Modified swarm intelligence algorithms for the pharmacy duty scheduling problem

https://doi.org/10.1016/j.eswa.2022.117246Get rights and content

Highlights

  • A new variant of PDSP for on-duty pharmacies under distance restriction.

  • Binary and Random-Key versions of PSO, GWO, DA, and HHO are adapted for PDSP.

  • Modified GWO performs the best in computational time and efficiency.

Abstract

Assigning duties to pharmacies to serve the public on weekends, nights, and public holidays is known as the pharmacy duty scheduling problem (PDSP). In this study, a p-median model is proposed with a case-specific constraint that provides the distance restriction between on-duty pharmacies. We adapted Binary and Random-Key versions of the most recent and popular Swarm Intelligence (SI) algorithms, which are Grey Wolf Optimizer (GWO), Particle Swarm Optimization (PSO), Dragonfly Algorithm (DA), and Harris Hawks Optimization (HHO) to solve it in a reasonable amount of time since the PDSP is known to be an NP-Hard problem. We also proposed several enhancements to the algorithms and conducted computational tests on real cases generated instances using Geographic Information System (GIS) tools to compare the performances of the proposed algorithms with the state-of-the-art general-purpose solver. Two real-world datasets, called DS1 and DS2, are built considering the demand points. To validate the proposed algorithms, the CPLEX results, obtained with the exact solution for the DS1 dataset, are used. Compared to the MIP results for 800 meters distance constraint as an instance, maximum relative errors for Binary and Random-Key GWO, PSO, DA, and HHO are 0.07, 0.08, 0.08, and 0.08, respectively. The success of the algorithms is similar to the other distance criteria as well. Moreover, the results for the large DS2 dataset show that the proposed Binary GWO is more effective and offers efficient solutions compared to the other swarm intelligence algorithms and CPLEX solvers in terms of convergence to optimization and solution time.

Introduction

The recent increase in population and global mobility around the world has led to the rapid spread of epidemics, thus increasing the importance of drug access and effective medical interventions. Pharmacies, as an integral part of the health care system, play an important role in public health by providing all kinds of medications and medical equipment. However, in order for this system to meet the desired demand in the most effective way, accurate planning is required at both strategic and operational levels. Strategic plans are those that require high investment and cost to change in the long term, such as the selection of facility location. Operational planning, on the other hand, is related to the process, which is easier to change in the medium or short term. Pharmacy duty scheduling is, in this sense, an operational planning. One thing that should be taken into account while making these plans is the fact that pharmacies are private businesses, although they are a system integrated into the public health system. For this reason, in addition to the goal of providing an effective public service, attention should be paid to equal opportunities for pharmacies in terms of revenue sharing during the schedule. With perfect demand-based planning, patients will have easy and fast access to health care supplies and medicines. In Turkey pharmacy system, a significant number of pharmacies are located, especially near hospitals or health care facilities. Thus, the patients can easily access the medication within working hours after hospital care. However, access to such needs becomes more difficult outside working hours. Therefore, in this study, it is planned that pharmacies will provide uninterrupted and rapid health support services to the public also outside of working hours.

Pharmacies in Turkey are open Monday to Friday from 8 a.m. to 6:30 p.m. However, according to the need for uninterrupted service, pharmacies are planned to be on duty at night and all weekend. To provide this uninterrupted service, the Chamber of Pharmacies plans the pharmacy duty schedule for each region in certain periods. In the existing literature, related problems and their variants are defined as the Pharmacy Duty Scheduling Problem.

The Pharmacy Duty Scheduling Problem was first introduced to the literature by Özpeynirci and Ağlamaz (2016). In their work, they developed the PDSP model based on the simple p-median model. Simple p-median is one of the main location models. A p-median model is used to locate p-facilities among a number of candidate ones to satisfy some problem-specific constraints while minimizing the total demand-weighted distance. The difference of PDSP from p-median is due to the purpose of having a certain number of facilities open among the facilities available in each period, rather than simply locating the p-facility. In other words and more explicitly, PDSP aims to find the perfect combination of a sufficient number of pharmacies scheduled to be on duty to meet the existing demand in each time period. PDSP differs from the location models because it is a multi-period p-median model, meaning it considers which facility will open when. In this study, we introduced a new version of the PDSP based on the distance restriction between pharmacies on duty by the Chamber of Pharmacies. In addition to the new variant of PDSP, the computational efficiency of the proposed solution methods is an important contribution to the literature of this study.

Even if the problem is not large-size, solving p-median and its variants using exact methods are difficult and not computationally efficient. Therefore, heuristics and meta-heuristics have been widely used to solve such problems. However, the computational efficiency and convergence properties of the methods are still under investigation, depending on the size and complexity of the problem.

Recently Swarm intelligence (SI) algorithms have been widely implemented for a wide range of real-world and engineering optimization problems. There are some popular SI methods such as Grey Wolf Optimizer (GWO), Particle Swarm Optimization (PSO), Dragonfly Algorithm (DA), and Harris Hawks Optimization (HHO). In addition, Binary and Random-Key versions of these algorithms are introduced for binary and combinatorial search optimization. SI algorithms need some tools and objective functions to implement for a different specific problem such as scheduling (Han et al., 2019, Han et al., 2020, Meng and Pan, 2021), vehicle routing (Hannan et al., 2018, Ouaarab et al., 2015, Son and Tan, 2021), and facility location problems (Gölcük and Ozsoydan, 2020, Wang and Watada, 2012). In this study, our motivation is that distance-constrained PDSP is practically hard to solve with fair access of the public and equitable allocation of pharmacies. We adapted Binary and Random-Key versions of the most recent and popular SI algorithms, which are Particle Swarm Optimization (PSO), Grey Wolf Optimizer (GWO), Dragonfly algorithm (DA), and Harris Hawks Optimization (HHO) to solve distance-constrained PDSP. The contributions of this study to the literature can be summarized in 3 highlights:

(1) A new variant of PDSP is suggested for distance restriction between on-duty pharmacies.

(2) A repair operator is adapted to modified Binary and Random-Key versions of PSO, GWO, DA, and HHO to avoid infeasible solutions for PDSP.

(3) The Euclidean distance metric and Geographic Information System (GIS) functions are used to generate two datasets. The first dataset was created assuming each site center is a demand node. In contrast, the latter was created by dividing sites into equal distances as demand nodes to distribute the population across the site with equal probability.

The rest of the paper is organized as follows. The next section presents the related literature. Section 3 introduces the proposed mathematical models for PDSP. In Section 4, the proposed algorithms are explained. The results and discussions are presented in Section 5. Finally, Section 6 presents the conclusions and future trends.

Section snippets

Literature review

In this section, facility location problems related to p-median and their variants are described. Also, the popular solution approaches for p-median and PDS problems are mentioned. We emphasized a significant number of studies on solving such models by using meta-heuristic methods.

The traditional p-median is one of the common network-based location models that was first proposed by Hakimi (1964). P-median has many applications on choosing the locations of p-facilities among several candidate

Single-duty PDSP

PDSP has been studied in two different sub-cases as single-duty and multiple-duty. These problems are defined based on the maximum number of duties that are planned to be given to each pharmacy in the planning horizon. Single-duty means to assign only one duty to each pharmacy. In the context of fairness, single-duty PDSP is modeled to determine the on-duty pharmacies that are given only one duty in the entire planning horizon by minimizing the total demand-weighted distance.

Solution algorithms

In this section, we present the exact and SI solution methods in solving the distance constrained single-duty PDSP.

Numerical experiments and comparisons

In this section, computational experiments are presented and the performance of the MB-GWO, MRK-GWO, MB-PSO, MRK-PSO, MB-DA, MRK-DA, MB-HHO, MRK-HHO, and CPLEX methods for the PDSP are evaluated. All solution methods are employed for two datasets, DS1 and DS2. In addition, 10 independent runs are conducted for each dataset in all algorithms. The sensitivity of the model is observed by changing the distance constraint between pharmacies on duty.

The algorithms are written in Matlab. All the

Conclusion

This study presents a new version of PDSP and modified Binary and Random-Key versions of GWO, PSO, DA, and HHO algorithms for solving PDSP, which is an NP-hard problem. In this study, PDSP is extended to a new version as a distance constrained single-duty PDSP. We ensure serving the public out of working hours with the same service level as in regular working hours. Although pharmacies are important for public health, pharmacies have owners and they are private systems. From the view of

CRediT authorship contribution statement

Fatih Kılıç: Conceptualization, Investigation, Validation, Methodology, Software, Data curation, Writing – original draft, Writing – review & editing, Visualization. Nuşin Uncu: Conceptualization, Investigation, Validation, Methodology, Software, Data curation, Writing – original draft, Writing – review & editing, Visualization.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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