1. Introduction

2. Literature Review

Table 1. related literatures.

3. Problem Formulation

3.1. Problem Description

3.1.1. Take-Out Delivery Flow

The take-out delivery process involves four stakeholders: customers, AGV, and restaurants. The connection among them is established by the take-out platform. Figure 1 shows the flow chart of take-out delivery. Customers place orders on the take-out platform through mobile APP (Step 1, Information flow). After the platform receives orders, it sends the order information to the restaurant (Step 2, information flow). The restaurants provide feedback to the platform on whether to accept the order in a short period of time. (Step 3, information flow). After receiving feedback, the platform sends assigns orders to appropriate AGV (Step 4, information flow). The corresponding AGV goes to the restaurant nodes to collect the food based its working state (Step 5, physical flow). Then, the AGV delivers the food to the customer's location within a period (Step 6, physical flow). Finally, they provide feedback to the take-out scheduling platform (Step 7 information flow).

3.1.2. Real-Time Dynamic and Rolling Scheduling

As universally acknowledged, the order time in take-out delivery exhibits significant variability. The AGV scheduling platform needs to continuously receive and assign orders.

Therefore, it is necessary to adjust the AGV's path constantly. In other words, when a new order is inserted, the algorithm should intervene to re-plan the allocation path of the AGVs. The insertion time of the new order is called the decision time of rolling scheduling. AGV should not immediately change the current action when performing delivery operations upon receipt of a new order to ensure consistency of operation. This action being carried out by the AGV is called inertia action. Therefore, after each rolling schedule, the starting time of the readjusting path should be the end time of the inertial action. It is important to note that the completion of the current inertia of the AGV does not equal the completion of the current order. Figure 2 shows an example of the order task sequence and routing combination based on the rolling scheduling framework.

Figure 2. Rolling pick-up and delivery path sequence and routing combination.

Figure 2. Rolling pick-up and delivery path sequence and routing combination.

3.1.3. Uncertain Preparation Times

The process of AGV completing delivery tasks involves numerous uncertain factors, including traffic resistance, meal preparation time of restaurants, time of AGV serving customers, etc. Among them, the uncertainty of meal preparation time of restaurants is the factor that has the greatest impact on the system [24]. Figure 3 shows the difference between the expected delivery time and the actual delivery time. PT1 represents the travel time for AGV to pick up order 1 from restaurant node 1.PT2 represents the travel time for AGV to pick up order 2 from restaurant node 2. DT1 represents the travel time of AGV to customer node 1 . DT2 represents the travel time of AGV to customer node 2 . WT1 represents the waiting time of AGV at restaurant node 1. WT2 represents the waiting time of AGV at restaurant node 2. ST represents the time of AGV serving customers. If the food arrives at the restaurant node too late, the waiting time of the AGV at the restaurant node will become longer, which will lead to the order timeout, that is, the actual arrival time of the AGV is later than the latest delivery time of the order, and the customer satisfaction will decline. Therefore, it is necessary to consider the uncertainty of the arrival time, which can better guarantee the robustness of the system.

3.1.4. Routing Combination

In terms of order delivery, for different orders with a small order time span and the same restaurants node, the AGV can wait slightly to pick up the food together; For different orders with the same customer involved, the AGV can unify the collection and delivery. This mode of operation is known as action consolidation (AC). Figure 4 shows the rolling pick-up and delivery path considering routing combination. The original path of the AGV is 1-2-3-4-5-6. A new order is received at customer node 2 (a new restaurant node 7 is inserted). Since restaurant node 7 and restaurant node 3 correspond to the same customer node 4, the AGV will merge food delivery at customer node 4. After the first rolling dispatching, the path is changed to 1-2-7-3-4-5-6 (the executed path is 1-2). A new order is received on restaurant node 7 (a new customer node 8 is inserted). Since customer node 8 and customer node 4 correspond to the same restaurant node 3, the AGV merges food pick-up at the customer nodes 3. After the second rolling scheduling, the path is changed to 1-2-7-3-4-8-5-6 (the executed path is 1-2-7). Although taking AC decision into account in route planning will increase the complexity of optimization, it can significantly improve delivery efficiency and customer satisfaction.

3.1.5. Look-Forward

In real life, route planning is often considered based on look-forward.AGV generally gives priority to the delivery of orders that are closer to their current location, and orders that are relatively far away will be slightly shelved. The purpose is to wait for future orders that may be able to combine with current orders.

3.2. Modeling

To solve the problem studied, the following multi-objective rolling pick-up and delivery route planning model (MO-RPDRP) is developed.

The model is based on the following assumptions:

• The inertial action currently being executed cannot be changed.
• The preparation time of the restaurant is subject to Gaussian distribution.
• The AGV's meal pick-up time occupies a small amount of time in the entire delivery process, so it is not considered.

In MO-RPDRP model, $N$ represents the order set involved in the current rolling period. $N_1$ represents the current set of orders that have been picked up but not delivered. If the inertial action of AGV is food picking up, the order involved is also included $N_1$. If the inertial action is food delivery, the order is considered completed, it is not included in $N_1$. $N_2$ represents the current order set that have not been picked up. $I$ represents the set of restaurant nodes. $J$ Represents the set of customer nodes. $A$ represents the set of all nodes. If the order is one-to-one mapping to both the restaurant nodes and the customers, $\left|N_1\right|+2\left|N_2\right|=\left|A\right|$. If there are multiple orders for the same restaurant node or multiple orders for the same customer, $\left|N_1\right|+2\left|N_2\right|>\left|A\right|$. If AC strategy is not executed, the length of pick-up and delivery path is $\left|N_1\right|+2\left|N_2\right|$. Otherwise, the actual length of pick-up and delivery path is in the interval $\left[\left|A\right|,\left|N_1\right|+2\left|N_2\right|\right]$. To make orders and restaurant nodes, orders and customers are mapped one-to-one, the virtual restaurant nodes and virtual customer nodes need to be set. The set of virtual restaurant nodes is represented as $I^{virtual}$, it and the set $I$ forms the set of restaurant node expansion $I\text{'}$, $I\text{'}=I\cup I^{virtual}$. The number of the virtual restaurant node set $I^{virtual}$ continues with the number of the restaurant node set $I$. The set of virtual customer nodes is represented as $J^{virtual}$, it and the set $J$ forms the set of customer node expansion $J\text{'}$, $J\text{'}=J\cup J^{virtual}$, the number of virtual customer node set $J^{virtual}$ continues with the number of the customer node set $J$. The set of all node expansion is represented as $A\text{'}=I\text{'}\cup J\text{'}$. For $n\in N_1$, it has a unique customer node corresponding to it. For $n\in N_2$，the only restaurant node can be found, and it has a customer node corresponding to it. The length of the virtual open chain (pick-up and delivery path) is $\left|A\text{'}\right|$.

The essence of MO-RPDRP decision is the sequence of all nodes in the set $A\text{'}$ and the identification of AC strategy. The decision variable $x_{ij}$ is set as 0-1 variable. If the AGV travels from node $i$ to node $j$,the decision variable $x_{ij}$ is equal to 1. Otherwise, it is equal to 0. Decision variable $x_{ij}$ can execute AC strategy while deciding the sequence of virtual nodes. The identification of AC strategy is reflected by 0-1 variables $p_l^1$ and $p_l^2$. Other sets, parameters and variables are shown in Table 2 for details.

Objective function 1 minimizes the expected order timeout rate, $\omega$ is the timeout rate for all orders.

$$\min F_1=E\left(\omega \right)$$

(1)

Objective function 2 minimizes the expected total customer waiting time. From the perspective of the customer, the customer needs the AGV to deliver the meals as soon as possible. The derivation of the completion time $g_n$ of the order $n$ is shown in (14).

$$\min F_2=E\left({\displaystyle \sum _{n\in N}\left(g_n-r_n\right)}\right)$$

(22)

Objective function 3 maximizes scheduling after-effect. A look-forward objective is introduced to adjust the impact of this decision on subsequent scheduling. See 3.1.5 for detailed description. In terms of scheduling, orders are completed intensively in the early stage, while orders are completed relatively dispersed in the later stage. In the objective function, it is expressed as (3). Figure 5 shows the relationship between the number of remaining nodes and the number of unfinished orders.

$$\min F_3=E\left({\displaystyle \sum _{l=1}^{|A\text{'}|}{\lambda }_l(a_l-a_{l-1})}\right)$$

(3)

Figure 5. Changes in the number of orders and nodes to be delivered.

Figure 5. Changes in the number of orders and nodes to be delivered.

Constraints (4) and (5) delineate deduction constraints governing the sequential visitation order of order nodes. The access sequence $s_l$ of the $l$th node is deduced by decision variables $x_{ij}$. Obviously, $s_{|A\text{'}|}$ is the last access node.

$$s_1={\displaystyle \sum _{j\in A\text{'}}j{x}_{s_{0}j}}$$

(4)

$$s_{l+1}={\displaystyle \sum _{j\in A^{\text{'}}}j{x}_{s_{l}j}},\forall l\in \text{{}1,2,\dots ,|A\text{'}|-1\text{}}$$

(5)

Constraints (6) and (7) are constraints on the volume of take-out carried by the AGV. The volume of take-out carried upon arrival cannot exceed the maximum capacity of the AGV. If the AGV passes through the restaurant node $s_l$ (pick-up stage), the take-out volume is increased .The take-out volume $d_{l+1}$ carried by the AGV upon arrival $s_{l+1}$ is the sum of $d_l$ and $v_{\arg \alpha (s_l)}$. If the AGV passe through the customer node $s_l$ (the delivery stage), the take-out volume is reduced. The takeaway volume $d_{l+1}$ carried by the AGV on arrival $s_{l+1}$ is the difference between $d_l$ and $v_{\arg \alpha (s_l)}$.

$$\begin{array}{c}{d}_0={\displaystyle \sum _{n\in N_1}{v}_n}\\ d_{l+1}=\left\{\begin{array}{l}\begin{array}{cc}{d}_l+v_{\arg \alpha (s_l)}& s_l\in I\text{'}\end{array}\\ \begin{array}{cc}{d}_l-v_{\arg \beta (s_l)}& s_l\in J\text{'}\end{array}\end{array}\right.\forall l\in \text{{}1,\dots ,|A\text{'}|-1\text{}}\end{array}$$

(6)

$$d_l\le V,\forall l\in \text{{}1,\dots ,|A\text{'}|-1\text{}}$$

(7)

Constraint (8) imposes limitations on the visiting sequence of order nodes. For the same order, the AGV should visit the restaurant node ${\alpha }_n$ first, then the customer node ${\beta }_n$ is visited. It is expressed as that the time when the AGV arrive at the restaurant node $a_{\arg s({\alpha }_n)}$ should be less than the time when they arrive at the customer node $a_{\arg s({\beta }_n)}$.

$$a_{\arg s({\alpha }_n)}<a_{\arg s({\beta }_n)},\forall n\in N$$

(8)

Constraints (9), (10) and (11) are the AC identification constraints of order.

$$p_l^1=\left\{\begin{array}{l}1，ma{p}_{s_l}^1=ma{p}_{s_{l+1}}^1\\ 0\end{array}\right.s_l,s_{l+1}\in I\text{'},\forall l\in \{1,\dots |A\text{'}|-1\text{}}$$

(9)

$$p_l^2=\left\{\begin{array}{l}1，ma{p}_{s_l}^2=ma{p}_{s_{l+1}}^2\\ 0\end{array}\right.s_l,s_{l+1}\in J\text{'},\forall l\in \{1,\dots |A\text{'}|-1\text{}}$$

(30)

$$p_l^1+p_l^2\le 1，\forall l\in \{1,\dots |A\text{'}|-1\text{}}$$

(41)

Constraint condition (14) is the constraint of the completion time of the order, $g_n$ is the sum of $a_{\arg s({\beta }_n)}$ and $t_n^s$. The derivation of $a_l$ is shown in (13). If there is a combination of routings between $s_l$ and $s_{l-1}$ (merged meal pick-up or delivery), $a_{l-1}$ is equal to $a_l$. If $s_{l-1}$ is the restaurant node, which does not produce merged meal pick-up with the last node, $a_l$ is the sum of $a_{l-1}$ and $t_{s_{l-1}{s}_l}^d$. If $s_{l-1}$ is the restaurant node, which produces merged meal pick-up with the last node, $se{t}_l$ is expressed as the sequence set of $s_{l-1}$ and last merged meal restaurant node. The expression is (12). $se{t}_l$ is the sequence set of $s_{l-1}$ and last merged meal restaurant node. $a_{l-1}$ is the maximum sum of the ordering time in $se{t}_{l-1}$ and the meal preparation time of the restaurant. The restaurant's meal preparation time $t_{\arg \alpha (s_w)}^p$ follows the Gaussian distribution. In other words, for different restaurants, they obey the Gaussian distribution with different expectation ${\mu }_i$, and variance ${\sigma }_i^2$. If $s_{l-1}$ is the customer node, it does not produce merged meal delivery with the last node, $a_l$ is the sum of $a_{l-1}$, $t_{\arg \beta (s_l)}^s$, and $t_{s_{l-1}{s}_l}^d$.If $s_{l-1}$ is the customer node, it produces merged meal delivery with the last node, $a_{l-1}$ is equal to $a_{l-2}$. The service time of the same customer node is same, so the calculation of $a_l$ is not affected.

$$se{t}_l=\text{{}l\text{'}|{\displaystyle \prod _{l\text{'}=1}^{l-1}{p}_{l\text{'}}^1}=1\}\cup l$$

(52)

$$\begin{array}{l}{a}_1=T_0+t_{s_0{s}_1}^d\hfill \\ a_l=\left\{\begin{array}{l}\begin{array}{cc}{a}_{l-1}& p_{l-1}^1+p_{l-1}^2=1\end{array}\\ \begin{array}{cc}\max (a_{l-1},\underset{w\in se{t}_{{\hspace{0.25em}}_{l-1}}}{\max }(r_{\arg \alpha (s_w)}+t_{\arg \alpha (s_w)}^p))+t_{s_{l-1}{s}_l}^d& \begin{array}{l}{p}_{l-1}^1+p_{l-1}^2=0\\ s_{l-1}\in I\text{'}\end{array}\end{array}\\ \begin{array}{cc}{a}_{l-1}+t_{\arg \beta (s_{l-1})}^s+t_{s_{l-1}{s}_l}^d& \hspace{0.25em}\end{array}{p}_{l-1}^1+p_{l-1}^2=0,s_{l-1}\in J\text{'}\end{array}\right.\end{array}$$

(63)

$$g_n=a_{\arg s({\beta }_n)}+t_n^s，\forall n\in N$$

(74)

Constraint condition (15) is the quantity constraint of uncompleted orders.

$$\begin{array}{c}{\lambda }_1=\left|N\right|\\ {\lambda }_{l+1}=\left\{\begin{array}{l}\begin{array}{cc}{\lambda }_l& s_l\in I\text{'}\end{array}\\ \begin{array}{cc}{\lambda }_l-1& s_l\in J\text{'}\end{array}\end{array}\right.\end{array}$$

(85)

Constraint conditions (16) - (19) are open chain structure constraints. To maintain the flow balance between intermediate nodes, (16) - (17) traverses all restaurant nodes and customer nodes. (18) and (19) are open chain endpoint constraints, the two endpoints of the path are not connected. (20) is the constraint to eliminate subloop.

$${\displaystyle \sum _{i\in A\text{'}\cup s_0}{x}_{ij}}=1，\forall j\in A\text{'}$$

(96)

$${\displaystyle \sum _{j\in A\text{'}}{x}_{ij}}=1，\forall i\in \left(A\text{'}-s_{\left|A\text{'}\right|}\right)\cup s_0$$

(107)

$${\displaystyle \sum _{i\in A\text{'}\cup s_0}{x}_{s_{\left|\mathrm{A}\text{'}\right|}i}}=0$$

(118)

$${\displaystyle \sum _{i\in A\text{'}}{x}_{i{s}_0}}=0$$

(129)

$${\displaystyle \sum _{i\in H}{\displaystyle \sum _{j\in H}{x}_{ij}\le \left|H\right|-1,\forall H\subset A^{\text{'}}}},2\le \left|H\right|\le \left|A^{\text{'}}\right|-2$$

(20)

The established MO-RPDRP is a stochastic programming model. Three objective functions in the model are derived as follows.

$$\begin{array}{l}\min F_1=E(\omega )\\ =E({\displaystyle \frac{{\displaystyle \sum _{n\in N}^{\hspace{0.25em}}\max (0,\mathrm{sgn}(g_n-b_n))}}{\left|N\right|}})\\ ={\displaystyle \frac{1}{\left|N\right|}}{\displaystyle \sum _{n\in N}^{\hspace{0.25em}}E(\max (0,\mathrm{sgn}(g_n-b_n)))}\\ ={\displaystyle \frac{1}{\left|N\right|}}{\displaystyle \sum _{n\in N}^{\hspace{0.25em}}{\displaystyle {\int }_0^{+\infty }\mathrm{sgn}(g_n-b_n)f(g_n-b_n)d(g_n-b_n)}}\\ ={\displaystyle \frac{1}{\left|N\right|}}{\displaystyle \sum _{n\in N}^{\hspace{0.25em}}{\displaystyle {\int }_0^{+\infty }f(g_n-b_n)d(g_n-b_n)}}\end{array}$$

(213)

$f(g_n-b_n)$ represents the probability density function of $g_n-b_n$

$$\begin{array}{l}\min F_2=E\left({\displaystyle \sum _{n\in N}\left(g_n-r_n\right)}\right)\\ =E({\displaystyle \sum _{n\in N}{g}_n})-{\displaystyle \sum _{n\in N}{r}_n}\end{array}$$

(22)

$$\begin{array}{l}\min F_3=E\left({\displaystyle \sum _{l=1}^{|A\text{'}|}{\lambda }_l(a_l-a_{l-1})}\right)\\ ={\displaystyle \sum _{l=1}^{|A\text{'}|}{\lambda }_{l}E\left(a_l-a_{l-1}\right)}\\ ={\displaystyle \sum _{l=1}^{|A\text{'}|}{\lambda }_l\left(E(a_l)-E(a_{l-1})\right)}\end{array}$$

(23)

Obviously, according to (21) - (23), the core of the derivation of three objective functions is the derivation of $E(a_l)$. So we derive the expression（24）of$E(a_l)$ according to the recursive form of (13).

$$a_l=\left\{\begin{array}{l}\begin{array}{cc}{a}_{l-1}& p_{l-1}^1+p_{l-1}^2=1\end{array}\\ \begin{array}{cc}\max (a_{l-1},\underset{w\in se{t}_{s_{l-1}}}{\max }(r_{\arg \alpha (s_w)}+t_{\arg \alpha (s_w)}^p))+t_{s_{l-1}{s}_l}^d& \begin{array}{l}{p}_{l-1}^1+p_{l-1}^2=0\\ s_{l-1}\in I\text{'}\end{array}\end{array}\\ a_{l-1}+t_{\arg \beta (s_{l-1})}^s+t_{s_{l-1}{s}_l}^d{\hspace{0.25em}}_{\hspace{0.25em}}{s}_{l-1}\in J\text{'},p_{l-1}^1+p_{l-1}^2=0\end{array}\right.$$

(24)

Obviously, when $p_{l-1}^1+p_{l-1}^2=1$，$E(a_l)=E(a_{l-1})$ when $p_{l-1}^1+p_{l-1}^2=0$ and $s_{l-1}\in J\text{'}$, $E(a_l)=E(a_{l-1})+$$t_{\arg \beta (s_{l-1})}^s+t_{s_{l-1}{s}_l}^d$.

When $p_{l-1}^1+p_{l-1}^2=0$and $s_{l-1}\in I\text{'}$. $\forall l>1$，due to the different forms of $a_{l-1}$ under $l$. When the AGV did not pass through the restaurant node before arriving at $s_{l-1}$, $a_{l-1}$ is a definite constant. When the AGV passe through the restaurant node before arriving at $s_{l-1}$, $a_{l-1}$ is a random variable.

1) If the AGV did not passes through the restaurant node before arriving at $s_{l-1}$, $a_{l-1}$is a constant. The expectation of $a_l$ can be obtained by referring to the derivation of $E(a_2)$.

2) As can be seen from the above, if the AGV pass through the restaurant node before arriving at node $s_{l-1}$, $a_{l-1}$ is a random variable. Obviously, the original function of the $E(a_l)$integrand in the expression is a non-elementary function [29], it is difficult to solve this problem by directly finding the original function.

Therefore, we choose to use the discretization method to estimate the expectation of $a_l$[28]. In other words, the meal preparation time obeying the Gaussian distribution is transformed into a series of discrete time points. The expectation of $a_l$ is estimated by using the rules and linear operations in Table 3, Table 4 and Table 5. Obviously, the time when the AGV arrive at any node is a time set,and the time when they leave any node is also a time set.

By adopting the above discretization method, on the one hand, the difficulty of directly finding the original function is avoided in the process of finding the expectation of $a_l$; On the other hand, since this problem is a dynamic problem based on the rolling frame, the discretization method can reduce the time complexity. Besides, the calculation efficiency can be improved by adjusting the parameters $\tau$、$\gamma$.

Obviously, we need to modify the objective function. For objective function 1:

$$\begin{array}{l}\min F_1=E(\omega )\\ ={\displaystyle \frac{1}{\left|N\right|}}{\displaystyle \sum _{n\in N}^{\hspace{0.25em}}E(\max (0,\mathrm{sgn}(g_n-b_n)))}\\ ={\displaystyle \frac{1}{\left|N\right|}}{\displaystyle \sum _{n\in N}^{\hspace{0.25em}}{\displaystyle \sum _{a_{\arg s({\beta }_n)}^{arrival\text{'}}\in A_{\arg s({\beta }_n)}^{arrive\text{'}}}\frac{\max (0,\mathrm{sgn}(a_{\arg s({\beta }_n)}^{arrive\text{'}}+t_n^s-b_n))}{\left|A_{\arg s({\beta }_n)}^{arrive\text{'}}\right|}}}\end{array}$$

(25)

For objective function 2：

$$\begin{array}{l}\min F_2=E\left({\displaystyle \sum _{n\in N}\left(g_n-r_n\right)}\right)\\ ={\displaystyle \frac{1}{\left|N\right|}}{\displaystyle \sum _{n\in N}^{\hspace{0.25em}}{\displaystyle \sum _{a_{\arg s({\beta }_n)}^{arrive\text{'}}\in A_{\arg s({\beta }_n)}^{arrive\text{'}}}\frac{(a_{\arg s({\beta }_n)}^{arrive\text{'}}+t_n^s)}{\left|A_{\arg s({\beta }_n)}^{arrive\text{'}}\right|}}}-E({\displaystyle \sum _{n\in N}{r}_n})\\ ={\displaystyle \frac{1}{\left|N\right|}}{\displaystyle \sum _{n\in N}^{\hspace{0.25em}}{\displaystyle \sum _{a_{\arg s({\beta }_n)}^{arrive\text{'}}\in A_{\arg s({\beta }_n)}^{arrive\text{'}}}\frac{(a_{\arg s({\beta }_n)}^{arrive\text{'}}+t_n^s)}{\left|A_{\arg s({\beta }_n)}^{arrive\text{'}}\right|}}}-{\displaystyle \sum _{n\in N}{r}_n}\end{array}$$

(26)

For objective function 3：

$$\begin{array}{l}\min F_3=E\left({\displaystyle \sum _{l=1}^{|A\text{'}|}{\lambda }_l(a_l-a_{l-1})}\right)\\ ={\displaystyle \sum _{l=1}^{|A\text{'}|}{\lambda }_l\left(E(a_l)-E(a_{l-1})\right)}\end{array}$$

(27)

$$E(a_l)={\displaystyle \frac{1}{\left|A_l^{arrive\text{'}}\right|}}{\displaystyle \sum _{a_l^{arrive\text{'}}\in A_l^{arrive\text{'}}}{a}_l^{arrive\text{'}}}$$

4. The Proposed AC-NSGA-III Algorithm

4.2. Operator Design

4.2.1. IRC Coding

Chromosome encoding is a pivotal component of the AC-NSGA-III algorithm. To ensure that the generated chromosome encodings correspond to feasible solutions and to prevent infeasible solutions resulting from crossover operations that violate model constraints (such as delivery priority, vehicle capacity limits, etc.), a decoding rule based on a cellular array structure was designed.

The cellular array $Q$ represents a chromosome, $Q=[x,y]$. The design of the chromosome structure leverages the concept of constructive heuristic methods, constructing the solution on an order-by-order basis. The variable $x$ represents the construction sequence of the generated orders. The variable $\mathrm{y}$ is composed of two components, $y=\left[\begin{array}{l}{y}^1\\ y^2\end{array}\right]$ ,where the values indicate the insertion positions of the corresponding nodes. Specifically, $y^1$ denotes the periodic insertion positions of the restaurant nodes associated with $x$, and $y^2$ denotes the periodic insertion positions of the customer nodes associated with $x$. Both $x$ and $y^2$ are one-dimensional arrays of length |N|, while $y^1$ is a one-dimensional array of length |N2|.

The values in $y^1$ and $y^2$ exhibit periodic reducibility, meaning they are handled cyclically. When these values exceed the total number of available insertion positions, the insertion position for the corresponding order node is determined by the remainder of this value divided by the total number of insertion positions. If the remainder is zero, the insertion position is the last available slot.

The decoding process for the chromosome involves inserting the restaurant and customer nodes corresponding to the order into the delivery route according to the values specified in $y^1$ and $y^2$. This ensures that the resulting solutions adhere to the model constraints while facilitating effective order construction and scheduling.

1) In this process, AC strategy must be followed, and node insertion must not violate constraint (9), meaning that if the actual positions represented by adjacent order nodes in L are the same, they should be combined for pickup or delivery.

2) Node insertion must not violate constraints (6) - (8), meaning it should not breach the delivery volume constraints or the order node and visit sequence restrictions.

3) Since the decoded results are real number sequences, the open chain structure constraint and sub-loop elimination constraints (16) - (20) can be naturally avoided. The positions filtered by this restriction rule are the actual insertable positions.

Figure 7 shows an example of IRC, which includes three orders involving restaurant nodes 1, 2, 3 (denoted as M1, M2, M3) and customer nodes 1, 2, 3 (denoted as C1, C2, C3).$x=(2,1,3)$,$y=\left[\begin{array}{l}5，7，5\\ 3，5，8\end{array}\right]$.In the encoding, the process starts with the restaurant node 2 (M2) and customer node 2 (C2) associated with the first order, forming the path M2-C2. Then, according to the value 5 in $y^1$and the total number of available insertion positions is 2, 5 % 2 = 1, so the restaurant node M1 is inserted into the first available position. The visit sequence after insertion is M2-M1-C2.

Next, according to the value 5 in $y^2$, the customer node C1 associated with order 1 is inserted. Based on constraint 8, the restaurant node 1 must be visited before the customer node 1, so the total number of available insertion positions is 2. 5 mod 2 = 1, so the customer node C1 is inserted into the first available position. The visit sequence after insertion is M2-M1-C1-C2.

Finally, according to the value 5 in $y^1$, the restaurant node M3 associated with order 3 is inserted. Since M3 and M2 correspond to the same actual pickup location, they are merged for pickup. Similarly, C3 is inserted following the same principle, and thus the complete delivery and pickup route for all orders is formed.

4.2.2. Initial Population

The population size is$N$, the initial population$P_0$ is generated randomly.

$x$ is a randomly generated sequence. Every random positive integer in $y^1$、$y^2$ is not greater than $\left|A\text{'}\right|-1$.

4.2.3. Non-Dominant Hierarchical Ranking of Population

Non-dominated sorting effectively distinguishes individuals' relative advantages in multi-objective optimization, aiding in the selection and retention of high-quality individuals, thereby maintaining population diversity and evolutionary effectiveness. Multi-objective optimization algorithms generate selection pressure through population sorting, encouraging the population to continuously approach the Pareto front.

4.2.4. Reference Point Evaluation System

In order to evaluate the population diversity, the reference point selection method of NSGA-III is adopted [31].

4.2.5. Binary Tournament Selection

Two individuals from the population are randomly selected to compete: If the non-dominant ranks of them are different, low-ranking individual is selected to be put into the mating poor. Otherwise, the individual whose associated reference points with smaller number of niches is select to be put into the mating pool. The selection is repeated $N$times.

4.2.6. OBX&SBX Mixed Crossover

For $x$, order-based crossover (OBX) is used, ensuring that no elements are missing after crossover, with a crossover probability of $P_c$.It is showed in Figure 8. In order to better deal with the sparsity of individual space in the process of dealing with multi-objective optimization problems, simulated binary crossover (SBX) [32] is adopted for $y^1$、$y^2$.The non-integer solution generated by SBX was rounded in the way of rounding off.

4.2.7. Mutation Operator

Exchange mutation is adopted for $x$. When an exchange mutation is applied to chromosomes based on binary or integer, two genes are randomly selected. Then, their values are exchanged. When the random number is less than the given mutation rate$P_m$, chromosome mutation occurs [33]. Non-uniform variation was adopted for$y^1$、$y^2$. The original gene value was randomly disturbed. The result was taken as the new gene value. After every locus is calculated with the same probability, it is equivalent to the entire solution vector making a slight change in the solution space. In the operation of non-uniform mutation from $X=x_1{x}_2{x}_3\mathrm{...}{x}_k\mathrm{...}{x}_1$ to $X\text{'}=x_1{x}_2{x}_3\mathrm{...}x{\text{'}}_k\mathrm{...}{x}_1$, if the value range of mutation point $x_k$ is $\left[U_{\min }^k,U_{\max }^k\right]$,the new gene value $x{\text{'}}_k$ will be determined by (28), where $t$ is the current iteration number.

$$x{\text{'}}_k=\left\{\begin{array}{l}{x}_k+\Delta (t,U_{\max }^k-x_k),i{f}_{\hspace{0.25em}}^{\hspace{0.25em}}random(0,1)=0\\ x_k-\Delta (t,x_k-U_{\min }^k),i{f}_{\hspace{0.25em}}^{\hspace{0.25em}}random(0,1)=1\end{array}\right.$$

(28)

4.2.8. Elite Strategy

Parent population $P_t$ and offspring population $Q_t$ are merged into a population $R_t$

Fast non-dominant hierarchical sorting: individuals in $R_t$ are divided into ranks ($F_{P_{t}1},F_{P_{t}2},\dots$). A new population of $S_t$ is constructed from $F_{P_{t}1}$. When $F_{P_{t}L}$ is put in it, $S_t=N$ or $S_t>N$.

Association operation: the line between the origin and the reference point is taken as the reference line. The individuals in population $P_{t+1}(P_{t+1}=S_t/F_{P_{t}L})$ are associated with the reference points on the nearest reference line. The number of individuals associated with each reference point (i.e., the number of niches) is calculated , it is denoted as ${\rho }_j$ (the niches number of the $j$ th reference point is ${\rho }_j$).

Individual retention operation: elite individuals from $F_{P_{t}L}$ are selected. Then, they enter $P_{t+1}$. In order to maintain the diversity of the population, the reference point of ${\rho }_j$ with the minimum is denoted as $j$ (if ${\rho }_j$of multiple reference points is equal and minimum, one of them is denoted as $\overline{j}$ at random).

4.2.9. Termination Condition of the Algorithm

The convergence algebra threshold is set as $\theta$, the inverse generation distance is introduced to identify the convergence algebra of the algorithm [34]. IGD is the average value of the minimum distance between the point (feasible solution) on the front surface of each theoretical Pareto and the non-dominated solution set obtained by the algorithm. Similarly, the distance between the non-dominated solution set of two successive generations of populations can be expressed. Since the elite strategy of AC-NSGA-Ⅲ makes the non-dominated solution set of the progeny population not inferior to that of the parent population, the IGD value is calculated based on the non-dominated solution set of the progeny population. $\beta$is defined as threshold value. In successive generations $\theta$,when the IGD between the parent generation and the child generation is smaller than $\beta$, the algorithm converges. The calculation formula of IGD is shown in (29). Where $Q_t$ is the child non-dominant solution set, $P_t$ is the parent non-dominant solution set, and $d(v,Q_t)$ is the minimum distance from $v$to $P_t$ of the individual in $Q_t$.

$$IGD(Q_t,P_t)={\displaystyle \frac{1}{\left|Q_t\right|}}{\displaystyle \sum _{v\in Q_t}d(v,P_t)}$$

(29)

5. Experiments

6. Conclusions

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