A Decision-Making Tool
for Home Health Care
Nurses' Planning
Rym Ben Bachouch
Université de Lyon, LIESP, INSA-LyonURAII, INSAT, Tunis rym.ben-bachouch@insa-lyon.fr
Alain Guinet
Université de Lyon, LIESP, INSA-Lyon alain.guinet@insa-lyon.frSonia Hajri-Gabouj
URAII, INSAT, Tunis sonia.gabouj@insat.rnu.tnThis article deals with problems encountered be routing nurses through home health care services. It is difficult to assign patients to different care workers by taking into account their availability and their skills. If patients need several cares during a week, they may be treated by the same employee. We show that this problem is equivalent to a routing problem with some specific constraints. We propose an integer linear program for deciding (1) which human resource should be used and (2) when to execute the service during the planning horizon in order to satisfy the care plan for each patient served by the home health care providers.
Key words: scheduling, home health care, mathematical modeling, mixed linear program, resource planning
Introduction
Many industries have discovered the benefits of improving efficiency by working on the routing decisions involved in their activities. Better routing and scheduling allow these decision makers to achieve savings in their costs and to expand their service capabilities. Home health care (HHC) services are a growing service industry that must face scheduling and routing problems. HHC development is accelerated by several factors such as the pressures of government to reduce the cost of health care, the difficulties of adapting health care systems to meet the growing needs of an aging population, new illnesses and cures, and a severe shortage of nursing staff. Home health care services provide complex and coordinated medical and paramedical care to patients at their homes. They include nursing, therapy activities, medical and social services, house cleaning, and so on. There is an increasing need to develop innovative approaches to improve the efficiency of HHC organizations. The particularity of HHC organizations is that the
patient is a component of the health care supply chain. Therefore, we have to take into account additional constraints such as the time window to provide care to patients, the necessity to synchronize all resources (i.e., humans and materials) involved in the care delivery process and the necessity to take into account care workers skills for each care delivery. In this article, we are interested in the routing problem of care workers in HHC. In the first part, we propose a review of the existing literature concerning staff planning and resource allocation. In the second part, we describe the routing problem. In the third part, we present the proposed approach for scheduling care worker planning. In the fourth part, we discuss and analyse the obtained results. Finally, we suggest some prospects for future research.
Literature review
This section surveys various solution techniques that are available in HHC. There have been few articles related to HHC routing problems in the literature. Cheng
© Copyright BEM ISSN print 1625-8312 ISSN online1624-6039
An International Journal Supply Chain Forum
Acknowledgments
This work was supported by the National School of Social Security of Saint Etienne (France) in a PhD work studying home health care operational pro-blems. It is part of an OSAD (Organisation des Soins A Domicile) project that is sponsored
and Rich (1998) consider two types of nurses: part time and full time. The global problem is formulated as a vehicle routing problem with time windows (VRPTW). Two formulation approaches using mixed-integer linear programming are described: one using triple-indexed variables and the other using double-indexed variables. The implemented heuristics are a two-phase algorithm: the first stage builds several routes simultaneously and the second stage attempts to make improvements on these tours. Chahed et al. (2009) deal with the planning of operations related to chemotherapy at home. The considered problem is restricted to the analysis of the stages of production and delivery of anti-cancer drugs. An exact method based on linear programming with the objective of minimizing the total travel time for the nurse is proposed. Bertels and Fahle (2006) present another optimization and planning tool. Here, nurses have different skills and the objective is not only to minimize the total cost but also to provide a weighted sum of the total travelled distance plus a sum of penalties associated to the violation of time windows or patient preferences. The heuristic developed by the authors is divided in two parts: (1) to build a set of patients to assign to each nurse and (2) to find an optimal sequencing for each set of patients. The approach is based on a combination of linear programming, constraint programming, and metaheuristics. Eveborn et al. (2006) describe a decision support system called “Laps Care” to aid the planners by using a set-partitioning model. The system consists of a number of components including information data bases, maps, optimization routines, and report possibilities. A repeated matching approach is used for finding a solution. The visit plan proposed is evaluated according to two performance criteria: the efficiency of the plan and its quality (continuity of care). Borsani et al. (2006) develop another linear integer model. The authors are interested in the problem of deciding which human
resources should be used and when to execute the service during the planning horizon in order to satisfy the care plan for each patient. The weekly plans generated by the proposed models are compared with the real ones according to a set of performance indicators: care continuity, outsourced visits, preferential days, and geographic coherence. Begur et al. (1997) study the scheduling and routing of nurses in Alabama and develop a spatial decision support system for HHC providers. This system takes into account route construction, nurse availability, and patient needs with their availability. It develops for each nurse a list of patients to visit ranked in an order that maximizes their productivity.
Thomson (2006) investigates how methods from operational research can be applied to the HHC field and emphasizes the routing problem, which is formulated as a VRPTW. The aim is to minimize the travelled distance and maximize the number of visits. To solve the problem, an insertion heuristic is used. The solutions found via the insertion heuristic are used as initial solutions for a tabu search. Chiba et al. (2005) develop a decision support system for HHC using a multi-agent system. The decision is performed autonomously by negotiations among agents so that it is sufficient for customers and helpers to confirm schedules settled by the agents. They can confirm the schedules using PDAs,
which can be easily handled even by older individuals. Consequently, it is expected that the proposed decision support system reduces the total cost of an HHC service. De Angelis (1998) studies the problem of allocating resources (nurses, doctors, social assistance, etc.). To solve this problem, De Angelis formulates a stochastic linear programming model in order to maximize the total number of patient deliveries. Boldy and Howell (1980) present a case study of methods to allocate a given amount of home help resources to a number of geographical areas within a county social services department. The approach described could also be applied to other health or social services resources for which an equitable distribution between areas or between different customer groups is required. Blais et al. (2003) undertook another districting study for the Côte-des-Neiges local community health clinic in Montreal. The territory models the area where a particular clinic is responsible for the logistics of HHC visits. The area must be partitioned into several (six) districts by suitably grouping territorial basic units. Five distinctive criteria must be satisfied: indivisibility of basic units, respect for borough boundaries, connectivity, personnel mobility, and workload equilibrium. The problem is solved by means of a tabu search technique.
Because of the recent development of HHC organizations, the number of existing articles about HHC problems is quite modest. The main issues tackled are the districting problem and the human resources planning problem or more precisely the nurse planning problem. Other studies (Landry & Philippe, 2004; Beaulieu et al., 2001) focused on new management ideas in order to better understand the role and impact of logistics in health care, to contain health care costs, and to adapt the health care system to the changing demographics. They present examples of how to better integrate logistics activities through a unique combination of reengineering and
Home health care
(HHC) services are
a growing service
industry that must
face scheduling and
activity-based costing. Delivering care in HHC is not an easy task because of the large number of actors that participate in the process, the variety of clinical and organizational decisions, and the difficulty of synchronizing human and material resources at the tactical and operational levels. Our interest is on the operational level where time constraints have to be considered in a different way because some patients require simultaneous or sequential interventions involving multiple resources. Moreover, planning in HHC is made difficult for several practical reasons: the service involves patients whose clinical and social conditions may change quickly, the large number of procedures and protocols to be followed reduce the flexibility of providers' organization, patients may be spread in a wide area, the synchronization of resources is relevant to provide the service in an effective and efficient way, and so on. In this context, the development of a short-term planning support tool for HHC providers is quite interesting. In HHC, the visit plans for each care worker are established by the nurse coordinator. In this work, we aim to provide a decision support tool to establish a feasible planning
for care workers taking into account all the constraints concerning continuity of care, patient availability, and so on. Our purpose is to facilitate the nurse coordinator work knowing that there is no retail offering for routing problems in HHC. Most of French HHC uses dedicated software that offer different applications for managing the patient care plan and medical records but do not offer planning tools.
We propose an exact method to solve the routing problem in HHC. We aim to incorporate the proposed approach to the French HHC information system. Thus, French HHC will have only to interface their existing software by adding our tool for route planning. Table 1 illustrates the constraints studied in the papers discussed previously. In French HHC, we observe that there are many constraints to take into account in nurse planning. In addition, scheduling the care worker route is difficult because of the need to synchronize human and material resources. It is important to take into account all the studied constraints in order to provide an efficient planning tool. In Table 1, Boldy and Howell (1980) and Blais et al. (2003) do not appear because
the authors did not mention the constraints considered in establishing the nurse planning. We propose an integer linear program to minimize the travelled distance, taking into account the nurses' skills, patient availability, lunch breaks for nurses, shared visits (i.e., visits requiring more than one care worker), and nurses' time window. On the one hand, the proposed approach takes into account all the constraints considered in the previous works. We integrate all the constraints studied in Cheng and Rich (1998), Begur et al. (1997), De Angelis (1998), Thomson (2006), and Chiba et al. (2005) in order to make an efficient planning tool. Eveborn et al. (2006) use a set-partitioning model and Borsani et al. (2006) use a mathematical model but our mathematical formulation is totally different from the one presented in Borsani et al. (2006). The next section focuses on the nurse routing problem.
Problem description
All patients in HHC service have to be treated according to their care plans, which include, among other factors, the number, type, and sequence of visits that the patient should receive. To provide this
Table 1
service, an HHC structure has to coordinate its resources, especially the human ones. In most cases, the HHC structure provides a daily visit plan for all its care workers, pointing out which patients each operator has to assist, what kind of visit he or she has to realize, and possibly when, specifying the hour. Unfortunately, care worker planning is done by hand and there is no decision support tool to establish planning. Nurse coordinators assign the care to care workers considering skills, working hours, and the patient needs. The most important quality objective for HHC is “continuity of care”: the assigned care worker to a patient is considered as the patient reference operator during the whole duration of the care plan. The nurse coordinator defines the days of visit for each patient taking into account personal preferences. After establishing the care worker planning, the nurse coordinator must manage all the unpredictable events that may occur during the week. Planners could also ask the family to execute some simple care activities for the patient. These activities are called “outsourced visits” and in order to keep a high level of patient satisfaction, planners have to avoid these types of visits.
Assumptions and notations
For given week, a list of patients needing several cares is known and the care worker plans must be calculated for patient care delivery at home. The criterion to optimize is the weekly travelled distance minimization. A care can also require more than one care worker. Care workers (nurses, physical therapists, care assistants, etc.) are available to treat patients under terms of working hours or geographical allocations. Some cares have to be assigned to the same care worker in order to ensure continuity of care.
Notations
Let us consider a set
of care workers, Sij the skill of the
care worker i. Each care worker has
a list of cares to deliver per day
according to his or her skill and
working hours.Dur_max represents
the maximal duration of a working day for a care worker. We allow that each care worker works eight hours per day and has the right to take a lunch break. This break may be modeled as a preassigned care of one hour. We note
the set of breaks of one hour to be assigned to care workers. There is a set of patients to be treated at home. We introduce a dummy patient denoted by 0, which represents the HHC office. We allow that each care worker must leave and return to the office at the end of the care tour for information and feedback. Each care is characterized by a duration
of treatment pi, the number of care
workers
nc
j needed to perform thecare for the patient j, and a time
window during which the care
must begin. Let
dist
j,k denote thedistance (in minutes) between two
consecutives visits jand k. In order
to allocate a geographical area to care workers, we define the data
Dist_max to delimit the distance
separating two sequential visits. There is a set of days
representing the planning horizon. To ensure the continuity of routes, we define the set
to take into account breaks (m
breaks) and visits (nvisits) in the
assignment of care workers.
Indexes
Patients: j, k, h = 1,…, m with m
number of patients
Care workers: i = 1,…, n with n
number of care workers
Day: t = 1,…, d with d number of
working days in a week
General assumptions
- The care plan has a finite horizon of one week.
- Only one reference operator can be assigned to one patient.
- Only one visit by day must be done for each patient.
- Travel times are not included in the visit duration.
- Care worker skills are expressed
as Sij assuming the following
values: it is equal to 1if operator i
has the skill to treat the patient j;
0otherwise.
- Each patient has a time window of availability.
- Visits have different durations. - A visit can require more than one
care worker. These visits are called “shared visits” and we figure that care workers arrive at the same time to the patient home.
Decision variables
- The variable yijkt models a visit to
patient before the patient k
assigned to the operator i during
the day t, and it assumes the
following values: it is equal to 1if
the visit is carried out by the care
worker ito patient jbefore patient
k during the day t; 0otherwise.
- The variable aijt indicates the
arrival time of the care worker ito
the patient jon the day t.
Objective function
The objective function (1) minimizes the total travelled distance by the care workers.
(1)
Model constraints
- Care workers can visit patients only when they are available according to the time window. The time window indicates the earliest and latest date of a visit
beginning to the patient j k
(constraints (2) and (3)). (2) (3) - All visits must be planned and
performed by the needed number
of care workers nck. Constraints
(4) and (5) consider the case of shared visits when more than one care worker must perform the visit to treat the patient at home.
(4) (5) - In the case of shared visits, the
care workers have to be at the patient home at the same time. Constraint (6) calculates the arrival time to the patient home for the care workers who perform the shared visit.
(6)
{
n
}
C
=
1
,...,
{
b bn}
B= 1,...,{
m}
P= 0,...,[ ]
e
j,
l
j jk D t ijkt j P k P j C i dist y Min \ ikt k a e C i D t P k , , : = C i j Pk k ijkt nc y D t P k \ : , = C i k P j j ijkt nc y D t P j \ : , k ikt l a C i D t P k , , : = P k k P skt ikt k a a nc D t i s , , 2:{ }
d D=1,..,{
n m}
Q= 0,..., +- The care worker skills must satisfy the patient needs (constraint (7)).
The data Sij indicate if the care
worker ihas the skills to perform
the visit jand it assumes the value
1if the care worker can perform
the visit j; 0otherwise.
(7) - Constraint (8) ensures that the
arrival time of the care worker ito
the patient k must be calculated
taking into account the treatment duration and the arrival time to
the preceding visited patient j.
(8) - In order to assign each care
worker to an area, we define
a maximal distance Dist_max
separating two successive visits
jand k(HV is a notation to model
an infinite value) in constraints (9).
(9) - Each care worker has one lunch
break per day (constraints (10)
and (11)). We define the set Bof
breaks. The number of breaks is equal to the care worker number. We assume that breaks are fictitious cares of one hour with a time window in order to restrict the beginning and ending date of each break.
(10) (11) - To avoid overtime for care
workers, we limit their working days to a maximal duration
Dur_max (constraint (12)). After
each visit, the model calculates the tour length performed, which must not exceed dur_max including the HHC office return.
(12) - Constraint (13) ensures the
continuity of care worker routes: when a care worker visits a patient home, he or she has to finish this visit before beginning another one.
(13)
- Constraint (14) expresses the continuity of care: cares must be performed by the same care worker during the week.
(14) - Constraints (15) and (16) impose
that each care worker exits from the HHC office only once and comes back to it at the end of his or her working day.
(15) (16) - Constraints (17) and (18) denote
that yijkt and aijt are respectively
binary and non negative variables.
(17) (18)
Technical improvements
In order to improve the processing time, some cuts can be added according to the time windows. Two sequential cares cannot have two overlapping time windows. The earliest date for beginning the care j adding the treatment duration and the distance travelled must not exceed the latest beginning date of the next visit (constraint 19).
(19)
Analysis of results
The integer linear program has been tested on random instances of the indicated problem. The scheduling model was solved with two kinds of software: LINGO of LINDO Systems (2003) and ILOG OPL-CPLEX STUDIO of ILOG. We aim to compare the obtained results of the two kinds of software. Duration of care belongs to the interval [15mn, 60mn] and the length of the time window for patient availability is one hour. The computing processing time was limited to two hours.
It is well known that CPLEX is usually faster in providing results than LINGO. However in previous research (Trilling, 2007; Ben
Bachouch, 2007), LINGO provided better results than CPLEX. Thus, we use the two kinds of software to solve the proposed model. Table 2 presents the obtained results with the model described in the article for different problem sizes. In a second part, we compare the results obtained with another model in which we do not take into account cut constraints and care worker skills.
Based on randomly generated cases, we intended to study the applicability of the proposed approach and we investigated it on two planning horizons: daily planning and weekly planning. We manipulated the model for different problem sizes and in all cases the proposed approach provides a feasible nurse planning. CPLEX does not give a solution for all instances because of “memory usage,” that is, the computer did not have enough memory to solve the linear integer model. Even though LINGO provides a feasible solution, it is not optimal. However, in care worker planning a feasible solution is sufficient. In this approach, we show that a simple integer linear program is able to produce good solutions. Table 1 summarizes all the constraints considered. Our approach is similar to Borsani et al. (2006) but we added two important constraints: nurse's lunch breaks and shared visits. The proposed linear integer model takes into account the same constraints, but we use fewer variables and data than in Borsani et al. (2006) and our model solves different problems sizes: from 3 to 7 nurses and 7 patients to 20 patients. The largest instance in Cheng and Rich (1998) involved 4 nurses and 10 patients, which was solved with an integer linear program solver. Thus, the proposed approach is simple and useful to establish daily and weekly planning. In the case of planning 7 nurses and 20 patients, a feasible solution is found after 34 seconds with LINGO but we interrupt the resolution to reach the optimal solution.
For daily and weekly planning, the solution is optimal or at least
D t k j P k P j C i , , , , : D t , j k, i C, j P, k P: HV y dist p a
aikt ijt j _max+(1 ijkt)
max _ : , , t D j Pa p dist0 dur C i ijt+ j+ j = C i jQh iCkQh ihkt ijkti y i y Q h D t \ \ : , + = C i jPk iChPk t ihk ijkti y i y P k D t \ \ ) 1 ( : ,
{ }
01, : , , , , j P k P j k t D yijkt C i k j D t P k P j C i , , , , : HV y l dist p ej+ j+ jk k+( ijkt 1) : 0 : , , j P t D aijt C i HV y dist p aaikt ijt+ j+ jk+( ijkt 1)
: C i k P j ij ijkt s y D t P j \ 1 : , = B j k P ijkt y D t C i , : 1 = B k j P ijkt y D t C i , : 1 = P k kt i y D t C i , : 0 1 = P j t ij y D t C i , : 0 1
feasible in all cases with LINGO. CPLEX did not provide feasible solutions because of computer performances. For weekly planning, LINGO appears to be faster and more efficient than CPLEX. When we interrupt the resolution with LINGO, we can have the nurse-planning information put into an Excel file, whereas with CPLEX, we can get solution results only if the optimal solution is found.
Table 3 illustrates the comparison between the results obtained with the two different model versions. The model1 is issued from a previous version in which we do not take into account the qualification of nurses and the cut constraints. The model2 is the model described in the third section of this paper. Thus, the
model1 is equivalent to model2 without constraints (7) and (19). The cut constraints reduce the number of infeasible solutions. The qualifications constraints also reduce the computation times. In fact, when we take into account nurses' skills, we match each nurse to a set of patients, which reduces the solution number. The combination of qualification constraints and cut constraints reduces the computation time. The aim of the software comparison is to identify which is adequate for scheduling care worker planning. In our experiments, it appears that LINGO is more efficient and provides feasible solutions in all cases. With LINGO, we can interrupt the resolution at any time having a
feasible and good solution. This is not the case with CPLEX and it is its major drawback.
Conclusion
The French HHC service is increasing rapidly in size and the need for decision support tools to improve the planning and quality of service is important. In this article, we considered the problem of assigning patients to nurses in HHC services. The nurse short-term planning process for HHC providers requires the respect of a large number of constraints and objectives in terms of the efficiency and quality of care. We studied previous works that have studied routing problems and resource allocation in the HHC area. We noted that the amount of existing research in HHC is limited and the nurse routing problem is among the most studied issues. The proposed approach focuses on care worker routing problems and we have included a larger variety of constraints than those of previous contributions. Our contribution is in integrating all the constraints into a model that could be calculated by a solver and adding to the information support software of French HHC. To our knowledge, French HHC providers do not have any planning tools in their information systems and the nurse coordinators establish the daily planning by hand. In this context, we proposed a simple model that involves all the constraints needed in care worker planning such as patient availability, nurses' skills, shared visits, and so on. The proposed integer linear program provides a nurse planning for different sizes of application instances. Furthermore, we have compared two kinds of software: LINGO of LINDO Systems and CPLEX of ILOG. It is well known that CPLEX is faster than LINGO but in few cases, LINGO could be faster and more efficient than CPLEX. In our experiments, LINGO appears to be faster than CPLEX and provided good solutions. Besides, our aim was to provide a planning, not necessarily the optimal one, because a feasible one is often sufficient. For this
Table 2
Computational experiments
Table 3
reason, we suggest interrupting the solver after two hours when the optimal solution is not found; however, we always found a feasible nurse planning. The model proposed can be extended by taking into account additional constraints or by modifying the objective function such as balancing workloads between care workers. For this study, we used randomly generated cases; however, this analysis will be validated later with real data. We based our approach on mathematical programming because it is an efficient method that does not require software development to implement our model and because LINGO and CPLEX works with Excel sheets.
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About the authors
Rym Ben Bachouch is a PhD candidate in
industrial engineering in the LIESP Laboratory (Laboratory of Computer Science for the Enterprise and the Production Systems) in INSA (National Institute of Applied Science) in Lyon, France. Her research interests include logistics, supply chain management, and operational research.
Alain Guinet is a university professor in the
industrial engineering department of INSA. He received a PhD in 1983 and a tenure (habilitation to lead research activities) in 1992. He teaches operation research, staffing and scheduling, and business process engineering. His research activities are based on hospital management problems such as operating theater control, hospital regrouping management, emergency network reengineering, home care resource coordination, hospital supply chain, and so on. His scientific investigations include human and material resources sizing, planning, and scheduling; production network reengineering; logistics; and so on.
Sonia Hajri-Gabouj is a professor at the
Tunisian Institute of Applied Sciences and Technology (INSAT). She received her BS degree in electrical engineering from the National Engineering School of Monastir in Tunisia. She got her MS and PhD degrees in industrial computing and automatic from the University of Sciences and Technologies of Lille (USTL) in France. She received her qualification to direct research degree from the National Engineering School of Tunis (ENIT). She is the associate responsible at the URAII Research Unit in automatic and industrial computing of the Institute of Applied Sciences and Technology in Tunis. Her research interests include design, configuration and control of manufacturing systems, optimization, and decision making.
