## 3 The Proposed EATS-MOBFS Method for the 2L-VRPSC  
### Framework of the Solution Mechanism  
With the proven superior performance of the tabu search algorithm in a broad range of optimization problems [32; 33] and the fact that some packing heuristics have shown high efficiency for solving large-scale two-dimensional packing problems, an EATS-MOBFS is proposed in this study for efficiently solving the 2L-VRPSC, where the EATS searches for the optimal routing combination while the MOBFS checks the feasibility of each route and guides the EATS searches to feasible solutions. To deal with the stochastic nature of the 2L-VRPSC, Monte Carlo simulation and scenario analysis methods are integrated for the solution-searching process of the algorithm to calculate the total expected transportation cost.  
The framework of the solution approach is shown in Algorithm 2. Specifically, the solution approach first constructs a feasible initial solution (Algorithm 2 line 1) and iteratively improves the initial solution by performing a non-tabu move or a move that satisfies the aspiration criterion (i.e., better than an incumbent best solution) in each iteration. In each iteration, a neighborhood operator is selected based on the self-adaptive mechanism (Algorithm 2 line 6). Therein, each operator is associated with a weight and is selected  
Figure 1: Example of the prior solution and scenario. (**a**) A prior solution visits nine customers. (**b**) The scenario corresponding to the prior solution, where customers 2 and 7 do not require service (disconnected from the routes).  
periodically and probabilistically according to its performance in the previous iterations (\(p_{u}\) iterations). After \(p_{u}\) iterations (i.e., inner-layer iterations), a diversification mechanism based on the remove-reinsert strategy inspired by large neighborhood search [34] is called to avoid the algorithm being trapped at a local optimum (Algorithm 2 line 20). Moreover, Monte Carlo simulation is called to estimate the expected transportation cost in lines 2 and 11, and MOBFS is evoked to examine the loading feasibility of the solution in lines 1 and 9. In the following, the key components of the solution approach will be detailed, including initial solution construction in Section 3.1.1, neighborhood operators in Section 3.1.2, adaptive weight adjustment in Section 3.1.3, diversification mechanism in Section 3.1.4 and the proposed MOBFS in Section 3.2.  
```
Input: Dataset of the 2L-VRPSC Output: Solution of the 2L-VRPSC \(X^{*}\)
1 Construct an initial solution \(X_{0}\), set \(X=X0\), \(X^{*}=X_{0}\). (Section 3.1.1)
2 Calculate the expected transportation cost by Monte Carlo simulation.
3 whilemaximum out-layer iterations are not metdo
4 whilemaximum inner-layer iterations are not metdo
5 Select a neighborhood operator according to their possibilities.
6 Create a candidate solution list by the selected neighborhood operator. (Section 3.1.2)
7 Find the best solution \(X^{{}^{\prime}}\) from the candidate list.
8 ifcapacity constraint is satisfiedthen
9 Check the loading feasibility of the solution. (Section 3.2)
10 ifloading constraints is satisfiedthen
11 Calculate the expected transportation cost of \(X^{{}^{\prime}}\) (denoted by \(Ob(X^{{}^{\prime}})\)) by Monte Carlo simulation.
12 if\(Ob(X^{{}^{\prime}})\)<\(Ob(X)\)then
13 Update the current solution, set \(X=X^{{}^{\prime}}\).
14
15 end if
16
17 end for
18 Update the tabu list.
19
20 end for
21 Update the weights of neighborhood operators. (Section 3.1.3)
22 Call the diversification mechanism to generate new solution. (Section 3.1.4)
23 end for
```  
**Algorithm 2**The proposed solution approach for the 2L-VRPSC.