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doi: 10.3934/jimo.2021097

Designing an annual leave scheduling policy: Case of a financial center

Department of Industrial Engineering, Çankaya University, Ankara, 06790, Turkey

* Corresponding author: Gonca Yıldırım

Received  November 2020 Revised  March 2021 Published  May 2021

Providing annual leave entitlements for employees can help alleviate burnout since paid-time off work directly affects the health and productivity of workers as well as the quality of the service provided. In this paper, we develop realistic vacation scheduling policies and investigate how they compare from both the employer and the employees' perspectives. Among those policies, we consider one that is used in practice, another that we propose as a compromise which performs very well in most cases, and one that is similar to machine scheduling for benchmarking. Integer programming models are formulated and solved under various settings for workload distribution over time, substitution and unit of time for vacations. We use three performance measures for comparisons: penalty cost of unused vacation days, percent vacation granted and level of employee satisfaction. We provide a real-life case study at a bank's financial center. Numerical results suggest that an all-or-nothing type of vacation policy performs economically worse than the others. Attractive annual leave scheduling policies can be designed by administering vacation schedules daily rather than weekly, ensuring full cover for off-duty employees, and offering employees some degree of choice over vacation schedules.

Citation: Gonca Yıldırım, Ayyuce Aydemir-Karadag. Designing an annual leave scheduling policy: Case of a financial center. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021097
References:
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[2]

S. M. Al-Yakoob and H. D. Sherali, Mixed-integer programming models for an employee scheduling problem with multiple shifts and work locations, Annals of Operations Research, 155 (2007), 119-142.  doi: 10.1007/s10479-007-0210-4.  Google Scholar

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S. M. Al-Yakoob and H. D. Sherali, Multiple shift scheduling of hierarchical workforce with multiple work centers, Informatica, 18 (2007), 325-342.  doi: 10.15388/Informatica.2007.180.  Google Scholar

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S. M. Al-Yakoob and H. D. Sherali, A column generation approach for an employee scheduling problem with multiple shifts and work locations, Journal of the Operational Research Society, 59 (2008), 34-43.   Google Scholar

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H. K. Alfares, Four-day workweek scheduling with two or three consecutive days off, Journal of Mathematical Modelling and Algorithms, 2 (2003), 67-80.  doi: 10.1023/A:1023671623927.  Google Scholar

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C. S. AzmatT. Hürlimann and M. Widmer, Mixed integer programming to schedule a single-shift workforce under annualized hours, Annals of Operations Research, 128 (2004), 199-215.  doi: 10.1023/B:ANOR.0000019105.54898.a4.  Google Scholar

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C. S. Azmat and M. Widmer, A case study of single shift planning and scheduling under annualized hours: A simple three-step approach, European Journal of Operational Research, 153 (2004), 148-175.  doi: 10.1016/S0377-2217(03)00105-X.  Google Scholar

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R. Burns and R. Narasimhan, Multiple shift scheduling of workforce on four-day workweeks, The Journal of the Operational Research Society, 50 (1999), 979-981.  doi: 10.1057/palgrave.jors.2600726.  Google Scholar

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G. Cairncross and I. Waller, Not taking annual leave: What could it cost Australia?, Journal of Economic and Social Policy, 9 (2004), 1-17.   Google Scholar

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A. CorominasA. Lusa and R. Pastor, Using MILP to plan annualised working hours, The Journal of the Operational Research Society, 53 (2002), 1101-1108.  doi: 10.1057/palgrave.jors.2601309.  Google Scholar

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A. CorominasA. Lusa and R. Pastor, Characteristics and classification of the annualised working hours planning problems, International Journal of Services, Technology and Management, 5 (2004), 435-447.  doi: 10.1504/IJSTM.2004.006276.  Google Scholar

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A. CorominasA. Lusa and R. Pastor, Planning annualised hours with a finite set of weekly working hours and cross-trained workers, European Journal of Operational Research, 176 (2007), 230-239.  doi: 10.1016/j.ejor.2005.06.048.  Google Scholar

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A. CorominasA. Lusa and R. Pastor, Using a MILP model to establish a framework for an annualised hours agreement, European Journal of Operational Research, 177 (2007), 1495-1506.  doi: 10.1016/j.ejor.2005.04.017.  Google Scholar

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R. de la TorreA. LusaM. Mateo and E.-H. Aghezzaf, Determining personnel promotion policies in HEI, Journal of Industrial & Management Optimization, 16 (2020), 1835-1859.  doi: 10.3934/jimo.2019031.  Google Scholar

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A. Earle and J. Heymann, A comparative analysis of paid leave for the health needs of workers and their families around the world, Journal of Comparative Policy Analysis: Research and Practice, 8 (2006), 241-257.  doi: 10.1080/13876980600858465.  Google Scholar

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M. Elshafei and H. K. Alfares, A dynamic programming algorithm for days-off scheduling with sequence dependent labor costs, Journal of Scheduling, 11 (2008), 85-93.  doi: 10.1007/s10951-007-0040-x.  Google Scholar

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show all references

References:
[1]

S. AğralıZ. C. Taşkın and A. T. Ünal, Employee scheduling in service industries with flexible employee availability and demand, Omega, 66 (2017), 159-169.  doi: 10.1016/j.omega.2016.03.001.  Google Scholar

[2]

S. M. Al-Yakoob and H. D. Sherali, Mixed-integer programming models for an employee scheduling problem with multiple shifts and work locations, Annals of Operations Research, 155 (2007), 119-142.  doi: 10.1007/s10479-007-0210-4.  Google Scholar

[3]

S. M. Al-Yakoob and H. D. Sherali, Multiple shift scheduling of hierarchical workforce with multiple work centers, Informatica, 18 (2007), 325-342.  doi: 10.15388/Informatica.2007.180.  Google Scholar

[4]

S. M. Al-Yakoob and H. D. Sherali, A column generation approach for an employee scheduling problem with multiple shifts and work locations, Journal of the Operational Research Society, 59 (2008), 34-43.   Google Scholar

[5]

H. K. Alfares, Four-day workweek scheduling with two or three consecutive days off, Journal of Mathematical Modelling and Algorithms, 2 (2003), 67-80.  doi: 10.1023/A:1023671623927.  Google Scholar

[6]

C. S. AzmatT. Hürlimann and M. Widmer, Mixed integer programming to schedule a single-shift workforce under annualized hours, Annals of Operations Research, 128 (2004), 199-215.  doi: 10.1023/B:ANOR.0000019105.54898.a4.  Google Scholar

[7]

C. S. Azmat and M. Widmer, A case study of single shift planning and scheduling under annualized hours: A simple three-step approach, European Journal of Operational Research, 153 (2004), 148-175.  doi: 10.1016/S0377-2217(03)00105-X.  Google Scholar

[8]

K. R. Baker, Workforce allocation in cyclical scheduling problems: A survey, Operational Research Quarterly, 27 (1976), 155-167.  doi: 10.1057/jors.1976.30.  Google Scholar

[9]

K. R. Baker and M. J. Magazine, Workforce scheduling with cyclic demands and day-off constraints, Management Science, 24 (1977), 161-167.  doi: 10.1287/mnsc.24.2.161.  Google Scholar

[10]

J. F. Bard and H. W. Purnomo, A column generation-based approach to solve the preference scheduling problem for nurses with downgrading, Socio-Economic Planning Sciences, 39 (2005), 193-213.  doi: 10.1016/j.seps.2004.04.001.  Google Scholar

[11]

I. BerradaJ. A. Ferland and P. Michelon, A multi-objective approach to nurse scheduling with both hard and soft constraints, Socio-Economic Planning Sciences, 30 (1996), 183-193.  doi: 10.1016/0038-0121(96)00010-9.  Google Scholar

[12]

A. Billionnet, Integer programming to schedule a hierarchical workforce with variable demands, European Journal of Operational Research, 114 (1999), 105-114.  doi: 10.1016/S0377-2217(98)00182-9.  Google Scholar

[13]

J. O. BrunnerJ. F. Bard and J. M. Köhler, Bounded flexibility in days-on and days-off scheduling, Naval Research Logistics, 60 (2013), 678-701.  doi: 10.1002/nav.21561.  Google Scholar

[14]

R. Burns and R. Narasimhan, Multiple shift scheduling of workforce on four-day workweeks, The Journal of the Operational Research Society, 50 (1999), 979-981.  doi: 10.1057/palgrave.jors.2600726.  Google Scholar

[15]

R. N. BurnsR. Narasimhan and L. D. Smith, A set-processing algorithm for scheduling staff on 4-day or 3-day work weeks, Naval Research Logistics, 45 (1998), 839-853.  doi: 10.1002/(SICI)1520-6750(199812)45:8<839::AID-NAV5>3.0.CO;2-R.  Google Scholar

[16]

G. Cairncross and I. Waller, Not taking annual leave: What could it cost Australia?, Journal of Economic and Social Policy, 9 (2004), 1-17.   Google Scholar

[17]

A. CorominasA. Lusa and R. Pastor, Using MILP to plan annualised working hours, The Journal of the Operational Research Society, 53 (2002), 1101-1108.  doi: 10.1057/palgrave.jors.2601309.  Google Scholar

[18]

A. CorominasA. Lusa and R. Pastor, Characteristics and classification of the annualised working hours planning problems, International Journal of Services, Technology and Management, 5 (2004), 435-447.  doi: 10.1504/IJSTM.2004.006276.  Google Scholar

[19]

A. CorominasA. Lusa and R. Pastor, Planning annualised hours with a finite set of weekly working hours and joint holidays, Annals of Operations Research, 128 (2004), 217-233.  doi: 10.1023/B:ANOR.0000019106.52631.ff.  Google Scholar

[20]

A. CorominasA. Lusa and R. Pastor, Planning annualised hours with a finite set of weekly working hours and cross-trained workers, European Journal of Operational Research, 176 (2007), 230-239.  doi: 10.1016/j.ejor.2005.06.048.  Google Scholar

[21]

A. CorominasA. Lusa and R. Pastor, Using a MILP model to establish a framework for an annualised hours agreement, European Journal of Operational Research, 177 (2007), 1495-1506.  doi: 10.1016/j.ejor.2005.04.017.  Google Scholar

[22]

Coulthard Barnes and Perpetual Guardian, The four-day week is here, 2019, https://4dayweek.com/. Google Scholar

[23]

R. de la TorreA. LusaM. Mateo and E.-H. Aghezzaf, Determining personnel promotion policies in HEI, Journal of Industrial & Management Optimization, 16 (2020), 1835-1859.  doi: 10.3934/jimo.2019031.  Google Scholar

[24]

R. Denniss, Paid annual leave in Australia: An analysis of actual and desired entitlements, Labour & Industry: A Journal of the Social and Economic Relations of Work, 15 (2004), 1-16.  doi: 10.1080/10301763.2004.10669301.  Google Scholar

[25]

S. Dewess, Socially acceptable annual holiday planning for the crew of a local public transport company in Germany, Public Transport, 2 (2010), 25-49.  doi: 10.1007/s12469-010-0019-4.  Google Scholar

[26]

A. Earle and J. Heymann, A comparative analysis of paid leave for the health needs of workers and their families around the world, Journal of Comparative Policy Analysis: Research and Practice, 8 (2006), 241-257.  doi: 10.1080/13876980600858465.  Google Scholar

[27]

M. Elshafei and H. K. Alfares, A dynamic programming algorithm for days-off scheduling with sequence dependent labor costs, Journal of Scheduling, 11 (2008), 85-93.  doi: 10.1007/s10951-007-0040-x.  Google Scholar

[28]

H. Emmons and D.-S. Fuh, Sizing and scheduling a full-time and part-time workforce with off-day and off-weekend constraints, Annals of Operations Research, 70 (1997), 473-492.  doi: 10.1023/A:1018998826960.  Google Scholar

[29]

EU, European Working Time Directive, vol. 46, Official Journal of the European Union, L 299, 18 November, 2003. Google Scholar

[30]

H. J. Freudenberger, Staff burn-out, Journal of Social Issues, 30 (1974), 159-165.  doi: 10.1111/j.1540-4560.1974.tb00706.x.  Google Scholar

[31]

N. Ghosheh, Remembering Rest Periods in Law: Another Tool to Limit Excessive Working Hours, Conditions of Work and Employment Series 78, International Labour Office, Geneva, Switzerland, 2016, https://www.ilo.org/wcmsp5/groups/public/---ed_protect/---protrav/---travail/documents/publication/wcms_516123.pdf. Google Scholar

[32]

J. GohJ. Pfeffer and S. A. Zenios, The relationship between workplace stressors and mortality and health costs in the United States, Management Science, 62 (2015), 608-628.  doi: 10.1287/mnsc.2014.2115.  Google Scholar

[33]

F. Hadwiger and V. Schmidt, Negotiating for Decent Working Time - a Review of Practice, Fact Sheet 5, International Labour Office, Geneva, Switzerland, 2019, https://www.ilo.org/global/topics/collective-bargaining-labour-relations/publications/WCMS_732080/lang--en/index.htm. Google Scholar

[34]

L. V. Heinemann and T. Heinemann, Burnout research: Emergence and scientific investigation of a contested diagnosis, SAGE Open, 7 (2017), 1-12.  doi: 10.1177/2158244017697154.  Google Scholar

[35]

C. O. HenriquesM. LuqueO. D. Marcenaro-Gutierrez and L. A. Lopez-Agudo, A multiobjective interval programming model to explore the trade-offs among different aspects of job satisfaction under different scenarios, Socio-Economic Planning Sciences, 66 (2019), 35-46.  doi: 10.1016/j.seps.2018.07.004.  Google Scholar

[36]

A. HertzN. Lahrichi and M. Widmer, A flexible MILP model for multiple-shift workforce planning under annualized hours, European Journal of Operational Research, 200 (2010), 860-873.  doi: 10.1016/j.ejor.2009.01.045.  Google Scholar

[37]

R. Hung, Single-shift workforce scheduling under a compressed workweek, Omega, 19 (1991), 494-497.  doi: 10.1016/0305-0483(91)90067-4.  Google Scholar

[38]

R. Hung, A three-day workweek multiple-shift scheduling model, Journal of the Operational Research Society, 44 (1993), 141-146.  doi: 10.1057/jors.1993.26.  Google Scholar

[39]

R. Hung, Multiple-shift workforce scheduling under the 3-4 workweek with different weekday and weekend labor requirements, Management Science, 40 (1994), 280-284.  doi: 10.1287/mnsc.40.2.280.  Google Scholar

[40]

ILO, Paid Annual Leave, Fact Sheet WT-6, International Labour Office, Geneva, Switzerland, 2004, https://www.ilo.org/travail/info/fs/WCMS_170703/lang--en/index.htm. Google Scholar

[41]

D. Kim, Does paid vacation leave protect against depression among working Americans? A national longitudinal fixed effects analysis, Scandinavian Journal of Work, Environment & Health, 22-32, http://www.sjweh.fi/show_abstract.php?abstract_id=3751. doi: 10.5271/sjweh.3751.  Google Scholar

[42]

K. LiS. Xu and H. Fu, Work-break scheduling with real-time fatigue effect and recovery, International Journal of Production Research, 58 (2020), 689-702.  doi: 10.1080/00207543.2019.1598600.  Google Scholar

[43]

A. LusaA. Corominas and R. Pastor, An exact procedure for planning holidays and working time under annualized hours considering cross-trained workers with different efficiencies, International Journal of Production Research, 46 (2008), 2123-2142.  doi: 10.1080/00207540601080480.  Google Scholar

[44]

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Figure 1.  Daily and average task loads over the vacation horizon under different task load distributions
Figure 2.  Penalty cost paid in all problem instances
Figure 3.  Percent vacation granted in all problem instances
Figure 4.  Percent vacation granted in all problem instances (with substitution)
Figure 5.  Percent vacation granted in all problem instances (without substitution)
Figure 6.  Percent vacation granted from actual preferences in all problem instances
Figure 7.  Effect of $ \alpha $ with substitution
Figure 8.  Effect of $ \alpha $ without substitution
Figure 9.  Penalty cost paid versus percent vacation granted (overall and from preferences) in all problem instances (with substitution)
Figure 10.  Penalty cost paid versus percent vacation granted (overall and from preferences) in all problem instances (without substitution)
Table 1.  Sets, parameters and decision variables used in models
Sets:
$ T $ Vacation horizon, in days, indexed by $ t $, $ \tau $, $ h $ or $ \ell $
$ W $ Vacation horizon, in weeks, indexed by $ k $ or $ w $
$ J $ Set of tasks, indexed by $ j $
$ I $ Set of employees, indexed by $ i $
$ I_{j} $ Set of employees eligible to perform task type $ j $
Parameters:
$ N_{jt} $ Number of type $ j $ tasks on day $ t $
$ r_{ij} $ Time, in fraction of days, required for employee $ i $ to complete a type $ j $ task, $ r_{ij} \in (0, 1] $
$ A_{i} \; (A_{i}^{\prime}) $ Annual leave entitlement, in days (weeks), of employee $ i $
$ P_{i} $ Number of vacation periods in employee $ i $'s preferences, indexed by $ p $
$ s_{ip} \; (s_{ip}^{\prime}) $ Starting day (week) of the $ p $th vacation-period preference of employee $ i $
$ d_{ip} \; (d_{ip}^{\prime}) $ Duration, in days (weeks), of the $ p $th vacation-period preference of employee $ i $
$ m $ Upper bound on the number of times an entitlement can be split
$ f \; (f^{\prime}) $ Minimum length, in days (weeks), for at least one part of the entitlement
$ c_{i} $ Penalty cost per day for employee $ i $
$ \alpha $ Multiplier for rewarding days granted from employee preferences
Decision variables:
$ x_{ijt} $ Binary variable with value 1 if employee $ i $ does task $ j $ on day $ t $
$ y_{it} $ ($ z_{ik} $) Binary variable with value 1 if employee $ i $ is on vacation on day $ t $ (in week $ k $)
$ u_{ip} $ ($ v_{ip} $) Binary variable with value 1 if employee $ i $ uses the $ p $th preference period under daily (weekly) preferences
$ u_{ipth} $ ($ v_{ipkw} $) Binary variable with value 1 if employee $ i $ uses $ h $ days ($ w $ weeks) of the $ p $th preference period starting on day $ t $ (in week $ k $) of that period
$ u_{ith} $ ($ v_{ikw} $) Binary variable with value 1 if employee $ i $ starts a vacation period of $ h $ days ($ w $ weeks) on day $ t $ (in week $ k $)
$ q_{ip} $, $ q_{ipth} $, $ q_{ith} $ Binary variables for enforcing disjunctions
$ n_{i} $ Number of times employee $ i $ splits a vacation
Sets:
$ T $ Vacation horizon, in days, indexed by $ t $, $ \tau $, $ h $ or $ \ell $
$ W $ Vacation horizon, in weeks, indexed by $ k $ or $ w $
$ J $ Set of tasks, indexed by $ j $
$ I $ Set of employees, indexed by $ i $
$ I_{j} $ Set of employees eligible to perform task type $ j $
Parameters:
$ N_{jt} $ Number of type $ j $ tasks on day $ t $
$ r_{ij} $ Time, in fraction of days, required for employee $ i $ to complete a type $ j $ task, $ r_{ij} \in (0, 1] $
$ A_{i} \; (A_{i}^{\prime}) $ Annual leave entitlement, in days (weeks), of employee $ i $
$ P_{i} $ Number of vacation periods in employee $ i $'s preferences, indexed by $ p $
$ s_{ip} \; (s_{ip}^{\prime}) $ Starting day (week) of the $ p $th vacation-period preference of employee $ i $
$ d_{ip} \; (d_{ip}^{\prime}) $ Duration, in days (weeks), of the $ p $th vacation-period preference of employee $ i $
$ m $ Upper bound on the number of times an entitlement can be split
$ f \; (f^{\prime}) $ Minimum length, in days (weeks), for at least one part of the entitlement
$ c_{i} $ Penalty cost per day for employee $ i $
$ \alpha $ Multiplier for rewarding days granted from employee preferences
Decision variables:
$ x_{ijt} $ Binary variable with value 1 if employee $ i $ does task $ j $ on day $ t $
$ y_{it} $ ($ z_{ik} $) Binary variable with value 1 if employee $ i $ is on vacation on day $ t $ (in week $ k $)
$ u_{ip} $ ($ v_{ip} $) Binary variable with value 1 if employee $ i $ uses the $ p $th preference period under daily (weekly) preferences
$ u_{ipth} $ ($ v_{ipkw} $) Binary variable with value 1 if employee $ i $ uses $ h $ days ($ w $ weeks) of the $ p $th preference period starting on day $ t $ (in week $ k $) of that period
$ u_{ith} $ ($ v_{ikw} $) Binary variable with value 1 if employee $ i $ starts a vacation period of $ h $ days ($ w $ weeks) on day $ t $ (in week $ k $)
$ q_{ip} $, $ q_{ipth} $, $ q_{ith} $ Binary variables for enforcing disjunctions
$ n_{i} $ Number of times employee $ i $ splits a vacation
Table 2.  Parameter settings for employees and tasks
EG 1 2 3 4 5 6 7 8
NE 2 2 3 3 4 5 5 6
DC 8 7 6 5 4 3 2 1
TD 60 40 24 20 16 12 8 6
NT $ [5, 13] $ $ [9, 21] $ $ [17, 57] $ $ [21, 69] $ $ [29, 117] $ $ [37, 197] $ $ [61, 297] $ $ [80, 477] $
TT 845 1, 380 3, 342 4, 193 6, 791 10, 440 16, 083 25, 449
EG 1 2 3 4 5 6 7 8
NE 2 2 3 3 4 5 5 6
DC 8 7 6 5 4 3 2 1
TD 60 40 24 20 16 12 8 6
NT $ [5, 13] $ $ [9, 21] $ $ [17, 57] $ $ [21, 69] $ $ [29, 117] $ $ [37, 197] $ $ [61, 297] $ $ [80, 477] $
TT 845 1, 380 3, 342 4, 193 6, 791 10, 440 16, 083 25, 449
Table 3.  Comparison of models under uniform task load distribution
Vacation With Substitution No Substitution
Schedules Model $ \alpha = 0 $ $ \alpha = 0.5 $ $ \alpha = 1 $ $ \alpha = 1.5 $ $ \alpha = 0 $ $ \alpha = 0.5 $ $ \alpha = 1 $ $ \alpha = 1.5 $
Penalty Cost Paid
Daily M1 680 680 680 680 1404 1404 1404 1404
M2 475 475 475 475 1151 1151 1151 1151
M3 336 336 336 336 630 630 630 630
Weekly M1 427 427 427 427 1120 1120 1120 1120
M2 427 427 434 434 1120 1120 1120 1120
M3 336 336 336 336 630 630 630 630
% Vacation Granted
Daily M1 85.9 85.9 85.9 85.9 58.8 58.8 58.8 58.8
M2 90.5 90.5 90.5 90.5 68.0 68.0 68.0 68.0
M3 94.3 94.3 94.3 94.3 88.6 88.6 88.6 88.6
Weekly M1 91.4 91.4 91.4 91.4 67.6 67.6 67.6 67.6
M2 91.4 91.4 91.4 91.4 67.6 67.6 67.6 67.6
M3 94.3 94.3 94.3 94.3 88.6 88.6 88.6 88.6
% Satisfaction
Daily M1 85.9 85.9 85.9 85.9 58.8 58.8 58.8 58.8
M2 90.5 90.5 90.5 90.5 68.0 68.0 68.0 68.0
M3 27.6 59.7 59.7 59.7 22.6 50.7 53.1 54.4
Weekly M1 55.4 55.9 55.9 55.9 42.4 45.4 45.4 45.4
M2 54.0 55.9 56.2 56.2 38.4 46.0 46.0 46.0
M3 32.0 62.0 62.0 62.0 27.8 51.2 51.2 51.2
Vacation With Substitution No Substitution
Schedules Model $ \alpha = 0 $ $ \alpha = 0.5 $ $ \alpha = 1 $ $ \alpha = 1.5 $ $ \alpha = 0 $ $ \alpha = 0.5 $ $ \alpha = 1 $ $ \alpha = 1.5 $
Penalty Cost Paid
Daily M1 680 680 680 680 1404 1404 1404 1404
M2 475 475 475 475 1151 1151 1151 1151
M3 336 336 336 336 630 630 630 630
Weekly M1 427 427 427 427 1120 1120 1120 1120
M2 427 427 434 434 1120 1120 1120 1120
M3 336 336 336 336 630 630 630 630
% Vacation Granted
Daily M1 85.9 85.9 85.9 85.9 58.8 58.8 58.8 58.8
M2 90.5 90.5 90.5 90.5 68.0 68.0 68.0 68.0
M3 94.3 94.3 94.3 94.3 88.6 88.6 88.6 88.6
Weekly M1 91.4 91.4 91.4 91.4 67.6 67.6 67.6 67.6
M2 91.4 91.4 91.4 91.4 67.6 67.6 67.6 67.6
M3 94.3 94.3 94.3 94.3 88.6 88.6 88.6 88.6
% Satisfaction
Daily M1 85.9 85.9 85.9 85.9 58.8 58.8 58.8 58.8
M2 90.5 90.5 90.5 90.5 68.0 68.0 68.0 68.0
M3 27.6 59.7 59.7 59.7 22.6 50.7 53.1 54.4
Weekly M1 55.4 55.9 55.9 55.9 42.4 45.4 45.4 45.4
M2 54.0 55.9 56.2 56.2 38.4 46.0 46.0 46.0
M3 32.0 62.0 62.0 62.0 27.8 51.2 51.2 51.2
Table 4.  Comparison of models under cyclic-1 task load distribution
Vacation With Substitution No Substitution
Schedules Model $ \alpha = 0 $ $ \alpha = 0.5 $ $ \alpha = 1 $ $ \alpha = 1.5 $ $ \alpha = 0 $ $ \alpha = 0.5 $ $ \alpha = 1 $ $ \alpha = 1.5 $
Penalty Cost Paid
Daily M1 769 769 769 769 2000 2000 2000 2000
M2 480 480 480 480 1598 1598 1598 1598
M3 192 192 192 208 684 684 696 716
Weekly M1 532 532 539 539 1876 1876 1876 1876
M2 532 532 553 553 1757 1757 1757 1757
M3 336 336 336 336 1267 1267 1267 1267
% Vacation Granted
Daily M1 80.4 80.4 80.4 80.4 31.8 31.8 31.8 31.8
M2 89.3 89.3 89.3 89.3 51.3 51.3 51.3 51.3
M3 96.7 96.7 96.7 96.5 85.9 85.9 85.4 85.0
Weekly M1 87.6 87.6 87.6 87.6 39.0 39.0 39.0 39.0
M2 87.6 87.6 87.6 87.6 43.8 43.8 43.8 43.8
M3 94.3 94.3 94.3 94.3 67.6 67.6 67.6 67.6
% Satisfaction
Daily M1 80.4 80.4 80.4 80.4 31.8 31.8 31.8 31.8
M2 89.3 89.3 89.3 89.3 51.3 51.3 51.3 51.3
M3 22.3 60.8 60.8 60.4 24.2 45.3 46.0 45.4
Weekly M1 52.8 53.5 53.7 53.7 23.9 25.2 25.2 25.2
M2 51.0 53.5 54.6 54.6 25.9 28.0 28.0 28.0
M3 29.1 61.4 62.0 62.0 23.1 34.7 34.4 34.7
Vacation With Substitution No Substitution
Schedules Model $ \alpha = 0 $ $ \alpha = 0.5 $ $ \alpha = 1 $ $ \alpha = 1.5 $ $ \alpha = 0 $ $ \alpha = 0.5 $ $ \alpha = 1 $ $ \alpha = 1.5 $
Penalty Cost Paid
Daily M1 769 769 769 769 2000 2000 2000 2000
M2 480 480 480 480 1598 1598 1598 1598
M3 192 192 192 208 684 684 696 716
Weekly M1 532 532 539 539 1876 1876 1876 1876
M2 532 532 553 553 1757 1757 1757 1757
M3 336 336 336 336 1267 1267 1267 1267
% Vacation Granted
Daily M1 80.4 80.4 80.4 80.4 31.8 31.8 31.8 31.8
M2 89.3 89.3 89.3 89.3 51.3 51.3 51.3 51.3
M3 96.7 96.7 96.7 96.5 85.9 85.9 85.4 85.0
Weekly M1 87.6 87.6 87.6 87.6 39.0 39.0 39.0 39.0
M2 87.6 87.6 87.6 87.6 43.8 43.8 43.8 43.8
M3 94.3 94.3 94.3 94.3 67.6 67.6 67.6 67.6
% Satisfaction
Daily M1 80.4 80.4 80.4 80.4 31.8 31.8 31.8 31.8
M2 89.3 89.3 89.3 89.3 51.3 51.3 51.3 51.3
M3 22.3 60.8 60.8 60.4 24.2 45.3 46.0 45.4
Weekly M1 52.8 53.5 53.7 53.7 23.9 25.2 25.2 25.2
M2 51.0 53.5 54.6 54.6 25.9 28.0 28.0 28.0
M3 29.1 61.4 62.0 62.0 23.1 34.7 34.4 34.7
Table 5.  Comparison of models under cyclic-2 task load distribution
Vacation With Substitution No Substitution
Schedules Model $ \alpha = 0 $ $ \alpha = 0.5 $ $ \alpha = 1 $ $ \alpha = 1.5 $ $ \alpha = 0 $ $ \alpha = 0.5 $ $ \alpha = 1 $ $ \alpha = 1.5 $
Penalty Cost Paid
Daily M1 843 843 843 843 2079 2079 2079 2079
M2 549 549 549 549 1635 1635 1635 1635
M3 112 112 112 112 643 647 650 654
Weekly M1 602 602 602 602 2142 2142 2142 2142
M2 588 602 602 602 1946 1946 1946 1946
M3 336 336 336 336 1435 1435 1435 1435
% Vacation Granted
Daily M1 78.6 78.6 78.6 78.6 21.0 21.0 21.0 21.0
M2 87.6 87.6 87.6 87.6 37.6 37.6 37.6 37.6
M3 98.1 98.1 98.1 98.1 81.9 81.8 81.4 81.2
Weekly M1 84.8 84.8 84.8 84.8 20.0 20.0 20.0 20.0
M2 84.8 84.8 84.8 84.8 26.7 26.7 26.7 26.7
M3 94.3 94.3 94.3 94.3 47.6 47.6 47.6 47.6
% Satisfaction
Daily M1 78.6 78.6 78.6 78.6 21.0 21.0 21.0 21.0
M2 87.6 87.6 87.6 87.6 37.6 37.6 37.6 37.6
M3 27.9 58.0 58.2 58.2 21.0 36.5 37.3 38.1
Weekly M1 51.2 52.8 52.8 52.8 7.9 11.3 11.3 11.3
M2 49.4 53.2 53.2 53.2 13.7 17.0 17.0 17.0
M3 26.3 62.4 62.4 62.4 14.0 23.3 23.3 23.3
Vacation With Substitution No Substitution
Schedules Model $ \alpha = 0 $ $ \alpha = 0.5 $ $ \alpha = 1 $ $ \alpha = 1.5 $ $ \alpha = 0 $ $ \alpha = 0.5 $ $ \alpha = 1 $ $ \alpha = 1.5 $
Penalty Cost Paid
Daily M1 843 843 843 843 2079 2079 2079 2079
M2 549 549 549 549 1635 1635 1635 1635
M3 112 112 112 112 643 647 650 654
Weekly M1 602 602 602 602 2142 2142 2142 2142
M2 588 602 602 602 1946 1946 1946 1946
M3 336 336 336 336 1435 1435 1435 1435
% Vacation Granted
Daily M1 78.6 78.6 78.6 78.6 21.0 21.0 21.0 21.0
M2 87.6 87.6 87.6 87.6 37.6 37.6 37.6 37.6
M3 98.1 98.1 98.1 98.1 81.9 81.8 81.4 81.2
Weekly M1 84.8 84.8 84.8 84.8 20.0 20.0 20.0 20.0
M2 84.8 84.8 84.8 84.8 26.7 26.7 26.7 26.7
M3 94.3 94.3 94.3 94.3 47.6 47.6 47.6 47.6
% Satisfaction
Daily M1 78.6 78.6 78.6 78.6 21.0 21.0 21.0 21.0
M2 87.6 87.6 87.6 87.6 37.6 37.6 37.6 37.6
M3 27.9 58.0 58.2 58.2 21.0 36.5 37.3 38.1
Weekly M1 51.2 52.8 52.8 52.8 7.9 11.3 11.3 11.3
M2 49.4 53.2 53.2 53.2 13.7 17.0 17.0 17.0
M3 26.3 62.4 62.4 62.4 14.0 23.3 23.3 23.3
Table 6.  Comparison of models under random task load distribution
Vacation With Substitution No Substitution
Schedules Model $ \alpha = 0 $ $ \alpha = 0.5 $ $ \alpha = 1 $ $ \alpha = 1.5 $ $ \alpha = 0 $ $ \alpha = 0.5 $ $ \alpha = 1 $ $ \alpha = 1.5 $
Penalty Cost Paid
Daily M1 884 884 884 884 2647 2647 2647 2647
M2 561 561 561 561 2491 2491 2491 2491
M3 336 336 336 336 2214 2218 2221 2221
Weekly M1 609 609 616 616 2597 2597 2597 2597
M2 609 609 630 630 2597 2597 2597 2597
M3 336 336 336 336 2583 2583 2583 2583
% Vacation Granted
Daily M1 77.3 77.3 77.3 77.3 1.4 1.4 1.4 1.4
M2 87.5 87.5 87.5 87.5 9.3 9.3 9.3 9.3
M3 94.3 94.3 94.3 94.3 32.9 32.7 32.2 32.2
Weekly M1 83.8 83.8 83.8 83.8 3.8 3.8 3.8 3.8
M2 83.8 83.8 83.8 83.8 3.8 3.8 3.8 3.8
M3 94.3 94.3 94.3 94.3 4.8 4.8 4.8 4.8
% Satisfaction
Daily M1 77.3 77.3 77.3 77.3 1.4 1.4 1.4 1.4
M2 87.5 87.5 87.5 87.5 9.3 9.3 9.3 9.3
M3 26.8 59.3 59.3 59.3 8.2 15.6 16.1 15.6
Weekly M1 51.2 52.5 52.8 52.8 1.6 2.3 2.3 2.3
M2 49.0 52.5 53.6 53.6 2.2 3.3 3.3 3.3
M3 26.9 58.6 58.6 58.6 1.8 3.0 3.0 3.0
Vacation With Substitution No Substitution
Schedules Model $ \alpha = 0 $ $ \alpha = 0.5 $ $ \alpha = 1 $ $ \alpha = 1.5 $ $ \alpha = 0 $ $ \alpha = 0.5 $ $ \alpha = 1 $ $ \alpha = 1.5 $
Penalty Cost Paid
Daily M1 884 884 884 884 2647 2647 2647 2647
M2 561 561 561 561 2491 2491 2491 2491
M3 336 336 336 336 2214 2218 2221 2221
Weekly M1 609 609 616 616 2597 2597 2597 2597
M2 609 609 630 630 2597 2597 2597 2597
M3 336 336 336 336 2583 2583 2583 2583
% Vacation Granted
Daily M1 77.3 77.3 77.3 77.3 1.4 1.4 1.4 1.4
M2 87.5 87.5 87.5 87.5 9.3 9.3 9.3 9.3
M3 94.3 94.3 94.3 94.3 32.9 32.7 32.2 32.2
Weekly M1 83.8 83.8 83.8 83.8 3.8 3.8 3.8 3.8
M2 83.8 83.8 83.8 83.8 3.8 3.8 3.8 3.8
M3 94.3 94.3 94.3 94.3 4.8 4.8 4.8 4.8
% Satisfaction
Daily M1 77.3 77.3 77.3 77.3 1.4 1.4 1.4 1.4
M2 87.5 87.5 87.5 87.5 9.3 9.3 9.3 9.3
M3 26.8 59.3 59.3 59.3 8.2 15.6 16.1 15.6
Weekly M1 51.2 52.5 52.8 52.8 1.6 2.3 2.3 2.3
M2 49.0 52.5 53.6 53.6 2.2 3.3 3.3 3.3
M3 26.9 58.6 58.6 58.6 1.8 3.0 3.0 3.0
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