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Collaborative model for planning and scheduling caregivers’ activities in homecare
Proceedings of the 18th World Congress The International Federation of Automatic Control Milano (Italy) August 28 - September 2, 2011
Collaborative model for planning and scheduling caregivers’ activities in homecare Redjem Rabeh*. Kharraja Saïd* Marcon Eric* * LASPI, F-42334, IUT de Roanne, Saint Etienne France (Tel: +33 6 43 00 31 17; e-mail: [email protected]. said.kharraja @univ-st-etiennet.fr marcon @univ-st-etiennet.fr) Abstract: Home Health care Services (HCS), are structures that provide continuous and coordinated cares in patients’ homes. In this paper, we deal with planning and scheduling caregivers’ visits in HCS. The caregivers visit their assigned patients, while minimizing the travelled and waited times, within the patients’ availabilities constraints. The problem was solved using an original MILP that coordinate caregivers’ tours. The mathematical model was solved using the academic solver LINGO of LINDO SYSTEMS. Keywords: Resource planning, home care, mathematical modelling, operations scheduling, VRPTW.
1. INTRODUCTION
2. PROBLEM DESCRIPTION
The organizational and economic problems emerged in care systems have contributed to the appearance of Home Health care Services (HCS), to provide continuous and coordinated care for patients in their homes, as an efficient solution for reducing health care’s costs and maintaining a satisfactory quality of service. HCS are defined as “a mini network in a wider one” (Com-Ruelle et al., 2003), since it requires a coordination between multiple actors from varied skills. However, it operates itself in a larger network involving several modes. The HCS increasingly take an important place among healthcare structures. In France, the number of the HCS has tripled between 1999 and 2006 from 68 to 185, and the places’ number in homecare have increased from 3908 authorised beds in 1999 to 7355 in 2006 (Chahed et al., 2008).
The complexity of planning and scheduling the caregivers’ tours is due to the various constraints to take into account (Chahed et al., 2008). Our problem concerns establishing the best caregivers’ tours, while optimising the travelled and waited times. We consider two caregivers; each one follows a route to care a set of allocated patients, within their time windows. Waiting times are due to the caregivers’ arrival at the patient’s home prior to its earliest time of availability. Patients’ care durations are predefined. In the case of patients requiring visits from both caregivers, we have two situations, (i) the visits can be coordinated, i.e. caregivers should visit the patient within a predefined order; (ii) the visits may be sequenced without coordination (predefined order). The order of visits is defined in the patients’ care protocol.
A homecare process requires a variety of organizational and clinical decisions, coordination and synchronization between various human and material resources and the participation of an important number of actors from different skills (Bertels et al., 2006). Thus, it is interesting to develop tools that improve the performances of HCS, as tools that enable planning caregivers’ tours. In this paper we propose a decision support tool based on integer linear programming to schedule two caregivers’ visits for patients, taking into account various constraints, especially the “visits’ coordination” constraints. The following parts are structured as follows. First, we describe the problem. Second, we present works that have dealt with planning the caregivers’ tours. In a third part, we propose our mathematical formulation for the problem. Next, we will proceed to an analysis of results obtained by using two objective functions and different scenarios. Finally, we conclude with some prospects for future researches.
978-3-902661-93-7/11/$20.00 © 2011 IFAC
Our purpose is to provide a decision tool to schedule caregivers’ activities in HCS. Besides, this problem can be considered as Vehicle Routing Problem with Time Window “VRPTW” (Thomsen et al., 2006), while caregivers are vehicles, HCS is the depot, and patients are cities. The VRPTW problem, named “vehicle scheduling” by (Clarke et al., 1964), involves a fleet of vehicles to serve a number of customers, with various demands. The objective is to find routes for vehicle, while minimizing travelled durations, without violating customers’ time windows (El-Sherbeny et al., 2010). This problem is an NP-hard (Lenstra et al., 1981), i.e. there is no algorithm polynomial in time to solve the problem. The existent algorithms solve only small instances. A VRPTW can be solved by exact methods as linear programming or approximate methods as heuristics (Rego et al., 1994). Thus, in our model, we aim at minimizing the total sum of travelled and waited times for all caregivers. In the next section we describe the main works that have dealt with caregivers’ planning tours.
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
3. EXISTING WORKS In this section we present many works that has addressed the caregivers’ (or nurses’) tours problem, in home health care.
(paid by the hour). Thus, optimal schedule are generated, such as a nurse leaves/returns home, visits a set of patients within their time windows, and takes a lunch break, all within the nurses’ time window, while minimizing both, over time for salaried nurses and part-time nurses.
(Akjiratikarl et al., 2007) presents a novel application of a collaborative population to schedule home caregivers; using a method based on a meta-heuristic called Particle Swarm Optimization (PSO). They minimize the travelled distance, providing that the capacity and time windows constraints of services are not violated.
(Ben Bachouch et al., 2009) have proposed a tool, to planning nurses’ tours in homecare. Taking into account different constraints, as patients’ availabilities, lunch break for nurses, they have minimized total travelled distance.
(Eveborn et al., 2006) have developed visiting schedules for care providers that incorporate some restrictions and the objective is to reduce the transport time. Each visit has a particular task to be performed and/or nursing activities.
(Bertels et al., 2006) have proposed a method combined of linear programming, constraint programming, and metaheuristics for the home health care problem taking into account multiple criteria, as patients’ satisfaction, nurses’ qualifications, and travelling costs. They have minimized the travelling costs.
(Begur et al., 1997) construct the nurses’ tour schedules taking into account patients availability, needs constraints, and nurses’ availability. They fix in advance the days of visiting patients. (Cheng and Rich., 1998) have treated the nurses’ tour problem in a homecare, using the “VRPTW”. While considering two types of nurses: salaried; part-time nurses
(Borsani et al., 2006) propose a linear model, which deals with the problem of deciding (i) which human resource using in each home visit and (ii) when to execute the service, in order to satisfy the care plan for each patient served by the home care providers. The benefit of the system proposed is an increase in service quality and efficiency. In table 1, we summarize a comparison between some works quoted above.
Table 1. Comparison between methods Optimized Criterion Cheng et Rich 1998 Begur et al 1997 Borsani et al 2006 Ben Bachouch et al 2009 Akjiratikarl et al.2007 Our approach
Costs of working hours Travel duration Balancing work load Travel duration Travel duration Travel and wait time
Patients’ availabilities
Multiple visits for patients
Shared patients
Coordinated visits
X X X X X X
X X X X
In this section, we have presented works that deals with scheduling and planning activities for nurses, and other works in both scheduling activities and allocation of patients to nurses. We noted that all works presented use either exact methods like linear programming, or approximate ones like meta-heuristics. We noted also that all works cited above do not address the aspect of coordination between different providers. Our purpose in this paper is to solve this problem using exact methods. In next part we introduce the problem’s mathematical formulation.
X X X X
Exact methods
X X X X X
X
duration. Thus, visits’ durations for the patients not assigned to a caregiver are equal to “zero”. Caregivers’ tours starts and ends at the HCS (dummy patient), and the work horizon is the day. Based on “care protocol”, an order of care can be defined in advance for patients who need care from both caregivers. 4.2 Parameters and notations N, S: Respectively, the number of patients and caregivers (in this model S = 2),
4. MATHEMATICAL FORMULATION
tdij: Travelling time, between two patients #i and # j,
The problem can be described as a VRPTW problem. Given a set of patients, two caregivers and the HCS, find a set of routes for each caregiver, starting and ending at HCS, such as each one has his set of allocated patients, while each patient may be cared by both caregivers, within his availabilities. Each patient is visited by one caregiver at the same time, and the visits’ order can be predefined (coordinated).
pis: Care duration delivered to patient #i by the caregiver s,
4.1 Hypothesis
ri, di: Respectively, the release/due date to begin/end care for patient #i, ypij: Binary value that denotes a predefined order between caregivers i and j to visit the patient #p, ypij = 1 means that caregiver i must visits the patient #p before caregiver j, thus ypij = 0 and ypji = 0 means that there is no order between caregivers i and j for patient #p,
We consider that, each patient receives the same caregiver once a day, within his time window. Each care activity has 2878
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
Ais: Binary value which defines for each caregiver his set of assigned patients, such that Ais = 1 if patient #i is assigned to caregiver s, and Ais = 0 otherwise, NPs NSi: Respectively, the number of patients allocated to caregiver s, and the number of caregivers to care patient #i,
∀s ∈ [1, S ]∀i ∈ [1, N ]
xiis = 0
Constraints (3) - (6) ensure that each caregiver visits his assigned patients once a day. Constraints (7) ensure sequencing patients’ visits for each caregiver:
(
)
tis + pis + td ij − M 1 − xijs ≤ t js
M: Large constant,
(6)
∀s ∈ [1, S ]∀i, j ∈ [1, N ]
(7)
4.3 Decision variables
We enforce by constraints (8) a predefined order of care between the caregivers for the same patient:
To model the problem we use the following decision variables
tis + pis − M (1 − yiss ' ) ≤ tis '
xijs, ziss’: A binary variables, such as, xijs = 1 if caregiver s visits patient #i strictly before patient #j, xijs = 0 otherwise, and ziss’ = 1 when the caregiver s realizes his visit for the patient #i before caregiver s’, ziss’ = 0 otherwise,
∀s, s '∈ [1, S ]∀i ∈ [1, N ]
(8)
Constraints (9) - (13) ensure that the visits performed by caregivers for the same patient are sequenced: S
∑ ziss ' ≤ Ais '
∀s '∈ [1, S ]∀i ∈ [1, N ]
(9)
∀s ∈ [1, S ]∀i ∈ [1, N ]
(10)
∀i ∈ [1, N ]
(11)
ziss = 0
∀s ∈ [1, S ]∀i ∈ [1, N ]
(12)
tis + pis − M (1 − ziss ' ) ≤ tis '
∀s, s '∈ [1, S ]∀i ∈ [1, N ]
(13)
s =1
tis: A positive variable which corresponds to the date the care begins for the patient #i by caregiver s,
S
∑ ziss ' ≤ Ais s'
arrivis, waitis, uis: A positive variables, denote respectively, (i) the arrival date of the caregiver s at the patient’ #i home, (ii) the caregiver’s s waited time, before the visit for patient #i, and (iii) the patient’s #i rank, in the tour of caregiver s, 4.4 Model formulation To model our problem, we introduce a dummy patient denoted by 1 which represents the HCS, such as: t1s = r1 = D+, d1 = D-, p1s = 0, while D+ / D- respectively begin / end of the caregivers’ working day. The problem was formulated as the following integer linear program. The goal is to minimize the caregivers’ travelled and waited times. We have developed for that two objective functions (1) and (2), then we have compared results in section (5): N N S S N Min ∑ ∑ attis + ∑ ∑ ∑ td ij × xijs i =1 j =1s =1 s =1i =1
Min ∑ (End s − t1s ) s = 1
s =1 s '=1
Caregivers have to respect patients’ availabilities and the working hours. We ensure that by constraints (14) - (17):
tis + pis + td i1 − M (1 − Ais ) ≤ d1 ∀s ∈ [1, S ]∀i ∈ [1, N ]
∀s ∈ [1, S ]∀i ∈ [1, N ]
∑ xijs = A js
tis + pis − M (1 − Ais ) ≤ di
∀s ∈ [1, S ]∀i ∈ [1, N ]
(17)
(2)
(18), (19) are modified (Desrocher and Laport., 1991) subtours eliminating constraints. The rank of the patients not assigned to the caregiver is set to zero in the constraints (20): uis − u js + (N − 1)xijs + (N − 3)x jis ≤ N − 2
(2.1)
(3)
N N
∀s ∈ [1, S ]∀i , j ∈ [2, N ]
(18)
uis + M (1 − Ais ) ≥ 2 uis − M (1 − Ais ) ≤ NPs
∀s ∈ [1, S ]∀i ∈ [2, N ]
(19)
uis − M × Ais ≤ 0 uis + M × Ais ≥ 0
∀s ∈ [1, S ]∀i ∈ [2, N ]
(20)
Constraints (21), (22) allow the calculation of the caregivers’ waited times: ∀s ∈ [1, S ]∀j ∈ [1, N ]
(4)
∀s ∈ [1, S ]
(5)
( (
) )
tis + pis + td ij − M 1 − xijs ≤ arriv js ∀s ∈ [1, S ]∀i, j ∈ [1, N ] (21) tis + pis + td ij + M 1 − xijs ≥ arriv js
i =1
∑ ∑ xijs = NPs
(15)
(1)
j =1 N
(14)
(16)
The constraints are:
∑ xijs = Ais
∀s ∈ [1, S ]∀i ∈ [1, N ]
tis + M (1 − Ais ) ≥ r1
∀s ∈ [1, S ]∀i ∈ [1, N ]
Ends: denotes the caregiver’s s arrival date to the HCS at the end of his tour, such as:
N
S
tis + M (1 − Ais ) ≥ ri
S
tis + pis + td i1 − M (1 − xi1s ) ≤ End s ∀s ∈ [1, S ]∀i ∈ [2, N ] tis + pis + td i1 + M (1 − xi1s ) ≥ End s
S
∑ ∑ ziss ' = NS p − 1
i =1 j =1
attis = tis − arrivis
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∀s ∈ [1, S ]∀i ∈ [2, N ]
(22)
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
(23), (24) are binary or non-negativity constraints.
5.1 Data for simulations
xijs , ziss ' ∈ {0,1}
∀s, s '∈ [1, S ]∀i, j ∈ [1, N ]
(23)
tis , attis , arrivis ≥ 0
∀s ∈ [1, S ]∀i ∈ [1, N ]
(24)
5. RESULTS To solve the model, we used the academic solver LINGO_11.0 from LINDO SYSTEMS INC. In this part, we aim to present results based on an example of 2 caregivers and 8 patients. We also present results of comparison between both objective functions. We have varied the tests, using two situations based on patients’ availabilities. “First situation”: all patients are available during the caregivers’ working hours. “Second situation”: some patients are available only the morning, and others only the afternoon. Three scenarios based on patients’ locations were tested. “First scenario”: all patients live the same district and the travelling times are between 15 to 30 minutes. “Second scenario”: all patients live in the same district except for one isolated. “Third scenario”: two different districts, with travelling times in the same district between 15 and 30 minutes, and between 50 to 65 minutes between different districts.
We have tested the model on an example of two caregivers and eight patients, with four shared patients {#1(HCS), #3, #4, #6 and #8}. Patients {#4 and #6} require coordination between caregivers. In our case, caregiver 2 have to perform his activity for patient #4 before caregiver 1; for patient #6 caregiver 1 must begin before caregiver 2. The work horizon is the day (8 hours of work or 480 minutes). The working day begins and ends in the dummy patient whose release and due dates are respectively r1 = 1, d1 = 480. Tables 2 and 3 contain respectively, the patients’ allocation to caregivers and care durations of each patient. Table 2. Allocation table CG 1 CG 2
#1(HCS) 1 1
#2 1 0
#3 1 1
#4 1 1
#5 0 1
#6 1 1
#7 1 0
#8 1 1
#3 50 45
#4 35 25
#5 0 60
#6 40 55
#7 60 0
#8 45 30
Table 3. Care durations CG 1 CG 2
#1(HCS) 0 0
#2 30 0
Fig.1. First situation caregivers’ tours Figure 1 illustrates results obtained when model is solved using both objective functions, in the first situation. The caregivers’ tours are represented above. In this situation, we note that, patients’ availabilities and coordination between caregivers for patients #4 and #6 were respected. We remark also that: (i) If all patients live near to each others (first scenario); using both objective functions leads to choosing the minimal travel durations to generate caregivers’ tour. (ii) When patients are divided in two distinct groups (second and third scenarios), the caregivers’ tour will be generated by choosing the minimal travel duration between patients from the same location, and also between patients from different locations. (iii) Using both objective functions can lead to: identical caregivers’ tours, as in first and third scenarios. But
we may obtain different tours, as the caregiver’s 1 tour in second scenario. The difference in this case is in patients’ scheduling. For example, order of patient #2 is “1” using first objective function, and “2” using the second, in the caregiver’s 1 tour. But the total sum of travelled and waited times is identical using both objective functions (as showed in section 5.2). (iv) For patients, #4 and #6, who need coordination, the care dates are generated taking into account the visits’ order. Thus, the coordination has an immediate impact on the caregivers’ tours. We conclude from this situation that, optimal tours generation depend mainly on next factors: travel durations between patients from same and/or different locations and coordination between caregivers.
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
Figure 2 illustrates results for the second situation (two types of patients’ availabilities: [1,240] and [240,480]). We remark that, (i) When all patients are in the same location (first scenario), caregivers’ tour is generated depending on travel durations and patients’ availabilities, i.e. starting by patients available the morning (patients #2, #3, #4), and ending by those available in the afternoon (patients #5, #6, #7, #8). (ii) When patients are divided in two distinct locations (second and third scenarios), caregivers’ tours are built with the patients available in the morning from the first region (where the HCS is located), then all patients from second location (beginning by those available the morning), and finally with the patients available the afternoon from the first location. (iii) The caregivers’ tours can be identical using both
objective functions, as in second and third scenarios; but it may be different as caregiver’s 1 tour in first scenario. The difference in this case is in the patients’ visit order. For example, the order of patient #3 in the caregiver’s 1 tour, using the first objective function is “1”, and “3” using the second. But the total sum of travelled and waited times are identical using both objective functions (section 5.2); (iv) while coordinating visits for patients #4 and #6, the care dates are generated taking into account order between caregivers; thus coordination has an immediate impact on the caregivers’ tours. Thus we conclude that optimal tours generation depends on next factors: travel durations from same or different location, visits’ coordination and patients’ availabilities.
Fig.2. Second situation caregivers’ tours From these tests we note that, if patients are grouped into different locations and without patients’ availability constraints, using both objective functions in our model gives the best schedule: i.e. a caregiver is introduced by first location, and then he moves to the second one with a minimum travelled duration. In case of complex situations, as the second and third scenarios in the second situation, we have noted the big impact of patients’ availabilities on tours generation. We have also remarked that tours’ generation is based on the same strategy whatever the objective function (see remark 3).
In this situation patients are available during the working day, thus there are no waiting times, and minimizing the sum of travelled and waited times (1st objective function), or minimizing the caregivers’ worked duration (2nd objective function), leads to minimizing only travelled durations. Thus the results obtained using both objective functions are similar. In terms of calculation times, using first objective function is less expensive than the second. This is due to time elapsed in calculating Ends (caregiver’s return date to home care) for each schedule, when all schedules are feasible because of patients’ availabilities during the working day.
5.2 Comparison between objective functions
Table 4. Comparison of objective functions in first situation
The objective functions are formulated in different ways. The first one minimizes the sum of travelled and waited times. The second one was formalised in goal to compact all care activities. It minimizes worked duration in all the caregiver’s tour. In this part we introduce a comparative work between objective functions. First situation: Table 4 illustrates results using both objective functions, in first situation and different scenarios (S1, S2 and S3).
S1 S2 S3
CG 1 CG 2 CG 1 CG 2 CG 1 CG 2
* Computing Time.
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Objective function (1)
Objective function (2)
CT*
CT*
5s 5s 7s
Travel
Wait
120
0
110
0
170
0
145
0
195
0
185
0
10mn19s 10mn5s 1mn53s
Travel
Wait
120
0
110
0
170
0
145
0
195
0
185
0
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
Second situation: We illustrate in table 5, results in case of second situation and different scenarios (S1, S2 and S3). We note in this situation that using both objective functions reaches to similar travelled and waited times, even if it’s possible to reach to different values of waited or travelled times in each objective function. We note also that optimal solutions are generated with similar calculating times. Table 5. Comparison of objective functions in second situation
S1 S2 S3
CG 1 CG 2 CG 1 CG 2 CG 1 CG 2
Objective function (1) CT* Travel wait 145 44 3s 140 94 3s 4s
206
0
182
36
192 182
17 37
Objective function (2) CT* Travel Wait 140 49 3s 140 94 3s 4s
206
0
182
36
192 182
17 37
To minimize travelled and waited times, two objective functions were formalised in different ways. The first one depends on travelled and waited times, and the second minimize worked duration in caregiver’s tour. Our goal by the second objective function is to compact all care activities performed by the caregivers, such as the visits’ durations and patients’ availabilities cannot be violated (hard constraints). We have proved by this comparative work that both objective functions allow to minimal travelled and waited times. We conclude that minimizing caregivers’ work duration leads immediately minimal sums of waited and travelled times. 6. CONCLUSIONS AND PERSPECTIVES In this paper we focused on the caregivers’ tours problem, taking into account the visits coordination. Our principal purpose was to use exact methods based on ILP to solve this problem. We have formalized two objective functions to minimize the total sum of waited and travelled times, and tested in different contexts linked to patient’s availabilities and their geographical locations. We have showed by this study that minimizing caregiver’s worked duration leads to minimal values for travelled and waited times. We have noted also that districting the served area leads to excessive travelled and waited times. It may be interesting in this case to allocate a multidisciplinary care team to each sub-area. Besides, this work can be extended to take into account a larger number of patients and caregivers. It could be also interesting to add other important constraints in care process, namely synchronization between caregivers or caregivers and material resources. The caregivers’ travel, the care durations, and availability of material resources may be uncertain, so it can be interesting to resolve this problem in this context. REFERENCES Akjiratikarl, C. Yenradee, P. and Drake, P.R. (2007). PSObased algorithm for home care worker scheduling in the UK. Computers & Industrial Engineering, volume (53), 559-583.
Ben Bachouch, R. Fakhfakh, M. Guinet, A. and Gabouj, S H., (2009). Planification des tournées des infirmiers dans une structure de soins à domicile. Gestion et Ingénierie des SystEms Hospitaliers, volume (8), 3-5. Bertels, S. and Fahle, T.A. (2006). Hybrid setup for a hybrid scenario: combining heuristics for the home health care problem. Computers & Operations Research, volume (33), 2866-2890. Borsani, V. Matta, A. Beschi, G. and Sommaruga, F.A. (2006). Home care scheduling model for human resources. International Conference on Service Systems and Service Management, volume (1), 449-454. Begur, S.V. Miller, D.M. and Weaver, J.R. (1997). An Integrated Spatial DSS for Scheduling and Routing Home-Health-Care Nurses. Interfaces, volume (27), 3548. Clarke, G. and Wright, J.W. (1964). Scheduling of vehicles from a central depot to a number of delivery points. Operation research, volume (12), 568-581. Cheng, E. and Rich, J.L. (1998). A Home Health Care Routing and Scheduling Problem. Technical report TR98-04-Rice University, volume (25). Chahed, J.S. Dalery, Y. and Evren, S. (2008). Modélisation et analyse de l'organisation et du fonctionnement des structures d'hospitalisation à domicile. Chapter 7. Multidisciplinary thesis server. Com-Ruelle, L. and Lebrun, E. (2003). Indicateurs d’état de santé des patients hospitalisés à domicile (ENHAD 1999-2000). Demassey, S. (2003). Méthodes hybrides de PC et PL pour le problème d'ordonnancement de projet à contraintes de ressources. Chapter 2. Hyper Articles en Ligne Desrocher, M. Laporte, G. (1991). Improvements and extension to the MTZ sub-tour elimination constraints. Operation research letters, volume (10), 27-36. Eveborn, P. Flisberg, P. and Rönnqvist, M. (2006)., Laps Care-an operational system for staff planning of home care. European Journal of Operational Research, volume (171), 962-976. El-Sherbeny, N.A. (2010) Vehicle Routing with Time Windows: An overview of exact, heuristic and metaheuristic methods. Journal of King Saud UniversityScience, volume (22), 123-131. Lenstra, J.K. and Kan, A.H.G.R. (1981). A complexity of vehicle routing and scheduling problems. Networks, volume. (11), 221-227 Li, X. Tian, P. and Leung, S.C. (2010). Vehicle routing problems with time windows and stochastic travel and service times: Models and algorithm. Journal of Production Economics, volume (125), 137-145. Rego, C. and Roucairol, C. (1994). Le problème de tournées de véhicules: études et résolutions approchée. Projet Reflecs: « rapport de recherche N°2197 », volume (125), 3-86. Thomsen, K. (2006). Optimization on home care. (2006). Chapter 2. Technical university of danemark.
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Collaborative model for planning and scheduling caregivers’ activities in homecare Redjem Rabeh*. Kharraja Saïd* Marcon Eric* * LASPI, F-42334, IUT de Roanne, Saint Etienne France (Tel: +33 6 43 00 31 17; e-mail: [email protected]. said.kharraja @univ-st-etiennet.fr marcon @univ-st-etiennet.fr) Abstract: Home Health care Services (HCS), are structures that provide continuous and coordinated cares in patients’ homes. In this paper, we deal with planning and scheduling caregivers’ visits in HCS. The caregivers visit their assigned patients, while minimizing the travelled and waited times, within the patients’ availabilities constraints. The problem was solved using an original MILP that coordinate caregivers’ tours. The mathematical model was solved using the academic solver LINGO of LINDO SYSTEMS. Keywords: Resource planning, home care, mathematical modelling, operations scheduling, VRPTW.
1. INTRODUCTION
2. PROBLEM DESCRIPTION
The organizational and economic problems emerged in care systems have contributed to the appearance of Home Health care Services (HCS), to provide continuous and coordinated care for patients in their homes, as an efficient solution for reducing health care’s costs and maintaining a satisfactory quality of service. HCS are defined as “a mini network in a wider one” (Com-Ruelle et al., 2003), since it requires a coordination between multiple actors from varied skills. However, it operates itself in a larger network involving several modes. The HCS increasingly take an important place among healthcare structures. In France, the number of the HCS has tripled between 1999 and 2006 from 68 to 185, and the places’ number in homecare have increased from 3908 authorised beds in 1999 to 7355 in 2006 (Chahed et al., 2008).
The complexity of planning and scheduling the caregivers’ tours is due to the various constraints to take into account (Chahed et al., 2008). Our problem concerns establishing the best caregivers’ tours, while optimising the travelled and waited times. We consider two caregivers; each one follows a route to care a set of allocated patients, within their time windows. Waiting times are due to the caregivers’ arrival at the patient’s home prior to its earliest time of availability. Patients’ care durations are predefined. In the case of patients requiring visits from both caregivers, we have two situations, (i) the visits can be coordinated, i.e. caregivers should visit the patient within a predefined order; (ii) the visits may be sequenced without coordination (predefined order). The order of visits is defined in the patients’ care protocol.
A homecare process requires a variety of organizational and clinical decisions, coordination and synchronization between various human and material resources and the participation of an important number of actors from different skills (Bertels et al., 2006). Thus, it is interesting to develop tools that improve the performances of HCS, as tools that enable planning caregivers’ tours. In this paper we propose a decision support tool based on integer linear programming to schedule two caregivers’ visits for patients, taking into account various constraints, especially the “visits’ coordination” constraints. The following parts are structured as follows. First, we describe the problem. Second, we present works that have dealt with planning the caregivers’ tours. In a third part, we propose our mathematical formulation for the problem. Next, we will proceed to an analysis of results obtained by using two objective functions and different scenarios. Finally, we conclude with some prospects for future researches.
978-3-902661-93-7/11/$20.00 © 2011 IFAC
Our purpose is to provide a decision tool to schedule caregivers’ activities in HCS. Besides, this problem can be considered as Vehicle Routing Problem with Time Window “VRPTW” (Thomsen et al., 2006), while caregivers are vehicles, HCS is the depot, and patients are cities. The VRPTW problem, named “vehicle scheduling” by (Clarke et al., 1964), involves a fleet of vehicles to serve a number of customers, with various demands. The objective is to find routes for vehicle, while minimizing travelled durations, without violating customers’ time windows (El-Sherbeny et al., 2010). This problem is an NP-hard (Lenstra et al., 1981), i.e. there is no algorithm polynomial in time to solve the problem. The existent algorithms solve only small instances. A VRPTW can be solved by exact methods as linear programming or approximate methods as heuristics (Rego et al., 1994). Thus, in our model, we aim at minimizing the total sum of travelled and waited times for all caregivers. In the next section we describe the main works that have dealt with caregivers’ planning tours.
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10.3182/20110828-6-IT-1002.01043
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
3. EXISTING WORKS In this section we present many works that has addressed the caregivers’ (or nurses’) tours problem, in home health care.
(paid by the hour). Thus, optimal schedule are generated, such as a nurse leaves/returns home, visits a set of patients within their time windows, and takes a lunch break, all within the nurses’ time window, while minimizing both, over time for salaried nurses and part-time nurses.
(Akjiratikarl et al., 2007) presents a novel application of a collaborative population to schedule home caregivers; using a method based on a meta-heuristic called Particle Swarm Optimization (PSO). They minimize the travelled distance, providing that the capacity and time windows constraints of services are not violated.
(Ben Bachouch et al., 2009) have proposed a tool, to planning nurses’ tours in homecare. Taking into account different constraints, as patients’ availabilities, lunch break for nurses, they have minimized total travelled distance.
(Eveborn et al., 2006) have developed visiting schedules for care providers that incorporate some restrictions and the objective is to reduce the transport time. Each visit has a particular task to be performed and/or nursing activities.
(Bertels et al., 2006) have proposed a method combined of linear programming, constraint programming, and metaheuristics for the home health care problem taking into account multiple criteria, as patients’ satisfaction, nurses’ qualifications, and travelling costs. They have minimized the travelling costs.
(Begur et al., 1997) construct the nurses’ tour schedules taking into account patients availability, needs constraints, and nurses’ availability. They fix in advance the days of visiting patients. (Cheng and Rich., 1998) have treated the nurses’ tour problem in a homecare, using the “VRPTW”. While considering two types of nurses: salaried; part-time nurses
(Borsani et al., 2006) propose a linear model, which deals with the problem of deciding (i) which human resource using in each home visit and (ii) when to execute the service, in order to satisfy the care plan for each patient served by the home care providers. The benefit of the system proposed is an increase in service quality and efficiency. In table 1, we summarize a comparison between some works quoted above.
Table 1. Comparison between methods Optimized Criterion Cheng et Rich 1998 Begur et al 1997 Borsani et al 2006 Ben Bachouch et al 2009 Akjiratikarl et al.2007 Our approach
Costs of working hours Travel duration Balancing work load Travel duration Travel duration Travel and wait time
Patients’ availabilities
Multiple visits for patients
Shared patients
Coordinated visits
X X X X X X
X X X X
In this section, we have presented works that deals with scheduling and planning activities for nurses, and other works in both scheduling activities and allocation of patients to nurses. We noted that all works presented use either exact methods like linear programming, or approximate ones like meta-heuristics. We noted also that all works cited above do not address the aspect of coordination between different providers. Our purpose in this paper is to solve this problem using exact methods. In next part we introduce the problem’s mathematical formulation.
X X X X
Exact methods
X X X X X
X
duration. Thus, visits’ durations for the patients not assigned to a caregiver are equal to “zero”. Caregivers’ tours starts and ends at the HCS (dummy patient), and the work horizon is the day. Based on “care protocol”, an order of care can be defined in advance for patients who need care from both caregivers. 4.2 Parameters and notations N, S: Respectively, the number of patients and caregivers (in this model S = 2),
4. MATHEMATICAL FORMULATION
tdij: Travelling time, between two patients #i and # j,
The problem can be described as a VRPTW problem. Given a set of patients, two caregivers and the HCS, find a set of routes for each caregiver, starting and ending at HCS, such as each one has his set of allocated patients, while each patient may be cared by both caregivers, within his availabilities. Each patient is visited by one caregiver at the same time, and the visits’ order can be predefined (coordinated).
pis: Care duration delivered to patient #i by the caregiver s,
4.1 Hypothesis
ri, di: Respectively, the release/due date to begin/end care for patient #i, ypij: Binary value that denotes a predefined order between caregivers i and j to visit the patient #p, ypij = 1 means that caregiver i must visits the patient #p before caregiver j, thus ypij = 0 and ypji = 0 means that there is no order between caregivers i and j for patient #p,
We consider that, each patient receives the same caregiver once a day, within his time window. Each care activity has 2878
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
Ais: Binary value which defines for each caregiver his set of assigned patients, such that Ais = 1 if patient #i is assigned to caregiver s, and Ais = 0 otherwise, NPs NSi: Respectively, the number of patients allocated to caregiver s, and the number of caregivers to care patient #i,
∀s ∈ [1, S ]∀i ∈ [1, N ]
xiis = 0
Constraints (3) - (6) ensure that each caregiver visits his assigned patients once a day. Constraints (7) ensure sequencing patients’ visits for each caregiver:
(
)
tis + pis + td ij − M 1 − xijs ≤ t js
M: Large constant,
(6)
∀s ∈ [1, S ]∀i, j ∈ [1, N ]
(7)
4.3 Decision variables
We enforce by constraints (8) a predefined order of care between the caregivers for the same patient:
To model the problem we use the following decision variables
tis + pis − M (1 − yiss ' ) ≤ tis '
xijs, ziss’: A binary variables, such as, xijs = 1 if caregiver s visits patient #i strictly before patient #j, xijs = 0 otherwise, and ziss’ = 1 when the caregiver s realizes his visit for the patient #i before caregiver s’, ziss’ = 0 otherwise,
∀s, s '∈ [1, S ]∀i ∈ [1, N ]
(8)
Constraints (9) - (13) ensure that the visits performed by caregivers for the same patient are sequenced: S
∑ ziss ' ≤ Ais '
∀s '∈ [1, S ]∀i ∈ [1, N ]
(9)
∀s ∈ [1, S ]∀i ∈ [1, N ]
(10)
∀i ∈ [1, N ]
(11)
ziss = 0
∀s ∈ [1, S ]∀i ∈ [1, N ]
(12)
tis + pis − M (1 − ziss ' ) ≤ tis '
∀s, s '∈ [1, S ]∀i ∈ [1, N ]
(13)
s =1
tis: A positive variable which corresponds to the date the care begins for the patient #i by caregiver s,
S
∑ ziss ' ≤ Ais s'
arrivis, waitis, uis: A positive variables, denote respectively, (i) the arrival date of the caregiver s at the patient’ #i home, (ii) the caregiver’s s waited time, before the visit for patient #i, and (iii) the patient’s #i rank, in the tour of caregiver s, 4.4 Model formulation To model our problem, we introduce a dummy patient denoted by 1 which represents the HCS, such as: t1s = r1 = D+, d1 = D-, p1s = 0, while D+ / D- respectively begin / end of the caregivers’ working day. The problem was formulated as the following integer linear program. The goal is to minimize the caregivers’ travelled and waited times. We have developed for that two objective functions (1) and (2), then we have compared results in section (5): N N S S N Min ∑ ∑ attis + ∑ ∑ ∑ td ij × xijs i =1 j =1s =1 s =1i =1
Min ∑ (End s − t1s ) s = 1
s =1 s '=1
Caregivers have to respect patients’ availabilities and the working hours. We ensure that by constraints (14) - (17):
tis + pis + td i1 − M (1 − Ais ) ≤ d1 ∀s ∈ [1, S ]∀i ∈ [1, N ]
∀s ∈ [1, S ]∀i ∈ [1, N ]
∑ xijs = A js
tis + pis − M (1 − Ais ) ≤ di
∀s ∈ [1, S ]∀i ∈ [1, N ]
(17)
(2)
(18), (19) are modified (Desrocher and Laport., 1991) subtours eliminating constraints. The rank of the patients not assigned to the caregiver is set to zero in the constraints (20): uis − u js + (N − 1)xijs + (N − 3)x jis ≤ N − 2
(2.1)
(3)
N N
∀s ∈ [1, S ]∀i , j ∈ [2, N ]
(18)
uis + M (1 − Ais ) ≥ 2 uis − M (1 − Ais ) ≤ NPs
∀s ∈ [1, S ]∀i ∈ [2, N ]
(19)
uis − M × Ais ≤ 0 uis + M × Ais ≥ 0
∀s ∈ [1, S ]∀i ∈ [2, N ]
(20)
Constraints (21), (22) allow the calculation of the caregivers’ waited times: ∀s ∈ [1, S ]∀j ∈ [1, N ]
(4)
∀s ∈ [1, S ]
(5)
( (
) )
tis + pis + td ij − M 1 − xijs ≤ arriv js ∀s ∈ [1, S ]∀i, j ∈ [1, N ] (21) tis + pis + td ij + M 1 − xijs ≥ arriv js
i =1
∑ ∑ xijs = NPs
(15)
(1)
j =1 N
(14)
(16)
The constraints are:
∑ xijs = Ais
∀s ∈ [1, S ]∀i ∈ [1, N ]
tis + M (1 − Ais ) ≥ r1
∀s ∈ [1, S ]∀i ∈ [1, N ]
Ends: denotes the caregiver’s s arrival date to the HCS at the end of his tour, such as:
N
S
tis + M (1 − Ais ) ≥ ri
S
tis + pis + td i1 − M (1 − xi1s ) ≤ End s ∀s ∈ [1, S ]∀i ∈ [2, N ] tis + pis + td i1 + M (1 − xi1s ) ≥ End s
S
∑ ∑ ziss ' = NS p − 1
i =1 j =1
attis = tis − arrivis
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∀s ∈ [1, S ]∀i ∈ [2, N ]
(22)
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
(23), (24) are binary or non-negativity constraints.
5.1 Data for simulations
xijs , ziss ' ∈ {0,1}
∀s, s '∈ [1, S ]∀i, j ∈ [1, N ]
(23)
tis , attis , arrivis ≥ 0
∀s ∈ [1, S ]∀i ∈ [1, N ]
(24)
5. RESULTS To solve the model, we used the academic solver LINGO_11.0 from LINDO SYSTEMS INC. In this part, we aim to present results based on an example of 2 caregivers and 8 patients. We also present results of comparison between both objective functions. We have varied the tests, using two situations based on patients’ availabilities. “First situation”: all patients are available during the caregivers’ working hours. “Second situation”: some patients are available only the morning, and others only the afternoon. Three scenarios based on patients’ locations were tested. “First scenario”: all patients live the same district and the travelling times are between 15 to 30 minutes. “Second scenario”: all patients live in the same district except for one isolated. “Third scenario”: two different districts, with travelling times in the same district between 15 and 30 minutes, and between 50 to 65 minutes between different districts.
We have tested the model on an example of two caregivers and eight patients, with four shared patients {#1(HCS), #3, #4, #6 and #8}. Patients {#4 and #6} require coordination between caregivers. In our case, caregiver 2 have to perform his activity for patient #4 before caregiver 1; for patient #6 caregiver 1 must begin before caregiver 2. The work horizon is the day (8 hours of work or 480 minutes). The working day begins and ends in the dummy patient whose release and due dates are respectively r1 = 1, d1 = 480. Tables 2 and 3 contain respectively, the patients’ allocation to caregivers and care durations of each patient. Table 2. Allocation table CG 1 CG 2
#1(HCS) 1 1
#2 1 0
#3 1 1
#4 1 1
#5 0 1
#6 1 1
#7 1 0
#8 1 1
#3 50 45
#4 35 25
#5 0 60
#6 40 55
#7 60 0
#8 45 30
Table 3. Care durations CG 1 CG 2
#1(HCS) 0 0
#2 30 0
Fig.1. First situation caregivers’ tours Figure 1 illustrates results obtained when model is solved using both objective functions, in the first situation. The caregivers’ tours are represented above. In this situation, we note that, patients’ availabilities and coordination between caregivers for patients #4 and #6 were respected. We remark also that: (i) If all patients live near to each others (first scenario); using both objective functions leads to choosing the minimal travel durations to generate caregivers’ tour. (ii) When patients are divided in two distinct groups (second and third scenarios), the caregivers’ tour will be generated by choosing the minimal travel duration between patients from the same location, and also between patients from different locations. (iii) Using both objective functions can lead to: identical caregivers’ tours, as in first and third scenarios. But
we may obtain different tours, as the caregiver’s 1 tour in second scenario. The difference in this case is in patients’ scheduling. For example, order of patient #2 is “1” using first objective function, and “2” using the second, in the caregiver’s 1 tour. But the total sum of travelled and waited times is identical using both objective functions (as showed in section 5.2). (iv) For patients, #4 and #6, who need coordination, the care dates are generated taking into account the visits’ order. Thus, the coordination has an immediate impact on the caregivers’ tours. We conclude from this situation that, optimal tours generation depend mainly on next factors: travel durations between patients from same and/or different locations and coordination between caregivers.
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
Figure 2 illustrates results for the second situation (two types of patients’ availabilities: [1,240] and [240,480]). We remark that, (i) When all patients are in the same location (first scenario), caregivers’ tour is generated depending on travel durations and patients’ availabilities, i.e. starting by patients available the morning (patients #2, #3, #4), and ending by those available in the afternoon (patients #5, #6, #7, #8). (ii) When patients are divided in two distinct locations (second and third scenarios), caregivers’ tours are built with the patients available in the morning from the first region (where the HCS is located), then all patients from second location (beginning by those available the morning), and finally with the patients available the afternoon from the first location. (iii) The caregivers’ tours can be identical using both
objective functions, as in second and third scenarios; but it may be different as caregiver’s 1 tour in first scenario. The difference in this case is in the patients’ visit order. For example, the order of patient #3 in the caregiver’s 1 tour, using the first objective function is “1”, and “3” using the second. But the total sum of travelled and waited times are identical using both objective functions (section 5.2); (iv) while coordinating visits for patients #4 and #6, the care dates are generated taking into account order between caregivers; thus coordination has an immediate impact on the caregivers’ tours. Thus we conclude that optimal tours generation depends on next factors: travel durations from same or different location, visits’ coordination and patients’ availabilities.
Fig.2. Second situation caregivers’ tours From these tests we note that, if patients are grouped into different locations and without patients’ availability constraints, using both objective functions in our model gives the best schedule: i.e. a caregiver is introduced by first location, and then he moves to the second one with a minimum travelled duration. In case of complex situations, as the second and third scenarios in the second situation, we have noted the big impact of patients’ availabilities on tours generation. We have also remarked that tours’ generation is based on the same strategy whatever the objective function (see remark 3).
In this situation patients are available during the working day, thus there are no waiting times, and minimizing the sum of travelled and waited times (1st objective function), or minimizing the caregivers’ worked duration (2nd objective function), leads to minimizing only travelled durations. Thus the results obtained using both objective functions are similar. In terms of calculation times, using first objective function is less expensive than the second. This is due to time elapsed in calculating Ends (caregiver’s return date to home care) for each schedule, when all schedules are feasible because of patients’ availabilities during the working day.
5.2 Comparison between objective functions
Table 4. Comparison of objective functions in first situation
The objective functions are formulated in different ways. The first one minimizes the sum of travelled and waited times. The second one was formalised in goal to compact all care activities. It minimizes worked duration in all the caregiver’s tour. In this part we introduce a comparative work between objective functions. First situation: Table 4 illustrates results using both objective functions, in first situation and different scenarios (S1, S2 and S3).
S1 S2 S3
CG 1 CG 2 CG 1 CG 2 CG 1 CG 2
* Computing Time.
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Objective function (1)
Objective function (2)
CT*
CT*
5s 5s 7s
Travel
Wait
120
0
110
0
170
0
145
0
195
0
185
0
10mn19s 10mn5s 1mn53s
Travel
Wait
120
0
110
0
170
0
145
0
195
0
185
0
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
Second situation: We illustrate in table 5, results in case of second situation and different scenarios (S1, S2 and S3). We note in this situation that using both objective functions reaches to similar travelled and waited times, even if it’s possible to reach to different values of waited or travelled times in each objective function. We note also that optimal solutions are generated with similar calculating times. Table 5. Comparison of objective functions in second situation
S1 S2 S3
CG 1 CG 2 CG 1 CG 2 CG 1 CG 2
Objective function (1) CT* Travel wait 145 44 3s 140 94 3s 4s
206
0
182
36
192 182
17 37
Objective function (2) CT* Travel Wait 140 49 3s 140 94 3s 4s
206
0
182
36
192 182
17 37
To minimize travelled and waited times, two objective functions were formalised in different ways. The first one depends on travelled and waited times, and the second minimize worked duration in caregiver’s tour. Our goal by the second objective function is to compact all care activities performed by the caregivers, such as the visits’ durations and patients’ availabilities cannot be violated (hard constraints). We have proved by this comparative work that both objective functions allow to minimal travelled and waited times. We conclude that minimizing caregivers’ work duration leads immediately minimal sums of waited and travelled times. 6. CONCLUSIONS AND PERSPECTIVES In this paper we focused on the caregivers’ tours problem, taking into account the visits coordination. Our principal purpose was to use exact methods based on ILP to solve this problem. We have formalized two objective functions to minimize the total sum of waited and travelled times, and tested in different contexts linked to patient’s availabilities and their geographical locations. We have showed by this study that minimizing caregiver’s worked duration leads to minimal values for travelled and waited times. We have noted also that districting the served area leads to excessive travelled and waited times. It may be interesting in this case to allocate a multidisciplinary care team to each sub-area. Besides, this work can be extended to take into account a larger number of patients and caregivers. It could be also interesting to add other important constraints in care process, namely synchronization between caregivers or caregivers and material resources. The caregivers’ travel, the care durations, and availability of material resources may be uncertain, so it can be interesting to resolve this problem in this context. REFERENCES Akjiratikarl, C. Yenradee, P. and Drake, P.R. (2007). PSObased algorithm for home care worker scheduling in the UK. Computers & Industrial Engineering, volume (53), 559-583.
Ben Bachouch, R. Fakhfakh, M. Guinet, A. and Gabouj, S H., (2009). Planification des tournées des infirmiers dans une structure de soins à domicile. Gestion et Ingénierie des SystEms Hospitaliers, volume (8), 3-5. Bertels, S. and Fahle, T.A. (2006). Hybrid setup for a hybrid scenario: combining heuristics for the home health care problem. Computers & Operations Research, volume (33), 2866-2890. Borsani, V. Matta, A. Beschi, G. and Sommaruga, F.A. (2006). Home care scheduling model for human resources. International Conference on Service Systems and Service Management, volume (1), 449-454. Begur, S.V. Miller, D.M. and Weaver, J.R. (1997). An Integrated Spatial DSS for Scheduling and Routing Home-Health-Care Nurses. Interfaces, volume (27), 3548. Clarke, G. and Wright, J.W. (1964). Scheduling of vehicles from a central depot to a number of delivery points. Operation research, volume (12), 568-581. Cheng, E. and Rich, J.L. (1998). A Home Health Care Routing and Scheduling Problem. Technical report TR98-04-Rice University, volume (25). Chahed, J.S. Dalery, Y. and Evren, S. (2008). Modélisation et analyse de l'organisation et du fonctionnement des structures d'hospitalisation à domicile. Chapter 7. Multidisciplinary thesis server. Com-Ruelle, L. and Lebrun, E. (2003). Indicateurs d’état de santé des patients hospitalisés à domicile (ENHAD 1999-2000). Demassey, S. (2003). Méthodes hybrides de PC et PL pour le problème d'ordonnancement de projet à contraintes de ressources. Chapter 2. Hyper Articles en Ligne Desrocher, M. Laporte, G. (1991). Improvements and extension to the MTZ sub-tour elimination constraints. Operation research letters, volume (10), 27-36. Eveborn, P. Flisberg, P. and Rönnqvist, M. (2006)., Laps Care-an operational system for staff planning of home care. European Journal of Operational Research, volume (171), 962-976. El-Sherbeny, N.A. (2010) Vehicle Routing with Time Windows: An overview of exact, heuristic and metaheuristic methods. Journal of King Saud UniversityScience, volume (22), 123-131. Lenstra, J.K. and Kan, A.H.G.R. (1981). A complexity of vehicle routing and scheduling problems. Networks, volume. (11), 221-227 Li, X. Tian, P. and Leung, S.C. (2010). Vehicle routing problems with time windows and stochastic travel and service times: Models and algorithm. Journal of Production Economics, volume (125), 137-145. Rego, C. and Roucairol, C. (1994). Le problème de tournées de véhicules: études et résolutions approchée. Projet Reflecs: « rapport de recherche N°2197 », volume (125), 3-86. Thomsen, K. (2006). Optimization on home care. (2006). Chapter 2. Technical university of danemark.
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