October 2017, 13(4): 1759-1770. doi: 10.3934/jimo.2017017

Minimization of the coefficient of variation for patient waiting system governed by a generic maximum waiting policy

1. 

Department of Health Services and Outcomes Research, National Healthcare Group, 138543, Singapore

2. 

Department of Emergency Medicine, Tan Tock Seng Hospital, 308433, Singapore

* Corresponding author

Received  March 2016 Published  December 2016

Timely access of care has been widely recognized as an important dimension of health care quality. Waiting times can affect patient satisfaction and quality of care in the emergency department (ED). This study analyzes a general patient waiting policy such that ED patients who wait beyond a threshold have their wait shortened. Assuming that the policy is implemented to accelerate the long-waiting cases within a short time interval, we transform the original waiting distribution to a piecewise distribution. The objective of this paper is to examine the reliability of the induced waiting system by minimizing the coefficient of variation (CV) of waiting time. We convert the CV minimization problem to an approximation counterpart using the sampling technique. With patient waiting time data from an emergency department in Singapore, we derive the optimal values of parameters, such as the threshold and the length of the underlying time interval, needed in the policy. Numerical results show that CV and variance of new waiting time will be reduced remarkably by 38% and 58% respectively, in comparison with the original ones.

Citation: Fanwen Meng, Kiok Liang Teow, Chee Kheong Ooi, Bee Hoon Heng, Seow Yian Tay. Minimization of the coefficient of variation for patient waiting system governed by a generic maximum waiting policy. Journal of Industrial & Management Optimization, 2017, 13 (4) : 1759-1770. doi: 10.3934/jimo.2017017
References:
[1]

Australian Institute of Health and Welfare (AIHW), Australian Hospital Statistics 2003. 04. (Canberra: AIHW, 2005) (AIHW cat. no. HSE 37; Health Services series no. 23).

[2]

B. BurschJ. Beezy and R. Shaw, Emergency department satisfaction: What matters most?, Ann. Emerg. Med., 22 (1993), 586-591. doi: 10.1016/S0196-0644(05)81947-X.

[3]

L. G. Connelly and A. E. Bair, Discrete event simulation of emergency department activity: A platform for system-level operations research, Acad. Emerg. Med., 11 (2004), 1177-1185.

[4]

B. L. Conner-SpadyC. SanmartinG. H. JohnstonJ. J. McGurranM. Kehler and T. W. Noseworthy, The importance of patient expectations as a determinant of satisfaction with waiting times for hip and knee replacement surgery, Health Policy, 101 (2011), 245-252. doi: 10.1016/j.healthpol.2011.05.011.

[5]

P. DeJ. B. Ghosh and C. E. Wells, Scheduling to minimize the coefficient of variation, Int. J. Prod. Econ., 44 (1996), 249-253. doi: 10.1016/0925-5273(96)00056-4.

[6]

R. L. GardnerU. Sarkar and J. H. Maselli, Factors associated with longer ED lengths of stay, Am. J. Emerg. Med., 25 (2007), 643-650. doi: 10.1016/j.ajem.2006.11.037.

[7]

L. V. Green and S. Savin, Reducing delays for medical appointments: A queueing approach, Oper. Res., 56 (2008), 1526-1538. doi: 10.1287/opre.1080.0575.

[8]

A. Guttmann, M. J. Schull, M. J. Vermeulen and T. A. Stukel, Association between waiting times and short term mortality and hospital admission after departure from emergency department: Population based cohort study from Ontario, Canada, BMJ, 342 (2011). doi: 10.1136/bmj.d2983.

[9]

P. R. Harper and H. M. Gamlin, Reduced outpatient waiting times with improved appointment scheduling: A simulation modelling approach, OR Spectrum, 25 (2003), 207-222. doi: 10.1007/s00291-003-0122-x.

[10]

S. A. Hemaya and T. E. Locke, How accurate are predicted waiting times, determined upon a patient's arrival in the Emergency Department?, Emerg. Med. J., 29 (2012), 316-318. doi: 10.1136/emj.2010.106534.

[11]

E. T. Ho, Improving waiting time and operational clinic flow in a tertiary diabetes center BMJ Qual. Improv. Rep. , 2 (2014). doi: 10.1136/bmjquality.u201918.w1006.

[12]

M. A. KallenJ. A. TerrellP. Lewis-Patterson and J. P. Hwang, Improving wait time for chemotherapy in an outpatient clinic at a comprehensive cancer center, J. Oncol. Practice, 8 (2012), e1-e7. doi: 10.1200/JOP.2011.000281.

[13]

K. Kelly, Sample size planning for the coefficient of variation from the accuracy in parameter estimation approach, Behav. Res. Methods, 39 (2007), 755-766. doi: 10.3758/BF03192966.

[14]

D. Kozlowski and D. Worthington, Use of queue modelling in the analysis of elective patient treatment governed by a maximum waiting time policy, European Journal of Operational Research, 244 (2015), 331-338. doi: 10.1016/j.ejor.2015.01.024.

[15]

J. LibreroM. MarinS. Peiro and A. V. Munujos, Exploring the impact of complications on length of stay in major surgery diagnosis-related groups, Int. J. Qual. Health Care, 16 (2004), 51-57. doi: 10.1093/intqhc/mzh008.

[16]

F. MengK. L. TeowC. K. OoiB. H. Heng and S. Y. Tay, Analysis of patient waiting time governed by a generic maximum waiting time policy with general phase-type approximations, Health Care Manag. Sci., 18 (2015), 267-278. doi: 10.1007/s10729-014-9308-9.

[17]

F. MengJ. QiM. ZhangJ. AngS. Chu and M. Sim, A robust optimization model for managing elective admision in a public hospital, Oper. Res., 63 (2015), 1452-1467. doi: 10.1287/opre.2015.1423.

[18]

MOH (Ministry of Health) Singapore, 2012. Available from: http://www.moh.gov.sg/content/moh_web/home/statistics/healthcare_institutionstatistics/Waiting_Time_for_Admission_to_Ward.html.

[19]

Ministry of Health, MOH Statistics Bulletin, Singapore, 2009.

[20]

J. C. MowenJ. W. Licata and J. McPhail, Waiting in the emergency room: how to improve patient satisfaction, J. Health Care Mark., 13 (1993), 26-33.

[21]

H. C. NdukweS. Omale and O. O. Opanuga, Reducing queues in a Nigerian hospital pharmacy, Afr. J. Pharm. Pharmacol., 5 (2011), 1020-1026.

[22]

H. P. Phua, Waiting Times at Public Sector Emergency Departments, Working paper, Health Information Management Branch, Ministry of Health, Singapore, 2005.

[23]

P. K. PlunkettD. G. ByrneT. BreslinK. Bennett and B. Silke, Increasing wait times predict increasing mortality for emergency medical admissions, Eur. J. Emerg. Med., 18 (2011), 192-196. doi: 10.1097/MEJ.0b013e328344917e.

[24]

W. RaufJ. J. BlitzM. M. Geyser and A. Rauf, Quality improvement cycles that reduced waiting times at Tshwane district hospital emergency department, S. Afr. Fam. Pract., 50 (2008), 43-43e. doi: 10.1080/20786204.2008.10873781.

[25]

G. F. ReedF. Lynn and B. D. Meade, Use of coefficient of variation in assessing variability of of quantitative assays, Clin. Diagn. Lab. Immunol., 9 (2002), 1235-1239.

[26]

D. RuM. L. McCarthy and J. S. Desmond, Characterizing waiting room time, treatment time, and boarding time in the emergency department using quantile regression, Acad. Emerg. Med., 17 (2010), 813-823.

[27]

L. Siciliani and J. Hurst, Tackling excessive waiting times for elective surgery: A comparative analysis of policies in 12 OECD countries, Health Policy, 72 (2005), 201-215. doi: 10.1016/j.healthpol.2004.07.003.

[28]

Y. SunK. L. TeowB. H. HengC. K. Ooi and S. Y. Tay, Real time prediction of waiting time in the emergency department using quantile regression, Ann. Emerg. Med., 60 (2012), 299-308. doi: 10.1016/j.annemergmed.2012.03.011.

[29]

H. Tekiner and D. Coit, System reliability optimization considering uncertainty: minimization of a coefficient of variation measure, Proceedings of the 2008 Industrial Engineering Research Conference, (2008), 995-1000.

[30]

D. A. ThompsonP. R. YarnoldD. R. Williams and S. L. Adams, Effects of actual waiting time, perceived waiting time, information delivery, and expressive quality on patient satisfaction in the emergency department, Ann. Emerg. Med., 28 (1996), 657-665. doi: 10.1016/S0196-0644(96)70090-2.

[31]

D. A. Thompson and P. R. Yarnold, Relating patient satisfaction to waiting time perceptions and expectations: the disconfirmation paradigm, Acad. Emerg. Med., 2 (1995), 1057-1062. doi: 10.1111/j.1553-2712.1995.tb03150.x.

[32]

E. U. WeberS. Shafir and A.-R. Blais, Predicting risk sensitivity in humans and lower annimals: Risk as variance or coefficient of variation, Psychol. Rev., 111 (2004), 430-445.

[33]

L. Zhao and B. Lie, Modeling and simulation of patient flow in hospitals for resource utilization, 2008. Available from: http://www.scansims.org/sims2008/02.pdf.

[34]

Z. C. Zhu, B. H. Heng and K. L. Teow, Reducing consultation waiting time and clinic overtime in outpatient clinic: challenges and solution, in Management Engineering for Effective Healthcare Delivery: Principles and Applications, Medical Information Science Reference (eds. A. Kolker and P. Story), Hershey, Pennsylvania, (2011), 229-245.

show all references

References:
[1]

Australian Institute of Health and Welfare (AIHW), Australian Hospital Statistics 2003. 04. (Canberra: AIHW, 2005) (AIHW cat. no. HSE 37; Health Services series no. 23).

[2]

B. BurschJ. Beezy and R. Shaw, Emergency department satisfaction: What matters most?, Ann. Emerg. Med., 22 (1993), 586-591. doi: 10.1016/S0196-0644(05)81947-X.

[3]

L. G. Connelly and A. E. Bair, Discrete event simulation of emergency department activity: A platform for system-level operations research, Acad. Emerg. Med., 11 (2004), 1177-1185.

[4]

B. L. Conner-SpadyC. SanmartinG. H. JohnstonJ. J. McGurranM. Kehler and T. W. Noseworthy, The importance of patient expectations as a determinant of satisfaction with waiting times for hip and knee replacement surgery, Health Policy, 101 (2011), 245-252. doi: 10.1016/j.healthpol.2011.05.011.

[5]

P. DeJ. B. Ghosh and C. E. Wells, Scheduling to minimize the coefficient of variation, Int. J. Prod. Econ., 44 (1996), 249-253. doi: 10.1016/0925-5273(96)00056-4.

[6]

R. L. GardnerU. Sarkar and J. H. Maselli, Factors associated with longer ED lengths of stay, Am. J. Emerg. Med., 25 (2007), 643-650. doi: 10.1016/j.ajem.2006.11.037.

[7]

L. V. Green and S. Savin, Reducing delays for medical appointments: A queueing approach, Oper. Res., 56 (2008), 1526-1538. doi: 10.1287/opre.1080.0575.

[8]

A. Guttmann, M. J. Schull, M. J. Vermeulen and T. A. Stukel, Association between waiting times and short term mortality and hospital admission after departure from emergency department: Population based cohort study from Ontario, Canada, BMJ, 342 (2011). doi: 10.1136/bmj.d2983.

[9]

P. R. Harper and H. M. Gamlin, Reduced outpatient waiting times with improved appointment scheduling: A simulation modelling approach, OR Spectrum, 25 (2003), 207-222. doi: 10.1007/s00291-003-0122-x.

[10]

S. A. Hemaya and T. E. Locke, How accurate are predicted waiting times, determined upon a patient's arrival in the Emergency Department?, Emerg. Med. J., 29 (2012), 316-318. doi: 10.1136/emj.2010.106534.

[11]

E. T. Ho, Improving waiting time and operational clinic flow in a tertiary diabetes center BMJ Qual. Improv. Rep. , 2 (2014). doi: 10.1136/bmjquality.u201918.w1006.

[12]

M. A. KallenJ. A. TerrellP. Lewis-Patterson and J. P. Hwang, Improving wait time for chemotherapy in an outpatient clinic at a comprehensive cancer center, J. Oncol. Practice, 8 (2012), e1-e7. doi: 10.1200/JOP.2011.000281.

[13]

K. Kelly, Sample size planning for the coefficient of variation from the accuracy in parameter estimation approach, Behav. Res. Methods, 39 (2007), 755-766. doi: 10.3758/BF03192966.

[14]

D. Kozlowski and D. Worthington, Use of queue modelling in the analysis of elective patient treatment governed by a maximum waiting time policy, European Journal of Operational Research, 244 (2015), 331-338. doi: 10.1016/j.ejor.2015.01.024.

[15]

J. LibreroM. MarinS. Peiro and A. V. Munujos, Exploring the impact of complications on length of stay in major surgery diagnosis-related groups, Int. J. Qual. Health Care, 16 (2004), 51-57. doi: 10.1093/intqhc/mzh008.

[16]

F. MengK. L. TeowC. K. OoiB. H. Heng and S. Y. Tay, Analysis of patient waiting time governed by a generic maximum waiting time policy with general phase-type approximations, Health Care Manag. Sci., 18 (2015), 267-278. doi: 10.1007/s10729-014-9308-9.

[17]

F. MengJ. QiM. ZhangJ. AngS. Chu and M. Sim, A robust optimization model for managing elective admision in a public hospital, Oper. Res., 63 (2015), 1452-1467. doi: 10.1287/opre.2015.1423.

[18]

MOH (Ministry of Health) Singapore, 2012. Available from: http://www.moh.gov.sg/content/moh_web/home/statistics/healthcare_institutionstatistics/Waiting_Time_for_Admission_to_Ward.html.

[19]

Ministry of Health, MOH Statistics Bulletin, Singapore, 2009.

[20]

J. C. MowenJ. W. Licata and J. McPhail, Waiting in the emergency room: how to improve patient satisfaction, J. Health Care Mark., 13 (1993), 26-33.

[21]

H. C. NdukweS. Omale and O. O. Opanuga, Reducing queues in a Nigerian hospital pharmacy, Afr. J. Pharm. Pharmacol., 5 (2011), 1020-1026.

[22]

H. P. Phua, Waiting Times at Public Sector Emergency Departments, Working paper, Health Information Management Branch, Ministry of Health, Singapore, 2005.

[23]

P. K. PlunkettD. G. ByrneT. BreslinK. Bennett and B. Silke, Increasing wait times predict increasing mortality for emergency medical admissions, Eur. J. Emerg. Med., 18 (2011), 192-196. doi: 10.1097/MEJ.0b013e328344917e.

[24]

W. RaufJ. J. BlitzM. M. Geyser and A. Rauf, Quality improvement cycles that reduced waiting times at Tshwane district hospital emergency department, S. Afr. Fam. Pract., 50 (2008), 43-43e. doi: 10.1080/20786204.2008.10873781.

[25]

G. F. ReedF. Lynn and B. D. Meade, Use of coefficient of variation in assessing variability of of quantitative assays, Clin. Diagn. Lab. Immunol., 9 (2002), 1235-1239.

[26]

D. RuM. L. McCarthy and J. S. Desmond, Characterizing waiting room time, treatment time, and boarding time in the emergency department using quantile regression, Acad. Emerg. Med., 17 (2010), 813-823.

[27]

L. Siciliani and J. Hurst, Tackling excessive waiting times for elective surgery: A comparative analysis of policies in 12 OECD countries, Health Policy, 72 (2005), 201-215. doi: 10.1016/j.healthpol.2004.07.003.

[28]

Y. SunK. L. TeowB. H. HengC. K. Ooi and S. Y. Tay, Real time prediction of waiting time in the emergency department using quantile regression, Ann. Emerg. Med., 60 (2012), 299-308. doi: 10.1016/j.annemergmed.2012.03.011.

[29]

H. Tekiner and D. Coit, System reliability optimization considering uncertainty: minimization of a coefficient of variation measure, Proceedings of the 2008 Industrial Engineering Research Conference, (2008), 995-1000.

[30]

D. A. ThompsonP. R. YarnoldD. R. Williams and S. L. Adams, Effects of actual waiting time, perceived waiting time, information delivery, and expressive quality on patient satisfaction in the emergency department, Ann. Emerg. Med., 28 (1996), 657-665. doi: 10.1016/S0196-0644(96)70090-2.

[31]

D. A. Thompson and P. R. Yarnold, Relating patient satisfaction to waiting time perceptions and expectations: the disconfirmation paradigm, Acad. Emerg. Med., 2 (1995), 1057-1062. doi: 10.1111/j.1553-2712.1995.tb03150.x.

[32]

E. U. WeberS. Shafir and A.-R. Blais, Predicting risk sensitivity in humans and lower annimals: Risk as variance or coefficient of variation, Psychol. Rev., 111 (2004), 430-445.

[33]

L. Zhao and B. Lie, Modeling and simulation of patient flow in hospitals for resource utilization, 2008. Available from: http://www.scansims.org/sims2008/02.pdf.

[34]

Z. C. Zhu, B. H. Heng and K. L. Teow, Reducing consultation waiting time and clinic overtime in outpatient clinic: challenges and solution, in Management Engineering for Effective Healthcare Delivery: Principles and Applications, Medical Information Science Reference (eds. A. Kolker and P. Story), Hershey, Pennsylvania, (2011), 229-245.

Table 1.  Basic Statistics of PAC3 Waiting Time Data
Sample Size27,689
Min0 (minute)
25th Percentile21.1 (minutes)
Median40.1 (minutes)
Mean50.6 (minutes)
75th Percentile69.6 (minutes)
95th Percentile128.3 (minutes)
Max321.8 (minutes)
Standard Deviation39.1 (minutes)
Variance1528.7
Coefficient of Variation0.772
Skewness1.454
Sample Size27,689
Min0 (minute)
25th Percentile21.1 (minutes)
Median40.1 (minutes)
Mean50.6 (minutes)
75th Percentile69.6 (minutes)
95th Percentile128.3 (minutes)
Max321.8 (minutes)
Standard Deviation39.1 (minutes)
Variance1528.7
Coefficient of Variation0.772
Skewness1.454
Table 2.  Coefficient of Variation under Different Scenarios on $t$ and $l$
$t$the length $l$ of time interval
1012141516171819212325
1280.6240.6180.6130.6100.6060.6030.6000.5960.5910.5820.576
1270.6200.6140.6080.6050.6010.5980.5950.5910.5860.5790.571
1260.6150.6100.6040.6000.5970.5940.5900.5880.5790.5730.567
1250.6110.6060.5990.5960.5920.5890.5860.5830.5760.5680.562
1240.6070.6010.5940.5910.5870.5850.5820.5770.5700.5630.556
1230.6030.5960.5890.5860.5830.5800.5750.5730.5650.5580.551
1220.5980.5910.5840.5810.5790.5740.5710.5670.5600.5530.546
1210.5930.5860.5800.5770.5720.5700.5650.5620.5550.5480.541
1200.5880.5810.5750.5710.5680.5640.5600.5570.5490.5430.535
1190.5840.5770.5690.5660.5620.5590.5550.5510.5440.5380.528
1180.5780.5720.5650.5610.5570.5530.5500.5460.5390.5310.523
1170.5730.5660.5590.5550.5520.5480.5440.5410.5340.5250.515
1160.5690.5610.5540.5500.5460.5420.5390.5350.5280.5190.508
1150.5630.5560.5480.5440.5410.5370.5330.5300.5210.5120.503
1140.5580.5500.5430.5390.5350.5310.5280.5240.5150.5040.496
1130.5520.5450.5370.5330.5290.5260.5220.5170.5080.4990.489
1120.5470.5390.5310.5280.5240.5200.5150.5110.5000.4920.483
1110.5410.5330.5260.5220.5180.5130.5090.5040.4950.4850.473
1100.5350.5270.5200.5160.5110.5070.5020.4970.4880.4780.466
1090.5290.5220.5140.5100.5050.5000.4950.4900.4810.4690.461
1080.5240.5160.5080.5030.4980.4930.4880.4840.4740.4620.452
1070.5180.5100.5010.4960.4910.4860.4810.4760.4650.4560.444
1060.5120.5040.4940.4890.4840.4790.4740.4690.4570.4470.434
$t$the length $l$ of time interval
1012141516171819212325
1280.6240.6180.6130.6100.6060.6030.6000.5960.5910.5820.576
1270.6200.6140.6080.6050.6010.5980.5950.5910.5860.5790.571
1260.6150.6100.6040.6000.5970.5940.5900.5880.5790.5730.567
1250.6110.6060.5990.5960.5920.5890.5860.5830.5760.5680.562
1240.6070.6010.5940.5910.5870.5850.5820.5770.5700.5630.556
1230.6030.5960.5890.5860.5830.5800.5750.5730.5650.5580.551
1220.5980.5910.5840.5810.5790.5740.5710.5670.5600.5530.546
1210.5930.5860.5800.5770.5720.5700.5650.5620.5550.5480.541
1200.5880.5810.5750.5710.5680.5640.5600.5570.5490.5430.535
1190.5840.5770.5690.5660.5620.5590.5550.5510.5440.5380.528
1180.5780.5720.5650.5610.5570.5530.5500.5460.5390.5310.523
1170.5730.5660.5590.5550.5520.5480.5440.5410.5340.5250.515
1160.5690.5610.5540.5500.5460.5420.5390.5350.5280.5190.508
1150.5630.5560.5480.5440.5410.5370.5330.5300.5210.5120.503
1140.5580.5500.5430.5390.5350.5310.5280.5240.5150.5040.496
1130.5520.5450.5370.5330.5290.5260.5220.5170.5080.4990.489
1120.5470.5390.5310.5280.5240.5200.5150.5110.5000.4920.483
1110.5410.5330.5260.5220.5180.5130.5090.5040.4950.4850.473
1100.5350.5270.5200.5160.5110.5070.5020.4970.4880.4780.466
1090.5290.5220.5140.5100.5050.5000.4950.4900.4810.4690.461
1080.5240.5160.5080.5030.4980.4930.4880.4840.4740.4620.452
1070.5180.5100.5010.4960.4910.4860.4810.4760.4650.4560.444
1060.5120.5040.4940.4890.4840.4790.4740.4690.4570.4470.434
Table 3.  Estimates of CV, Mean and Variance Changes with $t=106$
length $l$121415161718192123
CV0.500.490.490.480.480.470.470.460.45
New mean52.2252.3652.4352.5652.6452.7152.7952.9553.10
New variance691.37669.14656.33647.02636.20624.37613.49586.38563.09
CV reduction34.8%36.1%36.8%37.4%38.0%38.7%39.3%40.8%42.2%
Mean increase3.1%3.4%3.5%3.6%3.9%4.0%4.2%4.4%5.0%
Var reduction54.1%55.4%56.2%57.1%58.0 % 58.4%59.2%61.0%62.0%
length $l$121415161718192123
CV0.500.490.490.480.480.470.470.460.45
New mean52.2252.3652.4352.5652.6452.7152.7952.9553.10
New variance691.37669.14656.33647.02636.20624.37613.49586.38563.09
CV reduction34.8%36.1%36.8%37.4%38.0%38.7%39.3%40.8%42.2%
Mean increase3.1%3.4%3.5%3.6%3.9%4.0%4.2%4.4%5.0%
Var reduction54.1%55.4%56.2%57.1%58.0 % 58.4%59.2%61.0%62.0%
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