METHOD FOR SOLVING FUZZY ASSIGNMENT PROBLEM USING ONES ASSIGNMENT AND ROBUSTS RANKING TECHNIQUEProject report submitted in partial fulfilment of the requirement of the SKILL BASED ELECTIVE COURSE-VI for the Degree of Bachelor of Science in MATHEMATICS. Submitted by1. F. Alfina 11M60022. J. Amudha 11M60033. C. Angel 11M60044. J. Banu Priya 11M60125. K. Irfana Afrin 11M60266. A. Joselin 11M60337. S. Martina Rani 11M60428. R. Rajathy 11M60549. C.Vincy Delfin 11M6069

PG AND RESEARCH DEPARTMENT OF MATHEMATICS HOLY CROSS COLLEGE (AUTONOMOUS) Affiliated to Bharathidasan University Nationally Re-accredited with A Grade By NAAC College with Potential for Excellence TIRUCHIRAPALLI-620 002 . April - 2014

Mrs. Lilly Robert, M.Sc., M.Phil., Ph.D.,Associate proffessor,PG and Research Department of MathematicsHoly cross College (Autonomous)Tiruchirappalli-620 002. CERTIFICATE This is to certify that the project work done under my guidance and the project report entitled A FUZZY ASSIGNMENT PROBLEM USING ONES ASSIGNMENT METHOD AND ROBUSTS RANKING TECHNIQUE. Submiitted by1. F. Alfina 11M60022. J. Amudha 11M60033. C. Angel 11M60044. J. Banu Priya 11M60125. K. Irfana Afrin 11M60266. A. joselin 11M60337. S. Martina Rani 11M60428. R. Rajathy 11M60549. C. Vincy Delfin 11M6069in partial fulfilment of the requirements of the Skill-Based Elective course VIfor the award of the Degree of Bachelor of Science in Mathematics for the academic year 2011-2014 is the original work of the candidates with the Specified Register Numbers1. 2. Signature of the Guide .3. 4.5.6. Signature of the External Examiner.7.8.9. Signaturer of the candidates. signature of the Head of the Department. Date :

ACKNOWLEDGEMENT NOTHING IS MORE HONOURABLE THAN GRATEFUL HEART

We are glad indeed to record or gratitude to ALMIGHTY FATHER, who has been an ineffable of strength and inspiriting in completing our project.

We are really grateful to Rev. Dr. Sr. JEUSIN FRANCIS, M.A., Ph.D., Principal , Holy Cross College (Autonomous), Tiruchirappalli, f or giving us a chance to be a member of this family and also rendering consistent help and innovation ideas to the PG and Research Department of Mathematics.

We thank with gratitude Miss MAHESWARI, M.Sc., M.Phil., Associate Professor , Head of the PG and Research Department of Mathematics for her constant encouragement, criticism and advice which kinded us to put our best in this project as well throughout the course. We take pleasure in Acknowledging our thanks to our guide Mrs. LILLY ROBERT, M.Sc., M.Phil., Ph.D., Associate Professor , PG and Research Department of Mathematics , who spent her valuable time amidst her busy schedule to make us understand and materialize this project.

Finally our most faithful thanks to our parents and we dedicated this project to them.

ABSTRACT

In this paper Ones Assignment Method is adopted to solve Fuzzy Assignment Problem(FLP). In this problem Cij denotes the cost for assigning the n jobs to the n workers and Cij has been considered to be triangular and trapezoidal number denoted by Cij which are more realistic and general in nature. For finding the optimal assignment ,we must optimize total cost this problem assignment. In this paper first the proposed fuzzy assignment problem is formulated to the crisp assignment problem in the linear programming problem (LPP) form and solved by Using Ones Assignment Method and using Robusts Raking Method for the fuzzy numbers . Numerical examples show that the Fuzzy ranking method offers and effective tool for handling the Fuzzy Assignment Problem (FAP) with imprecise render and requirement condition. The algorithm of this approach is presented , and explained briefly with numerical instance to show its efficiency.. KEY WORDS: Assignment problem, Ones Assignment Algorithm, Fuzzy number , Robusts ranking method.

CONTENTS SI.NO

TITLEPAGE NO.

1

INTRODUCTION

2

LITERATURE REVIEW

3

CHAPTER-IBASIC DEFINITIONS OF FUZZY SETS,FUZZY MEMBERS, -CUTS

4

CHAPTER-IIROBUSTS RANKING METHOD

5

CHAPTER-IIIUSING ONES ASSIGNMENT METHOD

6CHAPTER-IVNMERICAL EXAMPLE

7CONTRIBUTION TO THE SOCIETY

8CONCLUSION

9BIBLIOGRAPHY

INTRODUCTION

Fuzzy sets introduced by Zadeh in 1965.Provide us a new mathematical tool to deal with uncertainty of information. since then, fuzzy set theory has been rapidly developed. In this paper we will review basic concepts of fuzzy sets ,fuzzy number. The assignment problem is a special type of linear programming problem in which our objective is to assign a number of origins to the equal number of destinations at a minimum cost (maximum profit).The mathematical formulation of the problems suggest that this is a 0-1 programming problem and highly degenerate all the algorithms developed to find optimal solutions of assignment problem. However due to its highly degeneracy nature a specially designed algorithm, known as Ones Assignment Method. In this paper, we provided a method to solve Fuzzy Assignment Problem (FAP) with Fuzzy cost Cij. Since the objectives are to minimize the total cost or to maximize the total profit, subject to some crisp constraints, the objective function is considered also as a Fuzzy number. First to rank the objective vales of the objective function by Robusts ranking method for transform the Fuzzy Assignment problem to a Crisp one so that the conventional solution methods may be applied to solve assignment problem. This idea is to transforms a problem with Fuzzy parameters to a Crisp version ad solve it by the One Assignment Method.

LITERATURE REVIEW

CHAPTER-I BASIC DEFINITIONS OF FUZZY SETS, FUZZY NUMBERS AND -CUTS 1: DEFINITIONS:1.1.1: Fuzzy set: Let R be the space of real numbers. A fuzzy set is a continuous function (x) is called membership function of the fuzzy set.

1.1.2: Membership Function: The characteristic function A of a crisp set A X assign a value either 0 or 1 to each member in x. This function can be generalised to a function such that the value assigned to the element of the universal set X fall within a specified range . i.e :X[0,1]. The assigned value indicates the membership grade of the element in the set A. The function is called the membership function.

1.1.3: Normal Fuzzy Set: A fuzzy set A of the universe of discourse X is called a Normal Fuzzy set implying that there exists atleast one x X. Such that A (X) = 1.

1.1.4 : Convex Fuzzy Set: A convex fuzzy set , is a fuzzy set in which x, y R, [0,1]. (x + (1-) y) min [ (x), (y)]

1.1.5: Positive fuzzy set: A fuzzy set is called positive if its membership function is such that (x) 0, x0.

1.1.6: Triangular fuzzy number: For a triangular fuzzy number A(x), it can be represented by A(a,b,c;1) with membership function (x) given by

(x) =

W

G2

w G1 G3 O a b c X The triangular fuzzy number = (a, b, c) is called positive triangular fuzzy number if: 0 a b c.

1.1.7: Trapezoidal fuzzy number: For a trapezoidal fuzzy number A(x),it can be represented by A(a,b,c,d;1) with membership function (x) given by(x) = For convenience , TFN represented by four real parameters a,b,c,d (a b c d) will be denoted by a tetraploid.

1 (x)

x a b c d A trapezoidal fuzzy number = (a,b,c,d) is called positive trapezoidal fuzzy number if: 0a b c d

1.1.8: cut: The - cut of a fuzzy number A(x) , is defined as A() = {x/(x) , [0,1]}. 1.2 Operations of Trapezoidal Fuzzy Numbers:

The following are the operations in Trapezoidal fuzzy numbers.Let = (a1,b1,c1.d1) and =(a2,b2,c2,d2) be any two TFNs, then:

1.2.1 Fuzzy Addition : = (a1 + a2, b1 + b2, c1 + c2, d1 + d2)

1.2.2 Fuzzy Difference : = (a1 d2, b1 c2, c1 b2, d1 a2 )

1.2.3 Maximum : (1,) = (max(a1 ,a2),max(b1,b2),max(c1,c2),max(d1,d2))

1.2.4 Minimum : (1,) = (min(a1,a2),min(b1,b2),min(c1,c2),min(d1,d2))

1.3 Robusts Ranking Technique: Robust ranking technique which satisfy compensation, linearity, and additively properties and provides results which consist human intuition. If is a fuzzy number then the Robust Ranking is defined as R() =Where is the - level cut of the fuzzy number .

CHAPTER-II ROBUSTS RANKING TECHNIQUEALGORITHMS: The assignment problem can be stated in the form of nn cost matrix [aij] of real numbers as given in the following

Job1 Job2 Job 3 ..... Job jJob N

Person 1 a11 a12 a13 ....... a1j a1n

Person2 ....... a2j a2n

Person i ........ ........ ai1 ........ ........ ai2 ........ ........ ai3 ........ ........ ......... aij ............... ain

Person N an1 an2 an3 .......... anj ann

Mathematically assignment problem can be stated as Minimize Z = Subject to , xij [0,1] (1)Where xij =

is the decision variable denoting the assignment of the person i to job j. is the cost of assigning the jog to the person. The objective is to minimize the total cost of assigning all the jobs to the

available persons. (One job to one person).When the costs or time are fuzzy numbers, then the total cost becomes a fuzzy number. =

Hence it cannot be minimized directly. For solving the problem we defuzzify the fuzzy cost coeffients into crisp ones by a fuzzy number ranking method.

Robusts ranking technique which satisfies compensation, linearity, and additivity properties and provides results which are consistent with human intuition. Give a convex fuzzy number , the Robusts Ranking Index is defined by

R() = 0.5 (, ) d , where (, ) is the level cut of the fuzzy number . In this paper we use this method for ranking the objective values. The Robusts ranking index R() gives the representative value of the fuzzy number . It satisfies the linearity and additive property:

If = + m and = k - t, where , m, k and t are constants, then we have R() = R() + mR() and R() = k R() t R().

On the basis of this property the fuzzy assignment problem can be transformed into a crisp assignment problem in Linear Programming Problem form.

The ranking Technique of the Robust is If R() R(), then i.e., min {, } = For the assignment problem (1) , with fuzzy objective function Min = we apply Robusts ranking method (using the linearity and assisociative property) to get the minimum objective value * from the formulation R(*) = Subject to , xij [0,1] (2) Where xij = is the decision variable denoting the assignment of the person i to j. is the cost of designing the job to the person. The objective is to minimize the total cost of assigning all the jobs to the available persons. Since R() are crisp values, this problem (2) is obviously the crisp assignment problem of the form (1) which can be solved by the conventional methods, namely the Hungarian Method or Simplex method to solve the Linear Programming problem form of the problem. Once the optimal solution x* of model (2) is found, the optimal fuzzy objective value * of the original problem can be calculated as

=

CHAPTER III USING ONES ASSIGNMENT METHOD 3.1 Mathematical formulation of assignment problem: Mathematically an assignment problem can be stated as follows: Optimize (1) subject to = 1, i =1 ,,n =1, j =1 ,,n (2) xij =0 or 1, i =1 ,,n , j=1 ,,n. where is the cost or eectiveness of assigning job to machine, and is to be some positive integer or zero, and the only possible integer is one, so the condition of = 0 or 1, is automatically satised. Associated to each assignment problem there is a matrix called cost or eectiveness matrix [] where is the cost of assigning job to facility. In this paper we call it assignment matrix, and represent it as follows: 1 2 3 ...... n 1 c11 c12 c13 ....... c1n 2 c21 c22 c23 ....... c2n 3 c31 c32 c33 ...... c3n : : : : : : : : : : : : n cn1 cn2 cn3 ...... cnn

which is always a square matrix, thus each task can be assigned to only one machine. In fact any solution of this assignment problem will contain exactly m non-zero positive individual allocations. A customary and convenient method , termed as assignment algorithm has been developed for such problems. This iterative method is known as Hungarian assignment method. It is based on add or subtract a constant to every element of a row or column of the cost matrix in a minimization model , and create some zeros in the given cost matrix and then try to nd a complete assignment in terms of zeros. In fact our aim is to create ones in place of zeroes, and try to assign them in our problem.3.2 A new approach for solving assignment problem: This section presents a new method to solve the assignment problem which is dierent from the preceding method. We call it one- assignment method, because of making assignment in terms of ones. The new method is based on creating some ones in the assignment matrix and then try to nd a complete assignment in terms of ones. By a complete assignment we mean an assignment plan containing exactly m assigned independent ones, one in each row and one in each column. Now, consider the assignment matrix where is the cost or eectiveness of assigning job tomachine. 1 2 3 ...... n 1 c11 c12 c13 ....... c1n 2 c21 c22 c23 ....... c2n 3 c31 c32 c33 ...... c3n : : : : : : : : : : : : n cn1 cn2 cn3 ...... cnn

3.2.1 ONES ASSIGNMENT ALGORITHM: let (1-2) be an assignment problem in which the objective function can be minimized or maximized.step 1: In a minimization (maximization) case, nd the minimum (Maximum) element of each row in the assignment matrix (say ) and write it on the right hand side of the matrix. 1 2 3 ...... n c11 c12 c13 ....... c1n a1 c21 c22 c23 ....... c2n a2 c31 c32 c33 ...... c3n a3 : : : : : : : : : : : : cn1 cn2 cn3 ...... cnn an

Then divide each element of row of the matrix by . These operations create at least one ones in each rows. In term of ones for each row and column do assignment, otherwise go to step 2. step 2. Find the minimum(maximum) element of each column in assignment matrix (say), and write it belowcolumn. Then divide each element of jth column of the matrix by bj. These operations create at least one ones in each columns. Make assignment in terms of ones. If no feasible assignment can be achieved from step (1) and (2) then go to step 3.

1 2 3 ...... n 1 c11/a1 c12/a1 c13/a1 ....... c1n/a1 2 c21/a2 c22/a2 c23/a2 ....... c2n/a2 3 c31/a3 c32/a3 c33/a3 ...... c3n/a3 : : : : : : : : : : : : : : : : n cn1/an cn2/an cn3/an ...... cnn/an b1 b2 b3 ...... bnNote: In a maximization case, the end of step 2 we have a fuzzy matrix, which all elements are belong to [0,1], and the greatest element is one. step 3. Draw the minimum number of lines to cover all the ones of the matrix. If the number of drawn lines less than n, then the complete assignment is not possible, while if the number of lines is exactly equal to n, then the complete assignment is obtained.step 4. If a complete assignment program is not possible in step 3, then select the smallest (largest) element (say ) out of those which do not lie on any of the lines in the above matrix. Then divide by each element of the uncovered rows or columns, which lies on it. This operation create some new ones to this row or column. If still a complete optimal assignment is not achieved in this new matrix, then use step 4 and 3 iteratively. By repeating the same procedure the optimal assignment will be obtained. Priority plays an important role in this method, When we want to assign the ones. Priority rule, For maximization (minimization) assignment problem, assign the ones on the rows which have smallest (greatest) element on the right hand side, respectively.

CHAPTER- IV Numerical Example Let us consider a fuzzy assignment problem with rows representing and four workers A, B, C and D and columns representing jobs, job1, job2, job3, job4. The cost matrix is given whose elements are triangular fuzzy numbers and trapezoidal fuzzy numbers. The problem is to find the optimal assignment so that the total cost of job assignment becomes minimum. (10,20,30) (10,20,40) (10,30,40) (10,20,30) [] = (10,20,40) (10,30,40) (10,20,30) (20,30,40) (10,20,40) (20,30,40) (20,30,40) (10,20,30) (20,30,50) (10,20,30) (20,40,60) (20,30,50) Solution for triangular fuzzy number: The fuzzy assignment problem can be formulated in the following mathematical programming form.

Min{R(10,20,30) x11+ R(10,20,40) x12 +R(10,30,40) x13+ R(10,20,30) x14+ R(10,20,40) x21+ R(10,30,40) x22 + R(10,20,30) x23+ R(20,30,40) x24+ R(10,20,40) x31+ R(20,30,40) x32+ R(20,30,40) x33+ R(10,20,30) x34+ R(20,30,50) x41+ R(10,20,30) x42+ R(20,40,60) x43+R(20,30,50) x44 }.

Subject to , x11+x12+x13+x14 =1 x11+x21+x31+x41 =1 x21+x22+x23+x24 =1 (1) x12+x22+x33+x42 =1 x31+x32+x33+x34 =1 x13+x23+x33+x43 =1 x41+x42+x43+x44 =1 x14+x24+x34+x44 =1 xij [0,1]. Now we calculate the R(10,20,30) by applying Robusts Ranking method The membership function of the triangular fuzzy numbers (10,20,30) is (x) = The -cut of the fuzzy number (10,20,30) is (, ) = (10 +10, 30-10) for which R[] = (, ) d = d = 20 Proceeding similarly ,the Robusts Ranking indices for the fuzzy costs are calculated as: R(c12) = 22.5 ; R(c13) = 27.5 ; R(c14) = 20 ; R(c21) = 22.5 ; R(c22) = 27.5 ; R(c23) = 20 ; R(c24) = 30 ; R(c31) = 22.5 ; R(c32) = 30 ; R(c34) = 20 ; R(c41) = 32.5 ; R(c42) = 20 ; R(c43) = 40 ; R(c44) = 32.5 ;

We replace these values for their corresponding in (1)Which results in a convenient assignment problem in the linear programming problem we solve it by hngarian method to get the following optmal solution

=

with the optimal objective value R = 80

the optimal assignment A 1 , B 3, C 4, D 2

The optimal solution R = 20+20+20+20 = 80 The fuzzy optimal total cost =+

= R(10,20,30)+R(10,20,30)+R(10,20,30)+R(10,20,30)

= R(40,80,120) Also we find that R = 80

Solution for trapezoidal fuzzy number:

The fuzzy assignment problem can be formulated in the following mathematical programming form.

Min{R(10,20,30) x11+ R(10,20,40) x12 +R(10,30,40) x13+ R(10,20,30) x14+ R(10,20,40) x21+ R(10,30,40) x22 + R(10,20,30) x23+ R(20,30,40) x24+ R(10,20,40) x31+ R(20,30,40) x32+ R(20,30,40) x33+ R(10,20,30) x34+ R(20,30,50) x41+ R(10,20,30) x42+ R(20,40,60) x43+R(20,30,50) x44 }.

Subject to , x11+x12+x13+x14 =1 x11+x21+x31+x41 =1 x21+x22+x23+x24 =1 (1) x12+x22+x33+x42 =1 x31+x32+x33+x34 =1 x13+x23+x33+x43 =1 x41+x42+x43+x44 =1 x14+x24+x34+x44 =1 xij [0,1]. Now we calculate the R(10,20,30) by applying Robusts Ranking method The membership function of the trapezoidal fuzzy numbers (10,20,30) is

(x) =

The -cut of the fuzzy number (10,20,30) is (, ) = (10 +10, 30-10) for which R[] = (, ) d

= d

= 20 Proceeding similarly ,the Robusts Ranking indices for the fuzzy costs are calculated as: R(c12) = 22.5 ; R(c13) = 27.5 ; R(c14) = 20 ; R(c21) = 22.5 ; R(c22) = 27.5 ; R(c23) = 20 ; R(c24) = 30 ; R(c31) = 22.5 ; R(c32) = 30 ; R(c34) = 20 ; R(c41) = 32.5 ; R(c42) = 20 ; R(c43) = 40 ; R(c44) = 32.5 ;

We replace these values for their corresponding in (1)Which results in a convenient assignment problem in the linear programming problem we solve it by hungarian method to get the following optimal solution

=

with the optimal objective value R = 80 which represents a optimal assignment is A 1 , B 3, C 4, D 2

The fuzzy optimal total cost =+

= R(10,20,30)+R(10,20,30)+R(10,20,30)+R(10,20,30)

= R(40,80,120) Also we find that R = 80

2. Consider the following assignment problem using ones assignment problem by assign the five jobs o the three machines so as to minimize the total cost. 1 2 3 4 5

1 12 8 7 15 4 2 7 9 1 14 10 3 9 6 12 6 7 4 7 6 14 6 10 5 9 6 12 10 6 Solution: Find the minimum element of each row in the assignment matrix (say ) and write it on the right hand side of the matrix, as follows: 1 2 3 4 5 min 1 12 8 7 15 4 4 2 7 9 1 14 10 1 3 9 6 12 6 7 6 4 7 6 14 6 10 6 5 9 6 12 10 6 6 Then divide each element of ith row of the matrix by ai. These operations create ones to each rows ,and the matrix reduces in following matrix. 1 2 3 4 5 min 1 3 2 7/4 15/4 1 4 2 7 9 1 14 10 1 3 3/2 1 2 1 7/6 6 4 7/6 1 7/3 1 5/3 6 5 3/2 1 2 5/3 1 6 Now find the minimum element of each column is assignment matrix (say ) and write it below that column .then divide each element of column of the matrix by 1 2 3 4 5 min 1 3 2 7/4 15/4 1 4 2 7 9 1 14 10 1 3 3/2 1 2 1 7/6 6 4 7/6 1 7/3 1 5/3 6 5 3/2 1 2 5/3 1 6 Min 7/6 1 1 1 1 The minimum number of lines required to pass through all the ones of the matrix is 5. 1 2 3 4 5 1 18/7 2 7/4 5/4 1 2 6 9 1 14 10 3 18/4 1 2 1 7/6 4 1 1 7/3 1 5/3 5 18/4 1 2 5/3 1 so the complete assignment is possible. 1 2 3 4 5 1 18/7 2 7/4 5/4 2 6 9 1 14 10 3 18/4 1 2 1 7/6

5 18/4 1 2 5/3 1

CONTRIBTION TO THE SOCIETY

Fuzzy set theory has numerous applications in various fields. They are artificial, intelligence, automata theory, computer science, control theory, decision making, expert systems, medical diagnosis,Neural networks, pattern recognition, robotics, social sciences and etc.... Parents dream that they their children will work to benefit to the society. While every parent may not anticipate raising a doctor or lawyer, the desire for the child to contribute to society to is strong. Children with special needs may make a different in many ways and to many people. Including family, community, job and etc..... Medical decisions (medication, procedures, and surgeries, and medical equipment) can be large part of the responsibility of parents with a child who has special needs. The purpose of this section is not to influence medical decisions or to make medical recommendations. rather it is intented to offer some questions that may be useful for discussion with your child doctors.

COCLUSION

In this paper, a simple yet effective method was introduced to solve fuzzy assignment problem by using ranking of Fuzzy numbers. This method can be used for all kinds of fuzzy assignment problem, whether triangular and trapezoidal fuzzy numbers. The new method is a systematic procedure, easy to apply and can be utilized for all type of assignment problem whether maximize or minimize objective function.

BIBLIOGRAPHY:

1. Hadi Basirzadeh ones assignment method for solving assignment problems (2012).2. A.Solairaju and R.Nagarajan computing improved fuzzy optimal Hungarian assignment problems with fuzzy costs under robust ranking techniques(2010).3. Kadhirvel .k,Balamurugan .k.method for solving Hungarian assignment problem using triangular and trapezoidal fuzzy number(2012). 4. S.H Chen, Ranking fuzzy numbers with maximizing set and minimizing set,fuzzy sets and systems.5. X.Wang, fuzzy optimal assignment problem,fuzzy math-3(1987).