This section provides a brief overview of an analytical hierarchy process (AHP) through multi-criteria decision-making method (MCMD) that is used to prioritize alternatives based on criteria and strategic decisions discussed in the previous section. The AHP is chosen primarily for its familiarity, popularity, and user-friendly nature compared to other MCDM methods.
3.4 Steps in Fuzzy AHP Analysis
AHP is a multi-criteria decision analysis technique that employs a hierarchical and pairwise comparison matrix. This technique facilitates the resolution of multiple intricate decision problems. In utilization of AHP, a hierarchical structure that is created is typically subdivided into various layers, ranging from level 0 to level n. The aim of the decision problem is conventionally positioned at the top of the hierarchical model. The remaining levels of the structure comprise of categories, factors, alternatives, and other relevant components [70].
The application of fuzzy sets to AHP analysis was criticized by Saaty, in its own work [71]. However, due to many advancements on the application of fuzzy set for AHP problems, its application became a common endeavor in MCDM problems. This is mainly because a single element or value is insufficient to model real life problems due to the vagueness and uncertainty of human evaluation. The steps followed in the analysis of the Fuzzy AHP problem of the present study is briefly described below.
Step 1: Analyzing the Problem
The AHP model developed for the present study is composed of four successive levels, as shown in Table 1 and Fig. 1. The top level of the hierarchy (Level 0) is the goal of the model, which is “Prioritization of information requirements for decision support in strategic decisions”. The first level (level 1) of the hierarchy is the criteria (cost, risk, and business benefit/value) for selecting information categories on the basis of possible strategic decisions (improve/adapt, maintain/keep/preserve, and deconstruct/disassemble). The second level (Level 2) refers to strategic decisions made by the asset owner or its representative. The third level (Level 3) is the alternatives which consists of five categories of information requirement in asset information modelling (AIM).
Table 1
Level 0 (Goal) | Level 1 (Criteria) | Level 2 (Strategic Decisions) | Level 3 (Information Requirements) |
Prioritization of Information Requirements for decision support in strategic decisions | Cost | Improve/Adapt | Technical information Managerial information Financial information Legal information Commercial information |
Risk | Maintain/Keep |
Business Value/ Benefit | Deconstruct/ Disassemble |
Step 2: Construction of Hierarchical Model
Using the criteria, selected strategic decisions and alternatives, a hierarchal structure is constructed to apply the fuzzy-AHP (FAHP) approach. The relationship among criteria, strategic decisions and alternatives are defined and mapped as shown in Fig. 3. The first level of the hierarchy is represented by a goal to prioritize information requirements based on strategic decisions. The second level represent the criteria (Cost, risk, business benefit/value). The third level represent strategic decisions for which information are required (improve, maintain, deconstruct). Finally, the bottom level represents five alternative categories of information.
Step 3: AHP Questionnaire Design and Survey
An AHP questionnaire was structured, designed and pilot tested before use for data collection. A total of purposely selected 20 experts through snowballing method were requested to provide feedback on the survey, describing the goal, criteria, and alternatives as well as the procedures to be followed by their judgments. Of the invited 20 experts, 11 were responsive and voluntarily participated in the entire process. Experts with proven experience in the fields of facility/asset management, building maintenance management, asset ownership, building information modelling, property management and research were recruited for the study. The experts received a link to complete the questionnaire in Google Sheets. The characteristics of the participating experts are summarized in Table 2.
Table 2
Expert Profiles Participated in Decision-Making
No. | Position | Experience (Year) | Academic Achievement | Educational Background | Organization |
1 | General manager | 15 | PhD | Civil Engineer | Electro-Mechanical & Research |
2 | Ass. Professor | 15 | PhD | BIM and Architecture | Consultant Researcher |
3 | Ass. Professor | 18 | PhD | Construction Management | Consultant & researcher |
4 | Director | 16 | PhD | Construction Management | Owner research |
5 | Ass. Professor | 10 | PhD | Civil Engineering | |
6 | Lecturer | 6 | Master’s Degree | Construction Management | Owner research |
7 | Civil engineer | 7 | Master’s Degree | Civil Engineering | Client |
8 | Senior BIM manager | > 20 | Master’s Degree | Architecture | Consultant |
9 | Senior BIM manager | > 20 | Master’s Degree | BIM and Mechanical Engineering | Owner |
10 | Civil Engineer | 5 | Master’s Degree | Civil Engineering | Owner |
11 | Civil Engineer | 7 | Master’s Degree | Civil Engineering | Owner |
Step 4: Building the Hierarchy of the Problem
Based on the hierarchical structure developed in Table 1, the AHP questionnaire was formulated and sent to experts in order to make pairwise comparisons of the importance of criteria, strategic decisions, and alternatives. The matrix is based on a hierarchy with n criteria, so that each expert is expected to make a total of n (n-1)/2 pairwise comparisons in the three levels of the hierarchy [67], [72], [73].
Step 5: Establishing Comparison Matrices
The comparison matrix for each expert indicated in Eq. (1) &(2) are established as explained in [74].
$$\,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{A} =\left[ {\begin{array}{*{20}{c}} 1&{{a_{12}}}& \cdots &{{a_{1n}}} \\ {{a_{21}}}&1& \cdots &{{a_{2n}}} \\ \vdots & \vdots & \ddots & \vdots \\ {{a_{n1}}}&{{a_{n2}}}& \cdots &1 \end{array}} \right]$$
1
The matrix has reciprocal properties, which are:
$${a_{ij}}=\frac{1}{{{a_{ij}}}},\forall i,j=1,2,3....n$$
2
where,
\({\tilde {a}_{ij}}=\left\{ \begin{gathered} \tilde {1},\tilde {2},\tilde {3},\tilde {4},\tilde {5},\tilde {6},\tilde {7},\tilde {8},\tilde {9}\;Criterion\,i\,relative\;impor\tan ce\;to\;criterion\;j \hfill \\ 1,\forall i=j\,\,criterion\;i\;isequal\;impor\tan ce\;criterion\;j \hfill \\ \;{{\tilde {1}}^{ - 1}},{{\tilde {2}}^{ - 1}},{{\tilde {3}}^{ - 1}},{{\tilde {4}}^{ - 1}},{{\tilde {5}}^{ - 1}},{{\tilde {6}}^{ - 1}},{{\tilde {7}}^{ - 1}},{{\tilde {8}}^{ - 1}},{{\tilde {9}}^{ - 1}}\;criterion\;i\;is\;relative\;impor\tan ce\;to\;criterion\;j \hfill \\ \end{gathered} \right\}\;\)
The linguistic variables, obtained from experts, were transformed into a triangular fuzzy number based on Afolayan et al [75], as shown in Table 4. The pairwise comparison judgements are represented as a fuzzy triangular numbers represented by the equations (6) and (7).
Step 6: Computation of the Consistency Index and Ratio
After experts make their judgments, it is important to consider whether or not the data are consistent. Some degree of inconsistency in the experts' responses is to be expected and acceptable in AHP analysis. For the acceptable limit of inconsistency (0.1), Saaty's [70] model is used. To calculate the inconsistency ratio, the inconsistency index is computed using Eq. (3).
$$\begin{gathered} CI=\frac{{{\lambda _{\hbox{max} }} - n}}{{n - 1}} \hfill \\ \hfill \\ \end{gathered}$$
3
Where the \({\lambda _{\hbox{max} }}\)is the largest Eigen value of the comparison matrix and n is the dimension of the matrix.
The consistency ratio (CR) is defined as a ratio between the consistency of a given evaluation matrix and consistency of a random matrix (Eq. 4 and Table 3).
$$CR=\frac{{CI}}{{RI}}$$
4
Where, RI is random consistency index attained from a large number of simulations runs and varies depending on the number of criteria as shown in Table 3.
If CR is greater than 0.1, the value indicates inconsistent judgment. In such a case, the expert is encouraged to reconsider and revise the original values in the pairwise comparisons [76].
Table 3
Random consistency index (RI)
n | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
RI | 0 | 0 | 0.58 | 0.9 | 1.12 | 1.24 | 1.32 | 1.4 | 1.45 | 1.12 |
Step 7: Establishing Fuzzy Pairwise Comparison Matrices
The comparison matrix expressed in Eq. (1) is transformed to fuzzy set based on Table 4 and employing Eq. (5). Matrix A represents the fuzzy judgment matrix of nth decision maker’s preference for ith criterion, whereby the fuzzy transformation becomes one as indicated in Eq. (5).
$$\,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{A} =\left[ {\begin{array}{*{20}{c}} 1&{({l_{12}},{m_{12}},{u_{12}})}& \cdots &{({l_{1n}},{m_{1n}},{u_{1n}})} \\ {\left( {\frac{1}{{{u_{12}}}},\frac{1}{{{m_{12}}}},\frac{1}{{{l_{12}}}}} \right)}&1& \cdots &{({l_{2n}},{m_{2n}},{u_{2n}})} \\ \vdots & \vdots & \ddots & \vdots \\ {\left( {\frac{1}{{{u_{1n}}}},\frac{1}{{{m_{1n}}}},\frac{1}{{{l_{1n}}}}} \right)}&{\left( {\frac{1}{{{u_{2n}}}},\frac{1}{{{m_{2n}}}},\frac{1}{{{l_{2n}}}}} \right)}& \cdots &1 \end{array}} \right]$$
5
Table 4
Saaty’s Scale vs Fuzzy AHP [75]
Linguistic Variables | Sarty’s Scale | Fuzzy AHP Scale |
(Crisp AHP Scale) | TFN | Reciprocal TFS |
Equally important | 1 | (1,1,1) | (1,1,1) |
Equally to Moderately Important | 2 | (1,2,3) | (1/3,1/2,1) |
Moderately Important | 3 | (2,3,4) | (1/2/4,1/3,1/2) |
Moderately to Strongly Important | 4 | (3,4,5) | (1/3/5,1/4,1/3) |
Strongly Important | 5 | (4,5,6) | (1/4/6,1/5,1/4) |
Strongly to Very Strongly Important | 6 | (5,6,7) | (1/5/7,1/6,1/5) |
Very Strongly Important | 7 | (6,7,8) | (1/6/8,1/7,1/6) |
Very Strongly to Extremely Important | 8 | (7,8,9) | (1/7/9,1/8,1/7) |
Extremely Important | 9 | (8,9,9) | (1/9/9,1/9,1/8) |
$$\mu A({\widetilde {a}_{ij}}):R \to \left[ {0 - 1} \right]\,where\,{\mu _A}({a_{ij}})$$
6
$$\begin{gathered} {\mu _A}({a_{ij}})=\left\{ \begin{gathered} \,\,\,0,{\widetilde {a}_{ij}}<l \hfill \\ {\widetilde {a}_{ij}} - {l_{ij}}/{m_{ij}} - {l_{ij}},{l_{ij}} \leqslant {\widetilde {a}_{ij}} \leqslant {m_{ij}} \hfill \\ n - {\widetilde {a}_{ij}}/{u_{ij}} - {m_{ij}},{m_{ij}} \leqslant {\widetilde {a}_{ij}} \leqslant {u_{ij}} \hfill \\ 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\widetilde {a}_{ij}}({l_{ij}}\,or\,{a_{ij}}){u_{ij}} \hfill \\ \end{gathered} \right\} \hfill \\ \hfill \\ \end{gathered}$$
7
Where, l ij ,mij uij shows the minimum possible, most likely and the maximum possible value of a fuzzy number via TFN respectively [67], [75].
Step 8: Aggregation of Experts’ Judgement
In the aggregation of decisions of experts, the study used the most widely used fuzzy geometric mean approach of Buckley [77] as formulated in Eq. (8).
$${\tilde {r}_i}={\left( {\prod\limits_{{i=1}}^{n} {({{\tilde {a}}_{ij}}} )} \right)^{\frac{1}{n}}}$$
8
Where \({\tilde {r}_i}\)is the geometric mean of fuzzy comparison values.
Step 9: Determination of the Fuzzy Priority Weights
In the computation of the fuzzy weights of each criterion, strategic decision, and alternative, defuzzification is used to transform the fuzzy numbers into crisp values. Eq. (8) is used to transform the individual fuzzy numbers into non-fuzzy or crisp. Then the matrix Mi is transformed to F=[fi], Where the element matrix for fi is calculated using Eq. (9). The sum of local weights of criteria, decisions, and alternatives on the same hierarchy should be one (1) and the same is tested in this study
$${f_i}=\frac{{M{}_{i}}}{{\sum\limits_{{i=1}}^{n} {{M_i}} }}$$
9
Where, f i is the normalized non-fuzzy weights.
Step 10: Determination of Weights
At this stage, the components of weight vectors of the criteria are determined using the Eq. (10).
$${w_i}=\frac{{{f_i}}}{n}\,\,{\forall _i} \in \left\{ {1,2,...,n} \right\}$$
10