Neutrosophic sets have been introduced as a generalization of crisp sets, fuzzy sets, and intuitionistic fuzzy sets to represent uncertain, inconsistent, and incomplete information about a real world problem. For the first time, this paper attempts to introduce the mathematical representation of a transportation problem in neutrosophic environment. The necessity of the model is discussed. A new method for solving transportation problem with indeterminate and inconsistent information is proposed briefly. A real life example is given to illustrate the efficiency of the proposed method in neutrosophic approach.
In the present day, problems are there with different types of uncertainties which cannot be solved by classical theory of mathematics. To deal with the problems with imprecise or vague information, Zadeh [
In due course, any generalization of fuzzy set failed to handle problems with indeterminate or inconsistent information. To overcome this, Smarandache [
Study of optimal transportation model with cost effective manner played a predominant role in supply chain management. Many researchers [
Though many researchers worked on transportation problem in fuzzy environment, it was ÓhÉigeartaigh [
Sometimes the membership function in fuzzy set theory was not a suitable one to describe an ambiguous situation of a problem. So, in 1986, Atanassov [
In a supply chain optimization, transportation system was the most important economic activity among all the components of business logistics system. Apart from the vagueness or uncertainty in the constraints of the present day transportation model, there exists some indeterminacy due to various factors like unawareness of the problem, imperfection in data, and poor forecasting. Intuitionistic fuzzy set theory can handle incomplete information but not indeterminate and inconsistent information. Smarandache [
Even though many scholars applied the notion of neutrosophic theories in multiattribute decision making problems [
The aim of this paper is to obtain the optimal transportation cost in neutrosophic environment. This paper is well organized as follows. In Section
Let
A fuzzy number there exist at least one
A fuzzy number
Let
An intuitionistic fuzzy subset There exists
The membership and nonmembership functions of
where
A trapezoidal intuitionistic fuzzy number is denoted by
Let
But it is difficult to apply neutrosophic set theories in real life problems directly. So Wang introduced single valued neutrosophic set as a subset of neutrosophic set and the definition is as follows.
Let
Let
Let
One can compare any two single valued trapezoidal neutrosophic numbers based on the score and accuracy functions. Let score function accuracy function
Let If If
Let
In this model, a transportation problem is introduced in a single valued neutrosophic environment. Consider a transportation problem with “
Supply constraints: Demand constraints: Nonnegativity constraints: Now the mathematical formulation of the problem is given by
In this model the decision maker will not be sure about the unit transportation costs, supply, and the demand units. So the mathematical formulation of the problem becomes
Calculate the score value of each neutrosophic cost
For each row and column of the table obtained in Step
In the row/column, corresponding to maximum penalty, make the maximum allotment in the cell having the minimum transportation cost.
If the maximum penalty corresponding to more than one row, select the topmost row, more than one column, select the extreme left column.
Repeat the above procedure until all the supplies are fully exhausted and all the demands are satisfied.
Convert each neutrosophic cost into crisp value by the score function and obtain the classical transportation problem.
Choose the minimum in each row and subtract it from the corresponding row entries. Do the same procedure for each column. Now there will be at least one zero in each row and column in the resultant table.
Verify whether the demand of each column is less than the sum of supplies whose reduced costs are zero in that column and supply of each row are less than the sum of demands whose reduced costs in that row are zero. If so, go to Step
Draw the minimum number of horizontal and vertical lines that cover all the zeros in the reduced table and revise the table as follows: Find the least element from the uncovered entries. Subtract it from all the uncovered entries and add it to the entries at the intersection of any two lines.
Again check the condition at Step
Select a cell
Select a cell in
Revise the reduced table by omitting fully exhausted row and fully satisfied column and repeat Steps
Repeat the procedure until all the supply units are fully used and all the demand units are fully received.
Initial basic feasible solution and the optimum solution of Model II will be obtained by the procedure in Sections
Consider a transportation problem in which the peanuts are initially stored at three sources, namely, O1, O2, and O3 and are transported to peanut butter manufacturing company located at four different destinations, namely D1, D2, D3, and D4, with trapezoidal neutrosophic unit transportation cost and crisp demand and supply as given in Table
Input data for neutrosophic transportation problem.
D1 | D2 | D3 | D4 | Supply | |
---|---|---|---|---|---|
O1 | (3, 5, 6, 8); 0.6, 0.5, 0.4 | (5, 8, 10, 14); 0.3, 0.6, 0.6 | (12, 15, 19, 22); 0.6, 0.4, 0.5 | (14, 17, 21, 28); 0.8, 0.2.0.6 | 26 |
O2 | (0, 1, 3, 6); 0.7, 0.5, 0.3 | (5, 7, 9, 11); 0.9, 0.7, 0.5 | (15, 17, 19, 22); 0.4, 0.8, 0.4 | (9, 11, 14, 16); 0.5, 0.4, 0.7 | 24 |
O3 | (4, 8, 11, 15); 0.6, 0.3, 0.2 | (1, 3, 4, 6); 0.6, 0.3, 0.5 | (5, 7, 8, 10); 0.5, 0.4, 0.7 | (5, 9, 14, 19); 0.3, 0.7, 0.6 | 30 |
|
|||||
Demand | 17 | 23 | 28 | 12 |
Now by score function of trapezoidal neutrosophic number, calculate the score value of each neutrosophic cost to obtain the crisp transportation problem. (Here score values are rounded off to the nearest integer.) The results are given in Table
Crisp transportation problem.
D1 | D2 | D3 | D4 | Supply | |
---|---|---|---|---|---|
O1 | 2 | 3 | 7 | 10 | 26 |
O2 | 1 | 3 | 6 | 4 | 24 |
O3 | 5 | 2 | 3 | 3 | 30 |
|
|||||
Demand | 17 | 23 | 28 | 12 |
Tabular representation with penalties.
D1 | D2 | D3 | D4 | Supply | Penalty | |
---|---|---|---|---|---|---|
O1 | 2 | 3 | 7 | 10 | 26 | 1 |
O2 | 1 | 3 | 6 | 4 | 24 | 2 |
O3 | 5 | 2 | 3 | 3 | 30 | 1 |
|
||||||
Demand | 17 | 23 | 28 | 12 | ||
|
||||||
Penalty | 1 | 1 | 3 |
1 |
In Table
First allotment with penalties.
D1 | D2 | D3 | D4 | Supply | Penalty | |
---|---|---|---|---|---|---|
O1 | 2 | 3 | — | 10 | 26 | 1 |
O2 | 1 | 3 | — | 4 | 24 | 2 |
O3 | 5 | 2 |
|
3 | 2 | 1 |
|
||||||
Demand | 17 | 23 | — | 12 | ||
|
||||||
Penalty | 1 | 1 | — | 1 |
Proceeding the neutrosophic initial basic feasible solution algorithm and after few iterations we get the complete allotment transportation units as given in Table
Table with complete allotment.
D1 | D2 | D3 | D4 | Supply | |
---|---|---|---|---|---|
O1 | — |
|
— |
|
— |
O2 |
|
— | — |
|
— |
O3 | — | — |
|
|
— |
|
|||||
Demand | — | — | — | — |
The initial basic feasible solution is
Consider the neutrosophic optimal solution for the transportation problem given in Table
Table with zero point.
D1 | D2 | D3 | D4 | Supply | |
---|---|---|---|---|---|
O1 | 0 | 1 | 4 | 7 | 26 |
O2 | 0 | 2 | 4 | 2 | 24 |
O3 | 3 | 0 | 0 | 0 | 30 |
|
|||||
Demand | 17 | 23 | 28 | 12 |
Now Table
Modified table with zero point.
D1 | D2 | D3 | D4 | Supply | |
---|---|---|---|---|---|
O1 | 0 | 0 | 3 | 6 | 26 |
O2 | 0 | 1 | 3 | 1 | 24 |
O3 | 4 | 0 | 0 | 0 | 30 |
|
|||||
Demand | 17 | 23 | 28 | 12 |
As per allotment rules given in Steps
Table with complete allocation.
D1 | D2 | D3 | D4 | Supply | |
---|---|---|---|---|---|
O1 |
|
|
— | ||
O2 |
|
|
— | ||
O3 |
|
|
— | ||
|
|||||
Demand | — | — | — | — |
The optimal solution is
Consider a problem for Model II with single valued neutrosophic trapezoidal cost, demand, and supply given in Table
Input data for neutrosophic transportation problem.
D1 | D2 | D3 | D4 | Supply | |
---|---|---|---|---|---|
O1 | (3, 5, 6, 8); |
(5, 8, 10, 14); 0.3, 0.6, 0.6 | (12, 15, 19, 22); 0.6, 0.4, 0.5 | (14, 17, 21, 28); 0.8, 0.2, 0.6 | (22, 26, 28, 32); 0.7, 0.3, 0.4 |
O2 | (0, 1, 3, 6); |
(5, 7, 9, 11); 0.9, 0.7, 0.5 | (15, 17, 19, 22); 0.4, 0.8, 0.4 | (9, 11, 14, 16); 0.5, 0.4, 0.7 | (17, 22, 27, 31); 0.6, 0.4, 0.5 |
O3 | (4, 8, 11, 15); |
(1, 3, 4, 6); 0.6, 0.3, 0.5 | (5, 7, 8, 10); 0.5, 0.4, 0.7 | (5, 9, 14, 19); 0.3, 0.7, 0.6 | (21, 28, 32, 37); 0.8, 0.2, 0.4 |
|
|||||
Demand | (13, 16, 18, 21); |
(17, 21, 24, 28); 0.8, 0.2, 0.4 | (24, 29, 32, 35); 0.9, 0.5, 0.3 | (6, 10, 13, 15); 0.7, 0.3, 0.4 |
For Table
Neutrosophic transportation problem with crisp cost.
D1 | D2 | D3 | D4 | Supply | |
---|---|---|---|---|---|
O1 | 2 | 3 | 7 | 10 | (22, 26, 28, 32); |
O2 | 1 | 3 | 6 | 4 | (17, 22, 27, 31); |
O3 | 5 | 2 | 3 | 3 | (21, 28, 32, 37); |
|
|||||
Demand | (13, 16, 18, 21); |
(17, 21, 24, 28); |
(24, 29, 32, 35); |
(6, 10, 13, 15); |
Here, the arithmetic operations of single valued neutrosophic trapezoidal numbers are applied to modify the neutrosophic demand and supply in each iteration. Proceed the neutrosophic optimal solution method and after few iterations the optimal solution in terms of single valued neutrosophic trapezoidal numbers is obtained as follows:
In Section
In the optimum solution, the degrees of indeterminacy and falsity are the same. Hence, degree of indeterminacy and falsity for the minimum transportation cost are
Neutrosophic sets being a generalization of intuitionistic fuzzy sets provide an additional possibility to represent the indeterminacy along with the uncertainty. Though there are many transportation problems that have been studied with different types of input data, this research has investigated the solutions of transportation problems in neutrosophic environment. Two different models in neutrosophic environment were considered in the study. The arithmetic operations on single valued neutrosophic trapezoidal numbers are employed to find the solutions. The solution procedures are illustrated with day-to-day problems. Though the proposed algorithms concretely analyze the solutions of neutrosophic transportation problems, there are some limitations in predicting the solutions of qualitative and complex data. The computational complexity in handling higher dimensional problems will be overcome by genetic algorithm approach. In future, the research will be extended to deal with multiobjective solid transportation problems in environment. The researchers will be interested to overcome the above stated limitations. Further, the approaches of transportation problems on fuzzy and intuitionistic fuzzy logic may be extended to neutrosophic logic.
The authors mentioned that there is no competing interests regarding the publication of the paper.