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Using client-variance information to improve dynamic appointment scheduling performance
Omega 28 (2000) 293±302
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Using client-variance information to improve dynamic appointment scheduling performance Thomas R. Rohleder a,*, Kenneth J. Klassen b a The University of Calgary, Faculty of Management, 2500 University Drive N.W., Calgary, AB, Canada T2N 1N4 Department of Management Science, College of Business Administration and Economics, California State University, Northridge, 18111 Nordho Street, Northridge, CA 91330-8378, USA
b
Received 1 August 1998; accepted 1 June 1999
Abstract Clients of services expect short waiting times and servers desire short periods of non-productive time. One of the areas where this is most important is appointment scheduling systems. Recent research has indicated that using information about clients' service time variability can simultaneously reduce waiting times and server idle time. In this study, a more realistic, dynamic appointment-scheduling environment is developed and used to analyze several scheduling rules. Additional complexities considered in this study include: continuously distributed service-time variances, special client appointment requests, and appointment-scheduler uncertainty. Results show that rules using client-variance information are still best at minimizing waiting time and idle time with the additional complexities. In fact, these rules perform best when client variance is large. However, on measures related to clients requesting speci®c appointment slots the results are not as clear cut. A key factor for these measures is the distribution of the desired slots. When the desired slots are near the end of the appointment scheduling period, traditional rules like ®rst-call-®rst-appointment perform better on client appointment request measures. 7 2000 Elsevier Science Ltd. All rights reserved. Keywords: Appointment scheduling; Services management; Simulation
1. Introduction Service operations are encountering pressure to simultaneously improve customer service and become more ecient. In fact, the amount of time spent waiting for service has become one of the major concerns for consumers evaluating service operations [7,12].
* Corresponding author. Tel: +1-403-220-7159; fax: +1403-284-7902. E-mail address: [email protected] (Thomas R. Rohleder).
Even for operations like health care where service quality has many facets, waiting times are increasingly used by clients in selecting and staying with a service provider [5]. In a recent study of health-care clinic appointment scheduling, Klassen and Rohleder [8] suggest that by using variance information about clients it is possible to minimize a combined performance objective of server idle time and client waiting time. The Low Variance Beginning (LVBEG) scheduling rule performed the best of the ten rules tested. In that study it was reported that receptionists (the ``schedulers'') did have knowledge available to them concerning client service-
0305-0483/00/$ - see front matter 7 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 3 0 5 - 0 4 8 3 ( 9 9 ) 0 0 0 4 0 - 7
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time variability. Speci®cally, the schedulers knew that some appointment procedures (e.g. booster shots) had low variability while other appointment times would be very dicult to predict (e.g. a caller provides a list of complicated symptoms). However, the study made the simplifying assumption that service time variances came from two distinct distributions; high variance and low variance. This assumption may be practical from an implementation standpoint since schedulers are likely to only be able to make coarse distinctions among clients. However, it is not likely that client service-time variances would come from two distinct distributions. Rather, it is more realistic to assume variances would be continuously distributed in at least an approximately-normally distributed fashion, similarly to people's heights and I.Q.s. It is important to note that the assumption that client variances were from two discrete distributions clearly created a favourable bias for the LVBEG rule. Thus, comparing this rule with traditional rules with a continuous variance distribution may lead to very dierent results. Therefore, this study will test whether the LVBEG rule is still eective with more realistic assumptions and when considering alternative performance measures. Realism is added by modelling continuously distributed service-time variances, special client appointment requests, and appointment-scheduler uncertainty. Regarding this last issue, the Klassen and Rohleder study assumed that schedulers could perfectly specify patients as being high or low variance. This study examines how LVBEG (putting clients with expected low-variance appointments, ®rst) performs when client variances are imperfectly classi®ed. This will address more realistically the comments from schedulers in [8] that while they could dierentiate clients by service time variability, they recognized it was not a perfect process. Again, when mistakes are made in classi®cation LVBEG may not perform nearly as well, or even be the best choice of scheduling rule. In addition, this research tests a brand new rule, which is a hybrid combination of LVBEG and BAILEY2 (putting two clients into the the ®rst appointment slot) which is one of the classic rules in the research and often used in practice [2]. The intent is to ®nd a new rule that combines the high server utilization of BAILEY2 and the low patient waiting time of LVBEG.
2. Model and experiment description 2.1. General environment The scheduling environment in this study is similar to that of Klassen and Rohleder [8] which was based primarily on typical medical practitioner situations. A
3 1/2 hour session is used as representative of a typical appointment scheduling period. Appointments occur in ten minute intervals, resulting in 21 appointment slots. Appointment sessions are considered independent of each other and each session is considered an observation for performance evaluation. Two appointment slots (11 and 16) were left open to accommodate urgent patients who call in with a Poisson distribution with a rate of two per scheduling session. Open slot planning was studied previously [8] and is not a decision of interest in this study. It is included primarily for realism purposes, and because it does have some eect on overall performance. Taking into consideration the open slots, 19 regular clients are scheduled into the remaining slots. As in previous studies, clients arrive punctually for their appointments, and waiting does not begin until the actual client appointment time. For realism purposes, a 5% no show rate is included. This is modelled by assuming each client has a 5% chance of not arriving. Interviews with schedulers previously [8], showed that most people did show up early or on time, and thus the 5% value was deemed reasonable. This value was also supported by the empirical ®ndings of Brahami and Worthington [3]. 2.2. Waiting/idle-time performance measures The primary performance measure used to compare scheduling rules is the combined ``cost'' of server idle time and client waiting time. Formally, this is: C
n X pWi oIi ,
1
i1
where: C = cost per scheduling session, Wi=waiting time of client i(dierence between actual service start time and scheduled appointment time), Ii=server idle time for client i(dierence between client i's service start time and client i ÿ 1's service completion); for i = 0, this is the starting time of the scheduling period, p=idle cost of the clients (set at 1.0 for this study) and o=idle cost of the server (set at 1.0 for this study). In addition, performance on the client and server waiting and idle time components of the overall objective are separately tracked (denoted as W and I, respectively). As further indicators of service waiting, the average maximum waiting time (MAX) and proportion of waits less than 10 min (P10) are calculated. Also, as a check on server idle-time and overtime, average session ending time (E ) is tracked. 2.3. Scheduling rules Five scheduling rules are tested in this study. The
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®rst is First-come-®rst-appointment which simply assigns appointment slots by the order in which the clients call. This rule is a good benchmark since it is commonly used in practice. Additionally, when there is no dierence in client variances, rules using client variance information are approximately equivalent to FCFA. Thus, if LVBEG does not perform well with a continuous variance distribution and scheduler uncertainty we would expect it to perform similarly to FCFA. This is true because FCFA schedules arbitrarily with respect to a client's service time variability. Poor classi®cation and the continuous variance distribution could cause LVBEG to make appointment assignments almost arbitrarily with respect to client variance. BAILEY2 [2], is identical to FCFA except the ®rst two clients to call in are scheduled for the ®rst appointment slot and the last slot is left open. BAILEY2 is included as a benchmark for the server idle time measure since it performed well on this dimension in studies by Ho and Lau and Klassen and Rohleder [6,8]. A similar rule, BAILEY4 [2] is conceptually the same, except the ®rst four clients are scheduled in the ®rst slot. This rule was tested by Klassen and Rohleder [8] and while it minimized server idle time, average client waiting time was too excessive to be considered for today's service oriented operations. OFFSET was developed in [6] and adjusts the assigned appointment times depending on the assigned appointment slots. For this study the following form is used: Ti
As ÿ k1 s K ÿ s for i and s 1, 2, . . . , K As k2 s s ÿ K for i and s K 1, K 2, . . . 21 2
where Ti is the scheduled appointment time for client i, As is the regular appointment time for slot s(e.g. A1=0.0, A2=10.0, . . . ,A21=200.0), and s is the standard deviation of the service time. As compared to FCFA, OFFSET moves the client appointment times earlier for the ®rst 12 patients, and later for the remainder. This rule is included due to its performance on the client waiting-time measure as reported in previous studies [6,8]. However, as noted by Klassen and Rohleder [8], using the original parameters for the early/delay breakpoints and the associated multipliers (K = 5, k1=0.15, and k2=0.30) suggested by Ho and Lau [6] resulted in excessively late end times, which would be unacceptable to service providers. Thus, for this study the parameters were re-evaluated to ®nd the values K = 12 and k1=k2=0.20 actually used in Eq. (2). A formal pre-study was performed to evaluate a wide range of possible values for K, k1, and, k2; the values chosen achieved the best balance in end times and C values.
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Low-Variance Beginning is the ®rst rule using client variance information. It schedules clients by putting those with the service times the scheduler believes to have low variance ®rst and those with high variance last. Operationally this is done by assigning low variance clients to the lowest available appointment slot and assigning high variance clients to the highest available appointment slot. The schedule is ®lled when the low and high variance service time clients ``meet''. Since the calls are at random, the number of low and high variance patients will ¯uctuate in each scheduling period. This rule was the best of those using client variance information in Klassen and Rohleder's study [8], and therefore was viewed as the best rule to test with more realistic service time and scheduler assumptions of this study. However, if LVBEG does not perform well with a continuous variance distribution and scheduler uncertainty we would expect it to perform similarly to FCFA. The ®nal rule tested is a combination of BAILEY2 and LVBEG. The rule follows the same pattern as LVBEG except two clients are scheduled into the ®rst appointment slot and none in the ®nal slot. Based on results from Klassen and Rohleder [8], the combined rule should reduce the waiting time of BAILEY2 and reduce the idle time of LVBEG. 2.4. Client variance The client service time mean is set at ten minutes for all clients in this study. This assumption is similar to that used in previous studies (see, for example [6,8]). Unlike production scheduling, most clinics assume a constant mean service time for patients. The clinics interviewed in Klassen and Rohleder [8] did vary scheduling for appointments such as complete physical exams due to their increased length compared to a standard appointment. This issue has not been thoroughly studied although testing showed that having some longer appointment slots reduced the number of total appointments [8]. Fewer slots results in better performance on client waiting time and server idle time. Further research on this issue is warranted. The key dierence in this study is the way variance is handled for each patient. We assume that service time variances [which we will denote as client variance dispersion (CVD)] are drawn from a lognormal distribution with a mean of 7.5 and two experimental standard deviation values of 3.75 and 7.5 min [giving coecient of variations (cv) of 0.5 and 1.0]. Once a client's service time standard deviation was randomly chosen, call it sc, this was used to generate the service time, using the mean of 10 min and the lognormal distribution [i.e. client service time=Lognormal (10, sc)]. Use of the lognormal distribution for service times
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was based on both theoretical and empirical grounds. Law and Kelton [9] state that the lognormal distribution is a good distribution for modelling service times made up of a series of tasks in sequences (as is typical in medical appointments). Klassen and Rohleder [8] found that empirically, the lognormal distribution was the best approximation of what was experienced by the schedulers at the clinics. They also found that a mean variance of the midpoint between the values of ®ve and ten was appropriate, based on the empirical interviews with family medical practice schedulers. These schedulers found that most clients service times were plus or minus ®ve minutes from the mean time and sometimes as large as 20 min larger than the mean. From this information and the variance values a lognormal distribution was roughly ®tted. 2.5. Scheduler uncertainty The key assumption in using a rule like LVBEG is that clients can be strati®ed in some manner by the variability in their service times. This issue was addressed empirically by Klassen and Rohleder [8], where the medical receptionists felt they did have information that could be used to classify patients by service-time variability. Classi®cation can be aided by knowledge of the individual characteristics of the client, the nature of the visit, and other relevant factors. For the LVBEG rule, classi®cation requires deciding at which standard deviation value the cuto between low and high variance patients should occur. For simplicity, we use the median value. So, if m and s are the mean and standard deviation of the lognormally-distributed standard deviation values, the median is dep rived as m2 = s2 m2 : Therefore, the cuto values for the standard deviation values of 3.75 and 7.5 are 6.71 and 5.30 (the medians with a mean of 7.5). Instead of the median, the mean could be used as the cuto point, however, in a pre-test this approach led to slightly worse performance than the median. The result of using the mean was that more clients were classi®ed as low variance than with the median. However, the dierence in performance was not signi®cant and, therefore, the median was used in this study. To handle the scheduler uncertainty, we thought it was reasonable to assume clients that had true variance values farther from the population median would be easier to classify than those with variances close to the median. To model this we use a normal distribution with a mean of 0 and standard deviations of 5 and 10. The probability is then calculated as: jx i ÿ nv j PCCi F 3 su
where PCCi is the probability of correct classi®cation for client i, xi is the client's true variance, nv is the median of the variance distribution, su is the standard deviation of the uncertainty distribution, and F is the unit normal probability function. For standard deviation values of 5 and 10 the above relationship creates functions as plotted in Fig. 1. As Fig. 1 shows, as the uncertainty standard deviation increases, the probability of correct classi®cation is lower for any dierence value of a client's standard deviation and the median standard deviation for the population. Thus, classi®cation is less likely to be correct with higher standard deviation values. The ®gure also shows that the values of ®ve and ten for the uncertainty standard deviation provide a good contrast of ``easy'' and ``hard'' classi®cation levels. Clearly, with su=5 the schedulers make the correct selection with a signi®cantly higher probability for most clients except when the dierence between the client's actual service time variance and the median are very close. In practice, neither a speci®c client's variance (xi) nor the standard deviation of the uncertainty distribution (su) would be known. As such they are not, generally, of practical value to the scheduler. Rather, the dierent values used represent experimental conditions to test the performance of the LVBEG scheduling rule for this study. Fig. 2 shows the uncertainty situation graphically. The random variable yi represents the uncertainty ``noise'' faced by the scheduler. Note that as su increases, the probability of incorrectly classifying a client (high variance in the graph) would go up since the hatched area would become larger. Thus, su represents the factors that makes the scheduler more or less capable of correctly classifying clients. Such factors could include scheduler experience, the nature of the service, general client knowledge in the drawing population, etc.
Fig. 1. Classi®cation uncertainty functions.
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2.6. Client preferences In previous studies of appointment scheduling, it was assumed that schedulers could put clients in whatever time slots schedulers desired. However, in reality schedulers almost always have to deal with clients who ask for speci®c appointment times. (Note: These will be termed as special requests for this study.) In informal interviews, appointment schedulers reported that about 25% of clients ask for speci®c appointment times, or at least, a speci®c range of appointment times. For most scheduling rules this does not aect performance on waiting and idle time since clients are considered homogenous for scheduling purposes. However, rules using client-variance information (LVBEG, B2LVBEG) may be aected because special requests will limit the ability to slot clients as desired. Certainly, a low-variance client may ask for an appointment time late in the scheduling period. Such occurrences will compromise the eectiveness of these rules. Since dierent services will have varying numbers of special requests, this study will consider three levels of special requests (SR); 10%, 25%, and 40% of clients. This is implemented in the model by giving the caller the SR percentage chance of requiring a speci®c time slot. Since the process is random, the exact proportion of SR callers on any day will vary, but the average will be correct for a large number of days sampled. Given that special request callers usually have some ¯exibility, appointment slots plus or minus two from the requested one will be checked for availability if the desired slot is already taken. Related to the percentage of SR callers is the distribution of desired slots (DDS). Two distributions will be considered in this analysis: uniform (UNIF) and end of period (EOP). For the UNIF approach, when SR callers ask for a slot, it may be any of the 21 slots (times) with equal probability. The EOP distribution has callers request any of the last ®ve slots with equal probability. Of course, many other special request distributions could be considered, however, based on
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informal interviews with appointment schedulers the EOP approach was cited as the most common, and UNIF provides a good contrast for comparison purposes. With the addition of the experimental factors related to special requests, we decided to track several performance measures related to this concept. These measures are: the proportion of special request clients receiving the speci®c slot requested (PSR), and the proportion of special request clients not receiving any slot (PSN). These two measures represent opposite extremes in client satisfaction (PSR=happy, PSN=unhappy). In addition, by taking 1.0ÿ(PSN+PSR), the proportion of clients receiving one of the plus or minus two slots from their requested slot can be calculated. While this measure will not be speci®cally considered in the analysis, it does represent a measure of moderate client satisfaction. Note that any special request client not receiving their requested slot or a slot nearby was put back into the calling pool to ensure the number of appointment requests remained constant. 2.7. Computer model A computer simulation of this model was developed in the Arena programming environment [1]. Performance for 100 scheduling periods is averaged to represent a replication and 100 replications are run for each experimental treatment. Such an approach provides sucient samples to obtain the desired power and help assure normality of the output responses [4]. In addition, common random numbers for each replication are used to reduce variation and provide greater precision in the analysis.
3. Results 3.1. Experiment summary
Fig. 2. Graphical view of scheduler uncertainty.
Table 1 presents a summary of the experimental factors. Overall, there are ®ve scheduling rules tested at: two levels of client-variance dispersion, three levels of special request proportions, and two levels of special request distributions. In addition, LVBEG and B2LVBEG have the nested factor of scheduler uncertainty, with three levels: no uncertainty, moderate uncertainty (su=5.0), and high uncertainty (su=10.0). This creates a total of 108 treatments with 100 replications and 100 scheduling periods in each replication, creating 1,080,000 total scheduling sessions, or observations (10,800 replications in the data set).
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Table 1 Summary of experimental factors Factor (# of levels)
Description of levels
Scheduling rules (5) Client variance dispersion (2) Proportion of special requests (3) Distribution of special requests (2) Scheduler uncertainty (3) (Note: only aects rules LVBEG and B2LVBEG)
FCFA, BAILEY2, OFFSET, LVBEG, B2LVBEG cv=0.50 and cv=1.0 0.10, 0.25, and 0.40 Uniform (UNIF) and End of Period (EOP) None, su=5.0, and su=10.0
3.2. Analysis of main objective function (C) Table 2 is the source of variation summary from an analysis of variance with C[the mean of the objective function in Eq. (1)] as the dependent variable (all statistical analysis was performed using the SAS system [11]). Note that non-signi®cant interactions (a > 0.05) are excluded from the table. Also, for this analysis, the results for all levels of scheduler uncertainty (su) are pooled together for LVBEG and B2LVBEG scheduling rules. Analysis of the scheduler uncertainty factor will be separately discussed later. From Table 2, it is clear that the scheduling rule factor explains the majority of the variation and all main eects are signi®cant, however, there are several signi®cant interactions that need to be considered. The three-way interaction between scheduling rule, special requests, and distribution of desired slots is signi®cant, and was therefore analyzed ®rst (all other four-way and three-ways were insigni®cant). A moderate interaction occurs for some rules with SR and DDS, most signi®cantly, BAILEY2. With this rule and DDS at the UNIF level, the means for all levels of SR
are essentially constant. This makes sense since C should not be in¯uenced too much by special requests that do not aect the ordering of clients. However, when there is a high percentage of special requests and the desired slots are at the end of the scheduling period, this actually improves the waiting-time performance of BAILEY2 because it may actually put a client in the very last slot. Without special requests, this slot would normally be left open, but by ``moving'' a client from a busier period to the last slot, overall performance is improved. While the above three-way interaction does occur, its eect on the results is easier to understand by looking at the two-way interaction between the scheduling rules and distribution of desired slots. Fig. 3 shows this interaction graphically. Note that for most rules wait and idle-time performance is not aected much by the preference distribution, however, for the reason mentioned previously BAILEY2 and B2LVBEG have better performance when the clients' preferred slots are near the end of the scheduling period. We will further analyze the eects of SR and DDS when performance related to client preferences is considered.
Table 2 ANOVA Results for C. Key: SCH RULE=Scheduling rule; CVD=Client Variance Dispersion, SR=Special Requests; DDS=Distribution of Desired Slots Source
d.f.
Sum of squares
Mean square
Model Error Corrected total
158 10641 10799 R2 0.478546 99 4 1 4 2 8 1 4 2 8
8514700.64 9278151.73 17792852.37 C.V. 13.36141 3160110.78 4622183.74 293909.52 200904.35 7435.26 40834.11 61670.49 81014.51 9391.66 24543.79
53890.51 871.92
REPLICATION SCH RULE CVD SCH RULECVD SR SCH RULESR DDS SCH RULEDDS SRDDS SCH RULESRDDS
Root MSE 29.5284 31920.31 1155545.93 293909.52 50226.09 3717.63 5104.26 61670.49 20253.63 4695.83 3067.97
F value 61.81
36.61 1325.28 337.08 57.60 4.26 5.85 70.73 23.23 5.39 3.52
Pr > F 0.0001 Overall Mean 220.997 0.0001 0.0001 0.0001 0.0001 0.0141 0.0001 0.0001 0.0001 0.0046 0.0005
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Table 3 Scheduling rule comparisons for C
Fig. 3. Interaction: scheduling rule preference distribution.
The two-way interaction between the scheduling rules and the proportion of special requests was previously explained with reference to the signi®cant three-way interaction. The improvement for C with BAILEY2 and B2LVBEG with DDS=EOP is, not surprisingly, most noticeable when the proportion of special requests is high. However, this eect is fairly weak and not nearly as powerful as the interaction between the scheduling rules and client variance dispersion. This interaction is shown in Fig. 4. This interaction plot shows that, in general, rule performance for average waiting plus idle time is improved when client variance is high. Although for most rules the improvement is insigni®cant, LVBEG shows dramatically better performance when client dispersion is high, with an improvement of nearly 20 for C. This is due to the rule's design that keeps certainty high early in the scheduling session and allowing the high variability clients that have high and low service times to trade o against each other later in the session. With high CVD, the advantages of LVBEG are enhanced. B2LVBEG also shows signi®cant improvement for the same reasons. On the other hand, the OFFSET rule's performance is signi®cantly worse with high client variance. Note that this rule uses the standard deviation of client service times when setting appointment times. With high client variance, this tends to increase waiting for clients early in the scheduling session and increase server idle time overall. Further work on parameterization could
Fig. 4. Interaction: rule client variance dispersion.
a=0.05 d.f.=10641 Number of Means 2 Critical Range 2.024 Means with the same letter are not Duncan Grouping Mean
MSE=871.9248 3 4 5 2.131 2.202 2.255 signi®cantly dierent N RULE
A B C D E
1200 1200 3600 1200 3600
253.166 238.492 231.805 219.371 194.178
BAILEY2 OFFSET B2LVBEG FCFA LVBEG
help to alleviate this problem, however, this work was not performed for this study because we believe this rule is too complex to implement in practice. As previously discussed, the scheduling rules explain the largest portion of variability in the ANOVA. Therefore, it is informative to look at the overall mean performance for this factor. Table 3 shows the results of a means comparison using the Duncan Multiple Range Test (see [10]). This comparison method was chosen due to its power in detecting true dierences in means. As the mean comparisons show, LVBEG is clearly the best rule for the C measure. However, as discussed in [6] it is informative to consider the implicit tradeo between client waiting time and server idle time ``costs''. This can be done graphically, by considering an ecient frontier. Fig. 5(a) and 5(b) show two of the frontiers for this study for diverse sets of experimental factors. In these ®gures, the ecient frontier is identi®ed by the points connected by the line. Thus, the
Fig. 5. Ecient frontier. (a) Low client variance, uncertainty variance = none. (b) High client variance, high scheduler uncertainty.
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Table 4 Scheduling rule performance: waiting-time measures Average maximum waiting timea
Percentage waiting <10 min
Value
Rule
Value
Rule
26.32 26.51 29.14 29.17 29.79
(1) (1) (2) (2) (3)
75.9% 73.2% 69.8% 66.4% 65.5%
(1) (2) (3) (4) (5)
a
LVBEG FCFA B2LVBEG BAILEY2 OFFSET
LVBEG FCFA OFFSET B2LVBEG BAILEY2
The numbers in parentheses represent the statistical groupings for mean comparisons.
frontier moves from the upper left to the lower right. Those on the upper left minimize idle time, and those on the lower right minimize waiting time. By considering the relative values of the average client wait time and server idle time, the implicit ratio of p/o can be inferred. For example, a service provider wishing to minimize server idle time could consider using BAILEY2 which has about a 12 to 1.3 ratio [in Fig. 4(a)] of client wait time to server idle time. Note that services should generally not consider rules o the frontier regardless of their p/o ratios since an ecient rule will always provide a lower total cost. Also, while the exact frontier will dier for every combination of environmental factor levels, the general form of the frontier is always the same as Fig. 5(a) and 4(b) demonstrate. 3.3. Waiting-time measures Performance for the average maximum waiting-time per scheduling period and the percentage of client waits less-than-or-equal-to ten is reported in Table 4. Only results for the scheduling rule factor are reported because for each performance measure this factor explained the vast majority of the model variation net of variability due to replication number (78.8% and 85.6% for MAX and P10, respectively). The results on these measures support LVBEG as the best rule from a client-waiting perspective. In particular, LVBEG does a good job of minimizing the number of clients that must wait more than 10 min. 3.4. Analysis of end times for scheduling rules As a contrast to the waiting time measures, the session end time is of greater concern to the service provider. An excessively large end time suggest the service provider will need to stay ``overtime'' or cut into needed breaks and lunch times. Table 5 shows the relative mean and maximum end times for each scheduling rule. Recall that the sessions are scheduled for 210 min.
From the table, several key ®ndings stand out. First, the rules that put two patients into the ®rst slot have signi®cantly lower end times. On the other hand, OFFSET, even with the adjusted parameters to achieve better end times is still signi®cantly worse than the other rules. 3.5. Analysis of special request measures The performance of the scheduling rules for the proportion of clients not receiving any slot (PSN) and the proportion of clients receiving their requests slot (PSR) is important because service providers and clients are interested not only in idle/waiting time, but also in a service system's capability to handle special requests. In analyzing the PSN measure, it was immediately apparent that there was an important two-way interaction of the scheduling rule factor with the client request distribution. Fig. 6 clearly shows that the rules using client variance information have the opposite reaction to the client preference distribution than the other rules. This result was not entirely unforseen since BAILEY2, FCFA, and OFFSET build their schedules from the beginning to the end, their PSN performance should be better with EOP. What is somewhat surprising is the magnitude of advantage LVBEG and B2LVBEG have for the UNIF preference distribution
Table 5 Mean and maximum end times for scheduling rules Meana
Maximum
Value
Rule
Value
Rule
217.59 218.99 224.57 224.82 239.79
BAILEY2 B2LVBEG FCFA LVBEG OFFSET
234.73 286.08 293.05 295.46 314.57
BAILEY2 B2LVBEG FCFA LVBEG OFFSET
a
All means were signi®cantly dierent at a=0.05.
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Table 6 Analysis of scheduler uncertainty for C measure Scheduler Uncertainty
LVBEG
B2LVBEG
Meana
None su=5 su=10
183.77 195.15 203.61
225.30 233.17 236.94
204.53 214.16 220.27
a
All means are signi®cantly dierent for a=0.05 using the Duncan Multiple Range Test. Fig. 6. Interaction of preference distribution rule for PSN.
and how much worse they are for EOP. The advantage of these rules with UNIF occurs because they build their schedules from both ends. This means that a special request call coming when the schedule is nearly full will more likely ®nd open slots around the requested time than for the other rules. The results for the PSR (the client ``happiness'') measure are, not surprisingly, similar to those for PSN. Fig. 7 shows the same interaction plot as for PSN. Note that for the PSR measure, higher is better. For the UNIF distribution of client requests all the rules are fairly close, however, when the requests are grouped near the end of the session, LVBEG and B2LVBEG perform signi®cantly worse and the other rules perform signi®cantly better. 3.6. Analysis of scheduler uncertainty In all the previous analyses, the levels of the scheduler-uncertainty factor were pooled for LVBEG and B2LVBEG. It is also useful, however, to consider how this factor aects the performance of these rules. For this analysis, a comparison of means was performed on just the su levels. Table 6 shows the results for each of the aected scheduling rules and the overall average for the C performance measure. A couple of interesting points about scheduler uncer-
tainty should be noted. First, the inclusion of this factor certainly aects the performance of the scheduling rules using client variance information. As scheduler uncertainty goes from zero to any positive value, the performance of these rules signi®cantly worsens. However, because of the nonlinear nature of the scheduler uncertainty function, su's aect is diminishing as it increases. For the mean eect of su going from none to the low level, performance for C is 4.7% worse, but going from the low to high level, performance is only 2.9% worse. So the results show that the LVBEG rule is not exceptionally sensitive to scheduler uncertainty, but nonetheless, schedulers should do what they can to improve their capability to classify clients to minimize the costs of waiting and server idleness. (This, in eect, lowers the value of su.) The eect of scheduler uncertainty was also explored for the other performance measures and with the other factors for possible interactions. However, other than the signi®cant result discussed above, scheduler uncertainty had little or no eect.
4. Conclusions In answer to one of the major questions of this study, the LVBEG scheduling rule that uses client-variance information is still eective in minimizing waiting and idle time when the realistic complexities of continuously distributed service-time variances, special client appointment requests, and appointment-scheduler uncertainty are added. The study also showed that each of these realistic factors did aect the results and so are useful when evaluating appointment scheduling systems. The remainder of this section will discuss practical implications and future study directions suggested by this research. 4.1. Practical implications
Fig. 7. Proportion of special requests satis®ed interaction: rule preference distribution.
From the interviews with appointment schedulers discussed in Klassen and Rohleder [8], it is apparent these schedulers do have information about client variance. Services that wish to simultaneously minimize
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client waiting time and server idle time should consider using this information with an approach like LVBEG for scheduling. It dominated performance for this measure and those related to waiting time. Even when schedulers are not perfect at assigning clients as high or low variance, LVBEG was still best. In fact, even with the highest level of scheduler uncertainty, LVBEG was still about 8% better than the next best rule for the C measure. Also, the dispersion of client variances does aect results for LVBEG Ð the greater the dispersion, the better the rule performs. This suggests that services that have a high degree of variability with their clients should be even more motivated to use client-variance information when scheduling. The results when considering the special-request performance is not as clear cut. When requests are spread out across the scheduling period, LVBEG is still a good choice for services. However, when requests are concentrated near the end of the scheduling period, LVBEG and B2LVBEG perform poorly in meeting these requests, at least compared to other rules that have a beginning-to-end scheduling pattern. The end of period distribution was analyzed based on informal interviews that suggested it is common. It should be noted that if the special requests tended to group near the beginning of scheduling sessions the rules such as FCFA, OFFSET, and BAILEY2 would not perform well on special request measures (likely worse than LVBEG or B2LVBEG). Also, LVBEG and B2LVBEG will likely fair well if the distribution of special requests tend to occur near the middle of the scheduling session. Obviously, services will have to pay attention to the distribution of special requests to determine what approach is sensible. 4.2. Future research Certainly, many research issues on this topic remain open. While this study looked at individual client variances, interviews by Klassen and Rohleder [8] suggest that mean classi®cation may also be possible. The clinics studied already adjusted appointment slot length to account for procedures they knew were likely to exceed the standard slot. In addition, dierent uncertainty levels and functions should be considered. Finally, all previous appointment-scheduling studies considered scheduling periods to be independent. In many (most?) appointment-oriented services clients are scheduled over a horizon that may span several days or even weeks. Certainly, this ``rolling-horizon'' en-
vironment will have implications for scheduling. Speci®cally, the eects of scheduling special requests into future time periods, the use of overtime, and controlling the ``exactness'' of adherence to rules like LVBEG can be tested in this even-more-realistic environment.
Acknowledgements The authors would like to express their appreciation for the helpful comments of three anonymous reviewers and Dr Edward A. Silver. In addition, this work was partially supported by the Natural Sciences and Engineering Research Council of Canada under grant number 138047.
References [1] Arena Version 3.01, ``Systems Modeling'' 1997. Sewickly, PA. [2] Bailey NT. A study of queues and appointment systems in hospital outpatient departments, with special reference to waiting times. Journal of the Royal Statistical Society 1952;A14:185±99. [3] Brahami M, Worthington DJ. Queueing models for outpatient appointment systems Ð a case study. Journal of the Operational Research Society 1991;5:91±102. [4] Fishman GS. Principles of discrete event simulation. New York: Wiley, 1978. [5] Gopalakrishna P, Mummalaneni V. In¯uencing satisfaction for dental services. Journal of Health Care Marketing 1993;13:16±22. [6] Ho C, Lau H. Minimizing total cost in scheduling outpatient appointments. Management Science 1992;38:1750± 64. [7] Katz K, Larson B, Larson R. Prescription for the waiting in line blues: entertain, enlighten and engage. Sloan Management Review 1991;1991:44±53. [8] Klassen KJ, Rohleder TR. Scheduling outpatient appointments in a dynamic environment. Journal of Operations Management 1996;14:83±101. [9] Law AM, Kelton WD. Simulation modelling and analysis, 2nd ed. New York: McGraw-Hill, 1991. [10] Montgomery DC. Design and analysis of experiments. New York: Wiley, 1991. [11] SAS Institute. SAS/STAT user's guide. Cary, NC: SAS Institute, 1988. [12] Taylor S. Waiting for service: the relationship between delays and evaluations of service. Journal of Marketing 1994;58:56±69.
www.elsevier.com/locate/orms
Using client-variance information to improve dynamic appointment scheduling performance Thomas R. Rohleder a,*, Kenneth J. Klassen b a The University of Calgary, Faculty of Management, 2500 University Drive N.W., Calgary, AB, Canada T2N 1N4 Department of Management Science, College of Business Administration and Economics, California State University, Northridge, 18111 Nordho Street, Northridge, CA 91330-8378, USA
b
Received 1 August 1998; accepted 1 June 1999
Abstract Clients of services expect short waiting times and servers desire short periods of non-productive time. One of the areas where this is most important is appointment scheduling systems. Recent research has indicated that using information about clients' service time variability can simultaneously reduce waiting times and server idle time. In this study, a more realistic, dynamic appointment-scheduling environment is developed and used to analyze several scheduling rules. Additional complexities considered in this study include: continuously distributed service-time variances, special client appointment requests, and appointment-scheduler uncertainty. Results show that rules using client-variance information are still best at minimizing waiting time and idle time with the additional complexities. In fact, these rules perform best when client variance is large. However, on measures related to clients requesting speci®c appointment slots the results are not as clear cut. A key factor for these measures is the distribution of the desired slots. When the desired slots are near the end of the appointment scheduling period, traditional rules like ®rst-call-®rst-appointment perform better on client appointment request measures. 7 2000 Elsevier Science Ltd. All rights reserved. Keywords: Appointment scheduling; Services management; Simulation
1. Introduction Service operations are encountering pressure to simultaneously improve customer service and become more ecient. In fact, the amount of time spent waiting for service has become one of the major concerns for consumers evaluating service operations [7,12].
* Corresponding author. Tel: +1-403-220-7159; fax: +1403-284-7902. E-mail address: [email protected] (Thomas R. Rohleder).
Even for operations like health care where service quality has many facets, waiting times are increasingly used by clients in selecting and staying with a service provider [5]. In a recent study of health-care clinic appointment scheduling, Klassen and Rohleder [8] suggest that by using variance information about clients it is possible to minimize a combined performance objective of server idle time and client waiting time. The Low Variance Beginning (LVBEG) scheduling rule performed the best of the ten rules tested. In that study it was reported that receptionists (the ``schedulers'') did have knowledge available to them concerning client service-
0305-0483/00/$ - see front matter 7 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 3 0 5 - 0 4 8 3 ( 9 9 ) 0 0 0 4 0 - 7
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time variability. Speci®cally, the schedulers knew that some appointment procedures (e.g. booster shots) had low variability while other appointment times would be very dicult to predict (e.g. a caller provides a list of complicated symptoms). However, the study made the simplifying assumption that service time variances came from two distinct distributions; high variance and low variance. This assumption may be practical from an implementation standpoint since schedulers are likely to only be able to make coarse distinctions among clients. However, it is not likely that client service-time variances would come from two distinct distributions. Rather, it is more realistic to assume variances would be continuously distributed in at least an approximately-normally distributed fashion, similarly to people's heights and I.Q.s. It is important to note that the assumption that client variances were from two discrete distributions clearly created a favourable bias for the LVBEG rule. Thus, comparing this rule with traditional rules with a continuous variance distribution may lead to very dierent results. Therefore, this study will test whether the LVBEG rule is still eective with more realistic assumptions and when considering alternative performance measures. Realism is added by modelling continuously distributed service-time variances, special client appointment requests, and appointment-scheduler uncertainty. Regarding this last issue, the Klassen and Rohleder study assumed that schedulers could perfectly specify patients as being high or low variance. This study examines how LVBEG (putting clients with expected low-variance appointments, ®rst) performs when client variances are imperfectly classi®ed. This will address more realistically the comments from schedulers in [8] that while they could dierentiate clients by service time variability, they recognized it was not a perfect process. Again, when mistakes are made in classi®cation LVBEG may not perform nearly as well, or even be the best choice of scheduling rule. In addition, this research tests a brand new rule, which is a hybrid combination of LVBEG and BAILEY2 (putting two clients into the the ®rst appointment slot) which is one of the classic rules in the research and often used in practice [2]. The intent is to ®nd a new rule that combines the high server utilization of BAILEY2 and the low patient waiting time of LVBEG.
2. Model and experiment description 2.1. General environment The scheduling environment in this study is similar to that of Klassen and Rohleder [8] which was based primarily on typical medical practitioner situations. A
3 1/2 hour session is used as representative of a typical appointment scheduling period. Appointments occur in ten minute intervals, resulting in 21 appointment slots. Appointment sessions are considered independent of each other and each session is considered an observation for performance evaluation. Two appointment slots (11 and 16) were left open to accommodate urgent patients who call in with a Poisson distribution with a rate of two per scheduling session. Open slot planning was studied previously [8] and is not a decision of interest in this study. It is included primarily for realism purposes, and because it does have some eect on overall performance. Taking into consideration the open slots, 19 regular clients are scheduled into the remaining slots. As in previous studies, clients arrive punctually for their appointments, and waiting does not begin until the actual client appointment time. For realism purposes, a 5% no show rate is included. This is modelled by assuming each client has a 5% chance of not arriving. Interviews with schedulers previously [8], showed that most people did show up early or on time, and thus the 5% value was deemed reasonable. This value was also supported by the empirical ®ndings of Brahami and Worthington [3]. 2.2. Waiting/idle-time performance measures The primary performance measure used to compare scheduling rules is the combined ``cost'' of server idle time and client waiting time. Formally, this is: C
n X pWi oIi ,
1
i1
where: C = cost per scheduling session, Wi=waiting time of client i(dierence between actual service start time and scheduled appointment time), Ii=server idle time for client i(dierence between client i's service start time and client i ÿ 1's service completion); for i = 0, this is the starting time of the scheduling period, p=idle cost of the clients (set at 1.0 for this study) and o=idle cost of the server (set at 1.0 for this study). In addition, performance on the client and server waiting and idle time components of the overall objective are separately tracked (denoted as W and I, respectively). As further indicators of service waiting, the average maximum waiting time (MAX) and proportion of waits less than 10 min (P10) are calculated. Also, as a check on server idle-time and overtime, average session ending time (E ) is tracked. 2.3. Scheduling rules Five scheduling rules are tested in this study. The
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®rst is First-come-®rst-appointment which simply assigns appointment slots by the order in which the clients call. This rule is a good benchmark since it is commonly used in practice. Additionally, when there is no dierence in client variances, rules using client variance information are approximately equivalent to FCFA. Thus, if LVBEG does not perform well with a continuous variance distribution and scheduler uncertainty we would expect it to perform similarly to FCFA. This is true because FCFA schedules arbitrarily with respect to a client's service time variability. Poor classi®cation and the continuous variance distribution could cause LVBEG to make appointment assignments almost arbitrarily with respect to client variance. BAILEY2 [2], is identical to FCFA except the ®rst two clients to call in are scheduled for the ®rst appointment slot and the last slot is left open. BAILEY2 is included as a benchmark for the server idle time measure since it performed well on this dimension in studies by Ho and Lau and Klassen and Rohleder [6,8]. A similar rule, BAILEY4 [2] is conceptually the same, except the ®rst four clients are scheduled in the ®rst slot. This rule was tested by Klassen and Rohleder [8] and while it minimized server idle time, average client waiting time was too excessive to be considered for today's service oriented operations. OFFSET was developed in [6] and adjusts the assigned appointment times depending on the assigned appointment slots. For this study the following form is used: Ti
As ÿ k1 s K ÿ s for i and s 1, 2, . . . , K As k2 s s ÿ K for i and s K 1, K 2, . . . 21 2
where Ti is the scheduled appointment time for client i, As is the regular appointment time for slot s(e.g. A1=0.0, A2=10.0, . . . ,A21=200.0), and s is the standard deviation of the service time. As compared to FCFA, OFFSET moves the client appointment times earlier for the ®rst 12 patients, and later for the remainder. This rule is included due to its performance on the client waiting-time measure as reported in previous studies [6,8]. However, as noted by Klassen and Rohleder [8], using the original parameters for the early/delay breakpoints and the associated multipliers (K = 5, k1=0.15, and k2=0.30) suggested by Ho and Lau [6] resulted in excessively late end times, which would be unacceptable to service providers. Thus, for this study the parameters were re-evaluated to ®nd the values K = 12 and k1=k2=0.20 actually used in Eq. (2). A formal pre-study was performed to evaluate a wide range of possible values for K, k1, and, k2; the values chosen achieved the best balance in end times and C values.
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Low-Variance Beginning is the ®rst rule using client variance information. It schedules clients by putting those with the service times the scheduler believes to have low variance ®rst and those with high variance last. Operationally this is done by assigning low variance clients to the lowest available appointment slot and assigning high variance clients to the highest available appointment slot. The schedule is ®lled when the low and high variance service time clients ``meet''. Since the calls are at random, the number of low and high variance patients will ¯uctuate in each scheduling period. This rule was the best of those using client variance information in Klassen and Rohleder's study [8], and therefore was viewed as the best rule to test with more realistic service time and scheduler assumptions of this study. However, if LVBEG does not perform well with a continuous variance distribution and scheduler uncertainty we would expect it to perform similarly to FCFA. The ®nal rule tested is a combination of BAILEY2 and LVBEG. The rule follows the same pattern as LVBEG except two clients are scheduled into the ®rst appointment slot and none in the ®nal slot. Based on results from Klassen and Rohleder [8], the combined rule should reduce the waiting time of BAILEY2 and reduce the idle time of LVBEG. 2.4. Client variance The client service time mean is set at ten minutes for all clients in this study. This assumption is similar to that used in previous studies (see, for example [6,8]). Unlike production scheduling, most clinics assume a constant mean service time for patients. The clinics interviewed in Klassen and Rohleder [8] did vary scheduling for appointments such as complete physical exams due to their increased length compared to a standard appointment. This issue has not been thoroughly studied although testing showed that having some longer appointment slots reduced the number of total appointments [8]. Fewer slots results in better performance on client waiting time and server idle time. Further research on this issue is warranted. The key dierence in this study is the way variance is handled for each patient. We assume that service time variances [which we will denote as client variance dispersion (CVD)] are drawn from a lognormal distribution with a mean of 7.5 and two experimental standard deviation values of 3.75 and 7.5 min [giving coecient of variations (cv) of 0.5 and 1.0]. Once a client's service time standard deviation was randomly chosen, call it sc, this was used to generate the service time, using the mean of 10 min and the lognormal distribution [i.e. client service time=Lognormal (10, sc)]. Use of the lognormal distribution for service times
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was based on both theoretical and empirical grounds. Law and Kelton [9] state that the lognormal distribution is a good distribution for modelling service times made up of a series of tasks in sequences (as is typical in medical appointments). Klassen and Rohleder [8] found that empirically, the lognormal distribution was the best approximation of what was experienced by the schedulers at the clinics. They also found that a mean variance of the midpoint between the values of ®ve and ten was appropriate, based on the empirical interviews with family medical practice schedulers. These schedulers found that most clients service times were plus or minus ®ve minutes from the mean time and sometimes as large as 20 min larger than the mean. From this information and the variance values a lognormal distribution was roughly ®tted. 2.5. Scheduler uncertainty The key assumption in using a rule like LVBEG is that clients can be strati®ed in some manner by the variability in their service times. This issue was addressed empirically by Klassen and Rohleder [8], where the medical receptionists felt they did have information that could be used to classify patients by service-time variability. Classi®cation can be aided by knowledge of the individual characteristics of the client, the nature of the visit, and other relevant factors. For the LVBEG rule, classi®cation requires deciding at which standard deviation value the cuto between low and high variance patients should occur. For simplicity, we use the median value. So, if m and s are the mean and standard deviation of the lognormally-distributed standard deviation values, the median is dep rived as m2 = s2 m2 : Therefore, the cuto values for the standard deviation values of 3.75 and 7.5 are 6.71 and 5.30 (the medians with a mean of 7.5). Instead of the median, the mean could be used as the cuto point, however, in a pre-test this approach led to slightly worse performance than the median. The result of using the mean was that more clients were classi®ed as low variance than with the median. However, the dierence in performance was not signi®cant and, therefore, the median was used in this study. To handle the scheduler uncertainty, we thought it was reasonable to assume clients that had true variance values farther from the population median would be easier to classify than those with variances close to the median. To model this we use a normal distribution with a mean of 0 and standard deviations of 5 and 10. The probability is then calculated as: jx i ÿ nv j PCCi F 3 su
where PCCi is the probability of correct classi®cation for client i, xi is the client's true variance, nv is the median of the variance distribution, su is the standard deviation of the uncertainty distribution, and F is the unit normal probability function. For standard deviation values of 5 and 10 the above relationship creates functions as plotted in Fig. 1. As Fig. 1 shows, as the uncertainty standard deviation increases, the probability of correct classi®cation is lower for any dierence value of a client's standard deviation and the median standard deviation for the population. Thus, classi®cation is less likely to be correct with higher standard deviation values. The ®gure also shows that the values of ®ve and ten for the uncertainty standard deviation provide a good contrast of ``easy'' and ``hard'' classi®cation levels. Clearly, with su=5 the schedulers make the correct selection with a signi®cantly higher probability for most clients except when the dierence between the client's actual service time variance and the median are very close. In practice, neither a speci®c client's variance (xi) nor the standard deviation of the uncertainty distribution (su) would be known. As such they are not, generally, of practical value to the scheduler. Rather, the dierent values used represent experimental conditions to test the performance of the LVBEG scheduling rule for this study. Fig. 2 shows the uncertainty situation graphically. The random variable yi represents the uncertainty ``noise'' faced by the scheduler. Note that as su increases, the probability of incorrectly classifying a client (high variance in the graph) would go up since the hatched area would become larger. Thus, su represents the factors that makes the scheduler more or less capable of correctly classifying clients. Such factors could include scheduler experience, the nature of the service, general client knowledge in the drawing population, etc.
Fig. 1. Classi®cation uncertainty functions.
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2.6. Client preferences In previous studies of appointment scheduling, it was assumed that schedulers could put clients in whatever time slots schedulers desired. However, in reality schedulers almost always have to deal with clients who ask for speci®c appointment times. (Note: These will be termed as special requests for this study.) In informal interviews, appointment schedulers reported that about 25% of clients ask for speci®c appointment times, or at least, a speci®c range of appointment times. For most scheduling rules this does not aect performance on waiting and idle time since clients are considered homogenous for scheduling purposes. However, rules using client-variance information (LVBEG, B2LVBEG) may be aected because special requests will limit the ability to slot clients as desired. Certainly, a low-variance client may ask for an appointment time late in the scheduling period. Such occurrences will compromise the eectiveness of these rules. Since dierent services will have varying numbers of special requests, this study will consider three levels of special requests (SR); 10%, 25%, and 40% of clients. This is implemented in the model by giving the caller the SR percentage chance of requiring a speci®c time slot. Since the process is random, the exact proportion of SR callers on any day will vary, but the average will be correct for a large number of days sampled. Given that special request callers usually have some ¯exibility, appointment slots plus or minus two from the requested one will be checked for availability if the desired slot is already taken. Related to the percentage of SR callers is the distribution of desired slots (DDS). Two distributions will be considered in this analysis: uniform (UNIF) and end of period (EOP). For the UNIF approach, when SR callers ask for a slot, it may be any of the 21 slots (times) with equal probability. The EOP distribution has callers request any of the last ®ve slots with equal probability. Of course, many other special request distributions could be considered, however, based on
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informal interviews with appointment schedulers the EOP approach was cited as the most common, and UNIF provides a good contrast for comparison purposes. With the addition of the experimental factors related to special requests, we decided to track several performance measures related to this concept. These measures are: the proportion of special request clients receiving the speci®c slot requested (PSR), and the proportion of special request clients not receiving any slot (PSN). These two measures represent opposite extremes in client satisfaction (PSR=happy, PSN=unhappy). In addition, by taking 1.0ÿ(PSN+PSR), the proportion of clients receiving one of the plus or minus two slots from their requested slot can be calculated. While this measure will not be speci®cally considered in the analysis, it does represent a measure of moderate client satisfaction. Note that any special request client not receiving their requested slot or a slot nearby was put back into the calling pool to ensure the number of appointment requests remained constant. 2.7. Computer model A computer simulation of this model was developed in the Arena programming environment [1]. Performance for 100 scheduling periods is averaged to represent a replication and 100 replications are run for each experimental treatment. Such an approach provides sucient samples to obtain the desired power and help assure normality of the output responses [4]. In addition, common random numbers for each replication are used to reduce variation and provide greater precision in the analysis.
3. Results 3.1. Experiment summary
Fig. 2. Graphical view of scheduler uncertainty.
Table 1 presents a summary of the experimental factors. Overall, there are ®ve scheduling rules tested at: two levels of client-variance dispersion, three levels of special request proportions, and two levels of special request distributions. In addition, LVBEG and B2LVBEG have the nested factor of scheduler uncertainty, with three levels: no uncertainty, moderate uncertainty (su=5.0), and high uncertainty (su=10.0). This creates a total of 108 treatments with 100 replications and 100 scheduling periods in each replication, creating 1,080,000 total scheduling sessions, or observations (10,800 replications in the data set).
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Table 1 Summary of experimental factors Factor (# of levels)
Description of levels
Scheduling rules (5) Client variance dispersion (2) Proportion of special requests (3) Distribution of special requests (2) Scheduler uncertainty (3) (Note: only aects rules LVBEG and B2LVBEG)
FCFA, BAILEY2, OFFSET, LVBEG, B2LVBEG cv=0.50 and cv=1.0 0.10, 0.25, and 0.40 Uniform (UNIF) and End of Period (EOP) None, su=5.0, and su=10.0
3.2. Analysis of main objective function (C) Table 2 is the source of variation summary from an analysis of variance with C[the mean of the objective function in Eq. (1)] as the dependent variable (all statistical analysis was performed using the SAS system [11]). Note that non-signi®cant interactions (a > 0.05) are excluded from the table. Also, for this analysis, the results for all levels of scheduler uncertainty (su) are pooled together for LVBEG and B2LVBEG scheduling rules. Analysis of the scheduler uncertainty factor will be separately discussed later. From Table 2, it is clear that the scheduling rule factor explains the majority of the variation and all main eects are signi®cant, however, there are several signi®cant interactions that need to be considered. The three-way interaction between scheduling rule, special requests, and distribution of desired slots is signi®cant, and was therefore analyzed ®rst (all other four-way and three-ways were insigni®cant). A moderate interaction occurs for some rules with SR and DDS, most signi®cantly, BAILEY2. With this rule and DDS at the UNIF level, the means for all levels of SR
are essentially constant. This makes sense since C should not be in¯uenced too much by special requests that do not aect the ordering of clients. However, when there is a high percentage of special requests and the desired slots are at the end of the scheduling period, this actually improves the waiting-time performance of BAILEY2 because it may actually put a client in the very last slot. Without special requests, this slot would normally be left open, but by ``moving'' a client from a busier period to the last slot, overall performance is improved. While the above three-way interaction does occur, its eect on the results is easier to understand by looking at the two-way interaction between the scheduling rules and distribution of desired slots. Fig. 3 shows this interaction graphically. Note that for most rules wait and idle-time performance is not aected much by the preference distribution, however, for the reason mentioned previously BAILEY2 and B2LVBEG have better performance when the clients' preferred slots are near the end of the scheduling period. We will further analyze the eects of SR and DDS when performance related to client preferences is considered.
Table 2 ANOVA Results for C. Key: SCH RULE=Scheduling rule; CVD=Client Variance Dispersion, SR=Special Requests; DDS=Distribution of Desired Slots Source
d.f.
Sum of squares
Mean square
Model Error Corrected total
158 10641 10799 R2 0.478546 99 4 1 4 2 8 1 4 2 8
8514700.64 9278151.73 17792852.37 C.V. 13.36141 3160110.78 4622183.74 293909.52 200904.35 7435.26 40834.11 61670.49 81014.51 9391.66 24543.79
53890.51 871.92
REPLICATION SCH RULE CVD SCH RULECVD SR SCH RULESR DDS SCH RULEDDS SRDDS SCH RULESRDDS
Root MSE 29.5284 31920.31 1155545.93 293909.52 50226.09 3717.63 5104.26 61670.49 20253.63 4695.83 3067.97
F value 61.81
36.61 1325.28 337.08 57.60 4.26 5.85 70.73 23.23 5.39 3.52
Pr > F 0.0001 Overall Mean 220.997 0.0001 0.0001 0.0001 0.0001 0.0141 0.0001 0.0001 0.0001 0.0046 0.0005
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Table 3 Scheduling rule comparisons for C
Fig. 3. Interaction: scheduling rule preference distribution.
The two-way interaction between the scheduling rules and the proportion of special requests was previously explained with reference to the signi®cant three-way interaction. The improvement for C with BAILEY2 and B2LVBEG with DDS=EOP is, not surprisingly, most noticeable when the proportion of special requests is high. However, this eect is fairly weak and not nearly as powerful as the interaction between the scheduling rules and client variance dispersion. This interaction is shown in Fig. 4. This interaction plot shows that, in general, rule performance for average waiting plus idle time is improved when client variance is high. Although for most rules the improvement is insigni®cant, LVBEG shows dramatically better performance when client dispersion is high, with an improvement of nearly 20 for C. This is due to the rule's design that keeps certainty high early in the scheduling session and allowing the high variability clients that have high and low service times to trade o against each other later in the session. With high CVD, the advantages of LVBEG are enhanced. B2LVBEG also shows signi®cant improvement for the same reasons. On the other hand, the OFFSET rule's performance is signi®cantly worse with high client variance. Note that this rule uses the standard deviation of client service times when setting appointment times. With high client variance, this tends to increase waiting for clients early in the scheduling session and increase server idle time overall. Further work on parameterization could
Fig. 4. Interaction: rule client variance dispersion.
a=0.05 d.f.=10641 Number of Means 2 Critical Range 2.024 Means with the same letter are not Duncan Grouping Mean
MSE=871.9248 3 4 5 2.131 2.202 2.255 signi®cantly dierent N RULE
A B C D E
1200 1200 3600 1200 3600
253.166 238.492 231.805 219.371 194.178
BAILEY2 OFFSET B2LVBEG FCFA LVBEG
help to alleviate this problem, however, this work was not performed for this study because we believe this rule is too complex to implement in practice. As previously discussed, the scheduling rules explain the largest portion of variability in the ANOVA. Therefore, it is informative to look at the overall mean performance for this factor. Table 3 shows the results of a means comparison using the Duncan Multiple Range Test (see [10]). This comparison method was chosen due to its power in detecting true dierences in means. As the mean comparisons show, LVBEG is clearly the best rule for the C measure. However, as discussed in [6] it is informative to consider the implicit tradeo between client waiting time and server idle time ``costs''. This can be done graphically, by considering an ecient frontier. Fig. 5(a) and 5(b) show two of the frontiers for this study for diverse sets of experimental factors. In these ®gures, the ecient frontier is identi®ed by the points connected by the line. Thus, the
Fig. 5. Ecient frontier. (a) Low client variance, uncertainty variance = none. (b) High client variance, high scheduler uncertainty.
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Table 4 Scheduling rule performance: waiting-time measures Average maximum waiting timea
Percentage waiting <10 min
Value
Rule
Value
Rule
26.32 26.51 29.14 29.17 29.79
(1) (1) (2) (2) (3)
75.9% 73.2% 69.8% 66.4% 65.5%
(1) (2) (3) (4) (5)
a
LVBEG FCFA B2LVBEG BAILEY2 OFFSET
LVBEG FCFA OFFSET B2LVBEG BAILEY2
The numbers in parentheses represent the statistical groupings for mean comparisons.
frontier moves from the upper left to the lower right. Those on the upper left minimize idle time, and those on the lower right minimize waiting time. By considering the relative values of the average client wait time and server idle time, the implicit ratio of p/o can be inferred. For example, a service provider wishing to minimize server idle time could consider using BAILEY2 which has about a 12 to 1.3 ratio [in Fig. 4(a)] of client wait time to server idle time. Note that services should generally not consider rules o the frontier regardless of their p/o ratios since an ecient rule will always provide a lower total cost. Also, while the exact frontier will dier for every combination of environmental factor levels, the general form of the frontier is always the same as Fig. 5(a) and 4(b) demonstrate. 3.3. Waiting-time measures Performance for the average maximum waiting-time per scheduling period and the percentage of client waits less-than-or-equal-to ten is reported in Table 4. Only results for the scheduling rule factor are reported because for each performance measure this factor explained the vast majority of the model variation net of variability due to replication number (78.8% and 85.6% for MAX and P10, respectively). The results on these measures support LVBEG as the best rule from a client-waiting perspective. In particular, LVBEG does a good job of minimizing the number of clients that must wait more than 10 min. 3.4. Analysis of end times for scheduling rules As a contrast to the waiting time measures, the session end time is of greater concern to the service provider. An excessively large end time suggest the service provider will need to stay ``overtime'' or cut into needed breaks and lunch times. Table 5 shows the relative mean and maximum end times for each scheduling rule. Recall that the sessions are scheduled for 210 min.
From the table, several key ®ndings stand out. First, the rules that put two patients into the ®rst slot have signi®cantly lower end times. On the other hand, OFFSET, even with the adjusted parameters to achieve better end times is still signi®cantly worse than the other rules. 3.5. Analysis of special request measures The performance of the scheduling rules for the proportion of clients not receiving any slot (PSN) and the proportion of clients receiving their requests slot (PSR) is important because service providers and clients are interested not only in idle/waiting time, but also in a service system's capability to handle special requests. In analyzing the PSN measure, it was immediately apparent that there was an important two-way interaction of the scheduling rule factor with the client request distribution. Fig. 6 clearly shows that the rules using client variance information have the opposite reaction to the client preference distribution than the other rules. This result was not entirely unforseen since BAILEY2, FCFA, and OFFSET build their schedules from the beginning to the end, their PSN performance should be better with EOP. What is somewhat surprising is the magnitude of advantage LVBEG and B2LVBEG have for the UNIF preference distribution
Table 5 Mean and maximum end times for scheduling rules Meana
Maximum
Value
Rule
Value
Rule
217.59 218.99 224.57 224.82 239.79
BAILEY2 B2LVBEG FCFA LVBEG OFFSET
234.73 286.08 293.05 295.46 314.57
BAILEY2 B2LVBEG FCFA LVBEG OFFSET
a
All means were signi®cantly dierent at a=0.05.
T.R. Rohleder, K.J. Klassen / Omega 28 (2000) 293±302
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Table 6 Analysis of scheduler uncertainty for C measure Scheduler Uncertainty
LVBEG
B2LVBEG
Meana
None su=5 su=10
183.77 195.15 203.61
225.30 233.17 236.94
204.53 214.16 220.27
a
All means are signi®cantly dierent for a=0.05 using the Duncan Multiple Range Test. Fig. 6. Interaction of preference distribution rule for PSN.
and how much worse they are for EOP. The advantage of these rules with UNIF occurs because they build their schedules from both ends. This means that a special request call coming when the schedule is nearly full will more likely ®nd open slots around the requested time than for the other rules. The results for the PSR (the client ``happiness'') measure are, not surprisingly, similar to those for PSN. Fig. 7 shows the same interaction plot as for PSN. Note that for the PSR measure, higher is better. For the UNIF distribution of client requests all the rules are fairly close, however, when the requests are grouped near the end of the session, LVBEG and B2LVBEG perform signi®cantly worse and the other rules perform signi®cantly better. 3.6. Analysis of scheduler uncertainty In all the previous analyses, the levels of the scheduler-uncertainty factor were pooled for LVBEG and B2LVBEG. It is also useful, however, to consider how this factor aects the performance of these rules. For this analysis, a comparison of means was performed on just the su levels. Table 6 shows the results for each of the aected scheduling rules and the overall average for the C performance measure. A couple of interesting points about scheduler uncer-
tainty should be noted. First, the inclusion of this factor certainly aects the performance of the scheduling rules using client variance information. As scheduler uncertainty goes from zero to any positive value, the performance of these rules signi®cantly worsens. However, because of the nonlinear nature of the scheduler uncertainty function, su's aect is diminishing as it increases. For the mean eect of su going from none to the low level, performance for C is 4.7% worse, but going from the low to high level, performance is only 2.9% worse. So the results show that the LVBEG rule is not exceptionally sensitive to scheduler uncertainty, but nonetheless, schedulers should do what they can to improve their capability to classify clients to minimize the costs of waiting and server idleness. (This, in eect, lowers the value of su.) The eect of scheduler uncertainty was also explored for the other performance measures and with the other factors for possible interactions. However, other than the signi®cant result discussed above, scheduler uncertainty had little or no eect.
4. Conclusions In answer to one of the major questions of this study, the LVBEG scheduling rule that uses client-variance information is still eective in minimizing waiting and idle time when the realistic complexities of continuously distributed service-time variances, special client appointment requests, and appointment-scheduler uncertainty are added. The study also showed that each of these realistic factors did aect the results and so are useful when evaluating appointment scheduling systems. The remainder of this section will discuss practical implications and future study directions suggested by this research. 4.1. Practical implications
Fig. 7. Proportion of special requests satis®ed interaction: rule preference distribution.
From the interviews with appointment schedulers discussed in Klassen and Rohleder [8], it is apparent these schedulers do have information about client variance. Services that wish to simultaneously minimize
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client waiting time and server idle time should consider using this information with an approach like LVBEG for scheduling. It dominated performance for this measure and those related to waiting time. Even when schedulers are not perfect at assigning clients as high or low variance, LVBEG was still best. In fact, even with the highest level of scheduler uncertainty, LVBEG was still about 8% better than the next best rule for the C measure. Also, the dispersion of client variances does aect results for LVBEG Ð the greater the dispersion, the better the rule performs. This suggests that services that have a high degree of variability with their clients should be even more motivated to use client-variance information when scheduling. The results when considering the special-request performance is not as clear cut. When requests are spread out across the scheduling period, LVBEG is still a good choice for services. However, when requests are concentrated near the end of the scheduling period, LVBEG and B2LVBEG perform poorly in meeting these requests, at least compared to other rules that have a beginning-to-end scheduling pattern. The end of period distribution was analyzed based on informal interviews that suggested it is common. It should be noted that if the special requests tended to group near the beginning of scheduling sessions the rules such as FCFA, OFFSET, and BAILEY2 would not perform well on special request measures (likely worse than LVBEG or B2LVBEG). Also, LVBEG and B2LVBEG will likely fair well if the distribution of special requests tend to occur near the middle of the scheduling session. Obviously, services will have to pay attention to the distribution of special requests to determine what approach is sensible. 4.2. Future research Certainly, many research issues on this topic remain open. While this study looked at individual client variances, interviews by Klassen and Rohleder [8] suggest that mean classi®cation may also be possible. The clinics studied already adjusted appointment slot length to account for procedures they knew were likely to exceed the standard slot. In addition, dierent uncertainty levels and functions should be considered. Finally, all previous appointment-scheduling studies considered scheduling periods to be independent. In many (most?) appointment-oriented services clients are scheduled over a horizon that may span several days or even weeks. Certainly, this ``rolling-horizon'' en-
vironment will have implications for scheduling. Speci®cally, the eects of scheduling special requests into future time periods, the use of overtime, and controlling the ``exactness'' of adherence to rules like LVBEG can be tested in this even-more-realistic environment.
Acknowledgements The authors would like to express their appreciation for the helpful comments of three anonymous reviewers and Dr Edward A. Silver. In addition, this work was partially supported by the Natural Sciences and Engineering Research Council of Canada under grant number 138047.
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