Interval-Valued Triangular Neutrosophic Linear Programming Problem

In this paper, we have proposed an Interval-valued triangular neutrosophic number (IV-TNN) as a key factor to solve the neutrosophic linear programming problem. In the present neutrosophic linear programming problem IV-TNN is expressed in lower, upper truth membership function, indeterminacy membership function, and falsity membership function. Here, we try the compare our proposed method with existing methods.

Linear programming is most powerful technique, which occurs in decision-making. Bellman and Zadeh [17] introduced fuzzy optimization problems where they have stated that a fuzzy decision can be viewed as the intersection of fuzzy goals and problem constraints. Many researchers such as; Zimmermann [18], Tanaka, et al. [19], Campos and Verdegay [20], Rommelfanger et al. [21], Cadenas and Verdegay [22] who were dealing with the concept of solving fuzzy optimization problems, later studied this subject. Parvathi and Malathi [23] have done their work on intuitionist fuzzy linear optimization.
Then Abdel-Baset et al. [24] and pramanik [25] proposed neutrosophic linear programming methods based on the idea of neutrosophic sets. Also, Abdel-Baset et al. [26] introduced the neutrosophic linear programming models where their parameters are represented with the trapezoidal neutrosophic numbers and presented a technique for solving problems. Nafei et al. [27] presented a new method for solving interval neutrosophic linear programming problems. Abdel-Baset et al. [25] discussed a novel method for solving the fully neutrosophic linear programming problems. Khatter [39] proposed Neutrosophic linear programming using possibilistic mean according to Kirna [39] the proposed approach converts each triangular neutrosophic number in linear programming problem to weighted value using possibilistic mean to determine the crisp linear programming problem. Bera et al. [40] approach the application of neutrosophic linear programming problem to real life. They proposed an algorithm of the Big-M simplex method in this new climate and then it is applied to a real-life problem. Ye [41] studied on neutrosophic number linear programming method and its application under neutrosophic number environments.
In this paper, we proposed an interval-valued triangular neutrosophic number to solve the neutrosophic linear programming problem so that we could have a better result in comparison to other methods. The structure of the chapter is as follows: the next section is a preliminary discussion; the third section describes the interval-valued triangular neutrosophic number of the proposed model; the fourth section describes steps of the proposed model; the last section summaries the conclusion.

3.3.Neutrosophic Linear Programming Problem (NLPP)
Linear programming is an optimization technique widely used in practical problem In this section we generalize the LPP term as the interval-valued triangular neutrosophic programming problem, denoted as IV-TNLP problem and defined as Where ̃ , , are interval-valued triangular neutrosophic numbers.

Proposed IV-TNLP Method
Step1 Let the decision-makers insert their IV-TNLP problem. Because we always try to maximize truth membership function and minimize indeterminacy membership function and falsity membership, then inform decision-makers to apply the concept when entering triangular neutrosophic numbers of the IV-TNLP problem.
Step 2 Convert IV-TNLP problem to its crisp model by using the following method Step 3 By applying the proposed ranking function converts each interval-valued triangular neutrosophic number to its equivalent crisp value. This lead to convert the IV-TNLP problem to its crisp model.
Step 4 Solve the crisp model using the standard method and obtained the optimal solution to the problem.

Comparison of the proposed method with existing methods
Comparison of the proposed method with [23] Maximize Ζ = 3 + 4 Such that, According to [23] the optimal value of the objective function is Z=21.45. Now we assume the parameters of LPP according to the proposed method which are represented as follows -DOI: 10 Solving the problem the optimal solution is = 24.51; = 0 with optimal objective value 92.39.
By comparing the proposed model results with the results of [23] of the same problem, we observed that our proposed model results are better than the results of [23]. Also, if we see the optimal solution of the existing solution under the intuitionistic fuzzy system are = 7.15, = 0 and [ ] = 21.45 and from the optimal solution of the proposed method linear programming problem the objective function value equals 92.39 which is a problem of maximization. The proposed approach is smoother than the existing method in [23]. The existing method in [23] is only able to solve the problem but we can handle the situation in interval-valued neutrosophic number and due to this we can convert each interval-valued triangular neutrosophic number to its equivalent crisp value in a better way in comparison to existing method. In addition, due to the explanation of determining truth membership function, falsity membership function, and indeterminacy-membership function, the proposed model is more effective than their method in [23].
This clears that the proposed method shows a more convenient optimal value than the existing method in [27]. Though both the methods (existing and proposed) are the same when = = = = = , = = = = = , but due to having more chances for taking lower and upper triangular neutrosophic numbers for determining truth membership function, falsity membership function, and indeterminacy-membership function the proposed method shows the more suitable result. Hence our method is more applicable for solving real-life problems.

Conclusion:
In this paper, we have tried to discuss the neutrosophic linear programming problem concerning the interval-valued triangular neutrosophic number and compare the proposed method with some papers. In the proposed method intervalvalued triangular neutrosophic number is expressed in lower, upper truth membership function, indeterminacy membership function, and falsity membership function. It is seen that the proposed method shows better results in comparison to [23]. The second comparison with [27] shows almost the same result when = = = = = , = = = = = for the interval-valued triangular number that is when lower and upper truth membership function, indeterminacy membership function and falsity membership function, and interval-valued triangular number become the same. However, when we go through the proposed method it shows different in solution.
In the future study, we extend the IV-TNLP algorithm in an interval-valued Neutrosophic Linear Fractional Programming Problems. Hope our method would help in the future for getting better results in the decision-making system.