Multi-Criteria Group Decision-Making Models in a Multi-Choice Environment

Abstract

The best–worst method (BWM) has recently demonstrated its applicability in addressing various decision-making problems in a practical setting. The traditional BWM method is based on deterministic information gathered from experts as pairwise comparisons of several criteria. The advantage of BWM is that it uses fewer calculations and analyses while maintaining good, acceptable consistency ratio values. A multi-choice best–worst method (MCBWM), which considers several options for pairwise comparison of preferences between the criteria, has recently been developed. The experts are given the option to select values from several comparison scales. The MCBWM technique has been shown to be better. Presenting the options for which an optimal solution has been found simplifies the calculation and establishes the ideal weight values. This study proposes two different mathematical programming models for solving multi-criteria decision-making problems having multiple decision-makers. The two methods are proposed considering the multi-choice uncertainty assumption in pairwise criteria comparisons. Additionally, it considers the best–worst method as the base model. The multi-choice uncertainty is applied to determine the best choice out of multiple choices. It gives a real-life scenario to the decision-making problems. Although there are many other forms of uncertainty, such as fuzzy, intuitionistic fuzzy, neutrosophic, probabilistic, etc., it focuses on choices instead of ambiguity in terms of the probabilistic or fuzzy nature of parameters. The parameter considered as multi-choice is the pairwise comparison. These parameters are handled by applying the Lagrange interpolating polynomial method. The proposed models are novel in terms of their mathematical structure and group decision-making approach. The models are formulated and further validated by solving numerical examples. It provides a framework for solving mcdm problems where the weightage to the decision-makers is also incorporated. The CR values for all the models of example 1 and 2, and the case study has been found acceptable.

1. Introduction

Multiple-criteria decision-making (MCDM) plays an important role in real-life decision-making problems. The MCDM approach determines the ranking of the options and selects the best option using the appropriate approach based on certain criteria. There are an enormous number of applications of MCDM methods in real-life problems. These methods include elimination and choice expressing reality (ELECTRE) [1,2], data envelopment analysis (DEA) [3,4], analytic hierarchy process (AHP) [5,6], preference ranking organization method for enrichment evaluations (PROMETHEE) [7,8], VlseKriterijumska Optimizacija I Kompromisno Resenje (VIKOR) [9,10], decision-making trial and evaluation laboratory (DEMATEL), the technique for order of preference by similarity to ideal solution (TOPSIS) [7,11], analytic network process (ANP) [12,13], and best–worst method (BWM) [14,15]. The best suitable method to handle an MCDM problem is determined based on the structure of the problem of decision-making. In general, the process of MCDM consists of multiple steps from formulation to identification of criteria to decision metric and, finally, calculation of weights and rank of criteria.

Among the above-mentioned MCDM approaches, In 2015, Rezai [14] developed an MCDM method named as the best–worst method. As compared to the mostly applied MCDM approach, i.e., AHP, BWM has shown a more reliable approach as it takes fewer number pairwise comparisons leading to fewer calculations and, hence, low inconsistency of pairwise comparisons.

Let us consider for AHP [16], a matrix X having $x_{ij};i,j=1,2,3,\dots ,n$ as a pairwise importance comparison between the $i\text{-}\mathrm{th}$ criteria and the $j\text{-}\mathrm{th}$ criteria. from the n criteria $C_1,C_2,C_3,\dots ,C_n$, then the comparison matrix X will become as follows:

$$\begin{array}{c}\hfill X=\begin{array}{c}{c}_1\\ c_2\\ ⋮\\ c_n\end{array}\stackrel{\begin{array}{cccc}{c}_1& c_2& \cdots & c_n\end{array}}{\left[\begin{array}{cccc}{x}_{11}& x_{12}& \cdots & x_{1n}\\ x_{21}& x_{22}& \cdots & x_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ x_{n1}& x_{n2}& \cdots & x_{nn}\end{array}\right]}\end{array}$$

(1)

where $x_{ij}$ represents the degree of relative importance (or the relative preference) of criterion $C_i$ over criterion $C_j$, the values of $x_{ij}$ range between the 1/9 to 9 scale. For an equal preference of criterion $C_i$ over $C_j$, $x_{ij}$ will take value 1. $x_{ij}\ge 1$ shows that criterion $C_1$ is relatively more important than criterion $C_2$, with $x_{ij}=9$ exhibiting the extreme preference of criterion $C_i$ over $C_j$. The relative importance of $C_j$ over $C_i$ is given by $x_{ji}$. It is essential that the relative importance of $C_i$ over $C_i$ is ($x_{ii}=1$) and $x_{ij}=1/x_{ji}$, $\forall i,j$. For X to be perfectly consistent $x_{ik}\times x_{kj}=x_{ij},\forall i,j$.

In comparison to AHP, the BWM approach is based on pairwise comparisons between the best criterion to all other criteria and all others to the worst criterion. All the favorites are assigned to a number through a scale ranging from 1 to 9. Since the secondary comparisons are not performed, the BWM technique seems to be more convenient, very precise, and far less redundant [14]. Meanwhile, executing a pairwise comparison, a decision-maker expresses both direction and the strength of the preference of i over j. In most situations, the decision-maker has no problem in conveying the direction; however, representing the strength of the choice is a difficult task that is almost the primary source of inconsistency. Since the degree of preference relation is assigned with a numeric number, it may be possible that the provided value is insufficient to accommodate the uncertainty with the linguistic term. Based on the above discussion, in this paper, the pairwise reference comparisons are considered multi-choice comparisons [17,18]. While taking responses from decision-makers (DMs), uncertain information may come up in terms of options. When comparing criterion i over j pairwise, the DM may offer many responses.

Multi-choice programming problem [17,18] is a sort of mathematical programming in which the goal is to select the best alternative among multiple potential combinations to optimize an objective function under a number of constraints. A good review of multi-choice mathematical programming is presented in [19]. The situation of multiple choices for a parameter exists in many managerial decision-making problems. Choosing the optimum combination of parameter values from a variety of parameter values is aided by the multi-choice programming approach. Our goal is to better comprehend the supposed multi-choice parameter, which is multiple opinions in the pairwise comparison that forms the foundation of our proposed method. In the MCDM problem, the multi-choice parameters were essential when a specialist was unsure of the significance of a certain criterion. It is due to a lack of information, unclear criteria, unfavorable factors, and the decision-point makers of view and judgment. The reference comparison’s reference parameters with several choices will increase the decision-making problems’ flexibility. It prompts the decision-makers to use not only a single value but more than one benefit to compare criterion i over j. This approach of pairwise comparison results in inconsistency. The decision-makers are supported in their efforts to comprehend accurately and correctly with regard to their final decisions by the obtained value of inconsistency. This approach aids in examining the consistency of the assessments of the decision’s importance. Because there are so many options for criteria, it is possible to take into account every viewpoint when comparing one criterion to another in pairs. The multi-choice best–worst method is a novel MCDM strategy that is introduced in this research to supplement the BWM for multi-choice comparisons.

The BWM method is a well-established, tested, and verified method. Since its inception, it has been applied to various kinds of real-life problems. A lot of extensions, including uncertainty and hybrid methods, have also been proposed in recent years. Additionally, we have seen that there is no extension of BWM with multi-choice parameters in a group decision-making scenario. This motivated us to carry out this proposed work. In recent years, some new methods have also been developed for determining the weights of the criteria, such as the fully consistency method (FUCOM) [20], level-based weight assessment (LBWA) [21], and defining interrelationships between ranked (DIBR) [22]. In literature, it has been shown that these methods are better than the AHP and BWM. The reason behind it is the less number of pairwise comparisons, higher level of consistency, requires an $(n-1)$ number of pairwise comparisons, reliability of results, and simple algorithm. As a limited study has only been carried out using them, so we have incorporated the multi-choice concept for group decision-making in the best–worst method. This work is not yet explored by anyone. The key reasons for the motivation behind this work are as follows:

• Numerous group decision-making models have been put out in the literature, but no one has ever taken into account pairwise comparisons as multi-choice parameters.
• Multiple options for a parameter, such as pairwise comparisons, are another type of uncertainty that can be managed in real-world problems utilizing the multi-choice mathematical programming approach in MCDM.
• The decision that determines whether a solution is optimal is revealed in the solution that results from solving the proposed models. The goal is to reduce inconsistency. Therefore, multiple models can be solved to find the option with the least amount of inconsistency.

Following are the main objectives of this study:

• This study’s major goal is to provide group decision-making techniques that incorporate the freedom to select several options for pairwise comparisons.
• To validate the proposed model by applying it to experimental studies.

Rest of this paper is organized as follows: The next Section 2 is about the literature survey. Some preliminaries are presented in the next Section 3. The proposed models incorporating multi-choice in group decision-making are presented in Section 4. In Section 5, an experimental study conducted using the proposed approach is presented. In Section 6, a case study of the piping selection problem has been presented. Finally, the manuscript concludes in Section 7 and discusses the future directions of the work along with the limitations of the proposed work.

2. Literature Survey

When choosing, organizing, and prioritizing various actions, MCDM considers the decision-makers subjectivity. It also examines the acceptability of alternative options in light of the resources at hand. There are multiple kinds of MCDM methods for ranking and prioritization of criteria. In MCDM, the best–worst method [14] has played a key role in solving many kinds of real-life decision-making problems. Initially, it was applied to a mobile selection problem. Later, It was applied in linking supplier segmentation [23] for enhancing the supplier development model. Gupta and Barua [24] worked on micro-small and medium-sized enterprises (MSMEs) in India, where they found the most significant enablers of technological innovation. Recently [25], the flexibility of information granularity is integrated with the best–worst method along with interval and type-2 fuzzy sets in linguistic terms. A novel model by Tavana et al. [26] has proposed a model which combines the compromise solution with the best–worst method. Malakoutikhah et al. [27] have incorporated fuzzy uncertainty in best–worst method and cognitive map and further applied it in the modeling of criteria and subcriteria affecting unsafe behaviors in organizational, individual, and socio-economic domains. Kharola et al. [28] utilized the best–worst method in prioritizing factors associated with green waste management in the food supply chain. They prioritized a total of 5 criteria and 25 sub-criteria. Bilbao et al. [29] incorporated multiple reference point concepts in the best–worst method and showed application in the assessment of non-life insurance companies. Sadaghiani et al. [30] studied the importance of external forces on the supply chain sustainability in the oil and gas industry; Groenendijk et al. [31] applied it to improve the quality of public transportation, water scarcity management [32], failure mode and effects analysis [33], the judgment of investment projects [34] etc. A good review of BWM articles is presented in [35]. Combination of BWM with other MCDM methods, such as TOPSIS [36], MULTIMOORA [37], VIKOR [38], FDA [39], and ELECTRE [40].

Researchers incorporated uncertainties, such as fuzzy, intuitionistic fuzzy, neutrosophic fuzzy [41], hesitant fuzzy [42], spherical fuzzy [43], probabilistic [44], interval type-2 fuzzy [45], and Bayesian [46] in the best–worst method. A Fuzzy hybrid BWM along with the geographic information system has been applied for the power station selection problem [47]. Fuzzy BWM has been applied to assess the potential environmental impacts of the process of ship recycling [48]. A rough BWM has been applied to the problem of prioritizing recovery solutions to the tourism sector after COVID-19 [49]. BWM has been applied for the land valuation model in three different scenarios [50]. An integrated model of BWM with superiority and inferiority ranking applied in an environment of probabilistic dual hesitant fuzzy sets to a Green supplier selection problem [51]. In [52], a probabilistic-based hybrid model has been proposed for solving group decision-making problems by combining BWM and Bayesian approaches to assess the quality index of medical devices. In [53], blockchain technology has been assessed using Bayesian BWM. A cost–benefit analysis of shale development in India has been carried out using the best–worst method approach [54]. In [55], identification and prioritization of criteria to tackle the COVID-19 outbreak has been carried out. Ref. [56] applied fuzzy BWM in prioritizing factors affecting ad hoc wireless networks. There are many applications of BWM, along with uncertainty, that exists in the literature.

The BWM model is a kind of linear mathematical programming model [15]. Linear mathematical programming refers to mathematical models having mathematical equations for an objective function that is needed to be achieved—with an optimal value under some set of constraints. The condition is that all equations should be of linear nature [57]. Multi-choice mathematical programming is a kind of mathematical programming having multi-choice parameters [18]. In the work of Hasan et al. [58], they have assumed a situation where the pairwise comparison parameter is considered multi-choice in nature. Hasan et al. [58] incorporated a type of uncertainty in the multi-choice form in the best–worst method. They have shown that having multiple choices for pairwise comparison of two criteria can be chosen by the experts instead of using any other kind of uncertainty, such as probabilistic, fuzzy, neutrosophic, etc. They have handled it using Lagrange interpolation and chosen those choices for which inconsistency has been minimized. Their approach has shown a significant decrease in inconsistency. This approach has not yet been explored in the case of group decision-making problems. So, on the basis of the above discussion, it can be observed that the researchers have not considered the most critical point, which is employing a multi-choice mathematical programming model to the MCDM problems in a group decision-making scenario. This has motivated me to work on such models, where multi-choice uncertainty can be incorporated into group decision-making problems. The issue, as mentioned above, is vital in the decision-making of real-world problems. Therefore, the present work focuses on a multi-choice mathematical programming model for group decision-making problems. This work presents two mathematical models. Both models are different with respect to the constraints and objective function. The authors have shown the approaches to solving group decision-making problems.

3. Preliminaries

As preliminary, this section provides an overview of Multi-choice mathematical programming models and the Lagrange interpolation method as follows:

• Multi-choice mathematical programming:

It is a mathematical programming model to determine the solution set X, where X is $(x_1,x_2,x_3,\dots ,x_n)$. The objective is the maximization of the function denoted by Z. The model is as follows:

$$\begin{array}{cc}\hfill MaxZ=& \sum _{j=1}^n{C}_j{x}_j\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill subjectto& \sum a_{ij}{x}_j\le \{b_i^{\left(1\right)},b_i^{\left(2\right)},b_i^{\left(3\right)},\dots ,b_i^{\left(k_i\right)}\},i=1,2,\dots ,m,\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill & x_j\ge 0,j=1,2,\dots ,n.\hfill \end{array}$$

(4)

The choice set of parameter b is $\{b_i^{\left(1\right)},b_i^{\left(2\right)},b_i^{\left(3\right)},\dots ,b_i^{\left(k_i\right)}\}$ with $k_i$ number of choices. The solution set will consist of only one choice from the choice set, which is to be selected for the optimized model.

To determine the solution of the above model, it is necessary to convert it into the standard form of mathematical programming. There are many interpolating polynomial methods [17] for handling the multi-choice parameter b, such as using Lagrange, Newton’s divided differences, Newton’s forward difference, and Newton’s backward difference interpolating polynomial. For the proposed extended methods, the Lagrange interpolation method is utilized.

• Lagrange interpolation:

Let $0,1,2,\dots ,(k_i-1)$ be $k_i$ number of node points, where $b_i^{\left(1\right)},b_i^{\left(2\right)},\dots ,b_i^{\left(k_i\right)}$ are the associated functional values of the interpolating polynomial at $k_i$ different node points. A polynomial $P_{k_{i-1}}\left(z^{\left(i\right)}\right)$ of degree $(k_i-1)$ which interpolates the given data is:

$$\begin{array}{cc}\hfill P_{k_{i-1}}\left(z^{\left(i\right)}\right)=& \frac{(z^{\left(i\right)}-1)(z^{\left(i\right)}-2)\dots (z^{\left(i\right)}-k_i+1)}{{(-1)}^{k_i-1}(k_i-1)!}{b}_i^{\left(1\right)}+\frac{z^{\left(i\right)}(z^{\left(i\right)}-2)\dots (z^{\left(i\right)}-k_i+1)}{{(-1)}^{k_i-2}(k_i-2)!}{b}_i^{\left(2\right)}\hfill \\ \hfill & +\frac{z^{\left(i\right)}(z^{\left(i\right)}-1)(z^{\left(i\right)}-3)\dots (z^{\left(i\right)}-k_i+1)}{{(-1)}^{k_i-3}(k_i-3)!2!}{b}_i^{\left(3\right)}\hfill \\ \hfill & +\dots +\frac{z^{\left(i\right)}(z^{\left(i\right)}-1)(z^{\left(i\right)}-2)\dots (z^{\left(i\right)}-k_i+2)}{(k_i-1)!}{b}_i^{\left(k_i\right)}\forall i=1,2,3,\dots m.\hfill \end{array}$$

(5)

4. The Proposed Multi-Criteria Group Decision-Making Models in Multi-Choice Environment

Suppose k numbers of decision-makers are represented by the set $D=\{D_1,D_2,\dots ,D_k\}$, where $D_k(k\in 1,2,\dots ,m)$ denotes the k-th expert. There are n numbers of decision criteria $C=\{c_1,c_2,\dots ,c_n\}$ with $n\ge 2$, where $c_j(j\in \{1,2,\dots ,n\})$ indicates the $j\text{-}$th criteria. The expertise of the decision-makers in expressing their importance in comparing criteria may be identical or different. The criterion’s importance is described using finite, pre-specified, and ordered linguistic terms sets, as shown in

Table 1. Linguistic expression for reference comparison.

Value	Linguistic Expression	Value	Linguistic Expression
1	EI: Equal important	2	IEM: Intermediate b/w EI and MI
3	MI: Moderate important	4	IMI: Intermediate b/w MI and I
5	I: Important	6	IVI: Intermediate b/w I and VI
7	VI: Very important	8	IEI: Intermediate b/w VI and EXI
9	EXI: Extreme important

. The k-th decision-maker describes the multi-choice preference details $S_{ij}^k={\{1,2,\dots ,s_{ij}\}}^k$; $i,j\in 1,2,\dots ,n$, where $s_{ij}$ represents the linguistic term of the $i\text{-}$th criterion, over $j\text{-}$th criterion with $T_k$ cardinalities, i.e., $S^k=\{s_1^k,s_2^k,s_3^k,\dots ,s_{T_k}^k\}$.

The notations used in this paper are presented in

Table 2. Symbols/Notations used in this study.

Notations	Descriptions
n	Number of Criteria in group decision-making system
m	number of decision-makers in group decision-making system
$c_1,c_2,\dots ,c_n$	Criteria of group decision-making system with indices of $i,j=1,2,\dots ,n$
$D_1,D_2,\dots ,D_k$	Decision-makers of group decision-making system with index of $k=1,2,\dots ,m$
$S^k$	Number of multi-choices in the reference comparisons of k-th decision-maker
${\left\{x_{ij}^{s_{ij}}\right\}}^k$	Multi-choice preference comparison of criterion i with j having $s_{ij}\ge 1$ number of possibilities for decision maker k
$c_B$	Best criterion with index B
$c_W$	Worst criterion with index W
$X_B^k$	Multi-choice preference comparisons best-to-others vector for k-th decision-maker
$X_W^k$	Multi-choice preference comparisons others-to-worst vector for k-th decision-maker
$P^k$	Interpolation polynomial of k-th decision-maker
${\lambda }_k$	Weight of k-th decision-maker $({\lambda }_k\in [0,1]$ and ${\sum }_k^{}{\lambda }_k=1)$
${\xi }_k$	The consistency ratio for the k-th decision-maker
${\omega }_i^{}$	The weight of criterion i
$\xi$	The consistency ratio group decision-making
$CI$	Consistency index
$C{R}_k^{}$	Consistency ratio of k-th DM in the group decision-making problem
$C{R}_{}^G$	Overall consistency ratio of the group decision-making problem

. The steps involved in the proposed group multi-choice best–worst method (GMCBWM) to obtain the criteria weights are described as follows.

(I). 

Defining the system of decision-making mechanism.

The decision criteria system comprises a collection of decision criteria essential in making a judgment on diverse alternatives to consider—assuming n number of decision criteria, $\{c_1,c_2,\dots ,c_n\}$. Then, the decision matrix can be presented as follows.

$$\begin{array}{c}\hfill X^k=\begin{array}{c}{c}_1\\ c_2\\ ⋮\\ c_n\end{array}\stackrel{\begin{array}{cccc}{c}_1& c_2& \cdots & c_n\end{array}}{\left[\begin{array}{cccc}{\{x_{11}^{\left(1\right)},x_{11}^{\left(2\right)},\dots ,x_{11}^{\left(s_{11}\right)}\}}^k& {\{x_{12}^{\left(1\right)},x_{12}^{\left(2\right)},\dots ,x_{12}^{\left(s_{12}\right)}\}}^k& \cdots & {\{x_{1n}^{\left(1\right)},x_{1n}^{\left(2\right)},\dots ,x_{1n}^{\left(s_{1n}\right)}\}}^k\\ {\{x_{21}^{\left(1\right)},x_{21}^{\left(2\right)},\dots ,x_{21}^{\left(s_{21}\right)}\}}^k& {\{x_{22}^{\left(1\right)},x_{22}^{\left(2\right)},\dots ,x_{22}^{\left(s_{22}\right)}\}}^k& \cdots & {\{x_{2n}^{\left(1\right)},x_{2n}^{\left(2\right)},\dots ,x_{2n}^{\left(s_{2n}\right)}\}}^k\\ ⋮& ⋮& \ddots & ⋮\\ {\{x_{n1}^{\left(1\right)},x_{n1}^{\left(2\right)},\dots ,x_{n1}^{\left(s_{n1}\right)}\}}^k& {\{x_{n2}^{\left(1\right)},x_{n2}^{\left(2\right)},\dots ,x_{n2}^{\left(s_{n2}\right)}\}}^k& \cdots & {\{x_{nn}^{\left(1\right)},x_{nn}^{\left(2\right)},\dots ,x_{nn}^{\left(s_{nn}\right)}\}}^k\end{array}\right]}\end{array}$$

(6)

where, $X^k$ represents the pairwise multi-choice comparison matrix of the k-th decision-maker and ${\left\{x_{ij}^{\left(s_{ij}\right)}\right\}}^k$ denotes the relative importance of criteria i to criterion j with $\left(s_{ij}\right)$ multiple reference comparisons. Additionally, $X_i^k=({\{x_{11}^{\left(1\right)},\dots ,x_{11}^{\left(s_{11}\right)}\}}^k$, ${\{x_{12}^{\left(1\right)},\dots ,x_{12}^{\left(s_{12}\right)}\}}^k,\cdots ,{\{x_{1n}^{\left(1\right)},\dots ,x_{1n}^{\left(s_{1n}\right)}\}}^k)$ is the vector of the $i\text{-}$th index’s multi-choice preference relation to other indexes. Furthermore, the multi-choice pairwise comparison matrix $X^k$ is deemed entirely consistent if and only if the following conditions are met:

$${\left\{x_{ip}^{s_{ip}}\right\}}^k\times {\left\{x_{pj}^{s_{pj}}\right\}}^k={\left\{x_{ij}^{s_{ij}}\right\}}^k\forall i,j$$

(7)

(II). 

Finding the best and the worst criteria to use.

Decision-makers must recognize the best and worst, typically focused on the built-in decision criteria mechanism of the previous step. The best and worst selection by assigning $c_B^k$ to the best and $c_W^k$ to the worst criterion for the k-th decision-maker. Afterward, it can determine whether two criteria experts often used in the great majority, $c_B^k$ and $c_W^k$, are the best and worst criteria. When several experts make recommendations, the best and worst criteria are often subjective, resulting in various $c_B^k$ and $c_W^k$. To eventually incorporate the $c_B^k$ and $c_W^k$ of various experts, it is required to choose a set of widely accepted best and worst criteria in front.

(III). 

Conducting multi-choice preference comparisons to determine best-to-others and others-to-worst criteria.

This step defines pairwise multi-choice preference comparisons of best criteria B over other criteria $j(j=1,2,\dots ,n)$, designated by $X_B^k$, of the k-th decision-maker. The decision-makers pairwise multi-choice comparisons of all criteria j over the worst criterion W, represented by $X_W^k$, are termed others-to-worst. The decision-makers determine pairwise perceptions on the order of 1 to 9, as said in Table 1. Although there are several options possible when it comes to such judgments. Figure 1 depicts the best-to-others $X_B^k$ and others-to-worst $X_W^k$ vector multi-choice preferences with many alternatives. The resulting multi-choice best-to-others vector is denoted by $X_B^k$ in Equation (8) as follows:

$$\begin{array}{c}{X}_B^k=\left(x_{B1}^{S_{B1}},x_{B2}^{S_{B2}},\dots ,x_{Bn}^{S_{Bn}}\right),\mathrm{where}{x}_{Bj}^{S_{Bj}}={\{x_{Bj}^{\left(1\right)},x_{Bj}^{\left(2\right)},\dots ,x_{Bj}^{\left(s_{Bj}\right)}\}}^k\forall j\hfill \end{array}$$

(8)

where $x_{Bj}^{S_{Bj}}$ is the set of multi-choice reference comparisons for the k-th expert evaluating the best criteria B over other criteria j, such that $x_{Bj}^{s_{Bj}}\ge 1\forall j=1,2,\dots ,n$ and $s_{Bj}\ge 1$.

Similarly, the multi-choice preference comparisons of all criteria j over the worst criterion W are obtained. The multi-choice others-to-worst vector ($X_W^k$) for k-th decision-maker is expressed by Equation (9) as follows:

$$\begin{array}{c}{X}_W^k=\left(x_{1W}^{S_{1W}},x_{2W}^{S_{2W}},\dots ,x_{nW}^{S_{nW}}\right),\mathrm{where}{x}_{jW}^{S_{jW}}={\{x_{jW}^{\left(1\right)},x_{jW}^{\left(2\right)},\dots ,x_{jW}^{\left(s_{jW}\right)}\}}^k\forall j\hfill \end{array}$$

(9)

where $x_{jW}^{S_{jW}}$ is the multi-choice preference of the criterion $j(j=1,2,\dots ,n)$ over the worst criterion W, such that $x_{jW}^{s_{jW}}\ge 1\forall j=1,2,\dots ,n$ and $s_{jW}\ge 1$.

(IV). 

Incorporate multi-choice preference comparisons of decision-makers’ evaluations.

Due to the fact that this work makes use of the input-based consistency ratio, first examine the consistency of every decision-maker’s multiple-choice preference comparison vectors. After obtaining the assessment of each decision-maker, now create the decision-maker’s multi-choice pairwise comparison matrix. The multi-choice comparison matrix for the k-th decision-maker is presented below:

$$\begin{array}{c}\hfill X^k=\left[\begin{array}{cccccccc}-& -& \cdots & -& \cdots & {\{x_{1W}^{\left(1\right)},x_{1W}^{\left(2\right)},\dots ,x_{1W}^{\left(s_{1W}\right)}\}}^k& \cdots & -\\ -& -& \cdots & -& \cdots & {\{x_{2W}^{\left(1\right)},x_{2W}^{\left(2\right)},\dots ,x_{2W}^{\left(s_{2W}\right)}\}}^k& \cdots & \\ ⋮& ⋮& ⋮& -& ⋮& ⋮& ⋮& ⋮\\ {\{x_{B1}^{\left(1\right)},x_{B1}^{\left(2\right)},\dots ,x_{B1}^{\left(s_{B1}\right)}\}}^k& {\{x_{B2}^{\left(1\right)},x_{B2}^{\left(2\right)},\dots ,x_{B2}^{\left(s_{B2}\right)}\}}^k& \cdots & 1& \cdots & {\{x_{BW}^{\left(1\right)},x_{BW}^{\left(2\right)},\dots ,x_{BW}^{\left(s_{BW}\right)}\}}^k& ⋮& {\{x_{Bn}^{\left(1\right)},x_{Bn}^{\left(2\right)},\dots ,x_{Bn}^{\left(s_{Bn}\right)}\}}^k\\ ⋮& ⋮& ⋮& -& ⋮& ⋮& ⋮& ⋮\\ ⋮& ⋮& ⋮& -& ⋮& 1& ⋮& ⋮\\ ⋮& ⋮& ⋮& -& ⋮& ⋮& ⋮& ⋮\\ -& -& \cdots & -& \cdots & {\{x_{nW}^{\left(1\right)},x_{nW}^{\left(2\right)},\dots ,x_{nW}^{\left(s_{nW}\right)}\}}^k& \cdots & -\end{array}\right]\end{array}$$

(10)

There are several techniques for combining the multi-choices of all the decision-makers. The associated method covered is the Lagrange interpolation method.

The main goal of such methods is the aggregation to obtain accurate results from either of the suitable choices from the multi-choice pairwise comparison matrix. There exist various acceptable approaches for handling multi-choice parameters. For multi-choice parameters, interpolating polynomials (IP) are defined by obtaining integer quantities referred to as nodal points or nodes. Each node represents a single functional significance of a multi-choice attribute. If a component includes $s_{ij}$ possibilities, an exactly $s_{ij}$ amount of nodes are required. The proposed interpolating polynomials aggregate the multi-choice comparison vector and obtain the non-linear function precisely at all nodes for multiple-choice. Substitute a multi-choice component with an appropriate polynomial. Lagrange method, Newton’s divided difference method, Newton’s forward difference method, and Newton’s backward difference method are the four significant forms of interpolating polynomials-based methods.

• Lagrangian method of polynomial interpolation

Lagrange’s interpolating polynomial (LIP) is used to tackle the multi-choice preference comparisons vectors $X_B^k$ and $X_W^k$ from Equations (8) and (9), respectively. From Equation (8), for k-th decision-maker, the $x_{Bj}^{S_{Bj}}=\{x_{Bj}^{\left(1\right)},x_{Bj}^{\left(2\right)},\dots ,x_{Bj}^{\left(s_{Bj}\right)}\}\forall j$ where $s_{Bj}(s_{Bj}\ge 1)$ denotes the number of multiple choices in the comparison of criterion B to criterion $j(j=1,2,\dots ,n)$, assume the $z^{Bj}$ variable representing the number of node points whose values are $(0,1,2,\dots ,s_{Bj}-1)$. Derive a LIP $P_{LIP}^k\left(z^{Bj}\right)$ of degree $(s_{Bj}-1)$ as follows:

$$\begin{array}{cc}\hfill P_{LIP}^k\left(z^{Bj}\right)=& \frac{(z^{Bj}-1)(z^{Bj}-2)\dots (z^{Bj}-s_{Bj}+1)}{{(-1)}^{s_{Bj}-1}(s_{Bj}-1)!}{x}_{Bj}^{\left(1\right)}+\frac{z^{Bj}(z^{Bj}-2)\dots (z^{Bj}-s_{Bj}+1)}{{(-1)}^{s_{Bj}-2}(s_{Bj}-2)!}{x}_{Bj}^{\left(2\right)}\hfill \\ \hfill & +\frac{z^{Bj}(z^{Bj}-1)(z^{Bj}-3)\dots (z^{Bj}-k+1)}{{(-1)}^{s_{Bj}-3}(s_{Bj}-3)!2!}{x}_{Bj}^{\left(3\right)}\hfill \\ \hfill & +\dots +\frac{z^{Bj}(z^{Bj}-1)(z^{Bj}-2)\dots (z^{Bj}-s_{Bj}+2)}{(s_{Bj}-1)!}{x}_{Bj}^{\left(s_{Bj}\right)}\forall j\hfill \end{array}$$

(11)

In the same way, suppose $z^{jW}$ represents node points with values $(0,1,2,\dots ,(s_{jW}-1))$ with respect to $X_W^k$. From Equation (9), for the k-th decision-maker the $x_{jW}^{S_{jW}}=\{x_{jW}^{\left(1\right)},x_{jW}^{\left(2\right)},\dots ,x_{jW}^{\left(s_{jW}\right)}\}\forall j$ where $s_{jW}(s_{jW}\ge 1)$ denotes the number of multiple choices in the other-to-worst, assume $z^{jW}$ representing node points with values $(0,1,2,\dots ,(s_{jW}-1))$. Derive a LIP $P_{LIP}^k\left(z^{jW}\right)$ of degree $(s_{jW}-1)$ as follows:

$$\begin{array}{cc}\hfill P_{LIP}^k\left(z^{jW}\right)=& \frac{(z^{jW}-1)(z^{jW}-2)\dots (z^{jW}-s_{jW}+1)}{{(-1)}^{s_{jW}-1}(s_{jW}-1)!}{x}_{jW}^{\left(1\right)}+\frac{z^{jW}(z^{jW}-2)\dots (z^{jW}-s_{jW}+1)}{{(-1)}^{s_{jW}-2}(s_{jW}-2)!}{x}_{jW}^{\left(2\right)}\hfill \\ \hfill & +\frac{z^{jW}(z^{jW}-1)(z^{jW}-3)\dots (z^{jW}-k+1)}{{(-1)}^{s_{jW}-3}(s_{jW}-3)!2!}{x}_{jW}^{\left(3\right)}\hfill \\ \hfill & +\dots +\frac{z^{jW}(z^{jW}-1)(z^{jW}-2)\dots (z^{jW}-s_{jW}+2)}{(s_{jW}-1)!}{x}_{jW}^{\left(s_{jW}\right)}\forall j\hfill \end{array}$$

(12)

Using the interpolating polynomials, model the multi-choice mathematical programming model as described in step V.

(V). 

Determine the optimal weights $({\omega }_1^{*},{\omega }_2^{*},\dots ,{\omega }_n^{*})$ by utilizing the optimization model.

In this step, determine the optimal weights $({\omega }_1^{*},{\omega }_2^{*},\dots ,{\omega }_n^{*})$ corresponding to each criteria $(c_1,c_2,\dots ,c_n)$. Then the non-linear mathematical optimization models that minimize the sum of the inconsistency variance for all decision-makers in accordance with the original group BWM are proposed. Following group BWM, we present two optimization models, MD-1 and MD-2. Both of them are discussed in detail.

• MD-1: Mathematical model-1

The optimal weight for each criterion is where, for each multi-choice comparison, ${\omega }_B/{\omega }_j$ and ${\omega }_j/{\omega }_W$ should have ${\omega }_B/{\omega }_j={\{x_{Bj}^{\left(1\right)},x_{Bj}^{\left(2\right)},\dots ,x_{Bj}^{\left(s_{Bj}\right)}\}}^k$ and ${\omega }_j/{\omega }_W={\{x_{jW}^{\left(1\right)},x_{jW}^{\left(2\right)},\dots ,x_{jW}^{\left(s_{jW}\right)}\}}^k$ of k-th decision-maker. To identify the best criteria weights with the group decision-making, the maximal differences between the computed weights and the presentation of every decision-maker must be minimal. Every criterion j of the k-th decision-maker fulfills such requirements, the actual discrepancy $|{\omega }_B/{\omega }_j-X_B^k|$ and $|{\omega }_j/{\omega }_W-X_W^k|$ for all j could be established as follows.

$$\begin{array}{c}min\sum _{k\in D}{\lambda }_k\underset{j}{max}\left\{\left|\frac{{\omega }_B}{{\omega }_j}-{\{x_{Bj}^{\left(1\right)},x_{Bj}^{\left(2\right)},\dots ,x_{Bj}^{\left(s_{Bj}\right)}\}}^k\right|,\left|\frac{{\omega }_B}{{\omega }_j}-{\{x_{jW}^{\left(1\right)},x_{jW}^{\left(2\right)},\dots ,x_{jW}^{\left(s_{jW}\right)}\}}^k\right|\right\}\hfill \\ s.t.\left\{\begin{array}{c}{\displaystyle \sum _{j=1}^n}{\omega }_j=1\hfill \\ {\omega }_j\ge 0\hfill \\ j=1,2,\dots ,n\hfill \end{array}\right.\hfill \end{array}$$

(13)

In the MD-1 model, presented in Equation (13), the objective function includes a parameter ${\lambda }_k$ for $k\text{-}\mathrm{th}$ expert having a value in the range $[0,100]$ presenting the individual weight (importance) of the decision-makers. Further, we define ${\xi }_k={max}_j\left\{\left|\frac{{\omega }_B}{{\omega }_j}-{\{x_{Bj}^{\left(1\right)},x_{Bj}^{\left(2\right)},\dots ,x_{Bj}^{\left(s_{Bj}\right)}\}}^k\right|,\left|\frac{{\omega }_B}{{\omega }_j}-{\{x_{jW}^{\left(1\right)},x_{jW}^{\left(2\right)},\dots ,x_{jW}^{\left(s_{jW}\right)}\}}^k\right|\right\}$ to simplify the proposed MD-1. Therefore, the proposed MD-1 model is transformed as:

$$\begin{array}{c}min\sum _{k\in D}{\lambda }_k{\xi }_k\hfill \\ s.t.\left\{\begin{array}{c}\left|\frac{{\omega }_B}{{\omega }_j}-{\{x_{Bj}^{\left(1\right)},x_{Bj}^{\left(2\right)},\dots ,x_{Bj}^{\left(s_{Bj}\right)}\}}^k\right|\le {\xi }_k\forall j,k\hfill \\ \left|\frac{{\omega }_B}{{\omega }_j}-{\{x_{jW}^{\left(1\right)},x_{jW}^{\left(2\right)},\dots ,x_{jW}^{\left(s_{jW}\right)}\}}^k\right|\le {\xi }_k\forall j,k\hfill \\ {\displaystyle \sum _{j=1}}{\omega }_j=1\forall j\hfill \\ {\omega }_j\ge 0\forall j\hfill \\ {\xi }_k\ge 0\forall k\hfill \\ j=1,2,\dots ,n\hfill \\ k=1,2,\dots ,m\hfill \end{array}\right.\hfill \end{array}$$

(14)

The above mathematical model represents a non-linear multi-choice optimization problem with the multi-choice comparison vectors as multi-choice parameters in the constraints. To tackle the multi-choice pairwise comparison parameters, we apply the polynomial interpolation methods with the aim of integer nodal points by using the previous Step (IV). Thus, the model presented in Equation (14) is transformed as follows:

$$\begin{array}{c}min\sum _{k\in D}{\lambda }_k{\xi }_k\hfill \\ s.t.\left\{\begin{array}{c}\left|\frac{{\omega }_B}{{\omega }_j}-P^k\left(z^{Bj}\right)\right|\le {\xi }_k\forall j,k\hfill \\ \left|\frac{{\omega }_B}{{\omega }_j}-P^k\left(z^{jW}\right)\right|\le {\xi }_k\forall j,k\hfill \\ {\displaystyle \sum _{j=1}}{\omega }_j=1\forall j\hfill \\ {\omega }_j\ge 0\forall j\hfill \\ {\xi }_k\ge 0\forall k\hfill \\ z^{Bj}=0,1,2,\dots ,(s_{Bj}-1)\forall j\hfill \\ z^{jW}=0,1,2,\dots ,(s_{jW}-1)\forall j\hfill \\ j=1,2,\dots ,n\hfill \\ k=1,2,\dots ,m\hfill \end{array}\right.\hfill \end{array}$$

(15)

The optimal criteria weights $({\omega }_1^{*},{\omega }_2^{*},\dots ,{\omega }_n^{*})$ are evaluated by solving the above model that is calculated as the actual values. Note that the proposed MD-1 model (as in Equation (15)) not only obtains the optimal criteria weights but also finds the position vectors, that is, integer nodal points $z^{Bj}$, and $z^{jW}$ for all $j=1,2,\dots ,n$ of the polynomial $P^k$ for all $k=1,2,\dots ,m$. Thus, the final solution includes the best pairwise comparison among the multiple choices that are provided by the decision-makers.

• MD-2: Mathematical model-2

As described in MD-1, each decision-maker has an individual weight component, which is used to calculate their relative relevance in the proposed model. Consequently, the second mathematical model, MD-2, with a min–max objective, is presented as follows:

$$\begin{array}{c}min\underset{k}{max}{\lambda }_k{\xi }_k\hfill \\ s.t.\left\{\begin{array}{c}\left|\frac{{\omega }_B}{{\omega }_j}-{\{x_{Bj}^{\left(1\right)},x_{Bj}^{\left(2\right)},\dots ,x_{Bj}^{\left(s_{Bj}\right)}\}}^k\right|\le {\xi }_k\forall j,k\hfill \\ \left|\frac{{\omega }_B}{{\omega }_j}-{\{x_{jW}^{\left(1\right)},x_{jW}^{\left(2\right)},\dots ,x_{jW}^{\left(s_{jW}\right)}\}}^k\right|\le {\xi }_k\forall j,k\hfill \\ {\displaystyle \sum _{j=1}}{\omega }_j=1\forall j\hfill \\ {\omega }_j\ge 0\forall j\hfill \\ {\xi }_k\ge 0\forall k\hfill \\ j=1,2,\dots ,n\hfill \\ k=1,2,\dots ,m\hfill \end{array}\right.\hfill \end{array}$$

(16)

Here, similar to the mathematical model presented in MD-1 (as in Equation (14)), we simplify the MD-2 model in Equation (16) as $\xi ={max}_k{\lambda }_k{\xi }_k$. Additionally, the multi-choice comparisons in the constraints of the model having the multi-choice parameter are interpolated using the previous Step (IV). We transform the model MD-2 as follows:

$$\begin{array}{c}min\xi \hfill \\ s.t.\left\{\begin{array}{c}\xi \ge {\lambda }_k{\xi }_k\forall k\hfill \\ \left|\frac{{\omega }_B}{{\omega }_j}-P^k\left(z^{Bj}\right)\right|\le {\xi }_k\forall j,k\hfill \\ \left|\frac{{\omega }_B}{{\omega }_j}-P^k\left(z^{jW}\right)\right|\le {\xi }_k\forall j,k\hfill \\ {\displaystyle \sum _{j=1}}{\omega }_j=1\forall j\hfill \\ {\omega }_j\ge 0\forall j\hfill \\ {\xi }_k\ge 0\forall k\hfill \\ z^{Bj}=0,1,2,\dots ,(s_{Bj}-1)\forall j\hfill \\ z^{jW}=0,1,2,\dots ,(s_{jW}-1)\forall j\hfill \\ j=1,2,\dots ,n\hfill \\ k=1,2,\dots ,m\hfill \end{array}\right.\hfill \end{array}$$

(17)

The optimal criteria weights $({\omega }_1^{*},{\omega }_2^{*},\dots ,{\omega }_n^{*})$, integer nodal points $z^{Bj}$, and $z^{jW}$ for all $j=1,2,\dots ,n$ are evaluated by solving the above proposed MD-2 model in Equation (17).

Following the solution of the mathematical models, MD-1 and MD-2, the optimum values of ${\xi }_k$ are used to compute the consistency ratio $\left({CR}_k\right)$ for every k-th decision-maker that will apply to evaluate the group consistency ratio $\left({CR}^G\right)$ in a group decision-making system. If $x_{Bj}\times x_{jW}=x_{BW}\forall j$, where $x_{Bj}$ is the preference of best criteria B over the all other criteria j and $x_{jW}$ is the preference of criterion j over the worst criterion B, then comparisons are said to be entirely consistent. Thus, the consistency ratio of the proposed GMCBWM is calculated in the following step.

(VI). 

Consistency ratio for GMCBWM

The consistency ratio (CR) was used to demonstrate the validity of pairwise comparisons. The CR of the proposed GMCBWM of the k-th decision-maker $C{R}_k$ and further the group decision-making $C{R}^G$ are calculated using Equations (18) and (19) given as follows.

$$\begin{array}{cc}\hfill {CR}_k=& {\lambda }_k\frac{{\xi }_k^{*}}{CI}\forall k=1,2,\dots ,m\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill {CR}^G=& \underset{k}{max}\left\{C{R}_k\right\}\hfill \end{array}$$

(19)

In Equation (18), ${\xi }_k^{*}$ is the optimum value of inconsistency achieved by resolving mathematical models (MD-1 or MD-2) for the k-th decision-makers, and ${\lambda }_k$ is the individual weight vector assigned to the k-th decision-maker depending on their degree of knowledge. The consistency index (CI) is a constant value presented in

Table 3. CI [14].

$x_{BW}^{max}$	1	2	3	4	5	6	7	8	9
CI (max $\xi$)	0.00	0.44	1	1.63	2.30	3	3.73	4.47	5.23

for each decision-maker’s value $x_{BW}^{max}$. The conclusive $C{R}^G$ of the GMCBWM is the maximum $C{R}_k$ among all decision-makers as in Equation (19). If $C{R}^G$ is zero, the result in the form of optimal weights is totally consistent; nevertheless, as $C{R}^G$ grows, the consistency diminishes.

5. Numerical Studies

This section presents two numerical illustrations of group decision-making problems to demonstrate the applicability of the proposed GMCBWM in different circumstances and analyze the outcomes. The proposed models are formulated under each case and evaluated using the AMPL [59] and NEOS [60] to achieve the optimal weights criteria.

5.1. Numerical Example 1

In this example, the responses for four criteria $(c_1,c_2,c_3,c_4)$ with $n=4$ are taken from two decision-makers (DMs), i.e., $k=2$ as DM1 and DM2 [61]. The criteria $c_1$ and $c_3$ are considered best $\left(B\right)$ and worst $\left(W\right)$ criteria by both experts having ${\lambda }_1$ and ${\lambda }_2$ individual weights, respectively. The multi-choice best-to-others $X_B^k$ (Equation (8)) and others-to-worst $X_W^k$ (Equation (9)) vector preferences on a scale of 1 to 9 are provided by each DM using Table 1. As $n=4$, according to the basic BWM, we require at least $2n-3=2\ast 4-3=5$ preference responses from each DM [14]. The multi-choice pairwise preference comparing the best-to-others and others-to-worst criteria are presented in

Table 4. Multi-choice preference comparisons of four criteria with two decision-makers in Example 1.

$\mathit{DM}$	${\mathit{x}}_{12}$	${\mathit{x}}_{13}$	${\mathit{x}}_{14}$	${\mathit{x}}_{23}$	${\mathit{x}}_{43}$
$D{M}_1$	2	{8, 9}	3	4	2
$D{M}_2$	2	8	{2, 3}	4	2

for both DMs. In Table 4, the response $x_{13}$ comparing $c_1$ to $c_3$ from DM1 is a multi-choice comparison with two choices 8 and 9, that is, $x_{13}=\{8,9\}$ and for $x_{14}$ comparing $c_1$ to $c_4$ from DM2 is 2 and 3, that is, $x_{14}=\{2,3\}$ with $s_{13}=2$ and $s_{14}=2$, respectively.

Thus, the multi-choice best-to-others vector is $X_B^1=\{x_{12},\{x_{13}^{\left(1\right)},x_{13}^{\left(2\right)}\},x_{14}\}\}=\{2,\{8,9\},3\}$ and $X_B^2=\{x_{12},x_{13},\{x_{14}^{\left(1\right)},x_{14}^{\left(2\right)}\}\}=\{2,8,\{2,3\}\}$ and others-to-worst vector is $X_W^1=\{\{x_{13}^{\left(1\right)},x_{13}^{\left(2\right)}\},x_{23},x_{43}\}=\{\{8,9\},4,2\}$ and $X_W^2=\{x_{13},x_{23},x_{43}\}=\{8,4,2\}$ using Table 4 from Step (III) of the proposed GMCBWM. The objective of the proposed models is to determine the best choice or the best pairwise comparison out of all the choices of pairwise comparisons assigned by DM1 and DM2. However, before formulating the mathematical model of the proposed GMCBWM, we first have to derive the polynomial functions that interpolate the multi-choice reference parameters associated with each comparison vector.

Thus, interpolation polynomials (IPs) are derived for the $x_{13}=\{x_{13}^{\left(1\right)},x_{13}^{\left(2\right)}\}=\{8,9\}$ and $x_{14}=\{x_{14}^{\left(1\right)},x_{14}^{\left(2\right)}\}=\{3,4\}$ multi-choice reference comparison taking integral values for the nodal points $z^{13}$ and $z^{14}$. Each node will have $s_{13}=s_{14}=2$ number of integer values. The Lagrange interpolating polynomial (LIP) functions are derived using Step 4 of Section 4. We formulate LIPs $P_{LIP}$ for $\left(z^{13}\right)$ of degree $(s_{13}-1)$ and for $\left(z^{14}\right)$ of degree $(s_{14}-1)$ using Equations (11) and (12) as follows:

$$\begin{array}{cc}\hfill P_{LIP}\left(z^{13}\right)=& \frac{(z^{13}-1)}{{(-1)}^{s_{13}-1}(s_{13}-1)!}{x}_{13}^{\left(1\right)}+\frac{z^{13}}{{(-1)}^{s_{13}-2}(s_{13}-2)!}{x}_{13}^{\left(2\right)}\hfill \\ \hfill =& \frac{(z^{13}-1)}{{(-1)}^{2-1}(2-1)!}8+\frac{z^{13}}{{(-1)}^{2-2}(2-2)!}9\hfill \\ \hfill =& z^{13}+8\hfill \\ \hfill P_{LIP}\left(z^{14}\right)=& \frac{(z^{14}-1)}{{(-1)}^{s_{14}-1}(s_{14}-1)!}{x}_{14}^{\left(1\right)}+\frac{z^{14}}{{(-1)}^{s_{14}-2}(s_{14}-2)!}{x}_{14}^{\left(2\right)}\hfill \\ \hfill =& \frac{(z^{14}-1)}{{(-1)}^{2-1}(2-1)!}2+\frac{z^{14}}{{(-1)}^{2-2}(2-2)!}3\hfill \\ \hfill =& z^{14}+2\hfill \end{array}$$

Now, to obtain the optimal weights $({\omega }_1^{*},{\omega }_2^{*},{\omega }_3^{*},{\omega }_4^{*})$ of the criteria, we present the two proposed mathematical programming model MD-1 and MD-2 from Step (V) of Section 4 using Equations (15) and (17) as follows:

$$\begin{array}{cc}\hfill & MD-1\hfill \\ \hfill & min({\lambda }_1{\xi }_1+{\lambda }_2{\xi }_2)\hfill \\ \hfill & s.t.\left\{\begin{array}{c}\left|\frac{{\omega }_1}{{\omega }_2}-2\right|\le {\xi }_1\hfill \\ \left|\frac{{\omega }_1}{{\omega }_2}-2\right|\le {\xi }_2\hfill \\ \left|\frac{{\omega }_1}{{\omega }_3}-(z^{13}+8)\right|\le {\xi }_1\hfill \\ \left|\frac{{\omega }_1}{{\omega }_3}-8\right|\le {\xi }_2\hfill \\ \left|\frac{{\omega }_1}{{\omega }_4}-3\right|\le {\xi }_1\hfill \\ \left|\frac{{\omega }_1}{{\omega }_4}-(z^{14}+2)\right|\le {\xi }_2\hfill \\ \left|\frac{{\omega }_2}{{\omega }_3}-4\right|\le {\xi }_1\hfill \\ \left|\frac{{\omega }_2}{{\omega }_3}-4\right|\le {\xi }_2\hfill \\ \left|\frac{{\omega }_4}{{\omega }_3}-2\right|\le {\xi }_1\hfill \\ \left|\frac{{\omega }_4}{{\omega }_3}-2\right|\le {\xi }_2\hfill \\ {\omega }_1+{\omega }_2+{\omega }_3+{\omega }_4=1\hfill \\ {\omega }_1,{\omega }_2,{\omega }_3,{\omega }_4\ge 0\hfill \\ {\xi }_1,{\xi }_2\ge 0\hfill \\ z^{13},z^{14}=0,1.\hfill \end{array}\right.\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill & MD-2\hfill \\ \hfill & min\xi \hfill \\ \hfill & s.t.\left\{\begin{array}{c}\xi \ge {\lambda }_1{\xi }_1\hfill \\ \xi \ge {\lambda }_2{\xi }_2\hfill \\ \mathrm{Rest}\mathrm{of}\mathrm{constraints}\mathrm{are}\mathrm{same}\mathrm{as}\mathrm{in}\mathrm{Equation}\left(20\right)\hfill \end{array}\right.\hfill \end{array}$$

(21)

The non-linear mixed-integer mathematical programming models MD-1 and MD-2, as shown in Equations (20) and (21), are solved to obtain the optimal weights $({\omega }_1^{*},{\omega }_2^{*},{\omega }_3^{*},{\omega }_4^{*})$ and the nodal points $(z^{13},z^{14})$ corresponding to the multi-choice reference comparison. The optimal ${\xi }^{*}$ of the MDs are applied to evaluate the inconsistency of the group decision-making of Example 1. Further, the individual weights (${\lambda }_1$, ${\lambda }_2$) of DMs have a substantial effect on the resulting consistency ratio (CR). Therefore, we investigate five different pairs (0.1,0.9), (0.3,0.7), (0.5,0.5), (0.7,0.3), and (0.9,0.1) of group decision instances to obtain optimal weights of the criteria. The sensitivity of weight values has been evaluated based on these pairs. Using the data provided in Table 4 of Example 1, we have solved the two models with different combinations of individual weight vectors of DMs.

The optimal results of Example 1 after solving MD-1 and MD-2 are provided in

Table 5. Result of Example 1 solving with MD-1.

No.	$({\mathit{\lambda }}_1,{\mathit{\lambda }}_2)$	$({\mathit{\omega }}_1^{*},{\mathit{\omega }}_2^{*},{\mathit{\omega }}_3^{*},{\mathit{\omega }}_4^{*})$	${\mathit{z}}^{13},{\mathit{z}}^{14}$	${\mathit{\xi }}_{\mathit{k}}^{*}$	${\mathit{CR}}_{\mathit{k}}$	${\mathit{CR}}^{\mathit{G}}$
1	(0.1, 0.9)	(0.506667, 0.266667	0	0.0266667	0.000596570	0.005369134
		0.0666667, 0.160000)	0	0.0266667	0.005369134
2	(0.3, 0.7)	(0.506667, 0.266667	0	0.0266667	0.001789732	0.004176000
		0.0666667, 0.160000)	0	0.0266667	0.004176040
3	(0.5, 0.5)	(0.506667, 0.266667	0	0.0266667	0.002982886	0.002982886
		0.0666667, 0.160000)	0	0.0266667	0.002982886
4	(0.7, 0.3)	(0.506667, 0.266667	0	0.0266667	0.004176040	0.004176040
		0.0666667, 0.160000)	0	0.0266667	0.001789732
5	(0.9, 0.1)	(0.506667, 0.266667	0	0.0266667	0.005369195	0.005369195
		0.0666667, 0.160000)	0	0.0266667	0.000596577

and

Table 6. Result of Example 1 solving with MD-2.

No.	$({\mathit{\lambda }}_1,{\mathit{\lambda }}_2)$	$({\mathit{\omega }}_1^{*},{\mathit{\omega }}_2^{*},{\mathit{\omega }}_3^{*},{\mathit{\omega }}_4^{*})$	${\mathit{z}}^{13},{\mathit{z}}^{14}$	${\mathit{\xi }}_{\mathit{k}}^{*}$	${\mathit{CR}}_{\mathit{k}}$	${\mathit{CR}}^{\mathit{G}}$
1	(0.1, 0.9)	(0.506667, 0.266667	0	0.0266667	0.000596577	0.005368993
		0.0666667, 0.160000)	0	0.0266656	0.005368993
2	(0.3, 0.7)	(0.506667, 0.266667	0	0.0266667	0.001789732	0.004175727
		0.0666667, 0.160000)	0	0.0266652	0.004175727
3	(0.5, 0.5)	(0.506667, 0.266667	0	0.0266667	0.002982886	0.002982886
		0.0666667, 0.160000)	0	0.0266667	0.002982886
4	(0.7, 0.3)	(0.506667, 0.266667	0	0.0266652	0.004175727	0.004175772
		0.0666667, 0.160000)	0	0.0622189	0.004175772
5	(0.9, 0.1)	(0.506667, 0.266667	0	0.0266656	0.005368993	0.005368993
		0.0666667, 0.160000)	0	0.2399900	0.005368904

, respectively. The outcomes of five pairs of (${\lambda }_1$, ${\lambda }_2$) with optimal weights of critical, nodal points, inconsistency value (${\xi }_1$, ${\xi }_2$), and consistency ratio ($C{R}_1,C{R}_2$) are presented for DM1 and DM2 with $k=2$ in tables. Importantly, in the last column of Table 5 and Table 6, the group consistency ratio ($C{R}^G$) is presented. It has been found that the optimal weight of criteria is the same for all pairs of ${\lambda }_1$ and ${\lambda }_2$ for both MD-1 and MD-2 for this numerical illustration. However, the $C{R}_k$ of each DMs changes depending upon the values of ${\lambda }_1$ and ${\lambda }_2$. In other words, the ranking of the criteria is the same for each pair in both models and is $c_1>c_2>c_4>c_3$. Moreover, the solutions of both models MD-1 and MD-2 for different combinations of ${\lambda }_1$ and ${\lambda }_2$ shows that the best nodal values are $z^{13}=0$, i.e., $x_{13}=8$ for DM1 comparing $c_1$ to $c_3$ and $z^{14}=0$, i.e., $x_{14}=2$ for DM2 comparing $c_1$ to $c_4$. Thus, the best choice of $x_{BW}$, i.e., $x_{13}$ equals eight and remains unchanged for all models. Further, the group consistency ratio of Example 1 is equal to the minimum among all $C{R}^G$ obtained for different values of ${\lambda }_1$ and ${\lambda }_2$. Therefore, the minimum consistency value of Example 1 using MD-1 is $C{R}^G=0.002982886$ (see Table 5) and for MD-2 is $C{R}^G=0.002982886$ (see Table 6) for ${\lambda }_1=0.5$ and ${\lambda }_2=0.5$. It is worth mentioning that the $C{R}^G$ of the Example 1 with Group-BWM [61] with both model-1 and model-2 is equal to $0.096$ with ranking $c_1>c_2>c_4>c_3$. Hence, we can say that the proposed models of GMCBWM achieve the best consistency value for Example 1 and also validate the ranking of criteria.

5.2. Numerical Example 2

We have assumed three decision-makers (DM1, DM2, DM3), $k=3$ with four criteria $(c_1,c_2,c_3,c_4)$, $n=4$. The $c_1$ and $c_3$ criteria are assumed as best (B) and worst (W) criterion by the all three experts having ${\lambda }_1$, ${\lambda }_2$, and ${\lambda }_3$ individual weights, respectively. The pairwise comparisons with the multi-choice response sets are presented in

Table 7. Pairwise comparison of three decision-makers in Example 2.

DM	${\mathit{x}}_{12}$	${\mathit{x}}_{13}$	${\mathit{x}}_{14}$	${\mathit{x}}_{23}$	${\mathit{x}}_{43}$
DM1	2	9	(3, 4)	4	2
DM2	2	(8, 9)	4	4	2
DM3	2	8	4	(3, 4)	2

.

The multi-choice best-to-others $X_B^k$ (Equation (8)) and others-to-worst $X_W^k$ (Equation (9)) vector preferences on a scale 1 to 9 are provided by each DM using Table 1. Table 4 shows the multi-choice pairwise comparisons of the worst and the best criteria than the other criteria for the two DMs.

The response from DM1 for $x_{13}$ is a multi-choice parameter with two choices 8 and 9, that is, $x_{13}=\{8,9\}$ and for DM2, the response is 2 and 3, that is, $x_{14}=\{2,3\}$ with $s_{13}=2$ and $s_{14}=2$, respectively. The objective is to determine the best choice or the best pairwise comparison out of all the choices of pairwise comparisons assigned by DM1 and DM2.

Interpolating polynomials (IPs) are formulated for the $\{x_{13}^{\left(1\right)},x_{13}^{\left(2\right)}\}=\{8,9\}$ and $\{x_{14}^{\left(1\right)},x_{14}^{\left(2\right)}\}=\{3,4\}$ multi-choice reference comparison taking integral values for the nodal points $z^{13}$ and $z^{14}$. Each node will have $s_{13}=s_{14}=2$ number of integer values. The Lagrange IP (LIP) functions are formulated using step 4 for each multi-choice reference comparison. We derive LIPs $P_{LIP}$ for $\left(z^{13}\right)$ of degree $(s_{13}-1)$ and for $\left(z^{14}\right)$ of degree $(s_{14}-1)$ using Equations (11) and (12) as follows:

$$\begin{array}{cc}\hfill P_{LIP}\left(z^{13}\right)=& \frac{(z^{13}-1)}{{(-1)}^{s_{13}-1}(s_{13}-1)!}{x}_{13}^{\left(1\right)}+\frac{z^{13}}{{(-1)}^{s_{13}-2}(s_{13}-2)!}{x}_{13}^{\left(2\right)}\hfill \\ \hfill =& \frac{(z^{13}-1)}{{(-1)}^{2-1}(2-1)!}8+\frac{z^{13}}{{(-1)}^{2-2}(2-2)!}9\hfill \\ \hfill =& z^{13}+8\hfill \\ \hfill P_{LIP}\left(z^{14}\right)=& \frac{(z^{14}-1)}{{(-1)}^{s_{14}-1}(s_{14}-1)!}{x}_{14}^{\left(1\right)}+\frac{z^{14}}{{(-1)}^{s_{14}-2}(s_{14}-2)!}{x}_{14}^{\left(2\right)}\hfill \\ \hfill =& \frac{(z^{14}-1)}{{(-1)}^{2-1}(2-1)!}2+\frac{z^{14}}{{(-1)}^{2-2}(2-2)!}3\hfill \\ \hfill =& z^{14}+2\hfill \end{array}$$

Now, to obtain the optimal weights $({\omega }_1^{*},{\omega }_2^{*},{\omega }_3^{*},{\omega }_4^{*})$ of the criteria, we present the two proposed mathematical programming models MD-1 and MD-2 as follows:

$$\begin{array}{cc}\hfill & MD-1\hfill \\ \hfill & min({\lambda }_1{\xi }_1+{\lambda }_2{\xi }_2+{\lambda }_3{\xi }_3)\hfill \\ \hfill & s.t.\left\{\begin{array}{c}\left|\frac{{\omega }_1}{{\omega }_2}-2\right|\le {\xi }_1,\left|\frac{{\omega }_1}{{\omega }_2}-2\right|\le {\xi }_2,\left|\frac{{\omega }_1}{{\omega }_2}-2\right|\le {\xi }_3\hfill \\ \left|\frac{{\omega }_1}{{\omega }_3}-9\right|\le {\xi }_1,\left|\frac{{\omega }_1}{{\omega }_3}-(z^{13}+8)\right|\le {\xi }_2,\left|\frac{{\omega }_1}{{\omega }_3}-8\right|\le {\xi }_3\hfill \\ \left|\frac{{\omega }_1}{{\omega }_4}-(z^{14}+3)\right|\le {\xi }_1,\left|\frac{{\omega }_1}{{\omega }_4}-4\right|\le {\xi }_2,\left|\frac{{\omega }_1}{{\omega }_4}-4\right|\le {\xi }_3\hfill \\ \left|\frac{{\omega }_2}{{\omega }_3}-4\right|\le {\xi }_1,\left|\frac{{\omega }_2}{{\omega }_3}-4\right|\le {\xi }_2,\left|\frac{{\omega }_2}{{\omega }_3}-(z^{23}+3)\right|\le {\xi }_3\hfill \\ \left|\frac{{\omega }_4}{{\omega }_3}-2\right|\le {\xi }_1,\left|\frac{{\omega }_4}{{\omega }_3}-2\right|\le {\xi }_2,\left|\frac{{\omega }_4}{{\omega }_3}-2\right|\le {\xi }_3\hfill \\ {\omega }_1+{\omega }_2+{\omega }_3+{\omega }_4=1\hfill \\ {\omega }_1,{\omega }_2,{\omega }_3,{\omega }_4\ge 0\hfill \\ {\xi }_1,{\xi }_2,{\xi }_3\ge 0\hfill \\ z^{13},z^{14},z^{23}=0,1.\hfill \end{array}\right.\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill & MD-2\hfill \\ \hfill & min\xi \hfill \\ \hfill & s.t.\left\{\begin{array}{c}\xi \ge {\lambda }_1{\xi }_1\hfill \\ \xi \ge {\lambda }_2{\xi }_2\hfill \\ \mathrm{Rest}\mathrm{of}\mathrm{constraints}\mathrm{are}\mathrm{same}\mathrm{as}\mathrm{MD}\text{-}1\hfill \end{array}\right.\hfill \end{array}$$

(23)

The results of Models 1 and 2 are provided in

Table 8. Result of Example 2 for MD-1.

No.	${\mathit{\lambda }}_{\mathit{k}}$	$({\mathit{\omega }}_1^{*},{\mathit{\omega }}_2^{*},{\mathit{\omega }}_3^{*},{\mathit{\omega }}_4^{*})$	${\mathit{z}}^{13},{\mathit{z}}^{14},{\mathit{z}}^{23}$	${\mathit{\xi }}_{\mathit{k}}^{*}$	${\mathit{CR}}_{\mathit{k}}$	${\mathit{CR}}^{\mathit{G}}$
1	0.2	(0.533333, 0.266667	1	0.066667	0.002549	0.002549
	0.2	0.066667, 0.133333)	0	0.000000	0.000000
	0.6		1	0.000000	0.000000
2	0.2	(0.533333, 0.266667	1	0.066667	0.002549	0.002549
	0.6	0.066667, 0.133333)	1	0.000000	0.000000
	0.2		1	0.000000	0.000000
3	0.6	(0.538462, 0.261538	1	0.015385	0.001765	0.002065
	0.2	0.061539, 0.138462)	1	0.015385	0.000688
	0.2		0	0.046154	0.002065
4	0.3	(0.533333, 0.266667	1	0.066667	0.003824	0.003824
	0.3	0.066667, 0.133333)	1	0.000000	0.000000
	0.4		1	0.000000	0.000000
5	0.3	(0.533333, 0.266667	1	0.066667	0.003824	0.003824
	0.4	0.066667, 0.133333)	1	0.000000	0.000000
	0.3		1	0.000000	0.000000
6	0.4	(0.533333, 0.266667	1	0.066667	0.005099	0.005099
	0.3	0.066667, 0.133333)	1	0.000000	0.000000
	0.3		1	0.000000	0.000000

and

Table 9. Result of Example 2 for MD-2.

No.	${\mathit{\lambda }}_{\mathit{k}}$	$({\mathit{\omega }}_1^{*},{\mathit{\omega }}_2^{*},{\mathit{\omega }}_3^{*},{\mathit{\omega }}_4^{*})$	${\mathit{z}}^{13},{\mathit{z}}^{14},{\mathit{z}}^{23}$	${\mathit{\xi }}_{\mathit{k}}^{*}$	${\mathit{CR}}_{\mathit{k}}$	${\mathit{CR}}^{\mathit{G}}$
1	0.2	(0.528000, 0.272000	1	0.048000	0.001836	0.002148
	0.2	0.064000, 0.136000)	1	0.016000	0.000612
	0.6		1	0.016000	0.002148
2	0.2	(0.538462, 0.261538	0	0.046154	0.001765	0.002065
	0.6	0.061539, 0.138462)	1	0.015385	0.002065
	0.2		1	0.046154	0.002065
3	0.6	(0.538462, 0.261538	1	0.015385	0.001765	0.002065
	0.2	0.061539, 0.138462)	1	0.046154	0.001765
	0.2		1	0.046154	0.002065
4	0.3	(0.524444, 0.275556	1	0.035556	0.002040	0.002386
	0.3	0.062222, 0.137778)	1	0.026667	0.001530
	0.4		1	0.026667	0.002386
5	0.3	(0.527473, 0.272527	0	0.035165	0.002017	0.002360
	0.4	0.061539, 0.138462)	1	0.026374	0.002360
	0.3		1	0.035165	0.002360
6	0.4	(0.527473, 0.272527	1	0.026374	0.002017	0.002360
	0.3	0.061539, 0.138462)	1	0.035165	0.002017
	0.3		1	0.035165	0.002360

. In Table 8, the obtained weights are presented for six different sets of (${\lambda }_1$, ${\lambda }_2$, ${\lambda }_3$). The change in weight values of the criterion has been evaluated based on these sets. It has been found that for model 2, no change is visible for five sets of (${\lambda }_1$, ${\lambda }_2$, ${\lambda }_3$). Whereas, for (0.6, 0.2, 0.2), the weight values of the criterion are different from the other five sets. In Table 9, the inconsistency value (${\xi }_1$, ${\xi }_2$, ${\xi }_3$) and consistency ratio (CR1, CR2, CR3) for decision maker 1, 2, and 3 were presented, respectively. Finally, in the last column, the Group CR is tabulated. All these are evaluated for six sets of ${\lambda }_1$, ${\lambda }_2$, and ${\lambda }_3$. For model 1, the minimum CRG is 0.002065 for set 3, whereas the maximum is for set 6 of ${\lambda }_k$. For model 2, the minimum CRG is for sets 2 and 3, i.e., 0.002065. The CRG is maximum for set 4, i.e., 0.002386 of ${\lambda }_1=0.3$, ${\lambda }_2=0.3$, and ${\lambda }_3=0.4$. The ranks of criteria for all models remain the same for all sets of ${\lambda }_1$, ${\lambda }_2$, and ${\lambda }_3$.

After solving all models, for example, 2, we have found that for all models, the ranks of criteria remain unchanged, but there is a little difference in values of weights for some sets of ${\lambda }_1$, ${\lambda }_2$, and ${\lambda }_3$. Additionally, the CR ratio for MD-1 is 0 whereas for MD-2, it is nearby to zero for all values of ${\lambda }_1$, ${\lambda }_2$, and ${\lambda }_3$. The CR values for models are acceptable.

6. Case Study

We have considered a real case study from [61] having 10 decision-makers about a piping selection method. The study consists of four criteria, i.e., total cost, security, social cost, and environmental cost. The total cost is the best criterion, and environmental cost is the worst criterion. The considered criteria, along with respective descriptions, are presented in

Table 10. Criteria and their description for piping selection.

Title	Description
Total cost ($C_1$)	Direct costs employer pays to the contractor
Security ($C_2$)	Security of less damage to underground pipes
Social costs ($C_3$)	Problems due to noise and traffic limitations
Environmental costs ($C_4$)	Environmental pollution such as air and soil pollution.

. The pairwise comparison data for the case study is presented in

Table 11. Case study.

	${\mathit{x}}_{12}$	${\mathit{x}}_{13}$	${\mathit{x}}_{14}$	${\mathit{x}}_{24}$	${\mathit{x}}_{34}$
$DM1$	{6, 7}	6	7	4	{2, 3}
$DM2$	7	7	{7,9}	{4, 6}	5
$DM3$	7	{6, 7}	9	{8, 9}	8
$DM4$	{2, 3, 5}	4	{6,7}	4	3
$DM5$	3	4	8	{3,4,5}	5
$DM6$	5	{6, 7}	7	5	{2, 3, 5}
$DM7$	9	{4, 6}	9	9	8
$DM8$	9	{7, 9}	9	{6, 8}	2
$DM9$	{8, 9}	9	9	{3, 4}	1
$DM10$	{7, 9}	9	9	{3, 5}	{1, 2, 3}

. The multi-choice response sets were also presented in this table. The responses as multi-choice data are presented in the last row of Table 11. This multi-choice row consists of responses from all 10 decision-makers taken as choices. The objective is to determine the optimal weights of criteria by choosing that set of responses from the multi-choice sets, which will minimize the inconsistency. Two models, MD-1 and MD-2, are formulated and solved similarly to previous examples to determine the optimal weights of criteria.

Now modeling and solving, results are obtained. The criteria weights and their respective ranks obtained for MD-1 and MD-2 are presented in

Table 12. Optimal weights obtained for case study using MD-1, and -2.

Criteria	MD-1	Rank	MD-2	Rank
1	0.754335	1	0.743833	1
2	0.0780347	3	0.0867027	3
3	0.0982659	2	0.103489	2
4	0.0693642	4	0.0659758	4

. In Table 12, we can see that in models 1 and 2, although the ranks of criteria are the same, there is a change in the weight values of criteria.

The individual consistency ratio for all 10 experts and respective consistency ratio along with the CR of the group is presented in

Table 13. Result of case study for MD-1, and -2.

	MD-1				MD-2
Expert	Weight	${\mathit{x}}_{\mathit{BW}}$	CR	CRG	Weight	${\mathit{x}}_{\mathit{BW}}$	CR	CRG
1	0.268786	7	0.018015147		0.282003	7	0.018901005
2	0.476879	7	0.003835488		0.441103	7	0.003547745
3	0.476879	9	0.002735444		0.441103	9	0.002530228
4	0.364162	6	0.004855493		0.570428	6	0.007605707
5	0.520231	8	0.008146794	0.018015	0.483725	8	0.007575112	0.018901
6	0.364162	7	0.007810445		0.31032	7	0.006655657
7	0.546243	9	0.010444417		0.507079	9	0.009695583
8	0.754335	9	0.014423231		0.705007	9	0.013480057
9	0.130058	9	0.003730153		0.187564	9	0.005379465
10	0.33815	9	0.009698375		0.470004	9	0.013480038

. The CRG obtained for model 1 is 0.018015, and for model 2, it is 0.018901. There is a minor difference in obtained consistency ratios. Out of these two models, the consistency ratio is minimum for MD-1 and maximum for MD-2.

7. Limitations, Conclusions and Future Work

This study proposes two different mathematical programming models for solving group MCDM problems. Since there is a large number of variants of mcdm methods, where each method has its own suitability with respect to the assumptions and applications, the proposed methods also have some limitations. The proposed approaches can be applied when the pairwise comparisons collected are multi-choice in nature. The method has some difficulty in its application since no excel or easy-to-apply software is available. The decision-maker needs to code and solve it using optimization solvers, which is difficult for the management or non-technical persons. In the future, this can be rectified by developing easy to calculate excel file. The present work only proposes two models; in the future, these models can be extended to some more models by incorporating more information, such as the confidence level of experts, etc.

The two methods are proposed considering the assumption of multi-choice uncertainty in pairwise comparisons of criteria. The multi-choice uncertainty has been applied to determine the best choice out of multiple choices. It gives a real-life scenario to the decision-making problems. Although there are many other forms of uncertainty, such as rough set, fuzzy, intuitionistic fuzzy, neutrosophic, probabilistic, etc., it focuses on choices instead of the probabilistic or fuzzy nature of parameters. The parameters are considered multi-choice in the pairwise comparison. These parameters are handled by applying the Lagrange interpolating polynomial method. The proposed models are novel in terms of their mathematical structure and group decision-making approach. The models are formulated and further validated by solving numerical examples. It provides a framework for solving MCDM problems and provides weightage to the decision-makers as well. In the results of the numerical example 1 for MD-1 and MD-2, it can be seen that there is no effect on the weights obtained for the criteria due to the weights of the decision-makers, i.e., ${\lambda }_k$, also the ranking of criteria is similar to an earlier study conducted in [61]. For numerical example 2, It can be seen that there is no effect of ${\lambda }_k$ on the weights of criteria using MD-1. For all variations of ${\lambda }_k$, weights are the same. Whereas in the case of MD-2, there is a change in weights due to variation in ${\lambda }_k$. It has been found that for all models, the ranks of criteria remain unchanged, but there is a little difference in values of weights for some sets of ${\lambda }_1$, ${\lambda }_2$, and ${\lambda }_3$. The CR values for all the models of example 1 and 2 has been found acceptable.

This work is an extension of multi-choice MCDM models to group decision-making models. In the future, the proposed models can be extended to models incorporating other kinds of uncertain parameters, such as rough set theory, fuzzy set theory, probabilistic, etc. The confidence level of decision-makers as probability or weights can also be incorporated in future studies. This can be applied to any case study following similar assumptions of proposed models. The study has shown the real-life application of proposed models in piping selection problem. Similar to it, the proposed models can be applied to other problems.