# A Survey on Variable Neighborhood Search for Sustainable Logistics
## 4 Results and Findings
### Multiobjective Analysis
In this context, evolutionary algorithms are commonly used to explore the solution space. These algorithms can generate an approximation of the Pareto set, which consists of non-dominated solutions representing the best possible trade-offs between competing objectives. This approach is used significantly more frequently than any other in our context [27; 28; 30; 34; 71; 72; 73; 74; 75; 76; 77; 78; 79; 80; 81; 82; 83; 84; 85; 86]. This may be due to the fact that one of the key advantages of directly addressing a problem as multiobjective is that it provides a diverse set of solutions, giving decision-makers the flexibility to choose an option that best fits their specific priorities or constraints, rather than being constrained by predefined weights or scalarized objectives. This approach is especially beneficial in complex problems where the objectives may conflict or be difficult to reconcile through simple weighting or constraint-based methods. Another well-established method for directly addressing multiobjective problems is compromise programming, which aims to find solutions that are as close as possible to the ideal solution--a hypothetical point where all objectives are optimized simultaneously. In practice, such an ideal solution rarely exists, as objectives often conflict. However, compromise programming seeks to minimize the distance between feasible solutions and this ideal point by using distance metrics such as the L1 (Manhattan distance) or L2 (Euclidean distance) norms. The mathematical formulation is:  
\[L_{p}(x)=\left(\sum_{i=1}^{k}w_{i}\cdot\left|\frac{f_{i}(x)-f_{i}^{*}}{f_{i}^{ \text{max}}-f_{i}^{\text{min}}}\right|^{p}\right)^{1/p} \tag{3}\]  
where \(f_{i}^{*}\) represents the ideal value of objective \(i\). The chosen norm (\(p\)) reflects the decision-maker's tolerance for trade-offs between objectives. In this method, the decision-maker specifies a preferred balance between objectives, and the algorithm then searches for solutions that are closest to this ideal according to the chosen norm. Therefore, this approach is particularly effective when the decision-maker has a clear understanding of the relative importance of each objective but still wants to explore solutions that lie near the ideal trade-off between them. Regrettably, this is not a common situation, as reflected by the number of approaches that use it in our context [87; 88]. Finally, another technique for solving multiobjective problems is the lexicographic order method, which prioritizes the objectives according to their importance. In this approach, the objectives are ranked in hierarchical order, and the optimization process focuses first on the most important objective. Once an optimal solution is found for the highest-priority objective, the second most important objective is optimized, but only within the set of solutions that maintains optimality for the first. This process continues down the hierarchy, optimizing each subsequent objective without sacrificing the optimality of higher-priority ones. Therefore, the lexicographic approach is expressed as a sequence of optimization problems:  
\[\min f_{1}(x),\,x\in X, \tag{4}\] \[\min f_{2}(x),\,x\in X\text{ such that }f_{1}(x)=f_{1}^{*},\] (5) \[\ldots \tag{6}\]  
This method is particularly useful when one or more objectives are clearly more critical than others and there is a need to maintain strict priority among them. However, one limitation of this method is that it can disregard potentially good trade-offs for lower-priority objectives, as it strictly adheres to the ranked order. This, along with the need to establish the critical order, makes it less popular in the pool of studies analyzed [69; 89; 90]. Figure 6 shows the distribution of the collected studies according to the multiobjective method. All these methods balance theoretical rigor with practical applicability. For example, scalarization techniques are straightforward and computationally efficient, making them ideal for preliminary studies or contexts with limited computational resources.