## 2 Mathematical Formulation of the 2L-VRPSC
### Stochastic Programming Model of the 2L-VRPSC
#### 2.2.2 Formulation of the 2L-VRPSC  
In this study, the 2L-CRPSC is described as a complete undirected graph \(G=(V,E)\). Therein, \(V=V_{0}\cup 0\) consists of \(n\) customers (\(V_{0}=\{1,2,\cdots,n\}\)) and a depot (node 0). Each customer \(i(i\in V_{0})\) consists of \(m_{i}\) rectangular items with a total weight of \(q_{i}\). Each item of customer \(i\) is denoted by a pair of indices \((i,b)\), and its width and length are denoted by \(w_{ib}\) and \(l_{ib}(b=1,2,\cdots,m_{i})\), respectively. A fleet of \(K\) homogeneous vehicles is available, and each vehicle has a maximum capacity \(Q\) and a rectangular loading surface with width \(W\) and length \(L\). Additional notations are described as follows:  
**Sets and parameters:**  
\(V_{p},V_{a}\): Set of customers that require service and do not require service (\(V_{p}\cup V_{a}=V_{0}\), \(V_{p}\cap V_{a}=\emptyset\)). \(S\): Set of preserved scenarios, indexed by \(s\) and \(s^{{}^{\prime}}\) (\(s,s^{{}^{\prime}}\in S\)). \(R\): Set of feasible routes, where \(R^{k}\) represents a feasible route visited by vehicle \(k\). \(\sigma_{i}\): Parameter that indicates the order that customer \(i\) is visited along the route (\(i\in R^{k}\)). \(d_{ij}\): Distance (i.e., travel cost) between node \(i\) and node \(j\) (\((i,j)\in E\)). **Variables:**  
\(z_{ij}^{k}\): Binary variable, \(z_{ij}^{k}=1\) if and only if vehicle \(k\) travels from node \(i\) to node \(j\), \(0\) otherwise. \(x_{ib},y_{ib}\): Integer variables that define the coordinate of the bottom-left corner of item \(b\) belonging to customer \(i\) in a vehicle. \(\Omega_{ib}\): Integer variable that indicates whether item \(b\) of customer \(i\) has been rotated. Let \(p_{i}\) be the probability that customer \(i\) requires service (assuming independence between customers), then the probability of scenario \(s\in S\) can be expressed as follows:  
\[P(s)=\sum_{i\in V_{p}}p_{i}\sum_{j\in V_{a}}(1-p_{j}). \tag{1}\]  
The objective of the 2L-VRPSC problem is to find a prior solution with minimal expected length. Based on the customer uncertainty modeling method described in Section 2.2.1, we simulate solution \(\tau\) to construct a desired number of scenarios so as to calculate the expected length \(E(L_{\tau})\), i.e.,  
\[\text{Minimize }E(L_{\tau})=\sum_{s\in S}P(s)L_{\tau,V_{a}}(s), \tag{2}\]  
where \(L_{\tau,V_{a}}(s)\) is the length of solution structure corresponding to scenario \(s\), in which the customers \(V_{a}\) that do not require service are skipped. Figure 1 illustrates an example of the prior solution and scenario with nine customers. In the example, customers 2 and 7 do not require service, and thus, vehicles skip them and visit other customers in the same order as they appear in the prior solution. In the proposed approach, the prior solution is built satisfying the following constraints:  
\[\sum_{j\in V_{0}}z_{0 jk}=\sum_{i\in V_{0}}z_{i0k},\forall k\in K, \tag{3}\] \[\sum_{k\in K}\sum_{i\in V}z_{ijk}=1,\forall j\in V_{0},\] (4) \[\sum_{i\in V}z_{iuk}=\sum_{j\in V}z_{ujk},\forall u\in V_{0},k\in K,\] (5) \[\sum_{k\in K}\sum_{j\in V_{0}}z_{0 ik}\leq K,\] (6) \[\sum_{i:(i,j)\in E}q_{i}z_{ijk}\leq Q,\forall k\in K, \tag{7}\]  
Constraint (3) ensures that all vehicles departing the depot have to return to the depot. Constraint (4) enforces that each customer is visited exactly once. Constraint (5) is the flow conservation constraint. Constraint (7) ensures that each vehicle is not overloaded. Constraint (6) guarantees that the maximum number of vehicles is not exceeded. Additionally, the packing feasibility of every route must be guaranteed. In this paper, four different two-dimensional loading configurations with respect to sequential and rotation constraints are considered.