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Abstract — Home health care is provided to patients with specialconditions in which the assistance is required in their homes.Depending on the pathology, each patient receives specific home careservices from specialists, mainly doctors, therapists and nurses. Inthis context the Home Health Care Routing and Scheduling Problem(HHCRSP) is related with routing and scheduling of the qualified

personnel. It integrates the Nurse Rostering Problem (NRP) and theVehicle Routing Problem (VRP). The HHCRSP considersconstraints related with time windows, workload and attentioncapacity among other limitations associated with patients and staff.Due the cost and quality implications that this kind of servicesgenerates in health care companies, this article presents a mixedinteger linear programming model for planning the periodic scheduleof medical staff and the route planning for to patient visits.

Keywords — Home health care, mixed integer linear programming, qualified staff scheduling, staff routing.

I. I NTRODUCTION

OME health care is a service that medical institutions

provide to patients who, due to their health conditions,can be treated in their homes, in other cases it's an strategy toincreasing the capacity of rooms in hospitals. Taking intoaccount the availability of qualified personnel (doctors, nurses,and therapists), the health sector companies offer a variety oftreatments required by patients in which time, cost and qualityof the service are crucial; therefore personnel scheduling andthe routing of visits has great importance.

The optimization of home health care has long been a fieldof interest for the operations research, in this context the

This work was supported by the Master in Operations Management andthe International School of Economics and Management Sciences (EICEA) ofthe Universidad de La Sabana, Chía, Colombia.

Andrés Felipe Torres-Ramos is with the International School of Economicsand Management Science (EICEA), Universidad de La Sabana, Chía,Colombia (corresponding author to provide phone: 571-8615555 ext.: 25109;e-mail: [email protected]).

Edgar Hernán Alfonso-Lizarazo is with the Engineering Department,Universidad de La Sabana, Chía, Colombia (e-mail:[email protected]).

Lorena Silvana Reyes-Rubiano is with the International School ofEconomics and Management Sciences (EICEA), Universidad de La Sabana,Chía, Colombia (e-mail: [email protected]).

Carlos Leonardo Quintero-Araújo is with the International School ofEconomics and Management Sciences (EICEA), Universidad de La Sabana,Chía, Colombia (e-mail: [email protected]).

highlighted areas of study are Home Health Care Routing andScheduling Problem (HHCRSP), which covers aspects of

programming and planning of routes for the medical staff arethe Rostering Problem (RP), e.g. Nurse Rostering Problem(NRP), and the planning of routes for visits to patients isconsidered a Vehicle Routing Problem (VRP). These problemsare considered as NP-Hard [1], [2].

In this paper personnel scheduling aims to allocate medicalstaff members to patients; this scheduling must be performedaccording to the type of pathology of each patient and theavailability of time for patients and staff. In this paper the maintreatments studied are related with pathologies of palliativetype, chronic care, blood anticoagulation and domiciles forwound care. Depending on the treatment there are three typesof specialists who can provides these treatments: doctors,nurses, and therapists. Each patient requires a level of

personalized medical care, given that some patients are insevere condition and require fewer intervals betweentreatments, as opposed to patients presenting better health.

This generates a periodic planning of each specialist accordingto the type of treatment and health conditions of each patient.

In order to schedule and plan the routes of patient visits it isnecessary to consider each patient' time windows, which is atime slot in the day which the patient defines or requires themedical care. Another important aspect in the planning ofroutes is the starting and end point of each staff member ’ sdaily, for this paper a multi-depot problem is considered, inwhich case the home of each specialist (doctors, nurses, andtherapists) is the beginning and end of each route, this aspectincrease the complexity of the model depending on the numberof staff members.

The outline of the article is as follows. Section II presents aliterature review for the HHCRSP. The characteristics on staffand patients for home care are presented in section III. Amathematical model for the HHCRSP is presented in sectionIV. The results of the mathematical model are shown insection V. Conclusions and recommendations for futureresearch are presented in section VI.

II. LITERATURE R EVIEW

The home health care routing and scheduling problem(HHCRSP), as mentioned above, make up two problems

Mathematical Model for the Home Health CareRouting and Scheduling Problem with Multiple

Treatments and Time Windows

Andrés Felipe Torres-Ramos, Edgar Hernán Alfonso-Lizarazo, Lorena Silvana Reyes-Rubiano,Carlos Leonardo Quintero-Araújo

H

Mathematical Methods in Science and Engineering

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associated with each operation involved. In terms of staff planning or the allocation of medical staff to patients referredto the Nurse Rostering Problem (NRP) [3], and routes of visitsto patients have been considered under different variations ofthe Vehicle Routing Problem (VRP) [2], [4]. The most studiedVRP variation in the HHCRSP is the Vehicle Routing Problemwith Time Windows (VRPTW) [1], [5] – [8], which includesthe daily time slot that the patient has to receive medicalattention. Other variations of the VRP applied to the HHCRSPhave been studied independently are the Multi TravelingSalesman Problem with Time Windows (MTSPTW), theVehicle Routing Problem with Multi-Depot (VRPMD) and theVehicle Routing Problem with Multi-Period (VRPMP), whichintend to characterize multiple staff and multiple points inwhich the staff start and end each route respectively. [9] – [11].

Different methodologies have been used to solve theHHCRSP within operations research. Within the exactmethods is the study of Y. Kergosien, C. Lenté and J-C Billaut[9], which seeks to determine the routes of the medical staffvisiting patient's home. In order to do so they determine a

whole linear programming model. Another exact method usedis the Branch-and-Price algorithm, in the paper [6] the authorsuse this algorithm to assign staff and determine routes byconsidering visits per groups of patients. Similarly heuristicmethods have been used to solve the HHCRSP, as in thearticle of D. Mankowska, F. Meisel y C. Bierwirth [10], inwhich the authors develop a heuristic that determines visits to

patients through services interconnected by heterogeneousstaff. A. Coppi, P. Detti and J. Raffaelli [12] develop aheuristic based on a local search to determine the personnel

planning and routing of visits. Despite the good results thatgenerate the exact and heuristic methods, different authorshave used metaheuristics, which allow the problem to bedevelopment with more data in a reasonably short time. In the

paper [8] the authors present the application of themetaheuristics called Particle Swarm Optimization (PSO) inthe programming of the house medical staff. In the paper [13]the authors apply Genetic Algorithm (GA) and Tabu Search(TS) metaheuristics to the delivery of drugs and the collectionof biological samples. Simulated Annealing (SA) and TabuSearch (TS) metaheuristics are proposals in the paper [14] todetermine the schedule of therapists within the medicaltreatments of patients home.

On the other hand, the authors have focused their researchon various objective functions, the most common is theminimization of the costs of operation, where assignment,overtime and reassignment of staff costs are considered [7],[15], [16], and costs associate to staff and transport [3], [6],[11], [13], [17]. Another objective mainly associated with therouting of the staff is the minimization of time and distancetraveled from the operation [8], [10], [18] – [20], whichincludes minimizing the total journey undertaken by staff tomake visits to the patients.

This article focuses on minimizing the total time ofoperation, as a component of the level of patient satisfaction.

Additionally, the study of the scheduling of personal and the planning of routes of patient visits integrating the MTSPTW,the VRPMD and the VRPMP in a single problem: Multi-Traveling Salesman Problem with Time Windows, Multi-Depot and Multi-Period (MTSPTWMDMP) .

III. CHARACTERISTICS OF STAFF AND PATIENTS IN HOME

HEALTH CARE

In the most of the revised articles only one type of staff isconsidered. The home care system studied in this paper, theservices can be provided by nurses, doctors and therapists. Onthe other hand the HHCRSP considers the attention ofdifferent services or pathologies of the patients. In this articlefour types of services are considered (domicile, bloodanticoagulation, chronic care, palliative care), and according tothe treatment of these pathologies patients require more thanone type of staff. The legal and economic aspects related withthe working time of the medical staff are considered [5]. Asummary of the aspects considered in our model are shown inFigure 1, adapted from Bertels and Fahle [18] .

Fig. 1 Characteristics for the HHCRSP

Patients have characteristics that, in addition to thecharacteristics of the medical staff, delimit the operation; oneof the most important is the time window, which represents thetime slot that each patient defines or requires for the home

visit. There are also features associated with the pathology ofeach patient. One of them is the demand for personnel, asmentioned above, the type of pathology determines the type ofstaff required and the frequency between visits, which isdependent on the condition of each patient's health.

IV. MATHEMATICAL FORMULATION FOR THE HHCRSP

This article proposes a mathematical model for theHHCRSP with different types of specialized personnel(doctors, nurses and therapists), which starts and ends everyroute in their own homes (multi-depot), and is performed in a


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horizon of time (multi-period). The problem is defined as adirected graph G=(V,A) with a set V = CM ∪ CE ∪ CT ∪ PM of nodes, which refer to the sets of nodes corresponding to thedepot represented by the homes of doctors ( CM ), homes ofnurses ( CE ) and homes of therapists ( CT ), and nodes of

patients ( PM ). And the set of arcs A = {(i, j): i, j ϵ V, i ≠ j} .Every patient i ϵ PM suffers from a unique pathology, which

is classified into four services: domicile, bloodanticoagulation, chronic care and palliative care. Each service

s ϵ S is served by the type of personal p ϵ P required accordingto the matrix. Additionally, each patient has a demand forvisits according to the type of staff and service( , , ), s s s

i i i DM DE DT these visits are performed within a time

horizon in days ( d ϵ D) with a periodicity according to the patient and the type of personnel ( , , ).i i i KM KE KT On the

other hand each patient's time window is framed within alength of time per day ( , ),d d

i ie l in which staff must reach the

house of the patient in d ie minimum and maximum in d

il . As

mentioned earlier, each staff member starts and ends its routein their respective home and they have a maximum workingtime per day TM . In addition the travel times ( ) p

ijTV differ

according to the type of personnel, since doctors are mobilized by means of private transport that is faster than publictransport by which nurses and therapists are mobilized.

The parameters and decision variables used from modellingthe HHCRSP are shown below in Table I.

Table I Notation used from modelling the HHCRSP

Parameters p

ijTV Travel time of personal p from the patient i to the patient j. psiTS Time of treatment requiring the patient i of the personal p in

service s. si DM Number of visits required by the patient i of doctors in service

s. si DE Number of visits required by the patient i of nurses in service

s. si DT Number of visits required by the patient i of therapists in

service s. s

pSP Personal p attending the service s.TM Maximum working time of the day of the staff.

ie Start time of the time window of patient i on the day d .d il Closing time of the time window of patient i on the day d .

Large number.Index i size.

i KM Period of time between doctors’ visits required by the patienti.

i KE Period of time between nurses visits required by the patient i.i KT Period of time between therapists visits required by the

patient i.Planning horizon.

Decision variables pd ij

Binary: 1. If the personal p visit the patient i and then the patient j on the day d . 0. On the contrary.

pd iY Time of arrival of the personal p visit the patient i on the day

d.iU Auxiliary variable to avoid subtours to visit each patient i.

The proposed mixed integer linear programming model below is to solve the HHCRSP .

Objective function

Minimize * pd p ps sij ij i p

i V j V p P d D s S

Z X TV TS SP

(1)

Subject to

, , , pd ps

ij i j V s S X TS TM i V p P d D

(2)

, , ,d pd d i i ie Y l i PM p P d D (3)

1 , , , , , pd ps p pd pd i i ij j ij

s S

Y TS TV Y M X i j V i j p P d D

(4)

, , , pd pd ij ji

PM j PM

X X i PM p P d D

(5)

,

0, , , pd ij

V j i

X i V p P d D

(6)

1, / , , pd ij

PM

X i V PM p P d D (7)

1, / , , p

iji PM

X j V PM p P d D (8)

/ , ,

0, , pij

V PM j p p P p j

X i V d D

(9)

,

1, ,id KM

pf ij i

V j i p M f d

X i PM d H KM

(10)

,

1, ,id KE

pf ij i

V j i p E f d

X i PM d H KE

(11)

,

1, ,id KT

pf ij i

V j i p T f d

X i PM d H KT

(12)

, , pd s sij p i

j V d D p M

X SP DM i PM s S

(13)

, , pd s sij p i

j V d D p E

X SP DE i PM s S

(14)

, , pd s sij p i

j V d D p T

X SP DT i PM s S

(15)

1, , pd ij

j V p M

X i PM d D

(16)

1, , pij

V p E

X i PM d D

(17)

1, , pd ij

V p T

X i PM d D

(18)

1, , , , pd i j ijU U X N N i j PM p P d D (19)

0,1 , , , , , pd ij i j V i j p P d D (20)

0, , , pd iY i V p P d D (21)

,i i (22)

The model presents the routing and scheduling of the homemedical staff, minimizing the total time of operation(transportation and service) (1). Constraints (2) determines themaximum working load per day for each staff. Constraints (3)


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and (4) impose the time window per each patient each dayaccording to the staff. Constraints (5) ensures the flow of staff

patients every day. Constraints (6) avoid fictitious routes of thestaff. Constraints (7), (8) and (9) determine the medical staffevery day goes out and returns to their respective home.Constraints (10), (11) and (12) determine the period of time

between visits to each patient according to the type of staff and planning horizon. Constraints (13), (14) and (15) guarantee thefulfillment of the demand for visits of each patient accordingto the type of staff required. Constraints (16), (17) and (18)impose maximum one service with every visit to medical

personnel for each patient per day. Constraints (19) eliminatesthe subtours generated in the programming model. Finally,constraint (20) define the variable pd

ij and constraints (21)

and (22) determine non-negativity pd iY and

i variables.

V. R ESULTS

As mentioned in section II the model proposed in this articleis complex, and is based on a Multi-Traveling Salesman

Problem with Time Windows, Multi-Depot and Multi-Period(MTSPTWMDMP). For HHCRSP model validation tests with

information from a company's industry in Colombia, test has16 patients of services: 1. Domicile, 2. Blood anticoagulation,3. Chronic care and 4. Palliative care. Each service requiresattention of different types of staff (doctors, nurses andtherapists), according to the service required of one or another

personal type as shown in the matrix s pSP (see Table II). In

addition, Table II shows the number of people for each type of

staff, where a total of 19 staff members is determined.The 16 patients require a total of 101 visits of all medicalstaff in a two weeks ' time horizon, in addition to a periodicity

between visits as shown in Table III.

Table II Services handled by each type of personal

Type of staffNumber ofspecialists

Type of service

1 2 3 4Doctors 3 0 1 1 1

Nurses 8 1 1 1 1Therapists 8 0 0 1 1

Total 19

Table III Type of service, demand and periodicity of visits required by patients

Patient i Type of service Periodicity between visits Demand of visits

1 2 3 4Doctors

i KM Nurses

i KE Therapists

i

Doctors si DM

Nurses si DE

Therapists si DT

1 1 0 0 0 - 1 - - 3 -2 1 0 0 0 - 2 - - 3 -3 0 1 0 0 3 1 - 1 3 -4 0 0 1 0 4 3 4 1 2 25 0 0 1 0 4 2 2 2 3 36 0 0 1 0 3 1 1 2 3 27 0 0 1 0 5 2 1 1 3 28 0 0 1 0 4 2 2 2 3 39 0 0 1 0 4 1 2 1 3 3

10 0 0 1 0 3 1 2 2 3 211 0 0 0 1 3 2 3 2 3 212 0 0 0 1 4 3 2 2 2 313 0 0 0 1 2 1 2 3 3 314 0 0 0 1 4 3 2 1 2 315 0 0 0 1 2 1 1 3 3 216 0 0 0 1 3 3 2 2 2 2

Total 25 44 32

The model was implemented using GAMS commercialsoftware version 24.1.3, with a time limit of 4000 seconds in a

personal computer Intel(R) Core(TM) i5-4200U CPU with 1.6GHz with 8 GB of RAM. The solution of the modeldetermines the routes per day needed to meet the demand ofvisits that patients require. Each route is carried out by amember of the medical staff who begins and ends at home.Table IV shows the routes to perform on day 1, whichidentifies 3 routes that are performed by the nurse 6, nurse 8and therapist 5 respectively. For example, the nurse 6 routestarts in her home, then visit the patients 6, 5, 8 and 13 in that

respective order, and finally part of the last patient (13) to herhome as the end point of the route.

The model gives total of 27 routes divided into 11 days(Appendix: Results of the Model of the HHCRSP), routes arecarried out by a total of 12 staff members (doctor 1, doctor 2,doctor 3, nurse 2, nurse 6, nurse 8, therapist 1, therapist 3,therapist 4, therapist 5, therapist 6 and therapist 7), as the totalnumber of staff members mentioned above are 19, thereforecompliance is evidence with the route with 7 members less,

proving the optimization of human resources and the capacityto serve a greater number of patients.


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Table IV Routs for day 1 Day 1

Nurse 6 Nurse 8 Therapist 5From To From To From To NH6 6 NH8 11 TH5 4

6 5 11 15 4 135 8 15 3 13 128 13 3 2 12 TH5

13 NH6 2 NH8

The total operation time of the staff in all the time horizon is8346 minutes, for which the 64.1% of the time corresponds tothe time of travel, and the remaining 35.9% of timecorresponds to the time of service.

I. CONCLUSION

This article proposes a model for the problem of routing andscheduling of medical staff in the home health care systems,which considers characteristics as different types of staff,different services, multi-depot, time windows and multi-

period. These features facilitates the application of this modelin real conditions related with the home health care services.

The study of the HHCRSP can lead in many directions. Firstimplement heuristics and metaheuristics, which allow theanalysis of the problem with more data in less time. On theother hand the integration of other types of services as deliveryand pick-up of medicines and biological samples and theemergency services, which involve new constraints andconsiderations associated with uncertainty in the demand andthe availability of staff.

APPENDIX : R ESULTS OF THE MODEL OF THE HHCRSP

The results of the model of the HHCRSP determine a totalof 27 routes in 12 days (two weeks) of operation.

Day 1

Nurse 6 Nurse 8 Therapist 5From To From To From To NH6 6 NH8 11 TH5 4

6 5 11 15 4 135 8 15 3 13 128 13 3 2 12 TH5

13 NH6 2 NH8

Day 3

Therapist 3From ToTH3 5

5 TH3

Day 4Doctor 3 Nurse 2 Therapist 4 Therapist 6

From To From To From To From ToDH3 10 NH2 10 TH4 8 TH6 16

10 6 10 6 8 9 16 146 15 6 9 9 13 14 15

15 13 9 15 13 12 15 TH613 12 15 1 12 TH412 DH3 1 NH2

Day 5

Doctor 1 Nurse 8From To From ToDH1 11 NH8 11

11 8 11 58 16 5 8

16 7 8 77 DH1 7 13

13 1212 NH8

Day 6

Nurse 2From To NH2 10

10 99 16

16 1414 44 NH2

Day 7Doctor 1 Therapist 1 Therapist 5 Therapist 7

From To From To From To From ToDH1 5 TH1 7 TH5 11 TH7 10

5 3 7 TH1 11 14 10 63 13 14 13 6 5

13 DH1 13 12 5 812 TH5 8 9

9 TH7

Day 8

Doctor 1 Nurse 2From To From ToDH1 10 NH2 7

10 9 7 39 DH1 3 2

2 11 NH2


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Day 9

Doctor 3From ToDH3 11

11 66 15

15 DH3

Day 10Nurse 2 Therapist 6 Therapist 7

From To From To From To NH2 10 TH6 8 TH7 10

10 6 8 7 10 66 9 7 TH6 6 99 15 9 15

15 1 15 TH71 NH2

Day 11Nurse 8 Therapist 6

From To From To NH8 3 TH6 11

3 2 11 52 NH8 5 16

16 1414 44 TH6

Day 12Doctor 1 Doctor 2 Nurse 2 Nurse 8

From To From To From To From ToDH1 5 DH2 4 NH2 16 NH8 11

5 8 4 12 16 14 11 58 16 12 DH2 14 4 5 8

16 14 4 NH2 8 714 15 7 1315 13 13 1213 DH1 12 NH8

ACKNOWLEDGMENT

The authors thank the sponsorship of this project to theMaster in Operations Management and the InternationalSchool of Economics and Management Sciences (EICEA) of

the Universidad de La Sabana.

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