**2. The Problem**

The purpose of this work is to provide the company with the caregivers' schedules for the week. In the interest of achieving the best possible schedules, we consider two clearly differentiated objectives: the cost of the schedule and the users' welfare.

The cost of the schedule represents the expenses associated with the caregivers carrying out their routes, and it is composed of two elements:

1. The overtime of the caregivers. This is caused by allowing the caregivers to work more hours during the week than initially agreed, while still adhering to their daily maximum allowed working time, which results in an extra cost for the company.

**Citation:** Méndez-Fernández, I.; Lorenzo-Freire, S.; González-Rueda, A.M. A Bi-Objective Scheduling Problem in a Home Care Business. *Eng. Proc.* **2021**, *7*, 42. https://doi. org/10.3390/engproc2021007042


Academic Editors: Joaquim de Moura, Marco A. González, Javier Pereira and Manuel G. Penedo

Published: 20 October 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

2. The total working time of the caregivers. This is the sum of the daily working time of each caregiver according to the schedules, and by reducing them we are saving the company money while also optimizing the caregivers time.

The users' welfare represents the degree of well being and satisfaction that users present according to the schedule, and combines two elements:


For the schedules to be feasible, the company requires that: each service has to be carried out within its hard time window by one caregiver, the caregivers' daily scheduled time cannot surpass their maximum working hours, the largest break a caregiver has during the day will not be considered as working time.

#### **3. Resolution Methods**

The problem has two conflicting objectives, which means that there is not a single solution that optimizes both of them at the same time. In a multi-objective problem with *p* objectives we say that *x* dominates *y* if *fk*(*x*) ≤ *fk*(*y*) ∀*k* ∈ {1, ..., *p*} and *fk*(*x*) < *fk*(*y*) for at least one *k* ∈ {1, ..., *p*}. To solve our problem we look for the Pareto frontier, which is a set composed by the non-dominated solutions.

We modelled the problem as a Mixed Integer Programming (MIP) one and used it to obtain non-dominated solutions with the AUGMECON2 method [1]. However, because this problem is a complex one, we can only solve small instances with the AUGMECON2 method. Therefore, it is necessary to develop an heuristic algorithm to generate good approximations of the non-dominated set for instances of bigger size.

The algorithm presented in this problem is divided into three steps:


#### **4. Results**

In this section we present the preliminary computational results obtained to check the behaviour of the heuristic algorithm. The instances we solved are the ones presented in [3], which were adapted to the characteristics of our problem.

In Figures 1 and 2 we present the Pareto front obtained by the AUGMECON2 method (blue colour) as well as two approximations of the non-dominated set obtained by different configurations of the algorithm. These configurations depend of the number of iterations used, and are described in Table 1.

**Table 1.** Configurations of the experiments.


**Figure 1.** Approximation of the Pareto frontier—Configuration 1.

**Figure 2.** Approximation of the Pareto frontier—Configuration 2.

We can see that the approximation presented in Figure 2 (black) is better than the one in Figure 1 (green), because it is closer to the Pareto frontier. In fact, Figure 2 shows that our algorithm can provide good approximations of the Pareto frontier.

**Author Contributions:** I.M.-F., S.L.-F. and Á.M.G.-R. studied the problem and designed the algorithm; I.M.-F. implemented the algorithm and performed the experiments. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research/work has been supported by MINECO grant MTM2017-87197-C3-1-P, and by the Xunta de Galicia through the ERDF (Grupos de Referencia Competitiva ED431C-2016-015 and ED431C-2020-14 and Centro de Investigación del Sistema universitario de Galicia ED431G 2019/01).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

