An integrated route planning approach for driverless vehicle delivery system

Definition of entities

Firstly, five entities involved in the multistage model are introduced as follows.

• (i) DC: Each DC can be represented as a triplet, $DC=<dcl,edvn,on>$, where dcl represents the geographical location of DC located by GPS, edvn represents the number of EDV in DC and on represents the number of orders received by logistics companies from customers. DC is the starting node of the DVDS transportation network. Companies design reasonable distribution routes based on ESs that need to be distributed and then select the appropriate EDVs for delivery.

• (ii) EDV: Each EDV can represent a quad, $EDV=<vt,edvd,lw,lv>$, where vt represents the vehicle type, edvd represents the distance that the EDV can drive under the remaining battery capacity, lw represents the loading weight of EDV, lv represents the loading volume of EDV. The EDV types can be roughly divided into small, medium and large and they have different load capacities. Big EDV has a large load and a long distance to travel, so more ESs or farther ES can be arranged for distribution.

• (iii) ES: Each ES can be expressed as a binary group, $ES=<esl,t>$, where esl refers to the geographical location of ES located by GPS and t refers to the type of ES, which mainly includes express station and delivery locker. For each delivery, all the packages stored in the ES will be delivered together.

• (iv) PA: The package delivered by DVDS from DC to ES and can be expressed as a quintuple, $PA=<n,pal,es,pw,pv>$, where n represents the specific package number of each PA, pal represents the receiving place of PA, es represents the ES where the package is deposited, pw represents the weight of the package and pv represents the volume of the package. All this information is an important basis for package delivery.

• (v) TN: The transportation network of a DVDS, which is a key factor in logistics and transportation, can be represented as a triplet, $TN=<s,d,c>$, where s represents all nodes in the network (including ES and DC), d represents the distance between all nodes and c represents the road congestion coefficient between all nodes. We can use $d_{ij}$ to represents the distance between station i and station j and use $c_{ij}$ to represents the road congestion coefficient between station i and station j. In this network, the paths between nodes are bidirectional and symmetric.

The multistage model

Based on traditional VPR models and defined entities, we further proposed an multistage model to characterize the route planning problem of DVDS.

In general, the multistage model has three stages. The first stage is HFVRP model, which is mainly for ESs allocation; the second stage is VRP model, which is mainly to generate basic distribution routes; the third stage is the DVRP model, which is responsible for optimizing the distribution route.

• (i) Package Arrangement Model: DC has several different types of EDV available for delivery and each vehicle has different battery power and load capacity. Assuming that various types of EDVs are fully charged, there are the following rules: the battery capacity of EDV is positively correlated with the vehicle loading capacity, i.e., the farther the EDV travels, the more packages it can carry. This model mainly arranges EDV’s driving routes according to the driving distance of EDV. For describing this problem, we employ the HFVRP that is used to find a fine-grained arrangement plan.

• (ii) Static Route Planning Model: With the shared information including EDVs status and road condition, the optimal driving path between ESs (or end users) should be calculated. In this part, the driving route of each EDV can be obtained through solving the Static VRP model in a relatively short time.

• (iii) Dynamic Route Planning Model: There may be uncertain factors in the distribution process, such as rush hour, special weather, traffic control and other situations, which may cause road congestion. If the EDV still follows the route derived from static path planning and drives into a congested road, it will increase the travel time for delivery, which ultimately affects the efficiency of delivery and usrs’ satisfaction. Hence, a dynamic route planning model combined with Global Positioning System (GPS) is used to obtain the traffic situation between each ES and path planning, which enables the delivery path to be reasonably planned on the basis of new realistic conditions. When each EDV finishes the delivery for a certain ES and is about to drive to the next station, if congestion is found in the next road section, it will re-plan the route according to the current road conditions, so as to complete the delivery in the shortest time with small delivery distance. Since there are dynamic constraints, we can consider this problem as DVRP.

These three models are independent and correlated. Any single of them is not able to describe the VRP problem for DVDS. Consequently, multistage model is proper solution. For further detailing the formulation of this multistage model, there are several assumptions that are needed to introduce.

Assumption of the multistage model

Assumptions about DC:

• (i) Only consider the case of single DC for each delivery routing planning.

• (ii) The time for EDV to load packages in DC and unload packages in each ES are not considered, but the driving distance and driving time of EDV with its remnant power of batteries are mainly considered.

Assumptions about EDV:

• (i) Each route is planed for only one EDV.

• (ii) Ignore the power loss generated by each EDV when it stops or waits in congested roads.

• (iii) Temporary faults on EDV or delivery errors are not considered during delivery.

Assumptions about delivery routes:

• (i) The path between any two ESs can be regarded as an undirected arc, i.e., an EDV can drive from $E{S}_i$ to $E{S}_j$ or from $E{S}_j$ to $E{S}_i$.

• (ii) The path between any two ESs is bidirectional but the congestion coefficient only has one-way weight, i.e., the coefficient matrix is a symmetric matrix.

Our focus is to plan the route of EDVs. Under the above restrictions and assumptions, we should make reasonable distribution arrangements.

Problem formulation

The weighted network graph is used to describe the delivery network. Given a simple undirected graph $G=<v,e,c>$ with $N+1$ nodes, where $v=\{v_1,v_2,\dots ,v_n,v_{n+1}\}$ is the node set in the graph. $v_{n+1}$ represents DC and the other nodes ( $v_1->v_n$) present ESs. The e is the edge set of the graph and c is the traffic congestion coefficient matrix. Other symbols are described as follows: K is the number of EDV, $q_k$ represents the package quantity (including loading volume and loading weight) to be loaded by the kth EDV in DC and $Q_k$ refers to the rated maximum loading capacity of the kth EDV (also including loading volume and loading weight). $D_k$ represents the maximum driving distance of the kth EDV and $d{d}_k$ denotes the driving distance traveled by the kth EDV to complete all deliveries. $T_k$ represents the time taken for the kth EDV to complete all deliveries. $x_{ijk}$ refers to whether the kth EDV travels from $v_i$ to $v_j$, which is a variable of 0 and 1.

Aiming at minimizing the total route length and delivery time of delivery vehicles, we design a uniform mathematical model to describe the multistage model. The mathematical model established in this paper is as follows:

(1) $$minZ=\sum _{i=1}^K(d{d}_k+T_k)$$

(2) $$s.t.\sum _{k=1}^K\sum _{i=1}^{N+1}{x}_{ijk}=1(j=1,2,\dots ,N)$$

(3) $$\sum _{k=1}^K\sum _{j=1}^{N+1}{x}_{ijk}=1(i=1,2,\dots ,N)$$

(4) $$\sum _{j=1}^N{x}_{N+1jk}=\sum _{i=1}^N{x}_{iN+1k}<=1,k\in \{1,2,\dots ,K\}$$

(5) $$\sum _{k=1}^K\sum _{j=1}^N{x}_{N+1jk}<=K$$

(6) $$Q_k-q_k>=0,k\in \{1,2,\dots ,K\}$$

(7) $$D_k-d{d}_k>=0,k\in \{1,2,\dots ,K\}$$

(8) $$x_{ijk}=\{\begin{array}{cc}1,& ifthekthEDVdeliversfrom{v}_{i}to{v}_j\\ 0,& otherwise\end{array}$$

In the above model, the objective function (1) considers minimizing the length of delivery path and delivery time. Constraints (2) and (3) indicate that each ES’s packages can only be delivered by one EDV, i.e., there is no such thing as multiple vehicles delivering packages to the same ES. Constraint (4) means that EDV no longer return to DC to pick up packages when performing the delivery service but pick up all packages that need to be delivered at the beginning of EDV’s delivery. Constraint (5) indicates that the number of EDVs departing from DC doesn’t exceed the number of EDVs available for transport services. Constraint (6) refers to that the package quantity delivered by each EDV is less than the maximum loading capacity of the EDV, which includes loading weight and loading volume. Constraint (7) indicates that the remaining driving distance of each EDV is greater than the distance to be traveled for the distribution of all packages. Constraint (8) indicates that for the $k-th$ EDV, if it drives from $E{S}_i$ to $E{S}_j$, then $x_{ijk}=1$, otherwise $x_{ijk}=0$.

As we can see that the uniform mathematical model is more complicated and has a large-scale solution space. For obtaining a routing planing for all EVDs, there are at least three times of optimization of the objective function with different constraints. Of course, an effective optimization algorithm is needed. It is well known that GA can be used to all kinds of optimization problem. But it still has its own drawbacks, such as pre-mature and easily trapped into local optimization, especially with the large-scale solution space. Consequently, we further improve the GA to effectively solve the uniform model with different situations.

Improved GA

Compared with other heuristic algorithms searching from a single point, GA searches from multiple points at the same time. However, traditional GA is easy to pre-mature when solving large-scale problems and cannot solve the uniform model effectively, so the optimal results can be efficiently obtained only by improving the GA.

GA usually consists of four stages: initial population, chromosome evaluation selection, crossover and mutation. The latter three stages need many iterations, then the new population will be better than the previous generation and gradually approach the global optimal solution. Next, the improvements of GA will be introduced below.

Encoding Settings: Genetic coding mainly adopts the set of natural number to represent the driving routes of K EDV, numbers $1,2,\dots ,N$ represents the ESs to be accessed by K EDVs. In order to represent K routes with one chromosome, $K-1$ interruption points are added for route segmentation and these points are $N+2,\dots ,N+K$. Each interruption point represents DC, so that the chromosome can be split into K loops and each of which represents the driving route of an EDV. A chromosome code is assigned to three EDVs and delivery routes are shown in Fig. 2, where $1,2,\dots ,13$ represent ESs, 14 represents DC and 15, 16 represent interruption points.

Selection Operation: At the beginning of GA, some particularly excellent individuals are usually produced. If the optimal individuals are selected according to the fitness value from high to low or according to the proportion, these abnormal individuals will control the selection process because their competitiveness is too prominent. As a result, at the later stage of evolution, these abnormal individuals occupy the whole population and their potential for further optimization is reduced. In the end, we may not get the global optimal solution but only get the local optimal solution. Hence, we use the Tournament Selection to select individuals by randomly selecting n individuals from the entire population (usually n = 2) and pitting them against each other in a series of competitions, leaving only the best of them. This method doesn’t need to sort all the fitness values, is not easy to fall into the local optimum and has a small time complexity (Shukla, Pandey & Mehrotra, 2015).

Crossover Operation: GA crossover operation exchanges some genes between two chromosomes in some way to produce a new superior individual. In this paper, Partial-Mapped Crossover (PMX) (Goldberg & Lingle, 1985) combined with Order Crossover (OX) is used, where the probability of executing PMX is $\delta$ and the probability of executing OX is 1- $\delta$. Both methods involve setting up two intersections, intercepting the gene fragments between the two intersections and recombining the genes after the crossover. After the crossover operation, it can better enhance the ability of GA to open up new search space, so that the final obtained offspring will be closer to the optimal solution of the problem.

Mutation Operation: The mutation operation can maintain the diversity of the population to some extent, increase the local search ability of the algorithm and prevent the premature convergence of the algorithm. The mutation operations adopted in this paper include exchange mutation, inversion mutation and insertion mutation. Here we transform inversion mutation and insertion mutation into selective inversion mutation (SIM) and selective insertion mutation (SIM’) to change the search ability of the whole algorithm, where the probability of executing SIM is $\mu$ and the probability of executing SIM’ is $\sigma$.

Inversion mutation refers to the 180-degree turnover of a gene segment in the chromosome. When it is applied to VRP, it means to reverse the driving route from $E{S}_A$ to $E{S}_B$ to produce a new distribution route. For the operation of SIM, randomly select a $E{S}_A$ then find the last $E{S}_{AEnd}$ in the distribution route where the ES is located. Between $E{S}_A$ and $E{S}_{AEnd}$, select a location nearest to $E{S}_A$ and then reverse the whole distribution route from $E{S}_{ARight}$ to $E{S}_{A^{\mathrm{\prime }}}$ to generate a new distribution route. For example, the process of SIM is in Fig. 3.

We randomly select a $E{S}_2$ and we know $E{S}_6$ is the last station of the distribution route according to interruption point 15. If the distance between $E{S}_5$ and $E{S}_2$ is the shortest, the new delivery route after selective inversion mutation is ‘child’.

On the whole, selective inversion mutation can find the good evolutionary characteristics of the gene sequence without disrupting the good chromosome sequence.

Insertion mutation refers to randomly taking two positions on the chromosome and then the second gene is inserted after the first one. Corresponding to the VRP, $E{S}_B$ is taken as the next driving destination of $E{S}_A$. The newly proposed SIM’ is to randomly select two ESs ( $E{S}_A$, $E{S}_B$) and then find the last $E{S}_{BEnd}$ in the distribution route where $E{S}_B$ is located. Between $E{S}_B$ and $E{S}_{BEnd}$, select a $E{S}_A$’, which is closest to $E{S}_A$ and compare the distance between them and the distance between $E{S}_A$ and $E{S}_{ARight}$. If the former is less than the latter, remove A’ and insert it after A to form a new distribution route; otherwise, the original distribution route will not be changed. For example, the process of SIM’ is in Fig. 4.

We randomly select two ESs: $E{S}_4$ and $E{S}_{10}$. Behind $E{S}_{10}$ (including $E{S}_{10}$), we search which station is closest to $E{S}_4$. If the distance from $E{S}_9$ to $E{S}_4$ is closest and less than the distance from $E{S}_4$ to $E{S}_5$, the new delivery route after selective insertion mutation is ‘child’.

This method expands the diversity of chromosomes to a certain extent and helps to find more effective solutions.

Next we will introduce each sub model in the multistage model in turn. The first model is centered on the farthest distance traveled by the EDV to get the ESs that EDV needs to pass through and then gives a preliminary driving route. According to the route of each EDV, each vehicle will pick up the packages to be delivered at DC. The second model is based on the first model to carry out detailed static route planning for each EDV and work out a more reasonable route to complete delivery. The third model is to obtain dynamic traffic congestion to carry out route planning, develop a more time-saving route scheme and effectively avoid the delay of delivery due to entering the congested road section.

The application of improved GA to multistage model

Improved GA with clustering algorithm for package arrangement Here, in order to solve the delivery problem in a large area efficiently, the improved GA needs to generate a better initial population and complete path planning more quickly. Considering that ESs with similar locations can be arranged to the same EDV. We employ k-means++ clustering algorithm (Arthur & Vassilvitskii, 2006) to enhance the searching ability of the improved GA called “KGA-based allocation algorithm”. This algorithm can group stations with similar locations and generate better individuals, which make the package arrangement more excellent in accuracy and speed.

Under complex requirements, the GA usually adopts the scheduling strategy and objective that can satisfy all demands as soon as possible. Therefore, we choose to minimize the driving distance and minimize the gap between the actual driving distance and the maximum driving distance of the EDV. The purpose is to evenly distribute delivery routes of different lengths to each EDV, i.e., the longer delivery route is allocated to EDV with longer driving distance. So we transform the multi-objective optimization problem into a single-objective optimization problem, whose objective function is defined as follows:

(9) $$minZ={\omega }_1\ast \sum _{k=1}^{K}d{d}_k+{\omega }_2\ast \sum _{k=1}^K(D_k-d{d}_k)$$where ${\omega }_1$ can be called as the distance weight, ${\omega }_2$ can be called as the “gap” weight and ${\omega }_3$ can be called as time weight appearing in below. The KGA-based allocation algorithm can quickly obtain the ESs that each EDV should passes through and then we can arrange the delivery for each EDV.

Algorithm 1 shows the pseudocode of KGA-based allocation algorithm.

Where I is the maximum number of iteration, P is the size of population, c_rate and m_rate are the crossover and mutation rates, TDD is total driving distance, i.e., sum of distances traveled by five EDVs, bd[m] is the distance that the m-th EDV in the global optimal individual needs to drive and DR is the driving route of all EDVs. So far, the first stage of express delivery is completed and the detailed route planning will be realized by the second and third stages.

Improved GA for static route planning: In the phase of solving the static route planning model, stations that each EDV needs to pass through have been obtained. Next, we have to decide the routing plan for each EDV with assigned stations. Hence clustering algorithm is no longer used for population initialization. We take the driving routes of each EDV generated in the previous stage and shuffle the sequence of the routes to form a new individual. P individuals are combined into a new population, where each individual is represented as a different driving route of the EDV. It should be noted that $d_{ij}$ is the distance from the $E{S}_i$ to the $E{S}_j$ and the driving speed of vehicles on the road is not only affected by road speed limit but also by traffic congestion or special weather. Therefore, we introduce the traffic congestion coefficient c to estimate the driving time (DT) taken by vehicles from station i to station j, and the formula is as follows:

(10) $$D{T}_k=\sum _{i=1}^{N+1}\sum _{j=1}^{N+1}(c_{ij}\ast {\displaystyle \frac{x_{ijk}\ast {\mathrm{d}}_{ij}}{v})},k\in \{1,2,\dots ,K\}$$

We mainly focus on the urban logistic distribution, where v is the driving speed of EDV in the process of distribution in the city, c indicates the congestion coefficient of the road between different ESs and $c\in [1,5]$. The larger value of c is, the more serious the road congestion is and the calculated driving time is greater than the time that it drives at a constant speed v. For this circumstance, the routing plan should take the shortest driving time as the objective condition that EDV’s driving distance may be more longer for preventing driving into congested areas and ensuring that each EDV can spend the least time to complete delivery. Therefore, we also transform the multi-objective problem into a single-objective optimization problem in order to shorten the traveling time and distance at the same time and its objective function is defined as follows.

(11) $$minZ={\omega }_1\ast d{d}_k+{\omega }_3\ast \mathrm{D}{\mathrm{T}}_k$$

The improved GA is used to solve the static route planning model, the pseudocode of GA-based static route planning algorithm is shown in Algorithm 2.

Where pop represents the whole population. This algorithm obtains the traffic congestion coefficient between each ES in the transportation network after the vehicles complete packages loading and then makes the route planning. Once the route of each EDV is determined, it will not be changed.

Improved GA for dynamic route planning: According to solving the static route planning model, a route planning is obtained for each EDV. However, road information collected at a certain time point is time-sensitive. If we only follow the distribution route obtained before departure, it’s likely to cause vehicles to enter the congested road and delay the delivery time. Therefore, we want to establish dynamic route planning according to real-time traffic conditions. In order to search the distribution route dynamically, we need to consider how to deal with the dynamic model. Generally, there are two dynamic calculation methods. One is to calculate at a fixed time interval. Starting from the departure of ESs, every time interval, the system will automatically obtain the traffic information between stations in the transportation network and re-plan the route. The characteristics of this method are as follows: when the time interval is too large, the traffic information will be out of date; when the time interval is too short, the route planning will be more accurate but the calculation time will be longer.

The other method is to calculate at the node. After arriving at any node, the EDV will re-obtain the traffic information in the transportation network to re-planing the distribution route according to the remaining nodes until it finally returns to the DC. Here, we adopt the second method and make corresponding improvements.

When EDV arrives at a certain $E{S}_i$ and finishes unloading, it still follows the distribution route plan made at the previous node: drive to $E{S}_j$. At the same time, according to the comparison between $c_{ij}$ (represents the traffic congestion coefficient between $E{S}_i$ and $E{S}_j$) and $\beta$ (the threshold determined by experience), if $c_{ij}<\beta$, it means that the congestion between the two stations is not very serious and EDV can still drive according to the distribution route, i.e., from $E{S}_i$ to $E{S}_j$. Otherwise, it is necessary to re-planing the distribution route for the remaining distribution nodes. At this circumstance, according to the new distribution route, the EDV should drive from $E{S}_i$ to $E{S}_k$ (other node with a lower congestion coefficient). The objective function of this model is consistent with the static route planning model.

The improved GA is used to solve the dynamic route planning model, the pseudocode of GA-based static route planning algorithm is shown in Algorithm 3.

Where chrome represents an individual in the population. This algorithm re-plans the route when the vehicles encounter serious traffic jams until each EDV completes the delivery.