Introduction and literature of review
Introduction
Literature of review
Overview on interval valued neutrosophic set
-
If \( b = c \) in interval valued trapezoidal neutrosophic number then it becomes interval valued triangular neutrosophic number.
Proposed improved algorithm and score function
Improved algorithm to solve SPP under interval valued neutrosophic number
Proposed score function
Computation of shortest path using IVNNs
Edges | Interval valued neutrosophic distance | Edges | Interval valued neutrosophic distance |
---|---|---|---|
1–2 \( \left( {e_{1} } \right) \) |
\( \left( {\left[ {0.1,0.2} \right],\left[ {0.2,0.3} \right],\left[ {0.4,0.5} \right]} \right) \)
| 3–4 \( \left( {e_{5} } \right) \) |
\( \left( {\left[ {0.2,0.3} \right],\left[ {0.2,0.5} \right],\left[ {0.4,0.5} \right]} \right) \)
|
1–3 \( \left( {e_{2} } \right) \) |
\( \left( {\left[ {0.2,0.4} \right],\left[ {0.3,0.5} \right],\left[ {0.1,0.2} \right]} \right) \)
| 3–5 \( \left( {e_{6} } \right) \) |
\( \left( {\left[ {0.3,0.6} \right],\left[ {0.1,0.2} \right],\left[ {0.1,0.4} \right]} \right) \)
|
2–3 \( \left( {e_{3} } \right) \) |
\( \left( {\left[ {0.3,0.4} \right],\left[ {0.1,0.2} \right],\left[ {0.3,0.5} \right]} \right) \)
| 4–6 \( \left( {e_{7} } \right) \) |
\( \left( {\left[ {0.4,0.6} \right],\left[ {0.2,0.4} \right],\left[ {0.1,0.3} \right]} \right) \)
|
2–5 \( \left( {e_{4} } \right) \) |
\( \left( {\left[ {0.1,0.3} \right],\left[ {0.3,0.4} \right],\left[ {0.2,0.3} \right]} \right) \)
| 5–6 \( \left( {e_{8} } \right) \) |
\( \left( {\left[ {0.2,0.3} \right],\left[ {0.3,0.4} \right],\left[ {0.1,0.5} \right]} \right) \)
|
Node number \( (j) \) |
\( l_{i} \)
| IVNSP between jth and node 1 |
---|---|---|
2 |
\( \left\langle {\left[ {0.1,0.2} \right],\left[ {0.2,0.3} \right],\left[ {0.4,0.5} \right]} \right\rangle \)
|
\( 1 \to 2 \)
|
3 |
\( \left\langle {\left[ {0.37,0.52} \right],\left[ {0.02,0.06} \right],\left[ {0.12,0.25} \right]} \right\rangle \)
|
\( 1 \to 2 \to 3 \)
|
4 |
\( \left\langle {\left[ {0.6,0.67} \right],\left[ {0.004,0.018} \right],\left[ {0.048,0.125} \right]} \right\rangle \)
|
\( 1 \to 2 \to 3 \to 4 \)
|
5 |
\( \left\langle {\left[ {0.19,0.47} \right],\left[ {0.06,0.12} \right],\left[ {0.08,0.15} \right]} \right\rangle \)
|
\( 1 \to 2 \to 5 \)
|
6 |
\( \left\langle {\left[ {0.35,0.63} \right],\left[ {0.018,0.048} \right],\left[ {0.008,0.075} \right]} \right\rangle \)
|
\( 1 \to 2 \to 5 \to 6 \)
|
Edges | \( S_{\text{Ridvan }} \) [43] |
\( {\mathbb{S}}_{\text{Nagarajan}} \)
|
---|---|---|
1–2
| 0.1 |
0.125
|
1–3
| 0.175 |
0.2
|
2–3
| 0.325 |
0.17
|
2–5
| 0.125 |
0.11
|
3–4
| 0.05 |
0.325
|
3–5
| 0.45 |
0.32
|
4–6
| 0.35 |
0.43
|
5–6
| 0.125 |
0.26
|
The proposed algorithm based \( {\mathbb{S}}_{\text{Nagarajan}} \) | Crisp path length | Ranking |
---|---|---|
1 → 2 → 5 → 6 |
0.485
|
1
|
1 → 3 → 5 → 6
| 0.78 | 2 |
1 → 2 → 3 → 5 → 6
| 0.875 | 3 |
1 → 3 → 4 → 6
| 0.955 | 4 |
1 → 2 → 3 → 4 → 6
| 1.05 | 5 |
Algorithm: a new approach to find SPP using TpIVNN and TIVNN
Illustrative example to find the shortest path using TpIVNN
Edges | Trapezoidal interval valued neutrosophic distance | Edges | Trapezoidal interval valued neutrosophic distance |
---|---|---|---|
1–2
\( \left( {e_{1} } \right) \)
|
\( \left\langle {\left( {1,2,3,4} \right);\,\left[ {0.1,0.2} \right],\left[ {0.2,0.3} \right],\left[ {0.4,0.5} \right]} \right\rangle \)
| 3–4 \( \left( {e_{5} } \right) \) |
\( \left\langle {\left( {2,4,8,9} \right);\,\left[ {0.2,0.3} \right],\left[ {0.2,0.5} \right],\left[ {0.4,0.5} \right]} \right\rangle \)
|
1–3
\( \left( {e_{2} } \right) \)
|
\( \left\langle {\left( {2,5,7,8} \right);\,\left[ {0.2,0.4} \right],\left[ {0.3,0.5} \right],\left[ {0.1,0.2} \right]} \right\rangle \)
| 3–5 \( \left( {e_{6} } \right) \) |
\( \left\langle {\left( {3,4,5,10} \right);\,\left[ {0.3,0.6} \right],\left[ {0.1,0.2} \right],\left[ {0.1,0.4} \right]} \right\rangle \)
|
2–3
\( \left( {e_{3} } \right) \)
|
\( \left\langle {\left( {3,7,8,9} \right);\,\left[ {0.3,0.4} \right],\left[ {0.1,0.2} \right],\left[ {0.3,0.5} \right]} \right\rangle \)
| 4–6 \( \left( {e_{7} } \right) \) |
\( \left\langle {\left( {7,8,9,10} \right);\,\left[ {0.4,0.6} \right],\left[ {0.2,0.4} \right],\left[ {0.1,0.3} \right]} \right\rangle \)
|
2–5
\( \left( {e_{4} } \right) \)
|
\( \left\langle {\left( {1,5,7,9} \right);\,\left[ {0.1,0.3} \right],\left[ {0.3,0.4} \right],\left[ {0.2,0.3} \right]} \right\rangle \)
| 5–6 \( \left( {e_{8} } \right) \) |
\( \left\langle {\left( {2,4,5,7} \right);\,\left[ {0.2,0.3} \right],\left[ {0.3,0.4} \right],\left[ {0.1,0.5} \right]} \right\rangle \)
|
Available path |
\( {\mathbb{S}}\left( {\theta_{i} } \right) \)
| Ranking |
---|---|---|
\( P_{1} :1 \to 2 \to 5 \to 6 \)
| 4.18 | 1 |
\( P_{2} :1 \to 3 \to 5 \to 6 \)
| 8.25 | 2 |
\( P_{4} :1 \to 3 \to 4 \to 6 \)
| 12.43 | 3 |
\( P_{3} :1 \to 2 \to 3 \to 5 \to 6 \)
| 13.31 | 4 |
\( P_{5} :1 \to 2 \to 3 \to 4 \to 6 \)
| 17.5 | 5 |
Illustrative example to find the shortest path using TIVNN
Edges | Triangular interval valued neutrosophic distance | Edges | Triangular interval valued neutrosophic distance |
---|---|---|---|
1–2
\( \left( {e_{1} } \right) \)
|
\( \left\langle {\left( {1,2,3} \right);\,\left[ {0.1,0.2} \right],\left[ {0.2,0.3} \right],\left[ {0.4,0.5} \right]} \right\rangle \)
| 3–4 \( \left( {e_{5} } \right) \) |
\( \left\langle {\left( {2,4,8} \right);\,\left[ {0.2,0.3} \right],\left[ {0.2,0.5} \right],\left[ {0.4,0.5} \right]} \right\rangle \)
|
1–3
\( \left( {e_{2} } \right) \)
|
\( \left\langle {\left( {2,5,7} \right);\,\left[ {0.2,0.4} \right],\left[ {0.3,0.5} \right],\left[ {0.1,0.2} \right]} \right\rangle \)
| 3–5 \( \left( {e_{6} } \right) \) |
\( \left\langle {\left( {3,4,5} \right);\,\left[ {0.3,0.6} \right],\left[ {0.1,0.2} \right],\left[ {0.1,0.4} \right]} \right\rangle \)
|
2–3
\( \left( {e_{3} } \right) \)
|
\( \left\langle {\left( {3,7,8} \right);\,\left[ {0.3,0.4} \right],\left[ {0.1,0.2} \right],\left[ {0.3,0.5} \right]} \right\rangle \)
| 4–6 \( \left( {e_{7} } \right) \) |
\( \left\langle {\left( {7,8,9} \right);\,\left[ {0.4,0.6} \right],\left[ {0.2,0.4} \right],\left[ {0.1,0.3} \right]} \right\rangle \)
|
2–5
\( \left( {e_{4} } \right) \)
|
\( \left\langle {\left( {1,5,7} \right);\,\left[ {0.1,0.3} \right],\left[ {0.3,0.4} \right],\left[ {0.2,0.3} \right]} \right\rangle \)
| 5–6 \( \left( {e_{8} } \right) \) |
\( \left\langle {\left( {2,4,5} \right);\,\left[ {0.2,0.3} \right],\left[ {0.3,0.4} \right],\left[ {0.1,0.5} \right]} \right\rangle \)
|
Available path |
\( {\mathbb{S}}\left( {\theta_{i} } \right) \)
| Ranking |
---|---|---|
\( P_{1} :1 \to 2 \to 5 \to 6 \)
| 4.9 | 1 |
\( P_{2} :1 \to 3 \to 5 \to 6 \)
| 8.27 | 2 |
\( P_{4} :1 \to 3 \to 4 \to 6 \)
| 11.1 | 3 |
\( P_{3} :1 \to 2 \to 3 \to 5 \to 6 \)
| 12.86 | 4 |
\( P_{5} :1 \to 2 \to 3 \to 4 \to 6 \)
| 15.69 | 5 |
Comparative study of the proposed algorithm
Shortcoming of the existing method
Advantage of the proposed algorithm
Comparative study of algorithm
Algorithm of Broumi | Path | Crisp path length |
---|---|---|
\( S_{\text{Ridvan}} \) [43] |
\( 1 \to 2 \to 5 \to 6 \)
| 0.35 |
\( S_{\text{Nagarajan}} \)
|
\( 1 \to 2 \to 5 \to 6 \)
| 0.485 |
Possible path | Sequence of nodes | Neutrosophic shortest path length |
---|---|---|
Neutrosophic shortest path with interval valued neutrosophic numbers [43] |
\( 1 \to 2 \to 5 \to 6 \)
|
\( \left\langle {\left[ {0.35,0.60} \right],\left[ {0.01,0.04} \right],\left[ {0.008,0.075} \right]} \right\rangle \)
|
Proposed algorithm on
\( S_{\text{Nagarajan}} \)
|
\( 1 \to 2 \to 5 \to 6 \)
|
\( \left\langle {\left[ {0.35,0.60} \right],\left[ {0.01,0.04} \right],\left[ {0.008,0.075} \right]} \right\rangle \)
|