Abstract
In this paper, we introduce some set-theoretic operations and laws of the IV-CFSSs, such as interval-valued complex fuzzy soft complement, union, intersection, t-norm, s-norm, simple product, Cartesian product, probabilistic sum, simple difference, and the convex linear sum of min and max operators. We define the distance measure of two IV-CFSSs. This distance measure is then used to define the \(\delta \)-equality of IV-CFSSs. We establish some particular examples and basic results of these operations and laws. Moreover, we use IV-CFSSs in decision-making problems. We develop a new decision-making method using the interval-valued complex fuzzy distance measures under the environments of IV-CFSSs. We discuss the real-life case based on the proposed decision-making method. A real-life example demonstrates that the decision-making method developed in the paper can be utilized to deal with problems of uncertainty. Further, the comparative study of IV-CFSSs with complex fuzzy soft sets, interval-valued fuzzy soft sets, and fuzzy soft sets is established.
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1 Introduction
Many theories are proposed to cope with uncertainty and imprecision that manage in almost all the real-life problems, such as the theory of fuzzy sets (FSs) (Zadeh 1965), theory of rough sets (Pawlak 1982), theory of intuitionistic fuzzy sets (IFSs) (Atanassov 2016), theory of Pythagorean fuzzy sets (Peng and Yang 2015), theory of complex fuzzy sets (CFSs) (Ramot et al. 2002), theory of picture fuzzy sets (Hussain et al. 2022), theory of soft sets (Molodtsov 1999), theory of fuzzy soft sets (FSSs) (Maji 2001), and theory of complex fuzzy soft sets (SSs) (Thirunavukarasu et al. 2017). All these models have their own limitations, advantages, and characteristics. These models are used in many situations of uncertainties, such as engineering, computer science, decision-making problems, networking, pattern recognition, and many other fields of science.
FSs, IFSs, and PFSs, these three types of sets all other a more flexible and nuanced representation of uncertainty and imprecision than classical set theory and have been applied in D–M, Pa–Re, and image processing. However, these three models cannot handle two-dimensional problems. To discuss two-dimensional phenomena, Ramot et al. (2002) gave the idea of complex fuzzy sets (CFSs). A CFS is the generalization of an FS whose membership function is a complex-valued function, that is, its range is the unit disk in a complex plane. The complex-valued function describes the degree of membership of a member in the CFS using both magnitude and phase information. CFSs have found applications in various fields such as image processing, Pa–Re, control systems, and D–M. They provide a powerful tool for modeling and analyzing complex systems with uncertainty and imprecision. Liu et al. (2020) discussed a D–M method based on complex fuzzy Di-Me and complex fuzzy entropy measure. Hu et al. (2018) introduced some Di-Mes in the environment of CFSs. They discussed their applications to continuity problems. Zhang et al. (2009) developed a new complex fuzzy Di-Me based on the maximum operator. They proposed an algorithm for signal processing in the environment of complex fuzzy Di-Mes. Zeeshan and Khan (2022) defined new Di-Mes in the environment of CFSs. They constructed an algorithm for the high resemblance of a signal based on complex fuzzy Di-Mes. A new algorithm for signal processing method in the environment of CFSs was proposed by Ma et al. (2019). A new self-learning complex neuro-fuzzy system based on CFSs was developed by Li et al. (2012). They utilized them in the adaptive image noise canceling process.
Bipolar fuzzy sets (BFSs) are introduced by Zhang (1994). BFS is the generalization of FSs and IFSs. Bipolar fuzzy sets find applications in various fields, including decision-making, control systems, and artificial intelligence, where uncertainty and imprecision need to be considered in a more nuanced manner than traditional fuzzy sets allow (Akram et al. 2018; Sarwar and Akram 2017; Patrascu 2015; Zararsız and Riaz 2022; Sakr et al. 2023). Alkouri et al. (2020) gave the idea of bipolar complex fuzzy sets (BCFSs). They discussed their applications in multiple attributes decision-making (MADM) problems. BCFSSs have many applications in various field of science (Gwak et al. 2023; Qiyas et al. 2024; Akram et al. 2023).
Maji (2001) developed the notion of the FSSs by associating the ideas of SSs and FSs. FSS is the generalization of FSs. The FSSs have considerable application potential in engineering, decision-making problems, computer science, and optimization (Atef et al. 2021; Petchimuthu et al. 2020; Beg et al. 2018). Močkoř and Hurtík (2021) introduced fuzzy soft relations and fuzzy soft approximations. They discussed an image processing application based on fuzzy soft relations and fuzzy soft approximations. Močkoř (2020) proposed the powerset theory of FSSs. Bhardwaj and Sharma (2021) developed a decision-making method under the environments of FSSs. Memis et al. utilized Euclidean pseudo-similarity to propose a new classification algorithm, i.e., fuzzy parameterized fuzzy soft Euclidean classifier (Memiş et al. 2021a). A new fuzzy soft expert system to predict lung cancer disease was proposed by Khalil et al. (2020). Begam et al. (2020) introduced similarity measures between lattice ordered multi-fuzzy soft sets. They discussed the applications of similarity measures of lattice ordered multi-fuzzy soft sets in decision-making problems.
Yang et al. (2009) introduced the concept of interval-valued soft set (IV-FSS) by combining the interval-valued fuzzy set and soft set models. The IV-FSS is more reasonable to give interval-valued data to describe membership degree. The IV-FSSs have desirable applications in engineering, decision-making problems, computer science, and optimization. Ma et al. (2021) proposed a decision-making method under the environments of IV-FSSs. A novel interval-valued fuzzy soft decision-making method based on Combined Compromise Solution (CoCoSo) and Criteria Importance Through Inter-criteria Correlation (CRITIC) for intelligent healthcare management evaluation was developed by Peng et al. (2021). Mohanty and Tripathy (2021) proposed an algorithm that is used to recommend the best variety of turmeric under the environments of IV-FSSs. Qin et al. (2021) discussed a new decision-making method based on an interval-valued fuzzy soft set by means of the contrast table.
Maji (2009) gave the concept of intuitionistic fuzzy soft set (IFSS) by combining the intuitionistic fuzzy set and soft set models. The IFSS is a powerful tool that is extremely useful in multicriteria decision-making (MCDM) problems. Hayat et al. (2021) proposed new aggregation operators on generalized intuitionistic fuzzy soft sets. They developed a decision-making method based on the generalized intuitionistic fuzzy soft aggregation operators. A TOPSIS method based on the correlation coefficient of interval-valued intuitionistic fuzzy soft sets (IV-IFSSs) was proposed by Zulqarnain et al. (2021). They defined interval-valued IFS weighted average and interval-valued IFS weighted geometric operators and proposed decision-making techniques based on the defined operators. The complex intuitionistic fuzzy soft prioritized weighted averaging operator, the complex intuitionistic fuzzy soft prioritized ordered weighted averaging operator, the complex intuitionistic fuzzy soft prioritized weighted geometric operator, and complex intuitionistic fuzzy soft prioritized ordered weighted geometric operator were developed by Ali et al. (2021). They discussed a decision-making method on the developed operators. Memiş et al. (2021b) defined a classification method based on Hamming pseudo-similarity of intuitionistic fuzzy parameterized intuitionistic fuzzy soft matrices. Ghosh et al. (2021) proposed a new hybrid soft computing entrenched segmentation method for the detection of breast cancer in early stages. Singh discussed the non-linear programming (NLP) methods under the environment of an IV-IFSS for solving multiattribute decision-making (MADM) problems (Singh 2021). To improve the visual quality and highlight the local details in enhanced images, Ghosh and Ghosh discussed a colonogram enhancement approach utilizing IFSSs (Ghosh and Ghosh 2021). Ali introduced complex intuitionistic fuzzy power interaction aggregation operators and discussed their applications (Ali 2022).
FSSs, IV-FSSs, IFSSs, and IV-IFSSs cannot handle imprecise, inconsistent, and incomplete information of two-dimensional phenomena. These models are applicable to different areas of science, but there is one major deficiency in these models, that is, a lack of capability to model periodic nature. To overcome this difficulty, Thirunavukarasu et al. (2017) introduced the concept of a complex fuzzy soft set (CFSS). They discussed an application in decision-making under the complex fuzzy soft information. Zeeshan et al. (2022) defined new distance measure based on CFSSs and applied them in signal processing. Mahmood et al. (2021) explored the concept of complex fuzzy N-soft sets. They developed a decision-making method under the environments of complex fuzzy N-soft sets. Akram et al. (2021a) discussed an application of Complex Spherical Fuzzy N-Soft Sets for solving a multiattribute group decision-making (MAGDM) problem, namely, the selection of a firm for participation in a Saudi oil refinery project in Pakistan. Ahsan et al. (2021) proposed a new mathematical model for the treatment of HIV using complex fuzzy hypersoft mapping. A multiattribute group decision-making method under complex spherical fuzzy N-soft sets was proposed by Akram et al. (2021b).
Selvachandran and Salleh gave the notion of IV-CFSSs by combining complex fuzzy sets with type-2 fuzzy sets and soft sets (Selvachandran and Salleh 2017). They discussed the application of the proposed model in economic problems. A generalized interval-valued complex fuzzy soft set model was proposed by Selvachandran and Salleh (2017). They discussed an application of the proposed model in multiattribute decision-making problems. Rahman et al. (2021) presented the novel concept of a complex interval-valued fuzzy hypersoft set. The interval-valued complex fuzzy soft weighted arithmetic averaging operator and the interval-valued complex fuzzy soft weighted geometric averaging operator was defined by Fan et al. (2019). They developed a decision-making method based on the proposed operators. Greenfield et al. (2016) gave the notions of interval-valued complex fuzzy logic. Dai et al. (2019) introduced distance measures between the interval-valued complex fuzzy sets. They discussed their applications in decision-making problems.
1.1 Background and motivation
IV-CFSSs are an extension of classical fuzzy soft sets that incorporate the concept of intervals and complex numbers. Here are some motivations for considering IV-CFSSs and interval-valued complex fuzzy soft distance measure in decision-making problems:
-
(1)
In many real-world situations, information is uncertain or vague. IV-CFSS provide a framework to represent uncertainty using interval-valued fuzzy sets, allowing for a more flexible representation of imprecision in the data.
-
(2)
Interval-valued fuzzy sets are a generalization of classical fuzzy sets that allow for a range of possible membership values rather than a single point value. IV-CFSS extend this idea to complex fuzzy sets, enabling a more versatile representation of uncertainty in both the real and imaginary components.
-
(3)
Fuzzy soft sets, including their interval-valued complex counterparts, find applications in decision-making processes. The flexibility of representation allows decision-makers to consider various sources of uncertainty and vagueness when making decisions.
-
(4)
Soft sets are well suited for modeling situations where information is incomplete or partial. IVCFSS enhance this modeling capability by introducing intervals and complex numbers, providing a richer representation of the uncertainty associated with the elements in the set.
-
(5)
The mathematical formalism of IV-CFSS provides a systematic way to handle uncertainty through the use of intervals and complex numbers. This allows researchers and practitioners to develop algorithms and methodologies for analysis, manipulation, and decision-making based on this formalism.
1.2 Contribution of interval-valued complex fuzzy soft sets in decision-making problems
The application of IV-CFSS in decision-making problems offers several contributions, mainly due to their ability to handle complex and uncertain information. Here are some key contributions of IVCFSS in decision-making problems:
-
(1)
IVCFSS provide a flexible and comprehensive representation of uncertainty. By incorporating intervals and complex numbers, these sets can express not only imprecision but also capture the relationships between real and imaginary components of uncertain information. This representation is valuable in decision-making scenarios where uncertainty is inherent.
-
(2)
IVCFSS allow the integration of both quantitative and qualitative information through the use of complex numbers. Decision-making often involves multiple criteria or attributes, and IV-CFSS enable the representation of these multidimensional aspects in a unified framework. This is particularly beneficial in cases where decisions are based on a combination of numerical and non-numerical factors.
-
(3)
Real-world decision-making problems are often characterized by complex relationships and uncertainty. IV-CFSS provide a more accurate modeling approach by considering the interplay of interval-valued and complex fuzzy information, offering a richer representation of the underlying complexities in the decision environment.
-
(4)
Decision-making problems may involve inconsistent or ambiguous information. IV-CFSS can accommodate such inconsistencies and ambiguities, allowing decision-makers to explicitly model and manage situations where the available information may be conflicting or unclear.
1.3 Significance of interval-valued complex fuzzy soft sets
IV-CFSSs hold significance in various fields and applications due to their ability to effectively model and handle complex, uncertain, and imprecise information. Here are some key aspects of the significance of IV-CFSSs:
-
(1)
IVCFSS provide a powerful framework for modeling uncertainty and imprecision in information. The combination of intervals and complex numbers allows for a more nuanced representation of uncertain data, addressing real-world scenarios where precise information may be unavailable or difficult to obtain.
-
(2)
IVCFSS contribute to decision-making processes by providing a soft computing approach that accommodates the uncertainty inherent in decision environments. The soft set framework allows decision-makers to model their subjective judgments, and the interval-valued complex nature enhances the realism of decision models.
-
(3)
IVCFSS seamlessly integrate with interval-valued fuzzy sets, offering a unified approach to handling uncertainty in decision-making problems. This integration is valuable in applications where both interval-valued and complex fuzzy information need to be considered simultaneously.
-
(4)
Decision problems frequently require the consideration of multiple criteria or attributes. IVCFSS enable the construction of multidimensional decision models, where both real and imaginary components can be used to represent diverse aspects of the decision-making criteria. This is especially useful in complex decision environments.
-
(5)
IVCFSS distance measures and similarity measures enable a quantitative assessment of the relationships between decision alternatives. This allows decision-makers to compare and evaluate alternatives based on a more comprehensive understanding of the similarities and differences within the decision space.
In this paper, we introduce some set-theoretic operations and laws of the IV-CFSSs, such as interval-valued complex fuzzy soft complement, union, intersection, t-norm, s-norm, simple product, Cartesian product, probabilistic sum, simple difference, and the convex linear sum of min and max operators. We propose the distance measure of two IV-CFSSs. This distance measure is then used to define the \(\delta \)-equality of IV-CFSSs. We establish some particular examples and basic results of these operations and laws. Moreover, we develop a decision-making method utilizing the interval-valued complex fuzzy distance measures under the environments of IV-CFSSs. We discuss the real-life case and comparison analysis based on the proposed decision-making method.
2 Preliminaries
In this section, we will recall some basic notions of fuzzy sets theory.
Definition 1
(Ramot et al. 2002) A CFS \(\chi \) is characterized by a grade value \(\Im _{ \mathcal {L} }(x)=\ell _{ \mathcal {L} }(x)e^{iw_{ \mathcal {L} }(x)}\) on a universe of discourse \( \mathcal {L}\). In a CFS \( \mathcal {L} \), the GV \(\Im _{ \mathcal {L} }(x)\) of an element \(x\in \mathcal {L} \) is determined by a complex-valued function rather than a simple grade value. In the grade value of CFS \( \mathcal {L}\), the term \(\ell _{ \mathcal {L} }(x)\) is said to be an amplitude term and \(w_{ \mathcal {L} }(x)\) is called a phase term.
Note that both these functions are real-valued and \(\ell _{ \mathcal {L} }(x)\in [0,1].\)
Mathematically;
It can be written as
Motivation
In many real-world scenarios, information is not only uncertain but also possesses a complex nature, characterized by both magnitude and phase. Complex fuzzy sets provide a mathematical framework to represent this complex information, allowing for a more accurate modeling of the relationships between elements. Complex fuzzy sets are built on a mathematically rigorous foundation. The use of complex numbers allows for precise and systematic representation, making it suitable for applications where a high level of mathematical rigor is required. Complex fuzzy sets can model cognitive and linguistic aspects of decision-making and reasoning. The qualitative and quantitative aspects of information in complex fuzzy sets can represent human judgments and linguistic expressions in a more nuanced manner. In summary, the motivation for complex fuzzy sets arises from the need to represent complex, multidimensional, and uncertain information in a mathematically rigorous manner. Their application spans various domains, ranging from engineering and physics to decision-making and cognitive modeling, making them a valuable tool in addressing real-world problems characterized by both complexity and uncertainty.
Definition 2
(Molodtsov 1999) Let \(\chi \) be a universe of discourse, and \( \mathcal {F} \) be a set of parameters. Let \(P(\chi )\) denotes the power set of \(\chi \). Let \( \mathcal {L} \subset \mathcal {F} \). A pair \((\Im , \mathcal {L} )\) is called a soft set over \(\chi \), where \(\Im \) is a mapping given by \(\Im : \mathcal {L} \rightarrow P(\chi )\).
Definition 3
(Maji 2001) Let \(\Im (\chi )\) denotes the set of all fuzzy sets of \(\chi \). A pair \((\Im , \mathcal {L} )\) is called a fuzzy soft set over \(\Im (\chi )\), where \(\Im \) is a mapping given by \(\Im :\tau \rightarrow P(\Im (\chi )).\)
Definition 4
(Selvachandran and Salleh 2017) Let \(\chi \) be a universe of discourse, and \( \mathcal {F} \) be a set of parameters. Let \( \mathcal {L} \subset \mathcal {F} \) and \(P(\chi )\) represent the power set of \(\chi .\) Then, a pair \((\Im , \mathcal {L} )\) is called an interval-valued complex fuzzy soft set (IV-CFSS) over \(\chi ,\) where \(\Im \) is a mapping defined by \(\Im : \mathcal {L} \rightarrow P(\chi )\), and is specified by
for all \(\alpha \in \mathcal {L} \) and \(i=\sqrt{-1}.\) Both the amplitude terms \( \mathbb {C} _{\Im _{\alpha }}^{-}(x)\) and \( \mathbb {C} _{\Im _{\alpha }}^{+}(x)\) and the phase terms \(\circledS _{\Im _{\alpha }}^{-}(x)\) and \(\circledS _{\Im _{\alpha }}^{+}(x)\) are real-valued with \( \mathbb {C} _{\Im _{\alpha }}^{-}(x)< \mathbb {C} _{\Im _{\alpha }}^{+}(x)\) and the phase terms \(\circledS _{\Im _{\alpha }} ^{-}(x)<\circledS _{\Im _{\alpha }}^{+}(x),\) and \( \mathbb {C} _{\Im _{\alpha }}^{-}(x), \mathbb {C} _{\Im _{\alpha }}^{+}(x)\in [0,1],\) \( \mathbb {C} _{\Im _{\alpha }}^{-}(x), \mathbb {C} _{\Im _{\alpha }}^{+}(x)\in [0,2\pi ].\)
Note that complex fuzzy soft set is a special case of IV-CFSS by taking \( \mathbb {C} _{\Im _{\alpha }}^{-}(x)= \mathbb {C} _{\Im _{\alpha }}^{+}(x)\) and \(\circledS _{\Im _{\alpha }}^{-}(x)=\circledS _{\Im _{\alpha }}^{+}(x).\)
Definition 5
(Selvachandran and Salleh 2017) Let \((\Im , \mathcal {L} )\) and \((\Xi ,\aleph )\) be two IV-CFSSs over \(\chi .\) Then,
(i) \((\Im , \mathcal {L} )\) is said to be a subset of \((\Xi ,\aleph ),\) denoted by \((\Im , \mathcal {L} )\subset (\Xi ,\aleph )\) if and only if \( \mathbb {C} _{\Im _{\alpha }}^{-}(x)\le \mathbb {C} _{\Xi _{\alpha }}^{-}(x)\) and \( \mathbb {C} _{\Im _{\alpha }}^{+}(x)\le \mathbb {C} _{\Xi _{\alpha }}^{+}(x)\) for the amplitude terms, and \(\circledS _{\Im _{\alpha } }^{-}(x)\le \circledS _{\Xi _{\alpha }}^{-}(x)\) and \(\circledS _{\Im _{\alpha }} ^{+}(x)\le \circledS _{\Xi _{\alpha }}^{+}(x)\) for the phase terms for all \(x\in \chi .\)
(ii) \((\Im , \mathcal {L} )\) is said to be a equal to \((\Xi ,\aleph ),\) denoted by \((\Im , \mathcal {L} )=(\Xi ,\aleph )\) if and only if \( \mathbb {C} _{\Im _{\alpha }}^{-}(x)= \mathbb {C} _{\Xi _{\alpha }}^{-}(x)\) and \( \mathbb {C} _{\Im _{\alpha }}^{+}(x)= \mathbb {C} _{\Xi _{\alpha }}^{+}(x)\) for the amplitude terms, and \(\circledS _{\Im _{\alpha } }^{-}(x)=\circledS _{\Xi _{\alpha }}^{-}(x)\) and \(\circledS _{\Im _{\alpha }} ^{+}(x)=\circledS _{\Xi _{\alpha }}^{+}(x)\) for the phase terms for all \(x\in \chi .\)
Definition 6
(Selvachandran and Salleh 2017) The complement of \((\Im , \mathcal {L} )\) denoted by \((\Im , \mathcal {L} )^{c}\) is defined as
Definition 7
(Selvachandran and Salleh 2017) Let \((\Im , \mathcal {L} )\) and \((\Xi ,\aleph )\) be two IV-CFSSs over \(\chi .\) Then, the union of \((\Im , \mathcal {L} )\) and \((\Xi ,\aleph )\) is an IV-CFSSs \((\Re ,\Pi ),\) defined as \((\Im , \mathcal {L} )\cup (\Xi ,\aleph )=(\Re ,\Pi ),\) where \(\Pi = \mathcal {L} \cup \aleph ,\) and
Definition 8
(Selvachandran and Salleh 2017) Let \((\Im , \mathcal {L} )\) and \((\Xi ,\aleph )\) be two IV-CFSSs over \(\chi .\) Then, the intersection of \((\Im , \mathcal {L} )\) and \((\Xi ,\aleph )\) is an IV-CFSSs \((\Re ,\Pi ),\) defined as \((\Im , \mathcal {L} )\cap (\Xi ,\aleph )=(\Re ,\Pi ),\) where \(\Pi = \mathcal {L} \cap \aleph ,\) and
3 Interval-valued complex fuzzy soft sets
In this section, we will introduce some new operations on IV-CFSSs.
Definition 9
1). A function \(\top :(0,1]\times (0,1]\rightarrow [0,1]\) is a quasi-triangular norm that satisfies the following conditions:
- i).:
-
\(\top (1,1)=1;\)
- ii).:
-
\(\top (a,b)=\top (b,a);\)
- iii).:
-
\(\top (a,b)\le \top (c,d);\) \(a\le c,\) \(b\le d.\)
- iv).:
-
\(\top (\top (a,b),c)=\top (a,\top (b,c)).\)
2). A function \(\top :[0,1]\times [0,1]\rightarrow [0,1]\) is a triangular norm that satisfies the conditions (i)–(iv) and the following condition:
v). \(\top (0,0)=0.\)
We said \(\top \) is an s-norm, if a triangular norm \(\top \) satisfies
vi). \(\top (a,0)=a.\)
We said \(\top \) is an t-norm, if a triangular norm \(\top \) satisfies
vii). \(\top (a,1)=a.\)
We said a binary function \(\overset{\div }{\top }:\)
where \( \mathcal {L} (\chi )\) is a collection of IV-CFSSs.
The operator \(\overset{\div }{\top }\) is a triangular norm if \(\top _{1}\) is a triangular norm and \(\top _{2}\) is a quasi-triangular norm. \(\overset{\div }{\top }\) is an s-norm if \(\top _{1}\) is an s-norm and \(\overset{\div }{\top }\) is a t-norm if \(\top _{1}\) is a t-norm.
Definition 10
Let \((\Im , \mathcal {L} )\) and \((\Xi ,\aleph )\) be two IV-CFSSs over \(\chi .\) Then, the IV-CFS product of \((\Im , \mathcal {L} )\) and \((\Xi ,\aleph )\) is an IV-CFSSs \((\Re ,\Pi ),\) denoted as \((\Im , \mathcal {L} )\bullet (\Xi ,\aleph )=(\Re ,\Pi ),\) and is specified by a function
Example 1
Let \(\chi =\{x_{1},x_{2}\}\) be a universe of discourse and \( \mathcal {L} =\aleph =\{ \alpha _{1}=beautiful,\alpha _{2}=big\}\) be sets of parameters, and
Then, the IV-CFS product of \((\Im , \mathcal {L} )\) and \((\Xi ,\aleph )\) is
Definition 11
Let \((\Im _{n}, \mathcal {L} _{n});\) \(n=1,2,\ldots ,N\) be N IV-CFSSs over \(\chi . \) Then, the IV-CFS Cartesian product of \((\Im _{n}, \mathcal {L} _{n}),\) denoted \((\Im _{1}, \mathcal {L} _{1})\times (\Im _{2}, \mathcal {L} _{2})\times \cdots \times (\Im _{n}, \mathcal {L} _{n})=(\Im , \mathcal {L} ),\) and is specified by a function
Example 2
Consider \((\Im , \mathcal {L} )\) and \((\Xi ,\aleph )\) in Example 1. Then, the Cartesian product of \((\Im , \mathcal {L} )\) and \((\Xi ,\aleph )\) is
Definition 12
Let \((\Im , \mathcal {L} )\) and \((\Xi ,\aleph )\) be two IV-CFSSs over \(\chi .\) Then, the IV-CFS probabilistic sum of \((\Im , \mathcal {L} )\) and \((\Xi ,\aleph )\) is an IV-CFSSs \((\Re ,\Pi ),\) denoted as \((\Im , \mathcal {L} )\circledast (\Xi ,\aleph )=(\Re ,\Pi ),\) and is specified by a function
Example 3
Consider \((\Im , \mathcal {L} )\) and \((\Xi ,\aleph )\) in Example 1. Then, the probabilistic sum of \((\Im , \mathcal {L} )\) and \((\Xi ,\aleph )\) is
Definition 13
Let \((\Im , \mathcal {L} )\) and \((\Xi ,\aleph )\) be two IV-CFSSs over \(\chi .\) Then, the IV-CFS simple difference of \((\Im , \mathcal {L} )\) and \((\Xi ,\aleph )\) is an IV-CFSSs \((\Re ,\Pi ),\) denoted as \((\Im , \mathcal {L} )\diagdown (\Xi ,\aleph )=(\Re ,\Pi ),\) and is specified by a function
Example 4
Consider \((\Im , \mathcal {L} )\) and \((\Xi ,\aleph )\) in Example 1. Then, the complement of \((\Xi ,\aleph )\) is
The simple difference of \((\Im , \mathcal {L} )\) and \((\Xi ,\aleph )\) is
Definition 14
Let \((\Im , \mathcal {L} )\) and \((\Xi ,\aleph )\) be two IV-CFSSs over \(\chi .\) Then, the IV-CFS convex linear sum of min and max of \((\Im , \mathcal {L} )\) and \((\Xi ,\aleph )\) is an IV-CFSSs \((\Re ,\Pi ),\) denoted as \((\Im , \mathcal {L} )\oplus _{\lambda }(\Xi ,\aleph )=(\Re ,\Pi ),\) and is specified by a function
Definition 15
A distance measure of IV-CFSSs is a function \(\Gamma :\overset{\div }{\top } (\chi )\times \overset{\div }{\top }(\chi )\rightarrow [0,1]\) with the property: for any \(\xi _{1}=(\Im , \mathcal {L} ),\xi _{2}=(\Xi ,\aleph ),\xi _{3}=(\Re ,\Pi )\in \overset{\div }{\top }(\chi )\) are three IV-CFSSs over \(\chi \)
- i).:
-
\(\Gamma (\xi _{1},\xi _{2})\ge 0,\)
- ii).:
-
\(\Gamma (\xi _{1},\xi _{2})=0\) if and only if \(\xi _{1}=\xi _{2},\)
- iii).:
-
\(\Gamma (\xi _{1},\xi _{2})=\Gamma (\xi _{2},\xi _{1}),\)
- iv).:
-
\(\Gamma (\xi _{1},\xi _{3})\le \Gamma (\xi _{1},\xi _{2})+\Gamma (\xi _{2} ,\xi _{3}).\)
In the following, the distance measure \(\Gamma \) is defined as follows:
where l denotes the length of the interval.
Example 5
Consider \(\xi =(\Im , \mathcal {L} )\) and \(\eta =(\Xi ,\aleph )\) in Example 1. Then, the IV-CFS distance measure is
Definition 16
Let \(\xi =(\Im , \mathcal {L} )\) and \(\eta =(\Xi ,\aleph )\) be two IV-CFSSs over \(\chi .\) Then, the IV-CFSSs \(\xi \) and \(\eta \) is said to be \(\delta \)-equal if and only if
4 Main results
Proposition 1
The IV-CFS union on \(\overset{\div }{\top }(\chi )\) is an s-norm.
Proof
The conditions (i), (ii), (v), and (vi) can be easily verified from definition 6. Here, we only prove the condition (iii) and (iv).
iii). Let \((\Im , \mathcal {L} ),\) \((\Xi ,\aleph ),\) \((\Re ,\Pi )\) be three IV-CFSSs over \(\chi ,\) and \(\mu _{(\Im , \mathcal {L} )}=[ \mathbb {C} _{\Im _{\alpha }}^{-}(x),\) \( \mathbb {C} _{\Im _{\alpha }}^{+}(x)].e^{i[\circledS _{\Im _{\alpha }}^{-}(x),\circledS _{\Im _{\alpha }}^{+}(x)]},\) \(\mu _{(\Xi ,\aleph )}=[ \mathbb {C} _{\Xi _{\alpha }}^{-}(x), \mathbb {C} _{\Xi _{\alpha }}^{+}(x)].e^{i[\circledS _{\Xi _{\alpha }}^{-}(x),\circledS _{\Xi _{\alpha }}^{+}(x)]},\) and \(\mu _{(\Re ,\Pi )}=[ \mathbb {C} _{\Pi _{\alpha }}^{-}(x), \mathbb {C} _{\Pi _{\alpha }}^{+}(x)].e^{i[\circledS _{\Pi _{\alpha }}^{-}(x),\circledS _{\Pi _{\alpha }}^{+}(x)]}\) their membership functions, respectively. We suppose \( \mathbb {C} _{\Im _{\alpha }}^{-}(x)\le \mathbb {C} _{\Xi _{\alpha }}^{-}(x),\) \( \mathbb {C} _{\Im _{\alpha }}^{+}(x)\le \mathbb {C} _{\Xi _{\alpha }}^{+}(x),\) \(\circledS _{\Im _{\alpha }}^{-}(x)\le \circledS _{\Xi _{\alpha }}^{-}(x),\) \(\circledS _{\Im _{\alpha }}^{+}(x)\le \circledS _{\Xi _{\alpha }}^{+}(x)\), \(\forall x\in \chi .\) Then,
iv).
\(\square \)
Proposition 2
The IV-CFS intersection on \(\overset{\div }{\top }(\chi )\) is a t-norm.
Proof
The conditions (i), (ii), (v), and (vi) can be easily verified from definition 6. Here, we only prove the condition (iii) and (iv).
iii). Let \((\Im , \mathcal {L} ),\) \((\Xi ,\aleph ),\) \((\Re ,\Pi )\) be three IV-CFSSs over \(\chi ,\) and \(\mu _{(\Im , \mathcal {L} )}=[ \mathbb {C} _{\Im _{\alpha }}^{-}(x),\) \( \mathbb {C} _{\Im _{\alpha }}^{+}(x)].e^{i[\circledS _{\Im _{\alpha }}^{-}(x),\circledS _{\Im _{\alpha }}^{+}(x)]},\) \(\mu _{(\Xi ,\aleph )}=[ \mathbb {C} _{\Xi _{\alpha }}^{-}(x), \mathbb {C} _{\Xi _{\alpha }}^{+}(x)].e^{i[\circledS _{\Xi _{\alpha }}^{-}(x),\circledS _{\Xi _{\alpha }}^{+}(x)]},\) and \(\mu _{(\Re ,\Pi )}=[ \mathbb {C} _{\Pi _{\alpha }}^{-}(x), \mathbb {C} _{\Pi _{\alpha }}^{+}(x)].e^{i[\circledS _{\Pi _{\alpha }}^{-}(x),\circledS _{\Pi _{\alpha }}^{+}(x)]}\) their membership functions, respectively. We suppose \( \mathbb {C} _{\Im _{\alpha }}^{-}(x)\le \mathbb {C} _{\Xi _{\alpha }}^{-}(x),\) \( \mathbb {C} _{\Im _{\alpha }}^{+}(x)\le \mathbb {C} _{\Xi _{\alpha }}^{+}(x),\) \(\circledS _{\Im _{\alpha }}^{-}(x)\le \circledS _{\Xi _{\alpha }}^{-}(x),\) \(\circledS _{\Im _{\alpha }}^{+}(x)\le \circledS _{\Xi _{\alpha }}^{+}(x)\), \(\forall x\in \chi .\) Then
iv).
\(\square \)
Proposition 3
The IV-CFS product on \(\overset{\div }{\top }(\chi )\) is a t-norm.
Proof
The conditions (i), (ii), (v), and (vi) can be easily verified from definition 6. Here, we only prove the condition (iii) and (iv).
iii). Let \((\Im , \mathcal {L} ),\) \((\Xi ,\aleph ),\) \((\Re ,\Pi )\) be three IV-CFSSs over \(\chi ,\) and \(\mu _{(\Im , \mathcal {L} )}=[ \mathbb {C} _{\Im _{\alpha }}^{-}(x),\) \( \mathbb {C} _{\Im _{\alpha }}^{+}(x)].e^{i[\circledS _{\Im _{\alpha }}^{-}(x),\circledS _{\Im _{\alpha }}^{+}(x)]},\) \(\mu _{(\Xi ,\aleph )}=[ \mathbb {C} _{\Xi _{\alpha }}^{-}(x), \mathbb {C} _{\Xi _{\alpha }}^{+}(x)].e^{i[\circledS _{\Xi _{\alpha }}^{-}(x),\circledS _{\Xi _{\alpha }}^{+}(x)]},\) and \(\mu _{(\Re ,\Pi )}=[ \mathbb {C} _{\Pi _{\alpha }}^{-}(x), \mathbb {C} _{\Pi _{\alpha }}^{+}(x)].e^{i[\circledS _{\Pi _{\alpha }}^{-}(x),\circledS _{\Pi _{\alpha }}^{+}(x)]}\) their membership functions, respectively. We suppose \( \mathbb {C} _{\Im _{\alpha }}^{-}(x)\le \mathbb {C} _{\Xi _{\alpha }}^{-}(x),\) \( \mathbb {C} _{\Im _{\alpha }}^{+}(x)\le \mathbb {C} _{\Xi _{\alpha }}^{+}(x),\) \(\circledS _{\Im _{\alpha }}^{-}(x)\le \circledS _{\Xi _{\alpha }}^{-}(x),\) \(\circledS _{\Im _{\alpha }}^{+}(x)\le \circledS _{\Xi _{\alpha }}^{+}(x)\), \(\forall x\in \chi .\) Then
iv).
\(\square \)
Proposition 4
The IV-CFS probabilistic sum on \(\overset{\div }{\top }(\chi )\) is an s-norm.
Proof
The conditions (i), (ii), (v), and (vi) can be easily verified from definition 6. Here, we only prove the condition (iii) and (iv).
iii). Let \((\Im , \mathcal {L} ),\) \((\Xi ,\aleph ),\) \((\Re ,\Pi )\) be three IV-CFSSs over \(\chi ,\) and \(\mu _{(\Im , \mathcal {L} )}=[ \mathbb {C} _{\Im _{\alpha }}^{-}(x),\) \( \mathbb {C} _{\Im _{\alpha }}^{+}(x)].e^{i[\circledS _{\Im _{\alpha }}^{-}(x),\circledS _{\Im _{\alpha }}^{+}(x)]},\) \(\mu _{(\Xi ,\aleph )}=[ \mathbb {C} _{\Xi _{\alpha }}^{-}(x), \mathbb {C} _{\Xi _{\alpha }}^{+}(x)].e^{i[\circledS _{\Xi _{\alpha }}^{-}(x),\circledS _{\Xi _{\alpha }}^{+}(x)]},\) and \(\mu _{(\Re ,\Pi )}=[ \mathbb {C} _{\Pi _{\alpha }}^{-}(x), \mathbb {C} _{\Pi _{\alpha }}^{+}(x)].e^{i[\circledS _{\Pi _{\alpha }}^{-}(x),\circledS _{\Pi _{\alpha }}^{+}(x)]}\) their membership functions, respectively. We suppose \( \mathbb {C} _{\Im _{\alpha }}^{-}(x)\le \mathbb {C} _{\Xi _{\alpha }}^{-}(x),\) \( \mathbb {C} _{\Im _{\alpha }}^{+}(x)\le \mathbb {C} _{\Xi _{\alpha }}^{+}(x),\) \(\circledS _{\Im _{\alpha }}^{-}(x)\le \circledS _{\Xi _{\alpha }}^{-}(x),\) \(\circledS _{\Im _{\alpha }}^{+}(x)\le \circledS _{\Xi _{\alpha }}^{+}(x)\), \(\forall x\in \chi .\) Then
iv).
\(\square \)
Proposition 5
The IV-CFS simple difference on \(\overset{\div }{\top }(\chi )\) is an s-norm.
Proof
The proof is similar to the Proof of Proposition 2. \(\square \)
Proposition 6
\(\Gamma (\xi ,\eta )\) defined by the equality (12.1) is a distance function of IV-CFSSs on \(\chi .\)
Proof
(i). The condition \(\Gamma (\xi ,\eta )\ge 0\) obviously holds true. Next, consider
Therefore, \(0\le \Gamma (\xi ,\eta )\le 1.\)
Also
The condition (ii) is a straight forward. To prove (iii), we have
Therefore
\(\square \)
Proposition 7
Let \(\Gamma (\xi ,\eta )\) be a distance function on IV-CFSSs. Then, the following holds.
- i).:
-
\(\Gamma (\xi ,\eta ^{c})=\Gamma (\xi ^{c},\eta )=\Gamma (\xi ,\eta ),\)
- ii).:
-
\(\Gamma (\xi ^{c},\eta ^{c})=\Gamma (\xi ,\eta ),\)
- iii).:
-
\(\Gamma (\xi ,\eta )=\Gamma (\xi \cap \eta ,\xi \cup \eta ).\)
Proof
(i). For \(\Gamma (\xi ,\eta )=\max \left\{ \begin{array}{c} \underset{x}{\sup }|l([ \mathbb {C} _{\xi _{\alpha _{i}}}^{-}(x), \mathbb {C} _{\xi _{\alpha _{i}}}^{+}(x)])-l([ \mathbb {C} _{\eta _{\alpha _{i}}}^{-}(x), \mathbb {C} _{\eta _{\alpha _{i}}}^{+}(x)])|,\\ \frac{1}{2\pi }\underset{x}{\sup }|l([\circledS _{\xi _{\alpha }}^{-} (x),\circledS _{\xi _{\alpha }}^{+}(x)])-l([\circledS _{\eta _{\alpha }} ^{-}(x),\circledS _{\eta _{\alpha }}^{+}(x)])| \end{array} \right\} ,\) we have the following:
Similarly, \(\Gamma (\xi ^{c},\eta )=\Gamma (\xi ,\eta ).\)
ii).
iii). To prove (iii), two cases arise here.
Case 1.
Case 2.
\(\square \)
Proposition 8
Let \(\xi =(\Im , \mathcal {L} )\) and \(\eta =(\Xi ,\aleph )\) be two IV-CFSSs on \(\chi .\) Then
- i).:
-
\(\zeta =(0)\eta .\)
- ii).:
-
\(\zeta =(1)\eta \) for \(\zeta =\eta .\)
- iii).:
-
\(\zeta =(\delta )\eta \iff \eta =(\delta )\zeta .\)
- iv).:
-
\(\zeta =(\delta _{1})\eta \) and \(\delta _{2}\le \delta _{1}\implies \zeta =(\delta _{2})\eta .\)
v). If, for all \(\gamma \in I,\) \(\zeta =(\delta _{\gamma })\eta ,\) where I is an index set, then \(\zeta =(\underset{\gamma \in I}{\sup }\delta _{\gamma })\eta .\)
Proof
(i). Since \(\Gamma (\xi ,\eta )\le 1.\) It can be written as \(\Gamma (\xi ,\eta )\le 1-0.\) So, by definition of IV-CFS \(\delta \)-equality, we have \(\zeta =(0)\eta .\)
(ii). Since \(\Gamma (\xi ,\xi )=0.\) It can be written as \(\Gamma (\xi ,\xi )=1-1.\) Therefore, by definition of IV-CFS \(\delta \)-equality, we have \(\zeta =(1)\eta .\)
(iii). Let \(\zeta =(\delta )\eta \), then, by definition of IV-CFS \(\delta \)-equality, we have
Therefore, (3) implies that \(\eta =(\delta )\zeta .\)
Similarly, if \(\eta =(\delta )\zeta \), then \(\zeta =(\delta )\eta .\)
(iv). Let \(\zeta =(\delta _{1})\eta ,\) then
Since for \(\delta _{2}\le \delta _{1},\) the above inequality also holds, that is
Thus, \(\zeta =(\delta _{2})\eta .\)
(v). Let \(\zeta =(\delta _{\gamma })\eta \), then
Thus, it is easy to see that
Therefore
This implies that \(\Gamma (\xi ,\eta )\le 1-\underset{\gamma \in I}{\sup } \delta _{\gamma }\), and hence, \(\xi =(\underset{\gamma \in I}{\sup }\delta _{\gamma })\eta .\) \(\square \)
Proposition 9
If \(\zeta =(\delta _{1})\eta \) and \(\eta =(\delta _{2})\psi \), then \(\zeta =(\delta )\psi ,\) where \(\delta =\delta _{1}*\delta _{2}.\)
Proof
Let \(\zeta =(\delta _{1})\eta \) and \(\eta =(\delta _{2})\psi \), then we have
and
Therefore
Thus, \(\xi =(\delta )\psi .\) \(\square \)
5 Applications of interval-valued complex fuzzy soft sets
In this section, we will discuss a real-life application of IV-CFSSs. Especially, the IV-CFSS explains how to get a better and clear choice in decision-making problems.
We propose the following definitions utilized in decision-making algorithm taking the idea of an IV-CFSSs into account.
Definition 17
Let \(\xi =(\Im , \mathcal {L} )\in \) \(\overset{\div }{\top }(\chi )\) be an IV-CFSSs. Then, the cardinality set of \(\xi ,\) represented by \(\xi _{C},\) is defined by
is a fuzzy set over \( \mathcal {L},\) and the membership function \(\mu _{\xi _{C}}(x)\) of \(\xi _{C}\) is a mapping \(\mu _{\xi _{C}}(x): \mathcal {L} \rightarrow [0,1],\) defined by
where \(|\chi |\) is the cardinality of universe \(\chi ,\) and \(|l([ \mathbb {C} _{\Im _{\alpha }}^{-}(x), \mathbb {C} _{\Im _{\alpha }}^{+}(x)]).e^{il([\circledS _{\Im _{\alpha }}^{-}(x),\circledS _{\Im _{\alpha }}^{+}(x)])}|\) is the scalar cardinality of a fuzzy set.
Note that the set of all cardinality sets is denoted by \(\overset{\div }{\top }_{C}(\chi ).\)
Definition 18
Let \(\xi =(\Im , \mathcal {L} )\in \) \(\overset{\div }{\top }(\chi )\) be an IV-CFSSs and \(\xi _{C}\) be the cardinality of \(\xi .\) Assume that \( \mathcal {L} \subset \mathcal {F} \) be a subset of parameters, such that \( \mathcal {L} =\{ \alpha _{1},\alpha _{2},\ldots ,\alpha _{n}\}.\) Then, \(\xi _{C}\) can be denoted by the following table.
\(\mathcal {L}\) | \(\alpha _{1}\) | \(\alpha _{2}\) | . | . | . | \(\alpha _{n}\) |
|---|---|---|---|---|---|---|
\(\mu _{\xi _{C}^{(\alpha )}}(x)\) | \(\mu _{\xi _{C}^{(\alpha _{1})}}(x)\) | \(\mu _{\xi _{C}^{(\alpha _{2})}}(x)\) | . | . | . | \(\mu _{\xi _{C}^{(\alpha _{3})}}(x)\) |
The cardinal set \(\mu _{\xi _{C}^{(\alpha )}}(x)\) is uniquely represented by a matrix, \([a_{1j}]_{1\times n}=[a_{11}\) \(a_{12\text { }}\ldots a_{1n}],\) and is called the cardinal matrix of the cardinal set \(\mu _{\xi _{C}^{(\alpha )}}(x)\) over \( \mathcal {L}.\)
Definition 19
Let \(\xi _{1}=(\Im , \mathcal {L} ),\xi _{2}=(\Xi ,\aleph ),\xi _{3}=(\Re ,\Pi )\in \overset{\div }{\top }(\chi )\) be any three IV-CFSSs over \(\chi .\) Then, the weighted distance measure of IV-CFSSs is defined as follows:
where l denotes the length of the interval and \(w_{(\alpha )}\) is the weighted vector corresponding to the parameter \(\alpha \).
6 Algorithm
In this algorithm, we utilize the concepts of the cardinality of IV-CFSSs, interval-valued complex fuzzy soft distance measure, interval-valued complex fuzzy soft weighted distance measure, and cardinal matrix proposed in the previous section. Here, we solve the decision-making problems under the environments of IV-CFSSs. The detailed steps of the decision-making algorithm for finding the best choice in decision-making problems are described as follows:
Step 1.
Construct an IV-CFSSs \(\xi =(\Im , \mathcal {L} )\) over \(\chi .\)
Step 2.
Compute the cardinal set \(\xi _{C}\) of \(\xi \) for amplitude term and phase term separately.
Step 3.
Compute the distance measure of IV-CFSSs \(\xi (\alpha _{i})\) and cardinal set \(\xi _{C}.\)
Step 4.
Rank the alternatives based on the distance measures between the parameters and hence select the most desirable one(s).
7 A case analysis
Suppose a businessman decides to make an investment in four football clubs (Alternatives). Let \(\chi =\{x_{1}=\)Real Madrid, \(x_{2}=Barcelona,\) \(x_{3}=Manchester\) United, \(x_{4}=Juventus\}\) be a set of alternatives. Let \(C=\{ \alpha _{1}=Sponsorship\) Deal, \(\alpha _{2}=Social\) Media Income, \(\alpha _{2}=Players\) Worth, Annual \(Tax\}\) be a set of criteria. Football clubs are organizations that field teams to compete in football (soccer) competitions. They are typically associated with a specific city or region and have a fan base that supports them passionately. Investing in football clubs can take various forms, and individuals or entities may choose different strategies based on their goals and interests.
Sponsorship deal
Companies often invest in football clubs through sponsorship deals. These deals involve the company’s logo appearing on the club’s jerseys, stadium, or other promotional materials. Sponsorship agreements can be lucrative and provide visibility for the sponsoring brand.
Media and broadcasting rights
Investment in media and broadcasting rights is crucial for the financial success of football clubs. Investors may negotiate lucrative broadcasting deals, streaming rights, or digital media partnerships, contributing to the club’s revenue streams.
Partnerships and collaborations
Investors may seek strategic partnerships and collaborations with football clubs. This could involve joint ventures, co-branding opportunities, or collaborative initiatives that benefit both the investor and the club.
Here are four well-known football clubs from different parts of the world:
Real Madrid CF (Spain)
One of the most successful football clubs globally based in Madrid, Spain. Real Madrid has won numerous domestic and international titles, including the UEFA Champions League.
FC Barcelona (Spain)
Another Spanish powerhouse, based in Barcelona. FC Barcelona has a rich history and is known for its commitment to an attractive, possession-based style of play.
Manchester United FC (England)
Located in Manchester, England, Manchester United is one of the most popular and successful football clubs in the world. The club has a massive global fan base.
Juventus FC (Italy)
Located in Turin, Italy, Juventus is one of the most successful and popular football clubs in Italy. The club has a strong tradition of success in Serie A.
The expert team investigates and takes part in the evaluation of these three enterprises under the environments of IV-CFSSs. The information of the decision-makers is given in Table 1.
Step 1. The IV-CFSSs \(\xi =(\Im , \mathcal {L} )\) over \(\chi \) is written by
Step 2.
The cardinal set \(\xi _{C}\) of \(\xi \) for amplitude term and phase term is
Step 3.
The distance measure of \(\xi (\alpha _{i})\) and cardinal set \(\xi _{C}\) are given in Table 2.
If \(w=(0.2,0.3,0.2,0.3)\) is the weightage assigned to each parameter \(\alpha _{i};i=1,2,3,4,\) then
From the distance measure of \(\xi (\alpha _{i})\) and \(\xi _{C}\), we conclude that \(x_{3}>x_{2}>x_{4}>x_{1}.\) Therefore, \(x_{3}=Manchester\) United is the best football club for investment.
If we assigned the weighted vector \(w=(0.2,0.3,0.2,0.3)\) to the parameters \(\alpha _{1},\alpha _{2},\alpha _{3},\alpha _{4},\) then the rank of alternatives is \(x_{3}>x_{2}>x_{4}>x_{1}.\) Therefore, we get the same order of rank.
8 Comparison analysis
Here, we discussed the comparison of the proposed distance measure of the interval-valued complex fuzzy soft set with the complex fuzzy soft set and fuzzy soft set. The interval-valued complex fuzzy soft distance measure reduces the environments of complex fuzzy soft set and fuzzy soft set. The comparison is given in cases 1 and 2. The numerical data in other fuzzy environments are also considered. Moreover, we discussed the advantages of the proposed distance measure.
Case 1.
The interval-valued complex fuzzy soft distance measure reduces to the environment of complex fuzzy soft sets if \( \mathbb {C} _{\xi _{\alpha _{i}}}^{-}(x)= \mathbb {C} _{\xi _{\alpha _{i}}}^{+}(x),\) \( \mathbb {C} _{\eta _{\alpha _{i}}}^{-}(x)= \mathbb {C} _{\eta _{\alpha _{i}}}^{+}(x),\) \(\circledS _{\xi _{\alpha }}^{-}(x)=\circledS _{\xi _{\alpha }}^{+}(x),\) \(\circledS _{\eta _{\alpha }}^{-}(x)=\circledS _{\eta _{\alpha }}^{+}(x),\) & remove length from the definition, then we have
where \( \mathbb {C} _{\xi _{\alpha _{i}}}(x), \mathbb {C} _{\eta _{\alpha _{i}}}(x)\in [0,1],\) and \(\circledS _{\xi _{\alpha } }(x),\circledS _{\eta _{\alpha }}(x)\in [0,2\pi ].\)
Case 2.
The interval-valued complex fuzzy soft distance measure reduces to the environment of interval-valued fuzzy soft sets if \(\circledS _{\xi _{\alpha } }^{-}(x)=\circledS _{\xi _{\alpha }}^{+}(x)=\circledS _{\eta _{\alpha }} ^{-}(x)=\circledS _{\eta _{\alpha }}^{+}(x)=0,\) then we have
where \( \mathbb {C} _{\xi _{\alpha _{i}}}^{-}(x), \mathbb {C} _{\xi _{\alpha _{i}}}^{+}(x), \mathbb {C} _{\eta _{\alpha _{i}}}^{-}(x), \mathbb {C} _{\eta _{\alpha _{i}}}^{+}(x)\in [0,1].\)
Case 3.
The interval-valued complex fuzzy soft distance measure reduces to the environment of fuzzy soft sets if \( \mathbb {C} _{\xi _{\alpha _{i}}}^{-}(x)= \mathbb {C} _{\xi _{\alpha _{i}}}^{+}(x),\) \( \mathbb {C} _{\eta _{\alpha _{i}}}^{-}(x)= \mathbb {C} _{\eta _{\alpha _{i}}}^{+}(x),\) \(\circledS _{\xi _{\alpha }}^{-}(x)=\circledS _{\xi _{\alpha }}^{+}(x)=\circledS _{\eta _{\alpha }}^{-}(x)=\circledS _{\eta _{\alpha }}^{+}(x)=0,\) & remove length from the definition, then we have
where \( \mathbb {C} _{\xi _{\alpha _{i}}}(x), \mathbb {C} _{\eta _{\alpha _{i}}}(x)\in [0,1].\)
All the above distance measures can be applied to decision-making problems. Now, we consider the decision-making problem in the environment of complex fuzzy soft sets.
The complex fuzzy soft information about the index’s evaluation is given in Table 4.
The IV-CFSSs \(\xi =(\Im , \mathcal {L} )\) over \(\chi \) are written by
The cardinal set \(\xi _{C}\) is
Now, the complex fuzzy soft distance measure of \(\xi (x)\) and \(\xi _{C}\) is given in Table 5.
The rank of the scheme is \(x_{3}>x_{1}=x_{4}>x_{2}.\)
If the weighted vector w is assigned to each parameter \(\alpha _{i};i=1,2,3,4,\) that is, \(w=(0.2,0.3,0.2,0.3).\) Then, the distance measure is
Therefore, the rank is of the scheme is \(x_{3}>x_{2}>x_{1}=x_{4}.\)
Similarly, we consider the decision-making problem in the environment of interval-valued fuzzy soft sets.
The interval-valued fuzzy soft information about the index’s evaluation is given in Table 7.
The decision-making problem based on interval-valued fuzzy soft information can be solved using the distance measure proposed in case 2.
We consider the decision-making problem in the environment of fuzzy soft sets.
The fuzzy soft information about the index’s evaluation is given in Table 8.
The decision-making problem based on fuzzy soft information can be solved using the distance measure proposed in case 3.
9 Key features and benefits of proposed distance measure
The key features and benefits of the proposed distance measure are given below:
i) The interval-valued complex fuzzy soft distance measure can solve the problems based on complex fuzzy soft sets, interval-valued fuzzy soft sets, and fuzzy soft sets. On the other hand, complex fuzzy soft distance measure, interval-valued fuzzy soft distance measure, and fuzzy soft distance measure cannot solve the decision-making problems in the form of CPFSs.
ii) The amplitude term and phase term are any interval numbers; it shows more probability of occurrence in the interval D[0, 1]. Therefore, the IV-CFSSs are more faithful than a complex fuzzy set and complex fuzzy soft set.
iii) Complex numbers in interval-valued complex fuzzy soft distance measures enable the modeling of information with both real and imaginary components. This is particularly valuable in applications where the phase of information is significant.
iv) Interval-valued complex fuzzy soft distance measures can assess the distance between sets in a multidimensional space. This is beneficial in decision-making problems involving multiple criteria or attributes, where the relationships between elements may have both real and imaginary components.
v) Interval-valued complex fuzzy soft distance measures seamlessly integrate with interval-valued fuzzy sets, allowing for a unified approach to handling uncertainty. This is particularly advantageous when dealing with applications that involve both interval-valued and complex fuzzy information.
10 Conclusion
In this paper, we explored the further development of the theory of IV-CFSSs (IV-CFSSs). Some set-theoretic operations and laws of the IV-CFSSs were proposed. We defined the distance measure in the environment of IV-CFSSs. The concept of \(\delta \)-equality of IV-CFSSs was developed based on interval-valued complex fuzzy soft distance measures. We discussed some particular examples and basic results of these operations and laws. Moreover, we discussed a decision-making problem based on IV-CFSSs. We developed a new decision-making method using the interval-valued complex fuzzy distance measures under the environments of IV-CFSSs. We discussed the real-life case based on the proposed decision-making method. A real-life example demonstrated that the decision-making method developed in the paper could be utilized to deal with problems of uncertainty. Further, the comparative study of IV-CFSSs with complex fuzzy soft sets, interval-valued fuzzy soft sets, and fuzzy soft sets was established.
In the future, we will extend the elaborated work for interval-valued complex intuitionistic fuzzy soft sets, interval-valued complex Pythagorean fuzzy soft sets, interval-valued complex neutrosophic fuzzy soft sets, complex q-rung orthopair fuzzy sets, complex spherical fuzzy sets, and complex T-spherical fuzzy sets, etc., to improve the quality of the research works.
Data Availability:
Our manuscript has no associated data.
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Acknowledgements
This work is financially supported by the Higher Education Commission of Pakistan (Grant No. 7750/Federal/NRPU/R &D/HEC/2017).
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Zeeshan, M., Khan, M., Abid, M.A. et al. Decision-making method under the interval-valued complex fuzzy soft environment. Comp. Appl. Math. 43, 203 (2024). https://doi.org/10.1007/s40314-024-02686-7
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DOI: https://doi.org/10.1007/s40314-024-02686-7
Keywords
- Soft set
- Interval-valued complex fuzzy soft set
- Interval-valued complex fuzzy soft operations
- Decision-making method