This research focuses on the home health care optimization problem that involves staff routing and scheduling problems. The considered problem is an extension of multiple travelling salesman problem. It consists of finding the shortest path for a set of caregivers visiting a set of patients at their homes in order to perform various tasks during a given horizon. Thus, a mixed-integer linear programming model is proposed to minimize the overall service time performed by all caregivers while respecting the workload balancing constraint. Nevertheless, when the time horizon become large, practical-sized instances become very difficult to solve in a reasonable computational time. Therefore, a new Learning Genetic Algorithm for mTSP (LGA-mTSP) is proposed to solve the problem. LGA-mTSP is composed of a new genetic algorithm for mTSP, combined with a learning approach, called learning curves. Learning refers to that caregivers’ productivity increases as they gain more experience. Learning curves approach is considered as a way to save time and costs. Simulation results show the efficiency of the proposed approach and the impact of learning curve strategy to reduce service times.

During the last decades, many studies have focused on healthcare management [

Home Health Care (HHC) is defined as a set of workers serving the patients at their homes by providing the required services (e.g., nursing, cleaning, drug delivery, etc.). The major challenge for HHC companies is to improve the quality of services offered to patients, in particular the quality of care, the competence of the medical staff, the management of intervention and punctuality, etc. The main objectives of HHC Companies are to minimize costs and maximize patient satisfaction. This alternative is complex due to the integration of the patient’s home in the supply chain of care and the uncertainties related to the care process (i.e., demand, durations of care and travel time). HHC decision-makers are making complex decisions that include optimization of staff travel and assignment of patients to workers by considering several constraints, such as, workload balancing, arriving on time and patient preferences, etc. The constraints may vary depending on the type of service to be performed or the pathology to be treated. These optimization problems are known in the literature as routing and scheduling problems. Transportation cost is the biggest operating cost for the HHC companies. Thus, it is very important to optimize traveling routes for the medical staff. The main resource that influences operating costs is workforce. Therefore, a good staff management strategy should be implemented (Ben Houria et al., 2016). In HHC, a new caregiver is usually going through a learning phase. The learning process can vary from a couple of weeks to several months. The beginner caregiver will eventually obtain all the skills necessary for independent performance. This approach is named in literature, “Learning curves” (LC) or Experience Curves. Human learning is considered as one of the most important factors that affect workforce capacity. The impacts of employee learning on the scheduling problem has becoming very recently a research topic. Several researches have applied the learning curve strategy to the industrial field and more precisely to scheduling problems [

This work is organized as follows: Section 2 summarizes the recent works related to HHC routing and scheduling problems and describes briefly the learning curves theory. The elaborated mathematical model and the proposed solution approach are presented in Section 3 and Section 4 respectively. Experimental results are presented and discussed in Section 5, conclusion is given in Section 6.

During this decade, the healthcare sector is becoming one of the largest economic sectors in Europe and North America. Due to the problem of increasing global aging, the HHC industry is developing very rapidly [^{1}

Cplex Solver:

^{2}

GAMS Solver:

^{3}

GUROBI Solver:

The definition of learning in operation and production management is the improvement in performance when an individual is involved in a repetitive task [

This section addresses a HHCRSP and presents the proposed mathematical formulation. The considered problem is an extension of mTSP that can be defined as follows. Given a directed graph G = (P, A) with a set of nodes P = {0, 1, …, n} and a set of arcs A = {(i, j) |i, j ∈ P, i

Subject to

The objective function (1) aims to minimize the overall service times, including travel times, operating times and lunch break for all caregivers during a given horizon. Constraints (2) and (3) guarantee that all caregivers daily start and finish their services at the HHC company. Constraint (4) ensures that each patient is assigned to only one caregiver during the horizon and each caregiver crosses a path exactly once over the horizon. Constraint (5) forces each caregiver to visit a fixed number of sub-paths. Constraint (6) guarantees that each caregiver does not exceed its maximum daily workload. Constraint (7) removes the sub-tours. Constraint (8) indicates that the decision variable

In this sub-section, a LC model is integrated to the proposed MIP model in order to calculate the appropriate service time for each caregiver and prove the influence of experience on caregiver’s productivity. It is inspired by the Wright’s Log Linear [

where

The learning percentage is represented by

Resuming the above MIP model, we add the following equations in order to calculate service times

Constraints (14) and (15) calculate the service time for each caregiver. Constraint (16) indicates that the learning rate should be between −1 and 0.

This section presents the solution approach for the HHCRSP, called Learning Genetic Algorithm for mTSP (LGA-mTSP). The proposed solution approach is carried out in two stages. It is composed of a learning approach combined with a genetic algorithm for mTSP.

The learning process, proposed in this study, allows to generate learning curves for a given set of caregivers. Let’s take the example a set of beginner caregivers (c: 1,…, m) who start a new task T and have to repeat it n times. The service time records, performed by these caregivers, are saved in the database. Each caregiver has a fixed number of tasks to perform, generated by the solver. The learning process consists of extracting the learning rates

The extraction model is based on the following equation:

Predictive model:

The predictive model is based on

The optimization solver:

The solver consists of equitably assigning patients to caregivers, founding the shortest path for all caregivers and providing the optimal solution that is the minimized overall service times.

The execution-time increases with the problem instance size, and often only small or medium-sized instances can be solved with exact solver. Therefore, a metaheuristic method is proposed to solve the problem in order to find the best solution in accepted simulation time. Genetic algorithms is a good example of stochastic algorithms. The considered problem is an extension of mTSP that receive a big attention in the last years. mTSP is a NP-Hard that various approaches have been proposed to solve it, especially the Genetic Algorithms (GAs). They are successfully implemented to solve TSP and mTSP such as in [

The chromosome is a numeric vector. Each number inside the chromosome is called a gene. In the case of mTSP, each gene represents a city. There are many ways to represent the chromosome for the mTSP. In this work, chromosomes are represented by the multi-chromosome technique [

After the random generation of the initial population, each individual is evaluated according to the fitness function. In this work, the fitness value is the overall service times performed by all workers. The considered problem is a minimization problem; thus, the smallest value is the best. The fitness function is calculated as following:

The genetic operators consist of evolving the chromosomes. They influence the search ability and convergence speed. There are two basic types of operators, which are the crossover and the mutation operators [

Crossover operator: generates new solutions (offspring) by exchanging genes of two previous solutions (parents). The crossover rate is the probability that crossover reproduction will be performed. Various crossover methods are available, including single point and multi-point crossover, etc. In this work, a three-point crossover method is adopted. This method refers to using 3 randomly points in order to determine the order and the distance between the genes.

Mutation operator: changes one or more gene values in a chromosome from its original state. There are several mutation methods available in the literature. A two-stage mutation operator is used in this paper. This method combines two mutation operators, which are the random swap and the reverse swap operators. First, two genes are exchanged basing on two randomly selected points (

The iterative process is composed of four phases (evaluation, selection, reproduction and replacement). The best individuals are selected according to their fitness values and memorized in “the mating pool”. These individuals are called parents. The mating pool is a concept used in evolutionary computation that consists of expecting to get a better-quality offspring (children) than its parents. The mating pool and the population have the same size. The original solution is replaced by a new solution if its rank is worse than the rank of the new solution. The stop criteria is satisfied when the maximum number of generations and the maximum execution time are reached or no improvement is made compared to previous generations.

The proposed LGA-mTSP is implemented in Java language with the Integrated Development Environment Eclipse IDE^{4}

Eclipse IDE :

^{5}

Instance name | Horizon | Total # patients | # assigned |
Initial operating |
Learning |
Learning |
Max |
Break lunch |
---|---|---|---|---|---|---|---|---|

Eil51-m3 | One working day | 50 | 16/17 | 20 | 90% | −0.152 | 480 | 60 |

97% | −0.043 | |||||||

85% | −0.234 | |||||||

Berlin52-m3 | 51 | 17 | 25 | 80% | −0.321 | 540 | 30 | |

93% | −0.104 | |||||||

95% | −0.074 | |||||||

Eil76-m3 | 75 | 25 | 25 | 80% | −0.321 | 660 | 10 | |

93% | −0.104 | |||||||

95% | −0.074 | |||||||

Rat99-m3 | 98 | 32/33 | 20 | 94% | −0.089 | 720 | 10 | |

83% | −0.268 | |||||||

87% | −0.200 | |||||||

Modified-Eil51-m3 | 1 week: |
250 | 83/84 | 20 | 90% | −0.152 | 480 | 60 |

97% | −0.043 | |||||||

85% | −0.234 | |||||||

Modified-Berlin52-m3 | 3 months: 60 days | 3060 | 1020 | 25 | 80% | −0.321 | 540 | 30 |

93% | −0.104 | |||||||

95% | −0.074 | |||||||

Modified-Eil76-m3 | 2 weeks: |
750 | 250 | 25 | 80% | −0.321 | 660 | 10 |

93% | −0.104 | |||||||

95% | −0.074 | |||||||

Modified-Rat99-m3 | 2 months: |
3920 | 1280/1320 | 20 | 94% | −0.089 | 720 | 10 |

83% | −0.268 | |||||||

87% | −0.200 |

The benchmark is generated from TSPLIB^{6}

^{7}

^{8}

^{9}

^{10}

Instance name | Learning interval | Learning state | # patients assigned to each caregiver | Job Volume |
---|---|---|---|---|

Eil51-m3 | [10, 20] | [10, 13]: Fast |
16/17 | Short |

Berlin52-m3 | [12, 25] | [12, 15]: Fast |
17 | Short |

Eil76-m3 | [12, 25] | [12, 15]: Fast |
25 | Short |

Ratt99-m3 | [10, 20] | [10, 13]: Fast |
32/33 | Short |

Modified-Eil51-m3 | [10, 20] | [10, 13]: Fast |
83/84 | Medium |

Modified-Berlin52-m3 | [12, 25] | [12, 15]: Fast |
1020 | Long |

Modified-Eil76-m3 | [12, 25] | [12, 15]: Fast |
250 | Long |

Modified-Ratt99-m3 | [10, 20] | [10, 13]: Fast |
1280/1320 | Long |

The proposed GA-mTSP is implemented with fixed parameters, presented in

Parameter | Value |
---|---|

Population size | 100 |

Number of generations | 100 |

Crossover rate | 80% |

Mutation rate | 30% |

Selection | tournament selection |

Elitism rate | 30% |

Stop criteria | Max number of iterations/generations |

This section describes computational experiments carried out to investigate the performance of the proposed solution approach LGA-mTSP.

The results of one working day instances are shown in

Instance name | Caregiver | Route | Travel time | # repeated tasks | Operating times | Service time | Overall travel |
Overall operating times | Overall service |
Last task | Learning GAP | Learning state |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Eil51-m3 | 1 | 1–32–11–38–5–49–10–39–33–45–15–44–37–17–47–12–46–51–1 | 137 | 17 | 230 | 427 | 17 | 15 | Slow | |||

2 | 1–22–2–16–50–9–30–34–21–29–20–35–36–3–28–31–26–8–1 | 149 | 17 | 183 | 392 | 13 | 35 | Fast | ||||

3 | 1–48–23–7–43–24–14–25–13–41–40–19–42–4–18–6–27–1 | 178 | 16 | 162 | 400 | 10 | 50 | Fast | ||||

Berlin52-m3 | 1 | 1–22–18–3–17–21–42–7–2–30–29–16–46–44–50–20–23–31–1 | 291.92 | 17 | 226 | 547.92 | 12 | 50 | Fast | |||

2 | 1–49–32–45–19–41–8–9–10–33–43–4–6–5–15–40–39–36–1 | 230.49 | 17 | 296 | 556.49 | 19 | 24 | Medium | ||||

3 | 1–34–37–48–25–28–27–26–47–14–13–52–11–12–51–24–38–35–1 | 379.48 | 17 | 320 | 729.48 | 20 | 20 | Medium | ||||

Eil76-m3 | 1 | 1–63–16–3–44–32–49–50–25–9–39–72–58–38–65–10–31–55–18–24–56–23–41–64–42–43–1 | 194.52 | 25 | 214 | 418.52 | 12 | 50 | Fast | |||

2 | 1–33–51–40–12–26–67–7–35–53–11–66–59–14–54–19–8–46–34–4–76–75–17–68–6–73–1 | 219.28 | 25 | 284 | 513.28 | 18 | 28 | Medium | ||||

3 | 1–62–2–30–45–52–27–13–57–15–5–29–48–47–36–37–20–70–60–71–69–21–74–28–61–22–1 | 268.33 | 25 | 305 | 583.33 | 20 | 20 | Medium | ||||

Ratt99-m3 | 1 | 1–10–20–29–47–74–65–56–38–30–39–48–57–66–75–84–76–85–86–95–94–93–83–92–91–82–73–64–55–46–37–28–19–1 | 791.38 | 32 | 269 | 1070.3 | 15 | 25 | Medium | |||

2 | 1–2–12–13–22–31–49–59–50–41–32–23–24–33–42–51–60–69–78–87–88–89–90–99–98–97–96–77–68–67–58–40–21–11–1 | 681.69 | 33 | 210 | 901.69 | 10 | 50 | Fast | ||||

3 | 1–3–14–15–43–61–70–80–79–62–52–34–25–16–26–53–81–72–63–54–45–44–35–17–36–27–18–9–8–7–6–5–4–1 | 792.18 | 33 | 219 | 1021.1 | 10 | 50 | Fast |

Route assignment is based on both: the shortest path routing strategy and the workload balancing between caregivers which is defined by the number of patients assigned to caregivers.

The instances of multiple working days are inspired from those of one working day. The travelled path of each instance is repeated during a given horizon. For example, the horizon of “Modified Eil51-m3” corresponds to five days. So, the travelled path of “Eil51-m3” is repeated five times.

Instance |
Caregiver | Travel time | # repeated tasks | Operating times | Service times | Overall travel times | Overall service times | Last task | Learning GAP | Learning state |
---|---|---|---|---|---|---|---|---|---|---|

Modified Eil51-m3 | 1 | 745 | 84 | 1001 | 2046 | 10 | 50 | Fast | ||

2 | 685 | 84 | 1430 | 2415 | 16 | 20 | Medium | |||

3 | 890 | 83 | 879 | 2069 | 10 | 50 | Fast | |||

Modified |
1 | 17515.2 | 1020 | 12280 | 31595.2 | 12 | 50 | Fast | ||

2 | 13829.4 | 1020 | 14030 | 29659.4 | 12 | 50 | Fast | |||

3 | 22768.8 | 1020 | 16703 | 41271.8 | 15 | 40 | Fast | |||

Modified |
1 | 1945.2 | 250 | 3041 | 5086.2 | 12 | 50 | Fast | ||

2 | 2192.8 | 250 | 3872 | 6164.8 | 14 | 44 | Fast | |||

3 | 2683.3 | 250 | 4506 | 7289.3 | 17 | 32 | Medium | |||

Modified |
1 | 31655.2 | 1280 | 12284 | 44339.2 | 11 | 45 | Fast | ||

2 | 27267.6 | 1320 | 10244 | 37911.6 | 10 | 50 | Fast | |||

3 | 31687.7 | 1320 | 10313 | 42400.7 | 10 | 50 | Fast |

According to

Instance |
GA-mTSP | ACS [ |
PCI [ |
||||||
---|---|---|---|---|---|---|---|---|---|

Lower bound | Upper bound | Optimal cost | Lower bound | Upper bound | Optimal cost | Lower bound | Upper bound | Optimal cost | |

Eil 51-m3 | 16 | 17 | 464 | 15 | 20 | 464.11 | 15 | 20 | 492 |

Berlin 52-m3 | 17 | 17 | 9019.09 | 10 | 27 | 8106.85 | 10 | 27 | 8407 |

Eil76-m3 | 25 | 25 | 682.14 | 21 | 30 | 579.30 | 21 | 30 | 612 |

Ratt99-m3 | 32 | 33 | 2245.26 | 27 | 36 | 1519.49 | 27 | 36 | 1647 |

In this sub-section, a comparative study is provided in order to evaluate the effectiveness of GA-mTSP as well as the robustness of the elaborated MIP model. First, a comparative table is given by

Both works [

In order to evaluate the influence of caregiver’s experience (human learning) on caregiver’s productivity (service time). a performance study is given below.

When the learning rate tends to 50%. the caregiver gets more experience and become faster. Therefore. operating time decreases considerably during a given period. This phenomena is called “learning phase”. When the operating time becomes stable. this phase is called “steady state phase” i.e., “no learning” and so. the caregiver become expert. Regarding to

Instance name | One working day | Multiple working days | ||||||
---|---|---|---|---|---|---|---|---|

Overall |
GAP |
CPU |
Overall |
GAP |
CPU |
|||

With learning | Without learning | With learning | Without learning | |||||

Eil-51-m3 | 575 | 1000 | 42.5 | 2749 | 3310 | 5000 | 33.8 | 5316 |

Berlin52-m3 | 842 | 1275 | 33 .96 | 3086 | 43013 | 76500 | 43.77 | 12248 |

Eil76-m3 | 803 | 1875 | 57.17 | 3012 | 11419 | 18750 | 39.09 | 7841 |

Ratt99-m3 | 698 | 1960 | 64.38 | 2902 | 32841 | 78400 | 58.11 | 5040 |

Regarding to

This paper presents a new mixed-integer programming (MIP) model for a HHCRSP problem. The proposed MIP model aims to minimize the overall service time for a set of caregivers, visiting a set of patients in their homes, in order to perform medical tasks, during a given time horizon. This model was used to solve small size instances. Nevertheless, due to the NP hardness of this class of problems and in order to solve large size instances, a Learning Genetic Algorithm approach (LGA-mTSP) is proposed. This approach combines a genetic algorithm, designed for mTSP, with a learning approach, called learning curves approach. The obtained results prove the effectiveness of the LGA-mTSP in term of run-time and reliability of learning curves approach to minimize service times. Moreover, the proposed solution approach proves its ability to reduce service costs and balance the workload between caregivers using their experiences. In future works, it would be interesting to consider uncertainty in task time duration as in [