Introduction
Preliminaries
Proposed algorithm for fuzzy interval type 2 transportation problem
Mathematical statement
Proposed algorithm 1
Proposed Algorithm 2
Modified MODI method
Numerical illustration
Numerical Example I
Arc | IT2F Cost |
---|---|
\({\widetilde{C}}_{11}\) | (( 0.09, 1.50, 3.00, 4.62), (1.79, 2.28, 2.28, 2.81, 0.40)) |
\({\widetilde{C}}_{12}\) | (( 3.59, 4.75, 5.50, 6.91 ), (4.86, 5.03, 5.03, 5.14, 0.27 )) |
\({\widetilde{C}}_{13}\) | (( 4.38, 6.50, 8.25, 9.62), (7.19, 7.58, 7.58, 8.21, 0.37 )) |
\({\widetilde{C}}_{14}\) | (( 3.38, 5.50, 7.50, 9.62 ), (5.79, 6.50, 6.50, 7.21,0.41 )) |
\({\widetilde{C}}_{21}\) | (( 0.38, 1.50, 2.50, 4.62 ), (1.09, 1.83, 1.83, 2.21, 0.53 )) |
\({\widetilde{C}}_{22}\) | (( 3.59, 4.75, 5.50, 6.91 ), (4.86, 5.03, 5.03, 5.14, 0.27 )) |
\({\widetilde{C}}_{23}\) | ((5.98, 7.75, 8.60, 9.52 ), (8.03, 8.36, 8.36, 9.17, 0.57)) |
\({\widetilde{C}}_{24}\) | (( 0.38, 1.50, 2.50, 4.62 ), (1.09, 1.83, 1.83, 2.21, 0.53 )) |
\({\widetilde{C}}_{31}\) | (( 0.09, 1.50, 3.00, 4.62), (1.79, 2.28, 2.28, 2.81, 0.40)) |
\({\widetilde{C}}_{32}\) | (( 3.38, 5.50, 7.50, 9.62 ), (5.79, 6.50, 6.50, 7.21,0.41 )) |
\({\widetilde{C}}_{33}\) | ((7.37, 9.41, 10, 10), (8.72, 9.91, 10, 10,1 )) |
\({\widetilde{C}}_{34}\) | (( 0.38, 1.50, 2.50, 4.62 ), (1.09, 1.83, 1.83, 2.21, 0.53 )) |
Arc | IT2F Supply/Demand |
---|---|
\({\widetilde{S}}_{1}\) | (( 3.59, 4.75, 6.00, 7.41 ), (4.79, 5.30, 5.30, 5.71, 0.42 )) |
\({\widetilde{S}}_{2}\) | (( 5.38, 7.50, 8.75, 9.81), (7.79, 8.30, 8.30, 9.21, 0.53 )) |
\({\widetilde{S}}_{3}\) | (( 0.59, 2.00, 3.25, 4.41), (2.29, 2.70, 2.70, 3.21, 0.42)) |
\({\widetilde{D}}_{1}\) | (( 0.09, 1.25, 2.50, 4.62 ), (1.67, 1.92, 1.92, 2.21, 0.30 )) |
\({\widetilde{D}}_{2}\) | (( 0.09, 1.25, 2.50, 4.62 ), (1.67, 1.92, 1.92, 2.21, 0.30 ))) |
\({\widetilde{D}}_{3}\) | (( 2.17, 4.25, 6.00, 7.83), (4.79, 5.29, 5.29, 6.02, 0.41)) |
\({\widetilde{D}}_{4}\) | (( 3.38, 5.50, 7.50, 9.62 ), (5.79, 6.50, 6.50, 7.21,0.41 )) |
\({\widetilde{D}}_{1}\)
|
\({\widetilde{D}}_{2}\)
|
\({\widetilde{D}}_{3}\)
|
\({\widetilde{D}}_{4}\)
| Supply | |
---|---|---|---|---|---|
\( {\widetilde{S}}_{1}\)
| 2.32 | 5.19 | 7.25 | 6.50 | 5.41 |
\( {\widetilde{S}}_{2}\)
| 2.13 | 5.19 | 8.12 | 2.13 | 8.01 |
\( {\widetilde{S}}_{3}\)
| 2.32 | 6.50 | 9.30 | 2.13 | 2.59 |
Demand | 2.19 | 2.19 | 5.13 | 6.50 | 16.01 |
Compute optimal value of Example I using MMODI method
Result using LINGO | Result using our proposed algorithm |
---|---|
\(\text {Minimize}~~ {\tilde{Z}}= 67.0683\) | Total cost=67.0683 |
\({\widetilde{x}}_{12}=0.28\), \({\widetilde{x}}_{13}=5.13\), \({\widetilde{x}}_{21}=2.19 \) | Optimal transportation value are |
\({\widetilde{x}}_{22}=1.91\), \({\widetilde{x}}_{24}=3.91\), \({\widetilde{x}}_{34}=2.59 \) | \({\widetilde{x}}_{12}=0.28\), \({\widetilde{x}}_{13}=5.13\), \({\widetilde{x}}_{21}=2.19 \) |
\({\widetilde{x}}_{22}=1.91\), \({\widetilde{x}}_{24}=3.91\), \({\widetilde{x}}_{34}=2.59 \) |
Numerical Example II
Arc | IT2F Cost |
---|---|
\({\widetilde{C}}_{11}\)
| (( 0.38, 1.50, 2.50, 4.62), (1.09, 1.83, 1.83, 2.21, 0.53)) |
\({\widetilde{C}}_{12}\)
| (( 5.38, 7.50, 8.75, 9.81 ), (7.79, 8.22, 8.22, 8.81, 0.45 )) |
\({\widetilde{C}}_{13}\)
| (( 5.38, 7.50, 9.00, 9.81), (8.29, 7.56, 8.56, 9.21, 0.38 ) |
\({\widetilde{C}}_{14}\)
| (( 2.59, 4.00, 5.50, 7.62 ), (4.29, 4.75, 4.75, 5.21,0.38 )) |
\({\widetilde{C}}_{15}\)
| (( 3.59, 4.75, 5.50, 6.91 ), (7.86, 5.03, 5.03, 5.14,0.27 )) |
\({\widetilde{C}}_{16}\)
| (( 3.38, 5.50, 7.50, 9.62 ), (5.79, 6.50, 6.50, 7.21,0.41 )) |
\({\widetilde{C}}_{17}\)
| (( 4.38, 6.50, 8.00, 9.41 ), (6.79, 7.38, 7.38, 8.21,0.49 )) |
\({\widetilde{C}}_{18}\)
| (( 5.38, 7.50, 8.75, 9.83 ), (7.69, 8.19, 8.19, 8.81,0.47 )) |
\({\widetilde{C}}_{21}\)
| (( 4.38, 6.50, 8.00, 9.41 ), (6.79, 7.38, 7.38, 8.21,0.49 )) |
\({\widetilde{C}}_{22}\)
| (( 0.09, 1.50, 3.00, 4.62 ), (1.79, 2.28, 2.28, 2.81, 0.40 )) |
\({\widetilde{C}}_{23}\)
| ((0.38, 2.50, 5.00, 7.83 ), (2.88, 3.61, 3.61, 4.21, 0.35)) |
\({\widetilde{C}}_{24}\)
| (( 5.38, 7.50, 8.75, 9.81 ), (7.79, 8.30, 8.30, 9.21, 0.53 ) |
\({\widetilde{C}}_{25}\)
| (( 3.59, 4.75, 5.50, 6.91 ), (7.86, 5.03, 5.03, 5.14,0.27 )) |
\({\widetilde{C}}_{26}\)
| (( 3.59, 4.75, 6.00, 7.41 ), (4.79, 5.30, 5.30, 5.71,0.42 )) |
\({\widetilde{C}}_{27}\)
| (( 0.38, 1.50, 2.50, 4.62), (1.09, 1.83, 1.83, 2.21, 0.53)) |
\({\widetilde{C}}_{28}\)
| (( 3.59, 4.75, 6.00, 7.41 ), (4.79, 5.30, 5.30, 5.71,0.42 )) |
\({\widetilde{C}}_{31}\)
| (( 0.09, 1.50, 3.00, 4.62 ), (1.79, 2.28, 2.28, 2.81, 0.40 )) |
\({\widetilde{C}}_{32}\)
| (( 3.38, 5.50, 7.50, 9.62 ), (5.79, 6.50, 6.50, 7.21,0.41 )) |
\({\widetilde{C}}_{33}\)
| (( 5.38, 7.50, 8.75, 9.81 ), (7.79, 8.22, 8.22, 8.81, 0.45 )) |
\({\widetilde{C}}_{34}\)
| ((0.38, 2.50, 5.00, 7.83 ), (2.88, 3.61, 3.61, 4.21, 0.35)) |
\({\widetilde{C}}_{35}\)
| ((2.17, 4.25, 6.00, 7.83 ), (4.79, 5.29, 5.29, 6.02, 0.41)) |
\({\widetilde{C}}_{36}\)
| ((5.98, 7.75, 8.60, 9.52 ), (8.03, 8.36, 8.36, 9.17, 0.57)) |
\({\widetilde{C}}_{37}\)
| ((2.17, 4.25, 6.00, 7.83 ), (4.79, 5.29, 5.29, 6.02, 0.41)) |
\({\widetilde{C}}_{38}\)
| (( 4.38, 6.50, 8.00, 9.41 ), (6.79, 7.38, 7.38, 8.21,0.49 )) |
\({\widetilde{C}}_{41}\)
| ((0.38, 2.50, 5.00, 7.83 ), (2.88, 3.61, 3.61, 4.21, 0.35)) |
\({\widetilde{C}}_{42}\)
| (( 4.38, 6.50, 8.00, 9.41 ), (6.79, 7.38, 7.38, 8.21,0.49 )) |
\({\widetilde{C}}_{43}\)
| (( 1.17, 3.50, 5.50, 7.83 ), (4.09, 4.65, 4.65, 5.41,0.40 )) |
\({\widetilde{C}}_{44}\)
| (( 1.17, 3.50, 5.50, 7.83 ), (4.09, 4.65, 4.65, 5.41,0.40 )) |
\({\widetilde{C}}_{45}\)
| (( 2.59, 4.00, 5.50, 7.62 ), (4.29, 4.75, 4.75, 5.21,0.38 )) |
\({\widetilde{C}}_{46}\)
| (( 5.38, 7.50, 9.00, 9.81), (8.29, 7.56, 8.56, 9.21, 0.38 ) |
\({\widetilde{C}}_{47}\)
| (( 3.59, 4.75, 6.00, 7.41 ), (4.79, 5.30, 5.30, 5.71,0.42 )) |
\({\widetilde{C}}_{48}\)
| (( 0.09, 1.25, 2.50, 4.62 ), (1.67, 1.92, 1.92, 2.21,0.30 )) |
\({\widetilde{C}}_{51}\)
| (( 3.59, 4.75, 6.00, 7.41 ), (4.79, 5.30, 5.30, 5.71,0.42 )) |
\({\widetilde{C}}_{52}\)
| (( 0.59, 2.00, 3.25, 4.41 ), (2.29, 2.70, 2.70, 3.21,0.42 )) |
\({\widetilde{C}}_{53}\)
| (( 5.38, 7.50, 8.75, 9.81 ), (7.79, 8.22, 8.22, 8.81, 0.45 )) |
\({\widetilde{C}}_{54}\)
| (( 4.38, 6.50, 8.00, 9.41 ), (6.79, 7.38, 7.38, 8.21,0.49 )) |
\({\widetilde{C}}_{55}\)
| (( 3.38, 5.50, 7.50, 9.62 ), (5.79, 6.50, 6.50, 7.21,0.41 )) |
\({\widetilde{C}}_{56}\)
| (( 3.59, 4.75, 5.50, 6.91 ), (7.86, 5.03, 5.03, 5.14,0.27 )) |
\({\widetilde{C}}_{57}\)
| (( 4.38, 6.50, 8.00, 9.41 ), (6.79, 7.38, 7.38, 8.21,0.49 )) |
\({\widetilde{C}}_{58}\)
| (( 0.09, 1.50, 3.00, 4.62 ), (1.79, 2.28, 2.28, 2.81, 0.40 )) |
\({\widetilde{C}}_{61}\)
| (( 0.09, 1.25, 2.50, 4.62 ), (1.67, 1.92, 1.92, 2.21,0.30 )) |
\({\widetilde{C}}_{62}\)
| (( 0.09, 1.50, 3.00, 4.62 ), (1.79, 2.28, 2.28, 2.81, 0.40 )) |
\({\widetilde{C}}_{63}\)
| (( 0.09, 1.25, 2.50, 4.62 ), (1.67, 1.92, 1.92, 2.21,0.30 )) |
\({\widetilde{C}}_{64}\)
| (( 3.59, 4.75, 6.00, 7.41 ), (4.79, 5.30, 5.30, 5.71,0.42 )) |
\({\widetilde{C}}_{65}\)
| (( 4.38, 6.50, 8.00, 9.41 ), (6.79, 7.38, 7.38, 8.21,0.49 )) |
\({\widetilde{C}}_{66}\)
| (( 4.38, 6.50, 8.25, 9.62 ), (7.19, 7.58, 7.58, 8.21,0.37 )) |
\({\widetilde{C}}_{67}\)
| (( 3.38, 5.50, 7.50, 9.62 ), (5.79, 6.50, 6.50, 7.21,0.41 )) |
\({\widetilde{C}}_{68}\)
| (( 4.38, 6.50, 8.25, 9.62 ), (7.19, 7.58, 7.58, 8.21,0.37 )) |
Arc | IT2F Supply/Demand |
---|---|
\({\widetilde{S}}_{1}\)
| (( 18.68, 19.91, 20.00, 20.00 ), (19.61, 19.97, 20.00, 20.00, 1.00 )) |
\({\widetilde{S}}_{2}\)
| (( 17.37, 19.73, 20.00, 20.00), (19.34, 19.95, 20.00, 20.00, 1.00 )) |
\({\widetilde{S}}_{3}\)
| (( 17.37, 19.82, 20.00, 20.00), (19.37, 19.95, 20.00, 20.00, 1.00)) |
\({\widetilde{S}}_{4}\)
| (( 17.37, 19.82, 20.00, 20.00), (19.74, 19.98, 20.00, 20.00, 1.00)) |
\({\widetilde{S}}_{5}\)
| (( 17.37, 19.59, 20.00, 20.00), (18.95, 19.93, 20.00, 20.00, 1.00)) |
\({\widetilde{S}}_{6}\)
| (( 17.37, 19.41, 20.00, 20.00), (18.72, 19.91, 20.00, 20.00, 1.00)) |
\({\widetilde{D}}_{1}\)
| (( 15.38, 17.50, 18.75, 19.81 ), (17.79, 18.22, 18.22, 18.81, 0.45 )) |
\({\widetilde{D}}_{2}\)
| (( 14.38, 16.50, 18.25, 19.62 ), (17.19, 17.58, 17.58, 18.21,0.37 )) |
\({\widetilde{D}}_{3}\)
| (( 5.38, 7.50, 8.75, 9.81 ), (7.79, 8.22, 8.22, 8.81, 0.45 )) |
\({\widetilde{D}}_{4}\)
| (( 13.38, 15.50, 17.50, 19.62 ), (15.79, 16.50, 16.50, 17.21,0.41 )) |
\({\widetilde{D}}_{5}\)
| (( 13.59, 14.75, 15.50, 16.91 ), (17.86, 15.03, 15.03, 15.14,0.27 )) |
\({\widetilde{D}}_{6}\)
| (( 14.38, 16.50, 18.00, 19.41 ), (16.79, 17.38, 17.38, 18.21,0.49 )) |
\({\widetilde{D}}_{7}\)
| (( 13.59, 14.75, 15.50, 16.91 ), (17.86, 15.03, 15.03, 15.14,0.27 )) |
\({\widetilde{D}}_{8}\)
| (( 7.37, 9.41, 10.00, 10.00), (8.72, 9.91, 10.00, 10.00, 1.00)) |
Source | \({\widetilde{D}}_{1}\) | \({\widetilde{D}}_{2}\) | \({\widetilde{D}}_{3}\) | \({\widetilde{D}}_{4}\) | \({\widetilde{D}}_{5}\) | \({\widetilde{D}}_{6}\) | \({\widetilde{D}}_{7}\) | \({\widetilde{D}}_{8}\) | Supply |
---|---|---|---|---|---|---|---|---|---|
\({\widetilde{S}}_{1}\) | 2.13 | 7.90 | 8.03 | 4.95 | 5.19 | 6.50 | 7.16 | 7.91 | 19.69 |
\({\widetilde{S}}_{2}\) | 7.16 | 2.32 | 3.90 | 8.01 | 5.19 | 5.41 | 2.13 | 5.41 | 19.37 |
\({\widetilde{S}}_{3}\) | 2.32 | 6.50 | 7.90 | 3.90 | 5.13 | 8.12 | 5.13 | 7.16 | 19.38 |
\({\widetilde{S}}_{4}\) | 3.90 | 7.16 | 4.56 | 4.56 | 4.95 | 8.03 | 5.41 | 2.19 | 19.31 |
\({\widetilde{S}}_{5}\) | 5.41 | 2.59 | 7.90 | 7.16 | 6.50 | 5.19 | 7.16 | 2.32 | 19.34 |
\({\widetilde{S}}_{6}\) | 2.19 | 2.32 | 2.19 | 5.41 | 7.16 | 7.25 | 6.50 | 7.25 | 19.30 |
Demand | 17.90 | 17.25 | 7.90 | 16.50 | 15.19 | 17.16 | 15.19 | 9.30 | 116.39 |
Result using LINGO | Result using proposed algorithm |
---|---|
\(\text {Minimize}~~ {\tilde{Z}}= 378.2358\) | Total cost=378.2358 |
\({\widetilde{x}}_{11}=17.90\), \({\widetilde{x}}_{15}=1.79\), \({\widetilde{x}}_{22}=4.18\) | Optimal value = |
\({\widetilde{x}}_{27}=15.19\), \({\widetilde{x}}_{34}=16.50\), \({\widetilde{x}}_{35}=2.88\) | \({\widetilde{x}}_{11}=17.90\), \({\widetilde{x}}_{15}=1.79\), \({\widetilde{x}}_{22}=4.18\) |
\({\widetilde{x}}_{45}=10.52\), \({\widetilde{x}}_{48}=8.79\), \({\widetilde{x}}_{52}=1.67\) | \({\widetilde{x}}_{27}=15.19\), \({\widetilde{x}}_{34}=16.50\), \({\widetilde{x}}_{35}=2.88\) |
\({\widetilde{x}}_{56}=17.16\), \({\widetilde{x}}_{58}=0.51\), \({\widetilde{x}}_{62}=11.4\) | \({\widetilde{x}}_{45}=10.52\), \({\widetilde{x}}_{48}=8.79\), \({\widetilde{x}}_{52}=1.67\) |
\({\widetilde{x}}_{63}=7.90\) | \({\widetilde{x}}_{56}=17.16\), \({\widetilde{x}}_{58}=0.51\) |
\({\widetilde{x}}_{62}=11.4\),\({\widetilde{x}}_{63}=7.90 \) |