This case study focuses on the application of the newly proposed OR/MS tools for efficient manpower resource allocation. Some details associated with the Six Sigma project have been deliberately left out for reasons of confidentiality. However, per- tinent infomation relating to the analysis with the newly proposed OR/MS tool is provided to demonstrate its effectiveness.
Here, we give an illustration of how the Six Sigma framework is applied to reduce the waiting times for a retail pharmacy in a hospital. This investigation was prompted by complaints about the long waiting times for drug prescriptions to be dispensed.
Initially, continuous efforts were made to expedite the work flow by the pharmacy staff without a systemic examination of relevant work processes. These efforts turned out to be insufficient in achieving the waiting time target set by the management for the pharmacy.
In the Define phase, external Six Sigma consultants, together with the hospital management, selected this particular project to reduce the waiting times of patients in the central pharmacy given the urgency and proximity of the process to the customers.
Furthermore, as there were multiple satellite pharmacies with similar processes to those of the central pharmacy, the project would be able to reap benefits beyond this particular department. The success of this project would enable the Six Sigma team to garner more extensive buy-in and support from the management and other hospital staff. Such internal ‘marketing’ efforts are essential for sustainable implementation and successful execution of future Six Sigma projects that will deal with increasingly difficult problems.
Four major tasks were identified in the current process -- typing, packing, checking, and checking and dispensing (or dispensing for short). The arrival rates of prescrip- tions to the pharmacy were measured based on counts of arrivals each 10 minutes.
The profile of estimated arrival rates at each 10 minutes interval is shown in Figure 5.1. After accounting for outliers in the profile and reasons for high arrival rates in some particular instances as shown in Figure 5.1, the profile was discretized and two distinct arrival rates identified by visually examining the data. Although the profile can be more accurately discretized by having additional segments, only two distinct arrival rates were identified in preliminary investigations after accounting for practical considerations related to manpower allocations and sources of variations,
Case Study: Manpower Resource Planning 59
0 100 1 2
Estimated Arrival Rates (prescriptions/ min)
3 4 5 6
200 300
Time (mins)
400 500 600
Outlier Arrival Rates Profile
Figure 5.1 Profile of estimated arrival rates.
such as day-to-day variations, in the arrival rates. Furthermore, subsequent sensitivity analysis based on different arrival rates using queuing methodologies will allow us to select the most robust manpower configurations for each distinct arrival rate.
With reasonable estimates of data on the arrival rates extracted from the arrival rate profile shown in Figure 5.1, together with service and rework rates shown in Table 5.4 obtained in the Measure phase, the lead times and value-added times for each process can be computed. The entire process can in fact be represented by an open queuing network as shown in Figure 5.2.
Table 5.4 Service and rework rates.
Process Estimated service rates (min/job)
Typing 1.92
Packing 0.20
Checking 1.70
Checking and dispensing 0.19
Rework routing Estimated proprotion of rework
Packing→Typing 0.025
Checking→Typing 0.025
Checking→Packing 0.025
Checking and Dispensing→Typing 0.001
Checking and Dispensing→Packing 0.001
60 Fortifying Six Sigma with OR/MS Tools Source
Legend:
: Queues : Servers
: External
Typing
Packing
Checking
Dispensing
: Internal Flows Sink
Figure 5.2 Queuing network representation of drug dispensing process.
For the Analyze phase, estimates of average waiting times and queue lengths can be derived using basic queuing methodologies based on the patient arrival rates and service rates. Steady-state queuing analysis is adequate here as the arrival and service rates are fast enough to ensure the system reaches its steady state in a short time. Inter- arrival and service times were assumed to be exponentially distributed. The entire process can thus be represented as a system of interconnected M/M/1 and M/M/s ser- vice stations. M/M/1 and M/M/s are standard abbreviations to characterize service stations in queuing methodologies, M/M/1 denoting service stations with a single server, and M/M/s stations with a finite numbersof servers (s>1). Both of these types of stations experience Markovian arrival and service processes, with service processes in an M/M/s system being independently and identically distributed. Furthermore, the queuing buffer is assumed to be of infinite size and each server can only serve one customer at a time and selects waiting customers on a first come, first served basis.
Given the preceding assumptions, mean total waiting times for the entire drug dispensing process in the pharmacy can be computed by first computing the mean sojourn times using standard queuing formulas for each service station,28 and then summing these mean sojourn times. In the computations, the mean total waiting time for the entire process does not include the mean service time of the final dispensing and checking process. This is because the waiting time of a patient is defined as the period from the time he submits the prescription to the time when service by the dispensing pharmacist is initiated.
5.4.1 Sensitivity analysis
At the Improve phase, the impact of different manpower configurations on the over- all waiting times can be assessed by varying the number of packers and dispensing
Case Study: Manpower Resource Planning 61
9.0 19.0 29.0 39.0 49.0 59.0 69.0 79.0 89.0
8 9 10 11
Number of Packers
Waiting Times
8 Dispensing Pharmacists 9 Dispensing Pharmacists 10 Dispensing Pharmacists 11 Dispensing Pharmacists
View A
10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0
8 9 10 11
Number of Dispensing Pharmacists
Waiting Times
8 Packers 9 Packers 10 Packers 11 Packers
View B 84
72 60 48 36 24 12 Total Waiting Times (mins)
0
View A
8
8
9 10
11 View B
No. of Dispensing Pharmacists
9 10
No. of Packers 11
Figure 5.3 Sensitivity of total waiting times to variations in the numbers of packers and dispensing pharmacists.
pharmacists in a sensitivity analysis. Results for the lower arrival rate extracted from the estimated arrival rate profile shown in Figure 5.1 are presented here. A correspond- ing analysis for the higher arrival rate can be conducted. Subject to other practical con- siderations, manpower deployment can then be adjusted dynamically throughout the day.
With an arrival rate of 88 prescriptions an hour, at least 8 packers and 8 dispens- ing pharmacists were needed for the packing and checking dispensing processes, respectively, in order to ensure the finiteness of steady-state waiting times. Figure 5.3 shows the impact of varying the number of packers and dispensing pharmacists on the mean total waiting times. It was observed that waiting times would be increased significantly if the number of packers was reduced to 8. With more than 8 packers, the waiting times were observed to be relatively stable over the different numbers of dispensing pharmacists in the experiment. It was further observed that by having one additional dispensing pharmacist, the targeted 15 minutes of mean total waiting times could potentially be met.
In order to assess the robustness of each possible system configuration, arrival rates of patients at the pharmacy were varied in the model. Figure 5.4 shows the
62 Fortifying Six Sigma with OR/MS Tools
9.0 14.0 19.0 24.0 29.0 34.0 39.0 44.0
8 9 10 11
No. of Dispensing Pharmacists
Waiting Times
85 prescriptions/hour 86 prescriptions/hour 87 prescriptions/hour 88 prescriptions/hour 89 prescriptions/hour 90 prescriptions/hour 91 prescriptions/hour
View A
9.0 14.0 19.0 24.0 29.0 34.0 39.0 44.0
85 86 87 88 89 90 91
Mean Arrival Rates
Waiting Times
8 Dispensing Pharmacists 9 Dispensing Pharmacists 10 Dispensing Pharmacists 11 Dispensing Pharmacists
View B 40
35 30 Total Waiting Times (min) 25
20 15 10
85.587.088.590.091.5 View B View A
Arrival Rates (patients/hour) 5
08 9 10 11
No. of Dispensing Pharmacists
Figure 5.4 Sensitivity of total waiting times to variations in number dispensing pharmacists and patients arrival rates.
sensitivity of mean total waiting times subjected to small perturbations in arrival rates for different numbers of dispensing pharmacists. It was observed that the original configuration with only 8 dispensing pharmacists would result in large increases in waiting times when there were only small increases in the number of prescriptions arriving per hour (in the range 85--91 prescriptions per hour). This provided the management with insights into the frequently experienced phenomenon of doubling in waiting times on some ‘bad’ days. By having an additional dispensing pharmacist, the system would be expected to experience a more stable waiting time distribution for different arrival rates.
In order to elicit more improvement opportunities, additional root cause analysis for the long waiting times was performed for the present process. This was conducted with the aid of a fishbone (or Ishikawa) diagram, enabling. The potential of stream- lining manpower deployment in the pharmacy can be further studied. From the anal- ysis, it was suggested that a new process, that screening, be implemented at the point when the pharmacy receives prescriptions from patients (prior to the existing typing
Case Study: Manpower Resource Planning 63 Table 5.5 Comparisons of sojourn times for each process and mean total
waiting times.
Type of job Without screening With screening
Screening -- 2.6 (1 pharmacist)
Typing 0.6 (2 typists) 0.6 (2 typists)
Packing 5.9 (10 packers) 5.6 (10 packers)
Checking 6.2 (1 pharmacist) 5.0 (1 pharmacist)
Dispensing checking* 9.0 (8 pharmacists) 2.0 (7 pharmacists)
Mean total waiting time 21.7 15.8
*Service times only
process). It was expected that this would help to alleviate problems associated with errors in prescriptions and medicine shortages.
A pilot run was implemented with one dispensing pharmacist moved to the newly implemented screening process. Such a reconfiguration would not require additional pharmacists to be hired. Service and arrival rates estimates were obtained from the pilot runs and mean waiting times predicted for each process. The mean total waiting times for prescriptions can again be computed by summing all the mean waiting and service times of each process. The average waiting times computed were statistically validated with actual data obtained from pilot runs.
Table 5.5 shows the improvements in the average sojourn times of each subprocess and the mean total waiting time for processes before and after the addition of the screening process. Improvements can be observed in the new process because many interruptions that occurred during the dispensing process were effectively reduced by the screening process upfront. As a result, the productive time of pharmacists increased and the mean queue length in front of the dispensing process shortened from 13 to 3.
Various possible system configurations were again tried with different number of packers and dispensing pharmacists with the new model that considered the screening process. In order to ensure finiteness of steady-state waiting times, at least 8 packers and 7 dispensing pharmacists for the packing and “checking and dispensing processes respectively are needed. From the analysis, the proposed new configuration was found to be more robust to changes in manpower deployment over the packing and dispensing subprocesses (see Figure 5.5). Eventually, this new robust design was adopted to ensure waiting-time stability over possible variations in manpower deployment.
Process validations on results generated from the mathematical models depicting original and improved processes have been dealt with throughout the Analyze and Improve phases in the preceding discussions. At the final Control phase, standard operating procedures (SOPs) were put in place in order to ensure stability the of new processes. During the generation of these SOPs, several new control measures were proposed by the team and implemented with the understanding and inputs from the
64 Fortifying Six Sigma with OR/MS Tools
10.0 12.0 14.0 16.0 18.0 20.0 22.0 24.0 26.0
8 9 10 11
Number of Packers
Waiting Times
7 Dispensing Pharmacists With Screening 8 Dispensing Pharmacists With Screening 9 Dispensing Pharmacists With Screening 10 Dispensing Pharmacists With Screening 11 Dispensing Pharmacists With Screening
View A
10.0 12.0 14.0 16.0 18.0 20.0 22.0 24.0 26.0
7 9 10 11
Number of Dispensing Pharmacists
Waiting Times
8 Packers With Screening 9 Packers With Screening 10 Packers With Screening 11 Packers With Screening
View B View A
View B
8 24
18 12 6 0
8 9
7 8 9 10 11
No. of Dispencing Pharmacists
10 11
No. of Packers Total Waiting Times (mins)
Figure 5.5 Sensitivity of total waiting times to variations in number of packers and dispensing pharmacists in the new dispensing process with screening.
relevant stakeholders through targeted ‘Kaizen’ events. The SOP and validated results were communicated to the pharmacy staff in order to reinforce their confidence in the new processes.
5.4.2 Relaxing exponential assumptions on service times distribution
Further sensitivity analysis can be performed to assess the effect of the exponential assumptions on the distribution of service times. This analysis was conducted to estab- lish whether the assumption of exponentially distributed service times would result in more conservative system design choices. Results based on the model without the screening process are described here. A similar sensitivity analysis can be conducted on the model with the new screening process.
Each service stationi was assumed to experience a Poisson arrival process with mean arrival rateλi. Single-server service stationsiwere assumed to experience ser- vice processes that follow any general distribution with mean service rateμi. This is commonly known as the M/G/1 queuing system. For service stations with (finitely) many servers, the service processes of each server were assumed to follow general
Case Study: Manpower Resource Planning 65 distributions that were independently and identically distributed with mean service rate μi. Such a system is commonly known as the M/G/s queuing system (s>1 finite). The mean total waiting time for the entire drug dispensing process in the pharmacy can be computed by summing the mean waiting and mean service times at each service station.
In order to derive the mean waiting times at each service station, the overall arrival rateλiat each service station, for the queuing network shown in Figure 5.2 has to be computed. At statistical equilibrium,λiis given by
λi=λi0+ N
j=1
λjPji,
where Nis the number of queuing stations in the network, λi is the arrival rate at stationifrom external sources, andpjiis the probability that a job is transferred to the jth node after service is completed at theith node.
In order to compute the mean waiting times at each single-server queuing stations (si =1), the mean queue length of each single server-station, ¯LiM/G/1, is first computed with the well-known Pollaczek--Khintchine formula,28
L¯iM/G/1= ρi2 1−ρi
1+C Vi2
2 .
HereC Vi2is the squared cofficient of variation of service times,Ti, which follow any general random distribution:
C Vi2=Var(Ti) T¯i2 ,
In which Var(Ti) is the variance of the random service time of serveri, and ¯Ti is the mean of the service time of serveri (or reciprocal of the service rateμi).ρi is the utilization of serveri, which can be interpreted as the fracttion of time during which the server is busy and is given by
ρi= λi
μi.
Given ¯LiM/G/1, the mean waiting times at each single-server service station, ¯WiM/G/1, can then be computed using Little’s theorem28as follows:
W¯iM/G/1= L¯iM/G/1 λi .
To compute the waiting times of queuing stations with multiple server (si>1), we apply an approximation due Cosmetatos.29For this, we first compute the mean
66 Fortifying Six Sigma with OR/MS Tools waiting times of an M/M/s queuing system ( ¯WiM/M/s):
W¯iM/M/s = L¯Mi /M/s λi
where the mean queue length of the M/M/s system, ¯LiM/M/s, is given by:
L¯Mi /M/s= (siρi)siρiPi0
si!(1−ρi)2.
In this expressionPi0is the probability of stationibegin empty,
Pi0= si−1
n=0
(siρi)n
n! + (siρi)si si!(1−ρi)
−1
,
andρiis the utilization of service stationi, this time multiple servers, ρi= λi
siμi
.
Next, we compute the mean waiting times of an M/D/s queuing system, that is, for s servers with constant (i.e deterministic) service times ( ¯WiM/D/s), as follows:
W¯iM/D/s= 1 2
1 Ki
W¯iM/M/s, where
Ki =
1+(1−ρi)(si−1)
√4+5si−2 16ρisi
−1
.
The approximate mean waiting times for an M/G/s system can finally be computed from:
W¯iM/G/s ≈C Vi2W¯iM/M/s+
1−C Vi2W¯iM/D/s.
Figure 5.6 shows the difference in mean waiting times computed with and without exponential service times assumptions. It was observed that the mean total waiting time was higher when service times were assumed to be exponentially distributed than to when no such assumption was made. In many service processes, the exponen- tial assumption of the distribution of service times usually results in more conservative queuing system designs. This is because, given the same system configurations, the
Case Study: Manpower Resource Planning 67
9.0 19.0 29.0 39.0 49.0 59.0 69.0 79.0 89.0
8 9 10 11
Number of Packers
Waiting Times
8 Dispensing Pharmacists (M/M/S) 8 Dispensing Pharmacists (M/G/S) 9 Dispensing Pharmacists (M/M/S) 9 Dispensing Pharmacists (M/G/S) 10 Dispensing Pharmacists (M/M/S) 10 Dispensing Pharmacists (M/G/S) 11 Dispensing Pharmacists (M/M/S) 11 Dispensing Pharmacists (M/G/S)
(a)
10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0
8 9 10 11
Number of Dispensing Pharmacists
Waiting Times
8 Packers (M/M/S) 8 Packers (M/G/S) 9 Packers (M/M/S) 9 Packers (M/G/S) 10 Packers (M/M/S) 10 Packers (M/G/S) 11 Packers (M/M/S) 11 Packers (M/G/S)
(b)
Figure 5.6 Comparisons of mean total waiting times computed with and without assumptions of exponential service times.
expected waiting times predicted with queuing models assuming exponentially dis- tributed inter-arrival and service times will be higher than with models assuming any other distributional assumptions whose coefficient of variation is less than unity.
Decisions based on mean waiting times and queue lengths predicted from such models would thus err on the safe side.
68 Fortifying Six Sigma with OR/MS Tools