.42 564.82 567.02 569.11 571.13 580.08

541.57 543.47 545.43 547.24 549.36 551.02 552.53 554.80 556.58 558.85 560.56 562.72 564.19 566.34 568.29 569.90 572.06 573.46 575.61 577.68 579.87 588.77

550.22 552.17 554.03 555.75 557.95 559.61 561.16 563.47 565.27 567.51 569.20 571.43 572.88 574.93 576.94 578.59 580.70 582.10 584.16 586.35 588.43 597.53

558.81 560.80 562.66 564.37 566.56 568.30 569.74 572.11 573.83 576.17 577.85 580.00 581.46 583.61 585.58 587.15 589.31 590.79 592.80 594.96 597.02 606.11

567.38 569.41 571.26 573.04 575.26 576.92 578.38 580.64 582.43 584.77 586.42 588.62 590.13 592.18 594.29 595.75 597.93 599.35 601.37 603.57 605.68 614.77

576.03 578.02 579.98 581.68 583.73 585.46 587.02 589.28 591.06 593.42 595.04 597.15 598.78 600.79 602.92 604.36 606.54 607.94 609.95 612.28 614.25 623.46

584.65 586.57 588.64 590.17 592.39 594.09 595.70 597.94 599.65 601.99 603.60 605.80 607.40 609.48 611.53 612.92 615.12 616.57 618.67 620.94 622.86 632.15

593.26 595.22 597.28 598.79 601.10 602.72 604.35 606.61 608.26 610.63 612.13 614.43 616.08 618.12 620.16 621.49 623.74 625.19 627.28 629.64 631.54 640.76

601.84 603.86 605.91 607.40 609.65 611.25 612.89 615.16 616.97 619.25 620.81 622.98 624.73 626.81 628.83 630.11 632.33 633.81 635.92 638.19 640.18 649.45

610.48 612.47 614.55 616.09 618.23 619.91 621.43 623.82 625.64 627.81 629.37 631.61 633.28 635.39 637.46 638.77 640.97 642.44 644.62 646.78 648.80 658.17

619.10 621.03 623.07 624.57 626.84 628.51 630.13 632.40 634.09 636.44 637.98 640.31 641.88 643.99 646.06 647.31 649.52 651.15 653.32 655.35 657.49 666.98

627.74 629.68 631.70 633.20 635.43 637.15 638.72 641.04 642.79 645.07 646.66 648.88 650.54 652.57 654.67 656.03 658.07 659.71 661.87 663.92 665.98 675.67

636.41 638.34 640.29 641.82 643.97 645.67 647.32 649.69 651.46 653.79 655.24 657.40 659.15 661.24 663.18 664.58 666.63 668.36 670.49 672.63 674.59 684.41

644.93 646.80 648.86 650.48 652.65 654.28 655.90 658.28 660.11 662.39 663.81 665.94 667.67 669.79 671.75 673.18 675.25 677.02 679.21 681.23 683.27 693.21

653.50 655.44 657.53 659.02 661.37 662.85 664.43 666.85 668.75 670.94 672.46 674.54 676.34 678.33 680.37 681.88 683.83 685.61 687.84 689.91 692.00 701.99

662.16 664.15 666.12 667.60 669.95 671.45 673.04 675.46 677.35 679.61 681.05 683.12 685.00 686.88 688.98 690.34 692.43 694.22 696.44 698.42 700.55 710.79

670.72 672.66 674.71 676.21 678.54 680.04 681.60 684.11 685.92 688.18 689.68 691.74 693.54 695.48 697.64 698.95 701.08 702.81 705.06 707.07 709.17 719.58

679.31 681.22 683.34 684.71 687.17 688.69 690.19 692.71 694.51 696.69 698.29 700.34 702.08 704.09 706.25 707.54 709.69 711.40 713.71 715.56 717.68 728.49

687.97 689.82 691.90 693.22 695.78 697.35 698.75 701.32 703.19 705.25 706.82 708.92 710.73 712.76 714.86 716.23 718.22 720.01 722.36 724.26 726.28 737.42

696.51 698.47 700.48 701.80 704.29 705.95 707.34 709.90 711.81 713.84 715.50 717.43 719.39 721.44 723.44 724.77 726.87 728.60 730.91 732.90 734.89 746.28

705.05 707.06 709.07 710.37 712.90 714.50 716.00 718.39 720.35 722.31 724.13 725.99 728.01 730.07 732.10 733.36 735.48 737.29 739.46 741.41 743.47 755.18

713.63 715.65 717.77 719.00 721.45 723.09 724.46 726.99 728.97 730.83 732.63 734.61 736.55 738.60 740.75 742.01 744.01 745.84 748.09 749.96 752.08 764.28

722.22 724.26 726.23 727.58 730.05 731.68 733.10 735.61 737.59 739.50 741.18 743.12 745.17 747.28 749.28 750.52 752.59 754.51 756.71 758.53 760.66 773.31

730.81 732.81 734.88 736.23 738.57 740.29 741.74 744.23 746.17 748.10 749.66 751.75 753.68 755.83 757.85 759.13 761.11 763.01 765.29 767.13 769.28 782.36

739.44 741.45 743.45 744.78 747.17 748.70 750.29 752.85 754.85 756.70 758.30 760.31 762.21 764.42 766.45 767.76 769.70 771.55 773.99 775.73 777.94 791.38

748.06 750.06 751.98 753.38 755.76 757.32 758.80 761.42 763.40 765.32 766.85 768.81 770.78 772.98 775.00 776.32 778.25 780.14 782.57 784.37 786.53 800.59

756.46 758.51 760.60 761.91 764.35 765.92 767.38 769.98 772.00 773.82 775.50 777.45 779.40 781.52 783.50 784.87 786.83 788.80 791.16 792.95 795.15 809.90

765.00 767.01 769.15 770.55 772.88 774.53 775.97 778.53 780.53 782.36 784.06 786.03 787.97 790.27 792.07 793.48 795.32 797.44 799.81 801.50 803.75 819.07

773.51 775.61 777.81 779.10 781.40 783.08 784.42 787.18 789.10 790.90 792.62 794.64 796.53 798.81 800.63 802.07 803.91 805.97 808.38 810.06 812.37 828.37

782.13 784.23 786.39 787.65 789.93 791.71 793.09 795.72 797.66 799.55 801.16 803.16 805.16 807.35 809.20 810.66 812.47 814.49 817.01 818.68 820.97 837.60

790.67 792.89 794.93 796.23 798.65 800.26 801.68 804.26 806.29 808.16 809.75 811.77 813.73 815.92 817.82 819.28 821.00 823.06 825.59 827.33 829.57 846.99

799.27 801.40 803.50 804.70 807.14 808.76 810.30 812.85 814.87 816.66 818.36 820.37 822.30 824.51 826.36 827.85 829.59 831.61 834.09 835.90 838.22 856.25

807.85 809.85 812.02 813.24 815.69 817.39 818.94 821.38 823.39 825.30 826.92 829.00 830.82 833.00 834.84 836.42 838.11 840.22 842.72 844.52 846.85 865.69

816.46 818.38 820.57 821.86 824.28 825.93 827.54 829.93 831.94 833.88 835.52 837.55 839.41 841.57 843.39 844.99 846.75 848.83 851.32 853.14 855.45 874.94

824.96 826.97 829.15 830.47 832.84 834.55 836.12 838.46 840.50 842.37 844.05 846.05 847.96 850.13 852.00 853.62 855.41 857.46 859.79 861.76 864.11 884.28

833.58 835.57 837.69 839.06 841.41 843.11 844.77 846.94 848.99 851.03 852.58 854.64 856.61 858.74 860.53 862.19 864.04 866.02 868.38 870.35 872.79 893.72

842.19 844.14 846.22 847.58 850.00 851.71 853.30 855.45 857.55 859.54 861.15 863.26 865.17 867.40 869.11 870.77 872.58 874.50 876.95 878.98 881.42 903.10

850.76 852.77 854.79 856.08 858.53 860.32 861.98 864.10 866.12 868.13 869.71 871.83 873.71 875.92 877.65 879.27 881.13 883.09 885.48 887.59 890.13 912.50

859.24 861.22 863.23 864.62 867.18 868.94 870.53 872.70 874.72 876.61 878.33 880.45 882.29 884.39 886.19 887.84 889.72 891.66 894.06 896.22 898.75 921.85

867.77 869.76 871.79 873.17 875.67 877.46 879.07 881.26 883.20 885.18 886.88 888.99 890.83 893.08 894.73 896.39 898.37 900.18 902.68 904.85 907.35 931.22

876.43 878.33 880.30 881.70 884.18 885.97 887.76 889.77 891.77 893.68 895.40 897.63 899.49 901.51 903.26 905.04 906.82 908.76 911.27 913.47 916.02 940.69

884.90 886.89 888.82 890.32 892.77 894.59 896.28 898.32 900.24 902.23 903.87 906.24 908.06 910.07 911.75 913.61 915.34 917.37 919.79 922.09 924.73 950.13

893.53 895.49 897.34 898.93 901.42 903.16 904.86 906.95 908.77 910.73 912.33 914.80 916.59 918.60 920.30 922.08 923.88 925.92 928.33 930.75 933.46 959.51

902.05 904.05 905.81 907.41 910.01 911.71 913.37 915.54 917.33 919.29 920.87 923.34 925.15 927.14 928.87 930.56 932.52 934.44 936.96 939.26 942.12 969.07

910.56 912.60 914.39 915.98 918.58 920.32 921.89 924.09 925.93 927.85 929.39 931.88 933.76 935.76 937.50 939.16 941.03 942.94 945.49 947.91 950.87 978.54

919.07 921.10 922.93 924.59 927.15 928.86 930.48 932.65 934.50 936.38 937.90 940.50 942.35 944.29 946.08 947.66 949.62 951.52 954.08 956.46 959.58 987.99

927.68 929.64 931.39 933.07 935.65 937.36 939.03 941.19 943.03 944.86 946.50 948.99 950.88 952.77 954.59 956.12 958.15 960.12 962.66 965.11 968.36 997.42

936.12 938.26 940.10 941.67 944.26 945.88 947.51 949.71 951.68 953.53 955.03 957.57 959.35 961.23 963.16 964.64 966.74 968.67 971.28 973.74 977.10 1006.92

944.77 946.74 948.55 950.15 952.84 954.46 956.15 958.22 960.20 962.00 963.56 966.11 967.90 969.81 971.65 973.24 975.29 977.28 979.86 982.35 985.84 1016.40

953.31 955.34 957.10 958.71 961.36 963.03 964.64 966.71 968.59 970.57 972.09 974.58 976.52 978.38 980.15 981.79 983.85 985.81 988.44 990.98 994.50 1025.77

961.87 963.81 965.63 967.31 969.86 971.70 973.17 975.29 977.21 979.16 980.62 983.16 985.07 987.00 988.75 990.32 992.50 994.33 997.01 999.67 1003.25 1035.28

970.40 972.34 974.27 975.83 978.39 980.21 981.68 983.81 985.70 987.67 989.17 991.71 993.71 995.54 997.30 998.90 1001.10 1002.84 1005.64 1008.19 1012.01 1044.84

978.94 980.86 982.72 984.36 986.90 988.77 990.29 992.32 994.22 996.28 997.65 1000.23 1002.22 1004.03 1005.81 1007.43 1009.63 1011.43 1014.16 1016.79 1020.74 1054.32

987.53 989.43 991.20 992.89 995.41 997.32 998.84 1000.86 1002.78 1004.72 1006.21 1008.69 1010.77 1012.57 1014.45 1015.91 1018.13 1019.92 1022.74 1025.42 1029.53 1063.77

995.98 998.08 999.76 1001.41 1003.93 1005.94 1007.43 1009.37 1011.29 1013.26 1014.69 1017.25 1019.36 1021.18 1023.01 1024.51 1026.67 1028.52 1031.30 1034.09 1038.29 1073.18

1004.54 1006.58 1008.25 1010.04 1012.44 1014.48 1015.91 1017.90 1019.86 1021.90 1023.24 1025.75 1027.83 1029.72 1031.59 1033.08 1035.20 1037.11 1039.90 1042.71 1047.06 1082.67

1013.10 1015.07 1016.80 1018.53 1021.02 1022.97 1024.48 1026.44 1028.30 1030.40 1031.81 1034.43 1036.46 1038.25 1040.11 1041.59 1043.76 1045.66 1048.54 1051.38 1055.86 1092.13

1021.64 1023.56 1025.44 1027.06 1029.61 1031.38 1032.92 1034.92 1036.82 1039.06 1040.35 1042.95 1044.95 1046.81 1048.64 1050.17 1052.34 1054.19 1057.09 1060.15 1064.68 1101.70

1030.20 1032.09 1033.90 1035.64 1038.06 1039.96 1041.48 1043.49 1045.38 1047.53 1048.85 1051.43 1053.48 1055.37 1057.11 1058.79 1060.81 1062.82 1065.55 1068.81 1073.46 1111.19

1038.68 1040.77 1042.46 1044.12 1046.56 1048.57 1050.14 1052.00 1053.90 1056.04 1057.37 1060.03 1061.96 1063.93 1065.61 1067.18 1069.42 1071.38 1074.20 1077.49 1082.28 1120.76

1047.28 1049.32 1051.12 1052.62 1055.19 1057.18 1058.67 1060.48 1062.45 1064.64 1065.84 1068.56 1070.51 1072.46 1074.22 1075.74 1077.90 1079.86 1082.89 1086.17 1090.99 1130.31

1055.76 1057.71 1059.55 1061.11 1063.71 1065.76 1067.20 1069.01 1071.01 1073.11 1074.40 1077.16 1079.07 1081.01 1082.74 1084.23 1086.42 1088.45 1091.46 1094.79 1099.76 1139.82

70

80

90

100

10

20

30

40

50

60

Figure 4: Sojourn times (above) and departure times (below) as a function of the group and thetee interval for a golf course with unscaled P3 holes, but now with uneven (two-level) tee times.More specifically, the tee times start off small at 7.00 for the first 20 groups, then increases afterthe 20th group; the tee times shown on the horizontal axis labels in the figure above denotes thetee times after the 20th group.

19

3.1.3 Case 3: P3 holes with alternative tee schedule

Finally, the third pair of figures in Figure 4 show average sojourn times (top) and departure times

(bottom) of each group (averaged over n r replications) as a function of the group number (y-

axis) and the fixed tee interval (x-axis) for the base case with unscaled P3 holes – but now with

alternative tee schedule, as discussed in §5 of the main paper. As discussed in the caption for

Figure 4, the tee times are 7.00 for the first 20 groups, but increases for the later groups. Again,

the base case refers to the hole sequence 454—434—454—434—454—434, as discussed in §3 of the

main paper.

Similarly to Figures 2 and 3, the top and the bottom figures in Figure 4 are color-coded. The

top figure shows 7 different colors to indicate different ranges of sojourn times, again indicating the

same set of sojourn time ranges as indicated in the descriptions for two cases above. Sojourn times

that appear below the black lines indicate that the constraint is violated. Similarly, in the bottom

figure, the ivory and the red colors respectively indicate that the departure time occurs before and

after 840 minutes, and the red color indicates that the departure-time constraint is violated.

Because the first 20 groups in Figure 4 play with tee interval of 7.00 (regardless of the tee

interval for next 80 groups), we see that the sojourn times of the first 20 groups are the same

across the x-axis. Then, the sojourn times for each y-axis fall into one of the two categories: (i)

the sojourn times continue to increase as the golf course becomes over-loaded, or (ii) the sojourn

times decrease as the course becomes under-loaded. In the top figure, when the tee interval (for the

later 80 groups) is less than 8.0, no more than 27 groups can satisfy the sojourn time constraint,

where as the top figure in Figure 2 (unscaled P3 holes without the alternative tee schedule) shows

that 60 groups can satisfy the sojourn time constraint. While there appears to be a disadvantage

in having the alternative tee schedule because the sojourn times appear to be higher in Figure 4

overall, the alternative schedule does realize higher throughput levels for P3WU holes and SP3

holes. Please see §5 in the main paper for greater details on the impact of alternative tee schedule

on throughput levels.

20

4 Extension of the Waiting Time Study

In this section, we provide additional results on comparing alternative course designs to supplement

our findings in Section 3 of the main paper. In the main paper, we found that the order of the holes

has minimal to no impact on the throughput levels of a course. However, the throughput levels

vary depending on whether the course is overloaded or underloaded. More specifically, in Tables

1, 2, and 3 of the main paper, we showed that P3WU holes performs substantially better than

the P3 holes – the performance improvement is more evident when the tee interval is shorter, i.e.,

when the golf course is more heavily loaded. Then, we showed in Table 5 of the main paper that

the average waiting time of a playing group decreases substantially as the tee increases to 8.50, but

we also showed that P3 holes still remain as a bottleneck in a golf course.

Here, we extend our results over two sub-sections. The first sub-section (Section 4.1) provides

detailed statistics of the average waiting times of the 75th group with the longer tee interval 8.50.

The second sub-section (Section 4.2) provides these wait time statistics for groups other than the

75th group.

21

4.1 Additional Wait Time Statistics for Longer Tee Intervals

Table 1: Simulation estimates of the mean waiting times for group 75 in minutes before startingplay on each of the 18 holes for three course designs with the three kinds of par-3 holes: P3, P3WUand SP3. In all cases the tee interval is 8.50 minutes, which makes all three cases under-loaded.The course designs are (i) the base case, (ii) the par-5-holes first and (iii) the par-3 holes first.

base case par-5 first par-3 first

hole par P3 P3WU SP3 par P3 P3WU SP3 par P3 P3WU SP3

1 4 0.10 0.08 0.09 5 0.09 0.09 0.08 3 5.78 0.76 0.292 5 0.21 0.24 0.21 5 0.31 0.28 0.28 3 7.24 1.28 0.563 4 0.41 0.42 0.40 5 0.43 0.42 0.38 3 6.46 1.77 0.77

4 4 0.41 0.43 0.36 3 8.39 1.16 0.91 4 0.17 1 .44 0.435 3 8.50 1.14 0.87 4 0.17 1 .39 0.47 5 0.26 0.63 0.456 4 0.16 1 .22 0.43 3 8.16 1.30 0.98 4 0.46 0.81 0.59

7 4 0.33 0.58 0.51 4 0.15 1 .34 0.49 4 0.45 0.60 0.488 5 0.35 0.52 0.46 3 7.36 1.36 1.03 4 0.48 0.63 0.579 4 0.55 0.64 0.63 4 0.17 1 .33 0.59 4 0.46 0.55 0.54

10 4 0.49 0.54 0.51 4 0.29 0.66 0.58 4 0.44 0.59 0.5311 3 9.42 1.33 1.07 4 0.38 0.64 0.54 5 0.46 0.51 0.4712 4 0.15 1 .41 0.49 4 0.40 0.57 0.51 4 0.61 0.71 0.64

13 4 0.32 0.66 0.56 4 0.43 0.59 0.50 4 0.53 0.63 0.5814 5 0.37 0.53 0.49 4 0.44 0.58 0.51 4 0.52 0.59 0.5215 4 0.56 0.73 0.68 4 0.47 0.57 0.53 4 0.54 0.60 0.55

16 4 0.47 0.61 0.60 4 0.47 0.59 0.62 4 0.57 0.58 0.6217 3 8.51 1.30 1.19 4 0.46 0.57 0.60 4 0.52 0.55 0.5118 4 0.17 1 .35 0.53 4 0.49 0.56 0.60 5 0.65 0.72 0.64

sum 31.47 13.72 10.04 29.05 14.02 10.21 26.39 13.94 9.73

First, we include the duplicate of the Table 4 in the main paper (labeled Table 1 in this Online

Companion) for readers’ reference. As described in the main paper, Table 1 provides simulation

estimates of the mean waiting times for group 75 on each of the 18 holes. We provide the estimates

for three types of course designs and for three types of P3 holes. As mentioned in the main paper,

the main takeaway from Table 1 is that waiting times are substantially lower for a golf course with

P3 holes when the course is under-loaded (e.g. when tee is 8.50).

22

Table 2: Simulation estimates of the proportion of waiting times for group 75 (in %) before startingplay on each of the 18 holes for three course designs with the three kinds of par-3 holes: P3, P3WUand SP3. In all cases the tee interval is 8.50 minutes, under which all three cases are underloaded.The course designs are (i) the base case, (ii) the par-5-holes first and (iii) the par-3 holes first.

base case par-5 first par-3 first

hole par P3 P3WU SP3 par P3 P3WU SP3 par P3 P3WU SP3

1 4 0.3% 0.6% 0.9% 5 0.3% 0.6% 0.8% 3 21.9% 5.4% 3.0%2 5 0.7% 1.8% 2.1% 5 1.1% 2.0% 2.8% 3 27.4% 9.2% 5.7%3 4 1.3% 3.0% 3.9% 5 1.5% 3.0% 3.8% 3 24.5% 12.7% 7.9%

4 4 1.3% 3.1% 3.6% 3 28.9% 8.3% 8.9% 4 0.6% 10.3% 4.4%5 3 27.0% 8.3% 8.6% 4 0.6% 9.9% 4.6% 5 1.0% 4.5% 4.6%6 4 0.5% 8.9% 4.0% 3 28.1% 9.3% 9.6% 4 1.7% 5.8% 6.1%

7 4 1.1% 4.2% 5.1% 4 0.5% 9.6% 4.8% 4 1.5% 4.3% 4.9%8 5 1.1% 3.8% 4.5% 3 25.3% 9.7% 10.1% 4 1.7% 4.5% 5.9%9 4 1.7% 4.7% 6.3% 4 0.6% 9.5% 5.8% 4 1.7% 3.9% 5.5%

10 4 1.6% 3.9% 5.1% 4 1.0% 4.7% 5.6% 4 1.9% 4.3% 5.4%11 3 29.9% 9.7% 10.7% 4 1.3% 4.6% 5.2% 5 1.7% 3.7% 4.8%12 4 0.5% 10.3% 4.8% 4 1.4% 4.1% 5.0% 4 2.2% 5.1% 6.5%

13 4 1.0% 4.8% 5.5% 4 1.5% 4.2% 4.9% 4 2.0% 4.5% 5.9%14 5 1.2% 3.9% 4.9% 4 1.5% 4.1% 5.0% 4 1.8% 4.2% 5.4%15 4 1.8% 5.3% 6.7% 4 1.6% 4.1% 5.2% 4 1.9% 4.3% 5.7%

16 4 1.5% 4.5% 6.0% 4 1.6% 4.2% 6.1% 4 2.1% 4.2% 6.3%17 3 27.0% 9.5% 11.8% 4 1.6% 4.1% 5.9% 4 2.0% 3.9% 5.2%18 4 0.6% 9.8% 5.3% 4 1.7% 4.0% 5.9% 5 2.5% 5.2% 6.6%

sum 100% 100% 100% 100% 100% 100% 100% 100% 100%

Tables 2 and 3 of the appendix provide detailed statistics in addition to Table 1 (i.e. Table 4 of

the main paper). In fact, Tables 2 and 3 are analogs of Tables 2 and 3 of the main paper (please

see §3.1 of the main paper in greater details).

Table 2 provides the corresponding estimated proportion of the total waiting time at each hole.

The proportion in bold types indicate the proportion of each P3 hole with respect to the total

waiting times. From the Table 2, we see that the proportion of waiting times of the 75th group

is larger in the golf course with P3 holes than in the course with P3WU or SP3 holes. More

specifically, the three P3 holes together comprise of 83.9%, 82.3% and 72.3% of the total waiting

times in the three golf course designs respectively. On the other hand, the combined proportion of

waiting times for P3WU holes are 27.6%, 27.3% and 27.3%, and those for SP3 holes are 32.1%,

29.6% and 16.6%. Consequently, our observation tells us that the P3 holes, despite the golf course

being under-loaded, still remain as bottlenecks.

We further note that the waiting time proportions at the hole subsequent to P3WU holes are

23

higher than the proportions at the hole subsequent to SP3 holes. More specifically, the proportions

at the hole after P3WU holes are approximately 9 to 10 percent, while those at the hole after SP3

holes are 5.8% at most. The difference between the two can be primarily attributed to the difference

between the number of people permitted to play at a P3WU hole and at a SP3 hole. With the

wave-up policy, two playing groups will be able to simultaneously play at P3WU hole, making the

interval between the two groups smaller than the interval at the SP3 hole. Subsequently, it is more

likely that the distance between playing groups is shorter in P3WU than in SP3, and such close

distance may make it more likely for the group behind the other group to wait for a longer period

of time.

Let us compare the Table 2 with its analog, Table 2 in the main paper. In Table 2 of the

main paper, we see that the waiting time proportion of the first P3 hole is particularly high (i.e.

67.5%, 69.6% and 69.6% in the three course designs, respectively) compared to the proportion in

the second and third P3 holes (ranges between 9.6% and 14.6%). On the other hand, in the Table

2 of the Online Companion, we see that the proportions are spread more evenly across the three

P3 holes. Consequently, the first P3 hole alone becomes the primary bottleneck when the course

is over-loaded but does not when the course is under-loaded.

We also compare the proportions of waiting times for P3WU and SP3 holes. For both P3WU

and SP3 holes, we see that the proportions are spread evenly across the three holes in all three

different course designs. We note that P3WU and SP3 holes are not bottleneck holes because golf

courses remain balanced even when the tee interval is 7.50.

In summary, Table 2 suggests that setting the tee interval to 8.50 relieves the congestion in golf

courses with P3 holes but makes relatively small impact on golf courses with P3WU or SP3 holes.

Such finding is corroborated by (i) high waiting time proportions of P3 holes, and (ii) concentrated

proportion in the first P3 hole when the tee interval is 7.50.

24

Table 3: Simulation estimates of the standard deviations of the waiting times before starting toplay on each hole for group 75 for each of the three kinds of par-3 holes: P3, P3WU and SP3.In all cases the tee interval is 8.50 minutes, under which all courses are stable. The course designsare (i) the base case, (ii) the par-5-holes first and (iii) the par-3 holes first. The last two rows givesimulation estimates for the standard deviation of the sum of all waiting times and the half widthof the 95% confidence interval for the mean.

base case par-5 first par-3 first

hole par P3 P3WU SP3 par P3 P3WU SP3 par P3 P3WU SP3

1 4 0.39 0.38 0.41 5 0.45 0.40 0.36 3 5.77 0.92 0.992 5 0.68 0.71 0.70 5 0.82 0.81 0.80 3 6.69 1.57 1.443 4 0.97 0.96 1.04 5 0.99 1.03 1.00 3 6.28 2.07 1.62

4 4 0.88 0.97 1.12 3 6.85 1.39 1.76 4 0.55 2.23 1.035 3 6.89 1.43 1.71 4 0.59 2.06 1.01 5 0.79 1.28 0.936 4 0.64 1.91 1.02 3 6.84 1.51 1.75 4 1.21 1.43 1.26

7 4 0.86 1.18 1.09 4 0.56 2.08 1.17 4 1.11 1.33 1.218 5 0.99 1.12 1.11 3 6.73 1.68 1.91 4 1.07 1.22 1.169 4 1.10 1.44 1.31 4 0.64 2.15 1.17 4 1.13 1.28 1.07

10 4 1.04 1.20 1.19 4 0.85 1.33 1.02 4 1.11 1.28 1.1311 3 7.17 1.46 1.99 4 0.95 1.27 1.20 5 1.08 1.13 1.0912 4 0.59 2.14 1.20 4 0.96 1.27 1.22 4 1.24 1.49 1.34

13 4 0.82 1.30 1.18 4 1.01 1.30 1.15 4 1.19 1.24 1.2614 5 0.94 1.26 1.09 4 1.10 1.28 1.25 4 1.21 1.25 1.2215 4 1.11 1.49 1.32 4 1.09 1.26 1.19 4 1.10 1.31 1.15

16 4 1.08 1.41 1.22 4 1.11 1.24 1.28 4 1.16 1.28 1.1917 3 7.06 1.64 2.01 4 1.15 1.20 1.22 4 1.10 1.14 1.2018 4 0.62 2.12 1.19 4 1.16 1.18 1.19 5 1.36 1.37 1.33

sum 9.80 6.19 6.05 9.86 6.59 6.00 9.61 6.81 5.85

HW 0.43 0.27 0.27 0.43 0.29 0.26 0.42 0.30 0.26

Table 3 provides simulation estimates of the standard deviations of the waiting times of the

75th group; similarly as previous two tables Tables 2 and 1, the estimates are shown for three types

of Par-3 holes and three types of course designs.

In Table 3, standard deviation estimates for each of P3, P3WU and SP3 holes all exhibit

different behaviors. For P3 holes, we see that the standard deviations are noticeably higher than

those for P4 and P5 holes. One reason that may explain such wider spread in waiting times at P3

holes is the “snowball” effect of the waiting times; that is, the waiting times caused by the slow

pace of play of the earlier groups will likely grow larger for later groups (i.e. groups with larger

group number). In other words, the slower the earlier groups play, the more the 75th groups wait.

On the other hand, if the earlier groups play quickly, the waiting times of the 75th group will be

substantially reduced. Our observation holds for all three course designs.

25

For P3WU holes, the standard deviations are not the highest at the P3WU holes, but at

the holes after the P3WU holes. For SP3 holes, similarly as P3 holes, standard deviations are

somewhat higher at Par-3 holes, but not noticeably higher than other holes – and even lower for

certain instances. For instance, we see that the estimate for the SP3 hole in the “Par-3 first” course

design is 0.99, but the estimates for other holes in the course (except the 5th Par-5 hole) are higher

than 1.

Finally, we show confidence intervals (95%; two-sided) of the (i) individual hole waiting times

and (ii) total waiting times in Tables 4, 5 and 6. In all of the three tables, the tee interval is 8.50

minutes, and the confidence intervals are shown for each of the three kinds of par-3 holes: P3,

P3WU and SP3.

Table 4: Confidence interval (95%; two-sided) of the waiting times before starting to play on eachhole for group 75 for each of the three kinds of par-3 holes: P3, P3WU and SP3. In all casesthe tee interval is 8.50 minutes, under which all courses are stable. The course design used in thistable is “the base case”. The lower and the upper confidence intervals for the total waiting timesare located in the last row.

base case

hole par P3 P3WU SP3

Lower Upper Lower Upper Lower Upper

1 4 0.08 0.12 0.06 0.10 0.07 0.112 5 0.18 0.24 0.21 0.27 0.18 0.243 4 0.36 0.45 0.38 0.46 0.35 0.45

4 4 0.37 0.45 0.39 0.47 0.31 0.415 3 8.20 8.80 1.08 1.20 0.80 0.946 4 0.13 0.19 1.14 1.30 0.39 0.47

7 4 0.29 0.37 0.53 0.63 0.46 0.568 5 0.31 0.39 0.47 0.57 0.41 0.519 4 0.50 0.60 0.58 0.70 0.57 0.69

10 4 0.44 0.54 0.49 0.59 0.46 0.5611 3 9.11 9.73 1.27 1.39 0.98 1.1612 4 0.12 0.18 1.32 1.50 0.44 0.54

13 4 0.28 0.36 0.60 0.72 0.51 0.6114 5 0.33 0.41 0.48 0.59 0.44 0.5415 4 0.51 0.61 0.67 0.80 0.62 0.74

16 4 0.42 0.52 0.55 0.67 0.55 0.6517 3 8.20 8.82 1.23 1.37 1.10 1.2818 4 0.14 0.20 1.26 1.44 0.48 0.58

Sum 31.04 31.90 13.45 13.99 9.77 10.31

26

Table 5: Confidence Interval (95%; two-sided) of the waiting times before starting to play on eachhole for group 75 for each of the three kinds of par-3 holes: P3, P3WU and SP3. In all casesthe tee interval is 8.50 minutes, under which all courses are stable. The course design used in thistable is “par-5 first”. The lower and the upper confidence intervals for the total waiting times arelocated in the last row.

par-5 first

hole par P3 P3WU SP3

Lower Upper Lower Upper Lower Upper

1 5 0.07 0.11 0.07 0.11 0.06 0.102 5 0.27 0.35 0.24 0.32 0.24 0.323 5 0.39 0.47 0.37 0.47 0.34 0.42

4 3 8.09 8.69 1.10 1.22 0.83 0.995 4 0.14 0.20 1.30 1.48 0.43 0.516 3 7.86 8.46 1.23 1.37 0.90 1.06

7 4 0.13 0.17 1.25 1.43 0.44 0.548 3 7.07 7.65 1.29 1.43 0.95 1.119 4 0.14 0.20 1.24 1.42 0.54 0.64

10 4 0.25 0.33 0.60 0.72 0.54 0.6211 4 0.34 0.42 0.58 0.70 0.49 0.5912 4 0.36 0.44 0.51 0.63 0.46 0.56

13 4 0.39 0.47 0.53 0.65 0.45 0.5514 4 0.39 0.49 0.52 0.64 0.46 0.5615 4 0.42 0.52 0.51 0.63 0.48 0.58

16 4 0.42 0.52 0.54 0.64 0.56 0.6817 4 0.41 0.51 0.52 0.62 0.55 0.6518 4 0.44 0.54 0.51 0.61 0.55 0.65

Sum 28.62 29.48 13.73 14.31 9.95 10.47

4.2 Comparing Waiting Times of Other Groups

In the main paper, we showed that using P3 holes with the wave-up rule significantly reduces the

waiting times of the playing group. To illustrate this advantage, we displayed the waiting times

of the 75th group as one example. We now expand the results to include waiting times of other

groups to show that all of the other playing groups – not just the 75th group – also greatly benefit

from the wave-up advantage.

Tables 7 and 8 show the average waiting times and proportion of (waiting times at the Par-3

hole to) total waiting times spent on Par-3 holes with tee 7.50, while Tables 9 and 10 show those

with tee 8.50. Since the critical tee interval τ∗ is 8.39, 6.53, and 6.79 for P3, P3WU and SP3,

respectively, the tee interval of 7.50 minutes will make P3 over-loaded (but not P3WU and SP3),

while the tee interval of 8.50 minutes will make all cases under-loaded. Base case (“4 5 4” case) is

used as the hole sequence for all results.

27

Table 6: Confidence Interval (95%; two-sided) of the waiting times before starting to play on eachhole for group 75 for each of the three kinds of par-3 holes: P3, P3WU and SP3. In all casesthe tee interval is 8.50 minutes, under which all courses are stable. The course design used in thistable is “par-3 first”. The lower and the upper confidence intervals for the total waiting times arelocated in the last row.

par-3 first

hole par P3 P3WU SP3

Lower Upper Lower Upper Lower Upper

1 3 5.53 6.03 0.72 0.80 0.25 0.332 3 6.95 7.53 1.21 1.35 0.50 0.623 3 6.18 6.74 1.68 1.86 0.70 0.84

4 4 0.15 0.19 1.34 1.54 0.38 0.485 5 0.23 0.29 0.57 0.69 0.41 0.496 4 0.41 0.51 0.75 0.87 0.53 0.65

7 4 0.40 0.50 0.54 0.66 0.43 0.538 4 0.43 0.53 0.58 0.68 0.52 0.629 4 0.41 0.51 0.49 0.61 0.49 0.59

10 4 0.39 0.49 0.53 0.65 0.48 0.5811 5 0.41 0.51 0.46 0.56 0.42 0.5212 4 0.56 0.66 0.64 0.78 0.58 0.70

13 4 0.48 0.58 0.58 0.68 0.52 0.6414 4 0.47 0.57 0.54 0.64 0.47 0.5715 4 0.49 0.59 0.54 0.66 0.50 0.60

16 4 0.52 0.62 0.52 0.64 0.57 0.6717 4 0.47 0.57 0.50 0.60 0.46 0.5618 5 0.59 0.71 0.66 0.78 0.58 0.70

Sum 25.97 26.81 13.64 14.24 9.47 9.99

28

Table 7: Simulation estimates of the mean waiting times for a range of different playing groups (e.g.5, 10, 15, ...) before starting play on each of the Par-3 holes with the three kinds of par-3 holes:P3, P3WU and SP3. In all cases the tee interval is 7.50 minutes, which makes P3 over-loaded,but not P3WU and SP3.

base case - P3 base case - P3WU base case - SP3

group 1st 2nd 3rd 3-hole Sum 1st 2nd 3rd 3-hole Sum 1st 2nd 3rd 3-hole Sum

5 3.7 2.2 1.7 7.6 1.4 1.0 0.9 2.6 1.0 0.9 0.7 3.2

10 8.1 4.1 3.2 15.4 1.7 1.5 1.3 4.2 1.6 1.4 1.2 4.5

15 12.6 5.6 4.2 22.4 1.9 1.7 1.5 5.3 2.0 1.8 1.5 5.1

20 17.1 6.8 5.2 29.1 2.0 2.0 1.7 5.9 2.1 2.1 1.7 5.7

25 21.5 7.7 5.8 35.0 2.0 2.0 1.8 6.4 2.3 2.2 2.0 5.8

30 25.9 8.7 6.5 41.1 2.0 2.0 1.9 6.9 2.4 2.3 2.2 5.9

35 30.3 9.4 7.2 47.0 2.0 2.0 1.9 7.0 2.3 2.5 2.2 6.0

40 34.8 10.1 7.7 52.5 2.0 2.1 2.0 7.3 2.3 2.6 2.4 6.0

45 39.2 10.8 8.4 58.4 2.0 2.2 2.1 7.7 2.4 2.8 2.5 6.3

50 43.6 11.4 8.6 63.6 2.0 2.1 2.2 7.6 2.3 2.8 2.4 6.3

55 48.0 12.1 9.1 69.2 2.0 2.1 2.1 7.5 2.4 2.7 2.4 6.2

60 52.6 12.8 9.5 74.9 2.0 2.0 2.2 7.6 2.4 2.6 2.6 6.3

65 57.0 13.3 10.0 80.3 1.9 2.1 2.1 7.8 2.2 2.8 2.8 6.1

70 61.4 13.9 10.0 85.4 1.9 2.1 2.2 7.8 2.3 2.8 2.6 6.2

75 65.0 14.3 10.4 89.7 1.9 2.1 2.1 7.5 2.2 2.7 2.6 6.2

80 70.7 14.6 10.9 96.2 1.9 2.1 2.2 7.6 2.3 2.7 2.6 6.3

85 75.1 15.2 11.2 101.5 2.0 2.2 2.3 7.7 2.3 2.7 2.6 6.5

90 79.7 15.6 11.5 106.8 2.0 2.1 2.2 7.8 2.3 2.8 2.7 6.4

95 84.1 15.9 11.8 111.8 2.0 2.2 2.2 7.6 2.2 2.7 2.8 6.4

100 88.5 16.4 12.1 117.0 2.0 2.2 2.1 7.6 2.2 2.7 2.7 6.3

Table 7 shows the average waiting times spent on three types of Par-3 holes, P3, P3WU

and SP3, with tee 7.50. Since the critical tee interval τ∗ is 8.39, 6.53, and 6.79 for these holes

respectively, the tee interval of 7.50 minutes will make P3 over-loaded (but not P3WU and SP3),

while the tee interval of 8.50 minutes will make all cases under-loaded. Base case (“4 5 4” case) is

used as the hole sequence for all results.

Table 7 provides simulation estimates of the mean waiting times for a number of groups, from

the 5-th group to the 100-th group (i.e., the last group). For the purpose of brevity, Table 7 (i)

displays waiting times for the base case only, and (ii) does not display waiting times for all 100

groups; Table 7 shows waiting times for every 5th group from the 5-th group to the 100-th group,

skipping the groups in between. However, since we see that the waiting times increase monotonically

as the group number increases, we can roughly interpolate the estimates for the groups not shown

in the table.

As shown in Table 1 of the main paper, the waiting times at P3 holes are noticeably higher

29

than the waiting times at P3WU or SP3 holes. In addition, the gaps between the waiting times

at P3 holes and those at P3WU or SP3 holes widen as the group number increases. In particular,

we even see that the 100-th group (i.e. last group) waits for almost 2 hours.

With P3 holes, we see that the waiting times continue to increase until the 100-th group without

converging. The waiting times with P3 holes are increasing and concave. On the other hand, with

P3WU and SP3 holes, we see that the waiting times converge to roughly 7.6 and 6.3 minutes (with

minor variations), respectively, approximately at the 45-th group. The convergence that emerges

in the base case with P3WU and SP3 holes explains that these two courses are stable in terms of

traffic, unlike the base case with P3 holes.

30

Table 8: Simulation estimates of the proportion (in %) of the waiting times at Par-3 holes (to thewaiting time of the entire 18-hole course) for a range of different playing groups (e.g. 5, 10, 15, ...)before starting play on each of the Par-3 holes with the three kinds of par-3 holes: P3, P3WU andSP3. In all cases the tee interval is 7.50 minutes, which makes P3 over-loaded, but not P3WU andSP3.

base case - P3 (%) base case - P3WU (%) base case - SP3 (%)

group 3-hole 18-hole % 3-hole 18-hole % 3-hole 18-hole %

Sum Sum Sum Sum Sum Sum

5 7.6 12.4 61% 2.6 11.2 28% 3.2 9.4 27%

10 15.4 21.7 71% 4.2 17.2 26% 4.5 14.4 29%

15 22.4 28.9 78% 5.3 20.0 26% 5.1 17.3 31%

20 29.1 36.0 81% 5.9 21.7 26% 5.7 19.1 31%

25 35.0 41.9 84% 6.4 22.9 25% 5.8 20.1 32%

30 41.1 47.9 86% 6.9 23.5 25% 5.9 21.2 32%

35 47.0 53.9 87% 7.0 23.9 25% 6.0 21.5 33%

40 52.5 59.7 88% 7.3 24.2 25% 6.0 22.2 33%

45 58.4 65.3 89% 7.7 24.9 25% 6.3 22.7 34%

50 63.6 70.6 90% 7.6 25.2 25% 6.3 22.5 34%

55 69.2 76.2 91% 7.5 24.9 25% 6.2 22.8 33%

60 74.9 81.9 91% 7.6 24.9 25% 6.3 23.3 33%

65 80.3 87.3 92% 7.8 24.8 25% 6.1 23.3 33%

70 85.4 92.3 93% 7.8 24.8 25% 6.2 23.1 34%

75 90.7 97.7 93% 7.5 24.9 25% 6.2 23.1 32%

80 96.2 103.3 93% 7.6 24.8 25% 6.3 23.3 33%

85 101.5 108.3 94% 7.7 24.9 26% 6.5 23.1 33%

90 106.8 113.8 94% 7.8 25.0 25% 6.4 23.1 34%

95 111.8 118.7 94% 7.6 25.2 25% 6.4 23.2 33%

100 117.0 123.8 94% 7.6 25.1 25% 6.3 23.0 33%

Table 8 shows the proportion of waiting times spent at the Par-3 holes to the total waiting times

of each group. Again, for the purpose of brevity, the table only shows waiting time proportions for

every 5th group for the base case only. Similarly as Table 7, because the proportions for all three

cases are either increases as the group number increases (for P3 holes) or converges (for P3WU

and SP3 holes), we can roughly interpolate the proportions for the groups not displayed in the

table.

With P3 holes, the proportion of the waiting times ranges from 61% (5th group) to 94% (100th

group). For the 20th group and beyond, we see that the waiting times at P3 holes account for more

than 80% of the total waiting times, and for the 50th group and beyond, we see that the waiting

times at P3 holes account for more than 90% of the total. Overall, Table 8 shows that the impact

of P3 holes is significant across all groups.

31

On the other hand, we see that the proportion of waiting times remain constant across all

groups for P3WU and SP3 holes. The proportion ranges from 25% to 28% for P3WU holes, and

from 27% to 34% for SP3 holes. In addition, we see that the proportion of waiting times is lower

at P3WU holes than at SP3 holes. The primary reason is that, although the waiting times of the

playing groups at other holes (e.g. P4, P5) are roughly the same for golf courses with P3WU holes

and those with SP3 holes, the waiting times at holes subsequent to P3WU holes tend to be higher

than those at holes subsequent to SP3 holes.

Table 9: Simulation estimates of the mean waiting times for a range of different playing groups(e.g. 5, 10, 15, ...) before starting play on each of the Par-3 holes with the three kinds of par-3holes: P3, P3WU and SP3. In all cases the tee interval is 8.50 minutes, which makes all threecases (P3, P3WU , and SP3) under-loaded.

base case - P3 base case - P3WU base case - SP3

group 1st 2nd 3rd 3-hole Sum 1st 2nd 3rd 3-hole Sum 1st 2nd 3rd 3-hole Sum

5 2.5 1.9 1.5 5.8 1.0 0.9 0.8 2.8 0.7 0.7 0.6 2.0

10 4.0 3.4 2.6 9.9 1.1 1.2 1.1 3.4 0.8 1.0 0.9 2.7

15 4.9 4.2 3.7 12.8 1.1 1.3 1.3 3.7 0.9 1.1 1.0 3.0

20 5.7 5.1 4.4 15.1 1.1 1.3 1.3 3.7 0.9 1.2 1.1 3.2

25 6.2 5.8 4.9 16.8 1.2 1.2 1.3 3.8 0.9 1.1 1.2 3.1

30 6.5 6.2 5.3 18.0 1.2 1.3 1.4 3.9 0.9 1.0 1.1 3.0

35 6.9 6.6 5.9 19.5 1.1 1.3 1.3 3.6 0.9 1.0 1.1 3.1

40 7.2 7.1 6.3 20.7 1.2 1.3 1.4 3.8 0.9 1.0 1.1 3.1

45 7.5 7.7 6.7 22.0 1.1 1.3 1.4 3.8 0.9 1.1 1.1 3.1

50 7.7 8.0 7.1 22.8 1.1 1.3 1.3 3.7 0.9 1.0 1.2 3.0

55 7.7 8.4 7.3 23.5 1.2 1.3 1.4 3.9 0.8 1.0 1.1 3.0

60 8.0 8.4 7.7 24.1 1.2 1.4 1.3 3.8 0.9 1.0 1.2 3.1

65 8.3 8.5 8.1 25.0 1.1 1.3 1.3 3.8 0.9 1.0 1.1 3.0

70 8.6 8.9 8.1 25.6 1.1 1.3 1.4 3.8 0.9 1.0 1.1 3.1

75 8.7 9.1 8.5 26.3 1.2 1.2 1.4 3.8 0.9 1.1 1.1 3.1

80 8.8 9.4 8.7 26.9 1.1 1.4 1.4 3.9 0.9 1.1 1.1 3.1

85 8.8 9.6 8.9 27.3 1.2 1.3 1.4 3.8 0.9 1.2 1.1 3.2

90 9.1 9.9 9.0 27.9 1.1 1.3 1.4 3.7 0.9 1.1 1.1 3.1

95 9.1 10.1 9.2 28.4 1.2 1.3 1.4 3.8 0.9 1.1 1.1 3.1

100 9.2 10.4 9.4 28.9 1.2 1.3 1.3 3.7 0.9 1.1 1.1 3.1

32

Similarly, we also show the results of our waiting time study with a larger tee time interval,

8.50, in Tables 9 and 10. Table 9 shows the waiting times of different playing groups at Par-3 holes

(same list of groups as in Table 7). The results shown below also reflect the waiting time study

results for the base case (“4 5 4” case).

When we compare Table 7 with Table 9, we notice that the waiting times of the playing group

decrease significantly—especially for P3 holes. For instance, we see that the respective waiting

times at P3 holes for groups 25, 50 and 100 decrease from 35.0, 63.6 and 117.0 minutes (in Table 7)

to 16.8, 22.8 and 28.9 minutes (in Table 7). These translate to roughly 51%, 64% and 77% decrease

in percentage points, respectively. On the other hand, we see that the waiting times at P3WU

holes for groups 25, 50 and 100 decrease only slightly, from 6.4, 7.6 and 7.6 minutes to 3.8, 3.7 and

3.7 minutes. Similarly, the waiting times at SP3 holes also undergo only a slight decrease. We

note that, however, the waiting times with P3 holes still remain substantially higher than waiting

times with P3WU or SP3 holes.

33

Table 10: Simulation estimates of the proportion (in %) of the waiting times at Par-3 holes (to thewaiting time of the entire 18-hole course) for a range of different playing groups (e.g. 5, 10, 15, ...)before starting play on each of the Par-3 holes with the three kinds of par-3 holes: P3, P3WU andSP3. In all cases the tee interval is 8.50 minutes, which makes all three cases (P3, P3WU , andSP3) under-loaded.

base case - P3 (%) base case - P3WU (%) base case - SP3 (%)

group 3-hole 18-hole % 3-hole 18-hole % 3-hole 18-hole %

Sum Sum Sum Sum Sum Sum

5 5.8 9.3 63% 2.8 9.2 30% 2.0 7.0 29%

10 9.9 14.3 69% 3.4 12.4 28% 2.7 9.1 30%

15 12.8 17.4 74% 3.7 13.5 27% 3.0 9.8 31%

20 15.1 19.7 77% 3.7 13.6 27% 3.2 10.2 31%

25 16.8 21.7 78% 3.8 13.7 27% 3.1 10.1 31%

30 18.0 22.8 79% 3.9 13.9 28% 3.0 9.9 30%

35 19.5 24.2 80% 3.6 13.5 27% 3.1 10.0 31%

40 20.7 25.6 81% 3.8 13.7 28% 3.1 10.1 31%

45 22.0 27.1 81% 3.8 13.9 28% 3.1 9.9 31%

50 22.8 27.8 82% 3.7 13.8 27% 3.0 10.1 30%

55 23.5 28.5 82% 3.9 13.9 28% 3.0 9.8 30%

60 24.1 29.0 83% 3.8 13.8 28% 3.1 10.0 31%

65 25.0 29.9 83% 3.8 13.8 27% 3.0 9.8 31%

70 25.6 30.6 84% 3.8 13.8 27% 3.1 9.8 32%

75 26.3 31.2 84% 3.8 14.0 27% 3.1 10.1 31%

80 26.9 31.8 85% 3.9 14.2 28% 3.1 10.1 31%

85 27.3 32.3 85% 3.8 14.0 27% 3.2 10.0 32%

90 27.9 32.8 85% 3.7 13.7 27% 3.1 10.0 31%

95 28.4 33.5 85% 3.8 13.7 28% 3.1 10.0 31%

100 28.9 34.0 85% 3.7 13.5 28% 3.1 10.3 31%

On the other hand, Tables 8 and 10 suggest that there is little to no difference between pro-

portion of waiting times. From Table 8, we see that the waiting time proportions for groups 25, 50

and 100 are 84%, 90% and 94%, respectively; from Table 10, we see that the proportions are 78%,

82% and 85%. For P3WU and SP3 holes, when the tee interval is 7.50, we pointed out earlier that

the proportion for P3WU holes ranges from 25 to 28%, and the proportion for SP3 holes ranges

from 27 to 34%. When the tee interval is 8.50, as is the case in Table 10, the proportion of waiting

times for P3WU ranges from 27 to 30%, and the proportion for SP3 holes ranges from 29 to 31%,

suggesting that there is no significance in proportion of waiting times in Tables 8 and 10.

Overall, comparing Tables 9 and 10 with their earlier analogs Tables 7 and 8, we se a decrease in

waiting times of the playing groups for all three types of Par-3 holes when the tee interval becomes

longer. The difference is especially stark for P3 holes (and becomes more noticeable as the group

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Group / Par # 4 5 4 4 3 4 4 5 4 4 3 4 4 5 4 4 3 4 Sum

005 0.3 0.5 0.7 0.6 3.7 0.2 0.3 0.3 0.5 0.4 2.2 0.1 0.2 0.2 0.3 0.3 1.7 0.1 12.4

010 0.3 0.6 1.0 0.9 8.1 0.2 0.3 0.4 0.5 0.4 4.1 0.2 0.3 0.3 0.4 0.4 3.2 0.1 21.7

015 0.3 0.6 0.9 0.8 12.6 0.2 0.3 0.4 0.6 0.5 5.6 0.2 0.3 0.3 0.5 0.4 4.2 0.1 28.9

020 0.3 0.6 1.0 0.9 17.1 0.2 0.3 0.4 0.6 0.5 6.8 0.2 0.3 0.4 0.5 0.5 5.2 0.2 36.0

025 0.3 0.6 1.0 0.9 21.5 0.2 0.3 0.4 0.6 0.5 7.7 0.2 0.3 0.4 0.5 0.5 5.8 0.1 41.9

030 0.4 0.6 1.0 0.9 25.9 0.2 0.3 0.4 0.6 0.5 8.7 0.2 0.3 0.4 0.5 0.5 6.5 0.2 47.9

035 0.3 0.6 1.0 1.0 30.3 0.2 0.3 0.3 0.6 0.5 9.4 0.2 0.3 0.4 0.5 0.5 7.2 0.2 53.9

040 0.3 0.6 1.0 1.0 34.8 0.2 0.4 0.4 0.6 0.5 10.1 0.2 0.3 0.4 0.5 0.5 7.7 0.2 59.7

045 0.3 0.6 1.0 0.9 39.2 0.2 0.4 0.4 0.6 0.5 10.8 0.1 0.3 0.4 0.5 0.5 8.4 0.2 65.3

050 0.3 0.6 1.0 1.0 43.6 0.2 0.4 0.4 0.6 0.5 11.4 0.2 0.3 0.4 0.6 0.5 8.6 0.1 70.6

055 0.3 0.6 1.0 0.9 48.0 0.2 0.3 0.4 0.6 0.6 12.1 0.2 0.4 0.4 0.5 0.5 9.1 0.2 76.2

060 0.3 0.6 0.9 0.9 52.6 0.2 0.3 0.4 0.6 0.5 12.8 0.2 0.4 0.4 0.6 0.5 9.5 0.2 81.9

065 0.3 0.6 1.0 0.9 57.0 0.2 0.3 0.4 0.6 0.5 13.3 0.2 0.3 0.4 0.5 0.5 10.0 0.2 87.3

070 0.3 0.6 1.0 1.0 61.4 0.2 0.3 0.4 0.6 0.5 13.9 0.2 0.3 0.4 0.6 0.4 10.0 0.2 92.3

075 0.3 0.6 0.9 1.0 66.0 0.2 0.3 0.4 0.6 0.5 14.3 0.2 0.3 0.4 0.6 0.5 10.4 0.2 97.7

080 0.3 0.6 1.0 1.0 70.7 0.2 0.4 0.4 0.5 0.5 14.6 0.2 0.3 0.4 0.6 0.5 10.9 0.2 103.3

085 0.3 0.6 1.0 1.0 75.1 0.2 0.3 0.4 0.5 0.5 15.2 0.2 0.3 0.4 0.5 0.5 11.2 0.2 108.3

090 0.3 0.6 1.0 0.9 79.7 0.2 0.4 0.4 0.6 0.5 15.6 0.2 0.3 0.4 0.5 0.5 11.5 0.2 113.8

095 0.3 0.6 0.9 0.9 84.1 0.2 0.4 0.4 0.6 0.5 15.9 0.2 0.3 0.4 0.6 0.5 11.8 0.2 118.7

100 0.3 0.6 1.0 0.9 88.5 0.2 0.3 0.4 0.6 0.5 16.4 0.2 0.3 0.4 0.6 0.5 12.1 0.2 123.8

Figure 5: Above figure shows waiting times of different playing groups at different holes. P3 holesare used, with the base case chosen as the course design for this particular case. The redder thecolors are, the higher the waiting times are.

number increases), but not so much for P3WU and SP3 holes. On the other hand, we see that

there is little to no difference in proportion of waiting times for all three types of Par-3 holes,

implying that Par-3 holes, regardless of the type, still remain as a bottleneck.

Finally, Figure 5 presents a visual representation of the waiting times of different playing groups,

with unscaled P3 holes on the base case. Figure 5 suggests that a substantial proportion of the

total waiting times is concentrated on the three Par-3 holes—and more specifically on the first of

the three Par-3 holes. Furthermore, Figure 5 also suggests that, even though the three par-3 holes

shown in the figure all have equal capacity, the waiting times gradually thin out in the later par-3

holes. This kind of behavior is exactly what is expected with 3 bottleneck holes in standard queues

in series.

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5 Extension of the Simulation Optimization Study

In this section, we extend the results from Table 5 in §4 of the main paper, Simulation Optimization

of the Tee Schedule.

Table 11: the maximum number of groups that can play each day as a function of (i) the tee intervalon the first hole, τ (ii) the hole order, and (ii) the type of par-3 hole used. Two hole orders areused: the “base case”, and the “par-3 first” case. The optimal tee intervals are in bold, while thecritical tee interval τ∗ from Whitt (2015) for that kind of par-3 hole is shown at the bottom.

throughput for base case throughput for “3 3 3” case

tee interval P3 P3WU SP3 P3 P3WU SP3

5.00 11 10 16 11 10 165.50 13 12 19 13 12 206.00 15 15 26 15 15 266.50 18 21 41 19 21 407.00 23 42 87 23 41 87

7.10 24 63 87 25 62 87

7.20 25 84 86 27 84 87

7.30 27 84 86 29 84 867.40 30 83 85 31 83 867.50 33 82 85 33 82 857.60 34 82 84 36 82 847.70 37 81 83 40 81 837.80 41 80 82 45 80 827.90 46 79 81 51 79 818.00 53 78 81 59 78 81

8.10 62 78 80 71 78 808.20 74 77 79 74 77 798.30 74 76 78 74 76 788.40 74 75 77 74 76 77

8.50 74 75 76 74 75 769.00 71 71 72 71 71 729.50 68 68 69 68 68 69

τ∗ 8.39 6.53 6.79 8.39 6.53 6.79

5.1 Comparing Throughput Levels

Here, Table 11 is the duplicate of Table 5 in §4.1 of the main paper, and Table 12 is an extension of

the results in Table 5 in §4.1 of the main paper. Similarly as Table 11, Table 12 in this companion

shows maximum number of groups (i.e. throughput) that can play each day as a function of (i) the

tee interval on the first hole, τ , (ii) the hole order and (iii) the type of Par-3 hole used. In Table

11, we show throughput levels for two course designs, namely “base case” and “par-3 first case”.

36

Table 12: the maximum number of groups that can play each day as a function of (i) the teeinterval on the first hole, τ (ii) the hole order, and (ii) the type of Par-3 hole used. Three holeorders are used: “5 5 5 case”, “3 3 3 mid” case, and “3 3 3 end” case. The optimal tee intervalsare in bold, while the critical tee interval τ∗ from Whitt (2015) for that kind of par-3 hole is shownat the bottom.

throughput for “5 5 5” case “3 3 3”-mid case “3 3 3”-end case

tee interval P3 P3WU SP3 P3 P3WU SP3 P3 P3WU SP3

5.00 12 10 16 12 10 16 11 11 165.50 13 12 20 13 12 19 13 13 196.00 15 15 26 16 15 26 15 16 266.50 19 21 39 19 21 39 18 22 397.00 24 42 87 24 42 87 23 49 87

7.10 25 63 87 25 66 87 24 85 87

7.20 26 84 86 27 84 86 26 85 867.30 28 84 86 29 84 86 28 84 867.40 30 83 85 31 83 85 30 83 857.50 33 82 84 33 82 85 32 83 857.60 35 82 84 36 82 84 35 82 847.70 39 81 83 40 81 83 38 81 837.80 43 80 82 44 80 82 43 80 827.90 49 79 81 50 79 81 48 79 818.00 56 78 80 58 78 81 57 78 81

8.10 67 78 80 68 78 80 65 78 808.20 74 77 79 74 77 79 74 77 798.30 74 76 78 74 76 78 74 76 788.40 74 75 77 74 75 77 74 76 77

8.50 74 75 76 74 75 76 74 75 769.00 71 71 72 71 71 72 71 71 729.50 68 68 69 68 67 69 68 68 69

τ∗ 8.39 6.53 6.79 8.39 6.53 6.79 8.39 6.53 6.79

In Table 12, we show the throughput results for three additional course designs: “5 5 5 case”, “3 3

3 mid case” and “3 3 3 end case”.

The hole sequence for these three designs, as introduced in Fig 3 of the main paper, are:

“5 5 5” case: 555 − 343− 434 − 444− 444 − 444

“3 3 3 mid” case: 444− 454 − 444− 454 − 444− 454 (concentrated in the middle)

“3 3 3 end” case: 454− 444 − 454− 444 − 454− 333 (concentrated in the end)

We include Table 12 in this companion primarily to confirm that our key findings in the main

paper hold true for other golf course designs. In §4.2 of the main paper, we presented five insights

gathered from Table 11. We now describe how each of the five findings hold true for the extension

37

of our findings in Table 12.

The first (two-fold) conclusion that we drew in §4.2 of the main paper was that (i) the optimal

tee interval for the unbalanced course with three P3 holes – and for the balanced course with three

SP3 holes – is approximately equal to the limiting cycle (i.e. the course is critically loaded), (ii) the

optimal tee interval for the balanced course with P3WU holes is slightly greater than the limiting

cycle (i.e. the course is under-loaded). As noted in Table 11, the limiting cycle times for P3,

P3WU and SP3 holes are 8.39, 6.53 and 6.79, respectively. Here, in Table 12, we observe that the

optimal tee intervals for the “5 5 5 case”, “3 3 3 mid case” and “3 3 3 end case” are the same as the

optimal tee intervals in Table 11, in which the optimal tee intervals are approximately [8.20, 8.50],

[7.20, 7.30] and [7.00, 7.10] for the P3, P3WU and SP3 holes, respectively. We confirm that the

optimal tee for P3WU holes remains noticeably larger than its limiting cycle regardless of the hole

order, and we consequently confirm that the wave-up rule contributes to the deviation of the tee

interval from critical load.

Our second conclusion was that the wave-up rule indeed contributes to a higher level of through-

put. From Tables 11 and 12, we see that the respective maximum throughput levels are 74, 84 and

87 with P3, P3WU and SP3 holes in all of the five course designs (with the exception of “3 3 3

end case”, which shows that the max throughput with P3WU holes is 85). Thus, we confirm that

the advantage of the wave-up rule exists regardless of the hole sequence. Subsequently, we confirm

our third conclusion, in which we asserted that the SP3 holes allow 3 more groups to play than

P3WU holes. Additionally, because the maximum throughput level and its optimal tee interval for

P3, P3WU and SP3 holes are essentially the same across all five course designs in Tables 11 and

12, we confirm our fourth conclusion that the hole sequence is relatively unimportant.

Finally, we confirm our fifth and final conclusion: there is a greater penalty from having too

small of an interval (i.e. over-loaded) than having an interval that is too large (i.e. under-loaded).

We substantiate our assertion by noting the noticeable similarity among the throughput trends

across the three course designs in Table 12. More specifically, we see that when the tee is 5.00,

the throughput levels are approximately 12, 10 and 16 for P3, P3WU and SP3 holes. Then, the

throughput levels increase at a steep rate as the tee interval increases until they reach the maximum,

but the throughput levels decrease at a much slower pace after reaching their peak. Although the

throughput levels at each tee interval for P3, P3WU and SP3 holes are not exactly the same, we

see that the trends for all Par-3 hole types and course designs are essentially tantamount to each

other.

38

References

d

Choi M, Fu Q, Whitt W (2017) Using simulation to manage the pace of play in golf, the main paper.

Whitt W (2015) The maximum throughput on a golf course. Production and Operations Management

24(5):685–703.

39