1. Introduction

• We introduce and formally define a multi-trip drone routing problem, aiming to achieve operational cost minimization, where drone speeds are considered as decision variables rather than constants.
• We clearly consider drone energy consumption as a nonlinear function of both flight speed and payload rather than supposing that flight endurance and range are constants.
• We proposed a three-phased method integrating variable drone speed to solve DRP-VFS, which can improve the solution in terms of computational time to a greater extent than MILP.

Figure 1. The required power, energy consumption per unit distance, endurance, and range of a drone versus speed and payload referenced from [15].

Figure 1. The required power, energy consumption per unit distance, endurance, and range of a drone versus speed and payload referenced from [15].

2. Literature Review

3. Model Formulation

4. Solution Method

4.1. Initialization

There are numerous heuristic approaches that can efficiently search for a feasible solution for the VRP, such as CW savings algorithm and sweep algorithm. In our method, we first apply an improved k-means algorithm that includes capacity and Euclidean distance as constraints to partition customers into several subsets (Algorithm ). Next, a traveling salesman problem with time windows (TSPTW) is solved within each subset to determine the customer sequence a drone serves. This algorithm is executed while satisfying the capacity, energy, and time window constraints. The number of clusters is calculated by the total demands and capacity of the drone as

$$\mathcal{K}=\lceil \sum _{i=1}^n{q}_i/Q\rceil$$

(37)

The $\mathcal{K}$ centroids are randomly initialized in the customers’ coordinate range. The Euclidean distances between each customer and all $\mathcal{K}$ centroids are calculated. Sort customers by demand from largest to smallest and assign each customer to the nearest centroids. The total demands for each subset of clusters cannot exceed the drone’s capacity. The selected customer will be assigned to the next nearest centroid if the capacity constraint is not satisfied. The centroid of kth cluster $\left(x_k,y_k\right)$ is calculated based on its members using (38).

$$\begin{array}{c}\hfill x_k=\sum _{i=1}^{C_k}{x}_i/C_k,\\ \hfill y_k=\sum _{i=1}^{C_k}{y}_i/C_k.\end{array}$$

(38)

The iterative procedure is repeated until the cluster is unchanged or the maximum iteration is reached.

The TSPTW is a special case of DRP-VFS since only one drone is used. Thus, we just need to remove the reusability constraints (10)-(12) and modify constraints (3) to the following.

$$\sum _{r\in {\mathbf{r}}_k}{x}_{0r}=\sum _{r\in {\mathbf{r}}_k}{x}_{r,n+1}=1k=1,2,\dots ,\mathcal{K}.$$

(39)

Therefore, an initial feasible solution can be obtained quickly.

4.2. Local Search

In this section, we propose a speed optimization model to adjust the flight speed in a trip and calculate the cost of a DRP-VFS solution, and the Adaptive Large Neighborhood Search (ALNS) algorithm is used to enhance the solution quality.

4.2.1. Speed Optimization

The speed optimization problem (SOP) involves adjusting the flight speed in order to minimize energy consumption. The SOP is defined on a route $\mathbf{r}=\left\{r_0,r_1,\dots ,r_{C+1}\right\}$ consisting of customers served by a single drone, where $r_0$ and $r_{C+1}$ represent the depot corresponding to 0 and $n+1$, respectively. $v_i$ and $m_i$ denote the flight speed and payload carried by a drone between node $r_i$ and its successor. The SOP is formulated as follows:

$$min\sum _{i=0}^C{e}_i,$$

(40)

subject to

$$\begin{array}{cc}\hfill & m_i=\sum _{j=i+1}^C{q}_{j}i=0,\dots ,C,\hfill \end{array}$$

(41)

$$\begin{array}{cc}\hfill & m_0\le Q,\hfill \end{array}$$

(42)

$$\begin{array}{cc}\hfill & {\tau }_i+s_i+d_i/v_i\le {\tau }_{i+1}i=0,\dots ,C,\hfill \end{array}$$

(43)

$$\begin{array}{cc}\hfill & a_i\le {\tau }_i\le b_{i}i=1,\dots ,C,\hfill \end{array}$$

(44)

$$\begin{array}{cc}\hfill & e_i=E_0\left(v_i,m_i\right)d_{i}i=0,\dots ,C,\hfill \end{array}$$

(45)

$$\begin{array}{cc}\hfill & f_0=0,\hfill \end{array}$$

(46)

$$\begin{array}{cc}\hfill & f_i+e_i\le f_{i+1}i=0,\dots ,C,\hfill \end{array}$$

(47)

$$\begin{array}{cc}\hfill & f_{C+1}\le E.\hfill \end{array}$$

(48)

The nonlinear energy function is linearized using SOS2 variables introduced in Section 3.3. Since $m_i$ is calculated in advance using (42), the $E_0$ between node $r_i$ and its successor depends only on $v_i$. We introduce ${\gamma }_i^s$ to linearly approximate the energy consumption function. Hence, the value of $v_i$ and the approximate value of ${\tilde{e}}_i$ can be represented by

$$\begin{array}{cc}\hfill & v_i=\sum _{s=1}^S{V}_s{\gamma }_i^s,\hfill \end{array}$$

(49)

$$\begin{array}{cc}\hfill & {\tilde{e}}_i=\sum _{s=1}^S{E}_0\left(V_s,m_i\right){\gamma }_i^s.\hfill \end{array}$$

(50)

To deal with $1/v_i$, the same method can be used. However, to speed up the solving process, we introduce a convexification method, i.e., replace $1/v_i$ with an auxiliary variable ${\lambda }_i$ to avoid reintroducing integer variables:

$${\tau }_i+s_i+d_i{\lambda }_i\le {\tau }_{i+1}.$$

(51)

Meanwhile, the following inequality should be imposed to ensure ${\lambda }_i$ converges to $1/v_i$.

$${\lambda }_i\ge 1/v_i.$$

(52)

It can be seen that constraint (52) is a quadratic and convex constraint. Therefore, the SOP is modeled as a mixed-integer quadratically constrained programming (MIQCP) and can be solved by commercial solvers.

4.2.2. ALNS heuristics

In this subsection, an adaptive large neighborhood search algorithm is adopted to improve the initial solution obtained by Section 4.1. The ALNS framework has demonstrated successful results in solving multiple variants of vehicle routing problems ([38]). By removing selected nodes from the solution and reinserting them, we obtain a neighborhood of the given solution. The operators are dynamically chosen based on their previous performance. For this reason, each operator is given a score that increases if it enhances the current solution. A new solution is accepted if it meets the criteria specified by the local search scheme, such as simulated annealing.

In this paper, we apply two removal operators and two insertion operators to effectively destroy and repair a DRP-VFS solution. In addition, for every $\Gamma$ iteration, the TSPTW model is solved once within each trip to adjust the customer sequence. The framework of the ALNS algorithm is presented in Algorithm . Figure 2(a)-(c) depict the initial solution, the solution after applying two destroy operators and the solution after executing two repair operators.

The destroy operators involve primarily removing several nodes from the current solution and placing them in a removal list $\mathcal{L}$. We present the removal operators involved in our approach as follows:

(1)

Random removal: This operator removes several nodes at random from the current solution.

(2)

Worst removal: This operator removes the highest-cost node from the solution, where the cost is determined by solving SOP.

The insertion operators are applied to repair a partly destroyed solution by reinserting the nodes from $\mathcal{L}$ back into the solution. The insertion operators employed in our algorithm include:

(1)

Greedy insertion: This operator repeatedly removes a node from $\mathcal{L}$ and inserts it into the lowest cost position of a route.

(2)

Regret insertion: An obvious disadvantage of greedy insertion is that it defers node insertion to later iterations where few feasible moves are available. The regret operator in our algorithm uses a 2-regret criteria. Define $\Delta c_i^j$ as the cost change incurred by inserting node i into the route where the cost is jth cheapest. The 2-regret criteria inserts the node i based on $i^{*}=\underset{i\in \mathcal{L}}{\mathrm{argmax}}\left(\Delta c_i^2-\Delta c_i^1\right)$, where $\Delta c_i^1$ and $\Delta c_i^2$ are the best and second-best insertion of node i. We iterate the procedure until no more nodes in $\mathcal{L}$ can be inserted.

We describe the adaptive weight adjustment process below. The operators are chosen by the roulette-wheel method. In the beginning, all operators are equally likely. They are updated as $P_i^{t+1}=P_i^t\left(1-\mathsf{\Theta }\right)+\mathsf{\Theta }{\pi }_i/{\omega }_i$ during program execution, where $P_i^t$ is the probability of operator i at the tth iteration, $\mathsf{\Theta }$ is the parameter of roulette wheel, ${\pi }_i$ denotes the score of operator i and ${\omega }_i$ indicates times it was selected in the previous $\Gamma$ iterations. The score reflects the performance of each operator. When a new best solution is found, the related operators’ scores are increased by ${\sigma }_1$. If the new solution outperforms the current one, the score is increased by ${\sigma }_2$. If the new solution falls short of the current one but meets the acceptance criteria, the score is increased by ${\sigma }_3$.

Our ALNS algorithm applies the simulated annealing as an acceptance criterion. We calculate the probability in line to determine whether to update the current solution and decrease the initial temperature T according to $T=\mathsf{\Phi }T$, where $\mathsf{\Phi }$ is a constant indicating the cooling rate. The algorithm produces the best solution that has been found after $\mathsf{\Lambda }$ iterations.

4.3. Assignment

In this section, we describe a list schedule algorithm to minimize the number of drones. The delivery time, denoted by h, is the latest allowable departure time from the depot, whereas the arrival time, denoted by l, indicates when the drone reaches the depot upon finishing the delivery. The two times are calculated by using (53).

$$\begin{array}{cc}\hfill h& ={\tau }_1-d_0/v_0,\hfill \\ \hfill l& ={\tau }_C+s_C+d_C/v_C,\hfill \end{array}$$

(53)

where ${\tau }_1$ and ${\tau }_C$ are the starts of service time at the first and last nodes in a route, $d_0,v_0$ and $d_C,v_C$ is the distance and speed at the first and last flights in a route, and $s_C$ is the service time of the last node.

We allocate $\mathcal{K}$ routes to U drones using the list scheduling algorithm (Algorithm ). First, sort routes by delivery time in ascending order. A binary heap is used to get the route with minimum arrival time quickly. Next, traverse the $\mathcal{K}$ routes; if the delivery time of ${\mathbf{r}}_k$ is greater or equal to the arrival time of the heap top route, then ${\mathbf{r}}_k$ is appended to this route. Otherwise, a new drone should be added. The time complexity is $\mathcal{O}\left(\mathcal{K}log\mathcal{K}\right)$. In conclusion, our heuristic method finds a near-optimal solution ${\mathcal{R}}^{*}$ for DRP-VFS.

5. Results and Discussion

6. Conclusion and Future Work

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