Tree of fuzzy shortest paths with the highest quality

Transcription

1 Mathematical Sciences Vol. 4, No. 1 (2010) Tree of fuzzy shortest paths with the highest quality Esmaile Keshavarz a, Esmaile Khorram b,1 a Faculty of Mathematics, Islamic Azad University-Sirjan Branch, Sirjan, Iran. b Faculty of Mathematics and Computer Sciences, Amirkabir University of Technology, Tehran, Iran. Abstract In this paper we present a network with a finite set of nodes and a set of imprecise arc lengths (costs) instead of real numbers. The imprecise lengths (costs) are modeled as fuzzy intervals with increasing membership functions (based on the quality), whereas the total cost of the shortest paths is a fuzzy interval with a decreasing linear membership function. To obtain a tree of fuzzy shortest paths from a source node to all other nodes, an algorithm is developed. By the max-min criterion suggested by Bellman and Zadeh, the fuzzy shortest path (with highest quality) problem can be treated as a mixed integer nonlinear programming problem. We show that this problem can be simplified into a bi-level programming problem that is easily solvable. An efficient algorithm, based on the parametric shortest path, is proposed for solving the bi-level programming problem. An illustrative example is also included to demonstrate our proposed algorithm. Keywords: Shortest path; Fuzzy interval, Bi-level programming, Labeling algorithms, Parametric shortest path. c 2010 Published by Islamic Azad University-Karaj Branch. 1 Introduction Determination of the shortest distance and the shortest path between two nodes in a directed graph is one of the most fundamental problems in graph theory. This problem 1 Corresponding Author. Address: eskhor@aut.ac.ir

2 68 Mathematical Sciences Vol. 4, No. 1 (2010) is a classical and important network optimization problem appearing in many applications. It is certainly one of the basic components in the fields of transportation and communication networks (Ahuja et al. [1]). In a network, the lengths of arcs are assumed to represent transportation time or cost rather than the geographical distances. Consider an acyclic directed network G (N, A), consisting of a set of nodes N = {1, 2,..., n} and of m directed arcs A N N. Each arc is denoted by ordered pair (i, j) where i, j N. There is an arc length (or arc cost) c ij associated with each arc (i, j) A. It is supposed that there is only one directed arc (i, j) from i to j. The network has a distinguished node s, called the source. We define a directed path p ij as a sequence of alternating nodes and arcs from i to j. If c ij denotes a positive deterministic number associated with arc (i, j) corresponding to the cost necessary to traverse (i, j) from i to j, the cost (length) of a directed path is the sum of the costs of arcs in the path. The tree of shortest paths problem(tspp) is to determine for every non source node i N a shortest length directed path from node s to node i. Alternatively, we might view the problem as sending 1 unit of flow as cheaply as possible (with arc flow costs as c ij ) from node s to each of the nodes in N {s}in un-capacitated network. This viewpoint gives rise to the following linear programming formulation of the tree of shortest path problem: Minimize s.t {j:(i,j) A} (i,j) A x ij c ij x ij (1.1) {j:(j,i) A} n 1 x ij = 1 for i = s for i N {s} (1.2) x ij 0 for any (i, j) A. (1.3) In the crisp world, it is apparent to obtain the shortest length. However, the costs of many real world applications are not deterministic numbers. In these cases, using fuzzy numbers for modeling the problem is quite appropriate, and the fuzzy shortest path problem (FSPP) appears in a natural way (Belmant and Zadeh [4]; Dubois and Prade [9]; Hernandes, et al. [10]; Klein [11]). In the FSPP, the total costs are fuzzy numbers,

3 E. Keshavarz and E. Khorram 69 and what is difficult is to find a path being smaller than all the others. The shortest path problem involves addition and comparison of the edge weights. Since, addition and comparison between two interval numbers or between two fuzzy numbers are not like those between two precise real numbers, there is an important problem for these operations. So, there are different approaches to viewing the fuzzy shortest path problem. Numerous papers have been published on the FSPP (Blue et al. [5]; Boulmakoul [6];Chuang et al. [7]; Hernandes et al. [10]; Klein [11]; Kung and Chuang [12]; Moreno et al. [14]; Nayeem and Pal [15]; Okada [16]; okada et al. [17,18,19]; Sengupta [20]), and each of them has proposed and solved a model from a different viewpoint. Some of them generalized Dijkstras algorithm (Okada and gen [17,18]), in which the weights of the arcs are considered interval numbers and are defined by different orders between interval numbers. Sengupta and Pal [20] have proposed a methodology that considers fuzzy preference ordering of intervals, and have given a procedure for choosing a preferred minimum from a set of n intervals, then they extended Dijkstras algorithm by this ordering. Dubois and Prade [9] pointed out using extended sum,, for solving the classical fuzzy shortest path problem through, and also, extended min and max,(i.e. min and max), for solve the problem by Fords algorithm. Blue et al. [5] have presented the taxonomy of fuzzy graphs and in a distinct section provided formulations of FSPP in terms of fuzzy paths and level sets. Nayeem and Pal [15] proposed an algorithm based on the acceptance index, which gives a single fuzzy shortest path or a guideline for choosing the best fuzzy shortest path. Okada [16] introduced the concept of the degree of possibility of an arc being on the shortest path. Boulmakoul [6] proposed an algebraic structure to solve a pathfinding problem in a fuzzy graph and adapted this structure to solve the fuzzy shortest path problem. Kung and Chuang [12] defined the similarity degree between fuzzy length sets on a shortest path problem and discrete fuzzy arc lengths, and e solved this problem. Other works have looked at this problem from various viewpoints. However, owing to the high computational complexity of FSPP, there is some difficulty in solving the problem. Some

4 70 Mathematical Sciences Vol. 4, No. 1 (2010) suggested approaches find costs without an existing path, some of them characterizethe solution not as a shortest path, but as a fuzzy set solution, where each element is a non-dominated path or Parteo optimal path with fuzzy edge weights. However, theses algorithms donot provide the decisionmaker with any guidelines for choosing the best point of view. In this paper, following the approach of Lin and Wen [13] to the fuzzy assignment problem, the costs of arcs in the network are defined as fuzzy intervals and an FSPP is formulated. Then we reformulate this problem as a bi-level programming problem and propose an efficient algorithm to solve it. In our model there are two issues. First, the total cost is intended to be minimized, where the costs are fuzzy intervals, and the second issue is that we need to consider quality at the highest level.in fact, two objectives are followed each of which can be modified with the other. On the other hand, the model is designed based on fuzzy numbers, but its algorithm solves the problem with exactness and without involving the fuzzy ranking approaches, and find the optimal path with the highest quality and minimum total cost, explicitly. So decision makers can choose the best path without confusion. In this model we can modify the algorithm so that it specifies an order between paths based on quality. The paper is developed as follows. In the next section a mathematical formulation is presented for finding a tree of fuzzy shortest paths based on the concept of quality. In section 3, the optimality conditions are analyzed and then, we reformulate the fuzzy shortest path problem as a bi-level programming problem with crisp variables and feasible solutions. In section 4, an algorithm based on the parametric shortest path problem is proposed in order to solve the bi-level programming problem, and for the sake of illustration, an example is sloved to explain the algorithm. Finally, in section 5, some conclusions are pointed out.

5 E. Keshavarz and E. Khorram 71 2 Tree of fuzzy shortest path Assume that there are several ways for traversing an arc, where traversing costs vary according to some factors, such as quality, safety, convenience, etc. We denote these factors by the general term: quality. Hence, a minimum cost for traversing an arc is assumed, where spending higher costs results in higher quality until it reaches an upper bound, when an increase in cost does not increase the quality. In this case, the costs are no longer deterministic numbers and they will be denoted them by c ij s. Futhermore, α ij is defined as the lowest cost associated with traversing (i, j) from i to j, and β ij as the highest cost associated with traversing (i, j) from i to j with the highest quality. It is obvious that β ij > α ij 0. We also assume that the quality associated with traversing (i, j) is q ij (c ij ), 0 q ij 1. The highest quality of traversing (i, j) is denoted by Q ij. In most real cases, it is clear that, 0 < Q ij 1. This concept can be used to define c ij as a (subnormal) fuzzy interval. The membership function of c ij is considered as the linear monotonically increasing function shown in (2.1) and Figure 1, which shows that any expense exceeding β ij is not useful since the quality can no longer be increased at its upper limit, i.e. Q ij. µ ij (c ij ) = q ij (c ij ) = Q ij if c ij β ij, x ij > 0 Q ij (c ij α ij ) β ij α ij if α ij c ij β ij, x ij > 0 0 otherwise (2.1) Condition x ij > 0 is added to (2.1) because there is no real expense if x ij = 0 in any feasible solution x = (..., x ij,...) of ( ). notation α ij, β ij is used to denote fuzzy interval c ij. Furthermore, let z denote the total cost (length) of paths from the source to all other nodes. z 0 and z 1 are defined as the lower and upper bounds of z, respectively. Therefore, the membership function of z is defined as the linear monotonically decreasing function in (2.2) and Figure 2, and also, notation z 0, z 1 is applied to denote fuzzy interval z.

6 72 Mathematical Sciences Vol. 4, No. 1 (2010) Figure 1: Membership function of c ij. µ(z(x)) = 1 if z(x) z 0 z 1 z(x) z 1 z 0 if z 0 z(x) z 1 0 otherwise (2.2) Where z(x) = (i,j) A c ijx ij and x is a feasible solution of ( ). z 0 and z 1 are constants, and are subjectively dependent on the decision of decision-makers. A logical and wise choice is for z 0 to be the minimum total cost with respect to minimum costs (α ij s) and z 1 to be the maximum total cost with respect to highest quality costs (β ij s), which means: z 0 = min x X (i,j) A α ij x ij and z 1 = max x X (i,j) A β ij x ij. Where X is the set of feasible points satisfying constrains (1.2) and (1.3). To obtain a good selection of paths that minimizes cost while maximizing quality, we choose Bellman-Zadeh s criterion [4], which maximizes the minimum of the membership function corresponding to that solution, i.e., max{min x X {µ ij(c ij ), µ(z(x))}} = max{ min x ij >0 {µ ij(c ij ), µ(z(x))}}. (2.3) Then, the fuzzy shortest path problem (with the highest quality) is represented as

7 E. Keshavarz and E. Khorram 73 Figure 2: Membership function of z. follows: max s.t { min { µ ij(c ij ), µ(z(x))} } (2.4) x ij >0 x ij n 1 for i = s x ij = (2.5) 1 for i N {s} {j:(i,j) A} {j:(j,i) A} x ij 0 for any (i, j) A. (2.6) This problem finds a tree of paths from node, s to all other nodes that has the minimum total cost. Furthermore, it has the maximum possible quality of arcs. By membership functions (2.1) and (2.2) we can represent ( ) as the equivalent of model ( ), where c λ ij denotes the minimum of the λ-cut of c ij ( i.e. µ ij (c λ ij ) = λ ) and (i,j) A cλ ij x ij is the corresponding total cost of the tree of paths from the source node to all other nodes (by x X ). It is obvious that λ can be the quality of arcs in the tree.

8 74 Mathematical Sciences Vol. 4, No. 1 (2010) max s.t. λ (2.7) λx ij Q ij(c λ ij α ij) β ij α ij x ij for (i, j) A (2.8) λ z 1 (i,j) A cλ ij x ij z 1 z 0 (2.9) 0 λx ij Q ij x ij for (i, j) A (2.10) 0 λ 1 (2.11) c λ ijx ij β ij x ij for (i, j) A (2.12) x ij n 1 for i = s x ij = (2.13) 1 for i N {s} {j:(i,j) A} {j:(j,i) A} x ij 0 for (i, j) A (2.14) Since x ij, c λ ij and λ, are decision variables in ( ), the model can be treated as a nonlinear programming problem. Although, the classical methods in nonlinear programming can be applied to solve this model (Bazaraa et al. [3]), this problem can be converted to a new model and then be dealt with by an efficient algorithm. 3 Fuzzy shortest path problem on crisp feasible solutions In this section, fuzzy shortest path problem ( ) based on crisp feasible solutions is described. For convenience, firstly, define E = {(i, j) A x ij > 0}as a special case

9 E. Keshavarz and E. Khorram 75 feasible solution x of ( ), then ( ) can be simplified as follows: max λ (3.1) s.t. λ Q ij(c λ ij α ij) β ij α ij for (i, j) E (3.2) λ z 1 cλ ij x ij z 1 z 0 (3.3) 0 λ Q ij for (i, j) E (3.4) 0 λ 1 (3.5) c λ ij β ij for (i, j) E (3.6) Since c λ ij β ij then, λ Q ij and λ 1 become redundant constraints in before model. Further, let d ij = β ij c λ ij 0, then ( ) can be expressed as follows: max λ (3.7) s.t. λ Q ij(β ij α ij d ij ) β ij α ij for (i, j) E (3.8) λ z 1 (β ij d ij )x ij z 1 z 0 (3.9) λ, d ij 0 for (i, j) E (3.10) Obviously, (when x ij s are given) λ and d ij s are decision variables of ( ). Now, assuming that x is a feasible solution of problem, the following theorems find optimality conditions for this solution. then Theorem 3.1. Let λ x, the optimal value of ( ), and suppose: min Q ij > z 1 (β ij d ij )x ij z 1 z 0 (3.11) λ x = z 1 (β ij d ij )x ij z 1 z 0 = Q ij(β ij α ij d ij ) β ij α ij for any (i, j) E.

10 76 Mathematical Sciences Vol. 4, No. 1 (2010) Proof. Rewriting ( ) into the following linear programming model: max λ (3.12) s.t. d ij + (β ij α ij ) λ β ij α ij for (i, j) E (3.13) Q ij d ij x ij + (z 1 z 0 )λ z 1 β ij x ij (3.14) d ij 0 for (i, j) E (3.15) λ 0 (3.16) its dual problem is obtained as follows: min (β ij α ij )w ij + (z 1 β ij x ij ) (3.17) s.t. w ij x ij w 0 for (i, j) E (3.18) (β ij α ij ) w ij + (z 1 z 0 )w 0 Q ij (3.19) w ij 0 for (i, j) E (3.20) w 0 (3.21) Let s ij, (i, j) E and s be the slack variables of (3.13) and (3.14), respectively. Similarly, let u ij, (i, j) E, and u be the surplus variables of (3.18) and (3.19), respectively. Since we have (3.11) then by (3.9) we have λ < min Q ij and d ij > 0 (because if d ij = β ij c λ ij then β ij = c λ ij and this means λ = 1 and so we have 1 < min Q ij which is a contradiction). Furthermore, based on the complementary slackness theorem (Bazaraa et al [2]) u ij = 0 is obtained for (i, j) E. Therefore, it is easy to show w ij x ij w = 0 for all (i, j) E. If w ij = w = 0, there is a contradiction to (3.19), so w ij = wx ij > 0 is obtained. Hence, since x ij > 0 for (i, j) E then we have w > 0 and w ij > 0. Again by the complementary slackness theorem s ij = s = 0 is obtained for (i, j) E and the proof is completed.

11 E. Keshavarz and E. Khorram 77 Theorem 3.2. Suppose min Q ij z 1 Then (d ij, λ) = ((1 ( min Q ij ( ). Q ij (β ij d ij )x ij z 1 z 0 )(β ij α ij )), min Q ij), is the optimal solution of Proof. We first denote (d ij, λ) = ((1 ( min Q ij/q ij )(β ij α ij )), min Q ij) for (i, j) E which satisfies the nonnegative constraints and (3.8),by hypothesis of theorem,also, satisfies constraint (3.9), so it is a feasible solution of problem. Furthermore, suppose λ is another feasible solution of problem such that λ > By (3.8), we have Q kl < λ Q kl(β kl α kl d kl ) β kl α kl nonnegative constraint of d kl. Hence, proof is completed. min Q ij = Q kl, for (k, l) E., i.e. d kl < 0, which contradicts the min Q ij is the maximum value of λ and the 3.1 Bi-level programming model Since λ x is the maximum value of the objective function of ( ), by having a feasible solution x of ( ), the maximum, λ x, must be the optimal solution of ( ). However, a shortest path problem has many feasible solutions and it is hard to identify the feasible solution that finds the maximum value of λ. By theorem (3.1), if we have (3.11) then optimal value,λ x is as follows: λ x = Q ij(c λx ij α ij) for any (i, j) E. (3.22) β ij α ij λ x = z 1 c λx ij x ij z 1 z 0 (3.23) By (3.22) it is clear that c λx (β ij = α ij + λ ij α ij ) x Q ij, and by (3.23), λ x is the maximum iff c λx ij x ij is the minimum. So we can say, if c λx ij x ij is minimized on X, for some λ and this λ satisfies (3.23) regarding E = {(i, j) A x ij > 0}, and also if

12 78 Mathematical Sciences Vol. 4, No. 1 (2010) the value of x ij and that of min Q ij obtained from this process satisfy (3.11) then λ will be optimal, otherwise by theorem (3.2) min Q ij is the optimal value of the problem.but how can the value of λ is found? By the before mentioned analysis,the problem can be written as a bi-level programming problem(dempe[8]), so that solving the problem in its new form leads to finding a value of λ which satisfies (3.22) and (3.23).Then,(3.11) can be checked to find the optimal value of the original problem. Now, consider the following bi-level programming problem: max λ (3.24) z 1 (i,j) A c λ ijx ij s.t. λ = (3.25) z 1 z 0 { } (..., x ij,...) = x argmin c λ ijx ij : x X (3.26) (i,j) A where c λ ij = α ij + λ (β ij α ij ) Q ij, and X is the set of feasible points satisfying constraints (1.2) and (1.3).We call ( ) and (3.26),upper-level and lower-level problems, respectively. If λ is a fixed value,the lower-level problem is a classical shortest path problem, but in fact λ is a variable that satisfies the conditions of the upper-level problem and it should be maximized. In general, there are many different methods to solve bi-level programming problems (Dempe[8]).But the present problem is of a special form and can be solved efficiently with exactness. 4 Solving the bi-level programming problem In this section, a simple and efficient algorithm is established based on the parametric cost shortest path problem to solve the bi-level programming problem ( ).Lower-level problem (3.26) is a parametric cost shortest path problem (with λ as a parameter),which can be solved by an effective algorithm (Ahuja et al. [1])but the solutions of this problem are dependent on λ, and the solution which satisfies upper-

13 E. Keshavarz and E. Khorram 79 level problem ( )should be found by its associated λ. This algorithm gives us intervals such as [λ 1, λ 2 ] [0, 1], and for any such interval,a shortest path tree T (λ) is found which is optimal for all λ [λ 1, λ 2 ]. Since cost functions c λ ij of the network are linear in term of λ, then the variation of the total cost (of the shortest paths) is linear in terms of λ. Therefore, constraint (3.25) can be checked and some interval can be found in which the optimal λ lies. To specify such an interval, for any λ [λ 1, λ 2 ] and its associated optimal solution x of lower level problem (3.26), we define by (3.25): f(λ) = λ z 1 (i,j) A c λ ijx ij z 1 z 0 (4.1) If λ is optimal then f(λ) = 0, (it is clear that f is a linear function of λ). By this definition, any interval [λ 1, λ 2 ] over which the sign of f(λ) changes ( i.e., f(λ 1 ).f(λ 2 ) 0 ) contains the optimal value of λ and λ can be found using the linearity of c λ ij and f(λ). Furthermore, by theorem (3.2) if: min Q ij z 1 (β ij d ij )x ij or z 1 z 0 min Q ij λ (since f(λ) = 0) then the optimal value of the our problem is λ = min Q ij. Regarding this discussion, in the next section an algorithm is proposed to solve the problem. 4.1 Proposed algorithm For simplicity, assume that: β ij α ij Q ij = γ ij, (i, j) A. Then the parametric cost of (i, j) is c λ ij = α ij + γ ij.λ. A labeling algorithm (such as label setting or label correcting (Ahuja et al [1]))is utilized to solve the parametric shortest path problem with costs c λ ij. The distance label of node i N is denoted by dλ i and that associated with costs γ ij by d γ i. Remember that the labeling algorithms find all shortest paths from a source to all other nodes, and at optimality, distance label(or potential) d i of node i shows the length of the shortest path from the source to i. Furthermore, from the point of view of linear programming duality,d i is the dual variable associated with the constraint of

14 80 Mathematical Sciences Vol. 4, No. 1 (2010) i-th node (Ahuja et al [1]). For any arc (i, j) A the reduced cost of (i, j) associated with c λ ij is denoted by cλ ij such that, cλ ij = cλ ij + dλ i dλ j (it should be noted that the necessary and sufficient condition for a set of distance labels to represent the shortest path distances in the network with arc costs c λ ij is cλ ij = cλ ij + dλ i dλ j 0 (Ahuja et al [1.Theorem 5.1])) and the reduced cost of (i, j) associated with γ ij is shown by γ ij so that γ ij = γ ij + d γ i dγ j. The following algorithm, describes the details of our proposed method to find the optimal value of bi-level programming ( ). Algorithm. Input: Network G = (N, A, c λ ), where, c λ ij = α ij + γ ij.λ. Step 1: Let λ = 0. Step 2: Solve the shortest path problem on G = (N, A, c λ ) (by a labeling algorithm), find d λ i for any i N, and find the tree of the shortest paths, T (λ). Step 3: For any i N, find distance labels d γ i associated with T (λ) (with costs γ ij ). Step 4: For any arc (k, l) / T (λ), find reduced costs: c λ kl, γ kl, and λ kl where: cλ kl γ if γ λ kl = kl kl < 0 otherwise Step 5: Let λ = min λ kl : (k, l) / T (λ) + λ. Step 6: Let arc (p, q) be a non-tree arc for which λ pq + λ = λ, then adding arc (p, q) to T (λ) and dropping the unique tree arc entering node q gives an alternate tree of the shortest path at λ; update the distance labels of T (λ). Step 7: If f(λ).f(λ) > 0, then let λ = λ and go to step(3), else let λ λ.f(λ) λ.f(λ) = f(λ) f(λ) and go to step (8). Step 8: If λ > min Q ij = min Q (i,j) T (λ ij, then let λ = min Q ij and update the ) distance labels of T (λ ), else let T (λ ) = T (λ) ( because by parametric analysis, the shortest path for any value in interval (λ, λ) is T (λ) ) Output: The optimal values are λ and T (λ ). One of the classic labeling algorithms (such as Dijkstra s algorithm) can be applied to

15 E. Keshavarz and E. Khorram 81 solve SPP (Ahuja et al. [1]).Then d λ ij in step(2) is the optimal distance label of node i at the end of the labeling algorithm. This label is the length of the shortest path from the source node to node i in the network with arc costs c λ ij. λ in step(5) is the maximum value for which tree T (λ) remains a tree of shortest path as long as λ < λ, and so, a new tree T (λ) in step(6) can be found. λ in step(7) is the root of f(λ). In fact f(λ) is a piecewise linear function and f(λ ) = 0 ( the equation of f from point (λ, f(λ)) to (λ, f(λ)) is obtained and then λ is found so that f(λ ) = 0 ). The algorithm starts with λ = 0, and in any iteration, after specifying the tree of shortest paths, T (λ), finds λ λ such that for all λ [λ, λ), T (λ) remains the shortest paths tree of the network with costs c λ ij. Then in step (7), the algorithm checks that interval [λ, λ] contains the root of f(λ) and finds it; otherwise λ is replaced with λ and the steps of the algorithm are repeated. If [λ, λ] contains the root of f(λ), by the linearity of it, in step (7) the root is found. In step (8) the algorithm checks λ and min Q ij, if λ > min Q ij, by theorem (3.2), the optimal value is min Q ij, else by theorem (3.1) the optimal value is λ. After updating T (λ ) at the end of the algorithm, we have the optimal value of our problem and the associated shortest paths tree that is the shortest paths tree with the highest quality. In the next section for further elaboration, a numerical example is solved to explain the algorithm. 4.2 Numerical example Let us consider the fuzzy shortest path problem on the network shown in Figure 3, with fuzzy costs and qualities as given in Table 1. By solving the shortest path problem for c ij = α ij and the longest path problem for c ij = β ij,then z 0 = 29 and z 1 = 107 are respectively obtained, and also, by γ ij = βij α ij Q ij values γ 12 = 2, γ 13 = 3, γ 14 = 2, γ 23 = 25 4, γ 24 = 3, γ 25 = 30 7, γ 34 = 1, γ 36 = 40 9, γ 45 = 6, γ 46 = 30 7, γ 47 = 10 3, γ 56 = 50 9, γ 57 = 2, γ 67 = 2 are obtained. Table 2 shows the results of the first iteration of the algorithm.

16 82 Mathematical Sciences Vol. 4, No. 1 (2010) Figure 3: Network of numerical example Table 1: Arc information of the numerical example Arc Fuzzy cost Quality Arc Fuzzy cost Quality (i, j) α ij, β ij Q ij (i, j) α ij, β ij Q ij (1,2) 3, (3,6) 3, (1,3) 3, (4,5) 2, (1,4) 4, (4,6) 3, (2,3) 1, (4,7) 5, (2,4) 1, (5,6) 3, (2,5) 2, (5,7) 3, (3,4) 2, (6,7) 2, By Table 2, when λ = 0, the non - tree arcs are (2,3),(2,4),(3,4),(4,5),(4,6),(4,7),(5,6),(6,7), and by step(4) and step(5) we will have: λ 23 = λ 24 = λ 34 = λ 45 = λ 56 = λ 67 =, λ 46 = cλ 46 γ 46 = 63 73, λ 47 = Therefore, λ = min{, 63 73, } + λ = = After updating the distance labels of T (λ), arc (4,7) enters, the new shortest path tree and the unique tree arc entering node 7 (i.e. (5,7) ) leaves it. Table 3 shows the new results. By step (7), since f(λ).f(λ) = f(0).f( ) = ( 1).( 7254 ) = > 0, we should repeat the algorithm. By Table 3, the non-tree arcs are (2,3),(2,4),(3,4),(4,5),(4,6),(5,6),(5,7),(6,7), and by

17 E. Keshavarz and E. Khorram 83 Table 2: Results of the first iteration of numerical example λ=0 costs arc (i, j) (1,2) (1,3) (1,4) (2,3) (2,4) (2,5) (3,4) T (λ) : c λ ij = arc (i, j) (3,6) (4,5) (4,6) (4,7) (5,6) (5,7) (6,7) c λ ij = distance node i Total cost of tree labels of d λ i = Z(x) = 29. nodes d γ i = Table 3: Results of the second iteration of numerical example λ = costs arc (i, j) (1,2) (1,3) (1,4) (2,3) (2,4) (2,5) (3,4) T (λ) : c λ ij = arc (i, j) (3,6) (4,5) (4,6) (4,7) (5,6) (5,7) (6,7) c λ ij = distance node i Total cost of tree labels of d λ i = 0 nodes d γ i = Z(x) = 3611 = step(4) and step(5) the following values are obtained: λ 23 = λ 24 = λ 34 = λ 45 = λ 56 = λ 57 = λ 67, λ 46 = Therefore λ = λ = ( ).( 5694 ) < 0. and we have: 63 = 63 73, by step(7), f(λ).f(λ) = f(21 62 ).f(63 73 ) = λ = λ.f(λ) λ.f(λ) f(λ) f(λ) = = [ 21 62, 63 ]. 73 By step (8),T (λ ) = T (λ), which contains arcs (1,2), (2,5), (1,4), (4,7), (1,3), (3,6). Since λ > min (i,j) T (λ ) Q ij = Q 25 = 0.5 the optimal value of λ is λ = 0.5. So

18 84 Mathematical Sciences Vol. 4, No. 1 (2010) Table 4: Results of numerical example for the optimal solution λ = 0.5 costs arc (i, j) (1,2) (1,3) (1,4) (2,3) (2,4) (2,5) (3,4) T (λ) : c λ ij = arc (i, j) (3,6) (4,5) (4,6) (4,7) (5,6) (5,7) (6,7) c λ ij = distance node i Total cost of tree labels of d λ i = nodes d γ i = Z(x) = = the tree of shortest path with the highest quality is T (0.5), which contains arcs such as T ( ) ( because 0.5 ( 21 62, )). Table 4 shows the results of the problem for the optimal value λ = Conclusion In problems of graphs involving uncertainties, the FSPP is one of the most studied topics since it has a wide range of applications in different areas and therefore deserves special attention. This paper studied a fuzzy shortest paths tree model in a network. This problem can be interpreted as the tree of shortest path with the highest quality. Any arc in the network has a fuzzy interval cost. We suggested an algorithm based on the parametric (cost) shortest path problem, which solves our problem with exactness. The algorithm can yield a number belonging to (0,1) as the highest reliability and a 14 shortest path with highest possible quality. This algorithm is independent of fuzzy ranking. In fact, the FSPP is reduced to a crisp model. This problem can be extended to some interesting models; for example, ordering paths by quality or by cost. Also, this problem can be defined on a network with some other factors such as quality or a network with discrete fuzzy arc lengths. Development of our model to

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