A Recent Operations Research Public Service Project

Shift Scheduling in the Durham Police Department

Client: Captain Charles M. Tiffin, Commander, Planning and Research Division, Durham Police Department

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

Executive Summary

The Durham Police Department (DPD) would like to improve its Uniform Patrol Bureau (UPB) shift scheduling system. The UPB is currently organized into four platoons. Each platoon is made up of a squad from each of four stations. Platoons are scheduled so that each day one platoon works the day shift, one works the night shift and two are off duty. Shifts are 12 hours long, begin and end at 6 pm and 6 am, and a 12 hour rest period must take place between consecutive shifts. In a 28-day period, each platoon works seven day shifts, seven night shifts and has 14 days off duty. Officers do not work more than three consecutive shifts of either type, nor do they have more than three consecutive days off.

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:

The secondary goals are to: The study presents solutions to the primary problem, subject to satisfying the constraints: The primary goal is met using mathematical programming, a standard operations research technique that allows the user to achieve a goal in an "optimal" fashion while meeting all constraints. Analysis of this goal is augmented by a statistical analysis of calls for police service. The secondary goals are not dealt with in the mathematical program, but the study offers useful insights into their precise solution. The technique of integer programming, which further restricts its solutions to "whole numbers", is used in this study to achieve the primary goal. That goal, called the objective, is to minimize officer fatigue as measured by the fatigue index. Limits, called constraints, are imposed in the form of the work-hour and other requirements already described. Variables are defined for each platoon (A,B,C,D), the type of shift (d = day-shift, n= night-shift, o = off-duty), the shift start day (1...28 where day 1 is a Monday), and the duration (d and n range from 2 to 4 days, o ranges from 1 to 7). Each variable has two possible values: 1 indicating the condition represented by the variable does occur in the solution, 0 indicating it does not. For example, if Ad3,3 =1, then platoon A works three consecutive day shifts beginning on day 3. Fatigue is minimized in the problem by minimizing the fatigue index, the sum of fatigue penalties assigned to each variable with a value of 1 in the solution. A solution consists of a list of all variables and their values and is called optimal if it achieves the lowest possible value for the objective.

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.

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Last Modified: February 19, 1998