The primary objective of this research is to examine how the Russia–Ukraine war impacted the global economy. To achieve this goal, we propose operational laws, a score and accuracy function, and a divergence measure for complex dual hesitant fuzzy sets. Furthermore, we investigate the challenges associated with multi-criteria decision-making (MCR) using aggregation operators and the TODIM method with complex dual hesitant fuzzy (CDHF) information. Drawing inspiration from arithmetic aggregation operations, we introduce several aggregation operators for complex dual hesitant fuzzy information, including the complex dual hesitant fuzzy weighted average (CDHFWA) operator, complex dual hesitant fuzzy ordered weighted average (CDHFOWA) operator, and complex dual hesitant fuzzy hybrid average (CDHFHA) operator. We thoroughly analyze the unique characteristics of these proposed operators and use them in conjunction with the TODIM method to develop practical approaches for solving complex dual hesitant fuzzy multi-criteria decision-making problems. To validate our approach and demonstrate its applicability, we provide a practical example of how the Russia–Ukraine war affected the global economy. Our study confirms the effectiveness of our approach and highlights how it can be used to solve real-world problems.
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Introduction
The idea of fuzzy sets (FSs), which incorporate the degree of truth for a unit interval, was created by Zander [1]. However, the fuzzy set theory has fallen short in several circumstances. For instance, FS are capable of handling information supplied to decision-makers in the form of truth and untruth evolutions. In reality, these sets have offered a wide range of methods for assigning the components level’s of membership or non-membership to a certain set as indicated by different attributes. IFSs, also referred to as IVFSs [2], can be illustrated numerically with a given choice that characterise a range to reflect some sensitivity of component enrollment attempt. IVFSs are FS speculation that can show vulnerability to the requirement for data by relegating a closed subinterval [0, 1] to the participation degree. IFSs and IVFSs equipollent of FSs, as illustrated by Atanssove and Gargov [3], who developed the theory of IVFSs, which has been widely studied and used [4‐8]. Type-2FSs, which are presented by enrollment work with more boundaries, provide the fuzzy enlistment as a FS, which promotes the showing performance over the previous. In term of science, IFSs are a type of Type-2FS in which the participant’s work restores a large number of new stretches. Despite their widespread use [9‐12], Type-2FS face challenges in expanding optional enrollment capacity and maintaining management [13‐15]. This difficulty was resolved by rickard [16], who provides an elective definition that highlights the importance of such a commutative quality both a set action as well as \(\alpha \)-cut. The first HFSs were introduced by Tora [4]. The concept of HF data is also more dependable and efficient for dealing with complicated and ambiguous information in actual situation. Tora [4] examined IFSs and FMSs, drew illustration, and identified commonalities between them. Given that the final values from both sets assume the structure of a finite subset unit interval, theories of HF data and fuzzy mustiest are the same concept. When a hesitant fuzzy set is a nonempty shut span, HFSs are values as IFSs. A definition of HFS envelope was provide by Tora [4], in view of the relationship between IFSs and hesitant fuzzy sets. Amalgam parameters, separation, as well as comparability measurements for HFSs, were examined bu XP and Xi [18] and used to solve DM issues [19‐21].
Relative work
Furthermore, Thu et al. [22] investigate the DHFS, which include special instance such as FSs, IFSs, HFSs, and fuzzy multi sets. The membership hesitancy function and non-membership hesitancy function are the two functions used to describe DHFSs. Distance and similarity indicators of DHFSs were defined by Wang et al. [23] with their application to MADM. We have a more exemplary and flexible way to assign value to each value in the domain when we take into account such function. Instead of a collection of exact numbers, Farhadinia [24] presented a DHFSs and DIVHFSs, whose key characteristic is that the outcome of the member ship function and non-member ship function is a set of interval values. This was done in the spirit of what has been for IVFSs. The effect of converting the fuzzy ranges into a unit disc on the complex plane has been questioned by certain academics. Ramot et al. [25], introduce the idea of complex fuzzy set (CFS), which contain truth values given as complex numbers by member of unit discs in the complex plane. In a single set, CFS works with two dimensions. CFS is an effective method for demonstrating how people believe that grades are formed. Only the degree of membership is taken into account by the complex fuzzy set; the non-membership components of the data elements, which play an equivalent role in the decision-making process evaluation of the object are not taken into consideration. Although it is frequently challenging to describe the assessment of the member ship degree by an accurate number in a fuzzy collection in the real world. In certain circumstances, it could be simpler to represent ambiguity and the reality of uncertainty using two dimensions of information rather than just one. An expansion of the may be quite beneficial to display the doubts of his or her hesitant judgement in the complex decision-making issues. It has been used by many researcher in a variety of environments.
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In real life situation, we frequently encounter situation where we must assess the information degree of vulnerability to make the best decisions. Significant tools for dealing with shaky found in our daily life concerns include exponential-based similitude measurement and without exponential-based comparability measures. The data is processed using HFS a number of measures, including, similitude, separation, entropy, and disincorporation, which enable us to reach to conclusion. Due to their extensive application in several domains, including as design recognition, clinical determination, grouping examination, and image portion, these measure have recently attracted a lot of attention from countless creators. All existing chiefs approach, based on exponential-based proximity measurement and without exponential-based similarity measure in FS, CFS, and hesitant fuzzy hypotheses, manage involvement capabilities with a measurement were obtained is a subset in the idea.
A brief history related to our example
The ongoing confrontation between Russia and Ukraine is known as the "Russia and Ukraine war" (along with pro-Russian rebel troops). It was begun by Russia and focused initially on the status of Donbas and Crimea, which are both acknowledged by world organizations as being a part of Ukraine. It occurred after the Ukrainian movement of dignity in February 2014. Political tension, cyberwarfare, and marine incident, the invasion of Crimea by Russia in 2014, and fighting in the Donbas between Ukrainian force and separatists supported by Russia all happened during the first eight years of the conflict (2014–present). Following a force build-up on Russia–Ukraine line that started in late 2021, the conflict grew quickly when Russia started its war of Ukraine on 24 February, 2022.
Pro-Russian unrest broke out across the entire country of Ukraine in the wake of February 2014 Eurmaidan protest and revolution took place out side the office of pro-Russian president Viktor Yanukovych. The Crimean parliament and important infrastructure in the Ukraine part of Crimea were taken over by Russian force who were not wearing any insignia. Crimea joined Russia as a result of Russians sponsoring a contentious referendum. Crimea was consequently annexed. Pro-Russian protests that broke out in April 2014 in Donbas led to a clash between the Ukraine military and insurgents from the self-declared Donetsk and Luhansk republic, who had Russian support.
Russian military trucks that were unidentified entered Donetsk state in August 2014. A conflict broke out in Palestine between rebels and Russian troops on the one side and Ukrainian force on the other side, despite Russia’s considerable effort to conceal its involvement. Because of the failure of the peace agrement, the war evolved into a defensive conflict. As of 2019, the Ukrainian government had declared \(7\%\) of its land to be temporarily invaded.
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Russia significantly increased its military build presence along Ukraine’s border in 2021 and beginning of 2022. NATO claimed that Russia was getting ready to invade, but Russia denied this. Valdimar Putin, President of Russia, demanded that Ukraine never join the military alliance and criticised NATO expansion as a risk to his country. Further, he spread radical view points, cast doubt on Ukraine’s sovereignty, and made flimsy claims that Vladimir Lenin created the nation. Russia sent soldiers to the two self-declared separatist administration in Donbas on February 21,\(\ 2022\), to formally designate them. Following three days, Russia invaded Ukraine. The majority of theglobal community condemned Russia for in its action in Ukraine, accusing it of understanding that nation’s sovereignty and disobeying international law. Following the Russia invasion, many nation imposed economic sanctions against Russia, Russia citizens or Russia businesses.
Motivation of the study
The CDHFS theory challenges the traditional approaches to evaluating enrollment and dishonesty levels, proposing a more comprehensive and robust framework. In this theory, decision-makers provide information expressed as complex numbers restricted to the unit interval, with the real and imaginary parts added together. However, existing notations often fail to capture the complexities of this approach.
CDHFS offers several advantages over previous concepts, including its ability to convert into DHFS or CIFS depending on the scenario. By utilizing information measures such as similarity, distance, entropy, and inclusion, we can effectively manage and process ambiguous information in real-world decision-making scenarios.
The CDHFS theory provides valuable insights into how we can measure and evaluate enrollment and dishonesty levels more effectively. Its innovative approach challenges traditional thinking and offers a more nuanced understanding of complex decision-making processes.
Contribution of the study
The paper makes several significant contributions to the field of decision-making using complex dual hesitant fuzzy (CDHF) theory.
First, the paper proposes new CDHF operational laws, divergence measure, score, and accuracy function, which utilize more original information. These novel methods offer a more comprehensive and nuanced approach to handling complex and uncertain information.
Second, the paper explores new CDHF multi-criteria decision-making (MCDM) methods that use arithmetic aggregation operators. These methods provide a more efficient and effective way of evaluating multiple criteria and making informed decisions.
Third, the paper uses the TODIM technique to investigate new CDHFMCDM methods. TODIM is a powerful tool for representing dominance degrees, which is essential for decision-making in complex and uncertain environments.
Fourth, the paper applies these techniques in the context of the impact of the Russia–Ukraine war on the global economy. This demonstrates the real-world applicability and relevance of the proposed methods.
Fifth, the paper compares the new CDHFMCDM method to existing methods to assess their validity and reliability. This comparative analysis provides valuable insights into the strengths and weaknesses of the different methods.
Finally, the paper provides a detailed discussion of the benefits of the proposed methods, along with graphical illustrations and comparative analysis. This enhances the understanding and accessibility of the proposed methods for decision-makers and researchers alike.
The remainder of this paper is structured as follows to help us process our discussion: Section Basic concepts introduces the fundamental concepts of HFS, DHFS, CF, CIFS, CDHFS, and hesitant normalized Euclidean distance. Section Proposed operational laws, divergence measure, score and accuracy function examine the operational laws, divergence measure, score and accuracy function. In Section Proposed complex dual hesitant fuzzy aggregation operators, a new proposed complex dual hesitant arithmetic aggregation operators. Section The description of issue, introduces the new CDHFS, MCDM method based on arithmetic aggregation operators and TODIM method, and the sensitive analysis constructed. Section Application in Russia–Ukraine war’s impact on global economy developed a numerical example based on arithmetic aggregation operators and TODIM method for selection of the impact on the global economy due to Russia–Ukraine war’s. Section Results and discussion explore the comparison of proposed arithmetic aggregation operators and TODIM method with some existing MCDM method. Section Conclusion contains some conclusions.
Table 1
Abbreviation
Name
Abbreviation
Name
Abbreviation
Fuzzy set
FS
Complex fuzzy set
CFS
Intuitionistic fuzzy set
IFS
Complex intuitionistic fuzzy set
CIFS
Hesitant fuzzy set
HF
Complex dual hesitant fuzzy set
CDHFS
Dual hesitant fuzzy set
DHF
Multi-criteria decision-making
MCDM
Decision-making
DM
Multi-criteria group decision-making
MCGDM
Basic concepts
The definition of HFS, DHFS, CF, CIFS, CDHFS, and hesitant normalized Euclidean distance across on universal set \(\breve{G}\) are reviewed in this section.
Let \(\breve{G}\) be the universe of discourse [4]. Then,
is defined as the HFS of \(\breve{G}\), where \(\mu _{\mathring{a}}(\breve{g} ) \) is referred to as be the membership function of the component \(\breve{g} \in X\) to the set \(\mathring{A}\), respectively, under the boundary condition that \(\mu _{\mathring{a}}(\breve{g})\rightarrow \left[ 0,1\right] ,\forall \breve{g}\in \breve{G}\).
Assume that \(\breve{G}\) is the discourse of universe [22]. Then,
is defined as the DHFS of \(\breve{G}\), where \(\mu _{\mathring{a}}(\breve{g} )\) and \(\nu _{\mathring{a}}(\breve{g})\) are said to be the membership and non-member ship degree of the element \(\breve{g}\in \breve{G}\) to the set \( \mathring{A}\), respectively, with condition:
is said to be CFS of \(\breve{G}\), where \(\alpha _{a}(\breve{g}):\breve{G} \rightarrow \left( a:a\in \breve{G},\left| a\right| \le 1\right) \) is a complex valued membership function defined as \(\alpha _{a}(\breve{g} )=\mu _{\mathring{a}}(\breve{g})e^{i2\pi \varpi _{\mu _{a}(x)}}\) where \( i=\surd -1,0\le \mu _{\mathring{a}}(\breve{g}),\varpi _{\mu _{\mathring{a}}( \breve{g})}\le 1.\)
Let \(\breve{G}\) be universe of discourse [44]. Then,
is said to CIFS of \(\breve{G}\), where \(\alpha _{\mathring{a}}(\breve{g} ),\beta _{\mathring{a}}(\breve{g}):\breve{G}\rightarrow \left( a:a\in \breve{ G},\left| a\right| \le 1\right) \) define as
are termed as complex-valued membership function and non-membership functions respectively. The pair of \(\breve{G}=((\mu ,\varpi _{\mu }),(\nu ,\varpi _{\nu }))\) and called as CIFN. where \(0\le \mu ,\nu ,\mu +\nu ,\varpi _{\mu },\varpi _{\nu },\varpi _{\mu }+\varpi _{\nu }\le 1.\)
Assume that \(\breve{G}\) the discourse of universe [43]. Then,
is defined as the CDHFS of \(\breve{G}\), where \(\alpha _{a}(\breve{g} ),\beta _{a}(\breve{g}):\breve{G}\rightarrow \left( a:a\in \breve{G},\left| a\right| \le 1\right) \) define as
denoting the possible membership function and non-membership function of the element \(\breve{g}\in \breve{G},\) to the set \(\mathring{A}\) respectively, with a condition:
for \(k=(1,2,\ldots ,m)\), and \(l=(1,2,\ldots ,n)\). Further \(\mu _{\mathring{a}_{k}}( \breve{g})e^{i2\pi \varpi _{\mu _{\mathring{a}_{k}}(\breve{g})}},\nu _{ \mathring{a}_{l}}(\breve{g})e^{i2\pi \varpi _{v_{\mathring{a}_{l}}(\breve{g} )}}\) is called complex dual hesitant fuzzy number(CDHFN).
Suppose that \(\rho _{i1}=(\varphi _{\rho _{i}})\) and \(\rho _{i2}=(\xi _{\rho _{i}})\) be any two HFSs, on a universal set \(\breve{G}\). Then [44] a normalized hesitant Euclidean distance is determined as
where \(\varphi _{\rho _{i1}\sigma (j)}\), and \(\xi _{\rho _{i2}\sigma (j)}\) are the jth greatest values in \(\rho _{i1}\) and \(\rho _{i2}\), respectively.
Proposed operational laws, divergence measure, score and accuracy function
For complex dual hesitant fuzzy sets, we describe several operational laws, divergence measure, score function, and accuracy function in this section.
1.
Suppose that \(\rho _{i}=\left( \varphi _{\rho _{i}},\xi _{\rho _{i}}\right) (i=1,2)\) are any two complex dual hesitant fuzzy elements, where \(\varphi _{\rho _{i}}=\mu _{\rho _{i}}e^{i2\pi \varpi _{\mu _{\rho _{i}}}}\) and \(\xi _{\rho _{i}}=\nu _{\rho _{i}}e^{i2\pi \varpi _{v_{\rho _{i}}}}\) on a universal set \(\breve{G}\) and \(\lambda >0\). Then some basic operational laws are define as;
Let \(\rho _{i1}=(\varphi _{\rho _{i}},\xi _{\rho _{i}}),\) and \(\rho _{i2}=(\varphi _{\rho _{i}},\xi _{\rho _{i}})\) be any two CDHFSs, where \( \varphi _{\rho _{i}}=\mu _{\rho _{i}}e^{i2\pi \varpi _{\mu _{\rho _{i}}}}\) and \(\xi _{\rho _{i}}=\nu _{\rho _{i}}e^{i2\pi \varpi _{v_{\rho _{i}}}}\). Then divergence measure is define as
where \(\mu _{i1\sigma (j)}\), \(\mu _{i2\sigma (j)}\), \(\varpi _{\mu _{i1}\sigma (j)}\), \(\varpi _{\mu _{i2}\sigma (j)}\), \(\nu _{i1\sigma (j)}\), \( \nu _{i2\sigma (j)}\), \(\varpi _{\nu _{i1}\sigma (j)}\), \(\varpi _{\nu _{i2}\sigma (j)}\) are the jth largest values in \(\rho _{i1}\) and \(\rho _{i2}\) respectively.
Let \(\rho _{i}=\left( \varphi _{\rho _{i}},\xi _{\rho _{i}}\right) (i=1,2)\), be any two complex dual hesitant fuzzy elements, where \(\varphi _{\rho _{i}}=\mu _{\rho _{i}}e^{i2\pi \varpi _{\mu _{\rho _{i}}}}\), and \(\xi _{\rho _{i}}=\nu _{\rho _{i}}e^{i2\pi \varpi _{v_{\rho _{i}}}}\). Then Score of \( \rho _{i}\) is define as
is said to be complex dual hesitant fuzzy weighted average (CDHFWA) operator, while \(\Lambda =(\Lambda _{1},\Lambda _{2},\ldots ,\Lambda _{n})^{n}\) is a vector of weights for \(\left( \rho _{j}\right) (j=1,2,\ldots ,n)\), and \( \Lambda _{j}>0\), \(\underset{j=1}{\overset{n}{\sum }}\Lambda _{j}=1.\)
We can derive theorem 1, by using the operation of complex dual hesitant fuzzy values described.
Theorem 1
If \(\left( \rho _{j}\right) (j=1,2,\ldots ,n)\) is a set of CDHFEs, the aggregate of those values obtained by applying the (CDHFWA) operator is also a CDHFE, and
while \(\Lambda =(\Lambda _{1},\Lambda _{2},\ldots ,\Lambda _{n})^{n}\) is a vector of weights for \(\left( \rho _{j}\right) _{(j=1,2,\ldots ,n)}\), and \( \Lambda _{j}>0,{\sum }^n_{j=1}\Lambda _{j}=1.\)
The following characteristics of the CDHFWA operators are clearly provable.
Theorem 2
(Idempotency): If all \(\left( \rho _{j}\right) (j=1,2,\ldots ,n)\) are equal, that is, if \(\rho _{j}=\rho \) for all j. Then
(Monotonicity): Let \(\left( \rho _{j}\right) (j=1,2,\ldots ,n)\) and \( \left( \rho _{j}\right) (j=1,2,\ldots ,n)\) are two sets of CDHFEs, and let \( \rho _{j}\le \rho _{j}^{ {\acute{}} }\) for all j. Then
where \((\sigma (1),\sigma (2),\ldots ,\sigma (n))\) is a permutation of \( (1,2,\ldots ,n)\) such that
\(\rho _{\sigma (j-1)}\ge \rho _{\sigma (j)}\) for all \(j=1,2,\ldots ,n,\) and \( \Lambda =(\Lambda _{1},\Lambda _{2},\ldots ,\Lambda _{n})^{n}\) is the weight
vector of \(\left( \rho _{j}\right) (j=1,2,\ldots ,n)\) such that \(\Lambda _{j}>0, {\sum ^n_{j=1}}\Lambda _{j}=1.\)
We can derive theorem 5, by using the operation of complex dual hesitant fuzzy values described.
Theorem 5
Let \(\left( \rho _{j}\right) (j=1,2,\ldots ,n)\) be a collection of CDHFEs. Then their aggregated value by using the (CDHFOWA) operator is also a CDHFE, and
where \([\sigma (1),\sigma (2),\ldots ,\sigma (n)]\) is a permutation of \( [1,2,\ldots ,n],\) such that
\(\rho _{\sigma (j-1)}\ge \rho _{\sigma (j)}\) for all \(j=1,2,\ldots ,n,\) and \( \Lambda =(\Lambda _{1},\Lambda _{2},\ldots ,\Lambda _{n})^{n}\) is the weight
vector of \(\left( \rho _{j}\right) (j=1,2,\ldots ,n)\) such that \(\Lambda _{j}>0\), \({\sum ^n_{j=1}}\Lambda _{j}=1.\)
The following characteristics of the CDHFOWA operator are clearly demonstrable.
Theorem 6
(Idempotency) If \(\left( \rho _{j}\right) (j=1,2,\ldots ,n)\) are all equal, i.e. \(\rho _{j}=\rho \) for all j. Then
(Monotonicity) Let \(\left( \rho _{j}\right) (j=1,2,\ldots ,n)\) and \( \left( \rho _{j}\right) (j=1,2,\ldots ,n)\) are two sets of CDHFEs, and \(\rho _{j}\le \rho _{j}^{ {\acute{}} }\) for all j. Then
where \(\Lambda =(\Lambda _{1},\Lambda _{2},\ldots ,\Lambda _{n})\) is the weight vector with \(\Lambda _{j}\in [0,1],{\sum \nolimits _{j=1}^{n} } \Lambda _{j}=1\), and \(\overset{.}{\varphi _{\sigma (j)}}\) is the jth greatest element of the CDHF arguments \(\overset{.}{\rho _{\sigma (j)}} \left\{ \overset{.}{\rho _{\sigma (j)}}=n\Lambda _{j}\rho _{\sigma (j)}\right\} (j=1,2,\ldots ,n),\Lambda =(\Lambda _{1},\Lambda _{2},\ldots ,\Lambda _{n})\) is the weighted vector of CDHF arguments \(\left( \rho _{j}\right) _{(j=1,2,\ldots ,n)}\), with \(\Lambda _{i}\in [0,1],{{\sum \nolimits _{i=1}^{n} }}\Lambda _{i}=1,\) and n is the balancing coefficient. If \( \Lambda =(\frac{1}{n},\frac{1}{n},\ldots ,\frac{1}{n})^{T},\) then CDHFHA is reduced to the CDHFWA operator; if \(\Lambda =(\frac{1}{n},\frac{1}{n},\ldots , \frac{1}{n})\), then CDHFHA is reduced to the CDHFOWA operator.
The description of issue
Consider that \(\Re =\left\{ \Re _{1},\Re _{2},\ldots ,\Re _{n}\right\} \) is a collection of alternatives, \(C=\left\{ C_{1},C_{2},\ldots ,C_{n}\right\} \) is a discrete set of criteria, and \(\Lambda =\left\{ \Lambda _{1},\Lambda _{2},\ldots ,\Lambda _{n}\right\} \) is a vector of weights with \(\Lambda _{j} \in \left[ {0,1} \right] ,\mathop {\mathop \sum \nolimits _{{j = 1}} }\limits ^{n} \Lambda _{j} = 1 \). consider that the assessment of alternative \(\Re _{i}\) in term of criteria \( C_{j}\) is represented by complex dual hesitant fuzzy (CDHF) matrix \( P=(p_{ij})_{m\times n}=(\varphi _{p_{ij}},\xi _{p_{ij}})_{m\times n}\), which is presented in Table 2.
Table 2
The complex dual hesitant fuzzy MCDM matrix
\(C_{1}\)
\(C_{2}\)
\(\cdots \)
\(C_{n}\)
\(\Re _{1}\)
\((\varphi _{p_{11}},\xi _{p_{11}})\)
\((\varphi _{p_{12}},\xi _{p_{12}})\)
\(\cdots \)
\((\varphi _{p_{1n}},\xi _{p_{1n}})\)
\(\Re _{2}\)
\((\varphi _{p_{21}},\xi _{p_{21}})\)
\((\varphi _{p_{21}},\xi _{p_{21}})\)
\(\cdots \)
\((\varphi _{p_{2n}},\xi _{p_{2n}})\)
\(\vdots \)
\(\vdots \)
\(\vdots \)
\(\ddots \)
\(\vdots \)
\(\Re _{n}\)
\((\varphi _{p_{m1}},\xi _{p_{m1}})\)
\((\varphi _{p_{m2}},\xi _{p_{m2}})\)
\(\cdots \)
\((\varphi _{p_{mn}},\xi _{p_{mn}})\)
Complex dual hesitant fuzzy MCDM method based on CDHF arithmetic aggregation operators
The \(P_{ij}=(i=1,2,\ldots ,;j=1,2,\ldots ,n)\) is a representation of the preferred values of the ith alternative in accordance with the jth criterion. The value of \(\Lambda _{j}\) denotes the weight of the jth criteria. Furthermore, the set of cost and benefit criteria’s are represented by C and B.
Step-1 Determine the matrix \(L=(l_{ij})_{m\times n}\) of linguistic term by Table 3.
Table 3
For assessing alternatives the values of linguistic terms
where \(\Lambda _{j}\) represent the weight vector of each criteria, and \( \Lambda _{r}=\max (\Lambda _{1},\Lambda _{2},\ldots ,\Lambda _{n})\) and \(0\le \Lambda _{r}\le 1\).
Step-4 Using the following mathematical model, determine the dominance degree of \(\Re _{i}\) over alternative \(\Re _{j}\) using Eqs. (5.2) and (3.1) depending on each criterion \(C_{j}\).
Where \(d(\rho _{i1,}\rho _{i2})\) is a divergence measure between two CDHFS, \(\rho _{i1}\) and \(\rho _{i2}\) and \(\Phi >0\) describe the attention factor.
Step-5 Calculate the dominance degree matrix of each alternative \(\Re _{i}\) based on the preceding criteria \(C_{j}\) are following:
Step-6 Use the criteria \(C_{j}\) to calculate the overall dominance degree of each alternative \(\Re _{i}\) with regard to another alternative \( a_{p}(p=1,2,\ldots ,m)\):
Step-8 Rank the values according to the previous step.
Fig. 1
Flow chart
×
Application in Russia–Ukraine war’s impact on global economy
In this section, we discuss the impact on the global economy due to Russia and Ukraine war. Due to the war between Russia and Ukraine, the system in the countries of the world have been disrupted. As exports and imports, travel environments, raise prices,raise prices of many different things such as edible oil, sunflower oil, and many other system are effected. In this section, we use the proposed method and see which countries have suffered a lot due to the war between Russia and Ukraine. Now consider the decision makers to evaluate and select the most important impact on the global economy among five criteria are given as "Energy \((C_{1})\)", "Transport \((C_{2})\)", "Supply Chain \((C_{3})\)", "Edible oil \((C_{4})\)", and "Food Supplies \((C_{5})\)", in term of five alternatives such as "Australia \((\Re _{1})\)", " Europe\((\Re _{2})\)", "North America \((\Re _{3})\)", "South America \((\Re _{4})\)", and "Asia \((\Re _{5})\)", are decision events. we consider that weight vectors is "\(\Lambda _{1}=0.25\)", "\(\Lambda _{2}=0.25\)", "\(\Lambda _{3}=0.20\)", "\( \Lambda _{4}=0.20\)", and "\(\Lambda _{5}=0.10\)", for decision makers.
\(\mathbf {(1):}\) Energy
Through a number of crucial pipelines, several European nations are reliant on Russian energy, particularly gas. The harsh economic sanctions placed on Russia may make it extremely difficult for these countries to purchase gas, even if the crisis is resolved. Oil prices surged on February 2022, as supply disruption increased as a result of sanctions against Russian bank, and traders rushed to find other oil sources in a competitive market. Brent oil future rose by almost \(\$8\), peaking at \(\$113.02\) a barrel, the highest since June 2014, before falling to \(\$111.53\) up \(\$6.53\) or 6.3 percent at 0950 GMT. Additionally, US west Texas (WTI) oil futures increased by more than per \(\$8\) barrel, touching their maximum level since August 2013, until losing sum of their strength and trading up \(\$6.39\) or 6.2 percent to \(\$109.80\) per barrel.
The penamic’s aftermath has badly interrupted worldwide transportation, and the conflict is expected to make matter worse. Rail freight and maritime shipments are the likely effected modes of transportation. Rail transports only a small portion of total freight between Asia and Europe, but it has proven essential during recent interruption in transportation, and it is continually growing.
\(\mathbf {(3):}\)Supply Chain
Companies were struggling to locate sufficient raw resources and elements to make items to fulfil the soaring customer demand as a result of the surprisingly rapid global rebound from the pandemic recession. Lack of supplies, shipping delays, and increased costs have been caused by overburdened manufacturers, ports, and freight yards. Any restoration to normalcy could be hundered by disruptions to the Russian and Ukrainian economies.
\(\mathbf {(4):}\)Edible Oil
Nearly half of sunflower oil exports come from just Ukraine. If harvesting and processing are hindered in war-torn Ukraine or if export are outlawed, importer will struggle to replenish the supply. Due to the grave possibility of supply disruptions, businesses in Asia little choice but to think about raising the price of the daily-used edible oil within a few weeks. Leading domestic procedures of edible oil claim that imports satisfy more than \(70\%\) of India’s demand for crude edible oil.
\(\mathbf {(5):}\)Food Supplies
Ukraine and Russia provide \(20\%\) of the world corn, \(30\%\) of wheat, and \( 80\%\) of sunflower oil used in food processing. According to the associated press a large portion of Russian and Ukrainian rewards is sent to impoverished, unstable nation like Yemen and Libya. The threat to farms in eastern Ukraine and the restriction of export through black sea port could lead to a decrease in food supplies at a time when food price are at their highest level sice 2011, several nations are experiencing a food crisis.
The chart presented here encapsulates all of the proposed work discussed in this article. It provides a clear overview of the various components and methodologies presented in our research, allowing readers to easily grasp the scope and significance of our contributions (Fig. 1).
Table 6
The alternative values of CDHFWA, CDHFOWA, CDHFHA operator
\(\Re _{1}\)
\(\Re _{2}\)
\(\Re _{3}\)
\(\Re _{4}\)
\(\Re _{5}\)
CDHFWA
\( -0.0277\)
\( -0.1652\)
0.0382
0.0318
0.0578
CDHFOWA
\( -0.0124\)
\( -0.1554\)
0.0533
0.0182
0.1001
CDHFHA
0.1062
0.0715
0.1055
0.1461
0.3216
Table 7
The ranking of the alternative values
CDHFWA
\(\Re _{5}>\Re _{3}>\Re _{4}>\Re _{1}>\Re _{2}\)
CDHFOWA
\(\Re _{5}>\Re _{3}>\Re _{4}>\Re _{1}>\Re _{2}\)
CDHFHA
\(\Re _{5}>\Re _{4}>\Re _{1}>\Re _{3}>\Re _{2}\)
Fig. 2
Ranking of aggregation operators
Table 8
Dominance degree under criterion \(C_{1}\)
Criteria\((C_{1})\)
\(\Re _{1}\)
\(\Re _{2}\)
\(\Re _{3}\)
\(\Re _{4}\)
\(\Re _{5}\)
\(\Re _{1}\)
0
\( -0.1527\)
\( -0.1527\)
\( -0.1428\)
\( -0.1124\)
\(\Re _{2}\)
0.1909
0
0
\( -0.1124\)
\( -0.1124\)
\(\Re _{3}\)
0.1909
0
0
\( -0.1124\)
\( -0.1124\)
\(\Re _{4}\)
0.1786
0.1409
0.1405
0
0
\(\Re _{5}\)
0.1786
0.1409
0.1405
0.0000
0
×
Example \({\textbf{1}}\): CDHF-MCDM method based on arithmetic aggregation operator
Decision-maker(expert) can first use the linguistic term information, to analyse the relative performance of alternative with each criteria as shown in Table 4.
Step-1 Table 4 contains the linguistic terms matrix \( L=(l_{ij})_{5\times 5}\).
Step-2 Convert the matrix L of linguistic terms into CDHF matrix P are given in Table 5.
Step-3 There is no need to convert because it is a benefit attribute.
Step-4 Determine the alternative values of CDHFWA, CDHFOWA, CDHFHA operator.
Step-5 Choose the best option based on how well each of the alternatives \(\Re _{i}(i=1,2,\ldots ,m)\) ranked using the score function.
Our plot displays the values of the alternative on the x-axis and the boundary of the score function on the y-axis. This allows us to examine how the score function changes with different values of the alternative. By visualizing this relationship, we can gain a deeper understanding of the performance of different alternatives under varying score conditions (Fig. 2).
Example \({\textbf{2}}\): CDHF-TODIM method
Step-1 Table 4 contains the linguistic terms matrix \( L=(l_{ij})_{5\times 5}\).
Step-2 Convert the matrix L of linguistic terms into SF matrix P in Table 5.
Step-3 There is no need to convert because it is a benefit attribute.
Step-4 Determine the dominance degree of each criterion by Eqs. (5.2) and (3.1) as;
Step-5 Compute the totally dominance degree by Eqs. ( 5.5) and (5.6) as;
Step-6 For each alternative, determine the positive-ideal solution \( (PI^{+})\) as follows;
Table 9
Dominance degree under criterion \(C_{2}\)
Criteria \((C_{2})\)
\(\Re _{1}\)
\(\Re _{2}\)
\(\Re _{3}\)
\(\Re _{4}\)
\( \Re _{5}\)
\(\Re _{1}\)
0
0.1786
0.1786
0.1405
0
\(\Re _{2}\)
0.1786
0
0
0.1909
\( -0.1418\)
\(\Re _{3}\)
\( -0.1428\)
0
0
\( -0.1527\)
\( -0.1427\)
\(\Re _{4}\)
\( -0.1124\)
0.1909
0.1909
0
\( -0.1428\)
\(\Re _{5}\)
0
0.1786
0.1786
0.1405
0
Table 10
Dominance degree under criterion \(C_{3}\)
Criteria \((C_{3})\)
\(\Re _{1}\)
\(\Re _{2}\)
\(\Re _{3}\)
\(\Re _{4}\)
\( \Re _{5}\)
\(\Re _{1}\)
0
\( -0.1564\)
\( -0.1777\)
\( -0.1229\)
0
\(\Re _{2}\)
0.1564
0
0.1118
0.1509
0.1564
\(\Re _{3}\)
0.1777
0
0
0.1663
0.1777
\(\Re _{4}\)
0.1229
0.1229
0.0000
0
\( -0.1188\)
\(\Re _{5}\)
0
\( -0.1564\)
\( -0.1777\)
\( -0.1293\)
0
The x-axis in our plot represents the values of the positive ideal solution, while the y-axis represents the boundary condition. This plot provides a clear visualization of how the boundary condition varies with different values of the positive ideal solution (Fig. 3).
Comparative analysis
In this section, we compare our proposed MCDM method aggregation operators and TODIM method with the other existing MCDM method.
Comparison analysis of the proposed MCDM method based on Aggregation operators with the existing MCDM method The comparison analysis of proposed MCDM techniques based on aggregation operators with the existing MCDM method as shown in Table 15.
Table 11
Dominance degree under criterion \(C_{4}\)
Criteria \((C_{4})\)
\(\Re _{1}\)
\(\Re _{2}\)
\(\Re _{3}\)
\(\Re _{4}\)
\( \Re _{5}\)
\(\Re _{1}\)
0
\( -0.1509\)
\( -0.1613\)
0.1293
\( -0.1613\)
\(\Re _{2}\)
0.1509
0
0.1188
0.1564
0.1188
\(\Re _{3}\)
0.1613
\( -0.1188\)
0
0.1777
0
\(\Re _{4}\)
0
\( -0.1269\)
\( -0.1564\)
0.0000
\( -0.1777\)
\(\Re _{5}\)
0.1613
\( -0.1188\)
0
0.1777
0
Table 12
Dominance degree under criterion \(C_{5}\)
Criteria \((C_{5})\)
\(\Re _{1}\)
\(\Re _{2}\)
\(\Re _{3}\)
\(\Re _{4}\)
\( \Re _{5}\)
\(\Re _{1}\)
0
\( -0.2114\)
\( -0.1861\)
\( -0.2114\)
\( -0.1861\)
\(\Re _{2}\)
0.1057
0
\( -0.1680\)
0
\( -0.1680\)
\(\Re _{3}\)
0.0930
0.0840
0.0000
0.0840
0
\(\Re _{4}\)
0.1057
0
\( -0.1680\)
0.0000
\( -0.1680\)
\(\Re _{5}\)
0.0930
0.0840
0
0.0840
0
Step-7 Rank the values according to the previous step. We plot the values obtained using our Aggregation operators method against those obtained using other existing methods on the x-axis, while the y-axis represents the boundary condition. This plot allows us to visually compare the performance of our method with other approaches in terms of the boundary condition. Consequently, the proposed method is accurate, effective and more generalizable as a tool to solve MCDM problems (Fig. 4).
Comparison analysis of the proposed TODIM method with the existing MCDM method The comparison analysis of the proposed TODIM method with the existing MCDM method as shown in Table 16.
We plot the values obtained using our TODIM method against those obtained using other existing methods on the x-axis, while the y-axis represents the boundary condition. This plot allows us to visually compare the performance of our method with other approaches in terms of the boundary condition. Consequently, the proposed method is accurate, effective and more generalizable as a tool to solve MCDM problems (Fig. 5).
Results and discussion
The proposed aggregation technique focuses on the bounded rationality of decision-makers, which allows for optimal leverage of their benefits. In contrast, recent techniques depend solely on the performance metric of the criteria. The proposed approach has several advantages over other methods, including the prevention of information distortion or loss since the alternative ranking solely depends on the score function. The complex dual hesitant fuzzy sets (CDHFSs) used in the approach are an excellent choice for complex decision-making issues since they combine multiple categories of assessment outcomes to represent the decision maker’s views. Additionally, the CDHFSs are built on four features, resulting in improved results compared to those obtained using hesitant fuzzy sets (HFS) and dual hesitant fuzzy sets (DHFSs). Although the proposed method does not consider competing standards due to the bounded rationality of decision makers, it effectively deals with situations involving contradictory parameters where decision-makers have insufficient logic and believe certain solutions to be optimal. As a result, the resulting consensus solution or ranking values are more reasonable and common.
Similarly, the TODIM technique focuses on the bounded rationality of decision-makers, allowing for optimal leverage of their benefits compared to other recent techniques. The proposed approach has several advantages, including the use of a divergence measure of parameters and a peculiar between alternatives, which helps prevent information distortion or loss. The benefits of TODIM are fully utilized in this approach. Although the TODIM approach does not consider competing standards due to the bounded rationality of decision-makers, it effectively deals with situations involving contradictory parameters and results in more reasonable and common consensus solutions or ranking values. Furthermore, the proposed TODIM method was designed to avoid the shortcomings of current methods and achieve more reliable outcomes in real-world problems involving instability, inconsistent parameters, and incorrect knowledge.
This article introduces a novel approach that integrates a new Aggregation operator and the TODIM method under CDHFSs. The main purpose of this research is to improve upon existing methods and provide more accurate and reliable results. With the use of original information, we proposed a new complex dual hesitant fuzzy (CDHF) operational laws, divergence measure, and accuracy function. Our novel \(CDHF-MCDM\) approach based on Aggregation operators outperforms existing methods in terms of versatility and accuracy. We also utilized the TODIM method to represent dominance degrees and calculate the Global dominance degree, resulting in a more comprehensive and efficient approach. The comparative analysis shows that our proposed TODIM approach is more effective in dealing with complex assessment problems in the realm of complex dual hesitant fuzzy sets. This research is a significant contribution to decision-making theory and pattern recognition and has potential applications in various fields.
Comparison of proposed TODIM method with other existing method
×
In the future, the proposed complex dual hesitant fuzzy sets approach has vast potential for future research and development, with applications in diverse fields such as decision-making, emergency decision-making problems, and beyond. Future research can focus on real-world scenarios and combining with other methodologies. Overall, this approach offers innovative and practical solutions for complex problem
Acknowledgements
General Program of Natural Funding of Sichuan Province (No.2021JY0108) and scientific Research Project of Neijiang Normal University (2022ZD10, 2021YB21, 18TD08).
Declarations
Conflicts of Interest
The authors declare that they have no conflicts of interest.
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