# American Institute of Mathematical Sciences

July  2022, 18(4): 2927-2958. 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  July 2022 Early access  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 and Management Optimization, 2022, 18 (4) : 2927-2958. doi: 10.3934/jimo.2021097
##### References:
 [1] S. Ağralı, Z. 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. [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. [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. [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. [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. [6] C. S. Azmat, T. 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. [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. [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. [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. [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. [11] I. Berrada, J. 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. [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. [13] J. O. Brunner, J. 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. [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. [15] R. N. Burns, R. 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. [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. [17] A. Corominas, A. 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. [18] A. Corominas, A. 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. [19] A. Corominas, A. 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. [20] A. Corominas, A. 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. [21] A. Corominas, A. 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. [22] Coulthard Barnes and Perpetual Guardian, The four-day week is here, 2019, https://4dayweek.com/. [23] R. de la Torre, A. Lusa, M. 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. [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. [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. [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. [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. [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. [29] EU, European Working Time Directive, vol. 46, Official Journal of the European Union, L 299, 18 November, 2003. [30] H. J. Freudenberger, Staff burn-out, Journal of Social Issues, 30 (1974), 159-165.  doi: 10.1111/j.1540-4560.1974.tb00706.x. [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. [32] J. Goh, J. 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. [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. [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. [35] C. O. Henriques, M. Luque, O. 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. [36] A. Hertz, N. 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. [37] R. Hung, Single-shift workforce scheduling under a compressed workweek, Omega, 19 (1991), 494-497.  doi: 10.1016/0305-0483(91)90067-4. [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. [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. [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. [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. [42] K. Li, S. 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. [43] A. Lusa, A. 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. [44] A. Maye, No-Vacation Nation, Revised, Report May, Center for Economic and Policy Research, Washington, DC, 2019, https://cepr.net/report/no-vacation-nation-revised/. [45] H. E. Miller, Personnel scheduling in public systems: A survey, Socio-Economic Planning Sciences, 10 (1976), 241-249.  doi: 10.1016/0038-0121(76)90011-2. [46] R. Narasimhan, An algorithm for single shift scheduling of hierarchical workforce, European Journal of Operational Research, 96 (1997), 113-121.  doi: 10.1016/S0377-2217(96)00364-5. [47] R. Narasimhan, Algorithm for multiple shift scheduling of hierarchical workforce on four-day or three-day workweeks, INFOR Journal, 38 (2000), 14-32.  doi: 10.1080/03155986.2000.11732398. [48] C. Özgüven and B. Sungur, Integer programming models for hierarchical workforce scheduling problems including excess off-days and idle labour times, Applied Mathematical Modelling, 37 (2013), 9117-9131.  doi: 10.1016/j.apm.2013.04.006. [49] R. Pastor and A. Corominas, A bicriteria integer programming model for the hierarchical workforce scheduling problem, Journal of Modelling in Management, 5 (2010), 54-62.  doi: 10.1108/17465661011026167. [50] D. A. J. Salvagioni, F. N. Melanda, A. E. Mesas, A. D. González, F. L. Gabani and S. M. d. Andrade, Physical, psychological and occupational consequences of job burnout: A systematic review of prospective studies, PloS One, 12 (2017), e0185781-e0185781, https://www.ncbi.nlm.nih.gov/pubmed/28977041. doi: 10.1371/journal.pone.0185781. [51] H. N. Shoshan and S. Sonnentag, The effects of employee burnout on customers: An experimental approach, Work & Stress, 34 (2020), 127-147.  doi: 10.1080/02678373.2019.1577312. [52] N. Skinner and B. Pocock, Paid annual leave in Australia: Who gets it, who takes it and implications for work-life interference, Journal of Industrial Relations, 55 (2013), 681-698.  doi: 10.1177/0022185613491680. [53] S. Ulusam Seçkiner, H. Gökçen and M. Kurt, An integer programming model for hierarchical workforce scheduling problem, European Journal of Operational Research, 183 (2007), 694-699.  doi: 10.1016/j.ejor.2006.10.030. [54] J. Van den Bergh, J. Beliĕn, P. De Bruecker, E. 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##### References:
 [1] S. Ağralı, Z. 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. [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. [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. [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. [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. [6] C. S. Azmat, T. 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. [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. [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. [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. [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. [11] I. Berrada, J. 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. [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. [13] J. O. Brunner, J. 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. [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. [15] R. N. Burns, R. 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. [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. [17] A. Corominas, A. 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. [18] A. Corominas, A. 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. [19] A. Corominas, A. 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. [20] A. Corominas, A. 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. [21] A. Corominas, A. 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. [22] Coulthard Barnes and Perpetual Guardian, The four-day week is here, 2019, https://4dayweek.com/. [23] R. de la Torre, A. Lusa, M. 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. [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. [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. [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. [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. [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. [29] EU, European Working Time Directive, vol. 46, Official Journal of the European Union, L 299, 18 November, 2003. [30] H. J. Freudenberger, Staff burn-out, Journal of Social Issues, 30 (1974), 159-165.  doi: 10.1111/j.1540-4560.1974.tb00706.x. [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. [32] J. Goh, J. 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. [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. [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. [35] C. O. Henriques, M. Luque, O. 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. [36] A. Hertz, N. 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. [37] R. Hung, Single-shift workforce scheduling under a compressed workweek, Omega, 19 (1991), 494-497.  doi: 10.1016/0305-0483(91)90067-4. [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. [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. [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. [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. [42] K. Li, S. 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. [43] A. Lusa, A. 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. [44] A. Maye, No-Vacation Nation, Revised, Report May, Center for Economic and Policy Research, Washington, DC, 2019, https://cepr.net/report/no-vacation-nation-revised/. [45] H. E. Miller, Personnel scheduling in public systems: A survey, Socio-Economic Planning Sciences, 10 (1976), 241-249.  doi: 10.1016/0038-0121(76)90011-2. [46] R. Narasimhan, An algorithm for single shift scheduling of hierarchical workforce, European Journal of Operational Research, 96 (1997), 113-121.  doi: 10.1016/S0377-2217(96)00364-5. [47] R. Narasimhan, Algorithm for multiple shift scheduling of hierarchical workforce on four-day or three-day workweeks, INFOR Journal, 38 (2000), 14-32.  doi: 10.1080/03155986.2000.11732398. [48] C. Özgüven and B. Sungur, Integer programming models for hierarchical workforce scheduling problems including excess off-days and idle labour times, Applied Mathematical Modelling, 37 (2013), 9117-9131.  doi: 10.1016/j.apm.2013.04.006. [49] R. Pastor and A. Corominas, A bicriteria integer programming model for the hierarchical workforce scheduling problem, Journal of Modelling in Management, 5 (2010), 54-62.  doi: 10.1108/17465661011026167. [50] D. A. J. Salvagioni, F. N. Melanda, A. E. Mesas, A. D. González, F. L. Gabani and S. M. d. Andrade, Physical, psychological and occupational consequences of job burnout: A systematic review of prospective studies, PloS One, 12 (2017), e0185781-e0185781, https://www.ncbi.nlm.nih.gov/pubmed/28977041. doi: 10.1371/journal.pone.0185781. [51] H. N. Shoshan and S. Sonnentag, The effects of employee burnout on customers: An experimental approach, Work & Stress, 34 (2020), 127-147.  doi: 10.1080/02678373.2019.1577312. [52] N. Skinner and B. Pocock, Paid annual leave in Australia: Who gets it, who takes it and implications for work-life interference, Journal of Industrial Relations, 55 (2013), 681-698.  doi: 10.1177/0022185613491680. [53] S. Ulusam Seçkiner, H. Gökçen and M. Kurt, An integer programming model for hierarchical workforce scheduling problem, European Journal of Operational Research, 183 (2007), 694-699.  doi: 10.1016/j.ejor.2006.10.030. [54] J. Van den Bergh, J. Beliĕn, P. De Bruecker, E. 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Penalty cost paid in all problem instances
Percent vacation granted in all problem instances
Percent vacation granted in all problem instances (with substitution)
Percent vacation granted in all problem instances (without substitution)
Percent vacation granted from actual preferences in all problem instances
Effect of $\alpha$ with substitution
Effect of $\alpha$ without substitution
Penalty cost paid versus percent vacation granted (overall and from preferences) in all problem instances (with substitution)
Penalty cost paid versus percent vacation granted (overall and from preferences) in all problem instances (without substitution)
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
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
 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
 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
 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
 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