A Discrete and Improved Bat Algorithm for solving a medical goods distribution problem with pharmacological waste collection

Abstract

The work presented in this paper is focused on the resolution of a real-world drugs distribution problem with pharmacological waste collection. With the aim of properly meeting all the real-world restrictions that comprise this complex problem, we have modeled it as a multi-attribute or rich vehicle routing problem (RVRP). The problem has been modeled as a Clustered Vehicle Routing Problem with Pickups and Deliveries, Asymmetric Variable Costs, Forbidden Roads and Cost Constraints. To the best of authors knowledge, this is the first time that such a RVRP problem is tackled in the literature. For this reason, a benchmark composed of 24 datasets, from 60 to 1000 customers, has also been designed. For the developing of this benchmark, we have used real geographical positions located in Bizkaia, Spain. Furthermore, for the proper dealing of the proposed RVRP, we have developed a Discrete and Improved Bat Algorithm (DaIBA). The main feature of this adaptation is the use of the well-known Hamming Distance to calculate the differences between the bats. An effective improvement has been also contemplated for the proposed DaIBA, which consists on the existence of two different neighborhood structures, which are explored depending on the bat's distance regarding the best individual of the swarm. For the experimentation, we have compared the performance of our presented DaIBA with three additional approaches: an evolutionary algorithm, an evolutionary simulated annealing and a firefly algorithm. Additionally, with the intention of obtaining rigorous conclusions, two different statistical tests have been conducted: the Friedman's non-parametric test and the Holm's post-hoc test. Furthermore, an additional experimentation has been performed in terms of convergence. Finally, the obtained outcomes conclude that the proposed DaIBA is a promising technique for addressing the designed problem.

Introduction

Transportation and logistics are important issues for the society these days, both for citizens and the business sector. We are perfectly aware that public transportation is used by almost all the population, and that it directly affects the people quality of life. In addition, business logistics can also be considered as transportation problem, which requires optimization techniques to solve. Therefore, this paper will focus on the logistic problems concerning medical device distribution and pharmacological waste collection.

In the business world, the fast advance of technology has made the logistic increasingly important in this area. Additionally, anyone in the whole world can be well connected. This situation has led transport networks to be very demanding, something that was less important in the past. Nowadays, a competitive logistic network can make the difference between some companies, and can crucially contribute to their success.

This work is focused on the proper modeling and treatment of a real-world logistic problem. Specifically, the real-world situation tackled in this paper is related to the distribution of medical goods. In this case, we center our attention in a SME¹ medical distribution enterprise, with regional influence. This company has an established logistic philosophy, which needs to be followed when they perform the daily distribution planning. All the characteristics that integrate this philosophy are explained in the following section. Finally, despite the object of this study is a company physically placed on Bizkaia (Spain), the main objective of this study is to propose a model which can be applied to every similar company.

Hence, the main objective of this work is to tackle efficiently this Drugs Distribution System with Pharmacological Waste Collection (DDSPWC). For reaching this goal properly, we have modeled the DDSPWC as a Rich Vehicle Routing Problem (RVRP). Currently, this type of complex problems is catching the attention of the scientific community, as can be read in several works, such as [1] or [2]. As we can be found in these surveys, RVRPs are special cases of the conventional Vehicle Routing Problem (VRP) [3]. These special cases are characterized for having multiple variables and constraints, and a complex formulation.

The principal reasons for the importance and popularity of these problems are twofold: the social interest they generate, and their inherent scientific interest. Firstly, RVRPs are usually designed for dealing with a specific real-world situation related to transport or logistics. This is the reason why their efficient resolution entails a profit, either business or social one. Secondly, most of RVRPs have a great computational complexity, and their resolution represents a major challenge for the scientific community.

Specifically, we present in this paper a Clustered Vehicle Routing Problem with Pickups and Deliveries, Asymmetric Variable Costs, Forbidden Roads and Cost Constraints (C-VRP-P∗C) to tackle the proposed DDSPWC. As has been mentioned, RVRPs have caught the attention of the current community. In this sense, [4] and [5] are two examples of recently published RVRPs. The first of these works is related to the capillary transport of goods problem. The research project presented in that work was carried out for an important Spanish distribution company, and its main goal is to manage their resources in urban areas by reducing costs caused by inefficiency and ineffectiveness. The RVRP considered in that study comprises some constraints such as pick up and deliveries, backhauls, site-dependence, time-windows, capacities and openness. Authors proposed two different methods for its resolution: a variable neighborhood search (VSN) and a tabu search (TS). The second of the mentioned works presents also a VNS for the resolution of a dynamic RVRP. In that case, several real constraints have been considered, such as heterogeneous fleet of vehicles, multiple and soft time windows and customers priorities. Furthermore, it is worth mentioning that the software developed in that work has been incorporated into the fleet management system of a company in Spain. An additional example of recently developed RVRP is the one proposed in Ref. [6]. In this paper, the authors present an RVRP to deal with the perishable food management. The RVRP designed in this case is a heterogeneous fleet site-dependent VRP with multiple time windows.

Furthermore, in 2016, Mancini presented in Ref. [7] an interesting RVRP with multiple periods, multiple depots and heterogeneous fleet is presented. For tackling this challenging problem, the author developed an adaptive VNS based approach. The experimentation performed in that paper addresses 9 different datasets composed by 50 and 75 customers, and highlight the quality of the presented method. Besides that, in Ref. [8], Belchemeri el al. developed a particle swarm optimization (PSO) algorithm for solving a real-world based RVRP with heterogeneous fleet, time windows and mixed backhauls. In that paper, the results obtained by the presented approach are compared with a basic local search, and an ant colony optimization (ACO). For the experimentation, an ad-hoc modification of the well-known Solomon VRPTW Benchmark is used, with instances composed of 100 nodes. Finally, another interesting example was presented in Ref. [9], in which an electric fleet size and mix VRP was designed, with recharging stations and time windows. For solving this novel problem, a hybrid iterative local search was implemented.

There are several appropriate approaches to deal with such complex optimization problems. Anyway, the most successful techniques to address the resolution of RVRP are heuristics and metaheuristics. In this paper, our attention focuses on the second of these categories: metaheuristics. In line with this, we propose a nature-inspired metaheuristic for the resolution of the designed C-VRP-P∗C.

Lots of metaheuristics have been presented in the literature along the years [10]. The implementation of new and classical methods, and their proper application still forms a hot topic in the scientific community [[11], [12], [13], [14]]. In fact, many novel approaches have been presented in the last decade, such as the Firefly Algorithm, proposed by Yang [15], Charged System Search, presented by Kaveh and Talatahari in 2010 [16], or the Spider Monkey Optimization, proposed by Bansal et al. in 2014 [17]. Another kind of methods that have demonstrated a good performance applied to RVRPs are the memetic algorithms [18]. Some examples of this good performance are [19], in which a RVRP with clustered backhauls and 3D loading constraints is tackled, or [20], where a multiperiod VRP with profit is addressed. Additional works can be found in Refs. [21,22].

This way, we have highlighted some methods which have been already used in the literature for solving RVRP problems: VNS, TS, PSO, ACO, local search methods, and memetic algorithms. Additional approaches can be found in the current literature to properly addressing this kind of problems, such as the genetic algorithm [23], or the simulated annealing [24]. The Large Neighborhood Search has also been recently used in the literature for solving a RVRP [25]. As can be logical, each of these methods have their advantages and disadvantages. In this specific paper, and with the aim of properly addressing the designed RVRP, we propose a nature-inspired metaheuristic based on the Bat Algorithm (BA). The BA was firstly presented by Yang in 2010 [26], and it is based on the echolocation behavior of microbats, which can find their prey and discriminate different kinds of insects even in complete darkness. As has been highlighted in several studies, such as [27] or [28], the BA has been applied to different optimization fields and problems up to now. Furthermore, the fact that many research works focused on BA are being currently published proves that this approach is still interesting for the researchers, in different areas such as the continuous optimization [29], or the thermal engineering [30]. Furthermore, the algorithm itself is also the focus of recent research, such as the works presented in Refs. [31] and [32], in which the parameter adaptation of the algorithm is studied.

Focusing on routing problems, several recently published papers have shown that the BA is a promising technique also in this field. For example, in Ref. [33], which was published in 2015, an adapted variant of this algorithms for solving the well-known Capacitated VRP. The Adapted BA developed in that study allows a large diversity of the population and a balance between global and local search. Furthermore, in 2017, the same authors presented in Ref. [34] an adaptation of the same technique for solving the well-known VRP with Time-Windows. Another interesting work is the one work proposed in Ref. [35] by Zhou et al., in which the Capacitated VRP is faced. In that paper, a hybrid BA with path relinking is described. This approach is constructed based on the framework of the continuous BA, in which the greedy randomized adaptive search procedure and path relinking are effectively integrated. Additionally, with the aim of improving the performance of the technique, the random subsequences and single-point local search are operated with certain probability.

Besides that, the BA has also been applied to the famous Traveling Salesman Problem several times in recent years. In Ref. [36], Osaba et al. presented an improved adaptation of the BA for addressing both symmetric and asymmetric TSP. The results show that the improved version of BA could obtain promising results, in comparison with some reference techniques, such as an evolutionary simulated annealing, a genetic algorithm, a distributed genetic algorithm or an imperialist competitive algorithm. An additional example of this specific application is the one presented by Saji and Riffi in 2106 [37]. In that work, the performance of their discrete version of the BA is compared with three different meta-heuristics: a discrete particle swarm optimization (PSO) [38], a genetic simulated annealing ant colony system with PSO techniques and a discrete cuckoo search [39].

Nevertheless, despite this interest, the BA has never been applied before to any kind of RVRP. This lack of works is one of the motivations behind using the BA for our study. There are additional reasons for the choosing of this technique, such as the growing scientific interest shown by the community in recent years, or the proper balance between exploration and exploitation shown by the technique for solving complex problems. Anyway, and most importantly, the good performance demonstrated since its first proposal, along with its fast execution, its reduced number of parameters, and its easy implementation are the crucial reasons which have motivated the using of BA.

With all this, the main contributions and novelties of the work presented on this paper are twofold. On the one hand, we have used an RVRP for dealing with the proposed DDSPWC. As will be explained later, similar problems have been previously presented in the scientific community, but never using an RVRP as complete as the one proposed in this study. In this sense, the main originality is not only the application of the BA to the medical distribution problem. In fact, the designed problem itself presents also a novelty, being the first time that an RVRP with these features is proposed in the literature.

On the other hand, in order to address the proposed problem, we have developed a discrete and improved version of the classic BA, named DaIBA. As far as we know, this is the first time that a BA is applied to such a complex RVRP. Additionally, the proposed technique is an adaptation of a recently proposed discrete (BA) [36], which has only been applied for both Symmetric and Asymmetric Traveling Salesman Problem. With the intention of proving that the DaIBA is a promising method for solving the raised C-VRP-P∗C, we have compared its results with the ones obtained by an evolutionary algorithm (EA), an evolutionary simulated annealing (ESA) [40], and a Firefly Algorithm (FA) [41].

The structure of this paper is as follows. The following Section 2 is devoted to the problem formulation. In this section, first, we describe the real-world problem that motivated this study. After that, we present the proposed RVRP. In Section 3, the designed DaIBA is deeply described. Furthermore, the experimentation performed is detailed in Section 4, along with the proposed benchmark. Finally, we end this paper with the conclusions of the study, and our planned future work (Section 5).