Prabhu-Fuzzy Hungarian Method for Solving Assignment Problem Involving Trapezoidal Fuzzy Numbers

7/29/2019 Prabhu-Fuzzy Hungarian Method for Solving Assignment Problem Involving Trapezoidal Fuzzy Numbers

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Indian Journal of Science and Technology Vol. x No. x (xxx. 201x) ISSN: 0974- 6846

Fuzzy Hungarian Method for Solving Assignment Problem

involving trapezoidal fuzzy numbers

K. Prabakaran and K. GanesanDepartment of Mathematics, Faculty of Engineering and Technology,

SRM University, Kattankulathur, Chennai - 603203, INDIA.

Email: [email protected], [email protected], [email protected]

AbstractIn this paper we propose a new approach for the fuzzy optimal solution of assignment problems whose decision

parameters are trapezoidal fuzzy numbers. We develop a fuzzy version of Hungarian algorithm for the solution

of fuzzy assignment problems involving trapezoidal fuzzy numbers without converting them to classical

assignment problems. The proposed method is easy to understand and to apply for finding solution of fuzzy

assignment problems occurring in real life situations. To illustrate the proposed method numerical examples are

provided and the obtained results are discussed.

Keywords: Fuzzy numbers, Trapezoidal Fuzzy numbers, Fuzzy Assignment Problem, Fuzzy Hungarian Method.

1. INTRODUCTION

An assignment problem is a special type of linear programming problem which deals with assigning various

activities (jobs or tasks or sources) to an equal number of service facilities (men, machine, laborers etc) on one

to one basis in such a way so that the total time or total cost involved is minimized and total sale or total profit is

maximized or the total satisfaction of the group is maximized. It is well known that Assignment problems play

major role in various areas such as science, engineering and technology, social sciences and many others.

Examples of these types of problems may be the case of assigning men to machines, men to offices, drivers and

conductors to buses, trucks to delivery routes etc. In order to solve an assignment problem, the decision

parameters of the model must be fixed at crisp values. But to model real-life problems and perform

computations we must deal with uncertainty and inexactness. These uncertainty and inexactness are due to

measurement inaccuracy, simplification of physical models, variations of the parameters of the system,

computational errors etc. Also the decision parameters like time / cost for doing an activity by a service facility(machine / person) might vary due to different reasons. Consequently, we cannot successfully use traditional

classical assignment problems and hence the use of fuzzy assignment problems is more appropriate.

In 1965 Zadeh [28] introduced the concept of fuzzy sets to deal with imprecision, vagueness in real life

situations. In 1970 Bellman and Zadeh [4] proposed the concept of decision making under fuzzy environments.

Since then, tremendous development of numerous methodologies and their applications to various decision

problems under fuzzy environment have been proposed. Assignment problems with fuzzy parameters have

been studied by several authors, such as Balinski and Gomory [3], Chanas et al [5], Chi-Jen Lin and Ue-pyng

Wen [8], Dubois and Fortemps [10], Chen [7], Ganesan and Veeramani [11], Kuhn [12], Liu and Gao [15],

Manimaran and Ananthanarayanan [16], Majumdar and Bhunia [17], Pandian and Natarajan [19],Shanmugasundari and Ganesan [23], Sathi Mukherjee and Kajla Basu [22] etc. Chen [7] proved sometheorems and proposed a fuzzy assignment model that considers all individuals to have same skills. Wang [25]

solved a fuzzy assignment model using graph theory. Lin and Wen [13] investigated a fuzzy assignment problem

in which the cost depends on the quality of the job. Mukherjee and Basu [18] proposed a new method forsolving fuzzy assignment problems. Amit Kumar and Anila Gupta [2] proposed two new methods for solving

fuzzy assignment problems and fuzzy travelling salesman problems. Dubois and Fortemps [10] proposed a

flexible assignment problem, which combines with fuzzy theory, multiple criteria decision-making and

constraint-directed methodology. Long sheng Huang and Guang-hui Xu [14] proposed a solution procedure for

the assignment problems with restriction of qualification. In general, they have transformed the fuzzy

assignment problems into one or a series of classical assignment problems and then obtained an optimalsolution. But in this paper, we develop a Hungarian like fuzzy algorithm for solving fuzzy assignment problems

without converting them to classical assignment problems.


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The rest of this paper is organized as follows: In section 2, we recall the definition of a new type of arithmetic

operations, a linear order relation on trapezoidal fuzzy numbers and some related results. In section 3, we define

fuzzy assignment problem as an extension of the classical assignment problem and propose fuzzy Hungarian

algorithm. In section 4, numerical examples are provided and the obtained results are discussed.

2. PRELIMINARIES

The aim of this section is to present some notations, notions and results which are of useful in our further study.

Definition 2.1 A fuzzy set a defined on the set of real numbers R is said to be a fuzzy number if itsmembership function a : R [0,1] has the following

(i) a is convex, that is { } { }1 2 1 2 1 2a x +(1-)x = min a(x ),a(x ) , for all x , x R and [0,1]

(ii) a is normal i.e. there exists an x R such that ( )a x 1=

(iii) a is piecewise continuous.

Definition 2.2 A fuzzy number a in R is said to be a trapezoidal fuzzy number if its membership function

a : X [0,1] has the following characteristics:

( )

11 2

2 1

2 3a

44

4 3

x - a, a x a

a - a

1, a x axa - x

, a x a3a - a

0, otherwise

=

We denote this trapezoidal fuzzy number by ( )1 2 3 4a = a , a , a , a . We use F(R) to denote the set of all

trapezoidal fuzzy numbers.

2.1 Arithmetic Operations on trapezoidal fuzzy numbersFor any two trapezoidal fuzzy numbers

( ) ( )1 2 3 4 2 3 1 1a a , a , a , a (core of (a), left spread of (a), right spread of (a)) [a , a ], ,= = = and

( ) ( )1 2 3 4 2 3 2 2b b , b , b , b (core (b), left spread of (b), right spread of (b)) [b , b ], ,= = = and for { }, , , , = +

the arithmetic operations on a and b are defined as

( ){ }

( ){ }

1 2 1 2

2 3 2 3 1 2 1 2 2 3 2 3

a b core (a) core (b), max{ , }, max{ , } : core (a) a and core (b) b

[a ,a ] [b ,b ] , max{ , }, max{ , } : [a ,a ] a and [b ,b ] b

=

=

In particular,

( ) ( )

( )

1 2 1 2 2 3 2 3 1 2 1 2

2 2 3 3 1 2 1 2

a b core (a) core( b), max{ , }, max{ , } [a ,a ] [b ,b ] , max{ , }, max{ , }

[a b , a b ], max{ , }, max{ , }

+ = + = +

= + +

(i) Addition:

( ) ( )

( )

1 2 1 2 2 3 2 3 1 2 1 2

2 2 3 3 1 2 1 2

a b core (a) core (b), max{ , }, max{ , } [a ,a ] [b ,b ] , max{ , }, max{ , }

[a b , a b ], max{ , }, max{ , }

= =

=

(ii) Subtraction :

1 2 3 4a a a a

Figure. 1 Trapezoidal Fuzzy Number

( )a x


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1 1

1 1

( core (a), , ), 0a

( core (a), , ), 0

=

0, then ( ) ( )1 2 3 4 2 3 1 1a a ,a ,a , a [a ,a ], ,= = is

said to be a positive trapezoidal fuzzy number and is denoted by a 0 . Further if Mag(a) 0,= then a is said to

be a zero trapezoidal fuzzy number and is denoted bya 0.

Definition 2.5 Two trapezoidal fuzzy numbers ( ) ( )1 2 3 4 2 3 1 1a a ,a ,a , a [a ,a ], ,= = and

( ) ( )1 2 3 4 2 3 2 2b b ,b ,b ,b [b ,b ], ,= = in F(R) are said to be equivalent if and only if Mag (a ) = Mag ( b ). That

is a b if and only if Mag(a) = Mag(b). Two trapezoidal fuzzy numbers ( ) ( )1 2 3 4 2 3 1 1a a ,a ,a , a [a ,a ], ,= =

and ( ) ( )1 2 3 4 2 3 2 2b b ,b ,b ,b [b , b ], ,= = in F(R) are said to be equal if and only if core(a) core(b),=

1 2 1 2, = = or 2 3 2 3[a ,a ] [b ,b ],= 1 2 1 2, = = and is denoted by a b.= Note that 2 3[a ,a ] [0,0],=

1 10, 0 = = if and only if a 0 [0,0,0,0].= =

Definition 2.6 Let i{a , i 1,2, ...,n}= be a set of trapezoidal fuzzy numbers. If k iMag(a ) Mag(a ) , for all i,

then the fuzzy number ka is the minimum of i{a , i 1,2 ,...,n}.=

Definition 2.7 Let i{a , i 1,2, ...,n}= be a set of trapezoidal fuzzy numbers. If t iMag(a ) Mag(a ) , for all i, then

the fuzzy numberta is the minimum of i{a , i 1,2, ...,n}.=

3. MAIN RESULTS

Suppose there are n activities (jobs or tasks or sources) to be performed and n service facilities (men, machine,laborers etc) are available for doing these activities. Assume that each service facility can perform one

activity at a time. The objective of the problem is to assign these activities to the service facilities on one toone basis in such a way so that the total time or total cost involved is minimized and total sale or total profit is


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maximized or the total satisfaction of the group is maximized.

3.1 Mathematical Model of Fuzzy Assignment Problem

Let the ith

person is assigned to the jth

job and is denoted by ijx and ijc be the corresponding fuzzy cost of

assigning the ith

person to the jth

job. Since the objective is to minimize the overall fuzzy cost forperforming all jobs, the mathematical model of this fuzzy assignment problem is as follows:

n n

ij iji=1j=1

n

iji=1

n

ijj=1

ij

Minimize z = c x

subject to x 1, j = 1, 2, ..., n

x 1, i = 1, 2, ..., n

x 0 or 1, j = 1, 2, ..., n

th th

ij th th

1, if the i person is assigned to j jobwhere x

0, if the i person is not assigned to j job

This fuzzy Assignment problem can be stated in the form of (n n) fuzzy cost matrix ij[c ] of real numbers as

given in the following table:

Table 1: Fuzzy cost matrix of fuzzy assignment problem

The cost or time ijc are fuzzy numbers(1) (2) (3) (4)

ij ij ij ij ijc [c ,c ,c ,c ],= is the cost of assigning the jth job to theith person.

3.2 Fuzzy Hungarian method

We, now introduce a new algorithm called the fuzzy Hungarian method for finding a fuzzy optimal assignment

for fuzzy assignment problem.

Step 1: Determine the fuzzy cost table from the given problem. If the number of sources is equal to the number

of destinations go to step 3. If the number of sources is not equal to the number of destinations go to step 2.

Step 2: Add a dummy source or dummy destination, so that the fuzzy cost table becomes a fuzzy square matrix.

The fuzzy cost entries of dummy source/destinations are always fuzzy zero.

Step 3: Subtract the row minimum from each row entry of that row.

Step 4: Subtract the column minimum of the resulting fuzzy Assignment problem after using step 3 from each

column entry of that column.

Each column and row now has at least one fuzzy zero.

Step 5: In the modified fuzzy assignment table obtained in step 4, search for fuzzy optimal assignment as

follows.

(a) Examine the rows successively until a row with a single fuzzy zero is found. Assign the fuzzy zero andcross off all other fuzzy zeros in its column. Continue this for all the rows.

Jobs

persons

1 2 3 - j - n

1 11c 12c

13c

1jc

1nc

2 21c 22c

23c

2 jc 2nc

i i1c i2c i3c ijc inc

n n1c n2c

n3c

njc nnc


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(b) Repeat the procedure for each column of reduced fuzzy assignment table.(c) If a row and / or column have two or more fuzzy zeros assign arbitrary any one of these fuzzy zeros and

cross off all other fuzzy zeros of that row/column. Repeat (a) through (c) above successively until the chain

of assigning or cross ends.

Step 6: If the number of assignments is equal to n, the order of the fuzzy cost matrix, fuzzy optimal solution is

reached. If the number of assignments is less than n, the order of the fuzzy zeros of the fuzzy cost matrix, go to

the step 7.

Step 7: Draw the minimum number of horizontal and / or vertical lines to cover all the fuzzy zeros of the

reduced fuzzy assignment matrix. This can be done by using the following:

(i) Mark rows that do not have any assigned fuzzy zero.(ii) Mark columns that have fuzzy zeros in the marked rows.(iii)Mark rows that do have assigned fuzzy zeros in the marked columns.(iv)Repeat ii) and iii) above until the chain of marking is completed. Draw lines through all the unmarked

rows and marked columns. This gives the desired minimum number of lines.

Step 8: Develop the new revised reduced fuzzy cost matrix as follows:

Find the smallest entry of the reduced fuzzy cost matrix not covered by any of the lines. Subtract this entry from

all the uncovered entries and add the same to all the entries lying at the intersection of any two lines.

Step 9: Repeat step 6 to step 8 until fuzzy optimal solution to the given fuzzy assignment problem is attained.

4. NUMERICAL EXAMPLES

Example 4.1 Consider a fuzzy assignment problem with rows representing three persons P 1, P2, P3 and columns

representing the three jobs J1, J2, J3 discussed in [21]. The cost matrix ijC is given whose elements are

trapezoidal fuzzy numbers. The problem is to find the optimal assignment so that the total cost of job

assignment becomes minimum.

(1,2,3,4) (1,3,4,6) (9,11,12,14)

(0,1,2,4) ( 1,0,1,2) (5,6,7,8)

(3,5,6,8) (5,8,9,12) (12,15,16,19)

The given problem is a balanced one. Since Mag(1,2,3,4), Mag(-1,0,1,2) and Mag(3,5,6,8) is minimum in the 1st,2nd and 3rd row respectively, using step 3 of the fuzzy Hungarian method we obtain

( ) ( )

( ) ( )

( ) ( )

(0) [1,2],2,1 [9,9], 2,2

[0,1],1,2 (0) [6,6],1,1

(0) [3,3],3,3 [10,10],3,3

Using step 4 of the fuzzy Hungarian method we obtain the following modified fuzzy assignment matrix.

( ) ( )

( )

( ) ( )

(0) [1,2], 2,1 [3,3], 2, 2

[0,1],1, 2 (0) (0)

(0) [3,3],3,3 [4, 4],3,3

Now using step 5 to step 7 of the fuzzy Hungarian method, we have the following fuzzy assignment matrix.

( ) ( )

( )

( ) ( )

[ (0) ] [1,2],2,1 [3,3], 2, 2

[0,1],1, 2 [ (0) ] ( 0 )

( 0 ) [3,3],3,3 [4,4],3,3


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Now using the step 8 of the fuzzy Hungarian method and repeating the procedure, we have the following fuzzy

optimal assignment matrix.

( )

( )

( ) ( )

( 0) [ (0) ] [2,1], 2,2

[1,3], 2,2 ( 0 ) [ (0) ]

[ (0) ] [2,1],3,3 [3,2],3,3

Therefore, the fuzzy optimal assignment for the given fuzzy assignment problem is 1 2 2 3 3 1P J ,P J ,P J .

The fuzzy optimal total cost is calculated as = ([3,5],2,1)+([6,7],1,1)+([5,6],2,2)

= (12, 14, 18, 20) Cost units.

Example 4. 2 Consider a fuzzy assignment problem with rows representing 4 persons P1, P2, P3, P4 and columns

representing the 4 jobs J1, J2, J3, J4 discussed in [16]. The cost matrix is given whose elements are trapezoidal

fuzzy numbers. The problem is to find the optimal assignment so that the total number of job assignment

becomes minimum.

(3,5,6,7) (5,8,11,15) (9,10,11,15) (5,8,10,11)

(7,8,10,11) (3,5,6,7) (6,8,10,12) (5,8,9,10)

(2,4,5,6) (5,7,10,11) (8,11,13,15) (4,6,7,10)

(6,8,10,12) (2,5,6,7) (5,7,10,11) (2,4,5,7)

The given problem is a balanced one. Since Mag(3,5,6,7), Mag(3,5,6,7), Mag(2,4,5,6) and Mag(2,4,5,7) is

minimum in the 1st, 2

nd, 3

rdand 4

rdrow respectively, using step 3 of the fuzzy Hungarian method, we obtain

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

(0) [3,5],3,1 [5,5], 2, 4 [3,4],3,1

[3,4],2,1 (0) [3,4], 2, 2 [3,3],3,1

(0) [3,5], 2,1 [7,8],3, 2 [2,2], 2,3

[4,5],2, 2 [1,1],3,2 [3,5], 2,2 (0)

Using step 4 of the fuzzy Hungarian method we obtain the following modified fuzzy assignment matrix.

( ) ( ) ( )

( ) ( )

( ) ( ) ( )

( ) ( ) ( )

(0) [3,5],3,1 [2,1], 2, 4 [3, 4],3,1

[3,4], 2,1 (0) (0) [3,3],3,1

(0) [3,5], 2,1 [4,4],3, 2 [2,2], 2,3

[4,5], 2, 2 [1,1],3, 2 [0,1],2,2 (0)

Now using step 5 to step 7 of the fuzzy Hungarian method, we have the following fuzzy assignment matrix.

( ) ( ) ( )

( ) ( )

( ) ( ) ( )

( ) ( ) ( )

(0) [3,5],3,1 [2,1], 2, 4 [3,4],3,1

[3, 4], 2,1 (0) ( 0 ) [3,3],3,1

( 0 ) [3,5],2,1 [4,4],3, 2 [2, 2], 2,3

[4,5], 2, 2 [1,1],3, 2 [0,1],2, 2 (0)

Now using the step 8 of the fuzzy Hungarian method and repeating the procedure, we have the following fuzzy

optimal assignment matrix.


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( ) ( )

( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( 0 ) [1,4],3,4 (0) [1,3],3, 4

[5,5],2, 4 (0) ( 0 ) [3,3],3,1

(0) [1,4], 2, 4 [2,3],3, 4 [0,1], 2, 4

[6,6], 2, 4 [1,1],3, 2 [0,1], 2, 2 (0)

Therefore, the fuzzy optimal assignment for the given fuzzy assignment problem is

1 2 2 3 3 1 4 4, , , . P J P J P J P J

The fuzzy optimal total cost is calculated as = ([10,11],1,4) + ([5,6],1,1) + ([4,5],2,1) + ([4,5],2,2)

= (21, 23, 27, 31) cost units.

5. CONCLUSIONWe have thus obtained an optimal assignment schedule for a fuzzy assignment problem using trapezoidal fuzzy

number by the proposed new algorithm called fuzzy Hungarian method. It can be seen that the fuzzy optimal

solution to the assignment problem given in example 4.1 is (12, 14, 18, 20) Cost units by the proposed fuzzy

Hungarian method, where as Sagaya Roseline et.al [21] got the fuzzy optimal total cost as (09, 14, 17, 22).

Also the fuzzy optimal solution to the assignment problem given in example 4. 2 is (21, 23, 27, 31) Cost units

by the proposed fuzzy Hungarian method, where as Manimaran et.al [16] got the fuzzy optimal total cost as

(16, 23, 27, 35). From these we see that the proposed fuzzy Hungarian method in this paper gives very sharp

results.ACKNOWLEDGMENT

The authors would like to thank the anonymous reviewers and Editors for their critical comments and valuable

suggestions.

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Prabhu-Fuzzy Hungarian Method for Solving Assignment Problem Involving Trapezoidal Fuzzy Numbers