An Integer Solution in Intuitionistic Transportation Problem with Application in Agriculture

In this paper, we investigate a Transportation problem which is a special kind of linear programming in which profits; supply and demands are considered as Intuitionistic triangular fuzzy numbers. The crisp values of these Intuitionistic triangular fuzzy numbers are obtained by defuzzifying them and the problem is formulated into linear programming problem. The solution of the formulated problem is obtained through LINGO software. If the obtained solution is non-integer then Branch and Bound method can be used to obtain an integer solution.


INTROdUCTION
Transportation problem is a spectial kind of linear programming problem in which goods are transported from a set of source to the set of destination subject to the supply and demands of the source and destinations.Hitchcock (1941) firstly introduced the Transportation Problem and after that it presented by Koopmans (1947).The first mathematical formulation of fuzziness was pioneered by Zadeh (1965).Orlovsky (1980) 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.Atanassov (1986) introduced Intuitionistic fuzzy sets as an extension of Zadeh's notion of fuzzy set.Intuitionistic fuzzy set is a powerful tool in solving real life problems and has a greater influence in solving Transportation problems to find optimal allocation.A new method for solving Transportation problems with Intuitionistic triangular fuzzy numbers was proposed by Paul et.al (2014).A balanced Intuitionistic fuzzy assignment problem was solved by Kumar et al.,(2014) .Intuitionistic fuzzy Transportation problem has been studied by many authors and with different approaches have been proposed such as (Ganiand Abbas (2013), Hussainand Kuma (2012), Hakim(2012) and Pramila and Ultra(2014) etc) .Ranking and defuzzification methods based on area compensation fuzzy sets and systems can be found in Fortemps and Roubens (1996).Ranking of trapezoidal Intuitionistic fuzzy numbers was presented De and Das.(2012).In this paper, the transportation problemconsidered in which the profits, availability and requirement are Intuitionistic triangular fuzzy numbers.By defuzzifying, the profits, availability and requirements are converted into crisp values.The problem is formulated into Linear programming problem and solution is obtained through LINGO Software.If the obtained solution is non-integer then we round the non integer value to the nearest integer value.But sometimes in practical situation by rounding, we get a solution which may be infeasible or impractical.Thus instead of rounding the non integer solution to the nearest integer value we use Branch and Bound to obtain integer solution.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

Preliminaries Fuzzy set
Let A be a classical set,

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 µ is a continuous mapping from R to the closed interval [0,1] and for all x∈R, the relation

Triangular intuitionistic fuzzy number (TrIFN):
A triangular intuitionistic fuzzy number is an intuitionistic fuzzy subset in R with the following membership function     b e ( I T F ) p r o f i t obtained from one hectare of ground sown by j th crop.Also, let the decision variable X ij denoting the number of hectares of i th ground sown by to the crop.The objective is to determine an optimal allocation of land (hectares) used for sowing so that over profit will be maximized.The crop problem can be found in Mitchell(2011)and Thornley and France (2006).The hypothetical data set in tabular form is shown below: The above problem can be formulated as a linear programming problem (LPP) and the solution can be obtained from the following given program in LINGO software.PROFIT( I, J) * VOLUME( I, J)); @FOR( crops( J): @SUM( grounds( I): VOLUME( I, J)) = mrcsa( J)); @FOR( grounds( I): @SUM( crops( J): VOLUME( I, J)) <= area( I)); END Global optimal solution found. Objective for all x∈X.For each x the member ship function membership and non-membership of the element x∈X to A⊂ X respectively.
fuzzy numbers of the quantity available and required quantity respectively.Let the decision variable X ij denoting the quantity transported a from i th IF orgion to the j th IF destination.Mathematically an IFTP is given below: n groundsand requirement of area for crop sowing with Intuitionistic tr iangular fuzzy numbers respectively.Let Using1, the above table can be replaced by their corresponding values as