Workforce Scheduling Evrim Didem Gunes Department of Industrial Engineering Bilkent University TR-06533 ANKARA
[email protected] April 15, 1999 Abstract
The workforce scheduling problem is in general, the problem of determining how many workers must be assigned to each of the planning periods of work time for an organization. In this study the workforce scheduling problem is de ned and its general characteristics are presented. The main focus is on the problems applicable to the service or manufacturing systems operating continuously. The relevant literature on the subject is also brie y reviewed.
1 Introduction The workforce scheduling problem is in general, the problem of determining how many workers must be assigned to each of the planning periods of work time for an organization. Since human resource is an essential component of almost all organizations whether in manufacturing or service sector, scheduling workforce is a common problem to all organizations. Especially for the service industry and continuous manufacturing settings, where the operations must be continued \round-the-clock", i.e. 24 hours a day, matching the personnel needs to available labor is an important planning task. In this report, an overview of \workforce scheduling" problem will be given with discussing the general characteristics, solution methods and the studies in the relevant literature.
2 Problem De nition The workforce scheduling problem can be de ned as \the problem of optimally matching available labor resources to the needs of an organization considering all applicable constraints" [9]. As a solution to this problem the work schedules are developed for each employee, specifying the working (on) and rest (o ) times through the planning period.
In partial ful llment of the requirements for the course IE 672 Spring 1999
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This problem arises in all organizations which has a human resource component, and the research concerning the subject matter is closely related with the industrial needs and applications. The research for this problem is extended in the service delivery system settings including nurses in hospitals, baggage handlers in airlines, operators in telephone companies, air crew at airline stations, console monitors in electrical generating facilities and patrol persons in a police department. The manufacturing systems with continuous manufacturing is also subject of research. The terminology used in workforce scheduling studies can be brie y de ned as follows [22]: Period: the smallest time interval for which the labor requirements are established, i.e. planning occurs, typically 15, 30 or 60 minutes. Shift: a work schedule for an employee for a speci c day Tour: a work schedule for an employee for a week, comprised of separate shifts The workforce planning process is a complex, multi-stage planning and control process. Depending on the particular conditions and the policies of the organization, there are several requirements that must be considered in the planning process. The environmental conditions for the workforce scheduling problem are brie y as follows: 1. Nature of demand for personnel - constant or varying over the days, cyclic or noncyclic. 2. Type of work schedule used. The most common is standard 5 day, 40 hour workweek. Other approaches that gain popularity are compressed workweeks and exitime, which can improve productivity, employee morale and absenteeism [17]. Examples for compressed workweeks are; 4 day, 10 hour/day week and alternating 3 and 4 day, 12 hours/day week. Flexitime allows workers to choose his/her starting and stopping times of work, as long as he/she works in the \core hours", usually 10 a.m. to 3 p.m. 3. Number and pattern of allowable daily shifts - There may be day, evening and night shifts, with alternative start-end times and break periods allowed. 4. Work rules concerning the days on/o patterns for the workers - The o days should be consecutive or not, the rotating shifts may be allowed or not etc. 5. The maximum workstretch (i.e. the number of consecutive work days) allowed 6. Workforce attributes - There may be di erent skills of workers, part-time workers may be considered etc. Considering all these factors, the task is to nd a feasible and minimum costly schedule. Besides these complicating factors, the main diculty of workforce scheduling problem is that it deals with human beings. So \cost" of a schedule is far more than the resulting total wage. The solution found has impacts on employee morale depending on how well it ts to the individual preferences, which a ects productivity. Moreover the training costs, 2
hiring- ring, overtime costs, health and safety factors should also be taken into account while preparing work schedules. The characteristics of a work schedule that should be considered while making the scheduling decision are: Coverage:The existence of minimum required number of workers for each time period. Quality:Measure of the desirability of the schedule for the worker who will work it. Stability:Measure of the extent to which the workers know their future days o and on duty. Flexibility:The ability to handle changes, such as from passing from full time to part time, and emergence of special requirements of workers. Fairness:The measure of extent that each worker is a ected same by the schedule, in terms of undesirable shifts etc. Hence, not all feasible and minimum costly schedules can be judged as best solutions for the workforce scheduling problem. The general steps involved in analyzing and solving a workforce scheduling problem are as follows [9]: 1. Determine quantity of work to be done. 2. Determine stang required to do the work (for each time period). 3. Determine total sta size enough to do the work. 4. Determine personnel availability. 5. Match personnel to stang requirements. Determine if labor needs and availability mismatches are signi cant. Investigate changing the work demand pattern. Investigate altering the time availability of personnel. Develop a work schedule. Develop a workforce scheduling management system. In the literature the rst two steps are generally skipped, assuming that the employee requirements for each planning period are given. Although the general tendency in the literature is emphasizing step three, most of the time the workforce size is given in an organization, while preparing the work schedules. However, nding the minimum size of the sta is essential for the systems like airline industry, where the crew cost comes second after fuel cost [24]. The fourth step is also not considered by the majority of researchers, assuming that there is unlimited personnel available. This is a reasonable approach considering the unemployment problems most of the countries face. The fth step (mainly fourth item) is the task of assigning personnel to the work and rest periods (shifts or days), satisfying all the feasibility requirements determined by the work rules and regulations, and maximizing the employee satisfaction. 3
3 General Approach to the Problem Workforce scheduling is a widely studied subject of research. There are many variants of the problem like air crew scheduling, audit scheduling, nurse scheduling etc. which can be considered as separate subject headings, and a considerable literature exists under these headings . In this study the most common properties of workforce scheduling problems are considered, which correspond most to the type of nurse scheduling. The workforce scheduling problems can be classi ed with respect to the following: 1. Nature of demand and stability of workforce-cyclic versus acyclic 2. Work time increment scheduled-shift, workweek (days-o ) and tour scheduling. Cyclic scheduling assumes a demand pattern that is constant within the speci ed time periods. For example if the demand may vary between days of the week and weekly pattern is constant, then cyclic scheduling can be applied. Once a schedule is determined each employee repeatedly cycles through the same set of weekly patterns. Cyclic scheduling assumes a constant number of personnel all of whom are available at all times. For the acyclic case, the problem should be solved each time the demand pattern and employee availability change, for a speci ed planning horizon. Shift scheduling problems are solved to develop a single day's work schedule. They arise when it is necessary to determine which shift the employees should be assigned each working day, especially when shifts are overlapping. In certain service organizations (e.g., operators in telephone companies) this is the only problem to be solved since the working days are xed by convention or management policy [19]. The workweek scheduling solutions nd days-o schedules, which determines the days each employee should be assigned o . This problem occurs when the length of an employee's workweek is di erent from the length of the business week. For example, 5-day workweek and 6-7 day business operation, as with nurses, toll collectors or retail salespersons. Dayso scheduling assumes a single shift for each day, which is not the case for organizations operating 24 hours a day. However, this is not as restrictive as it may appear, since many multiple shift operations sta and schedule each shift independently [14]. When both the daily shift for each working day, and days-o schedules are required, the two problems should be solved simultaneously, leading to more general tour scheduling problems. Tour scheduling involves the determination of work and nonwork days during the week, as well as the associated daily shift starting and nishing times (shift schedules) for each employee. Tour schedules for continuous operations mostly require workers to rotate through shifts. Shift work has many negative e ects on workers' health and social life, however it is sometimes unavoidable. In this case, rapid rotation between shifts is recommended, from day to evening to night shifts preferably [9], [15].
3.1 Solution Methods
In practice, a great portion of the work schedules can be developed by straightforward empirical approaches. These problems have the following characteristics: usually a xed 4
number of workforce is scheduled, meal and break scheduling is ignored, the shifts are nonoverlapping and the stang requirements are relatively constant. The objective in this case is to nd a schedule that would satisfy the non nancial parameter (number of work stations available, days of work, etc.) of the work situation. The empirical approach assumes that the number of employees is given. On the other hand, there are many practical situations where these conditions do not hold, for example demand may vary through the week. The researchers have been studying on such cases and developing di erent solution methods with two general approaches, optimization methods and heuristics. For workforce scheduling, an optimal solution means, a solution having the optimum number of workers and satisfying the work rules. But, as explained before, performance of a schedule should also be measured with respect to other characteristics such as exibility, quality etc. It is therefore useful to produce several good solutions for nal subjective judgment. The mathematical programming formulations for the problem are based upon the following set-covering formulation, originally proposed by Dantzig [11]:
Xc x n
Minimize s.t.
j =1
j
j
Xa x r n
i = 1; 2; :::; m x 0; and integer
j =1
ij
j
i
j
This formulation can be used for shift, o -day and tour scheduling problems if we de ne the notation used appropriately. For shift scheduling the notation is as follows:
x =number of employees assigned to daily shift pattern j . r =number of employees required to work in the ith time period. c =the cost of an employee assigned to the j th shift. n =number of daily shift types to be considered. m =number of time periods to be scheduled over a single day a =1 if time period i is a work period in the daily shift pattern j ; 0 otherwise. j
i
j
ij
All the possible alternative work schedules that obey the work rules should be determined at the beginning, to be an input to this program. As the solution, number of employees assigned to each pattern will be determined. After that it is an easy task to determine the work schedules for individuals, for which many algorithms are found in the literature. For days-o scheduling the alternatives considered are the days-o patterns, and time period is a day, where m is the number of days per week that the system operates. For tour scheduling, di erent tour patterns are represented by x 's, where m is the number of time periods to be scheduled over the week. The cost term c can represent the undesirability level of shift j , in addition to the wage rate, thus incorporating the intangible factors a ecting the quality of the schedule. With this formulation, individual preferences cannot be considered. But there are some models in the literature that can manage the individual preferences ([25],[12] for example.). ij
ij
5
When number of feasible alternatives is small, and the time period is large, this formulation can be used to nd the optimum size and allocation to the di erent patterns, of the personnel that can do the required work. But as the problem size increases, it becomes too hard to get an integer formulation. Bartholdi [4] has shown that the cyclic sta scheduling problem is NP-complete. So, it is highly-unlikely that a fast, optimal- nding solution technique will be found. Thus, heuristic algorithms are justi ed. Another method of optimization approach is to nd the optimum workforce size by deriving closed form expressions as a function of the parameters of the demand pro le (see [3],[2],[14],[17] for example.). This approach simply nds lower bounds for the workforce size, considering the work rules to be satis ed, such as number of days o . For example, a very simple case studied in [3] requires N employees each weekday, and n each weekend day, and there must be two o days for each employee. Than a fundamental lower bound for workforce size, W is: max(n,N). Secondly, the capability of workforce must be sucient to meet the total number of work days required in a week so; 5W 5N +2n. If there is no other restriction, than the optimal workforce size is found as: W = maxfn; N + [2n=5]g. After the optimal workforce size is found, the work schedules for individuals are developed using some procedures, which can be very simple manual ones or computerized search algorithms. The simplest one is, for the above example: just assign the rst (WN) employees Monday o ; for the remaining weekdays, assign the next (W-N) employees the next day o ; for the weekend days, assign the next (W-n) employees o ; assign the remaining o days arbitrarily so that each employee is left with ve workdays [3]. As the problem setting gets complicated, more complex algorithms have to be developed. Heuristic algorithms generally start with the LP relaxation of the integer programming formulation and use some rounding procedures to nd feasible integer solutions. The number of alternative patterns to be input to the mathematical program may be reduced as a heuristic approach. For the air crew scheduling problems the model used is a bit di erent. A tour of duty (TOD) is de ned as a feasible sequence of unbroken ights between two ports with given departure and arrival times. The coverage constraint is then that \for every ight a crew must be assigned", rather than \for every time period minimum required number of workers must be assigned". There are some additional complicating constraints related with the location of ight destinations (the duration of ights). For this reason, determining the alternative feasible work tours is also an important problem to solve for aircrew scheduling problems, in contrast with the classical workforce scheduling problems mentioned until now. For determination of the alternative TODs air crew scheduling problems are generally formulated as set partitioning integer programming problems where the columns correspond to alternative tours of duty, and rows correspond to ights, i.e. each ight must be assigned to a tour of duty exactly once. But since the problem size is generally very large, it is not practical to solve this model directly. Algorithms based on vehicle routing heuristics [24], and column generation approaches are some examples of solution approaches for this problem.
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4 Literature Review There is a considerable research for the workforce scheduling problem. The problem settings considered, and the solution methods used lie in a wide range. We can classify these studies according to the type of schedule considered, that is shift, days-o or tour scheduling. Other classi cations are also possible; as cyclic and noncyclic scheduling; heuristic and optimization approaches et. as can be seen in Table 1. Among these articles reviewed below, there are many case studies, which are directly applied to the organizations in US, Canada and UK, mostly hospitals. It is reported that substantial amount of saving from the planning time and labor cost is achieved as a result of these studies (see [25], [12], [18]). Days-O Scheduling
In the days-o scheduling literature, optimization approach is more popular. This can be due to the fact that closed form expressions for optimum workforce size are more easily determined for days-o schedules, days do not overlap as shifts may do. Another reason may be that the problem size is more manageable with the days-o scheduling. Tibrewala et al. [23] found a simple algorithm that provides optimal solution to cyclic scheduling of two consecutive o days, for a 7-day workweek. Baker [2] proposed a more ecient algorithm for the same problem. Baker and Magazine [3] studied this problem under a variety of days-o policies, including 2 days o , 2 consecutive days o , every other weekend of and 4 days o every two weeks. They assumed the employee requirement is N for each weekday, and n for weekend days. They have derived explicit formulas for the workforce size and developed algorithms to construct a feasible schedule in each case. Rosenbloom and Goertzen [20] considered the cyclic nurse scheduling problem under a variety of work rules, concerning workstretch and o -days. They have presented an optimal algorithm which can be implemented easily on a microcomputer. Their algorithm generates all feasible schedules for a week, and solves an integer program formulating the cyclic constraints and daily requirements constraints. In a recent study, Emmons and Fuh [14] considered the days-o problem for 7 day workweek where demand during the weekdays is constant. Their contribution is that they considered two types of part-time workers to supplement the full timers, one limited in numbers and less costly than full timers, the other more costly and available in any number. Moreover, each full timer must have two days o each week and A out of B weekends o . Under these work rules, they have developed formulas to nd the minimum costly worker mix, and an algorithm to nd a feasible schedule. Emmons and Burns [13] have studied cyclic scheduling in the case of hierarchical workforce, under the days o and workstretch constraints. They have given explicit formulations for the optimum workforce mix, and algorithms to nd feasible schedules. Billionnet [7] studied a hierarchical workforce, where a higher quali ed worker can substitute for a lower quali ed one, but not vice versa. Every worker must be assigned nconsecutive days o . They have formulated this problem as an integer program, and showed experimentally that its solution requires a few seconds of computation time. The assumption of pre-determined demand pattern, which is very common in the literature, was eliminated in a recent study by Alfares and Bailey [1]. They have considered the project management problem integrated with workforce scheduling, where the task 7
durations depend on the number of workers assigned to it. They have developed an integer linear programming model, which was impractical, and developed a heuristic solution procedure for this problem. As a result a signi cant improvement in total cost and labor cost was achieved . Shift Scheduling
Shift scheduling tries to nd a single day's schedule, so if we assume that the demand during di erent periods are constant through the week, then cyclic scheduling can be applied. The shift scheduling studies does not mention about the variability of demand through the week, but implicitly they assume that the schedule will be cyclic. Otherwise, the shif scheduling problem can be solved for each single day separately. Henderson and Berry [16] studied heuristic methods for shift scheduling for telephone operators. Their methods use the set covering formulation, with a smaller set of alternative shifts, selected heuristicly. Then, LP relaxation of this model is solved, and tried to be improved. Bechtold and Jacobs [5] have developed an implicit formulation for shift scheduling problem, which is superior to the traditional set covering formulation with respect to execution time, computer memory requirements and the ability to produce integer solutions to larger problems. They have also presented an experimental analysis. In a more recent paper [6], they have shown that their implicit integer programming formulation is equivalent to the set covering formulation. Tour Scheduling
Tour scheduling problems have gain more popularity in later studies since they represent a more realistic setting, and provide a more exible solution. Morris and Showalter [19] emphasized the need to integrate shift and days-o scheduling problems, and introduced a simple round-down heuristic for tour scheduling problem, starting with LP relaxation of the set covering formulation of Dantzig. Burns and Koop [10] introduced a modular algorithm to solve cyclic tour scheduling problems with days-o constraints. The number of workers required is calculated using the lower bounds with respect to feasibility constraints. The opportunities to apply the algorithm to di erent workforce scheduling problems are also discussed. Hung and Emmons [17] introduced a compressed workweek problem for the rst time, which assumes a hierarchical workforce. They have considered alternating 3 days,12hour/day and 4 days, 10 hour/day compressed workweek, where shifts may overlap. Each worker must have A out of every B weekends o , and maximum allowed workstretch is 5 days. For these requirements, they have developed optimal algorithms that can be implemented by hand. Brusco et al. [8] studied the tour-scheduling problem of ground personnel for airlines, where planning period is 15 minutes, and there are limitations on part time and full time shift ratio. They have designed a modular system; rst module is a column generation procedure that focuses on the selection of full time and part time shifts, and the second one improves an initial tour using simulated annealing. They have applied this system to 8
Solution Approaches Problem Types Optimization Heuristic Days-o /cyclic [2], [3], [23],[20],[13] Days-o /non-cyclic [7], [14] [1] Shift/cyclic [5], [6] [16] Tour/cyclic [17], [10] [19] Tour/non-cyclic [18], [25] [8], [12] Table 1: Classi cation of reviewed articles United Airlines, and achieved a potential saving of $8 million. As mentioned earlier, the quality of a work schedule is closely related with how well it ts to the individual preferences of the workers. One of the earliest studies taking this fact into account was done by Warner [25], for cyclical nurse scheduling, and the system developed has been applied in 20 hospitals by the time of study, and reduced the average amount of time to make a 6-week scheduling decision from 18-24 hours to about one hour. In this study, the system for learning and formulating the nurses' preferences are also explained in detail. A multiple-choice programming formulation is used and solved by block-pivoting method. Recent studies on nurse scheduling pay more attention to the employee's preferences. One example is a study by Jaumard et al. [18]. Since they have collaborated with a hospital in Canada, the problem setting is very realistic. The nurses have di erent skill levels, overtime and use of part time workers is allowed. There are seven shift types which may overlap. Each nurse indicate preference levels on the alternative tour patterns, which become input for the mathematical programming formulation. They have used a 0-1 column generation model with a resource constrained shortest path auxiliary problem for the solution. Another realistic study for nurse scheduling is done by Downslad [12] to be applied for a UK hospital. The workforce is hierarchical, and part-time workers are available. There are only day and night shifts, and for each di erent weekly pattern (tour) and nurse pair, a penalty value is determined depending on the overall quality of the pattern, the nurse's requests for days o and the recent history of shifts worked. The algorithm uses tabu search, and it was shown that its performance was very good. In all these studies a common assumption is that the number of employees required each period is readily given. However, as mentioned in section 2, rst step to be accomplished is determining the customer demand, and the second is translating this demand to employee requirements. Thompson [21] realized this fact and focused on the second step of workforce planning. In a later study [22] he studied the tour/shift models allowing control over the desired service levels to customer.
5 Conclusion In this study, a brief overview on the workforce scheduling problem is tried to be given. The problem is de ned and general characteristics are explained, in addition to the com9
plicating factors and the solution approaches in the literature. Then the relevant literature is reviewed. Workforce scheduling is a widely studied subject of research. There are many variants of the problem like air crew scheduling, audit scheduling, nurse scheduling etc. which can be considered as separate subject headings, and a considerable literature exists under these headings. In this report the most general form of the problem was investigated. Workforce scheduling problems are real life problems encountered in almost all organizations and problem environment changes in every organization because of the policies and regulations. For this reason, the practical situation that the system o ers should be taken into account while solving the workforce scheduling problem for an organization.
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[12] Downsland K., \Nurse scheduling with tabu search and strategic oscillation", European Journal of Operational Research, 106, 393-407, 1998. [13] Emmons H. and Burns R. N., \O -day scheduling with hierarchical worker categories", Operations Research, 39 (3), 484-495, 1991. [14] Emmons H. and Fuh D., \Sizing and scheduling a full-time and part-time workforce with o -day and o -weekend constraints", Annals of Operations Research, 70, 473492, 1997. [15] Folkard S. and Monk H. T. (eds.), Hours of Work, Wiley & Sons, New York, 1985, p197. [16] Henderson W. B. and Berry W. L., \Heuristic methods for telephone operator shift scheduling: An experimental analysis", Management Science, 22 (12), 1372-1380, 1976. [17] Hung R. and Emmons H., \Multiple-shift workforce scheduling under the 3-4 compressed workweek with a hierarchical workforce", IIE Transactions, 25 (5), 82-89, 1993. [18] Jaumard B., Semet F. and Vovor T., "A generalized linear programming model for nurse scheduling", European Journal of Operational Research, 107, 1-18, 1998. [19] Morris J. G. and Showalter M. J., \Simple approaches to shift, days-o and tour scheduling problems", Management Science, 29 (8), 942-951, 1983. [20] Rosenbloom E. S. and Goertzen N. F., \Cyclic nurse scheduling", European Journal of Operational Research, 31, 19-23, 1987. [21] Thompson G., \Accounting for the multi-period impact of service when determining employee requirements for labor scheduling", JOurnal of Operations Management, 11, 269-287, 1993. [22] Thompson G., \Labor stang and scheduling models for controlling service levels", Naval Research Logistics, 44, 719-740, 1997. [23] Tibrewala R., Philippe D. and Browne J., \Optimal scheduling of two consecutive idle periods", Management Science, 19 (1), 71-75, 1972. [24] Wark P., Holt J., Ronnqvist M. and Ryan D., \Aircrew schedule generation using repeated matching", 107, 21-35, 1997. [25] Warner D. M., \Scheduling nursing personnel according to nursing preference: A mathematical programming approach", Operations Research, 24 (5), 842-856, 1976.
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