FUZZY DIAGONAL OPTIMAL ALGORITHM TO SOLVE INTUITIONISTIC FUZZY ASSIGNMENT PROBLEMS

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International Journal of Civil Engineering and Technology (IJCIET)
Volume 9, Issue 11, November 2018, pp. 378–383, Article ID: IJCIET_09_11_037
Available online at http://www.iaeme.com/ijciet/issues.asp?JType=IJCIET&VType=9&IType=10
ISSN Print: 0976-6308 and ISSN Online: 0976-6316
©IAEME Publication Scopus Indexed
FUZZY DIAGONAL OPTIMAL ALGORITHM TO
SOLVE INTUITIONISTIC FUZZY ASSIGNMENT
PROBLEMS
S. Dhanasekar, A. Manivannan, V. Parthiban
VIT, Tamilnadu, Chennai, India
ABSTRACT
In this article, we propose a new approach to solve intuitionistic fuzzy assignment
problem. Classical assignment problem deals with deterministic cost. In practical
situations it is not easy to determine the parameters. The parameters can be modeled
to fuzzy or intuitionistic fuzzy parameters. This paper develops an approach based on
diagonal optimal algorithm to solve an intuitionistic fuzzy assignment problem. A new
ranking procedure based on combined arithmetic mean is used to order the
intuitionistic fuzzy numbers so that Diagonal optimal algorithm [22] can be applied to
solve the intuitionistic fuzzy assignment problem. To illustrate the effectiveness of the
algorithm numerical examples were given..
Mathematics Subject Classification: 90C08, 90C70, 90B06, 90C29, 90C90
Key words: Intuitionistic Fuzzy number, Intuitionistic Triangular fuzzy number,
Intuitionistic Trapezoidal fuzzy number, Intuitionistic Fuzzy arithmetic operations,
Intuitionistic Fuzzy assignment problems, Intuitionistic Fuzzy optimal solution.
Cite this Article: S. Dhanasekar, A. Manivannan, V. Parthiban, Fuzzy Diagonal
Optimal Algorithm to Solve Intuitionistic Fuzzy Assignment Problems, International
Journal of Civil Engineering and Technology (IJCIET) 9(11), 2018, pp. 378–383.
http://www.iaeme.com/IJCIET/issues.asp?JType=IJCIET&VType=9&IType=11
1. INTRODUCTION
Assignment Problem (AP) arises in the situations of assigning the things to other things for
example assigning of persons to jobs, or classes to rooms, operators to machines, drivers to
trucks, trucks to routes, or problems to research teams, etc. The assignment problem is a kind
of linear programming problem (LPP) where the objective is to minimize the assignment cost.
Various algorithms have been proposed to find solution to assignment problems, such as
linear programming [9,10,14, 18], Hungarian algorithm [16], neural network [13], genetic
algorithm [8]. But , in real life, the parameters of assignment problem are not crisp numbers.
The fuzzy set theory introduced by Zadeh [1] in 1965 has been applied successfully in various
fields. In 1970, Belmann and Zadeh [2] applied concepts of fuzzy set theory into the decision-
making problems with uncertainty and imprecision. Many authors [ 3,4,19,21,23] developed
many algorithms to solve fuzzy assignment problems. Different kinds of fuzzy assignment
problems are solved in the papers [5, 12, 13, 14, 21]. Atanassov [7] proposed the Intuitionistic

Fuzzy Sets (IFSs) in 1986 is found to more useful to overcome the vagueness than fuzzy set
theory. Here we investigate a intuitionistic fuzzy assignment problem. Senthil Kumar et
al..[24] applied the Hungarian method to solve Intuitionistic fuzzy assignment problem.
Nagoor Gani et al..[25] solved Intuitionistic fuzzy linear bottle neck assignment problem by
using the perfect matching algorithm. In this paper, ranking based on combined arithemetic
mean is applied for the comparison of the intuitionistic fuzzy numbers. Finally an
Intuitionistic Fuzzy diagonal optimal method may be applied to solve an IFAP.
In Section 2 gives some basic terminology and ranking of triangular and trapezoidal
intuitionistic fuzzy numbers, Section 3, provides the proposed algorithm. Numerical examples
are solved in Section 4. The concluding remarks are given in Section 5.
2. SECTION-1
Definition
The fuzzy set can be obtained by assigning to each element in the universe of discourse a
value representing its grade of membership in the fuzzy set.
Definition
The fuzzy number ̃ is a fuzzy set whose membership function ̃ ( ) is a piecewise
continuous, convex and normal.
Definition
The intuitionistic fuzzy set, ̃ in the universe of discourse X is the set of ordered triples,
̃ *〈 ̃( ) ̃( )〉 + Where ̃( ) ̃( ) is a function from X to [0,1] such that
̃( ) ̃( ) , ∀ . Where membership ̃( ) ̃( ) represent the degree
of membership and the degree of non – membership of the element to ⊂
respectively.
Definition
A intuitionistic fuzzy number ̃ ( ) ( )is called to be a
intuitionistic trapezoidal fuzzy number if its membership function is of the form
̃ ( ) =
{
and ̃ ( ) =
{
Where and
If then ̃ ( ) ( ) is called Intuitionistic triangular
fuzzy number.
Operations on Intuitionistic trapezoidal number and Intuitionistic triangular
number:
Addition: ( ) ( ) ( ) ( ) (
)( )
( )( ) ( ) ( ) ( )(
)

Fuzzy Diagonal Optimal Algorithm to Solve Intuitionistic Fuzzy Assignment Problems
Subtraction: ( ) ( ) ( ) ( ) (
)( )
( )( ) ( ) ( ) ( )(
)
Definition
If ̃ ( )( ) is a intuitionistic triangular fuzzy number then the ranking of ̃
is given by ̅ ( )
̅ ( )
, (̃)
̅ ̅
.
If (̃) (̃), then ̃ ̃. This ranking technique satisfies Compensation ,
Linearity and additive properties.
Definition
Intuitionistic Fuzzy assignment problem is defined by
̃ ∑ ∑ ̃
subject to ∑ ∑
where {
The above problem can also be depicted as follows
.
(
̃ ̃ ̃
̃ ̃ ̃
̃ ̃ ̃
̃ ̃ ̃ )
Where ̃ the cost per unit in transporting from i th place to j th place.
2. PROPOSED ALGORITHM
1) Find the intuitionistic fuzzy penalty by subtracting the minimum intuitionistic fuzzy cost
and the next minimum intuitionistic fuzzy cost and put it against the corresponding row. Do
the same procedure for the column too and put it against the corresponding column.
2) Choose the maximum intuitionistic fuzzy penalty among all. If it is along the row locate
the minimum intuitionistic fuzzy cost in that corresponding row and remove the
corresponding row and the corresponding column of the intuitionistic fuzzy element. If it is
along the column locate the minimum intuitionistic fuzzy cost in that corresponding column
and remove the corresponding row and the corresponding column of the intuitionistic fuzzy
element.
3) Let ̃ be the assigned cost for all the columns. Subtracting ̃ from each entry of ̃ the
corresponding column of assignment matrix.
4) For each unassigned cell, form a loop such that one corner contains negative intutionistic
Fuzzy penalty and remaining two corners are assigned intuitionistic cost values in
corresponding row and column. Find the sum of diagonal cells of unassigned element, say ̃

Choose the most negative
I
ijd
~
and exchange the assigned cell of diagonals. Repeat this
process until all ̃ ̃ . If any I
ij
I
d 0
~~
 , then exchange the cells of diagonals at the end.
SECTION-3
Example 1
Consider the following interval integer Assignment problem
.
(
( )( ) ( )( ) ( )( ) ( )( )
( )( ) ( )( ) ( )( ) ( )( )
( )( ) ( )( ) ( )( ) ( )( )
( )( ) ( )( ) ( )( ) ( )( )
)
Applying the proposed algorithm
(
( )( ) ( )( ) ( )( ) ( )( )
( )( ) ( )( ) ( )( ) ( )( )
( )( ) ( )( ) ( )( ) ( )( )
( )( ) ( )( ) ( )( ) ( )( )
( )( ) ( )( ) ( )( ) ( )( ))
( )( )
( )( )
( )( )
( )( )
Choose the maximum intuitionistic Fuzzy penalty which is in the first row. Choose the
minimum intuitionistic fuzzy value in that row. Strikeout the corresponding row and column.
(
( )( ) ( )( ) ( )( ) ( )( )
( )( ) ( )( ) ( )( ) ( )( )
( )( ) ( )( ) ( )( ) ( )( )
( )( ) ( )( ) ( )( ) ( )( )
)
Repeating the same procedure for the remaining matrix
(
( )( ) ( )( ) ( )( )
( )( ) ( )( ) ( )( )
( )( ) ( )( ) ( )( )
( )( ) ( )( ) ( )( )
)
( )( )
( )( )
( )( )
Choose the maximum intuitionistic Fuzzy penalty which is in the second row. Choose the
minimum intuitionistic fuzzy value in that row. Strike out the corresponding row and column
(
( )( ) ( )( ) ( )( )
( )( ) ( )( ) ( )( )
( )( ) ( )( ) ( )( )
)
Repeating the same procedure for the remaining matrix
(
( )( ) ( )( )
( )( ) ( )( )
( )( ) ( )( )
)
( )( )
( )( )
Choose the maximum intuitionistic Fuzzy penalty which is in the second row. Choose the
minimum intuitionistic fuzzy value in that row. Strike out the corresponding row and column
(
( )( ) ( )( )
( )( ) ( )( )
*
The assignments are
(
( )( ) ( )( ) ( )( ) ( )( )
( )( ) ( )( ) ( )( ) ( )( )
( )( ) ( )( ) ( )( ) ( )( )
( )( ) ( )( ) ( )( ) ( )( )
)
Checking the optimum of these assignments

Fuzzy Diagonal Optimal Algorithm to Solve Intuitionistic Fuzzy Assignment Problems
(
( )( ) ( )( ) ( )( ) ( )( )
( )( ) ( )( ) ( )( ) ( )( )
( )( ) ( )( ) ( )( ) ( )( )
( )( ) ( )( ) ( )( ) ( )( )
( )( ) ( )( ) ( )( ) ( )( ) )
Subtracting the each element of the column from the corresponding assignment
(
( )( ) ( )( ) ( )( ) ( )( )
( )( ) ( )( ) ( )( ) ( )( )
( )( ) ( )( ) ( )( ) ( )( )
( )( ) ( )( ) ( )( ) ( )( )
)
For the non-assigned cells
̃ (
( )( ) ( )( )
( )( ) ( )( )
* ( )( )
( )( ) ( )( ) ̃.
̃ = ̃ ̃ = ̃ ̃ = ̃ ̃ = ̃ ̃ = ̃ , ̃ = ̃
For all the non assigned cells ̃ ̃ . So the assignments are optimum.
The optimum solution is
(
( )( ) ( )( ) ( )( ) ( )( )
( )( ) ( )( ) ( )( ) ( )( )
( )( ) ( )( ) ( )( ) ( )( )
( )( ) ( )( ) ( )( ) ( )( )
)
( )( ) ( )( ) ( )( ) ( )( )
=(27,34,73)(15,44,67).
SECTION-4
3. CONCLUSIONS
In this paper, we applied diagonal optimal method to find a solution of an assignment problem
in which parameters are triangular and trapezoidal intuitionistic fuzzy numbers. The total
intuitionistic optimal cost obtained by this method remains same as that obtained by
converting the total intuitionistic fuzzy cost by applying the ranking method. Also the
membership and non-membership values of the intuitionistic fuzzy costs are derived. This
algorithm can also be used in solving other types of assignment problems like, unbalanced,
prohibited assignment problems.
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