## 4 The Proposed AC-NSGA-III Algorithm
### Operator Design
#### 4.2.1 IRC Coding
\begin{array}{ll}x_{k}+\Delta(t,U_{\max}^{k}-x_{k}),\;if\;random (0,1)=0\\ \\ x_{k}-\Delta(t,x_{k}-U_{\min}^{k}),\;if\;random(0,1)=1\end{array} \right. \tag{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 is when individuals in \(R_{t}\) are divided into ranks (\(F_{P_{1},F_{P_{2}},\ldots\)). A new population of \(S_{t}\) is constructed from \(F_{P_{1},1}\). When \(F_{P_{t},L}\) is put in it, \(S_{t}=N\) or \(S_{t}>N\). Association operation is when 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 [37; 38]. 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 is when elite individuals from \(F_{P,L}\) 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 \(\theta\), and the inverse generation distance is introduced to identify the convergence algebra of the algorithm [39]. 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

Figure 8: Order-based crossover, OBX. 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 the 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 \(Q_{t}\). \[IGD(Q_{t},P_{t})=\frac{1}{|Q_{t}|}\sum_{v\in Q_{t}}d(v,P_{t}) \tag{29}\]

## 5 Experiments

### Basic Data

In order to explore the effectiveness of the proposed model and algorithm, the following experiments were conducted. According to the one-month actual investigation of a delivery enterprise, it is calculated that the number of orders received by AGVs is the largest during the peak hours (11:00-13:00) in one day. Therefore, the information of the top four orders of peak hours in the month were selected as examples, and the details of the examples are shown in Appendix A. The design of fixed parameters in the model is shown in Table 3. The hardware environment of the numerical experiment is Intel i9-12900K/KS CPU, DDA4-3600 MH-32G(16G*2) memory, 970-EVO-Plus-NVMe-M.2-1T solid state disk, windows 10 operating system. The proposed algorithms are implemented in Python language. ### Experimental Design

(1) Comparison of optimization effects considering uncertainty factors. Under the same algorithm conditions, to verify the superiority of the proposed model, the uncertainty model considering the meal preparation time and the model without considering the meal preparation time are compared. (2) Comparison of order combination modes.