To the best of our knowledge, there is only one approach for solving neutrosophic cost minimization transportation problems. Since neutrosophic transportation problems are a new area of research, other researchers may be attracted to extend this approach for solving other types of neutrosophic transportation problems like neutrosophic solid transportation problems, neutrosophic time minimization transportation problems, neutrosophic transshipment problems, and so on. However, after a deep study of the existing approach, it is noticed that a mathematical incorrect assumption has been used in these existing approaches; therefore there is a need to modify these existing approaches. Keeping the same in mind, in this paper, the existing approach is modified. Furthermore, the exact results of some existing transportation problems are obtained by the modified approach.
In daily life problems several times there is a need to transport the product from various sources to different destinations. To find a way to transport the product in such a manner so that the total transportation cost is minimum is called the optimal way and the problem is called cost minimization transportation problems [
However, to assume these parameters as real numbers is not always valid according to real life situations; for example, the transportation cost depends upon the circumstances like price of petrol/diesel, weather, travel time, traffic jam, and so on. Similarly, availability of crops varies according to the monsoon, fertilizers, chemicals, and so on; demands of the various clothes depend on the season, fashion trends, discount offers, and so on. Furthermore, the opinions of the experts about these parameters indicate that they cannot always be represented as real numbers; for example, generally experts provide their opinion about these parameters in terms of linguistic variables like high, very high, low, very low, and so on.
Being one of the widely adopted ways in the literature, to deal with such situations is to represent these parameters as fuzzy numbers [
Since neutrosophic transportation problems are new area of research, others may be attracted to extend these approaches for solving other types of neutrosophic transportation problems like neutrosophic solid transportation problems, neutrosophic time minimization transportation problems, neutrosophic transshipment problems, and so on. However, after a deep study of these existing approaches, it is noticed that a mathematical incorrect assumption has been used in these existing approaches; therefore there is a need to modify these existing approaches. Keeping the same in mind, in this paper, these existing approaches are modified. Furthermore, the exact results of some existing transportation problems are obtained by the modified approaches.
To point out the mathematical incorrect assumptions in the approaches, proposed by Thamaraiselvi and Santhi [
Using the approach, proposed by Thamaraiselvi and Santhi [
Formulate the neutrosophic transportation problem as a neutrosophic linear programming problem
Transform the neutrosophic linear programming problem
Transform the crisp linear programming problem
Represent the crisp linear programming problem
Tabular representation of transformed crisp transportation problem.
Sources | Destinations | Supply | ||||
---|---|---|---|---|---|---|
|
|
|
… |
| ||
|
|
|
|
… |
|
|
|
|
|
|
… |
|
|
|
|
|
|
… |
|
|
|
|
|
|
|
|
|
|
|
|
|
… |
|
|
|
||||||
Demand |
|
|
|
… |
|
Find the crisp optimal solutions
Find the total minimum neutrosophic transportation cost by putting the optimal solution
Using the approach, proposed by Thamaraiselvi and Santhi [
Formulate the neutrosophic transportation problem as a neutrosophic linear programming problem
Transform the neutrosophic linear programming problem
Transform the neutrosophic linear programming problem
Represent the neutrosophic linear programming problem
Tabular representation of neutrosophic transportation problem of Type II.
Sources | Destinations | Supply | ||||
---|---|---|---|---|---|---|
|
|
|
… |
| ||
|
|
|
|
… |
|
|
|
|
|
|
… |
|
|
|
|
|
|
… |
|
|
|
|
|
|
|
|
|
|
|
|
|
… |
|
|
|
||||||
Demand |
|
|
|
… |
|
Find the neutrosophic optimal solution
Find the total minimum neutrosophic transportation cost by putting the optimal solution
In this section, the mathematical incorrect assumption, considered in Thamaraiselvi and Santhi approaches [
It is obvious from Steps
It is pertinent to mention that, to transform the problem
Let
Hence, the approaches for solving neutrosophic transportation problem, proposed by Thamaraiselvi and Santhi [
It is obvious from Section
Let
In this section, to resolve the flaws of the existing approaches, pointed out in Section
In this section, the existing approach is modified to solve neutrosophic transportation problem of Type I.
The steps of the modified approach are as follows.
Use Steps
Using the relation, obtained in Section
Represent the neutrosophic linear programming problem
Tabular representation of transformed crisp transportation problem.
Sources | Destinations | Supply | ||||
---|---|---|---|---|---|---|
|
|
|
|
| ||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||||||
Demand |
|
|
|
|
|
Find the crisp optimal solutions
Find the total minimum neutrosophic transportation cost by putting the optimal solution
In this section, the existing approach is modified to solve neutrosophic transportation problem of Type II.
The steps of the modified approach are as follows.
Use Steps
Using the relation, obtained in Section
Represent the neutrosophic linear programming problem
Tabular representation of neutrosophic transportation problem of Type II.
Sources | Destinations | Supply | ||||
---|---|---|---|---|---|---|
|
|
|
|
| ||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||||||
Demand |
|
|
|
|
|
Find the neutrosophic optimal solution
Find the total minimum neutrosophic transportation cost by putting the optimal solution
Thamaraiselvi and Santhi [
Thamaraiselvi and Santhi [
Input data for neutrosophic transportation problem.
|
|
|
|
Supply | |
---|---|---|---|---|---|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|||||
Demand |
|
|
|
|
In this section the exact solution of this problem is obtained by the modified approach. Using the modified approach the exact solution of neutrosophic transportation problem of Type I, presented in Table
Using Steps
Crisp transportation problem.
|
|
|
|
Supply | |
---|---|---|---|---|---|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|||||
Demand |
|
|
|
|
When solving the crisp transportation problem (Table
Using the optimal solution, the minimum total neutrosophic cost is
Thamaraiselvi and Santhi [
Input data for neutrosophic transportation problem.
|
|
|
|
Supply | |
---|---|---|---|---|---|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|||||
Demand |
|
|
|
|
In this section, the exact solution of this problem is obtained by the modified approach. Using the modified approach the exact solution of neutrosophic transportation problem of Type II, presented in Table
Using Steps
Neutrosophic transportation problem with crisp cost.
|
|
|
|
Supply | |
---|---|---|---|---|---|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|||||
Demand |
|
|
|
|
When solving the neutrosophic transportation problem with crisp cost (Table
Using the optimal solution the minimum total neutrosophic cost is
The score value of minimized neutrosophic transportation cost, obtained by existing approaches [
Existing and modified score values.
Minimum score value of neutrosophic transportation cost | ||
---|---|---|
Existing approaches [ |
Modified approaches | |
First problem | 140.118 | 127.25 |
Second problem | 182.981 | 170.437 |
It is obvious from the results, shown in Table
It is pointed out that it is not genuine to use the existing approaches [
The authors declare that there are no conflicts of interest regarding the publication of this paper.