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doi: 10.3934/jimo.2021097
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## 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 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 & Management Optimization, doi: 10.3934/jimo.2021097
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##### References:
Daily and average task loads over the vacation horizon under different task load distributions
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
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
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
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
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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