## 3 Problem Formulation
### Modeling
If \(s_{l-1}\) is the restaurant node, which produces a merged meal pick-up with the last node, \(set_{l}\) is expressed as the sequence set of \(s_{l-1}\) and the last merged meal restaurant node. The expression is (12). \(set_{l}\) is the sequence set of \(s_{l-1}\) and the last merged meal restaurant node. \(a_{l-1}\) is the maximum sum of the ordering time in \(set_{l-1}\) and the meal preparation time of the restaurant. The restaurant's meal preparation time \(t_{\text{arga}(s_{w})}^{p}\) follows the Gaussian distribution. In other words, for different restaurants, the Gaussian distribution is obeyed with a different expectation \(\mu_{i}\) and variance \(\sigma_{i}^{2}\). If \(s_{l-1}\) is the customer node, it does not produce a merged meal delivery with the last node; \(a_{l}\) is the sum of \(a_{l-1}\), \(t_{\text{arg}\beta(s_{l})}^{s}\), and \(t_{s_{l-1}s_{l}}^{d}\). If \(s_{l-1}\) is the customer node, it produces a 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 the same, so the calculation of \(a_{l}\) is not affected. \[set_{l}\text{= }\left\{l\prime|\prod\limits_{l=1}^{l-1}p_{l}^{1}=1\right\}\cup I \tag{12}\]\[a_{1}=T_{0}+t_{s_{0}s_{1}}^{d}\]  
\[a_{l-1}\ p_{l-1}^{1}+p_{l-1}^{2}=1\]  
\[a_{1}=T_{0}+t_{s_{0}s_{1}}^{d}\]  
\[a_{l-1}\ p_{l-1}^{1}+p_{l-1}^{2}=1\]  
\[a_{l-1}\ p_{l-1}^{1}+p_{l-1}^{2}=0\]  
\[\max(a_{l-1},\max_{w\in set_{l-1}}(r_{\text{arg}\alpha(s_{w})}+t_{\text{arg} \alpha(s_{w})}^{p}))+t_{s_{l-1}s_{l}}^{d}\]  
\[a_{l-1}+t_{\text{arg}\beta(s_{l-1})}^{s}+t_{s_{l-1}s_{l}}^{d}\ p_{l-1}^{1}+p_{ l-1}^{2}=0,s_{l-1}\in Jt\]  
\[g_{n}=a_{\text{args}(\beta_{n})}+t_{n}^{s},\forall n\in N \tag{14}\]  
Constraint condition (15) is the quantity constraint of uncompleted orders. \[set_{l}\text{= }\left\{l\prime|\prod\limits_{l=1}^{l-1}p_{l}^{1}=1\right\}\cup I \tag{12}\]0  
\[\lambda_{l+1}=\left\{ \begin{aligned} &\lambda_{l}\ s_{l}\in I \\ &\lambda_{l}-1\ s_{l}\in J\end{aligned} \right. \tag{15}\]  
Constraint conditions (16)-(19) are open chain structure constraints. To maintain the flow balance between intermediate nodes, (16)-(17) traverse 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 a sub-loop. \[\sum\limits_{i\in A\
u,\text{ls}_{0}}x_{ij}=1,\forall i\in A\prime \tag{16}\]  
\[\sum\limits_{j\in A\prime}x_{ij}=1,\forall i\in\left(A\prime-s_{|A\prime|} \right)\cup s_{0} \tag{17}\]  
\[\sum\limits_{i\in A\
u,\text{ls}_{0}}x_{s_{|A\prime|}i}=0 \tag{18}\]  
\[\sum\limits_{i\in A\prime}x_{is_{0}}=0 \tag{19}\]  
\[\sum\limits_{i\in A\prime}\sum\limits_{j\in H}x_{ij}\leq|H|-1,\forall H\subset A \prime,2\leq|H|\leq\left|A^{\prime}\right|-2 \tag{20}\]  
The established MO-RPDRP is a stochastic programming model. Three objective functions in the model are derived as follows. \[ \begin{array}{rl}\text{min}F_{1}=&E(\omega)\\ &=E(\frac{\sum\limits_{n\in N}\max(0,\text{sgn}(g_{n}-b_{n}))}{|N|})\\ &=\frac{1}{|N|}\sum\limits_{n\in N}E(\max(0,\text{sgn}(g_{n}-b_{n})))\\ &=\frac{1}{|N|}\sum\limits_{n\in N}\int_{0}^{+\infty}\text{sgn}(g_{n}-b_{n})f (g_{n}-b_{n})d(g_{n}-b_{n})\\ &=\frac{1}{|N|}\sum\limits_{n\in N}\int_{0}^{+\infty}f(g_{n}-b_{n})d(g_{n}-b_{ n})\end{array}  \tag{21}\]  
where \(f(g_{n}-b_{n})\) represents the probability density function of \(g_{n}-b_{n}\). \[ \begin{array}{rl}\text{min}F_{2}=&E(\sum\limits_{n\in N}(g_{n}-r_{n}))\\ &=E(\sum\limits_{n\in N}g_{n})-\sum\limits_{n\in N}r_{n}\end{array}  \tag{22}\]\[ \begin{array}{rl}\min F_{3}=&E\left(\sum\limits_{l=1}^{|A_{l}|}\lambda_{l}(a_{l}-a _{l-1})\right)\\ &=\sum\limits_{\begin{subarray}{c}l=1\\ l=1\end{subarray}}^{|A_{l}|}\lambda_{l}E(a_{l}-a_{l-1})\\ &=\sum\limits_{\begin{subarray}{c}|A^{l}|\\ l=1\end{subarray}}^{|A_{l}|}\lambda_{l}(E(a_{l})-E(a_{l-1}))\end{array}  \tag{23}\]  
Obviously, according to (21)-(23), the core of the derivation of the three objective functions is the derivation of \(E(a_{l})\).