Abstract

An optimized matter element decision method is proposed based on intuitionistic fuzzy set (IFS) and Choquet integral to solve the problems of equipment development scheme decision (e.g., the correlation of scheme attributes and the low decision reliability of evaluation results). First, the nonlinear programming problem of expert intuition fuzzy subjective and objective decision information difference is constructed using the Choquet integral method, and the measure values and related factors of decision attributes are solved using the optimization algorithm. Second, given the comprehensive contribution of attributes, the fuzzy measure Shapley value is employed to represent the attribute weight. Last, the matter element extension model is optimized using the attribute weight information and intuitionistic fuzzy score value, and a two-level evaluation strategy is formulated to determine the ranking and grade. On that basis, scientific decisions are made for several different schemes to be selected.

1. Introduction

The decision on equipment development plans has been confirmed as a vital task in equipment program demonstration. In general, it is performed by the research and development (R&D) units by conforming to the general requirements in terms of equipment development and the R&D contract for the development plan evaluation and selection. It is noteworthy that a strong correlation exists between the attributes in equipment development plan decision-making, and the feature descriptions of attributes are fuzzy and uncertain to a certain extent. Given the abovedescribed context, the fuzziness, uncertainty, and correlation of the attributes in the decision-making process should be considered. Accordingly, the equipment development plan decision-making in this study conforms to the intuitionistic fuzzy set, Shapley value, Choquet integral, and the optimized matter-element extension model.

Fuzzy set theory has been extensively applied since it was proposed by Zadeh. However, fuzzy sets cannot express the decision-making information of decision-makers fully for their membership degree-based description of the fuzziness of things. Atanassov extended fuzzy sets and introduced the concept of intuitionistic fuzzy sets that considers both membership and nonmembership information. The extended fuzzy sets are effective in processing fuzzy information. Atanassov et al. further extended intuitionistic fuzzy sets using interval numbers to represent membership and nonmembership in intuitionistic fuzzy sets [1]. On that basis, the concept of interval intuitionistic fuzzy sets was introduced. Existing research on intuitionistic fuzzy theory has placed a greater focus on several aspects (e.g., intuitionistic fuzzy operators, similarity measures, attribute weight determination, intuitionistic judgment matrix, extension of classical attribute decision-making methods, and score functions, as well as precise functions) [2].

In terms of MADM, attribute weights should be determined. Any two decision attributes are assumed to be independent and mutually exclusive at early stages [3]. However, decision attributes (criteria) are often not completely independent but interdependent in the decision-making problems of practical engineering projects, and the interactive features between attributes substantially affect the decision results. Sugeno [4] proposed the fuzzy measure method, which effectively expresses the interactions between attributes, to tackle down the abovementioned limitation. Next, Choquet [5] introduced the Choquet integral to combine the fuzzy information of attributes. Subsequently, the Choquet integral has been extensively applied to decision problems. Murofushi and Sugeno [6] explained the interactions between attributes using the nonadditivity of fuzzy measures and discussed the rationality of Choquet integral, proposing the Choquet integral of fuzzy measures. Marichal [7] classified the correlations between attributes and highlighted the discrete Choquet integral as a tool for aggregating correlated attributes and gave axioms. Xu [8] proposed the Intuitionistic Fuzzy Correlation Average operator (IFCA), the Intuitionistic Fuzzy Correlation Geometric operator (IFCG), and the Interval Intuitionistic Fuzzy Correlation Average operator (IVIFCA), as well as the Interval Intuitionistic Fuzzy Correlation Geometric operator (IVIFCG) based on the Choquet integral and apply them to a practical decision-making problem involving the prioritization of information technology improvement projects. Given the phenomenon of interdependence between decision attributes, Tan and Chen [9] proposed the Intuitionistic Choquet Integral operator based on the Choquet integral and developed a method of solving the decision-making problem with correlated multiple attributes. Wan and Dong [3] give the TIFN arithmetic integration operator and the TIFN Choquet integration operator in the case of attribute independence. The Choquet integral aggregation operator of TIFN uses fuzzy measure instead of additive set function to measure the weights of attributes and attribute sets. Tao and Ling [10] developed a fuzzy number intuitionistic fuzzy number information aggregation operator in accordance with the Choquet integral and used it to study decision-making problems with correlated multiple attributes. Dong et al. [11].

The Choquet integral is effective in addressing group decision-making problems with relevant attributes, and it is capable of effectively expressing the interaction between attributes based on intuitive uncertain linguistic information. However, it is limited to the interaction between adjacent attribute coalitions [12]. The Shapley value [13], a measure of the significance of individual attributes proposed by Shapley in cooperative game theory, is used to measure the significance of attributes to more accurately express the interaction between attribute coalitions under different combinations. Chang and Tan [14] proposed a multiattribute decision-making method based on the Martingale system and transformation, suggesting the use of Shapley values to replace attribute weights in solving fuzzy measures.

In practical engineering applications, the ranking of alternative solutions should be known, and their specific levels of superiority or inferiority in the evaluation set (i.e., to determine the specific levels of alternative solutions in the evaluation set) should be determined. The matter-element extension method [15] refers to a comprehensive evaluation and decision-making method that is capable of effectively solving complex and incompatible problems with multiple attributes. The matter-element extension method has been employed in a wide variety of fields (e.g., regional development, mechanical processing, risk assessment, and environmental governance). This method constructs the hierarchical evaluation index system of the system to be evaluated, divides the hierarchical evaluation level, determines the classical domain, the boundary region, and the evaluated object, and calculates the level membership degree of the evaluated object in combination with the attribute weight, such that a basis is laid for decision-making. The matter-element extension method often follows the analytic hierarchy process (AHP) to determine the attribute weights, whereas the credibility of the attribute weights determined by this method is poor when a correlation exists between attributes. Thus, the interactive features between attributes can be more effectively expressed using IFS-Choquet integral and Shapley value in combination with the matter-element extension method, and the superiority and inferiority evaluation levels and ranking of alternative solutions can be effectively determined.

The correlation between inherent attributes of a system takes on a critical significance to the correlation between indicator weights. For multiattribute decision-making problems, the overall contribution of attributes exerts subjective and objective effects on decision information. In this study, intuitionistic fuzzy sets and Choquet integral are adopted to process subjective and objective decision information of alternative solutions. The comprehensive decision information difference of individual solutions is calculated in accordance with the decision information difference of attribute values of alternative solutions. A nonlinear programming model is built based on attribute fuzzy measures and attribute correlation factors to invert the inherent measures of attributes under the indicator system. To be specific, the correlation factors are adopted to account for the actual guiding role of the attribute power set. Given the overall contribution of attributes, Shapley values are employed to characterize attribute weights. The physical element model is optimized using the attribute weight information and intuitionistic fuzzy score values. Moreover, a two-layer physical element evaluation method is developed to make decisions on alternative solutions and determine the ranking and level of the solutions. Based on IFS-Choquet integral, attribute weights and intuitionistic fuzzy score values aim at ensuring the consistency of the data source and reducing the effect of subjective decision errors and operational errors.

2. Construction of Decision Model for Equipment Development Scheme

To ensure the rationality of decision-making, a nonlinear programming problem which satisfies the minimum absolute deviation between subjective and objective preferences is constructed based on IFS and Choquet integral method, and the attribute measure value is solved using optimization algorithm.

2.1. Solution of Attribute Weight Based on IFS-CHOQUET Integral

Definition 1 [15]. Set as a nonempty set, calling an IFS, and are element in , which belong to the membership degree and nonmembership degree of , respectively, and . Meanwhile, they met the condition .
Set as intuitionistic fuzzy number, and the correlation functions of intuitionistic fuzzy number are as follows:(1) is the hesitation degree of element falling into .(2) is the accuracy of the element belonging to .(3)A scoring function [17] is introduced to compare the magnitudes of intuitionistic fuzzy numbers. is the commonly used scoring function, whereas it has limited practical applications. The shortcomings of the existing scoring function are analyzed from a voting perspective. To address these shortcomings, a novel scoring function is proposed, which considers the effects of supporters, opponents, and abstainers on decision-making.Assuming there are two intuitionistic fuzzy number and , their sizes can be determined by the commonly used scoring function and precision measures (e.g., and ). Accordingly, when the common scoring function serves as a criterion for evaluating solutions, the solution corresponding to outperforms the one corresponding to . However, when explained using a voting model, it is generally considered that the solution corresponding to outperforms that corresponding to , as the solution corresponding to indicates that, apart from 30% abstentions, the remaining 70% of the votes are in favor. Thus, the effect of the abstaining population on decision-making should be considered when evaluating the effectiveness of solutions. On that basis, Wang et al. proposed the following modified scoring function [18]:The previous equation considers the effect of supporters, opponents, and abstainers. However, when two intuitionistic fuzzy numbers and are set, equation (1) suggests that , , and . As revealed by the previous results, the scheme corresponding to outperforms that of , which is obviously not consistent with the actual situation.
After analysis, the abstaining population is assigned to three groups (i.e., those who tend to support, those who tend to oppose, and those who still tend to abstain). The proportion of the tendency is equivalent to that of supporters and opponents, and the hesitation level is distributed to the membership and nonmembership degree in proportion to . The optimized scoring function [19] through iterative loops to obtain the limit is expressed as follows:

Definition 2 [14]. Set as a power set of and fuzzy measure as the set function defined on .

Definition 3 [20]. Given , is the power set of , and for satisfies the following:(1);(2) and :(3) continuity;Subsequently, is termed a fuzzy measure defined on , which is a case of fuzzy measure Y. fuzzy measure on a single point set is termed fuzzy density, let satisfy the following:The marginal influence factors of attribute are considered in the research on attribute weight in equipment development scheme, and the attribute weight value is substituted with Shapley value, which is adopted to represent the comprehensive contribution value of attribute in the attribute set. In accordance with the definition of Shapley function [21] in multiplayer game and Grabisch’s generalized Shapley function, the calculation method of Shapley value corresponding to fuzzy measure is as follows:
If are fuzzy measures defined on , the Shapley value of the importance index of attribute Angstrom with respect to fuzzy measure is expressed as follows:In equation (5), is the number of elements in ; is the cardinality of ; is the comprehensive contribution value of attribute, and satisfies .

Definition 4 [15]. as a fuzzy measure defined on , satisfying the law of , the discrete Choquet fuzzy integral of on the set with respect to the fuzzy measure which is defined as follows:where is the number of elements in , is the i-th element of rearranged under the action of function so that , and , .

2.2. Establish a Nonlinear Programming Model to Calculate Attribute Measure and Shapley Value

If is a function defined on the evaluation attribute set , let , considering the correlation between attributes, the comprehensive score of scheme can be obtained by equation (6) as .

The subjective intuitionistic fuzzy number of scheme is , the objective intuitionistic fuzzy number of index under scheme is , and the corresponding subjective intuitionistic fuzzy number score value is and objective intuitionistic fuzzy number score value is from equation (1).

Under subjective and objective intuitionistic fuzzy numbers, there is a deviation in the comprehensive score value of the scheme, and the deviation is . Under scheme , the intuitionistic fuzzy number score of attribute is , and the intuitionistic fuzzy number score of subjective preference is , then .

Combined with equation (6), under scheme , the subjective and objective comprehensive score deviation based on intuitionistic fuzziness is expressed as follows:where represents the ith element of the rearrangement of angstrom at so that , and , .

can be generalized by equation (2), where

This satisfies , , and , and is the power set of set , and and are the number of elements in . Combined with equation (7), the final form of the sum of squares of subjective and objective score deviations of intuitionistic fuzzy set is as follows:

Attribute measure satisfies that the sum of squares of subjective and objective score deviations of decision makers is the smallest, even if takes the smallest value. Among them, is the number of programs in the program concentration. Therefore, a nonlinear programming model with multivariate constraints can be established as follows:

The optimization algorithm is used to solve the model, and the attribute measures and are determined. In the next step, the Shapley value of the respective attribute is solved by combining the obtained attribute measures and with the equation (5) as the corresponding attribute weight.

At the same time, the attribute score value under scheme is taken as the attribute matter element value in the next matter element extension evaluation model, which makes the solution weight consistent with the decision information source of matter element analysis and reduces the accumulated error caused by multiple artificial decisions.

2.3. Optimized Matter Element Model for Equipment Development Scheme Decision

The main idea of matter element extension method is to abstract the evaluation object into a matter element, i.e., an ordered triple including object name , feature attribute , and magnitude based on attribute evaluation function.

According to the matter element extension method, an optimized matter element extension model for equipment development scheme decision is established using intuitionistic fuzzy measure of attributes and Shapley value of measure. The logic diagram of the model is shown in Figure 1.

The specific method and steps of establishing optimized matter element extension model to perform equipment development scheme decision are elucidated as follows:

Step 1. The evaluation matter element, classical domain, and node domain matter element of equipment development scheme decision are constructed, the alternative scheme set of equipment development scheme is recorded as , the corresponding index attribute is recorded as the characteristic of and record it as , the value of about is recorded as , and is the score value of experts on the evaluation grade of the index. Considering the consistency of data structure, the score in this model has been obtained by solving the prenonlinear programming model and Shapley function, which is the standardized value of under scheme . Accordingly, the matter element matrix of equipment development scheme decision evaluation is constructed as follows:where is the matter element for equipment development scheme decision; is the number of index attributes; is the ith index attribute; is the score of , .
When optimizing the equipment development scheme, under normal circumstances, failure to set a reasonable evaluation grade dimension will lead to the same evaluation result grade, especially for multischeme attribute comprehensive evaluation problem. According to matter element extension theory, the fundamental reason for this kind of situation is that the dimension of evaluation grade is small when building the model, which makes the probability that the evaluation result is in the same interval of evaluation grade being high.
In view of the previous problems, the following solutions are proposed:

Step 2. Reasonably increase the dimension of evaluation grade, so that the probability of different evaluation results falling into the same evaluation grade is small enough and so that the evaluation results are distributed in different evaluation grade intervals and have stronger discrimination. In this step, the set evaluation grade is recorded as cross evaluation grade, and only the schemes are sorted, and the advantages and disadvantages of the schemes are not evaluated. The dimension of cross evaluation grade is recorded as .

Step 3. Construct the evaluation grade of advantages and disadvantages, set the reference table of advantages and disadvantages to meet the decision-making requirements, and evaluate the advantages and disadvantages of the scheme based on the results of Step 2. The dimension of the evaluation grade is recorded as , and generally, there is .
The classical domain represents the value range when the equipment development scheme is in each cross evaluation level, and the matter element matrix of the classical domain set in Step 2 is as follows:where is the classical domain matter element of the th cross-evaluation grade under the scheme , is the serial number of the cross-evaluation grade and ; is the value range of index attribute in the th cross-evaluation grade, and and are the upper and lower limits of the value range. Record as the th cross-evaluation grade.
The nodal domain represents the value range of the equipment development scheme in the whole domain of the quality evaluation grade, and the nodal domain matter element matrix is as follows:where denotes the nodal matter element, which is the total range of under the scheme ; is the total range of , and and are the upper and lower limits of the range; is set as the whole area of cross-evaluation grade.

Step 4. Calculate the correlation degree of the respective index attribute under the equipment development scheme using the correlation function,
where the correlation function is as follows:where is the correlation degree of attribute with respect to the th evaluation level. is the distance between and ; is the distance between and .
The equation is as follows:

Step 5. According to the expert intuitionistic fuzzy decision-making information and equations (1)–(10), using an optimization algorithm to solve the attribute measure and value, we calculate the Shapley values of the respective attribute to form a set , which is used as the weight of the corresponding attribute and is used for calculating the comprehensive correlation degree of the scheme.

Step 6. Calculate the comprehensive correlation degree of the scheme. On the basis of knowing the attribute weight and correlation degree of the respective index of the equipment development scheme, the comprehensive correlation degree vector of the development scheme to be evaluated is calculated, and the equation is as follows:where is the weight vector of the index attribute of the equipment development scheme; is the comprehensive correlation degree vector of scheme , ; is the correlation degree matrix of index attributes of equipment development scheme as given as follows:where denotes the correlation degree matrix of index attribute , , .

Step 7. Evaluate of the comprehensive cross-evaluation level of equipment development scheme. The comprehensive correlation degree vector is analyzed, and the comprehensive cross evaluation grade of equipment development scheme is determined following the principle of maximum membership degree, i.e., if , the comprehensive cross evaluation grade of equipment model development scheme is .

Step 8. Determine the evaluation level of equipment development scheme. Set the merits and demerits evaluation level of the overall evaluation of the optimal scheme. Set and the reference table of merits and demerits evaluation level through Step 3. The position of in the scale interval is analyzed, and the advantages and disadvantages of the optimized scheme are obtained if the grade number of the scheme optimized by cross-evaluation grade is , where denotes the scale upper limit of the reference table of the advantages and disadvantages evaluation grade. Table 1 lists the classification table of good and bad evaluation grades.

3. Descision-Making Case of Equipment Development Scheme

The equipment development scheme decision is an important link in the development of weapon and equipment system and an important system to avoid major decision-making mistakes and ensure the smooth progress of national defense scientific research, production and use. With the optimization of the equipment development scheme decision, the risk of weapon and equipment development can be effectively controlled, equipment quality can be improved, and military needs can be satisfied. This example conforms to a certain type of weapon equipment development scheme decision-making work, the purpose of which is to select the best scheme from the equipment development alternative scheme set . Specifically, it is evaluated from five first-level index attributes.

The attributes of evaluation index are as follows:-Satisfaction of strategic intention;-Equipment survivability;-Operational effectiveness;-System availability;-Cost.

3.1. Determine the Decision Information oF IFS

The objective intuitionistic fuzzy evaluation matrix for determining the evaluation attribute set under the alternative is shown in Table 2.

The subjective intuitionistic fuzzy evaluation matrix for determining the alternative is shown in Table 3.

According to the intuitionistic fuzzy score equation (1), the objective intuitionistic fuzzy score matrix is obtained as follows:

Taking as the matter element value of equipment development scheme decision, the evaluation matter element matrix can be obtained as follows:

The subjective intuitionistic fuzzy score matrix is written as

The subjective and objective intuitionistic fuzzy score deviation matrix is obtained from equation (7). To prevent positive and negative deviations from canceling each other in fuzzy measure integration operation and weaken deviation accumulation effect, absolute value treatment is pretended. The results are as follows:

3.2. Solution of Fuzzy Measure and Attribute Correlation Factor According to Intuitionistic Fuzzy Decision Information

According to equation (10), the nonlinear programming model based on IFS and attribute measures is

Through the optimization algorithm, the relevant fuzzy measure values are , , , , , among which .

The fuzzy measures for obtaining attribute power sets are shown in Table 4.

Here, represents the set representation of attributes, expresses the number of elements in the set, and the elements in the attribute set of each column of the previous list are arranged in sequence according to the subscript serial number. For instance, column refers to in turn.

For fuzzy measure , the correlation between attributes is significant, thus leading to super additivity and positive cooperation between attributes. For the fuzzy measure corresponding to , the measure of is 0.4443, which reaches the minimum in the column of , whereas the measure corresponding to is 1, suggesting that the marginal contribution produced by adding attribute to the set reaches the maximum as compared with that of other attributes. Thus, the experts involved in decision-making consider that the development cost is a relatively vital factor for the equipment development scheme decision. The measure structure of the attribute power set in this example is shown in Figure 2.

According to the integral property of Choquet fuzzy measure, the rearrangement matrix of the respective attribute value of under the scheme set is determined as follows:

By equation (8), the corresponding fuzzy measure matrix of Choquet fuzzy integral of under scheme set is obtained as follows:

Table 5 lists the corresponding values of the final subjective and objective intuitionistic fuzzy score deviation under the scheme set in accordance with the intuitionistic fuzzy score deviation matrix and fuzzy measure matrix combined with equation (6).

Shapley value of the respective attribute under the scheme set could be solved by equation (5), as shown in Table 6.

The corresponding result serves as the attribute weight value in the matter element extension optimized model.

In this case, the cross-evaluation grade dimension and the good-and-bad evaluation grade dimension are determined through experimental analysis.

With as the matter element value of equipment development scheme decision, the classical domain matter element matrix is expressed as

The node domain matter element matrix is expressed as

3.3. Equipment Development Scheme Decision

Based on equations (14) and (15), the distribution diagram of correlation degree of the respective index attribute of equipment development scheme decision under different schemes with cross evaluation level under scheme set is shown in Figures 37

Based on equations (16) and (17), the comprehensive correlation degree of matter element matrix of equipment development schemes in the classical domain of cross-evaluation under scheme set is obtained.

Table 7 lists the comprehensive correlation degree of equipment development.

The trend of comprehensive correlation degree of equipment development scheme with the cross evaluation level is shown in Figure 8.

Through the classical domain matter element matrix and cross-evaluation grade, the comprehensive correlation degree of the total evaluation value of model scheme is sorted. Because the cross-evaluation level of scheme is , the cross-evaluation level of scheme is , the cross-evaluation level of scheme is , the cross-evaluation level of scheme is , and the cross-evaluation level of scheme is .

Since , scheme is at the highest cross-evaluation level in the scheme set, i.e., , and the optimal result is scheme .

Subsequently, the advantages and disadvantages of scheme are selected, and the final optimal scheme is obtained as “qualified” in combination with Table 1.

4. Conclusion

The conventional matter-element extension evaluation method typically utilizes methods such as the Analytic Hierarchy Process (AHP), Entropy Weight Method, Simple Correlation Function Method, and Expert Survey Method to determine attribute weights. However, the abovedescribed methods either involve significant subjective factors or do not consider the differences in the significance of evaluation indicators to the problem at hand. Besides, the methods overlook the interactive effects of evaluation attributes. In practical evaluation problems, the significance of indicator factors is objectively presented and affected by the subjective intentions of decision-makers. On that basis, both subjective and objective decision information should be effectively aggregated and the interactive effects of evaluation attributes should be fully considered to obtain fair decision results.

In this study, a nonlinear programming problem with subjective and objective fuzzy information discrepancies is proposed to build a comprehensive evaluation model for matter-element extension evaluation. The measurement values and relevant factors of evaluation attributes are calculated using Choquet integral and optimization algorithms, and the attribute weights are determined based on the Shapley value method. The proposed model optimizes the conventional matter-element extension decision-making model by incorporating fuzzy information and attribute weights, such that the consistency and strong correlation of parameters are ensured, and the limitations of conventional weight determination methods are addressed. Moreover, a two-level evaluation rank model is introduced to cope with the roughness of conventional matter-element extension models, such that scheme rankings and grades can be effectively determined. Furthermore, the effectiveness of the proposed model is verified based on a multiattribute decision-making problem under the context of equipment development.

The main innovations of this study are elucidated as follows:(1)Using Choquet integral method and the score function considering the influence of preference to find the intuitionistic fuzzy score value of experts, the accuracy and consistency of using expert decision information are optimized.(2)For the comprehensive evaluation of multischeme attributes, during the building of the matter element extension evaluation model, the unreasonable setting of evaluation grade dimension often results in the problem of low discrimination of evaluation results, and the reasonable adjustment of evaluation grade dimension makes the correlation degree decline to a relatively accurate interval, such that the accuracy and discrimination of conclusions are enhanced.(3)Based on small sample data, the expert intuitionistic fuzzy score value and attribute measure Shapley value of the score function considering preference influence being transmitted to the evaluation matter-element matrix of the matter-element extension model and the attribute weight required for solving the comprehensive correlation degree, respectively. On that basis, the consistency of data usage is enhanced, and the intuitionistic fuzzy measure integration method and matter element extension are well fused to avoid error accumulation.

Data Availability

The calculation data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This research was funded by the Domain Fund (no. 145BZB17006).