**Student Consultant:** Joseph R. Sherman,
Operations Research Department, UNC-CH

**Faculty Project Advisor:** Prof. Mark Hartmann,
Operations Research Department, UNC-CH

**Course Coordinator:** Prof. George Fishman,
Operations Research Department, UNC-CH

**Date:** April 1997

This scheduling arrangement produces a great deal of officer fatigue, primarily because of the number of times an officer completes a sequence of night shifts, forcing a change back to a daytime routine, and the short off-duty sequences required by the current schedule. In addition, shifts now change at a time of peak demand for police service and the rotating shift assignments fail to accommodate those officers with special family needs who would benefit from a weekday-only work schedule. Not only do officers point out the difficulties with the present schedule, but a recently concluded management audit of the DPD recommends "reconsideration of the shift rotation schedule and that consideration be given to developing a schedule that achieves efficiency and effectiveness of operations, while meeting the needs of the staff in a reasonable manner." That is the underlying purpose of this study.

The primary goal, which is solved mathematically, is to:

- Determine a rotating shift schedule that will reduce officer fatigue

- Develop a method to accommodate those officers with special scheduling needs
- Incorporate shift changes into the schedule at off-peak demand times.

- costs must remain at the present level
- officers must not be expected to work more than 168 hours per month
- each officer must work the same number of day and night shifts as every other
- each officer must be guaranteed two weekends off per month
- officers must be able to schedule court appearances during their regular shifts between 9 - 11 am or 2 - 4 pm on weekdays
- supervisors must be paired with the same patrol officers for every shift because of chain of command requirements.

The study considers and solves 20 different scenarios which differ in the number of consecutive shifts or days off they allow. The first group of five scenarios allows only two and three consecutive day or night shifts and from (1) one to three, (2) one to four, (3) one to five, (4) one to six, and (5) one to seven consecutive days off. Five more allow two, three, or four consecutive shifts and the same off-day sequences as the first five. The second group of ten is identical to the first, except a penalty is added to prevent one undesirable sequence found in the first group's solutions, namely a sequence of night shifts followed by a single off day followed by a sequence of day shifts. In all cases, the study assumes the UPB is still organized into four platoons working 12-hour shifts with shift changes at 6 am and 6 pm daily. These characteristics were dictated by the cost constraints mentioned previously.

Solutions to the 20 scenarios fit into three classes. All those in a class achieved the same value for their fatigue indices (objective functions). In particular, 12 achieved the value of 732 which matches the fatigue index of the schedule now in use, six achieved the value of 608, and two achieved the value of 488, a value that matches a well-liked schedule used several years ago by the DPD known as the DuPont schedule. In principle, those with smallest fatigue index, namely 488, are preferable to the remaining scenarios. They include the maximum number of consecutive shifts (4) and days off (7) allowed in any of the scenarios. All schedules meet all staffing, work-hour and cost requirements while giving the department a group of schedules to choose from to match community policing and public image needs.

Based upon its fatigue index and the evalution of Captain Tiffin, this study recommends one of these shift schedules which is very similar to the DuPont schedule:

Some aspects of the solution method deserve attention. First, optimality implies that the solution technique used by the computer is able to find a solution to each model that achieves the lowest possible objective value. However, it does not provide all optimal solutions if more than one exists. It simply produces the first optimal solution it has found. Second, solutions to the primary goal are highly influenced by the values of the fatigue penalties assigned to the work and off-duty sequences. Changes in the penalties assigned to those sequences alter the value of the fatigue index, which in turn changes the shift schedule that results. To determine the extent of this change, a sensitivity analysis was performed in which the penalties were varied. This analysis determined that a change to any penalty might alter the schedule found by the computer, but each penalty has an allowable range within which the solution remained optimal. (In each case, there were a number of optimal schedules with the same fatigue index.) That meant the fatigue index changed by the amount of the penalty change in the same schedule or, equivalently, the schedule had the same work and off-duty sequences, but they were rearranged throughout the 28-day period. Outside of that range, however, even which work and off-duty sequences were present in the schedule changed: increasing a penalty out of range caused that work sequence to disappear from the resulting schedule, while decreasing it out of range caused additional occurances of the same sequence.

The study offers insights into the solution of the secondary goals, although they are not incorporated into a mathematical formulation that could provide quantitative answers. Statistical analysis of calls for police service serves as the starting point for useful answers. For example, this analysis indicates that the call rate is very low between 4 am and 9 am. Therefore, a two-phase evening shift change in which a portion, perhaps half, of the night shift arrives two hours early would reduce the strain of a single evening shift change, when demand is high, and reduce excess staffing in the early morning hours. The morning changeover would occur normally at 6 am; any earlier would be contrary to the goal of reducing fatigue. For those officers with special scheduling needs, the study proposes they simply be removed from the shift rotation and placed on permanent day shift during the week. This realizes the benefit of increasing staffing during the day, allowing for court appearances and vehicle maintenance during normal working hours, and causes only a small staffing shortfall overnight. Obviously, enrollment and duration in this special program must be limited by the department.

In the future, as the DPD expands or priorities change, the department may consider several avenues to improve service. Abandoning chain of command requirements, which have the effect of maintaining constant staffing levels in the face varying demand, would allow the department to employ only those officers needed to meet demand. Alternatively, a budget climate that allows costs to increase suggests a larger number of platoons that could more easily deal with a permanent day shift contingent.

Return to the OR Home Page or back to the UNC Home Page.

Last Modified: February 19, 1998