A hybrid genetic algorithm for a home health care routing problem with time window and fuzzy demand

Highlights

• The uncertain quantity of drugs for patients in home health care service is considered.

• A fuzzy chance constraint programming for vehicle scheduling problem is proposed.

• A hybrid genetic algorithm integrated with stochastic simulation method is designed.

• 3 series of experiments are conducted to test the efficiency of the algorithm.

• Cost changes for difference value of Dispatcher Preference Index are discussed.

Abstract

Home Health Care (HHC) companies are widespread in European countries, and aim to serve patients at home to help them recover from illness and injury in a personal environment. Since transportation costs are among the biggest sources of expenditure in company activities, it is of great significance to optimize this in the Home Health Care industry. From the perspective of optimizing the cost of transportation, this paper studies the Vehicle Routing Scheduling problem as it applies to HHC companies. According to a survey of the HHC companies, during the process of delivering medication drugs, the quantity of drugs required for each patient is non-deterministic when the company makes planned routes. This paper considers uncertain demand as a fuzzy variable, which is closer to a potential real life scenario. A Home Health Care Scheduling Problem with fuzzy demand is considered and a fuzzy chance constraint model is designed. We propose a hybrid genetic algorithm integrated with stochastic simulation methods to solve the proposed model. Firstly, the problem is reduced to the classical vehicle routing problem within a time window. Experimental results for Solomon’s and Homberger’s benchmark instances show that the proposed algorithm performs efficiently. Then other experiments on the fuzzy version model are undertaken with the variable value of the Dispatcher Preference Index (DPI) parameter between [0, 1]. Finally, the influence of DPI on the final objective and the indicators of the problem are discussed using stochastic simulation, and the best value of DPI is obtained. This research will help HHC companies to make appropriate decisions when arranging their vehicle scheduling routes.

Introduction

The World Health Organization has announced that the rate of care-dependent elderly people in Europe will strongly increase in the coming decades (Mankowska, Meisel, & Bierwirth, 2014). In recent years, the health care industry has become one of the largest sectors of the economy in developed countries such as France, Germany, Australia, etc. Due to the issue of global aging and the improvement of treatment conditions, there is no doubt this trend will continue (Golden, Raghavan, & Wasil, 2008).

Home health care (HHC) is a growing medical service in France and other developed countries. This service is provided by the Home Health Care companies, which aim to serve the patients at home to help them recover from illness or injury in a personal environment (Liu, Xie, & Garaix, 2014). Each day a HHC company carries out various logistics activities, including the delivery of drugs or medical instruments from a pharmacy to the patients, and the transportation of biological samples from patients’ home to the laboratory (Liu, Xie, Augusto, & Rodriguez, 2013). A large number of customers is distributed in a given area (such as in a town or a village). Each patient has a service time horizon which is called a time window, and a certain quantity of the drugs is required according to the patient’s degree of recovery. Transportation cost is one of the largest forms of expenditure in company activities, so it is of great significance to optimize the vehicle scheduling, which aims to provide the patients a high quality service and reduce costs at the same time.

According to a survey of the HHC companies (Harris, 2015, Liu, Xie, Augusto, Rodriguez, 2013, Liu, Xie, Garaix, 2014), the main operation process of HHC companies can be summarized as follows:

• In the first stage, the HHC company collects information from patients. This information may include: the patient’s name, sex, age, address, types of diseases, symptoms, anticipated service time, and other necessary information.

• Then the HHC company designs the planned routes by considering the information collected. At the same time as ensuring the provision of a high quality home health care service (for instance, assisting the patients within their allocated time horizon and delivering them the right quantity of drugs), the planned route should also be designed to minimize the transportation cost. After all, HHC company is a for-profit organization.

• In the last stage, the nurses are scheduled to visit the patients. Each nurse is assigned to a planned route, and he/she has to carry out all of the service-related activities for the route. This nurse will drive the vehicle to visit the patients one by one according to the designed route. In case of a lack of drugs, the nurse has to go back to the depot, load more drugs into the vehicle and continue to attend to the remaining patients until all the patients are attended to.

Specifically, the HHC company delivers drugs from depot to patients, picks up biological samples from patients to the laboratory, and observes capacity and time window constraints, and this is firmly related to the Vehicle Routing Problem with Time Window (VRPTW) or Vehicle Routing Scheduling Problem (VRSP) (Solomon, 1987). VRPTW is a challenging and difficult problem in the family of Vehicle Routing Problem (VRP) (Toth & Vigo, 2014). In the classical VRPTW, a vehicle starts from a depot, serves a given number of customers in a correct order, and ends at the depot. Each customer is served once and only once. Each vehicle is used no more than once. The objective of the VRPTW is to service all the customers without violating the vehicle capacity constraints with a minimum travel distance or scheduling time.

In our HHC scheduling, each vehicle starts its journey from a depot (i.e. the pharmacy), and terminates its journey at a laboratory. The specific structure of the Home Health Care supply chain can be seen in Fig. 1. This graph consists of one depot, one laboratory and numerous patients (or customers). Additionally, we assume that the drugs have their own volume, so the vehicle capacity must be taken into consideration. However, the samples are vials of blood, or temperature record sheets, which could be assumed to be negligible and will not be considered with respect to the capacity of the vehicle. Our problem is neither like the classical VRP nor like the Open Vehicle Routing problem (OVRP); all the variations of VRP can be seen in the literature (Golden, Raghavan, Wasil, 2008, Pillac, Gendreau, Guéret, Medaglia, 2013, Toth, Vigo, 2014). In the classical VRP, each vehicle needs to return to the depot again, while in the OVRP, each vehicle does not return to the depot after servicing the last customer on a route, but may end at a different location.

Moreover, in the second stage of the operational process, the demand for drugs for each patient is non-deterministic, because the exact quantity of drugs cannot be known unless the nurse arrives at the patient’s home. Only by diagnosing the patient for a certain time (called the working time) can the nurse know the exact quantity of drugs required. However, the HHC company has to design the planned route before the patients are visited. We assume that the demand of each patient is a fuzzy variable.

In this paper we consider the home health care scheduling problem with fuzzy demand, and a Fuzzy Chance Constraint Programming (FCCP) (Liu & Liu, 2002) is constructed based on the fuzzy credibility theory (Liu & Liu, 2002). Due to the capacity constraint of each vehicle, the nurse cannot load unlimited drugs in the vehicle. Unavoidably, sometimes when the nurse reaches the patient, more drugs may be required than planned. In this situation, the nurse has to drive back to the depot, load the drugs into the vehicle, and then drive back to this patient. Then, the nurse will continue to attend to the remaining patients until all the patients are attended to. In this case, the visiting process results in the failure route. There is no doubt that the failure route results in the creation of additional routes, and this may also cause a delay in the service time or can even result in violation of the time window constraint.

We assume that: (1) the vehicles are homogeneous; (2) each vehicle starts from the depot, then visit the patients, and ends at the lab; (3) each vehicle will not be used more than once; (4) each patient cannot be served before the left boundary of its time window; that is, if the nurse arrives at the patient’s home in advance, he/she has to wait; (5) the demand of each patient i is a triangle fuzzy variable which can be described as $\widetilde{d_i}=(d_{1,i},d_{2,i},d_{3,i})$; (6) the distance between ith patient and jth patient is c_ij; (7) each nurse carries out one route in the process of delivery and pickup; in the event that remaining drugs are not sufficient for the next patient, she must return back to the depot, and fill up the capacity of her vehicle.

In this paper, we initially investigate a variety of literature on the VRPTW, HHC and fuzzy demand, propose mathematical formulations of our problem, and then a hybrid genetic algorithm and stochastic simulation method are integrated to solve the proposed model. The rest of the paper is organized as follows: Section 2 introduces the relevant literature. The fuzzy credibility theory is introduced in Section 3, and then the mathematical formulation is built based on the credibility theory in Section 4. Section 5 proposes a hybrid genetic algorithm for solving the problem. Computational experiments are described in Section 6. Section 7 concludes the paper.