An Evolutionary Algorithm Coupled to an Outranking Method for the Multicriteria Shortest Paths Problem

In this article, we are interested in solving a combinatorial optimization problem, the shortest path problem in a multi-attribute graph, by the outranking methods. A multi-attribute graph has simultaneously qualitative and quantitative criteria. This situation gives rise to incomparable paths thus forming the Pareto front. Outranking methods in Multi-criteria Decision Making (MCDM) are the only methods that can take into account this situation (incomparability of actions). After presenting the categories of Mul-ti-criteria Decision Making (MCDM) and the difficulties related to the problems of the shortest paths, we propose an evolutionary algorithm based on the outranking methods to solve the problem of finding “best” paths in a mul-ti-attribute graph with non-additive criteria. Our approach is based on the exploration of induced subgraphs of the outranking graph. Properties have been established to serve as algorithmic basis. Numerical experiments have been carried out and the results presented in this article.


Introduction
The concept of relations was born out of difficulties encountered with diverse concrete outranking problems [1]. They can handle simultaneously qualitative and quantitative criteria. Criteria scores can be left in their own units, which is In view of the above, we thought it would be useful to find a way to overcome the weakness of limiting alternatives using the outranking method. In our work, we want to show that it is possible to remove this limitation. Instead of searching for the "best" solutions directly in the outranking graph, which is impossible if there are a very large number of alternatives, we show how to iteratively explore its induced subgraphs. To test our approach, we applied it to the shortest paths problem in a multi-attribute graph.
In the first part, we present the basic concepts related to Multi-criteria Decision Making, multi-attribute graphs, difficulties in finding the shortest multicriteria path and the SPARTE (Solution au PARadoxe de voTE-Solution to the voting paradox) outranking method which is very close to the ELECTRE (ELimination Et Choix Traduisant la REalité-elimination and choice expressing reality) outranking method. In the second part, we introduce our method by describing some properties that we will use as algorithmic basis. Finally, we present in the third part our numerical experiments and the results obtained.

Multi-Attribute Decision Making
Multi-criteria Decision Making (MCDM) is classified into two general categories: multi-objective decision making (MODM) and multi-attribute decision making (MADM) [10] [11], based on the different purposes and different data types. In the MADM problems, the decision-maker must choose from among a

1) Definitions and notation
A graph G is a pair of sets (V, E), where V is the set of vertices and E is the set of edges, formed by pairs of vertices. In order to avoid any possible confusion, the vertex set of a graph G is denoted by V(G) and its edge set by E(G A kernel represents a set of winning positions in a graph [15]. 2) Multi-Attribute graph Let G = (V, E) be a directed and connected graph. Without loss of generality, we only considered the graph in which there exists at most one edge between a pair of ordered nodes. For a given graph, each edge connecting two nodes u and v is specified by a weight vector cording to the criterion (or point of view) k. m represent the number of criteria.
Each criterion is associated with a value k δ representing its weight.
( ) can be expressed in words (for example, "very bad", ..., "average", ..., "very good") or letters with which numbers are associated to insist on the order that exists between these words/letters. In this case, they are ordinal numbers.
They do not have an algebraic structure, which prohibits the use of arithmetic operations (for example addition "+", multiplication "×") on such numbers.

Shortest Paths Problem in Multi-Attribute Graph
Given a directed graph G = (V, E), an origin s V ∈ and a destination t V ∈ , the shortest-path problem (SPP) aims to find the minimum distance path in G from s to t. The Multi-Attribute Shortest Paths Problem is an extension of the traditional shortest path problem and is concerned with finding a set of efficient paths with respect to two or more criteria that are usually in conflict and incommensurable. In MODM, the problem is known to be NP-hard [ In general a multi-criteria shortest path is a kind of vector optimization problem, which can be described as follows: on an appropriate scale. The way the aggregation problem is addressed in this approach leads to define a complete pre-order on the set ( ) , P s t . Most often, the formal rules consist in mathematical formulas that lead to an explicit definition of a unique criterion synthesizing the m criteria. This is the case with MAVT, MAUT, SMART, TOPSIS, MACBETH, AHP, etc. The complete preorder on the set ( ) , P s t can be established. We usually find a better solution than all the others. In anyway, this approach does not allow any incomparability [20]. However, in a real traffic situation, the decision maker is not content with one solution but with a set of more effective solutions to make his own choice.
The second approach is based on pairwise comparison of actions (paths). The approach can be reduced to a single binary relation. This second operational approach has led to various methods, most of which are covered by the label of

Outranking Method SPARTE
The SPARTE outranking method [21] is based on the general approach of the ELECTRE outranking methods. Let p and q be two paths belonging to the set ( ) , P s t . It is possible to build the following sets of indices: , the set of criteria for which the path p is preferred to the path q.
, the set of criteria for which the path q is preferred to the path p.
, the set of criteria for which the path q is equal to the path p.
The preference index is similar to the global concordance index of the ELECTRE methods. It is calculated as follows: There is a difference only in the formulation of the discordance matrix. The idea is based on the assumption that outranking can be accepted even if the minority shows strong opposition [21]. It is enough for the majority to impose the option with a determination at least as important as that expressed by the minority to reject it. To formulate this idea, we define an adhesion indicator, denoted ( ) , m p q , at any point symmetrical to the opposition indicator given by [21]: The outranking relations S of which meaning is at least as good as is defined as follow:

Our Contribution: Kernel Search in Combinatorial Set
The search for the kernel of a graph is an NP-complete problem [22]. If the graph is cyclic, the kernel set is not unique and may not exit [23]. However, when a graph is directed and acyclic, there is always a single kernel [24] and can be computed in polynomial time [25].
This means that there is not a dominated solution that outranked u. Let ϕ be a function which at a digraph G matches its condensed graph This transformation can be carried out in polynomial time by the Kosaraju algorithm [26] and the Tarjan algorithm [27]. The condensed graph

Gc V Gc E Gc
= is an acyclic digraph [28].
Property 1: If the digraph G is acyclic then Gc G = . The condensed graph Gc and G are the same.
Proof: Let ( ) V Gc the set of strongly connected components of an acyclic graph ( ) is a way from u to u′ and another way from u′ to u. Thus, the way from u to u′ connected to the way from u′ to u gives a cycle. Absurd because Gc is an Similarly ( ) Theorem 1: In an acyclic digraph, there exist at least one source (a vertex of which in-degree is zero) and at least one sink (a vertex of which out-degree is zero) [28].
In this method, we reduce the search for "best solutions" in search of the subset Kr K ⊆ which is actually the set of source vertices in the outranking graph.
be the condensed graph of an outranking graph. Let Kr[Gc] denote its reduced kernel. We have: Proof: a) The subgraph of an acyclic graph is acyclic and therefore admits at least one  Step 1: Let ...
Step n: Construct the induced subgraph

Performance of Actions in a Strongly Connected Component
When a fictional alternative is a strongly connected component and belongs to a reduced kernel, we evaluate the performance of each real alternative that composes it in order to keep only the alternatives that are really significant in the component. We proceed as follows for evaluation.     , , , is not a singleton whose performance (see Table 1) of each alternative in the strongly connected component must be calculated. Depending on the problem, one can either retain the best performer or retain only those who have a positive performance.

Using the Evolutionary Algorithm for the Problem
An evolutionary algorithm operates on a set of candidate solutions that is subsequently modified by two basic operators: selection and variation. Selection is used to model the reproduction mechanism among living beings, while variation mimics the natural capability of creating new living things by means of recombination and mutation [29]. In our proposed evolution algorithm, the selection

Convergence of the Algorithm
We used two performance indices [30] to measure the convergence of our multi-attribute optimization method. The Error Ratio (ER) checks the proportion of non true Pareto points in the approximation front over the population size and the Generational Distance (GD) measures how far the evolved solution set is from the true Pareto front. We can observe the evolution of these two metrics in Figure   2. The algorithm is applied on a graph of 50 vertices having 734,455 paths.

Test Environment and Implementation
The programming environment used for implementation of the approach is Percentage of paths found (%) and Generation) were chosen to study the convergence of our method.

Tables of Results and Discussion
Example 1: Multi-attribute graph with nine vertices.
In Figure 3, we present a graph with two criteria to be minimized simultaneously. The first is an additive criterion and the second is a non-additive, but a bottleneck criterion. In Figure 4, An acyclic graph with 50 vertices of 10% density and two attributes was randomly generated for simulation. The first criterion is additive, the second is a bottleneck criterion. At first, we generated all the paths in the graph. The exhaustive search time was 9 h 20 m 10 s. The number of paths found is 734,455.
The second step was the generation of Pareto-efficient paths. Table 2 shows the CPU time and size of Pareto-efficient paths. The symbol "−" (resp. "+") means that the criterion must be minimized (resp. maximized). Thus for example (+, −) means that the first criterion is maximized while the second is minimized. Third, we ran our method and compared the "best solutions" obtained with the Pareto front. For each sense of optimizing criteria, we ran our method ten times. Tables 3-5 present the results obtained by applying our method respectively to PROMETHEE,   In Table 3 to Table 5, we have on the first row the execution time of our algorithm; on the second row the number of the best solutions found; on the third and fourth rows the Error Ratio and Generational Distance performance metrics. On the fifth row the percentage of distinct solutions found during the search for best solutions. And finally, the number of generations reached iteratively by the genetic algorithm. On the columns, for the ten executions, we display the minimum (min), the average (avg) and the maximum (max) value of each parameter.
In Figure 5, we observe all the solutions forming the reduced kernel and the set of combinatorial alternatives. This is the case where the first criterion is maximized while the second criterion is minimized.
By observing these results, the PROMETHEE method has better Error Ratio

Conclusions and Perspectives
In this paper, we have shown that it is possible to use the outranking methods for solving a multi-attribute combinatorial problem. This approach is based on searching for the reduced kernel of the outranking graph. Theorem 2, which we have established on the reduced kernel of induced subgraphs, has served us as an algorithmic basis. Thus, by iteratively searching for the reduced kernel of the induced subgraphs, it is possible to extract the solutions from a combinatorial optimization problem. Three outranking methods: PROMETHEE, SPARTE and ELECTRE were each coupled to an evolutionary algorithm. The resulting solutions were compared to the Pareto front. The results were very satisfactory for the SPARTE and ELECTRE methods compared to the PROMETHEE method.
As perspectives we intend to use this method with other metaheuristic such as simulated annealing and Taboo search, and apply it to other combinatorial optimization problems like Minimum Spanning Tree Problem multi-attributes that have real applications in the network coverage domain.

Conflicts of Interest
The authors declare no conflicts of interest regarding the publication of this paper.