2.1. Interval-valued picture fuzzy set

With the advent of the information age and the rapid development of computer technology, huge and complex fuzzy datasets have appeared in many fields of daily life [19]. To address practical uncertainties, numerous scholars have formulated classical theories that align with real-world requirements. For example, since the emergence of fuzzy sets (FS), it has quickly become a hot spot in the study of uncertainty theory, and has generated research and discussion by scholars [20]. To facilitate specific applications, various extended forms of fuzzy sets have been put forth. For example, Yager first proposed theory of Pythagorean fuzzy set to describe the uncertainty issues of experts in MCDM. Subsequently, numerous scholars conducted research on the Pythagorean fuzzy set theory. A responsible enterprise not only focuses on its own development, but also on the protection of the ecological environment. Green supplier selection contributes positively to environmental protection. Therefore, Mostafa et al.[21] conducts research on Python fuzzy sets, they apply Pythagorean fuzzy set and TOPSIS method to determined the best supplier. Morsy [22] proposed a method that incorporates Pythagorean Fuzzy Number into risk return rate, portfolio risk amount, and expected return rate. Finally, an example was used to verify the effectiveness of the proposed method. In 1986, Atanassov [23] first proposed the theory of IFS, which mainly solves the indecision phenomenon in which experts may not know enough about the decision object or other factors in the decision-making process. IFS has three parts: a membership degree, a non-membership degree and an uncertainty degree. Because of its neutrality, IFS is more widely used in daily decision-making than fuzzy set theory. As the decision-making object becomes more complex, IFS theory cannot be solved. Taking expert voting as an example, there may be four possible outcomes: support, neutrality, opposition, and abstention. Therefore, in 2013 Cuong et al.[24] proposed an extended and new fuzzy set-PFS to deal with the above situation.

However, in MCDM problems, due to the limitations of experts’ understanding of decision-making objects and the ambiguity of the decision-making environment, experts can give an interval number rather than a specific real number when making a decision. Therefore, in 2013, Cuong [24] proposed IVPFS theory, so that membership, neutrality, non-membership and abstention can be represented by interval numbers to enhance the credibility of decision-making results. In certain real-world applications, IVPFS theory, an extension of PFS, are more adept at handling and modeling inconsistent, indeterminate, and incomplete data [25–27]. Therefore, it becomes very important to study the group decision problem under the IVPFS.

2.2 Interval-valued picture fuzzy ordered weighted interactive averaging operator (IVPFOWIA)

The aggregation operator is a particularly efficient technique for solving MCDM problems. To clarify concerns with decision-making, many scholars have studied aggregation operators in recent years. For example, Wei [28] put forward some aggregation operators under picture fuzzy set, then used these operators to deal with MCDM problems. In final, a typical example was used to demonstrate the effectiveness of the proposed operators. Due to the advantages of Archimedes operator and Dombey operator in providing appropriate generalization ability and flexibility in aggregating information, respectively. Saha et al. [29] concentrate on the use of these operators in the classifier’s aggregate similarity calculation. Finally, by comparing with previous operators, it is verified that the proposed operator has higher accuracy. Senapati et al. [30] introduced the Aczel-Alsina (AA) weighted geometric operator and the AA ordered weighted geometric operator for IFS, the effectiveness of the proposed operator was verified through two typical cases. Liu et al.[31] first introduced the Archimedean Heronian (AH) operator and the weighted Archimedean Heronian (WAH) operator for IFS. Then, they put in multiple attribute group decision making approach based on these operators. Similarly, verify the effectiveness of the proposed operator through its application in typical cases. Due to the stability and flexibility advantages of the Frank aggregation operator, Zeng et al. [32] proposed a Frank operator for the q-rung orthopair fuzzy set. Besides, they discuss the some properties,such as idempotency, commutativity, monotonicity and boundedness. Batool et al. [33] proposed a novel decision-making approach based on aggregation operators under Pythagorean probabilistic hesitant fuzzy number. In addition, they also conducted a detailed discussion on some properties of the proposed operator. Regarding the issue of defects in some operators, Luo and Long [34] proposed a weighted geometric operator and an ordered weighted geometric operator for a picture fuzzy set (PFS). Finally, they use the aggregation operators to deal with MCDM problems. Garg [35] put forward various fuzzy picture aggregation operations, and combined them with classic methods to solve practical problems. Nevertheless, the aforementioned aggregate operator merely takes the attributes’ independence into account; in reality, many attributes will be connected in varying degrees like complementary relationships, redundancy, preference relationships, etc. In our study, we applied the OWIA operator to IVPFS, taking advantage of OWIA to eliminate the correlation of indicators in the process of MCDM.

2.3. Improved TOPSIS method

MCDM is a decision-making approach used to evaluate and select the best option from a set of alternative solutions when multiple criteria or factors need to be considered [36]. MCDM methods help decision-makers rank and prioritize options based on these criteria, making it easier to make informed and objective decisions [37–40]. This approach is widely applied in various fields such as engineering, management, environmental planning, and medical decision-making. Through MCDM, decision-makers can better balance various factors to make wiser decisions.

MCDM methods include the weighted summation method, weighted product method, TOPSIS method [41], the analytical hierarchy process (AHP) [42], the grey relation projection method (GRP) method [43], utility theory, graphical tools, quality function deployment (QFD) and fuzzy logic method [44]. In 1981, the TOPSIS technique was initially introduced as a prevalent method for MCDM. TOPSIS method is used to select the best choice among a set of schemes. The basic idea of the TOPSIS method is to assume positive and negative ideal solutions, calculate the distance between each sample and the positive and negative ideal solutions, obtain their relative closeness to the ideal solution (i.e., the closer they are to the positive ideal solution, the farther they are to the negative ideal solution), sort each solution, and ultimately determine the optimal solution. TOPSIS is widely applied in engineering, management, and decision sciences to effectively address multi-criteria decision problems. However, TOPSIS method only considers the Euclidean distance between indicators, and cannot reflect the dynamic changes of evaluation index series. There are shortcomings in decision analysis, especially for dynamic series. Grey correlation analysis can reflect the internal changes of each scheme, making up for the shortcomings of the TOPSIS method. In our research, the TOPSIS model is combined with the grey correlation method to improve TOPSIS, defining a new set of positive and negative ideal points and a formula for relative closeness. The improved TOPSIS model solves the problem of large fluctuations in data by calculating the degree of correlation between evaluation objects. When calculating Euclidean distance, it avoids the phenomenon of disordered equidistant sorting between sample points and positive and negative ideal points, and has strong stability and objectivity.

3.1. Picture fuzzy set

As an important extension of IFS, PFS has received increasing academic attention in recent years. Compared to FS [20] and IFS, PFS is more flexible and practical for expressing and processing fuzzy information. PFS takes into account the support, uncertainty, opposition and abstention of decision makers when evaluating alternatives in real-life scenarios, and expresses the ambiguity and uncertainty of decision makers about the decision problem they face in terms of four aspects: positive membership degree, neutral membership degree, negative membership degree and refusal membership degree [24].

The following are significant definitions and related properties of PFS.

Definition 1 [24] Let X be a non-empty set, then it is said that: (1)

Where the terms μ_A: A, η_A: A, v_A: A∈[0,1] denote the degrees of positive, neutral, and negative membership of an element x in X, respectively. Additionally, μ_A, η_A and v_A meet the following requirement: (2)

And is said to be refusal membership degree of an element x in X. If ξ_A(x) = 0, in this case A returns to the IFS.

Definition 2 [24] Suppose that θ₁ = (μ₁, η₁, v₁) and θ₂ = (μ₂, η₂, v₂) are two different PFNs, then

1. ;
2. ;
3. ;
4. .

3.2. Interval-valued picture fuzzy set

Cuong et al. (2013) proposed IVPFS theory based on PFS [24]. This theory allows for a more flexible representation of uncertainty and ambiguity in data, making it highly suitable for decision-making in complex systems [45, 46]. To formalize this concept, let X be a given domain of discourse. Let G[0,1] be the subintervals set of interval [0,1] and x≠0 be the given set. Thus, an IVPFS can be described as follows: (3)

Where . The intervals μ_A(x), η_A(x), ν_A(x) represent positive, negative and neutral membership degrees of A, and represent the lower and upper end points. Then, an interval-valued picture fuzzy set A can be written as: (4)

Where and . Refusal membership degree expressed by π_A can be determined using the formula below (5).

(5)

Definition 3 For two IVPFNs A = (μ_A(x), η_A(x), ν_A(x)) and B = (μ_B(x), η_B(x), ν_B(x)). α as a scalar value α>0. The following shows the basic and significant operations of IVPFS:

The operations based on IVPFS can give the following properties:

Suppose that M, N & H are the three IVPFNs, where M = (μ_M(x), η_M(x), ν_M(x)) and N = (μ_N(x), η_N(x), ν_N(x)) and H = (μ_H(x), η_H(x), ν_H(x)) and α>0, therefore:

Definition 4 Let that be the IVPFN, Ω is the set of whole IVPFNs. ω = (ω₁, ω₂,⋯,ω_n)^T as the weight vector of them, a mapping IVPFOWIA: Ωⁿ→Ω of dimension n is an interval-valued picture fuzzy ordered weighted interactive averaging (IVPFOWIA) operator, with . Then, (6)

Let be a collection of IVPFNs, then by Eq (6) and the weighted interaction operational laws in Definition 4, the aggregated result by using the IVPFOWIA operator is also an IVPFN, and then: (7)

Where ω = (ω₁, ω₂,⋯,ω_n)^T is the weight of A_i(i = 1,2,⋯,n), such that .

Appendix A provides the proof for this theorem.

Definition 5 Let be an IVPFN, then the score function S(A_i) and the accuracy function H(N_i) of the IVPFNs can be described as: (8)

In the light of the equation of the score function S(A_i) and the accuracy function H(A_i) given in Definition 5, the size of the two IVPFNs can be compared by the following equation:

1. Supposing that S(A₁)<S(A₂), so that A₁<A₂
2. Supposing that S(A₁) = S(A₂), so that:
   1. Supposing that H(A₁)<H(A₂) so that A₁<A₂
   2. Supposing that H(A₁) = <H(A₂) so that A₁ = A₂

The rule can be drawn from the above equation: the larger the score function S(A_i) is, the larger the IVPFN is.

Definition 6 Let be two IVPFNs, so the Hamming distance of A₁ and A₂ is as follows: (9)

The Euclidean distance of A₁ and A₂ is as follows: (10)

3.3. The entropy of interval-valued picture fuzzy set

3.3.1. Shannon’s entropy.

As a prevalent technique for weight determination, Shannon’s entropy objectively assigns weights to indicators based on the amount of information they provide and the correlation between them [2, 47, 48], and it has been used in areas such as product design, environmental protection and medicine and health [48]. In order to mitigate the influence of human-induced discrepancies, Shannon’s entropy is employed to compute weights by quantifying the extent of divergence among individual index values. According to Wang and Lee [49], The specific calculation of Shannon’s entropy is as follows:

Suppose that a team of decision makers has been given the task of weighing m alternatives against n design criteria. Let D = [y_ij] be the decision matrix, where 1<i<m and 1<j<n.

Step 1: Standardize the decision matrix D, and eliminate differences in scale between criteria.

(11)

Step 2: Compute the information entropy matrix of criterion E = (e_j) by Eq (12).Where e_j denotes j^th criterion’s entropy value.

(12)

In Eq (12): k is a constant, we can use formula to calculate it.

Step 3: Determine the weight () of the criteria element, which is calculated as follows: (13)

In Eq (13), .

3.3.2. The entropy of IVPFS.

Due to the properties of the information environment, Shannon entropy has various extensions. For example, Burillo and Bustince [50] introduced entropy of IFS, and Chen [2] employed entropy of rough set (RS) in design concept evaluation. Joshi and Kumar [51] defined the entropy of interval-valued intuitionistic hesitant fuzzy set (IVIHFS), by using the entropy of IVIHFS to calculate the weights of criteria. Ye [52] introduced a fuzzy cross entropy of the interval-valued intuitionistic fuzzy sets (IVIFS) to determine the weights of schemes. Qiyas et al. [53] proposed the entropy of generalized interval-valued picture fuzzy linguistic set (GIVPFLS), which is defined in detail in Definition 7. Based on the literature review on entropy weight mentioned above, we can see that many scholars have provided entropy measures for RS, IFS, IVIHFS, IVIFS, and GIVPFLS. In this manuscript, we utilize the entropy weighting method proposed by Qiyas et al. In MCDM, IVPFS exhibits a notable advantage in addressing decision-maker hesitancy and uncertainty. A key feature of IVPFS is its incorporation of abstention intervals, which proves invaluable in handling decision-maker hesitancy [54]. When decision-makers encounter uncertainty regarding certain elements or attributes, or when they cannot definitively ascertain the membership and non-membership degrees of elements, they can employ abstention intervals to signify their hesitancy. These abstention intervals allow decision-makers to express the extent of their uncertainty within a specified interval range, without the necessity of making explicit decisions. This flexibility is particularly pertinent for complex real-world decision problems where decision-makers regularly confront uncertainty and conflicting information. The entropy of IVPFS is a useful method, especially in circumstances when decision-makers are unable to convey their preferences about criteria during the early phases of the decision-making process or are unsure of how to do so. This approach accounts for the degree of uncertainty pertaining to the alternatives offered by decision-makers and the divergence among various aspects within the IVPFS, permitting the straightforward computation of the relative relevance of the criteria.

Definition 7 [53] let be a separate element in an IVPFLS . For this, entropy is defined as.

According to the Definition 7, we derive the calculation formula for the entropy method of IVPFS.

(14)

Subsequently, the subsequent formula is employed for the computation of attribute weights.

(15)

for all j = 1,2,⋯,n.

4.1 Prepare phase

It is vital to determine each of the assessment criteria and sub-criteria during the design concept formulation and selection process [55]. The evaluation indicators for proposing design concepts can be determined from the perspectives of product economic benefits, technology, and environmental protection. The design concepts evaluation index system has been constructed from four aspects: aesthetic evaluation, technical evaluation, economic evaluation, and environmental evaluation. We have collected and summarized literature on design concept evaluation in recent years, and found that certain indicators have appeared multiple times in the design concept evaluation indicator system. In order to identify the most appropriate indicators for the evaluation of design concepts, a panel of decision-makers from relevant fields such as industrial design and design concepts was invited to this study. By communicating with the expert group, analyzing and determining the evaluation indicators of design concept, a design concept evaluation indicator system is constructed, as shown in Fig 2. For each standard layer and its sub-standard layers, literature sources are provided in Table 1.

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Fig 2. Design concept evaluation index system.

The design evaluation index system is mainly divided into goal layer and criteria layer. The criteria layer includes first level criteria and second level sub-criteria layer.

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Table 1. Design concept evaluation index system.

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4.2 Determine comprehensive evaluation decision matrix

According to the index system in Table 1, for such decision problems as design concept evaluation, the indicators are portrayed by qualitative methods, which have uncertainty and imprecision. In this regard, there are generally two types of measurement methods: a quantitative measurement method, where each indicator is artificially assigned a deterministic value within a specified range of values; and a qualitative level measurement method, where each indicator is assigned a certain level within a specified range of levels. Therefore, the latter method is often used to avoid extreme evaluation events and reflect the fuzzy characteristics of fuzzy indicators, such as using A, B, C, and D levels. Of course, there are also extreme events, some are assigned A and some are assigned D. However, the dispersion of such ratings is obviously much smaller for different ratings given by multiple decision-makers. If we can adopt a scientific approach to normalize the statistics and apply them scientifically without losing the information contained in them, the latter must be better than the former. To this end, this research proposes a new index measurement method based on the connotation of IVPFNs and its generation mechanism.

For the MCDM problem of design concept evaluation, we assume D = {D₁, D₂,⋯,D_k} is the set of k DMs, C = {C₁, C₂,⋯,C_n} is the set of n design criteria, A = {A₁, A₂,…,A_m} is the set of m design schemes. The weights of design criteria are presented by w = (w₁, w₂,⋯,w_j), where . The next sections discuss the specifics of the established design concept alternative evaluation model based on these assumptions.

Step 1: Construct interval-valued picture fuzzy decision matrices belonging to the criteria.

Based on the views of all decision makers, under the l^th sub-criteria with respect to criterion C_j, IVPF evaluation values of alternatives

(16)

The specific calculation steps are as follows:

First, for sub-criteria with respect to criteria C_j, each decision-maker evaluates n design concept alternatives. The evaluation opinions are provided by the questionnaire survey in Table 2 and Fig 3.

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Fig 3. IVPFN composition graph.

This figure includes four evaluation options (High、Medium、Low and Difficult to judge). Each evaluation option has two attitudes, namely affirmative and hesitate.

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Table 2. Sub-criteria evaluation questionnaire.

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Second, for data statistics, for the second level evaluation indicator with respect to criteria C_j, calculate the evaluation opinions of all decision-makers on n alternatives. can be calculated by Eq (17).

(17)

Where P represents the number of all evaluation experts. denotes the number of evaluation experts who hold a high (affirmative) attitude towards the design concept alternative A_i relative to the sub-criterion denotes the number of evaluation experts who hold a high (hesitate) attitude towards the design concept alternative A_i relative to the sub-criterion denotes the number of evaluation experts who hold a medium (affirmative) attitude towards the design concept alternative A_i relative to the sub-criterion denotes the number of evaluation experts who hold a medium (hesitate) attitude towards the design concept alternative A_i relative to the sub-criterion denotes the number of evaluation experts who hold a low (affirmative) attitude towards the design concept alternative A_i relative to the sub-criterion denotes the number of evaluation experts who hold a low (hesitate) attitude towards the design concept alternative A_i relative to the sub-criterion .

By using Eq (17), building the IVPF evaluation matrix corresponding to sub-criteria belonging to the criterion C_j.

(18)

Step 2: Construct comprehensive IVPF decision matrix.

the weight of the sub-criteria can be computed using Eqs (14)–(15). Through applying Eq (19), compute the aggregated value for design concept scheme A_i relative to criterion C_j by employing IVPFOWIA.

(19)

Finally, the comprehensive IVPF decision matrix can be determined by using the IVPFOWIA operator given in Eq (20): (20)

4.3 Determine and choose the best design concept alternative(s)

In this phase, we employ the improved TOPSIS technique based on IVPFS to rank the best design concept scheme. Following are the detailed steps:

Step 1: Standardize the decision-making assessment matrix. To counteract the impact of varying criteria scales on the decision-making procedure, it is essential to render the initial matrix dimensionless.Therefore, in this step, we transform the initialized judgement matrix into a normalized decision matrix .

(21)

Where i = 1,2,⋯m, j = 1,2⋯,n.

Step 2: Utilizing the standardized judgment matrix, ascertain the IVPF-positive ideal solution (R⁺) and the IVPF-negative ideal solution (R⁻): (22) (23)

Where: (24) (25)

Step 3: Compute the grey correlation degree between alternatives and positive (negative) ideal points, respectively.

1. i. Compute the grey correlation coefficients and between alternatives and the IVPF-PIS (NIS) , respectively:

(26)

Where ρ is called the resolution coefficient. When ρ = 0, there is no surrounding environment. Likewise, when ρ = 1, nothing changes in the surrounding environment. Usually, is the distance between and which can be obtained by Eq (9).

1. ii. Compute the grey correlation degree and between alternatives and IVPF-PIS (NIS).

(27)

(28)

Step 4: Compute the Euclidean distance between alternatives and IVPF-PIS (NIS).

(29)

Normalize and separately, and merge the normalized , and combinations to obtain the closeness of and between alternatives and the IVPF-PIS (NIS) .

(30)

Step 5: Sort and select the best design concept scheme.

Compute the relative closeness D_i of design concept schemes: (31)

The design concept alternatives are ranked and preferred according to their relative closeness D_i, with higher values of D_i indicating higher design effectiveness of the alternative. Conversely, the lower the value.

5.1. Construct the collective IVPF decision matrix

According to the design concept evaluation index system in Table 1, taking leisure yachts design A₁ as an example. This study invited 30 experts to participate in the evaluation. There are 7 experts who have a high (affirmative) attitude towards the leisure yachts alternative A₁ relative to the sub-criterion , 5 experts hold a high (hesitate) attitude towards the leisure yachts alternative A₁ relative to to the sub-criterion , 5 experts hold a medium (affirmative) attitude towards the leisure yachts alternative A₁ relative to the sub-criterion , 3 experts hold a medium (hesitate) attitude towards the leisure yachts alternative A₁ relative to the sub-criterion , 4 experts hold a low (affirmative) attitude towards the leisure yachts alternative A₁ relative to the sub-criterion , 5 experts hold a low (hesitate) attitude towards the leisure yachts alternative A₁ relative to the sub-criterion , and then, taking P = 30 in Eq (17), IVPF evaluation value can be calculated for leisure yachts alternative A₁ relative to sub-criteria :

Therefore . In this study, we can also calculate the leisure yachts alternative A₁ relative to the sub-criteria relative to criterion C₂. The above calculation process is based on A₁ as an example And then, the IVPF evaluation value for the leisure yachts alternatives A₂−A₅ relative to the sub-criteria , which are shown in Table 3, is determined using the same calculation procedure. Finally, we can also compute the IVPF evaluation matrix relative to the sub-criteria belonging to criteria (C₁, C₃, C₄) using the same procedure.

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Table 3. Judge matrices belonging to the sub-criteria of Technology (C2).

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The entropy of IVPFS can be used to computed the local weights of sub-criteria, as shown by Eqs (14)–(15):

By applying the IVPFOWIA, we can obtain the collective decision matrix as shown in Table 4.

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Table 4. The comprehensive IVPF decision matrix.

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With the help of Eqs (14)–(15), we can determine the entropy weights of IVPFS of C = {C₁, C₂, C₃, C₄} is w = (0.245,0.244,0.251,0.260)^T.

5.2. Determine and choose the best design concept alternative(s)

Step 1: Due to the four criteria are all benefits (not costs), according to Eq (21), therefore the normalized evaluation decision matrix is consistent with Table 5.

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Table 5. The standardized IVPF decision matrix.

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Step 2: The IVPF-PIS and IVPF-NIS of the standardized decision matrix are calculated through Eqs (22)–(23).

Step 3: Determine the grey correlation degree between alternatives and positive (negative) ideal points, respectively.

Determine the grey correlation coefficients and of each alternative with the positive (negative) ideal point sets and by using Eq (26), and then calculate the grey correlation degree and of each alternative with the positive (negative) ideal point sets and based on Eqs (27)–(28). The results obtained are shown in Table 6.

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Table 6. Grey correlation degree .

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Step 4: Compute the Euclidean distance between alternatives and IVPF-PIS (NIS) by Eq (29). The results obtained are shown in Table 7.

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Table 7. Euclidean distance.

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Step 5: Select the best design concept scheme.

Compute the relative closeness D_i of each design concept alternative through Eqs (30)–(31). The specific parameters and alternatives are listed in Table 8.

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Table 8. The order of the five design concept alternatives.

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5.3. Sensitivity analysis

In this section, we conduct the sensitivity analysis of the resolution coefficient ρ in the proposed IVPF-Improved TOPSIS method. When ρ = 0.5, the ranking of the five design concept solutions is A₃>A₁>A₂>A₅>A₄. Fig 4 shows the results of the ranking of the alternatives for different values of the resolution coefficients ρ. According to Fig 4, it can be seen that in addition to ρ = Beyond 0.1, A₃ is always the best of the five design concept alternatives. Sensitivity analysis shows the stability and reliability of the model proposed in this study.

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Fig 4. Results of sensitivity analysis.

This figure presents the sensitivity analysis results for IVPF-improved TOPSIS method at different ρ values (ρ = 0.1, 0.3, 0.5, 0.7, 0.9). When ρ = 0.1, due to ρ The value of is relatively small, which may make the model less sensitive to changes in parameters. Therefore, even slight changes in parameters may result in different sorting results. This is because the smaller ρ The value reduces the sensitivity threshold to changes in weights and evaluation values.

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5.4. Comparative analysis

In order to evaluate and verify the effectiveness of the proposed algorithm in this study, comparison studies are done in addition to the case study using interval-valued intuitionistic fuzzy (IVIF)-Extended TOPSIS method and IVPF-TOPSIS method. Fig 5 compares several models, while Table 9 and Fig 6 present the findings of a comparison of various methodologies.

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Fig 5. Comparison of different models.

This figure illustrates the comparison between various models and their performance in the evaluation process.

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Fig 6. Relative closeness of different MCDM methods.

This figure depicts the relative closeness and comparative analysis of various methods used in the study.

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Table 9. The results of comparisons between different methods.

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From Fig 6, we can see that the proposed model and the other two models obtain the same optimal solution. The differences are reflected in the entire process of design optimization or some data processing steps. The details are as follows:

1. IVIF improved TOPSIS method. The difference between this method and our model is that the fuzzy environment is different. The IVPFS proposed in this research considers IVIFS at the same time. In addition, it also takes into account the neutrality of expert evaluation information, which is closer to the real evaluation process.
2. IVPF-TOPSIS method. The difference between this and our proposed method is that this research improves the TOPSIS method. The improved TOPSIS model solves the problem of large data fluctuations by calculating the degree of correlation between evaluation objects. From Table 9, it can also be seen that the difference in relative closeness between the five design alternatives obtained by the method proposed in this research is smaller, and the difference in relative closeness values between the five design alternatives obtained using IVPF-TOPSIS is significant. Thus, the method proposed in this manuscript has strong sensitivity.

The Spearman correlation coefficients among the results of the three methods are shown in Fig 7. It can be clearly seen from the Fig 7 that the correlation between the IVIF-Extended TOPSIS and IVPF-TOPSIS methods is not very high (Spearman correlation coefficient is less than 0.8), because the IVIF-Extended TOPSIS method did not consider neutrality, and the IVPF-TOPSIS method did not consider the shortcomings of TOPSIS method in its evaluation. The Fig 7 also shows that the correlation between the IVPF-TOPSIS method and the proposed IVPF-improved TOPSIS is very high, up to 0.900. However, the sorting order between these two methods is not exactly the same, indicating the effectiveness of improved TOPSIS method based on IVPFS.

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Fig 7. Spearman correlation coefficients between the results of the different methods.

This figure presents the Spearman correlation coefficients that measure the relationships between the results obtained from various evaluation methods.

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Appendix A

Proof. By applying mathematical induction to positive integer n, we demonstrate that Eq (7) holds.

When n = 1, we have:

Thus, Eq (7) holds for n = 1.

Assuming that the Eq (7) also holds when n = k, then:

Then, when n = k+1, by inductive assumption and Definition 4, we have

The conclusion that Eq (7) is true for positive integer n = k+1 follows from this. After that, it is true for all positive integers n.