Hybrid Intuitionistic Fuzzy Entropy-SWARA-COPRAS Method for Multi-Criteria Sustainable Biomass Crop Type Selection

Abstract

To select a biomass crop type of the highest sustainability for the purpose of producing biofuel is recognized as a problem of the multi-criteria decision analysis (MCDA) type, as it comprises different conflicting criteria. To effectively address this problem, the present paper introduces a novel integrated approach using the complex proportional assessment (COPRAS) method under the intuitionistic fuzzy sets (IFSs). The proposed approach works based on the IFSs operators as well as an innovative process utilized in evaluating the attributes’ weights. To evaluate these weights, the subjective weights using the step-wise weight assessment ratio analysis (SWARA) model are integrated with the objective weights achieved using an entropy-based approach in order to attain more realistic weights. As MCDA problems inevitably suffer from different degrees of uncertainty, the proposed approach could be of great help to those who are required to make decisions in uncertain settings. The paper took into consideration a sustainable biomass crop selection problem to exemplify the effectiveness of the presented approach in handling real MCDA problems. Moreover, a sensitivity analysis with respect to the diverse values of the attributes is presented in order to assess the stability of the introduced model. This study reveals that the combination of the objective and subjective weights enhances the stability of the introduced approach with diverse attribute weights. Finally, the results of the introduced model are compared to some existing intuitionistic fuzzy information-based methods. The findings of the comparison confirm the efficiency of the present approach in performing the defined tasks under uncertain environments.

1. Introduction

As fossil-based forms of energy are naturally confined and impose many negative impacts on the environment, it is therefore of high importance to explore the best sources for generating energy with high degrees of sustainability. In this regard, ethanol, which is utilized in numerous countries, can be used as a substitution for fossil-based fuel. Though, as different countries have different biomass sources, a key challenge is how to select the most sustainable source for producing ethanol.

In general, sustainability, in an attempt to achieve long-term development, takes into consideration three key dimensions: continuous economic growth, enhanced social welfare, and environmental friendliness [1,2]. To consider the economic aspects of sustainability while selecting the sustainable biomass crop type (SBCT), the present study takes into account the quantity of yield, the rate of biofuel conversion, and the input cost for various farm operations. In addition, there are some other economic factors that cause the biomass crop selection to be an operational and tactical decision, including the length of crop life, availability of technology, and equipment requirements. Moreover, to well consider the environmental aspects of sustainability in selecting SBCT, there is a need for considering different factors such as CO₂ sequestration, soil erosion, biodiversity, and water pollution [3]. To consider the social aspects of sustainability in the selection of SBCT, there is a need to consider the working conditions, unemployment rate, and the society welfare level, amongst other criteria [4,5].

The presence of economic, social, and environmental dimensions makes it is essential to develop effective multi-criteria decision analysis (MCDA) models applicable to the selection of biomass crop types with the highest sustainability. Generally, MCDA problems arise when a set of options/alternatives is available with different (and even conflicting) criteria in a decision-making process. Criteria may be associated with various units, e.g., time, expense, and dimensions, which could make the achievement of convenient data very exclusive, and also make it difficult to compare different criteria with each other. To address such challenges, the literature offers different MCDA methods that attempt to solve such complex problems with different approaches [6,7]. The recent study discusses a collective MCDA technique for the selection of SBCT, which will be found particularly applicable to those cases involving multiple alternatives as well as uncertainty.

Accordingly, conventional decision-making methods typically fail to effectively solve complex problems such as the evaluation of SBCT. In such conditions, the fuzzy set (FS) [8] significantly helps to model uncertain information with higher precision. The FS is characterized by the membership function, whose values lie between 0 and one. The literature consists of different FST-based frameworks for the assessment of the energy and food crops in the domain of bioenergy production from a sustainability perspective [9,10,11,12]. In an MCDA problem, the data made available for an alternative using different attributes might be qualitative, inaccurate, or imperfect in nature. To deal with the inaccurate and vague data and address certain limitations of the FS, Atanassov [13] recommended the “intuitionistic fuzzy set (IFS)” theory. In an IFS, an element is represented by “membership grade (MG)”, “non-membership grade (NG)”, and “indeterminacy grade (IG)”; the grades of all three of these functions are real numbers between 0 and one. Unlike the FS that only considers the MF of an element, the IFS takes into account the MF, NF, and HF of an element; in this new system, the addition of MF and NF is bounded to one. Due to the IFS’s advancement in treating with uncertain data, this paper proposes an MCDA approach based on IFS.

In recent years, Du [14] suggested the subtraction and division operations for IFSs using optimization algorithms. They investigated the derivative and continuity on IFSs. In another study, a novel MCDA with the IFSs and the “combinative distance-based assessment (CODAS)” framework called CODAS (IF-CODAS) was presented by Mishra et al. [15], which was applicable for assessing the low-carbon sustainable suppliers (LCSS). Bas et al. (2021) developed a new intuitionistic fuzzy inference system using time series mappings and applied for time series forecasting. Moreover, Alkan and Kahraman [16] proposed a hybridized decision-making methodology through the combination of CRITIC and the DEVADA approaches on IFSs to select the best place for waste disposal. Zhao et al. [17] combined the cumulative prospect model with the “multi-attributive border approximation area comparison (MABAC)” model on IFSs. Zavadskas et al. [18] planned a novel extension of the simple weighted sum product model under an intuitionistic fuzzy environment. Tripathi et al. [19] designed a decision-making method, called IF-CoCoSo, using the combined compromise solution (CoCoSo) with parametric divergence measures on IFSs to deal with the medical MCDA problem. An MCDA model was discussed by Deb et al. [20], called the “intuitionistic fuzzy improved MARCOS (IF-IMARCOS)”, using the double normalization processes, with the aim of solving the problem of enterprise resource planning system selection. Mishra et al. [21] developed a multi-attribute decision analysis (MADA) tool for evaluating and prioritizing the sustainable urban transportation (SUT) options under the IFSs settings. Mishra et al. [22] presented a hybrid tool with interval-valued intuitionistic fuzzy (IVIF)-distance measure and IVIF-relative closeness coefficient (RCC) approaches to develop associations between the sustainable development barriers (SDBs) to identify the significant SDBs.

The idea of COPRAS was pioneered by Zavadskas et al. [23] to effectively solve MCDA problems. The key outcomes of the COPRAS approach are as follows: (1) its application is simple; (2) it considers the ratios to the option and the ideal solution, at the same time; and (3) it returns the results in a short time. Such advantages have motivated the researchers to use the COPRAS approach in several MCDA problems [24,25,26,27]. As MCDA problems are becoming more and more complex, the original COPRAS is extended from a hesitant linguistic perspective in order to evaluate the medical MCDA problem [28]. In addition, an analytical COPRAS approach was proposed by Mishra et al. [29] for assessment of MCDA problems on HFSs. In Alipour et al.’s [30] study, the evaluation and prioritization of hydrogen components suppliers were assessed with COPRAS and PFSs. To assess the vendors in supply chain management, in the study of Deretarla et al. [31], an MCDA tool was introduced through the integration of two methods, i.e., an analytic hierarchy process (AHP) and COPRAS. Kusakci et al. [32] proposed a hybrid AHP-COPRAS method on “interval type-2 fuzzy sets (IT2FSs)” to evaluate the metropolitan cities from sustainable viewpoints. Mishra et al. [33] gave rise to the notion of using interval-valued hesitant Fermatean fuzzy set (IVHFFS) in the solve and selection process of desalination technology for treating feed water. In a study of Hezam et al. [34], an integrated MCDA model was developed by integrating the “method based on the removal effects of criteria (MEREC)”, SWARA, and COPRAS tools with single-valued neutrosophic information. They applied their methodology for assessing of “bioenergy production technologies (BPTs)” from sustainability viewpoints.

The SWARA model was pioneered by Kersuliene et al. [35] for finding the subjective weights of criteria. Comparing to the AHP, the SWARA is computationally less complex and free from pairwise comparisons. Until now, the SWARA has been applied to several decision-making situations. For instance, Agarwal et al. [36] presented a hybrid tool based on the SWARA and WASPAS methods to explore the obstacles of “humanitarian supply chain management (HSCM)” and evaluate solutions for avoiding these obstacles. Bouraima et al. [37] incorporated the SWARA model with the CoCoSo method and “interval rough numbers (IRN)”, named as IRN-SWARA-CoCoSo framework. They used their framework for assessing alternative railway systems from a sustainable development perspective. Debnath et al. [38] combined the SWARA model with the WASPAS tool for assessing suppliers in healthcare testing facilities. They applied the SWARA tool for deriving the sustainable attribute weights. Saraç et al. [39] used the SWARA-WASPAS method for determining the best vegan cake sample. The literature comprises a number of methods integrated with SWARA [40,41,42,43,44,45,46,47].

In some studies [1,3,9,10,11,12,48,49,50], the decision-making techniques applied to the solution of the SBCT assessment problem have been designed using crisp, fuzzy, and intuitionistic fuzzy numbers. The weakness of prior studies is their limited capability of expressing the fuzzy information. This has made it tricky to evaluate the SBCT assessment problem in conditions associated with uncertainty and complexity. IFS theory offers an effective way to make real-life applications less uncertain and vague; as a result, the present paper is concentrated on the evaluation of the SBCT evaluation problem under an IFS setting. In this context, the contributions of the paper are explained:

• This study conducts a survey through review of the existing literature and holding of interviews with experts with the aim of finding the key attribute for the assessment of the SBCT alternatives.
• It develops an integrated intuitionistic fuzzy entropy-SWARA-COPRAS model to evaluate the SBCT evaluation problem when the available information is uncertain.
• This paper proposes a novel combined weighting approach with entropy and SWARA to evaluate the criteria weights under an intuitionistic fuzzy environment.
• This study empirically studies the SBCT selection problem to evaluate the practicability and efficacy of the developed IF-entropy-SWARA-COPRAS methodology.
• This paper also involves comparative study and sensitivity investigation to assess the effectiveness and strength of the presented approach.

In the rest of the study, Section 2 presents the fundamental concepts and shows a new IF-Entropy-SWARA-COPRAS framework. Afterwards, Section 3 discusses a case study of the SBCT assessment problem and performs the comparison and sensitivity analysis. Lastly, Section 4 concludes the paper and also recommends some situations for further research in the future.

2. Proposed Methodology

2.1. Preliminaries

The present section discusses a number of fundamental ideas about IFSs.

Definition 1

([13]). An IFS M on a fixed set $Y=\left\{y_1, y_2, \dots , y_n\right\}$ is defined as

$$M=\left\{\langle y_i, {\mu }_M(y_i), {\nu }_M(y_i)\rangle : y_i\in Y\right\}$$

(1)

where ${\mu }_M:Y\to [0, 1]$ denote the MG and ${\nu }_M: Y\to [0, 1]$ denote the NG of an element $y_i$ to M in Y, with the condition

$$0\le {\mu }_M\left(y_i\right)\le 1, 0\le {\nu }_M\left(y_i\right)\le 1\mathrm{and}0\le {\mu }_M\left(y_i\right)+ {\nu }_M\left(y_i\right)\le 1, \forall y_i\in Y$$

(2)

The IG of an object $y_i \in Y$ to M is given as

${\pi }_M\left(y_i\right)=1- {\mu }_M\left(y_i\right)-{\nu }_M\left(y_i\right)$ and $0\le {\pi }_M\left(y_i\right)\le 1, \forall y_i \in Y.$

Next, Xu [51] considered the “intuitionistic fuzzy number (IFN)” $\varsigma = \left({\mu }_{\varsigma }, {\nu }_{\varsigma }\right),$ which satisfies ${\mu }_{\varsigma }, {\nu }_{\varsigma } \in \left[0, 1\right]$ and $0 \le {\mu }_{\varsigma }+ {\nu }_{\varsigma }\le 1.$

Definition 2

([51]). Let ${\varsigma }_k=\left({\mu }_k, {\nu }_k\right),$ $k=1, 2, \dots ,n,$ be the collection of IFNs and $\psi ={\left({\psi }_1, {\psi }_2,\dots , {\psi }_n\right)}^T$ be the weight of ${\varsigma }_k, k=1, 2, \dots ,n,$ with ${\displaystyle {\sum }_{k=1}^n{\psi }_k=1}$ and ${\psi }_k\in \left[0, 1\right].$ Then the IFWA and IFWG operators are defined as

$${IFWA}_w\left({\varsigma }_1,{\varsigma }_2,\dots ,{\varsigma }_n\right)=\underset{k=1}{\overset{n}{\oplus }}{\psi }_k {\varsigma }_k=\left[1-{\displaystyle \prod _{k=1}^n{\left(1- {\mu }_k\right)}^{{\psi }_k}}, {\displaystyle \prod _{k=1}^n{\nu }_k{}^{{\psi }_k}}\right]$$

(3)

$${IFWG}_w\left({\varsigma }_1, {\varsigma }_2,\dots , {\varsigma }_n\right)=\underset{k=1}{\overset{n}{\otimes }} {\varsigma }_k^{{\psi }_k}=\left[{\displaystyle \prod _{k=1}^n{\mu }_k{}^{{\psi }_k}}, 1-{\displaystyle \prod _{k=1}^n{\left(1-{\nu }_k\right)}^{{\psi }_k}}\right]$$

(4)

Definition 3

([51]). The score and accuracy values of an IFN $\varsigma = \left({\mu }_{\varsigma }, {\nu }_{\varsigma }\right)$ are given by

$$S\left(\varsigma \right)=\left({\mu }_{\varsigma }- {\nu }_{\varsigma }\right)$$

(5)

$$A\left(\varsigma \right)=\left({\mu }_{\varsigma }+ {\nu }_{\varsigma }\right)$$

(6)

respectively. Here, $S\left(\varsigma \right)\in \left[-1,1\right]$ and $A\left(\varsigma \right)\in \left[0,1\right].$ As $S\left(\varsigma \right)\in \left[-1,1\right],$ then Xu et al. [52] gave an improved score value as follows:

Definition 4

([52]). Consider $\varsigma =\left({\mu }_{\varsigma }, {\nu }_{\varsigma }\right)$ to be an IFN. Then,

$$S^{*}\left(\varsigma \right)=\frac{1}{2}\left(S\left(\varsigma \right)+1\right)$$

(7)

is defined as the normalized score function for IFN$\varsigma .$ Here, $S^{*}\left(\varsigma \right)\in \left[0,1\right].$

2.2. New IF-Entropy-SWARA-COPRAS Model

This section proposes a collective IF-entropy-SWARA-COPRAS method based on the concept of IFSs for aiming to find the objective and subjective weights of attributes and evaluating the ranking order of options. In the following, the computational steps of the introduced model are as follows (see Figure 1):

Step 1: Create an IF-decision-matrix.

Assume options set $P=\left\{P_1, P_2,\dots , P_m\right\}$ and criteria set $L=\left\{L_1, L_2,\dots ,L_n\right\}.$ Form a group of $\ell$ “decision experts (DEs) $E=\left\{E_1, E_2,\dots , E_{\ell }\right\}$ to assess the options over given attributes. Suppose $\Omega ={\left({\theta }_{ij}^{(k)}\right)}_{m\times n}$ to be a “linguistic decision matrix (LDM)” by k^th DE, where ${\theta }_{ij}$ determines the performance of i^th option P_i over j^th attribute L_j.

Step 2: Evaluate the DEs’ weights

Let $\xi =\left({\mu }_k, {\nu }_k\right)$ be the intuitionistic fuzzy significance value of k^th DE. For the purpose of presenting the significance of each expert in the MCDA process, numeric weight of DE is given as

$${\omega }_k=\frac{{\mu }_k\left(2-{\mu }_k-{\nu }_k\right)}{{\displaystyle \sum _{k=1}^{\ell }\left[{\mu }_k\left(2-{\mu }_k-{\nu }_k\right)\right]}}$$

(8)

Step 3: Aggregate the individual decision-matrices.

For the construction of the “aggregated IF-decision-matrix (AIF-DM)”, there is a need to integrate all the individual matrices into one group with respect to DEs’ ideas. To make this process easier, the IFWA operator (Xu, 2007) is applied. Let $ℝ={\left({\xi }_{ij}\right)}_{m\times n}$ be the AIF-DM, where:

$${\xi }_{ij}={IFWA}_{\omega } \left({\theta }_{ij}^{(1)}, {\theta }_{ij}^{(2)},\dots , {\theta }_{ij}^{(\ell )}\right)= \left(1-{\displaystyle \prod _{k= 1}^{\ell }{\left(1- {\mu }_{ij}\right)}^{{\omega }_k}} , {\displaystyle \prod _{k=1}^{\ell }{\left({\nu }_{ij}\right)}^{{\omega }_k}}\right)$$

(9)

Step 4: Compute the weights of criteria.

Assume that $\lambda ={\left({\lambda }_1, {\lambda }_2,\dots , {\lambda }_n\right)}^T$ satisfying ${\displaystyle {\sum }_{j=1}^n{\lambda }_j=1}$ and let ${\lambda }_j\in \left[0, 1\right]$ be a weight of attribute set. Then, to obtain $\lambda ,$ the following procedures are applied:

Case 1: Determine the objective weight ${\lambda }_j^o$ of j^th criterion using the entropy measure given by Mishra and Rani (2019).

$${\lambda }_j^o=\frac{1-{\displaystyle \sum _{i=1}^m\left(E\left({\xi }_{ij}\right)\right)}}{n-{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{i=1}^m\left(E\left({\xi }_{ij}\right)\right)}}}, j=1,2,\dots ,n$$

(10)

where:

$$E\left({\xi }_{ij}\right)=1-\frac{1}{n}{\displaystyle \sum _{i=1}^n[\left({\mu }_{ij}\left(y_i\right)-{\nu }_{ij}\left(y_i\right)\right)I_{\left[{\mu }_{ij}\left(y_i\right)\ge {\mu }_{ij}\left(y_i\right)\right]}+\left({\nu }_{ij}\left(y_i\right)-{\mu }_{ij}\left(y_i\right)\right)I_{\left[{\mu }_{ij}\left(y_i\right)<{\mu }_{ij}\left(y_i\right)\right]}]}$$

(11)

Case 2: Derive the subjective weight ${\lambda }_j^s$ of j^th criterion using the SWARA tool.

The starting point of the SWARA weighting method is to determine the rank of each criterion. The steps of the IF-SWARA tool are as follows:

Step 4.1: From Equation (7), find the IF-score value of each attribute.

Step 4.2: Prioritize the attributes using the DEs’ ratings in the form of IF-score values in descending order.

Step 4.3: Find comparative importance ($s_j$) of attribute from the attribute that is preferred in second place by comparing j^th attribute (j-1)^th attribute.

Step 4.4: Estimate the comparative coefficient with Equation (12) as

$$k_j=\{\begin{array}{l}1, j=1\\ s_j+1, j>1\end{array}$$

(12)

Step 4.5: Calculate the weights $\left(p_j\right)$ of the attribute as

$$p_j=\{\begin{array}{l}1, j=1\\ \frac{p_{j-1}}{k_j}, j>1\end{array}$$

(13)

Step 4.6: Obtain the normalized weight of the attribute as

$${\lambda }_j^s=\frac{p_j}{{\displaystyle {\sum }_{j=1}^n{p}_j}},j=1,2,\dots ,n$$

(14)

Case 3: Compute the integrated weight j^th attribute as follows:

$${\lambda }_j= \tau {\lambda }_j^s+\left(1- \tau \right) {\lambda }_j^o$$

(15)

where $\tau \in \left[0, 1\right]$ is the aggregated coefficient.

Step 5: Sum the attribute value for benefit-type and cost-type attributes.

Assume that $\Delta =\left\{1,2,\dots , l\right\}$ is a set of benefit-type, i.e., the max ratings represent appropriate option. After that, for each of the alternatives, calculate the index value using Equation (16):

$${\Lambda }_i=\underset{j=1}{\overset{\ell }{\oplus }}{\lambda }_j {\xi }_{ij}, i=1,2,\dots ,m$$

(16)

Assume that $\nabla =\left\{l+1,l+2,\dots , n\right\}$ is cost-type attributes, i.e., the smallest value shows the better option. Next, for each of the alternatives, calculate the index value as

$${{\rm T}}_i=\underset{j=\ell +1}{\overset{n}{\oplus }}{\lambda }_j {\xi }_{ij}, i=1,2,\dots ,m$$

(17)

Step 6: Compute the “relative grade (RG)” with Equation (18) as

$${\gamma }_i=S^{*}\left({\Lambda }_i\right)+\frac{\underset{i}{\min }{S}^{*}\left({{\rm T}}_i\right){\displaystyle \sum _{i=1}^m{S}^{*}\left({{\rm T}}_i\right)}}{S^{*}\left({{\rm T}}_i\right){\displaystyle \sum _{i=1}^m\frac{\underset{i}{\min }{S}^{*}\left({{\rm T}}_i\right)}{S^{*}\left({{\rm T}}_i\right)}}}, i=1\left(1\right)m$$

(18)

Step 7: Evaluate the priority order.

The alternatives’ preference relation is shown based on the RD. The alternative that has the highest RD is assigned the best rank and is recognized as the optimum (desired) alternative.

$$P^{*}= \underset{i}{\max } {\gamma }_i, i=1,2,\dots ,m$$

(19)

Step 8: Determination of the utility grade (UG).

The illustrated alternative is compared with the prominent alternative in order to evaluate the UG$\left({\lambda }_i\right)$. The UG ranges between 0 and 100%. Equation (20) is used to compute the UG as

$${\lambda }_i=\frac{{\gamma }_i}{{\gamma }_{\max }}\times 100\%, i=1,2,\dots ,m$$

(20)

Here, ${\gamma }_i$ and ${\gamma }_{\max }$ stand for the importance of the alternatives presented in Equation (19).

Step 9: End.

3. Result and Discussion

3.1. Case Study: SBCT Selection Problem

The current section applies the developed IF-Entropy-SWARA-COPRAS method to a problem of selecting biomass crop. MCDA problems typically involve three key factors of environmental, economic, and social criteria. Under IFSs, 16 criteria are taken into account for assessment. In this study, 10 experts from Kansas were interviewed to determine the most suitable biomass crop type in Kansas. They were chosen strategically since they had proficiency in various aspects of agricultural sustainability and biomass-to-bioenergy production. Now, a team of DEs {E₁, E₂, E₃} has been constructed for implementing the developed model to choose the appropriate SBCT. These DEs are from numerous disciplines, comprising the environmental, agricultural, and sustainability fields, with 20–25 years of experience. A widespread list of attributes is considered in SBCT with significant extant literature distinguishing different dimensions of attributes for SBCT [11,12,49,50,53,54,55,56,57,58,59]. According to the literature survey, the details of 25 crucial attributes are sorted with the DEs opinions, and in conclusion, 16 attributes are taken on 3 pillars (economic, environmental, and social) of the sustainability perspective in selecting an appropriate SBCT for biofuel production, and are depicted in Figure 2. Each DE provided his/her opinion relating each two features of the same section and at the same level of hierarchy.

Table 1. List of the selected criteria and alternatives for SBCT selection.

Aspects	Criteria	Type	SBCT
Economic	Seeding (L1)	Benefit	Corn (P1)
	Biomass yield (L2)	Benefit
	Production and harvesting (L3)	Benefit
	Transportation and storage (L4)	Benefit	Wheat (P2)
	Conversion rate (L5)	Benefit
	Robustness to risks (L6)	Cost
	Equipment and knowledge (L7)	Benefit	Miscanthus (P3)
Environmental	Soil quality impact (L8)	Benefit
	Carbon emissions (L9)	Cost
	Water quality/requirement (L10)	Benefit	Switchgrass (P4)
	Biodiversity and wildlife (L11)	Benefit
	Invasiveness (L12)	Cost
Social	Technological development (L13)	Benefit	Sugarcane (P5)
	Workforce requirement (L14)	Benefit
	Energy safety and welfare (L15)	Benefit
	Food competition (L16)	Benefit

and Figure 2 describe the considered criteria for the selection of an optimal SBCT alternative.

Table 2. The LRs for significance rating DEs.

Linguistic Rating	IFN
Extremely skilled (ES)	(1.00, 0.00, 0.00)
Very very skilled (VVS)	(0.95, 0.03, 0.02)
Very skilled (VS)	(0.80, 0.15, 0.05)
Skilled (S)	(0.60, 0.30, 0.10)
Less skilled (LS)	(0.40, 0.55, 0.05)
Very less skilled (VLS)	(0.30, 0.65, 0.05)
Extremely less skilled (ELS)	(0.10, 0.85, 0.05)

and

Table 3. LRs for evaluating SBCT alternatives and criteria.

Linguistic Rating	IFN
Absolutely good (AG)	(0.95, 0.05)
Very very good (VVG)	(0.85, 0.10)
Very good	(0.80, 0.15)
Good (G)	(0.70, 0.20)
Pretty good (PG)	(0.60, 0.30)
Medium (M)	(0.50, 0.40)
Pretty poor (PP)	(0.40, 0.50)
Poor (P)	(0.30, 0.60)
Very poor (VP)	(0.20, 0.70)
Very very poor (VVP)	(0.10, 0.80)
Absolutely poor (AP)	(0.05, 0.95)

show the LRs and related IFNs to evaluate the performance of DEs’, criteria and SBCT alternatives with respect to considered criteria, respectively. The weights of DEs are calculated using Equation (8) and given in

Table 4. DEs’ weights in the evaluation of SBCT alternatives.

DEs	E1	E2	E3
Linguistic ratings	Very skilled (VS) (0.80, 0.15, 0.05)	Skilled (S) (0.60, 0.30, 0.10)	Less skilled (LS) (0.40, 0.55, 0.05)
Weight	0.4286	0.3367	0.2347

. Then,

Table 5. The LDMs given by Des for evaluating SBCT.

	P1	P2	P3	P4	P5
L1	(PP, P, PG)	(G, PG, M)	(VG, G, PG)	(G, PG, VG)	(M, M, PP)
L2	(PG, PG, M)	(VG, G, G)	(PG, G, PG)	(VG, VG, PG)	(G, PG, PG)
L3	(VP, P, PP)	(M, P, PP)	(G, M, P)	(VG, PG, PP)	(VVP, VP, P)
L4	(PP, M, M)	(M, PP, PP)	(P, M, PP)	(VG, G, PP)	(PP, VP, VVP)
L5	(P, VP, VP)	(PP, P, P)	(VG, M, PP)	(VG, VVG, PG)	(PP, P, P)
L6	(VVG, G, G)	(G, P, M)	(PP, PP, PG)	(M, G, P)	(VVG, G, P)
L7	(G, M, M)	(PG, PP, VP)	(M, P, VP)	(PG, P, M)	(VG, M, P)
L8	(PG, VG, VP)	(M, VP, P)	(VG, PP, P)	(VG, PG, VP)	(VG, P, M)
L9	(G, PP, P)	(P, VVP, PP)	(P, PP, VVP)	(P, PG, P)	(VG, PP, P)
L10	(VVG, M, G)	(G, PP, P)	(G, P, PP)	(VG, G, P)	(VVG, M, PG)
L11	(M, P, VVP)	(M, VVP, P)	(VP, PP, P)	(VG, P, PP)	(G, VP, PG)
L12	(VVG, PG, PP)	(PG, PP, VP)	(PG, VP, PP)	(PG, P, PP)	(VVG, G, M)
L13	(G, PG, P)	(P, PP, VP)	(G, VG, PP)	(G, PP, VG)	(VG, PG, M)
L14	(VG, G, PP)	(PG, M, VP)	(M, P, VVP)	(G, VP, PP)	(VVG, MP, P)
L15	(PG, PP, P)	(PG, P, PP)	(G, PP, P)	(VG, G, P)	(VG, M, P)
L16	(PG, G, P)	(M, M, P)	(PG, P, G)	(M, P, M)	(G, PG, M)

presents each alternative’s performance value regarding each selected criterion, given by a panel of DEs in the form of LDMs. With the use of DEs’ weights and Equation (10), the AIF-DM is created and shown in

Table 6. AIF-DM for SBCT evaluation.

	P1	P2	P3	P4	P5
L1	(0.568, 0.326)	(0.627, 0.270)	(0.730, 0.194)	(0.699, 0.214)	(0.478, 0.422)
L2	(0.578, 0.321)	(0.748, 0.177)	(0.637, 0.262)	(0.765, 0.176)	(0.646, 0.252)
L3	(0.285, 0.614)	(0.416, 0.483)	(0.565, 0.327)	(0.673, 0.251)	(0.185, 0.715)
L4	(0.459, 0.440)	(0.445, 0.454)	(0.397, 0.502)	(0.703, 0.219)	(0.273, 0.625)
L5	(0.244, 0.655)	(0.345, 0.555)	(0.648, 0.277)	(0.786, 0.154)	(0.345, 0.555)
L6	(0.777, 0.149)	(0.550, 0.341)	(0.454, 0.444)	(0.544, 0.348)	(0.728, 0.192)
L7	(0.598, 0.297)	(0.460, 0.435)	(0.375, 0.523)	(0.491, 0.405)	(0.635, 0.289)
L8	(0.627, 0.290)	(0.366, 0.531)	(0.612, 0.311)	(0.650, 0.272)	(0.622, 0.301)
L9	(0.538, 0.352)	(0.222, 0.677)	(0.295, 0.604)	(0.420, 0.475)	(0.612, 0.311)
L10	(0.735, 0.188)	(0.538, 0.352)	(0.530, 0.359)	(0.692, 0.229)	(0.717, 0.206)
L11	(0.357, 0.540)	(0.340, 0.556)	(0.612, 0.311)	(0.605, 0.317)	(0.553, 0.335)
L12	(0.711, 0.211)	(0.460, 0.435)	(0.444, 0.450)	(0.469, 0.427)	(0.749, 0.175)
L13	(0.584, 0.308)	(0.314, 0.585)	(0.692, 0.225)	(0.637, 0.271)	(0.687, 0.238)
L14	(0.703, 0.219)	(0.493, 0.403)	(0.357, 0.540)	(0.509, 0.378)	(0.657, 0.262)
L15	(0.477, 0.419)	(0.469, 0.427)	(0.538, 0.352)	(0.692, 0.229)	(0.623, 0.300)
L16	(0.586, 0.308)	(0.459, 0.440)	(0.549, 0.344)	(0.440, 0.459)	(0.627, 0.270)

.

Based on Equations (10) and (11), and Table 6, the objective weight is estimated as follows:

${\lambda }_j^o=$ (0.0581, 0.0492, 0.0597, 0.0705, 0.0554, 0.0598, 0.0716, 0.0612, 0.0647, 0.0546, 0.0664, 0.0674, 0.0555, 0.0651, 0.0687, 0.0721).

For the calculation of each criterion’s weight using the IF-SWARA, the experts played a significant role. Based on DEs’ opinions, we aggregated the criterion’s performance value by means of an IFWA operator. Then, we computed the score value of each aggregated IFN as shown in

Table 7. IF-score grades of attributes for SBCT evaluation.

Attributes	E1	E2	E3	Aggregated IFN	Score Value
L1	VVP	VP	VP	(0.159, 0.741)	0.209
L2	M	M	PP	(0.478, 0.422)	0.528
L3	VP	VP	VP	(0.200, 0.700)	0.250
L4	VVP	VVP	VVP	(0.100, 0.800)	0.150
L5	P	VP	P	(0.268, 0.632)	0.318
L6	P	PP	P	(0.335, 0.564)	0.386
L7	PP	P	PP	(0.368, 0.532)	0.418
L8	PG	G	PG	(0.637, 0.262)	0.688
L9	VVP	VVP	VP	(0.125, 0.775)	0.175
L10	G	VG	G	(0.738, 0.182)	0.778
L11	VP	VP	P	(0.225, 0.675)	0.275
L12	PG	M	PG	(0.569, 0.331)	0.619
L13	PP	M	PP	(0.436, 0.464)	0.486
L14	M	M	PG	(0.526, 0.374)	0.576
L15	PG	M	G	(0.597, 0.301)	0.648
L16	VG	G	VVG	(0.786, 0.150)	0.818

. Each criterion’s importance was selected by each expert. Then, the expert ranked all the criteria from first to last.

On the basis of SWARA, given in Equations (12)–(14), the criterion of the highest significance was given as higher rank, while that of the lowest significance was shown as lower rank.

Table 8. Weight of attributes by IF-SWARA method for SBCT evaluation.

Attributes	Score Values	sj	kj	pj	${\mathit{\lambda }}_{\mathit{j}}^{\mathit{s}}$
L16	0.818	-	1.000	1.000	0.0869
L10	0.778	0.040	1.040	0.9615	0.0836
L8	0.688	0.090	1.090	0.8821	0.0767
L15	0.648	0.040	1.040	0.8482	0.0737
L12	0.619	0.029	1.029	0.8243	0.0716
L14	0.576	0.043	1.043	0.7903	0.0687
L2	0.528	0.048	1.048	0.7541	0.0655
L13	0.486	0.042	1.042	0.7237	0.0629
L7	0.418	0.068	1.068	0.6776	0.0589
L6	0.386	0.032	1.032	0.6566	0.0571
L5	0.318	0.068	1.068	0.6148	0.0534
L11	0.275	0.043	1.043	0.5895	0.0512
L3	0.250	0.025	1.025	0.5751	0.0501
L1	0.209	0.041	1.041	0.5524	0.0480
L9	0.175	0.034	1.034	0.5342	0.0464
L4	0.150	0.025	1.025	0.5212	0.0453

presents the weight of attributes by IF-SWARA tool. The criteria final weights are given by:

${\lambda }_j^s=$ (0.0480, 0.0655, 0.0501, 0.0453, 0.0534, 0.0571, 0.0589, 0.0767, 0.0464, 0.0836, 0.0512, 0.0716, 0.0629, 0.0687, 0.0737, 0.0869).

Based on (15), the integrated weight of attributes by objective and subjective weights of attributes (with $\tau$ = 0.5) are computed as:

${\lambda }_j=$ (0.053, 0.057, 0.055, 0.058, 0.054, 0.058, 0.065, 0.069, 0.056, 0.069, 0.059, 0.070, 0.059, 0.067, 0.071, 0.080).

The ratings of index $\left({\Lambda }_i\right)$ and $\left({{\rm T}}_i\right)$ for each SBCT option are computed in

Table 9. The UG of option of the IF-Entropy-SWARA-COPRAS model.

Options	${\Lambda }_{\mathit{i}}$	${\mathit{S}}^{*}\left({\Lambda }_{\mathit{i}}\right)$	${{\rm T}}_{\mathit{i}}$	${\mathit{S}}^{*}\left({{\rm T}}_{\mathit{i}}\right)$	${\mathit{\gamma }}_{\mathit{i}}$	${\mathit{\lambda }}_{\mathit{i}}$
P1	(0.482, 0.421)	0.530	(0.195, 0.757)	0.219	0.635	80.342
P2	(0.411, 0.492)	0.459	(0.098, 0.867)	0.116	0.658	83.198
P3	(0.493, 0.413)	0.540	(0.091, 0.877)	0.107	0.756	95.556
P4	(0.573, 0.342)	0.616	(0.113, 0.850)	0.132	0.791	100.000
P5	(0.504, 0.405)	0.550	(0.202, 0.753)	0.224	0.653	82.574

, which are estimated through Equations (16) and (17). Next, the RG $\left({\gamma }_i\right),$ preference order, and the UG $\left({\lambda }_i\right)$ for each option are calculated using Equations (18)–(20) and are depicted in Table 9. Hence, the option Switchgrass (P₄) is the best SBCT alternative and the preference order of SBCT alternatives is $P_4\succ P_3\succ P_2\succ P_5\succ P_1.$

3.2. Comparison and Sensitivity Investigation

This paper also compares the outcomes of the IF-Entropy-SWARA-COPRAS approach and some prior developed approaches. To demonstrate the efficacy and show the unique merits of the IF-Entropy-SWARA-COPRAS model, the IF-TOPSIS and IF-SWARA-TOPSIS methods previously proposed by Mishra [60] and Saeidi et al. [47] were implemented to handle the same MCDA problem.

The computational procedure of IF-TOPSIS [60] for solving the SBCT alternative selection problem is as follows:

Steps 1–4: Similar as the proposed approach.

Step 5: The IF-IS and IF-AIS are determined as ${\varphi }^{+}=$ {(0.730, 0.194), (0.765, 0.176), (0.673, 0.251), (0.703, 0.219), (0.786, 0.154), (0.454, 0.444), (0.635, 0.289), (0.650, 0.272), (0.222, 0.677), (0.735, 0.188), (0.612, 0.311), (0.444, 0.450), (0.692, 0.225), (0.703, 0.219), (0.692, 0.229), (0.627, 0.270)}

${\varphi }^{-}=$ {(0.478, 0.422), (0.578, 0.321), (0.185, 0.715), (0.273, 0.625), (0.244, 0.655), (0.777, 0.149), (0.375, 0.523), (0.366, 0.531), (0.612, 0.311), (0.530, 0.359), (0.340, 0.556), (0.749, 0.175), (0.314, 0.585), (0.357, 0.540), (0.469, 0.427), (0.440, 0.459)}

Step 6: The values of weighted similarity of each option $P_i\left(i=1\left(1\right)m\right)$ and the IF-IS ${\varphi }^{+}$ are $Sim\left(P_1, {\varphi }^{+}\right) = 0.745,$ $Sim\left(P_2, {\varphi }^{+}\right) = 0.727,$ $Sim\left(P_3, {\varphi }^{+}\right) = 0.836,$ $Sim\left(P_4, {\varphi }^{+}\right) = 0.906$ and $Sim\left(P_5, {\varphi }^{+}\right) = 0.764.$ The values of weighted similarity of each option $P_i\left(i=1\left(1\right)m\right)$ and the IF-AIS ${\varphi }^{-}$ are $Sim\left(P_1, {\varphi }^{+}\right) = 0.812,$ $Sim\left(P_2, {\varphi }^{+}\right) = 0.827,$ $Sim\left(P_3, {\varphi }^{+}\right) = 0.721,$ $Sim\left(P_4, {\varphi }^{+}\right) = 0.654$ and $Sim\left(P_5, {\varphi }^{+}\right) = 0.797.$

Step 7: The “relative closeness coefficient (RCC)” of each SBC option is $ℂ\left(P_1\right) = 0.478,$ $ℂ\left(P_2\right) = 0.468,$ $ℂ\left(P_3\right) = 0.537,$ $ℂ\left(P_4\right) = 0.581$ and $ℂ\left(P_5\right) = 0.489.$

By means of the RCC, the option Switchgrass (P₄) is the most suitable SBCT option, and the preference order of sustainable biomass crop type alternatives is $P_4\succ P_3\succ P_5\succ P_1\succ P_2.$ The preference order of sustainable biomass crop type alternatives introduced by the IF-Entropy-SWARA-COPRAS approach show a high consistency with the IF-TOPSIS model. From both methods, we obtained the same optimal sustainable biomass crop type alternative (P₄), whereas the preference orders of the proposed method and the IF-TOPSIS method are slightly different. However, our proposed method provides a more reliable and accurate decision due to the involvement of DEs’ weights and integrated objective-subjective weights of criteria. The comparison outcomes of the presented method and the IF-TOPSIS method are depicted in Figure 3.

Furthermore, a sensitivity investigation is implemented, which is associated with various values of parameter. The different values of τ were taken into consideration for this analysis. The variation of parameter ratings helps to consider the sensitivity of the developed model.

Table 10. Sensitivity analysis of weights of the criteria with different values of parameter.

$\mathit{\tau }$	0.0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1.0
L1	0.058	0.057	0.056	0.055	0.054	0.053	0.052	0.051	0.050	0.049	0.048
L2	0.049	0.051	0.052	0.054	0.056	0.057	0.059	0.061	0.062	0.064	0.066
L3	0.060	0.059	0.058	0.057	0.056	0.055	0.054	0.053	0.052	0.051	0.050
L4	0.070	0.068	0.065	0.063	0.060	0.058	0.055	0.053	0.050	0.048	0.045
L5	0.055	0.055	0.055	0.055	0.055	0.054	0.054	0.054	0.054	0.054	0.053
L6	0.060	0.060	0.059	0.059	0.059	0.058	0.058	0.058	0.058	0.057	0.057
L7	0.072	0.070	0.069	0.068	0.067	0.065	0.064	0.063	0.061	0.060	0.059
L8	0.061	0.063	0.064	0.066	0.067	0.069	0.071	0.072	0.074	0.075	0.077
L9	0.065	0.063	0.061	0.059	0.057	0.056	0.054	0.052	0.050	0.048	0.046
L10	0.055	0.058	0.060	0.063	0.066	0.069	0.072	0.075	0.078	0.081	0.084
L11	0.066	0.065	0.063	0.062	0.060	0.059	0.057	0.056	0.054	0.053	0.051
L12	0.067	0.068	0.068	0.069	0.069	0.070	0.070	0.070	0.071	0.071	0.072
L13	0.056	0.056	0.057	0.058	0.058	0.059	0.060	0.061	0.061	0.062	0.063
L14	0.065	0.065	0.066	0.066	0.067	0.067	0.067	0.068	0.068	0.068	0.069
L15	0.069	0.069	0.070	0.070	0.071	0.071	0.072	0.072	0.073	0.073	0.074
L16	0.072	0.074	0.075	0.077	0.078	0.080	0.081	0.082	0.084	0.085	0.087

presents 11 different criteria weight sets. Through this procedure, an elegant choice of attribute weights was formed for analyzing of the introduced model’s sensitivity to the attributes’ weight variations. Figure 4 displays the ranking orders with the parameter values. This figure shows that in all the sets, the SBCT option switchgrass has the highest ranking, P₃ ranks the second, P₅ ranks the third, P₁ ranks the fourth, whereas P₂ is ranked the worst. Therefore, the findings showed the high stability of the developed method with the various parameter values. As a result, the utilization of various parameter values improved the steadiness of the developed method.

The prioritizations of SBCTs discussed by the above-mentioned models are portrayed in Figure 3, which exemplifies that the final preferences trend of the worst possible options in the discussed five models are similar and mentioned in

Table 11. Comparative discussion of diverse models with several parameters.

Parameters	Mishra [60]	Buyukozkan & Gocer [61]	Cobuloglu and Buyuktahtakin [12]	Mishra et al. [49]	Proposed Method
Model	TOPSIS	COPRAS	AHP-based LLSM	ARAS	COPRAS
Evaluation settings	IFSs	PFSs	FSs	PFSs	IFSs
AOs	Averaging	Averaging	Averaging	Averaging	Averaging
Prioritization scheme	Compromise degree	Compromise degree	Outranking degree	Utility theory	Utility theory
Criteria weights	Objective weight assessment	Subjective weight by AHP	Subjective weight by AHP	Objective weight assessment	Integrated weight by objective and subjective weights
MCDA	Group	Group	Group	Group	Group
Does the preference tool consider nature of attributes	Yes	Yes	Yes	Yes	Yes
DE’s weight	Calculated	Calculated	Calculated	Calculated (Using scoring tool)	Calculated (Using scoring tool)
Types of normalization	Vector	Vector	Linear	Linear	Linear, vector
Best SBCT option	P4	P4	P4	P4	P4

. Henceforward, The SBCT option (Switchgrass) is the suitable SBCT option by the IF-TOPSIS tool. Similarly, we have compared the IF-Entropy-SWARA-COPRAS tool with the Cobuloglu & Buyuktahtakın [12], Mishra et al. [49], and Buyukozkan & Gocer [61] models, as discussed in Table 11. By way of comparison, the IF-Entropy-SWARA-COPRAS tool is more inclusive in treating the SBCT problem under IFSs. In the assessment with numerous extant tools, the IF-Entropy-SWARA-COPRAS model has the following benefits:

• In the IF-Entropy-SWARA-COPRAS model, both criteria types, i.e., benefit and cost, were taken in the evaluation. Taking these attributes with complex proportions provides more accurate data than only treating with the benefit or cost criteria. Hence, it improves the usefulness of the initial information and the correctness of the outcomes.
• In the IF-TOPSIS model [47,60], it is required to find the similarity/distance between each alternative and the IF-IS, which is a difficult process that diminishes the accuracy of the results, whereas the PF-COPRAS [61] can be obtained from the association of complex relations between options of the PF-DM with the PFWA operator. Here, the integrated IF-Entropy-SWARA-COPRAS model uses an easy and instinctive procedure that produces suitable, sensible, and comparatively exact results in assessing and choosing various SBCT options based on the performance tools. The IF-Entropy-SWARA-COPRAS model offers a notion of UGs to prefer the best options.
• In the Cobuloglu & Buyuktahtakın [12] and Buyukozkan & Gocer [61] models, the AHP tool is implemented for finding attributes’ weights, which is an intricate process [62]. In the Mishra et al. [49] model, the similarity-based weighting tool is utilized to compute the objective weights of attributes. In the presented IF-Entropy-SWARA-COPRAS model, an integrated weighting model—by joining the entropy-based tool for objective weight and the SWARA model for subjective weight—is discussed to obtain the attribute weights.

4. Conclusions

The present study focused on providing an effective solution to the problem of choosing the suitable SBCT for biofuel production. Different economic, environmental, and social factors are considered when evaluating the sustainability of a biomass crop type. The current paper proposed a novel integrated approach using COPRAS, IFNs, and an integrated weighting tool for attributes in addressing MCDA problems. The weights of criteria in the proposed tool are computed using a combination of the subjective weight computed using IF-SWARA and the objective weight attained using an entropy-based tool. A case study of SBCT selection problems has been implemented. To assess the efficiency and applicability of the developed model in solving MCDA problems and to certify the outcomes, the performance of the proposed approach was compared to that of some methods previously proposed in the literature. This study also involved a sensitivity investigation with various attributes to evaluate the proposed approach stability. The findings confirmed the stability of the presented IF-Entropy-SWARA-COPRAS model as well as its reliability with other methods. The presented IF-Entropy-SWARA-COPRAS model has two key benefits: (1) it simply performs the required computations in an intuitionistic fuzzy environment; and (2) it uses a special process to achieve more realistic weights of attributes, hence improving the approach’s steadiness. The proposed approach is applicable to any problem with the collective process of MCDA problems, which uses IFNs. Although the present study was concentrated on a problem with subjective assessment, the proposed approach is also valid in general situations.

Furthermore, the presented IF-Entropy-SWARA-COPRAS tool can be appropriately implemented for other MCDA problems such as hospital cite selection, zero-carbon measures assessment for sustainable transport, medical waste disposal technique selection, and others. We will generalize this presented Entropy-SWARA-COPRAS model to IVq-ROFSs, IVFFSs, and picture fuzzy sets. Also, we will utilize several other approaches, such as GLDS, MAIRCA, and WISP models to deal with the SBCT selection problem in diverse uncertain settings.