Combining the Fuzzy Analytic Hierarchy Process Method with the Weighted Aggregated Sum Product Assessment Method to Address Internet Platform Selection Problems in an Environment with Incomplete Information

Abstract

With the advancement of information technology, the Internet is pivotal in today’s society, serving as a global connectivity platform. Leveraging Internet technology within an enterprise can improve operational efficiency and curtail costs. However, traditional Internet platform selection methods cannot simultaneously handle quantitative and qualitative information, fuzzy semantics, and incomplete expert-provided information. To address these limitations, this study integrated the fuzzy analytic hierarchy process (FAHP) and the weighted aggregated sum product assessment (WASPAS) approaches to tackle Internet platform selection problems within an incomplete information environment. To demonstrate the validity of this research approach, this study utilized a construction industry Internet platform selection case to confirm the efficacy of the proposed novel fuzzy analytic hierarchy process-based method. Comparative analysis against the weighted sum model (WSM), weighted product model (WPM), FAHP, and typical WASPAS approaches was conducted with numerical verification, revealing that the proposed method in this study effectively manages comprehensive information and yields more rational outcomes for construction industry Internet platforms.

1. Introduction

Advancements in science and technology and social changes have propelled the Internet to serve as a medium for information dissemination and global interconnectedness. It overcomes geographical restrictions, enabling seamless communication and sharing across nations while promoting broader opportunities for business cooperation. However, choosing an Internet platform is crucial to obtaining accurate and useful information. Smart choices foster effective information sharing and communication and help reduce misunderstandings, improve project development efficiency, and optimize resource management. However, caution must be exercised regarding the potential risks posed by the Internet. The process of Internet platform selection involves considering factors such as credibility, efficiency, reliability, and cost. Different Internet platforms possess unique characteristics and advantages; choosing the appropriate Internet platform is a complex multi-criteria decision-making (MCDM) problem. However, traditional Internet platform selection methods have shortcomings, including the inability to handle quantitative and qualitative information problems, the incapacity to handle expert-provided fuzzy information, reliance solely on a single arithmetic model for decision-making problems, and the inability to handle incomplete information provided by experts. Durao et al. [1] used the AHP method to determine the relative weights of five assessment criteria (reliability, security, business, mobility, and heterogeneity) for handling Internet of Things process selection. Mashal et al. [2] combined the AHP method with the simple additive weighting method to address IoT application system selection problems. However, the methods proposed by both Durao et al. [1] and Mashal et al. [2] cannot handle the fuzzy information provided by experts, nor can they process incomplete expert-provided information.

The widely used weighted sum model (WSM) and weighted product model (WPM) approach, characterized by its simple mathematical calculations and ease of understanding, is one of the most commonly applied methods to resolve various MCDM issues. However, this approach cannot process qualitative, incomplete, and fuzzy expert-provided information, leading to challenges in selecting an Internet platform. To simultaneously handle both quantitative and qualitative information and consider the hierarchical structure of the assessment criteria, Saaty [3] proposed the analytical hierarchy process (AHP) method. This method uses paired comparisons of different criteria to handle quantitative and qualitative information in decision making [4]. Over time, the AHP approach has been extended to address diverse decision-making challenges across various fields, such as sustainable forest management [5], military simulation training systems [6], solid waste management [7], wind farm site selection [8], ecotourism suitability [9], and food safety [10].

To address the complexities of MCDM problems involving fuzzy semantics, Zadeh [11] introduced the concept of a fuzzy set, which employs membership degrees to represent human cognitive information. Today, numerous scholars have expanded fuzzy sets to solve decision-making problems involving various forms of semantic information, such as intuitionistic fuzzy information [12], picture fuzzy information [13], neutrosophic fuzzy information [14,15], Pythagorean fuzzy information [16], spherical fuzzy information [17], and hesitant fuzzy linguistic information [18].

Building upon the concept of the AHP approach and integrating the benefits of fuzzy sets, the fuzzy analytic hierarchy process (FAHP) approach establishes a hierarchical structure to calculate the relative weights of different criteria and utilizes fuzzy sets to handle natural language information from experts. Since subjective judgments in pairwise comparisons of evaluation criteria may contain inaccurate information, Liu et al. [19] believe that combining fuzzy sets with the AHP approach is a reasonable reflection of the actual situation. They used the FAHP method to address supplier selection issues. Vahidnia et al. [20] believe that combining AHP and fuzzy sets can achieve more flexible judgments and handle complex decision-making problems, especially in managing multiple evaluation criteria and processing both qualitative and quantitative information. Gungor et al. [21] believe that the traditional analytic hierarchy process cannot fully reflect human thinking methods and that incorporating fuzzy information can enhance the judgment of decision makers. Due to its enhanced capability in handling fuzzy semantic information compared to traditional AHP approaches, FAHP finds widespread application in addressing decision-making issues across different fields, such as material circularity in building construction [22], safety zone site selection problem [23], risk assessment of coal mine gas explosions [24], stock portfolio selection [25], and soil conservation [26].

Traditional MCDM methods often rely on a single arithmetic model for solving decision-making problems. Zavadskas et al. [27] combined the concept of the WSM and the WPM to develop the weighted aggregated sum product assessment (WASPAS) approach. The WASPAS approach provides more accurate results than WSM and WPM and allows for adjusting the lambda (λ) value for different problems and situations. By employing different combinations of WSM and WPM and utilizing different lambda (λ) values to represent the expert’s preference parameters, the WASPAS approach determines the total relative importance of different alternatives [28]. Due to its straightforward calculation process and accurate results, the WASPAS approach has gained significant attention from decision makers across various sectors of society, establishing itself as an effective decision-making tool [29]. To date, the WASPAS approach has been successfully applied in addressing decision-making issues across different fields, such as marine risk assessment [30], supply chain network design [31], urban transportation [32], wastewater reuse applications [33], and solar water heating systems [34].

Recently, Dede and Zorlu [35] employed the entropy-based WASPAS approach to conduct a geoheritage assessment of the Karcal Mountains. While the method introduced by Dede and Zorlu [35] effectively determines the priority values of geological sites, enhancing the accuracy of decision-making results, this method cannot handle fuzzy and incomplete expert-provided information. In response to the limitations of traditional Internet platform selection approaches, this study combined the FAHP approach and the WASPAS approach to handle Internet platform selection problems within an incomplete information environment. Moreover, the proposed approach employed data-filling techniques to handle incomplete expert-provided information, making the calculation results more reasonable and aligned with real-world situations. The theoretical and practical contributions of the proposed novel fuzzy analytic hierarchy process-based method are as follows: First, it can simultaneously handle quantitative and qualitative information. Second, it can process fuzzy semantic messages provided by experts. Third, it can simultaneously consider the characteristics of two arithmetic models. Fourth, it can effectively process incomplete information. Fifth, it evaluates the relative importance of evaluation criteria through pairwise comparisons.

The remainder of this article is organized as follows. Section 2 briefly introduces the basic definitions and calculation rules of WSM, WPM, FAHP, and WASPAS approaches. Section 3 proposes a novel FAHP-based approach to handle Internet platform selection problems within an incomplete information environment. Section 4 illustrates the proposed novel FAHP-based approach through an actual construction industry Internet platform selection and compares it with other algorithm methods to verify its reasonableness and effectiveness. Section 5 summarizes this paper’s conclusions and provides future research directions.

2. Preliminaries

This section briefly reviews some basic introductions and operational rules of the WSM, WPM, FAHP, and WASPAS approaches.

2.1. Weighted Sum Model (WSM) and Weighted Product Model (WPM)

Due to the WSM and WPM approach’s simple calculation process, which does not require complex calculation skills, it has emerged as a common solution method for decision-making problems. In this approach, the summation of all target values for each alternative is performed, with the alternative yielding the largest sum value being recommended as the top-ranked solution. The operational procedure of the WSM and WPM approach is as follows [27,36,37].

(1)

Creating the initial decision matrix ($A_{ij}$)

$$A_{ij}=\left[\begin{array}{cccc}{a}_{11}& a_{12}& \cdots & a_{1n}\\ a_{21}& a_{22}& \cdots & a_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ a_{m1}& a_{m2}& \cdots & a_{mn}\end{array}\right], i=1,2,\dots , m, j=1,2,\dots ,n$$

(1)

where i is the number of alternatives and j represents the number of criteria.

(2)

Calculating the WSM and WPM values of alternative i

$P_i^1$ represents the relative importance of the ith alternative of the WSM approach, and $P_i^2$ represents the relative importance of the ith alternative of the WPM approach.

$$P_i^1={\sum }_{j=1}^n{A}_{ij}{·w}_j$$

(2)

$$P_i^2={\prod }_{j=1}^n{A_{ij}}^{w_j}$$

(3)

where $w_j$ is the weight of the evaluation criterion j.

2.2. Fuzzy Analytic Hierarchy Process Approach

Saaty [3] introduced the AHP approach, leveraging mathematics and psychology concepts to solve complex MCDM problems featuring hierarchical structures. However, the typical AHP method cannot process the fuzzy semantic information experts provide. Zadeh [11] pioneered the fuzzy set approach for processing cognitive information to solve the fuzzy semantics and uncertain information of decision-making problems. The FAHP approach combines the strengths of fuzzy sets and AHP approaches, can handle uncertainty and fuzzy information when evaluating decision-making problems, and uses pairwise comparisons of different criteria to compute the relative weight of each criterion.

The fuzzy set often uses triangular fuzzy numbers (TFNs) for operations. The operation rules of TFN are as follows.

Definition 1 [38].

Considering that two TFNs are $A_1=(l_1,m_1,u_1$) and $B_1=(l_2,m_2,u_2)$, the basic operational laws are as follows:

$$A_1\oplus B_1=\left(l_1,m_1,u_1\right)+\left(l_2,m_2,u_2\right)=(l_1+l_2, m_1+m_2,u_1+u_2)$$

(4)

$$A_1⊝B_1=\left(l_1,m_1,u_1\right)\ominus \left(l_2,m_2,u_2\right)=(l_1-u_2,m_1-m_2, u_1-l_1)$$

(5)

$${{A_1}^{-1}=\left(l_1, m_1,u_1\right)}^{-1}=\left(\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$u_1, $}\right.\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$m_1, $}\right.\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$l_1$}\right.\right)$$

(6)

where $l_1$ and $l_2$ denote the lower bounds of the TFN, $m_1$ and $m_2$ are the middle values of the TFN, and $u_1$ and $u_2$ are the upper bounds of the TFN.

The calculation procedure of the FAHP approach is as follows [37].

• Step 1. Build the hierarchy structure of decision problems and determine criteria.

Build the hierarchical structure based on the assessment questions, and then determine the assessment criteria.

• Step 2. Build the pairwise comparison matrix.

The relative importance of the assessment criteria is evaluated through pairwise comparison.

Table 1. Relative importance of assessment criteria.

Relative Importance Scale	Linguistic Variable	TFN
1	Equally important	(1, 1, 3)
3	Weakly important	(1, 3, 5)
5	Strongly important	(3, 5, 7)
7	Very strongly important	(5, 7, 9)
9	Extremely important	(7, 9, 9)
2, 4, 6, 8	Between each semantic

shows a pairwise comparison of the relative importance of the evaluation criteria [39].

Based on Table 1, experts determine the pairwise comparison matrix of the assessment criteria using their past experience.

• Step 3. Defuzzify the TFN values.

Let $A_1=(l_1,m_1,u_1$) be a TFN, then defuzzify the $A_1$ value as follows [38].

$$\mathrm{D}\mathrm{F}(A_1)=\frac{1}{3}(l_1+m_1+u_1)$$

(7)

• Step 4. Perform expert opinion consistency check.

The FAHP approach applies the consistency ratio (CR) to perform an expert opinion consistency check. The expert opinions are consistent if the CR value is less than 0.1. The CR value is calculated using the maximum eigenvalue (${\lambda }_{max}$) of the pairwise comparison matrix (M), the random index (RI) value, and the consistency index (CI) value. Here, w denotes the relative weight of assessment criteria and n denotes the number of square matrices.

Table 2. The RI values of different square matrices.

n	1	2	3	4	5	6	7	8	9	10
RI	0.00	0.00	0.58	0.90	1.12	1.24	1.32	1.41	1.45	1.49

shows the RI values of different square matrices [3].

$$Mw={\lambda }_{max}w$$

(8)

$$\mathrm{CI}=\frac{{\lambda }_{max}-n}{n-1}$$

(9)

$$\mathrm{CR}=\frac{CI}{RI}$$

(10)

2.3. The Weighted Aggregated Sum Product Assessment Approach

Zavadskas et al. [27] introduced the WASPAS approach, a hybrid model of WSM and WPM for decision making. Renowned for its efficiency, mathematical simplicity, and consideration of expert preference flexibility, this approach has gained significant recognition. The associated calculation steps of the WASPAS approach are as follows [40]:

• Step 1: Create the initial decision matrix.

Based on the different alternatives, create the initial decision matrix between the criterion and the alternatives, as shown in Equation (1).

• Step 2: Normalize the initial decision matrix $N_{ij}$.

Normalize the initial decision matrix using Equations (11) and (12).

$$N_{ij}=\frac{a_{ij}}{\underset{i}{\max }{a}_{ij}}$$

(11)

$$N_{ij}=\frac{\underset{i}{\min }{a}_{ij}}{a_{ij}}$$

(12)

Equation (11) is used when the assessment criterion belongs to the benefit criterion. On the contrary, Equation (12) is used when the assessment criterion belongs to the non-benefit criterion.

• Step 3: Compute the values of WSM and WPM for different alternatives.

Compute the value of WSM and WPM using Equations (13) and (14).

$$Q_i^1={\sum }_{j=1}^n{N}_{ij}{·w}_j$$

(13)

$$Q_i^2={\prod }_{j=1}^n{N_{ij}}^{w_j}$$

(14)

• Step 4: Compute the total relative importance of different alternatives.

The total relative importance of the ith alternative is expressed as $Q_i$. The value of the preference parameter (λ) is usually determined subjectively by experts. When experts cannot determine a parameter, 0.5 is often used as a baseline.

$$Q_i=\mathsf{\lambda }{Q}_i^1+(1-\mathsf{\lambda })Q_i^2$$

(15)

3. The Proposed Novel Fuzzy Analytic Hierarchy Process-Based Approach

With the rapid advancement of science and technology, combining information and Internet platforms has created unprecedented opportunities for upgrading and transforming numerous traditional industries and enterprises. However, traditional Internet platform selection methods cannot handle benefit and non-benefit data simultaneously, nor can they simultaneously handle expert-provided fuzzy semantic information. Moreover, expert-provided data may contain both complete and partly incomplete information, making it more difficult to correctly select the most suitable Internet platform. To address the shortcomings of traditional Internet platform selection approaches, this study combined the FAHP and WASPAS approaches to handle the Internet platform selection issues within an incomplete information environment. The proposed approach utilized the FAHP approach to calculate the subjective weights between different criteria. Defuzzification and data-filling technologies were used to completely consider all expert-provided information for fuzzy semantic and unclear expert-provided information. Finally, the WASPAS approach was applied to calculate the total relative importance of the alternatives and rank the potential alternatives based on the scoring results. The execution process of the proposed novel FAHP-based approach includes the following ten steps.

• Step 1. Confirm the possible alternatives.

Experts discussed and confirmed all of the possible alternatives.

• Step 2. Confirm the evaluation criteria.

Experts discussed and confirmed all possible evaluation criteria.

• Step 3. Determine the rating values of possible alternatives under different evaluation criteria.

Experts determined the rating values of possible alternatives under different evaluation criteria based on their past experience.

• Step 4. Determine the pairwise comparison matrix of different evaluation criteria.

Experts used the TFN value to determine the pairwise comparison matrix for different evaluation criteria.

• Step 5. Data filling of incomplete information.

To address the issue of incomplete information in traditional data collection methods, incomplete information was filled in using the average of the information provided by experts who furnish complete data.

• Step 6. Calculate the relative weight of evaluation criteria and perform consistency checks.

Based on Step 4 and Step 5, Equations (8)–(10) were used to perform consistency checks and to calculate the relative weights of different criteria. The expert opinions were consistent when the CR value was less than 0.1.

• Step 7. Calculate the normalized value of the initial decision matrix.

According to the initial decision matrix of Step 5, Equations (11) and (12) were used to calculate the normalized value of the initial decision matrix.

• Step 8. Calculate the value of WSM and WPM of different alternatives.

Based on the results of Step 7, Equations (13) and (14) were used to calculate the $Q_i^1$ value and $Q_i^2$ value.

• Step 9. Subsequently, the total relative importance of different alternatives was computed.

According to the $Q_i^1$ value and $Q_i^2$ value, Equation (15) was used to calculate the $Q_i$ value. The value of λ is usually set to 0.5 when the expert cannot decide the λ value.

• Step 10. Sort the $Q_i$ value and provide decision-making suggestions.

The possible alternative sorting results were determined by sorting the $Q_i$ value from largest to smallest.

4. Case Study

4.1. Overview

With the evolution of information technology, the Internet serves as an invisible bridge, facilitating universal access to diverse information anytime and anywhere while promoting new business models and economic growth. Network technology helps the construction industry improve work efficiency, reduce costs, and promote digital transformation. To illustrate the proposed approach, this paper utilized an illustrative example of construction industry Internet platform selection (adapted from Li et al. [41]) to test and confirm the rationality and accuracy of the proposed novel FAHP-based method.

The development trajectory of China’s Internet is divided into three development stages, short-term, medium-term, and long-term development, providing a framework for the development policy of construction industry Internet platforms. The evaluation outcomes derived from the abovementioned construction industry Internet platform selection serve as a benchmark for the establishment sequence of Asian construction industry Internet platforms. Comprising experts from different fields, the construction industry Internet platform selection evaluation team consists of four experts (E1, E2, E3, and E4). The construction industry Internet platform selection includes four evaluation criteria and twenty-six possible alternatives. The four evaluation criteria include technology maturity (TM), urgency of need (UN), policy feasibility (PF), and completeness of standard (CS), all of which belong to performance indicators. The twenty-six possible alternatives include geography investigation platform (P1), synergism design platform (P2), building information modeling platform (P3), digital entrustment platform (P4), real-name registration system for labor (P5), supply procurement platform (P6), machine administration platform (P7), human–machine interaction platform (P8), resource administration platform (P9), component management platform (P10), smart site management platform (P11), smart site supervision platform (P12), information consolidation platform (P13), examination and detection platform (P14), building waste management platform (P15), green building platform (P16), digital twin running platform (P17), intelligent decision-making platform (P18), energy surveillance platform (P19), adviser platform (P20), cost estimation administration platform (P21), business approval platform (P22), bid administration platform (P23), integrity monitoring platform (P24), supply chain administration platform (P25), and financial transaction platform (P26).

Based on the four evaluation criteria, each expert utilized a seven-point Likert-type questionnaire, where Level 1 = not important, Level 2 = very slightly important, Level 3 = slightly important, Level 4 = moderately important, Level 5 = important, Level 6 = strongly important, and Level 7 = extremely important. This scale was used to rate 26 construction industry Internet platforms.

Table 3. Four experts selected evaluation scores for 26 construction industry Internet platforms.

Platform	Expert	TM	UN	PF	CS	Platform	Expert	TM	UN	PF	CS
P1	E1	4	4	1	2	P14	E1	4	5	6	2
	E2	3	3	3	1		E2	3	6	5	2
	E3	5	3	2	3		E3	4	4	4	3
	E4	4	2	1	2		E4	1	5	3	3
P2	E1	6	6	4	4	P15	E1	2	2	3	2
	E2	4	4	5	2		E2	2	3	5	3
	E3	3	5	5	3		E3	4	3	4	1
	E4	4	4	6	3		E4	2	2	5	2
P3	E1	5	7	7	5	P16	E1	2	3	5	4
	E2	7	6	6	5		E2	2	2	6	5
	E3	5	5	5	5		E3	1	2	5	2
	E4	6	7	6	4		E4	2	2	4	3
P4	E1	2	5	4	2	P17	E1	5	3	5	3
	E2	2	3	3	2		E2	4	3	6	3
	E3	3	3	5	1		E3	5	4	4	5
	E4	4	3	4	1		E4	3	2	6	2
P5	E1	5	6	7	4	P18	E1	2	1	1	2
	E2	5	7	6	4		E2	3	2	3	2
	E3	6	5	7	4		E3	1	3	2	2
	E4	7	7	5	4		E4	2	3	1	2
P6	E1	6	6	3	1	P19	E1	1	2	4	2
	E2	5	6	4	2		E2	2	3	5	2
	E3	5	4	2	2		E3	2	1	6	3
	E4	4	4	2	3		E4	2	2	3	2
P7	E1	4	5	2	5	P20	E1	7	4	6	5
	E2	4	3	6	2		E2	6	3	4	6
	E3	3	4	4	2		E3	5	5	4	5
	E4	4	3	3	3		E4	6	5	5	4
P8	E1	3	2	2	3	P21	E1	4	5	4	2
	E2	1	2	3	4		E2	4	4	3	3
	E3	2	3	4	4		E3	5	4	2	3
	E4	4	2	3	3		E4	**	**	**	**
P9	E1	3	4	2	1	P22	E1	5	6	6	2
	E2	3	3	2	1		E2	4	5	5	1
	E3	2	4	1	2		E3	5	7	4	2
	E4	4	3	1	2		E4	**	**	**	**
P10	E1	5	2	3	4	P23	E1	5	7	4	2
	E2	3	6	2	3		E2	5	7	4	2
	E3	5	4	1	3		E3	4	7	4	1
	E4	2	3	3	2		E4	6	7	5	2
P11	E1	5	7	5	6	P24	E1	6	6	5	6
	E2	4	4	7	4		E2	6	6	4	5
	E3	3	5	6	5		E3	6	6	5	4
	E4	3	4	6	5		E4	7	7	6	4
P12	E1	4	4	4	3	P25	E1	1	2	1	2
	E2	3	5	5	5		E2	2	2	2	1
	E3	3	4	4	5		E3	1	3	3	2
	E4	2	3	4	3		E4	1	2	2	1
P13	E1	2	4	3	1	P26	E1	2	3	3	4
	E2	3	5	2	2		E2	3	3	2	5
	E3	3	4	3	2		E3	3	2	2	4
	E4	3	3	4	1		E4	2	2	3	4

outlines the rating results. The symbol ** indicates non-existent information.

Based on their experience, each expert employed the TFN to determine the pairwise comparison matrix of the evaluation criterion, which is given in

Table 4. Pairwise comparison matrix of four evaluation criteria.

	Expert	TM	UN	PF	CS
TM	E1	(1, 1, 1)	(1/3, 1/2, 1/2)	(2, 2, 3)	(3, 4, 4)
	E2	(1, 1, 1)	(1, 1, 1)	(3, 3, 3)	(4, 6, 6)
	E3	(1, 1, 1)	(1, 1, 1)	(1, 2, 2)	(5, 5, 6)
	E4	(1, 1, 1)	(1/3, 1/2, 1/2)	(2, 4, 4)	(3, 4, 4)
UN	E1	(2, 2, 3)	(1, 1, 1)	(3, 4, 4)	(3, 4, 6)
	E2	(1, 1, 1)	(1, 1, 1)	(2, 2, 3)	(5, 6, 6)
	E3	(1, 1, 1)	(1, 1, 1)	(5, 6, 7)	(4, 4, 5)
	E4	(2, 2, 3)	(1, 1, 1)	(3, 4, 4)	(3, 4, 4)
PF	E1	(1/3, 1/2, 1/2)	(1/4, 1/4, 1/3)	(1, 1, 1)	(2, 2, 2)
	E2	(1/3, 1/3, 1/3)	(1/3, 1/2, 1/2)	(1, 1, 1)	(1, 1, 2)
	E3	(1/2, 1/2, 1)	(1/7, 1/6, 1/5)	(1, 1, 1)	(2, 2, 3)
	E4	(1/4, 1/4, 1/2)	(1/4, 1/4, 1/3)	(1, 1, 1)	(1, 1, 1)
CS	E1	(1/4, 1/4, 1/3)	(1/6, 1/4, 1/3)	(1/2, 1/2, 1/2)	(1, 1, 1)
	E2	(1/6, 1/6, 1/4)	(1/6, 1/6, 1/5)	(1/2, 1, 1)	(1, 1, 1)
	E3	(1/6, 1/5, 1/5)	(1/5, 1/4, 1/4)	(1/3, 1/2, 1/2)	(1, 1, 1)
	E4	(1/4, 1/4, 1/3)	(1/4, 1/4, 1/3)	(1, 1, 1)	(1, 1, 1)

.

4.2. Solution of the Weighted Sum Model and the Weighted Product Model

Both the WSM approach [37] and the WPM approach [35] are widely applied in decision analysis known for its simplicity and accessibility, requiring no complex mathematical calculations and skills. It involves calculating a weighted sum and a weighted product of the evaluation results based on various criteria. However, the WSM approach [37] and WPM approach [35] are limited to handling complete information and cannot process the expert-provided incomplete information. Due to expert E4 providing partially incomplete information for alternatives P21 and P22 (see Table 3), only the information provided by experts E1, E2, and E3 were utilized for calculation.

For example, experts E1, E2, and E3 determined the evaluation criteria TM score in platform P1 to be 4, 3, and 5, respectively; the aggregated expert-provided information is shown as follows.

$$\left(1/3\right)\times 4+\left(1/3\right)\times 3+\left(1/3\right)\times 5=4.000$$

Based on the evaluation scores presented in Table 3, the experts employed the arithmetic average to calculate the evaluation scores for each construction industry Internet platform. The calculation results of the WSM and WPM approach are shown in

Table 5. Calculation results of the WSM and WPM approach.

Platform	TM	UN	PF	CS	WSM Approach		WPM Approach
					${\mathit{P}}_{\mathit{i}}^1$ Value	Rank	${\mathit{P}}_{\mathit{i}}^2$ Value	Rank
P1	4.000	3.333	2.000	2.000	2.833	19	2.702	20
P2	4.333	5.000	4.667	3.000	4.250	8	4.173	6
P3	5.667	6.000	6.000	5.000	5.667	1	5.651	1
P4	2.333	3.667	4.000	1.667	2.917	18	2.748	18
P5	5.333	6.000	6.667	4.000	5.500	2	5.405	2
P6	5.333	5.333	3.000	1.667	3.833	12	3.453	14
P7	3.667	4.000	4.000	3.000	3.667	13	3.642	12
P8	2.000	2.333	3.000	3.667	2.750	22	2.677	22
P9	2.667	3.667	1.667	1.333	2.333	24	2.159	24
P10	4.333	4.000	2.000	3.333	3.417	15	3.279	15
P11	4.000	5.333	6.000	5.000	5.083	4	5.030	4
P12	3.333	4.333	4.333	4.333	4.083	10	4.058	8
P13	2.667	4.333	2.667	1.667	2.833	19	2.677	21
P14	3.667	5.000	5.000	2.333	4.000	11	3.824	11
P15	2.667	2.667	4.000	2.000	2.833	19	2.746	19
P16	1.667	2.333	5.333	3.667	3.250	16	2.953	16
P17	4.667	3.333	5.000	3.667	4.167	9	4.109	7
P18	2.000	2.000	2.000	2.000	2.000	25	2.000	25
P19	1.667	2.000	5.000	2.333	2.750	22	2.497	23
P20	6.000	4.000	4.667	5.333	5.000	5	4.944	5
P21	4.333	4.333	3.000	2.667	3.583	14	3.501	13
P22	4.667	6.000	5.000	1.667	4.333	6	3.908	9
P23	4.667	7.000	4.000	1.667	4.333	6	3.842	10
P24	6.000	6.000	4.667	5.000	5.417	3	5.384	3
P25	1.333	2.333	2.000	1.667	1.833	26	1.795	26
P26	2.667	2.667	2.333	4.333	3.000	17	2.912	17

.

4.3. Solution of the Fuzzy Analytic Hierarchy Process Approach

The FAHP approach combines the fuzzy set approach [11] and the typical AHP approach [3] to effectively navigate uncertain and fuzzy decision-making environments, providing more accurate and credible results. The FAHP approach could not handle the incomplete information provided by expert E4. Based on the data in Table 4, experts E1, E2, and E3 provided the aggregated information to establish the pairwise criteria comparison matrix. The calculation results are outlined in

Table 6. The pairwise comparison matrix of different criteria of the FAHP approach.

Criteria	TM	UN	PF	CS
TM	(1.000, 1.000, 1.000)	(0.778, 0.883, 0.833)	(2.000, 2.333, 2.667)	(4.000, 5.000, 5.333)
UN	(1.333, 1.333, 1.667)	(1.000, 1.000, 1.000)	(3.333, 4.000, 4.667)	(4.000, 4.667, 5.667)
PF	(0.389, 0.444, 0.611)	(0.242, 0.306, 0.344)	(1.000, 1.000, 1.000)	(1.677, 1.677, 2.333)
CS	(0.194, 0.206, 0.261)	(0.178, 0.222, 0.261)	(0.444, 0.667, 0.667)	(1.000, 1.000, 1.000)

.

According to Table 6, Equation (9) was used to defuzzify the TFN and to calculate the value of ${\lambda }_{max}$ as 4.190. Equations (11) and (12) were used to calculate the values of CI and CR. The calculation process is as follows.

$$\mathrm{C}\mathrm{I}=\frac{{\lambda }_{max-n}}{n-1}=\frac{4.190-4}{4-1}=0.063$$

$$\mathrm{C}\mathrm{R}=\frac{\mathrm{C}\mathrm{I}}{\mathrm{R}\mathrm{I}}=\frac{0.063}{0.900}=0.070$$

The CR value is <0.1 in the consistency test, which is considered acceptable. According to the data in Table 6, Equation (10) was used to calculate the weights of the four evaluation criteria (TM, UN, PF, and CS). The weight values are 0.335, 0.446, 0.139, and 0.080, respectively.

Based on the data in Table 5, the four evaluation criteria weighted values and the aggregated value of the FAHP approach were calculated. The results are shown in

Table 7. Calculation results of the FAHP approach.

Platform	TM Weighted Value	UN Weighted Value	PF Weighted Value	CS Weighted Value	Aggregated Values	Rank
P1	1.340	1.487	0.278	0.160	3.265	17
P2	1.452	2.230	0.649	0.240	4.570	9
P3	1.898	2.676	0.834	0.400	5.808	1
P4	0.782	1.635	0.556	0.133	3.106	18
P5	1.787	2.676	0.927	0.320	5.709	3
P6	1.787	2.379	0.417	0.133	4.716	8
P7	1.228	1.784	0.556	0.240	3.808	14
P8	0.670	1.041	0.417	0.293	2.421	23
P9	0.893	1.635	0.232	0.107	2.867	19
P10	1.452	1.784	0.278	0.267	3.780	15
P11	1.340	2.379	0.834	0.400	4.953	6
P12	1.117	1.933	0.602	0.347	3.998	13
P13	0.893	1.933	0.371	0.133	3.330	16
P14	1.228	2.230	0.695	0.187	4.340	10
P15	0.893	1.189	0.556	0.160	2.799	20
P16	0.558	1.041	0.741	0.293	2.634	22
P17	1.563	1.487	0.695	0.293	4.038	11
P18	0.670	0.892	0.278	0.160	2.000	25
P19	0.558	0.892	0.695	0.187	2.332	24
P20	2.010	1.784	0.649	0.427	4.869	7
P21	1.452	1.933	0.417	0.213	4.015	12
P22	1.563	2.676	0.695	0.133	5.068	5
P23	1.563	3.122	0.556	0.133	5.375	4
P24	2.010	2.676	0.649	0.400	5.735	2
P25	0.447	1.041	0.278	0.133	1.899	26
P26	0.893	1.189	0.324	0.347	2.754	21

.

4.4. Solution of the Typical Weighted Aggregated Sum Product Assessment Approach

The WASPAS approach combines WSM and WPM models, which help experts make decisions under the influence of multiple attributes or criteria [35]. The main advantage of the WASPAS approach is that it is highly resistant to ranking reversals of the alternatives considered [42].

The typical WASPAS approach can only handle complete information provided by the expert. However, expert E4 provided information containing some incomplete information; thus, only the information of experts E1, E2, and E3 was used for calculation. Using the data in Table 5, Equation (2) was used to normalize the evaluation scores of 26 construction industry Internet platforms; then, Equations (4) and (5) were used to calculate the values of $Q_i^1$ and $Q_i^2$. The results are shown in

Table 8. The computation results of the typical WASPAS approach.

Platform	TM	UN	PF	CS	${\mathit{Q}}_{\mathit{i}}^1$ Value	${\mathit{Q}}_{\mathit{i}}^2$ Value	${\mathit{Q}}_{\mathit{i}}\mathbf{V}\mathbf{a}\mathbf{l}\mathbf{u}\mathbf{e}$	Rank
P1	4.000	3.333	2.000	2.000	0.454	0.435	0.445	20
P2	4.333	5.000	4.667	3.000	0.675	0.671	0.673	6
P3	5.667	6.000	6.000	5.000	0.910	0.909	0.909	1
P4	2.333	3.667	4.000	1.667	0.456	0.442	0.449	18
P5	5.333	6.000	6.667	4.000	0.874	0.869	0.872	2
P6	5.333	5.333	3.000	1.667	0.603	0.556	0.579	13
P7	3.667	4.000	4.000	3.000	0.586	0.586	0.586	12
P8	2.000	2.333	3.000	3.667	0.451	0.431	0.441	21
P9	2.667	3.667	1.667	1.333	0.367	0.347	0.357	24
P10	4.333	4.000	2.000	3.333	0.555	0.527	0.541	15
P11	4.000	5.333	6.000	5.000	0.817	0.809	0.813	4
P12	3.333	4.333	4.333	4.333	0.659	0.653	0.656	8
P13	2.667	4.333	2.667	1.667	0.444	0.431	0.437	22
P14	3.667	5.000	5.000	2.333	0.628	0.615	0.622	11
P15	2.667	2.667	4.000	2.000	0.450	0.442	0.446	19
P16	1.667	2.333	5.333	3.667	0.525	0.475	0.500	16
P17	4.667	3.333	5.000	3.667	0.673	0.661	0.667	7
P18	2.000	2.000	2.000	2.000	0.324	0.322	0.323	25
P19	1.667	2.000	5.000	2.333	0.438	0.402	0.420	23
P20	6.000	4.000	4.667	5.333	0.818	0.795	0.807	5
P21	4.333	4.333	3.000	2.667	0.573	0.563	0.568	14
P22	4.667	6.000	5.000	1.667	0.674	0.629	0.652	9
P23	4.667	7.000	4.000	1.667	0.673	0.618	0.645	10
P24	6.000	6.000	4.667	5.000	0.874	0.866	0.870	3
P25	1.333	2.333	2.000	1.667	0.292	0.289	0.290	26
P26	2.667	2.667	2.333	4.333	0.497	0.468	0.483	17

. Then, using Equation (13), the λ coefficient value was set to 0.5, and the weights of four evaluations were set to an equal weight to calculate the $Q_i$ value, as expressed in Table 8.

4.5. Solution of the Proposed Novel Fuzzy Analytic Hierarchy Process-Based Approach

In the context of construction industry Internet platform selection, traditional MCDM methods exhibit limitations in simultaneously addressing fuzzy semantic information, neglecting relative weights between evaluation criteria, and handling incomplete expert-provided information. To effectively address these shortcomings, this study integrated the FAHP and the WASPAS approaches to handle the challenges of selecting Internet platforms for the construction industry within an incomplete information environment. In the selection process of the construction industry Internet platforms, the experts initially conferred to determine the possible alternatives and evaluation criteria. Subsequently, they assigned rating values to the potential alternatives under different evaluation criteria (Table 3) and established the pairwise comparison matrix for various evaluation criteria (Table 4). This sequence of steps constitutes the proposed approach’s initial stages (Steps 1 to 4).

• Step 5. Data filling of incomplete information.

In Table 3, it was noted that expert E4 provided information containing partial data for alternatives P21 and P22; thus, the average of the information provided by the other experts was used to fill in the incomplete information.

• Step 6. Determine the relative weight of the evaluation criteria calculation and perform consistency checks.

Based on the data in Table 4, which aggregated the evaluation opinions of the four experts, the results of the pairwise comparison matrix for evaluation criteria were determined, as expressed in

Table 9. Pairwise comparison matrix for evaluation criteria of the proposed approach.

Criteria	TM	UN	PF	CS
TM	(1.000, 1.000, 1.000)	(0.667, 0.750, 0.750)	(2.000, 2.750, 3.000)	(3.750, 4.750, 5.000)
UN	(1.500, 1.500, 2.000)	(1.000, 1.000, 1.000)	(3.250, 4.000, 4.500)	(3.750, 4.500, 5.250)
PF	(0.354, 0.396, 0.583)	(0.244, 0.292, 0.342)	(1.000, 1.000, 1.000)	(1.500, 1.500, 2.000)
CS	(0.208, 0.217, 0.279)	(0.196, 0.229, 0.279)	(0.583, 0.750, 0.750)	(1.000, 1.000, 1.000)

.

According to the information in Table 9, Equation (7) was used to defuzzify the TFN and calculate the value of ${\lambda }_{max}$ as 4.202. Equations (9) and (10) were used to calculate the values of CI and CR. The calculation process is as follows.

$$\mathrm{C}\mathrm{I}=\frac{{\lambda }_{max-n}}{n-1}=\frac{4.202-4}{4-1}=0.067$$

$$\mathrm{C}\mathrm{R}=\frac{\mathrm{C}\mathrm{I}}{\mathrm{R}\mathrm{I}}=\frac{0.067}{0.900}=0.075$$

After confirming that the CR value was <0.1 in the consistency test, it was deemed acceptable. Based on the data in Table 9, Equation (8) was employed to calculate the weights of the four evaluation criteria (TM, UN, PF, and CS). The weighted values are 0.329, 0.453, 0.132, and 0.086, respectively.

• Step 7. Calculate the normalized value of the initial decision matrix.

The initial decision matrix was normalized using Equation (11).

• Step 8. Calculate the value of WSM and WPM for different alternatives.

According to the data in Step 7, Equations (13) and (14) were used to calculate the $Q_i^1$ value and $Q_i^2$ value, as shown in

Table 10. The $Q_i^1$, $Q_i^2$, and $Q_i$ values of the proposed approach.

Platform	TM	UN	PF	CS	${\mathit{Q}}_{\mathit{i}}^1$ Value	${\mathit{Q}}_{\mathit{i}}^2$ Value	${\mathit{Q}}_{\mathit{i}}\mathbf{V}\mathbf{a}\mathbf{l}\mathbf{u}\mathbf{e}$	Rank
P1	4.000	3.000	1.750	2.000	0.476	0.460	0.468	18
P2	4.250	4.750	5.000	3.000	0.688	0.687	0.687	8
P3	5.750	6.250	6.000	4.750	0.916	0.915	0.915	2
P4	2.750	3.500	4.000	1.500	0.482	0.474	0.478	17
P5	5.750	6.250	6.250	4.000	0.908	0.907	0.907	3
P6	5.000	5.000	2.750	2.000	0.679	0.662	0.670	9
P7	3.750	3.750	3.750	3.000	0.571	0.570	0.570	14
P8	2.500	2.250	3.000	3.500	0.401	0.389	0.395	22
P9	3.000	3.500	1.500	1.500	0.442	0.429	0.435	19
P10	3.750	3.750	2.250	3.000	0.539	0.533	0.536	15
P11	3.750	5.000	6.000	5.000	0.734	0.722	0.728	7
P12	3.000	4.000	4.250	4.000	0.575	0.568	0.572	13
P13	2.750	4.000	3.000	1.500	0.493	0.485	0.489	16
P14	3.000	5.000	4.500	2.500	0.620	0.608	0.614	11
P15	2.500	2.500	4.250	2.000	0.418	0.408	0.413	20
P16	1.750	2.250	5.000	3.500	0.404	0.370	0.387	23
P17	4.250	3.000	5.250	3.250	0.585	0.565	0.575	12
P18	2.000	2.250	1.750	2.000	0.322	0.321	0.322	25
P19	1.750	2.000	4.500	2.250	0.355	0.333	0.344	24
P20	6.000	4.250	4.750	5.000	0.777	0.759	0.768	5
P21	4.333	4.333	3.000	2.667	0.618	0.613	0.616	10
P22	4.667	6.000	5.000	1.667	0.768	0.748	0.758	6
P23	5.000	7.000	4.250	1.750	0.836	0.807	0.821	4
P24	6.250	6.250	5.000	4.750	0.921	0.918	0.920	1
P25	1.250	2.250	2.000	1.500	0.279	0.273	0.276	26
P26	2.500	2.500	2.500	4.250	0.419	0.405	0.412	21

.

• Step 9. Calculate the total relative importance of different alternatives.

According to the data in Step 8, the value of λ was set to 0.5. Subsequently, Equation (15) was used to calculate the $Q_i$ value, as expressed in Table 10.

• Step 10. Sort the $Q_i$ value and provide decision-making suggestions.

Sorting $Q_i$ values from largest to smallest yields all possible alternative sorting results, as expressed in Table 10.

4.6. Comparison and Discussion

With the continuous advancement of network technology, the construction industry is undergoing digital transformation. The choice of network platform for the construction industry has garnered heightened significance, as leveraging network technology can significantly enhance operational efficiency and reduce required costs. However, traditional construction industry Internet platform selection approaches cannot handle quantitative and qualitative information simultaneously, nor can they simultaneously handle the fuzzy semantic information that experts provide, often relying on a single arithmetic model. Moreover, expert-provided data containing both clear and incomplete information cause serious difficulties for the construction industry Internet platforms. To address these limitations, this study combined FAHP and WASPAS approaches. By integrating these approaches, this study aims to address the construction industry Internet platform selection problem within an incomplete information environment. This method applied the FAHP approach to calculate the relative weight of different criteria and then used the two arithmetic models of the WASPAS approach to improve the accuracy of the calculation results. Subsequently, the data-filling method was used to process the expert-provided incomplete and unclear information.

In Section 4, a case study focusing on the construction industry Internet platform selection was used to illustrate and prove the effectiveness of the proposed approach. The calculation results were also compared with the WSM approach [37], WPM approach [35], FAHP approach [43], and a typical WASPAS approach [35]. The calculation results of the four different calculation approaches are shown in

Table 11. The ranking results obtained by four different algorithm approaches.

	Score					Ranking
Platform	WSM Approach	WPM Approach	FAHP Approach	WASPAS Approach	Proposed Approach	WSM Approach	WPM Approach	FAHP Approach	WASPAS Approach	Proposed Approach
P1	2.833	2.702	3.265	0.445	0.468	19	20	17	20	18
P2	4.250	4.173	4.570	0.673	0.687	8	6	9	6	8
P3	5.667	5.651	5.808	0.909	0.915	1	1	1	1	2
P4	2.917	2.748	3.106	0.449	0.478	18	18	18	18	17
P5	5.500	5.405	5.709	0.872	0.907	2	2	3	2	3
P6	3.833	3.453	4.716	0.579	0.670	12	14	8	13	9
P7	3.667	3.642	3.808	0.586	0.570	13	12	14	12	14
P8	2.750	2.677	2.421	0.441	0.395	22	22	23	21	22
P9	2.333	2.159	2.867	0.357	0.435	24	24	19	24	19
P10	3.417	3.279	3.780	0.541	0.536	15	15	15	15	15
P11	5.083	5.030	4.953	0.813	0.728	4	4	6	4	7
P12	4.083	4.058	3.998	0.656	0.572	10	8	13	8	13
P13	2.833	2.677	3.330	0.437	0.489	19	21	16	22	16
P14	4.000	3.824	4.340	0.622	0.614	11	11	10	11	11
P15	2.833	2.746	2.799	0.446	0.413	19	19	20	19	20
P16	3.250	2.953	2.634	0.500	0.387	16	16	22	16	23
P17	4.167	4.109	4.038	0.667	0.575	9	7	11	7	12
P18	2.000	2.000	2.000	0.323	0.322	25	25	25	25	25
P19	2.750	2.497	2.332	0.420	0.344	22	23	24	23	24
P20	5.000	4.944	4.869	0.807	0.768	5	5	7	5	5
P21	3.583	3.501	4.015	0.568	0.616	14	13	12	14	10
P22	4.333	3.908	5.068	0.652	0.758	6	9	5	9	6
P23	4.333	3.842	5.375	0.645	0.821	6	10	4	10	4
P24	5.417	5.384	5.735	0.870	0.920	3	3	2	3	1
P25	1.833	1.795	1.899	0.290	0.276	26	26	26	26	26
P26	3.000	2.912	2.754	0.483	0.412	17	17	21	17	21

. Considering the differences in factors among the four different research approaches (as shown in

Table 12. Differences in solution characteristics and information processing considered by four different algorithmic approaches.

Research Approach	Handle Quantitative and Qualitative Information Simultaneously	Process Fuzzy Semantic Messages Provided by Experts	Simultaneously Consider the Characteristics of Two Arithmetic Models	Handle Incomplete Information	Evaluate the Relative Importance of Evaluation Criteria through Pairwise Comparisons
The WSM approach [37]	No	No	No	No	No
The WPM approach [35]	No	No	No	No	No
The FAHP approach [43]	Yes	Yes	No	No	Yes
The WASPAS approach [35]	No	No	Yes	No	No
Proposed approach	Yes	Yes	Yes	Yes	Yes

), several advantages of the proposed approach are explained below.

(1)

Handling of quantitative and qualitative information simultaneously

The WSM, WPM, and typical WASPAS approach lack the ability to handle qualitative information problems. Conversely, both the proposed approach and the FAHP approach utilize the computational properties of the AHP to effectively quantify experts’ subjective ratings of the evaluation criteria for complex decision-making problems.

(2)

Processing fuzzy semantic messages provided by experts

The WSM, WPM, and typical WASPAS approach are confined to handling crisp information and cannot handle expert-provided fuzzy semantic messages. Both the proposed approach and the FAHP approach can consider the interval value of the fuzzy semantic messages simultaneously, including lower bound, center bound, and upper bound information, to allow for more objective consideration of fuzzy expert-provided information.

(3)

Simultaneously considering the characteristics of two arithmetic models

The WSM, WPM, and FAHP approach rely solely on a single arithmetic model to rank the possible construction industry Internet platforms. These approaches are susceptible to data-specific biases. Conversely, both the proposed approach and the typical WASPAS approach leveraged the combination of WSM and WPM values to perform a corresponding ranking evaluation of possible alternatives, providing a more robust decision suggestion.

(4)

Handling of incomplete information

When the assessment information of expert-provided data contains some unclear information, the WSM, WPM, FAHP, and typical WASPAS approaches struggle to handle the ambiguity, disregarding some useful information. To process unclear information, the proposed approach applied information from experts who provided complete information for data-filling and fully considered all experts, to provide useful information.

(5)

Evaluate the relative importance of evaluation criteria through pairwise comparisons

The WSM, WPM, and typical WASPAS approaches do not evaluate the relative importance of evaluation criteria through pairwise comparisons, leading to erroneous and out-of-focus evaluation results. Both the proposed approach and the FAHP approach applied the AHP approach’s solution characteristics to calculate the evaluation table’s relative subjective weight, making the evaluation results more reasonable and accurate.

5. Conclusions

With the continuous evolution of various information technologies, the Internet has evolved into a platform that integrates various advancements, such as big data, cloud computing, and mobile Internet. This integration extends across various industries, including finance, manufacturing, and healthcare, promoting innovative smart application services. However, the choice of Internet platforms has emerged as a crucial concern, garnering significant attention across various industries and academia. Scholars, particularly in the construction industry, have shown research interest in Internet platform selection to improve operational efficiency and reduce required costs. However, traditional calculation approaches often overlook critical factors. They do not consider the relative weight of assessment criteria, cannot simultaneously handle quantitative and qualitative information, and cannot handle fuzzy and uncertain information. Moreover, these approaches often rely on a single arithmetic model for evaluation in the context of the construction industry Internet platforms selection problem. To address these shortcomings and improve the accuracy of the assessment results, this study integrated the FAHP approach and WASPAS approach to process Internet platform selection problem within an incomplete information environment. The proposed approach uses the triangular fuzzy set and the AHP approaches to deal with qualitative, quantitative, and fuzzy information in the relative weight comparison assessment criteria. On the other hand, the proposed method combines the WSM and WPM approach to calculate the total relative importance score of different alternatives. The proposed FAHP-based approach offers several advantages, including the following:

(1)

The proposed FAHP-based approach can simultaneously handle quantitative and qualitative information.

(2)

The proposed FAHP-based approach can process fuzzy semantic messages provided by experts.

(3)

The proposed FAHP-based approach can simultaneously consider the characteristics of two arithmetic models.

(4)

The proposed FAHP-based approach can effectively process incomplete information.

(5)

The proposed FAHP-based approach evaluates the relative importance of evaluation criteria through pairwise comparisons.

Moving forward, subsequent researchers may explore alternative research approaches for evaluating alternative performance, such as the VlseKriterijumska Optimizacija I Kompromisno Resenje (VIKOR), the technique for order preference by similarity to the ideal solution (TOPSIS), and the combined compromise solution (CoCoSo) approaches. Additionally, enhancing the consideration of objective weights for evaluation factors can broaden the scope of research and further enrich the analysis.