Research

18 pages, 726 KiB

Open AccessArticle

Toward Effective Uncertainty Management in Decision-Making Models Based on Type-2 Fuzzy TOPSIS

by Elissa Nadia Madi, Zahrahtul Amani Zakaria, Aceng Sambas and Sukono

Mathematics 2023, 11(16), 3512; https://0-doi-org.brum.beds.ac.uk/10.3390/math11163512 - 14 Aug 2023

Cited by 2 | Viewed by 794

Over the past century, there has been a dramatic increasing interest in the multi-criteria group decision-making (MCGDM) technique, with a considerable amount of studies published regarding it. One of the well-known approaches in the MCGDM paradigm is Technique for Order Preference by Similarity [...] Read more.

Over the past century, there has been a dramatic increasing interest in the multi-criteria group decision-making (MCGDM) technique, with a considerable amount of studies published regarding it. One of the well-known approaches in the MCGDM paradigm is Technique for Order Preference by Similarity to Ideal Solution (TOPSIS). The integration of the TOPSIS method with fuzzy set theory has proven to be successful in various applications. Recently, a wide array of publications has proposed implementing a type-2 fuzzy set with TOPSIS. However, the additional degree of uncertainty represented by type 2 has largely been ignored, especially in a few specific mathematical operations in the model. We propose constructing interval type-2 fuzzy membership functions (IT2 MFs) using interval-based data gathered from a survey, where this is used to generate a new scale to represent ratings for each alternative. This procedure utilized all information gathered from decision makers. In addition, we present a complete algorithm for TOPSIS based on IT2 fuzzy sets (IT2 FSs) which preserve the interval-based form output. The output in the form of intervals offers decision makers (DMs) with more detailed information, enabling them to make more nuanced decisions. This can include cautious decisions when intervals are wider and overlapping. Although understanding the exact meaning of these intervals and their widths in a decision-making context is challenging, this paper introduces a systematic method for connecting input uncertainty to output uncertainty in the TOPSIS technique. This approach establishes a solid foundation for future research. Thus far, no other researchers have suggested a data-driven method that combines TOPSIS with fuzzification and provides intervals as the final output. Full article

(This article belongs to the Special Issue Computing Mathematics with Fuzzy Sets)

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27 pages, 3126 KiB

Open AccessArticle

Seaport Network Efficiency Measurement Using Triangular and Trapezoidal Fuzzy Data Envelopment Analyses with Liner Shipping Connectivity Index Output

by Dineswary Nadarajan, Saber Abdelall Mohamed Ahmed and Noor Fadiya Mohd Noor

Mathematics 2023, 11(6), 1454; https://0-doi-org.brum.beds.ac.uk/10.3390/math11061454 - 17 Mar 2023

Cited by 1 | Viewed by 1639

Abstract

Seaport network efficiency is very crucial for global maritime economic trades and growth. In this work, data of three years (2018–2020) with input variables (time in port, age of vessels, size of vessels, cargo carrying capacity of vessels) and output variables (Liner Shipping [...] Read more.

Seaport network efficiency is very crucial for global maritime economic trades and growth. In this work, data of three years (2018–2020) with input variables (time in port, age of vessels, size of vessels, cargo carrying capacity of vessels) and output variables (Liner Shipping Connectivity Index (LSCI) and Gross Domestic Product (GDP)) are collected. Few screening tests are performed to ensure the data are fit for further analyses. Since none of the existing studies has ever considered LSCI as an output variable, the main purpose of this study is to measure seaport network efficiency based on LSCI using data envelopment analysis (DEA), both classical and fuzzy. In fuzzy DEA, triangular fuzzy number (TrFN) and trapezoidal fuzzy number (TpFN) are used to construct the fuzzy sets of efficiency scores with DEA. The comparison between DEA and triangular fuzzy data envelopment analysis (TrFDEA) shows the range of differences in the results ranges from −0.0274 to 0.0105, while the comparison between DEA and trapezoidal fuzzy data envelopment analysis (TpFDEA) yields the differences within the range of −0.0307 to 0.0106. Using DEA as the relative reference, it is further revealed that the TpFDEA has smaller standard deviations and variances than the TrFDEA in 2018 and 2019, whereas the opposites hold true during the pandemic year of 2020. With the use of fuzzy numbers, the uncertainty levels in the seaport network efficiency measurement can further be investigated as the minimum, mean, median and maximum values are taken into consideration. Moreover, the proposed TrFDEA and TpFDEA lead new insights on the boundedness concept of the efficiency scores, which were never reported before by any researcher, especially in the maritime industry research. Fuzzy regression modelling based on the Possibilistic Linear Regression Least Squares (PLRLS) method was also performed to determine the interval of minimum and maximum connectivity efficiencies, which gave a better estimation than the regular regression model. Full article

(This article belongs to the Special Issue Computing Mathematics with Fuzzy Sets)

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19 pages, 1434 KiB

Open AccessArticle

On the Number of Finite Fuzzy Subsets with Analysis of Integer Sequences

by Rajesh Kumar Mohapatra and Tzung-Pei Hong

Mathematics 2022, 10(7), 1161; https://0-doi-org.brum.beds.ac.uk/10.3390/math10071161 - 03 Apr 2022

Viewed by 1461

Abstract

This paper solves the issues of determining the number F_n of fuzzy subsets of a nonempty finite set

X

. To solve this, this paper incorporates the equivalence relation on the collection of all fuzzy subsets of

X

. We derive two [...] Read more.

This paper solves the issues of determining the number F_n of fuzzy subsets of a nonempty finite set

X

. To solve this, this paper incorporates the equivalence relation on the collection of all fuzzy subsets of

X

. We derive two closed explicit formulas for

F_{n}

, which is the sum of a finite series in the product of binomial numbers or the sum of

k

-level fuzzy subsets

F_{n, k}

by introducing a classification technique. Moreover, these explicit formulas enable us to find the number of the maximal chains of crisp subsets of

X .

Further, this paper presents some elementary properties of

F_{n, k}

and

F_{n}

. Full article

(This article belongs to the Special Issue Computing Mathematics with Fuzzy Sets)

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6 pages, 247 KiB

Open AccessArticle

On Triangular Multisets and Triangular Fuzzy Multisets

by Apostolos Syropoulos

Mathematics 2022, 10(5), 726; https://0-doi-org.brum.beds.ac.uk/10.3390/math10050726 - 25 Feb 2022

Viewed by 1274

Abstract

The basic set operations between fuzzy sets are defined using the min and max functions; however, later on, new operators were introduced that used other functions, which, nevertheless, had similar properties to functions min and max. The resulting fuzzy set theories are more [...] Read more.

The basic set operations between fuzzy sets are defined using the min and max functions; however, later on, new operators were introduced that used other functions, which, nevertheless, had similar properties to functions min and max. The resulting fuzzy set theories are more suitable for the description and processing of specific data sets. Crisp and fuzzy multisets have found numerous applications but still the basic operations are based on functions min and max. It is straightforward to replace these functions in the fuzzy part of fuzzy multisets; however, it is not as easy but is feasible to do the same with the multisets and the “crisp” part of fuzzy multisets. The new mathematical structures are called triangular multisets and triangular fuzzy multisets, respectively. The aim is to facilitate the processing of certain data sets so they can be used in multi-criteria decision making and computing. Full article

(This article belongs to the Special Issue Computing Mathematics with Fuzzy Sets)

14 pages, 295 KiB

Open AccessArticle

Between the Classes of Soft Open Sets and Soft Omega Open Sets

by Samer Al Ghour

Mathematics 2022, 10(5), 719; https://0-doi-org.brum.beds.ac.uk/10.3390/math10050719 - 24 Feb 2022

Cited by 13 | Viewed by 1240

Abstract

In this paper, we define the class of soft

ω^{0}

-open sets. We show that this class forms a soft topology that is strictly between the classes of soft open sets and soft

ω

-open sets, and we provide some sufficient conditions [...] Read more.

In this paper, we define the class of soft

ω^{0}

-open sets. We show that this class forms a soft topology that is strictly between the classes of soft open sets and soft

ω

-open sets, and we provide some sufficient conditions for the equality of the three classes. In addition, we show that soft closed soft

ω

-open sets are soft

ω^{0}

-open sets in soft Lindelof soft topological spaces. Moreover, we study the correspondence between soft

ω^{0}

-open sets in soft topological spaces and

ω^{0}

-open sets in topological spaces. Furthermore, we investigate the relationships between the soft

α

-open sets (respectively, soft regular open sets, soft

β

-open sets) of a given soft anti-locally countable soft topological space and the soft

α

-open sets (respectively, soft regular open sets, soft

β

-open sets) of the soft topological space of soft

ω^{0}

-open sets generated by it. Finally, we introduce

ω^{0}

-regularity in topological spaces via

ω^{0}

-open sets, which is strictly between regularity and

ω

-regularity, and we also introduce soft

ω^{0}

-regularity in soft topological spaces via soft

ω^{0}

-open sets, which is strictly between soft regularity and soft

ω

-regularity. We investigate relationships regarding

ω^{0}

-regularity and soft

ω^{0}

-regularity. Moreover, we study the correspondence between soft

ω^{0}

-regularity in soft topological spaces and

ω^{0}

-regularity in topological spaces. Full article

(This article belongs to the Special Issue Computing Mathematics with Fuzzy Sets)

26 pages, 1147 KiB

Open AccessArticle

On PSO-Based Simulations of Fuzzy Dynamical Systems Induced by One-Dimensional Ones

by Jiří Kupka and Nicole Škorupová

Mathematics 2021, 9(21), 2737; https://0-doi-org.brum.beds.ac.uk/10.3390/math9212737 - 28 Oct 2021

Cited by 1 | Viewed by 972

Abstract

Zadeh’s extension principle is one of the elementary tools in fuzzy set theory, and among other things, it provides a natural extension of a real-valued continuous self-map to a self-map having fuzzy sets as its arguments. The purpose of this paper is to [...] Read more.

Zadeh’s extension principle is one of the elementary tools in fuzzy set theory, and among other things, it provides a natural extension of a real-valued continuous self-map to a self-map having fuzzy sets as its arguments. The purpose of this paper is to introduce a new algorithm that can be used for simulations of fuzzy dynamical systems induced by interval maps. The core of the proposed algorithm is based on calculations including piecewise linear maps, and consequently, an implementation of an optimization algorithm (in our case, particle swarm optimization) was prepared to obtain necessary piecewise linear approximations. For all parts of this algorithm, we provide detailed testing and numerous examples. Full article

(This article belongs to the Special Issue Computing Mathematics with Fuzzy Sets)

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13 pages, 2820 KiB

Open AccessArticle

Stability Analysis and Robust Stabilization of Uncertain Fuzzy Time-Delay Systems

by Chun-Tang Chao, Ding-Horng Chen and Juing-Shian Chiou

Mathematics 2021, 9(19), 2441; https://0-doi-org.brum.beds.ac.uk/10.3390/math9192441 - 01 Oct 2021

Cited by 3 | Viewed by 1142

Abstract

New sufficient conditions for delay-independent and delay-dependent robust stability of uncertain fuzzy time-delay systems based on uncertain fuzzy Takagi-Sugeno (T-S) models are presented by using the properties of matrix and norm measurements. Further sufficient conditions are formulated, in terms of the linear matrix [...] Read more.

New sufficient conditions for delay-independent and delay-dependent robust stability of uncertain fuzzy time-delay systems based on uncertain fuzzy Takagi-Sugeno (T-S) models are presented by using the properties of matrix and norm measurements. Further sufficient conditions are formulated, in terms of the linear matrix inequalities (LMIs) of robust stabilization, and are developed via the technique of parallel distributed compensation (PDC), and then the simplification of the conditions for the controller design of uncertain fuzzy time-delay systems. The proposed methods are simple and effective. Some examples below are presented to illustrate our results. Full article

(This article belongs to the Special Issue Computing Mathematics with Fuzzy Sets)

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13 pages, 291 KiB

Open AccessArticle

Weaker Forms of Soft Regular and Soft T₂ Soft Topological Spaces

by Samer Al Ghour

Mathematics 2021, 9(17), 2153; https://0-doi-org.brum.beds.ac.uk/10.3390/math9172153 - 03 Sep 2021

Cited by 21 | Viewed by 1853

Abstract

Soft

ω

-local indiscreetness as a weaker form of both soft local countability and soft local indiscreetness is introduced. Then soft

ω

-regularity as a weaker form of both soft regularity and soft

ω

-local indiscreetness is defined and investigated. Additionally, soft

ω

[...] Read more.

Soft

ω

-local indiscreetness as a weaker form of both soft local countability and soft local indiscreetness is introduced. Then soft

ω

-regularity as a weaker form of both soft regularity and soft

ω

-local indiscreetness is defined and investigated. Additionally, soft

ω

-

T_{2}

as a new soft topological property that lies strictly between soft

T_{2}

and soft

T_{1}

is defined and investigated. It is proved that soft anti-local countability is a sufficient condition for equivalence between soft

ω

-locally indiscreetness (resp. soft

ω

-regularity) and soft locally indiscreetness (resp. soft

ω

-regularity). Additionally, it is proved that the induced topological spaces of a soft

ω

-locally indiscrete (resp. soft

ω

-regular, soft

ω

-

T_{2}

) soft topological space are (resp.

ω

-regular,

ω

-

T_{2}

) topological spaces. Additionally, it is proved that the generated soft topological space of a family of

ω

-locally indiscrete (resp.

ω

-regular,

ω

-

T_{2}

) topological spaces is soft

ω

-locally indiscrete and vice versa. In addition to these, soft product theorems regarding soft

ω

-regular and soft

ω

-

T_{2}

soft topological spaces are obtained. Moreover, it is proved that soft

ω

-regular and soft

ω

-

T_{2}

are hereditarily under soft subspaces. Full article

(This article belongs to the Special Issue Computing Mathematics with Fuzzy Sets)

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