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The New Ranking Method using Octagonal Intuitionistic Fuzzy Unbalanced Transportation Problem

This document presents a new ranking method for solving the intuitionistic fuzzy unbalanced transportation problem using octagonal intuitionistic fuzzy numbers for costs, supplies, and demands. The proposed method aims to derive optimal solutions and is illustrated with a numerical example. It emphasizes the importance of intuitionistic fuzzy sets in handling vagueness in unbalanced transportation scenarios.

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International Journal of Trend in Scientific Research and Development (IJTSRD) Volume 4 Issue 4, June 2020 @ IJTSRD | Unique Paper ID – IJTSRD31675 The New Ranking Method Fuzzy Unbalanced Dr. 1Research Guide, Assistant Professor, Department of Mathematics Chikkanna Govt Arts College, Tirupur 2Research scholar, Assistant Professor, Department of Mathematics Sri Shakthi Institute of Engineering and Technology, ABSTRACT In this paper a new ranking method is proposed for finding an optimal solution for intuitionistic fuzzy unbalanced transportation problem, in which the costs, supplies and demands are octagonal intuitionistic fuzzy numbers. The procedure is illustrated with a numerical example. KEYWORDS: Unbalanced transportation problems, Octagonal Intuitionistic fuzzy numbers, Ranking method, Modi method, Initial Basic Feasible Solution, Optimal Solution 1. INTRODUCTION In general, Unbalanced transportation problems are solved with assumptions that the cost, supply and demand are specified in precise manner. However, in many cases the decision maker has no precise information about the coefficient belonging to the unbalanced transportation problem. Intuitionistic fuzzy set is a powerful tool to deal with such vagueness. L. A. Zadeh introduced fuzzy set theory in 1965. Different types of fuzzy sets are defined in order vagueness of the existing problems. The concept of Intuitionistic Fuzzy Sets (IFSs), proposed by Atanassov in [1] and [2], has been found to be highly useful to deal with vagueness. Many authors discussed the solutions of Fuzzy Transportation Problem (FTP) using various techniques. In 2016, Mrs. Kasthuri. Pentagonal intuitionistic fuzzy. In 2015, A. and R. Santhi [3] introduced Hexagonal Intuitionistic Fuzzy Numbers. In 2015, Thangaraj Beaula Priyadharshini [4] proposed A New Algorithm for Finding a Fuzzy Optimal Solution. K. Prasanna Devi, M. Devi Durga [5] and G. Gokila, Juno Saju [6] introduced Octagonal Fuzzy Number. end in Scientific Research and Development (IJTSRD) 2020 Available Online: www.ijtsrd.com e-ISSN: 2456 31675 | Volume – 4 | Issue – 4 | May-June 2020 The New Ranking Method using Octagonal Intuitionistic Unbalanced Transportation Problem Dr. P. Rajarajeswari1, G. Menaka2 Research Guide, Assistant Professor, Department of Mathematics Chikkanna Govt Arts College, Tirupur, Tamil Nadu, India ch scholar, Assistant Professor, Department of Mathematics Sri Shakthi Institute of Engineering and Technology, Coimbatore, Tamil In this paper a new ranking method is proposed for finding an optimal zy unbalanced transportation problem, in which the costs, supplies and demands are octagonal intuitionistic fuzzy numbers. The procedure is illustrated with a numerical example. Unbalanced transportation problems, Octagonal Intuitionistic y numbers, Ranking method, Modi method, Initial Basic Feasible Solution, How to cite this paper Rajarajeswari | G. Menaka "The New Ranking Method using Octagonal Intuitionistic Fuzzy Unbalanced Transportation Problem" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456 6470, Volume Issue-4, June 2020, pp.1735 www.ijtsrd.com/pa Copyright © 20 International Journal of Trend in Scientific Research and Development Journal. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (CC BY 4.0) (http://creativecommons.org/licenses/ by/4.0) In general, Unbalanced transportation problems are solved with assumptions that the cost, supply and demand are specified in precise manner. However, in many cases ormation about the coefficient belonging to the unbalanced transportation problem. Intuitionistic fuzzy set is a powerful tool to deal Zadeh introduced fuzzy set theory in 1965. Different types of fuzzy sets are defined in order to clear the The concept of Intuitionistic Fuzzy Sets (IFSs), proposed by Atanassov in [1] and [2], has been found to be highly useful to deal with vagueness. Many authors discussed the n Problem (FTP) using Kasthuri. B introduced Pentagonal intuitionistic fuzzy. In 2015, A. Thamaraiselvi and R. Santhi [3] introduced Hexagonal Intuitionistic Thangaraj Beaula – M. proposed A New Algorithm for Finding Prasanna Devi, M. Devi Durga Gokila, Juno Saju [6] introduced Octagonal Fuzzy In this paper, a new method is proposed for finding an optimal solution for intuitionistic transportation problem, in which the costs, supplies and Demands are octagonal intuitionistic fuzzy numbers. Using Octagonal Intuitionistic fuzzy unbalanced transportation problem we get best minimum value of optimal solution. The paper is organized as follows, In section 2 introduction with some basic concepts of fuzzy and Intuitionistic fuzzy definitions, In section 3 introduced Octagonal Intuitionistic Fuzzy Definition and proposed algorithm followed by a Numerical example using Modi method and finally the paper is concluded in section 4. 2. PRELIMINARIES 2.1. Definition (Fuzzy set Let X be a nonempty set. A fuzzy set as‫ܣ‬̅=ሼ൏ ‫,ݔ‬ ߤ஺ ഥ ሺxሻ ൐/‫ݔ‬ ∈ ܺሽ. Where membership function, which maps each element of X to a value between 0 and 1. 2.2. Definition (Fuzzy Number A fuzzy number is a generalization of a regular real number and which does not refer to a single value but rather to a connected a set of possible values, where each end in Scientific Research and Development (IJTSRD) ISSN: 2456 – 6470 2020 Page 1735 sing Octagonal Intuitionistic Transportation Problem Research Guide, Assistant Professor, Department of Mathematics, ch scholar, Assistant Professor, Department of Mathematics, Coimbatore, Tamil Nadu, India How to cite this paper: Dr. P. Rajarajeswari | G. Menaka "The New Ranking Method using Octagonal Intuitionistic Fuzzy Unbalanced ortation Problem" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456- 6470, Volume-4 | 4, June 2020, pp.1735-1739, URL: www.ijtsrd.com/papers/ijtsrd31675.pdf Copyright © 2020 by author(s) and International Journal of Trend in Scientific Research and Development Journal. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (CC BY 4.0) p://creativecommons.org/licenses/ In this paper, a new method is proposed for finding an optimal solution for intuitionistic fuzzy Unbalanced transportation problem, in which the costs, supplies and Demands are octagonal intuitionistic fuzzy numbers. Using Octagonal Intuitionistic fuzzy unbalanced transportation problem we get best minimum value of optimal solution. s organized as follows, In section 2 introduction with some basic concepts of fuzzy and Intuitionistic fuzzy definitions, In section 3 introduced Octagonal Intuitionistic Fuzzy Definition and proposed algorithm followed by a Numerical example using Modi thod and finally the paper is concluded in section 4. ሾࡲࡿሿ)[3] Let X be a nonempty set. A fuzzy set ‫ܣ‬̅ of Xis defined ሽ. Where ߤ஺ ഥ (x) is called membership function, which maps each element of X to a Definition (Fuzzy Numberሾࡲࡺሿ) [3] number is a generalization of a regular real number and which does not refer to a single value but rather to a connected a set of possible values, where each IJTSRD31675
International Journal of Trend in Scientific Research and Development (IJTSRD) @ www.ijtsrd.com eISSN: 2456-6470 @ IJTSRD | Unique Paper ID – IJTSRD31675 | Volume – 4 | Issue – 4 | May-June 2020 Page 1736 possible value has its weight between 0 and 1. The weight is called the membership function. A fuzzy number ‫ܣ‬̅ is a convex normalized fuzzy set on the real line R such that There exists at least one x ∈R with ߤ஺ ഥ ሺxሻ = 1. ߤ஺ ഥ ሺxሻߤ஺ ഥ ሺxሻ is piecewise continuous. 2.3. Definition (Intuitionistic Fuzzy Set [IFS]) [3] Let X be a non-empty set. An Intuitionistic fuzzy set ‫ܣ‬̅ூ of X is defined as ‫ܣ‬̅ூ = {<x,ߤ஺̅಺ሺ‫ݔ‬ሻ, ߴ஺̅಺(x) > / x∈ ܺ}. Where ߤ஺̅಺ሺ‫ݔ‬ሻ ܽ݊݀ ߴ஺̅಺(x) are membership and non-membership function. Such that ߤ஺̅಺ሺ‫ݔ‬ሻ, ߴ஺̅಺(x): X→ [0, 1] and 0≤ ߤ஺̅಺ሺ‫ݔ‬ሻ + ߴ஺̅಺(x) ≤ 1 for all x∈ ܺ. 2.4. Definition (Intuitionistic Fuzzy Number [IFN]) [3] An Intuitionistic Fuzzy Subset ‫ܣ‬̅ூ = {< x, ߤ஺̅಺ሺ‫ݔ‬ሻ, ߴ஺̅಺(x) > / x∈ ܺ} of the real line R is called an Intuitionistic Fuzzy Number, if the following conditions hold, There exists m ∈ R suct that ߤ஺̅಺ሺ݉ሻ = 1 ܽ݊݀ ߴ஺̅಺(m) = 0. ߤ஺̅಺ߤ஺̅಺ሺ‫ݔ‬ሻ is a continuous function from R → [0,1] such that 0≤ ߤ஺̅಺ሺ‫ݔ‬ሻ + ߴ஺̅಺(x) ≤1 for all x∈ ܺ. The membership and non- membership functions of ‫ܣ‬̅ூ are in the following form ߤ஺̅಺ሺ‫ݔ‬ሻ = ‫ە‬ ۖ ‫۔‬ ۖ ‫ۓ‬ 0 ݂‫ݎ݋‬ − ∞ < ‫ ݔ‬ ≤ ܽଵ ݂ሺ‫ݔ‬ሻ ݂‫ܽ ݎ݋‬ଵ ≤ ‫ ݔ‬ ≤ ܽଶ 1 ݂‫ݔ ݎ݋‬ = ܽଶ ݃ሺ‫ݔ‬ሻ ݂‫ܽ ݎ݋‬ଶ ≤ ‫ ݔ‬ ≤ ܽଷ 0 ݂‫ܽ ݎ݋‬ଷ ≤ ‫ݔ‬ < ∞ ۙ ۖ ۘ ۖ ۗ ߴ஺̅಺(x) = ‫ە‬ ۖ ‫۔‬ ۖ ‫ۓ‬ 1 ݂‫ ݎ݋‬ − ∞ < ‫ ݔ‬ ≤ ܽଵ ′ ݂′ሺ‫ݔ‬ሻ ݂‫ܽ ݎ݋‬ଵ′ ≤ ‫ ݔ‬ ≤ ܽଶ 0 ݂‫ݔ ݎ݋‬ = ܽଶ ݃′ሺ‫ݔ‬ሻ ݂‫ܽ ݎ݋‬ଶ ≤ ‫ ݔ‬ ≤ ܽଷ′ 1 ݂‫ܽ ݎ݋‬ଷ′ ≤ ‫ݔ‬ < ∞ ۙ ۖ ۘ ۖ ۗ Where݂,݂′ , ݃, ݃′ are functions from R → [0,1] . ݂ and ݃′ are strictly increasing functions and ݃ and ݂′ are strictly decreasing functions with the conditions 0 ≤ ݂ሺ‫ݔ‬ሻ+ ݂′ሺ‫ݔ‬ሻ ≤ 1 and 0 ≤ ݃ሺ‫ݔ‬ሻ + ݃′ሺ‫ݔ‬ሻ ≤ 1. 3. OCTAGONAL INTUITIONISTIC FUZZY NUMBER 3.1. Definition (Octagonal Intuitionistic Fuzzy Number [OIFN]) An Octagonal Intuitionistic Fuzzy Number is specified by ‫ܣ‬̅௢௖ ூ =ሺܽଵ, ܽଶ ,ܽଷ , ܽସ , ܽହ, ܽ଺, ܽ଻,଼ܽሻ , (ܽଵ′, ܽଶ ′,ܽଷ ′ , ܽସ , ܽହ, ܽ଺ ′ , ܽ଻ ′ , ଼ܽ′ሻ. Whereܽଵ, ܽଶ ,ܽଷ , ܽସ , ܽହ, ܽ଺, ܽ଻,଼ܽ,ܽଵ′, ܽଶ ′,ܽଷ ′ , ܽ଺ ′ , ܽ଻ ′ ܽ݊݀ ଼ܽ ′ and its membership and non-membership functions are given below ߤ஺ ഥ ሺxሻ = ‫ە‬ ۖ ۖ ۖ ۖ ۖ ‫۔‬ ۖ ۖ ۖ ۖ ۖ ‫ۓ‬ 0 ݂‫ݔ ݎ݋‬ < ܽଵ ݇ ൬ ‫ ݔ‬ − ܽଵ ܽଶ − ܽଵ ൰ ݂‫ܽ ݎ݋‬ଵ ≤ ‫ ݔ‬ ≤ ܽଶ ݇ ݂‫ܽ ݎ݋‬ଶ ≤ ‫ ݔ‬ ≤ ܽଷ ݇ + ሺ1 − ݇ሻ ሺ ‫ ݔ‬ − ܽଷ ܽସ − ܽଷ ሻ ݂‫ܽ ݎ݋‬ଷ ≤ ‫ ݔ‬ ≤ ܽସ 1 ݂‫ܽ ݎ݋‬ସ ≤ ‫ ݔ‬ ≤ ܽହ ݇ + ሺ1 − ݇ሻ ൬ ܽ଺ − ‫ݔ‬ ܽ଺ − ܽହ ൰ ݂‫ܽ ݎ݋‬ହ ≤ ‫ ݔ‬ ≤ ܽ଺ ݇ ݂‫ܽ ݎ݋‬଺ ≤ ‫ ݔ‬ ≤ ܽ଻ ݇ሺ ଼ܽ − ‫ݔ‬ ଼ܽ − ܽ଻ ሻ ݂‫ܽ ݎ݋‬଻ ≤ ‫ ݔ‬ ≤ ଼ܽ 0 ݂‫ݔ ݎ݋‬ > ଼ܽ ۙ ۖ ۖ ۖ ۖ ۖ ۘ ۖ ۖ ۖ ۖ ۖ ۗ ߴ஺̅಺ሺxሻ = ‫ە‬ ۖ ۖ ۖ ۖ ۖ ۖ ‫۔‬ ۖ ۖ ۖ ۖ ۖ ۖ ‫ۓ‬ 1 ݂‫ܽ ݎ݋‬ଵ ′ < ‫ݔ‬ ݇ + ሺ1 − ݇ሻ ቆ ܽଶ′ − ‫ݔ‬ ܽଶ′ − ܽଵ′ ቇ ݂‫ܽ ݎ݋‬ଵ′ ≤ ‫ ݔ‬ ≤ ܽଶ′ ݇ ݂‫ܽ ݎ݋‬ଶ′ ≤ ‫ ݔ‬ ≤ ܽଷ′ ݇ሺ ܽସ − ‫ݔ‬ ܽସ − ܽଷ′ ሻ ݂‫ܽ ݎ݋‬ଷ′ ≤ ‫ ݔ‬ ≤ ܽସ 0 ݂‫ܽ ݎ݋‬ସ ≤ ‫ ݔ‬ ≤ ܽହ ݇ ൬ ‫ ݔ‬ − ܽହ ܽ଺′ − ܽହ ൰ ݂‫ܽ ݎ݋‬ହ ≤ ‫ ݔ‬ ≤ ܽ଺′ ݇ ݂‫ܽ ݎ݋‬଺′ ≤ ‫ ݔ‬ ≤ ܽ଻′ ݇ + ሺ1 − ݇ሻ ሺ ‫ ݔ‬ − ܽ଻′ ଼ܽ′ − ܽ଻′ ሻ ݂‫ܽ ݎ݋‬଻′ ≤ ‫ ݔ‬ ≤ ଼ܽ′ 1 ݂‫ݔ ݎ݋‬ > ଼ܽ′ ۙ ۖ ۖ ۖ ۖ ۖ ۖ ۘ ۖ ۖ ۖ ۖ ۖ ۖ ۗ Graphical representation of Octagonal Intuitionistic Fuzzy Numbers ___ Membership Functionߤ஺ ഥ ሺxሻ ---- Non-Membership Functionߴ஺̅಺ሺxሻ 3.2. Arithmetic operations on Octagonal Intuitionistic Fuzzy Numbers Let ‫ܣ‬̅௢௖ ூ =ሺܽଵ, ܽଶ ,ܽଷ , ܽସ , ܽହ, ܽ଺, ܽ଻,଼ܽሻ (ܽଵ′, ܽଶ ′,ܽଷ ′ , ܽସ ′ , ܽହ ′ , ܽ଺ ′ , ܽ଻ ′ , ଼ܽ′ሻ and ‫ܤ‬ത௢௖ ூ = ሺܾଵ, ܾଶ ,ܾଷ , ܾସ , ܾହ, ܾ଺, ܾ଻,଼ܾሻ (ܾଵ′, ܾଶ ′,ܾଷ ′ , ܾସ ′ , ܾହ ′ , ܾ଺ ′ , ܾ଻ ′ , ଼ܾ′ሻ be two Octagonal Intuitionistic Fuzzy Numbers, then the arithmetic operations are as follows. 3.2.1. Addition: ‫ܣ‬̅௢௖ ூ + ‫ܤ‬ത௢௖ ூ = ሺ ܽଵ + ܾଵ , ܽଶ + ܾଶ , ܽଷ + ܾଷ, ܽସ + ܾସ, ܽହ + ܾହ , ܽ଺ + ܾ଺ , ܽ଻ + ܾ଻ , ଼ܽ + ଼ܾሻ ሺܽଵ ′ + ܾଵ ′ ,ܽଶ ′ + ܾଶ ′ , ܽଷ ′ + ܾଷ ′ ,ܽସ ′ + ܾସ ′ , ܽହ ′ + ܾହ ′ , ܽ଺′+ܾ଺ ′ , ܽ଻ ′ + ܾ଻ ′ , ଼ܽ ′ + ଼ܾ ′ሻ 3.2.2. Subtraction: ‫ܣ‬̅௢௖ ூ − ‫ܤ‬ത௢௖ ூ = ሺ ܽଵ − ଼ܾ , ܽଶ − ܾ଻ , ܽଷ − ܾ଺, ܽସ − ܾହ, ܽହ − ܾସ , ܽ଺ − ܾଷ , ܽ଻ − ܾଶ , ଼ܽ − ܾଵሻ
International Journal of Trend in Scientific Research and Development (IJTSRD) @ www.ijtsrd.com eISSN: 2456-6470 @ IJTSRD | Unique Paper ID – IJTSRD31675 | Volume – 4 | Issue – 4 | May-June 2020 Page 1737 ሺܽଵ ′ − ଼ܾ ′ ,ܽଶ ′ − ܾ଻ ′ , ܽଷ ′ − ܾ଺ ′ ,ܽସ ′ − ܾହ ′ , ܽହ ′ − ܾସ ′ , ܽ଺′- ܾଷ ′ , ܽ଻ ′ − ܾଶ ′ , ଼ܽ ′ − ܾଵ ′ሻ 3.2.3. Multiplication: ‫ܣ‬̅௢௖ ூ ∗ ‫ܤ‬ത௢௖ ூ = ሺ ܽଵ ∗ ܾଵ , ܽଶ ∗ ܾଶ , ܽଷ ∗ ܾଷ, ܽସ ∗ ܾସ, ܽହ ∗ ܾହ , ܽ଺ ∗ ܾ଺ , ܽ଻ ∗ ܾ଻ , ଼ܽ ∗ ଼ܾሻ ሺܽଵ ′ ∗ ܾଵ ′ ,ܽଶ ′ ∗ ܾଶ ′ , ܽଷ ′ ∗ ܾଷ ′ ,ܽସ ′ ∗ ܾସ ′ , ܽହ ′ ∗ ܾହ ′ , ܽ଺′ ∗ ܾ଺ ′ , ܽ଻ ′ ∗ ܾ଻ ′ , ଼ܽ ′ ∗ ଼ܾ ′ሻ 3.3. RANKING OF OCTAGONAL INTUITIONISTIC FUZZY NUMBERS The ranking function of Octagonal Intuitionistic Fuzzy Number (OIFN) ‫ܣ‬̅௢௖ ூ =ሺܽଵ, ܽଶ ,ܽଷ , ܽସ , ܽହ, ܽ଺, ܽ଻,଼ܽሻ (ܽଵ′, ܽଶ ′, ܽଷ ′ , ܽସ ′ , ܽହ ′ , ܽ଺ ′ , ܽ଻ ′ , ଼ܽ′ሻ maps the set of all Fuzzy numbers to a set of real numbers defined as R[‫ܣ‬̅௢௖ ூ ] = Max [ ‫݃ܽܯ‬ఓሺ‫ܣ‬̅௢௖ ூ ),‫݃ܽܯ‬ణሺ‫ܣ‬̅௢௖ ூ )] And similarly ܴ[ ‫ܤ‬ത௢௖ ூ ] = ‫݃ܽܯ [ ݔܽܯ‬ఓሺ‫ܤ‬ത௢௖ ூ ሻ, ‫݃ܽܯ‬ణሺ‫ܤ‬ത௢௖ ூ )], Where ‫݃ܽܯ‬ఓ (‫ܣ‬̅௢௖ ூ ሻ = ଶ௔భାଷ௔మ ାସ௔య ାହ ௔రାହ௔ఱାସ௔లାଷ௔ళାଶ௔ఴ ଶ଼ ‫݃ܽܯ‬ణ(‫ܣ‬̅௢௖ ூ ሻ= ଶ௔భ′ାଷ௔మ ′ାସ௔య′ାହ ௔ర′ାହ௔ఱ′ାସ௔ల′ାଷ௔ళ′ାଶ௔ఴ′ ଶ଼ And similarly ‫݃ܽܯ‬ఓ (‫ܤ‬ത௢௖ ூ ሻ = ଶ௕భାଷ௕మ ାସ௕య ାହ ௕రାହ௕ఱାସ௕లାଷ௕ళାଶ௕ఴ ଶ଼ ‫݃ܽܯ‬ణ(‫ܤ‬ത௢௖ ூ ሻ= ଶ௕భ′ାଷ௕మ ′ାସ௕య′ାହ ௕ర′ାହ௕ఱ′ାସ௕ల′ାଷ௕ళ′ାଶ௕ఴ′ ଶ଼ 3.4. MODI METHOD There are many methods to find the basic feasible solution, Modi method is heuristic method. The advantage of this method is that it gives an initial solution which is nearer to an optimal solution. Here in this paper Modi method is suitably modified and used to solving Intuitionistic Fuzzy transportation problem. PROPOSED ALGORITHM Step –1: In Octagonal Intuitionistic Fuzzy transportation problem (OIFN) the quantities are reduced into an integer using the ranking method called accuracy function. Step – 2: For an initial basic feasible solution with m + n -1 occupied cells, calculate ‫ݑ‬௜ and ‫ݒ‬௝ for rows and columns. The initial solution can be obtained by any of the three methods discussed earlier. To start with, any of‫ݑ‬௜ ’s or‫ݒ‬௝ ’sis assigned the value zero. It is better to assign zero for a particular ‫ݑ‬௜ or ‫ݒ‬ ௝ . Where there are maximum numbers of allocations in a row or column respectively, as it will reduce arithmetic work considerably. Then complete the calculation of ‫ݑ‬௜ ’s and‫ݒ‬௝ ’s for other rows and columns by using the relation.‫ܥ‬௜௝ = ‫ݑ‬௜ + ‫ݒ‬௃ for all occupied cells (i,j). Step – 3: For unoccupied cells, calculate opportunity cost by using the relationship ݀௜௝ = ‫ܥ‬௜௝ - (‫ݑ‬௜ + ‫ݒ‬௃ሻ for all i and j. Step – 4: Examine sign of each ݀௜௝ . If ݀௜௝ >0, then current basic feasible solution is optimal. If ݀௜௝ = 0, then current basic feasible solution will remain unaffected but an alternative solution exists. If one or more ݀௜௝ <0, then an improved solutions can be obtained by entering unoccupied cell (i,j) in the basis. An unoccupied cell having the largest negative value of ݀௜௝ is chosen for entering into the solution mix (new transportation schedule). Step – 5: Construct a closed path (or loop) for the unoccupied cell with largest negative opportunity cost. Start the closed path with the selected unoccupied cell and mark a plus sign (+) in this cell, trace a path along the rows (or columns) to an occupied cell, mark the corner with minus sign (-) and continue down the column (or row) to an occupied cell and mark the corner with plus sign (+) and minus sign (-) alternatively, close the path back to the selected unoccupied cell. Step – 6: Select the smallest quantity amongst the cells marked with minus sign on the corners of closed loop. Allocate this value to the selected unoccupied cell and add it to other occupied cells marked with plus signs and subtract it from the occupied cells marked with minus signs. Step – 7: Obtain a new improved solution by allocating units to the unoccupied cell according to step – 6 and calculate the new total transportation cost. Step – 8: Test the revised solution further for optimality. The procedure terminates when all ݀௜௝ ≥ 0, for unoccupied cells. 3.5. Numerical Example: Consider a 3×3 Octagonal Intuitionistic Fuzzy Number. D1 D2 D3 Supply S1 (1,2,3,4,5,6,7,8) (0,1,2,3,4,5,6,7) (3,4,5,6,7,8,9,10) (1,2,3,4,5,6,7,8) (6,7,8,9,10,11,12,13) (3,4,5,6,7,8,9,10) (3,4,5,6,7,8,9,10) (8,9,10,11,12,13,14,15) S2 (4,5,6,7,8,9,10,11) (1,2,3,4,5,6,7,10) (8,9,10,11,12,13,14,15) (3,4,5,6,7,8,9,10) (3,6,7,8,9,10,12,13) (2,3,4,5,6,7,8,9) (3,4,5,6,7,8,9,10) (6,7,8,9,10,11,12,13) S3 (5,6,7,8,9,10,11,12) (0,1,2,3,4,5,6,7) (7,8,9,10,11,12,13,14) (3,6,7,8,9,10,12,13) (4,5,6,7,8,9,10,11) (1,2,3,5,6,7,8,10) (1,2,3,5,6,7,8,13) (0,1,2,3,4,7,9,10) Demand (7,8,9,10,11,12,13,14) (3,6,7,8,9,10,12,13) (2,4,6,7,8,9,10,11) (1,2,3,4,5,6,7,10) (1,2,3,5,6,7,8,10) (4,5,6,7,8,9,10,11) ΣDemand = Σ Supply
International Journal of Trend in Scientific Research and Development (IJTSRD) @ www.ijtsrd.com eISSN: 2456-6470 @ IJTSRD | Unique Paper ID – IJTSRD31675 | Volume – 4 | Issue – 4 | May-June 2020 Page 1738 The problem is a balanced transportation problem. Using the proposed algorithm, the solution of the problem is as follows. Applying accuracy function on Octagonal Intuitionistic Fuzzy Number [(1,2,3,4,5,6,7,8) (0,1,2,3,4,5,6,7)], we have R (‫ܣ‬̅௢௖ ூ ሻ = Max [ ‫݃ܽܯ‬ఓሺ‫ܣ‬̅௢௖ ூ ),‫݃ܽܯ‬ణሺ‫ܣ‬̅௢௖ ூ )] = Max [ ଶା଺ାଵଶାଶ଴ାଶହାଶସାଶଵାଵ଺ ଶ଼ , ଴ାଷା଼ାଵହାଶ଴ାଶ଴ାଵ଼ାଵସ ଶ଼ ] = Max [4.5, 3.5] R (‫ܣ‬̅௢௖ ூ ሻ = 4.5 Similarly applying for all the values, we have the following table after ranking Table 1 - Reduced Table ࡮૚ ࡮૛ ࡮૜ Supply ‫ܣ‬ଵ 4.5 6.5 9.5 11.5 ‫ܣ‬ଶ 7.5 11.5 8.5 9.5 ‫ܣ‬ଷ 8.5 10.5 7.5 5.25 Demand 10.5 7.25 8.5 26.25 Applying VAM method, Table corresponding to initial basic feasible solution is Table 2 - Reduced Table of VAM Method ࡮૚ ࡮૛ ࡮૜ Supply ‫ܣ‬ଵ [4.25] 4.5 [7.25] 6.5 9.5 11.5 ‫ܣ‬ଶ [6.25] 7.5 11.5 [3.25] 8.5 9.5 ‫ܣ‬ଷ 8.5 10.5 [5.25] 7.5 5.25 Demand 10.5 7.25 8.5 26.25 Since the number of occupied cell m+n-1=5 and are also independent. There exists a non-negative basic feasible solution. The initial transportation cost is [(4.25 ×4.5) + (7.25×6.5) + (6.25×7.5) + (3.25×8.5) + (5.25×7.5)] = 180.125 Applying MODI method, table corresponding to optimal solution is Table 3 – Reduced Table Modi Method ࡮૚ ࡮૛ ࡮૜ Supply ࡭૚ [4.25] 4.5 [7.25] 6.5 (4) 9.5 11.5 ࡭૛ [6.25] 7.5 (2) 11.5 [3.25] 8.5 9.5 ࡭૜ (2) 8.5 (0.5) 10.5 [5.25] 7.5 5.25 Demand 10.5 7.25 8.5 26.25 Since all ݀௜௝ ≥ 0 the solution in optimum and unique. The solution is given by‫ݔ‬ଵଵ= 4.25, ‫ݔ‬ଵଶ = 7.25 , ‫ݔ‬ଶଵ= 6.25, ‫ݔ‬ଶଷ= 3.25, ‫ݔ‬ଷଷ= 5.25 The problem is a unbalanced transportation problem. Using the proposed algorithm, the solution of the problem is as follows. Applying accuracy function on Octagonal Intuitionistic Fuzzy Number [(1,2,3,4,5,6,7,9)(3,4,5,6,7,8,10,12)], we have R (‫ܣ‬̅௢௖ ூ ሻ = Max [ ‫݃ܽܯ‬ఓሺ‫ܣ‬̅௢௖ ூ ),‫݃ܽܯ‬ణሺ‫ܣ‬̅௢௖ ூ )] = Max [ ଶା଺ାଵଶାଶ଴ାଶହାଶସାଶଵାଵ଼ ଶ଼ , ଺ାଵଶାଶ଴ାଷ଴ାଷହାଷଶାଷ଴ାଶସ ଶ଼ ] = Max [4.5, 6.75] R (‫ܣ‬̅௢௖ ூ ሻ = 6.75 The optimal solution is = [ (4.25 × 4.5 ) +(7.25×6.5)+(6.25×7.5)+(3.25×8.5)+(5.25×7.5)] = 180.125 Using Octagonal Intuitionistic Fuzzy Numbers, Cost values are same but Supply and Demand values are not equal. Hence this is considered as Unbalanced Transportation Problem. The values are compared to find the minimum transportation cost.
International Journal of Trend in Scientific Research and Development (IJTSRD) @ www.ijtsrd.com eISSN: 2456-6470 @ IJTSRD | Unique Paper ID – IJTSRD31675 | Volume – 4 | Issue – 4 | May-June 2020 Page 1739 D1 D2 D3 Dummy Supply S1 (1,2,3,4,5,6,7,8) (0,1,2,3,4,5,6,7) (3,4,5,6,7,8,9,10) (1,2,3,4,5,6,7,8) (6,7,8,9,10,11,12,13) (3,4,5,6,7,8,9,10) (0,0,0,0,0,0,0,0) (0,0,0,0,0,0,0,0) (3,4,5,6,7,8,9,10) (8,9,10,11,12,13,14,15) S2 (4,5,6,7,8,9,10,11) (1,2,3,4,5,6,7,10) (8,9,10,11,12,13,14,15) (3,4,5,6,7,8,9,10) (3,6,7,8,9,10,12,13) (2,3,4,5,6,7,8,9) (0,0,0,0,0,0,0,0) (0,0,0,0,0,0,0,0) (3,4,5,6,7,8,9,10) (6,7,8,9,10,11,12,13) S3 (5,6,7,8,9,10,11,12) (0,1,2,3,4,5,6,7) (7,8,9,10,11,12,13,14) (3,6,7,8,9,10,12,13) (4,5,6,7,8,9,10,11) (1,2,3,5,6,7,8,10) (0,0,0,0,0,0,0,0) (0,0,0,0,0,0,0,0) (1,2,3,5,6,7,8,9,10) (0,1,2,3,4,5,6,7,8,13) Demand (1,2,3,4,5,6,7,9) (3,4,5,6,7,8,10,12) (4,5,6,7,8,9,10,11) (0,1,2,3,4,5,6,7) (1,2,3,4,5,6,7,8) (-1,0,1,2,3,4,5,6) (1,2,3,5,6,7,8,10) (4,5,6,7,8,9,10,11) Σ Demand ≠ Σ Supply Similarly applying for all the values, we have the following table after ranking Table 4 - Reduced Table ࡮૚ ࡮૛ ࡮૜ Dummy Supply ‫ܣ‬ଵ 4.5 6.5 9.5 0 11.5 ‫ܣ‬ଶ 7.5 11.5 8.5 0 9.5 ‫ܣ‬ଷ 8.5 10.5 7.5 0 5.25 Demand 6.75 7.5 3.5 8.5 26.25 Applying VAM method, Table corresponding to initial basic feasible solution is Table 5- Reduced Table of VAM Method ࡮૚ ࡮૛ ࡮૜ Dummy Supply ‫ܣ‬ଵ [4] 4.5 [7.5] 6.5 9.5 0 11.5 ‫ܣ‬ଶ [2.75] 7.5 11.5 [3.5] 8.5 [3.25] 0 9.5 ‫ܣ‬ଷ 8.5 10.5 7.5 [5.25] 0 5.25 Demand 6.75 7.5 3.5 8.5 26.25 Since the number of occupied cell m+n-1=5 and are also independent. There exists a non-negative basic feasible solution. The initial transportation cost is [(4 ×4.5) + (7.5×6.5) + (2.75×7.5) + (3.5×8.5) + (3.25×0)+(5.25× 0ሻ] = 117.125 Applying MODI method, table corresponding to optimal solution is Table 6 - Reduced Table of MODI Method ࡮૚ ࡮૛ ࡮૜ Dummy Supply ࡭૚ [4] 4.5 [7.5] 6.5 (4) 9.5 (3) 0 11.5 ࡭૛ [2.75] 7.5 (2) 11.5 [3.5] 8.5 ↓ [3.25] ← 0 9.5 ࡭૜ (1) 8.5 (1) 10.5 (-1) 7.5 → [5.25] 0 ↑ 5.25 Demand 6.75 7.5 3.5 8.5 26.25 Table 7 - Reduced Table of MODI Method ࡮૚ ࡮૛ ࡮૜ Dummy Supply ‫ܣ‬ଵ [4] 4.5 [7.5] 6.5 9.5 0 11.5 ‫ܣ‬ଶ [2.75] 7.5 11.5 8.5 [6.75] 0 9.5 ‫ܣ‬ଷ 8.5 10.5 [3.5] 7.5 [8.75] 0 5.25 Demand 6.75 7.5 3.5 8.5 26.25 Since all ݀௜௝ ≥ 0 the solution in optimum and unique. The solution is given by‫ݔ‬ଵଵ= 4, ‫ݔ‬ଵଶ = 7.5 , ‫ݔ‬ଶଵ= 2.75, ‫ݔ‬ଶସ= 6.75, ‫ݔ‬ଷଷ= 3.5, ‫ݔ‬ଷସ= 8.75 The optimal solution is = [ (4 × 4.5) +(7.5×6.5) +(2.75×7.5) +(6.75×0)+(3.75×7.5)+ (8.75×0)] = 113.625 4. CONCLUSION: In this paper, we discussed finding optimal solution for Octagonal Intuitionistic Fuzzy Balanced Transportation problem and Unbalanced Transportation problem. It is concluded that Unbalanced Transportation method proves to be cost effective. REFERENCES: [1] Fuzzy sets and K.Atanassov.1989. More on Intuitionistic Fuzzy sets, Fuzzy sets and systems, 33, pp.37-46. [2] Atanassov. K. T. “Intuitionistic Fuzzy Sets”, Fuzzy sets and systems, Vol.20 (1), pp: 87-96, (1986) [3] A. Thamaraiselvi and R. Santhi, “On Intuitionistic Fuzzy Transportation Problem Using Hexagonal Intuitionistic Fuzzy Numbers”, International Journal of Fuzzy Logic systems (IJFLS) Vol.5, No.1, January 2015. [4] Thangaraj Beaula – M. Priyadharshini, “ A New Algorithm for Finding a Fuzzy Optimal Solution for Intuitionistic Fuzzy Transportation Problems, International Journalof Applications of Fuzzy Sets and Artificial Intelligence ( ISSN 2241-1240), Vol.5(2015),183- 192. [5] Dr. S. Ismail Mohideen, K. Prasanna Devi, Durga, “Fuzzy Transportation Problem of Octagon Fuzzy Numbers with ߙ-Cut and Ranking Technique”, Dr.Ismail Mohideen et al, Journal of Computer – JoC, Vol.1 Issue.2, July-2016, pg-60-67. [6] Dr. Chandrasekaran, G. Gokila, Juno Saju, “ Ranking of Octagonal Fuzzy Numbers for Solving Multi Objective Fuzzy Linear Programming Problem with Simplex Method and Graphical Method, International Journal of Scientific Engineering and Applied Science (IJSEAS) – Volume -1, Issue-5, August-2015. [7] Dr. M. S. Annie Christi, Mrs. Kasthuri. B “Transportation Problem with Pentagonal Intuitionistic Fuzzy Numbers Solved Using Ranking Technique and Russell’s Method” , Journal of Engineering Research and Applications, ISSN: 2248 – 9622,Vol.6.Issue 2, (part-4), Feb 2016,pp.82- [8] L. A. Zadeh, “Fuzzy sets,Information and Control”, 8, 338-353, 1965.

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The New Ranking Method using Octagonal Intuitionistic Fuzzy Unbalanced Transportation Problem

  • 1.
    International Journal ofTrend in Scientific Research and Development (IJTSRD) Volume 4 Issue 4, June 2020 @ IJTSRD | Unique Paper ID – IJTSRD31675 The New Ranking Method Fuzzy Unbalanced Dr. 1Research Guide, Assistant Professor, Department of Mathematics Chikkanna Govt Arts College, Tirupur 2Research scholar, Assistant Professor, Department of Mathematics Sri Shakthi Institute of Engineering and Technology, ABSTRACT In this paper a new ranking method is proposed for finding an optimal solution for intuitionistic fuzzy unbalanced transportation problem, in which the costs, supplies and demands are octagonal intuitionistic fuzzy numbers. The procedure is illustrated with a numerical example. KEYWORDS: Unbalanced transportation problems, Octagonal Intuitionistic fuzzy numbers, Ranking method, Modi method, Initial Basic Feasible Solution, Optimal Solution 1. INTRODUCTION In general, Unbalanced transportation problems are solved with assumptions that the cost, supply and demand are specified in precise manner. However, in many cases the decision maker has no precise information about the coefficient belonging to the unbalanced transportation problem. Intuitionistic fuzzy set is a powerful tool to deal with such vagueness. L. A. Zadeh introduced fuzzy set theory in 1965. Different types of fuzzy sets are defined in order vagueness of the existing problems. The concept of Intuitionistic Fuzzy Sets (IFSs), proposed by Atanassov in [1] and [2], has been found to be highly useful to deal with vagueness. Many authors discussed the solutions of Fuzzy Transportation Problem (FTP) using various techniques. In 2016, Mrs. Kasthuri. Pentagonal intuitionistic fuzzy. In 2015, A. and R. Santhi [3] introduced Hexagonal Intuitionistic Fuzzy Numbers. In 2015, Thangaraj Beaula Priyadharshini [4] proposed A New Algorithm for Finding a Fuzzy Optimal Solution. K. Prasanna Devi, M. Devi Durga [5] and G. Gokila, Juno Saju [6] introduced Octagonal Fuzzy Number. end in Scientific Research and Development (IJTSRD) 2020 Available Online: www.ijtsrd.com e-ISSN: 2456 31675 | Volume – 4 | Issue – 4 | May-June 2020 The New Ranking Method using Octagonal Intuitionistic Unbalanced Transportation Problem Dr. P. Rajarajeswari1, G. Menaka2 Research Guide, Assistant Professor, Department of Mathematics Chikkanna Govt Arts College, Tirupur, Tamil Nadu, India ch scholar, Assistant Professor, Department of Mathematics Sri Shakthi Institute of Engineering and Technology, Coimbatore, Tamil In this paper a new ranking method is proposed for finding an optimal zy unbalanced transportation problem, in which the costs, supplies and demands are octagonal intuitionistic fuzzy numbers. The procedure is illustrated with a numerical example. Unbalanced transportation problems, Octagonal Intuitionistic y numbers, Ranking method, Modi method, Initial Basic Feasible Solution, How to cite this paper Rajarajeswari | G. Menaka "The New Ranking Method using Octagonal Intuitionistic Fuzzy Unbalanced Transportation Problem" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456 6470, Volume Issue-4, June 2020, pp.1735 www.ijtsrd.com/pa Copyright © 20 International Journal of Trend in Scientific Research and Development Journal. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (CC BY 4.0) (http://creativecommons.org/licenses/ by/4.0) In general, Unbalanced transportation problems are solved with assumptions that the cost, supply and demand are specified in precise manner. However, in many cases ormation about the coefficient belonging to the unbalanced transportation problem. Intuitionistic fuzzy set is a powerful tool to deal Zadeh introduced fuzzy set theory in 1965. Different types of fuzzy sets are defined in order to clear the The concept of Intuitionistic Fuzzy Sets (IFSs), proposed by Atanassov in [1] and [2], has been found to be highly useful to deal with vagueness. Many authors discussed the n Problem (FTP) using Kasthuri. B introduced Pentagonal intuitionistic fuzzy. In 2015, A. Thamaraiselvi and R. Santhi [3] introduced Hexagonal Intuitionistic Thangaraj Beaula – M. proposed A New Algorithm for Finding Prasanna Devi, M. Devi Durga Gokila, Juno Saju [6] introduced Octagonal Fuzzy In this paper, a new method is proposed for finding an optimal solution for intuitionistic transportation problem, in which the costs, supplies and Demands are octagonal intuitionistic fuzzy numbers. Using Octagonal Intuitionistic fuzzy unbalanced transportation problem we get best minimum value of optimal solution. The paper is organized as follows, In section 2 introduction with some basic concepts of fuzzy and Intuitionistic fuzzy definitions, In section 3 introduced Octagonal Intuitionistic Fuzzy Definition and proposed algorithm followed by a Numerical example using Modi method and finally the paper is concluded in section 4. 2. PRELIMINARIES 2.1. Definition (Fuzzy set Let X be a nonempty set. A fuzzy set as‫ܣ‬̅=ሼ൏ ‫,ݔ‬ ߤ஺ ഥ ሺxሻ ൐/‫ݔ‬ ∈ ܺሽ. Where membership function, which maps each element of X to a value between 0 and 1. 2.2. Definition (Fuzzy Number A fuzzy number is a generalization of a regular real number and which does not refer to a single value but rather to a connected a set of possible values, where each end in Scientific Research and Development (IJTSRD) ISSN: 2456 – 6470 2020 Page 1735 sing Octagonal Intuitionistic Transportation Problem Research Guide, Assistant Professor, Department of Mathematics, ch scholar, Assistant Professor, Department of Mathematics, Coimbatore, Tamil Nadu, India How to cite this paper: Dr. P. Rajarajeswari | G. Menaka "The New Ranking Method using Octagonal Intuitionistic Fuzzy Unbalanced ortation Problem" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456- 6470, Volume-4 | 4, June 2020, pp.1735-1739, URL: www.ijtsrd.com/papers/ijtsrd31675.pdf Copyright © 2020 by author(s) and International Journal of Trend in Scientific Research and Development Journal. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (CC BY 4.0) p://creativecommons.org/licenses/ In this paper, a new method is proposed for finding an optimal solution for intuitionistic fuzzy Unbalanced transportation problem, in which the costs, supplies and Demands are octagonal intuitionistic fuzzy numbers. Using Octagonal Intuitionistic fuzzy unbalanced transportation problem we get best minimum value of optimal solution. s organized as follows, In section 2 introduction with some basic concepts of fuzzy and Intuitionistic fuzzy definitions, In section 3 introduced Octagonal Intuitionistic Fuzzy Definition and proposed algorithm followed by a Numerical example using Modi thod and finally the paper is concluded in section 4. ሾࡲࡿሿ)[3] Let X be a nonempty set. A fuzzy set ‫ܣ‬̅ of Xis defined ሽ. Where ߤ஺ ഥ (x) is called membership function, which maps each element of X to a Definition (Fuzzy Numberሾࡲࡺሿ) [3] number is a generalization of a regular real number and which does not refer to a single value but rather to a connected a set of possible values, where each IJTSRD31675
  • 2.
    International Journal ofTrend in Scientific Research and Development (IJTSRD) @ www.ijtsrd.com eISSN: 2456-6470 @ IJTSRD | Unique Paper ID – IJTSRD31675 | Volume – 4 | Issue – 4 | May-June 2020 Page 1736 possible value has its weight between 0 and 1. The weight is called the membership function. A fuzzy number ‫ܣ‬̅ is a convex normalized fuzzy set on the real line R such that There exists at least one x ∈R with ߤ஺ ഥ ሺxሻ = 1. ߤ஺ ഥ ሺxሻߤ஺ ഥ ሺxሻ is piecewise continuous. 2.3. Definition (Intuitionistic Fuzzy Set [IFS]) [3] Let X be a non-empty set. An Intuitionistic fuzzy set ‫ܣ‬̅ூ of X is defined as ‫ܣ‬̅ூ = {<x,ߤ஺̅಺ሺ‫ݔ‬ሻ, ߴ஺̅಺(x) > / x∈ ܺ}. Where ߤ஺̅಺ሺ‫ݔ‬ሻ ܽ݊݀ ߴ஺̅಺(x) are membership and non-membership function. Such that ߤ஺̅಺ሺ‫ݔ‬ሻ, ߴ஺̅಺(x): X→ [0, 1] and 0≤ ߤ஺̅಺ሺ‫ݔ‬ሻ + ߴ஺̅಺(x) ≤ 1 for all x∈ ܺ. 2.4. Definition (Intuitionistic Fuzzy Number [IFN]) [3] An Intuitionistic Fuzzy Subset ‫ܣ‬̅ூ = {< x, ߤ஺̅಺ሺ‫ݔ‬ሻ, ߴ஺̅಺(x) > / x∈ ܺ} of the real line R is called an Intuitionistic Fuzzy Number, if the following conditions hold, There exists m ∈ R suct that ߤ஺̅಺ሺ݉ሻ = 1 ܽ݊݀ ߴ஺̅಺(m) = 0. ߤ஺̅಺ߤ஺̅಺ሺ‫ݔ‬ሻ is a continuous function from R → [0,1] such that 0≤ ߤ஺̅಺ሺ‫ݔ‬ሻ + ߴ஺̅಺(x) ≤1 for all x∈ ܺ. The membership and non- membership functions of ‫ܣ‬̅ூ are in the following form ߤ஺̅಺ሺ‫ݔ‬ሻ = ‫ە‬ ۖ ‫۔‬ ۖ ‫ۓ‬ 0 ݂‫ݎ݋‬ − ∞ < ‫ ݔ‬ ≤ ܽଵ ݂ሺ‫ݔ‬ሻ ݂‫ܽ ݎ݋‬ଵ ≤ ‫ ݔ‬ ≤ ܽଶ 1 ݂‫ݔ ݎ݋‬ = ܽଶ ݃ሺ‫ݔ‬ሻ ݂‫ܽ ݎ݋‬ଶ ≤ ‫ ݔ‬ ≤ ܽଷ 0 ݂‫ܽ ݎ݋‬ଷ ≤ ‫ݔ‬ < ∞ ۙ ۖ ۘ ۖ ۗ ߴ஺̅಺(x) = ‫ە‬ ۖ ‫۔‬ ۖ ‫ۓ‬ 1 ݂‫ ݎ݋‬ − ∞ < ‫ ݔ‬ ≤ ܽଵ ′ ݂′ሺ‫ݔ‬ሻ ݂‫ܽ ݎ݋‬ଵ′ ≤ ‫ ݔ‬ ≤ ܽଶ 0 ݂‫ݔ ݎ݋‬ = ܽଶ ݃′ሺ‫ݔ‬ሻ ݂‫ܽ ݎ݋‬ଶ ≤ ‫ ݔ‬ ≤ ܽଷ′ 1 ݂‫ܽ ݎ݋‬ଷ′ ≤ ‫ݔ‬ < ∞ ۙ ۖ ۘ ۖ ۗ Where݂,݂′ , ݃, ݃′ are functions from R → [0,1] . ݂ and ݃′ are strictly increasing functions and ݃ and ݂′ are strictly decreasing functions with the conditions 0 ≤ ݂ሺ‫ݔ‬ሻ+ ݂′ሺ‫ݔ‬ሻ ≤ 1 and 0 ≤ ݃ሺ‫ݔ‬ሻ + ݃′ሺ‫ݔ‬ሻ ≤ 1. 3. OCTAGONAL INTUITIONISTIC FUZZY NUMBER 3.1. Definition (Octagonal Intuitionistic Fuzzy Number [OIFN]) An Octagonal Intuitionistic Fuzzy Number is specified by ‫ܣ‬̅௢௖ ூ =ሺܽଵ, ܽଶ ,ܽଷ , ܽସ , ܽହ, ܽ଺, ܽ଻,଼ܽሻ , (ܽଵ′, ܽଶ ′,ܽଷ ′ , ܽସ , ܽହ, ܽ଺ ′ , ܽ଻ ′ , ଼ܽ′ሻ. Whereܽଵ, ܽଶ ,ܽଷ , ܽସ , ܽହ, ܽ଺, ܽ଻,଼ܽ,ܽଵ′, ܽଶ ′,ܽଷ ′ , ܽ଺ ′ , ܽ଻ ′ ܽ݊݀ ଼ܽ ′ and its membership and non-membership functions are given below ߤ஺ ഥ ሺxሻ = ‫ە‬ ۖ ۖ ۖ ۖ ۖ ‫۔‬ ۖ ۖ ۖ ۖ ۖ ‫ۓ‬ 0 ݂‫ݔ ݎ݋‬ < ܽଵ ݇ ൬ ‫ ݔ‬ − ܽଵ ܽଶ − ܽଵ ൰ ݂‫ܽ ݎ݋‬ଵ ≤ ‫ ݔ‬ ≤ ܽଶ ݇ ݂‫ܽ ݎ݋‬ଶ ≤ ‫ ݔ‬ ≤ ܽଷ ݇ + ሺ1 − ݇ሻ ሺ ‫ ݔ‬ − ܽଷ ܽସ − ܽଷ ሻ ݂‫ܽ ݎ݋‬ଷ ≤ ‫ ݔ‬ ≤ ܽସ 1 ݂‫ܽ ݎ݋‬ସ ≤ ‫ ݔ‬ ≤ ܽହ ݇ + ሺ1 − ݇ሻ ൬ ܽ଺ − ‫ݔ‬ ܽ଺ − ܽହ ൰ ݂‫ܽ ݎ݋‬ହ ≤ ‫ ݔ‬ ≤ ܽ଺ ݇ ݂‫ܽ ݎ݋‬଺ ≤ ‫ ݔ‬ ≤ ܽ଻ ݇ሺ ଼ܽ − ‫ݔ‬ ଼ܽ − ܽ଻ ሻ ݂‫ܽ ݎ݋‬଻ ≤ ‫ ݔ‬ ≤ ଼ܽ 0 ݂‫ݔ ݎ݋‬ > ଼ܽ ۙ ۖ ۖ ۖ ۖ ۖ ۘ ۖ ۖ ۖ ۖ ۖ ۗ ߴ஺̅಺ሺxሻ = ‫ە‬ ۖ ۖ ۖ ۖ ۖ ۖ ‫۔‬ ۖ ۖ ۖ ۖ ۖ ۖ ‫ۓ‬ 1 ݂‫ܽ ݎ݋‬ଵ ′ < ‫ݔ‬ ݇ + ሺ1 − ݇ሻ ቆ ܽଶ′ − ‫ݔ‬ ܽଶ′ − ܽଵ′ ቇ ݂‫ܽ ݎ݋‬ଵ′ ≤ ‫ ݔ‬ ≤ ܽଶ′ ݇ ݂‫ܽ ݎ݋‬ଶ′ ≤ ‫ ݔ‬ ≤ ܽଷ′ ݇ሺ ܽସ − ‫ݔ‬ ܽସ − ܽଷ′ ሻ ݂‫ܽ ݎ݋‬ଷ′ ≤ ‫ ݔ‬ ≤ ܽସ 0 ݂‫ܽ ݎ݋‬ସ ≤ ‫ ݔ‬ ≤ ܽହ ݇ ൬ ‫ ݔ‬ − ܽହ ܽ଺′ − ܽହ ൰ ݂‫ܽ ݎ݋‬ହ ≤ ‫ ݔ‬ ≤ ܽ଺′ ݇ ݂‫ܽ ݎ݋‬଺′ ≤ ‫ ݔ‬ ≤ ܽ଻′ ݇ + ሺ1 − ݇ሻ ሺ ‫ ݔ‬ − ܽ଻′ ଼ܽ′ − ܽ଻′ ሻ ݂‫ܽ ݎ݋‬଻′ ≤ ‫ ݔ‬ ≤ ଼ܽ′ 1 ݂‫ݔ ݎ݋‬ > ଼ܽ′ ۙ ۖ ۖ ۖ ۖ ۖ ۖ ۘ ۖ ۖ ۖ ۖ ۖ ۖ ۗ Graphical representation of Octagonal Intuitionistic Fuzzy Numbers ___ Membership Functionߤ஺ ഥ ሺxሻ ---- Non-Membership Functionߴ஺̅಺ሺxሻ 3.2. Arithmetic operations on Octagonal Intuitionistic Fuzzy Numbers Let ‫ܣ‬̅௢௖ ூ =ሺܽଵ, ܽଶ ,ܽଷ , ܽସ , ܽହ, ܽ଺, ܽ଻,଼ܽሻ (ܽଵ′, ܽଶ ′,ܽଷ ′ , ܽସ ′ , ܽହ ′ , ܽ଺ ′ , ܽ଻ ′ , ଼ܽ′ሻ and ‫ܤ‬ത௢௖ ூ = ሺܾଵ, ܾଶ ,ܾଷ , ܾସ , ܾହ, ܾ଺, ܾ଻,଼ܾሻ (ܾଵ′, ܾଶ ′,ܾଷ ′ , ܾସ ′ , ܾହ ′ , ܾ଺ ′ , ܾ଻ ′ , ଼ܾ′ሻ be two Octagonal Intuitionistic Fuzzy Numbers, then the arithmetic operations are as follows. 3.2.1. Addition: ‫ܣ‬̅௢௖ ூ + ‫ܤ‬ത௢௖ ூ = ሺ ܽଵ + ܾଵ , ܽଶ + ܾଶ , ܽଷ + ܾଷ, ܽସ + ܾସ, ܽହ + ܾହ , ܽ଺ + ܾ଺ , ܽ଻ + ܾ଻ , ଼ܽ + ଼ܾሻ ሺܽଵ ′ + ܾଵ ′ ,ܽଶ ′ + ܾଶ ′ , ܽଷ ′ + ܾଷ ′ ,ܽସ ′ + ܾସ ′ , ܽହ ′ + ܾହ ′ , ܽ଺′+ܾ଺ ′ , ܽ଻ ′ + ܾ଻ ′ , ଼ܽ ′ + ଼ܾ ′ሻ 3.2.2. Subtraction: ‫ܣ‬̅௢௖ ூ − ‫ܤ‬ത௢௖ ூ = ሺ ܽଵ − ଼ܾ , ܽଶ − ܾ଻ , ܽଷ − ܾ଺, ܽସ − ܾହ, ܽହ − ܾସ , ܽ଺ − ܾଷ , ܽ଻ − ܾଶ , ଼ܽ − ܾଵሻ
  • 3.
    International Journal ofTrend in Scientific Research and Development (IJTSRD) @ www.ijtsrd.com eISSN: 2456-6470 @ IJTSRD | Unique Paper ID – IJTSRD31675 | Volume – 4 | Issue – 4 | May-June 2020 Page 1737 ሺܽଵ ′ − ଼ܾ ′ ,ܽଶ ′ − ܾ଻ ′ , ܽଷ ′ − ܾ଺ ′ ,ܽସ ′ − ܾହ ′ , ܽହ ′ − ܾସ ′ , ܽ଺′- ܾଷ ′ , ܽ଻ ′ − ܾଶ ′ , ଼ܽ ′ − ܾଵ ′ሻ 3.2.3. Multiplication: ‫ܣ‬̅௢௖ ூ ∗ ‫ܤ‬ത௢௖ ூ = ሺ ܽଵ ∗ ܾଵ , ܽଶ ∗ ܾଶ , ܽଷ ∗ ܾଷ, ܽସ ∗ ܾସ, ܽହ ∗ ܾହ , ܽ଺ ∗ ܾ଺ , ܽ଻ ∗ ܾ଻ , ଼ܽ ∗ ଼ܾሻ ሺܽଵ ′ ∗ ܾଵ ′ ,ܽଶ ′ ∗ ܾଶ ′ , ܽଷ ′ ∗ ܾଷ ′ ,ܽସ ′ ∗ ܾସ ′ , ܽହ ′ ∗ ܾହ ′ , ܽ଺′ ∗ ܾ଺ ′ , ܽ଻ ′ ∗ ܾ଻ ′ , ଼ܽ ′ ∗ ଼ܾ ′ሻ 3.3. RANKING OF OCTAGONAL INTUITIONISTIC FUZZY NUMBERS The ranking function of Octagonal Intuitionistic Fuzzy Number (OIFN) ‫ܣ‬̅௢௖ ூ =ሺܽଵ, ܽଶ ,ܽଷ , ܽସ , ܽହ, ܽ଺, ܽ଻,଼ܽሻ (ܽଵ′, ܽଶ ′, ܽଷ ′ , ܽସ ′ , ܽହ ′ , ܽ଺ ′ , ܽ଻ ′ , ଼ܽ′ሻ maps the set of all Fuzzy numbers to a set of real numbers defined as R[‫ܣ‬̅௢௖ ூ ] = Max [ ‫݃ܽܯ‬ఓሺ‫ܣ‬̅௢௖ ூ ),‫݃ܽܯ‬ణሺ‫ܣ‬̅௢௖ ூ )] And similarly ܴ[ ‫ܤ‬ത௢௖ ூ ] = ‫݃ܽܯ [ ݔܽܯ‬ఓሺ‫ܤ‬ത௢௖ ூ ሻ, ‫݃ܽܯ‬ణሺ‫ܤ‬ത௢௖ ூ )], Where ‫݃ܽܯ‬ఓ (‫ܣ‬̅௢௖ ூ ሻ = ଶ௔భାଷ௔మ ାସ௔య ାହ ௔రାହ௔ఱାସ௔లାଷ௔ళାଶ௔ఴ ଶ଼ ‫݃ܽܯ‬ణ(‫ܣ‬̅௢௖ ூ ሻ= ଶ௔భ′ାଷ௔మ ′ାସ௔య′ାହ ௔ర′ାହ௔ఱ′ାସ௔ల′ାଷ௔ళ′ାଶ௔ఴ′ ଶ଼ And similarly ‫݃ܽܯ‬ఓ (‫ܤ‬ത௢௖ ூ ሻ = ଶ௕భାଷ௕మ ାସ௕య ାହ ௕రାହ௕ఱାସ௕లାଷ௕ళାଶ௕ఴ ଶ଼ ‫݃ܽܯ‬ణ(‫ܤ‬ത௢௖ ூ ሻ= ଶ௕భ′ାଷ௕మ ′ାସ௕య′ାହ ௕ర′ାହ௕ఱ′ାସ௕ల′ାଷ௕ళ′ାଶ௕ఴ′ ଶ଼ 3.4. MODI METHOD There are many methods to find the basic feasible solution, Modi method is heuristic method. The advantage of this method is that it gives an initial solution which is nearer to an optimal solution. Here in this paper Modi method is suitably modified and used to solving Intuitionistic Fuzzy transportation problem. PROPOSED ALGORITHM Step –1: In Octagonal Intuitionistic Fuzzy transportation problem (OIFN) the quantities are reduced into an integer using the ranking method called accuracy function. Step – 2: For an initial basic feasible solution with m + n -1 occupied cells, calculate ‫ݑ‬௜ and ‫ݒ‬௝ for rows and columns. The initial solution can be obtained by any of the three methods discussed earlier. To start with, any of‫ݑ‬௜ ’s or‫ݒ‬௝ ’sis assigned the value zero. It is better to assign zero for a particular ‫ݑ‬௜ or ‫ݒ‬ ௝ . Where there are maximum numbers of allocations in a row or column respectively, as it will reduce arithmetic work considerably. Then complete the calculation of ‫ݑ‬௜ ’s and‫ݒ‬௝ ’s for other rows and columns by using the relation.‫ܥ‬௜௝ = ‫ݑ‬௜ + ‫ݒ‬௃ for all occupied cells (i,j). Step – 3: For unoccupied cells, calculate opportunity cost by using the relationship ݀௜௝ = ‫ܥ‬௜௝ - (‫ݑ‬௜ + ‫ݒ‬௃ሻ for all i and j. Step – 4: Examine sign of each ݀௜௝ . If ݀௜௝ >0, then current basic feasible solution is optimal. If ݀௜௝ = 0, then current basic feasible solution will remain unaffected but an alternative solution exists. If one or more ݀௜௝ <0, then an improved solutions can be obtained by entering unoccupied cell (i,j) in the basis. An unoccupied cell having the largest negative value of ݀௜௝ is chosen for entering into the solution mix (new transportation schedule). Step – 5: Construct a closed path (or loop) for the unoccupied cell with largest negative opportunity cost. Start the closed path with the selected unoccupied cell and mark a plus sign (+) in this cell, trace a path along the rows (or columns) to an occupied cell, mark the corner with minus sign (-) and continue down the column (or row) to an occupied cell and mark the corner with plus sign (+) and minus sign (-) alternatively, close the path back to the selected unoccupied cell. Step – 6: Select the smallest quantity amongst the cells marked with minus sign on the corners of closed loop. Allocate this value to the selected unoccupied cell and add it to other occupied cells marked with plus signs and subtract it from the occupied cells marked with minus signs. Step – 7: Obtain a new improved solution by allocating units to the unoccupied cell according to step – 6 and calculate the new total transportation cost. Step – 8: Test the revised solution further for optimality. The procedure terminates when all ݀௜௝ ≥ 0, for unoccupied cells. 3.5. Numerical Example: Consider a 3×3 Octagonal Intuitionistic Fuzzy Number. D1 D2 D3 Supply S1 (1,2,3,4,5,6,7,8) (0,1,2,3,4,5,6,7) (3,4,5,6,7,8,9,10) (1,2,3,4,5,6,7,8) (6,7,8,9,10,11,12,13) (3,4,5,6,7,8,9,10) (3,4,5,6,7,8,9,10) (8,9,10,11,12,13,14,15) S2 (4,5,6,7,8,9,10,11) (1,2,3,4,5,6,7,10) (8,9,10,11,12,13,14,15) (3,4,5,6,7,8,9,10) (3,6,7,8,9,10,12,13) (2,3,4,5,6,7,8,9) (3,4,5,6,7,8,9,10) (6,7,8,9,10,11,12,13) S3 (5,6,7,8,9,10,11,12) (0,1,2,3,4,5,6,7) (7,8,9,10,11,12,13,14) (3,6,7,8,9,10,12,13) (4,5,6,7,8,9,10,11) (1,2,3,5,6,7,8,10) (1,2,3,5,6,7,8,13) (0,1,2,3,4,7,9,10) Demand (7,8,9,10,11,12,13,14) (3,6,7,8,9,10,12,13) (2,4,6,7,8,9,10,11) (1,2,3,4,5,6,7,10) (1,2,3,5,6,7,8,10) (4,5,6,7,8,9,10,11) ΣDemand = Σ Supply
  • 4.
    International Journal ofTrend in Scientific Research and Development (IJTSRD) @ www.ijtsrd.com eISSN: 2456-6470 @ IJTSRD | Unique Paper ID – IJTSRD31675 | Volume – 4 | Issue – 4 | May-June 2020 Page 1738 The problem is a balanced transportation problem. Using the proposed algorithm, the solution of the problem is as follows. Applying accuracy function on Octagonal Intuitionistic Fuzzy Number [(1,2,3,4,5,6,7,8) (0,1,2,3,4,5,6,7)], we have R (‫ܣ‬̅௢௖ ூ ሻ = Max [ ‫݃ܽܯ‬ఓሺ‫ܣ‬̅௢௖ ூ ),‫݃ܽܯ‬ణሺ‫ܣ‬̅௢௖ ூ )] = Max [ ଶା଺ାଵଶାଶ଴ାଶହାଶସାଶଵାଵ଺ ଶ଼ , ଴ାଷା଼ାଵହାଶ଴ାଶ଴ାଵ଼ାଵସ ଶ଼ ] = Max [4.5, 3.5] R (‫ܣ‬̅௢௖ ூ ሻ = 4.5 Similarly applying for all the values, we have the following table after ranking Table 1 - Reduced Table ࡮૚ ࡮૛ ࡮૜ Supply ‫ܣ‬ଵ 4.5 6.5 9.5 11.5 ‫ܣ‬ଶ 7.5 11.5 8.5 9.5 ‫ܣ‬ଷ 8.5 10.5 7.5 5.25 Demand 10.5 7.25 8.5 26.25 Applying VAM method, Table corresponding to initial basic feasible solution is Table 2 - Reduced Table of VAM Method ࡮૚ ࡮૛ ࡮૜ Supply ‫ܣ‬ଵ [4.25] 4.5 [7.25] 6.5 9.5 11.5 ‫ܣ‬ଶ [6.25] 7.5 11.5 [3.25] 8.5 9.5 ‫ܣ‬ଷ 8.5 10.5 [5.25] 7.5 5.25 Demand 10.5 7.25 8.5 26.25 Since the number of occupied cell m+n-1=5 and are also independent. There exists a non-negative basic feasible solution. The initial transportation cost is [(4.25 ×4.5) + (7.25×6.5) + (6.25×7.5) + (3.25×8.5) + (5.25×7.5)] = 180.125 Applying MODI method, table corresponding to optimal solution is Table 3 – Reduced Table Modi Method ࡮૚ ࡮૛ ࡮૜ Supply ࡭૚ [4.25] 4.5 [7.25] 6.5 (4) 9.5 11.5 ࡭૛ [6.25] 7.5 (2) 11.5 [3.25] 8.5 9.5 ࡭૜ (2) 8.5 (0.5) 10.5 [5.25] 7.5 5.25 Demand 10.5 7.25 8.5 26.25 Since all ݀௜௝ ≥ 0 the solution in optimum and unique. The solution is given by‫ݔ‬ଵଵ= 4.25, ‫ݔ‬ଵଶ = 7.25 , ‫ݔ‬ଶଵ= 6.25, ‫ݔ‬ଶଷ= 3.25, ‫ݔ‬ଷଷ= 5.25 The problem is a unbalanced transportation problem. Using the proposed algorithm, the solution of the problem is as follows. Applying accuracy function on Octagonal Intuitionistic Fuzzy Number [(1,2,3,4,5,6,7,9)(3,4,5,6,7,8,10,12)], we have R (‫ܣ‬̅௢௖ ூ ሻ = Max [ ‫݃ܽܯ‬ఓሺ‫ܣ‬̅௢௖ ூ ),‫݃ܽܯ‬ణሺ‫ܣ‬̅௢௖ ூ )] = Max [ ଶା଺ାଵଶାଶ଴ାଶହାଶସାଶଵାଵ଼ ଶ଼ , ଺ାଵଶାଶ଴ାଷ଴ାଷହାଷଶାଷ଴ାଶସ ଶ଼ ] = Max [4.5, 6.75] R (‫ܣ‬̅௢௖ ூ ሻ = 6.75 The optimal solution is = [ (4.25 × 4.5 ) +(7.25×6.5)+(6.25×7.5)+(3.25×8.5)+(5.25×7.5)] = 180.125 Using Octagonal Intuitionistic Fuzzy Numbers, Cost values are same but Supply and Demand values are not equal. Hence this is considered as Unbalanced Transportation Problem. The values are compared to find the minimum transportation cost.
  • 5.
    International Journal ofTrend in Scientific Research and Development (IJTSRD) @ www.ijtsrd.com eISSN: 2456-6470 @ IJTSRD | Unique Paper ID – IJTSRD31675 | Volume – 4 | Issue – 4 | May-June 2020 Page 1739 D1 D2 D3 Dummy Supply S1 (1,2,3,4,5,6,7,8) (0,1,2,3,4,5,6,7) (3,4,5,6,7,8,9,10) (1,2,3,4,5,6,7,8) (6,7,8,9,10,11,12,13) (3,4,5,6,7,8,9,10) (0,0,0,0,0,0,0,0) (0,0,0,0,0,0,0,0) (3,4,5,6,7,8,9,10) (8,9,10,11,12,13,14,15) S2 (4,5,6,7,8,9,10,11) (1,2,3,4,5,6,7,10) (8,9,10,11,12,13,14,15) (3,4,5,6,7,8,9,10) (3,6,7,8,9,10,12,13) (2,3,4,5,6,7,8,9) (0,0,0,0,0,0,0,0) (0,0,0,0,0,0,0,0) (3,4,5,6,7,8,9,10) (6,7,8,9,10,11,12,13) S3 (5,6,7,8,9,10,11,12) (0,1,2,3,4,5,6,7) (7,8,9,10,11,12,13,14) (3,6,7,8,9,10,12,13) (4,5,6,7,8,9,10,11) (1,2,3,5,6,7,8,10) (0,0,0,0,0,0,0,0) (0,0,0,0,0,0,0,0) (1,2,3,5,6,7,8,9,10) (0,1,2,3,4,5,6,7,8,13) Demand (1,2,3,4,5,6,7,9) (3,4,5,6,7,8,10,12) (4,5,6,7,8,9,10,11) (0,1,2,3,4,5,6,7) (1,2,3,4,5,6,7,8) (-1,0,1,2,3,4,5,6) (1,2,3,5,6,7,8,10) (4,5,6,7,8,9,10,11) Σ Demand ≠ Σ Supply Similarly applying for all the values, we have the following table after ranking Table 4 - Reduced Table ࡮૚ ࡮૛ ࡮૜ Dummy Supply ‫ܣ‬ଵ 4.5 6.5 9.5 0 11.5 ‫ܣ‬ଶ 7.5 11.5 8.5 0 9.5 ‫ܣ‬ଷ 8.5 10.5 7.5 0 5.25 Demand 6.75 7.5 3.5 8.5 26.25 Applying VAM method, Table corresponding to initial basic feasible solution is Table 5- Reduced Table of VAM Method ࡮૚ ࡮૛ ࡮૜ Dummy Supply ‫ܣ‬ଵ [4] 4.5 [7.5] 6.5 9.5 0 11.5 ‫ܣ‬ଶ [2.75] 7.5 11.5 [3.5] 8.5 [3.25] 0 9.5 ‫ܣ‬ଷ 8.5 10.5 7.5 [5.25] 0 5.25 Demand 6.75 7.5 3.5 8.5 26.25 Since the number of occupied cell m+n-1=5 and are also independent. There exists a non-negative basic feasible solution. The initial transportation cost is [(4 ×4.5) + (7.5×6.5) + (2.75×7.5) + (3.5×8.5) + (3.25×0)+(5.25× 0ሻ] = 117.125 Applying MODI method, table corresponding to optimal solution is Table 6 - Reduced Table of MODI Method ࡮૚ ࡮૛ ࡮૜ Dummy Supply ࡭૚ [4] 4.5 [7.5] 6.5 (4) 9.5 (3) 0 11.5 ࡭૛ [2.75] 7.5 (2) 11.5 [3.5] 8.5 ↓ [3.25] ← 0 9.5 ࡭૜ (1) 8.5 (1) 10.5 (-1) 7.5 → [5.25] 0 ↑ 5.25 Demand 6.75 7.5 3.5 8.5 26.25 Table 7 - Reduced Table of MODI Method ࡮૚ ࡮૛ ࡮૜ Dummy Supply ‫ܣ‬ଵ [4] 4.5 [7.5] 6.5 9.5 0 11.5 ‫ܣ‬ଶ [2.75] 7.5 11.5 8.5 [6.75] 0 9.5 ‫ܣ‬ଷ 8.5 10.5 [3.5] 7.5 [8.75] 0 5.25 Demand 6.75 7.5 3.5 8.5 26.25 Since all ݀௜௝ ≥ 0 the solution in optimum and unique. The solution is given by‫ݔ‬ଵଵ= 4, ‫ݔ‬ଵଶ = 7.5 , ‫ݔ‬ଶଵ= 2.75, ‫ݔ‬ଶସ= 6.75, ‫ݔ‬ଷଷ= 3.5, ‫ݔ‬ଷସ= 8.75 The optimal solution is = [ (4 × 4.5) +(7.5×6.5) +(2.75×7.5) +(6.75×0)+(3.75×7.5)+ (8.75×0)] = 113.625 4. CONCLUSION: In this paper, we discussed finding optimal solution for Octagonal Intuitionistic Fuzzy Balanced Transportation problem and Unbalanced Transportation problem. It is concluded that Unbalanced Transportation method proves to be cost effective. REFERENCES: [1] Fuzzy sets and K.Atanassov.1989. More on Intuitionistic Fuzzy sets, Fuzzy sets and systems, 33, pp.37-46. [2] Atanassov. K. T. “Intuitionistic Fuzzy Sets”, Fuzzy sets and systems, Vol.20 (1), pp: 87-96, (1986) [3] A. Thamaraiselvi and R. Santhi, “On Intuitionistic Fuzzy Transportation Problem Using Hexagonal Intuitionistic Fuzzy Numbers”, International Journal of Fuzzy Logic systems (IJFLS) Vol.5, No.1, January 2015. [4] Thangaraj Beaula – M. Priyadharshini, “ A New Algorithm for Finding a Fuzzy Optimal Solution for Intuitionistic Fuzzy Transportation Problems, International Journalof Applications of Fuzzy Sets and Artificial Intelligence ( ISSN 2241-1240), Vol.5(2015),183- 192. [5] Dr. S. Ismail Mohideen, K. Prasanna Devi, Durga, “Fuzzy Transportation Problem of Octagon Fuzzy Numbers with ߙ-Cut and Ranking Technique”, Dr.Ismail Mohideen et al, Journal of Computer – JoC, Vol.1 Issue.2, July-2016, pg-60-67. [6] Dr. Chandrasekaran, G. Gokila, Juno Saju, “ Ranking of Octagonal Fuzzy Numbers for Solving Multi Objective Fuzzy Linear Programming Problem with Simplex Method and Graphical Method, International Journal of Scientific Engineering and Applied Science (IJSEAS) – Volume -1, Issue-5, August-2015. [7] Dr. M. S. Annie Christi, Mrs. Kasthuri. B “Transportation Problem with Pentagonal Intuitionistic Fuzzy Numbers Solved Using Ranking Technique and Russell’s Method” , Journal of Engineering Research and Applications, ISSN: 2248 – 9622,Vol.6.Issue 2, (part-4), Feb 2016,pp.82- [8] L. A. Zadeh, “Fuzzy sets,Information and Control”, 8, 338-353, 1965.