Cost Minimization Fuzzy Assignment Problem applying Linguistic Variables

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1 Inter national Journal of Pure and Applied Mathematics Volume 113 No , ISSN: (printed version); ISSN: (on-line version) url: ijpam.eu Cost Minimization Fuzzy Assignment Problem applying Linguistic Variables G.Uthra 1 K.Thangavelu 1 R.M.Umamageswari 2 1 P.G and Research Department of Mathematics, Pachaiyappas college, Chennai , India. 2 Jeppiaar Engineering College, Chennai , India. umabalaji30@gmail.com February 9, 2017 Abstract The paper affirms the solution of assignment problem whose decision parameters are generalized triangular fuzzy numbers. The linguistic variables are utilized to transform the qualitative data into quantitative data. A new ranking method is proposed to defuzzify the fuzzy numbers, followed by Hungarian technique for optimal cost. A numerical example is demonstrated to prove the proposed approach. AMS Subject Classification: 03E72 and 90BXX. Key Words and Phrases:Triangular Fuzzy Numbers, Generalized Triangular Fuzzy Numbers, Generalized Fuzzy Assignment Problem and Ranking of Generalized Fuzzy Numbers. 1 INTRODUCTION In real life situations, the data obtained is vague, imprecise and ambiguous. Conventional or crisp logic theory is insufficient to deal with uncertainty, imprecision and complexity of day-to-day practical problems. This inexact and uncertain environment explored the concept of fuzzy logic and fuzzy theory. ijpam.eu

2 Assignment problem is a special type of linear programming problem dealing with assigning various tasks (jobs or sources) to an equal number of service facilities (men or machine) on one to one basis such that total cost or time is minimized and total profit is maximized. Assignment problem plays a major role in various areas of science, engineering and management. In solving assignment problem, the decision parameters of the model are crisp values. But real life problems deals with uncertainty and complexity, which are due to inaccurate measurement, modeling, variations of the parameters, computational errors etc. Under these circumstances, fuzzy assignment problem is more efficient than the classical assignment problem. In 1965, L. A. Zadeh [10] formulated the concept of decision making under fuzzy environment. Over past 5 decades, many authors such as R. E. Bellman and L. A. Zadeh [1], S. H. Chen [2], H. W. Kuhn [5] and W. L. Gao and D. J. Buehrer [3] etc, explored the assignment problem with fuzzy parameters. Due to its tremendous applications recently researchers [4, 6, 9] proposed various techniques for solving generalized fuzzy assignment problem. G. Nirmala and R. Anju [7], K. Ruth Isabels and G. Uthra [8] formulated the application of linguistic variables in fuzzy assignment problem. The new ranking technique practiced in this paper yields optimal cost efficiently. This paper is organized as follows: Section II deals with the basic concepts of generalized triangular fuzzy numbers, Arithmetic addition and new ranking method for generalized triangular fuzzy numbers. In Section III, the fuzzy assignment problem and its mathematical formulation are displayed. Section IV, briefs the proposed algorithm. The numerical example illustrated in section V, reveals the efficiency and effectiveness of the proposed technique. Section VI exposes the concluding remarks. 2 Basic Definition Definition 2.1 Fuzzy Set: ijpam.eu

3 A Fuzzy set à characterized by a membership function mapping elements of a domain, space, or universe of discourse X to the unit interval [0, 1].(i.e) à = {(x, µ à (x)) ; x X}, here µ à : X [0, 1] is a mapping called the degree of membership function of the fuzzy set à and µ à (x) called the membership function value of x Xin the fuzzy set Ã.These membership grades are often represented by real numbers ranging from [0, 1]. Definition 2.2 Triangular fuzzy number: A fuzzy number à = (a, b, c) is said to be a triangular fuzzy number if its membership function is given by (x a), a x b (b a) (c x) µã (x) = b x c, where a, b, c R. (c b) 0, elsewhere Definition 2.3 Generalized Triangular Fuzzy Number: A generalized fuzzy number à = (a, b, c; ω) is said to be generalized triangular fuzzy number [GTFN] if its membership function is given ω (x a), a x b (b a) by µã (x) = ω (c x) b x c (c b). 0, elsewhere Definition 2.4 Arithmetic addition on Generalized Triangular Fuzzy Number: Let à = ((a 1, a 2, a 3 ) ; ωã) and B = ((b 1, b 2, b 3 ) ; ω B) be two GTFNs. Then Ã+ B = ((a 1 + b 1, a 2 + b 2, a 3 + b 3 ), ω) where ω = min {ωã, ω B}. Definition 2.5 Linguistic Variable: A linguistic variable is a variable whose values are linguistic terms. The concept of linguistic variable is applied in dealing with situations that are too complex or too ill defined to be reasonably described in conventional quantitative expressions [8]. Definition 2.6 Ranking Technique: Let à = ((a 1, a 2, a 3 ), ωã) be a generalized triangular fuzzy number and then we define the new ranking function of à as R(Ã) = ω à (a 1+3a 2 +a 3 ). 5 ijpam.eu

4 3 Mathematical Formulation of Generalized Fuzzy Assignment Problem The generalized fuzzy assignment problem can be mathematically stated as Minimize Z = n n C ij x ij, i = 1, 2,..., n. i=1 j=1 { 1, ifi Subject to x ij = th resource is assigned to j th job 0, otherwise n x ij = 1 (One job is performed by i th resource, i = 1, 2,..., n) i=1 and n x ij j=1 = 1 (Only one resource should be assigned to j th job, j = 1, 2,..., n) Where x ij specifies that j th job is assigned to the i th resource. 4 Proposed approach of Generalized Fuzzy Assignment Problem [GFAP] Step 1: Replace the cost matrix C ij with linguistic variables by generalized triangular fuzzy numbers. Step 2: Find the rank of each cell Cij of the chosen fuzzy cost matrix by using the ranking function as mentioned in section II. Step 3: Test whether the given GTF Assignment Problem is balanced or not. (i) If it is a balanced one (i.e., the number of resource are equal to the number of jobs) then go to step 5. (ii) If it is an unbalanced one (i.e., the number of resources are not equal to the number of jobs) then go to step 4. Step 4: Introduce dummy rows or dummy columns with zero fuzzy costs to form a balanced one. Step 5: Proceed by the Hungarian method to solve the fuzzy cost table to get optimal fuzzy assignment. Step 6: Add the optimal fuzzy assignment using fuzzy addition mentioned in section II, to optimize cost within minimum time. ijpam.eu

5 5 Numerical example Consider a fuzzy assignment problem with four persons A, B, C, D and four jobs P, Q, R and S whose costs varying between 0 to 50 dollars. The cost matrix is given with linguistic variables which are replaced by fuzzy numbers. The problem is to find the optimal assignment in an efficient way. P Q R S A Reasonably Low Distinctly Low Reasonably High Very High B Drastically Low Moderate Drastically High Reasonably High C Moderate Very Low Distinctly High Distinctly Low D Distinctly Low Reasonably High Reasonably Low Very Low Solution: The cost involving in executing a given job is considered as fuzzy quantifiers which characterize the linguistic variables are replaced by generalized triangular fuzzy numbers using the following table. As the cost varies between 0 to 50 dollars, the minimum possible value is taken as 0 and the maximum possible value is taken as 50. Drastically Low (0, 1, 2 ;0.8) Very Low (1, 3, 6 ;0.7) Distinctly Low (5, 8, 12 ;0.6) Reasonably Low (10, 14,16;0.5 ) Moderate (13, 17, 19;0.4) Reasonably High (18, 22, 26;0.3) Distinctly High (25, 29, 33;0.2) Very High (32, 38, 42;0.1) Drastically High (41, 46, 50;0.1) The linguistic variables are represented by generalized triangular fuzzy numbers Persons/Jobs P Q R S A (10,14,16;0.5) (5,8,12;0.6) (18,22,26;0.3) (32,38,42;0.1) B (0,1,2;0.8) (13,17,19;0.4) (41,46,50;0.1) (18,22,26;0.3) C (13,17,19;0.4) (1,3,6;0.7) (25,29,33;0.2) (5,8,12;0.6) D (5,8,12;0.6) (18,22,26;0.3) (10,14,16;0.5) (1,3,6;0.7) ijpam.eu

6 Using step 2, the rank of generalized triangular fuzzy cost matrix is given below Rank Table: Persons/Jobs P Q R S A B C D Proceeding by Hungarian method, the optimal allocations are: Persons/Jobs P Q R S A B C D Therefore, the assignment is A R, B P, C Q and D S. The cost allocations are (18, 22, 26;0.3), (0,1,2;0.8), (1,3,6;0.7) and (1,3,6;0.7). By fuzzy addition, the Minimum Cost:(20,29,40;0.3). Minimum Cost = Comparative Results K. Kadhirvel and K. Balamurugan [4] used Robust ranking method to convert FAP into classical AP and solved using Hungarian technique. G. Nirmala and R. Anju [7] proposed a new ranking technique for defuzzification and followed by ASM for optimal solution. K. Ruth Isabels and G. Uthra [8] applied Yagers ranking method and followed by Hungarian method. In this paper we have solved a GAP in terms of Linguistic variables by proposing a new defuzzification measure and proceeded by classical Hungarian technique for optimal solution. RANKING RESULT(cost) w(a+2b+c)/ w(a+b+c)/3 8.9 w(2a+3b+2c)/ ijpam.eu

7 6 Conclusion This paper yields an optimal solution for the fuzzy assignment problem with costs as fuzzy quantifiers characterized by linguistic variables and replaced by generalised triangular fuzzy numbers. The proposed new ranking technique would be efficient in dealing generalized fuzzy assignment problems. This new ranking method defined by us is easy and efficient for the use of decision makers dealing with supply chain and logistics management. As a future extension, the proposed algorithm may be used to solve assignments involving linguistic expressions applying intuitionistic assignment problem, generalized intuitionistic assignment problem, generalized transportation, generalized intuionistic transportation, etc. References [1] R. E. Bellman and L. A. Zadeh, Decision making in a fuzzy environment,management Sciences, vol.17, (1970), [2] S. H. Chen, Operations on fuzzy numbers with function principal, Tamkang J. Management Sci., 6, (1985), [3] W. L. Gau and D. J. Buehrer, Vague sets, IEEE transactions on Systems, Man and Cybernetics, vol. 23,2, (1993), [4] K. Kadhirvel and K. Balamurugan, Method for Solving Hungarian Assignment Problems Using Triangular And Trapezoidal Fuzzy Number, International Journal of Engineering Research and Applications, Vol. 2, Issue 5, (September- October 2012), [5] H. W. Kuhn, The Hungarian method for the assignment problem, Naval Research Logistics, Quarterly, Vol. 2, (1955), [6] A. Nagoor Gani and V. N. Mohamed, Solution of a Fuzzy Assignment Problem by Using a New Ranking Method, Intern. J. Fuzzy Mathematical Archive, Vol. 2, (2013), [7] G. Nirmala and R. Anju, Cost Minimization Assignment Problem Using Fuzzy Quantifier, International Journal of ijpam.eu

8 Computer Science and Information Technologies, Vol. 5 (6), (2014), [8] K. Ruth Isabels and Dr. G. Uthra, An Application of Linguistic Variables in Assignment Problem with Fuzzy Costs, International Journal of Computational Engineering Research, Vol. 2, Issue. 4, (August 2012). [9] K. Sangeetha, H. Haseena Begum and M. Pavithra, Ranking of Triangular Fuzzy Number method to solve an unbalanced Assignment Problem, Journal of Global Research in mathematical Archives, Volume 2, No. 8, (August 2014). [10] Zadeh. L. A, Fuzzy sets, Information and Control, 8, (1965), ijpam.eu

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