Neutrosophic variational inequalities with applications in decision-making

In this paper, we introduced some new concepts of a neutrosophic set such as neutrosophic convex set, strongly neutrosophic convex set, neutrosophic convex function, strongly neutrosophic convex function, the minimum and maximum of a function f with respect to neutrosophic set, min,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\min ,$$\end{document} and max\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\max $$\end{document} neutrosophic variational inequality, neutrosophic general convex set, neutrosophic general convex function, and min\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\min $$\end{document}, max\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\max $$\end{document} neutrosophic general variational inequality. We introduced some basic results on these new concepts. Moreover, we discussed the application of the neutrosophic set in optimization theory. We developed an algorithm using neutrosophic min\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\min $$\end{document} and max\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\max $$\end{document} variational inequalities and identified the maximum and minimum profit of the company.


Introduction
suggested the theory of fuzzy sets (FSs) to solve various forms of uncertainties. This theory has now been successfully implemented in different fields (Pedrycz 1990;Zadeh 1975). A single value μ A (x) ∈ [0, 1] is used by traditional FSs to describe the degree of membership of the fuzzy set A, which is specified on a universal scale, and they are unable to manage those instances where it is difficult to describe μ A by specific value one. Atanassov introduced intuitionistic fuzzy sets (IFSs) (Atanassov 2016), which are an extension of Zadeh's FSs, to address the lack of knowledge of non-membership degrees. Moreover, vague M. Khan, M. Zeeshan and S. Iqbal have contributed equally to this work. cal solutions of fuzzy Fredholm-Volterra integrodifferential equations. Moreover, Arqub and Al-Smadi (2020) proposed a new definition of fuzzy fractional derivative, so-called fuzzy conformable.
Although the theory of FSs has been developed and generalized, in various real-life problems, it does not deal with all uncertainty. For example, it is not possible to deal with certain kinds of uncertainty, such as indeterminate and inconsistent information. For example, when an expert is asked for his or her opinion about a certain statement, he or she may say that the possibility that the statement is true is 0.5, that it is false is 0.6 and the degree that he or she is not sure is 0.2 (Wang et al 2005b). This problem is beyond the reach of FSs and IFSs, so it needs some new theories. Smarandache (1999) suggested neutrosophic (NS) sets and neutrosophic logic. An NS is a set where each element of the universe has the degrees of truth, indeterminacy, and falsity and it lies in ]0 − , 1 + [, the non-standard unit interval. This is simply an extension to the standard interval [0, 1] of the IFSs. Moreover, the uncertainty presented here, i.e., the indeterminacy element, is independent of the values of true and falsity, while the incorporated uncertainty depends on the degree of belonging and non-belonging to IFSs. However, in practical situations, NSs are difficult to apply without a particular description. In various areas of knowledge, this theory is being used. See recent examples as Crespo Berti (2020) in modeling real-life problems; Hatip (2020), and Saqlain et al (2020) who developed extensions of it. A new framework for dealing with impreciseness is provided by Neutrosophic Theory. It is well known that statistical concepts and methods can be expanded using a neutrosophic point of view, see Smarandache (2013), Smarandache (2014), Schweizer (2020), Almeida et al (2020).
Single-valued neutrosophic sets (SVNSs), which are a variant of NSs, were proposed in Majumdar and Samanta (2014). Moreover, the information energy of SVNSs, their coefficient of correlation and correlation, and the process of decision-making used by them were proposed in Ye (2013). In addition, Ye (2014a) introduced the simplified neutrosophic sets (SNSs), which can be represented by three real numbers in the real unit interval [0, 1], and proposed a method of MCDM using SNS aggregation operators. Moreover, Majumdar and Samanta (2014) introduced a measure of SVNS entropy. The definition of interval neutrosophic sets (INSs) was proposed by Wang et al (2005a). In addition, Ye (2014b) proposed similarity measures between SVNSs and INSs based on the relationship between measures of similarity and distances.
There are different concepts developed from NS, neutrosophic probability, and neutrosophic statistics. Each of these is interactive. The sets derived from NS are intuitionistic set, paraconsistent set, paradoxist set, trivialist set, nihilist set, dialetheist set, and fallibilist set. Tautological proba-bility and statistics, intuitionistic probability and statistics, dialetheist probability and statistics, fallibilist probability and statistics, paraconsistent probability and statistics, trivialist probability and statistics, and nihilist probability and statistics are derived from neutrosophic probability and statistics. Nabeeh et al (2019) proposed an approach that would facilitate a personal selection process by incorporating the process of neutrosophic analytical hierarchy to demonstrate the ideal solution between various options similar to an ideal solution for order preference technique (TOPSIS). Abdel-Basset et al (2019b) developed a new kind of technique for neutrosophy called neutrosophic numbers of type 2. They suggested a novel T2NN-TOPSIS process, combining type 2 neutrosophic number and TOPSIS, which is very useful in group decision-making. A multi-criteria group decision-making method of the analytical network process method and the VIKOR method was investigated in a neutrosophical setting dealing with high-order imprecision and incomplete information (Abdel-Baset et al 2019b). M. A. Baset introduced a new technique for estimating the GDM selection process for smart medical devices in a vague decision-making environment. Neutrosophic with the TOPSIS strategy is used in decision-making processes to deal with incomplete information, vagueness, and ambiguity, taking into account the decision-making criteria in the information obtained by decision-makers in Abdel-Basset et al (2019a). They proposed a robust ranking method with NS to manage the performance of the supply chain management (GSCM) and methods that have been commonly used to promote environmental sustainability and achieve competitive advantages. The principle of the N.S. has been used to handle imprecision, linguistic imprecision, ambiguous details, and incomplete information (Abdel-Baset et al 2019a). Moreover, Abdel-Basset et al (2018) et al. used NS for evaluation techniques and decision-making to identify and analyze factors influencing the selection of suppliers for the supply chain management. Bera and Mahapatra (2018a) et al. characterized a neutrosophic norm for a soft linear space known as a neutrosophic soft linear space. They also explore the notion of neutrosophic soft (Ns) prime ideal over a ring. They introduced the idea of N's completely prime ideals, N's fully semi-primary ideals, and N's prime K-ideals Bera and Mahapatra (2018b). Moreover, Bera and Mahapatra (2018c) established the concept of connectedness and compactness in N's topological space along with its various characteristics. Shah and Hussain (2016) et al. studied the P-OR, P-intersection and P-union, and P-AND of neutrosophic cubic sets and their associated properties. Shah Shah and Hussain (2016) et al. discussed neutrosophic soft graphs. They proposed a connection between the neutrosophic soft sets and the graphs.
In decision-making problems, the use of optimization approaches is ubiquitous. The purpose of this article is twofold. The first half aims to present the theoretical foundations of neutrosophic in optimization such as neutrosophic variational inequalities and neutrosophic convex function, and the second half aims to present these theoretical foundations and key techniques in convex optimization, decision-making, and the principle of the neutrosophic variational inequalities in a coherent manner. The purpose of these innovative concepts is to provide a new approach with useful mathematical tools to address the fundamental problem of decision-making (e.g., maximization and minimization of the problem). The generality of the neutrosophic variational inequalities system is given special importance, illustrating how many interesting optimization decision-making problems can be formulated as a problem of neutrosophic variational inequalities. These applied contexts provide solid evidence of the wide applications of the neutrosophic variational inequality approach to model and research decision-making problems. This article will stimulate the interest in neutrosophic variational inequality and its application in optimization.
In this paper, we introduce some new concepts of a neutrosophic set such as neutrosophic convex set, strongly neutrosophic convex set, neutrosophic convex function, strongly neutrosophic convex function, the minimum and maximum of a function f with respect to neutrosophic set, min, and max neutrosophic variational inequality, neutrosophic general convex set, neutrosophic general convex function, and min, max neutrosophic general variational inequality. We study some basic results on these new concepts. Moreover, we discuss the application of the neutrosophic set in optimization theory. We propose a method using neutrosophic min and max variational inequality and identify the maximum and minimum profit of the company.

Preliminaries
We will define here some new concepts on the neutrosophic set and also discuss particular examples of these new concepts.
Definition 1 Smarandache (1999) Let X be a space of points, and let x ∈ X . A neutrosophic set N in X is characterized by a truth membership function T N , an indeterminacy membership function I N , and a falsity membership function F N . T N (x), I N (x), and F N (x) are real standard or non-standard subsets of 0 − , 1 + , and F N .
The neutrosophic set N can be represented as: There is no restriction on the sum of T N (x), I N (x), and Definition 2 Ali and Smarandache (2017) be a neutrosophic set. Then, the complement of a neutrosophic set N is denoted by N c and is defined by Definition 3 Ali and Smarandache (2017) Let N 1 and N 2 be two neutrosophic sets in a universe of discourse X . Then, the union of N 1 and N 2 is denoted by N 1 ∪ N 2 , which is defined by for all x ∈ X , and ∨, ∧ represent the max and min operators, respectively. (2017) Let N 1 and N 2 be two neutrosophic sets in a universe of discourse X . Then, the intersection of N 1 and N 2 is denoted by N 1 ∩ N 2 , which is defined by

Definition 4 Ali and Smarandache
for all x ∈ X , and ∨, ∧ represent the max and min operators, respectively.
Definition 5 Let N be a neutrosophic set on X , and μ N = for all x, y ∈ R n , and t ∈ [0, 1]. Note: Every neutrosophic convex set is a neutrosophic set, but the converse is not true.
Definition 6 Let N be a neutrosophic set on X , and μ N = Or for all x = y, x, y ∈ R n , and t ∈ [0, 1]. Note: The strongly neutrosophic convex set is the neutrosophic convex set, but the converse is not true.
Definition 7 Let N be a neutrosophic convex set on X = R n and is characterized by a membership function x ∈ X } denotes the collection of neutrosophic convex sets. Then, the function f on neutrosophic convex sets N is said to be a neutrosophic convex function if the following condition holds: for all x, y ∈ R n , and all t ∈ [0, 1]. The inequality (4) can be written as: Note that the neutrosophic convex function is more significant in optimization theory. They are used in models for optimization problems (maximization and minimization problems).

Example 2
The identity function on the neutrosophic convex set N is a neutrosophic convex function.
Definition 9 Let N be a neutrosophic convex set on X = R n and is characterized by a membership function μ N = x ∈ X } denotes the collection of neutrosophic convex sets. Then, the inequality is called neutrosophic min variational inequality. ] be two neutrosophic sets and f be a function defined by Now, From (6) and (7), we have Definition 11 Let N be a neutrosophic convex set on X = R n , and μ N = T N (x), x ∈ X } denotes the collection of neutrosophic convex sets. Then, the inequality is called neutrosophic max variational inequality. ] be two neutrosophic sets and f be a function defined by From (8) and (9), we have

Generalized convex sets and convex functions
In the problems, if the domain set may not be a convex set, in those situations, the non-convex set can be made a convex set with respect to an arbitrary function. These sets are called general convex sets, and the function defined on the general convex set is called general convex function.

Definition 12
Let N be a neutrosophic set on X , and μ N = T N (x), I N (x), F N (x) denotes their membership function. Then, N is said to be general convex if N (g(y)). Or for all g(x), g(y) ∈ R n g , and t ∈ [0, 1]. Definition 13 Let N be a neutrosophic general convex set on X = R n , and μ N = T N (g(x)), ) : x ∈ X } denotes the collection of neutrosophic general convex sets. Then, the function f on neutrosophic general convex sets N is said to be a neutrosophic general convex function if for all g(x), g(y) ∈ R n g , and t ∈ [0, 1]. Case 1 f g = I in inequality (10), then the neutrosophic general convex function is the neutrosophic convex function.

Definition 14
Let N be a neutrosophic general convex set on X = R n , and μ N = T N (g(x)), I N (g(x)), F N (g(x) denotes their membership function. Let f : τ → τ be a neutrosophic general convex function, where denotes the collection of neutrosophic general convex sets. Then, the inequality f (N i (g(x))), f (N i (g(x))) ∩ f (N j (g(x))) ≤ f (N i (g(x))) • f (N j (g(x))); i = j, ∀N i (g(x)), N j (g(x)) ∈ τ is called neutrosophic min general variational inequality.

Definition 15
Let N be a neutrosophic general convex set on X , and μ N = T N (g(x)), I N (g(x)), F N (g(x) , denotes their membership function. Let f : τ → τ be a neutrosophic general convex function, where denotes the collection of neutrosophic general convex sets. Then, the inequality f (N i (g(x))), f (N i (g(x))) ∪ f (N j (g(x))) ≥ f (N i (g(x))) • f (N j (g(x))); is called neutrosophic max general variational inequality.

collection of neutrosophic convex sets and N i (x) ∈ τ be a minimum of the neutrosophic convex function f on τ. Then, N i (x) satisfies the neutrosophic min variational inequality.
Proof Let N i (x) ∈ τ be the minimum of f . Then, Also, from inequality (11), we have The inequality (12) can be written as: Thus, N i (x) ∈ τ satisfies the neutrosophic min variational inequality.
x ∈ X } be a collection of neutrosophic convex sets and N i (x) ∈ τ be a maximum of the neutrosophic convex function f on τ. Then, N i (x) satisfies the neutrosophic max variational inequality.
Also, from inequality (13), we have The inequality (14) can be written as: Thus, N i (x) ∈ τ satisfies the neutrosophic max variational inequality.

be a collection of neutrosophic general convex sets and N i (g(x)) ∈ τ be a minimum of the neutrosophic general convex function f on τ. Then, N i (g(x)) satisfies the neutrosophic general variational inequality.
Proof Let N i (g(x)) ∈ τ be the minimum of f . Then, Also, from inequality (15), we have The inequality (16) can be written as: Thus, N i (g(x)) ∈ τ satisfies the neutrosophic general min variational inequality.
) : g(x) ∈ X } be a collection of neutrosophic general convex sets and N i (g(x)) ∈ τ be a maximum of the neutrosophic max general convex function f on τ. Then, N i (g(x)) satisfies the neutrosophic max general variational inequality.
Proposition 6 For any two neutrosophic convex sets N 1 and N 2 , N 1 ∪ N 2 is also a neutrosophic convex set.
Proof Since N 1 is a neutrosophic convex set, we have Also, N 2 is a neutrosophic convex set, then Thus, N 1 ∪ N 2 is a neutrosophic convex set.
x ∈ X } be a minimum of the neutrosophic convex function f . Then, From (19) and (20), we have x ∈ X } be a maximum of the neutrosophic convex function f . Then, Proof Assume that N i (x) ∈ τ be a maximum of f . Then, As From (21) and (22), we have Theorem 1 Let f : τ → τ be a mapping and "∼ be a relation defined in the following way " Show that the relation "∼ is an order relation.
Proof To prove the relation "∼ is an order relation, we have to show the following.
The inequality (23) implies that We will discuss the algorithm by using the neutrosophic max variational inequality and neutrosophic min variational inequality. In this algorithm, we will discuss how the trucking company gets maximum profit and minimum profit.

Algorithm
Suppose ABC Trucking is a company that operates 20 trucks for transport and logistics. When they are full and on the track, trucks make the most money for the company. ABC Trucking has the following vector entities or groups: (i). Truck Company(truck type, age, engine size).
(ii). Income ((Euro) 1 , (Euro) 2 , (Euro) 3 ). A neutrosophic set N 1 , N 2 , and N 3 in X = truck t ype, Y = age, Z = engine si ze is characterized by a truth membership function T N 1 , T N 2 , T N 3 , an indeterminacy membership function I N 1 , I N 2 , I N 3 , and a falsity membership T N 3 , I N 1 , I N 2 , I N 3 , and F N 1 , F N 2 , F N 3 are real standard or non-standard subsets of 0 − , 1 + . A neutrosophic set N 1 , N 2 , and N 3 in X = Euro, Y = Dollar, Z = Riyal is characterized by a truth membership function T N 1 , T N 2 , T N 3 , an indeterminacy membership function I N 1 , I N 2 , I N 3 , and a falsity member- The trucking company needs to optimize the use of its trucks and workers for the highest possible profits. To find the probability of the maximum or minimum profit of the trucking company, we define a relation f : τ → τ by Now, if the relation f satisfies the max variational inequality, that is, Taking the left-hand side of the inequality (39), we have which gives the maximum profit with a neutrosophic set N characterized by a truth membership function T N , an indeter-minacy membership function I N , and a falsity membership function F N .
If the relation f satisfies the min variational inequality, that is, Taking the left-hand side of the inequality (41), we have which gives the minimum profit characterized by a neutrosophic set N with a truth membership function T N , an indeterminacy membership function I N , and a falsity membership function F N .
Example 5 Suppose ABC Trucking is a company that operates 20 trucks for transport and logistics. When they are full and on the track, trucks make the most money for the company. ABC Trucking has the following vector entities or groups: The neutrosophic sets N 1 , N 2 , and N 3 in X , Y , and Z are: (0.4, 0.5, 0.8) x , x .
Let f be a function defined by x .
Now, neutrosophic max variational inequality is: Taking the left side of the inequality (43), we have (2014) discussed the fuzzy linear programming, and the proposed method is to maximize or minimize the total utility of the objective function, as an aggregated function of its intersection with the minimization and maximization sets. Shirin et al (2014) discussed the application of the optimization problem which belongs to a fuzzy environment. He used fuzzy linear programming for the setting of the company production plan. However, many researchers are using fuzzy linear programming for the applications of fuzzy optimization. Moreover, Chakraborty et al (2014) have proposed a new method for solving an intuitionistic fuzzy CCM using chance operators and discussed three different approaches to solve the intuitionistic fuzzy linear programming (IFLPP) using possibility, necessity, and credibility measures. The model presented in this paper for optimization problems (maximization and minimization problems) is unique than the methods previously developed. Here, we used max and min variational inequalities to develop an algorithm for further use in optimization problems. Moreover, through this model, we determined the maximum value and minimum value separately. However, our designed model is not a perfect one, and it stuck with a deficiency of theoretical support. The concept of neutrosophic variational inequalities may be useful for applications. Therefore, it will be significant for future work.

Conclusion
In this paper, we have introduced some new concepts of a neutrosophic set such as neutrosophic convex set, strongly neutrosophic convex set, neutrosophic convex function, strongly neutrosophic convex function, the minimum and maximum of a function f with respect to neutrosophic set, min, and max neutrosophic variational inequality, neutrosophic general convex set, neutrosophic general convex function, and min, max neutrosophic general variational inequality. We have discussed some basic results on these new concepts.
We proposed the application of the neutrosophic set in optimization theory. This work and further study of neutrosophic max and min variational inequalities will give a new direction of application in the field of optimization. Moreover, neutrosophic differential equation is important in applied science and engineering for modeling of uncertainty.

Author Contributions All authors contributed equally.
Funding This work is financially supported by the Higher Education Commission of Pakistan (Grant No: 7750/Federal/NRPU/R&D/HEC/ 2017).

Data Availability
Our manuscript has no associated data.

Conflict of interests
The authors declare that there is no conflict of interest regarding the publication of this article.
Ethical approval This article does not contain any studies with human participants or animals performed by any of the authors.