Neutrosophic Goal Programming Technique and its Application

This paper develops a multi-objective Neutrosophic Goal Optimization technique for optimizing the design of truss structure with multiple objectives subject to a specified set of constraints. In this optimum design formulation, the objective functions are weight and the deflection; the design variables are the cross-sections of the bar; the constraints are the stress in member. The classical three bar truss structure is presented here in to demonstrate the efficiency of the neutrosophic goal programming approach. The model is numerically illustrated by Neutrosophic Goal Optimization technique with different aggregation method. The result shows that the Neutrosophic Goal Optimization technique is very efficient in finding the best optimal solutions.


I. INTRODUCTION
In present day, problems are there with different types of uncertainties which cannot be solved by classical theory of mathematics.Fuzzy set (FS) theory has long been introduced to deal with inexact and imprecise data by Zadeh [2], Later on the fuzzy set theory was used by Bellman and Zadeh [3]to the decision making problem.A few works have been doneas an application of fuzzy set theoryon structural design.Several researchers like Wang et al. [5] first applied α-cut method to structural designs wherevarious design levels α were used to solve the non-linear problems.In this regard, a generalized fuzzy number has been usedDey et al. [6]in context of a non-linear structural design optimization.Dey et al. [8] developed parameterized t-norm based fuzzy optimization method for optimum structural design.
In such extension, Intuitionistic fuzzy set which is one of the generalizations of fuzzy set theory and was characterized by a membership, a nonmembership and a hesitancy function was first introduced by Atanassove [1].In fuzzy set theory the degree of acceptance is only considered but in case of IFS it is characterized by degree of membership and non-membership in such a way that their sum is less or equal to one.Dey et al. [7] solved two bar truss non-linear problem by using intuitionistic fuzzy optimization problem.Again,Dey et al. [9] used intuitionistic fuzzy optimization technique to solve multi objective structural design.Intuitionistic fuzzy sets consider both truth and falsity membership and can only handle incomplete information but not the information which is connected with indeterminacy or inconsistency.
In due course, any generalization of fuzzy set failed to handle problems with indeterminate or inconsistent information.To overcome this, Smarandache [4] proposed a new theory , namely, neutrosophic logic, by adding another independent membership function named as indeterminacy

 
Ix of neutrosophic sets are equal, then neutrosophic set will become the intuitionistic fuzzy set.The components of neutrosophic set, namely truth-membership degree, indeterminacy-membership degree and falsity-membership degree, were suitable to represent indeterminacy and inconsistent information.
Goal Programming (GP) models was originally introduced by Charnes and Copper [11] in early 1977.Multiple and conflicting goals can used in goal programming.Also, GP allows simultaneous solution of a system of complex objectives, and the solution of the problem requires ascertaining among these multiple objectives.In this case, the model must be solved in such a way, that each of the objective to be achieved.Dey et al. [10]proposed intuitionistic goal programming technique on nonlinear structural model.The Neutrosophic approach for goal programming in structural design is rare.This is the first time NSGO technique is in application to multiobjective structural design.The present study investigates computational algorithm for solving multi-objective structural problem by single valued generalized NSGO technique.The results are compared numerically for different aggregation method of NSGO technique.From our numerical result, it has been seen that the best result obtained for geometric aggregation method for NSGO technique in the perspective of structural optimization technique.

II. MULTI-OBJECTIVE STRUCTURAL MODEL
In the design problem of the structure i.e. lightest weight of the structure and minimum deflection of the loaded joint that satisfies all stress constraints in members of the structure.In truss structure system ,the basic parameters (including allowable stress etc.) are known and the optimization's target is that identify the optimal bar truss cross-section area so that the structure is of the smallest total weight with minimum nodes displacement in a given load conditions .The multi-objective structural model can be expressed as where 12 , ,...,

T n
A A A A    are the design variables for the cross section, n is the group number of design variables for the cross section bar ,   is the total weight of the structure ,  

A. Fuzzy Set
Let X be a fixed set.A fuzzy set where the function defined the truth membership of the element xX  to the set A .

B. Intuitionistic Fuzzy Set
Let a set X be fixed.An intuitionistic fuzzy set or IFS i A  in X is an object of the form There is no restriction on the sum of

D. Single Valued Neutrosophic Set
Let a set X be the universe of discourse.A single valued neutrosophic set n A  over X is an object having the form for all xX  .

E. Single Valued Generalized Neutrosophic Set
Let a set X be the universe of discourse.A single valued neutrosophic set n A  over X is an object having the form

G. Union of Neutrosophic Sets
The union of two single valued neutrosophic sets A and B is a single valued neutrosophic set C , written as C A B  ,whose truth membership, indeterminacy-membership and falsity-membership functions are given by

H. Intersection of Neutrosophic Sets
The intersection of two single valued neutrosophic sets A and B is a single valued neutrosophic set C , written as C A B  ,whose truth membership, indeterminacy-membership and falsity-membership functions are given by

., T n
A A A A    are the design variables for the cross section, n is the group number of design variables for the cross section bar.
To solve this problem we first calculate truth, indeterminacy and falsity membership function of objective as follows According to neutrosophic goal optimization technique using truth, indeterminacy and falsity membership function, MOSOP (7) With the help of truth, indeterminacy, falsity membership function the neutrosophic goal programming problem (8) based on geometric aggregation operator can be formulated as The multi-objective optimization problem can be stated as follows ,; 2 ,; 22 where P  applied load ;   material density ; A  Cross section of bar-1 and bar- 3; 2 A  Cross section of bar-2;  is deflection of loaded joint.
 Now these non-linear programming problems Model-I,II can be easily solved by an appropriate mathematical programming to give solution of multiobjective non-linear programming problem (12) by generalized neutrosophic goal optimization approach and the results are shown in the table 1 as follows.
Here we get best solution in geometric aggregation method for objective functions.

VII. CONCLUSIONS
The research study investigates that neutrosophic goal programming can be utilized to optimize a nonlinear structural problem. .The results obtained for different aggregation method of the undertaken problem show that the best result is achieved using geometric aggregation method.The concept of neutrosophic optimization technique allows one to define a degree of truth membership, which is not a complement of degree of falsity; rather, they are independent with degree of indeterminacy.As we have considered a non-linear three bar truss design problem and find out minimum weight of the structure as well as minimum deflection of loaded joint, the results of this study may lead to the development of effective neutrosophic technique for solving other model of nonlinear programming problem in different field.

Fx
membership functions.Neutrosophic set is a generalization of intuitionistic fuzzy sets.If hesitancy degree   Hx of intuitionistic fuzzy sets and the indeterminacy membership degree


are the bar length, cross section area and density of the th i group bars respectively.  A  is the stress constraint and    is allowable stress of the group bars under various conditions, min A and max A are the lower and upper bounds of cross section area A respectively.
Subject to xX  where i t are scalars and represent the target achievement levels of the objective functions that the decision maker wishes to attain provided, X is feasible set of constraints.The nonlinear goal programming problem can be written

Fig. 1 .
Fig.1.Truth Membership, Indeterminacy Membership and Falsity Membership Function of i z the help of truth, indeterminacy, falsity membership function the neutrosophic goal programming problem (4) based on geometric aggregation operator can be formulated as The multi-objective neutrosophic fuzzy structural model can be expressed as the help of truth, indeterminacy, falsity membership function the neutrosophic goal programming problem(8) based on arithmetic aggregation operator can be formulated as

Fig 2 .
Fig 2. Design of Three Bar Planer Truss allowable compressive stress for bar 3.This multi objective structural model can be expressed as neutrosophic fuzzy model as