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Fuzzy Control Systems: Past, Present and Future

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Toward Humanoid Robots: The Role of Fuzzy Sets pp 27–95 Cite as

Fuzzy Sets and Extensions: A Literature Review

• Eda Bolturk 4 &
• Cengiz Kahraman 4  
• First Online: 05 April 2021

239 Accesses

2 Citations

Part of the book series: Studies in Systems, Decision and Control ((SSDC,volume 344))

Humanoid robots generated by inspiring by human appearances and abilities have become essential in human society to improve the quality of their life. All over the world, there have been many researchers who have focused on humanoid robots to develop their capabilities. Generally, humanoid robot systems include mechanisms of decision making and information processing. Because of the uncertainty behind decision making and information processes, fuzzy sets can be used in humanoid systems efficiently. This study presents a comprehensive literature review on the recent developments and theories associated with fuzzy set models.

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1 Introduction

Fuzzy logic has been generally employed for the automatic navigation of robots in the literature. This is since the capability of fuzzy logic to process large quantities of incomplete and vague input signals is very high for the automatic navigation of robots under uncertainty. Robots carry several sensors on them for sensing environmental information. The outputs of these sensors serve as inputs to the fuzzy controller. Fuzzification, fuzzy inference, and defuzzification generate decisions that control the robots’ behaviors enabling robots to navigate automatically [ 1 ]. Fuzzy logic control systems provide an automatic navigation system very similar to the thinking style of humans.

Classical fuzzy logic is composed of only a membership degree where the non membership degree is the complement of this degree. Since human behaviour is the outcome of a complex thinking system, we need more parameters to define uncertainty, which the extensions of classical fuzzy logic aim at providing this capability. In the following, we will introduce these extensions since they are more suitable to model human behaviours than classical fuzzy logic. After the introduction of ordinary fuzzy sets (OFS) by Zadeh [ 2 ], fuzzy sets have been very popular in modeling the problems involving vagueness and impreciseness. Various researchers have proposed several extensions of ordinary fuzzy sets as given in Fig. 1 with a historical order. In this figure, large circles show that those extensions relatively have a larger impact than other extensions on the expansion of fuzzy set theory. In recent years, several researchers have used these extensions in the solution of various problems such as mathematical modeling and optimization, multicriteria decision making, data mining, and quality control. A classification of extensions of OFS is presented in the following.

Fuzzy sets extensions

1.1 Preliminaries: Extensions of Fuzzy Sets

In this section, the basic concepts and the mathematical operations of extensions of fuzzy sets have been briefly introduced.

1.1.1 Ordinary Fuzzy Sets

A way of describing a fuzzy set is to list ordered pairs: an object  x  and its membership degree \(\mu_{A} \left( x \right)\left[ {0, 1} \right]\) in a set \(\tilde{A}\) . To describe an ordinary fuzzy set, the following notation proposed by Zadeh [ 2 ] can be used:

where X is the discrete universe. The non-membership degree of any x is calculated by the subtraction \(1 - \mu_{{\tilde{A}}} \left( x \right)\) .

1.1.2 Type-2 Fuzzy Sets (T2FS)

The concept of a type-2 fuzzy set was introduced by Zadeh [ 3 ] as an extension of the concept of an ordinary fuzzy set. Such sets are fuzzy sets whose membership grades themselves are fuzzy. They are veryseful in circumstances where it is difficult to determine an exact membership function for a fuzzy set.

A type 2 fuzzy set \(\tilde{{\tilde{A}}}\) in the universe of discourse X can be represented by a type 2 membership function \(\mu_{{\tilde{{\tilde{A}}}{ }}}\) , shown as follows [ 3 ]:

where \(J_{x}\) . denotes an interval [0,1].

1.1.3 Interval-Valued Fuzzy Sets

Let us denote the set of all closed sub intervals in [0,1] by L([0,1]), that is, L([0,1]) = {x = [x L ,x U ] > (x L ,x U ) ∈ [0,1] 2 and x L  ≤ x U } (1) An interval-valued fuzzy set (IVFS) \(\tilde{A}\) on the universe ∪  ≠ ∅ is a mapping \(\tilde{A}\) : ∪  → L([0,1]), such that the membership degree of u ∈  ∪ is given by A(u) = [A L (u), A U (u)] ∈ L([0,1]), where A: ∪  → [0,1] and A: ∪  → [0,1] are mappings defining the lower and the upper bound of the membership interval A(u), respectively.

1.1.4 Intuitionistic Fuzzy Sets

Intuitionistic fuzzy sets introduced by Atanassov [ 4 ] enable defining both the membership and non-membership degrees of an element in a fuzzy set. Their sum can be equal to or less than 1. The difference from 1, if any, is called hesitancy. Let U be a universe of discourse. An IFS \(\tilde{I}\) is defined as follows:

Definition 1

Let \(X\) be a non-empty set. An intuitionistic fuzzy set \(I\) in \(X\) is given by:

where the function \(\mu_{I} :X \to \left[ {0,1} \right]\) and \(\upsilon_{I} :X \to \left[ {0,1} \right]\) defines the degree of membership and the degree of non-membership of element to the sets \(I\) , respectively, with the condition that

The degree of hesitancy is calculated as follows:

Definition 2

Let \({\tilde{\text{A}}} = \left( {\mu _{{\tilde{A}}} ,\nu _{{\tilde{A}}} } \right)\) and \({\tilde{\text{B}}} = \left( {\mu _{{\tilde{B}}} ,\nu _{{\tilde{B}}} } \right)\) be two IFSs, then the addition and multiplication operations on these two IFSs are calculated as follows:

Atanassov [ 4 ] introduced triangular intuitionistic fuzzy sets (IFS) by defining membership and non-membership functions independently with the constraint that their sum must be at most one, letting a hesitancy degree be included. A triangular intuitionistic fuzzy number is shown in Fig.  2 .

A triangular intuitionistic fuzzy number

Defuzzification of triangular intuitionistic fuzzy numbers (IFNs) can be made by two defuzzification methods given in the following: one for triangular IFNs and one for trapezoidal IFNs [ 5 ]:

Let \(I_{i} = \left( {a_{L} ,a_{M} ,a_{U} ;b_{L} ,b_{M} ,b_{U} } \right)\) be a triangular intuitionistic fuzzy number. Then, defuzzification is realized by using the function defined in Eq. ( 8 ).

where τ is a very large number, such as 100, indicating the effect of non-membership function on the IFN.

Let \(I_{i} = \left( {a_{L} ,a_{M1} ,a_{M2} ,a_{U} ;b_{L} ,b_{M1} ,b_{M2} ,b_{U} } \right)\) be a trapezoidal intuitionistic fuzzy number. Then, the defuzzification is realized by using the function defined in Eq. ( 9 ).

Definition 3

Let \(X\) be a non-empty set. An interval-valued intuitionistic fuzzy (IVIF) set in X is an object \(\tilde{A}\) given as in Eq. ( 10 ) [ 4 ]:

where \(0 \le {\upmu }_{{{\tilde{\text{A}}}}}^{ + } + {\upupsilon }_{{{\tilde{\text{A}}}}}^{ + } \le 1\) for every \({\text{x}} \in {\text{X}}.\)

Definition 4

Let \(\tilde{A} = \left( {\left[ {\mu_{{\tilde{A}}}^{ - } ,\mu_{{\tilde{A}}}^{ + } } \right],\left[ {v_{{\tilde{A}}}^{ - } ,v_{{\tilde{A}}}^{ + } } \right]} \right)\) and \(\tilde{B} = \left( {\left[ {\mu_{{\tilde{B}}}^{ - } ,\mu_{{\tilde{B}}}^{ + } } \right],\left[ {v_{{\tilde{B}}}^{ - } ,v_{{\tilde{B}}}^{ + } } \right]} \right){ }\) be two IVIF numbers [ 6 ]. Then,

Definition 5

Let \(\tilde{r}_{ij}^{k} = \left( {\left[ {\mu_{{\tilde{r}}}^{ - } ,\mu_{{\tilde{r}}}^{ + } } \right],\left[ {v_{{\tilde{r}}}^{ - } ,v_{{\tilde{r}}}^{ + } } \right]} \right)\) be the IVIF numbers where \(k = 1,2, \ldots ,n\) . Then the aggregated IVIF number \((\tilde{r}_{ij}^{A}\) ) is obtained by using interval-valued intuitionistic fuzzy hybrid geometric (IIFHG) operator as in Eq. ( 13 ) [ 7 ]:

where \(\omega_{k}\) is the weight vector of expert \(k\) where \(\mathop \sum \nolimits_{k = 1}^{n} \omega_{k} = 1\) .

Definition 6 Let \(\tilde{\user2{r}}_{1} = \left( {\left[ {\mu_{1}^{ - } ,\mu_{1}^{ + } } \right],\left[ {v_{1}^{ - } ,v_{1}^{ + } } \right]} \right)\) and \(\tilde{r}_{2} = \left( {\left[ {\mu_{2}^{ - } ,\mu_{2}^{ + } } \right],\left[ {v_{2}^{ - } ,v_{2}^{ + } } \right]} \right)\) be two IVIF numbers. The distance between these two IVIF numbers is obtained by Hamming distance as in Eq. ( 14 ) [ 8 ]:

Defitiniton 7 Let \(\tilde{r}_{1} = \left( {\left[ {\mu_{{\tilde{x}}}^{ - } ,\mu_{{\tilde{x}}}^{ + } } \right],\left[ {v_{{\tilde{x}}}^{ - } ,v_{{\tilde{x}}}^{ + } } \right]} \right)\) is an IVIF number. Defuzzification formula ( \({\mathfrak{D}}\left( {\text{x}} \right))\) for \(\tilde{r}_{1}\) is given in Eq. ( 15 ) [ 9 ].

1.1.5 Fuzzy Multi Sets

Yager [ 10 ] first discussed fuzzy multi-sets, although he uses the term of fuzzy bag,an element of X may occur more than once with possibly the same or different membership values. Assume X is a set of elements. Then a fuzzy bag A drawn from X can be characterized by a function Count.Mem A such that

where Q is the set of all crisp bags from the unit interval.

1.1.6 Neutrosophic Sets

Smarandache [ 11 ] developed neutrosophic logic and neutrosophic sets (NSs) as an extension of intuitionistic fuzzy sets. The neutrosophic set is defined as the set where each element of the universe has a degree of truthiness, indeterminacy and falsity. The sum of these degrees can be at most equal to 3 since each of them can be independently at most equal to 1. Let ∪ be a universe of discourse.

In neutrosophic sets literature, a common specific symbol for a neutrosophic set has not been used up to now. Bolturk and Kahraman [ 12 ] propose the symbol \(\widetilde{\dddot A}\) for the neutrosophic set A, that the three dots represent the elements of a neutrosophic set,T, I, F and tilde represents that it is also a fuzzy set.

Definition 8

Smarandache [ 11 ]. Let E be a universe. A neutrosophic set \(\widetilde{\dddot A}\) in E is characterized by a truth-membership function \(T_{A}\) , a indeterminacy-membership function \(I_{A}\) , and a falsity-membership function \(F_{A}\) .

\(T_{A}\) ( \(x\) ), \(I_{A}\) ( \(x\) ) and \(F_{A}\) ( \(x\) ) are real standart elements of [0,1]. A neutrosophic set \(\widetilde{\dddot A}\) can be given by Eq. ( 17 ):

There is no restriction on the sum of \(T_{A} { }\) ( \(x\) ), \(I_{A}\) ( \(x\) ) and \(F_{A}\) ( \(x\) ), so that \(0^{ - } \le T_{A} { }\left( x \right) + I_{A} { }\left( x \right) + F_{A} { }\left( x \right) \le 3^{ + } .\)

Definition 9

Li et al. [ 13 ] X be a universe of discourse. An interval-valued neutrosophic set \(\widetilde{\dddot N}\) in X is independently defined by a truth-membership function \(T_{N} \left( x \right)\) , an indeterminacy-membership function \(I_{N} \left( x \right)\) , and a falsity-membership function \(F_{N} \left( x \right)\) for each \(x \in X\) , where \(T_{N} \left( x \right) = \left[ {T_{N\left( x \right)}^{L} ,T_{N\left( x \right)}^{U} } \right] \subseteq \left[ {0,1} \right]\) , \(I_{N} \left( x \right) = \left[ {I_{N\left( x \right)}^{L} ,I_{N\left( x \right)}^{U} } \right] \subseteq \left[ {0,1} \right]\) , and \(F_{N} \left( x \right) = \left[ {F_{N\left( x \right)}^{L} ,F_{N\left( x \right)}^{U} } \right] \subseteq \left[ {0,1} \right]\) . They also meet the condition \(0 \le T_{N}^{L} \left( x \right) + I_{N}^{L} \left( x \right) + F_{N}^{L} \left( x \right) \le 3\) . So, the interval-valued neutrosophic set \(\widetilde{{{\dddot{\text {N}}}}}\) can be given by Eq. ( 18 ):

Definition 10

Bolturk and Kahraman [ 12 ] propose a new deneutrosophication function of an interval-valued neutrosophic number which is given below:

where \(\widetilde{{{\dddot{\text {x}}}}}_{{\text{j}}} = \left\langle {\left[ {{\text{T}}_{{\text{x}}}^{{\text{L}}} ,{\text{T}}_{{\text{x}}}^{{\text{U}}} } \right],\left[ {{\text{ I}}_{{\text{x}}}^{{\text{L}}} ,{\text{I}}_{{\text{x}}}^{{\text{U}}} } \right],\left[ {{\text{F}}_{{\text{x}}}^{{\text{L}}} ,{\text{F}}_{{\text{x}}}^{{\text{U}}} } \right]} \right\rangle .\)

Definition 11

Let \(\widetilde{\dddot a} = \left[ {{ }T_{a}^{L} ,T_{a}^{U} } \right],\left[ {{ }I_{a}^{L} ,I_{a}^{U} } \right],\left[ {{ }F_{a}^{L} ,F_{a}^{U} } \right]\) and \(\widetilde{\dddot b} = \left\langle {\left[ {{ }T_{b}^{L} ,T_{b}^{U} } \right],\left[ {{ }I_{b}^{L} ,I_{b}^{U} } \right],\left[ {{ }F_{b}^{L} ,F_{b}^{U} } \right]} \right\rangle\) be two interval-valued neutrosophic numbers. Their relations and arithmetic operations are given by Eqs. ( 20 )–( 24 ) [ 14 ]:

Definition 12

Subtraction operation of two interval-valued neutrosophic sets is given in Eq. ( 25 ) Karasan and Kahraman [ 15 ]:

where \({\widetilde{\dddot x}}=\langle \left[{\mathrm{T}}_{\mathrm{x}}^{\mathrm{L}},{\mathrm{T}}_{\mathrm{x}}^{\mathrm{U}}\right],\left[ {\mathrm{I}}_{\mathrm{x}}^{\mathrm{L}},{\mathrm{I}}_{\mathrm{x}}^{\mathrm{U}}\right],\left[{\mathrm{F}}_{\mathrm{x}}^{\mathrm{L}},{\mathrm{F}}_{\mathrm{x}}^{\mathrm{U}}\right]\rangle\) and \({\widetilde{\dddot y}}=\langle \left[{\mathrm{T}}_{\mathrm{y}}^{\mathrm{L}},{\mathrm{T}}_{\mathrm{y}}^{\mathrm{U}}\right],\left[ {\mathrm{I}}_{\mathrm{y}}^{\mathrm{L}},{\mathrm{I}}_{\mathrm{y}}^{\mathrm{U}}\right],\left[{\mathrm{F}}_{\mathrm{y}}^{\mathrm{L}},{\mathrm{F}}_{\mathrm{y}}^{\mathrm{U}}\right]\rangle\) .

Definition 13

Let \({{\widetilde{\dddot{\text {A}}}}} =\left\{\langle x,\left[{\mathrm{T}}_{N}^{\mathrm{L}}(x),{\mathrm{T}}_{\mathrm{x}}^{\mathrm{U}}(x)\right],\left[ {\mathrm{I}}_{\mathrm{x}}^{\mathrm{L}}(x),{\mathrm{I}}_{\mathrm{x}}^{\mathrm{U}}(x)\right],\left[{\mathrm{F}}_{\mathrm{x}}^{\mathrm{L}}(x),{\mathrm{F}}_{\mathrm{x}}^{\mathrm{U}}(x)\right]\rangle |x\in X\right\}\) be an interval-valued neutrosophic number. The following ranking formula is proposed for \({{\tilde{\dddot{\text {A}}}}}\) by Kahraman et al. [ 16 ]:

Definition 14

Let \(\stackrel{\sim }{\mathrm{A}}=\langle \left[{\mathrm{T}}_{1}^{\mathrm{L}},{\mathrm{T}}_{1}^{\mathrm{U}}\right],\left[ {\mathrm{I}}_{1}^{\mathrm{L}},{\mathrm{I}}_{1}^{\mathrm{U}}\right],\left[{\mathrm{F}}_{1}^{\mathrm{L}},{\mathrm{F}}_{1}^{\mathrm{U}}\right]\rangle ; \stackrel{\sim }{\mathrm{B}}=\langle \left[{\mathrm{T}}_{2}^{\mathrm{L}},{\mathrm{T}}_{2}^{\mathrm{U}}\right],\left[ {\mathrm{I}}_{2}^{\mathrm{L}},{\mathrm{I}}_{2}^{\mathrm{U}}\right],\left[{\mathrm{F}}_{2}^{\mathrm{L}},{\mathrm{F}}_{2}^{\mathrm{U}}\right]\rangle\) be IVNNs where \({\mathrm{T}}_{2}^{\mathrm{L}}>0; {\mathrm{T}}_{2}^{\mathrm{U}}>0; {\mathrm{I}}_{2}^{\mathrm{L}}>0; {\mathrm{I}}_{2}^{\mathrm{U}}>0; {\mathrm{F}}_{2}^{\mathrm{L}}>0;{\mathrm{F}}_{2}^{\mathrm{U}}>0.\) The division operation is proposed as in Eq. ( 27 ) [ 17 ]:

Definition 15

Let \({\text{a}}_{{\text{j}}} = \left\langle {\left[ {{\text{T}}_{{{\text{a}}_{{\text{j}}} }}^{{\text{L}}} ,{\text{T}}_{{{\text{a}}_{{\text{j}}} }}^{{\text{U}}} } \right],\left[ {{\text{ I}}_{{{\text{a}}_{{\text{j}}} }}^{{\text{L}}} ,{\text{I}}_{{{\text{a}}_{{\text{j}}} }}^{{\text{U}}} } \right],\left[ {{\text{F}}_{{{\text{a}}_{{\text{j}}} }}^{{\text{L}}} ,{\text{F}}_{{{\text{a}}_{{\text{j}}} }}^{{\text{U}}} } \right]} \right\rangle\) , \(j = 1,2, \ldots ,n\) be a collection of INNs. Based on the weighted aggregation operators of INNs, the interval neutrosophic number weighted average operator is given as below [ 14 ]

where \(w_{j} \left( {j = 1,2, \ldots ,n} \right)\) is the weight of \(a_{j} \left( {j = 1,2, \ldots ,n} \right)\) with \(w_{j} \in \left[ {0,1} \right]\,and \mathop \sum \nolimits_{j = 1}^{n} w_{j} = 1.\)

Definition 16

The weighted aggregation operation (INNWA) for interval-valued neutrosophic numbers is given in Eq. ( 29 ) [ 14 ]

where \(W = \left( {w_{1} ,w_{2} , \ldots ,w_{n} } \right)\) is the weight vector of \(A_{j} \left( {j = 1,2, \ldots ,n} \right)\) , with \(w_{j} \in \left[ {0,1} \right]\) and \(\mathop \sum \limits_{j = 1}^{n} w_{j} = 1\) .

There are few deneutrosophication methods to compare neutrosophic numbers [ 18 , 19 ]. Bolturk et al. [ 17 ] proposed a new deneutrosophication method in order to compare the interval-valued neutrosophic numbers. It is given in Definition 17 .

Definition 17

Let \({\text{A}} = \left\langle {\left( {{\text{T}}^{{\text{L}}} ,{\text{ T}}^{{\text{U}}} } \right),{ }\left( {{\text{I}}^{{\text{L}}} ,{\text{ I}}^{{\text{U}}} } \right),\left( {{\text{F}}^{{\text{L}}} ,{\text{ F}}^{{\text{U}}} } \right)} \right\rangle\) be an interval-valued neutrosophic number. The deneutrosophicated A value \(\left( {{\mathfrak{D}}\left( A \right)} \right)\) is proposed in Eq. ( 30 ):

1.1.7 Nonstationary Fuzzy Sets

Nonstationary fuzzy sets are introduced by Garibaldi and Ozen [ 20 ]. Let A denote a fuzzy set of a universe of discourse X characterized by a membership function \(\mu_{{\dot{A}}}\) . Let T be a set of time points t i (possibly infinite) and f: T →  < denote the perturbation function. Associates with each element \(\left( {t,x} \right)\) of \(T \times X\) a time specific variation of \(\mu_{A} \left( x \right)\) . The nonstationary fuzzy set \(\dot{A}\) is denoted by

1.1.8 Hesitant Fuzzy Sets

Hesitant fuzzy sets introduced by Torra [ 21 ] allow many potential degrees of membership of an element to be assigned to a set. These fuzzy sets force the membership degree of an element to be possible values between zero and one. A hesitant fuzzy set on X can be defined as in Eq. ( 49 ):

where h H (x) is a set of hesitant fuzzy elements whose membership values are in [0,1].

1.1.9 Pythagorean Fuzzy Sets

Pythagorean fuzzy sets (PFSs) are an extension of intuitionistic fuzzy sets and it allows researchers to assign membership and nonmembership degrees in a wider area. Atanassov’s intuitionistic fuzzy sets of second type (IFS2) or Yager’s Pythagorean fuzzy sets [ 22 ] are characterized by a membership degree and a non-membership degree satisfying that their squared sum is equal to or less than one, which is a generalization of intuitionistic fuzzy sets. This provides a larger area than IFS in order to assign membership and non-membership degrees [ 23 ]. Let ∪ be a universe of discourse. A PFS \(\tilde{P}\) is an object having the form,

where \(\mu_{P} :X \to \left[ {0,1} \right]\) is the membership degree and \(v_{P} :X \to \left[ {0, 1} \right]\) is the nonmembership degree. Then, Eq. ( 34 ) is valid:

The degree of indeterminancy is defined as follows:

A Single Valued Pythagorean Fuzzy Set (SVPFS) is defined as follows:

For two PFSs, \(\tilde{P}_{1} = \left\{ {\left. {x,P_{1} \left( {\mu _{{P1}} \left( x \right),v_{{P1}} \left( x \right)} \right)} \right|x \in X} \right\}\) and \(\tilde{P}_{2} = \left\{\left.x, {P}_{2}\left({\mu }_{P2}\left(x\right), {v}_{P2}\left(x\right)\right)\right|x\in X\right\}\) , the following arithmetic operations are valid:

Zhang and Xu [ 24 ] defined the Euclidean distance between two PFSs as in Eq. ( 40 ):

The Taxican distance between two PFSs is defined by Eq. ( 41 ):

Let \({p}_{1}=({\mu }_{1},{v}_{1})\) and \({p}_{2}=({\mu }_{2},{v}_{2})\) be two PFNs and \(\rho >0.\) The following operations are presented for PFNs [ 24 , 25 ].

Definition 18

Let Int([0,1]) denote the set of all closed subintervals of [0,1], and \(X\) be a universe of discourse. An interval-valued PFS (IVPFS) \(\stackrel{\sim }{P}\) in \(X\) is given by Eq. ( 44 ) [ 26 ].

where the functions \({\mu }_{p}\) : X  → Int([0, 1])( x  ∈  X  →  \({\mu }_{p}\) ( x ) ⊆ [0, 1]) and \({v}_{p}\) : X  → Int([0, 1]) ( x ∈ X  →  \({v}_{p}\) ( x ) ⊆ [0, 1]) denote the membership degree and non-membership degree of the element x ∈ X to the set \({\tilde{P}}\) , respectively, and for every x ∈ X , 0 ≤ sup \(\left\{{\left({\mu }_{p}(x)\right)}^{2}\right\}\) + sup \(\left\{{\left({v}_{p}\left(x\right)\right)}^{2}\right\}\) ≤ 1. Also, for each x ∈ X , \({\mu }_{p}\) ( x ) and \({v}_{p}\) ( x ) are closed intervals and their lower and upper bounds are denoted by \({\mu }_{p}^{L}\left(x\right)\) , \({\mu }_{p}^{U}\left(x\right)\) , \({v}_{p}^{L}\left(x\right)\) , \({v}_{p}^{U}\left(x\right)\) , respectively. Therefore, \(\tilde{P}\) can also be expressed as follows:

The degree of indeterminacy is given by Eq. ( 46 ).

Definition 19

Let \(\tilde{A} = \left\langle {\left[ {\mu_{A}^{L} ,\mu_{A}^{U} } \right],\left[ {{ }v_{A}^{L} ,{ }v_{A}^{U} } \right]} \right\rangle\) , \(\tilde{B} = \left\langle {\left[ {\mu_{B}^{L} ,\mu_{B}^{U} } \right],\left[ {v_{B}^{L} ,v_{B}^{U} } \right]} \right\rangle\) be two IVPF numbers, and \(\lambda > 0\) , then some operations of IVPF numbers are defined as follows [ 26 ].

Definition 20

An IVPF number \({\tilde{\text{A}}} = \left( {\left[ {{\text{a}},{\text{ b}}} \right],\left[ {{\text{c}},{\text{ d}}} \right]} \right)\) can be defuzzified by using Eq. ( 51 ).

In Eq. ( 51 ), the terms (1 − c) and (1 − d) convert non-membership degrees to membership degrees while the term \(\sqrt {\left( {1 - c} \right) \times \left( {1 - d} \right)}\) decreases the defuzzified value.

Definition 21

Score function for an IVPF number \({\tilde{\text{A}}} = \left( {\left[ {{\text{a}},{\text{ b}}} \right],\left[ {{\text{c}},{\text{ d}}} \right]} \right)\) is given by Eq. ( 52 ) [ 26 ]:

where \(S\left( {\tilde{A}} \right) \in \left[ { - 1, + 1} \right]\) .

PFSs introduced by Yager [ 22 ] can be defined by both membership and non-membership degrees here the square sum of membership degree and non-membership degrees should be equal or less than 1. PFSs can be considered as a generalization of IFSs [ 4 ]. In order to provide some basic knowledge of IFSs and PFSs, Table 1 and Fig.  3 are given. Figure  3 shows the relation between IFSs and PFSs based on Table 1 . PFS is a kind of dilation operations over IFSs. Alternatively, we can say that IFS is a kind of concentration operation over PFSs (Kahraman et al. [ 28 ]).

Limits of membership and nonmembership degrees in IFS (blue) and PFSs (grey)

Let \(\tilde{A} = \left\langle {[\upmu _{{\text{L}}} ,\upmu _{{\text{U}}} ],[{\text{V}}_{{\text{L}}} ,{\text{V}}_{{\text{U}}} ]} \right\rangle\) be an interval-valued PFN, \(\pi_{L}\) and \(\pi_{U}\) are the hesitancy degree of the lower and upper points of \(\tilde{A}\) , respectively, can be calculated as in Eq. ( 53 ) and ( 54 ):

We know that the score functions or defuzzifying procedures are efficient when we compare PFNs in MADM problems. However, the score functions in the literature are insufficient to indicate which PFN is larger than the other since they don’t associate the hesitancy properly. Motivated by the definition of hesitancy degree function of PFNs and defuzzification function for IFNs [ 29 ], a defuzzification function as in Definition 22.

Definition 22 [ 30 ] Let \(\tilde{A} = \left\langle {[\upmu _{{\text{L}}} ,\upmu _{{\text{U}}} ],[{\text{V}}_{{\text{L}}} ,{\text{V}}_{{\text{U}}} ]} \right\rangle\) be an interval-valued PFN and \(\pi_{L}\) , \(\pi_{U}\) are the hesitancy degree of the lower and upper points of \(\tilde{A}\) , then the defuzzifying procedure of this number is calculated by Eq. ( 55 ):

A larger value of \({\mathfrak{H}}\) indicates a larger \(\tilde{A}\) . Since \(0 \le \mu_{v}^{2} + v_{u}^{2} \le 1,\) \(\mathfrak{H}\left(\stackrel{\sim }{A}\right)\epsilon \left[\mathrm{0,1}\right]\) .

1.1.10 Picture Fuzzy Sets

Intuitionistic fuzzy set (IFS) theory has been introduced by Atanassov [ 4 , 23 ]. Although it has been used in many different fields, it has not provided to be a realistic approach for some situations. Hence, IFS has been extended to picture fuzzy sets (PiFS).

PiFS based approaches are more effective methods to meet different human views such as yes, abstain, no, and refusal. PiFS based models are successful in symbolizing uncertain information in different processes such as cluster analysis and pattern recognition [ 31 ]. Cuong [ 31 ]introduced picture fuzzy sets (PiFS) which are direct extensions of intuitonistic fuzzy sets. A picture fuzzy set

A picture fuzzy set à on the universe X is an object of the form

where μ à (x) ∈ [0, 1] is called the “degree of positive membership of Ô, η à (x) ∈ [0, 1] is called the “degree of neutral membership of Ô and ν à (x) ∈ [0, 1] is called the “degree of negative membership of Ô, and μ à (x), η à (x), and ν à (x) satisfy the following condition: 0 ≤ μ à (x) + η à (x) + ν à (x) ≤ 1, ∀ x ∈ X. Then for x ∈ X, π à (x) = 1 − μ à (x) − η à (x) − ν à (x)) could be called the degree of refusal membership of x in Ã. Thereafter, \(\langle x;{\mu }_{\tilde{\mathrm{A} }}\left(x\right),{\eta }_{\tilde{\mathrm{A} }}\left(x\right),{\vartheta }_{\tilde{\mathrm{A} }}\left(x\right)\rangle\) will be given as \(\left(x;{\mu }_{\tilde{\mathrm{A} }},{\eta }_{\tilde{\mathrm{A} }},{\vartheta }_{\tilde{\mathrm{A} }}\right)\) .

Voting can be a good illustration of such a situation as the human voters may be divided into four groups of those who: vote for, hesitant, and vote against, refusal of the voting [ 32 ]. Some definitions and theorems of PiFS are given in the following [ 33 ].

Definition 23

A PFSs on a \(\tilde{A}_{p}\) of the universe of discourse U is given by;

Then, for each \(u\) , the numbers \(\mu_{{\tilde{A}_{S} }} (u),{\kern 1pt} {\kern 1pt} \nu_{{\tilde{A}_{S} }} (u){\kern 1pt} {\kern 1pt}\) and \(\pi_{{\tilde{A}_{S} }} (u)\) are the degree of membership, non-membership and hesitancy of \(u\) to \(\tilde{A}_{S}\) , respectively. \(\rho =1-\left({\mu }_{{\stackrel{\sim }{A}}_{P}}\left(u\right)+{v}_{{\stackrel{\sim }{A}}_{P}}\left(u\right)+{\pi }_{{\stackrel{\sim }{A}}_{P}}\left(u\right)\right)\) is called as a refusal degree [ 34 ].

Definition 24

Let \(\stackrel{\sim }{\alpha }=\left({\mu }_{a},{\eta }_{\alpha },{\vartheta }_{\alpha }\right)\) be a picture fuzzy number (PiFN). Then, the score function S of a picture fuzzy number can be given as follows [ 35 ]:

Definition 25

Basic operators of Single-valued PFSs;

Picture fuzzy arithmetic aggregation operators are used for aggregating the different evaluations of multiexperts. Picture fuzzy weighted averaging (PiFWA) operator and picture fuzzy weighted geometric (PiFWG) operator are aggregation operators of PiFS as arithmetic and geometric aggregation operators which are developed by Guiwu [ 33 ].

Definition 26

Let α j ( j  = 1, 2, …, n ) be a collection of PFNs. The picture fuzzy weighted averaging (PFWA) operator is a mapping P n  →  P such that

where ω  = ( ω 1 , ω 2 , … , ω n ) T be the weight vector of α j ( j  = 1, 2, … , n ), and ω j  > 0, \(\sum_{j=1}^{n}{\omega }_{j}=1\)

The aggregated value by using PFWA operator is also a PiFN, where

Definition 27

Let α j (j = 1, 2, …, n) be a collection of PiFNs. The picture fuzzy weighted geometric (PiFWG) operator is a mapping P n  → P such that

where ω  = ( ω 1 , ω 2 , …, ω n ) T be the weight vector of α j ( j  = 1, 2, … , n ), and ω j  > 0, \(\sum_{j=1}^{n}{\omega }_{j}=1\)

The aggregated value by using PFWG operator is also a PFN, where

where ω  = ( ω 1 , ω 2 , …, ω n ) T be the weight vector of α j ( j  = 1, 2, …, n ), and ω j  > 0, \(\sum_{j=1}^{n}{\omega }_{j}=1\)

Definition 28

Single-valued Picture Fuzzy Weighted Averaging operator (PFWA) with respect to, \(w = (w_{1} ,w_{2} \ldots ,w_{n} );{\kern 1pt} {\kern 1pt} {\kern 1pt} w_{i} \in [0,1];{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \sum\limits_{i = 1}^{n} {w_{i} = 1}\) , is defined as;

Definition 29

Score functions and Accuracy functions of sorting picture fuzzy numbers are defined by;

Note that: \(\tilde{A}_{p} < \tilde{B}_{p} {\kern 1pt}\) if and only if

\(Score(\tilde{A}_{p} ) < Score(\tilde{B}_{p} )\) or and

\(Score(\tilde{A}_{p} ) = Score(\tilde{B}_{p} )\) and \(Accuracy(\tilde{A}_{p} ) < Accuracy(\tilde{B}_{p} )\)

Definition 30

Distance formulas for picture fuzzy numbers \(\left( {\tilde{A}_{p} ,{\kern 1pt} \tilde{B}_{p} {\kern 1pt} } \right)\) are defined as follows [ 36 ],

The normalized Hamming Distance;

The normalized Euclidean Distance;

1.1.11 Q-Rung Orthopair Fuzzy Sets (Q-RPFSs)

Q-ROFSs introduced by Yager [ 37 ] are represented with the degree of membership and non-membership. In q-ROFSs, the sum of the q th power of the membership and non-membership degrees must be at most equal to one [ 37 ]. In Fig.  1 , it is easily observed that q-ROFSs have an acceptable membership grade space larger than of IFSs and PFSs. Q-ROFSs are described as in Definition 31 and geometric space for IFNs, PFNs and q-ROFs are shown in Fig. 4 .

Geometric space range of IFNs, PFNs, and q-ROFNs

Definition 31

A q-ROFS Q in a finite universe of discourse X is defined as follows by Yager [ 37 ].

where the function \({\mu }_{Q}:X\to \left[\mathrm{0,1}\right]\) denotes the degree of membership and \({\upsilon }_{Q}:X\to \left[\mathrm{0,1}\right]\) ] denotes the degree of non-membership of the element x ∈ X to the set Q, respectively, with the condition that \(0\le {\mu }_{Q }\left(x\right)+ {\nu }_{Q }\left(x\right)\le 1,\) \(\left(q\ge 1\right)\) for every \(x \epsilon X\) . The degree of indeterminacy is given as \({\pi }_{P}\left(x\right)= \sqrt[q]{1-{{\mu }_{P}\left(x\right)}^{q}- {{\nu }_{P}\left(x\right)}^{q}}\) [ 38 ].

Definition 32

Let \(Q=\left({\mu }_{Q} ,{\nu }_{Q}\right)\) , \({Q}_{1}=\left({\mu }_{{Q}_{1}} ,{\nu }_{{Q}_{1}}\right)\) and \({Q}_{2}=\left({\mu }_{{Q}_{2}} ,{\nu }_{{Q}_{2}}\right)\) be three q-rung orthopair fuzzy numbers (q-ROFNs), then their operations can be defined as follows [ 37 ].

Definition 33

Let \(Q=\left({\mu }_{Q} ,{\nu }_{Q}\right)\) be a q-ROFN, then the score function \(s\left(\alpha \right)\) and accuracy function \(H\left(\alpha \right)\) of a can be defined as follows [ 37 ]:

Definition 34

Let \({Q}_{i}=\left({\mu }_{{Q}_{i}} ,{\nu }_{{Q}_{i}}\right)\) \(\left(i=1, 2, \dots ,n\right)\) be a set of q-ROFNs and \(w = {({w}_{1},{ w}_{2},\dots ,{w}_{n})}^{T}\) be weight vector of \({Q}_{i}\) with \(\sum_{i=1}^{n}{w}_{i}=1\) , then a q-rung orthopair fuzzy weighted average \(\left(\mathrm{q}-\mathrm{ROFWA}\right)\) operator is [ 37 ]:

Definition 35

Let \({Q}_{i}=\left({\mu }_{{Q}_{i}} ,{\nu }_{{Q}_{i}}\right)\) \(\left(i=\mathrm{1,2},\dots ,n\right)\) be a set of q-ROFNs and \(w = {({w}_{1},{ w}_{2},\dots ,{w}_{n})}^{T}\) be weight vector of \({Q}_{i}\) with \(\sum_{i=1}^{n}{w}_{i}=1\) , then a q-rung orthopair fuzzy weighted geometric \(\left(q-ROFWG\right)\) operator is [ 37 ]:

1.1.12 Fermatean Fuzzy Sets

[ 37 ] have introduced q-rung orthopair fuzzy sets being a general class of IFSs and PFSs. The sum of the q th power of membership degree and nonmembersip degree q-rung orthopair fuzzy sets is bounded by one. When q = 3, Senapati and Yager [ 39 ] have called q-rung orthopair fuzzy sets as fermatean fuzzy sets (FFSs). In Fig.  5 , the relation between IFSs, PFSs and FFSs is given

Comparision of IFSs, PFSs and FFSs

Definition 36

Let \(X\) be a universe of discourse. A Fermatean fuzzy sets \(\mathcal{F}\) in \(X\) is an object having the form Senapati and Yager [ 39 ]:

where \({\mu }_{F}:X\to \left[\mathrm{0,1}\right]\) and \(\nu _{F} :X \to [0,1]\)

for all \(x \epsilon X\) . The numbers \({\mu }_{F}\left(x\right)\) and \({\nu }_{F}(x)\) indicate, respectively, the degree of membership and the degree of non-membership of the element \(x\) in the set \(\mathcal{F}\) .

For any FFS \(\mathcal{F}\) and \(x \epsilon X\) , the degree of hesitancy is calculated as follows:

Definition 37

Let \(\mathcal{F}=\left({\mu }_{F} ,{\nu }_{F}\right)\) , \({\mathcal{F}}_{1}=\left({\mu }_{{F}_{1}} ,{\nu }_{{F}_{1}}\right)\) and \({\mathcal{F}}_{2}=\left({\mu }_{{F}_{2}} ,{\nu }_{{F}_{2}}\right)\) be three FFSs, then their operations are defined as follows [ 39 ]:

Definition 38

Let \(\mathcal{F}=\left({\mu }_{F} ,{\nu }_{F}\right)\) , \({\mathcal{F}}_{1}=\left({\mu }_{{F}_{1}} ,{\nu }_{{F}_{1}}\right)\) and \({\mathcal{F}}_{2}=\left({\mu }_{{F}_{2}} ,{\nu }_{{F}_{2}}\right)\) be three FFSs and \(\lambda >0\) , then the operations of these three FFNs are interpreted in these ways 39 :

Definition 39

Let \({\mathcal{F}}_{i}=\left({\mu }_{{F}_{i}} ,{\nu }_{{F}_{i}}\right)\) \(\left(i=\mathrm{1,2},\dots ,n\right)\) be a set of FFNs and \(w = {({w}_{1},{ w}_{2},\dots ,{w}_{n})}^{T}\) be weight vector of \({\mathcal{F}}_{i}\) with \(\sum_{i=1}^{n}{w}_{i}=1\) , then a fermatean fuzzy weighted average \(\left(\mathrm{FFWA}\right)\) operator is [ 40 ]:

Definition 40

Let \({\mathcal{F}}_{i}=\left({\mu }_{{F}_{i}} ,{\nu }_{{F}_{i}}\right)\) \(\left(i=\mathrm{1,2},\dots ,n\right)\) be a set of FFNs and \(w = {({w}_{1},{ w}_{2},\dots .,{w}_{n})}^{T}\) be weight vector of \({\mathcal{F}}_{i}\) with \(\sum_{i=1}^{n}{w}_{i}=1\) , then a fermatean fuzzy weighted geometric \(\left(\mathrm{FFWG}\right)\) operator is [ 40 ]:

Definition 41

Let \(X\) be an interval-valued fermatean fuzzy set (IVFFS). IVFFS \(\stackrel{\sim }{\mathcal{F}}\) in \(X\) is an object having the form

where \({\mu }_{\stackrel{\sim }{\mathcal{F}}}(x)\subseteq \left[\mathrm{0,1}\right]\) and \({\upsilon }_{\stackrel{\sim }{\mathcal{F}}}(x)\subseteq \left[\mathrm{0,1}\right]\) denote the membership degree and non-membership degree of the element \(x \epsilon X\) to the set \(\stackrel{\sim }{\mathcal{F}}\) , respectively. Also, for each \(x \epsilon X\) , \({\mu }_{\stackrel{\sim }{\mathcal{F}}}\left(X\right)\) and \({\upsilon }_{\stackrel{\sim }{\mathcal{F}}}\left(X\right)\) are closed intervals and their lower and upper bounds are denoted by \({\mu }_{\stackrel{\sim }{\mathcal{F}}}^{L}(x)\) , \({\mu }_{\stackrel{\sim }{\mathcal{F}}}^{U}(x)\) , \({\upsilon }_{\stackrel{\sim }{\mathcal{F}}}^{L}(x)\) , \({\upsilon }_{\stackrel{\sim }{\mathcal{F}}}^{U}(x)\) , respectively. Therefore, \(\stackrel{\sim }{\mathcal{F}}\) can also be expressed as follows:

where the expression is subject to the condition

For every \(x \epsilon X\) , \({\pi }_{\stackrel{\sim }{\mathcal{F}}}\left(x\right)=\left[{\pi }_{\stackrel{\sim }{\mathcal{F}}}^{L}\left(x\right),{\pi }_{\stackrel{\sim }{\mathcal{F}}}^{U}\left(x\right)\right]\) is called as the degree of hesitancy in IVFFSs, where \({\pi }_{\stackrel{\sim }{\mathcal{F}}}^{L}\left(x\right)=\sqrt[3]{1-{\left({\mu }_{\stackrel{\sim }{\mathcal{F}}}^{U}(x)\right)}^{3}-{\left({\upsilon }_{\stackrel{\sim }{\mathcal{F}}}^{U}(x)\right)}^{3}}\) and \({\pi }_{\stackrel{\sim }{\mathcal{F}}}^{U}\left(x\right)=\sqrt[3]{1-{\left({\mu }_{\stackrel{\sim }{\mathcal{F}}}^{L}(x)\right)}^{3}-{\left({\upsilon }_{\stackrel{\sim }{\mathcal{F}}}^{L}(x)\right)}^{3}}\) .

Definition 42

Let \(\stackrel{\sim }{\mathcal{F}}=\left(\left[{\mu }_{\stackrel{\sim }{\mathcal{F}}}^{L}, {\mu }_{\stackrel{\sim }{\mathcal{F}}}^{U}\right],\left[{\upsilon }_{\stackrel{\sim }{\mathcal{F}}}^{L},{\upsilon }_{\stackrel{\sim }{\mathcal{F}}}^{U}\right]\right),\) \({\stackrel{\sim }{\mathcal{F}}}_{1}=\left(\left[{\mu }_{{\stackrel{\sim }{\mathcal{F}}}_{1}}^{L}, {\mu }_{{\stackrel{\sim }{\mathcal{F}}}_{1}}^{U}\right] ,\left[{\upsilon }_{{\stackrel{\sim }{\mathcal{F}}}_{1}}^{L},{\upsilon }_{{\stackrel{\sim }{\mathcal{F}}}_{1}}^{U}\right]\right)\) and \({\stackrel{\sim }{\mathcal{F}}}_{2}=\left(\left[{\mu }_{{\stackrel{\sim }{\mathcal{F}}}_{2}}^{L}, {\mu }_{{\stackrel{\sim }{\mathcal{F}}}_{2}}^{U}\right] ,\left[{\upsilon }_{{\stackrel{\sim }{\mathcal{F}}}_{2}}^{L},{\upsilon }_{{\stackrel{\sim }{\mathcal{F}}}_{2}}^{U}\right]\right)\) be three FFSs and \(\lambda >0\) , then their operations are defined as follows:

Definition 43

Let \({\stackrel{\sim }{\mathcal{F}}}_{i}=\left(\left[{\mu }_{{\stackrel{\sim }{\mathcal{F}}}_{i}}^{L}, {\mu }_{{\stackrel{\sim }{\mathcal{F}}}_{i}}^{U}\right] ,\left[{\upsilon }_{{\stackrel{\sim }{\mathcal{F}}}_{i}}^{L},{\upsilon }_{{\stackrel{\sim }{\mathcal{F}}}_{i}}^{U}\right]\right)\left(i=\mathrm{1,2},\dots ,n\right)\) be a set of IVFFSs and \(w = {({w}_{1},{ w}_{2},\dots .,{w}_{n})}^{T}\) be weight vector of \({\mathcal{F}}_{i}\) with \(\sum_{i=1}^{n}{w}_{i}=1\) , then an interval-valued Fermatean fuzzy weighted average \(\left(\mathrm{IVFFWA}\right)\) operator is a mapping IVFFWA: \({\stackrel{\sim }{\mathcal{F}}}^{n}\to \stackrel{\sim }{\mathcal{F}}\) , where

Definition 44

Let \({\stackrel{\sim }{\mathcal{F}}}_{i}=\left(\left[{\mu }_{{\stackrel{\sim }{\mathcal{F}}}_{i}}^{L}, {\mu }_{{\stackrel{\sim }{\mathcal{F}}}_{i}}^{U}\right] ,\left[{\upsilon }_{{\stackrel{\sim }{\mathcal{F}}}_{i}}^{L},{\upsilon }_{{\stackrel{\sim }{\mathcal{F}}}_{i}}^{U}\right]\right)\left(i=\mathrm{1,2},\dots ,n\right)\) be a set of IVFFSs and \(w = {({w}_{1},{ w}_{2},\dots ,{w}_{n})}^{T}\) be weight vector of \({\stackrel{\sim }{\mathcal{F}}}_{i}\) with \(\sum_{i=1}^{n}{w}_{i}=1\) , then an interval-valued Fermatean fuzzy weighted geometric \(\left(\mathrm{IVFFWG}\right)\) operator is a mapping IVFFWG: \({\stackrel{\sim }{\mathcal{F}}}^{n}\to \stackrel{\sim }{\mathcal{F}}\) , where

Definition 45

Deffuzzification of \({\stackrel{\sim }{\mathcal{F}}}_{i}=\left(\left[{\mu }_{{\stackrel{\sim }{\mathcal{F}}}_{i}}^{L}, {\mu }_{{\stackrel{\sim }{\mathcal{F}}}_{i}}^{U}\right] ,\left[{\upsilon }_{{\stackrel{\sim }{\mathcal{F}}}_{i}}^{L},{\upsilon }_{{\stackrel{\sim }{\mathcal{F}}}_{i}}^{U}\right]\right)\left(i=\mathrm{1,2},\dots ,n\right)\) is given as in Eq. ( 106 ):

This defuzzification operation is based on Saaty’s classical 1–9 scale so that the defuzzification produces values between 1 and 9 for \(EI \le IVFFN \le CHI\) and 1/9 – 1 for \(SLI \le IVFFN \le CLI\) .

1.1.13 Spherical Fuzzy Sets

Spherical fuzzy sets (SFS) have been recently introduced by Kutlu Gundogdu and Kahraman (2019). These ets are based on the fact that the hesitancy of a decision maker can be assigned inde pendently satisfying the condition that the squared sum of membership, non-membership and hesitancy degrees is at most equal to 1. Thus, SFS are a mixture of PFS and NS theories. In the following, definition of SFS is presented:

Definition 46

Single valued Spherical Fuzzy Sets (SFS) \(\tilde{A}_{S}\) of the universe of discourse U is given by

For each \(u\) , the numbers \(\mu_{{\tilde{A}_{S} }} (u),{\kern 1pt} {\kern 1pt} \nu_{{\tilde{A}_{S} }} (u){\kern 1pt} {\kern 1pt}\) and \(\pi_{{\tilde{A}_{S} }} (u)\) are the degree of membership, non-membership and hesitancy of \(u\) to \(\tilde{A}_{S}\) , respectively.

The novel concept of Spherical Fuzzy Sets provides a larger preference domain for decision makers to assign membership degrees since the squared sum of the spherical parameters is allowed to be at most 1.0. DMs can determine their hesitancy degree independently under spherical fuzzy environment. Spherical fuzzy Sets (SFS) are a generalization of Pythagorean Fuzzy Sets, picture fuzzy sets and neutrosophic sets [ 41 – 45 ]. Neutrosophic sets are defined with three parameters truthiness, falsity, and indeterminacy. The values of these parameters are between 0 and 1, and the sum of these parameters can be between 0 and 3. The parameters can be defined independently in neutrosophic sets.

In spherical fuzzy sets, the squared sum of membership, nonmembership, and hesitancy parameters can be between 0 and 1, and each of them can be defined between 0 and 1 independently. Figure  6 illustrates the differences between IFS, PFS, NS, and SFS [ 46 – 56 ].

Geometric representations of IFS, PFS, NS, and SFS [ 43 ]

Definition 47

Basic operators of Single-valued SFS;

Definition 48

For these SFS \(\tilde{A}_{S} = (\mu_{{\tilde{A}_{S} }} ,v_{{\tilde{A}_{S} }} ,\pi_{{\tilde{A}_{S} }} )\) and \(\tilde{B}_{S} = (\mu_{{\tilde{B}_{S} }} ,v_{{\tilde{B}_{S} }} ,\pi_{{\tilde{B}_{S} }} )\) , the followings are valid under the condition \(\lambda ,\lambda_{1} ,\lambda_{2} \ge 0\) .

Definition 49

Single-valued Spherical Weighted Arithmetic Mean (SWAM) with respect to, \(w = (w_{1} ,w_{2} \ldots ,w_{n} );{\kern 1pt} {\kern 1pt} {\kern 1pt} w_{i} \in [0,1];{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \sum\limits_{i = 1}^{n} {w_{i} = 1}\) , SWAM is defined as;

Definition 50

Single-valued Spherical Weighted Geometric Mean (SWGM) with respect to, \(w = (w_{1} ,w_{2} \ldots ,w_{n} );{\kern 1pt} {\kern 1pt} {\kern 1pt} w_{i} \in [0,1];{\kern 1pt} \quad \sum\limits_{i = 1}^{n} {w_{i} = 1}\) , SWGM is defined as

Definition 51

Score functions and Accuracy functions of sorting SFS are defined by;

Note that: \(\tilde{A}_{S} < \tilde{B}_{S} {\kern 1pt}\) if and only if

\(Score(\tilde{A}_{S} ) < Score(\tilde{B}_{S} )\) or

\(Score(\tilde{A}_{S} ) = Score(\tilde{B}_{S} )\) and \(Accuracy(\tilde{A}_{S} ) < Accuracy(\tilde{B}_{S} )\)

1.2 Literature Review

In order to reach the appropriate number of samples, we have emphasized the two main features of the fuzzy sets articles available in the literature; the number of citations and been published in the recent years. To identify the articles, the most comprehensive academic search engines, SCOPUS and ScienceDirect, are used. The used search pattern consists of the keywords “Neutrosophic Sets”, “Pyhtagorean Fuzzy Sets”, “Spherical Fuzzy Sets” and etc. After the analysis, we introduce brief information about the studies which is given in Fig.  7 .

Distribution of ordinary fuzzy set papers with respect to years

1.2.1 Literature Review on Ordinary Fuzzy Sets

We search the “ Ordinary Fuzzy” Sets term in scopus and find 1031 results. The analysis of Ordinary fuzzy sets are given in Fig.  7 . In this figure, the distribution of papers by years is given.

Figure  7 shows that the distribution of “Ordinary fuzzy” sets papers with respect to years. It is seen that most of the studies have been published in 2020 with a rate of 5.04%.

In Fig.  8 , ordinaryfuzzy sets papers are summarized with their subject areas. The other subject areas with smaller percentages are Social Sciences, Earth and Planetary Sciences, Agricultural and Biological Sciences, Chemistry, Business, Management and Accounting, Economics, Econometrics and Finance, Energy, Biochemistry, Genetics and Molecular Biology and Medicine, etc.

Distribution of ordinary fuzzy set papers with respect to subject areas

Figure  9 illustrates the sources of the ordinary fuzzy set papers with their percentages.

Ordinary fuzzy set papers by their sources

Most of the publications on ordinary fuzzy sets have been published in Fuzzy Sets and Systems, Lecture Notes in Computer Science Including Subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics.

In Fig.  10 , the publication percentages of authors on ordinary fuzzy sets are presented. Kahraman and Belohlavek are the leader among these authors.

The distribution of publication percentages of authors on ordinary fuzzy sets

In Fig.  11 , the distribution of publications on ordinary fuzzy sets with respect to their source countries is illustrated. China is the leading country on ordinary fuzzy publications. United States and Turkey are the next two countries after China.

Distribution of ordinary fuzzy sets papers by their countries

In Fig.  12 , the document types on ordinary fuzzy sets are of seven types: articles with a percentage of 64.7, book chapters with a percentage of 2.2, conference papers with a percentage of 31.4 and conference review with a percentage of 0.3, review with a percentage of 0.6 and editorial with the percentage of 0.1.

Distribution of ordinary fuzzy sets papers by their types

1.2.2 Literature Review on Type 2 Fuzzy Sets

We search the “Type 2 Fuzzy” sets term in Scopus and find 4064 results. The analysis of type 2 fuzzy sets are given in Figs. 13 , 14 , 15 , 16 , 17 and 18 . In Fig.  13 , the distribution of type 2 papers by years is given.

Distribution of Type 2 fuzzy sets papers with respect to years

Distribution of type 2 fuzzy set papers with respect to subject areas

Type 2 fuzzy set papers by their sources

Distribution of publication percentages of authors on Type 2 fuzzy sets

Distribution of Type 2 fuzzy sets papers by their countries

Distribution of type 2 fuzzy sets papers by their types

Figure  13 shows that the distribution of “Type 2 fuzzy” set papers with respect to years. It is seen that most of the studies have been published in 2019 with a rate of 9.1%.

In Fig.  14 , type 2 fuzzy set papers are summarized by their subject areas. In Fig.  14 , the distribution of type 2 fuzzy sets papers by their subject areas is illustrated. The other subject areas with lower percentages are Materials Science, Business, Management and Accounting, Earth and Planetary Sciences, Medicine, Chemical Engineering, Agricultural and Biological Sciences, Biochemistry, Genetics and Molecular Biology, Chemistry, Neuroscience, Economics, Econometrics and Finance, Multidisciplinary, Arts and Humanities, Health Professions, Psychology, Immunology and Microbiology, and Nursing.

Figure  15 presents type 2 fuzzy set papers by their sources. Most of the publications on type 2 fuzzy sets have been published in IEEE International Conference on Fuzzy Systems. The total number of journals is 130.

In Fig.  16 , the publication percentages and the corresponding numbers of authors on type 2 fuzzy sets are presented. Mendel and Castillo are the leader among these authors.

In Fig.  17 , the distribution of publications on type 2 fuzzy sets with respect to their source countries is illustrated. China is the leading country on type 2 fuzzy publications. United States and India are the next two countries after China.

The percentages and the corresponding numbers of Type 2 fuzzy papers are illustrated in Fig.  18 . The document types on type 2 fuzzy sets are article with a percentage of 54.55%, book chapter with a percentage of 1.99, conference paper with a percentage of 40.03% and conference review with a percentage of 1.8, editorial with a percentage of 0.15, book with a percentage of 0.27.

1.2.3 Literature Review on Interval Valued Fuzzy Sets

We search the “Interval Valued Fuzzy Sets” term in Scopus and find 2989 results. The analysis of interval valued fuzzy sets is given in Figs. 19 and 20 . In Fig.  19 , the distribution of papers by years is given.

Distribution of interval valued fuzzy set papers with respect to years

Distribution of interval valued fuzzy sets papers with respect to subject areas

Figure  19 shows that the distribution of “Interval Valued fuzzy” sets papers with respect to years. It is seen that most of the studies have been published in 2018 with a rate of 11%.

In Fig.  20 , interval valued fuzzy set papers are summarized with respect to their subject areas. The other subject areas with lower percentages are Physics and Astronomy, Environmental Science, Social Sciences, Materials Science, Multidisciplinary, Economics, Econometrics and Finance, Earth and Planetary Sciences, Energy, Chemistry, Biochemistry, Genetics and Molecular Biology, Arts and Humanities, Neuroscience, Medicine, Chemical Engineering, Agricultural and Biological Sciences, Pharmacology, Toxicology and Pharmaceutics, Immunology and Microbiology.

Figure  21 illustrates interval valued fuzzy set papers by their sources. Most of the publications on interval valued fuzzy sets have been published in Fuzzy Sets and Systems and Journal of Intelligent and Fuzzy Systems. The total of other journals is 135.

Interval valued fuzzy set papers by their sources

In Fig.  22 , the publication percentages of authors on interval valued fuzzy sets are presented. Bustince and Xu are the leader among these authors.

Distribution of publication percentages of authors on interval valued fuzzy sets

In Fig.  23 , the distribution of publications on interval valued fuzzy sets with respect to their source countries is illustrated. China is the leading country on interval valued fuzzy publications. India and United States are the next two countries after China.

Distribution of interval valued fuzzy sets papers by their countries

The percentages and the corresponding numbers of interval valued fuzzy papers are illustrated in Fig.  24 . The document types on interval valued fuzzy sets are articles with a percentage of 67.21%, book chapters with a percentage of 1.84, conference papers with a percentage of 27.53% and conference review with a percentage of 1.84.

Distribution of interval valued fuzzy sets papers by their types

1.2.4 Literature Review on Intuitionistic Fuzzy Sets

We search the “intuitionistic fuzzy sets” term in Scopus and find 5893 results. The analysis of intuitionistic fuzzy sets is given in Figs. 25 , 26 , 28 , 29 and 30 . In Fig.  25 , the distribution of papers by years is given. It is seen that most of the studies have been published in 2019 with a rate of 12%.

Distribution of intuitionistic fuzzy set papers with respect to years

Distribution of intuitionistic fuzzy set papers with respect to subject areas

Intuitionistic fuzzy set papers by their sources

Distribution of publication percentages of authors on intuitionistic fuzzy sets

Distribution of intuitionistic fuzzy sets papers by their countries

Distribution of intuitionistic fuzzy sets papers by their types

In Fig.  26 , intuitionistic fuzzy set papers are summarized with their contents, types of picture and applied methods. In Fig.  26 , the distribution of intuitionistic fuzzy sets papers by their subject areas. The other subject areas with lower percentages are Materials Science, Social Sciences, Energy, Environmental Science, Multidisciplinary, Chemistry, Earth and Planetary Sciences, Biochemistry, Genetics and Molecular Biology Economics, Econometrics and Finance Medicine, Chemical Engineering, Agricultural and Biological Sciences, Arts and Humanities, Neuroscience, Pharmacology, Toxicology and Pharmaceutics, Psychology, Health Professions, and Immunology and Microbiology.

Figure  27 illustrates the sources of intuitionistic fuzzy papers. Most of the publications on intuitionistic fuzzy sets have been published in Journal of Intelligent and Fuzzy Systems and Advances in Intelligent Systems and Computing. The total of other journals is 143.

In Fig.  28 , the publication percentages and the corresponding authors on İntuitionistic fuzzy sets are presented. Xu and Garg are the leader among these authors.

In Fig.  29 , the distribution of publications on Intuitionistic fuzzy sets with respect to their source countries is illustrated. China is the leading country on Nonstationary fuzzy publications. India and Turkey are the next two countries after China.

The percentages and the corresponding numbers of intuitionistic fuzzy papers are illustrated in Fig.  30 . The document types on intuitionistic fuzzy sets are articles with a percentage of 68%, book chapters with a percentage of 2, conference papers with a percentage of 28% and conference review with a percentage of 1.

1.2.5 Literature Review on Fuzzy Multisets

We search the “Fuzzy Multisets” term in Scopus and find 129 results. The analysis of Fuzzy Multisets is given in Figs. 31 , 32 , 33 , 34 , 35 and 36 . In Fig.  31 , the distribution of papers by years is given. Figure  31 shows that the distribution of Fuzzy Multisets papers with respect to years. It is seen that most of the studies have been published in 2014 with a rate of 10%.

Distribution of fuzzy multisets papers with respect to years

Distribution of fuzzy multisets papers with respect to subject areas

Fuzzy multisets papers by their sources

The distribution of publication percentages of authors on Fuzzy Multisets

Distribution of fuzzy multisets papers by their countries

Distribution of Fuzzy Multisets papers by their types

In Fig.  32 , Fuzzy Multisets papers are summarized with respect to their subjects. The other subjects with lower percentages are Physics and Astronomy, Chemistry, Business, Management and Accounting, Agricultural and Biological Sciences, Chemical Engineering, Economics, Econometrics and Finance, Environmental Science, Materials Science, Social Sciences,Medicine, Chemical Engineering, Agricultural and Biological Sciences, Arts and Humanities, Neuroscience, Pharmacology, Toxicology and Pharmaceutics, Psychology, Health Professions, Immunology and Microbiology.

Figure  33 shows the sources of the fuzzy multisets publications. Most of the publications on Fuzzy Multisets have been published in Lecture Notes in Computer Science Including Subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics and Fuzzy Sets and Systems. The total of other journals is 51.

In Fig.  34 , the publication percentages and the corresponding numbers of authors on Fuzzy Multisets are presented. Miyamoto and Kudo are the leader among these authors.

In Fig.  35 , the distribution of publications on Fuzzy Multisets with respect to their source countries is illustrated. Japan is the leading country on Fuzzy Multisets publications. Spain and China are the next two countries after Japan.

The percentages and the corresponding numbers of Fuzzy Multisets papers are illustrated in Fig.  36 . The document types on Fuzzy Multisets are of five types: articles with a percentage of 49%, book chapters with a percentage of 2, conference papers with a percentage of 40%, conference review with a percentage of 9 and review with a percentage of 1.

1.2.6 Literature Review on Intuitionistic Fuzzy Sets of Second Type

We search the “Intuitionistic Fuzzy Sets of Second Type” sets term in Scopus and find 54 results. The analysis of intuitionistic Fuzzy Sets of Second Type is given in Figs. 37 , 38 , 39 , 40 , 41 and 42 . In Fig.  37 , the distribution of papers by years is given. It is seen that the largest percentage has been published in 2016 with a rate of 17%.

Distribution of intuitionistic fuzzy sets of second type papers with respect to years

Distribution of intuitionistic fuzzy sets of second type papers with respect to subject areas

Intuitionistic fuzzy sets of second type papers by their sources

Distribution of publication percentages of authors on Intuitionistic Fuzzy Sets of second type

Distribution of intuitionistic fuzzy sets of Second Type papers by their countries

Distribution of intuitionistic fuzzy set of second type by their types

In Fig.  38 , Intuitionistic Fuzzy Sets of Second Type papers are summarized with respect to their subject areas. The other subject areas with lower percentages are Arts and Humanities, Health Professions, Multidisciplinary, Neuroscience, Physics and Astronomy and Social Sciences.

Figure  39 shows the sources of the published papers. Most of the publications on Intuitionistic Fuzzy Sets of Second Type have been published in Information Sciences and Journal of Intelligent and Fuzzy Systems.

In Fig.  40 , the publication percentages and the corresponding numbers of authors on Intuitionistic Fuzzy Sets of Second Type are presented. Huang and Li are the leaders among these authors.

In Fig. 41 , the distribution of publications on Intuitionistic Fuzzy Sets of Second Type with respect to their source countries is illustrated. China is the leading country on intuitionistic fuzzy sets of second type publications. India and United States are the next two countries before Turkey.

The percentages and the corresponding numbers of intuitionistic fuzzy sets of second type papers are illustrated in Fig.  42 . The document types on intuitionistic fuzzy sets of second type are of three types: articles with a percentage of 78%, book chapters with a percentage of 2, conference papers with a percentage of 20%.

1.2.7 Literature Review on Neutrosophic Fuzzy Sets

We search the “Neutrosophic Sets” term in Scopus and find 456 results. The analysis of neutrosophic sets are given in Figs. 43 , 44 , 45 , 46 and 48 . In Fig.  43 , the distribution of papers by years is given.

Distribution of neutrosophic sets papers with respect to years

Distribution of neutrosophic sets papers with respect to subject areas

Neutrosophic sets papers by their sources

Distribution of publication percentages of authors on neutrosophic sets

Distribution of neutrosophic sets papers by their countries

Distribution of neutrosophic sets papers by their document types

In Fig.  44 , neutrosophic sets papers are summarized with respect to their subject areas. Other subject areas with lower percentages are Multidisciplinary Business, Management and Accounting Energy Economics, Econometrics and Finance Medicine Earth and Planetary Sciences Arts and Humanities Neuroscience Agricultural and Biological Sciences.

Figure  45 illustrates that the most of the publications on neutrosophic sets have been published in International Journal of Intelligent Systems and Journal of Intelligent and Fuzzy Systems.

In Fig.  46 , the publication percentages and the corresponding numbers of authors on neutrosophic sets are presented. Smarandache and Ye are the leader among these authors.

In Fig.  47 , the distribution of publications on neutrosophic fuzzy sets with respect to their source countries is illustrated. China and Turkey are the leading countries on neutrosophic fuzzy publications. India and United States are the next two countries after China.

The percentages and the corresponding numbers of neutrosophic papers are illustrated in Fig.  48 . The document types on neutrosophic sets are of five types: articles with a percentage of 79 book chapters with a percentage of 4, conference papers with a percentage of 14, conference review with a percentage of 1 and review with a percentage of 1.

1.2.8 Literature Review on Hesitant Fuzzy Sets

We search the “Hesitant Fuzzy” Sets term in Scopus and find 1583 results. The analysis results of Hesitant fuzzy sets are given in Figs. 49 , 50 , 51 , 52 , 53 and 54 . Figure  49 shows that the distribution of “Hesitant fuzzy” sets papers with respect to years.

Distribution of Hesitant fuzzy sets papers with respect to years

Distribution of Hesitant fuzzy sets papers with respect to subject areas

Hesitant fuzzy sets papers by their sources

Distribution of publication percentages of authors on Hesitant fuzzy sets

Distribution of hesitant fuzzy sets papers by their countries

Distribution of hesitant fuzzy sets papers by their document types

In Fig.  50 , Hesitant fuzzy sets papers are classified with respect to their subject areas. The other subject areas with lower percentages are Medicine, Earth and Planetary Sciences, Neuroscience, Biochemistry, Genetics and Molecular Biology, Arts and Humanities, Chemical Engineering, Agricultural and Biological Sciences, Pharmacology, Toxicology and Pharmaceutics, Health Professions, Immunology and Microbiology and Psychology.

Figure  51 shows the sources of hesitant fuzzy publications. Most of the publications on Hesitant fuzzy have been published in International Journal of Intelligent Systems and Journal of Intelligent and Fuzzy Systems.

In Fig.  52 , the publication percentages and the corresponding numbers of authors on Nonstationary fuzzy sets are presented. Xu and Liao are the leaders among these authors.

In Fig.  53 , the distribution of publications on Hesitant fuzzy sets with respect to their source countries is illustrated. China and Spain are the leading country on Hesitant fuzzy publications. Turkey and India are the next two countries after Turkey.

The percentages and the corresponding numbers of Hesitant fuzzy papers are illustrated in Fig.  54 . The document types on Hesitant fuzzy sets are of five types: articles with a percentage of 81, book chapters with a percentage of 1, conference papers with a percentage of 15 and conference review with a percentage of 2.

1.2.9 Literature Review on Nonstationary Fuzzy Sets

We search the Nonstationary Fuzzy Sets term in Scopus and find 456 results. The analysis of Nonstationary fuzzy sets is given in Figs. 55 , 56 , 57 , 58 , 59 and 60 .

Distribution of nonstationary fuzzy sets papers with respect to years

Distribution of nonstationary fuzzy sets papers with respect to subject areas

Nonstationary fuzzy sets papers by their sources

Distribution of publication percentages of authors on Nonstationary fuzzy sets

Distribution of nonstationary fuzzy sets papers by their countries

Distribution of nonstationary fuzzy sets papers by their document types

Figure  55 shows that the distribution of “Nonstationary fuzzy” sets papers with respect to years.

In Fig.  56 , the distribution of Nonstationary fuzzy sets papers is given by their subjects areas. The other subject areas with lower percentages are Medicine, Business, Management and Accounting, Decision Sciences, Social Sciences, Biochemistry, Genetics and Molecular Biology, Chemical Engineering, Health Professions, Immunology and Microbiology, Agricultural and Biological Sciences, Economics, Econometrics and Finance and Multidisciplinary.

Figure  57 shows the sources of Nonstationary fuzzy publications. Most of the publications on Nonstationary have been published in IEEE International Conference on Fuzzy Systems and IEEE Transactions on Fuzzy Systems.

In Fig.  58 , the publication percentages and the corresponding numbers of authors on Nonstationary fuzzy sets are presented. Gomide and Leite are the leaders among these authors.

In Fig.  59 , the distribution of publications on Nonstationary fuzzy sets with respect to their source countries is illustrated. China is the leading country on Nonstationary fuzzy publications. USA and Brazil are the next two countries after China.

The percentages and the corresponding numbers of Nonstationary fuzzy papers are illustrated in Fig.  60 . The document types on Nonstationary fuzzy sets are of five types: articles with a percentage of 49%, book chapters with a percentage of 3, conference papers with a percentage of 40 and conference review with a percentage of 7.

1.2.10 Literature Review on Pythagorean Fuzzy Sets

We search the “Pythagorean Fuzzy Sets” term in Scopus and find 456 results. The analyses of Pythagorean fuzzy sets are given in Figs. 61 , 62 , 63 , 64 , 65 and 66 .

Distribution of Pythagorean fuzzy sets papers with respect to years

Distribution of pythagorean fuzzy sets papers with respect to subject areas

Pythagorean fuzzy sets papers by their sources

Distribution of publication percentages of authors on Pythagorean fuzzy sets

Distribution of pythagorean fuzzy sets papers by their countries

Distribution of Pythagorean fuzzy sets papers by their types

In Fig.  61 , the distribution of papers by years is given. It shows that the distribution of Pythagorean fuzzy set papers with respect to years. It is seen that most of the studies have been published in 2020 with a rate of 35%.

In Fig.  62 , the distribution of Pythagorean fuzzy sets papers by their subjects areas is presented. The other subject areas with lower percentages are Business, Management and Accounting, Multidisciplinary, Energy, Economics, Econometrics and Finance, Medicine, Earth and Planetary Sciences, Neuroscience, Arts and Humanities, Agricultural and Biological Sciences.

Figure  63 shows the sources of Pythagorean fuzzy set publications. Most of the publications on Pythagorean fuzzy sets have been published in International Journal of Intelligent Systems and Journal of Intelligent and Fuzzy Systems.

In Fig.  64 , the publication percentages and the corresponding numbers of authors on Pythagorean fuzzy sets are presented. Kahraman and Garg are the leaders among these authors.

In Fig.  65 , the distribution of publications on Pythagorean fuzzy sets with respect to their source countries is illustrated. China is the leading country on Pythagorean fuzzy publications. Turkey and Pakistan are the next two countries after China.

The document types of Pythagorean fuzzy papers are illustrated in Fig.  66 . The document types on Pythagorean fuzzy sets are of five types: articles with a percentage of 85, book chapters with a percentage of 3, conference papers with a percentage of 11 and conference review with a percentage of 1.

1.2.11 Literature Review on Picture Fuzzy Sets

We search the “Picture Fuzzy Sets” term in Scopus and find 298 results. The analyses of Picture fuzzy sets are given in Figs. 67 , 68 , 69 , 70 , 71 and 72 .

Distribution of picture fuzzy sets papers with respect to years

Distribution of picture fuzzy sets papers with respect to subject areas

Picture fuzzy sets papers by their sources

Distribution of publication percentages of authors on picture fuzzy sets

Distribution of picture fuzzy sets papers by their countries

Distribution of picture fuzzy sets papers by their document types

Figure  67 shows that the distribution of “q-rung Orthopair fuzzy” set papers with respect to years. It is seen that most of the studies have been published in 2019 with a rate of 25.2%.

In Fig.  68 , Picture fuzzy sets papers are summarized by their subject areas. The other subject areas with lower percentages are Decision Sciences Environmental Science Energy Economics, Econometrics and Finance Multidisciplinary Agricultural and Biological Sciences Arts and Humanities Biochemistry, Genetics and Molecular Biology Medicine Chemical Engineering Neuroscience Earth and Planetary Sciences Health Professions Immunology and Microbiology Pharmacology, Toxicology and Pharmaceutics Psychology.

Figure  69 shows the sources of picture fuzzy set publications. Most of the publications on picture fuzzy sets have been published in Symmetry and International Journal of Intelligent Systems.

In Fig.  70 , the publication percentages and the corresponding numbers of authors on picture fuzzy sets are presented. Mahmood and Wei are the leader among these authors.

In Fig.  71 , the distribution of publications on picture fuzzy sets with respect to their source countries is illustrated. China and India are the leading countries on picture fuzzy publications. Pakistan and Vietnam are the next two countries after India.

The percentages and the corresponding numbers of picture fuzzy papers are illustrated in Fig.  72 . The document types on Picture fuzzy sets are of five types: articles with a percentage of 68.1%, book chapters with a percentage of 28, conference papers with a percentage of 10% and conference review with a percentage of 3.

1.2.12 Literature Review on q-rung Orthopair Fuzzy Sets

We search the q-rung Orthopair Fuzzy Sets term in Scopus and find 126 results. The analyses of q-rung Orthopair fuzzy sets are given in Figs. 73 and 74 . Figure  73 shows that the distribution of “q-rung Orthopair fuzzy” sets papers with respect to years. It is seen that most of the studies have been published in 2019 with a rate of 49.2%.

Distribution of q-rung orthopair fuzzy sets papers with respect to years

Distribution of q-rung Orthopair fuzzy sets papers with respect to subject areas

In Fig.  74 , q-rung Orthopair fuzzy set papers are summarized by their subject areas. The other subject areas are Economics, Econometrics and Finance Energy Environmental Science Multidisciplinary Agricultural and Biological Sciences Biochemistry, Genetics and Molecular Biology Social Sciences Arts and Humanities Business, Management and Accounting Chemical Engineering Medicine Neuroscience. In Fig. 75 , it is seen that most of the publications on q-rung Orthopair fuzzy sets have been published in International Journal of Intelligent Systems and Symmetry.

q-rung Orthopair fuzzy sets papers by their sources

Most of the publications on q-rung Orthopair fuzzy sets have been published in International Journal of Intelligent Systems and Symmetry.

In Fig.  76 , the publication percentages and the corresponding numbers of authors on q-rung Orthopair fuzzy sets are presented. Liu and Wei are the leader among these authors.

Publication percentages and numbers of authors on q-rung Orthopair fuzzy sets

In Fig.  78 , the distribution of publications on q-rung Orthopair fuzzy sets with respect to their source countries is illustrated. China and Pakistan are the leading countries on q-rung orthopair fuzzy publications. India and Saudi Arabia are the next two countries after Pakistan. In Fig. 77 , the distribution of publications on q-rung Orthopair fuzzy sets with respect to ther source countries is illustrated. China and Pakistan are the leading countries on q-rung Orthopair fuzzy publications. India and Saudi Arabia are the next two countries after Pakistan.

Distribution of q-rung orthopair fuzzy sets papers by their countries

Distribution of q-rung Orthopair fuzzy sets papers by their types

The percentages and the corresponding numbers of q-rung orthopair fuzzy papers are illustrated in Fig.  78 . The document types on q-rung Orthopair fuzzy sets are of four types: articles with a percentage of 92.1, book chapters with a percentage of 1.6, conference papers with a percentage of 4 and conference review with a percentage of 0.8.

1.2.13 Literature Review on Fermatean Fuzzy Sets

We search the Fermatean Fuzzy Sets term in Scopus and find 8 results. The analysis of fermatean fuzzy sets are given in Figs. 79 , 80 , 81 , 82 , 83 and 84 . Figure  79 shows the distribution of fermatean fuzzy set papers with respect to years. It is seen that most of the studies have been published in 2019 with a rate of 62.5%.

Distribution of Fermatean fuzzy sets papers with respect to years

Distribution of Fermatean fuzzy sets papers with respect to subject areas

Fermatean fuzzy sets papers by their sources

Publication percentages and numbers of authors on Fermatean fuzzy sets

Distribution of fermatean fuzzy sets papers by their countries

Distribution of Fermatean fuzzy sets papers by their types

In Fig.  80 , fermatean fuzzy set papers are summarized by their subject areas.

Figure  81 illustrates the sources of fermatean fuzzy sets. Most of the publications on fermatean fuzzy sets have been published in International Journal of Intelligent Systems.

In Fig.  82 , the publication percentages and the corresponding numbers of authors on fermatean fuzzy sets are presented. Senapati and Yager are the leader among these authors.

In Fig.  83 , the distribution of publications on fermatean fuzzy sets with respect to their source countries is illustrated. China, India and USA are the leading countries on fermatean fuzzy publications.

The percentages and the corresponding numbers of fermatean fuzzy set papers by their document types are illustrated in Fig.  85 . The document types on fermatean fuzzy sets are of four types: articles with a percentage of 59%, book chapters with a percentage of 28, conference papers with a percentage of 10% and conference review with a percentage of 3.

Distribution of spherical fuzzy sets papers with respect to years

1.2.14 Literature Review on Spherical Fuzzy Sets

We search the Spherical Fuzzy Sets term in Scopus and find 79 results. The analyses of spherical fuzzy sets are given in Figs. 85 , 86 , 87 , 88 , 89 and 90 .

Distribution of spherical fuzzy sets papers with respect to subject areas

Spherical fuzzy sets papers by their sources

Publication percentages and numbers of authors on Spherical fuzzy sets

Distribution of Spherical fuzzy sets papers by their countries

Distribution of spherical fuzzy set papers by their document types

Figure  85 shows the distribution of spherical fuzzy set papers with respect to years. It is seen that most of the studies have been published in 2020 with a rate of 37%.

In Fig.  86 , spherical fuzzy sets papers are summarized by their subjects areas.

Most of the publications on SFSs have been published in Studies in Fuziness and Soft Computing and Journal of Intelligent and Fuzzy Systems.

In Fig.  88 , the publication percentages and the corresponding numbers of authors on spherical fuzzy sets are presented. Cengiz Kahraman and Fatma Kutlu Gündoğdu are the leader among these authors.

In Fig.  89 , the distribution of publications on spherical fuzzy sets with respect to their source countries is illustrated. Turkey is the leading country on Spherical fuzzy publications. Pakistan and India are the next two countries after Turkey.

The percentages and the corresponding numbers of spherical fuzzy papers by their document types are illustrated in Fig.  90 . The document types on spherical fuzzy sets are of four types: articles with a percentage of 59, book chapters with a percentage of 28, conference papers with a percentage of 10 and conference review with a percentage of 3.

1.3 Conclusion

The overall aim of the fuzzy set extensions is to define membership functions with more parameters for more flexible and more realistic modeling. Intuitionistic fuzzy sets, Pythagorean fuzzy sets, and q-rung orthopair fuzzy sets are the members of the same class, each having two parameters, namely membership and non-membership degrees whereas neutrosophic sets, picture fuzzy sets, fermatean fuzzy sets, and spherical fuzzy sets are the members of another class, each having three independent parameters, namely membership, nonmembership, and indeterminacy degrees. Human behaviour has numerous features that are very complex, difficult to model, nonrigid and difficult to predict. There is a big difference between the happiness of a person having a success in his/her business life and the happiness of having a lunch with his/her best friend. Modeling these two kinds of happiness completely requires different membership functions of happiness. While the happiness in business success increases one’s self-confidence, the happiness he/she will have at lunch with his/her best friend increases his/her emotions. It is possible to model these features using fuzzy logic rather than rigid classical logic models. If the behavior of humanoid robots is intended to resemble that of humans, it is inevitable to use fuzzy set extensions in modeling.

The classical control methods cannot handle the differences between these kinds of happiness because of their rigid modelling techniques. Hence, the functions of humanoid robots’ mouth, face, eyes, and other body mimics must be modelled based on fuzzy logic rather than classical logic. Fuzzy sets composed of more parameters clearly present more opportunities to model human behaviours.

Chen, C.H., Wang, C.C., Wang, Y.T., Wang, P.T.: Fuzzy logic controller design for intelligent robots. Math. Probl. Eng. (2017)

Google Scholar  

Zadeh, L.A.: Information and control. Fuzzy sets 8 (3), 338–53 (1965)

Zadeh, L.A.: The concept of a linguistic variable and its application to approximate reasoning. Inf. Sci. 8 (3), 199–249 (1975)

Article   MathSciNet   Google Scholar  

Atanassov, K.T.: Intuitionistic fuzzy sets. Fuzzy Sets Syst. 20 (1), 87–96 (1986)

Article   Google Scholar  

Sari, I.U., Kahraman, C.: Intuitionistic Fuzzy Z-numbers. In: Kahraman, C., Cevik Onar, S., Oztaysi, B., Sari, I., Cebi, S., Tolga, A. (eds.) Intelligent and Fuzzy Techniques: Smart and Innovative Solutions. INFUS 2020. Advances in Intelligent Systems and Computing, vol. 1197. Springer, Cham (2021)

Xu, Z.: Methods for aggregating interval-valued intuitionistic fuzzy information and their application to decision making. Kongzhi Yu Juece/Control Decis. 22 (2), 215–219 (2007)

Wei, G., Wang, X.: Some geometric aggregation operators based on interval-valued intuitionistic fuzzy sets and their application to group decision making. In: 2007 International Conference on Computational Intelligence and Security (CIS 2007), pp. 495–499. Harbin (2007)

Abdullah, L., Ismail, W.K.: Hamming distance in intuitionistic fuzzy sets and interval- valued intuitionistic fuzzy sets: a comparative analysis. Adv. Comput. Math. Appl. 1 (1), 7–11 (2012)

Karasan, A., Kahraman, C.: Interval-valued neutrosophic extension of EDAS method, In: Kacprzyk, J., Szmidt, E., Zadrożny, S., Atanassov, K., Krawczak, M. (eds.) Advances in Fuzzy Logic and Technology 2017. EUSFLAT 2017, IWIFSGN 2017. Advances in Intelligent Systems and Computing, vol. 642, pp. 343–357. Springer, Cham (2017)

Yager, R.R.: On the theory of bags. Int. J. Gener. Syst. 13 (1), 23–37 (1986)

Smarandache, F.: Neutrosophy, Neutrosophic Probability, Set, and Logic. Amer. Res. Press, Rehoboth, 105 (1998)

Bolturk and Kahraman: A novel interval-valued neutrosophic AHP with cosine similarity measure. Soft. Comput. 22 , 4941–4958 (2018)

Li, Y., Wang, Y., Liu, P.: Multiple attribute group decisionmaking methods based on trapezoidal fuzzy two-dimension linguistic power generalized aggregation operators. Soft Comput. 20 (7), 2689–2704 (2016)

Zhang, H.Y., Wang J.Q., Chen, X.H.: Interval neutrosophic sets and their application in multicriteria decision-making problems. Sci. Worl J. 15 (2014)

Karasan, A., Kahraman, C.: A noval ıntuitionistic fuzzy dematel—anp—topsis ıntegrated methodology for freight village location selection. J. Intell. Fuzzy Syst., 1–18 (2017)

Kahraman, C., Otay, İ. (eds.).: Fuzzy Multi-criteria Decision-Making Using Neutrosophic Sets, Vol. 16, pp. 382–394. Springer, Berlin, Germany (2019)

Boltürk, E., Karasan, A., Kahraman, C.: Simple additive weighting and weighted product methods using neutrosophic sets. In: Kahraman, C., Otay, I. (eds.), Fuzzy Multicriteria Decision Making Using Neutrosophic Sets, Studies in Fuzziness and Soft Computing, vol. 369, pp. 647–676. © Springer Nature Switzerland AG 2019 (2019). https://doi.org/ https://doi.org/10.1007/978-3-030-00045-5_25

Biswas, P., Pramanik, S., Giri, B.C.: TOPSIS method for multi-attribute group decision-making under single-valued neutrosophic environment. Neural Comput. Appl. 27 (3), 727–737 (2016)

Wang, H., Smarandache, F., Sunderraman, R., Zhang, Y.Q.: Interval neutrosophic sets and logic: theory and applications in computing, 1–87 (2005)

Garibaldi, J.M., Ozen, T.: Uncertain fuzzy reasoning: a case study in modelling expert decision making. IEEE Trans. Fuzzy Syst. 15 (1), 16–30 (2007)

Torra, V.: Hesitant fuzzy sets. Int. J. Intell. Syst. 25 (6), 529–539 (2010)

MATH   Google Scholar  

Yager, R.R.: Pythagorean membership grades in multicriteria decision making, IEEE Trans. Fuzzy Syst. 22 , 958–965 (2014)

Atanassov, K.T.: More on intuitionistic fuzzy sets. Fuzzy Sets Syst. 33 (1), 37–45 (1989)

Zhang, X., Xu, Z.: Extension of TOPSIS to multiple criteria decision making with pythagorean fuzzy sets. Int. J. Intell. Syst. 29 (12), 1061–1078 (2014)

Peng, X., Yang, Y.: Some results for Pythagorean fuzzy sets. Int. J. Intell. Syst. 30 (11), 1133–1160 (2015)

Peng, X., Yang, Y.: Fundamental properties of interval-valued Pythagorean fuzzy aggregation operators. Int. J. Intell. Syst. 31 (5), 444–487 (2016)

Kahraman, C., Oztaysi, B., Onar, S.c.: Multicriteria Scoring Methods using Pythagorean Fuzzy Sets. Adv. Intell. Syst. Comput. 642 (2018)

Kahraman, C., Onar, S.C., Oztaysi, B: Present worth analysis using pythagorean fuzzy sets. In: Kacprzyk, J., Szmidt, E., Zadrożny, S., Atanassov, K., Krawczak, M. (eds.) Advances in Fuzzy Logic and Technology 2017. IWIFSGN 2017, EUSFLAT 2017. Advances in Intelligent Systems and Computing, vol 642. Springer (2018)

Karasan, A., Kahraman, C.: A novel intuitionistic fuzzy DEMATELANPTOPSIS integrated methodology for freight village location selection. J. Intell. Fuzzy Syst. Preprint, pp. 1–18 (2017)

Karasan, A., Ilbahar, E., Kahraman, C.: A novel pythagorean fuzzy AHP and its application to landfill site selection problem. Soft. Comput. (2018). https://doi.org/10.1007/s00500-018-3649-0

Cường, B.C.: Picture fuzzy sets. J. Comput. Sci. Cybernet. 30 (4), 409 (2014)

Cuong B.C., Kreinovich, V.: Picture fuzzy sets-a new concept for computational intelligence problems. In: 2013 Third World Congress on Information and Communication Technologies (WICT 2013), pp. 1–6 (2013)

Wei, G.W.: Picture fuzzy aggregation operators and their application to multiple attribute decision making. J. Intell. Fuzzy Syst. 33 , 1–12 (2017). https://doi.org/10.3233/JIFS-161798

Meksavang, P., Shi, H., Lin, S.M., Li, H.C.: An extended picture fuzzy VIKOR approach for sustainable supplier management and its application in the beef industry. Symmetry 11 (4), 468 (2019)

Chen, S.M., Tan, J.M.: Handling multicriteria fuzzy decision-making problems based on vague set theory. Fuzzy Sets Syst. 67 (2), 163–172 (1994)

Dutta, P.: Medical diagnosis based on distance measures between picture fuzzy sets. Int. J. Fuzzy Syst. Appl. (IJFSA) 7 (4), 15–36 (2018)

Yager, R.R.: Generalized orthopair fuzzy sets. IEEE Trans. Fuzzy Syst. 25 , 1222–1230 (2017)

Ju, Y., Luo, C., Ma, J., Gao, H., Santibanez Gonzalez, E.D., Wang, A.: Some interval-valued q-rung orthopair weighted averaging operators and their applications to multiple-attribute decision making. Int. J. Intell. Syst. 34 (10), 2584–260 (2019)

Senapati, T., Yager, R.R.: Fermatean fuzzy weighted averaging/geometric operators and its application in multi-criteria decision-making methods. Eng. Appl. Artif. Intell. 85 , 112–121 (2019)

Senapati, T., Yager, R.: Fermatean fuzzy sets. J. Amb. Intell. Humanized Comput. 11 (2019) https://doi.org/10.1007/s12652-019-01377-0

Kahraman, C., Gündoğdu, F.K., Onar, S.C., Oztaysi, B.: Hospital Location Selection Using Spherical Fuzzy TOPSIS. In: 2019 Conference of the İnternational Fuzzy Systems Association and the European Society for Fuzzy Logic and Technology (EUSFLAT 2019). Atlantis Press (2019)

Kutlu Gündoğdu, F., Kahraman, C.: A novel spherical fuzzy analytic hierarchy process and its renewable energy application. Soft Comput. (2019)

Kutlu Gündoğdu, F., Kahraman, C.: Extension of WASPAS with spherical fuzzy sets. Informatica 30 (2), 269–292 (2019)

Kutlu Gündoğdu, F., Kahraman, C.: A novel VIKOR method using spherical fuzzy sets and its application to warehouse site selection. J. Intell. Fuzzy Syst. 37 (1), 1197–1211 (2019)

Biswas, P., Pramanik, S., Giri, B.C.: A new methodology for neutrosophic multi-attribute decisionmaking with unknown weight information. Neutrosophic Sets Syst. 3 , 42–50 (2014)

Gündoğdu, FK., Kahraman, C.: Spherical Fuzzy Sets and Decision Making Applications. In International Conference on Intelligent and Fuzzy Systems, pp. 979–987. Springer, Cham

Karasan, A., Kahraman, C.: Interval-valued neutrosophic extension of EDAS method. Adv. Intell. Syst. Comput. 642 , 343–357 (2018)

Kutlu Gündoğdu, F., Kahraman, C.: Spherical fuzzy sets and spherical fuzzy TOPSIS method. J. Intell. Fuzzy Syst. 36 (1), 337–352 (2019)

Kutlu Gündoğdu, F., Kahraman, C.: A novel fuzzy TOPSIS method using emerging intervalvalued spherical fuzzy sets. Eng. Appl. Artif. Intell. 85 , 307–323 (2019)

Kutlu Gündoğdu, F., Kahraman, C.: A novel spherical fuzzy QFD method and its application to the linear delta robot technology development. Eng. Appl. Artif. Intell. 87 , 103348 (2020)

Kutlu Gündoğdu, F., Kahraman, C.: Spherical fuzzy sets and spherical fuzzy topsis method. J. Intell. Fuzzy Syst. 36 (1), 337–52 (2019)

Smarandache, F.: A unifying field in logics. neutrosophy: neutrosophic probability, set and logic (1999)

Ju, Y., Luo, C., Ma, J., Gao, H., Santibanez Gonzalez, E. D., Wang, A.: A Some interval‐valued q‐rung orthopair weighted averaging operators and their applications to multiple‐attribute decision making. Int. J. Intell. Syst.  34 (10), 2584-2606 (2019)

Wang, H., Smarandache, F., Zhang, Y.Q., Sunderraman, R.: Single valued neutrosophic sets. Multispace Multistruct. 4 , 410–413 (2010)

Wang, J., Gao, H., Wei, G.W., Wei, Y.: Methods for multiple-attribute group decision making with q-Rung interval-valued orthopair fuzzy information and their applications to the selection of green suppliers. Symmetry 11 , 56 (2019)

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Bolturk, E., Kahraman, C. (2021). Fuzzy Sets and Extensions: A Literature Review. In: Kahraman, C., Bolturk, E. (eds) Toward Humanoid Robots: The Role of Fuzzy Sets. Studies in Systems, Decision and Control, vol 344. Springer, Cham. https://doi.org/10.1007/978-3-030-67163-1_2

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A review of fuzzy methods in automotive engineering applications

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European Transport Research Review volume  7 , Article number:  29 ( 2015 ) Cite this article

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The paper addresses evolution of fuzzy systems for core applications of automotive engineering.

The presented study is based on the analysis of bibliography dedicated to fuzzy sets and fuzzy control for ground vehicles. A special attention is given to fuzzy approaches used in the following domains of automotive engineering: vehicle dynamics control systems, driver and driving environment identification, ride comfort control, and energy management of electric vehicles.

The bibliographical analysis, supplemented with statistics of relevant research publications, has allowed to allocate the most important fuzzy application cases for each domain. In particular, it concerns driver identification, human-machine interface, recognition of road conditions, and controllers of vehicle chassis and powertrain systems. It is found out that fuzzy methods have the primary use most of all for tasks requiring identification and forecasting procedures, especially in conditions of limited informational space. Additional observation that can be also derived from the presented survey points to reasonable integration of fuzzy technique with other control engineering methods to improve the performance of automotive control systems.

Conclusions

In the aggregate the performed review indicates that fuzzy computing can be considered as a versatile tool for automotive engineering applications of different nature.

1 Introduction

Fuzzy logic, among other computational intelligence methods, attracts increased attention of engineers and researchers involved in the development of complex control solutions for road vehicles and their subsystems. A possibility to overcome various nonlinear models and to use intuitive logical rules makes fuzzy control and fuzzy identification procedures useful tools by rapid system prototyping and on early design stages. It is of particular relevance for intelligent vehicle functions and systems requiring the prediction of manoeuvres dynamics, identification of driving environment parameters and objects, and interaction with the driver. For the mentioned topics, the development of models of objects and relevant controllers of on-board systems can meet essential problems with feasibility, sufficient level of complexity and other features demanded for real-time automotive applications. These issues can be efficiently handled with the computational intelligence methods, which are known as reasonable and powerful tools in solving non-well-posed analytical problems. Among different computational intelligence methods like neural networks, swarm intelligence computing et al. the fuzzy logic can be of special relevance for intelligent automotive systems. The reason is that the fuzzy logic has good applicability for identification tasks and pre-emptive control under strong presence of data and model uncertainty.

The above-listed arguments are motivated the study outlined in this paper. The main goal of the study is to give a survey of existing fuzzy methods and systems that are finding applications in the automotive area. Because the subject of the study is characterized by a high degree of interdisciplinarity, the discussed review can be of particular interest for researchers working on topics of Ground Vehicle Engineering, Control Engineering, and Computational Intelligence. Specific objectives of the presented work can be formulated as follows:

Define research topics that are most relevant to automotive fuzzy systems and tools;

Identify the main application cases for each defined topic;

Estimate development trends for each defined topic;

Analyse particular matters of automotive fuzzy systems from viewpoint of (i) industrial applications and (ii) integration with other control techniques.

The results of the performed survey are given in next sections in accordance with the listed objectives. The article includes also extensive bibliography aimed at the representation of various research schools around the world in context of the discussed topics.

2 Review background

The history of fuzzy systems in vehicle-related areas begins with several research works of founders of fuzzy logic dedicated to practical applications of fuzzy sets. An example is the intersection controller that reduced junction-related delay of vehicle driving proposed by Pappis and Mamdani in [ 1 ]. Later Sugeno and Murakami in [ 2 ] and Sugeno and Nishida in [ 3 ] have introduced the fuzzy realization of driver’s logic for the cases of an automated car parking system and handling the desired vehicle trajectory. Then, from early 1990s, more and more automotive systems with fuzzy logic are subjected both to academic and industrial areas. This resulted in a considerable amount of publications and patents.

An analysis of the published studies and works allows to allocate a number of domains, where fuzzy logic has found the most acceptance. These domains can be conditionally called as “Driver”, “Vehicle Dynamics”, “Ride comfort”, “Electric vehicles”, and “Driving environment”. Results of analysis of publications relevant to the fuzzy modeling and control of automotive systems in the listed engineering areas are further introduced in the paper.

The presented review uses the methodology proposed and partially implemented in [ 4 ]. The analysis uses only English-language scientific publications found in Scopus abstract and citation database of peer-reviewed literature ( http://www.elsevier.com/online-tools/scopus ). The following limitations of the analysis are accepted within the framework of this paper: The works are analyzed in relation to the ground and road vehicles only. Mobile robots and machines with non-holonomic constraints are excluded from the analysis due to their less relevance to automotive applications.

The overview is accompanied by statistics of peer-reviewed journal articles from two recent decades (dated 1994–2013) in each domain. (Conference papers are excluded from statistical analysis to avoid the consideration of duplicated content. The elimination has been also done for those papers, which content was already published in another journal. This is because it was observed that in particular cases the same content is repeated with minor variations in several conference papers published by the same authors.)

Additional section of the paper will analyse the publications of SAE International ( http://www.sae.org ) as one of the most recognized informational source worldwide for automotive engineering and presenting mainly the studies, which are providing experimental verification and close to industrial application.

3 Fuzzy methods for driver modelling and driver assistance systems

The domain “Driver”, proposed for the classification, implicates topics related to the driver models, driver behavior, and driver assistance systems. These topics are of special relevance for fuzzy applications because almost emotional and psychological facets of human behavior carry more semantic as numerical uncertainty. It complicates the formulation of the driver in simulation and control tasks through non-soft computing methods. The first applications of fuzzy sets to the modeling of the driver are arisen in 1980s. In particular, Kramer and Rohr in [ 5 ] and Kramer in [ 6 ] proposed to use a fuzzy model for the representation of perception characteristics of the driver and illustrated this approach on the driving simulator. Nagai, Kojima and Sato [ 7 ] suggested the fuzzy driver model describing the subjective recognition, judgment and control of the vehicle speed. This model was built from the accident analysis data. Then Kageyama and Pacejka [ 8 ] and Ehara and Suzuki [ 9 ] developed the first fuzzy models of the human reasoning for more complex driving situations. Analysis of relevant journal publications, Fig.  1 , and conference papers allows to allocate the most typical applications of fuzzy methods within the domain „Driver“:

Number of journal publications related to fuzzy methods in domain “Driver” and cited in Scopus database

Identifications and classifications of the drivers regarding fatigue, emotions and other human attributes, including the procedures of driver state recognition and forecasting through monitoring of various physiological parameters like electroencephalography-estimated brain activity, eye movement, gestures et al. [ 10 – 15 ],

Structure and controllers of driver assistance systems and devices of human machine interface [ 16 – 18 ],

Models of driver actions on vehicle controlling devices (brake and throttle pedals, steering wheel) for authentic simulation of vehicle maneuvers on driving simulators; controllers of pedal and steering wheel robots [ 19 – 22 ],

Simulation of driver reasoning for controllers of (semi-) automated and unmanned ground vehicles [ 23 – 25 ],

Understanding of subjective evaluation of vehicle dynamics; driver feeling of vehicle dynamics parameters like velocity, road friction and other [ 26 – 29 ],

Advisory functions of human-machine interface systems supporting the driver in eco- and low-emission vehicle operation [ 30 – 32 ].

From practical viewpoint, the fuzzy logic gives valuable opportunity to develop controllers of human machine interface and driver assistance, which can be integrated with other automotive systems responsible for active safety control through the correction of vehicle dynamics by active chassis and powertrain subsystems. The controllers of these subsystems can also implement different fuzzy methods that will be discussed in next section.

4 Fuzzy methods for vehicle dynamics control and ride comfort

Within the framework of the presented review the domain “Vehicle Dynamics” relates to the systems controlling the lateral and longitudinal vehicle motion. The corresponding representatives are anti-lock braking (ABS) and traction control systems, torque vectoring, electronic stability control, electronic differentials et al. The systems responsible for the vertical dynamics of the vehicle are allocated to another domain “Ride Comfort” due to considerable amount of relevant publications. Statistics of journal papers for both mentioned domains is depicted in Figs.  2 and 3 .

Number of journal publications related to fuzzy methods in domain “Vehicle Dynamics” and cited in Scopus database

Number of journal publications related to fuzzy methods in domain “Ride Comfort” and cited in Scopus database

The first applications of fuzzy methods to the vehicle dynamics were related to the formalization of tire parameters, which are used in the brake and acceleration control. It can be explained with the fact that automotive tires have nonlinear characteristics of friction and slip, which cannot be measured by on-board sensors and require therefore real-time estimation with the help of various numerical methods. Several early examples of fuzzy computing for tire parameters estimation are described in works of Stumpf, Arendt and Lux [ 33 ], and Madau, Yuan, Davis Jr. and Feldkamp [ 34 ]. Among vehicle dynamics control systems, ABS belongs to the classical examples of automotive control applications to verify and to define functional properties of different control methods, including fuzzy computing. In particular, Intel Corporation has proposed MCS 96 microcontroller family for the first fuzzy braking processors [ 35 ]. Later Siemens AG has applied fuzzy coprocessors C99A for brake-by-wire systems [ 36 ]. Progress in vehicle dynamics control (VDC) systems has given many opportunities for practical implementation of fuzzy logic. Whereas the first VDC systems have used only brakes and engine as actuators, the actual trend is to enable other chassis systems such as steering and suspension in an integrated control circuit. As a result, the domain “Vehicle Dynamics” includes a series of the research problems where fuzzy methods found use:

Brake control including architecture and algorithms of anti-lock braking systems [ 37 – 40 ]

Vehicle traction control including solutions for engine control, electronic differentials, powertrain and driveline control in general [ 41 – 43 ],

Control of lateral vehicle dynamics, in particular, in terms of yaw rate and vehicle sideslip control [ 44 – 48 ],

Steering control including solutions for electric power steering; active front and rear steering [ 49 – 51 ]

Estimation of vehicle state (linear velocity, yaw rate, vehicle sideslip angle et al.) from the sensors and experimental data [ 52 – 54 ],

Integrated sequential or parallel control on vehicle dynamics through independent subsystems (e.g. brakes, steering, suspension, driveline) [ 55 – 58 ],

Identification and estimation of parameters of tire-road interaction [ 59 – 63 ].

It was found during the analysis of research literature that studies of semi-active and active suspension control take up a larger share comparing with publications in other of automotive systems based on fuzzy methods. The reason can be that the ride control systems are characterized by complex nonlinear oscillating behaviour influencing simultaneously several properties like comfort, handling, or NVH (noise, vibration, harshness). The use of soft computing methods can certainly simplify the control on these vehicle properties. The earliest variants of fuzzy architecture of active suspension controllers were proposed by Yester and McFall [ 64 ], and Lin and Lu [ 65 ]. Then many other studies related to the fuzzy applications in the domain “Ride comfort” are arisen with the trend to continuous growth of research in this area. The main topics within the domain “Ride comfort” can be subjected as follows:

Methods of objective evaluation and control of the vehicle ride comfort [ 66 – 68 ],

Control algorithms of suspension actuators and shock absorbers with adaptation to environmental and operational conditions such as road roughness, vehicle dynamic variables and other [ 69 – 73 ]

Special control strategies of electrorheological, magnetorheological, and electrical dampers and specific suspension elements [ 74 – 77 ].

It should be noted that in spite of numerous publications in the domain “Ride comfort”, there are few reports about industrial variants of fuzzy suspension controllers. Overwhelming majority of analyzed investigations belongs to the fundamental or conceptual applied research.

Now many classes of ground vehicles require the mandatory installation of VDC systems. This fact can encourage more intensive investigations on fuzzy approaches in vehicle dynamics and ride control.

5 Fuzzy methods and vehicle—environment interaction

Both driver assistance and vehicle dynamics control systems, discussed in previous sections, can benefit from new information and communication technologies allowing closer interaction of the vehicle with the driving environment. Impactful control solutions improving vehicle safety and energy efficiency are being proposed now with the help of various on-board and on-road sensors as well road infrastructure services. All these aspects including the technologies for (semi-)autonomous driving are subjected to the domain “Driving Environment”, which has been also analysed in the presented study, Fig.  4 . It should be mentioned that specific topics of traffic control and architecture of Intelligent Transport Systems are excluded from the review because they are related mainly to Transport Engineering, but not to Automotive Engineering.

Number of journal publications related to fuzzy methods in domain “Driving Environment” and cited in Scopus database

The first research works relevant to fuzzy systems in the domain “Driving Environment” have investigated path planning algorithms for an autonomous vehicle supported by a navigation system or equipped with advanced set of sensors, which are able to percept external parameters like distance between the vehicles or to identify external objects. The studies of Hogle and Bonissone [ 78 ] and Freisleben and Kunkelmann [ 79 ] can be mentioned in this context. These and other topics are in scope of standing research interest from middle of 1990s and now the main subjects of fuzzy applications for the domain “Driving Environment” can be classified as follows:

Adaptive cruise control for the vehicles controlled by the driver; road following control for automated driving; collision prevention systems [ 80 – 82 ],

Processing sensor information; recognition of driving environment parameters; vehicle localization [ 83 – 89 ],

Coordination of road vehicle platoon systems [ 90 – 92 ],

Control architecture of autonomous vehicles [ 93 – 96 ];

Specific problems of brake, traction and steering control systems of autonomous vehicles [ 97 – 99 ];

Automated parking systems [ 100 – 102 ].

In accordance with the short- and long-term projection, the mass-produced cars with functions of autonomous driving will be stepwise introduced on the market in coming decade. This fact should stimulate more intensive research in various topics of the domain “Driving Environment”.

6 Fuzzy methods for electric vehicles

Hybrid electric vehicles, full electric vehicles, fuel cell electric vehicles are at the centre of permanent attention of researchers and developers of fuzzy systems. This tendency is growing now because of sweeping development of technologies for “Green Mobility”, Fig.  5 . Early studies have introduced fuzzy energy management (EM) of electric vehicles in general. In particular, Farrall and Jones in [ 103 ] and Cerruto, Consoli, Raciti and Testa in [ 104 ] have proposed different fuzzy EM systems for hybrid vehicle, which were responsible for efficient powertrain control. Then, comparing with other automotive applications, fuzzy methods found many-sided use in the domain “Electric vehicles”. An analysis of relevant research works allows to mention the following applicative areas for fuzzy methods:

Number of journal publications related to fuzzy methods in domain “Electric Vehicles” and cited in Scopus database

Global energy management of electric vehicles; hybrid powertrain control of operational modes (internal combustion engine / electric motor) [ 105 – 110 ];

Traction control, anti-lock braking and regenerative braking control of electric vehicles [ 111 – 113 ];

Internal controllers of electric motors, starters, inverters; electric propulsion controllers [ 114 – 116 ];

Forecast and optimization of driving range [ 117 , 118 ];

Estimation of battery performance and algorithms for battery charging [ 119 – 121 ].

A number of other aspects like interaction of electric vehicles with the road and urban infrastructure are among further promising fuzzy applications in this research area.

7 Fuzzy methods for miscellaneous applications

In addition to previously introduced domains, various unclassified or rarely studied examples of automotive fuzzy systems are also presented in analyzed research publications.

In particular, the most interesting variants cover the following topics: Transmission control including gear shifting algorithms and clutch control [ 122 ], Systems for diagnosing different vehicle systems and elements [ 123 – 125 ], Methods for the assessment of passive safety [ 126 , 127 ], Thermal and NVH comfort control systems [ 128 , 129 ], Vehicle body design [ 130 ], etc. Figure 6 shows the statistics of relevant journal publications. Hence, fuzzy methods can be considered as well-established research tools in different aspects of automotive engineering.

Number of journal publications related to fuzzy methods for miscellaneous automotive applications and cited in Scopus database

8 Fuzzy methods in applied and industrial research

One of the interesting results of the analysis is that a serious gap still persists between pure research and industrial applications of fuzzy methods in automotive engineering. To support this statement, Fig.  7 displays statistics of publications related to industrial or ready-to-installation specimens of road vehicle systems with fuzzy controllers or fuzzy models embedded into the processing units. This selection reflects the analysis of technical papers published by SAE International as one of the most recognized information sources worldwide for automotive engineering. The preliminary search has found only few publications every year with information about application of fuzzy methods in automotive area. The deep analysis detected neither the clear-cut growth of fuzzy applications nor their regularity in individual domains. The reason for the available gap between theoretical studies and industrial systems can be explained by the fact that most of traditional automotive control systems have well-established variants of system architecture and corresponding real-time functional realizations, which are most often based on rule-based, nonlinear, optimal and other control approaches that are not directly related to computational intelligence methods. Therefore, until recently the progress in the vehicle control systems has been related first of all to the implementation of more powerful and efficient hardware accompanied with cautious modification of software part. However an increased demand for on-board systems, communicating with the driver and performing adaptation and learning in accordance with human behaviour, shows certain limitations in traditional, non-soft computing control techniques.

Selected SAE publications related to fuzzy methods

Nevertheless, it should be noted that some applicative areas for fuzzy methods are especially most commonly addressed in published industrial research studies. This is particularly true for on-board systems like the cruise control, lane change assistant et al., which require identification of the driver behaviour. The research of Ford Motor on the driver characterization for the car following [ 131 , 132 ] and for the vehicle destination prediction [ 133 ] can be mentioned in this regard. A reasonable applicability of fuzzy methods to this topic has been also demonstrated in studies with the participation of Centro Richerce Fiat for the driver distraction modelling [ 134 ]. Other work [ 135 ], performed under coordination of researchers from Renault, has indicated an efficient implementation of fuzzy sets and fuzzy space windowing for the psycho-physiological characterization of drivers within the context of the car following.

An analysis of industrial white papers and analytical publications indicates trends to further consistent growth of intelligent automotive systems with the logic based on fuzzy methods [ 136 ]. This statement can be also confirmed with numerous industrial patents for fuzzy algorithms and controllers, claimed by automotive OEMs and suppliers. It allows to expect that the mentioned gap between pure research and industrial applications should be overcome in near future.

9 About compatibility of fuzzy methods in automotive engineering

Eventual barriers for more intensive dissemination of fuzzy methods in automotive engineering can be removed through their integration with other analytical and numerical methods. In particular, such integration can improve robustness and adaptivity of controllers with simultaneous keeping of straightforward formalization of control tasks. The fact that fuzzy logic remains one of the most inviting approaches in basic and applied sciences it can be supported by the important indicator of high compatibility of fuzzy methods with other computing and control techniques. The analyzed research publications offer many examples of integration of fuzzy sets and fuzzy logic that are summarized in Table 1 .

10 Conclusions

Analysis of research literature, especially results of published experimental works, demonstrates that fuzzy methods have solid background and good prospects for the implementation in automotive engineering applications. The following positions can be especially mentioned in this context:

For the selected topics each—Driver, Vehicle Dynamics Control, Electric Vehicles, Driving Environment—the analysis has discovered a series of fuzzy application cases of different nature that points to good versatile and flexible feasibility of fuzzy methods for automotive engineering tasks;

Relevant publication statistics indicates continuous growth of fuzzy-related research activities in the selected topics each;

Despite the currently available gap between theoretical research and industrial implementations, a strong demand in automotive systems with intelligent functions and involving more intuitive interaction with the driver and environment will stimulate a more wide adoption of fuzzy tools and systems on serial vehicles.

The most feasible breakthrough in further development of fuzzy systems and tools for automotive engineering applications can be expected from two kinds of “fusion”. From theoretical side, the fusion of fuzzy methods with other methods of soft computing or nonlinear control opens very promising prospects. From technological side, the fusion of many vehicle controllers and estimators on common fuzzy basis can get over numerical and linguistic uncertainties accompanying complex processes of vehicle dynamics and vehicle-driver-environment interaction. These trends would encourage the researchers and engineers in seeking for novel applicative fuzzy solutions in automotive engineering.

Pappis CP, Mamdani EH (1977) A fuzzy logic controller for a traffic junction. IEEE Trans Syst Man Cybernet SMC-7(10):707–717

Article   MATH   Google Scholar  

Sugeno M, Murakami K (1984) Fuzzy parking control of model car, Proc. IEEE Conf Decis Control, pp. 902–903

Sugeno M, Nishida M (1985) Fuzzy control of model car. Fuzzy Sets Syst 16:103–113

Article   Google Scholar  

Ivanov V (2001) Fuzzy methods in ground vehicle engineering : state-of-the-art and advanced applications, in Proc. 8th International Conference on Structural Dynamics, EURODYN 2011, Leuven, Belgium, paper MS19-270

Kramer U, Rohr G (1982) A model of driver behaviour. Ergonomics 25(10):891–907

Kramer U (1985) On the application of fuzzy sets to the analysis of the system driver-vehicle-environment. Automatica 21(1):101–107

Nagai M, Kojima Y, Sato H (1987) Driver’s decision and control process during right-turning at intersections (application of fuzzy linguistic model for accident analysis), SAE Technical Papers Series, paper no. 871250

Kageyama I, Pacejka HB (1991) On a new driver model with fuzzy control. Veh Syst Dyn 20(suppl 1):314–324

Google Scholar  

Ehara N, Suzuki K (1991) Estimation of human reasoning state in machine operation using fuzzy linear regression. Trans Jpn Soc Mech Eng Part C 57:3907–3912

Lin C-T, Tsai S-F, Ko L-W (2013) EEG-based learning system for online motion sickness level estimation in a dynamic vehicle environment. IEEE Trans Neural Netw Learn Syst 24(10):1689–1700, art. no. 6587600

Kuhn K-P, Heidinger A (1997) On-line driver type classification. Int J Veh Des 18(6):616–625

Lee D, Donnell ET (2007) Analysis of nighttime driver behavior and pavement marking effects using fuzzy inference system. J Comput Civ Eng 21(3):200–210

Wu J-D, Chen T-R (2008) Development of a drowsiness warning system based on the fuzzy logic images analysis. Expert Syst Appl 34(2):1556–1561

Boyraz P, Acar M, Kerr D (2008) Multi-sensor driver drowsiness monitoring. Proc IMechE Part D J Automob Eng 222(11):1857–1878

Wahab A, Quek C, Tan CK, Takeda K (2009) Driving profile modeling and recognition based on soft computing approach. IEEE Trans Neural Netw 20(4):563–582

Maeda M (1995) Fuzzy drive expert system for an automobile. Inf Sci Appl 4(1):29–48

Willumeit H-P (1997) Driver models: how to avoid critical yaw movements of vehicles. Int J Veh Des 18(6):652–667

Nigro J-M, Loriette-Rougegrez S, Rombaut M (2002) Driving situation recognition with uncertainty management and rule-based systems. Eng Appl Artif Intell 15(3–4):217–228

Hong C-W, Shio T-W (1996) Fuzzy control strategy design for an autopilot on automobile chassis dynamometer test stands. Mechatronics 6(5):537–555

Hong C-W (1997) Tuning the fuzzy control autopilot strategy for driving pattern simulation of road vehicles. Int J Veh Des 18(1):35–52

Lee MC, Park MK, Yoo WS, Son K, Han MC (2003) Sliding mode control of real-time PNU vehicle driving simulator and its performance eveluation. JSME Int J Ser C Mech Syst Mach Elem Manuf 46(2):557–564

Takahashi H, Ukishima D, Kawamoto K, Hirota K (2007) A study on predicting hazard factors for safe driving. IEEE0 Trans Ind Electron 54(2):781–789

Matthews ND, An PE, Roberts JM, Harris CJ (1998) A neurofuzzy approach to future intelligent driver support systems. Proc IMechE Part D J Automob Eng 212(1):43–58

Takahashi H, Kuroda K (1999) Intelligent vehicle control considering driver’s visual perception, Proc. IEEE Conference on Intelligent Transportation Systems, pp. 252–257

Chen G, Zhang W-G, Zhang X-N (2013) Fuzzy neural control for unmanned robot applied to automotive test. Ind Robot 40(5):450–461, art. no. 17094477

Laurence P, Basset M, Coutant P, Gissinger G (2000) Lateral vehicle behaviour: comparison of subjective/objective assessment using the Choquet integral. Veh Syst Dyn 34(5):357–379

Ma X, Andréasson I (2007) Statistical analysis of driver behavior data in different regimes of the car-following stage. Transportation Research Record, no. 2018, pp. 87–96

Moridpour S, Rose G, Sarvi M (2009) Modelling the heavy vehicle drivers’ lane changing decision under heavy traffic conditions. Road Transp Res 18(4):49–57

Hurwitz DS, Wang H, Knodler MA Jr, Ni D, Moore D (2012) Fuzzy sets to describe driver behavior in the dilemma zone of high-speed signalized intersections. Transport Res F: Traffic Psychol Behav 15(2):132–143

Vangi D, Virga A (2003) Evaluation of energy-saving driving styles for bus drivers. Proc IMechE Part D J Automob Eng 217(4):299–305

Syed FU, Filev D, Ying H (2007) Fuzzy rule-based driver advisory system for fuel economy improvement in a hybrid electric vehicle, Proc. of Annual Conference of the NAFIPS, no. 4271056, pp. 178–183

Luu HT, Nouvelière L, Mammar S (2010) Ecological and safe driving assistance system: Design and strategy, Proc. of IEEE Intelligent Vehicles Symposium, no. 5548012, pp. 129–134

Stumpf H, Arendt G, Lux F (1988) Linking evaluation of subjective handling tests on the road with objectively measured data by using fuzzy-set-theory, SAE Technical Papers Series, paper no. 885006

Madau DP, Yuan F, Davis Jr LI, Feldkamp LA (1993) Fuzzy logic anti-lock brake system for a limited range coefficient of friction surface, 1993 I.E. International Conference on Fuzzy Systems, pp. 883–888

Gullett C, Elting D, Kowalczyk R, Fennich M, Hellenthal B (1995) Intel fuzzy logic tool simplifies ABS design, Intel Corporation Application Note, No. 272595–001, pp. 1–8

Eichfeld H, Mertens A, Brandmeier T, Waidelich F, Graumann R, Schwab M (1998) Applications of SAE 81C99x fuzzy coprocessors. Proc IEEE Int Conf Fuzzy Syst 1:19–24

Mauer GF (1995) Fuzzy logic controller for an ABS braking system. IEEE Trans Fuzzy Syst 3(4):381–388

Chen F-W, Liao T-L (2000) Nonlinear linearization controller and genetic algorithm-based fuzzy logic controller for ABS systems and their comparison. Int J Veh Des 24(4):334–349

Kim S, Kim H, Sung Y-G (2001) Electronic control of braking force distribution for vehicles using a direct adaptive fuzzy controller. KSME Int J 15(1):66–80

Kuti I (2004) An adaptive fuzzy control to antilock non-straight line braking using three dimensional (finite element) vehicle model. Int J Veh Des 34(4):327–339

Klaus K, Hasemann M (1995) An embedded fuzzy anti-slippage system for heavy duty off road vehicles. Inf Sci Appl 4(1):1–27

Hiller M, Schmitz T, Schuster C (1996) Design of a fuzzy traction control system using spatial vehicle dynamics simulation. Heavy Veh Syst 3(1–2):240–248

Trebi-Ollennu A, Dolan JM, Khosla PK (2001) Adaptive fuzzy throttle control for an all-terrain vehicle. Proc IMechE Part I J Syst Control Eng 215(3):189–198

Hessburg T, Tomizuka M (1995) An adaptation method for fuzzy logic controllers in lateral vehicle guidance. Mechatronics 5(8):873–898

Boada BK, Boada MJL, Díaz V (2005) Fuzzy-logic applied to yaw moment control for vehicle stability. Veh Syst Dyn 43(10):753–770

Chadli M, El Hajjaji A, Oudghiri M (2008) Robust output fuzzy control for vehicle lateral dynamic stability improvement. Int J Model Identif Control 3(3):247–257

Du H, Zhang N, Smith W (2009) Robust yaw moment control for vehicle handling and stability. SAE Int J Passenger Cars Mech Syst 2(1):772–779

Ivanov V, Shyrokau B, Augsburg K, Gramstat S (2010) Advancement of vehicle dynamics control with monitoring the tire rolling environment. SAE Int J Passenger Cars Mech Syst 3(1):199–216

Szosland A (2000) Fuzzy logic approach to four-wheel steering of motor vehicle. Int J Veh Des 24(4):350–359

Huh K, Kim J (2001) Active steering control based on the estimated tire forces. J Dyn Syst Meas Control Trans ASME 123(3):505–511

Jalali K, Uchida T, McPhee J, Lambert S (2013) Development of an advanced fuzzy active steering controller and a novel method to tune the fuzzy controller. SAE Int J Passenger Cars Electron Electr Syst 6(1):241–254

Basset M, Zimmer C, Gissinger GL (1997) Fuzzy approach to the real time longitudinal velocity estimation of a FWD car in critical situations. Veh Syst Dyn 27(5–6):477–489

Hwang JK, Song CK (2005) Fuzzy estimation of vehicle speed using an accelerometer and wheel sensors. Int J Automot Technol 6(4):359–365

Cheli F, Sabbioni E, Pesce M, Melzi S (2007) A methodology for vehicle sideslip angle identification: comparison with experimental data. Veh Syst Dyn 45(6):549–563

Zeyada YF, Elbeheiry EM, Elarabi ME (2000) Fault diagnosis and μ-estimation for integrated chassis control of vehicles in emergency maneuvers. J Eng Appl Sci 47(4):665–681

Yoshimura T, Emoto Y (2003) Steering and suspension system of a full car model using fuzzy reasoning and disturbance observers. Int J Veh Auton Syst 1(3–4):363–386

March C, Shim T (2007) Integrated control of suspension and front steering to enhance vehicle handling. Proc IMechE Part D J Automob Eng 221(4):377–391

Song J (2012) Integrated control of brake pressure and rear-wheel steering to improve lateral stability with fuzzy logic. Int J Automot Technol 13(4):563–570

Ergun M (2007) Estimation of friction coefficient of asphalt concrete road surfaces using the fuzzy logic approach. Can J Civ Eng 34(9):1110–1118

Article   MathSciNet   Google Scholar  

Shyrokau BN, Ivanov VG (2008) Alterable fuzzy sets in automotive control applications. Int J Model Identif Control 3(3):305–317

Li L, Song J, Li H-Z, Shan D-S, Kong L, Yang CC (2009) Comprehensive prediction method of road friction for vehicle dynamics control. Proc IMechE Part D J Automob Eng 223(8):987–1002

Ivanov V, Shyrokau B, Augsburg K, Algin V (2010) Fuzzy evaluation of tyre-surface interaction parameters. J Terrramech 47(2):113–130

Ivanov V (2011) Fuzzy architecture of systems with alterable information: case study for tyre–ground friction estimators. Int J Reliab Saf 5(3/4):398–419

Yester JL, McFall RH (1992) Fuzzy logic controller for active suspension. Proc Soc Automot Eng P-260:259–270

Lin Y-J, Lu Y-Q (1993) Toward better ride performance of vehicle suspension systems via intelligent control. Am Soc Mech Eng Dyn Syste Control Div (Publ) DSC 48:105–112

Jianghong Z, Long T (1994) An evaluation of comfort of a bus seat. Appl Ergon 25(6):386–392

Eslaminasab N, Biglarbegian M, Melek WW, Golnaraghi MF (2007) A neural network based fuzzy control approach to improve ride comfort and road handling of heavy vehicles using semi-active dampers. Int J Heavy Veh Syst 14(2):135–157

Suzuki T, Takahashi M (2010) Robust active suspension control for vibration reduction of passenger’s body. Trans JSME Part C 76(762):380–387

Cherry AS, Jones RP (1995) Fuzzy logic control of an automotive suspension system. IEE Proc Control Theory Appl 142(2):149–160

Yoshimura T, Isari Y, Li Q, Hino J (1997) Active suspension of motor coaches using skyhook damper and fuzzy logic control. Control Eng Pract 5(2):175–184

Caponetto R, Diamante O, Fargione G, Risitano A, Tringali D (2003) A soft computing approach to fuzzy sky-hook control of semiactive suspension. IEEE Trans Control Syst Technol 11(6):786–798

Sie W-T, Lian R-J, Lin B-F (2006) Enhancing grey prediction fuzzy controller for active suspension systems. Veh Syst Dyn 44(5):407–430

Kaldas MMS, Çalişkan K, Henze R, Küçükay F (2013) Rule optimized fuzzy logic controller for full vehicle semi-active suspension. SAE Int J Passenger Cars Mech Syst 6(1):332–344

Choi S-B, Park D-W, Suh M-S (2002) Fuzzy sky-ground hook control of a tracked vehicle featuring semi-active electrorheological suspension units. J Dyn Syst Meas Control Trans ASME 124(1):150–157

Giuclea M, Sireteanu T, Stancioiu D, Stammers CW (2004) Model parameter identification for vehicle vibration control with magnetorheological dampers using computational intelligence methods. Proc IMechE Part I J Syst Control Eng 218(7):569–581

Zheng L, Li YN, Baz A (2009) Fuzzy-sliding mode control of a full car semi-active suspension systems with MR dampers. Smart Struct Syst 5(3):261–277

Lee S, Kim W-J (2010) Active suspension control with direct-drive tubular linear brushless permanent-magnet motor. IEEE Trans Control Syst Technol 18(4):859–870, art. no. 5286231

Hogle RA, Bonissone PP (1994) Fuzzy algorithm for path selection in autonomous vehicle navigation. Proc IEEE Conf Decis Control:898–900

Freisleben B, Kunkelmann T (1993) Combining fuzzy logic and neural networks to control an autonomous vehicle, 1993 I.E. International Conference on Fuzzy Systems, pp. 321–326

Blöchl B, Tsinas L (1994) Automatic road following using fuzzy control. Control Eng Pract 2(2):305–311

Chakroborty P, Kikuchi S (1999) Evaluation of the general motors based car-following models and a proposed fuzzy inference model. Transp Res Part C Emerg Technol 7(4):209–235

Naranjo JE, González C, Reviejo J, García R, De Pedro T (2003) Adaptive fuzzy control for inter-vehicle gap keeping. IEEE Trans Intell Transp Syst 4(3):132–142

Qiao MS, Takeda H (1995) Learning algorithm of environmental recognition in driving vehicle. IEEE Trans Syst Man Cybernet 25(6):917–925

Stover JA, Hall DL, Gibson RE (1996) A fuzzy-logic architecture for autonomous multisensor data fusion. IEEE Trans Ind Electron 43(3):403–410

Goebel KF, Agogino AM (2001) Sensor validation and fusion for automated vehicle control using fuzzy techniques. J Dyn Syst Meas Control Trans ASME 123(1):145–146

Rigatos G, Tzafestas S (2007) Extended Kalman filtering for fuzzy modelling and multi-sensor fusion. Math Comput Model Dyn Syst 13(3):251–266

Article   MATH   MathSciNet   Google Scholar  

Kim SY, Choi HC, Won WJ, Oh SY (2009) Driving environment assessment using fusion of in- and out-of-vehicle vision systems. Int J Automot Technol 10(1):103–113

Broggi A, Cerri P, Ghidoni S, Grisleri P, Jung HG (2009) A new approach to urban pedestrian detection for automatic braking. IEEE Trans Intell Transp Syst 10(4):594–605, art. no. 5290131

Filev DP, Kolmanovsky I (2012) Markov chain modeling and on-board identification for automotive vehicles. Lect Notes Control Inf Sci 418:111–128

Bauer M, Tomizuka M (1996) Fuzzy logic traction controllers and their effect on longitudinal vehicle platoon systems. Veh Syst Dyn 25(4):277–303

Lee GD, Kim SW, Yim YU, Jung JH, Oh SY, Kim BS (1999) Longitudinal and lateral control system development for a platoon of vehicles, IEEE Conference on Intelligent Transportation Systems, Proceedings, ITSC, pp. 605–610

Kim K, Kim J, Huh KS, Yi K, Cho D (2006) A real-time multi-vehicle simulator for longitudinal controller design. Veh Syst Dyn 44(5):369–386

Pasquier M, Quek C, Toh M (2001) Fuzzylot: a novel self-organising fuzzy-neural rule-based pilot system for automated vehicles. Neural Netw 14(8):1099–1112

Dai X, Li CK, Rad AB (2005) An approach to tune fuzzy controllers based on reinforcement learning for autonomous vehicle control. IEEE Trans Intell Transp Syst 6(3):285–293

Milanés V, Villagrá J, Pérez J, González C (2012) Low-speed longitudinal controllers for mass-produced cars: a comparative study. IEEE Trans Ind Electron 59(1):620–628, art. no. 5759771

Pérez J, Milanés V, Godoy J, Villagrá J, Onieva E (2013) Cooperative controllers for highways based on human experience. Expert Syst Appl 40(4):1024–1033

Kim HM, Dickerson J, Kosko B (1996) Fuzzy throttle and brake control for platoons of smart cars. Fuzzy Sets Syst 84(3):209–234

Naranjo JE, Reviejo J, González C, García R, De Pedro T (2004) A throttle and brake fuzzy controller: towards the automatic car. Lect Notes Comput Sci (Incl Subseries Lec Notes Artif Intell Lect Notes Bioinforma 2809:291–301

Naranjo JE, González C, García R, De Pedro T, Haber RE (2005) Power-steering control architecture for automatic driving. IEEE Trans Intell Transp Syst 6(4):406–415

Ryu Y-W, Oh S-Y, Kim S-Y (2008) Robust automatic parking without odometry using an evolutionary fuzzy logic controller. Int J Control Autom Syst 6(3):434–443

Chen C-Y, Feng H-M (2009) Hybrid intelligent vision-based car-like vehicle backing systems design. Expert Syst Appl 36(4):7500–7509

Shiao Y, Yu C-J (2012) Automatic parking control for vehicles with stereo vision. Adv Sci Lett 8:49–54

Farrall SD, Jones RP (1993) Energy management in an automotive electric/heat engine hybrid powertrain using fuzzy decision making, Proc. 1993 International Symposium on Intelligent Control, pp. 463–468

Cerruto E, Consoli A, Raciti A, Testa A (1994) Fuzzy logic based efficiency improvement of an urban electric vehicle. IECON Proc (Ind Electron Conf 2:1304–1309

Vermeiren L, Guerra TM, Paganelli G (1997) Application of fuzzy set theory to electric car control. Cybern Syst 28(8):675–693

Baumann BM, Washington G, Glenn BC, Rizzoni G (2000) Mechatronic design and control of hybrid electric vehicles. IEEE/ASME Trans Mechatron 5(1):58–72

Schouten NJ, Salman MA, Kheir NA (2002) Fuzzy logic control for parallel hybrid vehicles. IEEE Trans Control Syst Technol 10(3):460–468

Ippolito L, Loia V, Siano P (2003) Extended fuzzy C-means and genetic algorithms to optimize power flow management in hybrid electric vehicles. Fuzzy Optim Decis Making 2(4):359–374

Ahn H-S, Lee NS, Moon CW, Jeong G-M (2007) Fuel economy improvement for fuel cell hybrid electric vehicles using fuzzy logic-based power distribution control. Int J Automot Technol 8(5):651–658

Solano Martínez J, Mulot J, Harel F, Hissel D, Péra M-C, John RI, Amiet M (2013) Experimental validation of a type-2 fuzzy logic controller for energy management in hybrid electrical vehicles. Eng Appl Artif Intell 26(7):1772–1779

Yoshimura T, Uchida H, Nasu H, Hino J, Ueno R (1997) Traction force control of an electric vehicle in 2WS-4WD mode using fuzzy reasoning. Int J Veh Des 18(5):442–454

Pusca R, Ait-Amirat Y, Berthon A, Kauffmann JM (2002) Fuzzy logic based control for electric vehicle with four separate traction drives. Proc IEEE Veh Technol Conf 4:2089–2097

Kim D-H, Kim J-M, Hwang S-H, Kim H-S (2007) Optimal brake torque distribution for a four-wheel-drive hybrid electric vehicle stability enhancement. Proc IMechE Part D J Automob Eng 221(11):1357–1366

Huang H (1999) Electrical two-speed propulsion by motor winding switching and its control strategies for electric vehicles. IEEE Trans Veh Technol 48(2):607–618

Sepe RB Jr, Miller JM (2002) Real-time collaborative experimentation via the internet: fuzzy efficiency optimization of a hybrid electric vehicle starter/alternator. Int J Veh Auton Szstems 1(1):63–82

Ferreira AA, Pomilio JA, Spiazzi G, de Araujo Silva L (2008) Energy management fuzzy logic supervisory for electric vehicle power supplies system. IEEE Trans Power Electron 23(1):107–115

Quigley CP, Ball RJ, Jones RP (2000) Fuzzy modelling approach to the prediction of journey parameters for hybrid electric vehicle control. Proc IMechE Part D J Automob Eng 214(8):875–885

Cunningham I, Burnham K (2013) Online use of the fuzzy transform in the estimation of electric vehicle range. Meas Control (UK) 46(9):277–282

Chau KT, Wu KC, Chan CC, Shen WX (2003) A new battery capacity indicator for nickel-metal hydride battery powered electric vehicles using adaptive neuro-fuzzy inference system. Energy Convers Manag 44(13):2059–2071

Albertus P, Couts J, Srinivasan V, Newman J (2008) A combined model for determining capacity usage and battery size for hybrid and plug-in hybrid electric vehicles. J Power Sources 183(2):771–782

Fleischer C, Waag W, Bai Z, Sauer DU (2013) On-line self-learning time forward voltage prognosis for lithium-ion batteries using adaptive neuro-fuzzy inference system. J Power Sources 243:728–749

Hayashi K, Shimizu Y, Dote Y, Takayama A, Hirako A (1995) Neuro fuzzy transmission control for automobile with variable loads. IEEE Trans Control Syst Technol 3(1):49–52

Halfmann C, Ayoubi M, Holzmann H (1997) Supervision of vehicles’ tyre pressures by measurement of body accelerations. Control Eng Pract 5(8):1151–1159

Hissel D, Candusso D, Harel F (2007) Fuzzy-clustering durability diagnosis of polymer electrolyte fuel cells dedicated to transportation applications. IEEE Trans Veh Technol 56(5 I):2414–2420, art. no. 4305501

Kim MH, Lee S, Lee KC (2011) A fuzzy predictive redundancy system for fault-tolerance of x-by-wire systems. Microprocess Microsyst 35(5):453–461

Murè S, Demichela M, Piccinini N (2006) Assessment of the risk of occupational accidents using a “fuzzy” approach. Cognit Technol Work 8(2):103–112

Mon Y-J (2007) Airbag controller designed by adaptive-network-based fuzzy inference system (ANFIS). Fuzzy Sets Syst 158(24):2706–2714

Khayyam H, Nahavandi S, Hu E, Kouzani A, Chonka A, Abawajy J, Marano V, Davis S (2011) Intelligent energy management control of vehicle air conditioning via look-ahead system. Appl Therm Eng 31(16):3147–3160

Nopiah ZM, Junoh AK, Ariffin AK (2013) Noise annoyance fuzzy index in passenger car cabin. Int J Veh Noise Vib 9(3–4):216–233

Farkas L, Moens D, Donders S, Vandepitte D (2012) Optimisation study of a vehicle bumper subsystem with fuzzy parameters. Mech Syst Signal Process 32:59–60

Filev D, Lu J, Tseng F, Prakah-Asante K (2011) Real-time driver characterization during car following using stochastic evolving models, IEEE International Conference on Systems, Man and Cybernetics, pp. 1031–1036

Lu J, Filev D, Tseng F (2015) Real-time determination of driver’s driving behavior during car following. SAE Int J Passenger Cars Electron Electr Syst, vol. 8, no. 2

D. Filev, F. Tseng, J. Kristinsson, and R. McGee, Real-time determination of driver’s driving behavior during car following, IEEE Symposium on Computational Intelligence in Vehicles and Transportaion Systems, pp. 87–91, 2011

Tango F, Minin L, Tesauri F, Montanari R (2010) Field tests and machine learning approaches for refining algorithms and correlations of driver’s model parameters. Appl Ergon 41(2):211–224

Dubart D, Loslever P, Popieul JC, Moessinger M (2011) Multiple correspondence analysis of fuzzyfied task performance and psycho-physiological test data: Use in real car following situations, 12th IFAC, IFIP, IFORS, IEA Symposium on Analysis, Design, and Evaluation of Human-Machine Systems

Gusikhin O, Filev D, Rychtyckyi N (2008) Intelligent vehicle system: Applications and new trends, Lecture Notes in Electrical Engineering, col. 15, pp. 3–14

Dahmani H, Chadli M, Rabhi A, El Hajjaji A (2013) Vehicle dynamic estimation with road bank angle consideration for rollover detection: Theoretical and experimental studies. Veh Syst Dyn 51(12):1853–1871

Yeh EC, Tsao YJ (1994) Fuzzy preview control scheme of active suspension for rough road. Int J Veh Des 15(1–2):166–180

Cheok KC, Smid GE, McCune DJ (2000) Multisensor-based collision avoidance system with application to a military HMMWV, IEEE Conference on Intelligent Transportation Systems, Proceedings, ITSC, pp. 288–292

Economou JT, Colyer RE (2006) Fuzzy-hybrid modelling of an Ackerman steered electric vehicle. Int J Approx Reason 41(3):343–368

Yang JH, Yang CC (2008) Adaptive sliding mode control for vehicle brake systems with fuzzy composition. Int J Model Identif Control 4(2):112–126

Chou J-H, Chen S-H, Lee F-Z (1998) Grey-fuzzy control for active suspension design. Int J Veh Des 19(1):65–77

Li Y, Ling Z, Liu Y, Qiao Y (2005) Method of fuzzy-PID control on vehicle longitudinal dynamics system, Lecture Notes in Artificial Intelligence, no. 3613 (PART I), pp. 822–832

Huang S, Ren W (1999) Use of neural fuzzy networks with mixed genetic/gradient algorithm in automated vehicle control. IEEE Trans Ind Electron 46(6):1090–1102

Kataoka H, Sado H, Sakai I, Hori Y (2001) Optimal slip ratio estimator for traction control system of electric vehicle based on fuzzy inference. Electr Eng Jpn (Engl Transl Denki Gakkai Ronbunshi) 135(3):56–63

Guglielmino E, Stammers CW, Stǎncioiu D, Sireteanu T, Ghigliazza R (2005) Hybrid variable structure-fuzzy control of a magnetorheological damper for a seat suspension. Int J Veh Auton Syst 3(1):34–46

Senaratne R, Hardy D, Vanderaa B, Halgamuge S (2007) Driver fatigue detection by fusing multiple cues, Lecture Notes in Computer Science, 4492 LNCS (PART 2), pp. 801–809

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Ivanov, V. A review of fuzzy methods in automotive engineering applications. Eur. Transp. Res. Rev. 7 , 29 (2015). https://doi.org/10.1007/s12544-015-0179-z

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A Literature Review on the Fuzzy Control Chart; Classifications & Analysis

Quality control plays an important role in increasing the product quality. Fuzzy control charts are more sensitive than Shewhart control chart. Hence, the correct use of fuzzy control chart leads to producing better-quality products. This area is complex because it involves a large scope of industries, and information is not well organized. In this research, we provide a literature review of the control chart under a fuzzy environment with proposing several classifications and analysis. Moreover, our research considered both attribute and variable control chart by analyzing the related researches based on the content analysis method, to classify past and current developments in the fuzzy control chart. This work has included a distribution of articles according to the journal, the case studies related to fuzzy control chart, the percentage of types of fuzzy control charts used in the literature, performance evaluation of the fuzzy control chart and summary of key points of each revi...

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Fuzzy controlled humanoid robots: A literature review

• Department of Industrial Engineering
• Turkish National Defence University
• Istanbul Technical University
• Igdir University

Research output : Contribution to journal › Article › peer-review

Humanoid robots generated by inspiring by human appearances and abilities have became essential in human society to improve the quality of their life. All over the world, there have been many researchers who have focused on humanoid robots to develop the capabilities of humanoid robots. Generally, humanoid robot systems include mechanisms of decision making and information processing. Because of the uncertainty behind decision making and information processes, fuzzy sets are used most commonly. This study investigates a comprehensive literature review about humanoid robots that presents the recent technological developments and the theories associated with fuzzy set models. The basic principles and concepts of fuzzy sets for humanoid robots are presented.

Bibliographical note

• Classification
• Fuzzy control
• Humanoid robots

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• fuzzy logic INIS 100%
• robots INIS 100%
• reviews INIS 100%
• Humanoid Robot Keyphrases 100%
• Humanoid Robots Computer Science 100%
• Fuzzy sets Computer Science 42%
• Information Processing Keyphrases 33%
• Fuzzy Sets Keyphrases 33%

T1 - Fuzzy controlled humanoid robots

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AU - Kahraman, Cengiz

AU - Deveci, Muhammet

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N1 - Publisher Copyright: © 2020 Elsevier B.V.

PY - 2020/12

Y1 - 2020/12

N2 - Humanoid robots generated by inspiring by human appearances and abilities have became essential in human society to improve the quality of their life. All over the world, there have been many researchers who have focused on humanoid robots to develop the capabilities of humanoid robots. Generally, humanoid robot systems include mechanisms of decision making and information processing. Because of the uncertainty behind decision making and information processes, fuzzy sets are used most commonly. This study investigates a comprehensive literature review about humanoid robots that presents the recent technological developments and the theories associated with fuzzy set models. The basic principles and concepts of fuzzy sets for humanoid robots are presented.

AB - Humanoid robots generated by inspiring by human appearances and abilities have became essential in human society to improve the quality of their life. All over the world, there have been many researchers who have focused on humanoid robots to develop the capabilities of humanoid robots. Generally, humanoid robot systems include mechanisms of decision making and information processing. Because of the uncertainty behind decision making and information processes, fuzzy sets are used most commonly. This study investigates a comprehensive literature review about humanoid robots that presents the recent technological developments and the theories associated with fuzzy set models. The basic principles and concepts of fuzzy sets for humanoid robots are presented.

KW - Classification

KW - Fuzzy control

KW - Fuzzy sets

KW - Humanoid robots

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DO - 10.1016/j.robot.2020.103643

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SN - 0921-8890

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JF - Robotics and Autonomous Systems

M1 - 103643