Implementing QSPR modeling via multiple linear regression analysis to operations research: a study toward nanotubes

Abstract

Chemical graph theory significantly predicts multifarious physio-chemical properties of complex and multidimensional compounds when investigated through topological descriptors and QSPR modeling. The targeted compounds are widely studied nanotubes attaining exquisite nanostructures due to their distinguishable properties attaining numeric values. The studied nanotubes are carbon, naphthalene, boron nitride, V-phenylene, and titania nanotubes. In this research work, these nanotubes are characterized through their significance level by implementing highly applicable MCDM techniques. TOPSIS, COPRAS, and VIKOR are used to perform a comparative analysis between them through each optimal ranking. The criteria originated from multiple linear regression modeling established between degree-based topological descriptors and the physio-chemical properties of each nanotube.

1 Introduction to chemical graph theory and topological descriptors

Chemical graph theory, a sub-field of mathematical chemistry, employs graph theory to simulate chemical mechanisms mathematically. Theoretically, according to graph theory, a chemical graph features can provide important insights into chemicals. Iván Gutman, Alexandru Balaban, Ante Graovac, Milan Randić, Haruo Hosoya, and Nenad Trinajstić [1] are considered the founders of chemical graph theory. There were allegedly multiple hundred specialists who collaborated in this domain in 1988, and they contributed roughly 500 manuscripts each year. A Chemical diagram shows molecular entities in chemical graph theory. Atoms and bonds are represented by nodes and edges in a chemical network. In cheminformatics, chemical graphs are the primary data types used to depict chemical structures. A key area of cheminformatics is the assessment of (quantitative) structure–activity and structure–property. That can be made possible by the calculable attributes of graphs. The physical characteristics of molecules can then be reflected in these graphs as graph-theoretical descriptors or indices [2, 3].

A sort of molecular descriptor that is assessed using the molecular graph of a chemical compound is a topological index, sometimes referred to as a connectivity index. Topological indices are numerical variables that describe the topology of a graph and are typically graph invariant. The topological index can be thought of as simple, two-dimensional descriptors that can be determined from molecular networks regardless of how the graph is represented or tagged and without the necessity to minimize the energy of the chemical structure [4].Wiener introduced one of the earliest graph invariants for the linkage between molecular characteristics and topological aspects. The topological index was utilized in his study to estimate the boiling point of paraffin. The Hosoya index, which measures the number of edge fits in a graph, was first proposed by Hosoya in 1971. In 1975, Randić topological index was added to this study. Since bond weights formed the foundation of the calculation, it is also known as the connectivity index. After that, in 1998 Bollobás and Erdos expanded it as a generalized Randić index.

$${R}_{\alpha }=\sum_{uv\in E}{\left({d}_{u}{d}_{v}\right)}^{\alpha }$$

To study about the topological indices, their mathematical aspects and applications see [5,6,7,8,9,10,11,12,13,14]

2 Introduction to QSAR/QSPR

QSAR/QSPR analysis is a prediction model generated from statistical methods that correlate chemical compounds’ biological activity and property with descriptors indicative of their molecular structure or characteristics. In the chemical and biological sciences as well as engineering, regression models are used. While classification of QSAR/QSPR models ties the predictor variables to a categorical value of the response variable, QSAR/QSPR regression methods relate a collection of” predictor” variables (X) to the potency of the response variable (Y), like other regression models.

In [15], Topological descriptors are substantially linked with the physical characteristics of antituberculosis medications. It may be possible for chemists and others in the pharmaceutical business to forecast the qualities of antituberculosis medications without having to do experiments, thanks to theoretical analysis. Based on neighborhood degree sums of nodes, various new topological descriptors were suggested by Mondal et al. [16]. An algorithm was developed to facilitate the computation of the novel indices. An excellent statistical tool for examining pharmacological activity or binding mode for various receptors is the quantitative structure–property relationship (QSPR) study. Hayat et al.[17, 18] assessed the efficacy of spectrum-based distance descriptors for polycyclic aromatic hydrocarbons and tested the applicability of polycyclic aromatic hydrocarbons’ spectrum-based valency descriptors. Edge and Vertex degree-based topological descriptors were used in [19] to build a quantitative structure–property relationship (QSPR), and physio-chemical properties of phytochemicals tested against SARS − CoV − 23CL^pro were assessed.

2.1 Multi-criteria decision making

Since its inception in the 1970s, multi-criteria decision-making (MCDM) has drawn the interest of numerous scholars. MCDM can be classified into two main categories: multi-objective decision-making (MODM) and multi-attribute decision-making (MADM). Since its inception, more than 70 multi-criteria decision-making strategies have been identified [20]. With so many very similar possibilities, it is necessary to adopt a strategic move to find the best solution. The alternatives differ, but because they are all part of the same group of real-world entities, it can be difficult to decide which is best among the better alternatives. This kind of decision-making calls for some level of insight. It is necessary to have a basis for ranking the options and differentiating them to improve the decision-making process. A few qualitative and quantitative variables that could aid in differentiating the options must be chosen to do this. These variables are referred to as criteria. The choices are contrasted with the chosen criteria to make the best choice. These variables are referred to as criteria. The alternatives are assessed against the chosen criteria to make the best choice. As a result, MCDM is made up of a set of numerous criteria, a set of alternatives, and some sort of comparison between them.

According to Hwang and Yoon’s review of the available solutions to MCDM problems, there are numerous approaches. Three highly applicable MCDM approaches, TOPSIS, COPRAS, and VIKOR, are being used in the current investigation. Among MCDM techniques, the Technique for Order of Preference by Similarity to the Ideal Solution (TOPSIS) is the second most prevalent. Numerous academics have modified or expanded the TOPSIS method to address unique challenges, as well as adapted TOPSIS to tackle basic or complex problems in various fields.

A total of 105 peer-reviewed works developed, extended, proposed, and presented the TOPSIS technique for tackling DM problems. According to the study’s findings, from 2000 to 2015, 49 academics extended or developed the TOPSIS technique, and 56 scholars proposed or presented new modifications for the technique’s use in solving problems. The study’s findings also showed that 2011 saw more modifications to this technique than any other year [21]. The AHP, ELECTRE, PROMETHEE, VIKOR, or TOPSIS procedures were modified [22]. The level of the information society (IS) development in the member states of the European Union from 2005 to 2010 is analyzed using the TOPSIS approach with interval arithmetic, and the findings are reported. Due to the shortcomings of conventional TOPSIS, it has undergone two significant changes because of this paper. The ranking of the weapon selection problem is next analyzed using the thorough grey TOPSIS by Yunjie Wu and Wenguang Yang [23]. The numerical findings demonstrate the viability of the enhanced grey relational analysis as well as the grey comprehensive TOPSIS. L. Ocampo et.al [24]. utilized the technique of TOPSIS to resolve a warehouse site selection problem under a substantial number of decision criteria in a group decision-making setting. A case study at a company that distributes products was done to clarify the suggested approach.

When evaluating and comparing the viability of alternative energy strategies or renewable energy technologies, VlseKriterijumska Optimizacija I Kompromisno Resenje (VIKOR) is frequently used to provide decision support for choosing the most viable and sensible options. In the realms of sustainability and renewable energy, the VIKOR approach has been employed in several earlier pieces of research. Environmental management, life cycle sustainability assessment, energy resources, and environmental evaluation are only a few of the various sub-areas that fall under the umbrella of sustainability and renewable energy. The majority of which were extracted from the “Web of Science and Scopus” databases dealt with operational research, management sciences, decision-making, sustainability, and renewable energy. These papers were published from 2004 to 2015 in 83 prestigious journals. Because the VIKOR method’s premise is simple and easy to understand, as well as its computing procedure, which is effective and can gauge how well different decision alternatives perform relative to one another, the experiment can assist in making effective decisions [25]. Wei et al. [26] suggested using 2-tuple linguistic neutrosophic numbers to solve the multiple criteria group decision-making (MCGDM) problem (2TLNNs).

In [27], Jong and Byeong extended prior works that primarily take intervals or fuzzy under a flat structure of criteria and introduced the hierarchical VIKOR approach that incorporates incomplete alternatives’ values as well as incomplete criteria weights. It used the aggregated group utility and individual regret scores obtained from the linear programs to rank the alternatives. They used a global supplier selection issue that has an impact on the organization’s sustainable growth as an example to demonstrate how to apply our suggested methodology. In 1994, Zavadskas, Kaklauskas, and Sarka presented the COmplex PRoportional ASsessment (COPRAS) approach. Using this method, the maximizing and minimizing index values are evaluated, and the impact of the indices on the evaluation of the outcomes is considered independently. The COPRAS approach is used in various contexts, including risk assessment, investment project selection, and material selection [28]. A case study carried out at a public transportation corporation in the southern region of India provides evidence of the established model’s efficacy. AHP (Analytical Hierarchy Process), FARE (Factor Relationship) method, and entropy measurement are three alternative approaches used to calculate the weights of the design parameters [29]. How to choose wisely while buying CFLs was addressed by the authors in [30]. This was addressed using the COPRAS multi-criteria decision-making approach. Applying FANMA and SDV decision-making models, objective methodologies were utilized to assess the relative importance of various factors. Krishnakumar etal [31]. proposed a comprehensive method for resolving the decision-making issue with probabilistic-hesitant fuzzy information (PHFI), which is an expansion of the hesitant fuzzy set and provided the PHFI with a COPRAS mechanism for ranking the alternatives. A case study of choosing a cloud vendor was used to illustrate the provided method, and its viability was determined by contrasting the outcomes of the technique with those of many other existing algorithms.

To choose the finest electric motorcycles, Patel et al. [32] introduced CRITIC-COPRAS and CRITICTOPSIS multi-criteria decision-making procedures. To help decision-makers choose the best option from among a variety of accessible electric motorcycle possibilities on the market, this approach’s noteworthy notion compares the two methodologies to discover the one that is most applicable to the motorbike-choosing scenario. For the objective of evaluating the affordability of sustainable housing, a practical application and assessment of six distinct multiple criteria decision-making (MCDM) methodologies were given in [33]. The comparison analysis goal is to evaluate how various MCDM techniques stack up when applied to a sustainable housing affordability assessment model. This research compares the ranks, explores the commonalities among MCDM approaches, and assesses their robustness. The TOPSIS and COPRAS algorithms were used in [34] for multi-objective enhancement to find the best combination of performance characteristics, such as spindle speed and feed rate, for simultaneous minimization of all responses when drilling Mg AZ91 with carbide tools. A comparative comparison of the TOPSIS, VIKOR, and COPRAS techniques was carried out by Hezer etal. [35] for the COVID-19 Regional Safety Assessment, and it is been noted that the COPRAS approach yields results that are the most in line with those of the report, while the VIKOR method yields results that are the furthest from them. To choose the best flow release scenario, seven alternative MCDM approaches are used in [36] to analyze an existing small run-of-the-river HP plant. Investigated and statistically contrasted are the outcomes of applying various MCDM techniques. Also highlighted are the advantages and disadvantages of the various methodological approaches. According to the comparison findings and the ensuing analyses, SHARE MCA, WPM, and VIKOR seem to possess the most intriguing traits in terms of general viability. Three significant and practical MCDM techniques WSM, AHP, and TOPSIS have been succinctly discussed [37]. To choose the best option among the many options under discussion, the article outlines the fundamental phases of each technique.

According to the weights assigned to each manufacturing technique for AHP, TOPSIS, and VIKOR, sand casting came out on top. Compared to TOPSIS and AHP, the VIKOR technique outperformed both in terms of computing complexity. Additionally, the VIKOR and TOPSIS methods were more useful for choosing manufacturing processes for their flexibility during the decision-making process, the number of alternative processes and criteria, their suitability for supporting a group decision, and their ability to add or remove a criterion [38]. The significant applications in chemical graph theory are a major driving force behind this study. To implement the MCDM (TOPSIS, VIKOR, and COPRAS) approaches and get the most benefit out of them, QSPR analysis is done using correlation and multiple linear regression. Some recent researched works draw attention toward studied mathematical models, and targeted structures are mentioned.

3 Nanotubes

Extensively investigated nanotubes are the focus of the current study. A nanoscale substance with a tube-like structure is called a nanotube. In the broad subject of nanotechnology, a relatively new discipline with enormous potential but also some serious drawbacks, nanotube structures offer a wide range of uses. Carbon, boron, or silicon are just a few of the substances that can be used to create nanotubes. Each of these has been used for contemporary technologies.

A type of nanostructure with a cylindrical shape is a carbon nanotube (CNT). Due to the broad use of titanium in the creation of catalytic, gas-sensing, and corrosion-resistant materials, it is one of the nanostructures that have been most thoroughly explored. Boron nanotubes have intriguing electrical characteristics and multicenter bonding, which makes them appealing. Triangles and hexagons are the building blocks of tri-hexagonal boron nanotubes. The QSPR/QSAR studies can benefit from it. The molecular structures of V-Phenylenic nanotubes, H-Naphthalenic nanotubes, titania nanotubes, boron nitride nanotubes, and carbon nanotubes are depicted in Figs. 1, 2, 3, 4, 5.

Fig. 1

Graph of V-Phenylenic nanotube

Fig. 2

Graph of boron nitride nanotube

Fig. 3

Graph of H-Naphthalenic nanotube

Fig. 4

Graph of titania nanotube

Fig. 5

Graph of hexagonal carbon nanotube

In this work, we construct the QSPR modeling via multiple linear regression analysis and then, implement the conclusions generated under modeling into three highly applicable MCDM techniques. This study allows us to visualize the dominancy in targeted nanostructures keeping in concern their physio-chemical properties and their predicted values generated through topological descriptors. Each step and complete ideology fitting is explained in the relevant sections below and outclass conclusions developed under this whole study make its applicable broad not only in chemical graph theory but also opens up the idea to design algorithms that helps studying such arising questions and ahead research queries covered under computational chemical study. Y. Li et al. [13] target highly resisted anticancer drugs via topological descriptors, and MCDM techniques and generated outstanding combinations of drugs. Also, Hui et al.[14] investigated complex crystal structures by implementing topological descriptors and MCDM techniques.

We utilize three widely applied MCDM techniques TOPSIS, COPRAS, and VIKOR to undertake a comparative analysis between them by observing ideal rankings: Different nanotubes are classified in this research work based on their significance level. The standards are developed using multiple linear regression modeling between degree-based topological descriptors and the physio-chemical characteristics of each nanotube.

4 Materials and methods

This part offers the resources for QSPR analysis using correlation and multiple linear regressions as well as the evaluation approach. V-Phenylenic nanotubes (VPHX[$m,n$]) [39], H-Naphthalenic nanotubes (NPHX[$m,n$]) [40], titania nanotubes (TiO₂[$m,n$]) [41], boron nitride nanotubes (BN[$m,n$]) [42] and carbon nanotubes (C[$m,n$]) [42] are among the materials being considered. We denote the vertex set of each nanotube by the symbol V $[m,n]$, while the edge set is denoted by E $[m,n]$. The order and size of the considered nanotubes are shown in Table 1.

Table 1 Vertex and Edge set for each nanostructure

Degree of a node u is the number of edges incident to it, and here, we symbolized it by d_u. To compute the topological descriptors, we need to find the edge partition of nanotubes based on the degree of end vertices of each edge. This edge partition of each nanotube is depicted in Tables 2, 3, 4, 5, 6 . For some specific values of m and n, the values of generalized Randić index for $\alpha =\pm 1, \pm \frac{1}{2}$ are shown in Fig. 7. We have organized the alternatives (nanostructures) in a preferred order by computing the descriptor values for each nanostructure.

Table 2 VPHX [m, n]

Table 3 BN [m, n]

Table 4 NPHX [m, n]

Table 5 TiO₂[m, n]

Table 6 C [m, n]

5 Discussion and results

The QSPR response variable could be a chemical’s biological activity or property; in QSPR modeling, the predictors are the chemical’s physio-chemical properties or theoretical molecular descriptors. In the first place, QSPR models summarize an alleged correlation between chemical structures and biological activity in a chemical data set. Additionally, QSPR models forecast the behaviors of novel compounds. This part covers multiple linear regression analysis, the perspective of selecting a specific order, and the association between physio-chemical parameters and chemical invariants. The concept we developed in this study is heavily reliant on QSPR analysis to match the physical and chemical characteristics of nanotube components to a particular topological descriptor. To get the best results, we have examined the nanostructure using the designated topological descriptor values. Table 7 displays the adjustments we made to $[m,n]$ to determine the proximity and relationships of the vertex sets of the nanostructures. Ultimately, it is difficult to determine which mathematical value is dominant at a given structure by comparing its values to others. It is very important to view numerical data directly and arrive at a single structure when they are changing or reaching certain fluctuations. We must examine the component of nanostructures by their physical characteristics, which are their molar mass, density, and melting point, as shown in Table 8, to associate the topological descriptor with the nanostructure. The goal is to determine the best-ranging numerical values that have strong correlation and regression, matching certain values of the topological descriptor. As a result of these analyses, we now know which criteria are useful and which are not. The current study’s analysis demonstrates that all the criteria, as indicated in Table 9 as topological indices, are beneficial and advantageous. We have the correlations between the considered topological index and properties shown in Table 10 and Fig. 6.

Table 7 Considering equally distributed components

Table 8 Physical properties of each nanobody

Table 9 Topological descriptors of nanostructures

Table 10 Correlation ability of physio-chemical properties with Randić indices for nanobodies

Fig. 6

Correlation ability

6 Multiple regression analysis

The estimation of associations between a dependent variable and one or more independent variables is the goal of regression analysis, which consists of a range of statistical techniques. It can be used to model how strongly the link between variables will develop in the future and to gauge the strength of the relationship between variables. Multiple linear regressions are the name of a statistical method for predicting the outcome of a variable based on the values of two or more other variables. The factors we use to predict the value of the dependent variable are known as independent or explanatory variables, while the variable we seek to forecast is known as the dependent variable. To predict the result of a dependent variable, a statistical method known as multiple linear regression employs two or more independent variables. Analysts can use the technique to calculate the model’s variation as well as the relative contributions of each independent variable to the overall variance. For the nanostructures under consideration, we aim to forecast their physio-chemical properties. Here, we have dependent variables named molar mass, density, and melting point and generalized Randić index for $\alpha =\pm 1,\pm \frac{1}{2}$ independent variable. Each dependent variable is measured along with all independent factors to determine the link between the dependent and independent variables. To establish a connection between the dependent and independent variables, we have three regression models here for each considered property. For a multiple regression model, we typically have an equation.

$${d}_{i}= {\beta }_{0}+ {\beta }_{1}(Ii1) + {\beta }_{2}(Ii2) + .....{\beta }_{j}(Iij)$$

Here, ${d}_{i}$ is a dependent variable and β₀ is y-intercept and β₁, β₂ and β_p is the regression coefficient, respectively, for each independent variable(Iim). ε denotes the random error term. We have observed from the modeling that we have not obtained numerical values as P values because of the extreme contribution that covers each model for each index and from that observation it shows high significance. We have obtained from the evaluations and observations for Models 1, 2 and 3 in Table 11, and other outputs for each model are given in Tables 12, 13 and 14. At this point of discussion, we have observed that we cannot reach exactly how and at what measure each index is contributing under multiple linear regression for each property, but it produced the error zero and the value of adjusted ${R}^{2}$ is one in each case. So, we have concluded and observed through line fit plotting for each index we have observed and obtained linear regression modeling through multiple modeling of the line fits plotting as shown in Tables 15, 16 and 17.

Table 11 Regression statistics for models 1, 2 and 3

Table 12 Multiple regression results for model 1

Table 13 Multiple regression results for Model 2

Table 14 Multiple regression results for model 3

Table 15 Best fit line plotting through multiple linear regression for model 1

Table 16 Best fit line plotting through multiple linear regression for model 2

Table 17 Best fit line plotting through multiple linear regression for model 3

7 MCDM implementation

We have constructed the initial input matrix to apply the multi-criteria decision-making techniques TOPSIS, VIKOR and COPRAS to alternatives. Table 18 consists of alternatives and criteria and the weights assigned to them.

Table 18 Initial input matrix

8 The technique for order of preference by similarity to ideal solution (TOPSIS)

The Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) evaluates a group of options by assigning weights to each criterion, normalizing scores for each criterion, and figuring out how far apart each alternative is from the ideal alternative, which is the alternative with the highest score for each criterion.

The monotonic increase or decrease in the criteria is a TOPSIS presumption. In multi-criteria situations, normalization is frequently necessary due to the criterions frequently contrasting dimensions. We have the following methodology of TOPSIS to study the nanostructures.

Step 1 Firstly, we construct the evaluation matrix ${\left({r}_{ij}\right)}_{m\times n}$ containing $m$ alternatives and $n$ criteria and $i = 1,2,...,n\, {\text{and}}\, j = 1,2,...,m.$ as shown in Table 18.

Step 2 Normalize the matrix ${\left({r}_{ij}\right)}_{m\times n}$ to have $N = {\left({n}_{ij}\right)}_{m\times n}$ by using the formula given as:

$${\eta }_{ij}=\frac{{r}_{ij}}{\sqrt{\sum_{k=1}^{m}{x}_{kj}^{2}}}$$

See Table 19.

Table 19 Normalized decision matrix

Step 3 Establish the weighted normalized matrix $W$, where,

$${W}_{ij}= {n}_{ij} \times {w}_{j}$$

as ${w}_{1}+ {w}_{2}+ {w}_{3}+ {w}_{4}= 1$. See Table 20.

Table 20 Weighted normalized matrix

Step 4 Next, we determine the best alternative ${(A}_{b})$ and worst alternative ${(A}_{w})$

$$ A_{b} = {\text{min}}\left\{ {W_{{ij}} |j \in V^{ - } } \right\},\;\;{\text{max}}\left\{ {W_{{ij}} |j \in V^{ + } } \right\} \equiv W_{{bj}} $$

$$ A_{w} = {\text{max}}\left\{ {W_{{ij}} |j \in V^{ - } } \right\},\;\;{\text{min}}\left\{ {W_{{ij}} |j \in V^{ + } } \right\} \equiv W_{{wj}} $$

We have V ⁺ = $\{j = 1, 2,...n|j\}$ is concerned with best criteria chosen. And V ⁻ = $\{j = \mathrm{1,2},...n|j\}$ is concerned with worst criteria chosen shown in Table 21.

Table 21 Best and worst alternative

Step 5 Compute ${L}^{2}$ among the target $i$, and best alternative ${(A}_{b})$ and worst alternative ${(A}_{w})$. See Table 22.

Table 22 ${L}^{2}$ Distance from best and worst alternative and similarity to worst condition

$${P}_{ib} = {\sum_{i=1}^{n}\left({W}_{ij} - {W}_{bj}\right)}^{2}$$

$${P}_{iw} ={\sum_{i=1}^{n}\left({W}_{ij} - {W}_{wj}\right)}^{2}$$

Step 6 Evaluate the similarity to worst condition.

$${Q}_{iw}=\frac{{P}_{iw}}{{P}_{iw}+{P}_{ib}}$$

${Q}_{iw}= 0$ iff alternative contains best solution. ${Q}_{iw}= 1$ iff alternative contains worst solution.

Step 7 Ranked the alternatives according to the results obtained in Table 23.

Table 23 TOPSIS rank

9 VIseKriterijumska Optimizacija I Kompromisno Resenje (VIKOR)

All the choices are compared in accordance with the defined criteria, and we want to find the answer that is closest to the ideal. The compromise option that comes close to the ideal is identified by VIKOR after ranking the alternatives. The methodology of VIKOR we followed is:

Step 1 Compute ideal best ${(f}^{+})$ an ideal worst ${(f}^{-})$ among all the criterion functions, for $i = \mathrm{1,2}...,n$ presented in Table 24,

Table 24 Ideal Best and ideal worst

${f}^{+}= max\{{f}_{ij},j = 1,...J\},min\{{f}_{ij},j = 1,...J\}$ for beneficial ith function.

$f- = min\{{f}_{ij},j = 1,...J\},max\{{f}_{ij},j = 1,...J\}$ for disadvantageous ith function.

Step 2 In Table 25, calculation of weighted normalized Manhattan distance ${S}_{j}$ and weighted normalized Chebyshev distance ${R}_{j}$ is presented where $j=1,\cdots , J$ using the formulas.

Table 25 ${S}_{j}$ and ${R}_{j}$

$${S}_{j}=\sum_{j=1}^{m}[{w}_{i}\times \frac{\left({f}_{i}^{+}-{f}_{ij}\right)}{\left({f}_{i}^{+}-{f}_{i}^{-}\right)}]$$

$${R}_{j}=\mathrm{max}\left[{w}_{i}\times \frac{\left({f}_{i}^{+}-{f}_{ij}\right)}{\left({f}_{i}^{+}-{f}_{i}^{-}\right)}\right]$$

Step 3 Calculate ${Q}_{j}$ where $j=1,\cdots ,J$ using the formula.

$${Q}_{j}=\left[v\times \frac{\left({S}_{j}-{S}^{+}\right)}{\left({S}^{-}-{S}^{+}\right)}\right]+\left[\left(1-v\right)\times \frac{\left({R}_{j}-{R}^{+}\right)}{\left({R}^{-}-{R}^{+}\right)}\right]$$

See Table 26.

Table 26 Computation for ${Q}_{j}$

Step 4 Rank the alternatives using the results. See Table 27.

Table 27 VIKOR rank

9.1 Complex proportional assessment (COPRAS)

Methodology for COPRAS method is following.

Step 1 Firstly, construct the decision matrix as mentioned above.

Step 2 Compute the normalized matrix using formula below. See Table 28.

Table 28 Normalized matrix

$${\eta }_{ij}=\frac{{r}_{ij}}{\sum_{k=1}^{m}{x}_{kj}}$$

$$N={\left({\eta }_{ij}\right)}_{m\times n}$$

Step 3: Construct weighted normalized matrix. See Table 29.

Table 29 Weighted normalized matrix

Step 4 Computed ${B}_{i}$ and ${C}_{i}$ by using the following equations. See Table 30.

Table 30 Calculation of ${B}_{i}$ and ${C}_{i}$

$${B}_{i}=\sum_{j=1}^{k}{W}_{ij}.$$

$${C}_{i}=\sum_{j=k+1}^{k}{W}_{ij}.$$

Step 5 Compute ${Q}_{i}$ as shown in Table 31 by using the equation.

Table 31 Calculation of ${Q}_{i}$ and ${U}_{i}$

$${Q}_{i}={B}_{i}+\frac{\mathrm{min}\left({C}_{i}\right)\times \sum_{i=1}^{n}{C}_{i}}{{C}_{i}\times \sum_{i=1}^{n}(\frac{\mathrm{min}\left({C}_{i}\right)}{{C}_{i}})}$$

Step 6 Compute utility degree by using the formula as shown in Table 31.

$${U}_{i}=\frac{{Q}_{i}}{\mathrm{max}({Q}_{i})}\times 100$$

Step 7 Rank the alternatives. See Table 32.

Table 32 COPRAS rank

10 Conclusion

This research work observed the characteristic of highly five versatile nanotubes named as $VPHX[\mathrm{3,3}]$, $NPHX\left[\mathrm{2,2}\right]$,$Ti{O}_{2}\left[\mathrm{2,2}\right]$,$BN\left[\mathrm{4,4}\right]$,$Carbon[\mathrm{3,3}]$ via generalized Randić index for $\alpha =1, -1,\frac{1}{2}, -\frac{1}{2}$. Furthermore, the QSPR modeling under multiple linear regression analysis has been constructed for three physio-chemical properties for each nanotube, properties named as molar mass, density, and melting point, from which we have observed that error becomes zero when considered multiple topological indices and the value of P exceeds in high rations that becomes unreadable. But via statistical concepts, it is observed as highly significant: as the value of P is not numeric and this means we can predict the correlated ability in numerical terms. We have chosen and observed values of ${R}^{2}$ from the best line fit plotting, developed under each multiple linear regression analysis for QSPR model as shown in Tables 15, 16 and 17. We can conclude that only ${R}_{\frac{1}{2}}$ performed outclass by achieving ${R}^{2}$ value $0.872$ generated under multiple linear regression modeling for model 2 which we considered as density. This allowed us to conclude and observe the combine effect and individual effect of each topological index for a specified nano-bodies. After reaching toward the prediction ability of each index to predict any property, the question arise here is to overcome the solution by ranking them best under the considered study and how QSPR effects the rankings been investigated in this research paper. We cannot consider any value to discriminate or allocate weightage as we have obtained the standard error zero and the value of effectiveness of the model run is one in each modeling. This allows us to allocate equal weightage 0.25 for each topological index. To rank these nanostructures, we have implemented three highly effective MCDM techniques named as TOPSIS, COPRAS and VIKOR. The ranking obtained by implementing the outputs based on QSPR modeling under multiple linear regression analysis justifies the weight allocation which we have chosen equally. The rankings we obtained through this whole ideology are shown in Fig. 7. We have direct visualizations that VPHX [m, n], NPHX [m, n] and C[m, n] are ranked equally when they ordered by considering m = n and VPHX [m, n] ranked high. These are not just rankings, but this opens a new era to implement and link two highly distinguish and applicable branches of mathematics to visualize, read and measure the effects of QSPR when it comes to choose or rank structures under QSPR .

Fig. 7

Rankings through MCDM under QSPR modeling via multiple linear regression analysis