A Method for Solving a Bi-Objective Transportation Problem under Fuzzy Environment

— A bi-objective fuzzy transportation problem with the objectives to minimize the total fuzzy cost and fuzzy time of transportation without according priorities to them is considered. To the best of our knowledge, there is no method in the literature to find efficient solutions of the bi-objective transportation problem under uncertainty. In this paper, a bi-objective transportation problem in an uncertain environment has been formulated. An algorithm has been proposed to find efficient solutions of the bi-objective transportation problem under uncertainty. The proposed algorithm avoids the degeneracy and gives the optimal solution faster than other existing algorithms for the given uncertain transportation problem.


I. INTRODUCTION
certain class of mathematical programming problem arises very frequently in practical applications.For example, a product may be transported from factories to retail stores.The factories are the sources and the stores are the destinations.The amount of the product which is available and the demand are also known.The problem is that the different legs of the network joining the sources to the destinations have different costs associated with them.The aim is to find the minimum cost routing of products from supply point to destination, this problem is widely known as the cost minimizing transportation problem.The transportation problem with a single objective to minimize the duration of transportation has been studied in detail by many researchers [19]- [22].And also with multiple objectives has been discussed in [15]- [17].As in practical life, decision makers do not have the exact transportation cost and time; then, there exists the uncertainty about the cost and time.Therefore, it is very interesting to deal with the transportation problem in fuzzy environment.The idea of fuzzy set was first proposed by Zadeh [25], as a mean of handling uncertainty that is due to imprecision rather than to randomness.Bellmann and Zadeh [2] presented the technique of decision making process in fuzzy environment.After that, many authors have studied fuzzy linear programming problem techniques such as Fang [7], Rommelfanger [18] and Tanaka et al. [24] etc.
In literature, we find that there many transportation models where fuzzy linear programming have been applied or approaches to solve multi-objective fuzzy transportation problem.From this view point, Chanas [5] proposed a fuzzy programming in multi-objective linear programming solved by Sukhveer Singh is with the Thapar University, Patiala, India (e-mail: sukhveer.singh@thapar.edu).parametric approach.Tanakka and Asai [23] introduced fuzzy linear programming problem in fuzzy environment.Zimmermann [26] proposed a fuzzy multi-criteria decision making set, by using intersection of all fuzzy goals and constraints.Lai-Hawng [10] considered multi-objective linear programming problem with all parameters, having a triangular possibility distribution.Bit [3] considered fuzzy programming approach to a multi-criteria decision making transportation problem in which the constraints are of equality types.Later, Bit et al. [4] also considered a fuzzy programming approach to multi-objective solid transportation problem.And other several authors [1], [6], [8], [14] have proposed different models for solving fuzzy multi-objective transportation problems.
In this paper, we define an algorithm that has been proposed to find the fuzzy optimal value of a bi-objective fuzzy transportation problem.The technique gives the optimal solution faster than other existing techniques.It also reduces the computational work.

A. Basic Definitions
In this section some basic definitions are reviewed [9]., is said to be a triangular fuzzy number if its membership function is given by (

B. Arithmetic Operations
In this subsection, arithmetic operations between two trapezoidal fuzzy numbers, defined on a universal set of real numbers R, are reviewed [9].
Let 1 be two trapezoidal fuzzy numbers, then (i) ,

C. Ranking Function
A convenient method for comparing fuzzy numbers is by using ranking function [12], [13].A ranking function : ( ) F R R    , where F(R) set of all fuzzy numbers defined on set of real numbers defined on set of real numbers, maps each fuzzy number into a real number.Let A  and B  be two fuzzy numbers, then: is the variable assuming the value 0 or 1 according as the entire requirement of destination j is not met or met from source i .Let C  and T  denote the total fuzzy cost and fuzzy time of transportation respectively.The mathematical formulation of the problem is as follows.Determine ' ij x s which minimize the two-objective functions: without according priorities to them, subject to the constraints, 1 ; ( 1, 2,..., )

IV. SOLUTION PROCEDURE
The proposed algorithm has three subparts, as given below, to find the fuzzy efficient optimal solution of the fuzzy biobjective transportation problem.

A. Conversion of Two Objectives into a Sequence of Single Objective
Here we use the following process to convert bi-objective fuzzy transportation problem in single objective fuzzy transportation problem as follows, Step 1.The set ( Here subject to the constraints (3)-( 5).

B. Proposed Algorithm to Obtain Fuzzy Efficient Solution
Step I: The single objective fuzzy transportation problem, obtained in Section IV.A, is transformed into the tabular form.
Step II: Consider a set S having the cells ( , ) i j which has the fuzzy cost with minimum rank among each entries of its corresponding row and column in the obtained table.
Step III: Calculate ij P  for each cell ( , ) i j S  .whereStep V: Check whether the requirement of each destination is fulfilled or not.If not then repeat Step II-V, else, the obtained fuzzy solution is our fuzzy optimal solution of fuzzy transportation problem.

C. Procedure to Obtain 2 nd Subsequent and Efficient Fuzzy Solution
After finding the first efficient solution, (1)

ij
x has been obtained of given fuzzy transportation problem.The second efficient solution (2)   ij x is obtained by deleting all the cells ( , ) ( ) . The resultant problem is designated the second efficient solution (2) ij x .Further, the third efficient solution is obtained by deleting those cells ( , ) i j in the second cost-time trade-off fuzzy transportation problem in fuzzy environment, that correspond to the . Subsequent efficient solutions are obtained by proceeding exactly in the same way.

V. NUMERICAL EXAMPLE
In this section, a numerical problem is considered of four origins and five destinations and applies the algorithm as explained in Section IV.The tableau representation of the numerical problem is given in Table I.The upper entries denote the fuzzy cost of unit product which have to transport from i th origin to j th destination and the lower entries denote the fuzzy time of transportation from i th origin to j th destination.
In Table I, the upper entries of cell   , i j depicts the unit fuzzy cost and the lower entries of a cell   , i j depict fuzzy time of fuzzy transportation from origin i O to destination j D .In the last row and column, j b and i a depicts the units of the commodity required at the destinations and available at the origins, respectively.The numerical problem seeks to determine ' ij x s which minimize the two objective functions, (0, 1, (0, 0.5, 1.5, 2 ) (0, 0.5, 1.5, 2 ) ( (0, 0.5,  (0, 0.5,1.5, 2){ ( ) subject to the given constraints ( 3)-( 5) after assigning numerical values to all the parameters therein.
numbers which can be negative or zero or positive, has the same sign as the non-zero k  with the smallest subscript in it irrespective of the values of other ' symbol >> indicates that the quantity on its left side arbitrarily World Academy of Science, Engineering and Technology International Journal of Computer and Information Engineering Vol:11, No:6, 2017 773 International Scholarly and Scientific Research & Innovation 11(6) 2017 Digital Open Science Index, Computer and Information Engineering Vol:11, No:6, 2017 waset.org/Publication/10007493large compared to right hand side).Step 3.After this, the cost-time trade-off fuzzy transportation problem with C  and T  as the first and second priority objectives, respectively, is reduced to an equivalent single-objective fuzzy transportation problem seeking to determine '

S u m o
f fu zzy c o sts o f n e arest ad jacen t sid e s o f cell ( , ) N u m b er o f fu z z y c o s ts a d d e d Allocate the cell (i, j) for which rank of ij P  i.e. ( ) ij P   is maximum.If two or more ( )' ij P s   have the same values then allocate that cell which has least cost among all cells for which ( )' ij P s   are equal.Again, if the costs of these cells are equal then randomly allocate that cell for which