An application of neutrosophic hypersoft mapping to diagnose brain tumor and propose appropriate treatment

Abstract

Brain tumors are one of the leading causes of death around the globe. More than 10 million people fall prey to it every year. This paper aims to characterize the discussions related to the diagnosis of tumors with their related problems. After examining the side effects of tumors, it encases similar indications, and it is hard to distinguish the precise type of tumors with their seriousness. Since in practical assessment, the indeterminacy and falsity parts are frequently dismissed, and because of this issue, it is hard to notice the precision in the patient’s progress history and cannot foresee the period of treatment. The Neutrosophic Hypersoft set (NHS) and the NHS mapping with its inverse mapping has been design to overcome this issue since it can deal with the parametric values of such disease in more detail considering the sub-parametric values; and their order and arrangement. These ideas are capable and essential to analyze the issue properly by interfacing it with scientific modeling. This investigation builds up a connection between symptoms and medicines, which diminishes the difficulty of the narrative. A table depending on a fuzzy interval between [0, 1] for the sorts of tumors is constructed. The calculation depends on NHS mapping to adequately recognize the disease and choose the best medication for each patient’s relating sickness. Finally, the generalized NHS mapping is presented, which will encourage a specialist to extricate the patient’s progress history and to foresee the time of treatment till the infection is relieved.

Keywords

Tumor neutrosophic hypersoft mapping inverse mapping

1 Introduction

A brain tumor is regarded as a collection of abnormal brain cells in a brain. Based upon its nature, it is categorized as benign or malignant. As their development shows, benign tumors develop moderately and don’t scatter in the adjacent tissues. Malignant tumors thrive and cover areas rapidly. Magnetic resonance imaging (MRI) is the most common strategy for brain imaging. Some other imaging methods include Computerized tomography (CT), positron emission tomography (PET). CT provides information about dense tissues such as tissues with calcination and bones, while MRI provides information only about delicate tissues [46, 47].

CT is regarded as the first technique utilized in modern times for the imaging and detection of brain tumors or just tumors in general. In continuance with the imaging, MRI techniques are now used for primary imaging and detection of tumors. The part where CT imaging falls behind modern imaging techniques is the imaging of hemorrhage, hydrocephalus, and herniation. However, the masses in the brain where there is calcination, such as oligodendrogliomas and meningiomas, can be detected by the utilization of CT Scanning methods [37, 38].

Structural MRI is utilized in initial brain tumor evaluation to determine the tumor location (i.e., intra-axial or extra-axial), determination of location in the brain for biopsy and surgical preparations, for the evaluation of the mass effect of the tumors on the brain, imaging of ventricular systems. These factors mentioned, along with MRI’s of other physiologically affected areas, is used for the suggestion of a possible diagnosis. Extra-axial tumors like schwannomas and meningiomas cannot be differentiated can generally be differentiated from one another, but not in all cases. Factors such as patient age and another primary or secondary malignancy influence the differential diagnosis [39, 40].

All the generalized purposes of imaging techniques are summarized below: CT is used for the evaluation of Mass effect, calcification, and hemorrhages; Pre and post-contrast T1 is utilized to monitor enhancement characteristics and the extent of enhancement of each portion of the tumors; T1/T2 Flair is utilized for detection for peri-tumoral edema, and non-enhancing tumors, MR Spectroscopy (Magnetic Resonance) is used for the evaluation of the metabolic profile of the tissue under observation; fMRI (functional Magnetic Resonance Imaging) is used for highlighting pre-operative mappings of the brain, for studying the treatment effects, and research; while PET Scans (Positron Emission Tomography) and MR (Magnetic Resonance Imaging) are used to study new potential radiotracers and how they influence certain tumors [41].

There are various strategies accessible for image precision, be specific dark level thresholding, edge discovery, histogram thresholding, surface, grouping calculation, Huang et al. [42], Li and Shen [43], region, segmentation dependent on fractals, wavelets Farias et al. [44], thus on.

For solving multifaceted problems in robotics, engineering, economics, and the environment, the use of conventional means isn’t enough. Despite the variety of incomplete information, there are four theories specific to these problems Probability set theory (PST), Fuzzy set theory (FST) Zadeh [1], Rough set theory (RST) Pawlak [14] and Period mathematics (PM) that is assumed as a scientific apparatus for managing with lacking information. Every one of these apparatuses acquires the pre-determination of few parameters, to begin with, density function (DF) in PST, membership degree in FST, and a congruence relation in RST. Such a prerequisite, observed in the scrim of flawed or deficient information, escalate numerous issues. Simultaneously, fragmented information stays the most glaring attribute of humanitarian, organic, monetary, social, political, and large man-machine frameworks of different kinds. It was Zadeh [1] who introduced the notion of fuzzy sets to handle the uncertainties. Since then, the fuzzy set has been pragmatic in many commands such as medical diagnosis [56], decision-making [16] and pattern recognition [54]. Keeping in view the prominence of fuzzy sets, many generalizations of fuzzy sets have been familiarized like rough sets [14], soft sets [12], intuitionistic fuzzy set [29], bipolar logic and bipolar fuzzy logic [31], m-polar soft set [34], hypersoft set [4] and multi-polar neutrosophic soft sets with application [56]. Although all these generalizations have their benefits, the notion of neutrosophic set [30] has expanded more responsiveness.

Smarandache discussed neutrosophy in 1998 [30], 1999 [45] and 2005 [35]. He started the neutrosophic hypothesis as another numerical mechanism for dealing with imprecise, indeterminacy, and uncertain information. Broumi et al. [36] introduced mapping on neutrosophic soft expert sets through which studied the images and inverse images of neutrosophic soft expert sets. Heilpern [7] presented the idea of fuzzy mapping and demonstrated a fixed point theory for fuzzy contraction mappings, which speculates the fixed point hypothesis for multivalued mappings of Nadler [13]. Estruch and Vidal [21] refer to a fixed point theory for fuzzy contraction mappings on a complete metric space that generalizes the fixed point hypothesis given by Heilpern’s fixed point hypothesis. Yan and Xu [23] have examined convexity, quasiconvexity of fuzzy mappings by considering the idea of ordering due to Goetschel and Voxman [5]. Syau [15, 18] demonstrated the concept of convex and concave fuzzy mappings and presented the idea of differentiability, summed up convexity, e.g., pseudoconvexity and invexity for fuzzy mappings of a few factors. His methodology is equal to Goetschel and Voxman approach for fuzzy mapping of a single variable in which the arrangement of fuzzy numbers is embedded in a topological vector space.

Molodtsov [12] successfully implemented soft set (SS) theory in several directions likes the ease of function, Riemann integration (RI), Peran integration (PI), Probability theory (PT), Measurement theory (MT), and so on. Promising results have been found by Kovkov et al. [8]. The SS theory is used for the process of optimization in Optimization Theory (OPT), Game theory (GT), and Operations research (OR). Maji et al. [11] presented S-sets applications in decision-making problems. In [22], Yang et al. highlighted S-sets’ requirements in engineering extended applications. Maji et al. [10] presented the concept of fuzzy SS and its many features. They presented it as an attractive enlargement of S-sets, additional features to vagueness and ambiguity on the highest level of incompleteness. Present researchers have explained [9 , 16] how to combine the two ideas into a more flexible, high expression structure for modeling and refined foggy data in the information system. Saeed et al. [55, 57] extended soft set to soft expert set and fuzzy soft expert set and applied AHP decision making technique to get best solution. They gave the application of distance-based similarity measure: in pattern recognition of COVID-19 spread and its effects in Pakistan in intuitionistic fuzzy structure [54]; and to diagnose dengue fever in neutrosophic structure [56]. Mahmood et al. [58] presented the techniques of decision making and medical diagnosis based on the structure of spherical fuzzy sets.

Karaaslan [6] presented the soft class and its relevant operations. He applied its utilization in decision-making successfully. Kharal et al. [2, 3] presented the concept of mappings on fuzzy soft classes and mappings on soft classes in 2009 and 2011, respectively. They considered S-images’ properties, S-inverse images, fuzzy SS, fuzzy S-images, fuzzy S-inverse images of fuzzy S-psets and illustrated these concepts with examples and counterexamples.

Alkhazaleh [24] et al. defined the idea of a mapping on classes where the neutrosophic soft classes are gatherings of the neutrosophic soft sets. Additionally, they characterized and studied the properties of neutrosophic soft images and neutrosophic soft inverse images of neutrosophic soft sets. Sulaiman et al. [26] presented the idea of mappings on multi-aspect fuzzy soft classes. They explored a few characteristics related to the image and pre-image of multi-aspect fuzzy soft sets and further illustrate some numerical examples. Bashir and Salleh [25] characterized the notation of mapping on intuitionistic fuzzy soft classes with intuitionistic fuzzy soft images and inverse images. Borah and Hazarika [27] gave the idea of composite mappings on hesitant fuzzy soft classes in 2016 and discussed some interesting properties of this idea.

Samarandache [4] presented the hypersoft set (HSS) notion as a generalization of soft set in 2018. At that point, he made the differentiation between the sorts of initial universes, crisp, fuzzy, intuitionistic fuzzy, neutrosophic, and plithogenic, respectively. Thus, he also showed that a HS set could be crisp, fuzzy, intuitionistic fuzzy, neutrosophic, and plithogenic. Saeed et al. [49, 50] explained some basic concepts like HS subset, HS complement, not HS set, absolute set, union, intersection, AND, OR, restricted union, extended intersection, relevant complement, restricted difference, restricted symmetric difference, HS set relation, sub relation, complement relation, HS representation in matrices form, and different operations on matrices. He also extended this theory to complex multi-fuzzy hypersoft set and applied it decision making based on entropy and similarity measure [52]. The hypersoft mapping and neutrosophic hypersoft mapping have been introduced with the applications of selection of best alternative and selection of appropriate treatment for Hepatitis diseases [51, 53].

1.1 Motivation

The primary motivation of this paper is to simulate a real-life problem of clinical diagnosis of a tumors disease and it’s appropriate and effective treatment since it is hard to distinguish the precise kind of tumors with its seriousness via previous existing theories and methodologies [2 , 48] because these methodologies are limited to comprehensive models. In [2 , 48], the techniques presented are insufficient to study the information in a deep sense for a better understanding to get proper treatment when parameters are divided into sets of sub-parametric values. Also, they can only target the truthness (membership) of objects and do not evaluate the falsity (nonmembership) and indeterminacy parts. Whereas the model presented in [34] evaluates the data in multipolar and studies the data in neutrosophic nature; however it still has deficiencies when a parameter has sub-parametric values. To overcome this issue, we generalized these models to the hybrid of a hypersoft set which can deal with the parametric values of such disease in more detail considering/concerning the sub-parametric values, since this structure is based on the sub-parametric values of the different parameter and their order and arrangement. A hypersoft set can also arrange the information/data so that it can be easily studied and evaluated. The second hybrid of this model is a structure of neutrosophic which studies the data in all three possible dimensions of positive, indeterminant, and negative parts regarding patient’s disease in accordance with parametric values, where these dimensions are independent to each other. A mapping is an association between two or more domains under some specific rules that maps the entangled parameter to the associated fundamental parametric value by considering the similarity in the structure and basis of these parameters. With the help of this mapping, one can deal similar nature of parameters in a single associated fundamental parameter. The study aims to characterize the nearby diagnosis of tumors with their related symptoms. After investigating different side-effects of tumors, we see that these tumors viruses encase related symptoms, and it is hard to distinguish the dissimilarities of tumors. In practical diagnosis, the indeterminacy and falsity parts are frequently dismissed. Because of this issue, it is hard to observe the precision in the patient’s progress history and can’t foresee the period of medication. To remove these hurdles, we present the NHS and NHS mapping with its inverse mapping. With the help of our presented model of NHS and mapping, we can study the data of patient’s disease in more depth, and all possible directions under the definition of NHS and point out the fundamental symptoms of disease using mapping. These ideas are capable and essential to analyze the issue properly by interfacing it with scientific modeling. This investigation builds up a connection between symptoms and medicines, diminishing the proposed study’s multifaceted nature. To start, a table depending on a fuzzy interval [0, 1] to range the types of tumors was constructed. A calculation was set up dependent on NHS mapping to recognize the disease properly and choose the best medication for each patient’s relating infection. Finally, the generalized NHS-mapping is presented to predict patient’s progress reports and if the proposed treatment results in negative effects to the patient, then the inverse mapping is defined to retain the preceding stage of the patient’s report under consideration for the suggestion of new treatment; and encourages a specialist to save the patient’s progress history until the infection is relieved.

1.2 Paper presentation

The rest of the article is arranged as follows. Section 2 presents some fundamental definitions regarding fuzzy set (FS), SS, fuzzy SS, neutrosophic set, neutrosophic soft class, neutrosophic image, neutrosophic inverse image, HS set, NHS are re-imagined. In Section 3, mapping on NHS classes, NHS image, NHS inverse image, and its relevant theorems with proofs are characterized. In Section 4, a practical application, comparative analysis and advantages of proposed method are given to exhibit the proposed approach’s validity. In the last part, the concluding remarks are described.

2 Preliminaries

In this section, some basic definitions are presented over the universe S.

2.1 Definition [1]

The FS, Q = {(y, I (y)) | y ∈ S} such that $I : S \to [0, 1],$ where S is the collection of objects and I (y) specifies the percentage of membership of y ∈ S.

2.2 Definition [12]

A pair (I, H) is said to be SS over S, where I is a mapping given as $I : H \to P (S),$ In other words, a SS over S is a parameterized family of subsets of the universe S. For ε ∈ H, I (ε) represented the set of ε approximate elements of the SS (I, H).

2.3 Definition [16]

Let S and E be initial universe and set of parameters respectively. Let P (S) denotes the power set of all fuzzy subsets of S and H ⊆ E. A pair (I, H) is said to be fuzzy SS over S, where I : H → P (S).

2.4 Definition [30]

A set ϑ in S is said to be neutrosophic set if it can be represented by using the membership T, indeterminacy I and non-membership F, where T (y) , I (y) and F (y) are elements of ] 0^-, 1⁺ [ for the alternative y. It can be scripted as ϑ = {(y, 〈T (y) , I (y) , F (y) 〉) : y ∈ S ; T (y) , I (y) , F (y) ∈]0^-, 1⁺ [} satisfying the constraint 0^- ≤ T (y) + I (y) + F (y) ≤3⁺.

2.5 Definition [24]

Let S, κ be universal set and set of parameters respectively. Then the gathering of all neutrosophic soft sets over S having parameters from κ is called a neutrosophic soft class and is expressed as (S, κ) .

2.6 Definition [24]

Let $(R, κ)$ and $(P, κ^{'})$ be two classes of neutrosophic-sets over the universal set $R$ and $P$ respectively. Let $θ : R \to P$ and χ : κ → κ′ be mappings. Then a mapping $ξ = (θ, χ) : (R, κ) \to (P, κ^{'})$ is defined as, for neutrosophic set (ζ, λ) in $(R, κ)$ and ξ (ζ, λ) is neutrosophic set in $(P, κ^{'})$ gained as follows, For ϑ ∈ χ (κ) ⊆ κ′ and $y \in P$ , then

ξ (ζ, λ) (ϑ) (y) $= {\begin{matrix} \underset{x \in θ^{- 1} (y)}{\lor} (\underset{ϱ \in χ^{- 1} (ϑ) \land λ}{\lor} ζ (ϱ)) (x), if θ^{- 1} (y) \neq \emptyset, \\ χ^{- 1} (ϑ) \land λ \neq \emptyset \\ 0 if otherwise \end{matrix}$ (1)ξ (ζ, λ) is called a neutrosophic image of neutrosophic set (ζ, λ).

Now, let $(φ, H)$ a neutrosophic set in $(P, κ^{'})$ , where $H \subseteq κ^{'}$ then $ξ^{- 1} (φ, H)$ is a neutrosophic set in $(R, κ)$ given as follows, $ξ^{- 1} (φ, H) (ϱ) (x) = {\begin{matrix} φ (χ (ϱ) (θ (x) & if χ (ϱ) \in H \\ 0 & if otherwise \end{matrix}$ (2) , where $ϱ \in χ^{- 1} (H) \subset κ$ , then $ξ^{- 1} (φ, H)$ said to be the neutrosophic inverse image of $(φ, H)$ .

2.7 Definition [4]

Let o₁, o₂, o₃, ⋯ , o_n be the distinct attributes whose corresponding attribute values belongs to the sets κ₁, κ₂, κ₃, ⋯ , κ_n respectively, where κ_i ∧ κ_j =∅ for i ≠ j. A pair (ϒ, O) is called a HS set over the universal set S, where ϒ is the mapping given by ϒ : O ⟶ P (S), where O = κ₁ × κ₂ × κ₃ × . . . × κ_n. For more definition see [17 , 50].

2.8 Definition [4]

Let S, P (S) be the universal set and power set of S respectively. Let o₁, o₂, o₃, ⋯ , o_n be the distinct attributes whose corresponding attribute values belongs to the sets κ₁, κ₂, κ₃, ⋯ , κ_n respectively, where κ_i ∧ κ_j =∅ for i ≠ j, and their relation O = κ₁ × κ₂ × κ₃ × . . . × κ_n, then the pair (ϒ, O) is called a NHS over the universal set S, where ϒ is the mapping given by ϒ : O ⟶ P (S) and ϒ (O) = {〈x, T (ϒ (O)) , T (ϒ (O)) , T (ϒ (O)) 〉}, where T is the membership value of truthiness, I is the membership value of indeterminacy and F is the membership value of falsity such that T, I, F : S → [0, 1] also 0 ≤ T (ϒ (O)) + I (ϒ (O)) + F (ϒ (O)) ≤3.

3 Mappings on Neutrosophic Hypersoft classes

In this part, we present the idea of mapping on NHS classes. NHS classes are gathering of NHS. We moreover characterize the properties of NHS like NHS images, NHS inverse images of NHS, and backing them with models and theorems. Throughout this section, consider κ₁ × κ₂ × κ₃ × . . . × κ_n = O, $κ_{1}' \times, κ_{2}' \times κ_{3}' \times . . . \times κ_{n}' = K$ , λ₁ × λ₂ × λ₃ × . . . × λ_n = λ, $H_{1} \times H_{2} \times H_{3} \times . . . \times H_{n} = H$ , (ϱ₁, ϱ₂, ϱ₃, . . . , ϱ_n) = ϱ and (ϑ₁, ϑ₂, ϑ₃, . . . , ϑ_n) = ϑ.

3.1 Definition

Suppose S be an initial universe, let ε₁, ε₂, ε₃, ⋯ , ε_n be the distinct attributes whose attribute values belongs to the sets κ₁, κ₂, κ₃, ⋯ , κ_n respectively, where κ_i ∧ κ_j =∅, for i ≠ j, let Ω = {ϖ_i : i = 1, 2, . . . , n} be a gathering of decision makers. Indexed class of NHS {ξ_{ϖ
_i} : ξ_{ϖ
_i} : O → P (S) , ϖ_i ∈ Ω}, where O = κ₁ × κ₂ × κ₃ × ⋯ × κ_n is said to be NHS class and it can be symbolized in such a form ξ_Ω. If, for any ϖ_i ∈ Ω, ξ_{ϖ
_i} =∅, the NHS ξ_{ϖ
_i} ∉ ξ_Ω.

3.2 Example

Let $R = {a =$ Line Interactive, b = Standby-Ferro, c = Delta Conversion On-Line} be types of UPS (Uninterruptible Power Supply) is considered as universe of discourse. Let ε₁ = efficiency, ε₂ = size, ε₃ = colour, distinct attributes whose attribute values belong to the sets κ₁, κ₂, κ₃. Let κ₁ = {j₁ = Good, j₂ = Very Good }, κ₂ = {j₃ = medium, j₄ = small}, κ₃ = {j₅ = brown } and let Ω = {ϖ₁, ϖ₁, ϖ₁} be a set of decision makers. If we consider NHS ξ_{ϖ
₁}, ξ_{ϖ
₂}, ξ_{ϖ
₃} given as $\begin{matrix} ξ_{ϖ_{1}} (j_{1}, j_{3}, j_{5}) = {a (0.4, 0.3, 0.5), b (0.1, 0.2, 0.7), \underset{¸}{(} 0.8, 0.3, 0.1)}, \\ ξ_{ϖ_{2}} (j_{1}, j_{4}, j_{5}) = {a (0.6, 0.1, 0.2), b (0.3, 0.2, 0.5), \underset{¸}{(} 0.2, 0.6, 0.3)}, \\ ξ_{ϖ_{3}} (j_{2}, j_{3}, j_{5}) = {a (0.2, 0.3, 0.7), b (0.1, 0.2, 0.8), \underset{¸}{(} 0.3, 0.6, 0.9)}, \\ ξ_{ϖ_{3}} (j_{2}, j_{4}, j_{5}) = {a (0.1, 0.1, 0.2), b (0.8, 0.2, 0.5), \underset{¸}{(} 0.2, 0.9, 0.2)}, \end{matrix}$

then ξ_Ω = {ξ_{ϖ
₁}, ξ_{ϖ
₂}, ξ_{ϖ
₃}} is a NHS class. Now let $\begin{matrix} g_{ϖ_{1}} (j_{1}, j_{3}, j_{5}) = \\ {a (0.4, 0.3, 0.5), b (0.1, 0.2, 0.7), c (0.8, 0.3, 0.1)}, \\ g_{ϖ_{1}} (j_{1}, j_{4}, j_{5}) = {a (0.2, 0.9, 0.1), b (0.2, 0.4, 0.5), c (0.4, 0.2, 0.8)}, \\ g_{ϖ_{1}} (j_{2}, j_{3}, j_{5}) = {a (0.2, 0.2, 0.4), b (0.6, 0.4, 0.5), c (0.2, 0.7, 0.4)}, \\ g_{ϖ_{1}} (j_{2}, j_{4}, j_{5}) = \\ {a (0.6, 0.1, 0.8), b (0.4, 0.2, 0.7), c (0.1, 0.4, 0.3)}, \end{matrix}$ is also NHS class. Then NHS classes can be written as {ξ_{ϖ
₁}, ξ_{ϖ
₂}, ξ_{ϖ
₃}},{g_{ϖ
₁}, g_{ϖ
₂}, g_{ϖ
₃}}.

3.3 Definition

Let $(R, O)$ and $(P, K)$ be two classes of NHS over the universal set $R$ and $P$ respectively. Let $θ : R \to P$ and $χ : O \to K$ be mappings. Then a mapping $ξ = (θ, χ) : (R, O) \to (P, K)$ is defined as for NHS (ζ, λ) in $(R, O)$ and ξ (ζ, λ) is NHS in $(P, K)$ obtained in such a way, For $ϑ \in χ (O) \subseteq K$ and $y \in P$ , then

ξ (ζ, λ) (ϑ) (y) $\bar{{} \begin{matrix} \underset{x \in θ^{-} 1 (y)}{\lor} (\underset{ϱ \in χ^{- 1} (ϑ) \land λ}{\lor} ζ (ϱ)) (x), if θ^{-} 1 (y) \neq \emptyset, \\ χ^{-} 1 (ϑ) \land λ \neq \emptyset \\ 0 if otherwise \end{matrix}$ (3)ξ (ζ, λ) is called a NHS image of NHS (ζ, λ).

3.4 Definition

Let $(R, O)$ and $(P, K)$ be two classes of NHS over the universal set $R$ and $P$ respectively. Let $θ : R \to P$ and $χ : O \to K$ be mappings. Now, let $(φ, H)$ be a NHS in $(P, K)$ , where $H \subseteq K$ then $ξ^{- 1} (φ, H)$ is a NHS in $(R, O)$ defined as follows, $ξ^{- 1} (φ, H) (ϱ) (x) = {\begin{matrix} φ (χ (ϱ) (θ (x) & if χ (ϱ) \in H \\ 0 & if otherwise \end{matrix}$ (4) where $ϱ \in χ^{- 1} (H) \subset O$ , then $ξ^{- 1} (φ, H)$ said to be the NHS inverse image of NHS $(φ, H)$ .

3.5 Example

Let $R = {a =$ Line Interactive, b = Standby-Ferro, c = Delta Conversion On-Line} and $P = {x =$ Microwave cum Convection, y = Conventional oven, z = Convection oven } be types of UPS (Uninterruptible Power Supply) and OVENS respectively, considered as universes of discourses. A Marketing manger wants to know the relationship between UPS and OVENS characteristics which will be more effective regarding his marketing. Let ε₁ = efficiency, ε₂ = size, ε₃ = colour, ε₄ = price and b₁ = efficiency, b₂ = colour, b₃ = price, b₄ = size be the two types of distinct attributes whose attribute values belong to the sets κ₁, κ₂, κ₃, κ₄ and $κ_{1}^{'}, κ_{2}^{'}, κ_{3}^{'}, κ_{4}^{'}$ respectively. Let κ₁ = {j₁ = Good, j₂ = Very Good}, κ₂ = {j₃ = medium, j₄ = small }, κ₃ = {j₅ = brown } , κ₄ = {j₇ = low price }. Similarly, $κ_{1}^{'} = {j'_{1} =$ Good, j′₂ = effective }, $κ_{2}^{'} = {j'_{3} = black, j'_{4} =$ white }, $κ_{3}^{'} = {j'_{5} =$ low price } , $κ_{4}^{'} = {j'_{7} =$ medium } and $(R, κ_{1} \times κ_{2} \times κ_{3} \times κ_{4})$ and $(P, κ_{1}^{'} \times κ_{2}^{'} \times κ_{3}^{'} \times κ_{4}^{'})$ be the classes of HSS. Let $θ : R \to P$ , $χ : κ_{1} \times κ_{2} \times κ_{3} \times κ_{4} \to κ_{1}^{'} \times κ_{2}^{'} \times κ_{3}^{'} \times κ_{4}^{'}$ be mappings as follows: θ (a) = z, θ (b) = y, θ (c) = y and $\begin{matrix} χ (j_{1}, j_{3}, j_{5}, j_{7}) = (j_{1}', j_{3}', j_{5}', j_{7}'), \\ χ (j_{1}, j_{4}, j_{5}, j_{7}) = (j_{1}', j_{3}', j_{5}', j_{7}'), \\ χ (j_{2}, j_{3}, j_{5}, j_{7}) = (j_{2}', j_{3}', j_{5}', j_{7}'), \\ χ (j_{2}, j_{4}, j_{5}, j_{7}) = (j_{2}', j_{4}', j_{5}', j_{7}'), \end{matrix}$ choose two NHS in $(R, κ_{1} \times κ_{2} \times κ_{3} \times κ_{4})$ and $(P, κ_{1}^{'} \times κ_{2}^{'} \times κ_{3}^{'} \times κ_{4}^{'})$ respectively, $\begin{matrix} (ζ, λ_{1} \times λ_{2} \times λ_{3} \times λ_{4}) = \\ {{j_{1}, j_{3}, j_{5}, j_{7}} = {a (0.7, 0.2, 0.5), b (0.4, 0.2, 0.8), \underset{¸}{(} 0.4, 0.1, 0.3)}, \\ {j_{1}, j_{4}, j_{5}, j_{7}} = {a (0.8, 0.3, 0.6), b (0.2, 0.2, 0.9), \underset{¸}{(} 0.5, 0.3, 0.1)}, \\ {j_{2}, j_{4}, j_{5}, j_{7}} = {a (0.8, 0.5, 0.3), b (0.3, 0.1, 0.9), \underset{¸}{(} 0.8, 0.2, 0.6)}}, \\ (φ, H_{1} \times H_{2} \times H_{3} \times H_{4}) = \\ {{j_{1}', j_{3}', j_{5}', j_{7}'} = {x (0.7, 0.2, 0.8), y (0.2, 0.4, 0.6), z (0.1, 0.7, 0.1)}, \\ {j_{1}', j_{4}', j_{5}', j_{7}'} = {z (0.5, 0.9, 0.5), y (0.3, 0.5, 0.8), z (0.4, 0.3, 0.7)}, \end{matrix}$ For a NHS (ζ, λ₁ × λ₂ × λ₃ × λ₄) in $(R, κ_{1} \times κ_{2} \times κ_{3} \times κ_{4})$ , then the NHS image can be written as (ξ (ζ, λ₁ × λ₂ × λ₃ × λ₄) , κ₁ × κ₂ × κ₃ × κ₄) is a NHS in $(P, κ_{1}^{'} \times κ_{2}^{'} \times κ_{3}^{'} \times κ_{4}^{'})$ , where (κ₁ × κ₂ × κ₃ × κ₄) = χ (λ₁ × λ₂ × λ₃ × λ₄) = {(j₁′, j₃′, j₅′, j₇′) , (j₂′, j₄′, j₅′, j₇′))} obtained as follows: $\begin{matrix} = (\underset{x \in θ^{- 1} (x)}{\lor} (ϱ \in χ^{- 1} (j_{1}', j_{3}', j_{5}', j_{7}') \land (λ_{1} \times λ_{2} \times λ_{3} \times λ_{4}) \end{matrix} \lor ζ (ϱ))) = (\underset{x \in θ^{- 1} (x)}{\lor} (\underset{ϱ \in {(j_{1}, j_{3}, j_{5}, j_{7}), (j_{1}, j_{4}, j_{5}, j_{7})}}{\lor} ζ (ϱ))) = (\underset{x \in θ^{- 1} (x)}{\lor} (ζ (j_{1}, j_{3}, j_{5}, j_{7}) \lor ζ (j_{2}, j_{4}, j_{5}, j_{7})))$ $= (\underset{x \in \emptyset}{\lor} ({a (0.7, 0.2, 0.5), b (0.4, 0.2, 0.8), c (0.4, 0.1, 0.3)} \lor {a (0.8, 0.3, 0.6), b (0.2, 0.2, 0.9), c (0.5, 0.3, 0.1)})),$ $(\underset{x \in \emptyset}{\lor} {a (0.8, 0.25, 0.5), b (0.4, 0.2, 0.9), c (0.5, 0.2, 0.3)}) = {0, 0, 0}$ $\begin{matrix} (ξ (ζ, λ_{1} \times λ_{2} \times λ_{3} \times λ_{4}), κ_{1} \times κ_{2} \times κ_{3} \times κ_{4}) \\ (j_{1}', j_{3}', j_{5}', j_{7}') (x) = {0, 0, 0}, \end{matrix}$ Similarly, $\begin{matrix} (ξ (ζ, λ_{1} \times λ_{2} \times λ_{3} \times λ_{4}), κ_{1} \times κ_{2} \times κ_{3} \times κ_{4}) \\ (j_{1}', j_{3}', j_{5}', j_{7}') (y) = {0.5, 0.2, 0.3}, \end{matrix}$ $\begin{matrix} (ξ (ζ, λ_{1} \times λ_{2} \times λ_{3} \times λ_{4}), κ_{1} \times κ_{2} \times κ_{3} \times κ_{4}) \\ (j_{1}', j_{3}', j_{5}', j_{7}') (z) = {0.8, 0.25, 0.5}, \end{matrix}$ $\begin{matrix} (ξ (ζ, λ_{1} \times λ_{2} \times λ_{3} \times λ_{4}), κ_{1} \times κ_{2} \times κ_{3} \times κ_{4}) \\ (j_{2}', j_{4}', j_{5}', j_{7}') (x) = {0, 0, 0}, \end{matrix}$ $\begin{matrix} (ξ (ζ, λ_{1} \times λ_{2} \times λ_{3} \times λ_{4}), κ_{1} \times κ_{2} \times κ_{3} \times κ_{4}) \\ (j_{2}', j_{4}', j_{5}', j_{7}') (y) = {0.8, 0.5, 0.6}, \end{matrix}$ $\begin{matrix} (ξ (ζ, λ_{1} \times λ_{2} \times λ_{3} \times λ_{4}), κ_{1} \times κ_{2} \times κ_{3} \times κ_{4}) \\ (j_{2}', j_{4}', j_{5}', j_{7}') (z) = {0.8, 0.5, 0.3}, \end{matrix}$ The NHS image form can be written as, $\begin{matrix} (ξ (ζ, λ_{1} \times λ_{2} \times λ_{3} \times λ_{4}), κ_{1} \times κ_{2} \times κ_{3} \times κ_{4}) \bar{{} (j_{1}', j_{3}', j_{5}', j_{7}') = {x (0, 0, 0), y (0.8, 0.15, 0.3), z (0.8, 0.35, 0.3)}, \bar{(} j_{2}', j_{4}', j_{5}', j_{7}') = {x (0, 0, 0), y (0.8, 0.5, 0.6), z (0.8, 0.5, 0.3)}}, \end{matrix}$ Now for NHS inverse image, $\begin{matrix} φ (ξ (j_{1}, j_{3}, j_{5}, j_{7}) θ (a) \end{matrix}$ $\begin{matrix} = φ (j'_{1}, j'_{3}, j'_{5}, j'_{7}) z \bar{{} 0.1, 0.7, 0.1} \end{matrix}$ $\begin{matrix} ξ^{- 1} ((φ, H_{1} \times H_{2} \times H_{3} \times H_{4}), Θ_{1} \times Θ_{2} \times Θ_{3} \times Θ_{4}) \\ ((j_{1}, j_{3}, j_{5}, j_{7}) (b) = {0.2, 0.4, 0.6}, \end{matrix}$ $\begin{matrix} ξ^{- 1} ((φ, H_{1} \times H_{2} \times H_{3} \times H_{4}), Θ_{1} \times Θ_{2} \times Θ_{3} \times Θ_{4}) \\ ((j_{1}, j_{3}, j_{5}, j_{7}) (c) = {0.2, 0.4, 0.6}, \end{matrix}$ $\begin{matrix} ξ^{- 1} ((φ, H_{1} \times H_{2} \times H_{3} \times H_{4}), Θ_{1} \times Θ_{2} \times Θ_{3} \times Θ_{4}) \\ ((j_{1}, j_{4}, j_{5}, j_{7}) (a) = {0.1, 0.7, 0.1}, \end{matrix}$ $\begin{matrix} ξ^{- 1} ((φ, H_{1} \times H_{2} \times H_{3} \times H_{4}), Θ_{1} \times Θ_{2} \times Θ_{3} \times Θ_{4}) \\ ((j_{1}, j_{4}, j_{5}, j_{7}) (b) = {0.2, 0.4, 0.6}, \end{matrix}$ $\begin{matrix} ξ^{- 1} ((φ, H_{1} \times H_{2} \times H_{3} \times H_{4}), Θ_{1} \times Θ_{2} \times Θ_{3} \times Θ_{4}) \\ ((j_{1}, j_{4}, j_{5}, j_{7}) (c) = {0.2, 0.4, 0.6}, \end{matrix}$ and it can be written as, $\begin{matrix} ξ^{- 1} ((φ, H_{1} \times H_{2} \times H_{3} \times H_{4}), Θ_{1} \times Θ_{2} \times Θ_{3} \times Θ_{4}) \bar{{} (j_{1}, j_{3}, j_{5}, j_{7}) \\ hspace * - 1.3 pc = {a (0.1, 0.7, 0.1), b (0.2, 0.4, 0.6), c (0.2, 0.4, 0.6)}, = (j_{1}, j_{4}, j_{5}, j_{7}) \\ hspace * - 1.3 pc = {a (0.1, 0.7, 0.1), b (0.2, 0.4, 0.6), c (0.2, 0.4, 0.6)}}, \end{matrix}$ where $Θ_{1} \times Θ_{2} \times Θ_{3} \times Θ_{4} = χ^{- 1} (H_{1} \times H_{2} \times H_{3} \times H_{4}) = {(j_{1}, j_{3}, j_{5}, j_{7}), (j_{1}, j_{4}, j_{5}, j_{7})} .$

3.6 Definition

Let $ξ : (R, O) \to (P, K)$ be a mapping and (ζ, λ) and $(φ, H)$ be the two NHS in $(R, O)$ . Then for $ϑ \in K$ , NHS union and intersection of NHS images of (ζ, λ) and $(φ, H)$ in $(R, O)$ are defined as; $(ξ (ζ, λ) \lor ξ (φ, H)) (ϑ) = ξ (ζ, λ) (ϑ) \lor ξ (φ, H) (ϑ), (ξ (ζ, λ) \land ξ (φ, H)) (ϑ) = ξ (ζ, λ) (ϑ) \land ξ (φ, H) (ϑ) .$

3.7 Definition

Let $ξ : (R, O) \to (P, K)$ be a mapping and (ζ, λ) and $(φ, H)$ be the two NHS in $(R, O)$ . Then for ϱ ∈ O, NHS union and intersection of NHS inverse images of (ζ, λ) and $(φ, H)$ in $(R, O)$ are defined as; $(ξ^{- 1} (ζ, λ) \lor ξ^{- 1} (φ, H)) (ϱ) = ξ^{- 1} (ζ, λ) (ϱ) \lor ξ^{- 1} (φ, H) (ϱ), (ξ^{- 1} (ζ, λ) \land ξ^{- 1} (φ, H)) (ϱ) = ξ^{- 1} (ζ, λ) (ϱ) \land ξ^{- 1} (φ, H) (ϱ) .$

3.8 Definition

Let $M_{1} \in NHS (R \times P)$ and $M_{2} \in NHS (P \times S)$ , then max-min composition of M₁ and M₂ can be denoted as M₁oM₂ and defined as $M_{1} o M_{2} = {(W, W'), T_{M_{1} o M_{2}} (W, W'), I_{M_{1} o M_{2}} (W, W'), F_{M_{1} o M_{2}} (W, W') : W \in R, W' \in S}$ , where $T_{M_{1} o M_{2}} (W, W') = \max_{W' \in P} (T_{M_{1}} (W, W'), T_{M_{1}} (W', W'^{'}))$ , $I_{M_{1} o M_{2}} (W, W') = \min_{W' \in P} (I_{M_{1}} (W, W'), I_{M_{1}} (W', W'^{'}))$ , $F_{M_{1} o M_{2}} (W, W') = \min_{W' \in P} (F_{M_{1}} (W, W'), F_{M_{1}} (W', W'^{'}))$ .

3.9 Theorem

Let $ξ : (R, O) \to (P, K)$ be a NHS-mapping, let (ζ, λ), $(φ, H)$ be the two NHS in $(R, O)$ and with (ζ_i, λ_i) as the family of a NHS in $(R, O)$ we have,

ξ (∅) = ∅

$ξ (R) \subset P$

$ξ ((ζ, λ) \lor (φ, H)) = ξ (ζ, λ) \lor ξ (φ, H)$ . Generally, ξ (∨ _i (ζ_i, λ_i) = ∨ _iξ (ζ_i, λ_i).

$ξ ((ζ, λ) \land (φ, H)) \supseteq ξ (ζ, λ) \land ξ (φ, H)$ . Generally, ξ (∧ _i (ζ_i, λ_i)) ⊆ ∧ _iξ (ζ_i, λ_i).

If $(ζ, λ) \subseteq (φ, H)$ then $ξ (ζ, λ) \subseteq ξ (φ, H)$ .

Proof: Here, points 1 and 2 are trivial case,

3: For $ϑ \in K$ , we have to prove $ξ ((ζ, λ) \lor (φ, H)) (ϑ) = (ξ (ζ, λ) \lor ξ (φ, H)) (ϑ)$ .

Take, $ξ ((ζ, λ) \lor (φ, H)) (ϑ) = ξ (s, λ \lor H) (ϑ)$ $= {\begin{matrix} θ (\lor_{ϱ \in χ^{- 1} (ϑ) \land (λ \lor H)} s (ϱ)) if \\ {chi}^{- 1} (ϑ) \land (λ \lor H) \neq \emptyset, \\ \emptyset, Otherwise \end{matrix}$ (5) where, $s (ϱ) = {\begin{matrix} ζ (ϱ) & if ϱ \in (λ - H) \\ φ (ϱ) & if ϱ \in (H - λ) \\ ζ (ϱ) \lor φ (ϱ) & if ϱ \in (H \land λ) \end{matrix}$ (6) L.H.S: For a non trivial case when $χ^{- 1} (ϑ) \land (λ \lor H) \neq \emptyset$ , we have $ξ ((ζ, λ) \lor (φ, H)) (ϑ)$ $= θ (\lor {\begin{matrix} ζ (ϱ) & if ϱ \in (λ - H) \land χ^{- 1} (ϑ) \\ φ (ϱ) & if ϱ \in (H - λ) \land χ^{- 1} (ϑ) \\ ζ (ϱ) \lor φ (ϱ) & if ϱ \in (λ \land H) \land χ^{- 1} (ϑ) \end{matrix})$ (7) R.H.S: for non trivial case, we have $(ξ (ζ, λ) \lor ξ (φ, H)) (ϑ) =$ $\begin{matrix} u (\lor_{ϱ \in p^{- 1} (ϑ) \land λ} ζ (ϱ)) \lor u (\lor_{ϱ \in p^{- 1} (ϑ) \land \overset{˘}{H}} φ (ϱ)) u (\lor_{ϱ} \in p^{- 1} (ϑ) \land λ ζ (ϱ) \lor_{ϱ \in p^{- 1} (ϑ) \land \overset{˘}{H}} φ (ϱ)) \end{matrix}$ $= θ (\lor {\begin{matrix} ζ (ϱ) & if ϱ \in (λ - H) \land χ^{- 1} (ϑ) \\ φ (ϱ) & if ϱ \in (H - λ) \land χ^{- 1} (ϑ) \\ ζ (ϱ) \lor φ (ϱ) & if ϱ \in (λ \land H) \land χ^{- 1} (ϑ) \end{matrix})$ (8)

From (7) and (8), we have point 3.

4: For $ϑ \in K$ and y ∈ Y we have to show that $\begin{matrix} ξ ((ζ, λ) \land (φ, H)) (ϑ) \supseteq (ξ (ζ, λ) \land ξ (φ, H)) (ϑ) \end{matrix}$ $\begin{matrix} (ξ (ζ, λ) \land (φ, H)) (ϑ) = ξ (s, λ \land H) (ϑ) \end{matrix}$ $= {\begin{matrix} θ (\lor_{ϱ \in χ^{- 1} (ϑ) \land (λ \land H)} s (ϱ)) if χ^{- 1} (ϑ) \land (λ \land H) \neq \emptyset, \\ \emptyset, Otherwise, \end{matrix}$ (9) Where, s (ϱ) = ζ (ϱ) ∧ φ (ϱ).

For the non trivial case, $χ^{- 1} (ϑ) \land (λ \land H) \neq \emptyset$

$ξ (s, λ \land H) (ϑ) = θ (\lor_{ϱ \in χ^{- 1} (ϑ) \land (λ \land H)} s (ϱ)) = θ (\lor_{ϱ \in χ^{- 1} (ϑ) \land (λ \land H)} (ζ (ϱ) \land φ (ϱ)))$ $\begin{matrix} (ξ (ζ, λ) \land (φ, H)) (ϑ) \\ = θ (\lor_{ϱ \in χ^{- 1} (ϑ) \land (λ \land H)} (ζ (ϱ) \land φ (ϱ))) \end{matrix}$ on the other side, by using definition 12 we have, $\begin{matrix} (ξ (ζ, λ) \land (φ, H)) (ϑ) = (ξ (ζ, λ) (ϑ) \land ξ (φ, H) (ϑ)) \end{matrix}$ $= ({\begin{matrix} θ (\lor_{ϱ \in χ^{- 1} (ϑ) \land λ} ζ (ϱ)) if χ^{- 1} (ϑ) \land (λ \land H) \neq \emptyset \\ \emptyset, Otherwise, \end{matrix})$ (10) $\land ({\begin{matrix} θ (\lor_{ϱ \in χ^{- 1} (ϑ) \land H} φ (ϱ)) if χ^{- 1} (ϑ) \land (λ \land H) \neq \emptyset \\ \emptyset, Otherwise, \end{matrix})$ (11) Neglecting the trivial case, we obtain $\begin{array}{l} (ξ (ζ, λ) \land (φ, ℋ)) (ϑ) = \\ θ (\lor_{ϱ \in χ^{- 1} (ϑ) \land λ} ζ (ϱ)) \land θ (\lor_{ϱ \in χ^{- 1} (ϑ) \land ℋ} φ (ϱ)) \\ \supseteq θ (\lor_{ϱ \in χ^{- 1} (ϑ) \land (λ \land ℋ)} (ζ (ϱ) \land φ (ϱ))) \\ = ξ ((ζ, λ) \land (φ, ℋ)) (ϑ) \end{array}$

5: For a non-trivial case for $ϑ \in K$

ξ ((ζ, λ) (ϑ)

$= {\begin{matrix} θ (\lor_{ϱ \in χ^{- 1} (ϑ) \land λ} ζ (ϱ)) if χ^{- 1} (ϑ) \land λ \neq \emptyset \\ \emptyset, Otherwise . \end{matrix}$ (12) Then, $\begin{matrix} ξ ((ζ, λ) (ϑ) = θ (\lor_{ϱ \in χ^{- 1} (ϑ) \land λ} ζ (ϱ)) \\ subseteq θ (\lor_{ϱ \in χ^{- 1} (ϑ) \land H} φ (ϱ)) \\ = ξ (φ, H) (ϑ) . \end{matrix}$ This yields point 5. Reversible of points 2 and 4 do not hold. It can be explain in the following example.

3.10 Example

From Example 2, $\begin{matrix} P ⊄ ξ (R) = {{j_{1}', j_{3}', j_{5}', j_{7}'} = {x (0.7, 0.2, 0.8), y (0.2, 0.4, 0.6), z (0.1, 0.7, 0.1)}, {j_{1}', j_{4}', j_{5}', j_{7}'} \\ = {z (0.5, 0.9, 0.5), y (0.3, 0.5, 0.8), z (0.4, 0.3, 0.7)}} . \end{matrix}$ This indicate that reverse of point 2 does not hold. Now we are going to show reverse of point 4 also does not hold. For this purpose, we choose two NHS in $(R, κ_{1} \times κ_{2} \times κ_{3} \times κ_{4})$ as $\begin{matrix} (ζ, λ_{1} \times λ_{2} \times λ_{3} \times λ_{4}) = {{j_{1}, j_{3}, j_{5}, j_{7}} \\ = {a (0.8, 0.1, 0.5), b (0.8, 0.2, 0.8), \underset{¸}{(} 0.4, 0.1, 0.3)}, {j_{1}, j_{4}, j_{5}, j_{7}} \\ = {a (0.6, 0.3, 0.6), b (0.2, 0.2, 0.9), \underset{¸}{(} 0.5, 0.3, 0.1)}, {j_{2}, j_{4}, j_{5}, j_{7}} \\ = {a (0.8, 0.5, 0.3), b (0.7, 0.5, 0.9), \underset{¸}{(} 0.1, 0.2, 0.6)}, \end{matrix}$ $\begin{matrix} (φ, H_{1} \times H_{2} \times H_{3} \times H_{4}) = {{j_{1}, j_{3}, j_{5}, j_{7}} = {a (0.3, 0.1, 0.4), b (0.9, 0.3, 0.6), \\ hspace * - 1 pc c (0.2, 0.9, 0.3)}, {j_{1}, j_{4}, j_{5}, j_{7}} \\ = {a (0.5, 0.3, 0.7), b (0.1, 0.6, 0.9), c (0.2, 0.3, 0.8)}, \end{matrix}$ Then calculations indicate that $\begin{matrix} ξ (ζ, λ_{1} \times λ_{2} \times λ_{3} \times λ_{4}) \land ξ (φ, H_{1} \times H_{2} \times H_{3} \times H_{4}) \\ hspace * - 1.7 pc ⊄ ξ ((ζ, λ_{1} \times λ_{2} \times λ_{3} \times λ_{4}) \land (φ, H_{1} \times H_{2} \times H_{3} \times H_{4})) \end{matrix}$

3.11 Theorem

Let $ξ : (R, O) \to (P, K)$ be a mapping, (ζ, λ) and $(φ, H)$ be the two NHS in $(R, O)$ and the family of a NHS (ζ_i, λ_i) in $(R, O)$ we have,

ξ ^-1 (∅) = ∅

$ξ^{- 1} (P) = R$

$ξ^{- 1} ((ζ, λ) \lor (φ, H)) = ξ^{- 1} (ζ, λ) \lor ξ^{- 1} (φ, H)$ . Generally, ξ ^-1 (ζ_i, λ_i) = ∨ _iξ ^-1 (ζ_i, λ_i).

$ξ^{- 1} ((ζ, λ) \land (φ, H)) = ξ^{- 1} (ζ, λ) \land ξ^{- 1} (φ, H)$ . Generally, ξ ^-1 (ζ_i, λ_i)) = ∧ _iξ ^-1 (ζ_i, λ_i).

If $(ζ, λ) \subseteq (φ, H)$ then $ξ^{- 1} ((ζ, λ) \subseteq ξ^{- 1} (φ, H)$ .

Proof: Here, points 1 and 2 are trivial case,

3: For ϱ ∈ O, we have to prove $ξ^{- 1} ((ζ, λ) \lor (φ, H)) (ϱ) = (ξ^{- 1} (ζ, λ) \lor ξ^{- 1} (φ, H)) (ϱ)$ .

Take $\begin{matrix} ξ^{- 1} ((ζ, λ) \lor (φ, H)) (ϱ) \\ = & ξ^{- 1} (s, λ \lor H) (ϱ) \\ = & u^{- 1} (s (χ (ϱ)), χ (ϱ) \in λ \lor H \\ = & u^{- 1} (s (ϑ)), χ (ϱ) = ϑ \end{matrix}$ where,

$= θ^{- 1} ({\begin{matrix} ζ (ϑ) & if ϑ \in (λ - H) \\ φ (ϑ) & if ϑ \in (H - λ) \\ ζ (ϑ) \lor φ (ϑ) & if ϑ \in (H \land λ) \end{matrix})$ (13) Now by using definition 13, we have $\begin{matrix} (ξ^{- 1} (ζ, λ) \lor ξ^{- 1} (φ, H)) ϱ \\ = & ξ^{- 1} (ζ, λ) (ϱ) \lor ξ^{- 1} (φ, H) (ϱ) \\ = & u^{- 1} (ζ (χ (ϱ))) \lor u^{- 1} (φ (χ (ϱ))), χ (ϱ) \in λ \land H \end{matrix}$ $= θ^{- 1} ({\begin{matrix} ζ (ϑ) & if ϑ \in (λ - H) \\ φ (ϑ) & if ϑ \in (H - λ) \\ ζ (ϑ) \lor φ (ϑ) & if ϑ \in (H \land λ) \end{matrix})$ (14) Where ϑ = χ (ϱ). From (13) and (14), we get point 3.

Now, for point 4 we take ϱ ∈ O $\begin{matrix} ξ^{- 1} (ζ, λ) \land (φ, H) (ϱ) \\ = & (ξ^{- 1} (s, λ \land H) (ϱ) \\ = & θ^{- 1} (s (p (ϱ))), p (ϱ) \in λ \land H, \\ = & θ^{- 1} (ζ (ϑ) \land φ (ϑ)), p (ϱ) = ϑ, \\ = & θ^{- 1} (ζ (ϑ)) \land θ^{- 1} (φ (ϑ)) \\ = & (ξ^{- 1} (ζ, λ) \land ξ^{- 1} (φ, H)) (ϱ) \end{matrix}$ This gives point 4.

5: For ϱ ∈ O, consider ξ ^-1 (ζ, λ) (ϱ) $\begin{matrix} = & θ^{- 1} (ζ (χ (ϱ))) \\ = & θ^{- 1} (ζ (ϑ)), χ (ϱ) = ϑ \\ \subseteq & θ^{- 1} (φ (ϑ)) \\ = & θ^{- 1} (φ (χ (ϱ))) \\ = & ξ^{- 1} (φ, H) (ϱ) \end{matrix}$ This yields point 5.

4 Application of NHS mapping to Brain tumors and its relevant properties

In this part of the paper, we examine the brain tumor system and its relevant problems are evaluated. The reasons, symptoms, diagnosis, and treatment of brain tumors patients are analyzed and discussed with the comprehensive concept of the NHS and its relative mapping, inverse mapping. This section shows how the proposed mathematical model is suitable to set a plan for tumor patients.

4.1 Investigation of tumors and its related properties

The analytical investigation of tumors and mathematical modelling have eternal significance in the medical field. There are different sorts of tumors in medicine, but here, only four are considered for evaluation.

Astrocytomas

Meningiomas

Oligodendrogliomas

Ependymomas

4.1.1 Astrocytomas

A brain tumors that originates from a specific type of star-shaped brain cells called astrocytes is called an Astrocytoma. Other organs and are usually not affected by this kind of tumors. They are diagnosed as gliomas and can be found in several cerebral cortex fragments and the spinal cord. For more detail see Figs. 1, 2.

Fig. 1

Two PET images, one showing a normal brain and the other showing an astrocytoma at the bottom. Source:https://www-sciencedirect-com-443.web.bisu.edu.cn/science/article/pii/S0303846716302177.

Fig. 2

Astrocytoma MRI.

4.1.2 Meningiomas

A tumors formed on the membrane that encapsulates the brain and the spinal cord inside the skull. As the name implies, these tumors form on three layers of membranes called meninges. For more detail, see Fig. 3, 4.

Fig. 3

Contrast enlarged CT scan of the brain, showing the appearance of a meningioma.

Fig. 4

MRI image of a meningioma with contrast.

4.1.3 Oligodendrogliomas

Oligodendrogluomas is a type of glomeruli that are thought to be derived from oligodendrocytes of the brain or from glial antiseptic cells. They are diagnosed primarily in adults (9.4 in all essential brain system and tumors of the focal sensory system) but also develop in children (4 out of all essential brain tumors). For more detail, see Fig. 5, 6.

Fig. 5

Micrograph of an oligodendroglioma showing the characteristic branching, small, chicken wire-like blood vessels and fried egg-like cells, with clear cytoplasm and well-contagious cell boundaries. Source:https://support.google.com/legal/answer/3463239?hl=en.

Fig. 6

An oligodendroglioma as seen on MRI

4.1.4 Ependymomas

An epidemioma is a tumors that comes from the epithelium, a tissue of the focal sensory system. Usually, in the case of children, this area consists of letters, while in adults it is the backbone. The run of the mill position of intracranial ependymomas in the fourth ventricle. Scarcely, ependymomas can happen in the pelvic cavity. For more detail, see Fig. 7, 8.

Fig. 7

Micrograph of an ependymoma. H and E stain.

Fig. 8

Ependymoma of 4.ventricle in MRI. Left without, right with contrast-enhancement.

The patient is showing some common causes and symptoms of these tumors. We write here certain indications associated with these problems.

Headaches, which might be extreme and deteriorate with movement or in the early morning.

Seizures. Individuals may encounter various sorts of seizures. Certain medications can help forestall or control them

Personality or memory changes

Nausea or vomiting

Fatigue

Drowsiness

Sleep issues

Memory issues

Changes in capacity to walk or perform every day activities

Pressure or cerebral pain close to the tumors

In the following section, we discuss the technique, which we will implement for our scientific demonstrating. We develop a unique algorithm dependent on NHS-mapping to analyze the sickness, to provide good treatment, and progress of treatment scenes.

4.2 Methodology

4.2.1 Pre step

A specialist faces a few complexities when analyzing a tumors patient because of the similar symptoms of tumors. It is very difficult to get the distinction between these classifications. It implies that these sorts of difficulties consist of uncertainties and unclearness, so the NHS is appropriate to deal with such information. Initially, a fuzzy interval [0, 1] is established for different types of tumors to interface verbal data into numerical language. For various kinds of tumors, a graph is established to evaluate the unique sort of tumors. This graph is given in Table 1.

Table 1
Diagnosis chart of brain tumors

Types of tumors Different ranges of [0, 1]

Astrocytomas [0.6, 1]

Meningiomas [0.5, 0.6)

Oligodendrogliomas (0.2, 0.5)

Ependymomas [0.1, 0.2]

No tumors [0, 0.1)

Types of tumors	Different ranges of [0, 1]
Astrocytomas	[0.6, 1]
Meningiomas	[0.5, 0.6)
Oligodendrogliomas	(0.2, 0.5)
Ependymomas	[0.1, 0.2]
No tumors	[0, 0.1)

Since each issue has its concentration with the progression of time. To gather the more beneficial information history of a patient, each doctor wants to observe at least 2-3 days of information compared to the showing up side effects for well finding. Different graphs are set of conditions and their day after day fixation to analyze the tumors. This graph is given in Table 2 or Figure 9. There are three stages for each kind of turmoil; one is serious confusion, the second is moderate, and the third is a common issue. The flow chart of various ranges assigned to these restrictions is given in Figure 9.

Table 2

Related conditions and their week after week fixation to analyze tumors

conditions	below 1st week	included 2nd and third week	above 3rd week
serious Astrocytomas (SA)	0.72 ≤ γ < 0.8	0.8 ≤ γ < 1	= 1
moderate Astrocytomas (MA)	0.75 ≤ γ < 0.82	0.82 ≤ γ < 0.87	0.87 ≤ γ < 0.92
low Astrocytomas (LA)	0.6 ≤ γ < 0.65	0.65 ≤ γ < 0.69	0.69 ≤ γ < 0.74
serious Meningiomas (SM)	0.55 ≤ γ < 0.57	0.57 ≤ γ < 0.58	0.58 ≤ γ < 0.59
moderate Meningiomas (SM)	0.551 ≤ γ < 0.558	0.558 ≤ γ < 0.559	0.559 ≤ γ < 0.5596
low Meningiomas (LM)	0.557 ≤ γ < 0.559	0.559 ≤ γ < 0.5597	0.5597 ≤ γ < 0.5593
serious Oligodendrogliomas (SO)	0.2 ≤ γ < 0.3	0.3 ≤ γ < 0.4	0.4 ≤ γ < 0.49
moderate Oligodendrogliomas (MO)	0.23 ≤ γ < 0.25	0.25 ≤ γ < 0.27	0.27 ≤ γ < 0.4
low Oligodendrogliomas (LO)	0.22 ≤ γ < 0.23	0.23 ≤ γ < 0.235	0.235 ≤ γ < 0.37
serious Ependymomas (SE)	0.1 ≤ γ < 0.15	0.15 ≤ γ < 0.17	0.17 ≤ γ < 0.176
moderate Ependymomas (ME)	0.12 ≤ γ < 0.13	0.13 ≤ γ < 0.15	0.15 ≤ γ < 0.157
low Ependymomas (LE)	0.123 ≤ γ < 0.125	0.125 ≤ γ < 0.129	0.129 ≤ γ < 0.189
No tumors (NT)	0.00 ≤ γ < 0.01	0.01 ≤ γ < 0.06	0.06 ≤ γ < 0.08

Fig. 9

Flow chart of diferent ranges corresponding to the listed conditions of Brain tumors.

4.2.2 Algorithm

Step 1. We distinguish the tumor. Let R = {r₁, r₂, r₃, . . . , r_n} be collection of patients suffering from tumor and A = {s₁, s₂, s₃, . . . , s_v} be the gathering of symptoms of tumor whose corresponding attribute values belong to sets S_i’s, where $S = \prod_{i = 1}^{v} S_{i}$ . The administration assemble a "t" number of daily diagnostic chart (which can be fit out as NHS) by the assistance of a mathematician utilizing etymological terms. This chart will assist us to find the proper infection of the patient. The NHS chart gave by the specialist after essential evaluation at ɛth times can be fit out as $z_{S}^{ɛ} = {z_{p}^{ɛ} = {r, 〈 T_{p}^{ɛ} (r), I_{p}^{ɛ} (r), F_{p}^{ɛ} (r) 〉} : r \in R, p \in S}$ , where $T_{p}^{ɛ} (r)$ , $I_{p}^{ɛ} (r)$ and $F_{p}^{ɛ} (r)$ are membership, indeterminacy and non membership grades of Astrocytomas, Meningiomas, Oligodendrogliomas and Ependymomas for kth symptoms and lth patients respectively, (l = 1, 2, 3, . . . , n, k = 1, 2, 3, . . . , |S|, ɛ = 1, 2, 3, . . . , t). We take NHS union of all "t" number of daily diagnostic charts to gain maximum information of all patients.

Step 2. We expect that $B = {s_{1}^{'}, s_{2}^{'}, s_{3}^{'}, . . ., s_{w}^{'}}$ be the gathering of associated symptoms to A whose corresponding attribute values belong to sets $S_{i}^{'}$ ’s, where $S^{'} = \prod_{i = 1}^{w} S_{i}^{'}$ . We consider an NHS (specialist designating weights by remembering that patients ɛ number day assessment of fundamental symptoms) in view of major or introductory signs of the patients.

Step 3. We build a mapping defining as ζ : R → R and ρ : S → S′ characterized as follows; ζ (r_l) = r_l, $ρ (p_{k}) = (p_{k^{'}}^{'})$ , (l = 1, 2, 3, . . . , n, k = 1, 2, 3, . . . , |S|, k′ = 1, 2, 3, . . . , |S′|) depends upon the relationship among fundamental and primary symptoms.

Let NHS-mapping Ψ = (ζ, ρ) : NHS (R) → NHS (R) defined as

$\begin{matrix} T_{Ψ (z_{S})} (p^{'}) (r) = | T_{p_{k^{'}}^{'}} | \\ {\begin{matrix} \max_{r \in ζ^{- 1} (r)} (\max_{p \in ρ^{- 1} (p^{'}) \cap S} T_{z_{S}}) (r) if \\ ζ^{- 1} (r) \neq \emptyset, ρ^{- 1} (p^{'}) \cap S \neq \emptyset, \\ 0 if otherwise \end{matrix} \end{matrix}$ (15)

$\begin{matrix} I_{Ψ (z_{S})} (p^{'}) (r) = | I_{p_{k^{'}}^{'}} | \\ {\begin{matrix} \min_{r \in ζ^{- 1} (r)} (\min_{p \in ρ^{- 1} (p^{'}) \cap S} I_{z_{S}}) (r) if \\ ζ^{- 1} (r) \neq \emptyset, ρ^{- 1} (p^{'}) \cap S \neq \emptyset, \\ 1 if otherwise \end{matrix} \end{matrix}$ (16)

$\begin{matrix} F_{Ψ (z_{S})} (p^{'}) (r) = | F_{p_{k^{'}}^{'}} | \\ {\begin{matrix} \min_{r \in ζ^{- 1} (r)} (\min_{p \in ρ^{- 1} (p^{'}) \cap S} F_{z_{S}}) (r) if \\ ζ^{- 1} (r) \neq \emptyset, ρ^{- 1} (p^{'}) \cap S \neq \emptyset, \\ 1 if otherwise \end{matrix} \end{matrix}$ (17) where $T_{p_{k^{'}}^{'}}$ , $I_{p_{k^{'}}^{'}}$ and $F_{p_{k^{'}}^{'}}$ are related weights from z_S′. Get the image of $⊔ z_{S}^{ɛ}$ under the characterized mapping Ψ, which can be composed as $z_{S^{'}}^{'}$ .

Step 4.

Then define the after effects set with the values given in Table 2 and gather the pre-diagnosis table from which we can see the precision of final results.

Step 5.

Calculate the score function values of the acquiring NHS $z_{S^{'}}^{'}$ and gain average of every score value corresponding to related symptoms by using $| T_{s}^{ɛ} (r) - 2 I_{s}^{ɛ} (r) - F_{s}^{ɛ} (r) |$ .

Then, we carry out our final outcome from Table 1.

Step 6.

We suppose that $B = {s_{1}^{'}, s_{2}^{'}, s_{3}^{'}, . . ., s_{w}^{'}}$ be the collection of associated symptoms, where $k = \prod_{i = 1}^{w} | S_{i}^{'} |$ and C = {c₁, c₂, c₃, . . . , c_x} be a collection of possible medications (treatments) then we can build χ_S′, where χ is NHS function from S′ to P (C) which is the set of doctor’s recommendations with the appropriate treatment corresponding to the symptoms of tumors.

Step 7.

We utilize min-max composition over $z_{S^{'}}^{'}$ and χ_S′ and get $R_{C}^{1}$ by using the definition 14.

Step 8.

We choose the medications (treatments) having additional advantages and fewer side impacts. The following steps are necessary for the progress history of the patient.

Step 9.

We characterize a unique generalized mappings; ζ′ : R ^q-1 → R ^q and ρ′ : C ^q-1 → C ^q such that ζ′ (r_l) = r_l and ρ′ (c_x) = c_x . Then NHS-mapping can be written in such a way $Ψ^{'} = (ζ^{'}, ρ^{'}) : R_{C}^{q - 1} \to R_{C}^{q}$ and can be evaluated as: $R_{C}^{q} = Ψ^{'} (R_{C}^{q - 1}) (c) (r)$ $= \frac{1}{q} {\begin{matrix} ⊔_{π \in ζ^{' - 1} (r)} (⊔_{ϑ \in ρ^{' - 1} (c) \cap C} R_{C}^{q - 1}) (π) if \\ ζ^{' - 1} (r) \neq \emptyset, ρ^{' - 1} (c) \cap C \neq \emptyset, \\ 0 if otherwise \end{matrix}$ (18) where q = 2, 3, 4 . . . is the number of medications(treatments) episodes and c ∈ ρ′ (C) ⊆ C, r ∈ R ^q, π ∈ R ^q-1, ϑ ∈ (C) ^q-1.

Step 10.

Redo step 9 again and again till we get our required outcomes.

4.2.3 Limitation of the method

There are several limitations of the method that must be assured before implementing the algorithm as follows;

There must be a mapping that maps the entangled parameter to the associated fundamental parametric value by considering the similarity in the structure and basis of these parameters.

The two sets on which mapping or compositions are defined must be independent of each other and must be from the same class of structure (NHS).

A doctor’s recommendations are needed with the appropriate treatment concerning the symptoms of the disease should be known based on the historical record of the disease.

The different ranges of concern types of disease must be known with the help of a doctor.

If the proposed treatment results in negative effects to the patient, then the inverse mapping is needed to retain the preceding stage of the patient’s report under consideration for the suggestion of new treatment to save the patient’s progress history until the infection is relieved.

4.3 Proposed study and numerical example

This section of the article is devoted to the application of the suggested algorithm to a medical scenario. The aid of medical personnel is used to translate and gather the input samples in mathematical language. The next step involves selecting the group of patients that has tumor symptoms pointed out by the doctor. From here, a descriptive map was built under the supervision of the doctor for the conditions of different tumors (Table 1) and their daily conditions (Table 2) concerning the diagnosis. With these tables, the symptoms related to tumors can be analyzed in terms of the severity of the disorder. The best feature of the technique is that one can add the initial data in our proposed model to determine the particular form of the diseases. The algorithm will also recommend the best course of medication for the specific form of the diagnosed conditions. The algorithm will allow complete generalized mapping of the patient’s recovery with efficient recovery graphs for individual patients and comparative case analysis and suitable criteria that will help optimize the technique shortly. For the algorithm, information was collected from several immune-challenged individuals, and with the help of the proposed model, the data can be applied for the statistical formulation and modeling of descriptive concepts. Four patients have a particular type of tumor and needed to be diagnosed by a doctor. As there are multiple overlapping symptoms for a similar disease, it is challenging to pinpoint the right one. After a few patients’ examinations, the doctor rules out some dynamics based on the patient’s health, recent and past skin color changes, the patient’s history, genetic factors, etc. From there, the doctor devises a treatment and rehabilitation plan after the initial evaluation of the patient.

We have considered hypothetical data for the execution of the proposed algorithm. It is an algorithm and a procedure to show how one can compute the proposed model with numeric calculations. However, if this model is generated for real-time data, one can obtain the desired result by evaluating the real data.

Step 1.

Let R = {r₁, r₂, r₃, r₄} be collection of four patients. Let s₁ = Pain, s₂ = Intense pain s₃ = Confusion, be distinct attributes of symptoms whose corresponding attribute values belong to the sets S₁, S₂ and S₃ respectively. Let S₁ = {s₁₁ = Pain in forehead, s₁₂ = Pain in temples of head}, S₂ = {s₂₁ = Soreness}, S₃ = {s₃₁ = agitation, s₃₂ = anxiety}, which can be assessed by the doctor after complete checkup. As indicated by the underlying initial information of patients with the above defined symptoms, we can build a chart of two (ɛ = 2) days with the information gathered by the doctor given as $z_{S}^{ɛ} \in NHS (R)$ for the 1st and 2nd-day given as (Table 3) and (Table 4) respectively, which are in the form of NHS. Next, we take NHS-union over the $z_{S}^{1}$ and $z_{S}^{2}$ . The resultant NHS $⊔ z_{S}^{ɛ}$ is given as Table 5.

Table 3
Tabular representation of $z_{S}^{1}$ : Chart of first day of patients regarding symptoms from S

symptoms / patients r ₁ r ₂ r ₃ r ₄

(s₁₁, s₂₁, s₃₁) (0.4, 0.1, 0.5) (0.7, 0.1, 0.9) (0.5, 0.3, 0.5) (0.2, 0.3, 0.8)

(s₁₁, s₂₁, s₃₂) (0.7, 0.2, 0.8) (0.1, 0.2, 0.4) (0.4, 0.1, 0.6) (0.8, 0.2, 0.9)

(s₁₂, s₂₁, s₃₁) (0.4, 0.1, 0.5) (0.4, 0.1, 0.5) (0.4, 0.1, 0.5) (0.6, 0.1, 0.5)

(s₁₂, s₂₁, s₃₂) (0.5, 0.7, 0.3) (0.3, 0.8, 0.6) (0.6, 0.1, 0.3) (0.9, 0.4, 0.3)

symptoms / patients	r ₁	r ₂	r ₃	r ₄
(s₁₁, s₂₁, s₃₁)	(0.4, 0.1, 0.5)	(0.7, 0.1, 0.9)	(0.5, 0.3, 0.5)	(0.2, 0.3, 0.8)
(s₁₁, s₂₁, s₃₂)	(0.7, 0.2, 0.8)	(0.1, 0.2, 0.4)	(0.4, 0.1, 0.6)	(0.8, 0.2, 0.9)
(s₁₂, s₂₁, s₃₁)	(0.4, 0.1, 0.5)	(0.4, 0.1, 0.5)	(0.4, 0.1, 0.5)	(0.6, 0.1, 0.5)
(s₁₂, s₂₁, s₃₂)	(0.5, 0.7, 0.3)	(0.3, 0.8, 0.6)	(0.6, 0.1, 0.3)	(0.9, 0.4, 0.3)

Table 4

Tabular representation of $z_{S}^{2}$ : Chart of second day of patients regarding symptoms from S

symptoms / patients	r ₁	r ₂	r ₃	r ₄
(s₁₁, s₂₁, s₃₁)	(0.5, 0.3, 0.9)	(0.5, 0.2, 0.9)	(0.5, 0.7, 0.8)	(0.6, 0.8, 0.3)
(s₁₁, s₂₁, s₃₂)	(0.4, 0.6, 0.5)	(0.4, 0.1, 0.7)	(0.4, 0.3, 0.8)	(0.6, 0.7, 0.6)
(s₁₂, s₂₁, s₃₁)	(0.1, 0.7, 0.6)	(0.2, 0.7, 0.5)	(0.6, 0.1, 0.5)	(0.3, 0.3, 0.8)
(s₁₂, s₂₁, s₃₂)	(0.6, 0.1, 0.5)	(0.3, 0.1, 0.5)	(0.4, 0.1, 0.5)	(0.4, 0.1, 0.5)

Table 5

Tabular representation of $⊔ z_{S}^{ɛ}$ : NHS union of $z_{S}^{1}$ and $z_{S}^{2}$

symptoms / patients	r ₁	r ₂	r ₃	r ₄
(s₁₁, s₂₁, s₃₁)	(0.5, 0.2, 0.5)	(0.7, 0.15, 0.9)	(0.5, 0.5, 0.5)	(0.6, 0.55, 0.3)
(s₁₁, s₂₁, s₃₂)	(0.7, 0.4, 0.5)	(0.4, 0.15, 0.4)	(0.4, 0.2, 0.6)	(0.8, 0.45, 0.6)
(s₁₂, s₂₁, s₃₁)	(0.4, 0.4, 0.5)	(0.4, 0.4, 0.5)	(0.6, 0.1, 0.5)	(0.6, 0.2, 0.5)
(s₁₂, s₂₁, s₃₂)	(0.6, 0.4, 0.3)	(0.3, 0.45, 0.5)	(0.6, 0.1, 0.3)	(0.9, 0.25, 0.3)

Step 2.

Supposee s₁′ = Headaches, s₂′ = Nausea, s₃′ = Memory problems, be distinct attributes of connected symptoms of tumors whose corresponding attribute values belong to the sets $S_{1}^{'}, S_{2}^{'}, S_{3}^{'}$ . Let $S_{1}^{'} = {s_{11}^{'} =$ tightness sensation in the head, $s_{12}^{'} =$ stroke}, $S_{2}^{'} = {s_{21}^{'} =$ dizziness}, $S_{3}^{'} = {s_{31}^{'} =$ Trouble understanding, $s_{32}^{'} =$ Memory loss that disrupts daily life }. Doctor allot the weight to the associated symptoms regarding to the gathered information of patients and we convert verbal data into numerical terminology into the type of NHS z_S′ given as Table 6.

Table 6

Tabular representation of z_S′: Weights of the associated symptoms S′ regarding to each patient in NHS

symptoms / patients	r ₁	r ₂	r ₃	r ₄
$(s_{11}^{'}, s_{21}^{'}, s_{31}^{'})$	(0.4, 0.1, 0.7)	(0.6, 0.1, 0.3)	(0.5, 0.3, 0.2)	(0.6, 0.2, 0.1)
$(s_{11}^{'}, s_{21}^{'}, s_{32}^{'})$	(0.8, 0.3, 0.2)	(0.9, 0.2, 0.4)	(0.7, 0.1, 0.3)	(0.8, 0.2, 0.2)
$(s_{12}^{'}, s_{21}^{'}, s_{31}^{'})$	(0.9, 0.2, 0.3)	(0.7, 0.3, 0.4)	(0.9, 0.2, 0.3)	(0.6, 0.5, 0.4)
$(s_{12}^{'}, s_{21}^{'}, s_{32}^{'})$	(0.9, 0.1, 0.1)	(0.8, 0.2, 0.4)	(0.7, 0.3, 0.1)	(0.8, 0.2, 0.1)

Step 3.

Now, we characterize two mappings; ζ : R → R and ρ : S → S′ such that ζ (r₁) = r₁, ζ (r₂) = r₂, ζ (r₃) = r₃, ζ (r₄) = r₄, and $ρ (s_{11}, s_{21}, s_{31}) = (s_{11}^{'}, s_{21}^{'}, s_{31}^{'}), ρ (s_{11}, s_{21}, s_{32}) = (s_{11}^{'}, s_{21}^{'}, s_{32}^{'}), rho (s_{12}, s_{21}, s_{31}) = (s_{12}^{'}, s_{21}^{'}, s_{31}^{'}), rho (s_{12}, s_{21}, s_{32}) = (s_{12}^{'}, s_{21}^{'}, s_{32}^{'}) .$

Then NHS-mapping can be written in such a way Ψ = (ζ, ρ) : NHS (R) → NHS (R). Now, we evaluate the image of $⊔ z_{S}^{ɛ}$ given as $z_{S^{'}}^{'}$ in table 7 by utilizing the above mapping in Step 3 in algorithm.

Step 4.

Comparing Table 7 with the Table 2 to get the table of initial diagnosis (Table 8). We will utilize this table later to analyze the precision of our outcomes.

Table 7

Tabular representation of $z_{S^{'}}^{'}$ : The image of $⊔ z_{S}^{ɛ}$ under NHS mapping

symptoms / patients	r ₁	r ₂	r ₃	r ₄
$(s_{11}^{'}, s_{21}^{'}, s_{31}^{'})$	(0.2, 0.002, 0.245)	(0.42, 0.0015, 0.081)	(0.25, 0.045, 0.02)	(0.36, 0.022, 0.003)
$(s_{11}^{'}, s_{21}^{'}, s_{32}^{'})$	(0.32, 0.036, 0.02)	(0.36, 0.06, 0.08)	(0.42, 0.003, 0.45)	(0.48, 0.008, 0.08)
$(s_{12}^{'}, s_{21}^{'}, s_{31}^{'})$	(0.63, 0.016, 0.045)	(0.28, 0.013, 0.064)	(0.36, 0.008, 0.054)	(0.48, 0.112, 0.096)
$(s_{12}^{'}, s_{21}^{'}, s_{32}^{'})$	(0.54, 0.004, 0.003)	(0.24, 0.018, 0.08)	(0.42, 0.009, 0.003)	(0.72, 0.01, 0.003)

Table 8

Tabular representation of initial diagnosis chart to analyze the precision of outcomes.

symptoms / patients	r ₁	r ₂	r ₃	r ₄
$(s_{11}^{'}, s_{21}^{'}, s_{31}^{'})$	(SO, NT, MO)	(SO, NT, LE)	(LO, NT, NT)	(SO, NT, NT)
$(s_{11}^{'}, s_{21}^{'}, s_{32}^{'})$	(SO, NT, NT)	(SO, NT, LE)	(SO, NT, SO)	(SO, NT, LE)
$(s_{12}^{'}, s_{21}^{'}, s_{31}^{'})$	(LA, NT, NT)	(LO, NT, NT)	(SO, NT, NT)	(SO, LE, LE)
$(s_{12}^{'}, s_{21}^{'}, s_{32}^{'})$	(MM, NT, NT)	(SO, NT, NT)	(SO, NT, NT)	(SA, NT, NT)

Step 5.

Computing the score estimations of NHS from table 7 using $| T_{s}^{ɛ} (r) - 2 I_{s}^{ɛ} (r) - F_{s}^{ɛ} (r) |$ .

for each patient concerning to their associated symptoms. After the score estimation values, taking the average score for each patient. These values are drawn in Table 9. Now, the diagnosis chart (Table 1) of tumors is used to compare with the outcomes obtained in Table 9. Correlation shows that patients r₁, r₃ and r₄ are determined to have Meningiomas and patient r₂ is determined to have Ependymomas.

Table 9

Tabular representation of patient score values data related to connected symptoms

patients / symptoms	$(s_{11}^{'}, s_{21}^{'}, s_{31}^{'})$	$(s_{11}^{'}, s_{21}^{'}, s_{32}^{'})$	$(s_{12}^{'}, s_{21}^{'}, s_{31}^{'})$	$(s_{12}^{'}, s_{21}^{'}, s_{32}^{'})$	Average score
r ₁	0.049	0.336	0.14	0.313	0.209
r ₂	0.22	0.16	0.036	0.384	0.2
r ₃	0.553	0.19	0.29	0.16	0.29
r ₄	0.529	0.124	0.399	0.715	0.44

Step 6.

After a diagnosis of the genuine kind of illness of each patient, the specialist will propose some medicine to the patients. We developed the NHS according to the specialist’s recommendations with the proper treatment relating to the types of tumors. Consider C = {c₁ = Chemotherapy, c₂ = Targeted therapy, c₃ = Alternating electric field therapy (tumor treating fields)} be distinct possible medications (treatments) then we build χ_S′, which is the set of doctor’s recommendations with the appropriate treatment corresponding to the symptoms of tumors. The set χ_S′ ∈ NHS (R) given as Table 10. In Table 10 the evaluations are given as indicated by the historical backdrop of each patient. The positive effects of medication (treatment) for each kind can be seen from membership grades, the indeterminacy grades represent the impartial impacts of each type, and falsity grades speak to the side effects of medications (treatments) kind of tumors along with its symptoms.

Table 10

Tabular representation of χ_S′: Doctor’s recommendations with the appropriate treatment corresponding to the symptoms of tumors.

treatments / symptoms	$(s_{11}^{'}, s_{21}^{'}, s_{31}^{'})$	$(s_{11}^{'}, s_{21}^{'}, s_{32}^{'})$	$(s_{12}^{'}, s_{21}^{'}, s_{31}^{'})$	$(s_{12}^{'}, s_{21}^{'}, s_{32}^{'})$
c ₁	(0.5, 0.3, 0.1)	(0.9, 0.5, 0.8)	(0.2, 0.5, 0.2)	(0.6, 0.3, 0.4)
c ₂	(0.5, 0.3, 0.2)	(0.4, 0.6, 0.1)	(0.7, 0.3, 0.4)	(0.6, 0.3, 0.2)
c ₃	(0.6, 0.1, 0.4)	(0.8, 0.1, 0.6)	(0.6, 0.3, 0.2)	(0.8, 0.2, 0.3)

Step 7.

We calculate NHS min-max composition among χ_S′ and $z_{S^{'}}^{'}$ to acquire the connection among suggested medicines and patients as Neutrosophic soft set $χ_{S^{'}} \circ z_{S^{'}}^{'} = R_{C}^{1}$ , see Table 11.

Table 11

Tabular representation of $R_{C}^{1}$ : Composition among χ_S′ and $z_{S^{'}}^{'}$ to acquire the connection among suggested medicines and patients

patients / treatments	c ₁	c ₂	c ₃
r ₁	(0.5, 0.1, 0.1)	(0.5, 0.1, 0.2)	(0.6, 0.1, 0.4)
r ₂	(0.9, 0.1, 0.3)	(0.9, 0.2, 0.1)	(0.9, 0.1, 0.4)
r ₃	(0.9, 0.2, 0.2)	(0.9, 0.2, 0.3)	(0.9, 0.2, 0.2)
r ₄	(0.8, 0.2, 0.1)	(0.8, 0.2, 0.1)	(0.8, 0.2, 0.1)

Step 8.

The medication (treatment) is suitable for the patients having greater benefits and less side effects. In this way, we find the score esteems by using score function that is given in algorithm step 4 relating to the medicines for each patient (Table 12).

Table 12

Tabular representation of patient score values data related to recommended treatment

patients / treatments	c ₁	c ₂	c ₃	Maximum esteems	Selected treatment
r ₁	0.2	0.1	0	0.2	c ₁
r ₂	0.4	0.4	0.3	0.4	c₁ or c₂
r ₃	0.3	0.2	0.3	0.3	c₁ or c₃
r ₄	0.3	0.3	0.3	0.3	any one

From Table 12, it can be determined that the treatments c₁ is best fit for patient r₁; while c₁ or c₂ can be suggested for patient r₂ for the best medication; r₃ patient can be treated with c₁ or c₃; for r₄ any one medication (treatment) can be chosen. The final choice is relying on the state of the patient as per his past clinical past history and kind of sickness.

Step 9.

Every patient’s scenario depends on the type of illness and its history of the patient. One can repeat the episodes till the illness is healed totally. We can observe the improvement of each patient by utilizing the NHS-mapping by defining two mappings ζ′ : R ^q-1 → R ^q and ρ′ : C ^q-1 → C ^q such that ζ′ (r₁) = r₁, ζ′ (r₂) = r₂, ζ′ (r₃) = r₃, ζ′ (r₄) = r₄;

and ρ′ (c₁) = c₁, ρ′ (c₂) = c₂, ρ′ (c₃) = c₃ .

Then NHS-mapping can be written in such a way $Ψ^{'} = (ζ^{'}, ρ^{'}) : R_{C}^{q - 1} \to R_{C}^{q}$

The NHS-mapping is given as

$\begin{matrix} R_{C}^{q} = Ψ^{'} (R_{C}^{q - 1}) (c) (r) = \frac{1}{q} \\ {\begin{matrix} \lor_{π \in ζ^{' - 1} (r)} (\lor_{ϑ \in ρ^{' - 1} (c) \cap C} R_{C}^{q - 1} (π) if \\ ζ^{' - 1} (r) \neq \emptyset, ρ^{' - 1} (c) \cap C \neq \emptyset \\ 0 if otherwise \end{matrix} \end{matrix}$ (19) where q = 2, 3, 4, . . . represents the number of episodes of treatments and c ∈ ρ′ (C) ⊆ C, r ∈ R ^q, π ∈ R ^q-1, ϑ ∈ C ^q-1 and episodes of treatment can be written in Tables 13, 14, 15 and 16 are given for q = 2, 3, 4 and 5 respectively. The progressive report for every patient are depicted in Figure 10, 11,12,13.

Table 13

Tabular representation of $R_{C}^{2}$ : Progress report of patients after second episode of treatments

patients / treatments	c ₁	c ₂	c ₃
r ₁	(0.25, 0.05, 0.05)	(0.25, 0.05, 0.1)	(0.3, 0.25, 0.2)
r ₂	(0.45, 0.05, 0.15)	(0.45, 0.1, 0.25)	(0.45, 0.25, 0.2)
r ₃	(0.45, 0.1, 0.2)	(0.45, 0.2, 0.3)	(0.45, 0.1, 0.1)
r ₄	(0.4, 0.1, 0.05)	(0.8, 0.1, 0.05)	(0.8, 0.1, 0.05)

Table 14

Tabular representation of $R_{C}^{3}$ : Progress report of patients after third episode of treatments

patients / treatments	c ₁	c ₂	c ₃
r ₁	(0.083, 0.016, 0.016)	(0.083, 0.016, 0.003)	(0.1, 0.083, 0.067)
r ₂	(0.15, 0.016, 0.05)	(0.15, 0.003, 0.083)	(0.15, 0.083, 0.067)
r ₃	(0.15, 0.003, 0.003)	(0.15, 0.003, 0.3)	(0.15, 0.003, 0.003)
r ₄	(0.13, 0.003, 0.016)	(0.8, 0.003, 0.016)	(0.8, 0.003, 0.016)

Table 15

Tabular representation of $R_{C}^{4}$ : Progress report of patients after fourth episode of treatments

patients / treatments	c ₁	c ₂	c ₃
r ₁	(0.020, 0.004, 0.004)	(0.020, 0.004, 0.008)	(0.025, 0.020, 0.016)
r ₂	(0.0375, 0.004, 0.0125)	(0.9, 0.008, 0.020)	(0.9, 0.020, 0.016)
r ₃	(0.0375, 0.008, 0.2)	(0.0375, 0.008, 0.3)	(0.0375, 0.008, 0.008)
r ₄	(0.033, 0.008, 0.016)	(0.8, 0.008, 0.004)	(0.8, 0.008, 0.004)

Table 16

Tabular representation of $R_{C}^{5}$ : Progress report of patients after fifth episode of treatments

patients / treatments	c ₁	c ₂	c ₃
r ₁	(0.004, 0.0008, 0.0008)	(0.004, 0.0008, 0.001)	(0.005, 0.004, 0.003)
r ₂	(0.007, 0.0008, 0.0025)	(0.007, 0.001, 0.004)	(0.007, 0.004, 0.003)
r ₃	(0.007, 0.001, 0.001)	(0.007, 0.001, 0.3)	(0.007, 0.001, 0.001)
r ₄	(0.006, 0.001, 0.016)	(0.8, 0.001, 0.0008)	(0.8, 0.001, 0.0008)

Fig. 10

Progress chart of patient r₁.

Fig. 11

Progress chart of patient r₂.

Fig. 12

Progress chart of patient r₃.

Fig. 13

Progress chart of patient r₄.

Step 10.

We redo step 9 again and again till we get outcomes for the patients. Improvement record can be seen in first, second and third can be depicted as 14, 15, 16.

Fig. 14

First episode patients record.

Fig. 15

Second episode patients record.

Fig. 16

Third episode patients record.

4.4 Discussion and comparative analysis

The proposed idea of NHS mapping is generally broad and proper for these kinds of sicknesses. These problems can’t be dealt with by utilizing existing theories because of their limitations (see Table 17). Because of these shortcomings, the collection of initial data of the patient isn’t possible. Yet, our proposed structure can change the patient history into a mathematical format with no deficiency of data, and we obtain predominant outcomes for diagnosis and medication of the patient. In Table 17, we compare our proposed model with the existing theories. However, when the attributes are further sub-divided into attribute values, all current theories fail to manage. This need is fulfilled in the proposed NHS mapping. It shows that our structure is solid compared with existing procedures and can positively deal with these sorts of issues. Now, we discuss our recommended approach and its precision.

In this calculation, we add numerous days because the tumors patient can’t analyze totally after the first checkup. The NHS and its union tell all the patient’s statistics, and we can relate seriousness with its symptoms.

We see that the relationship among the related and essential signs with its allocated appropriate weights is important in each patient trial. Suppose if we choose just initial symptoms at that point, outcomes will be unspecific.

In the second stage, we select the treatment for the patients as indicated by their kind of tumors. The score function can be used to rank the selected treatments.

Thirdly, we utilize a more generalized form of NHS mapping to observe the patients’ improvement history. With each episode, all the evaluations are diminishing up to zero, which implies that the symptoms of tumors, neutral impacts of medicine with treatments, and side impacts are diminishing. This model represents the progress of patients with the progression of time.

If a patient does not progress in the first episode, then inverse NHS-mapping can be utilized to get him back on the preceding episode to start medication from here once more.

The proposed procedure helps an enormous number of patients with different illnesses and multiform criteria under the impact of parameterizations. This study is steady and consistent to deal with such problems in the medical field and MCDM.

The decision-making committee (Doctors) will evaluate the data in the form of NHS by considering the degree of the influence and the total time of the influence as a complex number; along with the deep evaluation of the information by taking sub parametric values of assigned attributes as hypersoft structure; where all the data can be taken in a numeric value between 0 (degree of zero percent match) and 1 (degree of hundred percent match).

The application of this model aims to characterize the nearby diagnosis of any diseases with their related symptoms. These ideas are capable and essential to analyze the issue properly by interfacing it with scientific modeling. This investigation builds up a connection between symptoms and medicines, which diminishes the difficulty of the narrative. The calculation depends on NHS mapping to adequately recognize the disease and choose the best medication for each patient’s relating sickness, and the generalized NHS mapping will encourage a specialist (Doctor) to predict the patient’s progress history and to foresee the time of treatment till the infection is relieved.

Table 17
Comparison of the proposed NHS with existing theories

SN References Disadvantage Ranking

1 [1] lack of indeterminacy and non-membership Not Possible

2 [28] lack of indeterminacy and non-membership Not Possible

3 [29] lack of indeterminacy Not Possible

4 [30] lack of multiple properties Not Possible

5 [31] lack of multiplicity and indeterminacy Not Possible

6 [32] Does not develop strong relation between falsity and indeterminacy Not Possible

7 [33] lack of multiplicity Not Possible

8 [34] fail to deal when attributes can be further sub-partitioned into attribute values Not Possible

9 [2] lack of indeterminacy and non-membership Not Possible

10 [3] lack of indeterminacy and non-membership Not Possible

11 [6] lack of indeterminacy and non-membership Not Possible

12 [7] lack of indeterminacy and non-membership Not Possible

13 [58] fail to deal when attributes can be further sub-partitioned into attribute values Not Possible

14 Proposed Method in this paper Lengthy and heavy calculations in decision-making This challenge can be sorted out with the help of computer program

SN	References	Disadvantage	Ranking
1	[1]	lack of indeterminacy and non-membership	Not Possible
2	[28]	lack of indeterminacy and non-membership	Not Possible
3	[29]	lack of indeterminacy	Not Possible
4	[30]	lack of multiple properties	Not Possible
5	[31]	lack of multiplicity and indeterminacy	Not Possible
6	[32]	Does not develop strong relation between falsity and indeterminacy	Not Possible
7	[33]	lack of multiplicity	Not Possible
8	[34]	fail to deal when attributes can be further sub-partitioned into attribute values	Not Possible
9	[2]	lack of indeterminacy and non-membership	Not Possible
10	[3]	lack of indeterminacy and non-membership	Not Possible
11	[6]	lack of indeterminacy and non-membership	Not Possible
12	[7]	lack of indeterminacy and non-membership	Not Possible
13	[58]	fail to deal when attributes can be further sub-partitioned into attribute values	Not Possible
14	Proposed Method in this paper	Lengthy and heavy calculations in decision-making	This challenge can be sorted out with the help of computer program

4.5 Advantages of proposed method

The NHS model studies the data in all three possible dimensions of positive, indeterminant, and negative parts regarding patient’s disease in accordance with parametric values, where these dimensions are independent to each other.

The NHS model can deal with the parametric values of such disease in more detail considering the sub-parametric values, since this structure is based on the sub-parametric values of the different parameter and their order and arrangement.

An NHS can also arrange the information/data so that it can be easily studied and evaluated.

With the help of NHS mapping, one can deal similar nature of parameters in a single associated fundamental parameter and point out the fundamental symptoms of disease.

The NHS mapping not just recognizes the disease properly as well as chooses the best medication for patient’s relating infection.

The generalized NHS-mapping can predict patient’s progress reports and if the proposed treatment results in negative effects to the patient, then the inverse mapping can be used to retain the preceding stage of the patient’s report under consideration for the suggestion of new treatment; and encourages a specialist to save the patient’s progress history until the infection is relieved.

5 Conclusions

In this article, we look at tumors and their related issues. We have proposed a comprehensive way to diagnose the patient’s underlying symptoms and analyze their tumors. That is why we have presented NHS mapping with its inverse mapping and few practical tasks with their characteristics. We have developed a calculation having three stages. In the first stage, the model was used to analyze the patients’ real sort of tumors. In the second stage, appropriate medicines for the patients were accessed as indicated by the seriousness of the disease by utilizing NHS mapping. Thirdly, generalized NHS mapping was developed to observe the patient’s progress history and anticipate the patient’s medication time until he reported its normal range in the nervous syndrome region. This procedure is valuable and successful in analyzing the infections. Correlation shows that the proposed calculation is predominant, simple to deal with, substantial, solid, and adaptable to solve MCDM problems. In the future, one can expand exploration in the domain of Pythagorean fuzzy uncertain environment, Neutrosophic Hypersoft Set, Plithogenic Crisp Hypersoft Set, Plithogenic Fuzzy Hypersoft Set, Plithogenic Intuitionistic Fuzzy Hypersoft Set, Plithogenic Neutrosophic Hypersoft Set, complex fuzzy hypersoft set, complex Intuitionistic Fuzzy Hypersoft Set, complex Neutrosophic Hypersoft Set, and their hybrid structures. We will apply them in medical imaging issues, pattern recognition, recommender frameworks, social; the monetary framework estimated thinking, image processing, and game theory.

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An application of neutrosophic hypersoft mapping to diagnose brain tumor and propose appropriate treatment

Abstract

Keywords

1 Introduction

1.1 Motivation

1.2 Paper presentation

2 Preliminaries

2.1 Definition [1]

2.2 Definition [12]

2.3 Definition [16]

2.4 Definition [30]

2.5 Definition [24]

2.6 Definition [24]

2.8 Definition [4]

3 Mappings on Neutrosophic Hypersoft classes

3.1 Definition

3.2 Example

3.3 Definition

3.6 Definition

3.7 Definition

3.8 Definition

3.9 Theorem

3.11 Theorem

4.1 Investigation of tumors and its related properties

4.1.1 Astrocytomas

4.2.1 Pre step

Table 1 Diagnosis chart of brain tumors Types of tumors Different ranges of [0, 1] Astrocytomas [0.6, 1] Meningiomas [0.5, 0.6) Oligodendrogliomas (0.2, 0.5) Ependymomas [0.1, 0.2] No tumors [0, 0.1)

4.3 Proposed study and numerical example

5 Conclusions

References

Table 1
Diagnosis chart of brain tumors

Types of tumors Different ranges of [0, 1]

Astrocytomas [0.6, 1]

Meningiomas [0.5, 0.6)

Oligodendrogliomas (0.2, 0.5)

Ependymomas [0.1, 0.2]

No tumors [0, 0.1)