Abstract
Based on the advantages of cloud theory and the connection number approach in handling uncertain information, this paper established a new quaternary connection number using the domain of interval corresponding to the centroid of a cloud model’s expectation curve. Consequently, based on the cloud model theory of transformation between qualitative and quantitative uncertainties, this study suggests a model describing uncertain information more precisely by combining the advantages of quaternary connection numbers with three digital aspects of the cloud model. We defined the weighted nearness degree of the new connection number and gave a solution for finding the weight of the weighted nearness degree given that a three-parameter interval number can more precisely represent the expert’s true intention than a classical interval number. The method of multiattribute group decision-making based on a cloud model’s quaternary connection number was developed using a novel methodology to find the evaluation-index weight. According to a comparative analysis, the existing membership cloud gravity center (MCGC) method is nothing more than an exception to our proposed decision-making technique. It was also demonstrated that the proposal may present a more complete picture of experts’ overall evaluations and communicate their preferences by adjusting the quaternion’s uncertain parameters, making the group decision-making approach more widely applicable to a certain extent. The method was used to test a vessel’s counterflooding capabilities to ensure its practicality and supremacy.
1. Introduction
Multiattribute decision-making (MADM) is one of the most important and ubiquitous real-life activities for selecting a suitable alternative among those required to achieve a certain goal. The information about accessing the alternatives has always been considered to be in the form of real numbers. However, in everyday life, ambiguity and insufficient knowledge make it difficult for a decision-maker to make an appraisal of an object. Zhao [1] created the set pair analysis (SPA) theory, in which certainty and uncertainty are examined as one system, to better handle uncertainties. The essential notions of SPA theory were addressed by Jiang et al. [2]. The fundamental idea behind SPA is to look at the characteristics of a set pair and create a connection number (CN) for them. To address MADM problems with a specified criteria weight and a criteria value that is an interval random variable, Wang and Gong [3] suggested a decision-making approach based on the SPA. Based on cumulative prospect theory and SPA, Hu and Yang [4] suggested a dynamic stochastic MADM. Using the TOPSIS approach, Xie et al. [5] demonstrated a CN under an interval-valued fuzzy set. Aside from this, some academics [6–9] used the SPA to handle fuzzy decision-making problems. Shen et al. [10] defined binary connection numbers (BCN) in SPA and utilized BCN to multiattribute decision-making issues in an interval-valued intuitionistic fuzzy set environment. Cloud theory, proposed by Li [11], an academician of the Chinese Academy of Engineering, is a method for resolving the fuzziness-randomness problem. The theory reveals the intrinsic relationship between fuzziness and randomness, which presented a unique algorithm architecture that simplifies the conversion between qualitative concepts and their quantitative representations. A cloud model is defined by three digital properties, in which is the primary, or most representative, value of a qualitative idea in the domain. The entropy measures a qualitative concept’s fuzziness and reflects the concept-accepted value range in the domain. As the entropy, the hyperentropy indicates the dispersion of cloud drops [11, 12].
A cloud model-based multiattribute decision-making (CM-MADM) with Monte Carlo technique is developed to address several challenges occurring in MADM problems [13]. Based on the probability theory and fuzzy set theory, the cloud model can study the detail of fuzziness of a qualitative concept membership function as well as the stochasticity by three numerical characteristics.
Currently, the cloud model is widely applied in the comprehensive evaluation, systemic decision-making, and data mining [14–16]. Many researchers have had positive outcomes in their research areas using the membership cloud gravity center (MCGC) [17], which is a common approach of evaluating a cloud model. In the study on the distribution of earnings from natural resource development, for example, Li and Zhou employed the MCGC model to show that a novel earning distribution strategy may make such distribution fair and square [18]. Combining the fuzzy extended analytical hierarchy process with the MCGC method, Zheng found a better way to manage enterprise knowledge [19]. To accurately evaluate the performance of portable devices, Peng et al. came up with an MCGC-based method [20]. Likewise, Zheng et al. put forward an MCGC-based approach to cybersecurity evaluation by introducing a cloud model to assess the cybersecurity situation [21]. Large-sample group decision-making is now made possible by the boom in big-data and computer technologies. Cloud theory is effective in presenting the opinions of large groups, whereas the dialectic set pair analysis can competently manage a matter’s uncertainties on a macrolevel.
The fact that CNs and interval numbers are isomorphic is one of the key reasons why SPA can be utilized frequently in MADM with interval numbers. The CN is characterized by both certainty and uncertainty. The interval number has upper and lower bounds that have been specified, as well as uncertainty that can be arbitrarily valued within its scope. In the CN, the uncertainty coefficient might range with in the interval [−1, 1].
The particle swarm optimization (PSO) is a relatively new notion of combinatorial metaheuristic algorithm which is based on a metaphor of social interaction, namely, bird flocking or fish schooling. In [22], Chatterjee and Siarry considered nonlinear inertia weight variation for dynamic adaptation in particle swarm optimization, where a new variation of PSO model is considered which introduced nonlinear variation of inertia weight along with a particle’s old velocity to improve the speed of convergence and fine-tune the search in the multidimensional space.
In this study, we developed cloud model’s connection numbers, by utilizing three parameters’ interval number containing upper and lower bounds as well as center of gravity. By integrating CNs in SPA, the production of a cloud model’s connection numbers reveals how cloud theory renders data expression understandable within a big-data context and exposes the laws governing the uncertainty of overall information. On the theoretical basis of a cloud model and the connection numbers in set pair analysis, we proposed the following: (1) a cloud model-based method determines quaternary connection numbers, which have three digital features of the cloud model and fully express the model’s fuzzy information; (2) we coined an arithmetic weighted average nearness of quaternary connection numbers and put forth an approach to determination of the nearness’s weight; (3) we raised a new method of determining the evaluation-index weight, given that the interval number expressed by three parameters can more accurately express an expert’s real intention; (4) we leveraged the results from the above three steps to develop a multiattribute group decision-making method based on the connection numbers of a cloud model and through comparative analysis proved that the MCGC model is merely an exception to our proposed decision-making technique; and (5) we verified our method’s practicality and superiority by adopting it in the evaluation of a vessel’s counterflooding capabilities.
To do so, the rest of the manuscript is summarized as follows: Section 2 gives some overview on the cloud model and CNs. In Section 3, a cloud model’s CN method for MADM has been presented under the SPA in which the assessments related to the attributes are taken in the form of quaternary connection numbers (QCNs). This section also reveals that the following: (i) the definition of weighted nearness degree of a QCN, (ii) determination method of weight of a QCN’s nearness degree, (iii) determination method of index weight based on three parameters interval number, and (iv) steps of multiattribute group decision-making based on cloud theory for QCNs; and Section 4 gave the comparison of the proposed model and Section 5 studied a numerical example as a case study.
2. Basic Theories on the Cloud Model and Connection Numbers
2.1. Cloud Model Basics
The cloud model is a type of model investigating the uncertain conversion between the qualitative concepts of natural language and their quantitative representations.
Definition 1. Assume that is a quantitative domain expressed by precise numerical values; is the linguistic value connected to ; the membership , which shows how the element in falls under the qualitative concept expressed by , represents a random number with steady tendency, and its distribution in the domain is called a membership cloud, or simply “cloud,” which can be expressed asAs a critical cloud model, the normal cloud is most effective in expressing linguistic values.
Definition 2. If the random variable satisfies and and satisfies , then the membership cloud of in the domain is normal. When , is known as the expectation curve for the normal cloud.
Given that the normal cloud follows the rule governing the normal distribution, more than 99.73% of cloud drops of the normal random number fall under the range, and the overall features of the cloud model are not affected by the cloud drops outside the range.
As is displayed in Figure 1 (where ), the digital features of a membership cloud have the following implications [1].(1)The domain value corresponding to the centroid in the membership cloud-covered area reflects the central value of information about fuzzy concepts.(2)The bandwidth of a membership cloud model’s expectation curve indicates the fuzziness of fuzzy concepts.(3)The randomly distributed variance of a membership corresponding to the point on a membership cloud-based expectation curve, or the hyperentropy , suggests the dispersion of the membership cloud.
2.2. Backward Cloud Generator
The backward cloud generator is designed to identify the three digital features of a cloud model by using the given cloud drops. A variety of ideas about the generator’s algorithm have been provided by scholars. Having considered the scale of data for multiattribute group decision-making in practical issues, we offered an optimized Algorithm 1 of the backward cloud generator.
|
2.3. Connection Number Basics
The connection number is a major instrument in set pair theory. It serves as a connector between a certain number and its range and between macroscopic certainties and microscopic uncertainties. Also, it is a structure function that reflects the relationships of studied subjects under various conditions. Its commonly used form is the ternary connection number.
In [22], Chatterjee and Siarry introduced the concept of an interval number a on real number R as follows:
, where aL and aU are the lower bound and upper bound of the interval, respectively. Sengupta and Pal [23] defined a three-parameter interval as , where aL is the lower bound, is the center of gravity (the number that has the highest possibility), and aU is the upper bound of the interval. If aL = = aU, then the three-parameter interval number is reduced to a real number.
Definition 3. Assuming that are real numbers and , thenwhich is called a ternary connection number and , , and denote the identity, difference, and opposite degrees of any two researched subjects.
Let be regarded as the ternary connection degree:Particularly, is called the binary or identical-different connection number.
In practical decision-making scenarios, the actual identity and difference degrees can be determined on a case-by-case basis, making possible the generation of all forms of the connection number, such as quaternary or quinary connection numbers.
3. Multiattribute Decision-Making Method Based on Cloud Model’s Connection Numbers
3.1. Creation of Cloud Model’s Connection Numbers
Various uncertain and fuzzy pieces of information must be dealt with in a practical decision-making process. In processing such information, many scholars have created connection numbers according to the expression of information [15]. However, the existing methods of establishing connection numbers find it hard to express the overall features of large-sample data. The creation of a cloud model’s connection numbers demonstrates how cloud theory renders data expression reasonable within a big-data context and shows the laws governing the uncertainty of overall information by combining connection numbers in set pair analysis. According to the three implications of a cloud model’s digital features mentioned in Section 2.1, the domain value corresponding to the centroid of the cloud model’s expectation curve reflects the central information of fuzzy concepts; that is, the domain value range can be used to develop quaternary connection numbers for the cloud model, which can express the model’s uncertain information more accurately. The specific process is as follows. Step 1: develop the equations for the internal and external envelopes by involving the standardized cloud model : Step 2: given the cloud model’s asymmetry, the right half of the internal and external envelopes was extracted for calculation of the corresponding abscissa value when : In light of the rule, the abscissa of the point was . Step 3: ensure that the quaternary connection number satisfies and make , where
Upon normalization, it was found that the quaternary identical-different connection number for the cloud model was and :
Built on the research on the abscissa interval of an expectation curve’s centroid, the method of creating the quaternary connection number fully takes into account the implications of the cloud model’s three digital features, thus expressing the model’s uncertain data in a more comprehensive way.
3.2. Definition of the Weighted Nearness Degree of a Quaternary Connection Number
As a concept in fuzzy mathematics, nearness degree measures the extent to which two fuzzy subsets are similar to each other.
Definition 4. Assume , if the mapping satisfies the following conditions:(1).(2).(3)If , then and .Then, is known as the nearness degree of and fuzzy sets, and is the nearness function in .
Based on the aforementioned definition of the fuzzy set nearness and the importance of fuzzy information contained in all parts of a quaternary connection number, the nearness of the quaternary connection number is defined.
Definition 5. Assuming that are quaternary connection numbers, and , where , , and ; then the nearness of the two connection numbers iswhere and . The definition can be straightforwardly proved to meet the three conditions in Definition 3.
3.3. Weight Determination in Definition of a Quaternary Connection Number’s Nearness Degree
This section provides a new method of determining the weight in the definition of a quaternary connection number’s nearness.
Assuming that, after standardizing the evaluation information of the evaluation index for all plans , the cloud model’s connection number is and , where . Let the ideal cloud model’s connection number of the indexes in the model be and its average cloud model’s connection number be , where
The optimized model is developed as follows:
The Lagrange function is created and solved:
Then, we obtainwhere
The weight determination method is designed to develop a nonlinear planning model using the objective function or the quadratic sum of the difference between the expected and ideal values. The smaller the objective function, the more reasonable the corresponding plan for assigning the weight. In this method, the difference between the expected and ideal values of group decision-making information is fully considered, which can objectively show how each part of the connection number is prioritized in group opinions by specific evaluation indices.
3.4. Method of Determining the Index Weight Based on the Interval Number Expressed by Three Parameters
In the practical decision-making process, decision-makers tend to evaluate information in the form of interval numbers, given that experts find it complicated and uncertain to describe the importance of indices. When a parameter is expressed by an interval number expressed by two parameters, the opportunity to obtain a value within the whole interval is believed to be equal, which would produce a result far from the real intention of experts. Hence, the section offers a novel method of determining the index weight based on the interval number expressed by three parameters.
Assume that, in the hundred-mark system, the score of the index given by the expert is ; then the estimated weighted value of the interval number expressed by three parameters can be obtained upon normalization through the following equation:where is the gravity center of the interval number. Let be the maximal measure of the upper and lower bounds of the index ’s estimated weights, where represents the minimal value of the lower bound of the index’s estimated values, while is the maximal value of the upper bound of the index’s estimated values.
Let the weighted preference deviation of the expert be :where is what the expert believes the real weight of the index is and is the ratio of the index ’s weight error to given by the expert . (Note: the expert’s weight error is measured by the difference between the gravity center of the interval number expressed by three parameters and the real weight.) Each evaluating expert may regard consistently with each evaluation index, expressed as . Therefore, the estimated value of ’s weight, expressed as , iswhere is an aggregation of and is an aggregation of involving all participating experts. The equation can be easily proved.
Definition 6. Assuming that is an -element function, is a weighted vector associated with and satisfies and . If , where is the number of in , a progression that goes from largest to smallest, then the function denotes the orderly weighted average (OWA) operator in the dimension.
The definition suggests that the weighted coefficient of the OWA operator is associated with the spot in , rather than the number . When , the OWA operator can be reduced to a simple arithmetic average operator. Thus, the aggregation can be conducted by using the following simplified equation.And, after they are substituted into (14), the estimated aggregated value of the index weight, expressed as , can be calculated.
3.5. Steps of Multiattribute Group Decision-Making Based on Cloud Model’s Quaternary Connection Number
Plan sorting has always been of critical importance in multiattribute decision-making methods. The section provides a method of this kind by ranking plans based on the calculation of the weighted arithmetic average nearness of the quaternary connection number and ideal connection number in different cloud models. The specific steps are as follows. Step 1: have experts score of all plans in light of the evaluation-index system and standards, generate standardized cloud drops by indices oriented to benefits and costs, translate these cloud drops into a cloud model by using the backward cloud generator constructed according to Section 2.2, and subsequently create digital features. Step 2: use the method of creating a cloud model’s connection numbers, as indicated in Section 3.1, to construct the quaternary connection number for a cloud model founded on scoring by experts, and obtain a connection number-based decision-making matrix. Step 3: develop the ideal cloud model by each index for the creation of the quaternary connection number, and, as instructed in Section 3.3, calculate the weight of each part of the cloud model’s quaternary connection number. Step 4: calculate the weight of all indices in the given index system as suggested in Section 3.4. Step 5: measure the nearness of all proposed plans and the ideal plan by using (8) and (12), respectively, and calculate , the overall nearness of all these plans, through the following equation, where ( is the total of plans): In this equation, T is the total number of evaluation indices, indicates the weight of each index, and denotes the quaternary connection number of each index for the ideal plan. Step 6: rank plans based on their overall nearness .
4. Comparative Analysis between the MCGC Model and the Decision-Making Method Based on Cloud Model’s Quaternary Connection Number
The MCGC method is typical for evaluating a system using a cloud model. Specifically, if the system status of evaluation indices is described by the -dimensional comprehensive cloud, then a changing system status would make the comprehensive cloud and its gravity center altered. Therefore, a possible evaluation way is through the weighted deviation of the comprehensive cloud gravity center and ideal cloud gravity center under a certain status. Also, proposed plans can be sorted based on the weighted deviation between their comprehensive cloud gravity center and that of the ideal plan. The detailed process is as follows [5].(1)For a system to be evaluated, calculate the gravity center vector of the -dimensional comprehensive cloud for the ideal plan through where is the position vector of the -dimensional comprehensive cloud gravity center and its component is the expectation value of each index; represents the altitude vector of the comprehensive gravity center, which is a determined value in a to-be-evaluated system and is commonly known as the weight value of each index [8].(2)For a given proposed plan , provide its corresponding -dimensional cloud gravity center vector, expressed as , through the equation , where .(3)Give the definition of deviation of and in the proposed plan , calculate the deviation using equation (17), and evaluate based on the weighted deviation.where
Building on the aforementioned MCGC logic and steps, we found the following.
Theorem 1. For a cloud model’s quaternary connection number and , when , the evaluation method based on a cloud model’s quaternary connection number is equivalent to the MCGC model.
Proof. As is known in the method of creating a cloud model’s quaternary connection number (see (7)), . When , the quaternion needs not to be normalized, meaning that it is reduced to . And, for the proposed plan , all its indices are reduced to the cloud model’s expectation values, expressed as
As to the cloud gravity center’s structure, ’s position vector is, in essence, the expected value of each index, and the altitude vector is the determined weighted value of each index. For a particular evaluation system, the index weight, once generated, is a determined value. Hence, each component of ’s cloud gravity vector is the constant multiple of an expected value or , where are weights of indices.
As shown in (18) and (21), is determined by investigating the extent to which the two weighted vectors, namely, and , deviate from each other. In the evaluation method based on a cloud model’s connection number, the overall nearness is determined by observing the deviation between and . Therefore, when , the two evaluation approaches are in practice equivalent.
The abovementioned theorem and proof indicate that the following.(1)The MCGC method is an exception to the evaluation method based on a cloud model’s connection number, where . For the latter, the quaternary identical-different connection number demonstrates how the expectation value plays its role in index evaluation information and takes into account the full impact of the model’s entropy and hyperentropy on evaluation.(2)The steps of evaluation based on a cloud model’s quaternary connection number showed that, in a practical decision-making process, expressing expert preferences through appropriate adjustments to can offset the evaluation error arising from how the quaternary connection number’s weight relies merely on objective numerical values rather than expert preferences.(3)Compared with the MCGC approach, the evaluation method based on a cloud model’s quaternary connection number collects more information from experts, thus producing more science-based results. Beyond that, in the connection number can be tailored to the needs of the application context, which makes results better aligned with the operating condition of the evaluation system. That is how the proposed method can be further applied across different scenarios.
5. Case Study
Among the damage control capabilities of a vessel, counterflooding is vital. There are four indices to evaluate a ship’s overall counterflooding performance, namely, the cabin’s capabilities of leak stoppage, bearing, drainage, and balancing. All these indices are oriented to benefits. Eight experts were invited to score the importance of these four evaluation indices in the hundred-mark system using the interval number expressed by three parameters, as is displayed in Table 1.
Table 1 is then normalized using (14), as shown in Table 2.
Using (13), we calculated the determined weight deviation degrees of experts, which were [−0.2186 −0.1976 −0.3100 −0.1524 −0.1993 −0.2141 −0.1706 −0.1857]. And upon the aggregation of the OWA operator, we input these values into (15) and obtained
Lastly, with (16), we measured the estimated weight value of these four counterflooding indices, which were
In addition, we had these experts’ score of the counterflooding performance of three vessels, and each of them was required to provide the score and corresponding membership degree of the four indices in ideal and actual scenarios. Then, the backward cloud generator, as instructed in Section 2.2, was employed to translate the cloud drops offered by experts into a cloud model, whose digital features are shown in Table 3. Among others, V0 represents the ship equipped with ideal counterflooding capabilities.
Adopting the method of establishing a cloud model’s connection number, as displayed in Section 3.1, we obtained the following decision-making tables, as shown in Tables 4–8.
The weights of the connection number’s identical and three difference degrees in each evaluation index were calculated, as instructed in Section 3.3, and are shown as follows.
According to (18), the nearness degrees between the three vessels’ counterflooding capabilities and their ideal performance were , respectively. That suggested that vessel V1 boasted the strongest counterflooding capabilities, followed by vessel V3 and vessel V2, consistently with the practical scenario.
6. Conclusions
The quantitative analysis of uncertain and fuzzy information has always been important in information processing. Quantifying such information in group decision-making would lead to reasonable and science-based results. Cloud theory and set pair analysis have made great contributions to the research area. In this paper, based on the strengths of a cloud model and the connection number method in processing uncertain data, we (1) proposed a new cloud model-based method of establishing a quaternary connection number, which has the model’s three digital features that render the expression of the model’s uncertain information more accurately and comprehensively; (2) we defined a new weighted arithmetic average nearness degree of the quaternary connection number and provided specific steps to determine the weight of such a nearness degree; (3) we offered a novel method of determining the weight of an index based on interval numbers expressed by three parameters, given the fact that the deviation degree remains unchanged when each expert determines the index weight; (4) we proposed an innovative multiattribute group decision-making method built on a cloud model’s connection number and through comparative analysis proved that the MCGC method is an exception to our proposed decision-making method. The proposed approach can provide a fuller picture of the overall evaluation made by experts and express their preferences through adjustments to the quaternary connection number’s uncertain parameters, which makes the group decision-making method more broadly applicable. Also, the technique can be applied to decision-making involving large-sample data and to the decision-making context where evaluation values are measured through modal particles. Nonetheless, when it comes to an application context, it awaits more research efforts to find out how to translate these particles into a cloud model reasonably and make sensible adjustments to the values in the model’s connection number, so that the resulting decisions better suit the context [24–29].
Abbreviations
| MCGC: | Membership cloud gravity center |
| MADM: | Multiattribute decision-making |
| SPA: | Set pair analysis |
| CN: | Connection number |
| BCN: | Binary connection number |
| : | Entropy |
| : | Hyperentropy |
| CM-MADM: | Cloud model-based multiattribute decision-making |
| PSO: | Particle swarm optimization |
| QCNs: | Quaternary connection numbers |
| : | Quantitative domain |
| : | Membership |
| F(U): | Set of fuzzy sets |
| : | Nearness degree of A and B |
| : | Total number of planes |
| V 0: | Ideal counterflooding capabilities. |
Data Availability
All data, models, and codes generated or used during the study are included within the submitted article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This word is supported by PhD Foundation of Wuchang Shouyi University (B20200206), National Key R&D Program of China (2021YFC3100202-1).