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Question1a.pdf

Police officer scheduling using goal programming

Dragana Todovic, Dragana Makajic-Nikolic, Milica Kostic-Stankovic and Milan Martic

Faculty of Organizational Sciences, University of Belgrade, Belgrade, Serbia

Abstract Purpose – The purpose of this paper is to develop a methodology for automatically determining the optimal allocation of police officers in accordance with the division and organization of labor. Design/methodology/approach – The problem is defined as the problem of the goal programming for which the mathematical model of mixed integer programming was developed. In modeling of the scheduling problem the approach police officer/scheme, based on predefined scheduling patterns, was used. The approach is applied to real data of a police station in Bosnia and Herzegovina. Findings – This study indicates that the determination of monthly scheduling policemen is complex and challenging problem, which is usually performed without the aid of software (self-rostering), and that it can be significantly facilitated by the introduction of scheduling optimization approach. Research limitations/implications – The developed mathematical model, in its current form, can directly be applied only to the scheduling of police officers at police stations which have the same or a similar organization of work. Practical implications – Optimization of scheduling significantly reduces the time to obtain a monthly schedule. In addition, it allows the police stations to experiment with different forms of organization work of police officers and to obtain an optimal schedule for each of them in a short time. Originality/value – The problem of optimal scheduling of employees is often resolved in other fields. To the authors knowledge, this is the first time that the approach of goal programming is applied in the field of policing. Keywords Goal programming, Police officers scheduling, Shift patterns Paper type Research paper

1. Introduction The problem of police officers who work per shift in terms of ergonomics and efficiency has been widely discussed in the literature. There are different views about the mode that best suits police officers and it can be said that the generally accepted opinion has not been established yet. However, even when a particular police station chooses a particular mode, the problem of determining the specific working schedule of police officers, in accordance with the needs of the police station and preferences of policemen, still remains. Such schedules are usually done on a monthly basis by commanders of police stations, who consider this sort of administrative – nonpolice work difficult and time-consuming. In general, the problem of scheduling of employees in other industries is often taken as the optimization problem which resulted in the automation of specific schedule production. However, in practice, the police solve this problem without the help of the software in a manner that is known in the literature under the name of self-rostering (Silvestro and Silvestro, 2000).

In this paper, the problem of scheduling of police officers is viewed as the optimization problem in which police officers should determine the schedules that fully or partially satisfy the conditions relating to the: required number of officers per shift, rules of shift work, number of working hours of police officers, the monthly working standard, etc. For the observed problem, a mathematical model of goal programming

Policing: An International Journal of Police Strategies & Management

Vol. 38 No. 2, 2015 pp. 295-313

©Emerald Group Publishing Limited 1363-951X

DOI 10.1108/PIJPSM-11-2014-0124

Received 21 November 2014 Revised 12 March 2015

Accepted 12 March 2015

The current issue and full text archive of this journal is available on Emerald Insight at: www.emeraldinsight.com/1363-951X.htm

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has been formulated. Such a problem allows the formulation, in a situation where it is not possible to simultaneously satisfy all the needs of the police station and preferences of police officers, to depart from some requests. The paper consists of six sections. After the introductory section, in the second section, an overview of the literature on the issue of policemen scheduling and different ways of working in shifts, in terms of ergonomics and efficiency, will be given. The third part addresses the solving of police officers scheduling problem as an optimization problem. In the fourth section, the specific scheduling problem is described, as well as the mathematical model by which the problem is modeled. In the fifth part, the application of the formulated model is illustrated using the example of “real-life” police station. The last chapter presents conclusions and directions for further research.

2. Literature review 2.1 Impacts of shift work on psycho-physical abilities of police officers Impacts of shift work on psycho-physical abilities of employees have long been studied in detail. It is known that police fatigue caused by a long-term shift work, biologically insensitive shift schemes and the use of double shifts, negatively affects the abilities of police officers, their health and safety, public relations and the quality of decision making (Vila et al., 2002). That difficulty of working in shifts is also reflected in the fact that police officers rank it among the top ten causes of stress (Hickman et al., 2011).

The concept of tolerance to shift work was first introduced in 1979 by Andlauer et al. They estimated on the basis of the presence of one of three types of health problems: digestive problems, sleeping problems and constant fatigue. The employees working in shifts for ten years or more, mostly have the low level of tolerance to shift work (Natvik et al., 2011). Also, tolerance to shift work differs in case of those who work consecutive night shifts compared to those who work in rotating shifts (Tamagawa et al., 2007).

Studies have shown that sleep deprivation lasting from 17 to 19 hours, experienced by police officers, reduces their ability as well as the blood alcohol content of 0.05 percent. Extension of awakeness up to 24 hours leads to a deterioration equivalent to 0.10 percent of alcohol (Vila et al., 2002). Such problems are more frequently present in situations where shift work requires a high level of commitment (Garbarino et al., 2002) and when officers are exposed to some traumatic events (Neylan et al., 2002).

Police work is extremely dangerous, and the risk of accidents is much higher than in the case of other occupations (Violanti et al., 2012). The possibility that the accident happen during the night shift is three times higher than during the morning shift (Folkard and Lombardi, 2006). Besides that, injuries that occur during night shifts are more severe and of longer duration (Ogiński et al., 2000). The risk of injury is increased during consecutive night shifts, so the fourth night shift carries 36 percent more risks than the first shift. Also, as the number of working hours increases, the risk of injury increases exponentially, so it is two times higher in the 12th hour of work than during the first eight hours (Folkard and Tucker, 2003). It was shown that the 12-hour shifts were associated with a 27 percent increased risk of accidents, and ten-hour shifts with a 13 percent risk (Folkard et al., 2005). A significant number of police officers’ injuries occur in situations of departing from the regular shift schedules (Vila, 2006).

However, not all shift schedules have negative impact on employees’ abilities. This largely depends on the design of schedule shifts (Kecklund et al., 2008). Design of shift schedules according to ergonomic recommendations will contribute to reduction of many problems faced by employees (Knauth, 1996; Knauth and Hornberger, 2003).

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It is recommended that the shift system should have a number of consecutive night shifts, fast rotating shifts and enough rest time between shifts. Shifts length should be consistent with the physical and mental burdens necessary for the assigned work tasks (Knauth et al., 1983).

Although there is no clear view among researchers about the most desirable shift type, majority of supervisors prefer eight-hour or 12-hour shifts, because it makes it easier to create a daily schedule, while police officers still favor compressed shifts more. Longer shifts can reduce the number of commuting days (Vila et al., 2002), but on the other hand they can lead to isolation and lack of communication between employees and supervisors (Gerber et al., 2010). In total, 20-hour shift system has many advantages and some of them are: increased productivity, higher level of job satisfaction, reduction of the total time of replacement and reduced absenteeism due to illness. Possible disadvantages of such working systems are: increased possibility of fatigue, reduction of employees’ abilities and increased risk of injuries (Baulk et al., 2009). It is recommended that more than four hours of additional work should not be added to the 12-hour working shift, with a minimum of eight to ten hours of rest after an extended work (Baulk et al., 2009).

By analyzing three general shift rotations, nonrotating shifts are evaluated as the best. Rotating shifts in rewind are the hardest for employees to adapt to, because the circadian rhythm of the body is predisposed to be rotated forward, and that is from daily to evening shift. Therefore, it takes about eight days to accommodate to the change from evening to night shifts, and about 12 days to accommodate to the change from daily to night shift. One of the advantages of rapid rotations of shifts is that employees can rest during more free days they have at their disposal. Generally, workers in these shifts are sensitive to issues such as shift maladaptation syndrome (Vila et al., 2002).

In research of Swedish police officers’ attitudes toward their working schedule, the shift schedule with rapid rotation forward, with at least 16 hours of rest between shifts and a few free days in a row, was best evaluated. The least popular shift system was the slowly rotating shift system with seven working days in a row, but with a lot of days off (Kecklund et al., 2008). When comparing the rapidly rotating shift systems with weekly rotating systems within 120 shift working schedules in Germany, police officers rated rapidly rotating shift system that provided them with more sleep and free time better (Knauth et al., 1983). Shift systems unfamiliar to employees had lower grades compared to the current system shifts, which may reflect the resistance and skepticism toward all other shift systems different from the present ones (Knauth, 2001). Shift system that was favored among police officers was usually similar to their current work schedule (Kecklund et al., 2008).

Employees’ satisfaction with the shift schedule is very important and affects their ability to cope with their shift schedule (Axellson et al., 2004). Today, the Swedish police stations increasingly present flexible shift systems in which employees choose an appropriate model of working time within certain limits. Although it was believed that such flexibility and impact on working time is a factor that facilitates coping with the shift work, 21 percent of the surveyed group had a negative attitude toward this schedule (Eriksen and Kecklund, 2007). Such work schedules are associated with high levels of job satisfaction and a positive attitude toward work schedule, but employees often choose a system of work that is inconsistent with ergonomic recommendations of shifts system defining, which can have negative effects on mental and physical health of employees (Kecklund et al., 2008).

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It is very important to design adequate shift systems, especially in the police, which will reduce the likelihood of various problems faced by the employees and which can be dangerous not only for police officers but also for the general public (Violanti et al., 2012). An important aspect of creating an adequate shift schedule in police departments is the need to provide 24-hour coverage of all areas of work (Baulk et al., 2009). It is necessary to fulfill expected service standards provided by the organizations of this type, and they can be reflected in the following: response time to incidents, ability to send a number of well-trained officers for various types of incidents, etc. The frequency of incidents varies at different times of the day, week or year. Increased demand for police officers is possible in certain areas, on Friday and Saturday night, as well as in tourist areas during the holiday season. In order to meet these service standards, changes in the frequency of incidents will result in changes in the number of police officers needed to provide the required levels of coverage (Ernst et al., 2004).

2.2 Employees scheduling problem optimization Schedules that are granted to employees may be fixed or rotating, while rotating schedules can be cyclic and noncyclic (Musliu et al., 2002). Regarding solution methods, employee scheduling problems can be classified into those that are based on heuristics or combinatorial optimization problems (Beaumont, 1997). In the case of combinatorial optimization, two modeling approaches are dominant in the literature (Cheang et al., 2003). The first approach (employee/day) considers each day of the observed period individually and, for each day, one of predefined on-duty or free shifts is assigned to each employee. Since the number of decision variables in this modeling approach is the product of: the number of employees, number of days, and number of shift types, it can be very large. Further, all scheduling regulations (day-off after night shift, etc.) must be additionally modeled as mathematical model constraints. The second approach (employee/pattern) is based on the predefined pattern (working schemes) in the observed period and one of them is assigned to each employee. Such a problem is modeled as a set covering problem in which the number of decision variables depends only on the number of employees and the number of predefined schemes (patterns). In addition, all scheduling regulations are already included into working schemes. However, the quality of the solution depends on the number and quality of predefined schemes (Clark and Walker, 2011). Dantzig (1954) first defined the problem of staff shift scheduling using the set covering formulation that is still very popular among researchers (Van den Bergh et al., 2013). This formulation is general and therefore many problems in staff scheduling can be modeled in this unified format (Ernst et al., 2004). In this paper, the scheduling problem is modeled as a set covering problem.

Although the set covering approach reduces computational time, the feasibility problem can appear. This impossibility of satisfying the conditions set by using the available staff and the defined patterns can be overcome in different ways. Belien and Demeulemeester (2008) introduced a procedure for generating new scheme after solving linear model. Bard and Purnomo (2005) introduced auxiliary variables representing the number of additional staff that needed to be engaged in order to fulfill all the constraints. However, we decided to use goal programming in this paper.

Recently, goal programming has been used for solving scheduling problems in different areas. The variety of applications of goal programming in nurse scheduling

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problem can be found in literature. Azaiez and Al Sharif (2005) developed 0-1 goal programming model for nurse scheduling which involves hospital objectives and nurses’ preferences. The solution for 13 nurses over a four-week period is presented. The problem of scheduling the marketing executive was solved using 0-1 goal programming in Mathirajan and Ramanathan (2007). Instead of the usual shifts assignment, the authors defined scheduling problem as a problem of assigning the visiting tours of the marketing executive to the customers. The developed model was presented on the case of an electronic manufacturing company where 20 customers were scheduled over a period of 21 working days. The special type of staff scheduling problem, the crew scheduling problem, was modeled using goal programming in Giachetti et al. (2013). Shahnazari-Shahrezaei et al. (2013) investigated manpower scheduling problem in production and service environments. Goal programming approach was illustrated by the case of scheduling of 20 employees during one month (28 days). Louly (2013) developed a goal programming approach for a staff scheduling problem over a six-week period at a telecommunications center with eight engineers. The final model with 2,352 decision variables and 1,999 constraints was solved using Lingo software after 26 minutes of computing time. Goal programming model of a Bank Information Technologies (IT) staff scheduling is presented in (Labidi et al., 2014). Two real-life problems: nine operators and five supervisors scheduling over a four-week period were solved using Lingo software.

The beginnings of the application of operational research methods in solving police officers scheduling problems can be found in Edie’s (1954) work in the case of traffic police. Further research on police scheduling continued in the late 1960s and early 1970s in the USA and UK, giving excellent bibliography on the scheduling of police officers (Taylor and Huxley, 1989).

Marangos (1993) in his paper presented the optimization problem which determined the number of police officers assigned to major shifts and the number of officers assigned to “swing” shifts. “Swing” shifts fit overlap periods from Ottawa shift system (Richbell et al., 1998). Police officers are assigned to three major shifts and two “swing” shifts. Swing shifts are shifts that overlap through the three major shifts (daily, evening and night shift) and they are used to compensate for the higher demand periods. Swing shift may begin at any time, which will allow at least one hour overlap with major shifts (Marangos, 1993). Edleston and Bartlett (2012) showed the problem of scheduling of the Leicestershire police force, which consisted of assigning a set of numbers in order to get best quantified demand. They used Tabu search algorithm for the research.

The use of goal programming in the optimization related to the police functioning can be found in two publications; both consider the problems that differ from the problem which is the subject of our study. Rabajante (2008) used Goal Programming for strategic spatial distribution of patrol locations and used the idea of Maximum Coverage Location Problem (MCLP) solved by Binary Integer Linear Programming. Taylor and Huxley (1989) modeled the problem of determining the number of officers starting the working week in the particular shift. The goal programming objective function minimizes the amount of shortages and then minimizes the maximum shortage for any given hour of the week.

Our problem can be characterized as a multi-day personnel scheduling problem, which includes two levels: the first level where the working days and free days are fixed for each employee, and the second level where scheduling problem is determined for every working day and every employee who works that day, has

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a work pattern granted (Brucker et al., 2011). Shift-based demand is the basis for scheduling policy which we analyzed. Our problem can also be observed as a problem of roster assignment which is the process of assigning lines of work to staff (Ernst et al., 2004).

3. Description of police officers scheduling problem In this paper we consider the problem of determining a monthly schedule of police officers at police stations on the territory of Bosnia and Herzegovina. At the head of each police station, there is a commander, who is responsible for the efficient performance of all officers of the station. One of the main tasks of the commander, at the end of each month, is to make a schedule of officers for the next month, and this is done in self-rostering way. In this case, station commander uses the established working schemes in determining the schedule. Therefore, in this paper, the approach police officer/scheme is used, i.e. the problem is modeled as a set covering problem. Due to the impossibility of satisfying all demands, the problem of the police officers scheduling is formulated as a goal programming problem.

According to the current organization of work, police officers work in two shifts of 12 hours. In this chapter, a mathematical model that represents the current state, will be shown first. Then, a mathematical model of working in three shifts will be introduced, with respect to the fact that management of the station is actually considering the possibility of transferring to this way of working.

When creating the schedule, the commander of the police station is trying to assign one of the established work patterns to each police officer. Basic schemes of working in two shifts are:

D-N-S-S N-S-S-N-S-S-D D-S-N-S-S S-S-N-S-S-D S-S-N-S-S-N D-D-N-S-S D-S-D-S-N-S-S D-S-D-N-S-S

where D is used to mark daily and N night shift while S represents the day-off. If working is organized in three shifts, basic working schemes would be (M

represents the inter-shift): D-D-M-M-N-S-S-N-S-S M-S-N-S-S-D D-M-N-S-S D-D-M-N-S-S D-M-M-N-S-S S-S-N-S-S-D-M D-S-M-S-N-S-S D-S-N-S-S-M-S D-N-S-S-M-S N-S-S-D-S-M D-M-N-S-S-N-S-S D-S-S-M-S-S-N-S-S

All of the above schemes meet the ergonomic rules that are set for the observed police station.

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In both cases, each scheme has as many offsets as the length of the scheme and the schemes (and their offsets) can be combined within one month. In this way, a set of all pre-defined schemes, which can be used during the month, is formulated. The assumptions of the model of working in two shifts are:

(1) police station to be analyzed has 11 officers;

(2) policemen are scheduled for the period of one month;

(3) the work is organized in two shifts (day and night) that last 12 hours;

(4) it is desirable to achieve the working standard of 180 hours per month;

(5) during the scheduling of police officers, the rule which means that after a day shift 12 hours of rest follow, is respected. Also, after a night shift 48 hours of rest follow; and

(6) employees’ requirements for free days and holidays are respected.

For the model of the working in three shifts, assumptions (1), (2), (4) and (6) of the model of working in two shifts are valid, while assumptions (3) and (5) are modified in the following manner:

(3)0 work is organized in three shifts (day, inter-shift and night) that last for nine hours each; and

(5)0 during the scheduling of police officers, the rule which means that 48 hours of rest follow after a night shift and minimum of nine hours of rest follows after daily and inter-shift.

During the month, a number of officers may be absent. For the purposes of modeling the scheduling problem, absences are divided into two groups, depending on the length of absence: one day absence and two or more days absence.

4. Mathematical model formulation

The problem of determining the working schedule of available police officers, can be formulated as follows: it is necessary to determine by what schedule (scheme) each of the officers should work during the month, so that the desired number of police officers can be in day and night shifts at the police station respecting the needs and preferences of police officers.

The notation used in the two-shifts mathematical model is as follows: Sets: R is the set of schedules which can be used during the month; Dd is the set of

daily shifts; Dn is the set of night shifts; D is the set of all shifts, D¼Dd∪Dn; and Z is the set of police officers at the observed police station.

Parameters:

akj ¼ 1 if schedule k includes shift j

0 otherwise ; jAD; kAR

�

pdj is the required number of police officers in the daily shift of the day j, j∈Dd; pnj is the required number of police officers in the night shift of the day j, j∈Dn; pdmj is the minimal number of police officers who must be engaged in the daily shift of the day j, j∈Dd; pnmj is the minimal number of police officers who must be engaged in the night shift of the day j, j∈Dn.

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cij ¼ 1 if the police officer i is absent in the shift j

0 otherwise ; iAZ ; jAD � for one� day absence;

�

bij ¼ 0 if the police officer i is absent in the shift j

1 otherwise ; iAZ ; jAD � for two or more days absence;

�

m is the total number of shifts during the month; pri is the percentage of policemen presence during the month, pri ¼

P jADbij=m; iAZ ; t is the the average number

of working hours during the month per police officer; and rt is the the length of shifts.

Variables:

xik ¼ 1 if schedule k is assigned to the police officer i

0 otherwise ; iAZ ; kAR;

�

d�i ; d þ i is the deviation from the average number of working hours during the

month for police officer i, i∈Z; ddj is the deviation from the required number of necessary police officers in the daily shift of the day j, j∈Dd; dnj is the deviation from the required number of necessary police officers in the night shift of the day j, j∈Dn.

Goal programming mathematical model of the two-shifts monthly scheduling of police officers of the observed police station can be formulated as follows:

min X iAZ

d�i þdþi � �þ X

jADd

ddjþ X jADn

dnj

st.: X kAR

xik ¼ 1; iAZ ð1Þ

X kAR

akj � X iAZ

bij � xikþddj ⩾ pdj; jADd ð2Þ

X kAR

akj X iAZ

bij � xikþdnj ⩾ pnj; jADn ð3Þ

X kAR

akj � X iAZ

bij � xik ⩾ pdmj; jADd ð4Þ

X kAR

akj � X iAZ

bij � xik ⩾ pnmj; jADn ð5Þ

X kAR

X jAD

akj � cij � xik ¼ 0; iAZ ð6Þ

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PIJPSM 38,2

X kAR

X jAD

akj � bij � xik � rtþd�i �dþi ¼ t � pri; iAZ ð7Þ

xikA 0; 1f g; iAZ ; kAR d�i ⩾ 0; d

þ i ⩾ 0; iAZ

ddj ⩾ 0; jADd dnj ⩾ 0; jADn

With the goal function, the deviation from the desired number of police officers during the reporting period and the minimum required number of police officers per shift and per day is minimized.

Constraint (1) ensures that every police officer has exactly one monthly working schedule (scheme) assigned. Constraint (2) refers to the required number of police officers in the daily shift, while constraint (3) refers to required number of police officers in the night shift. Constraints (2) and (3) can be unsatisfied. However, constraints (4) and (5) ensure the presence of a minimum number of police officers every day and in every night shift. Constraint (6) prevents the police officer from receiving working schemes with shifts on their one-day absence. Parameter bij does not prevent the allocation of schemes during periods of absence, but it ensures that in those periods officer who is absent does not participate in the total number of police officers scheduled in the shift (if the requirement that police officer must not obtain a scheme by which he would work throughout the period of absence is set, the problem would not have a fusible solution). On the other hand, parameter cij prevents the allocation of schemes during the day of absence. Constraint (7) refers to the number of working hours that each officer should complete during the month, in proportion to their monthly attendance.

In addition to the notation used in the two-shifts mathematical model, the notation in the three-shifts mathematical model is extended as follows:

Sets: R is the set of schedules which can be used during the month; Dm is the set of inter-shifts; D is the set of all shifts, D¼Dd∪Dm∪Dn.

Parameters:

akj ¼ 1 if schedule k includes shift j

0 otherwise ; jAD; kAR

�

pmj is the required number of police officers in the inter-shift of the day j, j∈Dm; pmmj is the minimal number of police officers who must be engaged in the inter-shift of the day j, j∈Dm.

Variables: dmj is the deviation from the required number of necessary police officers in daily shift of the day j, j∈Dm.

Goal programming mathematical model of the three-shifts monthly scheduling of police officers of the observed police station can be formulated as follows:

min X iAZ

d�i þdþi � �þX

jADd

ddjþ X jADm

dmjþ X jADn

dnj

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Police officer scheduling

st.: X kAR

xik ¼ 1; iAZ ð10Þ

X kAR

akj � X iAZ

bij � xikþddj ⩾ pdj; jADd ð20Þ

X kAR

akj X iAZ

bij � xikþdnj ⩾ pnj; jADn ð30Þ

X kAR

akj � X iAZ

bij � xik ⩾ pdmj; jADd ð40Þ

X kAR

akj � X iAZ

bij � xik ⩾ pnmj; jADn ð50Þ

X kAR

X jAD

akj � cij � xik ¼ 0; iAZ ð60Þ

X kAR

X jAD

akj � bij � xik � rtþd�i �dþi ¼ t � pri; iAZ ð70Þ

X kAR

akj X iAZ

bij � xikþdmj ⩾ pmj; jADm ð8Þ

X kAR

akj � X iAZ

bij � xik ⩾ pmmj; jADm ð9Þ

xikA 0; 1f g; iAZ ; kAR d�i ⩾ 0; d

þ i ⩾ 0; iAZ

ddj ⩾ 0; jADd dmj ⩾ 0; jADm dnj ⩾ 0; jADn

The goal function and the constraints (10-70) have the same meaning as in the two-shifts mathematical model. New constraints (8) and (9) refer to the required and minimal number of police officers present in the inter-shift, respectively.

Presented models do not include preferences that police officers have toward the working schemes. Police officers can have different perception of compacted or stretched working schemes, depending on their personal characteristics. Those preferences can be included into mathematical models as soft constraints, with additional deviation variables, that will be satisfied if it is possible.

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Another limitation of the models is the assumption that all the available police officers have similar skills level and can fulfill tasks with approximately equal efficiency. In the observed police station, this assumption is correct. However, police officers can differ regarding their qualifications or skills that can influence working schedule.

5. Numerical results The 11 police officers who work in shifts, on duty and patrol services in the police station were observed. On duty service, one officer is scheduled in every shift, and in patrol service, there is one police officer scheduled in the day shift and two police officers in the night shift. Taking into account that both services work in the same way, the total number of police officers in the day shift is two and in the night shift is three, as it is set in the model. If it is not possible to provide the required number of officers, it is necessary that at least one police officer is present in each shift. Police officers’ requests to use vacation days and free days are as follows: Policeman 1 – December 19, December 20 and December 31; Policeman 9 from December 13 to December 24; Policeman 10 – December 19 and December 20.

Table I shows the old schedule made by the person in charge in the police station, which was used for scheduling of police officers’ work during the monitored month. The first column shows the 11 police officers who were scheduled, while other columns represent 31 days in December. The shaded cells in the table represent the days when certain officers were on leave.

Based on data for December, 2013, the first mathematical model with 14,894 binary, 84 continuous (deviating) variables and 106 constraints has been formulated. Also, the second mathematical model with 56,991 binary, 115 continuous variables and 137 constraints has been formulated. The model was solved exactly, and, GNU Linear Programming Kit (GLPK), open source software for solving problems of linear and mixed integer programming was used for finding the optimal solution (GLPK, 2014).

Optimization results are summarized in Tables II-IV. Table II shows the schedule when police officers work in two shifts, which is obtained as the solution of the model (1)-(9). The optimal solution ensures that after each night shift, an employee has at least 48 hours of rest. In the original working schedule (Table I), it was deviated from this rule as much as 32 times. In two cases, after a night shift there was no rest: the police officer Number 9 – on 30th and 31st day and the police officer Number 10 – on 13th and 14th day.

Table III shows the scheduling whenpolice officers work in three shifts, which is obtained as the solution of the model (1’)-(9). Within this work schedule, it is also ensured that after each night shift, 48 hours of rest follow. This scenario was resolved so that superiors in the police station could be able to analyze how police officers would work if three shifts were introduced. After the analysis, it was found that, given the small number of police officers in the respective police station and the number of days that would have increased from an average of 15 to an average of 20 working days per officer, working in two shifts should be maintained.

Another advantage of optimal schedules compared to the old schedule refers to the total number of working hours of every police officer during the month. With the assumptions of the problem, in Chapter 3, it is stated that it is desirable to achieve the working standard of 180 hours per month. Table IV shows the comparison of the number of hours that police officers make on a monthly basis in relation to the schedule they are working by.

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Police officer scheduling

W ee k 1

W ee k 2

W ee k 3

W ee k 4

PO S

M T

W T

F S

S M

T W

T F

S S

M T

W T

F S

S M

T W

T F

S S

M T

P1 D

N D

N N

D N

N D

N N

D N

P2 N

D N

N D

N P3

D N

N N

D N

N N

D D

N N

D N

N N

P4 N

D N

N D

N P5

N D

D N

N D

N N

P6 D

N D

N N

P7 D

N D

N N

D N

N N

N P8

N D

N N

D N

N N

D N

N N

D N

D P9

D N

N N

D N

N N

N P1

0 N

N N

D N

N D

N N

D N

P1 1

D D

D N

N N

N D

D N

D D

D N

Table I. Police officers schedule – old solution

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PIJPSM 38,2

W ee k 1

W ee k 2

W ee k 3

W ee k 4

PO S

M T

W T

F S

S M

T W

T F

S S

M T

W T

F S

S M

T W

T F

S S

M T

P1 D

N N

D N

N D

D N

P2 D

N D

D N

D D

N D

D N

N N

P3 N

N D

N N

D N

P4 D

N N

D N

N D

P5 N

N N

N D

D N

D D

N D

D N

P6 N

D D

N D

D N

D D

N N

P7 D

D N

D D

N D

D N

N D

N N

D N

P8 N

D N

N D

N N

D N

P9 N

D N

N D

N N

N P1

0 N

D N

N D

D N

D D

N D

P1 1

D N

D D

N D

D N

N D

Table II. Police officers

schedule – two-shifts optimal solution

307

Police officer scheduling

W ee k 1

W ee k 2

W ee k 3

W ee k 4

PO S

M T

W T

F S

S M

T W

T F

S S

M T

W T

F S

S M

T W

T F

S S

M T

P1 M

N D

M M

N D

M M

N M

N D

M M

N D

M M

N P2

D D

M N

D D

M N

D D

M M

M N

N D

D M

M N

P3 D

M N

D M

N D

M N

D D

D M

N D

D M

N D

D M

P4 D

D M

N D

D M

N D

D M

N D

D M

M N

N D

D M

P5 D

M N

D M

N D

M N

D M

N D

D M

N D

D M

N P6

D M

N D

D M

N D

D M

N D

M N

D M

N D

M N

P7 N

D D

M N

D D

M N

D D

D M

N D

D M

N D

D P8

D D

M N

D D

M N

D D

M N

D D

M M

N N

D D

P9 M

N N

D D

M M

N N

D M

N D

M N

D M

N P1

0 D

M M

N D

M M

N D

M M

M N

D M

N D

M M

N P1

1 D

D M

M N

N D

D M

M N

D D

M N

D D

M N

D N ot es

:I n T ab le s II an d II Ii tc an

be se en

th at th e po lic e of fic er N um

be r9

ha s sh ift s fr om

D ec em

be r1

3 to 24

(s ha de d ar ea s) as si gn

ed to .H

ow ev er ,a s de sc ri be d in

Ch ap te r4

,p ar am

et er b i j en su re s th at in th is pe ri od

of fic er 9 do es

no tp

ar tic ip at e in th e to ta ln um

be ro

fp ol ic e of fic er s sc he du

le d in th e sh ift .T

hi s m ea ns

th at bo th

w or k sc he du

le s (in

tw o sh ift s an d in

th re e sh ift s) al lo w

th at

po lic e of fic er s ta ke

da ys

of f w he n th ey

ne ed

to be

ab se nt

fr om

w or k

Table III. Police officers schedule – three- shifts optimal solution

308

PIJPSM 38,2

When observing the old work schedule (column Old scheduling), it can be noted that achieved working standards significantly differ among police officers. In the obtained schedule of working in two shifts (column Optimal solution, two-shifts) all police officers who were present during the whole month, worked exactly 180 hours. Only the police officer Number 9, who was absent for 11 days during the month, had lower working standard. On the basis of the value of the parameter bij for the police officer Number 9, it was provided the value of the parameter pri ¼

P jADbij=m; iAZ that

represents the percentage of the presence of a police officer during the month: pr9 ¼ 2031 ¼ 0:645.

The parameter pri i∈Z in the constraint (7) provides that a police officer’s working standard is proportional to his presence during the month. By multiplying the obtained percentage with the standard of 180 hours, it appears that working standard of the police officer Number 9 is approximately 116 hours in December. Taking into account that shifts last 12 hours, this means that the police officer Number 9 cannot get exactly the standard of 116 hours, but 120 hours (12 hours per ten days) or 108 hours (12 hours per nine days), as was obtained in our case. When it comes to the schedule of working in three shifts (column Optimal solution, three-shift), two police officers received nine hours above the standard, while the policeman Number 9 received 117 hours of work.

6. Conclusion It is clear that there is no single view of researchers on what is the most effective way of shift work. Superiors often favor eight-hour or 12-hour shift schedules, while police officers prefer compressed schedules. Shift schedules with back rotation are considered to be the most difficult for employees’ adjustment. In some police stations, flexible work schedules are more frequently encountered, which have their own advantages and disadvantages. Also, in addition to the usual number of shifts (two or three), swing shifts that overlap throughout main shifts also appear in work schedules. When determining adequate work schedules, it is necessary to take into account the needs of the police station, ergonomic recommendations for shift work and the demands of policemen.

Even when the police station adopts the working mode that best suits it, it is necessary to make specific work schedules every month, taking into account the needs of the police station and the preferences of employees. The paper proposes a mathematical model that automates the scheduling of police officers. Mathematical

Police officer Old scheduling Optimal solution two-shifts Optimal solution three-shifts

P1 180 180 180 P2 192 180 180 P3 192 180 189 P4 192 180 189 P5 192 180 180 P6 204 180 180 P7 192 180 180 P8 180 180 180 P9 108 108 117 P10 156 180 180 P11 216 180 180

Table IV. Police officers

schedule – two-shifts optimal solution

309

Police officer scheduling

model was applied to the current situation, a work schedule in which police officers actually work and the potential distribution of working in three shifts. Changing of the current mode of the police station would require little intervention on the mathematical model, which means that the existing mathematical model could be adapted to other working modes.

Since the mathematical model was set and the initial base of employees and predefined schemes of work was made, making the specific schedule for the next months would require a definition of the parameters relating to the reporting month, including: absences of police officers and a required number of officers by days and shifts. In order to adapt the model to the needs of the person who makes schedules, the interface for the input of parameters for a particular month and loading the optimal schedule, should be made. In this case, the user would not have to possess knowledge of modeling, nor should know to use GLPK or any other solver.

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