Fuzzy Shortest Path Problem Based on Level 𝜆-Triangular LR Fuzzy Numbers

Algorithm 1

  Input:  𝐿_𝑖=(𝑚_𝑖,𝛼_𝑖,𝛽_𝑖;𝜆)_LR, 𝑖=1 to 𝑛 where 𝐿_𝑖 denotes the level 𝜆-triangular LR fuzzy path length.

  Output:  𝐿_min=(𝑚,𝛼,𝛽;𝜆)_LR, where 𝐿_min denotes the level 𝜆-triangular LR fuzzy shortest length.

  Step 1: Construct a Network 𝐺=(𝑉,𝐸) where 𝑉 is the set of vertices and 𝐸 is the set of edges. Here 𝐺 is an acyclic

      digraph and the arc length takes the level 𝜆-triangular LR fuzzy numbers.

  Step  2: Calculate all the possible paths 𝑃_𝑖 and the corresponding path lengths 𝐿_𝑖, using Definition 3. Set

      𝐿_𝑖=(𝑚_𝑖,𝛼_𝑖,𝛽_𝑖;𝜆)_LR, 𝑖=1 to 𝑛 and 0<𝜆≤1.

  Step  3: Calculate the fuzzy shortest length 𝐿_min using Definition 4 and set 𝐿_min=(𝑚,𝛼,𝛽;𝜆)_LR.

  Step 4: Calculate the level 𝜆-triangular LR intersection index between 𝐿_min and 𝐿_𝑖 using Definition 5 for 𝑖=1 to 𝑛.

  Step 5: The path having the highest level 𝜆-triangular LR intersection index is identified as the shortest path.