The Delphi method has been widely used to generate forecasts in technology, education, and other fields. The method is based on a structured process of collecting and distilling knowledge from a group of experts by means of a series of questionnaires interspersed with controlled opinion feedback. Kaufmanm and Gupta (1988) [5] developed a triangular fuzzy function for the purpose of studying the Delphi function. Ishikawa et al. (1993) [6] used the concepts of cumulative frequency distribution and fuzzy integration to convert expert opinions into fuzzy numbers, resulting in the fuzzy Delphi method. Jeng (2001) [7] modified the fuzzy Delphi method by using double triangular fuzzy numbers to integrate expert opinions and then testing for consistency among the opinions using the gray zone test.

This study employed the modified fuzzy Delphi method proposed by Jeng to identify assessment factors for construction projects and established evaluation criteria for candidate projects based on these factors. The relevance of each assessment factor was scored on a scale of 0 - 10. A higher score indicates a greater influence on project selection outcomes. The operational stages of the modified fuzzy Delphi method are as follows.

Stage 1: Collect the fuzzy interval values of relevance scores.

This study collected the interval values of the relevance scores assigned by experts to each evaluation criterion. The minimum interval value (most conservative cognitive value) is indicated by C, while the maximum value (most optimistic cognitive value) is indicated by O; a is the optimal representative value as determined by experts (subjective cognitive value). Cⁱ, Oⁱ and aⁱ respectively represent the minimum value, maximum value, and subjective cognitive value of the relevance score for evaluation criterion i.

Stage 2: Eliminate extreme values.

This study calculated the mean and standard deviation of the minimum value, maximum value, and subjective cognitive value of evaluation criterion i. Extreme values that fell beyond the range of mean ±2* standard deviation were eliminated.

Stage 3: Determine the triangular fuzzy number for the relevance scores of assessment factors.

This study calculated the following:

1) minimum value Cⁱ_L, geometric mean Cⁱ_M and maximum value Cⁱ_U of the most conservative cognitive value (Cⁱ) with respect to the relevance of evaluation criterion i;

2) minimum value Oⁱ_L, geometric mean Oⁱ_M and maximum value Oⁱ_U of the most optimistic cognitive value (Oⁱ) with respect to the relevance of evaluation criterion i;

3) minimum value aⁱ_L, geometric mean aⁱ_M and maximum value aⁱ_U of the subjective cognitive value (aⁱ) of experts.

Based on the above we established the triangular fuzzy numbers for the minimum value (Cⁱ_L, Cⁱ_M, Cⁱ_U) and maximum value (Oⁱ_L, Oⁱ_M, Oⁱ_U) of the relevance score of evaluation criterion i (see Figure 1).

Stage 4: Test for consistency.

The gray zone test was used to test for consistency among the opinions of experts (test value Zⁱ = (Oⁱ_M − Cⁱ_M) − (Cⁱ_U − Oⁱ_L)).

1) When Cⁱ_U > Oⁱ_L and Zⁱ ? 0, this indicates expert opinions are converging (consistent).

2) When Cⁱ_U < Oⁱ_L and Zⁱ < 0, this indicates expert opinions are inconsistent. At this point a second survey of this specific evaluation criterion must be conducted.

Stage 5: Calculate the value of consensus among experts.

In Figure 1, the cognitive value corresponding to the intersection point of Cⁱ and Oⁱ is the value of consensus among experts (indicated by Gⁱ) in relation to the relevance score of evaluation criterion i. The higher the value of G is, the higher the consensus among experts of the significance of this assessment factor.

Stage 6: Select evaluation criteria.

This study sets G = 6.0 as the threshold value for selecting evaluation criteria. Only those assessment factors with a G value exceeding the threshold value were selected.

Employing both qualitative and quantitative analysis, ANP provides a systemized process of analysis that determines the weight of each criterion, which in turn determines the relevance of each criterion and the potential for successfully reaching the objective.

Using a network approach, ANP structures a problem into a hierarchical format. Apart from its hierarchical analysis function, ANP can also be used to analyze feedback relationships between different levels and interdependent relationships among elements on the same level. The qualitative and quantitative procedures of ANP are described in [4]. This study conducts a questionnaire survey of experts, in our case a group of construction professionals, then conduct pairwise comparison. Using the nine-point priority scale developed by Saaty, we obtained the following three types of weighted values:

1) the relative importance of each evaluation criterion to the objective of decision-making;

2) the relative importance of each candidate project to specific criteria;

3) interdependent relationships among criteria to specific groups of elements.

Then, construct the initial supermatrix Ω with multiple sub-matrices comprising the priority vectors calculated and obtain the weighted supermatrix W by unitizing each column in the initial supermatrix. The sum of all elements in a column in a weighted supermatrix is 1. Calculate the limit supermatrix by raising the weighted supermatrix to the power of 2k+1 with k→∞ (i.e. lim_k→∞W^2k+1). The weighted values corresponding to each candidate project are derived from the first column of the limit supermatrix.

A case study was used to verify the feasibility of applying the ANP approach developed by Saaty to selection of construction projects. Four different construction projects, labelled Projects A, B, C and D, were considered in this study. The background variables of these four projects are provided in Table 1.

Using Jeng’s modified fuzzy Delphi method, this study identified key factors that influence the selection of construction projects and designated these factors as evaluation criteria for candidate projects. This study distributed 10 questionnaires to five industry professionals and five academics, which were all recovered as valid questionnaires. Six factors were identified as significant: environment, finance, operations, management, legislation and technology. These factors are described in Table 2.

Table 3 presents the statistical results of the modified FDM questionnaire survey, illustrating that the consensus values for the six assessment factors all exceeded the threshold of 6.0, i.e., experts agreed that these factors were sufficiently relevant to decision making.

Following interviews with experts confirming the mutual interdependence of the evaluation criteria, the qualitative processes of ANP are illustrated in the 3-tier framework presented in Figure 2. The first tier is the decision-making objective, the second the six mutually interdependent evaluation criteria, and the third comprises the candidate projects. The ANP structural model and corresponding initial supermatrix are shown in Figure 3. In the initial supermatrix, w1 indicates a 6x1 matrix; W2 is a 6x6 matrix; W3 is a 4x6 matrix, and I is the unit matrix.

Table 4 integrates the pairwise comparison matrix, consistency test and relative importance weights of the six evaluation criteria. The results in Table 4 show that when construction companies select projects, the evaluation criteria are weighted in order of importance as follows: environment (0.209), finance (0.187), operations (0.176), legislation (0.172), management (0.146) and technology (0.110). Interestingly, environment has the highest weight while technology presents the lowest. This means that environment is the most important factor.

The case study comprised 4 candidate projects (Projects A-D). Table 5 shows the pairwise comparison matrix, consistency test and relative importance weights of each project in relation to environment. The CR value of 0.05 is less than 0.1, indicating that the expert survey results were consistent. The last column of Table 5 shows the relative importance weights of the four candidate projects in relation to environment. The weight of environment was greatest in Project B and lowest in Project C, indicating that Project B has the lowest impact on environment and Project C the strongest. In addition, Project B had the highest weight for finance. Project B also had the highest weight for operations. The weight of the management factor was greatest in Project A. Project D had the highest weight for legislation. Lastly, the weight of the technology factor was greatest in Project A.

Table 6 shows the pairwise comparison weights of the interdependent relationships among evaluation criteria. From Tables 4-6 we can obtain the initial supermatrix (Figure 4). The values in the weighted supermatrix are column stochastic. Note that the values in each column of the weighted supermatrix add up to a value of 1. If the values in the weighted supermatrix are mutually interdependent, then by repeated squaring we can obtain a converged limit supermatrix and the priority weights for the four candidate projects, which are as follows: Project A (0.307), Project B (0.3), Project D (0.276), and Project C (0.117). Therefore, the construction firm should elect to undertake Project A.