## 1 Introduction

*et al.*, 2017a; Büyüközkan and Güleryüz, 2016), VIKOR (Vls Kriteriju miska Optimizacija I Kompromisno Resenje) (Hu

*et al.*, 2014; Mishra and Rani, 2019; Rani and Mishra, 2020a; Rani

*et al.*, 2019b), ELECTRE (ELimination and Choice Expressing REality) (Chen

*et al.*, 2018; Mishra

*et al.*, 2020a), WASPAS (Weighted Aggregates Sum Product Assessment) (Mishra

*et al.*, 2019a; Rani and Mishra, 2020b), PROMETHEE (Rani and Jain, 2017; Liao

*et al.*, 2018), MULTIMOORA (Wu

*et al.*, 2018) and GLDS (Wu and Liao, 2019) methods. From the literature, various MCDM approaches have been applied to identify the most desirable Smartphone (Hu

*et al.*, 2014; Akyene, 2012; Büyüközkan and Güleryüz, 2016; Wu

*et al.*, 2018). Hu

*et al.*(2018) proposed a procedure that can promote mobile-commerce improvement towards attaining the aspiration level in a fuzzy setting. They developed fusion model to conduct the feedback-effect and dependency among criteria, and it combined the DEMATEL, DANP, and GRA methods.

*et al.*, 2016) and programming language selection (Mishra

*et al.*, 2020d). Therefore, it is an attractive explorative way to implement the MABAC in the Smartphone selection. Atanassov (1986) developed the notion of IFSs which extends the fuzzy sets doctrine by accumulating the non-membership degree. As IFSs doctrine has widely been implemented by the researchers in various disciplines for handling uncertainties in the MCDM (Liu and Liao, 2017; Mishra and Rani, 2019), their analogous analysis is significant.

*et al.*, 2014; Tang and Liao, 2019). To evaluate the discrimination information between IFSs, first, Vlachos and Sergiadis (2007) proposed IF-discrimination measure, established relation between them and implemented it in various disciplines. Consequently, various prominent discrimination measures have been introduced for FSs and IFSs (Mishra

*et al.*, 2017b; Ansari

*et al.*, 2018; Rani

*et al.*, 2019a; Jiang

*et al.*, 2019; Liang

*et al.*, 2019b; Rani

*et al.*, 2020; Mishra

*et al.*, 2020b, 2020c; Kumari and Mishra, 2020).

- i. New IF-discrimination measures using the characteristics of IFSs are proposed and compared with other current discrimination measures under IFSs.
- ii. Considering the discrimination between alternatives, a procedure to assess the criteria weights is carried out.
- iii. After defining the border approximation area (BAA) matrix using the proposed discrimination measure, an integrated MCDM method, IF-MABAC, is developed for MCDM problems under intuitionistic fuzzy environment.
- iv. Considering a real-life smartphone selection problem, the IF-MABAC approach is implemented to choose the desirable smartphone. The usefulness of the introduced approach is examined by comparing it with existing approaches.

## 2 Literature Review

### 2.1 An Overview of MABAC Method

*et al.*(2019a) introduced MABAC technique to assess rockburst risks under triangular fuzzy numbers (TFNs). Xue

*et al.*(2016) proposed IVIF-MABAC approach to assess the material selection. Gigović

*et al.*(2017) presented a combined method with DEMATEL, MABAC, Geographic Information Systems (GIS) and ANP to select the location for the wind farms. Peng and Dai (2018) established a new model on single-valued neutrosophic (SVN) and similarity measure and distance measure to solve MADM problem based on MABAC and TOPSIS procedures. Yu

*et al.*(2017) proposed a method based on MABAC under interval type-2 fuzzy numbers (IT2FNs) for selecting the best hotel on a tourism website. Sun

*et al.*(2018) established a projection-based MABAC approach under hesitant fuzzy linguistic term sets (HFLTSs) to select and evaluate patients. The summary of other related papers is presented in Table 1.

##### Table 1

Authors | Method | Fuzzy and conventional environment | Application area |

Roy et al. (2016) | MABAC | Type-2 trapezoidal fuzzy sets environment | System analysis engineer selection |

Peng and Dai (2017) | MABAC, COPRAS, WASPAS, | HFSSs | Software development project |

Peng et al. (2017) | MABAC, EDAS | IVIFSs | Investment company |

Ji et al. (2018) | ELECTRE, MABAC | SVN linguistic sets | Outsourcing provider selection |

Nunić (2018) | MABAC, WASPAS, ARAS, FUCOM | Conventional MCDM | Manufacturer PVC carpentry |

Vesković et al. (2018) | Delphi, MABAC SWARA | Conventional MCDM | Railway management |

Bozanic et al. (2018) | Fuzzy MABAC, fuzzy Analytic Hierarchy Process (AHP) | Saaty’s fuzzy sets | Deep wading location selection |

Bojanic et al. (2018) | Fuzzy AHP, MABAC | Interval of fuzzy numbers | Military decision-making process |

Hu et al. (2019) | MABAC | Interval type-2 fuzzy numbers (IT2FNs) | Patient care assessment |

Jia et al. (2019) | MABAC | Intuitionistic fuzzy rough numbers | Medical devices supplier selection |

Božanić et al. (2019) | Full Consistency Method. (FUCOM), fuzzy MABAC | Triangular fuzzy number | Location selection for bridge construction |

Biswas and Das (2019) | MABAC, fuzzy AHP | Fuzzy sets | Commercially available electric vehicle |

Majchrzycka and Poniszewska-Maranda (2018) | MABAC | Conventional MCDM | Mobile access control |

Biswas and Das (2018) | MABAC | Conventional MCDM | Hybrid vehicle selection |

Luo and Liang (2019) | MABAC | Linguistic neutrosophic numbers | Roadway support schemes |

Liu (2019) | MABAC | IVIFSs | Radiation therapy assessment |

Božanić et al. (2016) | MABAC | Conventional MCDM | Defensive operation |

Pamučar and Božanić (2019) | MABAC | SVNSs | Logistics center selection |

Liang et al. (2019a) | MABAC | IFSs | Human resource management problem |

Shen et al. (2020) | MABAC | Z-number | Circular economy development selection |

Dorfeshan and Mousavi (2020) | MABAC, WASPAS | IT2FSs | Aircraft maintenance planning |

Mishra et al. (2020c) | MABAC | IVIFSs | Programming language assessment |

Wang et al. (2020) | MABAC | Q-rung orthopair fuzzy sets | Construction projects selection |

Wei et al. (2020) | MABAC | Uncertain probabilistic linguistic sets (UPLTSs) | Green supplier selection |

### 2.2 Review of Discrimination Measures of IFSs

*et al.*, 2015; Bao

*et al.*, 2017; Cavallaro

*et al.*, 2018, 2019; Kong

*et al.*, 2018; Lohrmann

*et al.*, 2018; Luo and Zhao, 2018; Ngan

*et al.*, 2018; Shen

*et al.*, 2018). Jia

*et al.*(2019) introduced a new IF-similarity measure of pattern recognition problem based on isosceles triangles. Bao

*et al.*(2017) presented a new approach according to evidential reasoning and prospect theory and extended new measures for IF-entropy and discrimination measure in the field of international shipping market. Shen

*et al.*(2018) generalized the IF-TOPSIS approach derived from similarity and distance measures for handling the risk assessment of MCDM issue. Luo and Zhao (2018) developed an IF-distance measure-based on a strictly increasing binary function and matrix norm for evaluating the medical diagnosis. Deng

*et al.*(2015) investigated monotonic similarity and geometrical relation measures under IFSs based on inclusion and entropy measures. Cavallaro

*et al.*(2018) and Cavallaro

*et al.*(2019) extended an IFs based on fuzzy Shannon entropy measure and extended IF-TOPSIS based on circular entropy weights vector for evaluating of the concentrated solar power (CSP).

## 3 Intuitionistic Fuzzy Sets and Existing Discrimination Measures

### 3.1 The Concepts Related to IFSs

##### Definition 1 *(Intutionistic fuzzy sets, see* Atanassov, 1986*).*

*E*on universe set $U=\{{u_{1}},{u_{2}},\dots ,{u_{n}}\}$ is described by

##### (1)

\[ E=\big\{\big\langle {u_{i}},{\mu _{E}}({u_{i}}),{\nu _{E}}({u_{i}})\big\rangle :{u_{i}}\in U\big\},\]*E*in

*U*, correspondingly, under the condition

*E*is defined by

##### (3)

\[ \mathbb{S}({\theta _{j}})=({\mu _{j}}-{\nu _{j}}),\hspace{2em}\hslash ({\theta _{j}})=({\mu _{j}}+{\nu _{j}}),\]*et al.*(2015) modified a concept of score values for IFN and given by

##### (4)

\[ {\mathbb{S}^{\ast }}({\theta _{j}})=\frac{1}{2}\big(\mathbb{S}({\theta _{j}})+1\big),\hspace{2em}{\hslash ^{{^{\circ }}}}({\theta _{j}})=1-\hslash ({\theta _{j}}),\]##### (5)

\[ {\textit{IFWA}_{w}}({\theta _{1}},{\theta _{2}},\dots ,{\theta _{n}})=\Bigg[1-{\prod \limits_{j=1}^{n}}{(1-{\mu _{j}})^{{\varpi _{j}}}},{\prod \limits_{j=1}^{n}}{\nu _{j}^{{\varpi _{j}}}}\Bigg],\]*et al.*(2015) demonstrated the discrimination measure is the more restrictive way when the comparison is performed with other measures and necessary for avoiding counter-intuitive situations.

##### Definition 2 *(Discrimination measure, see* Montes *et al.*, 2015*).*

*L*satisfies the following postulates:

### 3.2 Existing Discrimination Measures for IFSs

*et al.*(2019b):

*et al.*(2020d):

## 4 New IF-Discrimination Measure and Comparison

### 4.1 New Discrimination Measure for IFSs

##### Definition 3.

##### (6)

\[\begin{aligned}{}{L_{1}}(E,F)& =\frac{1}{2n\ln 2}{\sum \limits_{i=1}^{n}}\Bigg[\big(\big({\mu _{E}}({u_{i}})+{\mu _{F}}({u_{i}})\big)\big)\ln \bigg\{\frac{({\mu _{E}}({u_{i}})+{\mu _{F}}({u_{i}}))}{\frac{1}{2}{(\sqrt{{\mu _{E}}({u_{i}})}+\sqrt{{\mu _{F}}({u_{i}})})^{2}}}\bigg\}\\ {} & \hspace{1em}+\big(\big({\nu _{E}}({u_{i}})+{\nu _{F}}({u_{i}})\big)\big)\ln \bigg\{\frac{({\nu _{E}}({u_{i}})+{\nu _{F}}({u_{i}}))}{\frac{1}{2}{(\sqrt{{\nu _{E}}({u_{i}})}+\sqrt{{\nu _{F}}({u_{i}})})^{2}}}\bigg\}\Bigg].\end{aligned}\]##### (7)

\[\begin{aligned}{}{L_{2}}(E,F)& =\frac{1}{2n\ln 2}{\sum \limits_{i=1}^{n}}\Bigg[\bigg(\frac{({\mu _{E}}({u_{i}})+{\mu _{F}}({u_{i}}))+2-({\nu _{E}}({u_{i}})+{\nu _{F}}({u_{i}}))}{2}\bigg)\\ {} & \hspace{1em}\times \ln \bigg\{\frac{({\mu _{E}}({u_{i}})+{\mu _{F}}({u_{i}}))+2-({\nu _{E}}({u_{i}})+{\nu _{F}}({u_{i}}))}{\frac{1}{2}{(\sqrt{{\mu _{E}}({u_{i}})+1-{\nu _{E}}({u_{i}})}+\sqrt{{\mu _{F}}({u_{i}})+1-{\nu _{F}}({u_{i}})})^{2}}}\bigg\}\\ {} & \hspace{1em}+\bigg(\frac{({\nu _{E}}({u_{i}})+{\nu _{F}}({u_{i}}))+2-({\mu _{E}}({u_{i}})+{\mu _{F}}({u_{i}}))}{2}\bigg)\\ {} & \hspace{1em}\times \ln \bigg\{\frac{({\nu _{E}}({u_{i}})+{\nu _{F}}({u_{i}}))+2-({\mu _{E}}({u_{i}})+{\mu _{F}}({u_{i}}))}{\frac{1}{2}{(\sqrt{{\nu _{E}}({u_{i}})+1-{\mu _{E}}({u_{i}})}+\sqrt{{\nu _{F}}({u_{i}})+1-{\mu _{F}}({u_{i}})})^{2}}}\bigg\}\Bigg].\end{aligned}\]##### Definition 4.

*E*and

*F*with $\gamma >0$ $(\gamma \ne 1)$ is proposed as follows:

##### (8)

\[\begin{aligned}{}{L_{3}}(E,F)& =\frac{1}{n({2^{(1-\gamma /2)}}-1)}{\sum \limits_{i=1}^{n}}\Bigg[{\bigg(\frac{{({\mu _{E}}({u_{i}}))^{2}}+{({\mu _{F}}({u_{i}}))^{2}}}{2}\bigg)^{\gamma /2}}\\ {} & \hspace{1em}-\frac{{\mu _{E}^{\gamma }}({u_{i}})+{\mu _{F}^{\gamma }}({u_{i}})}{2}+{\bigg(\frac{{({\nu _{E}}({u_{i}}))^{2}}+{({\nu _{F}}({u_{i}}))^{2}}}{2}\bigg)^{\gamma /2}}\\ {} & \hspace{1em}-\frac{{\nu _{E}^{\gamma }}({u_{i}})+{\nu _{F}^{\gamma }}({u_{i}})}{2}+{\bigg(\frac{{({\pi _{E}}({u_{i}}))^{2}}+{({\pi _{F}}({u_{i}}))^{2}}}{2}\bigg)^{\gamma /2}}\\ {} & \hspace{1em}-\frac{{\pi _{E}^{\gamma }}({u_{i}})+{\pi _{F}^{\gamma }}({u_{i}})}{2}\Bigg].\end{aligned}\]##### Theorem 1.

*The functions*${L_{\alpha }}(E,F)$; $\alpha =1,2,3$

*, given by*(6)–(8)

*are IF-discrimination measures*:

**(P1).**${L_{\alpha }}(E,F)={L_{\alpha }}(F,E)$

*;*$\alpha =1,2,3$

*.*

**(**$P2$

**).**${L_{\alpha }}(E,F)=0\hspace{2.5pt}\textit{iff}\hspace{2.5pt}E=F$

*.*

**(P3).**${L_{\alpha }}(E\cup P,F\cup P)\leqslant {L_{\alpha }}(E,F)$

*for every*$P\in \textit{IFS}(Z)$

*.*

**(P4).**${L_{\alpha }}(E\cap P,F\cap P)\leqslant {L_{\alpha }}(E,F)$

*for every*$P\in \textit{IFS}(Z)$

*.*

**(P5).**${L_{\alpha }}(E,E\cup F)+{L_{\alpha }}(E,E\cap F)={L_{\alpha }}(E,F)$

*.*

**(P6).**${L_{\alpha }}(E,E\cap F)={L_{\alpha }}(E,E\cup F)$

*.*

**(P7).**${L_{\alpha }}(E,F)={L_{\alpha }}\big({E^{c}},{F^{c}}\big)$

*.*

**(P8).**${L_{\alpha }}\big(E,{F^{c}}\big)={L_{\alpha }}\big({E^{c}},F\big)$

*.*

**(P9).**${L_{\alpha }}\big(E,{E^{c}}\big)=1$

*iff E is a crisp set.*

### 4.2 Comparison with the Existing IF-Discrimination Measures

##### Table 2

Case 1 | Case 2 | Case 3 | Case 4 | Case 5 | |

$E=\langle {\mu _{E}},{\nu _{E}}\rangle $ | $E=\langle 0.3,0.3\rangle $ | $E=\langle 0.3,0.4\rangle $ | $E=\langle 0.5,0.5\rangle $ | $E=\langle 0.4,0.2\rangle $ | $E=\langle 0.4,0.2\rangle $ |

$F=\langle {\mu _{F}},{\nu _{F}}\rangle $ | $F=\langle 0.4,0.4\rangle $ | $F=\langle 0.4,0.3\rangle $ | $F=\langle 0.0,0.0\rangle $ | $F=\langle 0.5,0.3\rangle $ | $F=\langle 0.5,0.2\rangle $ |

${L_{ZJ}}(E,F)$ | 0.0000 | 0.0050 | 0.0000 | 0.0000 | 0.0013 |

${L_{WY}}(E,F)$ | 0.0226 | 0.0072 | NaN | 0.0233 | 0.0063 |

${L_{V{S_{1}}}}(E,F)$ | 0.0078 | 0.0026 | NaN | 0.0081 | 0.0023 |

${L_{V{S_{2}}}}(E,F)$ | 0.8385 | 0.7852 | NaN | 0.8052 | 0.7882 |

${L_{M{S_{2}}}}(E,F)$ | 0.4122 | 0.3581 | 1.0000 | 0.4122 | 0.3581 |

${L_{O}}(E,F)$ | 0.0000 | 0.0050 | 0.0000 | 0.0000 | $\boldsymbol{-}$0.0113 |

${L_{1}}(E,F)$ | 0.0000 | 0.0017 | 0.0000 | 0.0000 | 0.0005 |

${L_{2}}(E,F)$ | 0.0025 | 0.0025 | 0.2402 | 0.0027 | 0.0010 |

${L_{3}}(E,F)$ | 0.0555 | 0.0173 | 0.9353 | 0.0565 | 0.0157 |

## 5 The IF-MABAC Approach for MCDM Problem

### 5.1 The Extended IF-MABAC Method-Based on the Discrimination Measures

**Stage 1:**Determine weights of decision experts’ (DEs)

*ℓ*DEs to include decision making concerning various perspectives. Suppose the rating specified for each DE through experts is ${E_{k}}=({\mu _{k}},{\nu _{k}},{\pi _{k}})$, $\forall k$. According to Boran

*et al.*(2009), DEs weight is calculated by

##### (9)

\[ {\lambda _{k}}=\frac{({\mu _{k}}+{\pi _{k}}(\frac{{\mu _{k}}}{{\mu _{k}}+{\nu _{k}}}))}{{\textstyle\textstyle\sum _{k=1}^{\ell }}({\mu _{k}}+{\pi _{k}}(\frac{{\mu _{k}}}{{\mu _{k}}+{\nu _{k}}}))},\hspace{1em}k=1,2,\dots ,\ell ,\]**Stage 2:**Construct IF-aggregation decision matrix (IF-ADM) over DEs weights

**Stage 3:**Evaluate the criteria weights based on the IF-discrimination measures

##### (10)

\[ {w_{j}}=\frac{{\textstyle\textstyle\sum _{i=1}^{m}}{\textstyle\textstyle\sum _{k=1}^{m}}{L_{\alpha }}({\xi _{ij}},{\xi _{kj}})}{{\textstyle\textstyle\sum _{j=1}^{n}}{\textstyle\textstyle\sum _{i=1}^{m}}{\textstyle\textstyle\sum _{k=1}^{m}}{L_{\alpha }}({\xi _{ij}},{\xi _{kj}})},\hspace{1em}\forall j,\alpha =1,2,3.\]**Stage 4:**Build the normalized IF-ADM

##### (11)

\[ {\stackrel{\frown }{\xi }_{ij}}=\left\{\begin{array}{l@{\hskip4.0pt}l}{\xi _{ij}}=\langle {\mu _{ij}},{\nu _{ij}}\rangle ,\hspace{1em}& \text{for beneficial criterion},\\ {} {({\xi _{ij}})^{c}}=\langle {\nu _{ij}},{\mu _{ij}}\rangle ,\hspace{1em}& \text{for non-beneficial criterion}.\end{array}\right.\]**Stage 5:**Evaluate the weighted IF-ADM

##### (12)

\[ {\varsigma _{ij}}={w_{j}}{\stackrel{\frown }{\xi }_{ij}}=\left\langle \big[1-{(1-{\mu _{ij}})^{{w_{j}}}}\big],\big[{({\nu _{ij}})^{{w_{j}}}}\big]\right\rangle ,\]**Stage 6:**Compute the border approximation area (BAA) matrix

##### (13)

\[ {\stackrel{\frown }{g}_{j}}={\prod \limits_{i=1}^{m}}{({\varsigma _{ij}})^{1/m}}=\left\langle \Bigg[{\prod \limits_{i=1}^{m}}{({\stackrel{\frown }{\mu }_{ij}})^{1/m}}\Bigg],\right.\left.\Bigg[1-{\prod \limits_{i=1}^{m}}{(1-{\stackrel{\frown }{\nu }_{ij}})^{1/m}}\Bigg]\right\rangle ,\]**Stage 7:**Compute the discrimination values from the BAA

##### (14)

\[ {\vartheta _{ij}}=\left\{\begin{array}{l@{\hskip4.0pt}l}L({\varsigma _{ij}},{\stackrel{\frown }{g}_{j}}),\hspace{1em}& \text{if}\hspace{2.5pt}{\varsigma _{ij}}\geqslant {\stackrel{\frown }{g}_{j}},\\ {} 0,\hspace{1em}& \text{if}\hspace{2.5pt}{\varsigma _{ij}}={\stackrel{\frown }{g}_{j}},\\ {} -L({\varsigma _{ij}},{\stackrel{\frown }{g}_{j}}),\hspace{1em}& \text{if}\hspace{2.5pt}{\varsigma _{ij}}\leqslant {\stackrel{\frown }{g}_{j}}\end{array}\right.\]*L*being demonstrated by Eq. (7).

**Stage 8:**Derive the ranking order

## 6 Application of Smartphone Selection of IF-MABAC Method

##### Table 3

LTs | IFNs |

Very Significant (VS) | (0.90, 0.10) |

Significant (S) | (0.80, 0.15) |

Moderate (M) | (0.65, 0.30) |

Insignificant (IS) | (0.45, 0.50) |

Very Insignificant (VI) | (0.20, 0.70) |

##### Table 4

LTs | IFNs |

Extremely High (EH) | $(1.00,0.00)$ |

Very High (VH) | $(0.90,0.10)$ |

High (H) | $(0.70,0.20)$ |

Average (A) | $(0.60,0.30)$ |

Low (L) | $(0.40,0.50)$ |

Very Low (VL) | $(0.20,0.70)$ |

Extremely Low (EL) | $(0.10,0.80)$ |

##### Table 5

${E_{1}}$ | ${E_{2}}$ | ${E_{3}}$ | |

LTs | Very significant | Significant | Moderate |

IFNs | $(0.90,0.10)$ | $(0.80,0.15)$ | $(0.65,0.30)$ |

Weight | 0.3709 | 0.3470 | 0.2821 |

##### Table 6

Parameters | Smartphone | Experts | ||

${E_{1}}$ | ${E_{2}}$ | ${E_{3}}$ | ||

Price $({F_{1}})$ | ${M_{1}}$ | H | H | H |

${M_{2}}$ | L | L | A | |

${M_{3}}$ | H | H | H | |

${M_{4}}$ | H | VH | H | |

${M_{5}}$ | VL | H | H | |

${M_{6}}$ | VL | A | VH | |

${M_{7}}$ | H | VH | H | |

Battery power $({F_{2}})$ | ${M_{1}}$ | H | VH | VH |

${M_{2}}$ | A | VH | H | |

${M_{3}}$ | VH | VH | A | |

${M_{4}}$ | VH | H | H | |

${M_{5}}$ | A | VH | H | |

${M_{6}}$ | H | A | VH | |

${M_{7}}$ | H | VH | VH | |

Camera $({F_{3}})$ | ${M_{1}}$ | A | VH | VH |

${M_{2}}$ | H | A | A | |

${M_{3}}$ | A | A | H | |

${M_{4}}$ | A | VH | VH | |

${M_{5}}$ | L | VH | H | |

${M_{6}}$ | L | VH | H | |

${M_{7}}$ | A | VH | VH | |

Storage capacity and RAM $({F_{4}})$ | ${M_{1}}$ | L | A | VH |

${M_{2}}$ | VH | A | VH | |

${M_{3}}$ | L | H | VH | |

${M_{4}}$ | L | H | H | |

${M_{5}}$ | L | H | VH | |

${M_{6}}$ | H | L | H | |

${M_{7}}$ | VH | A | VH | |

Processor type $({F_{5}})$ | ${M_{1}}$ | A | H | H |

${M_{2}}$ | A | H | H | |

${M_{3}}$ | H | H | A | |

${M_{4}}$ | H | H | A | |

${M_{5}}$ | A | H | A | |

${M_{6}}$ | H | H | A | |

${M_{7}}$ | VH | VH | VH | |

Screen size $({F_{6}})$ | ${M_{1}}$ | VH | VH | H |

${M_{2}}$ | H | VH | A | |

${M_{3}}$ | H | H | VH | |

${M_{4}}$ | H | H | VH | |

${M_{5}}$ | H | VH | A | |

${M_{6}}$ | VH | VH | H | |

${M_{7}}$ | VH | VH | H | |

Ease of use $({F_{7}})$ | ${M_{1}}$ | A | H | VH |

${M_{2}}$ | A | H | VH | |

${M_{3}}$ | H | L | A | |

${M_{4}}$ | H | L | A | |

${M_{5}}$ | L | H | H | |

${M_{6}}$ | L | VH | H | |

${M_{7}}$ | VH | H | VH | |

Operating system $({F_{8}})$ | ${M_{1}}$ | A | A | H |

${M_{2}}$ | H | A | H | |

${M_{3}}$ | A | A | H | |

${M_{4}}$ | A | VH | A | |

${M_{5}}$ | A | VH | A | |

${M_{6}}$ | H | A | H | |

${M_{7}}$ | A | A | VH |

##### Table 7

${M_{1}}$ | ${M_{2}}$ | ${M_{3}}$ | ${M_{4}}$ | ${M_{5}}$ | ${M_{6}}$ | ${M_{7}}$ | |

${F_{1}}$ | (0.7000, 0.2000) | (0.4648, 0.4329) | (0.7000, 0.2000) | (0.7951, 0.1572) | (0.5684, 0.3183) | (0.6502, 0.3013) | (0.7951, 0.1572) |

${F_{2}}$ | (0.8493, 0.1293) | (0.7720, 0.1828) | (0.8521, 0.1363) | (0.8004, 0.1547) | (0.7720, 0.1828) | (0.7568, 0.1893) | (0.8493, 0.1293) |

${F_{3}}$ | (0.8328, 0.1503) | (0.6405, 0.2581) | (0.6312, 0.2676) | (0.8328, 0.1503) | (0.7350, 0.2209) | (0.7350, 0.2209) | (0.8328, 0.1503) |

${F_{4}}$ | (0.6856, 0.2660) | (0.8382, 0.1464) | (0.7154, 0.2310) | (0.6121, 0.2809) | (0.7154, 0.2310) | (0.6184, 0.2749) | (0.8382, 0.1464) |

${F_{5}}$ | (0.6662, 0.2325) | (0.6662, 0.2325) | (0.6746, 0.2242) | (0.6746, 0.2242) | (0.6380, 0.2606) | (0.6746, 0.2242) | (0.9000, 0.1000) |

${F_{6}}$ | (0.8637, 0.1216) | (0.7778, 0.1763) | (0.7799, 0.1645) | (0.7799, 0.1645) | (0.7778, 0.1763) | (0.8637, 0.1216) | (0.8637, 0.1216) |

${F_{7}}$ | (0.7552, 0.1912) | (0.7552, 0.1912) | (0.5862, 0.3082) | (0.5862, 0.3082) | (0.6121, 0.2000) | (0.7350, 0.2209) | (0.8536, 0.1272) |

${F_{8}}$ | (0.6312, 0.2676) | (0.6685, 0.2302) | (0.6312, 0.2676) | (0.7527, 0.2049) | (0.7527, 0.2049) | (0.6685, 0.2302) | (0.7295, 0.2201) |

##### Table 8

${M_{1}}$ | ${M_{2}}$ | ${M_{3}}$ | ${M_{4}}$ | ${M_{5}}$ | ${M_{6}}$ | ${M_{7}}$ | |

${F_{1}}$ | (0.2000, 0.7000) | (0.4329, 0.4648) | (0.2000, 0.7000) | (0.1572, 0.7951) | (0.3183, 0.5684) | (0.3103, 0.6502) | (0.1572, 0.7951) |

${F_{2}}$ | (0.8493, 0.1293) | (0.7720, 0.1828) | (0.8521, 0.1363) | (0.8004, 0.1547) | (0.7720, 0.1828) | (0.7568, 0.1893) | (0.8493, 0.1293) |

${F_{3}}$ | (0.8328, 0.1503) | (0.6405, 0.2581) | (0.6312, 0.2676) | (0.8328, 0.1503) | (0.7350, 0.2209) | (0.7350, 0.2209) | (0.8328, 0.1503) |

${F_{4}}$ | (0.6856, 0.2660) | (0.8382, 0.1464) | (0.7154, 0.2310) | (0.6121, 0.2809) | (0.7154, 0.2310) | (0.6184, 0.2749) | (0.8382, 0.1464) |

${F_{5}}$ | (0.6662, 0.2325) | (0.6662, 0.2325) | (0.6746, 0.2242) | (0.6746, 0.2242) | (0.6380, 0.2606) | (0.6746, 0.2242) | (0.9000, 0.1000) |

${F_{6}}$ | (0.8637, 0.1216) | (0.7778, 0.1763) | (0.7799, 0.1645) | (0.7799, 0.1645) | (0.7778, 0.1763) | (0.8637, 0.1216) | (0.8637, 0.1216) |

${F_{7}}$ | (0.7552, 0.1912) | (0.7552, 0.1912) | (0.5862, 0.3082) | (0.5862, 0.3082) | (0.6121, 0.2000) | (0.7350, 0.2209) | (0.8536, 0.1272) |

${F_{8}}$ | (0.6312, 0.2676) | (0.6685, 0.2302) | (0.6312, 0.2676) | (0.7527, 0.2049) | (0.7527, 0.2049) | (0.6685, 0.2302) | (0.7295, 0.2201) |

##### Table 9

${M_{1}}$ | ${M_{2}}$ | ${M_{3}}$ | ${M_{4}}$ | ${M_{5}}$ | ${M_{6}}$ | ${M_{7}}$ | |

${F_{1}}$ | (0.0590, 0.9073) | (0.1433, 0.8115) | (0.0590, 0.9073) | (0.0456, 0.9394) | (0.0992, 0.8573) | (0.0963, 0.8893) | (0.0456, 0.9394) |

${F_{2}}$ | (0.0708, 0.9237) | (0.0557, 0.9362) | (0.0715, 0.9256) | (0.0606, 0.9301) | (0.0557, 0.9362) | (0.0534, 0.9375) | (0.0709, 0.9237) |

${F_{3}}$ | (0.1963, 0.7933) | (0.1175, 0.8475) | (0.1148, 0.8512) | (0.1963, 0.7933) | (0.1498, 0.8315) | (0.1498, 0.8315) | (0.1963, 0.7933) |

${F_{4}}$ | (0.1573, 0.8221) | (0.2362, 0.7526) | (0.1696, 0.8052) | (0.1307, 0.8288) | (0.1696, 0.8052) | (0.1328, 0.8261) | (0.2362, 0.7526) |

${F_{5}}$ | (0.1645, 0.7874) | (0.1645, 0.7874) | (0.1680, 0.7828) | (0.1680, 0.7828) | (0.1533, 0.8023) | (0.1680, 0.7828) | (0.3142, 0.6858) |

${F_{6}}$ | (0.0850, 0.9103) | (0.0649, 0.9255) | (0.0653, 0.9227) | (0.0653, 0.9227) | (0.0649, 0.9255) | (0.0850, 0.9103) | (0.0850, 0.9103) |

${F_{7}}$ | (0.2148, 0.7526) | (0.2148, 0.7526) | (0.1407, 0.8169) | (0.1407, 0.8169) | (0.1502, 0.7584) | (0.2040, 0.7715) | (0.2811, 0.7017) |

${F_{8}}$ | (0.0375, 0.9508) | (0.0414, 0.9453) | (0.0375, 0.9508) | (0.0521, 0.9411) | (0.0521, 0.9411) | (0.0414, 0.9453) | (0.0488, 0.9437) |

##### Table 10

${F_{1}}$ | ${F_{2}}$ | ${F_{3}}$ | ${F_{4}}$ | ${F_{5}}$ | ${F_{6}}$ | ${F_{7}}$ | ${F_{8}}$ | ${\mathbb{C}_{i}}$ | Rank | |

${M_{1}}$ | −0.0001 | 0.00004 | 0.00033 | −0.00009 | −0.00005 | 0.0039 | 0.0004 | −0.0003 | 0.00413 | 2 |

${M_{2}}$ | 0.0026 | 0.00002 | 0.0003 | 0.0008 | 0.00005 | 0.00003 | 0.0001 | 0.0001 | 0.00400 | 3 |

${M_{3}}$ | 0.00005 | 0.00003 | 0.0004 | 0.000003 | 0.00002 | 0.00002 | 0.0006 | 0.0003 | 0.001423 | 5 |

${M_{4}}$ | 0.0007 | 0.0000 | 0.0003 | 0.0003 | 0.00002 | 0.00002 | 0.0006 | 0.00004 | 0.00198 | 4 |

${M_{5}}$ | 0.0005 | 0.00002 | 0.00002 | 0.00002 | 0.0002 | 0.00003 | 0.00005 | 0.00004 | 0.00088 | 6 |

${M_{6}}$ | 0.0001 | 0.00004 | 0.00002 | 0.0003 | 0.00002 | 0.00005 | 0.00001 | 0.0001 | 0.00064 | 7 |

${M_{7}}$ | 0.0007 | 0.00004 | 0.0003 | 0.0008 | 0.0028 | 0.00005 | 0.0014 | 0.00008 | 0.00617 | 1 |

### 6.1 Comparison with Other Works

*et al.*, 2016) and the Shapley discrimination measure VIKOR method (Mishra and Rani, 2019). Outcomes of the different approaches were obtained to certify the outcomes of the developed IF-MABAC method. Moreover, we implement the given case study to investigate the above methods and to show the effectiveness of the proposed approach. Figure 3 and Table 11 demonstrate the preference orders of the SPSs alternatives as achieved by applying the existing methods.

##### Table 11

Methods | Discipline | Benchmark | Criterion weights | Expert weights | Ranking order | Best Smartphone |

Yildiz and Ergul (2015) | FSs | ANP – Generalized choquet integral | ANP | Assumed | ${M_{7}}\succ {M_{2}}\succ {M_{1}}\succ {M_{4}}\succ {M_{3}}\succ {M_{5}}\succ {M_{6}}$ | ${M_{7}}$ |

Belbag et al. (2016) | FSs | Fuzzy ELECTRE | TFNs | Assumed | ${M_{7}}\approx {M_{1}}\succ {M_{4}}\succ {M_{3}}\succ {M_{2}}\succ {M_{5}}\succ {M_{6}}$ | ${M_{7}}$, ${M_{1}}$ |

Mishra and Rani (2019) | IFSs | IF-VIKOR | Shapley function with entropy method | Not considered | ${M_{7}}\succ {M_{1}}\succ {M_{4}}\succ {M_{2}}\succ {M_{3}}\succ {M_{5}}\succ {M_{6}}$ | ${M_{7}}$ |

Proposed method | IFSs | IF-MABAC | Discrimination measure | Computed | ${M_{7}}\succ {M_{1}}\succ {M_{2}}\succ {M_{4}}\succ {M_{3}}\succ {M_{5}}\succ {M_{6}}$ | ${M_{7}}$ |

- i. To tackle with uncertainty in MCDM problems, all the facets, namely, the alternative on the assessments criteria by various DEs, the DEs weights, and the criteria weights are taken in the form of IFNs.
- iii. The criteria weights of proposed IF-MABAC approach are obtained through the proposed IF-discrimination measure, which gives more precise weights, different from the randomly assumed criteria weights in Belbag
*et al.*(2016).