M-Parameterized N-Soft Topology-Based TOPSIS Approach for Multi-Attribute Decision Making

In this article, we presented the notion of M-parameterized N-soft set (MPNSS) to assign independent non-binary evaluations to both attributes and alternatives. The MPNSS is useful for making explicit the imprecise data which appears in ranking, rating, and grading positions. The proposed model is superior to existing concepts of soft set (SS), fuzzy soft sets (FSS), and N-soft sets (NSS). The concept of M-parameterized N-soft topology (MPNS topology) is defined on MPNSS by using extended union and restricted intersection of MPNS-power whole subsets. For these objectives, we define basic operations on MPNSSs and discuss various properties of MPNS topology. Additionally, some methods for multi-attribute decision making (MADM) techniques based on MPNSSs and MPNS topology are provided. Furthermore, the TOPSIS (technique for order preference by similarity to an ideal solution) approach under MPNSSs and MPNS topology is established. The symmetry of the optimal decision is illustrated by interesting applications of proposed models and new MADM techniques are demonstrated by certain numerical illustrations and well justified by comparison analysis with some existing techniques.


Introduction
The information in various complex real life problems is generally imprecise, ambiguous, and imperfect. Fuzzy modeling and fuzzy decision making are very helpful to capture these uncertainties. Conventionally, the information about an alternative is considered by the crisp numbers or linguistic numbers. The researchers have introduced various mathematical models to handle such realistic issues. Zadeh [1] innovated fuzzy set theory, rough set theory introduced by Pawlak in (1982) [2] and soft set theory established by Molodtsov [3] are powerful tools towards uncertainties. These theories are independent generalizations of classical sets or crisp sets. The notion of intuitionistic fuzzy set (IFS) innovated by Atanassov [4] is the extension of fuzzy set(FS) and a Pythagorean fuzzy set (PFS) established by Yager [5,6] is the expansion of IFS.
Soft set is the parametric representation of objects of universe that provides the binary evaluation to the objects. Numerous researchers have studied soft set to handle uncertainties. Fatimah et al. [7] invented the idea of N-soft set (NSS) to handle situations when non-binary assessments are expected to demonstrate the objects real importance. Recently, Riaz et al. [8] innovated the concept of N-soft topology (NS-topology) and its applications to MAGDM. Akram et al. [9,10] extended this concept to fuzzy N-soft sets (FNSSs) and hesitant N-soft sets (HNSSs) for MAGDM applications. Akram and Adeel [11] established the TOPSIS method for MAGDM with interval-valued hesitant fuzzy N-soft sets. Some researchers established various hybrid mathematical structures of soft sets (see [12][13][14][15][16][17][18]). Soft topology on soft sets was proposed by Cagman et al. [19], and Shabir and Naz [20]. Riaz and Tehrim [21] introduced bipolar fuzzy soft topology on bipolar fuzzy soft sets and developed an important application in medical diagnosis. Soft set theory and fuzzy set theory have been studied for decision-making and modeling uncertainties in recent decades (see [22][23][24][25][26][27][28][29][30][31]). Garg and Arora [32,33] introduced Dual hesitant fuzzy soft aggregation operators and Generalized intuitionistic fuzzy soft power aggregation operator. Pamucar and Jankovic [34] presented an application of the hybrid interval rough weighted Power-Heronian operators. Riaz et al. [35] introduced hesitant fuzzy soft topology and its applications to MADM. Riaz et al. [36,37] introduced soft rough topology and soft multi rough topology with new properties and applications to MADM. The concept of linear Diophantine fuzzy Set (LDFS) introduced by Riaz and Hashmi [38]. Kamaci [39] introduced new algebraic structures of LDFSs.
In N-soft set environments, the ranking, rating. or grading is assigned to alternatives/objects only. Meanwhile, there is a lack of independent non-binary evaluations to the attributes which may effect the decision analysis phenomena. To enhance the significance of attributes there is a need for non-binary grading positions given to the attributes. The main objective of this study is to handle these difficulties with M-parameterized N-soft set (MPNSS) and MPNS topology. The proposed model of MPNSS is very helpful to assign independent non-binary evaluations to both attributes and alternatives. Additionally, this model is useful for developing strong MADM techniques to select most convincible alternative and make a robust optimal decision.
To facilitate our discussion, the classification of the paper is presented as follows: In Section 2, a few basic concepts of soft set, NSS, and FPSS are given. In Section 3, the notion of M-parameterized N-soft set (MPNSS) is introduced. The concepts of empty, universal, bottom weak complements, top weak complements, weak complements, restricted intersection, and extended union of MPNSSs are presented. In Section 4, the construction of MPNS topology is defined on MPNSS by using MPNS-power whole subsets, extended union and restricted intersection of MPNSSs. Several key properties of MPNS topology, as well as their implications, are well identified. In Section 5, MPNS topology-based MADM methods and their corresponding Algorithms 1 and 2 are developed to estimate the losses, formed extensive damage, displaced and affecting several people in the most affected districts in Sindh province, south-east Pakistan, during historical flooding of August 2011. Section 6 develops and illustrates a robust MADM method of TOPSIS with MPNSSs and MPNS topology using a numerical illustration. Finally, in Section 7, we summarize the findings of this research study.

Preliminaries
In the section presented, we discuss some rudiments of soft set (SS), N-soft sets (NSS), fuzzy soft set (FSS) and FP soft sets (FPSS) that are helpful in understanding the contributions in rest of the paper.

Definition 1 ([3]
). Suppose Λ be the universal set, Υ = ∅ be the class of decision variables or parameters, and λ Υ. A soft set (SS) defined on Λ is a set of order pairs, denoted by (C, λ) and can be represented as, where C : λ → 2 Λ is a set valued mapping. In short, (C, λ) can also be denoted as C λ . Definition 2 ([7]). Let Λ be the universe of discourse, Υ =∅ be the collection of decision variables or parameters. Suppose H = {0, 1, 2, · · · , N − 1} is the grading set, where N ∈ {2, 3, · · · }. The N-soft set (NSS) over Λ is formalized by C N = (C, Υ, N) where C : Υ → 2 Λ×H in such a manner that for every ρ ∈ Υ there exist a specific ξ, I ρ (ξ) ∈ Λ × H for all ξ ∈ Λ, ρ ∈ Υ. Definition 3 ([7]). Let C N be NSS defined over Λ. The weak complement of NSS (C N ), specified as Cĉ N = (C c , Υ, N), where Cĉ N (ρ) C N (ρ) =∅, for each ρ ∈ Υ. Definition 4 ([7]). Let C N be NSS defined over Λ. The top weak complement of NSS (C N ) is a NSS, defined by Cˆt N where 40]). Let Λ be the collection of universal elements, 2 Λ is the aggregation of subsets of Λ, Υ is the collection of decision variables and K be fuzzy set over Υ. A Fuzzy Parameterized soft set (FPSS), denoted by C K on the universe Λ is defined as, where C K : Υ → 2 Λ is a set valued mapping and µ K : Υ → [0, 1] is called membership function.

M-Parameterized N-Soft Set (MPNSS)
Fatima et al. [7] presented the idea of NSS as an extension of SS to cope up with situations in which non-binary assessment is required. This section is devoted to the establishment of M-parameterized N-soft set (MPNSS) which is superior than NSS and SS. For M = 2 MPNSS becomes NSS and for M = N = 2 it reduces to SS. Definition 7. Let Λ be the universe, λ Υ is a collection of attributes. Consider two different sets for grading or rating H = {0, 1, 2, · · · N − 1} and = {0, 1, 2, · · · M − 1}, where M, N ∈ {2, 3, · · · }. Then the M-Parameterized N-soft set (MPNSS) over Λ, designated as Λ M N or Λ(M, N) and defined by Table 1 gives the matrix representation of MPNSS as follows.
Then MPNSS(Λ) represents the collection of all MPNSSs.
For alternatives, the evaluation scales are, Higly recommended = † † † † means 4 Recommended = † † † means 3 Moderately recommended = † † means 2 Slightly recommended = † means 1 Not recommended = • means 0 According to comprehensive properties of the cities, the public give assessment scores to the evaluation attributes and cities, presented in Table 2 and matrix form of 6P6S-set is presented in Table 3. Table 2. Evaluation of data provided by public. Table 3. Tabular representation of corresponding 6P6S (Λ 6 6 ).
Definition 9. Let Λ be universe, Υ is the set of attributes and λ Υ. Then the universal MPNSS, denoted by S M−1 N−1 or S(M − 1, N − 1) and defined as Definition 10. Let Λ be a set of universal elements and λ Υ is the set of attributes. The weak compliment of MPNSS (Λ M N ) over Λ, indicated by (Λ M N )ĉ and described as Consider a 6P6S-set(Λ 6 6 ) as given in Example 1. The weak compliment of Λ 6 6 is given in Table 4. Table 4. Tabular representation of (Λ 6 6 )ĉ.

M-Parameterized N-Soft Topology
The concept of M-parameterized N-soft topology (MPNS topology) based on MPNSS is introduced in this section. Certain properties of MPNS topology are expressed and their corresponding results are established.
Example 11. Consider a (Λ 8 8 ,T 1 ) is 8P8S-topological space as given in Example 8. Let 8P8S-subspace can be obtained as A basis is an assemblage of subsets of Λ M N , for a topology on Λ M N , which holds the following conditions, (1) There exists one or multiple elements β containing Λ (i) , for each Λ (i) ∈ Λ M N (2) If intersection of β 1 and β 2 contains Λ (i) then there must exist a β 3 containing Λ (i) in such a way that β 3 β 1 β 2 .
Proof. The proof is obvious.

Definition 23. Let
N 2 ) and characterized as,

MPNS-Topology Based MADM
MPNS topology is the generalization of soft topology and NS-topology. In this section, we execute the MPNS topology towards MADM to make a robust optimal decision. MPNS topology provides strong mathematical modeling towards uncertainty. The eminent characteristic of MPNS topology-based MADM is that the attributes and alternatives are analyzed by the decision makers (say) D 1 , D 2 , ..., D n and their evaluations are represented in terms of MPNS-open sets (say) Λ T 1 , Λ T 2 , · · · Λ T n . To meet these objectives, we present two algorithms named as Algorithms 1 and 2 and their corresponding real life applications. The flow chart of MPNS topology based method 1 is expressed by Algorithm 1 as follows.

Algorithm 1: (MPNS topology based method 1).
Step 1: Input Λ = {ξ 1 , ξ 2 , · · · , ξ n } as a collection of objects, Υ = {δ 1 , δ 2 , · · · , δ k } as a collection of attributes, and a team of decision makers Step 2: Compute MPNSSs according to opinion of each decision expert with the help of information systems which assign attributes with feasible number of ♦, non-zero grading with † and zero grading with • to the alternatives.
Step 6: Final the optimal alternative with maximum of aggregated values max Λ T 1 ⊕T 2 ···⊕T n (ξ i ).
The flow chart of Algorithm 1 is given in Figure 1. MPNS topology based method 2 is expressed by Algorithm 2 as follows.
Step 2: Compute information systems with feasible number of ♦ to attributes, † for non-zero grade and • for zero grade to alternatives corresponding to the opinion of each decision expert and compute MPNSSs.
Step 5: Find the aggregate MPNSSs by using the formula, where M Λ Tn , M t cΛ Tn and M Λ Tn are the matrices corresponding Λ T n , cΛ T n and Λ T n , respectively. The matrix M t cΛ Tn represent transpose of the matrix M cΛ Tn .
Flow chart of Algorithm 2 is given in Figure 2.

Numerical Example
Floods normally are short-lived and local incidences that can occur all of sudden, often with no alerts. They are generally occur due to exquisite storms that develop more drain than a region can stream or store may carry inside its normal channel. Floods can also occur when ice jams, when dams fail or landslides provisionally obstruct a channel or when snow melts swiftly. In a more comprehensive manner, usually floods occur in dry lands by high tides, by high levels of lakes or by waves directed in the ground by stiff breeze. Some floods occur seasonally due to monsoon rains, fill river basins, along with melting snows. Pakistan continued to face crisis situation due to the destructive flood of 2011. In Sindh province, a disastrous flood entered in August 2011, presumed as the most severe in the history, molded extensive devastation and crowd out thousands of people and millions were badly affected. The province persisted disabled over the end of 2011, as the affected communities and government accomplished with overburdened funds and enormous financial damage. About 4.8 million people, in which children were half of the number were badly affected by the floods in Sindh and according to estimates that some 72,000 people inhabited in relief camps. Sindh was the most affected province where monsoon rains, swamped 22 districts. According to, National Disaster Management Authority (NDMA), the floods in Sindh have caused 756 injuries and 466 deaths, 1.5 million houses were damaged and 6.6 million acres of land was affected. Provincial Disaster Management Authority (PDMA) and Pakistan Red Crescent Society (PRCS) had been dispatched evaluation teams to the area, which illustrated a sketch of huge destruction, although because of unavailability of roads, had problems to carry out a comprehensive evaluation. The provincial branch of PRCS in Sindh aligned with PDMA. Specifically, due to Pakistan's dreadful economic condition, is conviction that total devastation of standing crops was about 10 million acres of land. An evaluated loss of 7 billion to Pakistan's land economy was caused by floods in 2011.
Step 1: Step 2: PRCS Sindh branch sent two rapid evaluation teams to get estimation of immediate requirements to make urgent progressive scheme in most affected district firstly. We consider two decision-makers (DMs) and made two separate teams for assessment and analysis of consequences of flood. Both teams gave the report about the situation of badly affected districts in accordance with chosen subsets by team-T 1 and team-T 2 in terms of sets, in which grades are given to the attributes.
i.e A T 1 = 4 , respectively. After a complete research both teams construct 10P10SS's, Λ T 1 and Λ T 2 over Λ. First we construct a 10P10SS over Λ namely Λ 10 10 on the assessment of other departments of different institutions, feed back of people of affected areas and according to demand of assessment teams of PRCS. The information system corresponding to collected data from other resources and people of affected areas is given in Table 10 and its matrix form is given in Table 11.  Table 13. Tabular representation of Λ T 1 .
The information system corresponding to team of decision experts T 2 is shown in Table 14 and its matrix form is given in Table 15. Table 14. Information system provided by decision team T 2 in terms of 10P0SS. Table 15. Tabular representation of Λ T 2 .
Step 9: Now we find the final decision 10P10SS by adding Λ T 1 and Λ T 2 only because there is no need to add (P 0 0 ) and (Λ 10 10 ) . Step 10: The optimal decision is obtained by taking maximum of final aggregated values as, This implies that the district ξ 1 = Badin has highest grading value and according to Algorithms 1 and 2 ξ 1 = Badin is the badly affected district. 2011 floods mobile health units morbidity surveillance is shown in Figure 3.

TOPSIS Method under M-Parameterized N-Soft Topology
Many researchers thoroughly investigated the multi-attribute decision making (MADM). The established methods particularly relay on the nature of problem under consideration. There are large number of vague, imperfect and uncertain realistic issues. In this section, we discuss how MPNS topology is useful in MADM, to cope with such real life circumstances. We develop TOPSIS method under MPNSSs and MPNS topology for MADM. TOPSIS method is strong and powerful approach for critical decision analysis to estimate the losses, constructed extensive damage and moving thousands of people and millions of people in worst affected districts in Sindh province in the course of flooding of August 2011. The linguistic variables, according to importance of attribute and the condition of most affected areas/alternatives are given below.
Decision experts (D i ) assigned grades, row-wise to each parameter, represented by I ik by using the linguistic variables. In all matrices, the first row (in bold letters) represents the grading values, assigned to parameters by chairman of PRCS according to the surveyed data of teams of other departments, by using linguistic variables from Table 17.  Step 3: Creating normalized weighted parameterized matrix U, Step 4: Creating weight vector W = (W 1 , W 2 , W 3 , · · · , W m ) by using the expression Step 5: Constructing MPNS-decision matrices T i for each team such that all T i make MPNS topology, . . .V m 11 12 13 . . .
Here jk are MPNS-elements.
Step 6: The aggregated matrix can be calculated as, Step 7: Constructing the final weighted decision matrix, Step 8: Now finding positive ideal solution (PIS) and negative ideal solution (NIS).
Step 9: Calulating separation measurementsS + andS − of PIS and NIS, respectively, for each parameter by making use of Step 10: Calculating the relative closeness, Step 11: Ranking the alternatives in descending order. The optimal choice would be the alternative with largest value of R + j . Figure 5 shows the the flow chart of MPNS topology based TOPSIS.
Decision experts (D i ) of assessment teams of PRCS, assigned grades to each evaluation attribute, represented by I ik by using the linguistic variables. In all matrices, first row (in bold letters) represents the grading values, assigned to evaluation attributes by chairman of PRCS according to the information of teams of other departments, by using linguistic variables given in Table 17.
Step 3: The normalized weighted parameterized matrix U, by using Equation (4) is given as, Step 4: The weight vector by using Equation (5) is given as, W = (0.0920, 0.1578, 0.1468, 0.1639, 0.2056, 0.1190, 0.0687, 0.0458) Step 5: The 10P10S-decision matrices T i of two teams are given in which each row represents alternatives and each column represents evaluation attributes and all T i make 10P10S-topology. There is no need to write null matrix and universal matrix for 10P10S-topology.
Step 6: The Aggregated matrix obtained as, Step 7: Constructing final weighted decision matrix Ω as, Step 9: The separation measurements of PIS and NIS for each parameter by using the Equations (9) and (10) are given in Table 18. Step 10: The relative closeness to alternatives are given in Table 19 as follows, Step 11: The ranking order is ξ 1 ξ 6 ξ 7 ξ 8 ξ 4 ξ 5 ξ 2 ξ 3 . This shows, Badin is the most affected district. Figure 6 shows the ranking of alternatives obtained by TOPSIS method.  Provincial disaster management authority (PDMA) provide data about the losses in most affected districts in Sindh which is approximately same as we evaluate from Algorithms and MPNS-TOPSIS technique, as given in Figure 8. According to this data, Badin was the badly affected district. The bad condition of districts measured according to number of cattle head perished (CHP), affected villages (AV), affected people (AP), damaged houses (DH), affected area in acres (AA) and damaged crop area in acres (DCA).

Comparison Analysis
The proposed MPNS topology-based Algorithms 1 and 2 and TOPSIS are compared as indicated in Table 20. In the comparison analysis, it can be noted that the suitable alternative obtained by any one proposed technique endorses the authenticity and effectiveness of the proposed algorithms. The comparison analysis of final ranking is also shown by multiple bar chart in the Figure 9. Table 20. Comparison analysis of final ranking with existing methods in given numerical example.

Conclusions
We deal with vague, ambiguous, unclear, and imprecise data in various real world issues. Existing models of soft sets, fuzzy sets, intuitionistic fuzzy sets, and neutrosophic sets are helpful in capturing these uncertainties. However, all of these models have some limitations on membership and non-membership grades. Existing mathematical frameworks are unable to address realistic issues when non-binary assessments are required while modeling uncertainty. Non-binary assessments are absolutely essential in ranking, grading or rating systems. The ranking may be specified in terms of grades, dots, stars or any notation. To deal with the real situation in life when the grading/rating of both parameters and alternatives is desired, we have introduced the novel concept of the M-parameterized N-soft set (MPNSS). Various concepts including MPNS-empty, MPNSuniversal, MPNS-weak compliment, MPNS-top weak compliment, MPNS-bottom weak compliment, extended union, and restricted intersection of MPNSSs are defined. On the basis of these concepts, the idea of MPNS topology is established and various properties of MPNS topology are well established. MPNS topology is the extension of soft topology and N-soft topology. MPNS topology is a strong mathematical model of uncertainties that has a large number of applications in many fields like image processing, artificial intelligence, computational intelligence, forecasting, medical diagnosis. We developed algorithms for MADM applications of MPNSSs and MPNS topology. We established the TOPSIS method for multi attribute decision making by using MPNSSs and MPNS topology. The symmetry of the optimal decision is illustrated by interesting applications of proposed models and new MADM techniques. The viability and flexibility of the proposed MADM techniques are justified by comparison analysis them with existing MADM techniques.
Author Contributions: M.R., A.R. and M.A., conceived and worked together to achieve this manuscript, D.P. and M.A. construct the ideas and algorithms for data analysis and design the model of the manuscript, M.R., A.R. and D.P., processed the data collection and wrote the paper. All authors have read and agreed to the published version of the manuscript.