The Uncertainty in the Home Health Care Assignment Problem

Afrae Errarhout

, Saïd Kharraja

and Andréa Matta

Laboratoire d’Analyse des Signaux et des Processus Industriels, Roanne, France

Université Jean Monnet, Saint-Etienne, France

Dipartimento di Meccanica, Politecnico di Milano, Milano, Italy

Keywords: Home Health Care (HHC), Assignment Problem, Continuity Care, Mixed Integer Programming,

Uncertainty.

Abstract: This paper presents an assignment problem in the home health care structures. In this problem, we search to

assign caregivers to patients during a mid-term and long-term planning horizon while considering the

caregivers’ skills and capacity. Moreover, we take into account the randomness of the patients’ demands

due to a change in their profiles or to addition of new patients through the planning horizon. As aim we

balance the caregivers’ workload and secure the continuity of the care. We use the Monte Carlo method in a

deterministic way to represent the randomness of the patients’ demand in the mixed integer programming

model we developed.

1 INTRODUCTION

The home health structures (HHC) were created in

order to reduce the hospitals charges (Jones et

al.1999). Hence, it provides medical, psychological

and social services to the patients at home among

their families. During the last decade, the importance

grow of the HHC structures allowed them to add

more complex and technical cares such as chronic

and palliative cares, home chemotherapy and

rehabilitation (Tarricone and Tsouros, 2008);

(Durand et al., 2010).

Given the standing of the HHC organization,

several studies related to the operations management

and the organization of the care delivery process

were developed. Different works could be found in

the literature, some dealing with main processes and

decisions related to home care structures for a better

understanding of the HHC operations and their

management (Matta et al. 2012) and others related to

the human resources planning such as: districting

problems (Benzarti 2012), routing and scheduling

problems (Hertz and Lahrichi, 2009) and the

assignment problems (Lanzarone et al., 2012);

(Matta et al., 2012); (Yalcindag et al., 2012); (Hertz

and Lahrichi, 2009); (Lanzarone and Matta, 2009);

(Bredström and Rönnqvist, 2008); (Punnakitikashem

et al., 2008).

This paper deals with the assignment problems in

the HHC. The works related to the assignment in the

HHC structures help the decision maker to know the

best way to allocate nurses to patients while securing

the nurses’ workload equilibrium, which insure the

cares’ quality and reduce the costs of the nurses’

tours. Each patient represents a therapeutic project

composed of different cares during his/her treatment.

However, while assigning nurses to patients, we

should take into account the nurses’ capacity. It’s

important to prevent the nurses’ overload which will

cause fatigue and stress which could be followed by

errors. Thus, workload smoothing would secure the

quality of cares and enhance patients’ satisfaction.

The HHC assignment problems present a great

uncertainty (Lanzarone and Matta, 2012; 2009). In

fact, for example, through the planning horizon,

there is a great variability in the patients’ demand; it

may be explained by the admission of new patients

or the change or evolution in patients’ profile after a

short/mid/long period of the treatment beginning.

The patients’ demand represents the risk factor in

these problems as having overloaded nurses and

adding some nurses’ tours or reducing them which

involve additional costs. Hence, we considered that

the demand prevision would help to reduce this risk.

However, although the prevision provides an

overview of the future situation, it doesn’t annihilate

the uncertainty (Kouvelis and Yu, 1997) in the

problem.

453

Errarhout A., Kharraja S. and Matta A..

The Uncertainty in the Home Health Care Assignment Problem .

DOI: 10.5220/0004905504530459

In Proceedings of the 3rd International Conference on Operations Research and Enterprise Systems (ICORES-2014), pages 453-459

ISBN: 978-989-758-017-8

 2014 SCITEPRESS (Science and Technology Publications, Lda.)

Therefore in this paper, we present an

assignment problem. We assign nurses to patient

taking into account the variability of the patient

demand due to their profile’s change or due to new

patients’ admission through the planning horizon

which ensure the care’s continuity. The objective is

to “master the overload risk” by balancing the

nurses’ workload in order to reduce the excessive

assignment. We assume that the nurses can operate

in the whole territory but we limit the number of the

districts where they can be assigned and so we

reduce their travel workload. The nurses must have

the skills required to the care they are assigned to

and there’re some care that require the presence of

two nurses, called synchronized care (Bredström &

Rönnqvist 2008).

The article is organized as follow. We introduce

in section 2 a literature review about the assignment

problems in the HHC. In section 3, we define the

problem and present the mathematical model. Then,

Section 4 reports results from computational

experiment and Section 6 concludes the paper.

2 LITERATURE REVIEW

Different fields of studies deal with the assignment

problems, such as, the production systems (Bilgin

and Azizoglu, 2009), the telecommunication

networks (Dell’Amico et al., 2001), the resources

planning (Mkaouar et al., 2012), the health care

(Volgenant, 2004), etc. The assignment problem was

introduced, for the first time, in the fifties by Votaw

and Orden (1952). This problem searches to assign

one task per agent. Then in 1975, Ross and Soland

(1977) introduced the generalized assignment

problems (GAP) which allocate many tasks to set of

agents while respecting their capacity occurred.

From the GAP many other problems emerged

depending on the situation to model. The complexity

of the GAP is NP-hard (Diaz and Fernandez 2001);

(Yagiura et al., 2006); (Woodcock and Wilson,

2010) which conducted to develop many resolution

methods exact and based-heuristics approaches.

Several methods and heuristic algorithms have been

presented in the literature to solve the GAP as the

genetic algorithm (Liu et al., 2012), the Tabu search

(Diaz and Fernandez, 2001), simulated annealing

(Righini 1995), ant colony, local search (Bischoff

and Dächert, 2009) and ejection chains (Yagiura et

al., 2006). More recently, several researchers

develop hybrid heuristics (Woodcock and Wilson,

2010). These are to combine different heuristics or

combine elements of exact methods with heuristics.

In the HHC context, as mentioned previously,

there’re different works related to the human

resources planning. The districting problems

consider the territory’s repartition into districts in

order to reduce the nurses’ workload and travel

workload; Benzarti (2012) developed two

mathematical models of the districting problem. The

author considers the compactness, the care workload

balance and different patient profiles. The visits’

scheduling problem search to reduce the nurses’

travel during their visits; Ben Bachouch et al.,

(2008) developed mixed linear programming model

of vehicle routing problem with time windows to

minimize the total distance travelled by the nurses.

The assignment problem (see table 1) seeks to

allocate nurses to patients while considering their

skills and workload balance and ensure the

continuity care. Lanzarone et al., (2012) proposed

different mathematical programming models with

the aim to balance the workload of the operators

within specific categories. These models consider

the care’s continuity constraint, operator’s skills and

the districts where the patients and the operators

belong. The patients’ demands are considered either

in deterministic or stochastic way.

Hertz and Lahrichi (2009) developed two mixed

integer programming models. They solved the model

with non-linear constraints and a quadratic objective

function using a Tabu search algorithm and they

used CPLEX to solve the other linear model. By

comparing the two solution methods they confirmed

the effectiveness of the Tabu search approach. They

aim to balance the nurses’ workload within different

categories.

Yalçindag et al., (2012) coupled the assignment

and routing problems in the HHC structures. They

focused on the interaction between assignment and

routing, where the output of the assignment problem

is incorporated as an input into the routing problem,

with the assumption of one district.

Lanzarone and Matta (2012) developed a

structural policy to assign a newly admitted patient

while balancing the operators’ workload by

minimizing the cost function that penalizes the

operators’ overtime. They consider that the patients’

demands are either deterministic or stochastic.

Bertels and Fahle (2006) present a combination

of linear programming, constraints programming

and (meta) heuristics for a HHC problem that

consider the staff rostering and vehicle routing

components while minimizing transportation costs

and maximizing satisfaction of the patients and

nurses.

Lanzarone and Matta (2009) present an integer

programming model for workload balancing among

ICORES2014-InternationalConferenceonOperationsResearchandEnterpriseSystems

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Table 1: The main literature on HHC assignment problems.

Articles Objective Constraint / Creteria

Lanzarone et al

(2012)

 Allocating operators to patients

 Balancing the operators’ workload (visit time)



The patients’ demands are:

deterministic or stochastic

 Continuity of care

 Operators’ skills

 The districts where the patients and

the nurses belong

Hertz & Lahrichi

(2009)

 Allocating operators to patients

 Balancing the operators’ workload



Continuity of care

 Visit load (the weight of each visit)

 Travel load (the distance traveled)

 Case load (the number of patients

assigned)

 The operators’ capacity

Yalçindag et al

(2012)

 Analysing the interaction between assignment problem and

routing problem

 Balancing the operators’ workload (visit & travel

time)Minimizing the total distance travelled



The districts where the patients and

the nurses belong

 Continuity of care

 Operators’ skills

Bertels & Fahle

(2006)

 Designing rosters with staff rostering and vehicle routing

components

 Minimizing transportation costs

 Maximizing the patients and the nurses’ preferences

 Time windows

 The qualification for a job

 The nurses and patients’ preferences

Table 2: Literature about the risk & uncertainty in health care problems through time periods.

Articles

Objective Constraint / Creteria

Benzarti (2012)

 Partitioning of the area where the HCC structure

operates into districts



The workload balance

 The compactness,

 The compatibility and indivisibility of

basic units

 The continuity of care

Lanzarone & matta (2009)

 Evaluating the Expected Value of Perfect

Information (EVPI)

 Allocating operators to patients in the HHC

 Balancing the operators’ workload (visit time)



Patient demand variability

 Stochastic demand

 Continuity of care

 Operators’ skills

 The districts where the patients and the

nurses belong

 Sub-periods

Punnakitikashem et al.

(2008)

 Assigning nurses to patients in health care



Shift (time periods)

 Nurses qualification

 Continuity of care

 Direct care

 Indirect care (e.g. updates of the patients’

condition)

 Unanticipated patients

Lanzarone & Matta (2012)

 Assign newly admitted patients

 Balancing the operators’ care workload (visits

time)

 Minimizing the operators’ overtime penalty



Continuity of care

 Operators’ skills

 The districts where the patients and the

nurses belong

 The patients’ demands are: deterministic

or stochastic

The work presented in this

paper

 Assigning nurses to patients in the HHC structures.

 Balancing the nurses’ workload



Continuity of care

 Uncertainty of the patients’ demand

 Districts

 Synchronized cares

 The travel load

 The nurses’ skills

 The nurses’ capacity

 Sub-periods

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455

nurses, while preserving the continuity of care. They

take into consideration the high variability of the

patient demand and they assume that the future

demand to be stochastic.

Punnakitikashem et al., (2008) develop a

stochastic integer programming model to assign

nurses to patient, they proposed the Bender’

decomposition to solve the problem and they

employed a greedy algorithm to solve the recourse

sub-problem.

The uncertainty in the future makes the

decisions’ project, in long term, difficult. Hence, the

best way to handle the uncertainty and make

decision under it, is to accept it, make strong effort

to understand it and structure it, and finally, make it

part of the decision making reasoning (Kouvelis

1997). Table 2 summarizes some works about the

risk and uncertainty in the health care field.

In this paper, we present an assignment problem

where we assign nurses to patients with the objective

is to balance the nurses’ workload in order to reduce

the excessive assignment. The nurses can operate

over the whole territory. We consider the variability

of the patients demand due to their profile’s change

or due to new patients’ admission through the

planning horizon while securing the care’s

continuity. We use the Monte Carlo method in a

deterministic way to represent the randomness of the

patients’ demand in our integer programming model.

3 MODEL DESCRIPTION

The following assignment model objective is to

balance the nurses’ workload while considering their

skills to give each care and their capacity. During

the planning horizon, the patients’ profile may

progress/regress and/or new patients may be

admitted within the HHC structure, which require a

dynamic model taking into consideration those

changes. Therefore the planning horizon is divided

to several sub-periods, weeks in our case. The

patients are characterized by the weekly number of

visits needed during their treatment, while the nurses

are defined by their weekly workload that may

change in dependency on the admission or the leave

of the patients.

The patients’ demands through the planning

horizon (sub-periods) are random, which represents

an uncertainty factor in our work and may conduct

to irrelevant results. On that account, we used a

range of estimated values instead of a single guess in

order to create more accurate model. Hence, we

choose to use the Monte Carlo simulation (Vose

2008) which tells how likely the outcomes are, so,

we can have a better understanding of the

uncertainty and the risk in the model.

Indeed, we use the Monte Carlo simulation

(Vose, 2008) in a determinist way to the mixed

integer model. Hence, we add a number of scenarios

to the model by considering randomness of the

patients’ demand for each week.

The mathematical model is presented as follow:

Decision variables:

 X







1ifthenurseiisassignedtothecarej

ofthenewpatientpinthedistrictk

0otherwise







1ifthenurseiisassignedto

thedistrictk

0otherwise





 W

,



the nurse i workload in the week h of the scenarios

 





the difference between the nurse i and i’

workload in the week h for each scenario f.

Parameters:

 m the number of nurses I = {1,.., m}

 n the number of cares J = {1,…,n}.

 v the total number of the patients during the

planning horizon P = {1,…, v}. This set is the

sum of the groups of patients during each week t.

 z the number of districts Z = {1,…, z}

 t the number of weeks in the planning horizon

T= {1,…, t}

 s the number of scenarios F={1,…,s}

 P





= 1, if the patient p of the district k needs the

care j during the week h for each scenarios f,

otherwise 0.

 S





= 1, if the patient p of the district k needs a

care j that requires one nurse during the week h

for each scenario f.

 S





= 2, if the patient p of the district k needs a

care j that requires two nurses during the week h

for each scenarios f.

 N



, the limited number of district where a nurse

can be assigned.

 ∝

,



, the frequency of the cares of the patient p

of the district k during the week t for the

scenarios f.

 q



= 1, if the nurse i has the skill to give the

ICORES2014-InternationalConferenceonOperationsResearchandEnterpriseSystems

456

care j to patient p of the district k, otherwise 0.

 b



, the capacity that defines the nurse

availability.

 τ ∈ [0,1] which represents the travel time ratio in

the nurses’ availability b



 a



, the time required by the care j of the

patient p of the district k from the nurse i.

 t



, is the average time for the care j to be

provided to patient p

 M, a great constant

The mathematical formulation of the problem is

given below:

Objective functions:

The model allows balancing the workload among the

nurses in the planning horizon by minimizing the

difference between the nurses’ weekly workload.

Minimize



D

































(1)

The constraints’ set:

























j∈J,p∈P,k∈Z,h∈T,

∈



(2)



q



xP





i∈I,j∈J,p∈P,k∈Z,h∈T,

∈F



(3)

,



























∝

,



a



i∈I,h∈T,

∈F



(4)

W

,











1τ



∗b



 i∈I,

∈F





(5)





W

,



W









i∈I,i′∈I,h∈T,

∈F



(6)





W





,



W

,



i∈I,i′∈I,h∈T,

∈F





(7)









N



i∈I

(8)

X









0i∈I,p∈P,k∈Z

(9)



∑∑

















i∈I,k∈Z

(10)

The constraints (2) and (3) assume respectively that

a same care can be given by one or two nurses and

that a nurse should be qualified to give the care.

Constraints (4) define the workload of each nurse

during the week t for each scenario. The constraint

(5) secures the nurses availability. Constraints (6)

and (7) define the difference between the nurses

workload to be minimized in the objective function.

Finally, constraints (8), (9) and (10) limit the

number of districts for the nurses.

4 EXPERIMENTATION AND

COMPUTATIONAL RESULTS

To test our integer programming model, we used the

optimization software ILOG CPLEX 12.5 and the

experiments were implemented on 3.39 GHz, 3GB

RAM computers.

4.1 Data Generation

Our model takes into account the uncertainty of the

patients’ demand. So, we used a real life data from

the health home care benchmarks (Howard, 1997).

We have 75 patients through the horizon planning (8

weeks) and 6 nurses. The HHC proposes 10 cares

that request a time varying from 25 to 45 minutes

and each nurse is qualified to give at least 4 types of

cares. The travel time is about τ=30% of the whole

time capacity of the nurses (each nurse works 8

hours per day and a week =5 days). The patients’

demand is represented by the frequency of the cares

they need, 3 to 8 visits per week. For each patient

the frequency may change from a week to another

over the scenarios.

In order to generate scenarios, we suppose that

the patients’ demand follow a normal distribution

(i.e. we consider the deterministic demand as the

mean and the standard deviation is represented by

the root square of the deterministic demand (Verweij

et al. 2003), see figure 1).

4.2 Computational Results

In order to test the efficiency of our model, we

generated 3 tests of random demands starting with 5

scenarios, 10 then 20 (i.e. each scenario represents a

number of the patients’ demand that was generated

from the normal distribution, see fig. 1) and we

compared their results with the results from the

deterministic demands. For all the tests the program

runs over 3 hours before running out of memory

without finding new solutions. So, we terminated the

program after 3 hours of running to get the latest

solution found. We report in the follow table the

results found for all the tests (see table 3).

TheUncertaintyintheHomeHealthCareAssignmentProblem

457

Table 3: The results of the experiments showing the average workload per scenario and per nurse, and the Minimum and

maximum workload.

Tests Avg. workload

Avg. nurses

workload

Min. workload Max. workload

5 scenarios 68725 11454 9479 12722

10 scenarios 69201 11534 11259 11618

20 scenarios 69004 11501 11286 11687

Deterministic - 11114 11080 11155

Figure 1: The normal distribution of the patients demand.

Figure 2: The nurses’ workload for the second experiment

“10 scenarios.

The results obtained by the experiments are

different from each other and we notice that these

results converge while increasing the numbers of

scenarios. So in order to get better result it’s more

appropriate to use a large number of scenarios.

These scenarios may represent the worst case, the

best case and the most likely case. Hence, we notice

that the workload from the deterministic demand to

the 5, 10 and 20 scenarios increases. Moreover, the

comparison between the results obtained by using

different scenarios and the results of the

deterministic demand shows a great difference,

which concludes that the resulting assignment is far

away from being accurate (see table 1). Furthermore,

the experiments show that the nurses’ workload is

balanced over all the scenarios (see fig. 2).

5 CONCLUSIONS

In this paper, we modeled the assignment problem in

the HHC structures. The main objective is to balance

the nurses’ workload and define the variability in the

problem that’s caused by the randomness of the

patients demand and the changes in the patients

profile every week represent an uncertainty factor.

Hence, in order to represent the variability in our

model we used the Monte Carlo method which

consists of generating a great number of scenarios

that represent the randomness of the patients’

weekly demand. We tested our mixed integer model

on CPLEX with different experiments that use a

number of scenarios and the results shows a big

difference between the test using the deterministic

demand and the ones using a random demand. This

difference illustrates the uncertainty in the

assignment problem and gives more accurate

solutions.

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