A supplier selection method based on cobweb similarity and prospect theory for prefabricated components

Abstract

Selecting suppliers for prefabricated components (PCs) involves a complex decision-making process, frequently relying on ambiguous information and subjective judgment. However, most existing methods use precise values to portray indicator information and overlook the uncertainty of weights and the subjective preferences of decision-makers (DMs). In order to address these limits, this paper proposes a novel approach to select suppliers of PCs. Initially, an evaluation index system for suppliers is established through literature analysis and a questionnaire survey. The system comprises six layers: product quality, price, service level, comprehensive ability, supply ability, and environmental sustainability. The group decision matrix is then constructed using the set-valued statistical method and the prospect theory. The index weights are determined by a combination weighting method. Next, the cobweb model is introduced to analyze the disparity between the alternative and ideal solutions, describing their similarities in terms of area and shape. Lastly, cobweb similarity is employed instead of comprehensive distance, combined with the minimum sum of squares criterion, to improve the closeness algorithm and contrast the alternatives. The results demonstrate that this method facilitates a comprehensive evaluation of the benefits and drawbacks of various alternatives from diverse perspectives. Furthermore, it allows flexible adjustments based on the risk preferences of DMs, ensuring accurate and reliable decision results.

Keywords

Select suppliers risk preference prospect theory cobweb model cobweb similarity

1 Introduction

Due to rapid modernization and a growing population, there has been an increasing demand for buildings. Statistics indicate that since 2010, the average annual housing construction area average annual housing construction area in China the country has been 7.31 billion square meters, with a compound growth rate of 10.05%. The robust construction tendency will continue for decades. The substantial demand for construction has significantly transformed the appearance of the country. Nonetheless, the utterly new change is coupled with severe energy consumption and environmental pollution [16]. In order to ensure a stable urbanization process in China and successfully implement the sustainable development strategy, the Chinese government has consecutively released the 13th Five-Year Plan for Prefabricated Buildings (2017) and the Report of the 20th National Congress (2022). These measures clearly demonstrate China’s ambitious objective of astonishingly transforming the construction industry by utilizing through the utilization of prefabricated buildings.

The construction of prefabricated buildings involves relocating traditional on-site operations to factories, where components and accessories are manufactured, transported, and installed [9]. Compared to cast-in-place concrete structures, prefabricated buildings offer stable product quality, cost certainty, resource conservation, and enhanced safety [7 , 20]. Due to product differentiation, varying criteria standards, and complex subjects, supplier selection of PCs is regarded as a multi-attribute decision-making process [6]. Appropriately selecting suppliers and monitoring the quality of PCs from their origin is critical for optimizing the prefabricated building supply chain and enhancing operational efficiency in prefabricated projects.

Numerous researchers have presented one’s perspectives on the decision-making issues related to suppliers of PCs. Currently, academic research on supplier decision-making predominantly concentrates on indicators and selection methods.

1.1 Research on evaluation indicators

The complexity of the indicators system for suppliers of PCs arises from the need to consider various factors in the decision-making process. Currently, numerous scholars are trying to identify evaluation indicators for suppliers of PCs in various ways and have achieved some results. Song et al. [23] distributed 300 questionnaires to various businesses and employed the structural equation model to derive an assessment indicator system of suppliers for PCs, encompassing quality, pricing, and long-term cooperation. Masood et al. [22] emphasized the importance of suppliers in the housing project supply chain. They identified 19 indicators and classified them into six performance dimensions through an investigation of component companies and consultations with relevant experts. Liu et al. [8] noted that prior research on assessment indicators had mainly focused on technology, process, and performance. Therefore, their team examined 34 significant Chinese prefabricated firms and developed an index system comprising five aspects: procurement process, operational efficiency, relationship coordination, strategic fit, and social responsibility. Lin et al. [18] conducted a questionnaire survey involving 35 experts from the Australian construction industry and proposed that the salt content can serve as a criterion for identifying suppliers of PCs for low-rise buildings. A review and analysis of the literature revealed that, despite considerable progress in research on supplier evaluation indicators, researchers still tend to focus their studies on the manufacturing industry. Limited attention has been paid to the construction industry, particularly the prefabricated construction sector. Additionally, evaluating PC suppliers focuses primarily on product quality and price, with little consideration for after-sales service, environmental sustainability, and transport capacity. It is crucial to establish a reasonable and comprehensive evaluation index system for PC suppliers to assist DMs in selecting the most suitable supplier.

1.2 Research on selection methods

Indicators are a prerequisite for decision-making, and the efficiency of selection methods directly influences the final result of DMs. To mitigate subjectivity in the decision-making process, Pan et al. [21] introduced the entropy weight method to determine the weights. They then proposed a supplier selection method for PCs based on entropy weight and TOPSIS, considering four perspectives: production delivery, logistics cost, customer service, and development potential. Zhang et al. [13] asserted that PCs are the primary structural elements in prefabricated buildings, and their quality substantially influences the safety and usability of main structures. Consequently, they analyzed the characteristics of these components and developed a decision-making model employing grey relational analysis(GRA) and the VIKOR method. Luo et al. [24] argued that selecting reliable and suitable PC suppliers is crucial for ensuring the quality of the assembly project. In this study, the researchers analyzed the inherent requirements of green buildings and the characteristics of residential industrialization. They then proposed a method that combined catastrophe theory and the Kent index, which was applied to select and evaluate suppliers of prefabricated components. Pan et al. [14] discussed the characteristics of PCs, established an evaluation index system, and utilized a group decision-making method based on DEMATEL and BP neural networks to select suppliers of PCs. Arashpour et al. [11] contended that robust supply decision-making is critical to the advanced manufacturing of prefabricated products. In a subsequent study, they proposed the researchers proposed a supplier decision-making model targeting cost and verified its rationality via an actual prefabricated panel production project as a test bed. While the approaches described above could assist DMs in supplier selection, several issues still need to be considered. (1) The indicators are presented as precise values in most methods. Unfortunately, real-world difficulties often involve ambiguity and complexity, making it challenging for DMs to access precise values beforehand. (2) The weighting method used for indicators needs to be more reasonable. Specific research papers employ subjective weighting methods, such as the Analytic Hierarchy Process (AHP) and the Delphi method. Excessive reliance on expert experience in these weighting methods leads to pronounced subjectivity in the decision outcomes. Compared with the former, some scholars also attempted to determine indicator weights via an objective weighting method. However, this approach solely relies on the index value, disregarding the inherent importance of the indicators. These limitations necessitate in-depth research.

It is worth noting that most decision-making studies regarding selecting PC suppliers are based on the assumption of perfect rationality. DM’s behavior is bounded rationality towards multi-index choice issues due to knowledge and experience limits. When confronted with diverse PC suppliers, the apparent expression of bounded rationality is that DMs would have varying risk preferences. Prospect theory is a decision-making method under bounded rationality proposed by Kahneman and Tversky in 1979 [3]. It illustrates a systematic perceptual bias more in accordance with human decision-making tendencies [17]. Prospect theory has found widespread application in decision-making regarding emergencies, investments, and location problems [5 , 19], yielding favorable outcomes. However, prospect theory integration into PC suppliers’ decision-making practices has received limited research attention. This knowledge gap highlights a need for further investigation and reveals a potential avenue for future research.

To address these limitations, this paper proposes a group decision-making method for selecting suppliers of PCs in a complex environment. The method incorporates the number of intervals as decision information and accounts for the uncertainty in indicator weights and the DM’s risk preference. The remainder of this study is displayed below: Section 2 establishes the index system of prefabricated component suppliers through literature analysis and expert interviews. Section 3 proposes an improved prefabricated component supplier selection algorithm based on prospect theory and cobweb similarity. Section 4 analyzes the case using the proposed method, compares it with previous studies to further highlight the bright point of our research. Section 5 lists the conclusions of this paper and shows the prospects.

2 Indicator system

The literature review indicates that the studies on supplier evaluation indicators in other fields have achieved some fruit. However, pronounced disparities exist between suppliers of prefabricated components and other suppliers regarding product types, standards, and procurement management models. The current evaluation system fails to comprehensively capture the unique characteristics of prefabricated components and adequately cover the supply and demand relationship of the system. Consequently, the previous evaluation criteria cannot be entirely relied upon when establishing the supplier evaluation system.

2.1 Preliminary selection based on literature

Word frequency statistics and word cloud analysis can be used to identify criteria for PC suppliers.

Search for databases such as CNKI and WOS, 136 papers are excavated as preliminary screening materials.

Identify and standardize the supplier evaluation indicators.

Count the word frequency of the evaluation index, and select the words with a frequency greater than 20.

Eliminate repeated evaluation words.

According to the above steps, the word cloud of suppliers for PCs is drawn by MATLAB, as depicted in Fig. 1.

Fig. 1

Preliminary identification results of prefabricated component supplier evaluation index.

Based on the initial identification, 43 indicators are obtained to assess PC suppliers. The critical indicators include component quotation, cost, and qualification rate. However, the identification results indicate a significant overlap of indicators. To address the issue, our team seeks to re-select indexes via the Likert scale to ensure the logical and scientific validity of the assessment indicators.

2.2 Re-screening based on the Likert scale

This work aims to re-screen redundant indicators using the Likert scale and supplement the indicator system with professional knowledge.

Compare with the initial identification, and design a questionnaire of evaluation indicators. The semi-open questionnaire allows experts to select from preset options and modify the evaluation indicators according to their professional knowledge and engineering experience.

The problem is assessed using the Likert scale. For instance, the quotation of prefabricated components is a critical determinant in supplier selection. The Likert scale assigns scores to responses as follows: strongly agree (5), agree (4), neutral (3), disagree (2), and strongly disagree (1).

Statistics the score and calculate the mean value for each index to integrate the respondents’ opinions.

According to questionnaire feedback and guided by principles such as systematicness, comprehensiveness, and operability, 22 criteria are ultimately chosen. These criteria are presented in Table 1.

Table 1
The evaluation index system of prefabricated component suppliers

ID Criteria ID Criteria

Product quality Product prices

C1 Quality cost indices C12 Component quotation

C2 Rework and return rate C13 Component cost

C3 Technological level C14 Price volatility

C4 The quality rate of components Supply ability

Comprehensive ability C15 On-time delivery

C5 Similar project experience C16 Lead time

C6 Performance ability C17 Transport cost ratio

C7 Enterprise financial level C18 Supply radiation range

C8 R&D capability C19 Transport capacity of large components

C9 Application level of information technology C20 Product flexibility

Service level Environmental sustainability

C10 After-sales serviceability C21 Usage rate environmentally friendly materials

C11 Technical serviceability C22 Utilization rate of waste recycling

ID	Criteria	ID	Criteria
	Product quality		Product prices
C1	Quality cost indices	C12	Component quotation
C2	Rework and return rate	C13	Component cost
C3	Technological level	C14	Price volatility
C4	The quality rate of components		Supply ability
	Comprehensive ability	C15	On-time delivery
C5	Similar project experience	C16	Lead time
C6	Performance ability	C17	Transport cost ratio
C7	Enterprise financial level	C18	Supply radiation range
C8	R&D capability	C19	Transport capacity of large components
C9	Application level of information technology	C20	Product flexibility
	Service level		Environmental sustainability
C10	After-sales serviceability	C21	Usage rate environmentally friendly materials
C11	Technical serviceability	C22	Utilization rate of waste recycling

3 Materials and methods

3.1 Preliminaries

3.1.1 Interval number

Definition 1. Suppose a = [a^L, a^H] is a non-negative interval number. In particular, a is a real number when a^L = a^H.

Definition 2. a = [a^L, a^H] and b = [b^L, b^H] are two non-negative intervals, then

a + b = [a^L, a^H] + [b^L, b^H] = [a^L + b^L, a^H + b^H];

λa = λ [a^L, a^H] = [λa^L, λa^H]. when λ = 0, λa = 0.

Definition 3. Let a = [a^L, a^H] and b = [b^L, b^H], the distance between a and b can be described as:

Δ_a-b = [(a^L + a^H) - (b^L + b^H)]/2.

3.1.2 Prospect theory

Prospect theory commonly employs the value coefficient to reflect DMs’ subjective attitudes towards losses and gains. The prospect values can be described as: $v (Δ t) = {\begin{matrix} (Δ t)^{α}, (Δ t \geq 0) \\ - θ (- Δ t)^{β}, (Δ t < 0) \end{matrix}$

Where Δt = t - t₀ is the deviation between the value of the index and reference point. If Δt ≥ 0, the deviation is positive, indicating that the indicator relative to the reference point is income; conversely, when the deviation is negative, there is a relative loss. α, β are the sensitivity of DMs to income and losses, α, β ∈ [0, 1]. θ is the avoidance coefficient of loss. When θ > 1, the DMs are more sensitive to losses than income in the same situation.

3.2 Initial group decision matrix

Set-valued statistics is an effective method to deal with fuzzy assessment. Suppose a supplier selection issue of PCs has n evaluation indicators A = {a₁, a₂, ⋯ , a_n} and c experts E = {e₁, e₂, ⋯ , e_c}. For any indicator a_r (r = 1, 2, ⋯ , n), if the value given by the kth expert e_k (k = 1, 2, ⋯ , c) is $[a_{rk}^{L}, a_{rk}^{R}]$ , a c-dimensional set-valued statistical sequence ${[a_{r 1}^{L}, a_{r 1}^{R}], [a_{r 2}^{L}, a_{r 2}^{R}], \dots, [a_{rc}^{L}, a_{rc}^{R}]}$ can be obtained. Let $a_{r}^{min} = {min}_{k = 1}^{c} (a_{rk}^{L})$ and $a_{r}^{max} = {max}_{k = 1}^{c} (a_{rk}^{R})$ , Then the expert comments of a_r form a random distribution on the number axis, as shown in Fig. 2.

Fig. 2

Random distribution of expert comments.

Suppose that F_ar (a) is the sample shadow function for a_r, which quantifies the probability that a is present in each evaluation interval. Noting that various experts have knowledge differences and preferred solutions, which might influence the judgment. In this study, a drop shadow function based on expert weights is introduced to strike a balance between unified and heterogeneous expert judgments [1].

3.2.1 The weight of decision experts

The calculation of expert weights is crucial in group decision-making and can be indirectly conveyed through the evaluation matrix. Suppose that a = [a^L, a^H] and b = [b^L, b^H] are two interval numbers, and the deviation δ (a, b) is defined as Equation (1). $δ (a, b) = \frac{| a^{L} - b^{L} | + | a^{R} - b^{R} |}{b^{R} - b^{L}}$ (1)

Where the bigger δ (a, b), the greater δ (a, b). When δ (a, b) =0, a = b.

If expert e_k evaluates the indicator a_r as $[a_{rk}^{L}, a_{rk}^{R}]$ , the average evaluation result of a_r by all c experts can be defined as ${\bar{a}}_{r}$ . ${\bar{a}}_{r} = \frac{1}{c} \sum_{k = 1}^{c} a_{rk} = \sum_{k = 1}^{c} [\frac{1}{c} a_{rk}^{L}, \frac{1}{c} a_{rk}^{R}]$ (2)

Let δ_rk be a relative deviation between a_rk and ${\bar{a}}_{r}$ , which can be defined as Equation (3), according to the calculation rules of Equation (1). $δ_{rk} = \frac{| a_{rk}^{L} - \frac{1}{c} \sum_{k = 1}^{c} a_{rk}^{L} | + | a_{rk}^{R} - \frac{1}{c} \sum_{k = 1}^{c} a_{rk}^{R} |}{\frac{1}{c} \sum_{k = 1}^{c} a_{rk}^{R} - \frac{1}{c} \sum_{k = 1}^{c} a_{rk}^{L}}$ (3)

The deviation of the same expert can vary for different criteria due to disparities in knowledge and individual cognition. There is a convergence effect in the conclusions drawn by multiple experts, while inconsistencies are believed to result from various random factors. When the number of indicators toward infinity, the deviation of the kth expert δ_k = [δ_1k, δ_2k, ⋯ , δ_nk] is a normal distribution $δ_{k} \sim N (0, δ_{k}^{2})$ .

When μ = 0, the maximum likelihood estimation of $δ_{k}^{2}$ can be defined as the deviation between the kth expert and the expert group. The likelihood function can be defined as Equation (4). $L (δ_{k}^{2}) = \prod_{r = 1}^{n} \frac{1}{\sqrt{2 π} δ_{k}} exp [- \frac{δ_{rk}^{2}}{2 δ_{k}^{2}}]$ (4)

Then the maximum likelihood estimation of $δ_{k}^{2}$ can be denoted as Equation (5). $δ_{k}^{2} = \frac{1}{n} \sum_{r = 1}^{n} δ_{rk}^{2}$ (5)

Where $δ_{k}^{2}$ is the maximum likelihood estimation, which reflects the deviation between the kth expert and the group experts. The smaller $δ_{k}^{2}$ , the greater expert weight, and vice versa.

Let λ_k be the weight of the kth expert, which satisfies Equation (6). $λ_{k} = \frac{1}{δ_{k}^{2} + 1} / \sum_{k = 1}^{c} \frac{1}{δ_{k}^{2} + 1}$ (6)

3.2.2 The comprehensive results

λ_k is essential for figuring out the value of interval assessment criteria. As can be seen in Fig. 2, the drop shadow function of each indicator can be demonstrated as follows. $F_{ar} (a) = \sum_{k = 1}^{c} λ_{k} F_{[a_{rk}^{L}, a_{rk}^{R}]} (a)$ (7)

Here $F_{[a_{rk}^{L}, a_{rk}^{R}]} (a) = {\begin{matrix} 1, a \in [a_{rk}^{L}, a_{rk}^{R}] \\ 0, a \notin [a_{rk}^{L}, a_{rk}^{R}] \end{matrix}$ (8)

Suppose that $a_{r}^{*}$ is the center of gravity for random sets and that is used to signify the comprehensive evaluation result of criteria, as calculated by $a_{r}^{*} = \frac{\int_{a_{r}^{min}}^{a_{r}^{max}} {aF}_{ar} (a) da}{\int_{a_{r}^{min}}^{a_{r}^{max}} F_{ar} (a) da}$ (9)

After substituting Equations (7)–(8) into Equation(9), we can obtain the following expressions. $a_{r}^{*} = \frac{1}{2} \frac{\sum_{k = 1}^{c} λ_{k} [{(a_{rk}^{R})}^{2} - {(a_{rk}^{L})}^{2}]}{\sum_{k = 1}^{c} λ_{k} (a_{rk}^{R} - a_{rk}^{L})}$ (10)

Where λ_k denotes the weight of the kth expert, $\sum_{k = 1}^{m} λ_{k} = 1$ .

In conclusion, let A = [a_ij] _m×n be an initial group decision matrix, where a_ij is comprehensive result of the jth criteria of the ith alternative.

3.3 Prospect decision matrix

Transfer the initial decision matrix A = [a_ij] _m×n via the normalization formula $b_{ij} = a_{ij} / \sqrt{\sum_{i = 1}^{m} a_{ij}^{2}}$ , the decision matrix B = [b_ij] _m×n is produced.

Prospect theory requires measuring the profit and loss with the reference point. Assume that $S_{j}^{+}$ and $S_{j}^{-}$ are the positive and negative ideal solutions of jth criteria, respectively. They are expressed by ${\begin{matrix} S_{j}^{+} = {(b_{ij}^{+} | max_{1 \leq i \leq m} b_{ij}, b_{ij} \in J_{1}), (b_{ij}^{+} | min_{1 \leq i \leq m} b_{ij}, b_{ij} \in J_{2})} \\ S_{j}^{-} = {(b_{ij}^{-} | min_{1 \leq i \leq m} b_{ij}, b_{ij} \in J_{1}), (b_{ij}^{-} | max_{1 \leq i \leq m} b_{ij}, b_{ij} \in J_{2})} \end{matrix}$ (11)

Where J₁ and J₂ indicate that jth evaluation criteria for benefit-oriented criteria and cost criteria.

Collect $S_{j}^{+}$ and $S_{j}^{-}$ in Equation (11), the positive ideal solution set S⁺ and the negative ideal solution set S^- can be defined as Equation (12). ${\begin{matrix} S^{+} = [S_{1}^{+}, S_{2}^{+}, \dots, S_{n}^{+}] \\ S^{* -} = [S_{1}^{* -}, S_{2}^{* -}, \dots, S_{n}^{* -}] \end{matrix}$ (12)

Where $S_{n}^{* -}$ is the virtual worst solution, $S_{n}^{* -} = 2 S_{n}^{-} - S_{n}^{+}$ .

People frequently concern more about the difference between the alternative and the ideal solution when using the prospect theory rather than about the outcome itself. This paper utilizes the grey correlation coefficient in place of the absolute distance between options to improve the value function in prospect theory. Assume that $ζ_{ij}^{+}$ is the grey correlation coefficient between alternative and positive ideal solution, $ζ_{ij}^{-}$ is the grey correlation coefficient between alternative and negative ideal solution. They are defined as Equation (13). ${\begin{matrix} ζ_{ij}^{+} = \frac{min_{i} min_{j} | S_{j}^{+} - b_{ij} | + ρ min_{i} min_{j} | S_{j}^{+} - b_{ij} |}{| S_{j}^{+} - b_{ij} | + ρ min_{i} min_{j} | S_{j}^{+} - b_{ij} |} \\ ζ_{ij}^{-} = \frac{min_{i} min_{j} | S_{j}^{-} - b_{ij} | + ρ min_{i} min_{j} | S_{j}^{-} - b_{ij} |}{| S_{j}^{-} - b_{ij} | + ρ min_{i} min_{j} | S_{j}^{-} - b_{ij} |} \end{matrix}$ (13)

Where ρ is resolution coefficient, ρ = 0.5.

When the positive ideal solutions are selected as the reference point, the subjective attitude of DMs is defined as relative losses; conversely, in terms of relative gains.

Add the $ζ_{ij}^{+}$ and $ζ_{ij}^{-}$ to the value function in 3.1.2, and an improved algorithm is defined as Equation (14). ${\begin{matrix} z_{ij}^{+} = [ζ (S_{j}^{-}, b_{ij})] = {(1 - ζ_{ij}^{-})}^{α} \\ z_{ij}^{-} = - θ {[1 - ζ (S_{j}^{+}, b_{ij})]}^{β} = - θ {(1 - ζ_{ij}^{+})}^{β} \end{matrix}$ (14)

Where α and β measure sensitivity to gains and losses, α = β = 0.88. θ is the avoidance coefficient of loss, θ = 2.25 [16].

Let V = [v_ij] _m×n be a prospect decision matrix, where v_ij is a comprehensive prospect value of the jth criteria of the ith alternative. v_ij is obtained as: $v_{ij} = \frac{| z_{ij}^{+} |}{\sum_{j = 1}^{n} | z_{ij}^{-} |} = \frac{| {(1 - ζ_{ij}^{-})}^{α} |}{\sum_{j = 1}^{n} | - θ {(1 - ζ_{ij}^{+})}^{β} |}$ (15)

3.4 Combination weight

The determination of indicator weights is frequently engaged in supplier decision-making processes. Weights can be derived in both subjective and objective ways. The subjective weighing approach has the benefit of expressing the significance of decision indicators based on expert experience. In contrast, the objective approach highlights numerical relationships between indicators. Moreover, whether subjective or objective, the assigned weights exhibit significant differences. This phenomenon indicates that the weight of an indicator is uncertain. Consequently, this study proposes an information cloud combination weighting approach to consider the advantages of both subjective and objective weights and address weight uncertainty. The core idea underlying our approach involves conceptualizing the weight of each indicator as a collection of cloud droplets enveloping the actual weights.

Suppose that the c₁ subjective weighing and c₂ weighing methods are applied to endowed with weights of n evaluation indicators, c₁ + c₂ = c. Let W be a weight matrix, which is defined as $W = [ω_{j}^{i}]_{c \times n}$ . Where $ω_{j}^{i}$ is the weight assigned to the jth indicator by the ith method, i = 1, 2, ⋯ , n, j = 1, 2, ⋯ , c.

The weight matrix’s column values are treated as cloud drops, and the weight cloud C_ω,j is formed via an inverse cloud generator. Let Ex_j, En_j, and He_j denote the weight cloud’s expectation, entropy, and hyper entropy. They can be defined as Equations (16)–(19).

a. Calculate the average value of weight samples. ${\bar{ω}}_{j} = \frac{1}{c} \sum_{i = 1}^{c} ω_{j}^{i} (j = 1, 2, \dots, n)$ (16)

b. Calculate the center distance of weight samples. $D_{j} = \frac{1}{c} \sum_{i = 1}^{c} | ω_{j}^{i} - {Ex}_{j} | (j = 1, 2, \dots, n)$ (17)

c. Calculate the variance of weight samples. $S_{j}^{2} = \frac{1}{c - 1} \sum_{i = 1}^{c} {(ω_{j}^{i} - \frac{1}{c} \sum_{i = 1}^{c} ω_{j}^{i})}^{2} (j = 1, 2, \dots, n)$ (18)

d. Determine the digital characteristics of the weight cloud. ${\begin{matrix} {Ex}_{j} = {\bar{ω}}_{j} (j = 1, 2, \dots, n) \\ {En}_{j} = \sqrt{\frac{π}{2}} D_{j} (j = 1, 2, \dots, n) \\ {He}_{j} = \sqrt{| S_{j}^{2} - {En}_{j}^{2} |} (j = 1, 2, \dots, n) \end{matrix}$ (19)

Considering the variations in the digital characteristics of the weight cloud for each indicator, especially disparities in entropy, the weights are further modified based on the solution notion of the entropy weight method. Calculate the improved combination weight by the following equation: $ω_{j} = {\begin{matrix} \frac{{Ex}_{j}}{ln (1 + {En}_{j}) + 1} \frac{1}{\sum_{j = 1}^{n} \frac{{Ex}_{j}}{ln (1 + {En}_{j}) + 1}}, ({En}_{j} \neq 0) \\ \frac{{Ex}_{j}}{\sum_{j = 1}^{n} {Ex}_{j}}, ({En}_{j} = 0) \end{matrix}$ (20)

Where ω_j is the information cloud combination weight of the ith decision index, j = 1, 2, ⋯ , n.

3.5 Cobweb model

Suppose that a supplier decision problem has n evaluation criteria, each indication is represented by a ray, with an angle of 360/n between the rays. They split the plane into n areas, and the ends of all rays converge at the circle’s center. With the origin as the center, the ideal solution and alternative are indicated on the ray and joined at the start and finish to form a spider web.

The cobweb model’s core idea is to collect the values of indicators and build a closed spider web to evaluate the options based on the similarity between alternatives and ideal solutions. The spider web resembles itself in two ways: area and form.

3.5.1 Area similarity

Let r_i,j is the value of the jth indicator on the spider web for the ith alternative, and r_o,j is the value of the jth indicator for the ith ideal solution. The area similarity can be defined as Equations (21) and (22). $\begin{matrix} S_{i} = \frac{1}{2} sin \frac{360^{\circ}}{n} ω_{1} ω_{2} | r_{i, 1} r_{i, 2} - r_{o, 1} r_{o, 2} | \\ + \frac{1}{2} sin \frac{360^{\circ}}{n} ω_{2} ω_{3} | r_{i, 2} r_{i, 3} - r_{o, 2} r_{o, 3} | + \dots \\ + \frac{1}{2} sin \frac{360^{\circ}}{n} ω_{j} ω_{1} | r_{i, j} r_{i, 1} - r_{o, j} r_{o, 1} | \end{matrix}$ (21) $α_{i} = S_{i} / \frac{1}{2} sin \frac{360^{\circ}}{n} \sum_{j = 1}^{n} (w_{j} w_{j + 1} r_{o, j} r_{o, j + 1})$ (22)

Where S_i signifies the difference in spider web area between the alternate and perfect option. α_i is the area similarity, a smaller α_i implies that the alternative and the optimum solution are more similar.

3.5.2 Shape similarity

To further comprehensively assess the similarity between the alternative and the ideal solution, shape similarity should be considered in addition to area similarity. This study is different from the traditional methodology by introducing a GCT algorithm [15] for calculation.

To determine the shape similarity of the cobweb structure, the x axis is rotated counterclockwise θ degrees about its center, and the plane is split equally based on the number of indications in the decision system, as illustrated in Fig. 3. If the number of indexes is odd, the plane is divided into 2n regions before solving. The data curve intersects with the axis x at two locations. Suppose that the distance between the two locations and the center O is a_i and b_j. Taking them as parameters to construct a complex number a_i + b_j, the feature vector of complex space for the cobweb F = (a₁ + b₁i, a₂ + b₂i, ⋯ , a_n + b_ni) can be obtained. The technique of computing shape similarity is given below, based on the preceding transformation and description.

Fig. 3

Analysis figure of shape similarity.

a. Calculate the phase-specific sequence P $P = (\frac{b_{1}}{a_{1}}, \frac{b_{2}}{a_{2}}, \dots, \frac{b_{n}}{a_{n}})$ (23)

b. Calculate the similar phase-specific sequence δ_i $δ_{i} = \sum_{j = 1}^{n} | p_{i, j} - p_{o, j} | / \sum_{j = 1}^{n} p_{o, j}$ (24)

Where p_i,j is the jth element of P for the ith alternative; p_o,j is the jth element of P for the ideal solution.

c. Calculate the intensity sequence M $M = (\sqrt{a_{1}^{2} + b_{1}^{2}}, \sqrt{a_{2}^{2} + b_{2}^{2}}, \dots, \sqrt{a_{n}^{2} + b_{n}^{2}})$ (25)

d. Calculate the similar intensity sequence η_i $λ_{i} = \sum_{j = 1}^{n} M_{i, j} / \sum_{j = 1}^{n} M_{o, j}$ (26) $η_{i} = \sum_{j = 1}^{n} | M_{i, j} - λ_{i} M_{o, j} | / \sum_{j = 1}^{n} M_{o, j}$ (27)

Where M_i,j signifies the jth element of M for the ith alternative; M_o,j is the jth element of M for the ideal solution; λ_i is the scaling factor of the ith indicator.

e. Calculate the shape similarity ζ_i

$ζ_{i} = \frac{1}{2} (δ_{i} + η_{i})$ (28)

Where a smaller ζ_i implies that the alternative and the optimum solution are more similar.

3.5.3 Comprehensive similarity

The cobweb similarity is the consequence of area and shape similarity working together. Let $ψ_{i}^{+}$ be the cobweb similarity between alternative and positive ideal solution, $ψ_{i}^{-}$ be the cobweb similarity between alternative and negative ideal solution, define them as Equation (29). ${\begin{matrix} ψ_{i}^{+} = \frac{1}{2} (α_{i}^{+} + ζ_{i}^{+}) \\ ψ_{i}^{-} = \frac{1}{2} (α_{i}^{-} + ζ_{i}^{-}) \end{matrix}$ (29)

Where $α_{i}^{+}$ and $ζ_{i}^{+}$ represent the area and shape similarity between alternative and the positive ideal solution; $α_{i}^{-}$ and $ζ_{i}^{-}$ represent the area and shape similarity between alternative and the positive ideal solution.

3.6 Closeness algorithm

The closeness degree is an essential quantity index to describe the similarity of two fuzzy sets. In multi-attribute decision-making problems, the closeness degree plays a role in determining the similarity between alternatives and ideal solutions, aiding decision-makers in making optimal choices. The similarity between the two is typically assessed based on the comprehensive distance measured between the indicators, whereby a smaller distance indicates a higher degree of similarity. Hence, the classical closeness algorithm is commonly called the relative distance method. The comprehensive distance is the cumulative discrepancy between the indicators and the ideal solution. This implies that while most indicators may be close to the ideal solution, some indicators’ extreme values can also contribute to dissimilar outcomes [2]. Furthermore, classical algorithms encounter the issue of alternatives converging towards both positive and negative ideal solutions, resulting in a potential failure of the closeness algorithm. In order to address the issues above, this paper proposes utilizing ‘cobweb similarity’ as a substitute for ‘comprehensive distance’ to mitigate the adverse effects of extreme index values. Then, the closeness algorithm is improved based on the minimum sum of squares criterion.

Suppose that the alternative always converges to the positive ideal solution with y_i and the negative ideal solution with 1 - y_i. Construct the objective function of the closeness degree F (y_i) based on the minimum sum of squares criterion by Equation (30). $min F (y_{i}) = (y_{i} ψ_{i}^{+} - ψ_{i}^{+})^{2} + [(1 - y_{i}) ψ_{i}^{-} - ψ_{i}^{-}]^{2}$ (30)

Let dF (y_i)/dy_i = 0, when the objective function F (y_i) reaches the minimum, y_i fulfills the Equation (31), according to its monotonicity. $y_{i} = \frac{(ψ_{i}^{+})^{2}}{(ψ_{i}^{+})^{2} + (ψ_{i}^{-})^{2}}$ (31)

Where y_i determine the merit of alternatives, 0 ≤ y_i ≤ 1, i = 1, 2, ⋯ , m.

3.7 Procedure of the proposed selection approach

This study presents a group decision-making approach of prefabricated component suppliers in complex environments. The subsequent section describes the specific steps involved in the method.

Step 1 : Utilize interval numbers to capture the uncertainty of evaluation indices and construct the initial group decision matrix A = [a_ij] _m×n via the set-valued statistical method.

Step 2 : Transfer the initial decision matrix A = [a_ij] _m×n using the normalization formula, and form the decision matrix B = [b_ij] _m×n.

Step 3 : Identify the reference points of each indicator, and apply the prospect theory to generate the prospect decision matrix V = [v_ij] _m×n.

Step 4 : Determine the indicator weight by c (c ≥ 2) weighting methods, and obtain the weight matrix $W = [ω_{j}^{i}]_{c \times n}$ .

Step 5 : Employ the information cloud combination weighting method to fuse and modify the weights.

Step 6 : Introduce the cobweb model to calculate the similarity ( $ψ_{i}^{+}$ and $ψ_{i}^{-}$ ) between the alternatives and the ideal solutions.

Step 7 : Utilize the cobweb similarity instead of the comprehensive distance, and improve the closeness algorithm by combining the minimum square sum criterion to compare the alternatives.

4 Implementation

In this part, a prefabricated office building project in China is used as an example for study to verify the proposed approach’s efficacy. During the main structure’s construction, the corporation was required to acquire some concrete slabs from four suppliers (A_i, i = 1, 2, 3, 4). A survey of the target suppliers indicated that each had product quality, quote, and transport capacity strengths. A decision-making group of five experts was formed to select the best supplier of prefabricated components. The decision-making team comprises experts in construction, design, procurement, and scientific research. The group members comprehensively assessed the four suppliers based on product quality, pricing, delivery capability, corporate capability, and environmental sustainability.

4.1 Application of the proposed selection approach

The calculation process involves several steps. First, Equations (1)–(10) are utilized to calculate the normalized decision matrix. Then, Equations (11)–(15) are employed to determine the prospective decision matrix. Next, Equations (16)–(20) are utilized to generate the combination weights of decision indicators. Lastly, the cobweb model (refer to Equations (21)–(29)) is used to calculate the degree of similarity between the suppliers and the ideal solutions. The outcomes of the computations are presented in Tables 2 and 3.

Table 2
The similarity between alternatives and positive ideal solutions

Suppliers $α_{i}^{+}$ $ζ_{i}^{+}$ $ψ_{i}^{+}$

A₁ 0.2410 0.1903 0.2157

A₂ 0.2453 0.2295 0.2374

A₃ 0.2464 0.2002 0.2133

A₄ 0.2673 0.4199 0.3436

Suppliers	$α_{i}^{+}$	$ζ_{i}^{+}$	$ψ_{i}^{+}$
A₁	0.2410	0.1903	0.2157
A₂	0.2453	0.2295	0.2374
A₃	0.2464	0.2002	0.2133
A₄	0.2673	0.4199	0.3436

Table 3

The similarity between alternatives and negative ideal solutions

Suppliers	$α_{i}^{-}$	$ζ_{i}^{-}$	$ψ_{i}^{-}$
A₁	0.2201	0.2917	0.2559
A₂	0.2435	0.2443	0.2439
A₃	0.2905	0.3268	0.3086
A₄	0.2460	0.1172	0.1816

To facilitate the comparison and selection of alternatives, Equations (30), (31) are utilized for closeness calculations. Table 4 displays the outcomes of the computations.

Table 4

Calculation results of closeness for supplier selection

Suppliers	$ψ_{i}^{+}$	$ψ_{i}^{-}$	y _i	Ranking
A₁	0.2157	0.2559	0.4153	2
A₂	0.2374	0.2439	0.4865	3
A₃	0.2133	0.3086	0.3436	1
A₄	0.3436	0.1816	0.7817	4

In accordance with the closeness presented in Table 4, the outcomes for 4 suppliers are 0.4153, 0.4865, 0.3436, and 0.7817, meaning that A₃ > A₂ > A₁ > A₄. As a consequence, the corporation ought to choose 3rd supplier as a vendor for the concrete precast slabs based on a comprehensive evaluation.

4.2 Comparative analysis

4.2.1 Comparison with other methods

To further demonstrate the effectiveness and superiority of proposed method, we utilize the VIKOR, TOPSIS, and MOORA algorithms to re-evaluate and rank the four suppliers. The decision results of the four methods are presented in Table 5.

Table 5
The decision results of the four methods

Methods Suppliers

A₁ A₂ A₃ A₄

MOORA y _i 0.2446 0.2360 0.3023 0.0237

Ranking 2 3 1 4

TOPSIS y _i 0.5430 0.5281 0.5755 0.4209

Ranking 2 3 1 4

VIKOR S _i 0.4073 (3) 0.3398 (2) 0.2947 (1) 0.8568 (4)

R _i 0.0613 (3) 0.0454 (1) 0.0524 (2) 0.0778 (4)

Q _i 0.3455 (3) 0.0401 (1) 0.1081 (2) 1.0000 (4)

Ranking 1 1 1 4

Proposed method y _i 0.4153 0.4865 0.3436 0.7817

Rank 2 3 1 4

Methods		Suppliers
MOORA	y _i	0.2446	0.2360	0.3023	0.0237
	Ranking	2	3	1	4
TOPSIS	y _i	0.5430	0.5281	0.5755	0.4209
	Ranking	2	3	1	4
VIKOR	S _i	0.4073 (3)	0.3398 (2)	0.2947 (1)	0.8568 (4)
	R _i	0.0613 (3)	0.0454 (1)	0.0524 (2)	0.0778 (4)
	Q _i	0.3455 (3)	0.0401 (1)	0.1081 (2)	1.0000 (4)
	Ranking	1	1	1	4
Proposed method	y _i	0.4153	0.4865	0.3436	0.7817
	Rank	2	3	1	4

* S_i is the group utility; R_i is the individual regret; Q_i is the compromise value.

According to the decision results, the three methods show consistent rankings for four suppliers, with the order being A₃ > A₂ > A₁ > A₄, except when the VIKOR algorithm is used. The VIKOR algorithm is a compromise decision-making method that ranks multi-criteria alternatives based on maximizing group utility and minimizing individual regret. In this study, apart from A₂ and A₄ simultaneously satisfying Condition 1 (dominance criteria) and Condition 2 (stability criteria), A₂ and other suppliers solely fulfill Condition 2, aligning with the requirements for multiple optimal solutions of the VIKOR algorithm [4]. Therefore, VIKOR recommends A₁, A₂, and A₃ as the optimal suppliers and only helps DMs exclude one potential choice. Utilizing the VIKOR algorithm is inappropriate when the alternatives are limited and similar.

To facilitate the intuitive comparison of the proposed method with MOORA and TOPSIS, we compute the discrepancies of consecutive ranking schemes based on Table 5 and draw a corresponding difference diagram (see Fig. 4).

Fig. 4

Difference diagram of consecutive ranking schemes.

Despite the identical ranking results achieved by all three methods, this study demonstrates higher discrimination based on the comparison results. Particularly, the differences are more pronounced when confronted with similar alternatives. Using A₁ and A₂ as examples, the decision values for suppliers 1 and 2 are closely aligned by the MOORA and TOPSIS methods. The former shows a difference of 0.0086, while the latter yields a difference of 0.0149 in the decision results. In contrast, the method proposed in this paper yields a higher difference of 0.0712, highlighting the superiority of the proposed method when evaluating the merits and drawbacks of similar alternative. Furthermore, compared with the absolute rationality showed in the MOORA and TOPSIS algorithms during the decision-making process, an additional advantage of the proposed method lies in its consideration of DMs’ risk preferences when confronted with potential losses and benefits. This characteristic renders the decision-making method more widely applicable and realistic.

4.2.2 Comparison with robustness

The Euclidean distance is widely employed to calculate the similarity between the alternative and ideal solutions. However, due to the potential influence of extreme values on the Euclidean distance during the similarity calculation, this paper introduces the cobweb similarity to improve it. In order to compare the disparity between the cobweb similarity and the Euclidean distance when faced with extreme index values, we analyze four indicators related to product quality as samples. Considering that the rework and return rate C2 has the most significant weight among the four indicators, to exclude the effects of other indicators on product quality, only this indicator is incremented by 0.1 to form a control group. Two similarity algorithms are used to calculate the control group, and the corresponding results are presented in Table 6.

Table 6
Comparison results of cobweb similarity and Euclidean distance

Group Euclidean distance cobweb similarity

d Δd % ψ Δψ %

1 0.1 0.0 0.0 0.118 0.000 0.000

2 0.2 0.1 100 0.210 0.092 7804

3 0.3 0.2 200 0.284 0.166 140.9

4 0.4 0.3 300 0.344 0.226 191.7

5 0.5 0.4 400 0.393 0.275 233.4

Group	Euclidean distance	cobweb similarity
1	0.1	0.0	0.0	0.118	0.000	0.000
2	0.2	0.1	100	0.210	0.092	7804
3	0.3	0.2	200	0.284	0.166	140.9
4	0.4	0.3	300	0.344	0.226	191.7
5	0.5	0.4	400	0.393	0.275	233.4

Based on Table 7, when the indicator C2 is incremented by 0.1 each time, the variation rates of results measured by Euclidean distance are 100%, 200%, 300%, and 400%, whereas the variation rates of results measured by cobweb similarity are 78.35%, 140.91%, 191.68%, and 233.36%. Thus, when faced with extreme values for indicators, it is apparent that cobweb similarity is less affected compared to Euclidean distance. The same approach is applied to analyze other indicators.

Table 7

Calculation results of the classical algorithm

Suppliers	Classical algorithm
	d ⁺	d ^-	y _i	Ranking
A₁	0.0529	0.0629	0.5430	2
A₂	0.0521	0.0583	0.5281	3
A₃	0.0583	0.0790	0.5755	1
A₄	0.0808	0.0587	0.4209	4

4.2.3 Comparison of different closeness algorithms

This section conducts a comparative analysis with the classical algorithm to verify the efficacy of the closeness algorithm proposed in this paper. Table 7 describes the closeness results obtained by the classical algorithm.

Apart from the impact of extreme values on closeness, this deviation can be attributed to the failure of Euclidean distance. Observing the former, the Euclidean distances of A₁ and A₂ from the positive ideal solution are 0.0521 and 0.0529, indicating that A₁ is closer to the positive ideal solution than A₂. While in terms of their distances from the negative solution, A₁ also tends to the negative ideal solution. Notably, the phenomenon is not observed in the methodology described in this study. A₁ has a higher similarity to the positive ideal solution and a lower similarity to the negative solution when compared to A₂. This method effectively rectifies the inadequacy of the classical algorithm, which results in alternatives being close to both the positive and negative ideal solutions. Consequently, when ranking alternatives, the proposed method is more acceptable.

5 Conclusions and future work

(a) This study presents an evaluation index system for suppliers of prefabricated components based on a literature analysis and expert interviews. The system comprises six levels: product quality, price, service level, comprehensive ability, supply ability, and environmental sustainability.

(b) Nowadays, many decision-making issues depend on a substantial amount of ambiguous information and subjective judgments, making it difficult for decision-makers to obtain precise values beforehand. This paper utilizes interval numbers as substitutes for precise values and generates a decision-making prospect interval matrix by incorporating the prospect theory. The cobweb similarity measure replaces the Euclidean distance, and a novel algorithm that integrates the minimum square sum criteria is proposed to evaluate proximity. This algorithm not only analyzes decision-making process uncertainties but also overcomes the limitations of the classical algorithm, thereby enhancing decision-making effectiveness.

(c) This study utilizes an office building project as a case study to demonstrate that the proposed method enables the comparison of different schemes based on shape similarity, area similarity, and closeness. The decision results are reasonable and accurate. However, it is undeniable that the method has certain limitations, including insufficient coverage of indicators and a lack of consideration for their interrelationships. These deficiencies need further research. Furthermore, given its high practicality, it holds potential for future expansion into manufacturing, logistics, and other fields to help managers improve decision-making efficiency.

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A supplier selection method based on cobweb similarity and prospect theory for prefabricated components

Abstract

Keywords

1 Introduction

1.1 Research on evaluation indicators

1.2 Research on selection methods

2 Indicator system

2.1 Preliminary selection based on literature

3.1 Preliminaries

3.1.1 Interval number

3.1.2 Prospect theory

3.2 Initial group decision matrix

3.5.1 Area similarity

4 Implementation

4.1 Application of the proposed selection approach

Table 2 The similarity between alternatives and positive ideal solutions Suppliers α i + ζ i + ψ i + A1 0.2410 0.1903 0.2157 A2 0.2453 0.2295 0.2374 A3 0.2464 0.2002 0.2133 A4 0.2673 0.4199 0.3436

4.2.1 Comparison with other methods

Table 6 Comparison results of cobweb similarity and Euclidean distance Group Euclidean distance cobweb similarity d Δd % ψ Δψ % 1 0.1 0.0 0.0 0.118 0.000 0.000 2 0.2 0.1 100 0.210 0.092 7804 3 0.3 0.2 200 0.284 0.166 140.9 4 0.4 0.3 300 0.344 0.226 191.7 5 0.5 0.4 400 0.393 0.275 233.4

5 Conclusions and future work

References

Table 2
The similarity between alternatives and positive ideal solutions

Suppliers $α_{i}^{+}$ $ζ_{i}^{+}$ $ψ_{i}^{+}$

A₁ 0.2410 0.1903 0.2157

A₂ 0.2453 0.2295 0.2374

A₃ 0.2464 0.2002 0.2133

A₄ 0.2673 0.4199 0.3436

Table 6
Comparison results of cobweb similarity and Euclidean distance

Group Euclidean distance cobweb similarity

d Δd % ψ Δψ %

1 0.1 0.0 0.0 0.118 0.000 0.000

2 0.2 0.1 100 0.210 0.092 7804

3 0.3 0.2 200 0.284 0.166 140.9

4 0.4 0.3 300 0.344 0.226 191.7

5 0.5 0.4 400 0.393 0.275 233.4