A Novel Framework for Solving the Optimal Path Problem in Collaborative Consignment Delivery Systems Using Drones

Abstract

Drone based consignment delivery system is considered to be the future of retail marketing where unmanned aerial vehicles are employed for door delivery of items. This results in a fast, efficient and accurate delivery system which is not affected by geographic conditions and can be used for delivery of items to areas where human reachability is difficult. Also, it can ensure that the customers avoid close contact with strangers, especially in the era of a pandemic like COVID 19. However, the system poses a number of challenges too, the most important of them being the limited battery life of drones. Though newer batteries provide better flight times, the battery discharges faster with increased number of landings and take-offs. Hence the flight of the drone has to be kept optimal for efficient delivery of consignments. In this paper, we propose a delivery system using drones where both the delivery agency and the customers collaborate together to distribute the consignments and model a novel framework for finding the optimal path of delivery drones. We employ unsupervised learning approaches to find the landing locations for drones, based on the concentration of customers. We use a nature inspired optimization algorithm using the fundamental principles of particle physics to get rid of the practical limitations of k-means, a widely used centroid based clustering technique. Besides, we address the case where the customers are spread in such a way that their partitions are not well-separated. Finally, we model, experiment and evaluate our methods on two different datasets.

Introduction

Retail market system, whole across the globe, has seen revolutionary changes in the recent past. Earlier, people used to go to the nearby market and buy the items they needed. If an item was not available in a market, they had to go the place where it was available, causing unnecessary expenditure of money, time and human labour. Then came the era of online markets where items were available through online platforms. Shopping and payment happens through a few mouse clicks and the items reach the consumer through parcel delivery systems.

The final link in the above logistics is door to door delivery. Normally, the product selling company has tie ups with third parties to perform this. Though a speedy delivery increases customer satisfaction, this is a job which requires a considerable human effort. With the advent of new technologies, drones are nowadays in use for door delivery systems and this ensures speedy delivery and a considerable reduction in human labour. Reduction in $C{O}_2$ emissions is an added advantage [26]. The first civilian consignment delivery via drone was conducted by Deutsche Post in 2013 for distributing medical products. With technology being developed fast and the principle time is money getting emphasized in human life, we can doubtlessly say that drone (or similar technology) based consignment delivery is going to be the future of retail markets [61].

Fast delivery, reduction in human labour, ease of delivery etc. are the advantages of drone based delivery systems. In the time of COVID 19, it provides added safety as the customers can ensure that they are not getting in contact with an unknown delivery person, thereby reducing the chances of infection. In the context of social distancing, it is very much important to keep away from possible sources of infection. There are drones with self-disinfecting technology which can ensure that the drone does not carry harmful germs from a landing location to another. However, this type of a delivery system poses some serious challenges also [1]. The present day drones are notorious for consumption of battery. Adding large capacity batteries normally increases the weight of drones, thereby causing inefficiency in flight. New weightless, more efficient batteries may replace the existing ones in the near future. Still it is necessary that the delivery drones make an efficient flight by keeping the total distance travelled to be the minimum possible. Also, it has to make sure that the number of take-offs and landings is optimal, at the same time making sure that the customers won’t feel any difficulty in collecting the consignments.

In this paper, we introduce a method of drone delivery system where, instead of carrying the consignments to each door, a drone starts from a delivery station, lands in certain points and comes back. The customers travel to the nearest drone landing point and collect their consignments. Hence, this is a collaborative consignment delivery system where the delivery agency and the customers cooperate together to ensure efficient delivery. Indeed, there should be proper protocols, guidelines and security measures. In this paper, we concentrate on the algorithmic framework for determining the drone landing points and the path to be taken by the drone.

This turns out to be a double optimization problem where the drones try to minimize the number of landings and take-offs whereas the customers try to have a drone landed to their nearest location. We address this problem by making a set of clusters of customers. A drone landing at a place will serve the customers in a radius if they can reach the landing place without much difficulty, effort and time. Obviously, the number of clusters is to be kept minimal. This brings down the problem to standard clustering in data mining where we group similar items into the same cluster, the measure of similarity being the nearness of customers. Now the challenge reduces to finding the most optimal path connecting these cluster centers, which is addressed by the famous Travelling Salesperson Problem [37].

General Clustering Problem

Clustering is an unsupervised learning approach to divide the data into different groups so that similar members are grouped together. Fundamentally, it is an optimization problem that minimizes the difference between data members of the same cluster and maximizes that between data belonging to different clusters [12]. A few categories of clustering include

• hierarchical models

• centroid based models

• distribution models

• density models

In our case, we have to find a few places where we need to land the drone for consignment delivery. The place where the drone lands in a cluster has to be the cluster center (centroid) for assuring optimal accessibility for the members of the cluster. Hence, we turn our attention to centroid based models for clustering. In our case, the similarity measure is the distance between data points, where the similarity increases as the distance between them reduces. Hence clusters should be chosen so as to minimize the distance between customers in a single cluster and maximize that between customers in different clusters. This leads to the classical k-means clustering algorithm where we divide n observations into k clusters where each observation is assigned to the cluster with the nearest cluster center [41].

K-means is an NP-hard algorithm. However there are heuristics that make the algorithm converge to optima, though global optimum is not guaranteed always. The methods we adopt to overcome this limitation are presented in coming sections.

Travelling Salesperson Problem (TSP)

TSP is an optimization problem to find the shortest route for a sales person to visit all places in a set of cities exactly once and reach back to the original city where the tour was started. The problem assumes that we know the distance between each pair of cities. The problem is NP-hard. However we have algorithms that find solution to TSP using heuristic functions.

The remaining sections of this paper are organized as follows. Section 2 provides a review of various existing techniques to tackle the problem. Section 3 narrates the various preliminaries associated with the work. Section 4 explains about the methods that we use for developing the solution, Section 5 details our experiments and results using these methods. Section 6 provides a detailed discussion of the various aspects of the method and Section 7 explains a few present limitations of the approach.

Related Works

This section explains the existing knowledge about:

1. Conventional vehicle routing problem

2. Drone based delivery system

3. Travelling salesperson problem

4. K-means clustering

Conventional Vehicle Routing Problem

The conventional vehicle routing problem is associated with serving a group of customers using a fleet of vehicles. This includes determining the assignment between customers and vehicles (the assignment problem) and the order in which a vehicle serves its customers (the routing problem). A classical solution to the assignment problem is the technique of simulated annealing [10]. Nagta and Bräysy have introduced a route minimization heuristic, based on ejection pool, powerful insertion and guided local search strategies. Their experiments on Homberger’s benchmark yielded 18 new best known solutions. The above approach was used in [46] to reduce the number of vehicles.

Other approaches to the solution of the problem include nature inspired techniques such as Ant Colony Optimization (ACO) [13] and Particle Swarm Optimization (PSO) [30]. Later, clustering has been incorporated to the solution of the assignment problem, as discussed in [4]. Rizkallah et al. [51] introduce a clustering algorithm based on Clark and Wright saving method [8] to solve the assignment problem in polynomial time. Zhou et al. [64] provide a survey on the electric vehicle routing problem. They conclude that the traditional approaches (exact methods, heuristic algorithms, meta-heuristic algorithms and hybrid heuristic algorithms) are applicable in this case also. In [36], the authors extend the electric vehicle routing problem with drones and use ACO based meta-heuristic methods as the solution. They observe that the energy constraints make drone routing problem unique and techniques are to be devised to optimize the number of landing and take-off for drones.

Clustered Vehicle Routing Problem

A clustered vehicle routing problem is a variant of the conventional vehicle routing problem where the customers are grouped into clusters and a vehicle entering a cluster has to visit all the customers within the cluster before leaving out for the next. Moreover, each customer can be visited exactly once. The objective of the clustered vehicle routing problem is to determine a set of minimum cost vehicle routes, starting and ending at the depot, without violating the vehicle capacity. The problem is NP-hard. It was introduced by Sevaux and S$\ddot{o}$renson as an efficient solution to large-scale vehicle routing problems faced by parcel pick-up and delivery companies [53]. They formulate the shortest pre-Hamiltonian path problem and propose that it can be used as a solution to identify the order of visit of customers in a cluster.

There have been various studies on the clustered vehicle routing problem. Battarra et al. [9] provide two exact solutions of the problem using an integer programming formulation. The first is a Branch & Cut algorithm whereas the second is a Branch & Price & Cut algorithm, with the first outperforming the second. They have made a case study of their approach by testing it on a real world application of urban garbage collection. Pop and Chira [49] use a hybrid approach based on genetic algorithms to solve the clustered vehicle routing problem. They have experimented their methods on standard instances of clustered vehicle routing problems and the solutions that they have proposed are able to handle all the cluster structures in the instances. Marc et al. [42] combine a genetic algorithm applied to the clustered graph with a simulated annealing method to compute the shortest Hamiltonian path between the nodes in a single cluster. Their experiments reveal better quality of solutions in comparison with existing methods. Vidal et al. [55] present two hybrid meta-heuristic algorithms for the problem. The first is an iterated local search method and the second is a hybrid genetic search. They have conducted computational experiments on various benchmark instances and found that their solutions produce high quality results, with the second method performing better than the first. In the research paper by Expósito-Izquierdo et al. [23], the authors propose an approximate two level solution of the clustered vehicle routing problem. In the first level, they use a meta-heuristic approach to determine the routes targeted at visiting the clusters. The second level aims at finding a visiting order for each of the customers in a cluster. This approximate solution is experimentally evaluated to be better in performance over other existing methods.

There have been attempts to solve the soft-clustered vehicle routing problem also, where there is the additional constraint that all the customers in a single cluster are to be served by the same vehicle. Cosma et al. [17] propose a two level based genetic algorithm and the experiments in benchmark instances prove that their method provides high quality solutions within acceptable running time. More recently, Zhou et al. [65] have proposed a solution based on memetic bilevel search which explores the search level at both cluster and customer levels. They have conducted their experiments on 390 public benchmark instances. Their experiments reveal that the solution is more efficient in quality and computational performance compared to similar existing methods.

Drone Based Delivery System

Gatteschi et al. [25] give a detailed idea on drone based delivery systems. It has got advantages such as increased delivery speed (even in areas where road traffic is high), better reachability and automatic carriage of consignments. They propose a framework for placement and shipment of orders. The main purpose is timely drug delivery which is much needed for saving lives. The paper discusses in depth about the hardware of quadcopter vehicle based drone system and how automated delivery can be achieved using it.

Dorling et al. [21] discuss about the practical limitations of drone delivery such as its small capacity and limited flight time. The authors propose a solution that drones may use multi-trips. They suggest a linear approximation by introducing various parameters as a mixed integer linear program.

Yadav et al. [58] introduce a heuristic based approach to minimize the delivery time when using drones. The paper details about two heuristic approaches (1) cost optimized scheduling heuristic (COSH) and (2) cost optimized intermediate replenishment scheduling heuristic (COIRSH). COSH improves the delivery schedule whereas COIRSH provides an improvement on COSH by intermediate replenishment of drone from stations.

Yoo et al. [61] explain about various complexities in drone based delivery. The paper narrates the difficulties in fulfilling the issues of last mile distribution and how delivery can be done in shorter lead time. The concept of drone delivery using autonomous mobility (DDAM) is illustrated using research science guidelines and their results suggest that DDAM concept is more efficient, especially when the demand is high.

Murray et al. [45] discuss a general parcel delivery system using a combination of trucks and drones, inspired by the drone delivery programmes of major e-commerce companies. Here, drones are launched from parcel delivery trucks to ensure faster consignment delivery. They try to find the optimized path for parcel delivery system using mixed integer programming. They also define heuristic functions for solving the path problem. In [35], the authors discuss an almost similar problem with a modified solution, with a view to optimize the total time taken for delivery. They use mixed integer programming along with a solution based on insertion heuristics to achieve this. Another research in the same truck and drone parcel delivery system is by Moshraf-javadi et al. [44] and they solve the joint routing problem by a meta-heuristic approach. The real world impact of the solution is demonstrated with a case study in Sao Paulo, Brazil.

Solutions of TSP

The classic problem of TSP was formalized mathematically by W. R. Hamilton in 1800s. Later, Karl Menger devised a brute force algorithm to solve the puzzle [60]. It is generally an NP hard problem and there have been many studies on developing nature inspired algorithms for TSP. In this section, we look into some of the recent developments towards the solution of TSP.

Chen et al. [14] discuss about a heuristic based method to solve TSP using a genetic simulated annealing ant colony system with particle swarm optimization. The method initializes the population for n cities with ant colony system. To achieve better solutions, genetic algorithm is applied with simulated annealing mutation techniques. Pheromone information is exchanged between groups using the methods of particle swarm optimization. Experiments were conducted on 25 data sets from TSPLIB and the results show the improvements they have obtained, compared to the existing methods (Angeniol’s method, Somhom’s method, Pasti and Castro’s method and Masutti and Castro’s method).

A heuristic simulated annealing genetic algorithm (HSAGA) has been proposed in [20]. Here, genetic algorithm functions act as a global search strategy and a heuristic simulated annealing is used for local search strategy. The heuristic simulated annealing avoids local optima and improves the search effectiveness over the solution space, since simulated annealing can accept inferior solutions with a certain probability. The performance analysis was conducted on 5 randomly chosen examples from TSPLIB95. The experiments show that HSAGA method can achieve the best tours within less number of iterations. Liu et al. [40] present a case study of ANJI-CEVA AUTOMOTIVE LOGISTICS CO. LTD. They devise the problem as TSP and solve it using simulated annealing.

In [34], Kim et al. present a TSP with drone stations. They make analysis on number of customers to be serviced, velocity and number of drones. They establish that the model they put forward can be used to overcome the limitations of drone based delivery systems. The model assumes that the locations of customers are known already.

An approach to solving TSP using the cutting-plane method was suggested by Applegate et al. [5]. It uses an optimization method based on linear programming. An implementation using ANSI C is available (Concorde TSP Solver) and has been incorporated to many programming packages. It is known to be one of the fastest TSP solvers.

K-means Clustering and its Improvements

Stuart P. Lloyd was the first to put forward k-means clustering algorithm [41] and, since then, it has been one of the most widely used algorithms for unsupervised learning. Though it is devised as an optimization algorithm, there is a chance that the algorithm will converge to local optima, especially when there is a bad choice of initial centroids. Also, k-means results in non-deterministic outputs in different runs. There have been many studies about choosing the optimal initial cluster centers and Hamerly and Elkan [28] suggest the Forgy [48] and the random partition [29] methods for the initialization of initial cluster centers.

Another study [11] provides an algorithm for refining the initial cluster centers. The refinement algorithm takes a small sample of the entire data set so as to minimize the memory requirements. The experiments show that the results obtained are superior (in terms of speed and accuracy) to that generated by random initialization of centroids.

K-means++ [50] provides a new way of finding initial cluster centers. The method chooses the initial center in a random manner. The remaining centers are chosen using a probability distribution which is proportional to the square of the distance from the data point to the cluster.

Raykov et al. [50] introduce a general probabilistic variant of k-means that overcomes many drawbacks of k-means. It takes density of points into account for forming clusters. Besides, this method overcomes another drawback of k-means that the value of k has to be explicitly specified, by automatically finding it.

Observations

The routing problem for a fleet of drones is similar to that in the case of conventional vehicles. Therefore, we can explore the solutions based on integrated clustering for vehicle routing problem. Still, it is unique due to the energy constraints of drones and efficient methods are to be applied to optimize the number of landing and take-off. There have been researches on last mile parcel delivery using drones. However, the system we propose is new, where the customers travel to their nearest drone delivery center and collect the consignments. There is collaboration between the parcel delivery unit and customers for efficient delivery and collection of items. Hence, detailed research has to be conducted for the optimal path problem in such a scenario.

Preliminaries

Definitions

Definition 1

A tour in a two dimensional dataset $S=\{(u_i,v_i)|$ $1\le i\le n\}$ is a sequence starting at a data point, visiting each data point exactly once and then reaching back the initial data point.

Definition 2

An optimal tour in a two dimensional dataset $S=\{(u_i,v_i)|1\le i\le n\}$ is a tour that minimizes the total distance travelled. Distance travelled between two adjacent data points $d_k$ and $d_{k+1}$ in a tour is $||d_k-d_{k+1}||$.

Problem Definition

S is a dataset of the positions of n customers. S = {($latitud{e}_i,longitud{e}_i$) $|$ $i\in [1\dots n$]}, where ($latitud{e}_i$, $longitud{e}_i$) represents the two dimensional coordinate positions of the $i^{\mathit{th}}$ customer. We divide S into k partitions such that $S_{\mathit{clust}}=\{C_1,C_2,\dots ,C_k\}$, where $C_i\cap C_j=\varphi$, and ${\bigcup }_{c\in S_{\mathit{clust}}}c=S$. The centroid of the partition C is given by

$$\begin{array}{c}\hfill centroid(C)=\left(\frac{{\sum }_{c\in C}latitud{e}_c}{|C|},,,\frac{{\sum }_{c\in C}longitud{e}_c}{|C|}\right).\end{array}$$

We aim at

1. minimizing the value of k.

2. minimizing the within cluster sum of square distance, ${\sum }_{C_i\in S_{\mathit{clust}}}{\sum }_{j\in C_i}||j-centroid(C_i)||$

3. finding an optimal tour (shortest round trip) from a starting position, visiting each member of the set $\{centroid$ $(c)|c\in S_{\mathit{clust}}\}$ exactly once.

Hence, it is a hierarchical optimization problem.

Refined PSO (RPSO) Clustering

Clustering aims at maximizing the intra-cluster similarity and minimizing the inter-cluster similarity. For centroid based clustering which uses euclidean distance as the metric of similarity, the intra-cluster similarity is measured using the within cluster sum of square distance. K-means, one of the widely used and efficient centroid based clustering algorithms, is capable of achieving this optimization on normal datasets. However, k-means tends to produce clusters of equal radii and hence, when it comes to not well-separated data, it fails to produce optimal outputs.

Hence, we turn our attention to clustering based on Particle Swarm Optimization (PSO) method. It is an evolutionary optimization algorithm for non-linear functions, inspired from the particle swarm methodology [32]. PSO based clustering algorithms such as MCPSO [38], MOPSO [6] and ATPSO [57] generally use multiple objective functions to partition the data. In [54], the authors propose a refined PSO clustering algorithm that can efficiently and accurately partition even not well-separated data.

The Refined PSO (RPSO) clustering is a multi-objective, multi-phase clustering technique, where each phase optimizes a different cost function. It combines both Centroidal Distance Driven PSO clustering and Cohesion Metric Driven PSO clustering. While the Centroidal Distance Driven PSO clustering minimizes the distance between the data points and the centroid of a cluster, the Cohesion Metric Driven PSO clustering optimizes a cohesion metric that shows the intra-cluster similarity. The overall block diagram of RPSO is given in Fig. 1.

Fig. 1.

An outline of RPSO clustering

RPSO starts by applying k-means on the dataset. The clusters are further refined using Centroidal Distance Driven PSO clustering and Cohesion Metric Driven PSO clustering.

PSO Clustering

A data clustering method using PSO has been proposed by van der Merwe and Engelbrecht in [43]. PSO is a stochastic optimization technique which mimics the behavior of a bird flock. Here, a bird knows the distance to the food, but it has no idea about its position. Hence, the best strategy for each bird is to move to the bird which is the closest to the food.

The general PSO strategy maintains the following parameters for each particle.

• $x_i$, the current position of the particle.

• $v_i$, the current velocity of the particle.

• $y_i$, the best position, so far, for the particle.

A particle updates its position and velocity using the following set of equations.

$$\begin{array}{cc}\hfill v_{i,k}(t+1)=& w{v}_{i,k}(t)+c_1{r}_{1,k}(t)(y_{i,k}(t)-x_{i,k}(t))\hfill \\ \hfill & +c_2{r}_{2,k}(t)(y(t)-x_{i,k}(t))\hfill \\ \hfill x_i(t+1)=& x_i(t)+v_i(t+1)\hfill \end{array}$$ 1

Here, the parameter w is the inertia weight, $c_1$ and $c_2$ are the acceleration constants and $r_{1,k}$ and $r_{2,k}$ are two uniform random numbers sampled from the interval (0, 1). Also, k represents the dimension under consideration. The personal best position of the particle is updated using the equation

$$\begin{array}{c}\hfill y_i(t+1)=\left\{\begin{array}{cc}{y}_i(t)\hfill & \text{if}f(x_i(t+1))\ge f(y_i(t))\hfill \\ x_i(t+1)\hfill & \text{if}f(x_i(t+1))<f(y_i(t))\hfill \end{array}\right)\end{array}$$ 2

The PSO algorithm makes a continuous iteration of Eqs. 1 and 2.

The general PSO method is applied in clustering by visualizing particles as a solution space of k centroids. That is, a particle $x_i$ can be represented as

$$\begin{array}{c}\hfill x_i=(centroid(C_1),centroid(C_2),\dots ,centroid(C_k))\end{array}$$ 3

The fitness of particles is measured using the equation

$$\begin{array}{c}\hfill J_e={\displaystyle \frac{{\sum }_{j=1}^k\left[{\displaystyle \frac{{\sum }_{z\in C_j}d(z,centroid(C_j))}{|C_j|}}\right]}{k}}\end{array}$$ 4

where the function d is defined as

$$\begin{array}{c}\hfill d(z,m)=\sqrt{\sum _{p=1}^D{\left(z_p,-,m_p\right)}^2}\end{array}$$ 5

Here, D stands for the dimensionality of data points.

Centroidal Distance Driven PSO clustering

This is similar to the standard PSO clustering where the objective function is to minimize the distance between the centroid and data points in each cluster. Here, we initialize m particles with a common position. Each particle represents a set of centroids. That is,

$$\begin{array}{c}\hfill x_i=(x_{i1},x_{i2},\dots ,x_{\mathit{ik}})\end{array}$$

We form clusters of data points, $\{C_1,C_2,\dots ,C_k\}$, by allocating them to the nearest centroid in $x_i$. The objective function of each particle is evaluated as

$$\begin{array}{c}\hfill f(x_i)=\sum _{p=1}^k\sum _{y\in C_p}d(y,centroid(C_p))\end{array}$$ 6

where the function d is given in Eq. 5. The PSO method finds the personal best and the global best positions to output the set of centroids.

The detailed method is given in Algorithm 1.

Cohesion Metric Driven PSO clustering

Cohesion metric is a measure of the intra-cluster similarity and it evaluates the nearness of data items within a cluster. The data items within a cluster are divided into

1. near centroid data - the set of data points which lie close to the centroid of the cluster.

2. marginal data - the data points which lie far away from the centroid.

We compute the cohesion metric ($f^{\prime }$) separately for the above data points.

$$\begin{array}{c}\hfill f^{\prime }=f_c^{\prime }+f_m^{\prime }\end{array}$$ 7

where $f_c^{\prime }$ and $f_m^{\prime }$ are the cohesion metrics for the near centroid data and marginal data respectively. The cohesion metric for the near centroid data of a cluster is the radius of a circle with origin at the centroid that contains p near centroid data items, as shown in Fig. 2. On the other hand, the cohesion metric for the marginal data for a cluster will be the sum of the distances of each of the marginal data points with its farthest neighbour. Hence, for an entire dataset, the equation for the cohesion metric turns out to be

$$\begin{array}{c}\hfill f^{\prime }=\sum _{j=1}^k{R}_{p_j}+\sum _{j=1}^k\sum _{m_q\in marginal\_data}d(m_q,m_q^{\ast })\end{array}$$ 8

where $m_q^{\ast }$ is the farthest point from $m_q$ in the marginal data of that cluster. Needless to say, the cluster similarity will be more for smaller values of $f^{\prime }$. Hence, each particle in the PSO method tries to minimize the value of $f^{\prime }$. The procedure is similar to Algorithm 1, except that the optimization function is different.

Fig. 2.

$f_c^{\prime }$ for a cluster is the radius of the circle ($R_p$) with the cluster centroid as the origin that contains p data points from the near centroid data

Refined PSO (RPSO) Clustering Algorithm

As shown in Fig. 1, the refined PSO clustering algorithm applies k-means on the dataset in the first stage. The centroids obtained in the k-means phase are fed as the initial set of particles to the Centroidal Distance Driven PSO clustering and its output will serve as the initial particles for the Cohesion Metric Driven PSO clustering.

Other than resulting in better clustering, Refined PSO clustering has the added advantage that it can accurately partition even not well-separable data [54].

Solving Travelling Salesperson Problem (TSP)

TSP is mathematically formulated as an optimal Hamiltonian tour on a weighted undirected graph, specified as (N, d) where N is the set of nodes and d is the distance function such that d: ($n_1$, $n_2$) $\to$ $\mathbb{R}$ where $n_1$, $n_2$ $\in$ $\mathbb{N}$. The function d satisfies the following criteria.

1. d(i, j) = d(j, i)

2. d(i, j) $\ge$ 0

3. $d(i,j)+d(j,k)$ $\ge$ d(i, k)

In other words, given n cities and a travel cost vector c, a tour through all the n cities can be thought of as an incidence vector having ${\displaystyle \frac{n(n-1)}{2}}$ elements. Then TSP is expressed as the optimization problem

$$\begin{array}{c}\hfill minimize{c}^{T}xsubjecttox\in S\end{array}$$

where S is the set of incidence vectors of all tours.

The problem is NP-hard [52]. A well-known exact solution for TSP is the Concorde [5]. It works under the principle of cutting-plane method which iteratively refines the feasible set of objective functions using a set of linear inequalities. Dantzig et al. [19] have already shown that the above optimization problem can be represented as the inequality

$$\begin{array}{c}\hfill minimize{c}^{T}xsubjecttoAx\le b\end{array}$$

since the latter is a relaxation of the former. This type of an inequality is called a cutting plane or simply cut. As more and more cuts are added, the solution space becomes narrower and when it becomes sufficiently small, the method of branch and cut is applied. That is, when S = {$S_1$, $S_2$}, we solve the sub-problems

$$\begin{array}{c}\hfill minimize{c}^{T}xsubjecttox\in S_i\end{array}$$

The branching can continue by splitting one of the sub-problems into further sub-problems when the solution space of that gets sufficiently small. A binary tree of sub-problems is thus formed where leaf nodes contain the solutions.

Concorde has been known to be one of the best solvers of TSP. The power of this has been proved by solving all TSPLIB instances having a total of 15,112 cities. Concorde program contains roughly 130,000 lines of ANSI C code and follows the method suggested in [19]. Though other methods (mainly approximate) are also in practice, Concorde remains one of the most widely used methods for solving TSP.

Proposed Methods

This section describes the way in which we approach the problem of efficient delivery of consignments using drones. The model we put forward is that the drone starts with the items to be delivered and lands at certain points before coming back to the supply station and the customers can collect their consignments from the nearest drone landing point. We know that drones are battery operated unmanned aerial vehicles and a major share of their battery is used for the operations of landing and take-off [59]. The objective is to find the optimal geographical locations where the drone has to land and take off so that the number of such landings is minimized and, at the same time, customers won’t have to travel long distances to collect their items. Besides, the drone has to travel in an optimal tour connecting these landing points. The overall aim is to maximize the luggage carrying capacity and the flight time of the drone so as to maximize the efficiency and reliability of the drone based delivery system. In this research work, we do not consider the altitude of the location for drone landing and the reduction in luggage weight after distributing the consignments at a location.

We tackle the above problem by partitioning the customers into different clusters based on their distances, the cluster center being the point of landing for the drone for the people in that cluster, and then finding an optimal tour for the drone connecting these cluster centers. Since clustering tries to minimize the intra-cluster distance, customers in a cluster are close to each other. A drone landing at a cluster centroid ensures that it is the optimal nearest point for all customers in that cluster. The optimal tour of the drone makes sure that the battery of the drone is utilized efficiently. A visualization of the method is given in Fig. 3.

Fig. 3.

A drone carries the consignments to the customers from a source. Clusters of customers are identified and the drone lands at the centroid of each cluster to serve the customers there. The drone takes an optimal tour to reach back the source, after delivering items to all the customers

We have to choose a clustering algorithm that doesn’t discard any customer as noise. A popular, centroid based and efficient clustering algorithm that considers all data points is k-means. However, k-means is prone to produce spherical clusters and can work efficiently only on well-clusterable data [3]. However, this may not be the situation in practice. Hence, we have to select a clustering method that is able to work with not well-separable data. For this purpose, we use RPSO clustering (discussed in Section 3.3), which can take into account both distance and cohesion, for effectively clustering the customers based on their geographic locations.

The RPSO clustering applies k-means on the dataset, followed by the Centroidal Distance Driven and Cohesion Metric Driven PSO clustering methods. While the Centroidal Distance Driven PSO clustering maximizes the inter-cluster distance, the Cohesion Metric Driven PSO clustering maximizes the intra-cluster similarity. We use the following strategy to cluster the customers based on their geographic locations.

1. When we have a well-clusterable, spherical dataset of the geographic locations of the customers, we use k-means.

2. When we have a not well-separable dataset of the geographic locations, we use RPSO.

In both cases, k-means is an integral phase. In fact, it is one of the most widely used clustering algorithms [63]. However, it suffers from the following drawbacks.

• Though it is simple and straight forward, there is a great chance that the algorithm will converge to local optima, thereby yielding suboptimal clusters.

• K-means is non-deterministic in nature (it can give different results in different runs of the algorithm).

An Improved k-means

In [24], the authors suggest that k-means can be improved by better initialization of cluster centers and the performance of k-means depends completely on the initial centroids, especially when the clusters are well-separable. The main initialization techniques used are Forgy [48], random partition [29] and kmeans++ [56], as discussed in Section 2.5. A common way to find the clusters is by repeatedly executing k-means with random cluster center initialization until we find a good solution. However, this is computationally inefficient as the entire algorithm is executed multiple times. We try to overcome this drawback using a nature inspired method for the initialization of cluster centers.

Artificial Physics Optimization (APO)

A general framework of Artificial Physics Optimization has been proposed by Xie et al. [56]. It is a physics inspired optimization mechanism which invokes a gravitational metaphor. It was first proposed in [27] in 2003. It visualizes the solution as a set of particles from a feasible region of problem space. Particles use a fitness function to evaluate their fitness and cluster to a region were fitness is high. The movement of particles is regulated by the force of attraction between them. Force is governed by the mass of the particle as well as its velocity.

APO is initialized with the following parameters.

• $N_{\mathit{pop}}$ - Number of individuals.

• $v_0$ - Initial velocity of a particle (usually set to 0).

• $X_{\mathit{best}}$ - Position vector of the particle with the best fitness.

• f(X) - Objective function for fitness.

The most important development towards the objective function is the calculation of force of attraction between two particles i and j of mass $m_i$ and $m_j$ respectively and separated by a distance r. The force can be calculated as

$$\begin{array}{c}\hfill F=G\frac{m_i{m}_j}{r^p}\end{array}$$ 9

where p is a user defined power and in real world, it takes a value 2. If the distance between particles increases beyond a threshold, the force becomes repulsive. When particles lie in a hyper sphere, the force on a particle is the vector sum of forces exerted on that particle by all other particles. In other words, the force on a particle is the vector sum of forces on that particle in all directions. If the $k^{\mathit{th}}$ component of force between particles i and j is $F_{ij,k}$, then

$$\begin{array}{c}\hfill F_{ij,k}={\displaystyle \frac{G{m}_i{m}_j}{{(x_{j,k}-x_{i,k})}^2}}·\widehat{k}\end{array}$$ 10

where $x_{i,k}$ and $x_{j,k}$ are the $k^{\mathit{th}}$ dimension coordinates of i and j. The unit vector $\widehat{k}$ designates that $F_{ij,k}$ is a vector along the $k^{\mathit{th}}$ direction.

Centroid Based Celestial Clustering (CBCC)

The execution sequence of k-means depends a lot on the selection of initial cluster centers and can produce different results in different runs of the algorithm [33]. There is a possibility that k-means will converge to local optima on a bad choice of initial cluster centers. In [33], the authors suggest that the optimal distribution of initial cluster centers is in such a way that they occupy positions where the distance between each other is the maximum. If each centroid is considered as a particle on a hyper plane, they will be maximally distant from each other when the gravitational force of attraction between them is the least (force is inversely proportional to the square of distance). Hence, the optimal distribution of the initial cluster centers is achieved by the objective function

$$\begin{array}{c}\hfill minimize\sum _{x,y\in X}{F}_{x,y}\end{array}$$ 11

Here, X is set of cluster centers and $F_{x,y}$ is the force of attraction between x and y. However, $F_{x,y}$ is the vector sum of force between x and y in all directions and can be represented as

$$\begin{array}{c}\hfill F_{x,y}=\sum _{i\in Directions}{F}_{x_i,y_i}\end{array}$$ 12

Here $F_{x_i,y_i}$ is the force between x and y in the $i^{\mathit{th}}$ direction, found using Eq. 10. Also, we know that the force of attraction between two particles is inversely proportional to the square of the distance between them.

$$\begin{array}{c}\hfill F_{x_i,y_i}\propto \frac{1}{{(x_i-y_i)}^2}\end{array}$$ 13

We develop the algorithm ClusterCenters to find the initial cluster centers using the above concepts. It is a greedy algorithm which starts by initializing the first k data points as the cluster centers. We then calculate the fitness function (total force in the system) using Eq. 12. Then we iterate over the remaining data points to see if substitution of any member in the present cluster centers by the current data point will reduce the force in the system. The method is presented in Algorithm 2.

In each iteration, the algorithm tries to find the initial cluster centers which are mutually farther than those in the previous step. As a result, the initial cluster centers produced finally are well apart and may lie in different clusters on well-clusterable spherical datasets.

Once we find the initial cluster centers, clustering proceeds by applying k-means on the dataset, as given in Algorithm 3. The way in which the initial cluster centers are selected is analogous to the orbital equilibrium formation of celestial bodies under gravitational force and, hence, we name our clustering method Centroid Based Celestial Clustering (CBCC).

It is evident that both ClusterCenters and k-means contribute to the time complexity of CBCC. From algorithm 1, it is obvious that the time complexity of ClusterCenters is

$$\begin{array}{c}\hfill T_1(n)=O((n-k)·k·{\displaystyle \frac{k(k-1)}{2}}·d)\end{array}$$ 14

where d is the dimensionality of data points. The time complexity of k-means is known to be [18]

$$\begin{array}{c}\hfill T_2(n)=O(iknd)\end{array}$$ 15

where i is the number of iterations taken by k-means for convergence to solution. From Eqs. 14 and 15, it is clear that the time complexity of CBCC is linear, especially for large values of n.

CBCC ensures that the clustering doesn’t converge to local optima in spherical datasets [33]. Also, it is deterministic and requires the minimum number of rounds of iterations for convergence, which is experimentally validated in later sections.

Finding the Optimal Number of Drone Landing Points

One of the objectives of the drone based delivery system is that the distance to be travelled by the customers to pick the consignments should be the minimum. On the other hand, we cannot land the drone for each customer as it will drain out the drone battery easily and cause delay in consignment delivery. As the delivery drone is landing in cluster centers, the idea is to optimize the distortion, the sum of squared distances between the observations and their cluster centers. The elbow method is a classical way to find the optimal number of clusters [39] and uses this principle. It fits the model with a range of values of k. We plot a curve with the values of k on the x-axis and distortion on the y-axis. Distortion usually falls fast initially and then forms an elbow followed by a nearly flat curve. The value of k at which the curve forms the elbow is taken as the optimum.

Also, we have to make sure that two drone landing locations are not close to each other, i.e, the clusters have to be well apart from each other. A cluster metric that shows the quality of clustering using this principle is the silhouette score [15]. Its values range from -1 to 1 where 1 indicates clearly distinguishable clusters and 0 shows indifferent partitions. A value of -1 implies that the data points are wrongly assigned to clusters. The silhouette score is calculated as

$$\begin{array}{c}\hfill silhouette\_score={\displaystyle \frac{b-a}{max(a,b)}}\end{array}$$ 16

where a is the average intra-cluster distance and b is the average inter-cluster distance. We analyse both elbow method and silhouette score to find the number of drone landing points. We use the elbow method to get the initial estimate of k and then we refine it using the silhouette score. Besides, we use the same analysis to choose between CBCC and RPSO for data partitioning.

Finding the Optimal Tour

As shown in Fig. 3, a delivery drone starts at a source, lands in all cluster centers and reaches back the source in an optimal tour. The optimal path can be found by solving the Travelling Salesperson problem where the source and the centroids of different clusters form the vertices of the graph. We use the Concorde method, presented in Section 3.4, for this purpose.

Experiments and Evaluations

We evaluate the performance of our methods in two separate datasets. The first dataset (SET1) is based on the data presented in [22], where the coordinates of 104 cities in the former Indian state of Andhra Pradesh are taken. The data set contains names of the cities, their latitudes and longitudes. The raw dataset is plotted in Fig. 4. The figure shows that the dataset spans over a vast area and we scale the dataset as in Fig. 5. The second is a synthetic dataset of 250 two dimensional data points, based on the work presented in [2], generated using the Gaussian distribution with values of $\mu$ and $\sigma$ as given in Table 1. It has 5 not-well-separable sets of data points, each of which is generated using a Gaussian model. A plot of SET2 is given in Fig. 6.

Fig. 4.

A plot of SET1

Fig. 5.

The scaled dataset SET1

Table 1.

Parameters for SET2

Sl. No.	$\mu$	$\sigma$
1	(-20, -20)	4
2	(-10, -10)	4
3	(-5, -5)	4
4	(0, 0)	4
5	(19, 19)	4

Fig. 6.

A plot of the datapoints in SET2. It is hard to cluster the data points with k-means as they are not well-separated

We have developed our programs using R and Python. First, we compare the performance of CBCC against k-means in terms of the average number of rounds to converge. For this purpose, we select 200 random data points from SET2 and apply clustering with varying values of k. There are many occurrences where k-means falls into local optima, yielding incorrect clusters. We have carefully avoided such situations from being considered. Here, we use two variants of k-means, viz., Forgy and Random Partition. The results are shown in Table 2, suggesting the superiority of CBCC.

Table 2.

Number of rounds taken for convergence by different clustering algorithms for varying values of k

k	Forgy	Random Partition	CBCC
2	7	8	4
3	7	7	5
4	8	6	4
5	9	8	5
6	8	8	5

Experiments on SET1

As discussed in Section 4, we partition the dataset SET1 using CBCC. In order to find the optimal number of clusters (k), we apply the elbow method and the plot obtained is shown in Fig. 7. The potential values of k are 3 and 4. To get more accurate information on the optimal number of clusters, we plot the silhouette score curve as in Fig. 8. From the figure, it is evident that the optimal number of clusters is 4.

Fig. 7.

Graph plotted for finding the optimal value of k using elbow method on SET1 (scaled)

Fig. 8.

Silhouette score curve for SET1 (scaled) under CBCC

Now comes the question of selection of the clustering method. For this, we plot the silhouette score curve under the algorithm RPSO and it is as in Fig. 9. It is evident that for optimal number of clusters (k = 4), the silhouette scores under both CBCC and RPSO are identical and we use CBCC, as it is computationally more efficient. The resulting clusters are shown in Fig. 10. We can see that the clusters are almost well-separated and there is not much of an overlap between them.

Fig. 9.

Silhouette score curve for SET1 (scaled) under RPSO

Fig. 10.

SET1 has been partitioned into 4 clusters using CBCC

Finally, we find the optimal tour of the drone, starting from the point (0,0). For this, we solve the TSP as explained in Section 4.3 so that a drone starting at (0,0) reaches back the same location after serving all cluster centroids in an optimal way. The resulting tour is plotted in Fig. 11.

Fig. 11.

An optimal tour of the drone from position (0,0) for the customers in SET1

Experiments on SET2

As seen in Fig. 6, SET2 has a very high customer concentration in the bottom left quadrant. When we apply the elbow method, the curve obtained is as in Fig. 12. Similarly, the silhouette scores for both CBCC and RPSO are plotted in Figs. 13 and 14 respectively.

Fig. 12.

The curve obtained for the elbow method on SET2

Fig. 13.

Silhouette score curve for SET2 under CBCC

Fig. 14.

Silhouette score curve for SET2 under RPSO

All these diagrams show that the optimal number of clusters is 2. However, for a large set of customers, a drone delivery system with only two stops will necessitate the customers to travel large distances to collect their consignments. When we closely examine the silhouette curves, we can observe that the silhouette scores in RPSO rise after a point (after k = 6) whereas silhouette score curve by CBCC exhibits a falling trend. Hence, in this case, we prefer RPSO over CBCC. For k = 9, the silhouette score obtained by RPSO method is much better that that obtained by CBCC. The customer partitions obtained using RPSO for k = 9 is shown in Fig. 15.

Fig. 15.

SET2 partitioned using RPSO for k = 9. The centroids are marked with large circles

Importance of Clustering in the Proposed Solution

Finally, we would like to validate the importance of clustering in our solution. We have used clustering to group customers based on their density. Other than clustering, a possible solution is that the drone lands at the location of each individual customer. However, this is not practical, especially when the number of customers is high, not to mention the energy demands for frequent landing and take-off. Another possibility is to assign a set of random drone landing locations and direct the customers to collect consignments from their nearest landing location. This eliminates the need of frequent landing and take-off. However, this can seriously affect customer satisfaction, as the distance to be travelled by the customers in this case will be more (our method minimizes the within cluster sum of square distance). We experimentally validate this for the dataset SET1. The within cluster sum of square distance found using our method is 16.25 whereas that found using random allocation of same number of drone landing points is 21.34. Note that we have found the sum of square distance in the random allocation method by repeating the experiment 100 times and then taking the average value. We have carefully omitted the situations when two landing points were very close to each other as, otherwise, they would result in impractical solutions. Also, none of these trials could find a value better than that found using our method. Hence, clustering is an integral part in our solution.

Analysis of Results

The idea that both the logistic company and the customers collaborate and coordinate together for effective consignment delivery using drones is completely new. Therefore, it is hard to compare our methods with any of the existing techniques. Hence, we concentrate on the objectives discussed in Section 3.2. Our main objective is to minimize the drone landing locations and, at the same time, minimize the distance that the customers have to travel to collect the consignments. Obviously, the number of drone landing points is minimized when the value of k gets smaller. However, a small value of k indicates that the distance that the customers have to travel gets larger. In Figs. 7 and 12, the y-axis represents a measure of the total distance that the customers have to cover to collect the items. We can note that the total distance travelled by the customers decreases as the value of k increases. Therefore, we are able to choose an optimal number of drone landing points using these curves, which is later verified/adjusted using the silhouette score curves in Figs. 8 and 14.

Discussions

Collaborative consignment delivery using drones is a futuristic approach in retail marketing. We have the traditional delivery system where consignments to be delivered in a specific area are transported to a delivery hub and, from there, delivery persons carry them to the customers. However, this requires considerable man power and resources. In real life, a delivery manager has to manually fix the path for each delivery person based on the addresses of the customers. This manual computation may result in sub-optimal paths, leading to wastage of resources. Besides, there are chances that these delivery persons find themselves stuck in traffic jams, leading to delays in delivery, not to mention the extra traffic caused by them. More than the path planning problem, there are additional overheads in recruiting and maintaining delivery persons at each delivery hub. On the other hand, the delivery system that we have proposed calculates the optimal path automatically. The issues associated with traffic jams are completely eliminated in this case. Drones can easily reach places where delivery persons find it difficult to navigate to. Moreover, drones travel faster and this makes sure that the customers are serviced without any delay. Most of the time, drones travel in a linear path and this reduces the time and energy requirements. These factors make the proposed approach unique and more intelligent than the traditional methods.

A prime factor governing the success of the proposed method is the cost of operation. The cost of a drone based delivery system includes the expenses associated with procurement of machinery, energy needs and use of air space [31]. It is difficult to assess the cost of these factors at present, as the technology is still in an evolutionary stage and implementation of delivery systems using drones has not been common. A study closely related to this area is by Ochielng et al. [47], which compares the cost effectiveness of delivering laboratory samples using drones against the traditional way of using motor cycles. The study was conducted in West Africa and it concludes that the cost effectiveness increases at a rapid rate with increasing range and operational life span of drones. Another study [62] has found that the shipping cost of medical products in post-maternal emergencies is lower in drone aided transportation systems, compared to traditional road transport. It concludes that the present day road transportation system is incapable of meeting the demands in such emergencies. In our approach, the lion share of cost savings happen in the form of reduced man power, not to mention the savings in fuel costs and environmental impacts. In all possibilities, the operational cost of delivery systems using drones will come down with the advent of technology capable of building drones with longer flight time and better weight carrying capacity, as the technology costs usually drop over time [16].

Another aspect to be considered is the implementation and management of the proposed approach. The consignments to be delivered to an area may reach the nearest delivery hub. A delivery manager (can be an automated software system) calculates the cluster centers in the area where the drone has to land to deliver the items. The delivery manager calculates the expected time of arrival of the drone at each delivery point (cluster center) and informs the consumers. The managerial activities of auditing, book keeping and reporting may be done by the delivery manager. Loading the drone with consignments may be mechanized. Similarly, delivering the items to the customers can happen as in the case of a dispenser machine (for example, the drone reads the code in the mobile phone of a customer and delivers the corresponding item). This eliminates the need of a large amount of human resource at each delivery station and makes the process mostly automated.

Limitations

As the proposed system is new, there are some present limitations to be addressed in future.

• The framework proposes drones delivering consignments to the centroid of a user cluster rather than making door to door delivery. Therefore, the protocols, guidelines and security measures are to be developed to facilitate this.

• The present work focusses on algorithmic framework formulation. More parameters such as battery constraints, charging station availability, terrain, weather conditions etc. are to be incorporated to the framework.

• Though the basic framework uses a single drone for delivering consignments, practical situations may require multiple drones to operate in parallel. The framework may be modified for that in future.

Conclusions and Future Work

We have made a detailed study of the drone based delivery system which is the future of consignment transportation for door delivery. There are several advantages for drone based delivery like improved efficiency, reduced human intervention and fast and accurate distribution of items. However the bottlenecks are limited battery life and inability to carry large weights. As a result, landing and take-off operations are to be kept minimal, at the same time ensuring that the customers do not have to travel large distances to collect their consignments. We have developed a novel framework for solving the problem using Centroid Based Celestial Clustering, Refined PSO clustering and TSP. We have applied our methods to a dataset consisting of the geographic locations of 104 cities and have found the optimal tour for a delivery drone connecting them. In order to experiment our methods on a dataset with not-well-separated data points, we have used a synthetic dataset containing the locations of 250 customers.

The present work takes into account only latitude and longitude of the locations of the customers. In future, we would like to incorporate the altitude of the location also so that a more accurate path can be formulated for consignment delivery. Similarly, the distance that the user has to travel to the nearest centroid may not be in a straight line and more appropriate measures (eg: Google map distance) can be taken in future. Also, we would like to develop an objective function that takes into account more drone parameters such as battery and weight carrying capacity, along with the location of customers. Finally, the delivery system proposed in this paper is limited to use a single drone. However, we would like to expand the research into a drone based delivery system for a large geographical area where a number of drones are used to make delivery in parallel. We can introduce fault tolerance into the system where another drone takes the delivery of a drone having a mechanical failure.

Philip Samuel

holds a Masters Degree from Cochin University of Science and Technology and Ph.D. degree from Indian Institute of Technology (IIT Kharagpur). He is currently serving as the Professor and Head, Dept. of Computer Science, Cochin University of Science and Technology, Kochi, India. His research interests include Big Data Computing, Automated Software Engineering, Distributed Computing and Artificial Intelligence.

Declarations

Conflicts of interest

The authors declare that they have no conflict of interest.