Assignment problem is a well known and most suited for solving real world problems.Â  This paper deals with Intuitionistic fuzzy assignment problem whose costÂ  has been considered as an Intuitionistic triangular fuzzy numbers. By defuzzifying, the assignment costs are converted into crisp values and the optimum solution is obtained by using Branch and Bound method.


Ismail, Y.

Lone, M. A.

Majid, R.

Mir, S. A.

English

AgEcon Search: Research in Agricultural and Applied Economics

Give to AgEcon SearchThe World’s Largest Open Access Agricultural & Applied Economics Digital LibraryThis document is discoverable and free to researchers across the globe due to the work of AgEcon Search.Help ensure our sustainability.AgEcon Searchhttp://ageconsearch.umn.eduaesearch@umn.eduPapers downloaded from AgEcon Search may be used for non-commercial purposes and personal study only. No other use, including posting to another Internet site, is permitted without permission from the copyright owner (not AgEcon Search), or as allowed under the provisions of Fair Use, U.S. Copyright Act, Title 17 U.S.C.No endorsement of AgEcon Search or its fundraising activities by the author(s) of the following work or their employer(s) is intended or implied._____________________________________________________________________________________________________  *Corresponding author: E-mail: mushtaqstat11@gmail.com;    Asian Journal of Agricultural Extension,  Economics & Sociology 15(4): 1-6, 2017; Article no.AJAEES.31967 ISSN: 2320-7027  SCIENCEDOMAIN international              www.sciencedomain.org   Intuitionistic Fuzzy Assignment Problem: An Application in Agriculture  M. A. Lone1*, S. A. Mir1, Y. Ismail1 and R. Majid1  1Division of Agricultural Statistics, Sher-e-Kashmir University of Agricultural Sciences and Technology, India.  Authors’ contributions   This work was carried out in collaboration between all authors. All authors read and approved the final manuscript.  Article Information  DOI: 10.9734/AJAEES/2017/31967 Editor(s): (1) Zhao Chen, Department of Biological Sciences, College of Agriculture, Forestry and Life Sciences, Clemson University, USA. Reviewers: (1) A. Ashok Kumar, Achrya N. G. Ranga Agricultural University, Andhra Pradesh, India. (2) Radosław Jedynak, Kazimierz Pulaski University of Technology and Humanities, Poland. Complete Peer review History: http://www.sciencedomain.org/review-history/18263    Received 31 st January 2017  Accepted 24 th  February 2017 Published 18 th March 2017    ABSTRACT  Assignment problem is a well known and most suited for solving real world problems.  This paper deals with Intuitionistic fuzzy assignment problem whose cost ijC~has been considered as an Intuitionistic triangular fuzzy numbers. By defuzzifying, the assignment costs are converted into crisp values and the optimum solution is obtained by using Branch and Bound method.    Keywords: Assignment problem; intuitionistic triangular fuzzy numbers; branch and bound method; optimal allocation.  1. INTRODUCTION  Assignment problem is a special kind of transportation problem in which each source should have the capacity to fulfill the demand of any of the destinations. Assignment problem plays an important role in solving problems of engineering, agriculture and management sciences. The objective of assignment problem is to assign a number of origins (jobs) to the equal Original Research Article      Lone et al.; AJAEES, 15(4): 1-6, 2017; Article no.AJAEES.31967    2  number of destinations (persons) at a minimum cost or maximum profit. Various algorithms / methods like Hungarian method, neural networks, genetic algorithm have been developed to obtain the optimal solution of the assignment problems. The first mathematical formulation of fuzziness was pioneered by [1]. [2] made a numerous attempts to explore the ability of fuzzy set theory to become a useful tool for adequate mathematical analysis of real world problems. [3] introduced Intuitionistic fuzzy sets as an extension of Zadeh's notion of fuzzy set. Fuzzy methods have been developed in virtually all branches of decision making problems can be found in [4,5,6]. [7] solve the FLPP with fuzzy variables in parametric form. [8] proved some theorems and proposed a fuzzy assignment model that considers all individuals to have same skills. [9] introduce the concepts of fuzzy set theory into the decision-making problems involving uncertainty and imprecision. [10] investigated Fuzzy generalized assignment problem with credibility Constraints. Different kinds of fuzzy assignment problems and Intuitionistic Fuzzy assignment problems has been discussed by several authors (see, [11,12, 13,14]). Ranking and defuzzification methods based on area compensation fuzzy sets and systems can be found in [15]. Ranking of trapezoidal Intuitionistic fuzzy numbers was presented [16]. The Hungarian method for the assignment and transportation problems given by [17]. In this paper assignment costs has been considered as Intuitionistic fuzzy numbers which are more realistic and general in nature. The assignment costs are converted into crisp values by defuzzifying with the accuracy function and the optimum solution is obtained by using Branch and Bound method.   2. PRELIMINARIES  2.1 Fuzzy Set  Let A be a classical set, )(A xµ be a function from A to [0, 1]. A fuzzy set A  with the membership function )(A xµ is defined by]}1,0[)(,));( {(x,AA∈∈= xAxxA µµ .  2.1.1 Intuitionistic fuzzy set (IFS)  Let X denote universe of discourse, then an intuitionistic fuzzy set IA  in X is given                by ]},));(),( {(x, II AA XxxvxA ∈= µ  where, ))(),( II AA xvxµ : X→[0,1] are functions such that 0 ≤ ( ))()( II AA xvx +µ ≤ 1  for all x∈X. For each x the member ship function )(IA xµ  and )(IA xv  represent the degree of membership and non-membership of the element x∈X to A⊂ X respectively.  2.1.2 Intuitionistic fuzzy number (IFN)  An Intuitionistic fuzzy set of real line R is called an Intuitionistic fuzzy number if the following holds: (i). There exists Rxo ∈ , )(IA oxµ  = 1 and )(IA oxv  = 0, ox is called the mean value ofIA .  (ii) IAµ is a continuous mapping from R to the closed interval [0,1] and for all x∈R, the relation   1II AA ≤+≤ vo µ   holds.  2.1.3  Triangular intuitionistic fuzzy number (TrIFN)  A triangular Intuitionistic fuzzy number IA is an Intuitionistic fuzzy subset in R with the following membership function )(IA xµ and non-membership function )(IA xv . 3322332112110,)(,)()(,0axaxaaaxaaxaaaaxxaxIA>≤≤−−≤≤−−=<µ  ,3,322,322,1,122,11,)(,)()(,1axaxaaaaxaxaaaxaxvaxIA>≤≤−−≤≤−−=<  Where  ,3321,1 aaaaa ≤≤≤≤  and  ( )(IA xµ ,5.0))(IA ≤xv  ,  )(IA xµ  = )(IA xv  for  all  x∈R.     Lone et al.; AJAEES, 15(4): 1-6, 2017; Article no.AJAEES.31967    3  The TrIFN is given by),;,,( 13211321 aaaaaaAI = .   3. DEFUZZIFICATION  We define accuracy function to defuzzify a given triangular Intuitionistic fuzzy number is    H8)2()2()('32'1321 aaaaaaa I+++++=  3.1 Intuitionistic Fuzzy Assignment Problem (IFAP)  Suppose there are n jobs which is to be performed and n persons are available for doing these jobs. Assume that each person’s can do each job at a time, though with different. Let ijC~be an Intuitionistic fuzzy cost of assigning the person to the  job. Let the decision variable  denoting he assignment of the the person to the job.  The problem is to find an assignment (which job should be assigned to which person on one – one basis) so that total cost of performing all jobs is minimum. Problems of this kind are known as assignment problem. Mathematically an IFAP is given below:  Minimize ∑∑===njijijnjXCZ11~    Subject to   ....,,2,111....,,2,111njfornjijXniforniijX======∑∑  Where  =  1,  ℎ  the crop  is assigned to the paddock0,  ℎ  the crop  is  not assigned to the paddock !  IijC~= (#$ , #% , #& )(#$′ , #% , #&′)  Crops                                                   Paddocks 1          2           ⋯                 j                      ⋯                        n    1              11~a       12~a         ⋯                ja1~                  ⋯                       na1~     2 21~a      21~a          ⋯                ja2~                  ⋯                        na2~ ⋮       i   ⋮ M          M                                   M                                     M  1~ia      1~ia           ⋯                ija~                   ⋯                         jna~  M           M                            M                                            M      n 1~na     2~na            ⋯               nja~                  ⋯                          nna~  4. NUMERICAL ILLUSTRATION  Let us consider an IFAP, where a farmer intends to plant four different crops in each of four equal sized paddocks. Rows representing four different crops Crop1, Crop2, Crop3 and Crop4 and Columns representing the four equal sized paddocks like P1, P2, P3 and P4. The nutrient requirements required for different crops vary and the paddocks vary in soil fertility. Thus the cost of the fertilizers which must be applied depends on which crop is grown in which paddock. Let [ ijC~] be the cost matrix whose elements are given triangular Intuitionistic fuzzy numbers. The farmer’s objective is to find the optimal assignment of paddocks to crops in such a manner that the total fertilizer cost becomes minimum.         Lone et al.; AJAEES, 15(4): 1-6, 2017; Article no.AJAEES.31967    4     Paddocks P1        P2 P3 P4  Crops Crops1 (7,9,1)(5,9,13) (5,19,27)(1,19,32) (10,23,30)(6,23,30) (15,28,59)(11,28,59) Crops2 (7,8,15)(1,8,21) (6,8,12)(4,8,19) (4,14,28)(3,14,31) (3,14,26)(3,14,27) Crops3 (4,8,21)(1,8,24) (2,10,24)(1,10,27) (8,20,23)(5,20,26) (7,9,11)(5,9,13) Crops4 (2,13,25)(1,13,26) (7,11,24)(4,11,27) (11,12,19)(5,12,25) (10,23,30)(6,23,30)    Fig. 1. Various nodes of tree using  The Accuracy function is    H8)2()2()('32'1321 aaaaaaa I+++++=     Lone et al.; AJAEES, 15(4): 1-6, 2017; Article no.AJAEES.31967    5  #$$ = 9, #$% = 17.62, #$& = 21, #$/ = 32,  #%$ = 9.5, #%% = 9.12, #%& = 15.25, #%/ = 14.37  #&$ = 10.5, #&% = 11.75, #&& = 17.75, #&/ = 9,  #/$ = 13.25, #/% = 13.25, #/& = 13.50, #// = 21  Therefore, the cost matrix =21.0013.5013.2513.259.00017.7575.1110.5014.3715.2512.99.5000.2300.12 62.71  9.00  Sometimes we don’t know the exact cost that occurs by assigning any crop to any paddock. It may be around in any number. Then in such situation we can apply this in agricultural fields. If cost occurs due to various factors and one factor may occurs the same cost, the costs are so formed interns of INFNs.   Now the optimal solution can be obtained from the following Assignment problem after replacing the corresponding values of of aij with the obtained cost matrix as shown below:     Paddocks P1       P2 P3 P4  Crops Crops1 9.00 17.62 21.00 32.00 Crops2 9.50 9.125 15.25 14.37 Crops3 10.50 11.75 17.75 9.00 Crops4 13.25 13.25 13.50 21.00  In order to get the optimal allocation, we solve the above problem using Branch and Bound Method. The branch and bound tree is shown in Fig. 1.  The conventional assignment problem in the Linear programming problem form can be obtained by replacing above values for their corresponding values. After solving, we get the optimal solution 40.62 and the optimal allocation is  (1,1), (2,2), (3,4),and (4,3). It means that crop1 is assigned to P1 Paddock, crop2 is assigned to P2 Paddock, crop3 is assigned to P4 Paddock and crop4 is assigned to P3 Paddock, With optimal fertilizer cost 40.62.  5. CONCLUSION  In this paper, the assignment cost has been considered as an Intuitionistic fuzzy numbers. By defuzzifying with the accuracy function, the assignment costs are converted into crisp values and the optimum solution is obtained by using Branch and Bound method. This method can be utilized to all type of Intuitionistic fuzzy assignment problem and is easy to apply.  COMPETING INTERESTS  Authors have declared that no competing interests exist.  REFERENCES  1. Zadeh LA. Fuzzy sets. Information and Control. 1965;8:338-353. 2. Orlovsky SA. On formulation of a       general fuzzy mathematical programming problem. Fuzzy Sets and System. 1980;3: 311-321. 3. Atanassov K. Intuitionistic fuzzy sets. Fuzzy Sets and Systems. 1986;20:87-96. 4. Tamiz M. Multi-objective programming    and goal programming theories and applications. Germany: Springer-Verlag; 1996. 5. Zimmermann HJ. Fuzzy set theory and its applications. (2nd rev. Ed). Boston: Kulwer; 1991. 6. Ross TJ. 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Naval Research Logistics  Quarterly. 1955;2:83-97. _________________________________________________________________________________ © 2017 Lone et al.; This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.     Peer-review history: The peer review history for this paper can be accessed here: http://sciencedomain.org/review-history/18263 

Intuitionistic Fuzzy Assignment Problem: An Application in Agriculture

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Intuitionistic Fuzzy Assignment Problem: An Application in Agriculture

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