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

  • * Corresponding author: Gonca Yıldırım

    * Corresponding author: Gonca Yıldırım 
Abstract / Introduction Full Text(HTML) Figure(10) / Table(6) Related Papers Cited by
  • 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.

    Mathematics Subject Classification: Primary: 90B70, 90B90, 90B35, 90B50, 90-10; Secondary: 90C10, 90C90.

    Citation:

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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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV
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