The patient assignment problem in home health care: using a data-driven method to estimate the travel times of care givers

Abstract

Home health care is one of the recent service systems where human resource planning has a great importance. The assignment of patients to care givers is a relevant issue that the home health care service provider must address before generating the daily routes. The assignment decision is typically made without knowing the visiting sequence, which creates some uncertainties and disparities regarding the effective workload of care givers. However, taking into account travel times in the care giver workload while solving the assignment problem is not straightforward, because travel times can also be affected by clinical conditions of patients and their homes. Providing good travel time estimates that would be used in the assignment decision is the specific topic this paper focuses on. In particular, we propose a data-driven method to estimate the travel times of care givers in the assignment problem when their routes are not available yet. The method, based on the Kernel regression technique, uses the travel times observed from previous periods to estimate the time necessary for visiting a set of patients located in specific geographical locations. The main advantage offered by this technique is the empirical modelling of the travel routes generated by care givers. Numerical results based on realistic problem instances indicate that the proposed estimation method performs better than the average value and k-nearest neighbor search methods and can be successfully used in a two-stage approach that first assigns patients to care givers and then defines their routes.

Appendix: Details for the genetic algorithms

GA is an adaptive search procedure applied to a set of solutions that uses the properties from population genetics (i.e., crossover and mutation) to guide the search. At each iteration, GA discards some solutions (poor ones) and generate new ones based on superior members of the current set of solutions. Evaluation of the solutions (e.g., poor or good) is based on a problem specific function that is named as fitness function. The general representation of the GA is presented in the Algorithm 3 below and problem specific components are explained in the following parts according to the adopted GAs.

1.1 GA for the assignment problem with KR approach

Each solution in GA is represented as a chromosome with two parts as the size of number of patients and care givers ($N+|\mathcal {B}|$). The first part of each chromosome contains the information for the care giver-patient match which is represented by a permutation of integers from 1 to N. On the other hand, the second part is used to show the number of patients that each care giver needs to provide the service for and consists of $|\mathcal {B}|$ non-negative integers. For example, Fig. 4 illustrates the matching between 3 care givers and 9 patients. According to this figure, the first care giver is responsible to visit 4 locations (patients) of the chromosome patient 1, patient 4, patient 5 and patient 9, Then, the second one would visit the next 2 locations (i.e., patient 3 and patient 7) and the the last one would visit the 3 remaining locations (i.e., patient 2, patient 6 and patient 8).

Fig. 4

Chromosome representation for the assignment problem with KR approach

The fitness function is the objective function of the assignment model given in the Eq. (3), which is trying to balance the trade-off between care giver workload balancing and their total travel times. Since the fitness function is calculated for the assignment problem, the order of the patients in the chromosome that are matched with the care givers is not necessarily important. This is because travel time values are estimated with the use of the KR approach (or any other estimation method) by only using the patient IDs.

Selection process involves in choosing the chromosomes that would serve as parents for the next population generation. The tournament system (Miller and Goldberg 1995) is used in which q chromosomes are randomly selected from the population. Then, two chromosomes with the minimum fitness function values are selected among these q individuals to be used as the two parents. This process is performed several times to populate the next generation.

Once the patents are selected a crossover operation is performed with a certain probability ($p_{c}$). First the chromosome is splitted into two parts according to the patient and care giver information. Then, with the part related to the patient information, the order crossover operation is performed to populate two children chromosomes (offsprings). Within this procedure, two cut points are randomly chosen from parent and parts between these cut points are mapped into two offsprings chromosomes. From the second cut point in one parent, the remaining genes are filled in the order that they appear in the other parent. After the order crossover operation, second part of the parent chromosomes is swaped and copied into the offsprings as well (see Fig. 5 for an example of the crossover operation).

Fig. 5

Crossover operation

Once the crossover operation is finalized, the mutation operation is held for the offspring chromosomes with a probability of $p_{m}$. The mutated chromosomes are obtained by randomly choosing two points between 1 and N and simply changing their places from the first part of where the patient information are stored. No mutation operation is performed for the second part of the chromosome.

Since the population matrix is generated according to the constraint where each patient can only be assigned to single care giver, feasibility is always ensured throughout the whole procedure.

It is also important to note that elitist selection process is also considered where the best chromosome in a generation is carried over the next one without any change.

1.2 GA for the TSP

Since TSP only deals with the visiting sequences of a single care giver, in the GA the chromosome represents the visiting sequence of the corresponding care giver. Thus, the chromosome represented in Fig. 4 can also be used for this algorithm by only considering the first part which corresponds to the patient information.

The fitness function is the objective function of the TSP model which is trying to minimize the total travel time of the care giver. It is important to note that, for the fitness calculation, the order of patients must be considered as the they appear in any chromosome.

The population selection, crossover, mutation and elitism operations are the same as the previously described GA (see Appendix GA for the assignment problem with KR approach section).

Since the population matrix is generated according to the constraint where each patient can be visited only once, feasibility is always ensured through out the whole procedure.

1.3 GA for the VRP

The chromosome representation of the VRP is the same as the one provided for the first GA (see Appendix GA for the assignment problem with KR approach section).

The fitness function is the objective function of the VRP given in the Eq. (10), which is trying to balance the trade-off between care giver workload balancing and their total travel times. As in the previous algorithm, the order of the patients must be considered as they are in any chromosome.

The population selection, crossover, mutation, elitism and feasibility operations are the same as the previously described GA (see Appendix GA for the assignment problem with KR approach section).

We provide details about the performance of the implemented GA for the VRP. Remind that the VRP model that is used in this paper is the modified variant of the mTSP problem where instead of only minimizing the total travel time of care givers, we try to balance the trade-off between care giver workload balancing and their total travel times. To be able to analyze the performance of the GA, we only provide results based on total travel time minimization which is the basic model that is present in the literature. To do so, in addition to the h and $\gamma$ terms of the objective function [Eq. (10)] of the corresponding model, we also eliminate the constraints for the workload balancing [Eqs. (15) and (16)].

Table 8 shows the objective function values minimized by the implemented GA and the method used as benchmark (Vidal et al. 2014). All the results are obtained with same group of instances used in Sect. 5. The first group (instance B.1–B.5) corresponds to the first set of small problem instances with 15–25 patients (and a health care center) and 3–8 care givers. These ones are obtained from the benchmark instances that are used in the VRP literature (Gillett 1974; Christofides et al. 1981). The second set of small problem instances (instance A.1–A.5) are as the same size of Group S.1 and generated from real data as described in Sect. 5. The third and last group of instances are also generated from real data and the difference lays on the dimensions. Hence, the third group corresponds to the medium sized instances (instances A.6–A.10) where each instance have 56 patients and 7 care givers (and a health care center). The last group (instances A.11–A.15) corresponds to the larger problem instances that have 150 patients and 15 care givers (and a health care center). Here the solutions are obtained with the same procedure as provided in the numerical result section.

Table 8 Accuracy analysis of the GA for VRP

The solutions provided by the GA for the instances A.6–A.15 are compared with the Unified Hybrid Generic Search method (UHGS) presented in the paper of Vidal et al. (2014). On the other hand, solution of the GA for both group of small sized instance (instances B.1–B.5 and A.1–A.5) are directly compared with the ones that are executed by the ILOG CPLEX solver. It is observed that the maximum average error for all the groups of instances is less than $1.73 \;\%$.