NURSE SCHEDULING: FROM THEORETICAL MODELING TO PRACTICAL RESOLUTION

Transcription

1 NURSE SCHEDULING: FROM THEORETICAL MODELING TO PRACTICAL RESOLUTION Hocine Bouarab: Sophie Champalle: Martine Dagenais : martine-2.dagenais@polymtl.ca Nadia Lahrichi : nadia.lahrichi@cirrelt.ca Antoine Legrain : antoine.legrain@polymtl.ca Mehdi Taobane : mehdi.taobane@polymtl.ca École Polytechnique de Montréal, Département de mathématiques et de génie industriel MTH6953A : Optimisation des ressources en santé (Resource Optimization in Healthcare) Fall, 2010 Abstract Scheduling problems are complex and difficult to tackle. When studying practical problems, one can observe that this exercise is often manual, very time consuming, and does not always provide the best quality results. The impact of scheduling in the healthcare sector is major in must jurisdictions operating under tight budgets. The aim of this paper is to study and analyze the scheduling process in practice, and propose models and heuristics to improve both the process and the quality of the resulting schedule. Nurses should benefit from this study by having higher quality schedules while the employees in charge of scheduling should enjoy the positive benefits of an optimization tool which should guide their work and is certainly superior to trial and error. Introduction Hospital s doctors offer to pay nurses salaries titled the Globe and Mail in May This is the situation of the healthcare sector in Canada. The province of Quebec is subject to an ever-increasing demand due to both universal access to healthcare and the aging of the population. Meanwhile, resources tend to go in the opposite direction: budget cuts are unavoidable and human resources are increasingly scarce. The latter dimension and nursing resources in particular are the main interests of this paper. Nurses are responsible for a huge part of the medical activities, and account for approximately 25% of the total hospital operating budget and 44% of direct care costs [7]. Providing nurses with good work conditions is a primary objective, particularly in the Quebec context where unions are powerful and discussions about strikes punctuate the news. Offering flexible schedules encourages the stability of workforce and makes the profession more attractive in a context where there are chronic staff shortages. One way of addressing this challenge is by providing an appropriate nurse scheduling system. This is the problem tackled in the class project of the course Resource Optimization in Healthcare. The objective in this project is threefold: first conduct a practical study on the process of nurse scheduling in hospitals, second introduce a heuristic that can be easily implemented in these hospitals at no extra cost, and finally, use a commercial software to generate new schedules and compare them to the current ones. The scheduling problem to solve is the determination of monthly schedules for nurses. The semantic of this paper is organized as follows. The next section presents the problem of nurse scheduling and the specific constraints, and a brief literature review follows. The methodology section introduces new heuristics and presents a brief description of the commercial software. A computational results

2 section shows the benefits of the proposed approaches and highlights their advantages, follow conclusions. Problem Statement Nurse scheduling is a complex exercise with multiple and contradictory objectives: minimizing total costs while maximizing the nurses preferences and requests, and equally distributing workload between nurses. Work constraints imposed by collective agreements and unions as well as contracts have to be respected. Constraints in nurse scheduling relate to: demand for each shift; shifts that can be assigned to each particular nurse; maximum number of consecutive days of work; minimum amount of rest time between two shifts; isolated days of work or days-off. In practice, to measure how these constraints, preferences and equity in workload affect the scheduling process, it is essential to obtain first hand information within a hospital unit. Two different hospitals in Montreal were contacted, Notre-Dame Hospital and Sainte- Justine Hospital. These are two large public and university hospitals. The structures of these hospitals are standard. They are composed of regular nurse teams within each unit of the hospital, supported by a float team for the whole hospital, whose purpose is to absorb demand variations. Nurse scheduling is decentralized and each unit delegates this task to a clerk responsible for manually setting up a 28 daysschedule for each nurse. In this perspective, a regular team for Hospital 1 (with a subset of nurses following a rota system while others are assigned to one specific shift) has been selected. For Hospital 2, a float team was also selected where all nurses are assigned to a specific shift. Both workforce sizes are 30. The objective of this paper is to evaluate how operations research and lean thinking principles can be used as tools in the development of efficient schedules. This will be achieved with a proper analysis of current processes based on the comparison between a manual but standardized process and commercial software. Literature review Literature on nurse rostering and scheduling is extensive. One may refer to literature reviews on the subject that provide indepth studies on this problem such as Burke et al. [1] and Ernst et al. [2]. These provide methodology descriptions on scheduling, models as well as references to algorithms. A wide variety of methods have been used to tackle nurse scheduling: mathematical programming, constraint programming, heuristics and meta-heuristics, hybrid methods as well as simulation. Even creative methods such as auction systems have been applied to tackle nurses preferences [3]. Different objectives are studied in this literature: to decrease manual scheduling; to increase demand covering in terms of workforce size but also according to required skills; to obtain equity between the schedules. As highlighted earlier, the aim of this project is not to develop a new methodology for solving the problem, but to simplify the actual process while reaching the same objectives. One study is particularly pertinent and need to be referred to. Ferland et al. [4] introduce an assignment type problem modeling scheme to solve the scheduling problem. The principle is to consider a set of objectives constituted by the formal objectives of the problem as well as a set of constraints. They have used a tabu search [5] where at each iteration, two solutions are compared by considering their objectives in a lexical order. This prioritization of objectives is central in scheduling, and some inspiration will be drawn from it.

3 Modeling nurse scheduling problems Even though nurse scheduling is a well studied problem, according to Ernst et al. [2], there is room for further contributions in the field. Improving self-scheduling and automating scheduling are of the easiest [6]. Some of them will be addressed in the methodology section. Two practical examples will be studied, modeled, and different ways of improving the current process will be presented. In both studied hospitals, the clerk goes through the following steps to create a schedule: 1. Collect preferences; 2. Sketch the schedule; 3. Correct the schedule; 4. Post the schedule; 5. Adjust the schedule. The clerk uses five inputs in his/her process: the constraints related to work agreements, the demand coverage, the draft schedule, her knowledge of the recurrent preferences and her personal judgment. The data collection exercise to gather these inputs for this project was challenging: the principal reason being that most information is only in the clerks head. Work rules First of all, the collective agreements were reviewed to collect work rules. These rules concern the shift (or set of shifts) assigned to each person, the number of shifts per week (usually five), per two weeks and per four weeks, the definition of fixed days (usually week-ends), the length of work sequence (usually a maximum of five), and the rest length between two shifts (minimum of 16 hours). Demand coverage The demand for a regular unit is usually set in advance and referred to as a quota. Quotas differ from day to day and from shift to shift and are fixed regarding the budget and the units needs. In the case of regular nurses, at least one responsible nurse is usually needed. Generally, quotas vary (increase or decrease) following the units workload. At the opposite, the demand for the float team is not known since it is related to: 1. the variation in the workload of the units, and, 2. the variation on the real workforce size (absenteeism for example). Since no estimation of this demand is provided, a historical data of the last year has been traced to evaluate the average demand for each shift and each day as well as the variance. As the real quota requirement totally differs from the one that can be covered (sometimes even 50% of the workforce is missing), the potential demand is modified using the following formula: (Average Demand / Total Demand * Available shifts) * (Variance / Average Variance). Preferences Even though gathering the preferences requirements seems an easy exercise, it can be fastidious in practice. Preferences typically relate to whether or not nurses would like to work on a particular date. Since the schedule planning is performed every four weeks, the clerk usually allows few days for the nurses to indicate their preferences. Each nurse will annotate the schedule illustrated in Figure 1 to indicate her preferences. Figure 1: One period for a new schedule However, recurrent preferences are not specified: the clerk is already aware of these specifications. The process of gathering preferences is interesting to study. The first clerk is very well organized; she asks for preferences, and nurses have two weeks to complete the typical schedule defined by the recurring sequence of shifts for each nurse. The preferred schedule is that schedule which contains only preferences of nurses. The clerk in the second hospital has no defined process: he/she accepts changes to the schedule every single day while working

4 directly on the schedule, until the official schedule is posted. This process is very timeconsuming, not efficient and certainly not optimized. No recording of the preferences is thus available. This problem will be addressed in the next section. In the meanwhile, different official schedules have been analyzed to determine the preferences for the October November 2010 period using the number of shift combination per nurse, the seniority, the skills, the type of rotation or shift, and the weekends. Objectives and personal judgment As one can expect, modeling personal judgment is a difficult exercise since no objective criteria is available. However, it is not surprising to hear that demand coverage (because of lack of resources) has to be maximized as well as nurses preferences. The clerk has been interviewed in order to recreate real time situations and enhance the tradeoffs needed to satisfy both preferences and quota requirements. One can summarize the situation by stating that there is absolutely no rule: the win-win relationship between the clerk and nurses often leads to informal and subjective rules or constraints, which are very difficult to track. The clerks have mentioned they consider the equity between schedules, but once again, there is no indicator on how to evaluate this equity. The notations to model the problem are introduced below: Notation - N : set of nurses considered on this exercise; - N R : set of responsible nurses; - J : set of days of the period (28 days in our example with j = 1 referring to a Sunday and j = 28 to the last Saturday); - K : set of shifts for each day (8 hours each in both cases: Night, Day, and Evening). Parameters: - F ij : the typical schedule of the nurse i expressed as a vector of 0 (no work) and +1 (work) for each day j; - A ij : Matrix of days off of the nurse i expressed as a vector of 0 (no day off) and +1 (day off) for each day j; - P ij : preferences of the nurse i expressed as -1 (no work), 0 (no preference) and +1 (work) for each day j (preferences are only related to day and not shifts); - D jk : demand in nurses on day j and shift k; - Q ik : available shift k for the nurse i; - T : maximum of days worked by a nurses in one week; - c + : cost of an over-covering; - c - : cost of an under-covering; - β i : cost of a preference not respected of the nurse i; - γ ik : cost of switching assignment; - r i : benefit of the respect of the typical schedule of the nurse i (aggregation of seniority, experiences, skills...). Decision variables: - x ijk : 1 if the nurse i is assigned shift k on day j and 0 otherwise; - z+ jk : number of nurses over-covering the day j in the shift k; - z - jk : number of nurses under-covering the day j in the shift k; - M i : length of the last work sequence in the previous period. The model can be stated as follows: Subject to:

5 (10) (13) Equations (1) ensures at least one responsible nurse is present during day shift, equations (2) - (4) target the respect of days off, maximum number of work days in a week and in two weeks. Equations (5) - (6) set the maximum of working days to T for the five first days of the month and for the rest of the period respectively. Equation (7) ensures nurses are only assigned to shifts they are allowed to work, and work exactly one week-end out of two in (8). Equations (9) and (10) set the minimum length of rest between two work shifts. Finally equations (11) and (12) allow measuring the over-covering and the under-covering (gap between quota requirement and actual workforce size on the floor). Equations (13) and (14) ensure variables are binary and integers when needed. The objective function is constituted of three terms. The first one specifies that rotation from one shift to another is minimized. The second one is a quadratic term that ensures that penalty increase rapidly when moving away from quota requirements and finally the third handle that preferences are maximized. This model reflects exactly the problem of the first hospital. To view the second hospital model, one has only to remove the first term in the objective function, and equations (1), (9) - (10) since nurses do not rotate over shifts. The next section first presents improvement procedures to the actual process without introducing new methodology. Secondly, it focuses on presenting reasonable heuristics for both problems: the two have the particularity of not needing any programming or additional resources. Finally, the first problem will be solved using commercial software designed for solving personnel scheduling while solving the second one to optimality using CPLEX. This software is efficient for solving problems with quadratic structures (one can use a piecewise linear function to approximate the objective). Solving directly the first model with CPLEX is tedious and the commercial software is not designed for the second problem. Methodology There is only one hard constraint that is required to be respected when the clerks are constructing the schedules: the workload expressed in terms of days of work into each period. All other constraints are referred to as soft constraints. Since there are shortages of human resources, quota requirements have to be reached rather than fulfilled, and particular attention is paid to Mondays and Fridays shifts. Nurses often call off on those days and the clerk tries to have the exact quota needed for those specific two days (to avoid subsequent absenteeism due to expected increasing workload). Week-ends assignments in these examples are fixed and cannot be modified.

6 Both clerks follow a five step process to create an official schedule. Both hospitals use a software, based on Excel, designed to manage the human resources and linked to the payroll department. It does not optimize the schedule but is used as a visual tool. Figure 2 illustrates the mapping of this process. that best satisfies her requirements using the same flip movement. The Figure 3 below illustrates this move. Figure 3: Illustration of flip movements Figure 2: Scheduling process map Collect preferences. This step is in both cases done using an initial schedule defined by only fixed assigned week-ends for nurses while the second one contains also the typical schedule frame. This schedule is usually chosen by each nurse when accepting her position. Sketch schedule. In the case of the first hospital, the clerk has a constructive procedure: she uses the initial schedule and adds shifts in order to maximize the quota requirements. When all mandatory shifts are added, she will then try to improve the schedule one week at a time using a flip movement. She will move one shift assignment from one day to another in the general case, and from a shift to another in the case of nurses under the rota system. The second clerk has an improving procedure: she moves from a completed schedule to another Correct schedule. Basically, this step in both cases consists of asking for the nurses feedback on the fulfillment of their preferences. The negotiating phase follows to get them to accept the schedule and then try to implement the changes when possible. Post schedule. Two weeks before the beginning of the new period, the schedule is posted on the unit. The schedule is final and no major changes should be done. Adjust schedule. This step is out of the scope of mid-term planning and considers all non predicted daily changes, but constitutes the link between regular nurses units and the unit of float team. Improving the current process Two potential improvements are proposed, one for each clerk that addresses two major wastes in a human resources shortage context: waste of time and waste of energy. In the sketch schedule step for the first clerk, when constructing the schedule by adding shifts, the clerk has to verify if all assignments correspond to the quota requirements for each shift. The clerk has often to change spreadsheet to find out whether it is respected or not, and memorize it while moving to the next step. This task could be simplified by adding three lines that count the number of persons for each shift and day. For the second clerk, she should establish a clear rule on when the list of preferences is expected and a deadline from which no more changes are accepted. Currently,

7 the clerk spends a tremendous time cycling between the first and the third steps because she accepts a new preference, sketch the schedule, correct it, add another preference, sketch it, and so on. At least 50% of her time could be saved by implementing this very simple rule whereas the actual process is very time-consuming, not efficient and certainly not optimized. Standardizing the sketch schedule step This step is the most important in the scheduling process. This section introduces two heuristics, one for each of the two problems that respects the following criteria: standard, simple to use, requiring zero additional cost and resource, producing better quality and finally less time-consuming. The actual scheduling is person-based process while standardization should be a motto. The entire process is based on the clerks knowledge and relies on her capabilities to create the best schedule for her unit. Absenteeism is in this case of high negative impact. In a standardized context, the steps and rules are structured, organized, and clearly stated. Thus in case of nurses dissatisfaction, the standardization will lead to (continuous) improvements and knowledge transfer in the process, instead of conflicts between staff that negatively affect the quality of work and may cause departure of nurses. Furthermore, these steps are standard and simple enough to be used by a neophyte. Finally, they are manual and only need the use of pen and paper. For both problems, the following objectives are considered: maximize preferences, minimize deviation from quota requirements, minimize shift rotation in the case of nurses under rota system, and minimize shifting from the typical schedule while insuring a minimal change. This latter objective is introduced to ensure change resistances are eliminated by introducing each period at least one difference. The algorithms are designed to be close to the current working method of the clerks to facilitate their adaptation. For the first hospital, in order to balance the deficit between shifts, a score is calculated using the following formula: where D shift represents the monthly gap between the quota and the number of nurses assigned by shift. Algorithm 1 describes the mechanism of the first heuristic. The initial schedule should contain the typical schedule to which preferences and days-off are included. Using the score defined previously, flip movements are performed first for the nurses under rota system than to the set of nurses in general. Compared to the current approach used by the clerk, a largest neighborhood is considered. Flip movements are performed not only within one week period but on a larger horizon, for all nurses with fixed schedule (such as responsible nurses). The second heuristic developed for Hospital 2 is presented in the following Algorithm.

8 Figure 5.1: Intermediate step of the algorithm Algorithm 2 uses permutations to minimize the cost objective function. A permutation π i j1,j2 permutes days j 1 and j 2 of a nurse i. Π i represents the set of authorized permutations. Constraints (days worked in two weeks) make it possible to define a neighbourhood for the heuristic. A permutation π i j1,j2 is authorized if it respects the two last load constraints and x i j + x i k = 1 as illustrated in Figure 5. Figure 4: Illustration of authorized permutation Shifts can only be moved within a two weeks period, from a non-working shift to a working shift. This algorithm is implemented using Excel and if desired, it can be used manually. We have chosen Excel in particular since it is very basic and already used in the hospitals. The figure below illustrates how the permutations are performed. The permutation is implemented when changing a value from 1 to 0 (or the opposite): moving from an assignment to one particular shift on a particular day to none (or the opposite). Figure 5.2: Stop criteria of the algorithm Introduction of automated scheduling As one can see, these problems are very complex to solve and evaluating all solutions in the space of solutions is combinatorial. To consider all authorized movements and to create a schedule that maximizes the objectives while respecting all specified constraints, optimization softwares should be considered. Even though the main objective of the project is to develop simple and non automated heuristics, two specific softwares were used to: 1) show what can be done with available software, 2) compare the solutions to evaluate the performance of the proposed algorithms. The use of a software has a lot of advantages, including the automatization, standardization and simplification of the process. It also decreases the time of creation and improves the quality of the new schedule. Results All the described methods were compared using different criteria: standard, simple to use, requiring zero additional cost and resource, respecting quota requirements, less time-consuming exercise, total cost of producing the schedule, fulfillment of preferences, priority admitted to seniority, equity in terms of balance of the non-desired assignments, and ergonomic rules that have positive impact on their quality of life at work. All criteria in the quality field are ranked from the most to the least important one according to our comprehension of the problem and interpretation from our interviews with the clerks. One has to recall that this ranking is not a formal one since clerks use very subjective

9 criteria each time they consider one particular preference and nurse. Instead of presenting the total costs of schedules, a scale for scoring schedules is introduced. Tables 1 and 2 summarize all the results using the following scale: - -, and -, represent that compared to 0 the chosen method is performing less efficiently, and +, and + +, that the method performs better. Table 1: Results for Hospital 1 Rank Criteria Clerk Heuristic Commercial 1 Quota requirement 2 Time consumption 3 Cost Preferences Seniority ++ + N/A 6 Ergonomics Equity + - N/A 8 Standard Simple to use Table 2: Results for Hospital 2 Rank Criteria Clerk Heuristic Commercial 1 Typical Schedule 2 Preferences Time consumption 4 Cost Quota requirement 6 Seniority Equity Ergonomics Standard Simple to use For example, using quantitative results rather than the previous qualitative ones, the clerks schedule obtains a score of 22.1, while our heuristics and CPLEX obtain 50.1 and 55 respectively. The heuristic developed compares very well in terms of the solution quality. For both hospitals, the estimated time reduction by using the two standardized methods is important. Table 3 summarizes these reductions. Table 3: Time reduction for both hospitals each month using a standardized approach TIME Hospital 1 Hospital 2 Actual time 2 to 4 days 4 weeks Heuristic time 0.5 day 2 weeks REDUCTION: 1.5 to 3.5 days 2 weeks By simply introducing work reorganization, improvement methods and a standardized approach to create the official schedule, time is reduced and both hospitals achieve more efficient use of their resources. Both methods presented increase the quality of the schedule. An additional potential gain would be to establish connection between the departments in their scheduling processes. Services should create their official schedules using the proposed standardized approach. After this step, their official schedules should be analyzed by the float team to ensure a better forecast of the demand. Figure 6: New linked scheduling process Conclusion This project aims to apply learnings in operations research and optimizing resources to practical cases. Two problems occurring in two hospitals in Montreal have been studied, one implying a regular team and the other one the float team. Both problems have been modelled and solved. Developed heuristics have proved to be efficient in both cases. The models can also be solved by means of optimization

10 softwares. As shown in this paper, the current schedules can benefit from this work. At a more strategic level, the demand coverage constraint for the float team nurse is strictly correlated to the quota requirements and the needs in the other units. One clerk should consider adding this dimension when solving the scheduling problem. Finally, the last comment is again related to the demand coverage constraint: this value should always stay flexible for float team nurses since their role is to absorb the surplus rather than to offer regular work. However, it can probably be better modelled using time series techniques. Aknowledgments We thank Martine Rail and Diane Guertin from Notre-Dame Hospital and Martine Legault, Lisanne Vallée and Priska Melançon- Luthi from Sainte-Justine Hospital for their precious help and time. They provided us with the data we needed and explained their scheduling process to us. We also thank Camille Puisais from Omega Optimisation for providing her help and advices during countless hours. Finally, we thank our University, École Polytechnique de Montréal for its support. References 1. Burke, E.K., De Causmaecker, P., Vanden Berghe, G. and Van Landegem, H. (2004). The state of the art of nurse rostering, Journal of Scheduling, 7: Ernst, A.T., Jiang, H., Krishnamoorthy, M. and Sier, D. (2004). Staff scheduling and rostering: A review of applications, methods and models, European Journal of Operational Research, 153: De Grano, M.L, Medeiros, D.G and Eitel, D. (2009). Accommodating individual preferences in nurse scheduling via auctions and optimization, Health Care Management Science, 12: Ferland, J.A., Berrada, I., Nabli, I., Ahiod, B., Michelon, P., Gascon, V. and Gagné, É. (2001). Generalized assignment type goal programming problem : Application to nurse scheduling, Journal of Heuristics, 7: Glover, F. and Laguna, M. (1997). Tabu search, Kluwer, Boston 6. Rönnberg, E and Larsson, T. (2010). Automating the self-scheduling process of nurses in Swedish healthcare: a pilot study, Health Care Management Science, 13: Whelton, J.M., Fisher, M.H., DeGrace, S. and Zone-Smith, L. (2006). Hospital nursing costs, billing, and reimbursement, Nursing Economics. Biographical Sketches Sophie Champalle holds a B.A. in Operations Management at HEC Montréal. She is now finishing a DESS at École Polytechnique de Montréal. She is interested in healthcare supply chain and humanitarian logistics. Martine Dagenais and Mehdi Taobane hold a Bachelor Degree in Industrial Engineering. They are M.Sc.A. candidates in Industrial Engineering at École Polytechnique de Montréal. Martine is interested in Lean and Industrial Engineering in healthcare. She also works at the McGill University Health Center in Montreal as a process review intern. Mehdi s master project is on the optimization of the patient flows in oncology clinic by using the integration of simulation and operations research. Antoine Legrain holds a Bachelor degree in Engineering from France. He is now completing an M.Sc.A in Operations Research. He is interested in scheduling, healthcare and stochastic optimization. Hocine Bouarab holds a Bachelor degree in Operations Research Engineering from Algeria. He is an M.Sc.A. candidate at Polytechnique Montreal and is interested in operations research in transportation. Nadia Lahrichi holds a Ph.D. in operations research and taught the Resource Optimization in Healthcare course.