# The Sustainable Home Health Care Process Based on Multi-Criteria Decision-Support

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## Abstract

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## 1. Introduction

## 2. Literature Review

## 3. Optimization System

## 4. Multi-Criteria Approach

#### 4.1. Multi-Objective Problem Formulation

- the locations of all patients, where $L=\{{l}_{1},\dots ,{l}_{nl}\}$ is the set of $nl\in \mathbb{N}$ places for home care visits, such that $\{1,\dots ,nl\}$ is the corresponding index set;
- the trip’s duration between the different locations, where ${c}_{nl\times nl}$ is the time matrix between the $nl$ locations and ${c}_{ij}$ represents the travel time from node i to node j, for $i,j\in L$;
- the distance traveled between the different locations, where ${d}_{nl\times nl}$ is the distance matrix between the $nl$ locations and ${d}_{ij}$ represents the travelled distance from node i to node j, for $i,j\in L$;
- each patient’s treatment plan are known (given by HUB), where $T=\{{t}_{1},\dots ,{t}_{nt}\}$ is the set of $nt\in \mathbb{N}$ treatments that patients are expected to receive, such that $\{1,\dots ,nt\}$ is the corresponding index set;
- the available treatments and the average execution time are known, where Q is the vector with treatments duration, $Q\in {\mathbb{R}}^{nt}$;
- the number of patients assigned to days of HHC visits, where $P=\{{p}_{1},\dots ,{p}_{np}\}$ is the set of $np\in \mathbb{N}$ patients who need to receive home visits, such that $\{1,\dots ,np\}$ is the corresponding index set;
- the number of vehicles available, where $V=\{{v}_{1},\dots ,{v}_{nv}\}$ is the set of $nv\in \mathbb{N}$ vehicles (with nurse allocated) used as the travel resources in the HHC, such that $\{1,\dots ,nv\}$ is the corresponding index set.

- all visits start and end at the HUB (depot);
- patients with various profiles may reside in the HUB area;
- the patient’s profile is known;
- all patients admitted to home care visits must be assigned to a group of nurses or vehicles in order to ensure that all patients assigned to a working day are covered;
- the average travel time for different patients in the same location or residential area (defined by the health unit) is considered;
- the average travel distance for different patients at the same location or residential area (defined by the health unit) is considered;
- only one health professional is allocated to each vehicle.

- ${w}_{ik}$: equal to 1 if patient $i\in P$ can be visited by vehicle (matching between treatment requested and nurse skill) $k\in V$; 0 otherwise;
- $M{Q}_{t}$: maximum time duration of any vehicle route (maximum nurse shift duration);
- $M{Q}_{d}$: maximum distance of any vehicle route (limited distance per day).

- ${x}_{ijk}$—binary variable, 1 if the vehicle $k\in V$ goes from $i\in {L}_{0}$ to visit patient at location $j\in {L}_{0}$; 0 otherwise.
- ${y}_{ijk}$—real variable, used to quantify the time spent on travel from location i to location j by vehicle k.
- ${z}_{ijk}$—real variable, used to quantify the distance traveled from location i to location j by vehicle k.

#### 4.2. Multi-Objective Genetic Algorithm

## 5. Real-Case Scenario

- Treatment 1 (Curative)—Treatments for pressure ulcers, venous ulcers, surgical wounds, traumatic wounds, ligaments, suture removal, burns, assessment and dressing of wound dressings are a few examples. Average time of 30 min.
- Treatment 2 (Surveillance and Rehabilitation)—Evaluation, execution and patient surveillance. Average time of 60 min.
- Treatment 3 (Curative and Surveillance)—Wound care, bandage supervision, frequency, and tension monitoring, patient education regarding complications and pathologies. Average time of 75 min.
- Treatment 4 (Surveillance)—Evaluate patient habits, self-care needs, the risk of falls and the provider’s understanding. Height, stress and heart rate are all monitored. Dietary and medical routines of patients. Average time of 60 min.
- Treatment 5 (General)—Assess, encourage and impart knowledge on mourning. Average time of 60 min.

## 6. Results and Discussion

^{®}, that is a variant of the elitist NSGA-II, This function allows users to customize the random key parameters, algorithm properties and termination criteria. Even if they have a lower fitness value, a controlled elitist GA promotes individuals who can help improve the population diversity [38]. The default values for the population size and the maximum number of generations were set to 50 and 500, respectively. Due to the algorithm’s stochastic behavior, 30 separate runs with a random initial population were carried out, and the maximum number of generations was used as the stopping condition, which was set at 1000. Since the Pareto proportion is 0.35 by default, specific solutions are discovered for each run ($0.35\times \mathrm{population}\phantom{\rule{4.pt}{0ex}}\mathrm{size}$).

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Annual scientific production related to keywords “Multi-objective optimization”, “Health”, and “Scheduling/Planning”.

Reference | Approach | Objectives | Patient Needs | Nurse Skills | Available Vehicles | Matching/ Synchronization |
---|---|---|---|---|---|---|

Braekers et al. [16] | Local Search and Pareto based | 2 | ✓ | ✓ | ||

Bredström and Rönnqvist [17] | Heuristic Solution | 3 | ✓ | ✓ | ||

Decerle et al. [3] | Pareto based | 3 | ✓ | ✓ | ✓ | |

En-nahli et al. [18] | MILP using CPLEX | 4 | ✓ | ✓ | ||

Fathollahi-Fard et al. [6] | Heuristic and Red Deer Algorithm | 3 | ✓ | ✓ | ✓ | |

Hiermann et al. [19] | CP and metaheuristics | 13 | ✓ | ✓ | ||

Mankowska et al. [20] | MILP formulation and heuristic | 3 | ✓ | ✓ | ✓ | |

Nickel et al. [2] | CP and (meta-)heuristics | 4 | ✓ | ✓ | ||

Rasmussen et al. [21] | Branch-and-price algorithm | 3 | ✓ | ✓ | ||

Yang et al. [7] | Bee colony metaheuristic | 3 | ✓ | ✓ | ✓ | |

Yan Li et al. [22] | Fuzzy and Grey wolf optimizer | 2 | ✓ | ✓ | ✓ | |

Our approach | NSGA-II and Pareto based | 3 | ✓ | ✓ | ✓ | ✓ |

Patients | Avg. Time | |
---|---|---|

T.1 | 1, 2, 3, 6, 7, 8, 9, 17, 21, 22 | 30 |

T.2 | 4, 5 | 60 |

T.3 | 13, 14 | 75 |

T.4 | 11, 12, 15, 16, 18, 19, 20 | 60 |

T.5 | 10 | 60 |

Locations | |||||||||
---|---|---|---|---|---|---|---|---|---|

A | B | E | Ml | Mo | P | Rd | Sm | Sd | |

Patients | 11, 12 | 1, 2, 3, 5, 6, 7, 13, 15, 16, 19, 20 | 4 | 22 | 21 | 9 | 8, 10 | 14 | 17, 18 |

A | B | E | Ml | Mo | P | Rd | Sm | Sd | |
---|---|---|---|---|---|---|---|---|---|

A | 15 | 16 | 25 | 21 | 18 | 29 | 18 | 15 | 30 |

B | 16 | 15 | 17 | 18 | 16 | 29 | 15 | 15 | 29 |

E | 25 | 17 | 15 | 33 | 25 | 36 | 23 | 25 | 37 |

Ml | 21 | 18 | 33 | 15 | 23 | 35 | 26 | 21 | 36 |

Mo | 18 | 16 | 25 | 23 | 15 | 15 | 16 | 15 | 18 |

P | 29 | 29 | 36 | 35 | 15 | 15 | 24 | 24 | 31 |

Rd | 18 | 15 | 23 | 26 | 16 | 24 | 15 | 15 | 25 |

Sm | 15 | 15 | 25 | 21 | 15 | 24 | 15 | 15 | 26 |

Sd | 30 | 29 | 37 | 36 | 18 | 31 | 25 | 26 | 15 |

A | B | E | Ml | Mo | P | Rd | Sm | Sd | |
---|---|---|---|---|---|---|---|---|---|

A | 10 | 14 | 24 | 21 | 16 | 28 | 14 | 11 | 33 |

B | 14 | 10 | 16 | 21 | 19 | 31 | 15 | 11 | 36 |

E | 24 | 16 | 10 | 35 | 25 | 36 | 21 | 20 | 42 |

Ml | 21 | 21 | 35 | 10 | 30 | 30 | 28 | 22 | 47 |

Mo | 16 | 19 | 25 | 30 | 10 | 12 | 12 | 12 | 21 |

P | 28 | 31 | 36 | 30 | 12 | 10 | 19 | 24 | 23 |

Rd | 14 | 15 | 21 | 28 | 12 | 19 | 10 | 14 | 25 |

Sm | 11 | 11 | 20 | 22 | 12 | 24 | 14 | 10 | 29 |

Sd | 33 | 36 | 42 | 47 | 21 | 23 | 25 | 29 | 10 |

#Sol. | Tmax | Kmmax | Cmax |
---|---|---|---|

#1 | 309 | 96 | 5 |

#2 | 335 | 92 | 5 |

#3 | 345 | 90 | 5 |

#4 | 366 | 86 | 5 |

#5 | 389 | 124 | 4 |

#6 | 392 | 122 | 4 |

#7 | 399 | 116 | 4 |

#8 | 402 | 110 | 4 |

#9 | 419 | 85 | 5 |

#10 | 420 | 109 | 4 |

#11 | 424 | 105 | 4 |

#12 | 454 | 104 | 4 |

#13 | 455 | 102 | 4 |

#14 | 469 | 100 | 4 |

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**MDPI and ACS Style**

Alves, F.; Costa, L.A.; Rocha, A.M.A.C.; Pereira, A.I.; Leitão, P.
The Sustainable Home Health Care Process Based on Multi-Criteria Decision-Support. *Mathematics* **2023**, *11*, 6.
https://doi.org/10.3390/math11010006

**AMA Style**

Alves F, Costa LA, Rocha AMAC, Pereira AI, Leitão P.
The Sustainable Home Health Care Process Based on Multi-Criteria Decision-Support. *Mathematics*. 2023; 11(1):6.
https://doi.org/10.3390/math11010006

**Chicago/Turabian Style**

Alves, Filipe, Lino A. Costa, Ana Maria A. C. Rocha, Ana I. Pereira, and Paulo Leitão.
2023. "The Sustainable Home Health Care Process Based on Multi-Criteria Decision-Support" *Mathematics* 11, no. 1: 6.
https://doi.org/10.3390/math11010006