## 2 Overview of Multiobjective Path Problems and Algorithms
### Types of Multiobjective Path Problems
#### 2.2.1.2. Intr  
Interactive Methods  
Paper [36] describes an interactive method for obtaining an approximation to the efficient solution set for the BOPP. The goal of the method is to assist the DM in the selection of the preferred/best trade-off solution within the efficient solution set, working in two phases. The first phase recurs to a Non-Inferior Set Estimation (NISE)-like procedure (enabling a specific order of calculation of supported efficient solutions) [77]; in the second phase of the method, a constrained shortest path problem is solved, enabling a search in specified duality gaps, see [78]. In [37], an interactive method for the BOSPP, which obtains the optimal solutions for a quasi-concave and non-increasing utility function defined by the DM, is proposed. In the first step, the DM evaluates OF values for supported efficient paths. The search region is reduced by carrying out pairwise comparisons until it is defined by two adjacent solutions in terms of costs. Afterwards, a labeling algorithm is applied to the obtainment of the non-supported nondominated points, while the domination cones derived from the pairwise comparisons carried out by the DM cause the search area to shrink. Paper [38] proposes an interactive method by showing that, given an acyclic network and a preference-order relation, the principle of optimality is valid if and only if the preferences are linear functions with respect to the objectives. It is suggested to use an interactive method for assessment of a value function while using the principle of optimality in the search for an optimal solution. Furthermore, in this process, the DM is asked about the compromises between solutions or to sort paths according to his/her preferences. Figure 5: Two-phase techniques: (**a**) Phase 1: calculation of supported efficient points; (**b**) Phase 2: calculation of unsupported efficient points. Green bullets represent supported dominated points. Orange boxes represent unsupported dominated points; grey crosses represent dominated points; the numbers indicate the order of calculation of the different points. The authors in [39] describe an interactive method for finding efficient biobjective shortest paths which is based on the proposal in [36], but now, in phase 2, the ranking of paths is performed, instead of solving constrained shortest path problems. Paper [42] deals with an application of the MOSPP in the context of cycling routes, and it develops a method focused on finding a well-balanced trade-off path between the objectives. The algorithm follows the Best Compromise A* method, the goal of which is to find a path that minimizes a Chebyshev-weighted norm considering a given reference point. The algorithm follows a labeling approach and includes pruning techniques to enhance its computational performance. 2.2.1.3. APR--A Priori Aggregation of Preferences Methods  
Next, we summarize more recent or particularly relevant references on a priori aggregation of preferences methods for MOPPPs. Paper [40] proposes a method that uses different metrics as utility functions based on the norm value associated with each path. Two algorithms for solving the minimum-cost path problem for the proposed metrics are then described. The authors in [44] introduce an exact label-setting algorithm that obtains the subset of efficient optimal paths which satisfy a set of lexicographic goals, or the subset which minimizes deviation from goals whenever these cannot be fully satisfied. In particular, given an a priori set of goals, the work addresses the problem of finding the subset of Pareto optimal paths which satisfy the goals or, if these cannot be satisfied, finds the subset of Pareto optimal paths which minimize deviation from the goals. This is achieved by means of a multiobjective label-setting algorithm with lower bounds and label expansion that obtains such goal-optimal solutions.