Ifigenia - User contributions [en]

Generalized nets

2022-04-02T10:45:24Z

Nikolay Ikonomov:

'''Generalized nets''' (''GNs'') constitute a discrete tool for universal description of adaptable, flexible, structured and reusable models of complex systems with many different and interacting components, not necessarily of the homogeneous structure and origin, usually involved in parallel, simultaneous activities. Generalized nets represent a significant extension and generalization of the concept of [[Petri nets]], as well as of other [[Extensions of Petri nets|Petri nets extensions and modifications]].

== Components of a generalized net ==
[[Image:GN-transition-mxn.png|right|thumb|25oxp|A GN transition with ''m'' inputs and ''n'' outputs]]

A generalized net consists of:
* a static structure,
* a dynamic structure,
* temporal components.

The '''''static structure''''' consists of objects called ''[[transition]]s'', which have input and output ''[[place]]s''. Two transitions can share a place, but every place can be an input of at most one transition and can be an output of at most one transition.

The '''''dynamic structure''''' consists of ''[[token]]s'', which act as information carriers and can occupy a single place at every moment of the GN execution. The tokens pass through the transition from one input to another output place; such an ordered pair of places is called ''[[transition arc]]''. The tokens' movement is governed by conditions (''predicates''), contained in the [[predicate matrix]] of the transition.

The information carried by a token is contained in its [[token characteristics|characteristics]], which can be viewed as an associative array of characteristic names and values. The values of the token characteristics change in time according to specific rules, called ''[[characteristic function]]s''. Every place possesses at most one characteristic function, which assigns new characteristics to the incoming tokens. Apart from movement in the net and change of the characteristics, tokens can also split and merge in the places.

The '''''temporal components''''' describe the time scale of GN execution. ''Temporal conditions'' control the transitions' moments of activation and duration of active state. Various other tools for fine tuning of the GN functioning are provided in the form of ''priorities'' of separate transitions, places and tokens, as well as ''capacities'' of places and transitions arcs.

=== Graphic representation ===

== Formal description ==

Formally described, the generalized net is represented by the following four-tuple:

<center><math>
< \underbrace{ <A, \pi_{A}, \pi_{L}, c, f, \theta_{1}, \theta_{2} >}_{1.\ Static \ structure},
\underbrace{ <K, \pi_{K}, \theta_{K} >}_{2. \ Dynamic \ structure},
\underbrace{ <T, t^{0}, t^{*} >}_{3. \ Time},
\underbrace{ <X, \Phi, b >}_{4. \ Memory} >
</math></center>

where:
; ''1. Static structure''
* <math>A</math> is a ''set of transitions'' (see the [[Transition#Formal description|formal definition of a transition]])
* <math>\pi_{A}</math> is a function giving the ''priorities of the transitions'', i.e. <math>\pi_{A} : A \rightarrow N</math> where N = {0, 1, 2, ...} ∪ {∞}
* <math>\pi_{L}</math> is a function giving the ''priorities of the places'', i.e. <math>\pi_{L} : L \rightarrow N</math>
* <math>c</math> is a function giving the ''place capacities'', i.e. <math>c : L \rightarrow N</math>
* <math>f</math> is a function giving the ''truth value of the predicates''. In the basic case, it may obtain values "true" (1) and "false" (0). In [[fuzzy generalized net]]s its domain is the [0;1] interval (see [[fuzzy set]]) and in the [[intuitionistic fuzzy generalized net]]s its domain is the set [0;1]×[0;1] (see [[intuitionistic fuzzy set]]).
* <math>\theta_{1}</math> is a function giving the ''next time moment'' when a given transition will be fired (will become active). Hence, <math>\theta_{1}(t) = t'</math> where <math>t, t' \in [T, T+t^*]; t \le t'</math>. The value of this function is being recalculated in the moment when the transition's active state ceases.
* <math>\theta_{2}</math> is a function giving the ''duration of the transition's active state''. Hence, <math>\theta_{2}(t) = t'</math> where <math>t, t' \in [T, T+t^*]; t' \ge 0</math>. The value of this function is calculated in the moment when the transition's active state begins.

; ''2. Dynamic structure''
* <math>K</math> is the ''set of tokens'' of the generalized net. In certain cases it is more convenient to denote this set as <math>K = \bigcup_{l \in Q^I} K_{l} </math> where <math>K_{l}</math> is the set of all GN tokens which are waiting to enter place <math>l</math> and <math>Q^I</math> is the set of all input places in the net.
* <math>\pi_K</math> is a function giving the ''priorities of the tokens'', i.e. <math>\pi_{K} : K \rightarrow N</math>
* <math>\theta_K</math> is a function giving the ''moment of time when a given token may enter'' the GN, i.e. <math>\theta_K (\alpha)=t</math> where <math>\alpha \in K, t \in [T; T+t^*]</math>

; ''3. Time''
* <math>T</math> is the moment of time when the generalized net starts functioning. This moment is determined according to a fixed global timescale.
* <math>t^0</math> is the elementary time step of the fixed global timescale (the interval with which time increments in the timescale).
* <math>t^*</math> is the total duration of functioning of the net.

; ''4. Memory''
* <math>X</math> is the ''set of initial characteristics'', which tokens may exhibit when they enter the net for first.
* <math>\Phi</math> is a ''characteristic function'', which assigns a new characteristic to each token when it makes the transfer from an input to an output place of a given transition.
* <math>b</math> is a function giving the ''maximal number of characteristics'', which a given token may obtain during its movement throughout the net, i.e. <math>b : K \rightarrow N</math>. In general, <math>b</math> may possess four different values: 0, 1, <math>s</math> or <math>\infty</math> meaning that the token keeps, respectively: none of its characteristics, its last characteristic, its last <math>s</math> characteristics, or all of its characteristics obtained during its movement in the net.

=== Algorithms related to generalized nets ===
In a Petri net implementation, parallelism is reduced to a sequential firing of the net transitions and in general the order of their activation is probabilistic or dependent on the transitions' priorities, if any. The respective algorithms for generalized nets enable a more detailed modelling of the described process.

In the the book "[[Generalized Nets (World Scientific)|Generalized Nets]]" from 1991 year <ref>Atanassov K., "Generalized Nets", World Scientific, Singapore, 1991, ISBN 978-981-02-0598-0</ref>, there were formulated two major algorithms, related to generalized nets:
* '''[[Algorithm for transition functioning]]''', which is to some extent equivalent to the tokens transfer algorithm in Petri nets, and
* '''[[Algorithm for generalized net functioning]]'''.

The algorithm for transition functioning was [[Algorithm for transition functioning#Modified algorithm, 2007|later modified]] in 2007 <ref>Atanassov K., Tasseva V., Trifonov T., [[Issue:Modification of the algorithm for token transfer in generalized nets|]], Cybernetics and Information Technologies, Vol. 7, 2007, Np. 1, 62-66</ref>.

''Theorem 20'' in the book "[[On Generalized Nets Theory]]" from 2007 year <ref>Atanassov K., [[On Generalized Nets Theory]], "Professor Marin Drinov" Academic Publishing House, 2007, ISBN 978-954-322-237-7 </ref> states that the functioning and the results of the work of a given GN transition are equal for both algorithms, yet the difference between both algorithms is that in almost any case the modified algorithm yields results more quickly.

== Reduced generalized nets ==
== Extensions of generalized nets ==
Generalized nets have been subject of multiple different extensions. For each of these it has been proved that represents a ''conservative extension'', i.e. there exists an ordinary generalized net which describes the extended GN, i.e. both nets have the same modelling capabilities.

The following extensions of GNs have been determined so far:
* [[Intuitionistic fuzzy generalized nets of type 1]]
* [[Intuitionistic fuzzy generalized nets of type 2]]
* [[Intuitionistic fuzzy generalized nets of type 3]]
* [[Intuitionistic fuzzy generalized nets of type 4]]
* [[Colour generalized nets]]
* [[Generalized nets with interval activation time]]
* [[Generalized nets with complex structure]]
* [[Generalized nets with global memory]]
* [[Generalized nets with optimization components]]
* [[Generalized nets with additional clocks]]
* [[Opposite generalized nets]]
* [[Generalized nets with three-dimensional structure]]

== Theoretical aspects of generalized nets ==
=== Algebraic aspect ===
=== Topological aspect ===
=== Logical aspect ===
=== Functional aspect ===
* [[Global operators over generalized nets|Global operators]]
* [[Local operators over generalized nets|Local operators]]
* [[Hierarchical operators over generalized nets|Hierarchical operators]]
* [[Reducing operators over generalized nets|Reducing operators]]
* [[Extending operators over generalized nets|Extending operators]]
* [[Dynamic operators over generalized nets|Dynamic operators]]

== Construction of generalized nets ==
== Modelling and simulation ==
== Software implementation of generalized nets ==

== See also ==
* [[Theory of generalized nets]]
* [[Applications of generalized nets]]
* [[History of generalized nets]]
* [[List of GN terms]]

== References ==

[[Category:Generalized nets]]
{{stub}}

Generalized nets

2022-04-02T10:44:08Z

Nikolay Ikonomov:

'''Generalized nets''' (''GNs'') constitute a discrete tool for universal description of adaptable, flexible, structured and reusable models of complex systems with many different and interacting components, not necessarily of the homogeneous structure and origin, usually involved in parallel, simultaneous activities. Generalized nets represent a significant extension and generalization of the concept of [[Petri nets]], as well as of other [[Extensions of Petri nets|Petri nets extensions and modifications]].

== Components of a generalized net ==
[[Image:GN-transition-mxn.png|right|thumb|25oxp|A GN transition with ''m'' inputs and ''n'' outputs]]

A generalized net consists of:
* a static structure,
* a dynamic structure,
* temporal components.

[[File:Priroda 4 2021 cover 1.pdf|thumb]]

The '''''static structure''''' consists of objects called ''[[transition]]s'', which have input and output ''[[place]]s''. Two transitions can share a place, but every place can be an input of at most one transition and can be an output of at most one transition.

The '''''dynamic structure''''' consists of ''[[token]]s'', which act as information carriers and can occupy a single place at every moment of the GN execution. The tokens pass through the transition from one input to another output place; such an ordered pair of places is called ''[[transition arc]]''. The tokens' movement is governed by conditions (''predicates''), contained in the [[predicate matrix]] of the transition.

The information carried by a token is contained in its [[token characteristics|characteristics]], which can be viewed as an associative array of characteristic names and values. The values of the token characteristics change in time according to specific rules, called ''[[characteristic function]]s''. Every place possesses at most one characteristic function, which assigns new characteristics to the incoming tokens. Apart from movement in the net and change of the characteristics, tokens can also split and merge in the places.

The '''''temporal components''''' describe the time scale of GN execution. ''Temporal conditions'' control the transitions' moments of activation and duration of active state. Various other tools for fine tuning of the GN functioning are provided in the form of ''priorities'' of separate transitions, places and tokens, as well as ''capacities'' of places and transitions arcs.

=== Graphic representation ===

== Formal description ==

Formally described, the generalized net is represented by the following four-tuple:

<center><math>
< \underbrace{ <A, \pi_{A}, \pi_{L}, c, f, \theta_{1}, \theta_{2} >}_{1.\ Static \ structure},
\underbrace{ <K, \pi_{K}, \theta_{K} >}_{2. \ Dynamic \ structure},
\underbrace{ <T, t^{0}, t^{*} >}_{3. \ Time},
\underbrace{ <X, \Phi, b >}_{4. \ Memory} >
</math></center>

where:
; ''1. Static structure''
* <math>A</math> is a ''set of transitions'' (see the [[Transition#Formal description|formal definition of a transition]])
* <math>\pi_{A}</math> is a function giving the ''priorities of the transitions'', i.e. <math>\pi_{A} : A \rightarrow N</math> where N = {0, 1, 2, ...} ∪ {∞}
* <math>\pi_{L}</math> is a function giving the ''priorities of the places'', i.e. <math>\pi_{L} : L \rightarrow N</math>
* <math>c</math> is a function giving the ''place capacities'', i.e. <math>c : L \rightarrow N</math>
* <math>f</math> is a function giving the ''truth value of the predicates''. In the basic case, it may obtain values "true" (1) and "false" (0). In [[fuzzy generalized net]]s its domain is the [0;1] interval (see [[fuzzy set]]) and in the [[intuitionistic fuzzy generalized net]]s its domain is the set [0;1]×[0;1] (see [[intuitionistic fuzzy set]]).
* <math>\theta_{1}</math> is a function giving the ''next time moment'' when a given transition will be fired (will become active). Hence, <math>\theta_{1}(t) = t'</math> where <math>t, t' \in [T, T+t^*]; t \le t'</math>. The value of this function is being recalculated in the moment when the transition's active state ceases.
* <math>\theta_{2}</math> is a function giving the ''duration of the transition's active state''. Hence, <math>\theta_{2}(t) = t'</math> where <math>t, t' \in [T, T+t^*]; t' \ge 0</math>. The value of this function is calculated in the moment when the transition's active state begins.

; ''2. Dynamic structure''
* <math>K</math> is the ''set of tokens'' of the generalized net. In certain cases it is more convenient to denote this set as <math>K = \bigcup_{l \in Q^I} K_{l} </math> where <math>K_{l}</math> is the set of all GN tokens which are waiting to enter place <math>l</math> and <math>Q^I</math> is the set of all input places in the net.
* <math>\pi_K</math> is a function giving the ''priorities of the tokens'', i.e. <math>\pi_{K} : K \rightarrow N</math>
* <math>\theta_K</math> is a function giving the ''moment of time when a given token may enter'' the GN, i.e. <math>\theta_K (\alpha)=t</math> where <math>\alpha \in K, t \in [T; T+t^*]</math>

; ''3. Time''
* <math>T</math> is the moment of time when the generalized net starts functioning. This moment is determined according to a fixed global timescale.
* <math>t^0</math> is the elementary time step of the fixed global timescale (the interval with which time increments in the timescale).
* <math>t^*</math> is the total duration of functioning of the net.

; ''4. Memory''
* <math>X</math> is the ''set of initial characteristics'', which tokens may exhibit when they enter the net for first.
* <math>\Phi</math> is a ''characteristic function'', which assigns a new characteristic to each token when it makes the transfer from an input to an output place of a given transition.
* <math>b</math> is a function giving the ''maximal number of characteristics'', which a given token may obtain during its movement throughout the net, i.e. <math>b : K \rightarrow N</math>. In general, <math>b</math> may possess four different values: 0, 1, <math>s</math> or <math>\infty</math> meaning that the token keeps, respectively: none of its characteristics, its last characteristic, its last <math>s</math> characteristics, or all of its characteristics obtained during its movement in the net.

=== Algorithms related to generalized nets ===
In a Petri net implementation, parallelism is reduced to a sequential firing of the net transitions and in general the order of their activation is probabilistic or dependent on the transitions' priorities, if any. The respective algorithms for generalized nets enable a more detailed modelling of the described process.

In the the book "[[Generalized Nets (World Scientific)|Generalized Nets]]" from 1991 year <ref>Atanassov K., "Generalized Nets", World Scientific, Singapore, 1991, ISBN 978-981-02-0598-0</ref>, there were formulated two major algorithms, related to generalized nets:
* '''[[Algorithm for transition functioning]]''', which is to some extent equivalent to the tokens transfer algorithm in Petri nets, and
* '''[[Algorithm for generalized net functioning]]'''.

The algorithm for transition functioning was [[Algorithm for transition functioning#Modified algorithm, 2007|later modified]] in 2007 <ref>Atanassov K., Tasseva V., Trifonov T., [[Issue:Modification of the algorithm for token transfer in generalized nets|]], Cybernetics and Information Technologies, Vol. 7, 2007, Np. 1, 62-66</ref>.

''Theorem 20'' in the book "[[On Generalized Nets Theory]]" from 2007 year <ref>Atanassov K., [[On Generalized Nets Theory]], "Professor Marin Drinov" Academic Publishing House, 2007, ISBN 978-954-322-237-7 </ref> states that the functioning and the results of the work of a given GN transition are equal for both algorithms, yet the difference between both algorithms is that in almost any case the modified algorithm yields results more quickly.

== Reduced generalized nets ==
== Extensions of generalized nets ==
Generalized nets have been subject of multiple different extensions. For each of these it has been proved that represents a ''conservative extension'', i.e. there exists an ordinary generalized net which describes the extended GN, i.e. both nets have the same modelling capabilities.

The following extensions of GNs have been determined so far:
* [[Intuitionistic fuzzy generalized nets of type 1]]
* [[Intuitionistic fuzzy generalized nets of type 2]]
* [[Intuitionistic fuzzy generalized nets of type 3]]
* [[Intuitionistic fuzzy generalized nets of type 4]]
* [[Colour generalized nets]]
* [[Generalized nets with interval activation time]]
* [[Generalized nets with complex structure]]
* [[Generalized nets with global memory]]
* [[Generalized nets with optimization components]]
* [[Generalized nets with additional clocks]]
* [[Opposite generalized nets]]
* [[Generalized nets with three-dimensional structure]]

== Theoretical aspects of generalized nets ==
=== Algebraic aspect ===
=== Topological aspect ===
=== Logical aspect ===
=== Functional aspect ===
* [[Global operators over generalized nets|Global operators]]
* [[Local operators over generalized nets|Local operators]]
* [[Hierarchical operators over generalized nets|Hierarchical operators]]
* [[Reducing operators over generalized nets|Reducing operators]]
* [[Extending operators over generalized nets|Extending operators]]
* [[Dynamic operators over generalized nets|Dynamic operators]]

== Construction of generalized nets ==
== Modelling and simulation ==
== Software implementation of generalized nets ==

== See also ==
* [[Theory of generalized nets]]
* [[Applications of generalized nets]]
* [[History of generalized nets]]
* [[List of GN terms]]

== References ==

[[Category:Generalized nets]]
{{stub}}

Notes on Intuitionistic Fuzzy Sets/24/4

2018-12-17T13:58:04Z

Nikolay Ikonomov:

<noinclude>'''[[Notes on Intuitionistic Fuzzy Sets]], [[Notes on Intuitionistic Fuzzy Sets/24|Volume 24 (2018)]], Number 4.''' DOI: [https://doi.org/10.7546/nifs.2018.24.4 10.7546/nifs.2018.24.4]

== Journal contents ==</noinclude>
{{journal-contents/header}}
{{journal-contents/row
|number = 1
|title = [[Issue:In Memoriam: Professor Beloslav Riečan|In Memoriam: Professor Beloslav Riečan]]
|authors = [[Katarína Čunderlíková]]
|pages = 1—4
}}
{{journal-contents/row
|number = 2
|title = [[Issue:On some methods of study of states on interval valued fuzzy sets|On some methods of study of states on interval valued fuzzy sets]]
|authors = [[Alžbeta Michalíková]] and <span style="border:1pt solid black">[[Beloslav Riečan]]</span>
|pages = 5—12
}}
{{journal-contents/row
|number = 3
|title = [[Issue:A note on a family of multiplicative and additive mappings preserving the class IFS(X)|A note on a family of multiplicative and additive mappings preserving the class IFS(X)]]
|authors = [[Peter Vassilev]]
|pages = 13—19
}}
{{journal-contents/row
|number = 4
|title = [[Issue:Shrinking operators over interval-valued intuitionistic fuzzy sets|Shrinking operators over interval-valued intuitionistic fuzzy sets]]
|authors = [[Krassimir Atanassov]], [[Eulalia Szmidt]] and [[Janusz Kacprzyk]]
|pages = 20—28
}}
{{journal-contents/row
|number = 5
|title = [[Issue:Modified level operator Nγ1γ2 applied over interval-valued intuitionistic fuzzy sets|Modified level operator <math>N_{\gamma_1}^{\gamma_2}</math> applied over interval-valued intuitionistic fuzzy sets]]
|authors = [[Vassia Atanassova]]
|pages = 29—39
}}
{{journal-contents/row
|number = 6
|title = [[Issue:Upper and lower limits and m-almost everywhere convergence of intuitionistic fuzzy observables|Upper and lower limits and m-almost everywhere convergence of intuitionistic fuzzy observables]]
|authors = [[Katarína Čunderlíková]]
|pages = 40—49
}}
{{journal-contents/row
|number = 7
|title = [[Issue:Complex trapezoidal intuitionistic fuzzy numbers|Complex trapezoidal intuitionistic fuzzy numbers]]
|authors = [[R. Parvathi]] and [[J. Akila Padmasree]]
|pages = 50—62
}}
{{journal-contents/row
|number = 8
|title = [[Issue:Selection of the attributes in intuitionistic fuzzy models|Selection of the attributes in intuitionistic fuzzy models]]
|authors = [[Eulalia Szmidt]] and [[Janusz Kacprzyk]]
|pages = 63—71
}}
{{journal-contents/row
|number = 9
|title = [[Issue:Intuitionistic fuzzy Dirichlet problem|Intuitionistic fuzzy Dirichlet problem]]
|authors = [[S. Melliani]], [[I. Bakhadach]], [[M. Elomari]] and [[L. S. Chadli]]
|pages = 72—84
}}
{{journal-contents/row
|number = 10
|title = [[Issue:Radical structures of intuitionistic fuzzy polynomial ideals of a ring|Radical structures of intuitionistic fuzzy polynomial ideals of a ring]]
|authors = [[P. K. Sharma]] and [[Gagandeep Kaur]]
|pages = 85—96
}}
{{journal-contents/row
|number = 11
|title = [[Issue:On intuitionistic fuzzy prime submodules|On intuitionistic fuzzy prime submodules]]
|authors = [[P. K. Sharma]] and [[Gagandeep Kaur]]
|pages = 97—112
}}
{{journal-contents/row
|number = 12
|title = [[Issue:Difference and symmetric difference for intuitionistic fuzzy sets|Difference and symmetric difference for intuitionistic fuzzy sets]]
|authors = [[Taiwo Enayon Sunday]], [[Romuald Dzati Kamga]], [[Siméon Fotso]] and [[Louis Aimé Fono]]
|pages = 113—140
}}
{{journal-contents/row
|number = 13
|title = [[Issue:System of intuitionistic fuzzy differential equations with intuitionistic fuzzy initial values|System of intuitionistic fuzzy differential equations with intuitionistic fuzzy initial values]]
|authors = [[Ömer Akin]] and [[Selami Bayeğ]]
|pages = 141—171
}}
{{journal-contents/row
|number = 14
|title = [[Issue:Optimization of EOQ model with space constraint: An intuitionistic fuzzy geometric programming approach|Optimization of EOQ model with space constraint: An intuitionistic fuzzy geometric programming approach]]
|authors = [[Bappa Mondal]], [[Arindam Garai]] and [[Tapan Kumar Roy]]
|pages = 172—189
}}
{{journal-contents/row
|number = 15
|title = [[Issue:Interval-valued intuitionistic fuzzy sets as tools for evaluation of data mining processes|Interval-valued intuitionistic fuzzy sets as tools for evaluation of data mining processes]]
|authors = [[Krassimir Atanassov]]
|pages = 190—202
}}
{{journal-contents/footer}}
This issue of Int. Journal "Notes on Intuitionistic Fuzzy Sets" is published with the financial support of the Bulgarian National Science Fund, Grant Ref. No. DNP-06-8/2017.
<noinclude>
[[Category:Notes on IFS|24]]
</noinclude>

Issue:Modified level operator Nγ1γ2 applied over interval-valued intuitionistic fuzzy sets

2018-12-17T13:55:34Z

Nikolay Ikonomov:

[[Category:Publications on intuitionistic fuzzy sets|{{PAGENAME}}]]
[[Category:Publications in Notes on IFS|{{PAGENAME}}]]
[[Category:Publications in 2018 year|{{PAGENAME}}]]
{{issue/title
| title = Modified level operator <math>N_{\gamma_1}^{\gamma_2}</math> applied over interval-valued intuitionistic fuzzy sets
| shortcut = nifs/24/4/29-39
}}
{{issue/author
| author = Vassia Atanassova
| institution = Institute of Biophysics and Biomedical Engineering, Bulgarian Academy of Sciences
| address = Acad. G. Bonchev Str., Bl. 105, Sofia-1113, Bulgaria
| email-before-at = vassia.atanassova
| email-after-at = gmail.com
}}
{{issue/data
| issue = [[Notes on Intuitionistic Fuzzy Sets/24/4|Notes on Intuitionistic Fuzzy Sets, Volume 24 (2018), Number 4]], pages 29–39
| doi = https://doi.org/10.7546/nifs.2018.24.4.29-39
| file = NIFS-24-4-029-039.pdf
| format = PDF
| size = 158 Kb
| abstract = The recently proposed intuitionistic fuzzy level operator <math>N_\gamma</math> generates a subset of an intuitionistic fuzzy set <math>A</math>, where the elements of the subset are those elements of <math>A</math>, for which the ratio of their degrees of membership to their degrees of non-membership is greater than or equal to a given constant <math>\gamma > 0</math>. Here we propose a continuation of this idea from the case of intuitionistic fuzzy sets to the case of interval-valued intuitionistic fuzzy sets. This modification requires us to introduce a second constant, i.e. <math>\gamma_1,\gamma_2 > 0</math>. We show that there are twenty possible scenarios for the mutual position of the intervalized level operator <math>N_{\gamma_1}^{\gamma_2}</math> and the element of the interval-valued intuitionistic fuzzy set, and give the respective formulas which calculate in each case the membership and non-membership degrees with which the IVIFS element belongs to the set defined by the operator <math>N_{\gamma_1}^{\gamma_2}</math>. These twenty scenarios are graphically interpreted in the intuitionistic fuzzy interpretational triangle, and the respective formulas have been derived. In conclusion, further ideas of research have been suggested.
| keywords = Interval valued intuitionistic fuzzy sets, Intuitionistic fuzzy sets, Level operator, Decision making under uncertainty.
| ams = 03E72.
| references =

# Atanassov, K. T. (1983). Intuitionistic fuzzy sets. VII ITKR Session, Sofia, 20-23 June 1983 (Deposed in Centr. Sci.-Techn. Library of the Bulg. Acad. of Sci., 1697/84) (in Bulgarian). Reprinted: Int. J. Bioautomation, 2016, 20(S1), S1–S6.
# Atanassov K. T. (1989). Geometrical Interpretation of the Elements of the Intuitionistic Fuzzy Objects, Mathematical Foundations of Artificial Intelligence Seminar, Sofia, 1989, Preprint IM-MFAIS-1-89. Reprinted: Int J Bioautomation, 2016, 20(S1), S27–S42.
# Atanassov, K. T. (1999). Intuitionistic Fuzzy Sets: Theory and Applications. Physica- Verlag, Heidelberg.
# Atanassov, K. T., & Gargov, G. (1999). Interval valued intuitionistic fuzzy sets. Fuzzy Sets and Systems, 31(3), 343–349.
# Atanassova, V. (2017). New modified level operator Nγ over intuitionistic fuzzy sets. In: Christiansen H., Jaudoin H., Chountas P., Andreasen T., Legind Larsen H. (eds) Flexible Query Answering Systems. FQAS 2017. Lecture Notes in Computer Science, vol 10333. Springer, Cham, 209–214.
# Doukovska, L., Atanassova, V., Mavrov, D., & Radeva, I. (2018). Intercriteria Analysis of EU Competitiveness Using the Level Operator Nγ. 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 641. Springer, Cham, 631–647.
# Gottman, J. (1995). Why Marriages Succeed or Fail: And How You Can Make Yours Last. Simon and Schuster.

| citations =
| see-also =
}}