EAI Endorsed Transactions on Energy Web and Information Technologies Implementation of Fuzzy Intuitionistic Algorithm for Traveling Salesman Problem

Traveling Salesman Problem is one of the motivating problem in classical and advanced Optimization. In this work, theoretical analysis and relative study of Traveling Salesman Problem in Intuitionistic Fuzzy Optimization is examined with real examples.


Introduction
Traveling salesman problem is a NP hard problem in classical combinative Optimization that has intrigued mathematicians and computer scientists for years.In TSP a salesman have to sojourn m cities where the interval (or expensive or duration or some other factor) of journey within some two cities is determined to him.The salesman should originate from a certain city and journey over each and every city preceding as well as coming to his beginning city.Thus the salesman possibly will decide (m-1)!Distinctive probable way.The dispute is toward hypothesizing the best.The optimal result is unconstrained of the choice of the origin city [10,16 ].
By governing in a Fuzzy neighbourhood represent a solving procedure in that the design and/or the limitations, however not at all necessary that the method to manipulate are in Fuzzy manner [11].An index of literatures which was used Fuzzy set to determine the traveling salesman problem is specified in table 1. (The index is no more of abundant role).Applying Fuzzy multi-objective linear programming concept, Symmetric traveling salesman problem with multiple objectives with Fuzzy is transforming to a linear programming problem.The choice of path for the problem is completed by means of executing objective level factors.In Intuitionistic Fuzzy neighborhood the range of rejection as well as the range of uncertainty ought to examine to obtain the optimal solution for the traveling salesman problem.
The proposed Intuitionistic Fuzzy approach is an extent and composite from FTSP [4], [5] and Intuitionistic Fuzzy choice devising method [2].Angelov [2] introduced the Optimization method in Intuitionistic Fuzzy and it is widely studied.In that method, the range of acknowledgement is enhanced while the range of rejection is reduced.But in proposed Intuitionistic Fuzzy approach the range of acknowledgment is enhanced and the range of rejection is reduced other than also the range of uncertainty is reduced [7].
This effort is an explanation about the Intuitionistic Fuzzy approach and proposed Intuitionistic Fuzzy approach technique.An appropriate paradigm is studied for traveling salesman problem in Intuitionistic Fuzzy neighborhood.

Definition [11]
A membership function for a Fuzzy subset Ã of X is given as , that assigns to every X, a real number in the closed unit interval [0, 1], where the value of at x represents the grade of membership of in Ã.

Definition [8]
An Intuitionistic Fuzzy set A in X is given as ,wherever and with the stipulation 0 ≤ ( + ) ≤ 1, wherever and symbolize the grade of membership and non-membership correspondingly.It is obvious to for every Fuzzy set Ã , there exist an Intuitionistic Fuzzy As well, for every Intuitionistic Fuzzy set in X, there is a set = and it is called the grade of uncertainty or Intuitionistic index of in A. It is obvious that 0 ≤ ≤ 1 used for each in universal set.

Methodology of Multi Objective Traveling salesman Problem with Fuzzy Linear Programming
Bellman and Zadeh [11] primarily projected the choice devising notion in Fuzzy atmosphere with numerous objectives.The projection for converting multiple objective linear programming to a single objective linear programming in Fuzzy was inflicted by Zimmerman [20] where,  is on the whole approval level achieved with respect to the result.
Think the condition while choice maker have to decide most good result of TSP with reduced expensive, interval, duration.The city from i to j could be linked as a particular objective functions as x ij (7) Let the expensive of traveling from the city i the city j is defined as e ij and 0 1 z be the corresponding function for objective function for the reduction of expensive & t 1 be toleration, then Let the interval traveling from the city i to the city j is defined as i ij and 0 2 z be the corresponding function for objective function for the reduction of interval & t 2 be toleration, then Let the duration spent in traveling from the city i to the city j is defined as d ij and 0 3 z be the corresponding function for objective function for the reduction of total duration & t 3 be toleration, Let the fuel expenses spent in traveling between the city i to the city j is defined as fe ij and 0 4 z is the corresponding function for objective function for the reduction of total fuel expenses and t 4 is the toleration then A limitation is forced that every city ought to enter exactly one of its nearest city and vice versa, that is Order should not be chosen more than one duration, that is and non-negative limitations 0 x ij  .In vector mode, the membership functions in Fuzzy are defined by the limitations for being objective functions.

Methodology of Proposed method with Intuitionistic Fuzzy Optimization
In Intuitionistic Fuzzy set [8], [9] it is examine not simply the grade of membership to a definite set, however also the grade of rejection such that the summary of both these values is less than or equal to one.Complement of this sum to '1' is known as Intuitionistic Fuzzy index (grade of uncertainty).Using this approach the Fuzzy Optimization problem was reformulated by Angelov [2] using this notion.In intuiotionistic Fuzzy sets the grade of rejection (non-membership) is defined concurrently with the grade of acknowledgement (membership) and while together these grades are neither complementary to one another.[18] considered the third parameter (grade of uncertainty) while scheming the Euclidean interval for Intuitionistic Fuzzy sets.In the finishing choice process the minimal uncertainty will be considered.Here T stands for targets and L stands for limitations in Optimization problem, then the choice C denoted by C=T L= { }.

Szmidt and Kacprzyk
This operator can be easily generalized and applied to the Intuitionistic Fuzzy Optimization problem [2]: Now, α (0  α ≤ 1) stands for the least acknowledgement limit of grade of objectives and limitations and β (0  β ≤ 1) stands for the highest grade of rejection of objectives and limitations and  (0   ≤ 1) stands for the highest acceptable grade of uncertainty of objectives and limitations.
As a result, the problem in Intuitionistic Fuzzy Optimization is changed to the succeeding crisp Optimization problem which can be simply solved mathematically or else with means of some typical software: Maximize αβ Subject to the limitations α  , j=1,2,…,(p+q), β ≥ , j=1,2,…,(p+q), ( 16) α, and other crisp limitations and non-negativity boundaries.

Mathematical Computations
Symmetric Traveling Salesman Problem is determined in this work.The salesman have to enter the three cities exactly once and finally he must come back to his home city by take up a path with minimum expensive, interval covered, duration, fuel expenses and vehicle maintenance expenses.The cities are listed along with their expensive, interval, duration, interval, fuel expenses and vehicle maintenance expenses in a matrix below, where (e, d, i, fe, vme) represents: expensive (in hundreds), interval in kilo meters, duration in hours, fuel expenses and vehicle maintenance expenses (in hundreds) respectively for the corresponding set of cites.
The functions to solve each objective function in Traveling Salesman Problem are fixed as 65,14,12,14,51 and their equivalent tolerations level of acknowledgement or in other words the grades of acknowledgment are determined as 5, 4, 5, 6, 7 respectively (by the choice maker).The respective toleration levels for rejection or the grade of rejection of the objective functions are determined as 6, 5, 6, 8, 10.Therefore the certain membership and nonmembership of the objective functions 1 Therefore the formulation of the Intuitionistic Fuzzy Traveling Salesman problem to enhance the range of acknowledgement, to reduce the range of rejection and to reduce the range of uncertainty is given by Maximize αβ subject to the limitations x x 1,  It is eminent that the membership, non membership functions be a linear one.Initially we determine only the Fuzzy function for the Traveling Salesman problem with Fuzzy multiple objectives as a linear programming problem which follows as Maximize α Subject to the limitations

Discussions in Comparative Study
A comparative study is carried out with the existing models Fuzzy multi objective linear programming, Angelov's method and the proposed model in Fuzzy Intuitionistic method.In this Angelov's method the range of acknowledgement is enhanced, the range of rejection is In consequence the Intuitionistic Fuzzy Linear Traveling Salesman Problem is improved into a linear programming and by solving the above equations we obtain the value α = 0.43, β = 0.41,  = 0.16 and z 1 = 62, z 2 = 16, z 3 = 12.84, z 4 = 18, z 5 = 53 which is preferable than Fuzzy decision and Angelov's Intuitionistic Fuzzy decision.

Conclusion
This work has discussed the Symmetric Traveling Salesman Problem under Intuitionistic Fuzzy environment.Here the range of acknowledgement is more advanced in Angelov's method comparing to the proposed method in Intuitionistic environment.But the expensive is reduced in proposed Intuitionistic Fuzzy approach technique comparing to the Angelov's method.Proposed method is considered to be the appropriate model by considering the minimum cost, range of uncertainty and ignoring the small variations in range of acknowledgement.The tolerations are imported by choice maker to acclimate this ambiguity.A variety of results along distinct objective are obtained by regulating those tolerations.Among that choice deviser determines the one that foremost fit for the acceptable among the given tolerations.Hence the proposed technique can be considered to be an appropriate model for the decision analysis.

5 z
the vehicle maintenance expenses spent in traveling between the city i to the city j is defined as vme ij and 0 is the corresponding function for the reduction of total vehicle maintenance expenses and t 5 is the toleration then make the most of the grade of acknowledgement of Intuitionistic Fuzzy objectives along with limitations and to reduce the grade of rejection of Intuitionistic Fuzzy objectives and limitations along with to reduce the grade of uncertainty of Intuitionistic Fuzzy objectives along with limitations, the following limitations require being determined: EAI Endorsed Transactions on Energy Web and Information Technologies 04 2018 -06 2018 | Volume 5 | Issue 18 | e7 z , 5 z could be given as follows: EAI Endorsed Transactions on Energy Web and Information Technologies 04 2018 -06 2018 | Volume 5 | Issue 18 | e7 above equations we obtain the value of α = 1 and then z 1 = 63, z 2 =15.1, z 3 = 11, z 4 = 15, z 5 = 50.
Implementation of Fuzzy Intuitionistic Algorithm for Traveling Salesman Problem EAI Endorsed Transactions on Energy Web and Information Technologies 04 2018 -06 2018 | Volume 5 | Issue 18 | e7 reduced, ignoring the range of uncertaity entirely then the corresponding above equations we obtain the value of α = 0.50, β = 0.50 and z 1 = 64, z 2 =16.1, z 3 =12.84,z 4 =16.97,z 5 = 53.At last in proposed Intuitionistic Fuzzy approach technique by inclusion of the range of uncertainty with the range of acknowledgement and the range of rejection is the problem be reformed as

Table 1 .
Index of Literatures

Table 3 .
Comparison among the different methods 53 EAI Endorsed Transactions on Energy Web and Information Technologies 04 2018 -06 2018 | Volume 5 | Issue 18 | e7