## 3 Problem Description and Mathematical Models
### Mathematical Model  
Based on the traditional VRPTW, the JIT-RMOVRPTW adds uncertain time constraints and designs a new metric alpha based on uncertain time as an objective to measure the robustness of the solution. In addition, the JIT-RMOVRPTW considers JIT delivery. JIT time is used as the deadline, and the AGV adjusts the departure time according to the JIT time of the workstation to achieve the objective of the JIT. The JIT-RMOVRPTW can be formulated as follows:  
The three objectives for the JIT-RMOVRPTW, each of which needs to be minimized, can be stated as follows:  
\(f_{d}\) (Total travel distance):  
\[minf_{d}\ =\ \sum_{k=1}^{K}\sum_{i=0}^{N}\sum_{j=0}^{N}d_{ij}x_{ijk} \tag{1}\]  
\(f_{alpha}\) (Robustness metrics):  
\[minf_{alpha}\ =\ \frac{1}{\prod_{i=1}^{N}al\,pha_{i}} \tag{2}\]  
\(f_{JIT}\) (JIT penalty time):  
Workstation demand time: The JIT problem contradicts the robust problem, so the lead time is used as the safety time for the vehicle. JIT-RMOVRPTW eliminates the traditional time windows, with \(JIT_{i}\) serving as the deadline for the JIT problem (as shown in Figure 1). \(p\) is the slack time. As shown in Equation (17), for the station \(i\), the larger the \(q_{i}\), the higher the JIT requirement. \[b_{i}=min\{b_{iv}\} \tag{15}\]  
\[a_{i}=max\{a_{ie}\},(a_{ie}\leq b_{i}) \tag{16}\]  
\[JIT_{i}=b_{i}-(b_{i}-a_{i})*p*\left(1-\frac{q_{i}}{Q}\right) \tag{17}\]  
9. The arrival time of the AGVs: \[t_{j}=t_{i}+\frac{d_{ij}}{v}x_{ijk}+s_{i},(i=0,1,2,\ldots,n;\ j=0,1,2,\ldots,n)\] (18)
10. Uncertainty constraints: \(t_{i}^{\sim}\) is the uncertainty time of arriving at station \(i\). \(\zeta\) is the overall random disturbance. \(\overline{D}\) is the average distance between workstations. \(\widetilde{u_{ij}}\) is the characteristic coefficient between stations \(i\) and \(j\). \(\widetilde{u_{0i}}\) has unique characteristics, so it is discussed separately in Equation (19). \[\widetilde{u_{ij}}=\left\{ \begin{array}{l}d_{ij}/\overline{D},(i=1,2,\ldots,n;\ j=1,2,\ldots,n)\\ 1,(i=0;\ j=1,2,\ldots,n)\end{array} \right.\] (19)  
\[t_{j}^{\sim}=t_{j}+\widetilde{u_{ij}}\widetilde{\zeta},\left(\overline{\zeta }=\frac{\zeta}{n}\right) \tag{20}\]
11. Feasibility of workstations: To measure the robustness of the solution, the metric alpha is proposed in JIT-RMOVRPTW. Alpha is specifically defined according to different random distributions that different scenarios have. In this paper, \(\zeta\) is taken as a continuous probability distribution of a triangular distribution with a lower limit of \(min\), a peak location of \(mid\), and an upper limit of \(max\), which can be expressed as \([min,max,mid]\). Equation (21) represents the range of random arrival times of the station \(j\). Equation (22) represents the degree of uncertainty from station \(i\) to station \(j\). \[tj\sim = ti\sim+\mathrm{d}ijvxijk+si+\mathrm{u}ij\zeta,\ i=0,1,2,\ldots,\ n;\ j=0,1,2,\ldots,\ n \tag{21}\] \[interval_{ij}^{\sim}=max\left(\widetilde{u_{ij}}\overline{\zeta} \right)-min\left(\widetilde{u_{ij}}\overline{\zeta}\right) \tag{22}\]  
Figure 1: JIT time setting. \(alpha_{i}\) represents the feasibility of station \(i\). The value of \(alpha_{i}\) is shown by Equation (23). The maximum value of alpha is set to 2 to prevent too large an alpha from affecting the realization of the JIT. \[