This report presents an  implementation the Dijkstra algorithm applied to a type V fuzz graph. This new algorithm can find the  shortest path in a graph with edge costs are defined as positive triangular fuzzy number

Alonso, Sira

Ariza-Quintana, Alfonso

Joya-Caparrós, Gonzalo

Valdés, Lissette

English

Valdes Valdes, Lissette

Repositorio Institucional Universidad de Málaga

An implementation of the Dijkstra algorithm forfuzzy costs. (Technical Report 2018)Lissette Valdés Valdés, Sira M. Allende, Alfonso Ariza, Gonzalo Joya1 Dijkstra algorithm for type V fuzzy graphIn this section, we propose a modified Dijkstra algorithm applied to a type V fuzzy graph(Definition 1.1).Definition 1.1. (Type V fuzzy graph)Also known as a crisp graph with fuzzy costs, a type V fuzzy graph has known vertices and edges,but uncertainty is present in the costs on the edges, i.e.:This algorithm finds the shortest path between the source vertex r and any other vertexon G˜, but dealing with fuzzy weights defined on links. In order to introduce our fuzzyversion of Dijkstra algorithm, is important to define the arithmetic operations and a rank-ing method for fuzzy numbers. The method that we use for the arithmetic operations oftriangular fuzzy numbers is introduced in section 1.1, we define the ranking method insection 1.2, and in section 1.3 we give our proposal of a Fuzzy Dijkstra algorithm for a typeV fuzzy graph.1.1 Method for the arithmetic operations of normalized triangular fuzzynumbersAt first, we give the definition of the so-called fL-fR positive fuzzy number:Definition 1.2. (fL − fR Positive fuzzy number)Let C(u,v) = (a, b, c, d, ω) be the positive fuzzy number representing the cost of an edge e = 〈u, v〉with u, v ∈ V . Its membership function can be expressed as:µC(u,v)(x) =fL(x) a ≤ x ≤ bω b ≤ x ≤ cfR(x) c ≤ x ≤ d0 otherwise(1)where fL : [a, b] → [0, ω] and fR : [c, d] → [0, ω] are both strictly monotonic (increasing anddecreasing, respectively) and continuous functions with ω ∈ [0, 1].1We asume fL and fR to be linear functions. Thus, according to the values of ω the costof each edge is either normalized or generalized, and according to a, b, c and d a triangularor trapezoidal fuzzy number.• L(u,v) normalized triangular fuzzy number.(ω = 1, b = c)• L(u,v) normalized trapezoidal fuzzy number.(ω = 1, a 6= b 6= c 6= d)• L(u,v) generalized triangular fuzzy number.(ω 6= 1, b = c)• L(u,v) generalized trapezoidal fuzzy number. (ω 6= 1, a 6= b 6= c 6= d)For the arithmetic operations of fuzzy numbers, we use the α-cut method. This methodis quite simple (in comparison to the extension principle method) since the operation be-tween fuzzy numbers is reduced to the operation between ordinary intervals in R.First, we state two essential properties for the operations of triangular and trapezoidalfuzzy numbers:1. The results from addition and subtraction between triangular or trapezoidal fuzzynumbers are also a triangular o trapezoidal fuzzy number, respectively.2. The results from multiplication or division as in maximum or minimum operationsare not a triangular or trapezoidal fuzzy number.The α-cut interval of a normalized triangular fuzzy number A˜, denoted by A˜α, is an or-dinary crisp interval in R containing all the elements of A˜ whose membership degree isgreater than or equal to the specified value α ∈ [0, 1]. Let A˜ = (a, b, c) be a normalizedtriangular fuzzy number with membership function with shape as in expression 1, wherefL(x) =x− ab− a and fR(x) =c− xc− b , A˜α is defined as in equation 2.A˜α = [(b− a)α+ a,−(c− b)α+ c] α ∈ [0, 1] (2)The following two properties of fuzzy numbers are based on the representation of a fuzzyset and the definition of the α-cut of a fuzzy set,(a) Each fuzzy set, and thus also each fuzzy number, can fully and univocally be repre-sented by its α-cut.(b) α-cuts of each fuzzy number are closed intervals of real numbers for all α ∈ [0, 1]Definition 1.3 provides the necessary elements to define the α-cut of the fuzzy number(A˜ ∗ B˜) by the α-cuts of the fuzzy numbers A˜ and B˜. Thus, we can use the operation ofstandard crisp intervals to find (A˜ ∗ B˜)α.Definition 1.3.Let A˜ and B˜ denote fuzzy numbers and let ∗ denote any of the four basic arithmetic operations“+,−, ·,÷”. Then for any α ∈ (0, 1], the fuzzy set on R, (A˜ ∗ B˜), is defined through its α-cut(A˜ ∗ B˜)α:(A˜ ∗ B˜)α = A˜α ∗ B˜αWhen ∗ = “÷′′, clearly, 0 /∈ B˜α ∀α ∈ (0, 1].2According to First Decomposition Theorem, the fuzzy number (A˜ ∗ B˜) is interpreted asthe union of special fuzzy numbers α(A˜ ∗ B˜), ∀α ∈ [0, 1], i.e.,(A˜ ∗ B˜) =⋃α∈(0,1]α(A˜ ∗ B˜)where µα(A˜∗B˜) = α · χ(A˜∗B˜)αThe membership function of (A˜ ∗ B˜) is computed using the union and decomposition offuzzy sets, i.e.,µ(A˜∗B˜)(x) = maxα∈(0,1]{µα(A˜∗B˜)}= maxα∈(0,1]{α · χ(A˜∗B˜)α(x)}where χ(A˜∗B˜)α(x) is the charactaristic function of (A˜ ∗ B˜)α.As summary, based on the properties and arithmetic analysis stated above, we can performthe addition and subtraction of normalized triangular fuzzy numbers by using their α-cuts.We obtain in each case the expressions 3-4,Addition: A˜+ B˜ = (a1, a2, a3)⊕ (b1, b2, b3) = (a1 + b1, a2 + b2, a3 + b3) (3)Subtraction: A˜− B˜ = (a1, a2, a3)	 (b1, b2, b3) = (a1 − b1, a2 − b2, a3 − b3) (4)1.2 Ranking method of triangular fuzzy numbersFor the comparison of the fuzzy costs, we use a ranking criterion proposed by [1]. Thismethod compares the Total Integral of fuzzy numbers depending on a parameter α, alsocalled index of optimism. This index represents the degree of optimism of the decisionmaker. Let A˜ = (a, b, c, d) be a trapezoidal fuzzy number, for α taking values greater than0.5 the comparison is made giving considering priority to numbers greater than the centralvalue. The opposite occurs when α is lower than 0.5.Let C(u,v) = (a, b, c, d, ω) be a generalized trapezoidal fuzzy number representing the fuzzycost in a link joining the vertices u and v. For a fixed α, the Total Integral of C(u,v) is definedas:SαT (C(u,v)) =ω2[(1− α)(a+ b) + α(c+ d)− 2Xmin] (5)where Xmin ≤ a. For simplicity, since every cost in G˜ is assumed to be a positive fuzzynumber in our study, we set Xmin = 0 without affecting the process performance.By making the proper modifications on a, b, c and d, we can obtain the total integral fora normalized and generalized triangular fuzzy number as for normalized or generalizedtrapezoidal fuzzy number.We need to establish a ranking for the fuzzy costs used in our algorithm. To obtain thisranking, we must define an order relation on the set of fuzzy numbers. Let C1 and C2 betwo fuzzy weights (defined either on an edge or a path), for α ∈ [0, 1], each Ci, i = 1, 2 hastotal integral SαT (Ci), then,3• If SαT (C1) > (<)SαT (C2) then C1 is greater (smaller) than C2, denoted as C1  (≺)C2• If SαT (C1) = SαT (C2) and Me(C1) > (<)Me(C2) then C1  (≺)C2• If SαT (C1) = SαT (C2) and Me(C1) = Me(C2) then C1 = C2where Me(Ci) (i = 1, 2) denotes the median of the fuzzy number Ci.This relation, indeed, is an order relation because it meets the antisymmetry, reflexivityand transitivity properties. On the other hand, since the Total Integral satisfies the linearityproperty, we can apply this order relation for both the comparison between the cost of twolinks and the cost of two paths by means of the comparison between their Total Integrals.Therefore, we can apply the classic Dijkstra Algorithm.1.3 AlgorithmLet Ψ(r, v) denote the set of all paths between r and v (v, r ∈ V with v 6= r), and P ∗ ∈ Ψ(r, v)be the shortest path between r (source) and v. The cost of P ∗ is defined as:δ˜(r,v) = CP∗(r,v) = min∀P∈Ψ(r,v){C(P (r,v))} if Ψ(r, v) 6= ∅∞ if Ψ(r, v) = ∅Then, for each vertex v ∈ V , the algorithm defines an attribute d˜, which is an upper boundon the weight of the shortest path between the source r and v,d˜(v) ≥ δ˜(r,v)Given a vertex v ∈ V , we will denote by Γ (v) the set of neighbors of v, that is, adjacentvertices to v. Moreover, we will denote by w(v) the predecessor of v in P ∗. The algorithmconsiders w(v) to be either a vertex of Γ (v) or NIL. Actually, during the execution of thealgorithm, w(v) is the predecessor of v in the shortest path to v known so far. Only whenthe algorithm has ended, we can say that w(v) is the predecessor of v by the shortest pathfrom r to v.Every vertex has got assigned a label which adapts throughout the algorithm execution. Ateach stage, the label of vertex v ∈ V contains its predecessor in the (known so far) shortestpath from r to v, and the corresponding cost of this path, i.e.,Label(v) =[d˜(v), w(v)]with d˜(v) = d˜(w(v)) ⊕ C(w(v),v)w(v) : (predecessor vertex of v in the provisional path P ∗(r, v))Where C(w(v),v) is the cost of link (w(v), v). At the end of the algorithm, for each vertex v,d˜(v) will coincide with δ˜(r,v) and w(v) will be its predecessor in P ∗(r, v).The Initialization (algorithm 1) assigns a label to every vertex in G˜. The cost of theshortest path from r to any other vertex in v ∈ G˜ is inicialized as∞, except for r, which isset to be equal to 0.4Algorithm 1 Initialization1: function INITIALIZE-SINGLE-SOURCE(G˜, r,L)2: d˜(r)← (0, 0, 0)3: w(r)← NIL4: for each vertex v ∈ V − {r} do5: d˜(v)←∞6: end for7: end functionThe algorithm advances by selecting and extracting vertices from a set, denoted as H ,which is initially defined as V . At each iteration, the vertex uwith minimum cost is selectedin H according to the ranking method described in section 1.2. Then, the neighbors of thisvertex are analyzed in algorithm 2 (Relaxation). The label of each neighbor v is updated ifthe path from r to v through u has a cost less than the current one.Algorithm 2 Relaxation of v1: function RELAXATION(u, v, C(u,v), α)2: d˜new(v) := d˜(u)⊕ C(u,v)3: if SαT (d˜(v)) > SαT (d˜new(v)) then4: d˜(v)← d˜new(v)5: w(v)← u6: end if7: end functionThe algorithm ends when there are no vertices left in H . At this point, the label of eachvertex v contains δ˜(r,v) together with its predecessor w(v) on P ∗(r, v). Following a back-ward procedure, the shortest path between r and each vertex can be found. Algorithm 3shows the pseudocode of the Dijkstra algorithm proposed.Algorithm 3 Dijkstra(G˜, r,L)1: Initialize-single-source(G˜, r,L) . Initialization2: H ← V3: while H 6= ∅ do4: u← t|d˜(t) = min∀x∈H{d˜(x)}5: Update H ← H − {u}6: if Γ (u) ∩H 6= ∅ then7: for each vertex v ∈ Γ (u) ∩H do8: Relaxation(u, v, C(u,v), α) . Relaxation of v9: end for10: end if11: end while5References[1] Vincent F. Yu and luu Quoc Dat: An improved ranking method for fuzzy numbers withintegral values. Applied Soft Computing 14., 603-608 (2014)6

An implementation of the Dilkstra algorithm for fuzzy costs (Technical report 2018)

Repositorio Institucional de la Universidad de Málaga

https://riuma.uma.es/xmlui/bitstream/10630/17478/1/Fuzzy_Dijkstra_algorithm.pdf

An implementation of the Dilkstra algorithm for fuzzy costs (Technical report 2018)

Abstract

Similar works

Full text

Available Versions

Repositorio Institucional Universidad de Málaga

Repositorio Institucional de la Universidad de Málaga