1. Introduction
Home Healthcare (HHC) aims at supporting people with different degrees of dependencies to remain at home instead of receiving long-term care at a traditional medical institution. It encompasses a large variety of care provided at hospitals and gives rise to challenging planning and coordination problems. HHC is set to become an increasingly important issue in the years ahead with the longer human life expectancy. Research on these emerging problems has the intent to establish a fine coordination of human and material resources to provide an optimized planning that maximizes the quality of home healthcare while controlling logistics costs. Therefore, effective organization of HHC structures requires the use of optimization methods and decision support tools.
The aim in our study is to determine the routes to be performed by a fleet of vehicles (associated with a group of caregivers) available to serve a set of geographically-dispersed customers (corresponding to the patients), who have preferences associated with caregivers, and so that the activity is planned in the most effective way. The specificities rely on the fact that a home healthcare service must often be performed in a given time window and may require the intervention of several caregivers simultaneously. In addition, sometimes a patient needs several types of care linked by precedence constraints. These timing constraints impose the coordination of several caregivers. This problem is hence modeled as a variant of the vehicle routing problem called
VRPTW-SP for the Vehicle Routing Problem with Time Windows, Synchronization, and Precedence constraints. In a previous work (Ait Haddadene et al. [
1]), the minimization of the traveling cost and the sum of non-preferences related to customers were considered as a single-objective function. The obtained results showed the conflicting nature of these criteria, which is why, we are particularly interested in solving the bi-objective version of
VRPTW-SP.
The problem at hand is an extension of the Vehicle Routing Problem (VRP), which is one of the most-studied combinatorial optimization problems. First introduced by Dantzig and Ramser [
2], the aim of this NP-hard problem is to determine a set of minimal cost routes in which a set of customers is to be served by a fleet of vehicles based at a depot node. A detailed survey can be found in Toth and Vigo [
3]. Currently, one of the most effective metaheuristics for the VRP is the hybrid genetic algorithm introduced by Vidal [
4], which was latter generalized to solve more than 26 VRP variants.
Among the different classes of VRP, the VRP with Time Windows (VRPTW) stands as the basis variant of our studied problem. Heuristic and metaheuristic approaches have been widely used to solve the VRPTW such as those proposed by Kindervater and Savelsbergh [
5], Bräysy and Gendreau [
6], Nagata [
7], and Bräysy et al. [
8]. In the field of healthcare, most problems are modeled as a VRPTW. In fact, the origin of vehicle routing problems in HHC systems may be linked to the door-to door-transportation of elderly or disabled persons (Dial A Ride Problems (DARP)) proposed by Bodin and Sexton [
9]. A few years later, this variant was extended to the Home Healthcare (HHC) problem by Cheng and Rich [
10], where caregivers are assigned to patient homes to provide specific care. A mixed integer mathematical model was proposed, and a heuristic approach was developed. At this stage, the numerical results were given for only four caregivers and 10 patients.
Since then, many other studies were carried out, and they are classified in this paper according to: (1) optimization criterion, (2) constraints, and (3) solution approaches. For a more exhaustive review, see for instance [
11,
12].
Table 1 provides a classification of the main types of tackled objectives. Note that each category involves a broad class of possible objective functions.
Table 2 summaries the criteria considered in the reviewed papers, these last being listed in Column 1. Columns 2–9 correspond to each of the objectives enumerated in
Table 1. When a paper tackles one of the objectives, a symbol (×) is shown in the corresponding cell.
As shown in the last table, the objective of minimizing traveling costs, waiting costs, or patient preferences is undoubtedly the most widespread in the literature. However, when multiple objective functions are handled, they are often considered within a single criterion as a weighted sum.
Instead of being part of the objective function, some characteristics of a problem may sometimes be considered as constraints. In our study for instance, time is handled through time windows associated with patient care, as well as through interdependencies between the routes performed by caregivers. This interdependence is defined as operation synchronization in the classification suggested by Drexl [
58]. More specifically, in the context of home healthcare, Labadie et al. [
32] identified two kinds of synchronization:
Simultaneous synchronization: A patient may request several caregivers at the same time. This is for example the case when a bath is requested by the patients, but more than one caregiver is needed to provide this service (because of a heavy load for example). This is called simultaneous synchronization, and it means that staff should start simultaneously (within a tight time lag) performing a given number of tasks or a specific type of care.
Precedence synchronization: A patient can also request more ordered cares. This is the case when preparing a patient is necessary before care or another service (such as to be transported to a hospital).
Even if synchronization is common in real-life problems, it has not received a large amount of attention in comparison with other constraints. In fact, researches have been mainly focused on ensuring high service quality through constraints such as time windows and service types.
Table 3 presents the most common families of constraints taken into account in the HHC context, while
Table 4 summarizes the constraints handled in the reviewed papers.
Table 4 shows that time windows and qualification are considered in the majority of the reviewed papers. However, some characteristics such as simultaneous synchronization, precedence synchronization, and periodicity have attracted less interest in the literature. To the best of our knowledge, only our previous works and that of [
25] combine both types of synchronization constraints (simultaneous and precedence) in the same model.
Table 5 presents solution approaches that were used to solve the problems studied in the reviewed papers. These methods are classified into two general categories: exact approaches (Column 2) and heuristics/metaheuristics (Column 3).
From this literature review on vehicle routing problems related to the HHC context, It can be seen that most of proposed methods are metaheuristics and that most of papers present a mathematical method (mixed integer linear program) by mean of solvers. Furthermore, the majority of studies involving different objectives considered an aggregated function defined as a weighted sum of the considered criteria. Indeed, in all these works, the multi-criteria issue was managed through a linear combination of the different objectives. The use of this kind of approach has well-known drawbacks: (a) finding the appropriate weights is not trivial; (b) multiple runs are needed to provide the decision maker with enough solutions; (c) non-supported solutions cannot be found whatever the selected weights.
Regarding vehicle routing problems, several authors have been interested in algorithms based on Pareto optimality to solve different combinatorial optimization problems. For example, Jozefowiez et al. [
59] proposed a Non-dominated Sorting Genetic algorithm (
NSGAII) for solving the vehicle routing problem with route balancing. In the same context, Lacomme et al. [
60] tested several local search procedures within the
NSGAII framework for the capacitated arc routing problem, and more recently, Velasco et al. [
61] adapted this previous local search for solving the pick-up and delivery problem.
To the best of our knowledge, [
38] and the more recent papers [
51,
52] are the only ones that consider the routing and planning problem in HHC as a multi-objective variant. In the first one, the compromise relations were between minimizing travel costs and maximizing the level of satisfaction of patients in home care structures. A metaheuristic based on multi-directional local search was proposed to build an approximated front. The latter was compared to the optimal Pareto front obtained by using an
$\u03f5$-constraint method. The study of [
51] considered also a multicriteria problem with three objectives: (1) minimizing the total working time of the caregivers, (2) optimizing the quality of service by minimizing the patients time window and the synchronized visits non-satisfaction, (3) and minimizing the maximal working time difference among nurses and auxiliary nurses. To solve the problem, these authors designed a Memetic Algorithm for Multi-objective Optimization (MAMO), which uses an NSGAII structure combined with multi-directional local searches. In [
52], a bi-objective optimization model aimed to make a tradeoff between transportation cost and environmental pollution. The authors proposed various versions of heuristics, metaheuristic, and modified/hybrid methods. Metaheuristics are simulated annealing and swarm algorithms. Their evaluation is made through four assessment metrics and the Pareto-optimal solutions of the employed metaheuristics by the epsilon constraint method.
Overall, it can be seen that most implementations can be found in the category of metaheuristics, and the majority of them use local search procedures. However, the papers [
38,
44,
47,
48,
51] may be the only ones that proposed a population-based metaheuristic for solving a vehicle routing problem in the HHC field. In addition, only [
38,
51] tackled a multicriteria problem through this kind of solution approach.
The last trends seem to focus either on uncertainties or dynamic data ([
62] studied a pick-up and delivery version of HHC with travel time uncertainty; [
63] worked on the caregiver routing problem solved in a dynamic context with uncertainties and random events by a multi-agent system; [
53] proposed a dynamic method to handle sudden incidents in practice; and [
54] considered the dynamic arrival and departure of patients and nursing staff to solve the daily scheduling and routing problem in a long-term perspective). However, none of those dealt with multi-objective functions.
Summarizing the literature review, most works on vehicle routing problems raised in HHC, either considering simultaneous/precedence synchronization or not, have tackled the multi-criteria issue by a weighted sum of the considered criteria. The contribution of this paper is three-fold: (1) The most-used benchmark from Bredström and Rönnqvist [
17] is handled under a bi-objective optimization, allowing another analysis of the trade-off between both the cost and the non-preference values, which tends to bring numerical results close to zero when using a weighted sum. (2) Population-based methods are designed to solve the bi-objective VRP with time windows, preferences, and timing constraints. (3) Several local search strategies are tested within the non-dominated sorting methods proposed here.
The remainder of this article is structured as follows: The problem definition is exposed in
Section 2. The different component of the basic
NSGAII are detailed in
Section 3. Three algorithm designs are deduced from combining these components and are exposed in
Section 4. The results of the proposed methods, tested on instances from the literature, are presented in
Section 5. Finally, conclusion and future research direction are provided in
Section 6.
2. Problem Definition
To model the studied problem, patients were associated with customers and caregivers with vehicles. The problem can be seen as a variant of the vehicle routing problem with a time window and specific synchronization constraints encountered in the home healthcare system. In particular, the problem at hand focuses on cases in which some patients may ask for some services requiring more than one caregiver with different skills to be accomplished, either simultaneously or in a given order.
Before explaining how simultaneous synchronization/precedence constraints and objective function are modeled, let us introduce the following notation:
N is the set of customers.
$L=\{d,f\}$ is the set of initial and final depots denoted as d and f, respectively.
$V=N\cup L$ is the set of nodes.
S is the set of offered services, where ${S}_{i}$ is the subset of services requested by customer $i\in N$.
$[{a}_{i},{b}_{i}]$ denotes the service time window at customer i, which are hard in this study. Indeed, service at customer i must start only in this interval: if a caregiver arrives at i before ${a}_{i}$, a waiting time occurs, and the service after time ${b}_{i}$ is not allowed to be provided.
K is the set of available vehicles, where ${K}_{s}$ is the subset of vehicles performing service $s\in S$.
${C}_{ij}$ is the travel cost from customer $i\in V$ to $j\in V$.
$Pre{f}_{ik}$ is the non-preference value of customer $i\in N$ to the caregiver $k\in K$.
${t}_{ik}$ is a scheduling variable corresponding to the starting time of the service by vehicle k at customer $i\in N$.
${x}_{ijk}$ is equal to one if and only if the vehicle $k\in K$ goes from i to j, zero otherwise.
VRPTW-SP consists of building a set of routes, such that the following conditions are satisfied:
each route starts at the initial depot d and ends at the final depot f,
each vehicle route does not exceed the maximal imposed duration,
each customer time window must be respected,
each customer service request is satisfied,
each service must be accomplished by the qualified caregiver,
timing constraints linking the start time of services at customers asking for more than one service must be considered.
In a preliminary work [
1], the authors extended the constraints proposed by [
32] to deal with both types of synchronization (simultaneous and precedence) while retaining the idea of avoiding node duplication for customers requiring synchronized visits. In addition, several heuristics and local research procedures were proposed to solve the problem as a single-objective variant. Here, we propose to use these methods to solve the bi-objective problem.
Synchronization constraints are expressed for some customers
$i\in N$ by defining
$ga{p}_{isr}$ as the required time between the starting times of services
s and
r requested by customer
i. For a simultaneous pair of services, this gap is obviously set to zero. Then, synchronization/precedence constraints are defined as:
The considered objective function is formulated as follows:
with,
Here,
Z is regarded as a two-dimensional vector
$Z=({Z}_{1},{Z}_{2})$, where both objectives must be optimized simultaneously. For more details on the proposed mathematical model, readers are referred to Ait Haddadene et al. [
1] in which
$Z={Z}_{1}+{Z}_{2}$ was minimized. Note that patient preference for caregivers is quantified by a negative value. Thus, a large negative value means a highly-desired assignment, while a positive value means an undesired visit.
5. Computational Experiments
5.1. Implementation and Instances
Our algorithms were implemented in C language and executed on a 3.20-GHz Intel(R) Core(TM) i5-3470 CPU computer with 8 GB of RAM, running under Ubuntu. A set of HHC instances was designed by Bredström and Rönnqvist [
17]. These instances include most of the particular features of the VRPTW-SP studied here: time windows, synchronization, and preferences. They were specifically extended to fit the additional problem specifications related to the different types of services offered to and requested by customers. The benchmark was grouped into three different categories according to the number of customers and vehicles, which were equal to 18 clients with 4 vehicles in the first group (
$G1$), 45 clients with 10 vehicles in the second group (
$G2$), and 73 clients with 16 vehicles in the third one (
$G3$). In each group, the number of synchronizations ranged from 2-5, and the time windows could be small, medium, or large. In the extended version of instances, the customer demands and caregiver qualifications were randomly generated, but the rest of the information remained the same as in [
17]. Without loss of generality, we only treated the case when at most two services were needed. The extended benchmark can be obtained by a simple request to the authors.
The file names followed an n-m-t-nb format, with $n\in \{18,45,73\}$ clients, $m\in \{4,10,16\}$ vehicles, $t\in \{s,l,m\}$ referring to the width of the customer time windows (small, medium, or large,) and the last index $nb\in \{2,3,4,5\}$ corresponding to the maximum number of clients to be synchronized. These values were inspired from literature instances, where the number of synchronized customers did not exceed 10% of the total number of customers. All randomized algorithms were executed five times on each instance.
5.2. Evaluation Criteria
In this paper, the performance of the different versions of the algorithms was measured through five metrics:
5.3. Test Protocol and Parameters
For the set of instances, calibration with different configurations of parameters was used to see which version produced the best results. The best values that allowed obtaining a good compromise between solution quality and running time according to the performance measures mentioned above are shown in
Table 8.
5.4. Numerical Results
Average results for the 37 test instances are shown for the different versions in
Table 9, respectively for
NSGAII,
MS-NSGAII, and
NSILS. In this table, the minimum (min), maximum (max), and the average values (av.) for the number of solutions in the first front (
$|{F}_{0}|$), the hyper-volume metric (
$HV$), the spacing measure (
$SP$), the percentage of the deviation from the optimal solution, and the
$CPU$ time reported in seconds are given. As can been seen in
Table 9, all versions with the local search procedure were better than versions without it, at a cost of a computing time about three-times higher on average.
In terms of the $|{F}_{0}|$ indicator, globally, NSGAII-1 found more solutions in the non-dominated Pareto front, offering the decision maker a set of solutions with a good compromise between the total travel cost and the customer preferences. Regarding the $SP$ metric, all versions of the NSGAII and NSILS-2 methods were almost equivalent with a good coverage in the objective space by members from the non-dominated fronts.
According to the $HV$ metric, from NSILS, applying $LS1$ was better. However, MS-NSGAII was practically equivalent to NSGAII for both local search-based methods. Furthermore, NSGAII-1 was the worst one according to this performance indicator. It is important to mention that the position of the solutions in the search space did not have a direct impact on the $HV$ metric. Even when the $SP$ metrics and $|{F}_{0}|$ were better, this did not guarantee better values for the $HV$ metric.
According to the optimal deviation from optimal solutions ${Z}_{1}$ and ${Z}_{2}$ indicator, all versions found the optimal solution for ${Z}_{1}$ and/or better with a good solution quality of ${Z}_{2}$. This shows that the methods were more accurate for ${Z}_{1}$ than for ${Z}_{2}$. However, in general, NSILS can find better solutions than genetic methods according to $D{O}_{1}$, which is related to the minimization of the total travel cost. Therefore, it can be deduced that, even when the genetic aspect was not considered, but the non-dominated sorting was retained, the method obtained solutions of good quality, in a reasonable execution time. Once again, this metric allowed us to deduce the impact of the local search procedure in terms of convergence.
As can be expected, the running time ($CPU$) depended on the methods used. Those can be classified as follows: NSILS was more rapid than NSGAII, which was faster than MS-NSGAII. In accordance with the different variations of the proposed methods, NSGAII-1 and MS-NSGAII-1 required less computational time due to the way the local search was integrated. Note also that the mutation procedure was not used here, since preliminary tests showed that the crossover followed by local search led to better performance of the solution; thus, the mutation procedure was replaced by the latter.
5.5. Graphical Examples
Graphical representations are well suited to visualize the behavior of bi-objective optimization methods.
The two figures illustrate the impact of the local search procedure on the same instance. This is by comparing the efficient set computed by the basic version NSGAII-1 and MS-NSGAII-1 (no local search) with the other versions obtained local search strategies for each algorithm.
Figure 6 shows that hybridization with local search was necessary to obtain efficient solutions. Here for example,
NSGAII-2 decreased the best travel cost of
NSGAII-1 by approximately 73% (from ∼320 to ∼265) and the best non-preference by 5% (from ∼−200 to ∼−210). Thus,
NSILS decreased the best travel cost of
NSGAII−2 by approximately 5% (from ∼ 265 to ∼ 250). Moreover, several efficient solutions were obtained, and they were better spread, thus providing the decision maker with a wider choice.
For the same instance,
Figure 7 shows that all local search-based algorithms had a good convergence. However,
NSGILS confirmed the numerical results, obtaining smaller fronts.
Thus, from
Figure 7, it is clear that
MS-NSGAII-1 had not entirely converged and that
MS-NSGAII-2 and
MS-NSGAII-3 were more efficient, hence the interest in including local search procedures.