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In this article, we are interested in solving a combinatorial optimization problem, the shortest path problem in a multi-attribute graph, by the out-ranking 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 Multi-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 multi-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.

The concept of relations was born out of difficulties encountered with diverse concrete outranking problems [

Although this method is simple and allows a very thorough analysis of the robustness, this limitation makes several researchers prefer to use the MAUT (multiple attribute utility theory) method. This other approach tries to define a complete pre-order on the set of alternatives. Most often, the formal rules consist in mathematical formulas that lead to an explicit definition of a unique criterion synthesizing the m criteria. The weaknesses that result from this method are the reduction of the multicriteria problem into a unique criterion problem. The consequence of this reduction is that all the actions are comparable, a property that is not always desirable to have because one observes that for a human decision-maker, there are situations involving incomparable alternatives [

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.

G = ( V ( G ) , E ( G ) ) : graph G defined by the set of vertex V(G) and the set of edge E(G);

G_{c}_{:} condensed graph of the graph G;

P(s, t): set of all elementary (s, t)-paths;

Kr[G] or Kr: Reduced Kernel of a graph G;

K[G] or K: Kernel set of a graph G;

λ ( p ) : Performance of path p in a strongly connected component;

δ k : weight of criterion k;

w_{k} (u, v): Evaluation of edge (u, v) according to criterion k;

W_{k}(p): Evaluation of the path p on the criterion k;

W ( p ) = ( W 1 ( p ) , W 2 ( p ) , ⋯ , W m ( p ) ) : Evaluation of the path p on m criteria.

Multi-criteria Decision Making (MCDM) is classified into two general categories: multi-objective decision making (MODM) and multi-attribute decision making (MADM) [

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 directed graph (or just digraph) G consists of the directed edges. A digraph H is a subdigraph of a digraph G if V ( H ) ⊆ V ( G ) , E ( H ) ⊆ E ( G ) and every edge in E(H) has both end-vertices in V(H). If every edge of E(G) with both end-vertices in V(H) is in E(H), we say that H is induced by V(H) and H is called an induced subdigraph of G. A strongly connected component H of the digraph G is a directed subgraph of G (not a null graph) such that H is strongly connected, but if we add any vertex or edge to it, then it is not strongly connected anymore. The condensed graph G_{c} of the digraph G is obtained by contracting all the edges in every strongly connected component. A subset K [ G ] ⊆ V ( G ) is a kernel if it is independent and absorbing/dominating. K[G] is dominating if ∀ u ∈ K [ G ] , ∃ v / ( v , u ) ∈ E ( G ) . It is independent if ∀ u , v ∈ K [ G ] , ( v , u ) ∉ E ( G ) . A kernel represents a set of winning positions in a graph [

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 ω ( u , v ) = ( ω 1 ( u , v ) , ω 2 ( u , v ) , ⋯ , ω m ( u , v ) ) where ω k ( u , v ) , k = 1 , 2 , ⋯ , m is the evaluation of the edge e = ( u , v ) ∈ E according to the criterion (or point of view) k. m represent the number of criteria. Each criterion is associated with a value δ k representing its weight.

A criterion k can be quantitative or qualitative. If it is a qualitative criterion, ω k ( u , v ) 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.

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 [

Let P ( s , t ) denote the set of all s-t paths in G. If a path p = ( e 1 , e 2 , ⋯ , e L ) ∈ P ( s , t ) . Its evaluation is the vector W ( p ) = ( W 1 ( p ) , W 2 ( p ) , ⋯ , W m ( p ) ) where W k ( p ) is the path evaluation according to the criterion k. If k is a quantitative criterion, W k ( p ) = ∑ e ∈ p ω k ( e ) , k = 1 , 2 , ⋯ , m (additive criterion). When k is a qualitative criterion, W k ( p ) = o p t ( ω k ( e 1 ) , ω k ( e 2 ) , ⋯ , ω k ( e L ) ) (Non-additive criterion) where “opt” may be one of the following operator min, max, median, mean, etc.

In general a multi-criteria shortest path is a kind of vector optimization problem, which can be described as follows: “ Minimize ” p ∈ P ( s , t ) W ( p ) = ( W 1 ( p ) , W 2 ( p ) , ⋯ , W m ( p ) ) . To solve this problem, we have two large groups of approach.

The first one is called traditional approach. It can be characterized by assigning a well-defined degree to the m performances ( W 1 ( p ) , W 2 ( p ) , ⋯ , W m ( p ) ) values. Most often a numerical value v ( p ) = Ψ ( W 1 ( p ) , W 2 ( p ) , ⋯ , W m ( p ) ) is assigned to each p ∈ P ( s , t ) 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 [

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 methods (mainly PROMETHEE and ELECTRE methods). For the choice problematic ( P . α ) solving the problem consists of two parts: construction of an outranking graph, and exploitation of the graph corresponding to this relation. The subset searched is the kernel of the graph [

The SPARTE outranking method [

· J + ( p , q ) = { j / W j ( p ) > W j ( q ) } , the set of criteria for which the path p is preferred to the path q.

· J − ( p , q ) = { j / W j ( p ) < W j ( q ) } , the set of criteria for which the path q is preferred to the path p.

· J = ( p , q ) = { j / W j ( p ) = W j ( q ) } , 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: C ( p , q ) = ∑ k ∈ ( J + ∪ J = ) δ k ∑ i = 1 m δ i .

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 [

· m ( p , q ) = m e d i a n { ( W j ( q ) − W j ( p ) , δ j ) where j ∈ J + ( p , q ) } ;

· m ′ ( p , q ) = m e d i a n { ( W j ( q ) − W j ( p ) , δ j ) where j ∈ J − ( p , q ) } ;

· D ( p , q ) = { 1 if m ( p , q ) ≥ m ′ ( p , q ) 0 otherwise .

The outranking relations S of which meaning is at least as good as is defined as follow:

S ( p , q ) = { 1 if C ( p , q ) ≥ c * and D ( p , q ) = 1 0 otherwise where c * ∈ [ 0.5 , 1 ] is the concordance threshold.

The search for the kernel of a graph is an NP-complete problem [

A subset Kr[G], denoted simply Kr when there is no possibility of confusion, of the kernel in a digraph G = ( V , E ) is a reduced kernel of G if it is independent, absorbing/dominating and not dominated. Kr is not dominated if ∀ u ∈ K r , ¬ ∃ v ∈ ( V \ K r ) / ( v , u ) ∈ E . 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 G c = ( V ( G c ) , E ( G c ) ) . Where V ( G c ) is the set of strongly connected components of G and E ( G c ) = { ( X , Y ) ∈ V ( G c ) × V ( G c ) / ∃ ( x , y ) ∈ X × Y ∧ ( x , y ) ∈ E } . This transformation can be carried out in polynomial time by the Kosaraju algorithm [

Property 1: If the digraph G is acyclic then G c = G . The condensed graph Gc and G are the same.

Proof: Let V ( G c ) the set of strongly connected components of an acyclic graph G = ( V , E ) . Let U ∈ V ( G c ) . Suppose that | U | > 1 ⇒ ∃ ( u , u ′ ) ∈ U there 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 acyclic digraph. So ∀ U ∈ V ( G c ) , | U | = 1 ⇒ | V ( G c ) | = | V | . Then V ( G c ) = V . Similarly E ( G c ) = E . By definition

E ( G c ) = { ( X , Y ) ∈ V ( G c ) × V ( G c ) / ∃ ( x , y ) ∈ X × Y ∧ ( x , y ) ∈ E } = { ( x , y ) ∈ V / ( x , y ) ∈ E }

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) [

In this method, we reduce the search for “best solutions” in search of the subset K r ⊆ K which is actually the set of source vertices in the outranking graph. Let G c = ( V ( G c ) , E ( G c ) ) be the condensed graph of an outranking graph. Let Kr[Gc] denote its reduced kernel. We have:

Property 2: Let G i = ( V i , E i ) be an induced subgraph of G c = ( V ( G c ) , E ( G c ) ) then:

a) K r [ G i ] ≠ ϕ .

b) If a vertex p ∈ K r [ G c ] ∩ V i then p ∈ K r [ G i ] .

Proof:

a) The subgraph of an acyclic graph is acyclic and therefore admits at least one source. So K r [ G i ] ≠ φ .

b) If p ∈ K r [ G c ] then ¬ ∃ q ∈ V i that ( q , p ) ∈ E i then p ∈ K r [ G i ] .if p ∈ V i .

Property 3: Let G i = ( V i , E i ) and G j = ( V j , E j ) be two graphs induced of Gc, respectively having Kr[G_{i}] and Kr[G_{j}] as a reduced kernel. If we construct a third induced subgraph G k = ( ( K r [ G i ] ∪ V j ) , E k ) then:

a) K r [ G k ] ⊆ K r [ G i ] ∪ K r [ G j ] .

b) If V i ∪ V j = V ( G c ) then K r [ G k ] ⊇ K r [ G c ] .

Proof:

a) Let p ∈ K r [ G k ] ⇒ p ∈ K r [ G i ] by definition. Let’s show that p ∈ K r [ G j ] . Let’s assume that p ∉ K r [ G i ] ⇒ ∀ q ∈ ( K r [ G i ] ∪ V j ) , ( q , p ) ∉ E j because p ∈ K r [ G k ] . So p ∈ K r [ G j ] . Conlusion K r [ G k ] ⊆ K r [ G i ] ∪ K r [ G j ] ._{ }

b) Let p ∈ K r [ G c ] ⇒ ∀ q ∈ V ( G c ) , ( q , p ) ∉ E ( G c ) . We have V i ∪ V j = V ( G c ) ⇒ E i ⊆ E ( G c ) and E j ⊆ E ( G c ) then ( q , p ) ∉ E i and ( q , p ) ∉ E j for all q ∈ V i ∪ V j . Then ∀ q ∈ K r [ V i ] ∪ V j , ( q , p ) ∉ E k . Hence p ∈ K r [ G k ] ._{ }

Theorem 2: Let G = ( V , E ) be an acyclic graph, and be the subsets V i , i = 1 , 2 , ⋯ , n of vertices V such that V = ∪ i = 1 n V i then the reduced kernel Kr of G(V, E) can be obtained recursively from its n subsets by the following recursion relation:

Step 1: Let G 1 = ( V 1 , E 1 ) the induced subgraph with its reduced kernel Kr_{1}.

Step 2: Construct the induced subgraph G 2 = ( ( V 2 ∪ K r 1 ) , E 2 ) and extract its reduced kernel Kr_{2}.

...

Step n: Construct the induced subgraph G n = ( ( V n ∪ K r ( n − 1 ) ) , E n ) and extract its reduced kernel Kr_{n}.

So K r ⊆ K r n ⊆ ∪ i = 1 n k r [ G ′ i ] . Where G ′ i = ( V i , E ( G ′ i ) ) are induced subgraphs formed only of V_{i} vertices.

Proof: Follows from property 3:

- We have K r ⊆ K r n from proprety 3b.

- We have K r n ⊆ ∪ i = 1 n k r [ G ′ i ] from proprety 3a.

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.

In a first time we determine for each alternative p all of its successors N + ( p ) = { p ′ / p S p ′ andnot ( p ′ S p ) } , all of its predecessors N − ( p ) = { p ′ / p ′ S p andnot ( p S p ′ ) } , the set of alternatives that are indifferent to it I + ( p ) = { p ′ / p ′ S p and p S p ′ } and the set of actions that are incomparable to it I − ( p ) = { p ′ / not ( p ′ S p ) andnot ( p S p ′ ) } .

In a second time we calculate the performance λ ( p ) of each alternative in the strongly connected component as follow:

λ ( p ) = | N + ( p ) | − | N − ( p ) | + ∑ q ∈ I + ( p ) ( | N + ( q ) | − | N − ( q ) | ) | I + ( p ) | − | I − ( p ) | n where n is the size of the strongly connected component.

In

N + ( x ) | N − ( x ) | I + ( x ) | I − ( x ) | | N + ( x ) | − | N − ( x ) | | λ ( x ) | |
---|---|---|---|---|---|---|

x_{4} | {x_{8}} | {x_{6}} | {x_{5}, x_{7}} | { } | 0 | 2 |

x_{5} | {x_{2}, x_{6}, x_{7}} | { } | {x_{4}} | { } | 3 | 3 |

x_{6} | {x_{4}} | {x_{5}} | { } | {x_{7}} | 0 | −0.75 |

x_{7} | {x_{2}, x_{8}} | {x_{5}} | {x_{4}} | {x_{6}} | 1 | 0.75 |

subset { x 4 , x 5 , x 6 , x 7 } is not a singleton whose performance (see

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 [_{r}, this other path is chosen from the same subset and chosen in another subset with the probability 1 − p_{r}. The mutation operation will consist in deleting or inserting an edge from a randomly selected kernel path. The algorithm is as follows:

1) Generate random initial population Pop with size N;

2) Perform crossover and mutation to members of the mating pool;

3) Form a new population PopNew made up of Pop and the offspring resulting from crossing and mutation;

4) Calculate the matrix performance of PopNew be MatPop;

5) Find the outranking graph;

6) Find the condensed outranking graph from the main outranking graph and look for its reduced kernel;

7) Form two subsets: KernelPop population from the kernel of condensed outranking graph and NotKernelPop population not belonging to KernelPop;

8) Assign Pop the population members of KernelPop or remove in Pop all population members of NotKernelPop;

9) Return Pop and stop if maximum number of generations is reached. Else consider go to Step 3.

We used two performance indices [

The programming environment used for implementation of the approach is

MATLAB (Matrix Laboratry). All experiments were performed on a computer with a 2.3 GHz Intel(R) Core(TM) i3-2348M CPU and 4.00 GB of RAM. Three different outranking methods are implemented. Five parameters: CPU time (second), Size of “best” paths, Error Ratio (ER), Generational Distance (GD), Percentage of paths found (%) and Generation) were chosen to study the convergence of our method.

Example 1: Multi-attribute graph with nine vertices.

In

Path 1 (10; 30): S®C®G®T; Path 2 (15; 25): S ® B ® C ® T; Path 3 (13; 27): S ® B ® C ® G ® T.

Path 4 (40; 10): S ® A ® E ® T; Path 5 (20; 15): S ® A ® D ® C ® T.

Example 2: Multi-attribute graph with 50 vertices.

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.

Sense of optimizing criteria | (−, −) | (−, +) | (+, −) | (+, +) |
---|---|---|---|---|

CPU time (m:s) | 6 m 27 s | 2 m 39 s | 13 m 47 s | 18 m 35 s |

Size of Pareto front | 4 | 35 | 35 | 8 |

ELECTRE SPARTE methods.

In

In

By observing these results, the PROMETHEE method has better Error Ratio (ER) and GD. However the number of solutions provided is not significant compared to the other two methods. The SPARTE and ELECTRE methods yielded similar results. In these two methods, the five observation parameters are very satisfactory. On the four senses of optimizing criteria, the SPARTE method is better in execution time. The time does not exceed 4 minutes and the error ratio is always very close to 0. The distance of the set of “best” solutions found in relation to the Pareto front is equal to 0 for the best execution cases. The percentage of paths found never exceeded 2%. This shows that our method does not need to explore a very large number of alternatives to converge to the Pareto front.

Sense of optimizing criteria | (−, −) | (−, +) | (+, −) | (+, +) | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Statistical parameters | Min | Avg | Max | Min | Avg | Max | Min | Avg | Max | Min | Avg | Max |

CPU time (second) | 51.7 | 53.1 | 55.5 | 88.5 | 89.5 | 91.2 | 95.3 | 97.4 | 98.9 | 137 | 144 | 153 |

Size of “best” paths | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 |

Error Ratio (ER) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.4 | 1 |

Generational Distance (GD) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 28.3 | 71.6 |

Percentage of paths found (%) | 1.53 | 1.54 | 1.56 | 1 | 1.04 | 1.07 | 1.02 | 1.19 | 1.49 | 1.64 | 1.81 | 2.11 |

Generation | 2000 | 2000 | 2000 | 2000 | 2000 | 2000 | 2000 | 2000 | 2000 | 2000 | 2000 | 2000 |

Sense of optimizing criteria | (−, −) | (−, +) | (+, −) | (+, +) | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Statistical parameters | Min | Avg | Max | Min | Avg | Max | Min | Avg | Max | Min | Avg | Max |

CPU time (second) | 0.56 | 5.15 | 14.6 | 13.5 | 113 | 152 | 140 | 146 | 151 | 2.26 | 55.3 | 138 |

Size of “best” paths | 4 | 4 | 4 | 32 | 34 | 35 | 28 | 33 | 35 | 7 | 8 | 8 |

Error Ratio (ER) | 0 | 0 | 0 | 0 | 0.01 | 0.03 | 0 | 0.37 | 0.62 | 0 | 0.03 | 0.14 |

Generational Distance (GD) | 0 | 0 | 0 | 0 | 7.4 | 37 | 0 | 138 | 210 | 0 | 14.3 | 71.6 |

Percentage of paths found (%) | 0.03 | 0.08 | 0.13 | 0.43 | 1.21 | 1.57 | 1.33 | 1.45 | 1.68 | 0.11 | 0.75 | 1.31 |

Generation | 16 | 65 | 137 | 243 | 1547 | 2000 | 2000 | 2000 | 2000 | 45 | 883 | 2000 |

Sense of optimizing criteria | (−, −) | (−, +) | (+, −) | (+, +) | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Statistical parameters | Min | Avg | Max | Min | Avg | Max | Min | Avg | Max | Min | Avg | Max |

CPU time (second) | 0.26 | 1.21 | 5.69 | 46.3 | 151 | 185 | 153 | 165 | 175 | 5.23 | 113 | 157 |

Size of “best” paths | 4 | 4 | 4 | 32 | 33 | 35 | 26 | 32 | 36 | 7 | 8 | 8 |

Error Ratio (ER) | 0 | 0 | 0 | 0 | 0.02 | 0.09 | 0 | 0.15 | 0.61 | 0 | 0 | 0 |

Generational Distance (GD) | 0 | 0 | 0 | 0 | 24.9 | 106 | 0 | 75.1 | 211 | 0 | 0 | 0 |

Percentage of paths found (%) | 0.02 | 0.05 | 0.15 | 0.79 | 1.31 | 1.48 | 1.35 | 1.5 | 1.62 | 0.18 | 1.01 | 1.35 |

Generation | 5 | 30 | 149 | 586 | 1792 | 2000 | 1792 | 1979 | 2000 | 85 | 1505 | 2000 |

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.

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

Gazawa, F.G., Kolyang and Damakoa, I. (2019) An Evolutionary Algorithm Coupled to an Outranking Method for the Multicriteria Shortest Paths Problem. American Journal of Operations Research, 9, 114-128. https://doi.org/10.4236/ajor.2019.93007