Interval-valued intuitionistic fuzzy AROMAN method and its application in sustainable wastewater treatment technology selection

Abstract

In the theory of interval-valued intuitionistic fuzzy set (IVIFS), the rating/grade of an element is the subset of the closed interval [0, 1], therefore the IVIFS doctrine is more useful for the decision expert to present their judgments in terms of intervals rather than the crisp values. The present work develops an integrated decision-making methodology for evaluating sustainable wastewater treatment technologies within the context of IVIFS. The proposed decision-making framework is divided into three stages. First, some Yager weighted aggregation operators and their axioms are developed to combine the interval-valued intuitionistic fuzzy information. These operators can offer us a flexible way to solve the realistic multi-criteria decision-making problems under IVIFS context. Second, an extension of Symmetry Point of Criterion model is introduced to determine the criteria weights under IVIFS environment. Third, an integrated alternative ranking order model accounting for two-step normalization (AROMAN) approach is proposed from IVIF information perspective. Next, the practicability and efficacy of the developed model is proven by implementing it on a case study of sustainable wastewater treatment technologies evaluation problem with multiple criteria and decision experts. Finally, comparative analysis is discussed to illustrate the consistency and robustness of the obtained outcomes.

Keywords

Interval-valued intuitionistic fuzzy set sustainability waste water treatment Yager aggregation operators score function AROMAN

1 Introduction

Wastewater treatment (WWT) is of growing interest within the water–energy–food nexus because it permits the recovery of all three resources-water, energy and nutrients for crop production [1]. It is one of the most imperative necessities of the current scenario that can provide wider environmental and societal benefits including improved ambient water quality, protect wildlife, reduced greenhouse gas emissions and so on [2]. Treated wastewater also brings economic benefits through reuse in different sectors like agriculture, groundwater recharge, vehicle washing, golf course irrigation, toilet flushes, cooling purposes in thermal power plants and building construction activities. WWT solutions can recuperate valued resources from wastewater namely biodiesel, electricity, recycled water and nutrients and serve as the components of fertilizer [3].

Choosing the best suitable wastewater treatment technology (WWTT) is a difficult issue due to involvement of multiple sustainability aspects of criteria such as economic, environment, social, technical, cultural and political aspects of the society wherein it is implemented. Thus, the multi-attribute decision-making (MCDM) techniques are more appropriate to systematically deal such type of decision-making problem [3 –7]. Due to process of solving multi-criteria WWTT selection problem, ambiguity arises in some form due to the subjectivity of human mind and imprecise information. The notion of fuzzy set (FS) and their extensions have widely been used to handle uncertain decision-making problems [8 –12]. Several generalizations of FS theory have been introduced and applied for several purposes [13 –18].

The theory of FS is characterized by the membership degree, which takes all the values between 0 and 1, and the non-membership degree is defined as one minus the membership degree. However, it may not always be true that the degree of non-membership of an element is equal to one minus the membership degree because there may be some hesitation degree. To overcome the limitation of FS, Atanassov [19] introduced the concept of intuitionistic fuzzy set (IFS), which is characterized by the degrees of membership, non-membership and hesitancy. In IFS, the degree of hesitancy is defined as one minus the sum of membership and non-membership degrees. The theory of IFS is more powerful as compared to FS as it deals with membership, non-membership, and hesitant degrees. After the pioneering work of Atanassov [19], various significant results have been achieved based on intuitionistic fuzzy set (IFS) theory [20 –22]. In the IFS theory, the degrees of membership and non-membership are exact numbers, which is hard for the experts to define their exact value in several decision-making problems. To conquer this issue, Atanassov and Gargov [23] gave the idea of interval-valued intuitionistic fuzzy set (IVIFS), which deals with uncertainty in practical decision making problems. Its basic feature is that both membership and non-membership functions of an element to a given set are considered and taken as interval values rather than exact numbers.

As an extended version of IFS, the IVIFS theory provides a more effective and reasonable way to cope with imprecise and uncertain information. Because of its higher flexibility in dealing with uncertain data, the IVIFS doctrine has been broadly explored in the literature [24 –26]. Due to the broad range of information coverage, the theory of IVIFS is used in our proposed work. For instance, Rani and Jain [27] proposed a hybrid interval-valued intuitionistic fuzzy information-based decision support system using entropy and divergence measures and presented its application in MCDM problems. Kumar and Chen [28] developed a hybrid MCDM approach with the combination of score function of connection numbers and the set pair analysis theory in the context of IVIFS. Rathnasabapathy and Palanisami [36] proposed a cosine similarity measure to quantify the degree of similarity between IVIFSs and discussed its application. Guo and Xu [30] presented a novel mathematical framework for knowledge measure from FSs through IVIFSs. Al-Barakati and Rani [31] proposed a novel IVIF-DNMA method for choosing the healthcare waste treatment methods from IVIF information perspective. Rani et al. [32] identified the most suitable sustainable recycling partner using a hybrid IVIF-MULTIMOORA approach. Mishra et al. [22] proposed an interval-valued intuitionistic fuzzy Hellinger distance measure-based multi-attribute ideal-real comparative analysis (MAIRCA) for solving WWTTs assessment problem from multiple criteria and sustainability perspectives.

To deal with the MCDM problems, several novel methods have been developed under the context of IVIFS [32, 33] based on one type of normalization. However, using only one normalization process could lead to inaccurate decision. Recently, Bošković et al. [34] developed the AROMAN model to solve the decision-making problem with crisp number. In this method, two types of normalization are conducted, and the obtained normalized values are aggregated into the averaged normalized decision-making matrix. The main difference between the AROMAN model and other existing MCDM models is in the way the data are normalized and how the final ranks of alternatives are determined. Nikolic et al. [35] presented an interval type-2 fuzzy generalization of AROMAN method and applied to improve the sustainability of the postal network in rural areas. There is no research which develops the AROMAN method from IVIF information perspective. Until now, there is no study which extends the classical AROMAN method into interval-valued intuitionistic fuzzy environment.

Next, we present research gaps and motivations behind this study, which as

Previously developed aggregation operators on IVIFSs [36 –39] have restrictions in several applications. However, Yager AOs make the decision outcomes more exact and precise than basic averaging and geometric AOs due to involvement of a parameter. Some authors [40 –43] have generalized the Yager’s class of operations from different fuzzy perspectives but these studies are unable to combine the interval-valued intuitionistic fuzzy data.

The AROMAN method has developed in the context of crisp set and interval type-2 FS [34, 35]. The methods presented by [34, 35] ignore the significance of criteria and experts’ weights. However, the determination of criteria and experts’ weights plays an important role in making accurate decision during the process of decision-making. In addition, their studies are not able to deal with interval-valued intuitionistic fuzzy information.

Based on the literature, we found that few authors [3–5 , 45] have developed some decision-making approaches to deal with WWTTs assessment problem. As a generalized version of IFS, the concept of IVIFS has the advantage that both membership and non-membership degrees are interval values and can used to characterize the uncertain information more flexibly due to its constraint condition. Unfortunately, existing decision support models are unable to deal with the interval-valued membership and non-membership degrees.

Inspired by the notions of AROMAN and IVIFS, the present study proposes a generalized version of AROMAN method under IVIFS environment. The key contributions of the study are as

To combine the individual decision information, some IVIF Yager weighted AOs are presented with their elegant properties.

To solve the MCDM problems, an integrated AROMAN method is developed based on Yager AOs and Symmetry point of criterion (SPC) model with IVIF information.

To derive the criteria weights, a novel generalization of SPC model is proposed under IVIFS environment.

The proposed method is applied on a case study of WWTTs selection problem to prove the applicability and feasibility of the proposed method.

Other sections are presented in the following way. Section 2 presents the extant studies related to this work. Section 3 presents the fundamental concepts. Section 4 proposes various weighted averaging and geometric AOs using Yager’s class and IVIFSs with their characteristics. Section 5 develops a hybrid IVIF-SPC-AROMAN method for solving MCDM problems from IVIF information perspective. Section 6 presents the implementation of introduced model on a sustainable WWTTs assessment. This section further discusses the comparative analysis. Section 7 shows the concluding remark and recommends the future research directions.

2 Related works

Many authors have presented their works on the development new technologies and also reviewed the current systems for wastewater treatment. For instance, Choudhury et al. [46] highlighted the working strategies of Microbial fuel cell (MFC) technology for WWT and reuse of wastewater for power generation. In a study, Tarpani and Azapagic [47] assessed the life cycle environmental impacts of advanced WWT methods for removal of pharmaceuticals and personal care products. Arroyo and Molinos-Senante [48] presented a choosing-by-advantages approach to evaluate seven WWT alternatives. Based on different aspects of sustainability, they selected the most suitable WWT alternative. A systematic review has presented to explain the advantages and disadvantages of different membrane technologies for water treatment [49]. Based on the Web of Science core collection database, Chen et al. [50] provided a comprehensive review on wastewater treatment and emerging contaminants research from 1998 to 2021. Munoz-Cupa et al. [51] highlighted the benefits and technical barriers of current MFC systems for WWT. Moreover, they have presented the effects of different reaction conditions on chemical oxygen demand removal and electricity generation from MFCs. Saravanan et al. [52] reviewed several WWT technologies and presented their remarkable power for toxic pollutants removal from wastewater. In addition, they discussed the difficulties related to commercial development of WWT technologies and suggested the future research directions. Saravanan et al. [2] presented sustainable strategy on MFC technology to treat the wastewater for the green energy production. For this purpose, the authors have reviewed diverse MFC technologies and their core performance in the direction of waste management and energy conversion. Zhang et al. [53] stated the current trends of WWT plants in China. Their study provided some useful implications to the authorities and policy makers by analyzing the industries’ current status in the WWT process. As per the existing studies, several technologies, ranging from conventional to advanced treatment processes, are available to treat the wastewater. Considering the preliminary, primary, secondary and tertiary treatment, Ullah et al. [5] established a MCDM framework for the selection of WWTT. Srivastava and Singh [4] established a decision support system to choose the most suitable WWTT from multiple criteria perspective in which the criteria weights are determined through Full consistency method. Salamirad et al. [4] suggested an integrated MCDM model using BWM and TOPSIS models. In that study, the authors have evaluated seven WWTTs from different aspects of sustainability. With the use of analytic hierarchy process (AHP), Ćetković et al. [44] reviewed the current situation and problems related to WWTTs and evaluated the optimal variant of WWTTs. Pennelilini et al. [45] presented a novel utility interval-based evidential reasoning approach to evaluate and prioritize the WWTT alternatives for agricultural reuse.

Uncertainty is an inherent feature of information. Though, the ordinary sets are not good enough to express uncertain information of real-life situations. To deal with uncertainty of WWTTs evaluation problem, Dursun [10] suggested a hybridized MCDM method by incorporating the DEMATEL and TOPSIS methods with 2-tuple fuzzy linguistic set and applied to evaluate the WWTT candidates. Attri et al. [54] presented the combined use of three MCDM methods such as the MOORA, SWARA and TOPSIS under fuzzy environment. In addition, they applied fuzzy SWARA model to evaluate the significance values of the considered criteria, while the fuzzy MOORA and TOPSIS approaches have used to determine the rank of six WWTTs. A hybrid decision support system has been developed using the PIPRECIA and TODIM approaches with linear diophantine fuzzy information to handle the multi-criteria WWTT selection problem [11]. Kamal et al. [55] proposed a single-valued neutrosophic information-based VIsekriterijumska optimizacija i KOmpromisno Resenje (VIKOR) method to deal with the multi-criteria WWTT selection problem from indeterminate, inconsistent and uncertainty perspectives. Till now, no authors have proposed a hybrid interval-valued intuitionistic fuzzy AROMAN method for dealing with sustainable WWTTs assessment problem.

To avoid the shortcomings of existing studies, the present work proposes an interval-valued intuitionistic fuzzy MCDM approach to evaluate and rank the sustainable WWTTs from multiple criteria perspective. The proposed approach does not only evaluate the considered alternatives but also aggregate the individual decision experts’ opinions through Yager’s aggregation operators. In addition, the proposed approach computes the decision experts and criteria weights to avoid the biasness in the decision-making process. The proposed approach can help the policy makers to get more confident about the ranking of sustainable WWTTs under uncertain environment.

3 Preliminaries

In this section, we discuss the fundamental definitions related to the present work.

Definition 3.1 [23]. An IVIFS G on a fixed set V = { ι₁, ι₂, . . . , ι_t } is defined as $G = {〈 v_{i}, ([μ_{G}^{-} (ι_{i}), μ_{G}^{+} (ι_{i})], [v_{G}^{-} (ι_{i}), v_{G}^{+} (ι_{i})]) 〉 : ι_{i} \in V},$ where $0 \leq μ_{G}^{-} (ι_{i}) μ_{G}^{+} (ι_{i}) 1$ , $0 \leq ν_{G}^{-} (ι_{i}) ν_{G}^{+} (ι_{i}) \leq 1$ and $0 \leq \leq μ_{G}^{+} (ι_{i}) + ν_{G}^{+} (ι_{i}) \leq 1$ . Here, $μ_{G} (ι_{i}) = [μ_{G}^{-} (ι_{i}), μ_{G}^{+} (ι_{i})]$ and $ν_{G} (ι_{i}) = [ν_{G}^{-} (ι_{i}), ν_{G}^{+} (ι_{i})]$ denote the degrees of membership and non-membership of an element ι_i in G, respectively, where sup(μ_G (ι_i)) + sup(ν_G (ι_i)) ≤ 1.

The function $π_{G} (ι_{i}) = [π_{G}^{-} (ι_{i}), π_{G}^{+} (ι_{i})]$ characterizes the hesitancy degree of an element ι_i to G, where $π_{G}^{-} (ι_{i}) = 1 - μ_{G}^{+} (ι_{i}) - ν_{G}^{+} (ι_{i})$ and $π_{G}^{+} (ι_{i}) = 1 - μ_{G}^{-} (ι_{i}) - ν_{G}^{-} (ι_{i})$ . For the simplicity, the term $([μ_{G}^{-} (ι_{i}), μ_{G}^{+} (ι_{i})], [ν_{G}^{-} (ι_{i}), ν_{G}^{+} (ι_{i})])$ is defined as “interval-valued intuitionistic fuzzy number/value (IVIFN/IVIFV)” and symbolized as $ω = ([μ_{ω}^{-}, μ_{ω}^{+}], [ν_{ω}^{-}, ν_{ω}^{+}])$ which fulfills $0 (μ_{ω}^{+}) + (ν_{ω}^{+}) 1$ .

Further, Xu [56] presented some basic operations on any two IVIFNs $ω_{1} = ([μ_{1}^{-}, μ_{1}^{+}], [ν_{1}^{-}, ν_{1}^{+}])$ and $ω_{2} = ([μ_{2}^{-}, μ_{2}^{+}], [ν_{2}^{-}, ν_{2}^{+}])$ , given as follows:

ω₁ ⊆ ω₂ iff $μ_{1}^{-} (ι_{i}) \leq μ_{2}^{-} (ι_{i})$ , $μ_{1}^{+} (ι_{i}) \leq μ_{2}^{+} (ι_{i})$ , $ν_{1}^{-} (ι_{i}) \leq ν_{2}^{-} (ι_{i})$ and $ν_{1}^{+} (ι_{i}) ν_{2}^{+} (ι_{i}), \forall ι_{i} \in V$ ,

ω₁ = ω₂ iff ω₁ ⊆ ω₂ and ω₁ ⊇ ω₂,

$ω_{1}^{c} = {(ι_{i}, [ν_{1}^{-} (ι_{i}), ν_{1}^{+} (ι_{i})], [μ_{1}^{-} (ι_{i}), μ_{1}^{+} (ι_{i})]) | ι_{i} \in V}$ ,

$ω_{1} \cup ω_{2} = {(\begin{matrix} ι_{i}, [μ_{1}^{-} (ι_{i}) \lor μ_{2}^{-} (ι_{i}), μ_{1}^{+} (ι_{i}) \lor μ_{2}^{+} (ι_{i})], \\ [ν_{1}^{-} (ι_{i}) \land ν_{2}^{-} (ι_{i}), ν_{1}^{+} (ι_{i}) \land ν_{2}^{+} (ι_{i})] \end{matrix}) | ι_{i} \in V}$ ,

$ω_{1} \cap ω_{2} = {(\begin{matrix} ι_{i}, [μ_{1}^{-} (ι_{i}) \land μ_{2}^{-} (ι_{i}), μ_{1}^{+} (ι_{i}) \land μ_{2}^{+} (ι_{i})], \\ [ν_{1}^{-} (ι_{i}) \lor ν_{2}^{-} (ι_{i}), ν_{1}^{+} (ι_{i}) \lor ν_{2}^{+} (ι_{i})] \end{matrix}) | ι_{i} \in V}$ .

To discriminate the different IVIFNs, Xu et al. [39] defined the score and accuracy functions. For any IVIFN $ω = ([μ_{ω}^{-}, μ_{ω}^{+}], [ν_{ω}^{-}, ν_{ω}^{+}])$ , Xu et al. [39] discussed the score and accuracy ratings, given by Equations (1) and (2), respectively. $𝕊 (ω) = \frac{1}{2} (\frac{1}{2} (μ_{ω}^{-} + μ_{ω}^{+} - ν_{ω}^{-} - ν_{ω}^{+}) + 1),$ (1) $ℍ (ω) = \frac{1}{2} (μ_{ω}^{-} + μ_{ω}^{+} + ν_{ω}^{-} + ν_{ω}^{+}) .$ (2)

Definition 3.2. For IVIFNs ω ={ ω₁, . . . , ω_t }, where $ω_{k} = ([μ_{k}^{-}, μ_{k}^{+}], [ν_{k}^{-}, ν_{k}^{+}]), k = 1, 2, \dots, t$ , Xu [56] defined the IVIFWA and IVIFWG operators as $\begin{matrix} \oplus_{k = 1}^{t} σ_{k} ω_{k} = ([1 - \prod_{k = 1}^{t} {(1 - μ_{k}^{-})}^{σ_{k}}, 1 - \prod_{k = 1}^{t} {(1 - μ_{k}^{+})}^{σ_{k}}], . \\ [\prod_{k = 1}^{t} {(ν_{k}^{-})}^{σ_{k}}, \prod_{k = 1}^{t} {(ν_{k}^{+})}^{σ_{k}}]), \end{matrix}$ (3) $\begin{matrix} \otimes_{k = 1}^{t} σ_{k} ω_{k} = ([\prod_{k = 1}^{t} {(μ_{k}^{-})}^{σ_{k}}, \prod_{k = 1}^{t} {(μ_{k}^{+})}^{σ_{k}}], \\ [1 - \prod_{k = 1}^{t} {(1 - ν_{k}^{-})}^{σ_{k}}, 1 - \prod_{k = 1}^{t} {(1 - ν_{k}^{+})}^{σ_{k}}]) . \end{matrix}$ (4) where σ_k denotes the weight of ω_k such that σ_k ∈ [0, 1] and $\sum_{k = 1}^{t} σ_{k} = 1$ .

4 Proposed IVIF-aggregation operators

The present section introduces some IVIF AOs with the Yager t-norm and t-conorm operations [57] and discusses their properties.

4.1 Basic concepts related to IVIF-Yager aggregation operators

Definition 4.1. For two real numbers α and β, the Yager t-norm (⊗) is defined as $\begin{matrix} T (α, β) = α \otimes β = max \\ (1 - {({(1 - α)}^{p} + {(1 - β)}^{p})}^{1 / p}, 0), \end{matrix}$ (5) where p > 0.

This function is increasing and the following particular cases are obtained as

If p→ ∞, then T (α, β) → min(α, β),

If p = 1, then T (α, β) = max(0, α + β - 1) (Lukasiewicz’ t-norm).

Any two real numbers α and β, Yager s-norm (or t-conorm) (⊗) is defined as $S (α, β) = α \oplus β = min ({(α^{p} + β^{p})}^{1 / p}, 1),$ (6)

This function is decreasing and the following particular cases are obtained

If p→ ∞, then S (α, β) → max(α, β),

If p = 1, then S (α, β) = min(1, α + β) (Lukasiewicz’ t-conorm).

According to Definition 4.1, we present following definition:

Definition 4.2. For any two IVIFNs $ℑ_{1} = ([μ_{1}^{-}, μ_{1}^{+}], [ν_{1}^{-}, ν_{1}^{+}])$ and $ℑ_{2} = ([μ_{2}^{-}, μ_{2}^{+}],$ $[ν_{2}^{-}, ν_{2}^{+}])$ and p > 0, then Yager’s operations on IVIFNs are defined as

$ℑ_{1} \oplus ℑ_{2} = (\begin{matrix} [\begin{matrix} min ({({(μ_{1}^{-})}^{p} + {(μ_{2}^{-})}^{p})}^{1 / p}, 1), \\ min ({({(μ_{1}^{+})}^{p} + {(μ_{2}^{+})}^{p})}^{1 / p}, 1) \end{matrix}], \\ [\begin{matrix} max (1 - {({(1 - ν_{1}^{-})}^{p} + {(1 - ν_{2}^{-})}^{p})}^{1 / p}, 0), \\ max (1 - {({(1 - ν_{1}^{+})}^{p} + {(1 - ν_{2}^{+})}^{p})}^{1 / p}, 0) \end{matrix}] \end{matrix})$ ,

$ℑ_{1} \otimes ℑ_{2} = (\begin{matrix} [\begin{matrix} max (1 - {({(1 - μ_{1}^{-})}^{p} + {(1 - μ_{2}^{-})}^{p})}^{1 / p}, 0), \\ max (1 - {({(1 - μ_{1}^{+})}^{p} + {(1 - μ_{2}^{+})}^{p})}^{1 / p}, 0) \end{matrix}], \\ [\begin{matrix} min ({({(ν_{1}^{-})}^{p} + {(ν_{2}^{-})}^{p})}^{1 / p}, 1), \\ min ({({(ν_{1}^{+})}^{p} + {(ν_{2}^{+})}^{p})}^{1 / p}, 1) \end{matrix}] \end{matrix})$ ,

$ρ ℑ_{1} = (\begin{matrix} [\begin{matrix} min ({(ρ {(μ_{1}^{-})}^{p})}^{1 / p}, 1), \\ min ({(ρ {(μ_{1}^{+})}^{p})}^{1 / p}, 1) \end{matrix}], \\ [\begin{matrix} max ((1 - {(ρ {(1 - ν_{1}^{-})}^{p})}^{1 / p}), 0), \\ max ((1 - {(ρ {(1 - ν_{1}^{+})}^{p})}^{1 / p}), 0) \end{matrix}] \end{matrix}), ρ \in$ ,

$ℑ_{1}^{ρ} = (\begin{matrix} [\begin{matrix} max ((1 - {(ρ {(1 - μ_{1}^{-})}^{p})}^{1 / p}), 0), \\ max ((1 - {(ρ {(1 - μ_{1}^{+})}^{p})}^{1 / p}), 0) \end{matrix}], \\ [\begin{matrix} min ({(ρ {(ν_{1}^{-})}^{p})}^{1 / p}, 1), \\ min ({(ρ {(ν_{1}^{+})}^{p})}^{1 / p}, 1) \end{matrix}] \end{matrix})$ .

Theorem 4.1. (a) For $μ_{1}^{-}, μ_{2}^{-} \in [0, 1]$ and $μ_{1}^{+}, μ_{2}^{+} \in [0, 1]$ , the s-norm given by operational law (i) in Definition 4.2 can be presented as $S^{-} (μ_{1}^{-}, μ_{2}^{-}) max (μ_{1}^{-}, μ_{2}^{-})$ and $S^{+} (μ_{1}^{+}, μ_{2}^{+}) max (μ_{1}^{+}, μ_{2}^{+})$ , where ‘max’ is the maximum operator.

Proof. See the Appendix.

Theorem 4.2. If p = 1, then the operations given in Definition 4.2 reduces to Lukasiewicz form of operations, i.e.,

$ℑ_{1} \oplus ℑ_{2} = (\begin{matrix} [min (μ_{1}^{-} + μ_{2}^{-}, 1), min (μ_{1}^{+} + μ_{2}^{+}, 1)], \\ [\begin{matrix} max (1 - ((1 - ν_{1}^{-}) + (1 - ν_{2}^{-})), 0), \\ max (1 - ((1 - ν_{1}^{+}) + (1 - ν_{2}^{+})), 0) \end{matrix}] \end{matrix})$ ,

$ℑ_{1} \otimes ℑ_{2} = (\begin{matrix} [\begin{matrix} max (1 - ((1 - μ_{1}^{-}) + (1 - μ_{2}^{-})), 0), \\ max (1 - ((1 - μ_{1}^{+}) + (1 - μ_{2}^{+})), 0) \end{matrix}], \\ [min (ν_{1}^{-} + ν_{2}^{-}, 1), min (ν_{1}^{+} + ν_{2}^{+}, 1)] \end{matrix})$ ,

$ρ ℑ_{1} = (\begin{matrix} [min (ρ μ_{1}^{-}, 1), min (ρ μ_{1}^{+}, 1)], \\ [\begin{matrix} max ((1 - (ρ (1 - ν_{1}^{-}))), 0), \\ max ((1 - (ρ (1 - ν_{1}^{+}))), 0) \end{matrix}] \end{matrix}), ρ \in$ ,

$ℑ_{1}^{ρ} = (\begin{matrix} [\begin{matrix} max ((1 - (ρ (1 - μ_{1}^{-}))), 0), \\ max ((1 - (ρ (1 - μ_{1}^{+}))), 0) \end{matrix}], \\ [min (ρ ν_{1}^{-}, 1), min (ρ ν_{1}^{+}, 1)] \end{matrix}), ρ \in$ .

Theorem 4.3. Consider ℑ = ([μ^-, μ⁺] , [ν^-, ν⁺]), $ℑ_{1} = ([μ_{1}^{-}, μ_{1}^{+}], [ν_{1}^{-}, ν_{1}^{+}])$ and $ℑ_{2} = ([μ_{2}^{-}, μ_{2}^{+}], [ν_{2}^{-}, ν_{2}^{+}])$ be three IVIFNs and ρ, ρ₁, ρ₂ > 0. Then

ℑ₁ ⊕ ℑ ₂ = ℑ ₂ ⊕ ℑ ₁,

ℑ₁ ⊗ ℑ ₂ = ℑ ₂ ⊗ ℑ ₁,

ρ (ℑ ₁ ⊕ ℑ ₂) = ρ ℑ ₁ ⊕ ρ ℑ ₂,

${(ℑ_{1} \otimes ℑ_{2})}^{ρ} = ℑ_{1}^{ρ} \otimes ℑ_{2}^{ρ}$ ,

(ρ₁+ ρ₂) ℑ = ρ₁ ℑ + ρ₂ ℑ,

ℑ^λ₁ ⊗ ℑ ^λ₂ = ℑ ^{λ₁ +λ₂}.

4.2 IVIF Yager weighted averaging operators

Suppose Θ be the set of all IVIFNs. Then, corresponding to the Definition 4.2, we have

Definition 4.3. Suppose $ℑ_{i} = 〈 [μ_{i}^{-}, μ_{i}^{+}],$ $[ν_{i}^{-}, ν_{i}^{+}] 〉$ be IVIFNs. Then the IVIF Yager weighted averaging (IVIFYWA) operator is a function IVIFYWA : Θ^m → Θ given as $IVIFYWA (ℑ_{1}, ℑ_{2}, . . ., ℑ_{m}) = \oplus_{i = 1}^{m} ϖ_{i} ℑ_{i},$ (7) wherein ϖ_i characterizes weight of ℑ_i such that ϖ_i ∈ [0, 1] and $\sum_{i = 1}^{m} ϖ_{i} = 1$ .

Theorem 4.4. The aggregated value with an IVIFYWA is also an IVIFN and defined by $IVIFYWA (ℑ_{1}, ℑ_{2}, . . ., ℑ_{m}) = \oplus_{i = 1}^{m} ϖ_{i} ℑ_{i}$ $= (\begin{matrix} [\begin{matrix} min (\sum_{i = 1}^{m} {(ϖ_{i} {(μ_{i}^{-})}^{p})}^{1 / p}, 1), \\ min (\sum_{i = 1}^{m} {(ϖ_{i} {(μ_{i}^{+})}^{p})}^{1 / p}, 1) \end{matrix}], \\ [\begin{matrix} max ((1 - \sum_{i = 1}^{m} {(ϖ_{i} {(1 - ν_{i}^{-})}^{p})}^{1 / p}), 0), \\ max ((1 - \sum_{i = 1}^{m} {(ϖ_{i} {(1 - ν_{i}^{+})}^{p})}^{1 / p}), 0) \end{matrix}] \end{matrix}) .$ (8)

Proof. See the Appendix.

Definition 4.4. Let $ℑ_{i} = 〈 [μ_{i}^{-}, μ_{i}^{+}], [ν_{i}^{-}, ν_{i}^{+}] 〉$ be the IVIFNs. Then the IVIF Yager weighted geometric (IVIFYWG) operator is a function IVIFYWG : Θ^m → Θ such that $IVIFPWG (ℑ_{1}, ℑ_{2}, . . ., ℑ_{m}) = \otimes_{i = 1}^{m} ℑ_{i}^{ϖ_{i}},$ (9) wherein ϖ_i symbolizes weight of ℑ_i with ϖ_i ∈ [0, 1] and $\sum_{i = 1}^{m} ϖ_{i} = 1 .$

Theorem 4.5. The aggregated grade with an IVIFYWG is also an IVIFN and discussed as $IVIFPWG (ℑ_{1}, ℑ_{2}, . . ., ℑ_{m}) = \otimes_{i = 1}^{m} ℑ_{i}^{ϖ_{i}},$ $(\begin{matrix} [\begin{matrix} max ((1 - \sum_{i = 1}^{m} {(ϖ_{i} {(1 - μ_{i}^{-})}^{α})}^{1 / α}), 0), \\ max ((1 - \sum_{i = 1}^{m} {(ϖ_{i} {(1 - μ_{i}^{+})}^{α})}^{1 / α}), 0) \end{matrix}], \\ [\begin{matrix} min (\sum_{i = 1}^{m} {(ϖ_{i} {(ν_{i}^{-})}^{α})}^{1 / α}, 1), \\ min (\sum_{i = 1}^{m} {(ϖ_{i} {(ν_{i}^{+})}^{α})}^{1 / α}, 1) \end{matrix}], \end{matrix}) .$ (10)

Proof. Follow as Theorem 4.4.

Definition 4.5. Let $ℑ_{i} = 〈 [μ_{i}^{-}, μ_{i}^{+}], [ν_{i}^{-}, ν_{i}^{+}] 〉$ be the IVIFNs. Then the IVIF Yager ordered weighted averaging (IVIFYOWA) operator is a function IVIFYOWA : Θ^m → Θ such that $IVIFYOWA (ℑ_{1}, ℑ_{2}, . . ., ℑ_{m}) = \oplus_{i = 1}^{m} ϖ_{i} ℑ_{σ (i)},$ (11) where ϖ_i denotes weight of ℑ_i with ϖ_i ∈ [0, 1] and $\sum_{i = 1}^{m} ϖ_{i} = 1,$ and (σ (1) , σ (2) , . . . , σ (m)) is permutation of (i = 1, 2, . . . , m) with ℑ_σ(i-1) ℑ _σ(i), ∀ i.

Theorem 4.6. The aggregated grade with an IVIFYOWA is an IVIFN and is discussed as $IVIFYOWA (ℑ_{1}, ℑ_{2}, . . ., ℑ_{m}) =$ $(\begin{matrix} [\begin{matrix} min (\sum_{i = 1}^{m} {(ϖ_{i} {(μ_{σ (i)}^{-})}^{p})}^{1 / p}, 1), \\ min (\sum_{i = 1}^{m} {(ϖ_{i} {(μ_{σ (i)}^{+})}^{p})}^{1 / p}, 1) \end{matrix}], \\ [\begin{matrix} max ((1 - \sum_{i = 1}^{m} {(ϖ_{i} {(1 - ν_{σ (i)}^{-})}^{p})}^{1 / p}), 0), \\ max ((1 - \sum_{i = 1}^{m} {(ϖ_{i} {(1 - ν_{σ (i)}^{+})}^{p})}^{1 / p}), 0) \end{matrix}] \end{matrix}) .$ (12)

Proof. Same as Theorem 4.4.

Remark 4.1. IVIFYWA and IVIFYOWA AOs are particular cases of IVIFYHWA AO.

4.3 Proposed IVIF Yager weighted geometric operators

Definition 4.6. Let $ℑ_{i} = 〈 [μ_{i}^{-}, μ_{i}^{+}], [ν_{i}^{-}, ν_{i}^{+}] 〉$ the IVIFNs. Then the IVIF Yager ordered weighted geometric (IVIFYOWG) operator is a function IVIFYOWG : Θ^m → Θ such that $IVIFYOWG (ℑ_{1}, ℑ_{2}, . . ., ℑ_{m}) = \otimes_{i = 1}^{m} ℑ_{σ (i)}^{ϖ_{i}},$ (13) where ϖ_i symbolizes weight of ℑ_i with ϖ_i ∈ [0, 1] and $\sum_{i = 1}^{m} ϖ_{i} = 1,$ and (σ (1) , σ (2) , . . . , σ (m)) is permutation of (i = 1, 2, . . . , m) with ℑ_σ(i-1) ℑ _σ(i), ∀ i.

Theorem 4.7. The aggregated grade with an IVIFYOWG is an IVIFN and is given by $IVIFYOWG (ℑ_{1}, ℑ_{2}, . . ., ℑ_{m}) =$ $(\begin{matrix} [\begin{matrix} max ((1 - \sum_{i = 1}^{m} {(ϖ_{i} {(1 - μ_{σ (i)}^{-})}^{α})}^{1 / α}), 0), \\ max ((1 - \sum_{i = 1}^{m} {(ϖ_{i} {(1 - μ_{σ (i)}^{+})}^{α})}^{1 / α}), 0) \end{matrix}], \\ [\begin{matrix} min (\sum_{i = 1}^{m} {(ϖ_{i} {(ν_{σ (i)}^{-})}^{α})}^{1 / α}, 1), \\ min (\sum_{i = 1}^{m} {(ϖ_{i} {(ν_{σ (i)}^{+})}^{α})}^{1 / α}, 1) \end{matrix}], \end{matrix}) .$ (14)

Proof. Follow as Theorem 4.4.

Definition 4.7. Suppose $ℑ_{i} = 〈 [μ_{i}^{-}, μ_{i}^{+}],$ $[ν_{i}^{-}, ν_{i}^{+}] 〉$ be the IVIFNs. Then the IVIF Yager hybrid weighted averaging (IVIFYHWA) operator is a function IVIFYHWA : Θ^m → Θ such that $IVIFYHWA (ℑ_{1}, ℑ_{2}, . . ., ℑ_{m}) = \oplus_{i = 1}^{m} ϖ_{i} {\bar{ℑ}}_{σ (i)},$ (15) where ϖ_i denotes the weight of ℑ_i such that ϖ_i ∈ [0, 1] and $\sum_{i = 1}^{m} ϖ_{i} = 1,$ and ${\bar{ℑ}}_{σ (i)}$ is the i^th largest weight of ${\bar{ℑ}}_{i} ({\bar{ℑ}}_{i} = m ϖ_{i} ℑ_{i}, i = 1, 2, . . ., m)$ and ‘m’ is the balancing parameter.

Theorem 4.8. The aggregated grade with an IVIFYHWA is an IVIFN and is given by $IVIFYHWA (ℑ_{1}, ℑ_{2}, . . ., ℑ_{m}) =$ $(\begin{matrix} [\begin{matrix} min (\sum_{i = 1}^{m} {(ϖ_{i} {({\bar{μ}}_{σ (i)}^{-})}^{p})}^{1 / α p}, 1), \\ min (\sum_{i = 1}^{m} {(ϖ_{i} {({\bar{μ}}_{σ (i)}^{+})}^{p})}^{1 / p}, 1) \end{matrix}], \\ [\begin{matrix} max ((1 - \sum_{i = 1}^{m} {(ϖ_{i} {(1 - {\bar{ν}}_{σ (i)}^{-})}^{p})}^{1 / p}), 0), \\ max ((1 - \sum_{i = 1}^{m} {(ϖ_{i} {(1 - {\bar{ν}}_{σ (i)}^{+})}^{p})}^{1 / p}), 0) \end{matrix}] \end{matrix}) .$ (16)

Proof. Follow as Theorem 4.4.

Remark 1. IVIFYWA and IVIFYOWA AOs are particular cases of IVIFYHWA AO.

Definition 4.8. Suppose $ℑ_{i} = 〈 [μ_{i}^{-}, μ_{i}^{+}],$ $[ν_{i}^{-}, ν_{i}^{+}] 〉$ be the IVIFNs. Then the IVIF Yager hybrid weighted geometric (IVIFYHWG) operator is a function IVIFPHWG : Θ^m → Θ given as $IVIFYHWG (ℑ_{1}, ℑ_{2}, . . ., ℑ_{m}) = \otimes_{i = 1}^{m} {\bar{ℑ}}_{σ (i)}^{ϖ_{i}},$ (17) where ϖ_i symbolizes weight of ℑ_i with ϖ_i ∈ [0, 1] and $\sum_{i = 1}^{m} ϖ_{i} = 1,$ and ${\bar{ℑ}}_{σ (i)}$ is the i^th largest weight of ${\bar{ℑ}}_{i} ({\bar{ℑ}}_{i} = ℑ_{i}^{m ϖ_{i}}, i = 1, 2, . . ., m)$ and ‘m’ is the balancing factor.

Theorem 4.9. The aggregated grade with an IVIFYHWG is an IVIFN and is discussed as $IVIFYHWG (ℑ_{1}, ℑ_{2}, . . ., ℑ_{m}) =$ $(\begin{matrix} [\begin{matrix} max ((1 - \sum_{i = 1}^{m} {(ϖ_{i} {(1 - {\bar{μ}}_{σ (i)}^{-})}^{p})}^{1 / p}), 0), \\ max ((1 - \sum_{i = 1}^{m} {(ϖ_{i} {(1 - {\bar{μ}}_{σ (i)}^{+})}^{p})}^{1 / p}), 0) \end{matrix}], \\ [\begin{matrix} min (\sum_{i = 1}^{m} {(ϖ_{i} {({\bar{ν}}_{σ (i)}^{-})}^{p})}^{1 / p}, 1), \\ min (\sum_{i = 1}^{m} {(ϖ_{i} {({\bar{ν}}_{σ (i)}^{+})}^{p})}^{1 / p}, 1) \end{matrix}], \end{matrix}),$ (18)

Proof. Follow as Theorem 4.4.

Remark 2. IVIFYWG and IVIFYOWG AOs are particular cases of IVIFYHWG AO.

5 A Hybrid IVIF-SPC-AROMAN model for MCDM problems

In the section, we propose an extended AROMAN model for solving MCDM problems under IVIF environment. Consider the set of options U ={ U₁, U₂, . . . , U_m } and criteria V ={ V₁, V₂, . . . , V_n }. A group of decision experts (DEs) E ={ E₁, E₂, . . . , E_t } is created to choose the most suitable option by means of considered criteria set. The DEs creates a linguistic decision matrix $D = {(δ_{ij}^{(k)})}_{m \times n}$ in which $δ_{ij}^{(k)} = 〈 [μ_{ij}^{- (k)}, μ_{ij}^{+ (k)}], [ν_{ij}^{- (k)}, ν_{ij}^{+ (k)}] 〉$ denotes the linguistic assessment of option U_i by means of V_j, where i = 1,2, . . . ,m; j = 1,2, . . . ,n. To determine the solution of group decision-making problems, the proposed model comprises the following procedure (see Fig. 1):

Fig. 1

Decision-making framework to evaluate WWTT selection.

Step 1: Compute the DEs’ significance values.

Consider the linguistic assessments of DEs’ significance values and convert them into IVIFNs. Let $E_{k} = ([μ_{k}^{-}, μ_{k}^{+}], [ν_{k}^{-}, ν_{k}^{+}]), k = 1, 2, . . ., t$ be the performance of kth DE. Then the procedure for estimating the numeric significance value of kth DE is as follows: $ϑ_{k} = \frac{(μ_{k}^{-} + μ_{k}^{+}) (2 + π_{k}^{-} + π_{k}^{+})}{\sum_{k = 1}^{t} ((μ_{k}^{-} + μ_{k}^{+}) (2 + π_{k}^{-} + π_{k}^{+}))},$ (19) where $ϑ_{k} \in [0, 1] and \sum_{k = 1}^{t} ϑ_{k} = 1$

Step 2: Aggregate the individual decision information.

The entire individual DEs’ opinions are combined to create an AIVIF-DM. For this purpose, the proposed IVIFYWA AO is used to create the AIVIF-DM Z = (z_ij) _m×n, wherein $\begin{matrix} z_{ij} = ([μ_{ij}^{-}, μ_{ij}^{+}], [ν_{ij}^{-}, ν_{ij}^{+}]) \\ = {IVIFYWA}_{ϑ_{k}} (δ_{ij}^{(1)}, δ_{ij}^{(2)}, . . ., δ_{ij}^{(k)}) . \end{matrix}$ (20)

Step 3: Normalize the given AIVIF-DM.

We convert the AIVIF-DM Z = (z_ij) _m×n into the normalized matrix Ψ = (ψ_ij) _m ×n, wherein $ψ_{ij} = {\begin{matrix} z_{ij}, for benefit criterion, \\ z_{ij}^{c}, for cost criterion . \end{matrix}$ (21)

Step 4: Compute the criteria weights.

In the step, we are considering that each criterion has different significance. Let w = (w₁, w₂, . . . , w_n) ^T be the weight of attribute with w_j ∈ [0, 1] and $\sum_{j = 1}^{n} w_{j} = 1$ . To determine the weight of each criterion, an extended IVIF-SPC model is described in the following steps:

Step 4.1: Compute the score value of each aggregated element of normalized AIVIF-DM using Equation (22). $α_{ij} = \frac{1}{2} (𝕊 (ψ_{ij}) + 1), j = 1, 2, . . ., n, i = 1, 2, . . ., m .$ (22)

Step 4.2: Find the symmetry point (SP) of each attribute.

Let (ℏ ₁₁, ℏ ₂₁, . . . , ℏ _i1) ^T, ∀ i be a column vector of V₁ criterion values, with respect to a set of options. If the lower value of the interval [φ₁, φ₂] is defined as a = min(ℏ ₁₁, ℏ ₂₁, . . . , ℏ _i1) ^T and the upper value of the interval [φ₁, φ₂] is defined as b = max(ℏ ₁₁, ℏ ₂₁, . . . , ℏ _i1) ^T, then the SP is given as $φ_{j} = \frac{min {ℏ_{ij}} + max {ℏ_{ij}}}{2}, \forall i, j .$ (23)

Step 4.3: Determine the matrix of absolute distance using Equation (24). $\begin{matrix} D = {(| d_{ij} |)}_{m \times n} = \\ {(\begin{matrix} | ℏ_{11} - φ_{1} | & | ℏ_{12} - φ_{2} | & . . . & | ℏ_{1 n} - φ_{n} | \\ | ℏ_{21} - φ_{1} | & | ℏ_{22} - φ_{2} | & . . . & | ℏ_{2 n} - φ_{n} | \\ . . . & . . . & . . . & . . . \\ | ℏ_{m 1} - φ_{1} | & | ℏ_{m 2} - φ_{2} | & . . . & | ℏ_{mn} - φ_{n} | \end{matrix})}_{m \times n}, \end{matrix}$ (24) wherein | ℏ _ij - φ_j| represents the absolute difference from ℏ_ij to each SP of attribute φ_j.

Step 4.4: Determine the matrix of moduli of symmetry (MoS).

Let D_i1 = { d₁₁, d₂₁, . . . , d_i1 } , ∀ i be the column values of absolute differences over attribute V₁. Then, the matrix of MoS is estimated as $H = {(| h_{ij} |)}_{m \times n} = {(\begin{matrix} \frac{\sum_{i = 1}^{m} d_{i 1}}{m . ℏ_{11}} & \frac{\sum_{i = 1}^{m} d_{i 2}}{m . ℏ_{12}} & . . . & \frac{\sum_{i = 1}^{m} d_{im}}{m . ℏ_{1 n}} \\ \frac{\sum_{i = 1}^{m} d_{i 1}}{m . ℏ_{21}} & \frac{\sum_{i = 1}^{m} d_{i 2}}{m . ℏ_{22}} & . . . & \frac{\sum_{i = 1}^{m} d_{in}}{m . ℏ_{2 n}} \\ . . . & . . . & . . . & . . . \\ \frac{\sum_{i = 1}^{m} d_{i 1}}{n . ℏ_{m 1}} & \frac{\sum_{i = 1}^{m} d_{i 2}}{n . ℏ_{m 2}} & . . . & \frac{\sum_{i = 1}^{m} d_{in}}{s . ℏ_{mn}} \end{matrix})}_{m \times n} .$ (25)

Step 4.5: Compute the modulus of symmetry of attribute using Equation (26). $Q = (q_{j}) = (\frac{\sum_{i = 1}^{m} h_{i 1}}{m} \frac{\sum_{i = 1}^{m} h_{i 2}}{m} . . . . \frac{\sum_{i = 1}^{m} h_{in}}{m}) .$ (26)

Step 4.6: Determine the weight of j^th criterion based on Equation (27). $w_{j} = (\frac{q_{1}}{\sum_{j = 1}^{n} q_{j}} \frac{q_{2}}{\sum_{j = 1}^{n} q_{j}} . . . . \frac{q_{j}}{\sum_{j = 1}^{n} q_{j}}), \forall j .$ (27)

Step 5: Normalize the AIVIF-DM.

The AIVIF-DM is normalized through linear and vector normalization tools as

Step 5.1 (Linear normalization). It eliminates dimensions of attributes using the principle of max-min operator. A linear normalized AIVIF-DM $ℕ^{(1)} = {(ɛ_{ij}^{(1)})}_{m \times n}$ is created using Equation (28), where $ɛ_{ij}^{(1)} = ({\bar{μ}}_{ij}^{(1)}, {\bar{η}}_{ij}^{(1)}, {\bar{ν}}_{ij}^{(1)}) = \frac{z_{ij}}{{max}_{i} 𝕊 (z_{ij})} .$ (28)

Step 5.2 (Vector normalization). A vector normalized AIVIF-DM $ℕ^{(2)} = {(ɛ_{ij}^{(2)})}_{m \times n}$ is constructed using Equation (29), where $ɛ_{ij}^{(2)} = ({\bar{μ}}_{ij}^{(2)}, {\bar{η}}_{ij}^{(2)}, {\bar{ν}}_{ij}^{(2)}) = \frac{z_{ij}}{\sqrt{\sum_{i = 1}^{m} {(𝕊 (z_{ij}))}^{2}}}, \forall i, j .$ (29)

Step 5.3. Find the averaged normalized AIVIF-DM.

The averaged normalized AIVIF-DM $ℕ = {(ɛ_{ij})}_{m \times n}$ , where $ɛ_{ij} = ({\bar{μ}}_{ij}, {\bar{η}}_{ij}, {\bar{ν}}_{ij})$ is done by applying the following expression as follows: $ɛ_{ij} = β ɛ_{ij}^{(1)} + (1 - β) ɛ_{ij}^{(2)},$ (30) where ɛ_ij denotes the averaged normalized AIVIF-DM. β is a normalization parameter changing from 0 to 1. Here, we take β = 0.5.

Step 6: Calculate the weighted averaged normalized AIVIF-DM.

Corresponding to Equation (31), the weighted averaged normalized AIVIF-DM $ℕ_{w} = {({\overset{⌢}{ς}}_{ij})}_{m \times n}$ , where ${\overset{⌢}{ς}}_{ij} = ({\overset{⌢}{μ}}_{ij}, {\overset{⌢}{η}}_{ij}, {\overset{⌢}{ν}}_{ij})$ is constructed, where ${\overset{⌢}{ς}}_{ij} = ({\overset{⌢}{μ}}_{ij}, {\overset{⌢}{η}}_{ij}, {\overset{⌢}{ν}}_{ij}) = w_{j} ɛ_{ij} =$ $= (\begin{matrix} [\begin{matrix} min (\sum_{j = 1}^{n} {(w_{j} {(μ_{ij}^{-})}^{p})}^{1 / p}, 1), \\ min (\sum_{j = 1}^{n} {(w_{j} {(μ_{ij}^{+})}^{p})}^{1 / p}, 1) \end{matrix}], \\ [\begin{matrix} max ((1 - \sum_{j = 1}^{n} {(w_{j} {(1 - ν_{ij}^{-})}^{p})}^{1 / p}), 0), \\ max ((1 - \sum_{j = 1}^{n} {(w_{j} {(1 - ν_{ij}^{+})}^{p})}^{1 / p}), 0) \end{matrix}] \end{matrix}) .$ (31)

Step 7: Evaluate the IVIF-score ratings of weighted averaged normalized AIVIF-DM of each option.

From weighted averaged normalized AIVIF-DM, we find the weighted normalized rating (L_i) for benefit-type criterion and the weighted normalized rating (B_i) for cost-type criterion as $L_{i} = \sum_{j \in V_{b}} 𝕊 ({\overset{⌢}{ς}}_{ij}),$ (32) $B_{i} = \sum_{j \in V_{c}} 𝕊 ({\overset{⌢}{ς}}_{ij}), i = 1, 2, . . ., m,$ (33) where $𝕊 ({\overset{⌢}{ς}}_{ij})$ symbolizes the IVIF-score function of each rating of weighted averaged normalized AIVIF-DM.

Step 8: Calculate the overall assessment degree (OAD) of option.

The OAD (s_i) of each option is obtained using Equation (34) as $s_{i} = λ L_{i} + (1 - λ) B_{i},$ (34) wherein λ symbolizes the parameter of changing the attribute type. If we involve both types of criteria, then we take λ = 0.5. Though, there is a choice to change the parameter λ by taking the different criterion type.

Step 9: Based on the OAD (s_i), where i = 1, 2, . . . , m, prioritize the options.

6 Results and discussion

This section firstly presents a case study of WWTTs assessment under the context of IVIFSs and further presents the sensitivity and comparative studies to exemplify the robustness of obtained results.

6.1 Case study: Evaluation of wastewater treatment technologies

In this section, we apply the developed tool on an empirical study of WWTTs assessment from IVIF information perspective. With the aggregate involvedness, time limitation and lack of exact knowledge/information, it is reasonably solid to assess the WWTTs over given sustainability aspects and criteria in realistic circumstances. In this section, a team of four DEs is designed to recognize the sustainability attributes and assess the WWTTs with considered criteria. These DEs are having 12 years of experience in their respective fields (see Fig. 2). Two of them are from the environmental engineering department and the others are from sustainable planning and management. Based on the literature review and online questionnaire, we have considered five WWTT alternatives and eight criteria. Description of the alternatives is presented as follows:

Fig. 2

Hierarchical framework to evaluate sustainable WWTT selection.

Microbial fuel cell (MFC) (U₁): It is rapidly growing eco-friendly and green technology, which uses bacteria to clean wastewater and generate electricity while removing the pollutants from the wastewater.

Membrane filtration (MF) (U₂): As a physical separation method, MF is used to treat wastewater by removing the harmful substances. It has the capability to separate the molecules of different sizes and characteristics.

Automatic variable filtration (AVF) (U₃): It is a simple water filtration way in which the downward flow of filter media cleans the upward flow of influent. It consumes less power, minimum operating and maintenance costs during the wastewater treatment.

Natural treatment methods (NTM) ( U₄): It is a biological treatment method to treat the wastewater naturally by removing contaminants from wastewater. These methods are eco-friendly, cost-effective, and can be mutually driven by public bodies and societies.

Advanced photo-oxidation process (APOP) (U₅): It involves a set of chemical procedures that oxidizes organic molecules in wastewater that are hard to accomplish biologically.

Furthermore, an online review has been arranged with the aim of defining the weight of attributes to select the WWTTs alternative. In addition, the criteria that may have an effect on the WWTT alternatives’ evaluation are assembled through literature survey. Table 1 gives the description of each considered attribute. Here, V₁, V₃ and V₄ are cost-type attributes and others are benefit attributes.

The developed IVIF-SPC-AROMAN model is applied on given WWTTs selection decision-making problem and the required implementation process is presented as follows:

Table 1

Sustainability indicators and criteria for WWTTs evaluation

Dimension	Criteria	Type	Sources
Economic (E)	Maintenance and operation cost (V₁)	Cost	[58, 59]
	Economic growth (V₂)	Benefit	[10 , 59]
Environmental (EN)	Energy consumption (V₃)	Cost	[2 , 58]
	Sludge production (V₄)	Cost	[2, 58]
Social (S)	Public acceptance (V₅)	Benefit	[2, 59]
	Aesthetic (V₆)	Benefit	[2, 60]
Technical (T)	COD removal capacity (V₇)	Benefit	[4, 60]
	BOD removal capacity (V₈)	Benefit	[4 , 61]

Steps 1-2: Table 2 presents the linguistic variables and the consequent IVIFVs [32]. Utilizing Table 2 and Equation (19), the weight of each DE is estimated and presented in Table 3.

Table 2

Linguistic scales and its corresponding IVIFNs

LVs	IVIFNs
Absolutely significant (AS)	([0.9,0.95], [0,0.05])
Very significant (VS)	([0.8,0.9], [0.05,0.1])
Significant (S)	([0.7,0.8], [0.1,0.15])
Slight significant (SS)	([0.65,0.7], [0.15,0.2])
Fair (F)	([0.55,0.65], [0.2,0.35])
Slight insignificant (SI)	([0.4,0.5], [0.4,0.45])
Insignificant (I)	([0.25,0.4], [0.45,0.5])
Very insignificant (VI)	([0.15,0.2], [0.6,0.75])
Extremely insignificant (EI)	([0.05,0.1], [0.8,0.9])

Table 3

Weights of chosen DEs

	E ₁	E ₂	E ₃	E ₄
LVs	AS	SS	F	VS
IVIFNs	([0.9,0.95], [0,0.05])	([0.65,0.7], [0.15,0.2])	([0.55,0.65], [0.2,0.35])	([0.8,0.9], [0.05,0.1])
Weight	0.2911	0.2327	0.2023	0.2739

Considering the linguistic scales into mind, the DEs provide their opinions for each WWTT alternative with respect to considered criteria and the required LDM is given in Table 4. To create the AIVIF-DM, the IVIFYWA operator (20) is applied on Table 4 and the required outcomes are presented in Table 5.

Table 4

LDM given by a group of DEs for WWTTs evaluation

Criteria	U ₁	U ₂	U ₃	U ₄	U ₅
V ₁	(VI,VI,SI,I)	(EI,I,VI,SI)	(I,VI,SI,VI)	(VI,EI,F,SI)	(F,VI,SI,SI)
V ₂	(S,S,VS,SS)	(S,VS,F,SS)	(F,SS,S,VS)	(S,S,SS,SI)	(F,F,SS,S)
V ₃	(VI,I,SI,F)	(SI,F,I,F)	(SI,VI,I,I)	(VI,SI,I,VI)	(VI,F,VI,SI)
V ₄	(F,I,VI,SI)	(I,VI,SI,EI)	(I,VI,F,I)	(I,SI,SI,I)	(VI,SI,SI,EI)
V ₅	(S,F,SS,VS)	(VS,SI,S,SI)	(F,F,S,SS)	(F,S,VS,S)	(VS,SS,VS,S)
V ₆	(VS,SS,F,SI)	(S,F,SS,SS)	(VS,SS,S,SI)	(S,S,VS,SS)	(AS,S,F,SS)
V ₇	(SS,AS,S,S)	(F,F,SS,VS)	(SS,F,F,S)	(VS,SI,S,SI)	(VS,F,SI,SI)
V ₈	(F,F,S,VS)	(S,S,VS,S)	(S,S,S,F)	(F,VS,F,F)	(SI,VS,F,SI)

Table 5

Aggregated matrix for WWTTs evaluation

Criteria	U ₁	U ₂	U ₃	U ₄	U ₅
V ₁	[0.2280,0.3155], [0.5185,0.6208]	[0.2126,0.2996], [0.5685,0.6533]	[0.2297,0.3189], [0.5159, 0.6165]	[0.2761,0.3499], [0.5108, 0.6218]	[0.3855,0.4739], [0.3883, 0.4907]
V ₂	[0.7065,0.7928], [0.1036, 0.1536]	[0.6792,0.7655], [0.1223, 0.1925]	[0.6721,0.7605], [0.1270, 0.2062]	[0.6077,0.6976], [0.1923, 0.2423]	[0.6113,0.7012], [0.1625, 0.2649]
V ₃	[0.3334,0.4305], [0.4151, 0.5216]	[0.4456,0.5558], [0.3088, 0.4095]	[0.2704,0.3826], [0.4704, 0.5436]	[0.2284,0.3103], [0.5231, 0.6296]	[0.3116,0.3869], [0.4521, 0.5748]
V ₄	[0.3582,0.4597], [0.3939, 0.4932]	[0.2023,0.2915], [0.5707, 0.6576]	[0.2874,0.4040], [0.4343, 0.5278]	[0.3153,0.4435], [0.3735, 0.4509]	[0.2314,0.3031], [0.5678, 0.6606]
V ₅	[0.6824,0.7723], [0.1197, 0.1930]	[0.5771,0.6771], [0.2374, 0.2874]	[0.6077,0.6940], [0.1661, 0.2685]	[0.6766,0.7766], [0.1190, 0.1981]	[0.7377,0.8261], [0.0870, 0.1370]
V ₆	[0.6050,0.6933], [0.1995, 0.2697]	[0.6413,0.7175], [0.1471, 0.2204]	[0.6353,0.7237], [0.1793, 0.2292]	[0.7065,0.7928], [0.1036, 0.1536]	[0.7142,0.7859], [0.1048, 0.1750]
V ₇	[0.7320,0.8058], [0.0913, 0.1413]	[0.6387,0.7286], [0.1488, 0.2512]	[0.6202,0.7056], [0.1581, 0.2516]	[0.5771,0.6771], [0.2374, 0.2874]	[0.5513,0.6513], [0.2516, 0.3248]
V ₈	[0.6488,0.7488], [0.1387, 0.2411]	[0.7202,0.8202], [0.0899, 0.1399]	[0.6589,0.7589], [0.1274, 0.2048]	[0.6082,0.7082], [0.1651, 0.2918]	[0.5234,0.6234], [0.2781, 0.3483]

Step 3: Since the given criteria set is the mixture of cost (V₁, V₃, V₄) and benefit criteria (V₂, V₅, V₆, V₇, V₈), we have created the normalized AIVIF-DM using Equation (21) and shown in Table 6.

Table 6

Normalized matrix for WWTTs evaluation

Criteria	U ₁	U ₂	U ₃	U ₄	U ₅
V ₁	([0.5185,0.6208], [0.2280,0.3155])	([0.5685,0.6533], [0.2126,0.2996])	([0.5159,0.6165], [0.2297,0.3189])	([0.5108,0.6218], [0.2761, 0.3499])	([0.3883,0.4907], [0.3855, 0.4739])
V ₂	[0.7065, 0.7928], [0.1036, 0.1536]	[0.6792, 0.7655], [0.1223, 0.1925]	[0.6721, 0.7605], [0.1270, 0.2062]	[0.6077, 0.6976], [0.1923, 0.2423]	[0.6113, 0.7012], [0.1625, 0.2649]
V ₃	([0.4151,0.5216], [0.3334, 0.4305])	([0.3088,0.4095], [0.4456, 0.5558])	([0.4704,0.5436], [0.2704, 0.3826])	([0.5231,0.6296], [0.2284, 0.3103])	([0.4521,0.5748], [0.3116, 0.3869])
V ₄	([0.3939,0.4932], [0.3582, 0.4597])	([0.5707,0.6576], [0.2023, 0.2915])	([0.4343,0.5278], [0.2874, 0.4040])	([0.3735,0.4509], [0.3153, 0.4435])	([0.5678,0.6606], [0.2314, 0.3031])
V ₅	[0.6824, 0.7723], [0.1197, 0.1930]	[0.5771, 0.6771], [0.2374, 0.2874]	[0.6077, 0.6940], [0.1661, 0.2685]	[0.6766, 0.7766], [0.1190, 0.1981]	[0.7377, 0.8261], [0.0870, 0.1370]
V ₆	[0.6050, 0.6933], [0.1995, 0.2697]	[0.6413, 0.7175], [0.1471, 0.2204]	[0.6353, 0.7237], [0.1793, 0.2292]	[0.7065, 0.7928], [0.1036, 0.1536]	[0.7142, 0.7859], [0.1048, 0.1750]
V ₇	[0.7320, 0.8058], [0.0913, 0.1413]	[0.6387, 0.7286], [0.1488, 0.2512]	[0.6202, 0.7056], [0.1581, 0.2516]	[0.5771, 0.6771], [0.2374, 0.2874]	[0.5513, 0.6513], [0.2516, 0.3248]
V ₈	[0.6488, 0.7488], [0.1387, 0.2411]	[0.7202, 0.8202], [0.0899, 0.1399]	[0.6589, 0.7589], [0.1274, 0.2048]	[0.6082, 0.7082], [0.1651, 0.2918]	[0.5234, 0.6234], [0.2781, 0.3483]

Step 4: Applying Equation (22), the IVIF-score rating of each aggregated element of normalized A-IVIFDM is computed in Table 7. Next, the SP of each criterion is computed with Equation (23) and presented in Table 7. Based on Equation (24), the matrix of absolute distance is computed in Table 8. Table 9 shows the matrix of the MoS and estimated through Equation (25) and the MoS of attribute through Equation (26). The last column of Table 9 denotes the final criteria weights, which is determined using Equation (27).

Table 7

IVIF score values and SP of criterion

Criteria	U ₁	U ₂	U ₃	U ₄	U ₅	min{ ℏ _ij }	max{ ℏ _ij }	φ _j
V ₁	0.351	0.323	0.354	0.373	0.495	0.323	0.495	0.409
V ₂	0.811	0.782	0.775	0.718	0.721	0.718	0.811	0.764
V ₃	0.457	0.571	0.410	0.346	0.418	0.346	0.571	0.459
V ₄	0.483	0.316	0.432	0.484	0.327	0.316	0.484	0.400
V ₅	0.785	0.682	0.717	0.784	0.835	0.682	0.835	0.759
V ₆	0.707	0.748	0.738	0.811	0.805	0.707	0.811	0.759
V ₇	0.826	0.742	0.729	0.682	0.657	0.657	0.826	0.741
V ₈	0.754	0.828	0.771	0.715	0.630	0.630	0.828	0.729

Table 8

Matrix of absolute distance

Criteria	U ₁	U ₂	U ₃	U ₄	U ₅
V ₁	0.058	0.086	0.055	0.036	0.086
V ₂	0.046	0.018	0.011	0.046	0.043
V ₃	0.002	0.112	0.049	0.112	0.041
V ₄	0.083	0.084	0.032	0.084	0.073
V ₅	0.027	0.076	0.042	0.025	0.076
V ₆	0.052	0.011	0.021	0.052	0.046
V ₇	0.085	0.000	0.012	0.059	0.085
V ₈	0.026	0.099	0.043	0.014	0.099

Table 9

Results of IVIF-SPC weighting tool

Criteria	U ₁	U ₂	U ₃	U ₄	U ₅	q_j	w_j
V ₁	0.183	0.199	0.181	0.172	0.130	0.173	0.216
V ₂	0.041	0.042	0.043	0.046	0.046	0.043	0.054
V ₃	0.138	0.111	0.154	0.182	0.151	0.147	0.184
V ₄	0.147	0.225	0.165	0.147	0.218	0.180	0.225
V ₅	0.063	0.072	0.069	0.063	0.059	0.065	0.082
V ₆	0.051	0.049	0.049	0.045	0.045	0.048	0.060
V ₇	0.058	0.065	0.066	0.071	0.074	0.067	0.084
V ₈	0.074	0.068	0.073	0.078	0.089	0.076	0.095

Figure 3 discusses the variation of weight of different criteria for WWTTs assessment. Sludge production (V₄) (0.225) is the most important factor during the assessment of WWTT alternatives. Maintenance and operation cost (V₁) (0.216) is the second most criteria for assessing the WWTTs. Energy consumption (V₃) (0.184) is third, BOD removal capacity assessment (V₈) (0.095) is fourth, COD removal capacity (V₇) (0.084) is fifth most significant factor for WWTTs evaluation and others are considered important factors in the evaluation of WWTT alternatives.

Fig. 3

Graphical description of criteria weights for WWTT selection.

Step 5: From Table 5 and Equations (28)–(29), linear normalized AIVIF-DM and vector normalized AIVIF-DM are constructed and shown in Tables 10, 11. With the use of Equation (30), the averaged normalized AIVIF-DM is constructed and mentioned in Table 12.

Table 10

Linear normalization matrix for WWTTs evaluation

Criteria	U ₁	U ₂	U ₃	U ₄	U ₅
V ₁	([0.120, 0.171], [0.722, 0.790])	([0.112, 0.162], [0.756, 0.810])	([0.121, 0.173], [0.721, 0.787])	([0.148, 0.192], [0.717, 0.790])	([0.214, 0.272], [0.626, 0.703])
V ₂	([0.630, 0.721], [0.159, 0.219])	([0.602, 0.691], [0.182, 0.263])	([0.595, 0.686], [0.188, 0.278])	([0.532, 0.621], [0.263, 0.317])	([0.535, 0.624], [0.229, 0.341])
V ₃	([0.207, 0.275], [0.605, 0.690])	([0.286, 0.371], [0.511, 0.601])	([0.165, 0.241], [0.650, 0.706])	([0.138, 0.191], [0.691, 0.768])	([0.192, 0.244], [0.636, 0.729])
V ₄	([0.193, 0.257], [0.637, 0.710])	([0.104, 0.154], [0.762, 0.817])	([0.151, 0.221], [0.668, 0.734])	([0.167, 0.247], [0.621, 0.680])	([0.120, 0.160], [0.761, 0.818])
V ₅	([0.616, 0.709], [0.170, 0.253])	([0.513, 0.611], [0.301, 0.353])	([0.542, 0.628], [0.223, 0.334])	([0.610, 0.714], [0.169, 0.259])	([0.673, 0.768], [0.130, 0.190])
V ₆	([0.529, 0.616], [0.271, 0.346])	([0.564, 0.641], [0.212, 0.294])	([0.558, 0.647], [0.248, 0.303])	([0.630, 0.721], [0.159, 0.219])	([0.638, 0.713], [0.161, 0.243])
V ₇	([0.663, 0.742], [0.138, 0.199])	([0.569, 0.660], [0.207, 0.319])	([0.551, 0.636], [0.218, 0.320])	([0.509, 0.607], [0.305, 0.357])	([0.484, 0.581], [0.320, 0.395])
V ₈	([0.579, 0.681], [0.195, 0.308])	([0.652, 0.758], [0.136, 0.196])	([0.589, 0.692], [0.182, 0.269])	([0.540, 0.639], [0.225, 0.361])	([0.458, 0.554], [0.347, 0.418])

Table 11

Vector normalization matrix for WWTTs evaluation

Criteria	U ₁	U ₂	U ₃	U ₄	U ₅
V ₁	([0.199, 0.278], [0.569, 0.664])	([0.186, 0.263], [0.616, 0.694])	([0.201, 0.281], [0.567, 0.660])	([0.242, 0.309], [0.562, 0.665])	([0.342, 0.424], [0.444, 0.543])
V ₂	([0.876, 0.932], [0.021, 0.041])	([0.856, 0.916], [0.028, 0.060])	([0.851, 0.912], [0.030, 0.068])	([0.797, 0.870], [0.060, 0.089])	([0.800, 0.872], [0.045, 0.104])
V ₃	([0.333, 0.430], [0.416, 0.522])	([0.445, 0.555], [0.309, 0.410])	([0.270, 0.382], [0.471, 0.544])	([0.228, 0.310], [0.524, 0.630])	([0.311, 0.386], [0.453, 0.575])
V ₄	([0.337, 0.435], [0.421, 0.519])	([0.189, 0.274], [0.594, 0.678])	([0.270, 0.381], [0.461, 0.553])	([0.296, 0.419], [0.401, 0.478])	([0.217, 0.285], [0.592, 0.681])
V ₅	([0.859, 0.920], [0.027, 0.060])	([0.770, 0.855], [0.086, 0.119])	([0.797, 0.867], [0.047, 0.106])	([0.854, 0.922], [0.027, 0.063])	([0.898, 0.949], [0.016, 0.034])
V ₆	([0.795, 0.867], [0.064, 0.107])	([0.826, 0.884], [0.038, 0.076])	([0.821, 0.888], [0.053, 0.081])	([0.876, 0.932], [0.021, 0.041])	([0.882, 0.928], [0.021, 0.051])
V ₇	([0.883, 0.931], [0.020, 0.041])	([0.810, 0.881], [0.045, 0.105])	([0.794, 0.864], [0.049, 0.105])	([0.754, 0.842], [0.096, 0.131])	([0.729, 0.821], [0.105, 0.160])
V ₈	([0.824, 0.899], [0.038, 0.094])	([0.879, 0.942], [0.018, 0.038])	([0.832, 0.906], [0.033, 0.072])	([0.789, 0.871], [0.050, 0.129])	([0.708, 0.802], [0.119, 0.174])

Step 6: From Equation (31) and Table 12, we determine the weighted averaged normalized AIVIF-DM using proposed Yager weighted averaging (or geometric) operator and criteria weights. The required result is presented in Table 13.

Table 12

Averaged normalization matrix for WWTTs evaluation

Criteria	U ₁	U ₂	U ₃	U ₄	U ₅
V ₁	([0.161, 0.226], [0.641, 0.724])	([0.149, 0.214], [0.682, 0.750])	([0.162, 0.229], [0.639, 0.721])	([0.196, 0.253], [0.635, 0.725])	([0.281, 0.353], [0.527, 0.618])
V ₂	([0.786, 0.862], [0.058, 0.095])	([0.761, 0.839], [0.071, 0.126])	([0.754, 0.834], [0.075, 0.137])	([0.692, 0.778], [0.126, 0.168])	([0.695, 0.781], [0.102, 0.188])
V ₃	([0.273, 0.357], [0.502, 0.600])	([0.371, 0.471], [0.398, 0.496])	([0.219, 0.315], [0.553, 0.620])	([0.184, 0.253], [0.601, 0.696])	([0.254, 0.319], [0.536, 0.648])
V ₄	([0.269, 0.352], [0.518, 0.607])	([0.147, 0.216], [0.673, 0.744])	([0.213, 0.306], [0.555, 0.637])	([0.235, 0.339], [0.499, 0.570])	([0.169, 0.225], [0.671, 0.746])
V ₅	([0.767, 0.847], [0.067, 0.124])	([0.665, 0.762], [0.161, 0.205])	([0.695, 0.778], [0.102, 0.188])	([0.762, 0.851], [0.067, 0.128])	([0.817, 0.892], [0.045, 0.080])
V ₆	([0.689, 0.774], [0.132, 0.192])	([0.725, 0.796], [0.090, 0.149])	([0.719, 0.802], [0.115, 0.157])	([0.786, 0.862], [0.058, 0.095])	([0.793, 0.856], [0.059, 0.112])
V ₇	([0.802, 0.867], [0.053, 0.090])	([0.714, 0.799], [0.096, 0.183])	([0.696, 0.777], [0.104, 0.183])	([0.653, 0.751], [0.171, 0.216])	([0.626, 0.726], [0.183, 0.251])
V ₈	([0.728, 0.821], [0.086, 0.170])	([0.795, 0.882], [0.050, 0.087])	([0.738, 0.830], [0.077, 0.139])	([0.688, 0.784], [0.106, 0.216])	([0.602, 0.703], [0.203, 0.269])

Table 13

Weighted averaged normalization matrix for WWTTs evaluation

Criteria	U ₁	U ₂	U ₃	U ₄	U ₅
V ₁	([0.037, 0.054], [0.908, 0.933])	([0.034, 0.051], [0.921, 0.940])	([0.037, 0.055], [0.908, 0.932])	([0.046, 0.061], [0.906, 0.933])	([0.069, 0.090], [0.871, 0.901])
V ₂	([0.080, 0.102], [0.857, 0.880])	([0.075, 0.094], [0.866, 0.894])	([0.073, 0.093], [0.869, 0.898])	([0.062, 0.078], [0.894, 0.908])	([0.062, 0.079], [0.883, 0.913])
V ₃	([0.057, 0.078], [0.881, 0.910])	([0.082, 0.111], [0.844, 0.879])	([0.045, 0.067], [0.897, 0.916])	([0.037, 0.052], [0.911, 0.935])	([0.053, 0.068], [0.892, 0.923])
V ₄	([0.068, 0.093], [0.862, 0.894])	([0.035, 0.053], [0.915, 0.935])	([0.052, 0.079], [0.876, 0.903])	([0.058, 0.089], [0.855, 0.881])	([0.041, 0.056], [0.914, 0.936])
V ₅	([0.112, 0.142], [0.803, 0.843])	([0.085, 0.110], [0.862, 0.879])	([0.092, 0.115], [0.830, 0.873])	([0.110, 0.144], [0.802, 0.846])	([0.129, 0.166], [0.777, 0.814])
V ₆	([0.068, 0.085], [0.886, 0.906])	([0.074, 0.091], [0.866, 0.892])	([0.073, 0.092], [0.879, 0.895])	([0.088, 0.112], [0.843, 0.868])	([0.090, 0.110], [0.844, 0.877])
V ₇	([0.127, 0.155], [0.782, 0.818])	([0.099, 0.125], [0.822, 0.868])	([0.095, 0.118], [0.827, 0.868])	([0.085, 0.110], [0.863, 0.880])	([0.079, 0.103], [0.868, 0.891])
V ₈	([0.117, 0.151], [0.791, 0.845])	([0.140, 0.184], [0.752, 0.792])	([0.120, 0.155], [0.783, 0.829])	([0.105, 0.136], [0.808, 0.864])	([0.084, 0.109], [0.859, 0.882])

Steps 7–9: With the use of Equation (32), the weighted normalized rating (L_i) for benefit-type criteria V₂, V₅, V₆, V₇ and V₈ are computed and shown in Table 14. Similarly, the weighted normalized rating (M_i) for cost-type criteria V₁, V₃ and V₄ are determined using Equation (33) and presented in Table 14. In accordance with Equation (34), the OADs (s_i) value is calculated for each WWTT alternative. Table 14 shows the IVIF-score values of weighted averaged normalized AIVIF-DM to estimate the weighted normalized ratings and OADs (s_i). Thus, the prioritization of WWTTs option is U₁ (0.4658)> U₂ (0.4398)> U₃(0.4226)> U₄(0.4220)> U₅(0.4177). Thus, the MFC (U₁) is most suitable option among the others for the given data sets.

Table 14

Results of AROMAN method for WWTTs evaluation

Criteria	U ₁	U ₂	U ₃	U ₄	U ₅
V ₁	0.062	0.056	0.063	0.067	0.097
V ₂	0.111	0.102	0.100	0.085	0.086
V ₃	0.086	0.117	0.075	0.061	0.076
V ₄	0.101	0.060	0.088	0.103	0.062
V ₅	0.152	0.114	0.126	0.152	0.176
V ₆	0.090	0.102	0.098	0.122	0.120
V ₇	0.170	0.134	0.129	0.113	0.106
V ₈	0.158	0.195	0.166	0.142	0.113

6.2 Sensitivity investigation

Here, we analyze the effect of changing values of parameter ‘λ’. The varying values of λ help us to analyze the sensitivity of developed IVIF-SPC-AROMAN tool to the prominence of normalization types. Table 15 and Fig. 4 exemplify the sensitivity results for assessing the WWTTs option over different values of normalization parameter λ. For λ = 0.0 to λ = 1.0, we find the same preference order of WWTT alternatives, which is U₁ ≻ U₂ ≻ U₃ ≻ U₄ ≻ U₅ and thus, the option “MFC (U₁)” is the most suitable choice among all the WWTTs. Thus, it is observed that the obtained outcomes by introduced approach are stable with respect to varied values of normalization parameter.

Table 15
The OADs of WWTTs over normalization parameter (λ)

Options λ = 0.0 λ = 0.1 λ = 0.2 λ = 0.3 λ = 0.4 λ = 0.5 λ = 0.6 λ = 0.7 λ = 0.8 λ = 0.9 λ = 1.0

U ₁ 0.6039 0.5768 0.5495 0.5219 0.4940 0.4658 0.4374 0.4087 0.3798 0.3505 0.3210

U ₂ 0.5700 0.5445 0.5187 0.4926 0.4664 0.4398 0.4131 0.3860 0.3588 0.3312 0.3034

U ₃ 0.5492 0.5243 0.4992 0.4739 0.4483 0.4226 0.3966 0.3704 0.3439 0.3172 0.2903

U ₄ 0.5489 0.5240 0.4988 0.4734 0.4479 0.4220 0.3960 0.3697 0.3432 0.3165 0.2896

U ₅ 0.5424 0.5179 0.4932 0.4682 0.4430 0.4177 0.3921 0.3662 0.3402 0.3139 0.2874

Options	λ = 0.0	λ = 0.1	λ = 0.2	λ = 0.3	λ = 0.4	λ = 0.5	λ = 0.6	λ = 0.7	λ = 0.8	λ = 0.9	λ = 1.0
U ₁	0.6039	0.5768	0.5495	0.5219	0.4940	0.4658	0.4374	0.4087	0.3798	0.3505	0.3210
U ₂	0.5700	0.5445	0.5187	0.4926	0.4664	0.4398	0.4131	0.3860	0.3588	0.3312	0.3034
U ₃	0.5492	0.5243	0.4992	0.4739	0.4483	0.4226	0.3966	0.3704	0.3439	0.3172	0.2903
U ₄	0.5489	0.5240	0.4988	0.4734	0.4479	0.4220	0.3960	0.3697	0.3432	0.3165	0.2896
U ₅	0.5424	0.5179	0.4932	0.4682	0.4430	0.4177	0.3921	0.3662	0.3402	0.3139	0.2874

Fig. 4

Sensitivity test on normalization parameter (λ) for assessing the WWTTs.

6.3 Comparative analysis

In this subsection, we present the comparative study for demonstrating the usefulness of the proposed approach. Comparisons are taken among the IVIF-information based MCDM approaches suggested by Mishra & Rani [62], Nguyen [63] and Wang et al. [64] in the context of IVIFS.

Fig. 5

Comparison of proposed with extant methods for WWTTs assessment.

6.3.1 Mishra & Rani’s method [62]

The IVIF-WASPAS model [62] is executed on aforesaid WWTTs evaluation problem. Using this model, the measures obtained through WSM are $C_{1}^{(1)} = ([0.649, 0.738]$ , [0.149, 0.220]), $C_{2}^{(1)} = ([0.568, 0.663]$ , [0.206, 0.292]), $C_{3}^{(1)} = ([0.543, 0.636]$ , [0.212, 0.309]), $C_{4}^{(1)} =$ ([0.539, 0.640], [0.223, 0.311]), $C_{5}^{(1)} =$ ([0.541, 0.642], [0.242, 0.320]). In addition, the IVIF-score values of WSM measures are $(C_{1}^{(1)})$ = 0.754, $(C_{2}^{(1)}) =$ 0.683, $(C_{3}^{(1)}) = 0.664$ , $(C_{4}^{(1)}) = 0.661$ and $(C_{5}^{(1)}) = 0.655$ . The measures obtained through WPM are $C_{1}^{(2)} = ([0.621, 0.709]$ , [0.179, 0.251]), $C_{2}^{(2)} = ([0.535, 0.630], [0.238, 0.327])$ , $C_{3}^{(2)} = ([0.528, 0.619]$ , [0.223, 0.322]), $C_{4}^{(2)} =$ ([0.517, 0.615], [0.238, 0.328]), $C_{5}^{(2)} =$ ([0.516, 0.619], [0.268, 0.346]). The IVIF-score values of WPM measures are $(C_{1}^{(2)}) =$ 0.725, $(C_{2}^{(2)}) =$ 0.650, $(C_{3}^{(2)}) =$ 0.651, $(C_{4}^{(2)}) =$ 0.641 and $(C_{5}^{(2)}) =$ 0.630. Next, the utility degree of each WWTT option is computed as C₁ = 0.7396, C₂ = 0.6664, C₃ = 0.6575, C₄ = 0.6513 and C₅ = 0.6429. Then the ranking order of WWTTs is U₁ ≻ U₂ ≻ U₃ ≻ U₄ ≻ U₅ and the most appropriate alternative is Microbial fuel technology (U₁).

6.3.2 Nguyen’s method [63]

The IVIF-CoCoSo model is applied on the WWTT alternatives evaluation problem, given in Subsection 6.1. In this model, the balanced compromise degrees of WWTT candidates are determined as $Q_{1}^{(1)} =$ 0.2203, $Q_{2}^{(1)} =$ 0.1985, $Q_{3}^{(1)} =$ 0.1958, $Q_{4}^{(1)} =$ 0.194, $Q_{5}^{(1)} =$ 0.1915, $Q_{1}^{(2)} =$ 2.3008, $Q_{2}^{(2)} =$ 2.0729, $Q_{3}^{(2)} =$ 2.0457, $Q_{4}^{(2)} =$ 2.0263, $Q_{5}^{(2)} =$ 2.0, $Q_{1}^{(3)} =$ 1.0, $Q_{2}^{(3)} =$ 0.901, $Q_{3}^{(3)} =$ 0.889, $Q_{4}^{(3)} =$ 0.8806 and $Q_{5}^{(3)} =$ 0.8692. Next, the OCDs of WWTTs are estimated as Q₁ = 1.9710, Q₂ = 1.7758, Q₃ = 1.7523, Q₄ = 1.7357 and Q₅ = 1.7133. Then the ranking order of WWTTs is U₁ ≻ U₂ ≻ U₃ ≻ U₄ ≻ U₅ and the “Microbial fuel cell (U₁)” is the most suitable alternative among the others.

6.3.3 Wang et al.’s method [64]

The IVIF-COPRAS model is applied on the WWTT alternatives evaluation problem given in Subsection 6.1. The sum of maximization criteria is obtained as ℘₁ = ([0.412, 0.494], [0.377, 0.465]), ℘₂ = ([0.393, 0.477], [0.399, 0.482]), ℘₃ = ([0.379, 0.457], [0.411, 0.504]), ℘₄ = ([0.377, 0.461], [0.421, 0.507]), ℘₅ = ([0.373, 0.453], [0.432, 0.513]) and the sum of minimization criteria is estimated as ℑ₁ = ([0.154, 0.209], [0.690, 0.759]), ℑ₂ = ([0.145, 0.201], [0.711, 0.773]), ℑ₃ = ([0.129, 0.188], [0.713, 0.771]), ℑ₄ = ([0.135, 0.189], [0.706, 0.769]), ℑ₅ = ([0.154, 0.199], [0.710, 0.779]). The relative degrees of WWTT alternatives are ℓ₁ = 0.3602, ℓ₂ = 0.3570, ℓ₃ = 0.3522, ℓ₄ = 0.3485 and ℓ₅ = 0.3433. Finally, the UD of each option is obtained as η₁ = 100, η₂ = 99.09, η₃ = 97.77, η₄ = 96.75 and η₅ = 95.29. Then the ranking order of WWTTs is U₁ ≻ U₂ ≻ U₃ ≻ U₄ ≻ U₅ and the “Microbial fuel cell (F₁)” is considered to be the best choice among the others.

The ranking results obtained by the developed and extant MCDM model are shown in Fig. 3. From Fig. 3, it can easily be seen that the most suitable choice “Microbial fuel cell (U₁)” is same for all the MCDM approaches, namely IVIF-WASPAS [62], IVIF-CoCoSo [63] and IVIF-COPRAS [64]. The main advantages of the developed IVIF-SPC-AROMAN methodology are as follows:

In the introduced approach, the individual opinions are combined through IVIF Yager weighted AOs, which are generalized versions of the algebraic AOs used by extant IVIF-models. Thus, the developed approach has more accuracy than existing methods.

Existing IVIF-COPRAS model considers direct weight of criteria, while the SPC model used in the proposed model has good effectiveness during the evaluation of criteria weights.

The proposed IVIF-SPC-AROMAN method uses the linear and vector normalization models to aggregate the information, therefore, it provides more accurate decision than existing methods.

7 Conclusion

In this study, we have proposed a hybrid MCDM methodology for solving WWTTs assessment under the context of IVIFSs. For this purpose, some weighted AOs have presented to aggregate the IVIF information. In this regard, we have defined the fundamental operational laws of Yager’s t-norm and t-conorm for IVIFNs and proposed a series of Yager weighted AOs including IVIFYWA, IVIFYOWA, IVIFYHWA, IVIFYWG, IVIFYOWG and IVIFYHWG operators with their desirable properties. Further, an integrated AROMAN method has developed based on the combination of Yager weighted AOs and SPC model. Moreover, sensitivity analysis has presented over the different values of normalization parameter. Furthermore, we have compared the obtained outcomes with some of the existing methods and found that the results are similar and compatible. The advantages of the proposed work include the development of new AOs in order to aggregate the uncertain information and criteria weighting through IVIF-SPC model. Thus, the developed model is more competent and accurate while making decisions under IVIFS context. In future, we can extend the MAIRCA model under different disciplines namely “interval-valued hesitant q-rung orthopair fuzzy sets (IVHq-ROFSs)”, “q-rung orthopair rough sets (q-ROFRSs)” and “picture fuzzy sets (PiFSs)”.

Funding

This research was conducted under a project titled “Researchers Supporting Project”, funded by King Saud University, Riyadh, Saudi Arabia under grant number (RSP2024R323).

Conflicts of interest

All authors declare that there is no conflict of interest.

10 Appendix

Proof of the Theorem 4.1: For $ν_{1}^{-}, ν_{2}^{-} \in [0, 1]$ , we have $T^{-} (ν_{1}^{-}, ν_{2}^{-}) = max (1 -$ ${({(1 - ν_{1}^{-})}^{p} + {(1 - ν_{2}^{-})}^{p})}^{1 / p}, 0)$ . If $({(1 - ν_{1}^{-})}^{p}$ ${+ {(1 - ν_{2}^{-})}^{p})}^{1 / p} 1$ , then $T^{-} (ν_{1}^{-}, ν_{2}^{-}) = max (1 - {({(1 - ν_{1}^{-})}^{p} + {(1 - ν_{2}^{-})}^{p})}^{1 / p}, 0) = 0 min (ν_{1}^{-}, ν_{2}^{-})$ . If ${({(1 - ν_{1}^{-})}^{p} + {(1 - ν_{2}^{-})}^{p})}^{1 / p} < 1$ , then $T^{-} (ν_{1}^{-}, ν_{2}^{-}) = [max (1 - ({(1 - ν_{1}^{-})}^{p}$ ${+ {(1 - ν_{2}^{-})}^{p})}^{1 / p}, 0) [= 1 - ({(1 - ν_{1}^{-})}^{p} +$ ${{(1 - ν_{2}^{-})}^{p})}^{1 / p}$ . For $ν_{1}^{-}, ν_{2}^{-} \in [0, 1]$ , $1 - ({(1 - ν_{1}^{-})}^{p}$ ${+ {(1 - ν_{2}^{-})}^{p})}^{1 / p} ν_{1}^{-}$ and $1 - ({(1 - ν_{1}^{-})}^{p}$ ${+ {(1 - ν_{2}^{-})}^{p})}^{1 / p} ν_{2}^{-}$ , therefore, we have $T^{-} (ν_{1}^{-}, ν_{2}^{-}) min (ν_{1}^{-}, ν_{2}^{-})$ . Correspondingly, we can verify that $T^{+} (ν_{1}^{+}, ν_{2}^{+}) min (ν_{1}^{+}, ν_{2}^{+})$ for $ν_{1}^{+}, ν_{2}^{+} \in [0, 1]$ .

(c) If p = ∞, then the operational laws (i) and (ii) given in Definition 4.2 reduces to Min-Max form of operations, i.e.,

$ℑ_{1} \oplus ℑ_{2} = (\begin{matrix} [max (μ_{1}^{-}, μ_{2}^{-}), max (μ_{1}^{+}, μ_{2}^{+})], \\ [min (ν_{1}^{-}, ν_{2}^{-}), min (ν_{1}^{+}, ν_{2}^{+})] \end{matrix})$ ,

$ℑ_{1} \otimes ℑ_{2} = (\begin{matrix} [min (μ_{1}^{-}, μ_{2}^{-}), min (μ_{1}^{+}, μ_{2}^{+})], \\ [max (ν_{1}^{-}, ν_{2}^{-}), max (ν_{1}^{+}, ν_{2}^{+})] \end{matrix})$ .

Proof of the Theorem 4.4: Theorem 4 can be verified with the mathematical induction principle.

For m = 2, Equation (8) becomes $ϖ_{1} ℑ_{1} =$ $(\begin{matrix} [\begin{matrix} min ({(ϖ_{1} {(μ_{1}^{-})}^{p})}^{1 / p}, 1), \\ min ({(ϖ_{1} {(μ_{1}^{+})}^{p})}^{1 / p}, 1) \end{matrix}], \\ [\begin{matrix} max ((1 - {(ϖ_{1} {(1 - ν_{1}^{-})}^{p})}^{1 / p}), 0), \\ max ((1 - {(ϖ_{1} {(1 - ν_{1}^{+})}^{p})}^{1 / p}), 0) \end{matrix}] \end{matrix}),$ $\begin{matrix} ϖ_{2} ℑ_{2} = \\ (\begin{matrix} [\begin{matrix} min ({(ϖ_{2} {(μ_{2}^{-})}^{p})}^{1 / p}, 1), \\ min ({(ϖ_{2} {(μ_{2}^{+})}^{p})}^{1 / p}, 1) \end{matrix}], \\ [\begin{matrix} max ((1 - {(ϖ_{2} {(1 - ν_{2}^{-})}^{p})}^{1 / p}), 0), \\ max ((1 - {(ϖ_{2} {(1 - ν_{2}^{+})}^{p})}^{1 / p}), 0) \end{matrix}] \end{matrix}) . \end{matrix}$

Therefore, we have $ϖ_{1} ℑ_{1} + ϖ_{2} ℑ_{2} =$ $(\begin{matrix} [\begin{matrix} min (\sum_{i = 1}^{2} {(ϖ_{i} {(μ_{i}^{-})}^{p})}^{1 / p}, 1), \\ min (\sum_{i = 1}^{2} {(ϖ_{i} {(μ_{i}^{+})}^{p})}^{1 / p}, 1) \end{matrix}], \\ [\begin{matrix} max ((1 - \sum_{i = 1}^{2} {(ϖ_{i} {(1 - ν_{i}^{-})}^{p})}^{1 / p}), 0), \\ max ((1 - \sum_{i = 1}^{2} {(ϖ_{i} {(1 - ν_{i}^{+})}^{p})}^{1 / p}), 0) \end{matrix}] \end{matrix}) .$

Thus, Equation (8) holds for m = 2. Let Equation (8) holds for m = t, then $IVIFYWA (ℑ_{1}, ℑ_{2}, . . ., ℑ_{t}) =$ $(\begin{matrix} [\begin{matrix} min (\sum_{i = 1}^{t} {(ϖ_{i} {(μ_{i}^{-})}^{p})}^{1 / p}, 1), \\ min (\sum_{i = 1}^{t} {(ϖ_{i} {(μ_{i}^{+})}^{p})}^{1 / p}, 1) \end{matrix}], \\ [\begin{matrix} max ((1 - \sum_{i = 1}^{t} {(ϖ_{i} {(1 - ν_{i}^{-})}^{p})}^{1 / p}), 0), \\ max ((1 - \sum_{i = 1}^{t} {(ϖ_{i} {(1 - ν_{i}^{+})}^{p})}^{1 / p}), 0) \end{matrix}] \end{matrix}) .$

Now, for m = t + 1, we have $IVIFYWA (ℑ_{1}, ℑ_{2}, . . ., ℑ_{t}, ℑ_{t + 1})$ $= \oplus_{i = 1}^{t} ϖ_{i} ℑ_{i}, \oplus ϖ_{t + 1} ℑ_{t + 1}$ $(\begin{matrix} [\begin{matrix} min (\sum_{i = 1}^{t} {(ϖ_{i} {(μ_{i}^{-})}^{p})}^{1 / p}, 1), \\ min (\sum_{i = 1}^{t} {(ϖ_{i} {(μ_{i}^{+})}^{p})}^{1 / p}, 1) \end{matrix}], \\ [\begin{matrix} max ((1 - \sum_{i = 1}^{t} {(ϖ_{i} {(1 - ν_{i}^{-})}^{p})}^{1 / p}), 0), \\ max ((1 - \sum_{i = 1}^{t} {(ϖ_{i} {(1 - ν_{i}^{+})}^{p})}^{1 / p}), 0) \end{matrix}] \end{matrix}) \oplus$ $(\begin{matrix} [\begin{matrix} min ({(ϖ_{t + 1} {(μ_{t + 1}^{-})}^{p})}^{1 / p}, 1), \\ min ({(ϖ_{t + 1} {(μ_{t + 1}^{+})}^{p})}^{1 / p}, 1) \end{matrix}], \\ [\begin{matrix} max ((1 - {(ϖ_{t + 1} {(1 - ν_{t + 1}^{-})}^{p})}^{1 / p}), 0), \\ max ((1 - {(ϖ_{t + 1} {(1 - ν_{t + 1}^{+})}^{p})}^{1 / p}), 0) \end{matrix}] \end{matrix}) .$

This implies that $IVIFYWA (ℑ_{1}, ℑ_{2}, . . ., ℑ_{t}, ℑ_{t + 1}) = \oplus_{i = 1}^{t + 1} ϖ_{i} ℑ_{i}$ $= (\begin{matrix} [\begin{matrix} min (\sum_{i = 1}^{t + 1} {(ϖ_{i} {(μ_{i}^{-})}^{p})}^{1 / p}, 1), \\ min (\sum_{i = 1}^{t + 1} {(ϖ_{i} {(μ_{i}^{+})}^{p})}^{1 / p}, 1) \end{matrix}], \\ [\begin{matrix} max ((1 - \sum_{i = 1}^{t + 1} {(ϖ_{i} {(1 - ν_{i}^{-})}^{p})}^{1 / p}), 0), \\ max ((1 - \sum_{i = 1}^{t + 1} {(ϖ_{i} {(1 - ν_{i}^{+})}^{p})}^{1 / p}), 0) \end{matrix}] \end{matrix}) .$

It means that Equation (8) is valid for m = t + 1. Thus, Equation (8) is true for all m.

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