The objective of the paper is to introduce a novel approach using the multi-attribute border approximation area comparison (MABAC) approach under intuitionistic fuzzy sets (IFSs) to solve the smartphone selection problem with incomplete weights or completely unknown weights. A novel discrimination measure of IFSs is proposed to calculate criteria weights. In view of the fact that the ambiguity is an unavoidable feature of multiple-criteria decision-making (MCDM) problems, the proposed approach is an innovative process in the decision-making under uncertain settings. To express the utility and strength of the developed approach for solving problems in the area of MCDM, a smartphone selection problem is demonstrated. To validate the IF-MABAC approach, a comparative discussion is made between the outcomes of the developed and those of the existing methods. The outcomes of analysis demonstrate that the introduced method is well-ordered and effective with the existing ones.

Until now, users’ attentiveness in mobile communication is increasing and is analogous to this concern; Smartphone developers have manufactured various latest models. As per the survey report of the International Telecommunication Union (ITU) (ITU, 2017), the utilization of communication technologies is increased, while communication costs are reduced. Due to quick evolution in Smartphone models, the subscribers have faced decision making complexity when acquiring the most desirable Smartphone. Moreover, the young generations are using Smartphones not only for phone calls, but also for numerous functions, viz., internet access, camera, music, and video players, and so on. For that reason, the customers desire to select the Smartphone by considering different qualitative and quantitative criteria. Quantitative criteria contain camera quality, RAM size, battery capacity, built-in memory, screen dimension, processor type, and cost, while qualitative criteria contain durability, user-friendliness, and brand.

Nowadays, MCDM approaches are extensively applied to elucidate the problems, namely Smartphone selection problem. However, MCDM problems differ according to the solution status and the approaches’ implementation. Up until now various MCDM approaches have been proposed in the literature, like the TOPSIS (Akyene,

The MABAC is an original MCDM approach pioneered by Pamučar and Ćirović (

Discrimination and entropy and measures are prominent tools for tackling the ambiguous information in the various fields. Entropy measures, measurement of the degree of fuzziness for FSs and IFSs have gained huge concentration from scholars in various disciplines (Liao

Nevertheless, from the literature, it is examined that all the measures do not incorporate the decision expert opinion of the preferences into the measure. In addition, the existing measure is in linear order and does not show the accurate behaviour of alternatives. As a result, by concentrating the standards of flexibility and proficiency of IFSs, this study proposes novel parametric discrimination measures. It has been observed from the literature that the existing discrimination measures are the special cases of the developed one. Next, to estimate the weights criteria, the developed IF-discrimination measures have been applied. Using this procedure for weighting criteria, an intuitionistic fuzzy MABAC (IF-MABAC) approach is developed to deal with MCDM problems. Now, we implement only subjective considerations of options; however, developed method is appropriate for ordinary MCDM circumstances with objective and/or subjective evaluations. Further, a Smartphone selection problem is considered to elucidate the procedure and interpret the performance of IF-MABAC approach in real case decision-making issues. To illustrate the reliability of the results, a comparative discussion between our developed approach and the other current approaches is performed to determine the validity of the results.

However, according to the above motivations, the main contributions of the paper are pointed out as

New IF-discrimination measures using the characteristics of IFSs are proposed and compared with other current discrimination measures under IFSs.

Considering the discrimination between alternatives, a procedure to assess the criteria weights is carried out.

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.

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.

The organization of this paper is as follows. In Section

For the first time, Pamučar and Ćirović (

Summary of the related works of MABAC method.

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

Roy |
MABAC | Type-2 trapezoidal fuzzy sets environment | System analysis engineer selection |

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

Peng |
MABAC, EDAS | IVIFSs | Investment company |

Ji |
ELECTRE, MABAC | SVN linguistic sets | Outsourcing provider selection |

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

Vesković |
Delphi, MABAC SWARA | Conventional MCDM | Railway management |

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

Bojanic |
Fuzzy AHP, MABAC | Interval of fuzzy numbers | Military decision-making process |

Hu |
MABAC | Interval type-2 fuzzy numbers (IT2FNs) | Patient care assessment |

Jia |
MABAC | Intuitionistic fuzzy rough numbers | Medical devices supplier selection |

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

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

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

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

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

Liu ( |
MABAC | IVIFSs | Radiation therapy assessment |

Božanić |
MABAC | Conventional MCDM | Defensive operation |

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

Liang |
MABAC | IFSs | Human resource management problem |

Shen |
MABAC | Z-number | Circular economy development selection |

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

Mishra |
MABAC | IVIFSs | Programming language assessment |

Wang |
MABAC | Q-rung orthopair fuzzy sets | Construction projects selection |

Wei |
MABAC | Uncertain probabilistic linguistic sets (UPLTSs) | Green supplier selection |

In the current decade, the applications of IFSs and information measures, namely, discrimination, entropy, and similarity, have been investigated by various scholars in different regions (Deng

This part of the paper presents some basic information of IFSs and the IF-discrimination measures.

Atanassov (

An IFS

The hesitancy degree of an element

For effortlessness, an intuitionistic fuzzy number (IFN) is characterized by

Let

The discrimination measure is a recognized device to measure the discrimination degree in IFSs. Later on, Montes

A mapping

Various existing IF-discrimination measures are analysed from the literature. The details of recent discrimination measures are listed as follows:

Maheshwari and Srivastava (

Verma and Sharma (

Garg (

Srivastava and Maheshwari (

Ohlan (

Mishra

Mishra

From the above discussions, it has been examined that all measures do not incorporate decision experts’ preferences into the measure. Keeping in mind the flexibility and efficiency of criteria for IFSs, this paper develops generalized discrimination measure to evaluate the fuzziness degree of a set.

In the following sub-section, to evade the drawbacks of current discrimination measures, novel IF-discrimination measures are developed.

Here, we have proposed some flexible and generalized parametric IF-discrimination measures. Various attractive properties of developed ones are being studied.

Let

Let

A parametric symmetric IF-discrimination measure between IFSs

To indicate the superiority of the developed IF-discrimination measures, we compared the developed IF-discrimination and the current discrimination measures. A comparison is employed based on the extensively utilized counter-intuitive cases. Table

Comparison of IF-discrimination measures (counter-intuitive cases are in bold type).

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

0.0050 | 0.0013 | ||||

0.0226 | 0.0072 | 0.0233 | 0.0063 | ||

0.0078 | 0.0026 | 0.0081 | 0.0023 | ||

0.8385 | 0.7852 | 0.8052 | 0.7882 | ||

0.4122 | 0.3581 | 0.4122 | 0.3581 | ||

0.0050 | |||||

0.0017 | 0.0005 | ||||

0.2402 | 0.0027 | 0.0010 | |||

0.0555 | 0.0173 | 0.9353 | 0.0565 | 0.0157 |

From Table

Here, the MABAC method is explored for solving the MCDM issues under IFSs.

Let

A graphical presentation of the proposed IF-MABAC algorithm.

Construct a group

Aggregate the individual DEs assessment matrices

Let

The normalized IF-ADM

When the weight

The matrix for BAA

With the proposed IF-discrimination measure, the degree of discriminations of the alternative from the BAA are determined by

The degrees of performance function for an alternative are determined to add the discriminations from the BAA for each alternative and are specified by

Next, the preference order of their degree of performance function for the alternative is evaluated and the desirable Smartphone for the given SPS problem can be demonstrated.

In the present section, the developed IF-MABAC approach is implemented to solve SPS problem. Seven Smartphones as alternatives are considered as follows: Apple

Hierarchical configuration of Smartphone selection problem.

Here, Table

The LTs to rate the significant criteria and DEs.

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) |

The LTs to rate the Smartphones selection.

LTs | IFNs |

Extremely High (EH) | |

Very High (VH) | |

High (H) | |

Average (A) | |

Low (L) | |

Very Low (VL) | |

Extremely Low (EL) |

The significance of the weights by experts.

LTs | Very significant | Significant | Moderate |

IFNs | |||

Weight | 0.3709 | 0.3470 | 0.2821 |

The linguistic variable for Smartphones rating.

Parameters | Smartphone | Experts | ||

Price |
H | H | H | |

L | L | A | ||

H | H | H | ||

H | VH | H | ||

VL | H | H | ||

VL | A | VH | ||

H | VH | H | ||

Battery power |
H | VH | VH | |

A | VH | H | ||

VH | VH | A | ||

VH | H | H | ||

A | VH | H | ||

H | A | VH | ||

H | VH | VH | ||

Camera |
A | VH | VH | |

H | A | A | ||

A | A | H | ||

A | VH | VH | ||

L | VH | H | ||

L | VH | H | ||

A | VH | VH | ||

Storage capacity and RAM |
L | A | VH | |

VH | A | VH | ||

L | H | VH | ||

L | H | H | ||

L | H | VH | ||

H | L | H | ||

VH | A | VH | ||

Processor type |
A | H | H | |

A | H | H | ||

H | H | A | ||

H | H | A | ||

A | H | A | ||

H | H | A | ||

VH | VH | VH | ||

Screen size |
VH | VH | H | |

H | VH | A | ||

H | H | VH | ||

H | H | VH | ||

H | VH | A | ||

VH | VH | H | ||

VH | VH | H | ||

Ease of use |
A | H | VH | |

A | H | VH | ||

H | L | A | ||

H | L | A | ||

L | H | H | ||

L | VH | H | ||

VH | H | VH | ||

Operating system |
A | A | H | |

H | A | H | ||

A | A | H | ||

A | VH | A | ||

A | VH | A | ||

H | A | H | ||

A | A | VH |

According to DEs weights obtained by Eq. (

The IF-ADM for Smartphones.

(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) | |

(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) | |

(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) | |

(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) | |

(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) | |

(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) | |

(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) | |

(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) |

The normalized IF-ADM for Smartphones.

(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) | |

(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) | |

(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) | |

(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) | |

(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) | |

(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) | |

(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) | |

(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) |

Using Eqs. (

The weighted IF-ADM for Smartphone selection.

(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) | |

(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) | |

(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) | |

(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) | |

(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) | |

(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) | |

(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) | |

(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) |

The BAA

Next, the discrimination matrix of SPSs option from BAA is evaluated by Eq. (

The discrimination matrix of all alternatives from the BAA for Smartphones.

Rank | ||||||||||

−0.0001 | 0.00004 | 0.00033 | −0.00009 | −0.00005 | 0.0039 | 0.0004 | −0.0003 | 0.00413 | 2 | |

0.0026 | 0.00002 | 0.0003 | 0.0008 | 0.00005 | 0.00003 | 0.0001 | 0.0001 | 0.00400 | 3 | |

0.00005 | 0.00003 | 0.0004 | 0.000003 | 0.00002 | 0.00002 | 0.0006 | 0.0003 | 0.001423 | 5 | |

0.0007 | 0.0000 | 0.0003 | 0.0003 | 0.00002 | 0.00002 | 0.0006 | 0.00004 | 0.00198 | 4 | |

0.0005 | 0.00002 | 0.00002 | 0.00002 | 0.0002 | 0.00003 | 0.00005 | 0.00004 | 0.00088 | 6 | |

0.0001 | 0.00004 | 0.00002 | 0.0003 | 0.00002 | 0.00005 | 0.00001 | 0.0001 | 0.00064 | 7 | |

0.0007 | 0.00004 | 0.0003 | 0.0008 | 0.0028 | 0.00005 | 0.0014 | 0.00008 | 0.00617 | 1 |

Here, we illustrate a comparative evaluation with the existing method to show the validity and usefulness of the IF-MABAC approach based on IF-discrimination measures. We have implemented the same numerical example applying the developed approach for comparing with the existing approaches.

The above Smartphone selection problem is also solved by the ANP-Generalized Choquet integral method (Yildiz and Ergul,

Rankings order comparison of Smartphones with different methods.

Discussion of the developed method with current methods.

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

Yildiz and Ergul ( |
FSs | ANP – Generalized choquet integral | ANP | Assumed | ||

Belbag |
FSs | Fuzzy ELECTRE | TFNs | Assumed | ||

Mishra and Rani ( |
IFSs | IF-VIKOR | Shapley function with entropy method | Not considered | ||

Proposed method | IFSs | IF-MABAC | Discrimination measure | Computed |

The outcomes show that the optimal preference of SPSs is the same, i.e.

The key distinctive outcomes of the developed IF-MABAC framework are as follows:

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.

The developed approach utilizes IFSs to develop the procedure, different from the methods in Yildiz and Ergul (

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

Multiple DEs have been selected in the developed method whose weights are given in terms of IFNs, while the methodology proposed in Yildiz and Ergul (

Criteria weights in the developed IF-MABAC method are provided as IFNs, whereas in Belbag

With the use of technology, human life becomes more comfortable, and therefore it becomes a requisite for users. Several brands or products materialize on the business world with fast-growing technology and Smartphones are one of these products. A desirable Smartphone selection from the available options is a complex problem since it has different types of processors, RAM in GB, screens with HD resolution, O/S, etc. Several interesting criteria affect the SPS, as similar to various products. Hence, MCDM approaches can facilitate to evaluate SPS problem. Here, an integrated approach based on MABAC under IFSs was developed to assess the SPS problem. To compute the weight of the vector, new IF-discrimination measures were developed, and some useful properties were presented. The novel developed discrimination measure based on IFSs is verified, it would solve the problem of some current distance measures. The assessment of each SPSs alternative over different criteria was assessed on IFSs, and a new IF-MABAC framework was applied to prefer the most desirable Smartphone. To investigate the usefulness of the IF-MABAC method, comparative analyses with existing approaches were presented. The computational findings found that the ranking outcomes achieved based on the IF-MABAC method were reliable with existing ones; and hence, the developed method was sound to the SPSs under uncertainty. By employing the integrated IF-MABAC approach, a more consistent and best ranking findings of SPS case would be obtained, which help to make the accurate decision for selection of smartphone.

Further, we will integrate the MABAC framework with various other procedures, viz., CRITIC, AHP and SWARA, in the MCDM process. Also, the introduced approach would be employed for deciphering the several real-world problems, namely, supplier or material selection, and electric vehicles charging station selection to elucidate its strength and usefulness.