**5. Case Study**

#### *5.1. Background*

In this case of selecting construction program manager, there are five candidates A = {A1, A2, A3, A4, A5}, and three decision makers D = {D1, D2, D3}. According to the selection attribute system in this paper, C = {C1 = Personality charm, C2 = Management ability, C3 = Communication and coordination, C4 = Professional skills, C5 = Risk control, C6 = Strategic vision}. Using the language assessment scale S = {S−<sup>4</sup> = Extreme 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 = Extreme good} to evaluate the five candidates, the linguistic evaluation matrix for the group decision makers can be obtained as shown in Tables 3–5. The significant impact between different attributes, such as personality charm, having a significant impact on communication and coordination, has also been taken into consideration. Then, the three decision makers used {no influence "0", very weak influence "1", weak influence "2", strong influence "3", very strong influence "4"} to analyze the influence relationship between the attributes and use the language assessment scale to get the direct correlation matrix between attributes as shown in Tables 6–8.

**Table 3.** Linguistic Multi-attribute Evaluation Matrix X1 of decision maker D1.



**Table 4.** Linguistic Multi-attribute Evaluation Matrix X2 of decision maker D2.

**Table 5.** Linguistic Multi-attribute Evaluation Matrix X3 of decision maker D3.


**Table 6.** Direct correlation matrix Y1 between indicators of decision maker D1.



**Table 7.** Direct correlation matrix Y2 between indicators of decision maker D2.

**Table 8.** Direct correlation matrix Y3 between indicators of decision maker D3.


#### *5.2. Decision-Making Steps*

(1) Given the parameters μ = 0.88 and δ = 0.3 [17,18], Formulas (4)–(9) are used to process the linguistic multi-attribute evaluation matrix X given by each decision maker to obtain the perceived utility function pij t , as shown in Tables 9–11.


**Table 9.** Perceived utility value of decision maker D1.

**Table 10.** Perceived utility value of decision maker D2.


**Table 11.** Perceived utility value of decision maker D3.


(2) The direct correlation matrix Y between indicators is processed according to Formulas (10)–(20), and the attribute weights are obtained as follows.

$$\mathbf{w}\_{\mathbf{j}}(1) = (0.1737, 0.2083, 0.1950, 0.1118, 0.1647, 0.1464)^{\mathbf{T}}$$

$$\mathbf{w}\_{\mathbf{j}}(2) = (0.1778, 0.2076, 0.1751, 0.1108, 0.1727, 0.1559)^{\mathbf{T}}$$

$$\mathbf{w}\_{\mathbf{j}}(\boldsymbol{\beta}) = (0.1689, 0.2112, 0.1886, 0.0970, 0.1823, 0.1519)^{\mathrm{T}}$$

(3) According to Formulas (21)–(27), the attribute weight wj(t) is processed, and the weight of decision makers is obtained as follows.

$$\mathbf{W}(\mathbf{t}) = (0.\mathbf{3287}, 0.\mathbf{3448}, 0.\mathbf{3265})$$

(4) According to Formula (28), the preference coefficient τ = 0, which means the process only focus on the weight of decision makers based on the minimization of deviation. So, the perceived utility value of each alternative construction program manager is p∗ <sup>i</sup> = (5.3858, 5.1525, 5.4812, 5.3175, 5.2502)T. Thus, the order of construction program managers is A3 > A1 > A4 > A5 > A2, that is, the construction unit chooses A3 as the optimal construction program manager.
