Abstract
Aggregation operators (AOs) are the most commonly used tools for the intuitionistic fuzzy (IF) information fusion, research into their construction methods and associated mathematical properties has already yielded abundant results. Indeed, the quality of information fusion is of paramount importance during the fusion process, as it directly affects the efficacy of the fusion task. However, the issue has drawn virtually no attention. Intuitively, an effective fusion tool should significantly reduce the uncertainty of fuzzy information and minimize information loss as much as possible. Moreover, information diffusion theory can, by means of a properly chosen diffusion function, produce a more accurate estimate of the relationships among fuzzy observations generated from an incomplete population than conventional statistical approaches. Inspired by these ideas to address this gap, we investigate measures for the quality of IF information fusion that integrate diffusion theory, and then explore a novel fusion method based on optimizing fusion quality. Empirical analysis demonstrates that our proposed approach significantly outperforms traditional AOs in enhancing the quality of information fusion.
Introduction
In the domains of artificial intelligence, pattern recognition, and decision making, the complexity of the objective world and the limitations of human cognitive abilities lead to the generation of a significant amount of uncertain and inaccurate information. To achieve the work objectives, integrating such information is unavoidable. Distinct from other fuzzy sets (FSs), IFS become a more appropriate tool for describing uncertain and inaccurate information by considering both the membership and non-membership degrees of the information [1]. Meanwhile, AOs are the most commonly used tools for aggregating fuzzy information. Therefore, many scholars have conducted extensive research on the AOs of IF information. Drawing on a variety of ideas, such as weighted averaging, t-norms and t-conorms, and the Choquet integral, numerous AOs have been proposed, and their mathematical properties have been analyzed.
Among AOs based on weighted averaging, the IF weighted geometric operator(IFWG) [2], the IF weighted averaging(IFWA) operator [3], the generalized IF weighted averaging operator [4], the IF geometric weighted Heronian mean (IFGWHM) operators [5], the scaled prioritized IF interaction AO and the scaled prioritized IF weighted interaction AO [6] belong to ordinary weighted-averaging operators. Their computation is straightforward, yet they ignore the relative importance of the fused data. To remedy this drawback, the AOs such as the IF ordered weighted geometric operator, the IF ordered weighted averaging (IFOWA) operator [3], the generalized IF ordered weighted averaging operator [4], the IFS generalized ordered weighted averaging (IFSGOWA) operator [7], the induced generalized IF ordered weighted averaging operator [8], the continuous interval-valued IF ordered weighted arithmetic averaging operator, the continuous interval-valued IF ordered weighted geometric averaging operator, the interval-valued IF ordered weighted averaging operator [9, 10], the intuitionistic trapezoidal fuzzy heavy ordered weighted averaging operator [11] and the triangular IF generalized ordered weighted averaging operator [12] adopt ordered weighted averaging method. The IF hybrid averaging (IFHA) operator [3], the IF hybrid geometric operator (IFHG) [2], the generalized IF hybrid averaging operator [4], the IF hybrid weighted arithmetic and geometric AO, the IF hybrid ordered weighted arithmetic and geometric AO [13] and various Einstein hybrid AOs [14, 15] go one step further by integrating both approaches. All these AOs possess desirable mathematical properties such as idempotency, boundary conditions, monotonicity, and commutativity.
Since t-norms and t-conorms inherently satisfy commutativity and associativity while also possessing monotonicity, they have naturally attracted scholars’ interest in using them to construct AOs. This also ensures that AOs such as the IF Einstein weighted geometric operator [16], the IF Einstein weighted averaging operator [17], the Aczel Alsina prioritized average (CPFAPA), the Aczel Alsina prioritized geometric (CPFAPG) operators, the Aczel Alsina prioritized weighted average (CPFAPWA) and the Aczel Alsina prioritized weighted geometric (CPFAPWG) operators for complex picture fuzzy information [18], the IF Aczel Alsina Heronian mean operator [19], various Hamacher AOs of interval-valued IF numbers, the Archimedean Bonferroni operators and Sugeno-Weber AOs for q-rung orthopair fuzzy numbers [20], the complex Pythagorean fuzzy yager AOs [21], the complex IF Hamacher AOs [22] and IF Aczel Alsina Hamy mean operators [23] also exhibit excellent mathematical properties.
Given that real-world decision criteria are frequently interdependent or interactive, directly employing traditional AOs based on additive measures to fuse fuzzy information is inadequate; hence, scholars have proposed operators that utilize Choquet integrals to deal with this issue, such as the IF Choquet integral (IFCCI) operator [24, 25], the Einstein-based IF Choquet averaging operator [14], the Einstein-based IF Choquet geometric operators [5], the generalized Shapley IF hybrid Choquet averaging (GSGIFHCA) operator [26], the Choquet integral-based IF arithmetic aggregation (CIIFAA) operator and the Choquet integral-based IF hybrid arithmetic aggregation (CIIFHAA) operator [27].
Constructed from distinct perspectives, these operators are virtually neck and neck when judged solely by their mathematical properties. Consequently, within an identical working scenario, rigorously judging whether the fusion output of one AO is superior or inferior to another is likely to be difficult.
Research Gap
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As stated above, existing studies have concentrated solely on constructing AOs and investigating their mathematical properties, completely neglecting the significance of information fusion quality. To date, no scholar has formally raised or examined this issue. Theoretically, the quality of information fusion determines the efficiency of fusion task. A fusion operator that yields poor fusion quality is meaningless, regardless of how desirable its mathematical properties may be.
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(2)
Efficient evaluation metrics are fundamental to analyzing fusion quality, yet existing research still lacks any discussion on measurable indicators for assessing the fusion quality of operators and the methods of fuzzy information fusion from the perspective of fusion quality optimization.
Our Contribution
How to obtain high quality fusion values is an important issue that needs to be addressed in information fusion. On the one hand, high-quality fusion requires reducing the uncertainty of information. The adoption of fusion methods that can reduce the uncertainty of complex information is beneficial for extracting valuable information more effectively, which means that the fusion method is effective for information processing. On the other hand, fusion should not lead to massive loss of original information. Otherwise, it may lead to significant deviation between the fusion results and the original information, resulting in serious inferential errors.
The main contributions of this article are:
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1.
This article examines the fusion quality of IF information from a fresh perspective, focusing on reducing both information uncertainty and information loss.
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2.
Drawing on the information diffusion theory, we have established evaluation indicators for IF information fusion quality from the perspectives of reducing uncertainty and minimizing information loss.
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3.
From the perspective of optimizing fusion quality, we have explored a new IF information fusion method.
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4.
Empirical analysis shows that our proposed method outperforms existing operators in terms of information fusion quality in multi-objective decisionmaking in IF environments.
The rest of the paper is organized as follows. Section “Preliminaries” reviews the requisite preliminaries. To mitigate information loss and uncertainty, Section “Quality Evaluation for IF Information Fusion” explores the measures for the quality of IF information fusion integrated with information-diffusion theory. Section “Proposed IF Information Fusion Method for Fusion Quality Optimization” proposes a novel information fusion method based on the optimization of fusion quality for IFVs information. Section “Comparative Analysis of Various AOs” validates the method through a randomized simulation, demonstrating its superior fusion quality relative to all existing AOs. Section “Application” provides a case study that confirms the superiority of the method over existing AOs in terms of fusion quality in multi-objective IF decision-making. Finally, Section “Conclusion and Future Research” concludes the paper and outlines future research directions.
Preliminaries
Intuitionistic Fuzzy Sets
Let \(X=\left\{ {{x}_{1}},\ldots ,{{x}_{n}} \right\} \) be a finite set. then an IFS on X is expressed by a mathematical symbol [1]:
Where the functions \({{\mu }_{{\tilde{A}}}}(x):X\rightarrow [0,1]\) and \({{\gamma }_{{\tilde{A}}}}(x):X\rightarrow [0,1]\) define the degree of membership and the degree of non-membership of the element \(x\in X\) to the IFS \(\tilde{A}\), and for every \(x\in X\):\(0\le {{\mu }_{{\tilde{A}}}}(x)+{{\gamma }_{{\tilde{A}}}}(x)\le 1\).
The IF index of the element x, \({{\pi }_{{\tilde{A}}}}(x)=1-{{\mu }_{{\tilde{A}}}}(x)-{{\gamma }_{{\tilde{A}}}}(x)\) shows the degree of hesitation of x to the IFS \(\tilde{A}\). The IFS will collapse into the fuzzy set when the IF index of each element is equal to 0. For convenience, we call \(\tilde{a}=({{\mu }_{{\tilde{a}}}},{{\gamma }_{{\tilde{a}}}})\) with \(0\le {{\mu }_{{\tilde{a}}}},{{\gamma }_{{\tilde{a}}}}\le 1\) and \({{\mu }_{{\tilde{a}}}}+{{\gamma }_{{\tilde{a}}}}\le 1\) an intuitionistic fuzzy value (IFV).
Information Diffusion Theory
Commonly in decision-making contexts, the information available is frequently insufficient. This insufficiency is attributed to the limitations of the decision-making framework and the cognitive constraints of those making the decisions. As a result, conventional statistical techniques often find it challenging to deliver precise predictions and assessments regarding the goals of decision-making when faced with such scant data. In fact, methods that focus on information diffusion have demonstrated their efficacy in overcoming these limitations. The principle of information diffusion guarantees the existence of reasonable diffusion functions to yield better estimates than non-diffusion when the given samples are incomplete.
From the perspective of information diffusion theory, fuzziness often arises from the incompleteness of a sample. In other words, when a given sample X is incomplete, there must exist some approach to extract fuzzy information from X for more accurate estimation of a relationship as a function approximation. Accordingly, applying information diffusion to fuzzy information can significantly enhance the accuracy of characterizing the distribution of such information. Scholarly evidence has established that these information diffusion methods excel in estimation accuracy compared to traditional approaches, including kernel estimation, particularly when information is incomplete.
Let \(X=\{{{x}_{1}},\cdots ,{{x}_{n}}\}\) be a sample and V be a subset of U. A mapping from \(X\times V\) to [0, 1]
is called an information diffusion of X on V, if it is decreasing: \(\forall {{x}_{i}}\in X\),\({{v}^{'}},{{v}^{''}}\in V\) , if \(\left\| {{v}^{'}}-{x}_{i} \right\| \le \left\| {{v}^{''}}-{x}_{i} \right\| \) then \(\mu (x,{{v}^{'}})\ge \mu ({x}_{i},{{v}^{''}})\). \(\mu \) is called a diffusion function and V is called a monitoring space. In short, \(\mu \) is called a diffusion.
Let \(X=\left\{ {{x}_{1}},\cdots {{x}_{n}} \right\} \) be a r-dimension random sample, and \(U=\left\{ {{u}_{1}},\cdots {{u}_{m}} \right\} \) be the chosen monitoring space, where \({{u}_{j}}=({{u}_{1j}},\cdots ,{{u}_{rj}}),j=1,\cdots ,m\). Then, the gained information \(p({{x}_{i}},{{u}_{j}})\) at the monitoring point \({{u}_{j}}\in U\) diffused from \({{x}_{i}}\in X\), in the normal diffusion process can be calculated by the diffusion function
Where \({{h}_{k}}\) is called k-th diffusion coefficient [28].
Correspondingly, for any \(x=({{x}_{1}},\cdots ,{{x}_{k}})\in {{R}^{k}}\) the normal diffusion estimation of the sample density function is:
Where
\({{h}_{k}}=\left\{ \begin{array}{ll} 0.6841{{\Delta }_{k}} & n=5 \\ 0.5404{{\Delta }_{k}} & n=6 \\ 0.4482{{\Delta }_{k}} & n=7 \\ 0.3839{{\Delta }_{k}} & n=8 \\ \frac{2.6851{{\Delta }_{k}}}{n-1} & n\ge 9 \\ \end{array} \right. \), \({{\Delta }_{k}}=\underset{1\le i\le n}{{\max }}\,\{{{x}_{ki}}\}-\underset{1\le i\le n}{{\min }}\,\{{{x}_{ki}}\}\).
Quality Evaluation for IF Information Fusion
In this section, as outlined in the introduction, we investigate evaluation indicators for assessing the quality of IFVs information fusion in terms of two critical dimensions: information loss and uncertainty.
Measurement of Information Loss in IFVs Information Fusion
The process of information fusion requires the refinement and condensation of raw information, which inevitably alters the quantity and structure of the original information, leading to the loss of information. If this loss is substantial, the fusion process might be considered suboptimal. Intuitively, if the fusion value deviates significantly from the original information, it implies that there is a considerable loss of information during the fusion process. Therefore, to gauge the extent of this information loss, we suggest employing the deviation between the fusion value and the original information as a metric.
Numerous academics have explored metrics for measuring distances within IFSs. Hung and Yang introduced the concept of the Hausdorff distance for IFSs [29]. Jiang, Huang, and their colleagues, as well as Garg and Rani, have developed innovative distance metrics grounded in triangular transformations [30, 31]. Mahanta and Panda proposed a distance metric for IFSs that accounts for the nonlinearity of the hesitation margin [32]. Notably, Garg and Rani have introduced a novel approach to distance measurement for IFSs, offering four distinct metrics that leverage the properties of triangular fuzzy numbers (TFNs) and the geometric centers of isosceles triangles, including centroids, circumcenters, incenters, and orthocenters. The superior discriminative power of the similarity measures derived from this distance, as compared to those established by other distance measures, underscores its superiority [33]. Motivated by this idea, we introduce a novel distance metric for IFVs. This approach is based on the transformation of TFNs and the construction of equilateral triangles, since in such triangles all four geometric centers coincide.
Let \(\tilde{a}=(u,v)\) be an IFV, we define the vertices of the transformed TFN \({{L}_{{\tilde{a}}}}\) relate to \(\tilde{a}\) as follows: \({{A}_{{{L}_{{\tilde{a}}}}}}=(u,v)\), \({{B}_{{{L}_{{\tilde{a}}}}}}=(u+\frac{\pi }{2},v+\tfrac{\sqrt{3}}{2}\pi )\),\({{C}_{{{L}_{{\tilde{a}}}}}}=(u+\pi ,v)\). Geometrically, these three vertices form an equilateral triangle with side length \({\pi }\). We can easily find the coordinates of centroids, circumcenters, incenters, and orthocenters:
Assume the original sequence is \(\underset{\scriptscriptstyle -}{\tilde{a}}=\left\{ {{{\tilde{a}}}_{1}}=({{u}_{1}},{{v}_{1}}),\cdots ,{{{\tilde{a}}}_{n}}=({{u}_{n}},{{v}_{n}}) \right\} \), and let the fusion value be \(\tilde{a}=\oplus {{\tilde{a}}_{i}}=(u,v)\). For each \(i=1,\cdots ,n\), we define the distance between \(\tilde{a}\) and \(\tilde{a}_{i}\) as
where \(\pi _{i}=1-\mu _{i}-v_{i}\) is the hesitation index of \({{\tilde{a}}_{i}}\) and \(\pi =1-\mu -v\) is the hesitation index of \({\tilde{a}}\).
As previously mentioned, from the perspective of information diffusion theory, fuzziness often arises from the incompleteness of sample, and the theory can, by means of a properly chosen diffusion function, yield a more accurate estimate of the relationships among pieces of fuzzy information generated from an incomplete population than conventional statistical approaches. Therefore, applying it to distance measures for fuzzy information should significantly enhance the accuracy of characterizing the distribution of such information.
Thus, we will use the original sequence to set m intervals with width \({{q}_{1}}\) and \({{q}_{2}}\) based on the variation range of \({u_{i}}\) and \({v_{i}}\), respectively. Then, we take the center point of each interval as the monitoring point \({{z}_{1l}}\) and \({{z}_{2l}}\), \(l=1,\cdots ,m\). Using (3.3) and taking the original IFV information as a sample of the population, we can obtain a normal diffusion estimation of“joint density function”
Set
for normalization. Then we can give the “probability” distance between the fusion value and original IFVs information
it is evident that \(0\le d(\oplus {{\tilde{a}}_{i}},\underset{-}{{{\tilde{a}}}}\,)\le 1\).
The degree of deviation and the extent of information loss are directly proportional to the “probability” distance between the fusion value and the original information. When this distance is greater, the deviation degree is higher, leading to more significant information loss. Conversely, a smaller distance corresponds to a lower degree of deviation and less information loss. Thus, the function
is referred to as the deviation degree of fusion between \(\oplus {{\tilde{a}}_{i}}\) and \(\underset{\scriptscriptstyle -}{\tilde{a}}\).
Measurement of Entropy Reduction Effect in IFVs Information Fusion
From an informational standpoint, entropy is a tool for measuring the level of uncertainty within a system. Regarding the entropy of IFS, Burillo and Bustince were pioneers in introducing an entropy measure that incorporates intuitionism [34]. Fan and Xie later presented an entropy measure based on the concept of divergence in fuzzy sets [35]. Szmidt and Kacprzyk introduced an entropy measure for IFS that is grounded in geometric interpretations [36]. Hung and Yang investigated various entropy measures that are based on probabilistic theories [37]. Szmidt and Kacprzyk, along with Zhang, introduced entropies that are based on distance metrics [36, 38]. Szmidt et al. defined a knowledge measure along with entropies that consider the degree of hesitation [39]. Nguyen and Guo further developed entropies based on knowledge measures [40, 41]. It is a well-established fact that the uncertainty of an IFS is determined by the gap between the degrees of membership and non-membership, represented as \({{\Delta }_{{\tilde{A}}}}({{x}_{i}})=\left| {{u}_{{\tilde{A}}}}({{x}_{i}})-{{v}_{{\tilde{A}}}}({{x}_{i}}) \right| \), and its hesitation degree \({{\pi }_{\tilde{A}}}\) , which are referred to as the intuitionistic and fuzzy factors, respectively. Each of these factors can generate an entropy measure. Building on this, Vlachos and Sergiadis proposed a divergence-based fuzzy entropy that includes both intuitionistic and fuzzy factors, a concept that was later expanded upon by Song et al. [42, 43].
For any \(\tilde{A}\in IFS\left( U \right) \), Song et al. proposed the following IF entropy measure
Similar to Section “Measurement of Information Loss in IFVs Information Fusion”, to capture the relationships among intuitionistic fuzzy information more accurately, (3.4) has enabled us to derive an estimated “joint density function” that correlates with the original IF information population. Moreover, from an information fusion standpoint, it is more logical to incorporate “probabilistic” information into the divergence measure-based entropy. Consequently, for the original IF information set \(\underset{\scriptscriptstyle -}{\tilde{a}}=\left\{ {{{\tilde{a}}}_{1}}=({{u}_{1}},{{v}_{1}}),\cdots ,{{{\tilde{a}}}_{n}}=({{u}_{n}},{{v}_{n}}) \right\} \), we propose the uncertain measure
Additionally, we introduce the indicator Entropy Reduction (ER) to quantify the entropy reduction effect of the fusion result \(\underset{i=1}{{\overset{n}{{\oplus }}\,}}\,{{\tilde{a}}_{i}}\) with respect to the original information \(\underset{\scriptscriptstyle -}{\tilde{a}}\)
where \(\begin{aligned}En(\underset{i=1}{{\overset{n}{{\oplus }}\,}}\,{{\tilde{a}}_{i}})&=\frac{1}{2\ln 2}\pi (\underset{i=1}{{\overset{n}{{\oplus }}\,}}\,{{\tilde{a}}_{i}})\ln (1+\pi (\underset{i=1}{{\overset{n}{{\oplus }}\,}}\,{{\tilde{a}}_{i}}))\\& \quad +(\Delta (\underset{i=1}{{\overset{n}{{\oplus }}\,}}\,{{\tilde{a}}_{i}})-1)\ln \frac{1+\Delta (\underset{i=1}{{\overset{n}{{\oplus }}\,}}\,{{\tilde{a}}_{i}})}{2}\end{aligned}\).
This approach allows us to assess the effectiveness of the fusion process in reducing uncertainty.
Quality Evaluation Indicators for IF Information Fusion
Intuitively, the greater the entropy reduction, the smaller the bias, and thus the higher the quality of information fusion. Conversely, poorer information fusion quality corresponds to weaker entropy reduction and larger bias. On the other hand, we argue that in the process of information fusion, efforts to significantly reduce the uncertainty of original information may lead to substantial changes in its structural relationships, thereby increasing information loss. Similarly, to minimize information loss, it is necessary to preserve the relationships within the original data structure, which may hinder the enhancement of entropy reduction. Therefore, the evaluation of information fusion quality must take both factors into account comprehensively. To this end, we propose the definition of an Information Fusion Quality Index(FQI), with larger values corresponding to higher fusion quality.
Definition 1
For original IFV information \(\underset{-}{\tilde{a}}=\left\{ {{{\tilde{a}}}_{1}}=({{u}_{1}},{{v}_{1}}),\right. \)\(\left. \cdots ,{{{\tilde{a}}}_{n}}=({{u}_{n}},{{v}_{n}}) \right\} \) and the fused value \(\underset{i=1}{{\overset{n}{{\oplus }}\,}}\,{{\tilde{a}}_{i}}\), the FQI \(f(ER(\underset{i=1}{{\overset{n}{{\oplus }}\,}} {\tilde{a}_i},\underset{-}{\tilde{a}}), Dev(\underset{i=1}{{\overset{n}{{\oplus }}\,}}\,{{\tilde{a}}_{i}},\underset{-}{\tilde{a}})):[ - 1,1] \times [0,1] \rightarrow [0,1]\), is an increasing function of entropy reduction \(ER(\underset{i=1}{{\overset{n}{{\oplus }}\,}}\,{{\tilde{a}}_{i}},\underset{\scriptscriptstyle -}{\tilde{a}})\) and the decreasing function of deviation \(Dev(\underset{i=1}{\overset{n}{{\oplus }}}\,{{\tilde{a}}_{i}},\underset{\scriptscriptstyle -}{\tilde{a}})\). Additionally, to normalize the quantity and ensure comparability across different datasets or scenarios, the following requirements must also be met:
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\(f(1,0)=1\).
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\(f(-1,1)=0\).
Here it should be clarified that, to ensure comparability across datasets of different scales, we constrain the range of the indicator to [0, 1]. A value of \(ER=1\) indicates that uncertainty has been maximally reduced by the fusion, while \(Dev = 0\) signifies that no information loss has occurred-together representing the best possible fusion quality. Consequently, the maximum value of the composite index is 1. Conversely, when \(ER=-1\) and \(Dev=1\), the worst-case fusion quality is attained, and the index attains its minimum value of 0.
Consider now the following function:
where \(x\in [-1,1]\) and \(y\in [0,1]\). Obviously, it is an increasing function of x and a decreasing function of y, and satisfies (1) and (2). Since when \(x=1,y=0\), \(f(x,y) = \frac{{\ln (x + 2)}}{{\ln (y + 3)}} = \frac{{\ln 3}}{{\ln 3}} = 1\), and when \(x=-1,y=1\),\(f(x,y) = \frac{{\ln (x + 2)}}{{\ln (y + 3)}} = \frac{{\ln 1}}{{\ln 4}} = 0\).
Thus, we introduce the FQI
Proposed IF Information Fusion Method for Fusion Quality Optimization
Assume that \(\underset{-}{{{\tilde{a}}}}\,=\left\{ {{{\tilde{a}}}_{1}}=({{u}_{1}},{{v}_{1}}),\cdots , \right. \) \(\left. {{{\tilde{a}}}_{n}}=({{u}_{n}},{{v}_{n}}) \right\} \) represent the original IFV information sequence, and \(\tilde{a}=(u,v)\) is the result generated by its fusion, if \(\tilde{a}\) is the optimal fusion outcome, then its FQI should be the best. Consequently, we propose employing optimization techniques to derive the fusion values from the original information, implying that \(\tilde{a}\) must fulfill the following conditions:
The program for our proposed fusion method is as follows (see Fig. 1):
Comparative Analysis of Various AOs
To validate the superiority of our approach, we conduct a reproducible experiment: intuitionistic fuzzy value information sets are first generated via statistical simulation, and then their fusion quality is benchmarked against that of existing AOs IFWA, IFWG, IFHG, IFCCI and CIIFAA.
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Generating IFV Datasets Randomly generate 20 IFVs and repeat this process 30 times. Each IFV (\(\mu ,\gamma \))satisfies:
$$ \mu \sim Uniform (0,1) $$$$ \gamma \sim Uniform (0,1-\mu ) $$(The membership degree \(\mu \) is drawn from a uniform distribution on [0, 1]: \(\mu \sim \text {Uniform}(0,1)\). For each generated \(\mu \), the non-membership degree \(\gamma \) is drawn uniformly from \([0, 1-\mu ]\): \(\gamma \sim \text {Uniform}(0, 1-\mu )\). This two-step uniform sampling ensures that every pair \((\mu , \gamma )\) satisfying the intuitionistic fuzzy constraint \(\mu + \gamma \le 1\) is equally likely to be generated. A standard fuzzy value corresponds to \(\mu + \gamma = 1\) (i.e., hesitation margin \(\pi = 0\)). Since \(\gamma \) is continuous, the probability of \(\gamma = 1-\mu \) is zero; thus, with probability 1, we have \(\mu +\gamma < 1\) and \(\pi = 1 - \mu - \gamma > 0\). This means that the probability of randomly generating a degenerate IFV with such a method is zero. Furthermore, even if a degenerate intuitionistic fuzzy value (i.e., one that reduces to a standard fuzzy value) appears, our method remains valid. Because our method is applicable to all IFVs, including special cases (degenerate ones)).
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Running the Operator and Collecting Results For each generated IFV dataset, in order to compare the fusion quality of IFVs information among those AOs and our proposed method (abbreviated as FQO), we calculate relevant indicators Dev,ER and FQI using (3.3) to (3.8), among which the result of FQO was obtained using the python software and particle swarm optimization algorithm. Then, take the averages of these indicators over the 30 repetitions to obtain \(\overline{Dev}\), \(\overline{ER}\) and \(\overline{FQI}\). The outcomes of these calculations are summarized in Table 1.
Table 1 indicates that our operator is slightly inferior to IFCCI and CIIFAA on Dev, but holds a marked advantage on ER and FQI. Among them, only the ER values of FQO, IFWA and IFHG are positive, exhibiting an entropy reduction effect, while the ER values for the other operators are all negative, signalling that their fusion results not only fail to reduce the uncertainty of the original information but actually increase it. In other words, the overall superiority in fusion quality achieved by our method stems primarily from entropy reduction. Concurrently, from the fusion quality of existing operators, we can also discover a general trend: stronger deviation suppression inevitably weakens entropy reduction, implying that fuzzy fusion cannot simultaneously minimise both information loss and uncertainty. This is consistent with our judgment in Section 3 regarding the relationship between information loss and the entropy reduction effect in the process of information fusion, and also demonstrates the rationality of our proposed fusion quality metric, which takes into account both factors.
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Statistical Analysis To assess the statistical significance of fusion quality differences between IFWA, IFWG, IFHG, IFCCI, and CIIFAA and our proposed operator, we perform five pairwise one-sided Wilcoxon tests on the 30 FQI values independently generated by each operator. The results are shown in Table 2, in which the alternative hypothesis of every test is that the column operator achieves a higher FQI than the row operator.
Table 2 confirms that our method clearly outperforms all other AOs in terms of the quality of information fusion.
Application
To further demonstrate the effectiveness of our proposed method in terms of fusion quality, this section presents a case study on software-development risk assessment by Wu et al. [44]. We apply our method to aggregate the results of the various alternatives reported in that study and subsequently identify the best option.
Risk evaluation in software development inevitably involves multiple, often conflicting criteria. Meanwhile, the available information about the risk factors is usually uncertain, vague, or imprecise, making fuzzy-set theory a natural modelling choice. Yet, so far, only a few researchers such as Lee et al. and Wu et al. have been found to turn to fuzzy set theory for software-development project risk assessment [44, 45].
A software company ABC conducted risk assessments on six software development projects, i.e., the alternative set can be written as \(Y=\left\{ {{y}_{1}},{{y}_{2}},{{y}_{3}},{{y}_{4}},{{y}_{5}},{{y}_{6}} \right\} \) in [44], and the results have a significant impact on the company’s development. This case involves three main risk factors: Product engineering risks (1); Development environment risks (2); Program constraints related risks (3). Product engineering risks typically arise from system engineering and software engineering activities, such as requirements analysis, software design, implementation, integration, and system testing. Development environment risks are related to the project environment and the processes used to develop the software product, such as the development process and system, management methods, and working environment. Project constraints related risks include factors that are beyond the control of the project, such as resources, contracts, and program interfaces. The IFV decision matrix is presented in Table 3.
The IFV decision matrix and the overall evaluations aggregated by IFWA, IFWG IFHG, and IFCCI were shown in Table 3 and Table 4 in [44]. The integrated values obtained for each scheme using FQO are shown in Table 4.
To sort two IFVs by order relationships reasonably in an intuitionistic fuzzy decision-making environment, the widely accepted order relationships \({{\le }_{sh}}\) in decision-making will be used [44].
To compare the fusion quality of the AOs IFWA, IFWG,IFHG, IFCCI and CIIFAA with our proposed method FQO, the relevant indicators Dev,ER and FQI are calculated. The results are shown in Table 5.
Statistically, dispersion analysis is the primary method for assessing the variability within a dataset. Commonly used indicators of dispersion include the standard deviation S.D, the coefficient of variation CV, and the average difference and so on(Here, it should be clarified that, due to the possibility of the average value of ER being negative, the denominator of the CV of ER uses the absolute value of the average value). In a similar vein, from the perspective of information fusion, to evaluate the stability of the fusion quality across various fusion methods, we explore the average levels and dispersion of Dev,ER and FQI for various fusion methods, and the results are shown in Tables 6, 7 and 8 and Fig. 2.
Table 6 shows that in terms of the average level of Dev, the smallest is FQ O with a value of 0.0859, followed by CIIFAA, IFCCI, IFHG, and IFWG, with the worst being IFWA, having a value of 0.1440. Regarding the dispersion \(C{{V}_{Dev}}\) of Dev, the lowest is also FQ O, with a value of 0.2831, followed by IFHG, IFWG, CIIFAA and IFCCI, with the highest being IFWA, with a value of 0.5251. In summary, from the perspective of information loss in information fusion, our method is significantly superior to existing operators in both absolute performance and stability.
Table 7 indicates that in terms of the average level of ER, the highest is FQ O with a value of 0.1038, followed by IFWA, CIIFAA, IFCCI, IFHG, with the worst being IFWG. Among them, IFWG, IFHG, IFCCI, and CIIFAA have negative values, indicating that these AOs not only fail to reduce the uncertainty of the original information during the information fusion process but actually increase the uncertainty. Regarding the dispersion of ER, the lowest is also FQ O, followed by IFWG, IFCCI, CIIFAA, and IFHG, with the highest being IFWA, having a value of 28.3359. In summary, from the perspective of the entropy reduction effect, our method clearly outperforms existing operators in both absolute level and stability.
Our method benefits from its good performance in both Dev and ER, especially its excellent performance in ER. Therefore, in Table 8 you can see that the FQI value for FQO is 0.6597, which is significantly better than the level of around 0.6 for existing AOs.
To compare the statistical significance of differences in fusion quality between the operators IFWA, IFWG,IFCI, IFCCI and CIIFAA with our proposed method FQO, we conduct a total of 15 paired Wilcoxon one-way tests on Dev, ER and FQI. The results are shown in Tables 9, 10 and 11, in which the alternative hypotheses are that the Dev, ER and FQI values of column- vector operator are lower than those of row-vector operator.
At the 0.05 significance level, The p values in Table 9 show that the null hypotheses holds, i.e., our method controls information loss better than every existing operator, with the exception of IFHG. Table 10 reveals that the entropy-reduction gain delivered by our operator is markedly larger than that of any alternative AO. Table 11 further confirms that our approach yields the highest fusion quality, an advantage that stems chiefly from its superior ability to reduce information uncertainty.
In summary, our method outperforms all existing operators in all aspects except that it is slightly weaker than IFHG in controlling information loss.
The aforementioned analysis has powerfully shown the superiority of our method in terms of information fusion quality compared to existing operators. Therefore, we will utilize FQO to integrate the decision information formed by the six options in [44], and subsequently re-rank them under the order relations \({{\le }_{sh}}\). The results are as follows:
The results show that option \({y}_{5}\) has a significantly better score than the other alternatives. Therefore, we conclude that the best option is \({y}_{5}\).
Conclusion and Future Research
Numerous scholars have conducted extensive research on fuzzy-information integration operators and their applications in decision-making, artificial intelligence, and pattern recognition, achieving remarkable results. However, few scholars have focused on the quality of information fusion. Since the principle of information diffusion provides an effective approach to extract fuzzy information to more accurately estimate a relationship as a function approximation, this paper introduces a novel fusion method for IFV information. Aiming to mitigate the uncertainty and information loss inherent in IF information, we propose evaluation indicators for the quality of information fusion, including the Entropy Reduction function ER, the Deviation function Dev, and the Fusion Quality Index FQI integrated with information diffusion theory. Case analysis shows that our method is superior to existing operators in terms of the quality of information fusion. This also indicates that the approach we propose, which is based on the idea of information diffusion, helps to improve the quality of information fusion. It presents a research direction with significant academic value for the field of information fusion.
This paper proposes several evaluation indicators for the quality of fuzzy-information fusion, offering a preliminary gauge of fusion reliability. Nevertheless, the structure of these indicators is relatively simple, and the information quality they capture remains limited. Moreover, quality standards for fuzzy information are intrinsically multifaceted, extending beyond the two criteria we have introduced. Enriching the connotation of existing indicators and selecting more scientifically grounded evaluation criteria are therefore open questions of considerable academic value, yet they also constitute challenging research topics for the future.
Optimizing the quality of fuzzy-information fusion has become crucial in artificial-intelligence and decision-science applications. In modern AI domains like AI-based visual-quality assessment, fuzzy-neural crowd counting for traffic control, and fusion image quality evaluators for autonomous systems, the shared needs are (1) to fuse heterogeneous fuzzy evidence, (2) to quantify fusion quality with a diffusion-consistent distance, (3) to optimize parameters until quality peaks, and (4) to feed the high-quality fused result to downstream AI models. Likewise, in engineering management, finance, or environmental governance, fuzzy decision information provided by experts is heterogeneous, high-dimensional, and often conflicting; the optimized fusion quality directly determines the final ranking of alternatives such as investment portfolios or green-engineering schemes. Consequently, developing more effective methods for applying fuzzy-information fusion quality optimization to these scenarios represents an important future research direction.
Data Availability
No datasets were generated or analysed during the current study.
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We are grateful to the editor and the anonymous referees for their valuable comments and suggestions that helped us to improve the earlier versions of this paper.
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Sun rong presented the ideal, conceptualization, methodology finished the writing-Original draft, and Wen yahan finished the investigation, supervision, and editing of the original manuscript. In the revision stage, Luo yan undertook substantial work, independently performing the case-section coding, empirical analysis, and finalization of the manuscript.
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Rong, S., Yan, L. & Yahan, W. Intuitionistic Fuzzy Information Fusion Optimized Through Entropy Reduction and Information Loss Measures Integrated with Diffusion Theory. Cogn Comput 18, 104 (2026). https://doi.org/10.1007/s12559-026-10626-2
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DOI: https://doi.org/10.1007/s12559-026-10626-2