4.2.1. Model Preparation

Assuming the alternative construction program manager set A = {A1, A2, A3, ... , An}, where Ai denotes the ith program manager, i ∈ N, N = {1, 2, 3, ... , n}. Property (index) set C = {C1, C2, C3, ... , Cm}, where Cj denotes the j property, j ∈ M, M = {1, 2, 3, ... , m}. The decision maker (expert) set D = {D1, D2, D3, ... , Dp}, where Dt denotes the tth decision maker, t ∈ P, P = {1, 2, 3, ... , p}. The linguistic multi-attribute evaluation matrix Xt = (xij t )n×m, where xij <sup>t</sup> denotes the linguistic evaluation value of decision maker Dt on the selection attribute Cj of construction program manager Ai.

In order to make better use of the knowledge and personal experience of the decision makers, let the decision makers to evaluate the degree of the interaction between the attributes, and then construct an attribute correlation matrix, so the direct correlation matrix Y<sup>t</sup> = (ykl t )m×m, where ykl <sup>t</sup> represents the impact assessment value of the t decision maker on the kth attribute and the lth attribute.

#### 4.2.2. Evaluation Language Based on Regret Theory

#### (1) Basic definition

This paper uses the non-uniform language scale and are defined as follows: The linguistic term set of *a* on the right side of numerical zero is

$$\mathcal{S}^{+} = \left\{ S\_a \, \middle| \, a = \frac{2(i-1)}{\sigma + 2 - i}, \, i = 2, \, \dots, \, \sigma - 1, \, \sigma \right\} \tag{1}$$

The linguistic term set of *a* on the left side of numerical zero is

$$S^{-} = \left\{ S\_a \middle| a = \frac{2(i-1)}{\sigma + 2 - i}, i = \sigma, \sigma - 1, \dots, 2 \right\} \tag{2}$$

Therefore, the language assessment scale is

$$\mathbf{S} = \left\{ \mathbf{S}\_{\mathbf{a}} \Big| \alpha = -(\sigma - 1), -\frac{2(\sigma - 2)}{3}, \dots, 0, \dots, \frac{2(\sigma - 2)}{3}, (\sigma - 1) \right\} \tag{3}$$

In particular, S−(σ−1) and S(σ−1) denote the lower and upper limits of the linguistic terms actually used by decision makers, σ is a positive integer, and the number of linguistic terms (2σ − 1) is called the granularity of the term set, and Sa satisfies the following properties: If α > β, then S<sup>α</sup> > Sβ; there exists a negative operator neg(Sa)=S−a.

For example, when σ is 5, the granularity of the linguistic term set is 9, and then S = {S−<sup>4</sup> = extremely poor, S−<sup>2</sup> = very poor, S−<sup>1</sup> = poor, S−0.4 = slightly poor, S0 = general, S0.4 = slightly good, S1 = good, S2 = very good, S4 = extremely good}.

(2) Utility perception value based on language information

The language identification process used in this study is as follows:

**Definition 1.** *Let Sa* ∈ *S be a term for a language evaluation set, then, a subscript conversion function that converts any language assessment into an exact number is used as shown in function (4).*

$$\mathbf{H(S\_a) = a}\tag{4}$$

**Definition 2.** *Let V*(*X*) *be a classical utility function, which is a monotone increasing concave function, namely V* (*X*) > 0*, V*(*X*) < 0*, indicating that the decision makers are risk aversion, then, a language utility function is formed as shown in function (5), in which, σ* − *1 is called the cardinality and satisfies 0* ≤ *(a + σ* − *1)/(2(σ* − *1))* ≤ *1.*

$$\text{UV}(\text{S}\_{\text{a}}) = \text{V}\left(\frac{\text{H}(\text{S}\_{\text{a}}) + \sigma - 1}{2(\sigma - 1)}\right) = \text{V}\left(\frac{\text{a} + \sigma - 1}{2(\sigma - 1)}\right) \tag{5}$$

From Definition 2, when S*<sup>a</sup>* takes the maximum value, the language utility function is also the largest. When S*<sup>a</sup>* takes the minimum value, the language utility function is the smallest. Therefore, it will ensure the accuracy of the results without information loss.

**Definition 3.** *Let Sa*, *Sa* ∗, *Sa* −, *Sa* <sup>+</sup> *be the current selected construction program manager, the ideal construction program manager, the negative ideal construction program manager and the positive ideal construction program manager, respectively. R(y) is a classical regret-happy function, which is also a monotone increasing concave function, where R (y) > 0, R(y) < 0, and to meet the intuitive judgment result R(0) = 0, then, there is function (6) that helps the decision maker to chooses the current project construction program manager Ai and abandons the ideal construction program manager A\*.*

$$\mathbf{TR}\_{\mathbf{i}} = \mathbf{R}\left(\mathbf{V}\left(\frac{\mathbf{a} + \sigma - 1}{2(\sigma - 1)}\right) - \mathbf{V}\left(\frac{\mathbf{a}^\* + \sigma - 1}{2(\sigma - 1)}\right)\right) \tag{6}$$

*When Sa* ∗ *= Sa* −*, expressed as the negative ideal construction program manager language evaluation value, namely*

$$\mathbf{TR}\_{\mathbf{i}}^{-} = \mathbf{R}\left(\mathbf{V}\left(\frac{\mathbf{a} + \sigma - 1}{2(\sigma - 1)}\right) - \mathbf{V}\left(\frac{\mathbf{a}^{-} + \sigma - 1}{2(\sigma - 1)}\right)\right) \tag{7}$$

*When Sa* ∗ *= Sa* <sup>+</sup>*, expressed as the positive ideal construction program manager language evaluation value, namely*

$$\mathrm{TR}\_{\mathrm{i}}^{+} = \mathrm{R}\left(\mathrm{V}\left(\frac{\mathrm{a} + \sigma - 1}{2(\sigma - 1)}\right) - \mathrm{V}\left(\frac{\mathrm{a}^{+} + \sigma - 1}{2(\sigma - 1)}\right)\right) \tag{8}$$

**Definition 4.** *Suppose that the language utility function of decision maker Dt for the evaluation value Sa of Ai of the selected project construction group manager is UV*(*Sa*)*, and the language regret-happiness function is TR i, then, decision maker Dt chooses the language perception utility function of the construction program manager Ai.*

$$\mathrm{Tl}\_{l}^{\mathrm{d}} = \mathrm{UV}(\mathrm{S}\_{\mathrm{h}}) + \mathrm{TR}\_{\mathrm{l}} = \mathrm{UV}(\mathrm{S}\_{\mathrm{h}}) + \mathrm{TR}\_{\mathrm{l}}^{-} + \mathrm{TR}\_{\mathrm{l}}^{+} = \mathrm{V}\left(\frac{\mathrm{a} + \sigma - 1}{2(\sigma - 1)}\right) + \mathrm{R}\left(\mathrm{V}\left(\frac{\mathrm{a} + \sigma - 1}{2(\sigma - 1)}\right) - \mathrm{V}\left(\frac{\mathrm{a} + \sigma - 1}{2(\sigma - 1)}\right) - \mathrm{V}\left(\frac{\mathrm{a}^{+} + \sigma - 1}{2(\sigma - 1)}\right)\right) = \mathrm{V}\left(\frac{\mathrm{a} + \sigma - 1}{2(\sigma - 1)}\right) + \mathrm{TR}\_{\mathrm{l}} = \mathrm{T}\_{\mathrm{h}} + \mathrm{TR}\_{\mathrm{l}} + \mathrm{TR}\_{\mathrm{r}} = \mathrm{T}\_{\mathrm{h}} + \mathrm{TR}\_{\mathrm{r}}$$

*Through the function TF<sup>t</sup> <sup>i</sup> , the language perception utility value of any construction program manager of any decision maker can be obtained.*

*In this paper, we take the function R(y) = 1* − *exp(δ*·*y), where the parameter δ*∈ *[0, +*∞*] is the regret aversion coefficient of the decision makers. The greater the parameter δ, the greater the regret aversion degree of the decision makers, and vice versa. In addition, the power function V(x) = Xμ(0 < μ < 1) is used as the utility function, where μ denotes the risk aversion coefficient of decision makers, and the greater the μ, the smaller the degree of risk aversion of decision makers.*

#### 4.2.3. Determination of Index Weights Based on Fuzzy-DEMATEL

The key to traditional DEMATEL method is to invite experts to evaluate the mutual influence of each attribute based on their knowledge and experience to form a direct correlation matrix. Due to the uncertainty of practical problems, the complexity of evaluation and the differences between invited experts, most of the evaluations given by experts are not accurate but are similar to fuzzy semantic expressions such as "important" or "satisfied". Therefore, this paper introduces the triangular fuzzy number method to process the initial matrix to improve the accuracy and the steps are as follows [44]:

Step 1: The construction program manager influencing factors system is constructed, denoted as F1, F2, ... , F6. The semantic scale assessed by experts is designed and divided into five levels according to the degree of influence, namely: no influence "0", very weak influence "1", weak influence "2", strong influence "3" and very strong influence "4", as shown in Table 2.

**Table 2.** Semantic transformation table.


Step 2: Invite experts to use language operators to evaluate the influencing factors of the construction program manager, and convert the evaluation semantics into the corresponding triangular fuzzy number Wij <sup>t</sup> = (β1ij t , β2ij t , β3ij t ) according to the semantic transformation table, which means that the t experts believe that the factor i has an effect on the factor j, where β1ij <sup>t</sup> is a conservative value, β2ij <sup>t</sup> is the closest to the reality value, and β3ij <sup>t</sup> is an optimistic value.

Step 3: Using the CFCS method to de-fuzzify the triangular fuzzy number, the direct influence matrix Z is obtained. Z reflects the direct effect between factors, and the steps are the following three steps.

(1) Standardization of triangular fuzzy numbers:

$$\propto \beta\_{1\text{ij}}^{\text{t}} = \left(\beta\_{1\text{ij}}^{\text{t}} - \min \beta\_{1\text{ij}}^{\text{t}}\right) / \Delta\_{\text{max}}^{\text{min}} \tag{10}$$

$$\mathbb{I}\propto\mathbb{I}\_{2\text{ij}}^{\text{t}} = \left(\mathbb{I}\_{2\text{ij}}^{\text{t}} - \min\mathbb{I}\_{1\text{ij}}^{\text{t}}\right) / \Delta\_{\text{max}}^{\text{min}} \tag{11}$$

$$\times \mathfrak{B}\_{\text{3ij}}^{\text{t}} = \left( \mathfrak{B}\_{\text{3ij}}^{\text{t}} - \min \mathfrak{B}\_{\text{1ij}}^{\text{t}} \right) / \Delta\_{\text{max}}^{\text{min}} \tag{12}$$

where Δmin max = maxβ3ij <sup>t</sup> − minβ1ij t , and xβ1ij t , xβ2ij t , xβ3ij <sup>t</sup> are calculated in turn. (2) Standardize left (ls) and right (rs) values:

$$\mathbf{x} \mathbf{l} \mathbf{s}\_{\mathrm{ij}}^{\mathrm{t}} = \mathbf{x} \mathfrak{G}\_{2\mathrm{ij}}^{\mathrm{t}} / \left( \mathbf{1} + \mathbf{x} \mathfrak{G}\_{2\mathrm{ij}}^{\mathrm{t}} - \mathbf{x} \mathfrak{G}\_{1\mathrm{ij}}^{\mathrm{t}} \right) \tag{13}$$

$$\mathbf{x} \mathbf{r} \mathbf{s}\_{\mathrm{ij}}^{\mathrm{t}} = \mathbf{x} \beta\_{3\mathrm{ij}}^{\mathrm{t}} / \left( \mathbf{1} + \mathbf{x} \beta\_{3\mathrm{ij}}^{\mathrm{t}} - \mathbf{x} \beta\_{1\mathrm{ij}}^{\mathrm{t}} \right) \tag{14}$$

(3) Calculate the clarity value after defuzzification:

$$\mathbf{x}\_{\text{ij}}^{\text{t}} = \left[ \mathbf{x} \text{ls}\_{\text{ij}}^{\text{t}} \left( 1 - \mathbf{x} \text{ls}\_{\text{ij}}^{\text{t}} \right) + \mathbf{x} \text{rs}\_{\text{ij}}^{\text{t}} \times \mathbf{x} \text{rs}\_{\text{ij}}^{\text{t}} \right] / \left[ 1 - \mathbf{x} \text{ls}\_{\text{ij}}^{\text{t}} + \mathbf{x} \text{rs}\_{\text{ij}}^{\text{t}} \right] \tag{15}$$

$$Z\_{\rm ij}^{\rm t} = \min \beta\_{1\rm ij}^{\rm t} + \mathbf{x}\_{\rm ij}^{\rm t} \times \Delta\_{\rm max}^{\rm min} \tag{16}$$

Step 4: Standardize the direct impact matrix Z<sup>t</sup> to get the standardized direct impact matrix *<sup>G</sup><sup>t</sup>* <sup>=</sup> (*gkl*)*m*×*m*, where

$$\mathbf{g\_{kl}^{t}} = \mathbf{z\_{kl}} / \max\_{1 \le i \le m} \sum\_{j=1}^{m} \mathbf{z\_{kl}} \tag{17}$$

Step 5: Measure the comprehensive influence matrix Tt , namely

$$\begin{aligned} \mathbf{T}^{\mathbf{t}} &= \lim\_{\mathbf{m} \to \infty} (\mathbf{G}^{\mathbf{t}} + \mathbf{G}^{\mathbf{2}} + \dots + \mathbf{G}^{\mathbf{m}}) \\ &= \mathbf{Z} (\mathbf{E} - \mathbf{Z})^{-1} \end{aligned} \tag{18}$$

where E is the unit matrix, when m <sup>→</sup> <sup>∞</sup>, G<sup>m</sup> = 0 is satisfied.

Step 6: Calculate the importance of influencing factors ε<sup>t</sup> j .

Note that each row of the elements in T<sup>t</sup> is added to the influence degree rk t , indicating the combined influence value. The addition of each column element in T<sup>t</sup> is the affected degree di t , indicating the comprehensive influence value of this element by other elements. Let k = l = j, then the importance of the influencing factor of ε<sup>t</sup> <sup>j</sup> is

$$\varepsilon\_{\rm j}^{\rm t} = \sqrt{\left(\mathbf{r}\_{\rm k}^{\rm t} + \mathbf{d}\_{\rm i}^{\rm t}\right)^2 + \left(\mathbf{r}\_{\rm k}^{\rm t} - \mathbf{d}\_{\rm i}^{\rm t}\right)^2} \tag{19}$$

Step 7: Determine the attribute weight wj(t). Normalize the importance of the influencing factor ε<sup>t</sup> <sup>j</sup> in step 6 to obtain the index weight as

$$\mathbf{w}\_{\mathbf{j}}(\mathbf{t}) = \varepsilon\_{\mathbf{j}}^{\mathbf{t}} / \sum\_{\mathbf{j}=1}^{\mathbf{m}} \varepsilon\_{\mathbf{j}}^{\mathbf{t}} \tag{20}$$

In the formula, 0 < *wj*(*t*) < 1, and satisfy ∑*<sup>m</sup> <sup>j</sup>*=<sup>1</sup> w*j*(*t*) = 1, for generality, let *wj*(*t*) = (w1(t), w2(t), ··· , wm(t))T, wj(t) represents the attribute weight of the t-th decision maker based on Fuzzy-DEMATEL on the j-th attribute.

#### 4.2.4. Weight Determination of Decision Makers Based on Deviation Minimization

The attribute weights are obtained by Fuzzy-DEMATEL based on the attribute evaluation of each decision maker. Based on the idea that smaller differences mean larger weights, this paper uses the deviation minimization to obtain the weight of decision makers [45]. For attribute weight wj(t), the difference value of attribute weight between decision maker Dt and other decision makers is Ej(t):

$$\mathbf{E}\_{\mathbf{j}}(\mathbf{t}) = \sum\_{\mathbf{t}'=1}^{\mathbf{P}} \left\{ \mathbf{w}\_{\mathbf{j}}(\mathbf{t}) - \mathbf{w}\_{\mathbf{j}}(\mathbf{t}') \right\}^2 \left(\mathbf{t}' = 1, 2, \cdots, \mathbf{P} \right) \tag{21}$$

Then define the attribute weight difference E(t) of the decision maker Dt with respect to all attributes compared to other decision makers as:

$$\mathbf{E(t) = \sum\_{j=1}^{m} \sum\_{t'=1}^{p} \left\{ \mathbf{w}\_{\mathbf{j}} \mathbf{t} - \mathbf{w}\_{\mathbf{j}}(\mathbf{t'}) \right\}^2 \left( \mathbf{t'} = \mathbf{1}, \mathbf{2}, \cdots, \mathbf{P} \right) \tag{22}$$

The selection of decision maker's weighting vector ϕ(t) should minimize the difference value of the total attribute weight of all decision makers for all attributes. Therefore, an objective weighting model for decision makers is constructed:

$$\text{minE} = \sum\_{\mathbf{t}=1}^{\mathbf{p}} \varphi(\mathbf{t})^2 \mathbf{E}(\mathbf{t}) \tag{23}$$

$$\sum\_{\mathbf{t}=1}^{\mathbf{p}} \varphi(\mathbf{t}) = 1 \; \; \; \varphi(\mathbf{t}) > 0 \; \; \mathbf{t} = 0 \; \; \; 1 \; \; \cdots \; \; \; \mathbf{p}$$

Introduce the Lagrange function to solve the above model:

$$\mathcal{L}(\boldsymbol{\varphi}(\mathbf{t}), \boldsymbol{\theta}) = \sum\_{\mathbf{t}=1}^{\mathbf{P}} \boldsymbol{\varphi}(\mathbf{t})^2 \mathbf{E}(\mathbf{t}) + 2\boldsymbol{\theta} \left[ \sum\_{\mathbf{t}=1}^{\mathbf{P}} \boldsymbol{\varphi}(\mathbf{t}) - 1 \right] \tag{24}$$

The derivations of ϕ(t) and θ are obtained:

s .t

$$\begin{cases} \frac{\partial \mathcal{L}}{\partial \boldsymbol{\varrho}(\mathbf{t})} = 2\boldsymbol{\varrho}(\mathbf{t})\mathbf{E}(\mathbf{t}) + 2\boldsymbol{\Theta} = \mathbf{0} \\ \qquad \sum\_{\mathbf{t}=1}^{\mathbf{P}} \boldsymbol{\varrho}(\mathbf{t}) = 1 \end{cases} \tag{25}$$

Thus:

$$\varphi(\mathbf{t}) = \frac{1}{\mathcal{E}(\mathbf{t})} \frac{1}{\sum\_{\mathbf{t}=1}^{\mathcal{P}} \frac{1}{\mathcal{E}(\mathbf{t})}} \tag{26}$$

The decision maker weight vector ϕ(t) is normalized to get the decision maker weight:

$$W(\mathbf{t}) = \boldsymbol{\varphi}(\mathbf{t}) / \sum\_{t=1}^{\mathbf{P}} \boldsymbol{\varphi}(\mathbf{t}) \tag{27}$$

In the formula, 0 < w(t) < 1, and satisfies ∑<sup>p</sup> <sup>t</sup>=<sup>1</sup> w(t) = 1, for generality, let w(t) = (w(1), w(2), ··· , w(p))T, w(t) represents decision maker Dt is based on the weight of decision maker with minimum deviation.

#### 4.2.5. Comprehensive Perceived Utility Value Calculation and Decision-Making

Let that pij <sup>t</sup> (i ∈ N, j ∈ M, t ∈ P) is the perceptual utility value calculated by decision maker Dt for the linguistic assessment value xij of alternative construction program manager Ai for attribute Cj, wj(t) is the attribute weight of decision maker Dt to attribute Cj based on Fuzzy-DEMATEL, wj(t) is the decision maker Dt's decision maker weight based on deviation minimization, then the comprehensive perceptual utility value of alternative construction program manager Ai is:

$$\mathbf{p}\_{\mathbf{i}}^{\*} = \sum\_{\mathbf{t}=1}^{\mathbf{p}} \sum\_{\mathbf{t}=1}^{\mathbf{m}} \left[ \tau \mathbf{w}\_{\mathbf{j}}(\mathbf{t}) + (1 - \tau) \mathbf{w}(\mathbf{t}) \right] \mathbf{p}\_{\mathbf{i}\mathbf{j}}(\mathbf{t}) \tag{28}$$

In the formula, the parameter τ ∈ [0, 1] is the weight preference adjustment coefficient, the larger the value of τ, indicating that the group decision makers pay more attention to the attribute weight based on Fuzzy-DEMATEL.
