Shift work disrupts the sleep-wake cycle, leading to sleepiness, fatigue, and performance impairment, with implications for occupational health and safety. For example, aircraft maintenance crew work a 24-hour shift rotation under the job stress of sustaining the flight punctuality rate. If an error occurs during the aircraft maintenance process, this error may become a potential risk factor for flight safety. This paper focuses on optimal work shift scheduling to reduce the fatigue of shiftworkers. We proposed a conditional exponential mathematical model to represent the fatigue variation of workers. The fatigue model is integrated with the work shift scheduling problem with considerations of workers’ preferences of days off, company or government regulations, and manpower requirements. The combined problem is formulated as a mixed-integer program, in which the shift assignments are described by binary variables. Using the proposed method, we can find a feasible work shift schedule and also have a schedule that minimizes the peak fatigue of shiftworkers while satisfying their days off demands. Several examples are provided to demonstrate the effectiveness of the proposed approach.

Fatigue-related problems cost America an estimated $18 billion per year in terms of lost productivity, and fatigue-related drowsiness on the highways contributes to more than 1,500 fatalities, 100,000 crashes, and 76,000 injuries annually [

The mechanism behind fatigue is complicated. Previous research has shown that fatigue increases along with the length of shift length, with associated increases in the likelihood of accidents occurring. Åkerstedt [

To automatically arrange a feasible shift table, the scheduling problem is usually formulated as a constraint program. Berrada et al. [

This paper is structured as follows. In Section

To support employees being at proper alertness levels when working shifts, we predict the fatigue level of each employee. Fatigue is generally accepted to be under the influence of previous sleep history, time spent at work, and length of time spent awake [

To manage fatigue, we need to know how much fatigue people experience. Therefore, a fatigue model to represent and quantify the predicted fatigue levels is required before constructing the optimization problem. Our fatigue model is inspired by the FAID system. The FAID model is constructed by a linear component (length of work period) and a sinusoidal component (circadian timing of work period), which is nonlinear and, hence, is not suitable for constructing constraints in the shift-scheduling problem. The primary concept of FAID is to view a duty schedule as a time-varying function by which an individual is considered in one of the two states: work or nonwork. Each state can be considered as an input, from which a continuously varying fatigue/recovery result is the output. Figure

The simulation results using FAID [

To address the aforementioned findings in the FAID result, in our previous work, we assume that the fatigue state of a worker is formulated as a first-order time-varying system as the following:

Although the fatigue model (

In the revised dynamics, we conditionally modify the increment/decrement rate of fatigue when the fatigue state is higher than some prespecified state

Comparison of different fatigue models.

Then, the same workload coefficient is used repeatedly in the following days. The data of the original model are generated from (

Because the dynamic model shown in (

Then,

Because of manpower and other requirements, employees usually will not be able to have their desired days off schedule. To make the scheduling problem feasible while trying to satisfy employees’ desired days off schedule, we first find all the possible days off schedule in a week and allow each employee to place the weightings on these days off selections based on their preferences. A lower weighting means that the selection is much preferred by the employee than the selections with higher weightings.

To formulate the optimal shift-scheduling problem, we first estimate the workload coefficients for each employee working for all types off work shifts. This can be done by first using FAID to generate the fatigue variation results and then use the least square error method to estimate these coefficients and construct the work shift coefficient matrix

Consider a weekly scheduling problem with the following notations used in the formulation of the optimal weekly shift-scheduling problem.

The number of employees to be scheduled

The number of days off schedule from which the employee can choose

The total number of shift schedule including days off that can be assigned in one day

The total number of sampling points of fatigue states for each employee

The sampling period of the fatigue state

The least upper bound of the natural log of all employees’ fatigue state

A row vector whose elements all equal

A zero vector

The maximum allowed work hours per week

The hours of work when the employee is assigned the

The manpower requirement during the work hour of the

The set of shift assignments that has contribution to the manpower during the work hour of the

A binary variable indicating whether employee

A binary variable indicating whether employee

The weighting coefficient ranging from

A matrix of binary value. Each column lists one disallowed combination of shifts in a week.

Incorporating the ideas shown in Section

The terms

The elements of

where

If

If

If

If

In this section, we use two different scenarios to demonstrate the effectiveness of the proposed algorithm. The following examples are run on a personal computer with an Intel Core i5-2400 CPU and 4 GB of RAM. The solver we used is Xpress IVE Version 1.14.36. All the examples can be solved within 5 seconds.

Air traffic control is the process by which aircraft is directed by air traffic controllers (ATCs) to separate air traffic to prevent collisions. ATCs need to provide information to pilots while aircrafts are taxiing on the ground or flying in the air to keep the air traffic unimpeded and to enhance aviation safety. On the basis of the above requirements, air traffic control must be provided for 24 hours per day in many airports, which leads to controllers working in shifts.

In this simulation, we consider the scenario with the following conditions.

There are eight people to be scheduled.

The simulation period is seven days long.

Every employee has two days off during each week.

The morning shift cannot be scheduled right after a night shift.

All employees prefer to have days off during the weekend. If not possible, a consecutive two days off is preferred.

The optimized schedule has the same manpower requirement as the original schedule.

The possible shift times are as follows:

The original ATC shift table.

ATC | Mon. | Tue. | Wed. | Thu. | Fri. | Sat. | Sun. |
---|---|---|---|---|---|---|---|

1 | I | A | O | C | O | C | I |

2 | C | F | C | I | O | O | I |

3 | J | O | O | A | F | C | I |

4 | C | G | F | I | O | O | C |

5 | O | E | E | O | A | A | J |

6 | O | J | O | J | H | H | E |

7 | J | O | A | G | O | J | J |

8 | I | O | B | H | O | I | I |

The fatigue states using the original ATC shift table.

We then apply the proposed algorithm to reschedule the shift table. The optimized shift table is shown in Table

The optimized ATC shift table.

ATC | Mon. | Tue. | Wed. | Thu. | Fri. | Sat. | Sun. |
---|---|---|---|---|---|---|---|

1 | J | O | O | I | F | C | I |

2 | A | I | E | O | O | C | I |

3 | A | B | A | J | O | O | I |

4 | J | O | O | G | A | J | I |

5 | C | G | B | I | O | O | J |

6 | H | F | F | O | O | I | J |

7 | C | E | C | C | O | O | C |

8 | H | O | O | I | H | I | E |

The fatigue states using the optimized ATC shift table.

In this simulation, we extract a work-study student shift table from the Internet and test the proposed method with the existing shift table. The following contains the settings of the simulation.

There are eight students to be scheduled.

The simulation period is seven days long.

Every student has two days off during each week.

All students desire to have consecutive days off during the week.

The optimized schedule has the same manpower requirement as the original schedule.

The possible shift times are as follows:

Table

The original work-study student shift table.

Student | Mon. | Tue. | Wed. | Thu. | Fri. | Sat. | Sun. |
---|---|---|---|---|---|---|---|

1 | G | G | E | O | A | O | E |

2 | F | E | G | G | G | O | O |

3 | O | G | E | A | G | B | O |

4 | E | O | E | G | G | O | G |

5 | A | O | E | G | G | O | G |

6 | G | G | G | E | F | O | O |

7 | G | A | E | O | C | G | O |

8 | G | O | O | E | F | A | A |

The optimized work-study student shift table.

Student | Mon. | Tue. | Wed. | Thu. | Fri. | Sat. | Sun. |
---|---|---|---|---|---|---|---|

1 | G | A | E | E | G | (O) | (O) |

2 | E | G | C | G | C | (O) | (O) |

3 | A | G | E | A | (G) | (O) | O |

4 | G | F | E | E | (G) | (O) | O |

5 | F | G | E | (G) | (F) | O | O |

6 | G | E | O | (O) | (A) | A | G |

7 | G | O | (G) | (O) | G | G | A |

8 | O | O | (G) | (G) | F | B | B |

The fatigue states using the original work-study student shift table.

The fatigue states using the optimized work-study student shift table.

In this paper, we proposed a modified fatigue model, which is shown to be much accurate than the model used in the previous work. We also demonstrated how to incorporate the proposed fatigue model with an optimal scheduling problem. The proposed formulation also allows employees to choose the weightings of their preferred days off schedules so that the solver can select the most appropriate days off arrangement to the employees’ likings. Other constraints, such as manpower requirements and government/company regulations, are also considered in our formulation so that the scheduling problem being considered is closer to the applications in the real world.

There are some issues not being considered in the proposed fatigue model. We did not consider the case in which the employee may work in different time zones. The circadian rhythm change effect is also not present in the current formulation. How to combine these effects in the fatigue model while making it possible to be integrated in the optimization problem formulation will be pursued in the future.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported by the National Science Council of Taiwan under Grant NSC-99-2221-E-006-057.