A Selection Model of Compositions and Proportions of Additive Lime Mortars for Restoration of Ancient Chinese Buildings Based on TOPSIS

Abstract

To improve the accuracy of choosing restoration materials for repairing ancient Chinese buildings and to mitigate the risk of decision-making, this paper establishes a novel selection model of compositions and proportions of additive lime mortars for the restoration of ancient Chinese buildings. The selection process is influenced by multi-criteria and determined by a group of experts through comprehensive judgment. Thus, it is a multi-criteria group decision-making (MCGDM) problem. Firstly, considering subjective and objective criteria simultaneously, establish a selection index system for compositions and proportions of additive lime mortars in the restoration of ancient Chinese buildings. Secondly, applying a neutrosophic set to characterize experts’ evaluation information and quantify the evaluation information. Thirdly, the best–worst method (BWM) is implemented to obtain criteria weights, and the entropy weight method is utilized to obtain index weights. Finally, obtaining the priority of each alternative solution by using the TOPSIS (Technique for Order Preference by Similarity to Ideal Solution) ranking technique. The practicality of the proposed model was demonstrated through a specific case of the selection of repair materials for a decorative window in one ancient Chinese building. The comparative analysis was carried out to verify the reliability and validity of the model.

1. Introduction

Lime usage has a long history in China [1]. The advantages of lime mortar added with glutinous rice slurry are high strength, reasonable toughness, and sound anti-seepage performance. This method is widely used in the construction of ancient tombs, water conservancy projects, buildings, bridges, and so on. However, since the early 19th century, the invention and widespread use of cement have greatly reduced the demand for lime. In the field of cultural heritage protection, cement has side effects such as excessive strength, low porosity, incompatibility with ancient architectural materials, and the introduction of soluble salts during use, which was gradually pointed out by cultural heritage protection experts. Given the outstanding advantages of traditional lime mortar technology in durability, strength, and harmony with ancient buildings, using traditional lime mortars instead of modern cement materials to restore ancient Chinese buildings is of great significance to ensure the sustainability of ancient building protection and restoration.

The selection of repair materials is a crucial step in the restoration process of ancient Chinese buildings [2,3]. Therefore, it is necessary to develop an appropriate method to select the most suitable compositions and proportions of additive lime mortars based on the actual situation of ancient Chinese buildings restoration. The selection of compositions and proportions of additive lime mortars is influenced by factors such as the basic situation of the repaired components, the performance of the additive lime mortars, and the possible repair results. It is usually determined by a group of experts through comprehensive judgment. Thus, it is a multi-criteria group decision-making (MCGDM) problem.

Although some progress has been made in existing research, there are still some issues that require further consideration. Firstly, the selection of materials for the restoration of ancient buildings is complex, and many factors need to be considered. However, the evaluation information tends to be partial, and the standardization research on the selection process of repair materials is insufficient [4]. Secondly, in practice, experts usually select compositions and proportions of additive lime mortars based on experience, and the comprehensiveness of information judgment is limited by the expert’s personal experience [5]. In this case, there exists relatively little research on using quantitative methods to determine the decision-making process of compositions and proportions of additive lime mortars for the restoration of ancient Chinese buildings.

To sum up, the research methods of this study are as follows:

• Regarding the complexity of evaluating information, the evaluation of compositions and proportions of additive lime mortars includes both subjective and objective criteria. Some studies have used the objective criteria to quantitatively evaluate the restoration of ancient Chinese buildings [6,7]. Subjective criteria, such as possible repair outcomes, have also been mentioned in Lourenço, P.B.’s research [8]. However, in the index system of the proposed model, subjective criteria are combined with objective ones.
• To address the issue of partial use of evaluation information, it is appropriate to use a neutrosophic set to describe evaluation information. Smarandache, F. [9] introduced neutrosophy, whose main argument is that every concept contains a certain degree of truth, as well as a certain degree of falsehood and uncertainty, all of which need to be considered independently of each other [10]. Zhang et al. [11,12] believed that neutrosophic sets can effectively evaluate fuzzy information. Thus, it is necessary to convert the evaluation results into neutrosophic numbers. For example, if the aesthetics of the evaluation repair result are “high”, then the corresponding value of the membership degree of truth is 0.8, the membership degree of falsehood is 0.2, and the membership degree of uncertainty is 0.15.
• In order to prioritize the compositions and proportions of additive lime mortars based on MCGDM, the TOPSIS method could be introduced. According to relevant scholars’ research, TOPSIS, as a practical technique for ranking and selecting many externally determined alternative solutions through distance measurement, is related to multi-criteria group decision-making (MCGDM) [13,14].

The expected results of this study might be (1) establishing an index system for the selection of compositions and proportions of additive lime mortars to repair ancient Chinese buildings; (2) using multi-granularity language terms to represent experts’ information, convert them into single-valued neutrosophic sets, and quantify such information to improve the comprehensiveness and reliability of the evaluation results; (3) combining BWM method and entropy weight method to obtain the criteria and index weights; (4) utilizing the TOPSIS method to determine the ranking of available solutions; and (5) demonstrating the process of the proposed model through empirical research by analyzing a particular case.

Based on the above, this paper aims to propose a new model for selecting compositions and proportions of additive lime mortars in the restoration of ancient Chinese buildings, which can improve the accuracy and standardization of material selection in the restoration process of ancient Chinese buildings, and protect the heritage of ancient Chinese buildings from a decision-making perspective. The content of this paper is organized as follows. A brief review of previous research is provided in Section 2. Section 3 introduces some basic concepts and definitions of neutrosophic sets. In Section 4, compositions and proportions of additive lime mortars selection model for the restoration of ancient Chinese buildings based on MCGDM are constructed. Subsequently, a case study is presented in Section 5, aiming to verify the proposed model through comparative analysis. In the end, Section 6 briefly summarizes the content of this paper and proposes some practical directions for further research.

2. Literature Review

2.1. Research on Lime Mortars

In recent years, many scholars have studied the effects of different additives on the performance of traditional lime mortars and explored the application of lime mortars with different additives in the restoration of ancient buildings [15,16,17]. These achievements have promoted the development of heritage conservation practices and provided more choices for materials used in the restoration of ancient Chinese buildings. Research on lime mortars has been continuously developed. For example, the comparative analysis of the performance evaluation of additive lime mortars provides an experimental basis for the selection of restoration materials for ancient Chinese buildings. Usually, mass spectrometry thermogravimetric analysis, X-ray powder diffraction (XRD), and scanning electron microscopy (SEM) are used to test the mechanical properties and microstructure of additive lime mortars [18,19]. Maria Apostolopoulou et al. [20] used artificial neural networks (ANN) to draw a development chart of mortar characteristics, which can help reveal the impact of parameter settings on each mortar property. B.A. Silva et al. [21] evaluated that different maintenance conditions will have an impact on the carbonation process of lime-based mortars and slurry. Maria Apostolopoulou et al. [22] proved that different proportions of binder and aggregate can significantly alter the mechanical and physical properties of mortars. The most important direction of research on lime mortars is the changes in the properties of lime mortars with different additives and their comparative analysis, especially the improvement of lime mortar properties by organic additives. Hee-Young Hwang et al. [23] conducted a comparative analysis of the performance improvement of lime mortars using three traditional natural organic additives commonly used in East Asia: starch, glutinous rice, and seaweed slurry. Yang, F. et al. [1] conducted a systematic study on the technology of glutinous rice lime mortar to help determine appropriate action plans for restoring ancient buildings. Other lime mortar additives, such as paraffin [24], biological additives [25,26], pozzolan [27], etc., have also been tested for their effects on various properties of lime mortars. Despite the continuous deepening of research on lime mortar materials, the research results have not yet been reasonably applied in practical engineering. In actual projects, experts need to consider not only the performance of lime mortar materials but also the basic situation of the ancient buildings that need to be restored and the possible restoration results. The principles of authenticity, recoverability, and minimal impact should be followed when selecting restoration materials for ancient Chinese buildings. If inappropriate compositions and proportions of additive lime mortars are used, it will lead to the failure of the restoration, even further damaging the ancient buildings.

2.2. Research on Related Algorithm Models

At present, relevant factors that affect the selection of repair materials for traditional building heritages have been considered. For example, some studies explore the current situation of ancient buildings that need to be restored. The existing theory suggests that the best way to protect and maintain ancient buildings is to combine active and passive maintenance strategies [28]. Gao et al. [29] introduced a fuzzy element theory based on asymmetric proximity improvement for efficient evaluation of the health of timber structures. The performance of compositions and proportions of lime mortars is also a factor that affects the selection of repair materials, mainly the improvement of durability, crack resistance, freeze–thaw resistance, corrosion resistance, adhesion, etc. [16]. There have also been studies analyzing the results of the restoration of ancient buildings. Lourenço [8] believes that the continuation of the original structural and material features of ancient buildings is of great significance for the value of architectural restoration and the authenticity of the restoration results. In the research on factors mentioned above, there are subjective judgments about the influencing factors of materials used in the restoration of ancient buildings, as well as objective judgments based on experiments. However, there is a lack of consideration for both subjective and objective standards, making it impossible to comprehensively evaluate the selection of repair material types. However, in practical cases, the factors that affect the selection of compositions and proportions of additive lime mortars are both subjective and objective [30]. Therefore, when constructing the index system, it is necessary to comprehensively consider the subjective and objective criteria of the basic conditions of the components, the performance of the lime mortars, and the repair effect.

Usually, we use subjective weighting and objective weighting methods to determine weights. Rezaei [31] proposed the best–worst method (BWM) for calculating subjective weights to solve multi-criteria decision-making problems. Compared to other methods, such as AHP, that determine subjective weights, the BWM method excels in having fewer data comparisons and higher consistency. However, if only subjective weights are used, the objectivity of the evaluation results cannot be guaranteed, which will bring difficulty and the possibility for decision-makers to make mistakes in their analysis. Therefore, the entropy weight method [32,33] will be used to calculate objective weights. Entropy value helps determine the degree of dispersion of the index by displaying their randomness and disorder. Hence, in order to make the results of index weights more reliable, it is necessary to combine the BWM method with the entropy weight method to calculate the weights of the index, not only the intrinsic characteristics of the index data are considered, but also the experience of experts is taken into account [34].

The neutrosophic set has a certain degree of similarity with human thinking, as it reflects the incompleteness of knowledge, the possibility of acquiring incorrect knowledge, or the uncertainty caused by random guessing [9]. Neutrosophic sets have also been recommended in a medical diagnosis system, as shown in the study of Abdel et al. [35]. In the decision-making process of selecting compositions and proportions of additive lime mortars for ancient building heritage restoration, due to the complexity of actual restoration projects and the multifaceted performance of additive lime mortars, expert evaluation information is often difficult to be comprehensive, and evaluation information often represents experiential preferences, making it difficult to express evaluation information in an accurate data format [36]. Hence, introducing neutrosophic numbers to describe the evaluation results is a suitable strategy.

In summary, it can be seen that fuzzy group decision theory and quantitative analysis have not yet been applied to solve the problem of materials selection in the restoration of ancient Chinese buildings. Therefore, this paper aims to design a model to assist experts in selecting compositions and proportions of additive lime mortars in the restoration process of ancient Chinese architecture. The relevancy between the main literature and the algorithm of the proposed model is shown in

Table 1. The relevancy between the main literature and algorithm of the proposed model.

Literature Source	Research Findings	Relevancy
Naciri, K.; Jayasingh, S. et al. [15,16,17,18,19,20,21,22,23,24,25,26,27]	Introduce the research progress of additive lime mortars, mainly including performance testing, composition and proportion comparison, application directions, etc.	This is the research background of this study, providing experts with multiple possible alternative solutions.
Lourenço, P.B.; Gao, Z.; Jayasingh, S. et al. [8,16,29,30]	Research on the factors affecting the selection of repair materials for ancient buildings restoration mainly includes the current state of the building, material properties, and possible restoration outcomes.	Providing reference for the establishment of an index system that combines subjective and objective criteria in this study.
Rezaei, J. [31]	Proposed the best–worst method (BWM) for calculating subjective weights to solve multi-criteria decision-making problems.	Applying the BMW method to calculate criteria weights.
Majumdar, P. et al. [32,33]	Suggest using entropy weight method to calculate objective weights.	Utilizing the entropy weight method to calculate index weights.
Smarandache, F.; Abdel-Basset, M. [9,35]	Introduced neutrosophy, whose elementary sentence is that every concept contains a certain degree of truth, a certain degree of falsehood and uncertainty, all of which need to be considered independently of each other.	Using neutrosophic numbers to represent expert evaluation information.
Afshar, A.; Shih, H.S. et al. [13,14]	TOPSIS, as a practical technique for ranking and selecting many externally determined alternative solutions through distance measurement, is related to multi-criteria group decision-making (MCGDM).	Ranking alternative solutions using TOPSIS method based on MCGDM.

.

3. Preliminaries

In this section, some concepts and definitions that contribute to the development of compositions and proportions of additive lime mortar selection models are introduced.

Definition 1.

Neutrosophic set [37]. Let X be a space of points (objects), with a generic element in X denoted by x. Then, a neutrosophic set A in X is characterized by three membership functions, including a truth membership function T_A, an indeterminacy membership function I_A, and a falsity membership F_A and is defined as A = {<x, T_A(x), I_A(x), F_A(x)>|x ∈ X}, where T_A(x), I_A(x) and F_A(x) are real standard or nonstandard subsets of ]0⁻, 1⁺[, that is, T_A(x):X→]0⁻, 1⁺[, I_A(x):X→]0⁻, 1⁺[, F_A(x):X→]0⁻, 1⁺[, and satisfying 0⁻ ≤ T_A(x) + I_A(x) + F_A(x) ≤ 3⁺.

Definition 2.

Single-valued neutrosophic set [38]. Let X be a space of point (object), with a generic element in X denoted by x. A single-valued neutrosophic set (SVNS) A in X is characterized by truth membership function T_A, indeterminacy membership function I_A, and falsity membership function F_A with T_A, I_A, F_A ∈ [0, 1] for all x in X. The sum of three memberships of an SVNS A, for all x ∈ X, 0 ≤ T_A(x) + I_A(x) + F_A(x) ≤ 3.

Definition 3

([39]). Let A = {(x₁|<T_A(x₁), I_A(x₁), F_A(x₁)>), …, (x_n|<T_A(x_n), I_A(x_n), F_A(x_n)>)}, and B = {(x₁|<T_B(x₁), I_B(x₁), F_B(x₁)>), …, (x_n|<T_B(x_n), I_B(x_n), F_B(x_n)>)} be two SVNSs for x_i ∈ X (i = 1, 2, …, n). Then, the normalized Euclidean distance between A and B can be defined as follows:

$$D(A,B)=\sqrt{{\displaystyle \frac{1}{3n}}{\displaystyle \sum _{i=1}^n\left\{{\left(T_A(x_i)-T_B(x_i)\right)}^2+{\left(I_A(x_i)-I_B(x_i)\right)}^2+{\left(F_A(x_i)-F_B(x_i)\right)}^2\right\}}}$$

(1)

Definition 4

([32]). Through the research of Majumdar et al., for single-valued neutrosophic set A = {<x, T_A(x), I_A(x), F_A(x)>|x ∈ X}, an entropy on neutrosophic set A can be calculated using the following equation:

$$E(A)=1-{\displaystyle \frac{1}{n}}{\displaystyle \sum _{x_i}\left(T_A(x_i)+F_A(x_i)\right)\otimes |I_A(x_i)-I_{A_C}(x_i)|}$$

(2)

Definition 5

([30]). According to the study by Tan et al., the entropy weight of a neutrosophic set is computed as follows:

$$W_j=\left(1-E(x_j)\right)/{\displaystyle {\sum }_j^n\left(1-E(x_j)\right)}$$

(3)

Definition 6.

In the study by Biswas et al. [40], fuzzification of SVNS $\tilde{N}$ = {(x|<$T_{\tilde{N}}$(x), $I_{\tilde{N}}$(x), $F_{\tilde{N}}$(x)>|x ∈ X} can be shown as follows:

$${\mathrm{\mu }}_{\tilde{F}}(x)=1-\sqrt{{\displaystyle \frac{1}{3}}\left\{{\left(1-T_{\tilde{N}}(x)\right)}^2+I_{\tilde{N}}{(x)}^2+F_{\tilde{N}}{(x)}^2\right\}}$$

(4)

Definition 7

([41]). According to Ye’s research, the single-valued neutrosophic weighted averaging (SVNWA) aggregation operator is defined as follows:

$$F_{A_i}={\psi }_1{A}_1\oplus {\psi }_2{A}_2\oplus \dots \oplus {\psi }_n{A}_n=\langle 1-{\displaystyle \prod _{i=1}^n{(1-T_{A_i})}^{{\psi }_i}},{\displaystyle \prod _{k=1}^n(I_{A_i}}{)}^{{\psi }_i},{\displaystyle \prod _{k=1}^n(F_{A_i}}{)}^{{\psi }_i}\rangle$$

(5)

In which Ψ = (Ψ₁, Ψ₂, …, Ψ_n) is the weight vector of A_i (i = 1, 2, …, n), Ψ_i ∈ [0, 1], and satisfying ${\displaystyle \sum _{i=1}^n{\psi }_i}=1$.

4. Methodology

In the case of ancient Chinese buildings restoration, the selection of compositions and proportions of additive lime mortars should be determined by a panel of experts and is the result of collective decision-making. In the decision-making process, there are many factors to consider, and many indices cannot be evaluated with accurate values. Therefore, a single-valued neutrosophic set is adopted to express the characteristics of each index. Figure 1 shows the detailed process of the proposed model. The rest of this section will explain the detailed information of the model. The process of the proposed model and its corresponding algorithm and principle of TOPSIS method are shown in Figure 2.

4.1. The Establishment of Index System

The selection of compositions and proportions of additive lime mortars in the restoration of ancient Chinese buildings is generally determined by a panel of k experts. According to the literature analysis, experimental results, and expert opinions, the determination of compositions and proportions of additive lime mortars can be described by three categories, represented by three criteria A_i (i = 1, 2, 3): basic condition of component (A₁), performance of different compositions and proportions of additive lime mortars (A₂), and restoration outcomes (A₃). In addition, there are several subcriteria under each criterion that affect the selection of lime mortar materials. Thus, we establish an index system, as shown in

Table 2. Index system of compositions and proportions of additive lime mortars selection for ancient Chinese buildings restoration.

Criteria	Index	Definition	Index Type
Basic condition of component (A1)	Contact surface material (a11)	The degree to which the contact surface material is suitable for the mortar material	Benefit
	Contact surface morphology (a12)	The degree to which the contact surface morphology is suitable for the mortar material	Benefit
	Component structure (a13)	The degree to which the component structure is suitable for the mortar material	Benefit
	Degree of damage (a14)	The degree of damage to the component is suitable for the mortar material	Benefit
Performance of different compositions and proportions of additive lime mortars (A2)	Hardness (a21)	The hardness of the mortar material is suitable for repairing the component	Benefit
	Compressive strength (a22)	The compressive strength of the mortar material is suitable for repairing component to a certain extent	Benefit
	Freeze–thaw resistance (a23)	The degree to which the freeze–thaw resistance of the mortar material is suitable for repairing component	Benefit
	Shrinkage (a24)	The degree to which the shrinkage of the mortar material is suitable for repairing component	Benefit
Restoration outcomes (A3)	The degree of preservation of historical information (a31)	The degree of preservation of historical information by the repaired component using the mortar material	Benefit
	The possibility of irreversible impact (a32)	The possibility of irreversible effects caused by the repair of component using the mortar material	Cost
	Durability (a33)	The durability of the repaired component with the mortar material	Benefit
	Aesthetics (a34)	The aesthetic appearance of the repaired component with the mortar material	Benefit
	Maintenance difficulty (a35)	The difficulty of maintenance after repairing the component with the mortar material	Cost

.

4.2. The Acquisition of Evaluation Matrix

The selection of compositions and proportions of additive lime mortars to repair ancient Chinese buildings requires consideration of actual situation on site, experts, building components to be restored, and various other factors. Experts may provide different evaluation results based on their knowledge structure, experimental data, work experience, different evaluation criteria, and performance of different compositions and proportions of additive lime mortars. Due to the diversity, ambiguity, and complexity of information sources and types, the evaluation values of index are often difficult to represent accurately with numbers. To address this issue, decision-makers typically use language terms such as fuzzy numbers with multiple granularities to evaluate index. Therefore, we convert the evaluation results acquired from experts into single-valued neutrosophic numbers to obtain the evaluation matrix R = (r_ij). The detailed process in this section can be shown as follows:

Step 1. Identifying language terms with multi-granularity. Experts usually use language terms according to their own habits, which may result in different experts having different evaluation values due to semantic differences in language terms. In order to unify the evaluation values of index provided by expert groups in the form of multi-granularity language terms [42], it is necessary to preset a language term set containing ordered language terms. Language term sets can characterize different membership function features. For example, the language term set {n₀, n₁, n₂, n₃, n₄, n₅, n₆, n₇, n₈} can be expressed as {extremely bad, very bad, bad, medium bad, medium, medium good, good, very good, extremely good}, while the language term set {l₀, l₁, l₂, l₃, l₄, l₅, l₆} can be donated as {very low, low, medium low, medium, medium high, high, very high} in linguistic terms. Obviously, different experts may have varying interpretations of the degree to which these terms are expressed. Therefore, it is necessary for experts to provide evaluation values for different compositions and proportions of additive lime mortar selection schemes based on a predetermined language term set.

Step 2. Based on the symmetrical language evaluation scale, convert the evaluation information from expert’s questionnaire into single-valued neutrosophic numbers. The language term set used in this research is {extremely high, extremely high, high, medium high, medium low, low, extremely low}, which includes nine granularity language terms, as shown in reference [43].

Table 3. Language terms for rating alternative solutions using SVNNs.

Language Terms	SVNNs
Extremely high	<1.00, 0.00, 0.00>
Very high	<0.90, 0.10, 0.05>
High	<0.80, 0.20, 0.15>
Medium high	<0.65, 0.35, 0.30>
Medium	<0.50, 0.50, 0.45>
Medium low	<0.35, 0.65, 0.60>
Low	<0.20, 0.75, 0.80>
Very low	<0.10, 0.85, 0.90>
Extremely low	<0.05, 0.90, 0.95>

defines language terms and SVNNs to evaluate each alternative based on each index.

Step 3. Calculating experts’ weights. Different experts have different backgrounds and experiences; therefore, it is necessary to determine the importance of experts in the decision-making process. In this article, the decision-making ability of experts can be represented by a set of language terms {very important (VI), important (I), medium (M), unimportant (UI), very unimportant (VUI)}.

Table 4. Language terms for rating the importance of experts with SVNNs.

Language Terms	SVNNs
VI	<0.90, 0.10, 0.05>
I	<0.80, 0.20, 0.15>
M	<0.50, 0.40, 0.45>
UI	<0.35, 0.60, 0.70>
VUI	<0.10, 0.80, 0.90>

shows language terms and SVNNs to evaluate the importance of experts. Let the weight of expert be represented as e_k (k = 1, 2, …, n). Based on Equation (4), we can calculate the weights of experts using Equation (6) [40]:

$$\begin{gathered} e_k=u_k/{\displaystyle {\sum }_{k=1}^l{u}_k} \\ ={\displaystyle \frac{1-\sqrt{\left\{{(1-T^k)}^2+{(I^k)}^2+{(F^k)}^2\right\}/3}}{{\displaystyle {\sum }_{k=1}^l\left(1-\sqrt{\left\{{(1-T^k)}^2+{(I^k)}^2+{(F^k)}^2\right\}/3}\right)}}} \end{gathered}$$

(6)

Step 4. Aggregating the neutrosophic numbers. The single-valued neutrosophic weighted average (SVNWA) aggregation operator shown in Equation (5) expressed in Section 2 is used to aggregate neutrosophic numbers. Hence, the evaluation matrix R = (r_ij) shall be obtained.

4.3. The Allocation of Index Weights

In this section, a combination of subjective and objective methods is utilized to determine index weights. Firstly, we use an effective BWM method to compute the criteria subjective weights. Then, the entropy weight of each index is calculated based on the principle of objectivity. Finally, according to subjective–objective method, the comprehensive weight of each index is determined. The specific steps are as follows.

4.3.1. Calculating Criteria Weights

Raize proposed BWM method to solve multi-criteria decision-making problems [44]. Defining a set of decision criteria {A₁, A₂, …, An}, then, the BWM method is applied it to calculate the three criteria weights in this section [45].

Step 1. The best (e.g., the most ideal and important) and worst (e.g., the least ideal and least important) criteria are determined by decision-makers. Based on the BWM questionnaire, experts should choose a number between 1 and 9 to determine the preference of the best criteria relative to all other criteria. The resulting best-to-others vector would be A_B = (a_B1, a_B2, …, a_Bn), among them, a_Bj indicates the preference of the best criteria B over criteria j. This can be obtained that a_BB = 1. Then, we can compute the preference of all the criteria over the worst criteria by choosing a number between 1 and 9. Then, we can obtain the resulting others-to-worst vector A_W = (a_1W, a_2W, …, a_nW), in which a_jW represents the preference of the criteria j over the worst criteria W, satisfying a_WW = 1.

Step 2. Obtaining the optimal weights {w₁*, w₂*, …, w_n*} by solving (7). The optimal weight for the criteria is the one where for each pair of w_B/w_j and w_j/w_W, we have w_B/w_j = a_Bj and w_j/w_W = a_jW. In order to meet these requirements for j, we shall find a method to meet the maximum absolute differences $\left|{\displaystyle \frac{w_B}{w_j}}-a_{Bj}\right|$ and $\left|{\displaystyle \frac{w_j}{w_w}}-a_{jw}\right|$ for all j are minimized. Given that the nonnegativity and condition for the weights, the following problem is the result:

$$\begin{array}{l}\min {\max }_j\left\{\left|{\displaystyle \frac{w_B}{w_j}}-a_{Bj}\right|,\left|{\displaystyle \frac{w_j}{w_w}}-a_{jw}\right|\right\}\\ s.t.\\ {\displaystyle \sum _j{w}_j}=1\\ wj\ge 0,for\hspace{0.33em}all\hspace{0.33em}j.\end{array}$$

(7)

Equation (7) is equivalent to Equation (8):

$$\begin{array}{l}\min \xi \\ s.t.\\ \left|{\displaystyle \frac{w_B}{w_j}}-a_{Bj}\right|\le \xi ,for\hspace{0.33em}all\hspace{0.33em}j\\ \left|{\displaystyle \frac{w_j}{w_w}}-a_{jw}\right|\le \xi ,for\hspace{0.33em}all\hspace{0.33em}j\\ {\displaystyle \sum _j{w}_j}=1\\ w_j\ge 0,for\hspace{0.33em}all\hspace{0.33em}j.\end{array}$$

(8)

The optimal weights (w₁*, w₂*, …, w_n*) and ξ* (consistency index) can be obtained through Equation (8). Then, we can calculate the consistency ratio. When consistency ratio is between [0, 1] and the value is close to 0, it indicates better consistency; conversely, if the value is close to 1, it means less consistency.

Hence, the weight vector {w₁, w₂, w₃} of basic condition of component A₁, performance of different compositions and proportions of additive lime mortars A₂, restoration outcomes A₃ is obtained.

4.3.2. Obtaining Index Weights

We calculate the weights of index based on the principle of combining subjective and objective factors in this section. First of all, the entropy weight method is used to calculate each index entropy weight. Then, the criteria weights are combined with entropy weights to obtain the comprehensive weights of the index. The specific process is as follows:

Step 1. Normalizing the evaluation index. We obtain the normalization matrix R = (β_ij) using Equation (9), an equation for normalizing [46] the neutrosophic numbers.

$$r_{ij}=\left\{\begin{array}{c}{T}_{ij},I_{ij},F_{ij}\hspace{1em}\hspace{1em}\hspace{1em}if\hspace{0.33em}j\hspace{0.33em}is\hspace{0.33em}benefit\hspace{0.33em}index\hfill \\ 1-T_{ij},1-I_{ij},1-F_{ij}\hspace{1em}\hspace{1em}if\hspace{0.33em}j\hspace{0.33em}is\hspace{0.33em}cost\hspace{0.33em}index\hfill \end{array}\right.$$

(9)

Step 2. Computing the entropy weights of indices. We can calculate the entropy value of each index E(x_j) using Equations (2) and (3) mentioned in Section 2; thus, the entropy weight w_j of each index can be acquired.

Step 3. Acquiring combination weights. The weights of evaluation index of selecting compositions and proportions of additive lime mortars are crucial, as they can affect the correctness of the evaluation results. The subjective weight method mainly depends on expert experience, which often leads to arbitrary evaluation results. However, the objective weighting method can compensate for this drawback, as it is based on objective reality and can greatly reduce subjective arbitrariness. Therefore, combining subjective and objective methods is an appropriate approach. According to the explanation of BWM in Section 4.3.1, the optimal weight vector of criteria is w_i*. Then, we calculate a comprehensive index weight H_ij based on the following Equation (10) [47].

$$H_{ij}={\displaystyle \frac{w_i^{\ast }\times w_j}{{\displaystyle \sum _{i=1}^n{w}_i^{\ast }\times w_j}}}$$

(10)

4.4. The Decision Process Based on TOPSIS

Hwang and Yoon [48] proposed TOPSIS method to deal with multicriteria decision-making problems. It obtains the best alternative by calculating the choice that is as close as possible to the optimal solution. The basic operation process of TOPSIS method is introduced as the following [49]:

Step 1. Aggregate the combination weights to the decision matrix by making V_ij = H_ijr_ij.

Step 2. Define the positive ideal solution (PIS), v_j⁺, and the negative ideal solution (NIS), v_j⁻, for each criterion.

v_j⁺ = <T_j⁺, I_j⁺, F_j⁺> = <maxT_ij^wj, minI_ij^wj, minF_ij^wj>, i = 1, 2, …, m, j = 1, 2, …, n
v_j⁻ = <T_j⁻, I_j⁻, F_j⁻> = <minT_ij^wj, maxI_ij^wj, maxF_ij^wj>, i = 1, 2, …, m, j = 1, 2, …, n

(11)

Step 3. Calculate the distance between the alternative solution and the positive ideal solution and the distance between the alternative solution and the negative ideal solution according to Equations (12) and (13).

$$S_i^{+}=\sqrt{\sum _{j=1}^n{(v_j^{+}-v_{ij})}^2},\hspace{1em}i=1,\hspace{0.33em}2,\dots ,m$$

(12)

$$S_i^{-}=\sqrt{\sum _{j=1}^n{(v_j^{-}-v_{ij})}^2},\hspace{1em}i=1,\hspace{0.33em}2,\dots ,m$$

(13)

Step 4. Calculate the proximity coefficient between each alternative solution and the ideal solution according to Equation (14).

$$C{C}_i={\displaystyle \frac{S_i^{-}}{S_i^{-}+S_i^{+}}}$$

(14)

Step 5. Rank alternative solutions based on CC_i. The bigger CC_i is, the better the alternative A_i will be.

5. Empirical Study

In this section, we apply the proposed model to solve the decision problem of repairing materials for ancient Chinese buildings. A decorative exterior window of an ancient building called “Dafudi” located in the central region of Hunan, China, needs to be restored. Dafudi is a national cultural heritage site built during the Qing Dynasty over 150 years ago. The exterior windows of Dafudi are made of cedar wood skeleton, which is decorated with mortar and depicted in different shapes, with strong decorative aesthetics and high historical, cultural, and artistic value (as shown in Figure 3). According to local residents, the decorative mortar for the exterior windows is made by mixing lime and glutinous rice. The management agency of Dafudi entrusted company R (Loudi, China) to repair it. A decorative exterior window to be repaired is donated by W₁. There are four experts, donated by DM₁, DM₂, DM₃, and DM₄, who conducted on-site inspections to prepare for obtaining the composition of lime mortar. The experts extracted exterior window decorative mortar material for XRD analysis on site. Through XRD (Instrument model: Bruker AXS X-ray diffractometer D8 Advance (Bruker, Billerica, MA, USA)), it was found that the proportion of calcium carbonate in the exterior window mortar was 69.12%, and there were also compounds of calcium (as shown in Figure 4). Then, considering the improvement of mortar material properties by additives, experts provided four possible repair materials, including non-added glutinous rice lime mortar, 3% tung oil glutinous rice lime mortar, 3% paper-reinforced glutinous rice lime mortar, and 6% aluminum sulfate glutinous rice lime mortar, which are donated by X₁, X₂, X₃, and X₄. In order to assist the experts in making more scientific decisions, the company R used experimental methods to produce samples of glutinous rice mortar with four different additives and observed the microstructure of the repair materials using SEM (Instrument model: Thermo Scientific TM Apreo SEM (Thermo Scientific, Waltham, MA, USA)) (shown in Figure 5). Experts evaluated the four types of repair materials through several evaluation indices (as shown in Figure 2) to determine their compatibility with the building components to be repaired.

According to the decision model proposed in Section 3, firstly, the evaluation matrix through the basic condition of components, performance of different compositions and proportions of additive lime mortars, and restoration outcomes can be obtained. Then, we calculate the index weights by combining criteria weights and entropy weights. Finally, the TOPSIS method can be used to obtain the ranking order of the four repair materials. In addition, the effectiveness and reliability of the proposed model are verified through a comparative analysis.

5.1. The Acquisition of Evaluation Matrix

Obtain an evaluation matrix from experts based on the evaluation index system. According to the evaluation method mentioned in Section 4.2, experts evaluate using the index evaluation values, which are represented by language terms (LT) with multi-granularity to evaluate the four repair materials. According to Table 3, we convert the language terms into single-valued neutrosophic numbers (SVNNs). Then, by assessing the project experience and educational background of the experts, the importance of the four experts is shown in

Table 5. Importance of experts expressed with SVNNs.

	DM1	DM2	DM3	DM4
LT	M	M	I	VI
SVNS	<0.5, 0.5, 0.45>	<0.5, 0.5, 0.45>	<0.8, 0.2, 0.15>	<0.9, 0.1, 0.05>

. The importance of each expert was determined using the language terms and their related SVNNs in Table 4, as shown in Table 5. In light of Equation (6), the weights of experts (e_k) are calculated separately: e₁ = 0.187, e₂ = 0.187, e₃ = 0.295, e₄ = 0.331. Next, it is necessary to aggregate the evaluation values of these indices. According to Equation (4), the single-value neutrosophic weighted averaging (SVNWA) aggregation operator should be used to aggregate these indices. Finally, based on the single-valued neutrosophic sets, we obtained the evaluation matrix R = (r_ij), outlined in

Table 6. The aggregated SVNNs decision matrix.

Criteria	Index	Alternatives
		X1	X2	X3	X4
A1	a11	<0.504, 0.496, 0.445>	<0.484, 0.516, 0.463>	<0.759, 0.241, 0.189>	<0.606, 0.394, 0.34>
	a12	<0.411, 0.589, 0.539>	<0.436, 0.554, 0.522>	<0.759, 0.241, 0.189>	<0.65, 0.35, 0.297>
	a13	<0.689, 0.311, 0.061>	<0.532, 0.468, 0.417>	<0.77, 0.23, 0.171>	<0.733, 0.267, 0.215>
	a14	<0.625, 0.368, 0.072>	<0.546, 0.448, 0.406>	<0.791, 0.209, 0.15>	<0.65, 0.35, 0.3>
A2	a21	<0.236, 0.732, 0.512>	<0.341, 0.639, 0.627>	<0.733, 0.267, 0.215>	<0.857, 0.143, 0.088>
	a22	<0.126, 0.824, 0.746>	<0.236, 0.732, 0.744>	<0.886, 0.114, 0.061>	<0.404, 0.596, 0.546>
	a23	<0.125, 0.825, 0.795>	<0.125, 0.825, 0.875>	<1, 0, 0>	<0.309, 0.678, 0.653>
	a24	<0.182, 0.768, 0.691>	<0.182, 0.768, 0.818>	<0.475, 0.525, 0.475>	<0.778, 0.222, 0.171>
A3	a31	<0.824, 0.176, 0.023>	<0.527, 0.473, 0.421>	<0.55, 0.45, 0.399>	<0.504, 0.496, 0.445>
	a32	<0.23, 0.73, 0.632>	<0.509, 0.491, 0.44>	<0.404, 0.596, 0.546>	<0.532, 0.468, 0.417>
	a33	<0.168, 0.782, 0.751>	<0.231, 0.736, 0.751>	<0.716, 0.284, 0.232>	<0.556, 0.444, 0.394>
	a34	<0.2, 0.75, 0.667>	<0.375, 0.618, 0.582>	<0.764, 0.236, 0.184>	<0.663, 0.337, 0.284>
	a35	<0.54, 0.46, 0.189>	<0.397, 0.597, 0.558>	<0.315, 0.65, 0.666>	<0.267, 0.695, 0.718>

.

5.2. The Calculation of Index Weights

In this part, we compute the subjective weights of the three criteria using the BWM method described in Section 4.3.1. Firstly, according to the expert’s opinion, the criteria in Table 2, denoted by A₁ to A₃, the performance of different compositions and proportions of additive lime mortars (A₂) is considered the most important criteria, while the restoration outcomes (A₃) is considered the least important criteria. Then, we can obtain the pairwise comparison vectors for the best and the worst criteria, as shown in

Table 7. Pairwise comparison vector of the most important criteria.

Best Criteria (A1 or A2 or A3)	A1	A2	A3
A2	3	1	5

and

Table 8. Pairwise comparison vector of the least important criteria.

Worst Criteria (A1 or A2 or A3)	A1	A2	A3
A3	3	5	1

. Table 7 represents that the preference values of the best criteria (A₂) over criteria (A₁) and criteria (A₃) are 3 and 5. The preference values of criteria (A₁) and criteria (A₂) over criteria (A₃) are 3 and 5, respectively. Finally, the weight vector of criteria w* = {0.218, 0.652, 0.13} is calculated by Equation (8).

Afterward, we can compute the weights of the index according to the entropy weight method introduced in Section 4.3.2. Firstly, the evaluation index can be normalized by using Equation (9), we obtain the normalized evaluation matrix R = (β_ij), as shown in

Table 9. The normalized evaluation matrix.

Criteria	Index	Alternatives
		X1	X2	X3	X4
A1	a11	<0.504, 0.496, 0.445>	<0.484, 0.516, 0.463>	<0.759, 0.241, 0.189>	<0.606, 0.394, 0.34>
	a12	<0.411, 0.589, 0.539>	<0.436, 0.554, 0.522>	<0.759, 0.241, 0.189>	<0.65, 0.35, 0.297>
	a13	<0.689, 0.311, 0.061>	<0.532, 0.468, 0.417>	<0.77, 0.23, 0.171>	<0.733, 0.267, 0.215>
	a14	<0.625, 0.368, 0.072>	<0.546, 0.448, 0.406>	<0.791, 0.209, 0.15>	<0.65, 0.35, 0.3>
A2	a21	<0.236, 0.732, 0.512>	<0.341, 0.639, 0.627>	<0.733, 0.267, 0.215>	<0.857, 0.143, 0.088>
	a22	<0.126, 0.824, 0.746>	<0.236, 0.732, 0.744>	<0.886, 0.114, 0.061>	<0.404, 0.596, 0.546>
	a23	<0.125, 0.825, 0.795>	<0.125, 0.825, 0.875>	<1, 0, 0>	<0.309, 0.678, 0.653>
	a24	<0.182, 0.768, 0.691>	<0.182, 0.768, 0.818>	<0.475, 0.525, 0.475>	<0.778, 0.222, 0.171>
A3	a31	<0.824, 0.176, 0.023>	<0.527, 0.473, 0.421>	<0.55, 0.45, 0.399>	<0.504, 0.496, 0.445>
	a32	<0.77, 0.27, 0.368>	<0.491, 0.509, 0.56>	<0.596, 0.404, 0.454>	<0.468, 0.532, 0.0.583>
	a33	<0.168, 0.782, 0.751>	<0.231, 0.736, 0.751>	<0.716, 0.284, 0.232>	<0.556, 0.444, 0.394>
	a34	<0.2, 0.75, 0.667>	<0.375, 0.618, 0.582>	<0.764, 0.236, 0.184>	<0.663, 0.337, 0.284>
	a35	<0.46, 0.54, 0.0.811>	<0.603, 0.403, 0.442>	<0.685, 0.35, 0.334>	<0.733, 0.305, 0.282>

. Then, we can obtain the entropy weight of the index according to the normalized evaluation matrix.

Finally, through a combination of the subjective weights of criteria and the entropy weights of the index, index weights can be acquired, which is shown in

Table 10. The weights of criteria and index for repair materials selection.

Criteria	Weight	Index	Weight
A1	0.218	a11	0.051
		a12	0.046
		a13	0.081
		a14	0.071
A2	0.652	a21	0.156
		a22	0.166
		a23	0.211
		a24	0.132
A3	0.13	a31	0.021
		a32	0.009
		a33	0.022
		a34	0.019
		a35	0.018

.

5.3. The Sequence of Repair Materials

We prefer the TOPSIS method based on MCGDM expressed to solve the repair materials selection problem according to Section 4.4. First, determine the positive ideal solution v_j⁺ and negative ideal solution v_j⁻ according to Equation (11). Then, calculate the distances S_i⁺ and S_i⁻ of each alternative solution to v_j⁺ and v_j⁻ according to Equations (12) and (13). Finally, calculate the proximity coefficient CC_i of each alternative solution (X_i) to the ideal solution according to Equation (14), the result is obtained as in

Table 11. Ranking orders of compositions and proportions of additive lime mortars.

Alternatives	Si+	Si−	CCi	Ranking Orders
X1	0.1209	0.0154	0.113	3
X2	0.121	0.0097	0.0744	4
X3	0.0227	0.1172	0.8379	1
X4	0.0821	0.0674	0.4511	2

. Hence, as CC₃ > CC₄ > CC₁ > CC₂, the most appropriate compositions and proportions of additive lime mortars for W₁ is “X₃”, namely, 3% paper-reinforced glutinous rice mortar.

5.4. Comparison Analysis

To verify whether the model proposed in Section 4 can be effectively and practically used to determine the most suitable compositions and proportions of additive lime mortars for a specific ancient Chinese building, considering the basic condition of the component to be repaired, the performance of different compositions and proportions of additive lime mortars and restoration outcomes, we employ the TODIM [50,51] method to make comparative analysis of the above empirical study. When θ = 2.25, the final dominance (ε_i) of the four alternatives X₁, X₂, X₃, and X₄ are 0.081, 0, 1, and 0.685, respectively, the ranking of the proposed model computed by the TODIM method is X₃ > X₄ > X₁ > X₂ (shown in

Table 12. Ranking comparison with TODIM.

Alternatives	Values εi	Ranking Orders
X1	0.081	3
X2	0	4
X3	1	1
X4	0.685	2

), which is consistent with the result computed by the proposed model. That is to say, the effectiveness of the proposed model can be verified. Compared with existing methods for selecting compositions and proportions of additive lime mortars for the restoration of ancient Chinese buildings, the advantages of the model proposed in this study can be summarized as follows:

• The proposed model combines fuzzy set theory and quantitative analysis and comprehensively considers subjective and objective criteria to develop an index system, which makes the selection of compositions and proportions of additive lime mortars for ancient Chinese buildings restoration more in line with the actual situation.
• The evaluation information is evaluated using a single-valued neutrosophic set that has a certain degree of similarity to human thinking [9], effectively solving the problems of diversity, ambiguity, and complexity in the sources and types of evaluation information, making compositions and proportions of additive lime mortars selection more accurate and reliable.
• The TOPSIS method is used to determine the superiority of alternative restoration materials, which are more flexible and operable in solving multi-criteria group decision-making problems [49]. Therefore, the proposed model can select the most suitable compositions and proportions of additive lime mortars for ancient Chinese building restoration.

6. Conclusions

This thesis establishes a selection model for compositions and proportions of additive lime mortars in the restoration of ancient Chinese buildings, which facilitates professionals to choose suitable restoration materials. When establishing the index system, both subjective and objective criteria were used. Furthermore, using single-valued neutrosophic sets represents evaluation information provided by experts using multi-granularity language terms, in which such information can be quantified. Moreover, by combining the BWM method and entropy weight method, the index weights can be obtained. In the end, the ranking of alternative solutions was determined using the TOPSIS method. Empirical research has been conducted on the proposed model in the selection of compositions and proportions of additive lime mortars for Dafudi to determine the type of mortar materials selection for a specific decorative exterior window, whose process was elaborated in detail. In addition, comparative analysis has demonstrated the effectiveness and reliability of the proposed model.

In summary, the model proposed in this study promotes the existing methods in the field of material selection for ancient Chinese architectural restoration and provides reasonable quantitative support for experts in the decision-making process. Therefore, this study embodies both theoretical and practical value. The contributions of this paper can be summarized as follows: (1) firstly, the proposed model fully considers the complexity of material selection in the restoration process of ancient building heritages, as well as the limitations of experts’ decision-making based on experience, which may come across practical issues and challenges; (2) the proposed model enriches the quantitative research methods for the sustainability of ancient building heritages protection. By optimizing the selection of restoration material, we believe more efficient and effective protection of antithetical heritages can be realized; and (3) this model could play a supporting role in the decision-making, which might be conducive to assisting experts in making the most appropriate decisions, improving the efficiency of ancient buildings restoration, avoiding potential damages, and promoting the sustainable protection of ancient building heritages.

However, the proposed model still shows some limitations. Firstly, the proposed model is aimed at the selection of mortar materials in the restoration of ancient buildings. However, in fact, there are various types of materials available in the restoration of ancient buildings, and surely adjustments are needed according to the actual situation in specific tasks. Secondly, there is a wide variety of compositions and proportions of lime mortars, but the limited use of samples during the research process makes it difficult to fully compare the differences between different types of lime mortar material. Thirdly, the proposed model should also adopt different methods for evaluating information processing based on the different varieties of data obtained in specific cases. This study can be further improved from the following aspects. First, the proposed model could consider the selection information of more types of restoration materials and further improve the index system with the development of related studies in future research. Second, decision methods such as TODIM, VIKOR, and other algorithms [52] can also be considered in materials selection when the project situation and data type are appropriate. Third, the model could be more applicable to the selection of compositions and proportions of additive lime mortars and could extend to the selection of other types of restoration materials for ancient Chinese buildings to ensure the sustainability of ancient buildings’ protection and restoration.