New algorithm for solving mixed intuitionistic fuzzy assignment problem
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This document discusses the mixed intuitionistic fuzzy assignment problem, emphasizing the challenges of using imprecise cost coefficients in assignment problems compared to traditional methods. The authors develop a systematic approach to derive optimal solutions for such problems through an intuitionistic fuzzy ranking method and numerical examples. The paper concludes with a proposed procedure that transforms the mixed intuitionistic fuzzy assignment problem into a balanced intuitionistic fuzzy assignment problem for better solvability.
![P. Senthil Kumar and R. Jahir Hussain/ Elixir Appl. Math. 73 (2014) 25971-2597725971
Introduction
Assignment Problem(AP) is used worldwide in solving real world problems. An assignment problem plays an important role in
an 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 special type of linear programming problem (LPP) in which our objective is to assign n
number of jobs to n number of machines (persons) at a minimum cost. To find solution to assignment problems, various algorithm
such as linear programming [8,9,13,17], Hungarian algorithm [15], neural network [12], genetic algorithm [6] have been developed.
However, in real life situations, the parameters of assignment problem are imprecise numbers instead of fixed real numbers
because time/cost for doing a job by a facility (machine/persion) might vary due to different reasons. The theory of fuzzy set
introduced by Zadeh[25] in 1965 has achieved successful applications in various fields. In 1970, Belmann and Zadeh introduce the
concepts of fuzzy set theory into the decision-making problems involving uncertainty and imprecision[7]. Amit Kumar et al
investigated Assignment and Travelling Salesman Problems with cost coefficients as LR fuzzy parameters[1], Fuzzy linear
programming approach for solving fuzzy transportation problems with transshipment[2], Method for solving fully fuzzy assignment
problems using triangular fuzzy numbers[3]. In [18], Sathi Mukherjee et al presented an Application of fuzzy ranking method for
solving assignment problems with fuzzy costs. Lin and Wen [16] proposed an efficient algorithm based an labeling method for
solving the linear fractional programming case. Y.L.P.Thorani and N.Ravi Sankar did Fuzzy assignment problem with generalized
fuzzy numbers [23].Different kinds of fuzzy assignment problems are solved in the papers [1,3,10,11,12,24].
The concept of Intuitionistic Fuzzy Sets (IFSs) proposed by Atanassov[5] in 1986 is found to be highly useful to deal with
vagueness. In [14], Jahir Hussian et all presented An Optimal More-for-Less Solution of Mixed Constrains Intuitionistic Fuzzy
Transportation Problems. P. Senthil Kumar et al did a systematic approach for solving mixed intuitionistic fuzzy transportation
problems, A method for solving balanced intuitionistic fuzzy assignment problem [20,21]. Here we investigate a more realistic
problem, namely mixed intuitionistic fuzzy assignment problem. Let ࢉ
ࡵ be the intuitionistic fuzzy cost of assigning the jth
job to the ith
machine. We assume that one machine can be assigned exactly one job; also each machine can do at most one job. The problem is to
find an optimal assignment so that the total intuitionistic fuzzy cost of performing all jobs is minimum or the total intuitionistic fuzzy
profit is maximum.
Assignment problem with crisp, fuzzy and intuitionistic fuzzy numbers as cost coefficients is called mixed intuitionistic fuzzy
assignment problem. Here the objective function is considered with crisp, fuzzy and intuitionistic fuzzy numbers. Then there is no
New algorithm for solving mixed intuitionistic fuzzy assignment problem
P. Senthil Kumar*
and R. Jahir Hussain
Department of Mathematics, Jamal Mohamed College, Tiruchirappalli – 620 020. India.
ABSTRACT
In conventional assignment problem, cost is always certain. In this paper, Assignment
problem with crisp, fuzzy and intuitionistic fuzzy numbers as cost coefficients is
investigated. There is no systematic approach for finding an optimal solution for mixed
intuitionistic fuzzy assignment problem. This paper develops an approach to solve a mixed
intuitionistic fuzzy assignment problem where cost is not in deterministic numbers but
imprecise ones. The solution procedure of mixed intuitionistic fuzzy assignment problem
is proposed to find the optimal assignment and also obtain an optimal value in terms of
triangular intuitionistic fuzzy numbers. Numerical examples show that an intuitionistic
fuzzy ranking method offers an effective tool for handling an intuitionistic fuzzy
assignment problem.
© 2014 Elixir All rights reserved.
ARTICLE INFO
Article history:
Received: 4 April 2014;
Received in revised form:
20 July 2014;
Accepted: 29 July 2014;
Keywords
Intuitionistic Fuzzy Set,
Triangular Fuzzy Number,
Triangular Intuitionistic Fuzzy
Number,
Mixed Intuitionistic Fuzzy Assignment
Problem, Optimal Solution.
Elixir Appl. Math. 73 (2014) 25971-25977
Applied Mathematics
Available online at www.elixirpublishers.com (Elixir International Journal)
Tele:
E-mail addresses: senthilsoft_5760@yahoo.com
© 2014 Elixir All rights reserved](https://image.slidesharecdn.com/newalgorithmforsolvingmixedintuitio-180706170752/85/New-algorithm-for-solving-mixed-intuitionistic-fuzzy-assignment-problem-2-320.jpg)
![P. Senthil Kumar and R. Jahir Hussain/ Elixir Appl. Math. 73 (2014) 25971-2597725972
systematic approach for finding an optimal solution for mixed intuitionistic fuzzy assignment problem but when we search our
literature it has demonstrated intuitionistic fuzzy assignment problems[19,21,22] only. In this paper, ranking procedure of Annie
Varghese and Sunny Kuriakose [4] is used to rank the intuitionistic fuzzy numbers and also compare the minimum and maximum of
it. The proposed method is used to transform the mixed intuitionistic fuzzy assignment problem into balanced intuitionistic fuzzy
assignment problem so that an intuitionistic fuzzy Hungarian method may be applied to solve the AP.
This paper is organized as follows: Section 2 deals with some basic terminology, In section 3, provides the definition of
intuitionistic fuzzy assignment problem and its mathematical formulation, Section 4, consists of solution procedure for mixed
intuitionistic fuzzy assignment problem. In section 5, to illustrate the proposed method a numerical example with results and
discussion is discussed and followed by the conclusions are given in Section 6.
Preliminaries
Definition 2.1 Let A be a classical set, ࣆሺ࢞ሻ be a function from A to [0,1]. A fuzzy set כ with the membership function ࣆሺ࢞ሻ is
defined by
כ
ൌ ൛൫࢞, ࣆሺ࢞ሻ൯; ࢞ א ࢇࢊ ࣆሺ࢞ሻ א ሾ, ሿൟ.
Definition 2.2 Let X be denote a universe of discourse, then an intuitionistic fuzzy set A in X is given by a set of ordered triples,
෩ࡵ
ൌ ሼ൏ ,ݔ ࣆሺ࢞ሻ, ࣖሺ࢞ሻ ; ݔ א ܺሽ
Whereࣆ, ࣖ: ࢄ ՜ ሾ, ሿ, are functions such that ࣆሺ࢞ሻ ࣖሺ࢞ሻ , ࢞ א ࢄ. For each x the membership
ࣆሺ࢞ሻ ࢇࢊ ࣖሺ࢞ሻ represent the degree of membership and the degree of non – membership of the element ࢞ א ࢄ to ؿ ࢄ
respectively.
Definition 2.3 A fuzzy number A is defined to be a triangular fuzzy number if its membership functions ࣆ:ℝ→ [0, 1] is equal to
ࣆሺܠሻ ൌ
ە
ۖ
۔
ۖ
ۓ ࢞ െ ࢇ
ࢇ െ ࢇ
ࢌ ࢞ א ሾࢇ, ࢇሿ
ࢇ െ ࢞
ࢇ െ ࢇ
ࢌ ࢞ א ሾࢇ, ࢇሿ
࢚ࢎࢋ࢙࢘࢝ࢋ
where ࢇ ࢇ ࢇ. This fuzzy number is denoted by ሺࢇ, ࢇ, ࢇሻ.
Definition 2.4 A Triangular Intuitionistic Fuzzy Number (ÃI
is an intuitionistic fuzzy set in R with the following membership
function ࣆሺܠሻ and non membership function ࣖሺܠሻ: )
ࣆሺܠሻ ൌ
ە
ۖ
ۖ
۔
ۖ
ۖ
ۓ
ࢌ࢘ ࢞ ൏ ࢇ
࢞ െ ࢇ
ࢇ െ ࢇ
ࢌ࢘ ࢇ ࢞ ࢇ
ࢌ࢘ ࢞ ൌ ࢇ
ࢇ െ ࢞
ࢇ െ ࢇ
ࢌ࢘ ࢇ ࢞ ࢇ
ࢌ࢘ ࢞ ࢇ
ࣖሺܠሻ ൌ
ە
ۖ
ۖ
۔
ۖ
ۖ
ۓ
ࢌ࢘ ࢞ ൏ ࢇ
ᇱ
ࢇ െ ࢞
ࢇ െ ࢇ
ᇱ ᇱ ࢌ࢘ ࢇ
ᇱ
࢞ ࢇ
ࢌ࢘ ࢞ ൌ ࢇ
࢞ െ ࢇ
ࢇ
ᇱ ᇱ
െ ࢇ
ࢌ࢘ ࢇ ࢞ ࢇ
ᇱ
ࢌ࢘ ࢞ ࢇ
ᇱ
Where ࢇ
ᇱ
ࢇ ࢇ ࢇ ࢇ
ᇱ and ࣆሺܠሻ, ۯሺܠሻ . for ࣆሺܠሻ ൌ ۯሺܠሻ ࢞ א ࡾ. This TrIFN is denoted by ෩ࡵ =
ሺࢇ, ࢇ, ࢇሻሺ ࢇ
ᇱ
, ࢇ, ࢇ
ᇱ
ሻ
Particular Cases
Let ෩ࡵ = ሺࢇ, ࢇ, ࢇሻሺ ࢇ
ᇱ
, ࢇ, ࢇ
ᇱ
ሻ be a TrIFN. Then the following cases arise
Case 1: If ࢇ
ᇱ
ൌ ࢇ, ࢇ
ᇱ
ൌ ࢇ , then ෩ࡵ represent Tringular Fuzzy Number
(TFN).It is denoted by ෩ ൌ ሺࢇ, ࢇ, ࢇሻ.
Case 2: If ࢇ
ᇱ
ൌ ࢇ ൌ ࢇ ൌ ࢇ ൌ ࢇ
ᇱ
ൌ , then ෩ࡵ represent a real number .
Definition 2.5 Let ෩ࡵ and ෩ࡵ be two TrIFNs. The ranking of ෩ࡵ and ෩ࡵ by the ℜ (.) on E, the set of TrIFNs is defined as follows:
i. ℜ (෩ࡵ)> ℜ (෩ࡵ) iff ෩ࡵ≻ ෩ࡵ
ii. ℜ (෩ࡵ)< ℜ (෩ࡵ) iff ෩ࡵ≺ ෩ࡵ](https://image.slidesharecdn.com/newalgorithmforsolvingmixedintuitio-180706170752/85/New-algorithm-for-solving-mixed-intuitionistic-fuzzy-assignment-problem-3-320.jpg)
![P. Senthil Kumar and R. Jahir Hussain/ Elixir Appl. Math. 73 (2014) 25971-2597725973
iii. ℜ (෩ࡵ)= ℜ (෩ࡵ) iff ෩ࡵ≈ ෩ࡵ
iv. ℜ (෩ࡵ
෩ࡵ)= ℜ (෩ࡵ)+ ℜ (෩ࡵ)
v. ℜ (෩ࡵ
െ ෩ࡵ
ሻ ൌ ℜ൫෩ࡵ
൯ െ ℜ ሺ෩ࡵ
ሻ
Arithmetic Operations
Let ෩ࡵ
ൌ ሺࢇ, ࢇ, ࢇሻሺ ࢇ
ᇱ
, ࢇ, ࢇ
ᇱ
ሻ and ෩ࡵ
ൌ ሺ࢈, ࢈, ࢈ሻሺ ࢈
ᇱ
, ࢈, ࢈
ᇱ
ሻbe any two TrIFNs then the arithmetic operations as
follows:
Addition: ෩ࡵ
ْ ෩ࡵ=ሺࢇ ࢈, ࢇ ࢈, ࢇ ࢈ሻሺࢇ
ᇱ
࢈
ᇱ
, ࢇ ࢈, ࢇ
ᇱ
࢈
ᇱ
ሻ
Subtraction: ÃI
Θ BI
=ሺࢇ െ ࢈, ࢇ െ ࢈, ࢇ െ ࢈ሻሺࢇ
ᇱ
െ ࢈
ᇱ
, ࢇ െ ࢈, ࢇ
ᇱ
െ ࢈
ᇱ
ሻ
Ranking of triangular intuitionistic fuzzy numbers
The Ranking of a triangular intuitionistic fuzzy number ෩ࡵ
ൌ ሺࢇ, ࢇ, ࢇሻሺ ࢇ
ᇱ
, ࢇ, ࢇ
ᇱ
ሻ is defined by Annie Varghese and Sunny
Kuriakose [4].
ࡾ൫෩ࡵ
൯ ൌ
ሺࢇ
ᇱ
െ ࢇ
ᇱ
ሻሺࢇ െ ࢇ
ᇱ
െ ࢇ
ᇱ ሻ ሺࢇ െ ࢇሻሺࢇ ࢇ ࢇሻ ሺࢇ
ᇱ
െ ࢇ
ᇱ
ሻ
ࢇ
ᇱ
െ ࢇ
ᇱ
ࢇ െ ࢇ
൩
The ranking technique [4] is:
If ℜ(෩ࡵ) ≤ ℜ(෩ࡵ), then ෩ࡵ
෩ࡵ i.e., min {෩ࡵ
, ෩ࡵ
ሽ ൌ ෩ࡵ
Example: Let ෩ࡵ
ൌ ሺૡ, , ሻሺ, , ሻ and ෩ࡵ
ൌ ሺ, , ૡሻሺ, , ሻ be any two TrIFN, then its rank is defined by ℜ൫෩ࡵ
൯ ൌ ,
ℜ(෩ࡵ
ሻ ൌ . this implies ෩ࡵ≻ ෩ࡵ
Intuitionistic Fuzzy Assignment Problem and its Mathematical Formulation
Consider the situation of assigning n machines to n jobs and each machine is capable of doing any job at different costs. Let ࢉ
ࡵ be
an intuitionistic fuzzy cost of assigning the jth
job to the ith
machine. Let ࢞ be the decision variable denoting the assignment of the
machine i to the job j. The objective is to minimize the total intuitionistic fuzzy cost of assigning all the jobs to the available machines
(one machine per job) at the least total cost. This situation is known as balanced intuitionistic fuzzy assignment problem.
(IFAP)Minimize ࢆ෩ࡵ
ൌ ∑ ∑ ࢉ
ࡵ
࢞
ୀ
ୀ
Subject to,
࢞
ୀ
ൌ , ࢌ࢘ ൌ , , … ,
∑ ࢞
ୀ ൌ , ࢌ࢘ ൌ , , … ,
࢞ א ሼ, ሽ
ࢃࢎࢋ࢘ࢋ ࢞ ൌ ቊ
, ࢌ ࢚ࢋܐ ܑܐܜ
ࢇࢉࢋܐ ࢙ ࢇ࢙࢙ࢍࢋࢊ ࢚ ࢚ܐ
࢈
, ࢌ ܑܐܜ
ࢇࢉࢋܐ ࢙ ࢚ ࢇ࢙࢙ࢍࢋࢊ ࢚ ࢚ܐ
࢈
ࢉ
ࡵ
ൌ ሺࢉ
, ࢉ
, ࢉ
ሻሺࢉ
ᇲ
, ࢉ
, ࢉ
ᇲ
ሻ
The above IFAP can be stated in the below tabular form as follows:
Table 1
1 2 … j … n
1 ࢉ
ࡵ
ࢉ
ࡵ … ࢉ
ࡵ
ࢉ
ࡵ
2 ࢉ
ࡵ
ࢉ
ࡵ … ࢉ
ࡵ … ࢉ
ࡵ
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
i ࢉ
ࡵ
ࢉ
ࡵ … ࢉ
ࡵ … ࢉ
ࡵ
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
n ࢉ
ࡵ
ࢉ
ࡵ … ࢉ
ࡵ … ࢉ
ࡵ](https://image.slidesharecdn.com/newalgorithmforsolvingmixedintuitio-180706170752/85/New-algorithm-for-solving-mixed-intuitionistic-fuzzy-assignment-problem-4-320.jpg)
![P. Senthil Kumar and R. Jahir Hussain/ Elixir Appl. Math. 73 (2014) 25971-2597725975
Example: Let us consider an mixed intuitionistic fuzzy assignment problem with rows representing 3 machines ࡹ, ࡹ, ࡹ
and columns representing the 3 jobs ࡶ, ࡶ, ࡶ. The cost matrix [ࢉࡵ] is given whose elements are different types of real, fuzzy and
intuitionistic fuzzy numbers. The problem is to find the optimal assignment so that the total cost of job assignment becomes
minimum.
Solution:
The corresponding balanced intuitionistic fuzzy assignment problem is
An intuitionistic fuzzy assignment problem can be formulated in the following mathematical programming form
Min[(3,5,8)(1,5,10)࢞+(10,10,10)(10,10,10) ࢞+(10,15,20)(10,15,20) ࢞+(3,3,3)(3,3,3) ࢞+(1,3,8)(1,3,8) ࢞+(4,5,7)(0,5,11)
࢞+(1,2,3)(1,2,3) ࢞+ℜ(3,5,8)(2,5,9) ࢞+(6,6,6)(6,6,6) ࢞
Subject to ࢞ ࢞ ࢞ ൌ , ࢞ ࢞ ࢞ ൌ ,
࢞ ࢞ ࢞ ൌ , ࢞ ࢞ ࢞ ൌ ,
࢞ ࢞ ࢞ ൌ , ࢞ ࢞ ࢞ ൌ ,
࢞ א ሼ, ሽ.
Now, using the step 5 of the proposed method, we have the following reduced intuitionistic fuzzy assignment table
Now, using the step 6 of the proposed method, we have the following reduced intuitionistic fuzzy assignment table
Now, using step 7 to step 10, we have the following optimum assignment table
The optimal solution is
࢞
כ
ൌ ࢞
כ
ൌ ࢞
כ
ൌ , ࢞
כ
ൌ ࢞
כ
ൌ ࢞
כ
ൌ ࢞
כ
ൌ ࢞
כ
ൌ ࢞
כ
ൌ ,
With the optimal objective value ℜ൫ࢆ෩ࡵ
൯ ൌ . which represents the optimal total cost. In other words the optimal assignment is
ࡹ ՜ ࡶ, ࡹ ՜ ࡶ, ࡹ ՜ ࡶ
The intuitionistic fuzzy minimum total cost is calculated as
ࢉ
ࡵ
ࢉ
ࡵ
ࢉ
ࡵ
ൌ ሺ, , ૡሻሺ, , ሻ ሺ, , ૡሻሺ, , ૡሻ
ࡶ ࡶ ࡶ
ࡹ (3,5,8)(1,5,10) 10 (10,15,20)
ࡹ 3 (1,3,8) (4,5,7)(0,5,11)
ࡹ (1,2,3) (3,5,8)(2,5,9) 6
ࡶ ࡶ ࡶ
ࡹ (3,5,8)(1,5,10) (10,10,10)(10,10,10) (10,15,20)(10,15,20)
ࡹ (3,3,3)(3,3,3) (1,3,8)(1,3,8) (4,5,7)(0,5,11)
ࡹ (1,2,3)(1,2,3) (3,5,8)(2,5,9) (6,6,6)(6,6,6)
ࡶ ࡶ ࡶ
ࡹ (-5,0,5)(-9,0,9) (2,5,7)(0,5,9) (2,10,17)(0,10,19)
ࡹ (0,0,0)(0,0,0) (-2,0,5)(-2,0,5) (1,2,4)(-3,2,8)
ࡹ (-2,0,2)(-2,0,2) (0,3,7)(-1,3,8) (3,4,5)(3,4,5)
ࡶ ࡶ ࡶ
ࡹ (-5,0,5)(-9,0,9) (-3,5,9)(-5,5,11) (-2,8,16)(-8,8,22)
ࡹ (0,0,0)(0,0,0) (-7,0,7)(-7,0,7) (-3,0,3)(-11,0,11)
ࡹ (-2,0,2)(-2,0,2) (-5,3,9)(-6,3,10) (-1,2,4)(-5,2,8)
ࡶ ࡶ ࡶ
ࡹ (-5,0,5)(-9,0,9) (-7,3,10)(-13,3,16) (-6,6,17)(-16,6,27)
ࡹ (-1,2,4)(-5,2,8) (-7,0,7)(-7,0,7) (-3,0,3)(-11,0,11)
ࡹ (-2,0,2)(-2,0,2) (-9,1,10)(-14,1,15) (-5,0,5)(-13,0,13)](https://image.slidesharecdn.com/newalgorithmforsolvingmixedintuitio-180706170752/85/New-algorithm-for-solving-mixed-intuitionistic-fuzzy-assignment-problem-6-320.jpg)
![P. Senthil Kumar and R. Jahir Hussain/ Elixir Appl. Math. 73 (2014) 25971-2597725976
ሺ, , ሻሺ, , ሻ= ሺ, , ሻሺૡ, , ሻ
Also we find that ℜ൫ࢆ෩ࡵכ
൯ ൌ ℜሾሺ, , ሻሺૡ, , ሻሿ ൌ .ܛ܀ .
In the above example it has been shown that the total optimal cost obtained by our method remains same as that obtained by
converting the total intuitionistic fuzzy cost by applying the ranking method Annie Varghese and Sunny Kuriakose [4].
Results and discussion
The minimum total intuitionistic fuzzy assignment cost is
ࢆ෩I
= ሺ, , ሻሺૡ, , ሻ ………… (1)
Figure 1 Graphical Representation of IFAC
The result in (1) can be explained (Refer to figure1) as follows:
(i) Assignment cost lies in [10,22].
(ii) 100% expect are in favour that an assignment cost is 14 as ࣆࢆ෩ࡵ ሺ࢞ሻ ൌ ,࢞ ൌ .
(iii) Assuming that ࣆ is a membership value and ࣖ is a non-membership value at c. Then ࣆ% experts are in favour and ࣖ%
experts are opposing but ሺ െ ࣆ െ ࣖሻ% are in confusion that an assignment cost is ࢉ.
Values of ࣆࢆ෩ࡵ ሺࢉሻ and ࣖࢆ෩ࡵ ሺࢉሻ at different values of c can be determined using equations given below.
ߤሺxሻ ൌ
ە
ۖ
ۖ
۔
ۖ
ۖ
ۓ
0 ݂ݎ ݔ ൏ 10
ݔ െ 10
4
݂ݎ 10 ݔ 14
1 ݂ݎ ݔ ൌ 14
22 െ ݔ
8
݂ݎ 14 ݔ 22
0 ݂ݎ ݔ 22
ߴሺxሻ ൌ
ە
ۖ
ۖ
۔
ۖۖ
ۓ
1 ݂ݎ ݔ ൏ 8
14 െ ݔ
6
݂ݎ 8 ݔ 14
0 ݂ݎ ݔ ൌ 14
ݔ െ 14
10
݂ݎ 14 ݔ 24
1 ݂ݎ ݔ 24
Conclusion
In this paper, Assignment problem with crisp, fuzzy and intuitionistic fuzzy numbers as cost coefficients is discussed. The
proposed method is a systematic approach for solving an assignment problem under mixed intuitionistic fuzzy environment. The total
optimal cost obtained by our method remains same as that obtained by converting the total intuitionistic fuzzy cost by applying the
ranking method of Annie Varghese and Sunny Kuriakose[4].Also the membership and non-membership values of an intuitionistic
fuzzy costs are derived. This technique can also be used in solving other types of problems like, project schedules, transportation
problems and network flow problems.
References
[1] Amit Kumar and Anila Gupta, Assignment and Travelling Salesman Problems with Coefficients as LR Fuzzy Parameters,
International Journal of Applied Science and Engineering 2012.10,3:155-170.
[2] Amit Kumar, Amarpreet Kaur, Anila Gupta, Fuzzy Linear Programming Approach for Solving Fuzzy Transportation problems
with Transshipment, J Math Model Algor (2011) 10:163-180.
[3] Amit Kumar, Anila Gupta and Amarpreet Kumar,Method for Solving Fully Fuzzy Assignment Problems Using Triangular Fuzzy
Numbers, International Journal of Computer and Information Engineering 3:4 2009.](https://image.slidesharecdn.com/newalgorithmforsolvingmixedintuitio-180706170752/85/New-algorithm-for-solving-mixed-intuitionistic-fuzzy-assignment-problem-7-320.jpg)
![P. Senthil Kumar and R. Jahir Hussain/ Elixir Appl. Math. 73 (2014) 25971-2597725977
[4] Annie Varghese and Sunny Kuriakose, Notes on Intuitionistic Fuzzy Sets Vol.18, 2012, No.1, 19-24.
[5] K.T.Atanassov, Intuitionistic fuzzy sets, fuzzy sets and systems, Vol.20, no.1.pp.87- 96, 1986.
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New algorithm for solving mixed intuitionistic fuzzy assignment problem
- 1. Elixir International Journal Publication August 2014 ELIXIR INTERNATIONAL JOURNAL (Available online at www.elixirpublishers.com) Index Copernicus, Poland Value (ICV) is 5.79 in 2012 (5.09 in 2011) August 2014 Issue
- 2. P. Senthil Kumar and R. Jahir Hussain/ Elixir Appl. Math. 73 (2014) 25971-2597725971 Introduction Assignment Problem(AP) is used worldwide in solving real world problems. An assignment problem plays an important role in an 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 special type of linear programming problem (LPP) in which our objective is to assign n number of jobs to n number of machines (persons) at a minimum cost. To find solution to assignment problems, various algorithm such as linear programming [8,9,13,17], Hungarian algorithm [15], neural network [12], genetic algorithm [6] have been developed. However, in real life situations, the parameters of assignment problem are imprecise numbers instead of fixed real numbers because time/cost for doing a job by a facility (machine/persion) might vary due to different reasons. The theory of fuzzy set introduced by Zadeh[25] in 1965 has achieved successful applications in various fields. In 1970, Belmann and Zadeh introduce the concepts of fuzzy set theory into the decision-making problems involving uncertainty and imprecision[7]. Amit Kumar et al investigated Assignment and Travelling Salesman Problems with cost coefficients as LR fuzzy parameters[1], Fuzzy linear programming approach for solving fuzzy transportation problems with transshipment[2], Method for solving fully fuzzy assignment problems using triangular fuzzy numbers[3]. In [18], Sathi Mukherjee et al presented an Application of fuzzy ranking method for solving assignment problems with fuzzy costs. Lin and Wen [16] proposed an efficient algorithm based an labeling method for solving the linear fractional programming case. Y.L.P.Thorani and N.Ravi Sankar did Fuzzy assignment problem with generalized fuzzy numbers [23].Different kinds of fuzzy assignment problems are solved in the papers [1,3,10,11,12,24]. The concept of Intuitionistic Fuzzy Sets (IFSs) proposed by Atanassov[5] in 1986 is found to be highly useful to deal with vagueness. In [14], Jahir Hussian et all presented An Optimal More-for-Less Solution of Mixed Constrains Intuitionistic Fuzzy Transportation Problems. P. Senthil Kumar et al did a systematic approach for solving mixed intuitionistic fuzzy transportation problems, A method for solving balanced intuitionistic fuzzy assignment problem [20,21]. Here we investigate a more realistic problem, namely mixed intuitionistic fuzzy assignment problem. Let ࢉ ࡵ be the intuitionistic fuzzy cost of assigning the jth job to the ith machine. We assume that one machine can be assigned exactly one job; also each machine can do at most one job. The problem is to find an optimal assignment so that the total intuitionistic fuzzy cost of performing all jobs is minimum or the total intuitionistic fuzzy profit is maximum. Assignment problem with crisp, fuzzy and intuitionistic fuzzy numbers as cost coefficients is called mixed intuitionistic fuzzy assignment problem. Here the objective function is considered with crisp, fuzzy and intuitionistic fuzzy numbers. Then there is no New algorithm for solving mixed intuitionistic fuzzy assignment problem P. Senthil Kumar* and R. Jahir Hussain Department of Mathematics, Jamal Mohamed College, Tiruchirappalli – 620 020. India. ABSTRACT In conventional assignment problem, cost is always certain. In this paper, Assignment problem with crisp, fuzzy and intuitionistic fuzzy numbers as cost coefficients is investigated. There is no systematic approach for finding an optimal solution for mixed intuitionistic fuzzy assignment problem. This paper develops an approach to solve a mixed intuitionistic fuzzy assignment problem where cost is not in deterministic numbers but imprecise ones. The solution procedure of mixed intuitionistic fuzzy assignment problem is proposed to find the optimal assignment and also obtain an optimal value in terms of triangular intuitionistic fuzzy numbers. Numerical examples show that an intuitionistic fuzzy ranking method offers an effective tool for handling an intuitionistic fuzzy assignment problem. © 2014 Elixir All rights reserved. ARTICLE INFO Article history: Received: 4 April 2014; Received in revised form: 20 July 2014; Accepted: 29 July 2014; Keywords Intuitionistic Fuzzy Set, Triangular Fuzzy Number, Triangular Intuitionistic Fuzzy Number, Mixed Intuitionistic Fuzzy Assignment Problem, Optimal Solution. Elixir Appl. Math. 73 (2014) 25971-25977 Applied Mathematics Available online at www.elixirpublishers.com (Elixir International Journal) Tele: E-mail addresses: [email protected] © 2014 Elixir All rights reserved
- 3. P. Senthil Kumar and R. Jahir Hussain/ Elixir Appl. Math. 73 (2014) 25971-2597725972 systematic approach for finding an optimal solution for mixed intuitionistic fuzzy assignment problem but when we search our literature it has demonstrated intuitionistic fuzzy assignment problems[19,21,22] only. In this paper, ranking procedure of Annie Varghese and Sunny Kuriakose [4] is used to rank the intuitionistic fuzzy numbers and also compare the minimum and maximum of it. The proposed method is used to transform the mixed intuitionistic fuzzy assignment problem into balanced intuitionistic fuzzy assignment problem so that an intuitionistic fuzzy Hungarian method may be applied to solve the AP. This paper is organized as follows: Section 2 deals with some basic terminology, In section 3, provides the definition of intuitionistic fuzzy assignment problem and its mathematical formulation, Section 4, consists of solution procedure for mixed intuitionistic fuzzy assignment problem. In section 5, to illustrate the proposed method a numerical example with results and discussion is discussed and followed by the conclusions are given in Section 6. Preliminaries Definition 2.1 Let A be a classical set, ࣆሺ࢞ሻ be a function from A to [0,1]. A fuzzy set כ with the membership function ࣆሺ࢞ሻ is defined by כ ൌ ൛൫࢞, ࣆሺ࢞ሻ൯; ࢞ א ࢇࢊ ࣆሺ࢞ሻ א ሾ, ሿൟ. Definition 2.2 Let X be denote a universe of discourse, then an intuitionistic fuzzy set A in X is given by a set of ordered triples, ෩ࡵ ൌ ሼ൏ ,ݔ ࣆሺ࢞ሻ, ࣖሺ࢞ሻ ; ݔ א ܺሽ Whereࣆ, ࣖ: ࢄ ՜ ሾ, ሿ, are functions such that ࣆሺ࢞ሻ ࣖሺ࢞ሻ , ࢞ א ࢄ. For each x the membership ࣆሺ࢞ሻ ࢇࢊ ࣖሺ࢞ሻ represent the degree of membership and the degree of non – membership of the element ࢞ א ࢄ to ؿ ࢄ respectively. Definition 2.3 A fuzzy number A is defined to be a triangular fuzzy number if its membership functions ࣆ:ℝ→ [0, 1] is equal to ࣆሺܠሻ ൌ ە ۖ ۔ ۖ ۓ ࢞ െ ࢇ ࢇ െ ࢇ ࢌ ࢞ א ሾࢇ, ࢇሿ ࢇ െ ࢞ ࢇ െ ࢇ ࢌ ࢞ א ሾࢇ, ࢇሿ ࢚ࢎࢋ࢙࢘࢝ࢋ where ࢇ ࢇ ࢇ. This fuzzy number is denoted by ሺࢇ, ࢇ, ࢇሻ. Definition 2.4 A Triangular Intuitionistic Fuzzy Number (ÃI is an intuitionistic fuzzy set in R with the following membership function ࣆሺܠሻ and non membership function ࣖሺܠሻ: ) ࣆሺܠሻ ൌ ە ۖ ۖ ۔ ۖ ۖ ۓ ࢌ࢘ ࢞ ൏ ࢇ ࢞ െ ࢇ ࢇ െ ࢇ ࢌ࢘ ࢇ ࢞ ࢇ ࢌ࢘ ࢞ ൌ ࢇ ࢇ െ ࢞ ࢇ െ ࢇ ࢌ࢘ ࢇ ࢞ ࢇ ࢌ࢘ ࢞ ࢇ ࣖሺܠሻ ൌ ە ۖ ۖ ۔ ۖ ۖ ۓ ࢌ࢘ ࢞ ൏ ࢇ ᇱ ࢇ െ ࢞ ࢇ െ ࢇ ᇱ ᇱ ࢌ࢘ ࢇ ᇱ ࢞ ࢇ ࢌ࢘ ࢞ ൌ ࢇ ࢞ െ ࢇ ࢇ ᇱ ᇱ െ ࢇ ࢌ࢘ ࢇ ࢞ ࢇ ᇱ ࢌ࢘ ࢞ ࢇ ᇱ Where ࢇ ᇱ ࢇ ࢇ ࢇ ࢇ ᇱ and ࣆሺܠሻ, ۯሺܠሻ . for ࣆሺܠሻ ൌ ۯሺܠሻ ࢞ א ࡾ. This TrIFN is denoted by ෩ࡵ = ሺࢇ, ࢇ, ࢇሻሺ ࢇ ᇱ , ࢇ, ࢇ ᇱ ሻ Particular Cases Let ෩ࡵ = ሺࢇ, ࢇ, ࢇሻሺ ࢇ ᇱ , ࢇ, ࢇ ᇱ ሻ be a TrIFN. Then the following cases arise Case 1: If ࢇ ᇱ ൌ ࢇ, ࢇ ᇱ ൌ ࢇ , then ෩ࡵ represent Tringular Fuzzy Number (TFN).It is denoted by ෩ ൌ ሺࢇ, ࢇ, ࢇሻ. Case 2: If ࢇ ᇱ ൌ ࢇ ൌ ࢇ ൌ ࢇ ൌ ࢇ ᇱ ൌ , then ෩ࡵ represent a real number . Definition 2.5 Let ෩ࡵ and ෩ࡵ be two TrIFNs. The ranking of ෩ࡵ and ෩ࡵ by the ℜ (.) on E, the set of TrIFNs is defined as follows: i. ℜ (෩ࡵ)> ℜ (෩ࡵ) iff ෩ࡵ≻ ෩ࡵ ii. ℜ (෩ࡵ)< ℜ (෩ࡵ) iff ෩ࡵ≺ ෩ࡵ
- 4. P. Senthil Kumar and R. Jahir Hussain/ Elixir Appl. Math. 73 (2014) 25971-2597725973 iii. ℜ (෩ࡵ)= ℜ (෩ࡵ) iff ෩ࡵ≈ ෩ࡵ iv. ℜ (෩ࡵ ෩ࡵ)= ℜ (෩ࡵ)+ ℜ (෩ࡵ) v. ℜ (෩ࡵ െ ෩ࡵ ሻ ൌ ℜ൫෩ࡵ ൯ െ ℜ ሺ෩ࡵ ሻ Arithmetic Operations Let ෩ࡵ ൌ ሺࢇ, ࢇ, ࢇሻሺ ࢇ ᇱ , ࢇ, ࢇ ᇱ ሻ and ෩ࡵ ൌ ሺ࢈, ࢈, ࢈ሻሺ ࢈ ᇱ , ࢈, ࢈ ᇱ ሻbe any two TrIFNs then the arithmetic operations as follows: Addition: ෩ࡵ ْ ෩ࡵ=ሺࢇ ࢈, ࢇ ࢈, ࢇ ࢈ሻሺࢇ ᇱ ࢈ ᇱ , ࢇ ࢈, ࢇ ᇱ ࢈ ᇱ ሻ Subtraction: ÃI Θ BI =ሺࢇ െ ࢈, ࢇ െ ࢈, ࢇ െ ࢈ሻሺࢇ ᇱ െ ࢈ ᇱ , ࢇ െ ࢈, ࢇ ᇱ െ ࢈ ᇱ ሻ Ranking of triangular intuitionistic fuzzy numbers The Ranking of a triangular intuitionistic fuzzy number ෩ࡵ ൌ ሺࢇ, ࢇ, ࢇሻሺ ࢇ ᇱ , ࢇ, ࢇ ᇱ ሻ is defined by Annie Varghese and Sunny Kuriakose [4]. ࡾ൫෩ࡵ ൯ ൌ ሺࢇ ᇱ െ ࢇ ᇱ ሻሺࢇ െ ࢇ ᇱ െ ࢇ ᇱ ሻ ሺࢇ െ ࢇሻሺࢇ ࢇ ࢇሻ ሺࢇ ᇱ െ ࢇ ᇱ ሻ ࢇ ᇱ െ ࢇ ᇱ ࢇ െ ࢇ ൩ The ranking technique [4] is: If ℜ(෩ࡵ) ≤ ℜ(෩ࡵ), then ෩ࡵ ෩ࡵ i.e., min {෩ࡵ , ෩ࡵ ሽ ൌ ෩ࡵ Example: Let ෩ࡵ ൌ ሺૡ, , ሻሺ, , ሻ and ෩ࡵ ൌ ሺ, , ૡሻሺ, , ሻ be any two TrIFN, then its rank is defined by ℜ൫෩ࡵ ൯ ൌ , ℜ(෩ࡵ ሻ ൌ . this implies ෩ࡵ≻ ෩ࡵ Intuitionistic Fuzzy Assignment Problem and its Mathematical Formulation Consider the situation of assigning n machines to n jobs and each machine is capable of doing any job at different costs. Let ࢉ ࡵ be an intuitionistic fuzzy cost of assigning the jth job to the ith machine. Let ࢞ be the decision variable denoting the assignment of the machine i to the job j. The objective is to minimize the total intuitionistic fuzzy cost of assigning all the jobs to the available machines (one machine per job) at the least total cost. This situation is known as balanced intuitionistic fuzzy assignment problem. (IFAP)Minimize ࢆ෩ࡵ ൌ ∑ ∑ ࢉ ࡵ ࢞ ୀ ୀ Subject to, ࢞ ୀ ൌ , ࢌ࢘ ൌ , , … , ∑ ࢞ ୀ ൌ , ࢌ࢘ ൌ , , … , ࢞ א ሼ, ሽ ࢃࢎࢋ࢘ࢋ ࢞ ൌ ቊ , ࢌ ࢚ࢋܐ ܑܐܜ ࢇࢉࢋܐ ࢙ ࢇ࢙࢙ࢍࢋࢊ ࢚ ࢚ܐ ࢈ , ࢌ ܑܐܜ ࢇࢉࢋܐ ࢙ ࢚ ࢇ࢙࢙ࢍࢋࢊ ࢚ ࢚ܐ ࢈ ࢉ ࡵ ൌ ሺࢉ , ࢉ , ࢉ ሻሺࢉ ᇲ , ࢉ , ࢉ ᇲ ሻ The above IFAP can be stated in the below tabular form as follows: Table 1 1 2 … j … n 1 ࢉ ࡵ ࢉ ࡵ … ࢉ ࡵ ࢉ ࡵ 2 ࢉ ࡵ ࢉ ࡵ … ࢉ ࡵ … ࢉ ࡵ . . . . . . . . . . . . . . . . . . . . . i ࢉ ࡵ ࢉ ࡵ … ࢉ ࡵ … ࢉ ࡵ . . . . . . . . . . . . . . . . . . . . . n ࢉ ࡵ ࢉ ࡵ … ࢉ ࡵ … ࢉ ࡵ
- 5. P. Senthil Kumar and R. Jahir Hussain/ Elixir Appl. Math. 73 (2014) 25971-2597725974 The Computational Procedure for Mixed Intuitionistic Fuzzy Assignment Problem Step1. Construct an assignment problem with crisp, fuzzy and an intuitionistic fuzzy numbers as cost coefficients. Step 2.Check whether the given mixed intuitionistic fuzzt cost matrix of a mixed intuitionistic fuzzy assignment problem is a balanced one or not. If it is a balanced one (i.e., number of jobs is equal to the number of machines) then go to step 4. If not, it is an unbalanced one (i.e., number of jobs is not equal to the number of machines) then go to step 3. Step 3. Introduce the required number of dummy rows and/or columns with zero intuitionistic fuzzy costs. Step4. Convert BMIFAP into balanced intuitionistic fuzzy assignment problem (BIFAP) using the following steps i. If any one or more in the costs (profits) of an assignment problem having a crisp number say ࢇ that can be expanded as a TrIFN ࢇ ൌ ሺࢇ , ࢇ , ࢇ ሻሺࢇ , ࢇ , ࢇ ሻ. ii. If any one or more in the costs (profits) of an assignment problem having a triangular fuzzy number say ሺࢇ, ࢇ, ࢇሻ that can be expanded as a TrIFN ሺࢇ, ࢇ, ࢇሻ ൌ ሺࢇ, ࢇ, ࢇሻሺࢇ, ࢇ, ࢇሻ. iii If any one or more in the costs (profits) of an assignment problem having a TrIFN say ሺࢇ, ࢇ, ࢇሻሺࢇ ᇱ , ࢇ, ࢇ ᇱ ሻ that can be kept as it is. That is ሺࢇ, ࢇ, ࢇሻሺࢇ ᇱ , ࢇ, ࢇ ᇱ ሻ ൌ ሺࢇ, ࢇ, ࢇሻሺࢇ ᇱ , ࢇ, ࢇ ᇱ ሻ Step 5. In the given intuitionistic fuzzy cost matrix, subtract the smallest element in each row from every element of that row by using ranking procedure as mentioned in section II. Step 6. In the reduced intuitionistic fuzzy cost matrix, subtract the smallest element in each column from every element of that column by using ranking procedure as mentioned in section II. Step 7. Make the assignment for the reduced intuitionistic fuzzy cost matrix obtained from Step 6 in the following way: a. Examine the rows successively until a row with exactly one unmarked intuitionistic fuzzy zero is found. Enclose this intuitionistic fuzzy zero in a box () as an assignment will be made there and cross (×) all other intuitionistic fuzzy zeros appearing in the corresponding column as they will not be considered for further assignment. Proceed in this way until all the rows have been examined. b. After examining all the rows completely, examine the columns successively until a column with exactly one unmarked intuitionistic fuzzy zero is found. Make an assignment to this single intuitionistic fuzzy zero by putting a box () and cross out (×) all other intuitionistic fuzzy zeros in the corresponding row. Proceed in this way until all columns have been examined. c. Repeat the operation (a) and (b) until all the intuitionistic fuzzy zeros are either marked () or crossed (×). Step 8. If there is exactly one assignment in each row and in each column then the optimum assignment policy for the given problem is obtained. Otherwise go to Step-9. Step 9. Draw minimum number of vertical and horizontal lines necessary to cover all the intuitionistic fuzzy zeros in the reduced intuitionistic fuzzy cost matrix obtained from Step-7 by inspection or by adopting the following procedure i. Mark ( ) all rows that do not have assignment ii. Mark ( ) all columns (not already marked) which have intuitionistic fuzzy zeros in the marked rows iii. Mark ( ) all rows (not already marked) that have assignments in marked columns, iv. Repeat steps 9(ii) and 9(iii) until no more rows or columns can be marked. v. Draw straight lines through all unmarked rows and marked columns. Step 10. Select the smallest element among all the uncovered elements. Subtract this least element from all the uncovered elements and add it to the element which lies at the intersection of any two lines. Thus, we obtain the modified matrix. Go to Step 7 and repeat the procedure. Numerical Example:
- 6. P. Senthil Kumar and R. Jahir Hussain/ Elixir Appl. Math. 73 (2014) 25971-2597725975 Example: Let us consider an mixed intuitionistic fuzzy assignment problem with rows representing 3 machines ࡹ, ࡹ, ࡹ and columns representing the 3 jobs ࡶ, ࡶ, ࡶ. The cost matrix [ࢉࡵ] is given whose elements are different types of real, fuzzy and intuitionistic fuzzy numbers. The problem is to find the optimal assignment so that the total cost of job assignment becomes minimum. Solution: The corresponding balanced intuitionistic fuzzy assignment problem is An intuitionistic fuzzy assignment problem can be formulated in the following mathematical programming form Min[(3,5,8)(1,5,10)࢞+(10,10,10)(10,10,10) ࢞+(10,15,20)(10,15,20) ࢞+(3,3,3)(3,3,3) ࢞+(1,3,8)(1,3,8) ࢞+(4,5,7)(0,5,11) ࢞+(1,2,3)(1,2,3) ࢞+ℜ(3,5,8)(2,5,9) ࢞+(6,6,6)(6,6,6) ࢞ Subject to ࢞ ࢞ ࢞ ൌ , ࢞ ࢞ ࢞ ൌ , ࢞ ࢞ ࢞ ൌ , ࢞ ࢞ ࢞ ൌ , ࢞ ࢞ ࢞ ൌ , ࢞ ࢞ ࢞ ൌ , ࢞ א ሼ, ሽ. Now, using the step 5 of the proposed method, we have the following reduced intuitionistic fuzzy assignment table Now, using the step 6 of the proposed method, we have the following reduced intuitionistic fuzzy assignment table Now, using step 7 to step 10, we have the following optimum assignment table The optimal solution is ࢞ כ ൌ ࢞ כ ൌ ࢞ כ ൌ , ࢞ כ ൌ ࢞ כ ൌ ࢞ כ ൌ ࢞ כ ൌ ࢞ כ ൌ ࢞ כ ൌ , With the optimal objective value ℜ൫ࢆ෩ࡵ ൯ ൌ . which represents the optimal total cost. In other words the optimal assignment is ࡹ ՜ ࡶ, ࡹ ՜ ࡶ, ࡹ ՜ ࡶ The intuitionistic fuzzy minimum total cost is calculated as ࢉ ࡵ ࢉ ࡵ ࢉ ࡵ ൌ ሺ, , ૡሻሺ, , ሻ ሺ, , ૡሻሺ, , ૡሻ ࡶ ࡶ ࡶ ࡹ (3,5,8)(1,5,10) 10 (10,15,20) ࡹ 3 (1,3,8) (4,5,7)(0,5,11) ࡹ (1,2,3) (3,5,8)(2,5,9) 6 ࡶ ࡶ ࡶ ࡹ (3,5,8)(1,5,10) (10,10,10)(10,10,10) (10,15,20)(10,15,20) ࡹ (3,3,3)(3,3,3) (1,3,8)(1,3,8) (4,5,7)(0,5,11) ࡹ (1,2,3)(1,2,3) (3,5,8)(2,5,9) (6,6,6)(6,6,6) ࡶ ࡶ ࡶ ࡹ (-5,0,5)(-9,0,9) (2,5,7)(0,5,9) (2,10,17)(0,10,19) ࡹ (0,0,0)(0,0,0) (-2,0,5)(-2,0,5) (1,2,4)(-3,2,8) ࡹ (-2,0,2)(-2,0,2) (0,3,7)(-1,3,8) (3,4,5)(3,4,5) ࡶ ࡶ ࡶ ࡹ (-5,0,5)(-9,0,9) (-3,5,9)(-5,5,11) (-2,8,16)(-8,8,22) ࡹ (0,0,0)(0,0,0) (-7,0,7)(-7,0,7) (-3,0,3)(-11,0,11) ࡹ (-2,0,2)(-2,0,2) (-5,3,9)(-6,3,10) (-1,2,4)(-5,2,8) ࡶ ࡶ ࡶ ࡹ (-5,0,5)(-9,0,9) (-7,3,10)(-13,3,16) (-6,6,17)(-16,6,27) ࡹ (-1,2,4)(-5,2,8) (-7,0,7)(-7,0,7) (-3,0,3)(-11,0,11) ࡹ (-2,0,2)(-2,0,2) (-9,1,10)(-14,1,15) (-5,0,5)(-13,0,13)
- 7. P. Senthil Kumar and R. Jahir Hussain/ Elixir Appl. Math. 73 (2014) 25971-2597725976 ሺ, , ሻሺ, , ሻ= ሺ, , ሻሺૡ, , ሻ Also we find that ℜ൫ࢆ෩ࡵכ ൯ ൌ ℜሾሺ, , ሻሺૡ, , ሻሿ ൌ .ܛ܀ . In the above example it has been shown that the total optimal cost obtained by our method remains same as that obtained by converting the total intuitionistic fuzzy cost by applying the ranking method Annie Varghese and Sunny Kuriakose [4]. Results and discussion The minimum total intuitionistic fuzzy assignment cost is ࢆ෩I = ሺ, , ሻሺૡ, , ሻ ………… (1) Figure 1 Graphical Representation of IFAC The result in (1) can be explained (Refer to figure1) as follows: (i) Assignment cost lies in [10,22]. (ii) 100% expect are in favour that an assignment cost is 14 as ࣆࢆ෩ࡵ ሺ࢞ሻ ൌ ,࢞ ൌ . (iii) Assuming that ࣆ is a membership value and ࣖ is a non-membership value at c. Then ࣆ% experts are in favour and ࣖ% experts are opposing but ሺ െ ࣆ െ ࣖሻ% are in confusion that an assignment cost is ࢉ. Values of ࣆࢆ෩ࡵ ሺࢉሻ and ࣖࢆ෩ࡵ ሺࢉሻ at different values of c can be determined using equations given below. ߤሺxሻ ൌ ە ۖ ۖ ۔ ۖ ۖ ۓ 0 ݂ݎ ݔ ൏ 10 ݔ െ 10 4 ݂ݎ 10 ݔ 14 1 ݂ݎ ݔ ൌ 14 22 െ ݔ 8 ݂ݎ 14 ݔ 22 0 ݂ݎ ݔ 22 ߴሺxሻ ൌ ە ۖ ۖ ۔ ۖۖ ۓ 1 ݂ݎ ݔ ൏ 8 14 െ ݔ 6 ݂ݎ 8 ݔ 14 0 ݂ݎ ݔ ൌ 14 ݔ െ 14 10 ݂ݎ 14 ݔ 24 1 ݂ݎ ݔ 24 Conclusion In this paper, Assignment problem with crisp, fuzzy and intuitionistic fuzzy numbers as cost coefficients is discussed. The proposed method is a systematic approach for solving an assignment problem under mixed intuitionistic fuzzy environment. The total optimal cost obtained by our method remains same as that obtained by converting the total intuitionistic fuzzy cost by applying the ranking method of Annie Varghese and Sunny Kuriakose[4].Also the membership and non-membership values of an intuitionistic fuzzy costs are derived. This technique can also be used in solving other types of problems like, project schedules, transportation problems and network flow problems. References [1] Amit Kumar and Anila Gupta, Assignment and Travelling Salesman Problems with Coefficients as LR Fuzzy Parameters, International Journal of Applied Science and Engineering 2012.10,3:155-170. [2] Amit Kumar, Amarpreet Kaur, Anila Gupta, Fuzzy Linear Programming Approach for Solving Fuzzy Transportation problems with Transshipment, J Math Model Algor (2011) 10:163-180. [3] Amit Kumar, Anila Gupta and Amarpreet Kumar,Method for Solving Fully Fuzzy Assignment Problems Using Triangular Fuzzy Numbers, International Journal of Computer and Information Engineering 3:4 2009.
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