Dematel-Based Completion Technique Applied for the Sustainability Assessment of Bridges Near Shore

2.1 AHP

The Analytic Hierarchy Process (AHP) is a MCDM technique first defined by Saaty back in 1980 [8]. This method is nowadays widely used to determine the weights to be assigned to each criterion involved in decision-making problems of any kind. This technique requires an expert to compare pairwise the relevance that each criterion shall have with respect to each other when taking the decision. By doing so, a square so-called comparison matrix A_nxn is constructed, where n is the number of criteria involved in the decision-making process. The comparison between two criteria is based on the Saaty’s fundamental scale, which is used to transform linguistic judgements into numerical values (Table 1).

Table 1. Saaty’s fundamental scale [8]

The elements of the resulting comparison matrix A_nxn correspond to values of the Saaty’s fundamental scale. It shall be noted that if criterion A is considered, for example, extremely more relevant than criterion B, then criterion B is considered extremely less relevant than criterion A. This results in the comparison matrix A_nxn to be reciprocal by definition, i.e., a_ij = 1/a_ji. The AHP allows to extract the relevance of each criterion from a so-built comparison matrix as the values of the eigenvector associated to the greatest eigenvalue of the matrix (λ_max).

The resulting criteria weights are considered valid only of the comparison matrix A_nxn is consistent, i.e., the judgements of the decision maker should have been coherent. If a perfect consistency of a comparison matrix was achieved, that would result in a_ij × a_jk = a_ik ∀ i, j, k.

The consistency shall be evaluated by means of the so-called Consistency Index CI, of the comparison matrix A_nxn, as:

$C I=\left(\lambda_{\max }-n\right) /(n-1)$         (1)

where, n is the total number of criteria involved in the decision-making process. The resulting weights are then valid only if the Consistency Ratio CR = CI/RI falls below a limiting value CR_lim which depends on the number of criteria n (Table 2). In the above presented equation RI stands for the so-called Random Index, which indicates the consistency of a fully randomized nxn comparison matrix.

Table 2. Values for RI and CR_lim depending on the number of criteria involved

2.2 DEMATEL-based completion technique

The weights obtained through the application of the above presented AHP technique are widely used as a basis for many other decision-making techniques, such as TOPSIS, COPRAS or VIKOR. However, one of the most criticised aspects of the AHP technique is that the resulting criteria weights are highly subjective. In fact, the ability of the decision makers to adequately reflect their view on a problem diminishes as the complexity of the problem increases. There exist several approaches to attempt reducing the subjectivity of the decisions based on the application of MCDM techniques. Research has been conducted on the application of the fuzzy theory to mathematically model the diffusivity of human thinking and include it as a source of relevant information for the decision-making process [9, 10].

Another approach which is also in the spotlight of many researchers is to reduce the complexity of the problem by reducing the number of judgements to be made by the decision maker when filling the AHP comparison matrix [11, 12]. To do so, the DEMATEL method can be used.

2.2.1 DEMATEL method

The goal of the MCDM DEMATEL technique is to transform intricate cause-and-effect connections among diverse elements into a well-structured and easy-to-understand visual model. This involves grouping factors into cause-and-effect categories, as explained in reference [13]. The conventional approach involves four distinct stages:

Stage 1: To generate a Direct Influence Matrix (DIM), experts are requested to complete a comparison matrix using a process similar to the Analytic Hierarchy Process (AHP). In this process, each expert estimates the degree of influence that factor i has on factor j using a four-level scale of integers ranging from 0 to 3. The scores represent "no influence," "low influence," "medium influence," and "high influence," respectively. The non-negative influence matrix DIM_k = {z_ij} is created for each expert k, where z_ij denotes the assigned influence score based on the aforementioned scale. The diagonal elements in the matrix are set to zero. Finally, the Direct Influence Matrix DIM is derived by averaging the matrices DIM_k obtained from all the experts.

Stage 2: The direct influence matrix shall now be normalized to the so-called NIM by dividing each element z_ij by p, where:

$\mathrm{p}=\max \left(\max _{1 \leq i \leq n} \sum_{j=1}^n z_{i j}, \max _{1 \leq i \leq n} \sum_{i=1}^n z_{i j}\right)$                            (2)

Stage 3: A total relation matrix TRM is now constructed by aggregating both direct and indirect influential effects as:

$T R M=N I M+N I M^2+N I M^3+\cdots+N I M^{\infty}=N I M(I-N I M)^{-1}$             (3)

In the equation above, I stands for an identity n×n matrix.

Stage 4: To determine the influential factors R_i and C_i, the sum of each row and column of the TRM must be calculated. If R_i - C_i is positive, a particular factor i is classified as a cause, whereas if it is negative, it is considered an effect.

2.2.2 DEMATEL-based AHP restoration

Zhou et al. [14] proposed a technique inspired by DEMATEL to restore incomplete AHP comparison matrices and ensure their initial state and consistency. This is relevant because DEMATEL is designed to uncover non-evident relationships among a group of factors, and both DEMATEL and AHP rely on the analysis of comparison matrices. This restoration technique can be used to reduce the number of pairwise comparisons to be made by the decision-maker, this reducing the complexity of the assessment and increasing the accuracy of the resulting weights. The DEMATEL-based completion technique consists in the following stages:

Stage 1: From an incomplete AHP comparison matrix A_nxn^* = {a_ij}, a DIM matrix can be derived. The elements of the DIM matrix {z_ij} are set equal to a_ij, but are set to zero, when the comparison element a_ij is unknown.

Stage 2: To generate the normalized influence matrix (NIM), each element z_ij of the matrix shall be divided by p as in the classical DEMATEL method.

Stage 3: The total relation matrix TRM = {g_ij} shall now be computed as in classical DEMATEL.

Stage 4: Based on the relationships between factors identified in the total relation matrix (TRM), a fully reciprocal pairwise comparison matrix A_nxn^’ = {a^’_ij} as:

$g_{i j} / a_{i j}^{\prime}=g_{j i} / a_{j i}^{\prime}$      (4)

It is important to ensure that the synthetic comparison matrix A'_nxn includes reciprocal central elements. To achieve this and considering the equation above that describes the relationship between factors, any missing entry a_ij in the original incomplete comparison matrix A_nxn^* can be calculated.

2.3 Scoring MCDM techniques

2.3.1 TOPSIS

The TOPSIS method was introduced by Hwang and Yoon [15] and is recognised as the most popular MCDM method used in civil engineering [16]. TOPSIS has been used to analyse the sustainability performance of wide variety of infrastructures, from bridges [17, 18] to buildings [2]. TOPSIS technique is applied following several steps. The first step consists in constructing a decision matrix R = [r_ij] and obtaining the weight w_i of each criterion i considered in the problem. Weights are usually derived using the Analytical Hierarchy Process (AHP) [19]. The decision matrix R is then normalized as:

$r_{i j}^{\prime}=\frac{r_{i j}}{\sqrt{\sum_{j=1}^n r_{i j}^2}}$      (5)

In the equation above, n is the total number of criteria. Now, the normalized decision matrix is weighted as:

$v_{i j}=w_i \cdot r_{i j}^{\prime}$        (6)

The ideal positive and negative solutions (PIS or NIS) are derived for each criterion. These solutions are constructed by maximising the utility criteria and minimising the cost criteria in the PIS case and vice versa in the NIS case. After that, the distance of each alternative to the PIS and NIS is obtained as:

$d_j^{+}=\sqrt{\sum_{i=1}^n\left(v_{i j}-v_i^{+}\right)^2}$       (7)

$d_j^{-}=\sqrt{\sum_{i=1}^n\left(v_{i j}-v_i^{-}\right)^2}$        (8)

In the equations above, v_i⁺ and v_i^- are the elements of the PIS and NIS respectively, d_j⁺ and d_j^- are the distances of alternative j to the PIS and NIS, respectively. Finally, a score Q_j is obtained that evaluates the relative distance of each alternative j to the PIS as:

$Q_j=\frac{d_j}{d_j^{-}+d_j^{+}}$        (9)

2.3.2 VIKOR

VIKOR is a MCDM method introduced by Opricovic [20] overcome the limitations of existing MCDM techniques where decision problems involve conflicting criteria. VIKOR is also a popular assessment tool, that has been used as well to solve the evaluation performance of a variety of infrastructures, such as bridges [21-23], airport infrastructures [24] or logistic centers [25].

VIKOR shares the same first step as TOPSIS: a decision matrix must be constructed R = [r_ij] the criteria weights w_i must be determined. The second step consists in finding the best and worst criteria values, namely r_i⁺ and r_i^-, so that the decision matrix R can be normalized as:

$r_{i j}^{\prime}=\frac{r_i^{+}-r_{i j}}{r_i^{+}-r_i^{-}}$            (10)

The third step requires to determine two distance measures S_j and R_j for each alternative j as follows:

$S_j=\sum_{i=1}^n w_i \cdot r_{i i}^{\prime}$          (11)

$R_j=\max \left\{w_i \cdot r_{i j}^{\prime}\right\}$        (12)

At last, the VIKOR measure index Q_j for each alternative j is calculated as:

$Q_j=v \cdot \frac{S_j-\min \left\{S_j\right\}}{\max \left\{S_j\right\}-\min \left\{S_j\right\}}+(1-v) \cdot \frac{R_j-\min \left\{R_j\right\}}{\max \left\{R_j\right\}-\min \left\{R_j\right\}}$    (13)

It is usual to compromise both distance metrics S_j and R_j by setting v = 0.5. The alternative that results in the greatest score Q_j will be the best performing one according to this decision-making technique.

2.3.3 COPRAS

COPRAS technique was defined by Zavadskas et al. [26] and has been also applied in a wide range related decision-making situations related with sustainability issues, such as the design of buildings [27, 28], the choice of construction materials [29] and others [30] due to its simplicity. As usual in other decision-making methods, COPRAS requires a decision matrix R = [r_ij] and the obtention of the criteria relevancies w_i as a starting point Then, the decision matrix must be normalized:

$r_{i j}^{\prime}=\frac{r_{i j}}{\sum_{i=1}^n \;r_{i j}}$        (14)

The second step consists in normalizing the decision matrix elements as:

$v_{i i}=w_i \cdot r_{i i}^{\prime}$         (15)

Then, the sum of the weighted normalized scores for both cost and benefit criteria for each alternative j are obtained as:

$S_{+j}=\sum_{i=1}^n w_i \cdot r_{i j,+}^{\prime}$        (16)

$S_{-j}=\sum_{i=1}^n w_i \cdot r_{i j,-}^{\prime}$        (17)

In the equations above, r’_ij,+ and r’_ij,- represent respectivelyfor the normalized scoring for the benefit and cost criteria. After doing so, the final score Q_j of each alternative j is calculated:

$Q_j=S_{+j}+\frac{\sum_{k=1}^m \;S_{-k}}{S_{-j} \cdot \sum_{k=1}^m \;S_{-k}}$        (18)

The alternative that reaches the greatest value of the index Q_j results in the best performance according to COPRAS method.

Table 6. Criteria weights resulting from the application of the AHP technique

Decision Criterion	Weight of the Criterion
C1 - Construction Costs	0.158
C2 - Maintenance Costs	0.079
C3 - Human Health	0.149
C4 - Ecosystem	0.181
C5 - Resources	0.287
C6 - Employment	0.027
C7 - Wealth	0.031
C8 - Users	0.049
C9 - Externalities	0.040

4.2.2 Completion results

Three distinct incompleteness levels of the baseline matrix are considered to assess the effectiveness of the presented completion technique. For each scenario, a varying number of entries are randomly chosen and considered as missing. In particular, for scenario 1, 5 entries are removed, 8 for scenario 2 and 12 for scenario 3, which implies the elimination of 33% of the judgements required to the decision maker when completing a conventional 9×9 comparison matrix. 1000 simulations are run to generate in each of them a unique random incomplete comparison matrix based on the baseline one presented above.

Figure 1 shows the dispersion of the weights resulting for each criterion depending on the number of entries missing in the baseline comparison matrix. Although 3 scenarios have been evaluated, results are shown for scenarios 1 and 3, being scenario 2 enveloped by these. It can be observed that the maximum deviation from the baseline is obtained, as expected, in scenario 4, where 33% of the judgements are omitted. The average relative deviation between these weights and the baseline weight set is 4.9%. It can be noticed that as the number of missing entries increases, there is an increase in the dispersion of the results, even though the mean fits well. However, the maximum deviation with respect to the baseline is 7.6%.

tu_1.png

Figure 1. Weights of the different criteria considering incompleteness scenarios 3 and 4

Considering the outcomes, the root mean square error (RMSE) is utilized to assess the overall effectiveness of the completion model. The RMSE measures the disparities between the forecasted estimations and the baseline weights. Figure 2 shows the normalized RMSE obtained for each criterion and incompleteness scenario. It can be observed that the greatest estimation errors are associated, as expected, for the incompleteness scenario with the greatest number of entries to be restored, namely scenario 3, for which the mean normalized RMSE reaches 24.2%.

tu_2.png

Figure 2. Normalized root mean square error for each criterion

It can be concluded from the results presented that the weight estimation is robust even if 8 entries are missing (incompleteness scenario 2). Removing more than 8 would lead to greater error values and results dispersion, and a so-called rank reversal phenomenon could occur.

4.2.3 MCDM results

The TOPSIS, COPRAS and VIKOR methods have been adopted to calculate a relative score for each design alternative, providing insight into its sustainability performance during its life cycle. The results obtained assuming these weights are presented in Table 7. These have been obtained considering the baseline weighting set.

Table 7. Alternative scores for the different MCDM techniques considering the baseline weights set

Alternative	TOPSIS	VIKOR	COPRAS
CC50	0.329	0.521	0.715
W/C35	0.665	0.689	0.260
PMC10	0.101	0.485	1.000
SF5	0.913	0.906	0.013
FA20	0.888	0.854	0.035

It can be observed that the best solution in terms of its sustainability life cycle performance is the alternative based on the use of silica fume as an addition to a conventional concrete mix (SF5), closely followed by alternative SF20. The outstanding performance of these solutions when used in chloride-laden environments is explained by the fact that the use of such additions result in a drastic reduction of the chloride diffusivity of the concrete cover, thus hindering the advance of the chloride front into the reinforcing bars. The reduced maintenance of these solutions, together with the fact that it allows a reduction in the cement content, results in the best sustainability scores among the rest of the alternatives.

The sensitivity of the results is now checked against the different defined incompleteness scenarios. Results are presented in Tables 8, 9 and 10. Mean weights for each of these scenarios are considered as an input to get the alternative scoring.

Table 8. TOPSIS scores considering different incompleteness scenarios

Table 9. VIKOR scores considering different incompleteness scenarios

Table 10. VIKOR scores considering different incompleteness scenarios

It can be concluded that, although slight differences can be observed depending on the incompleteness scenario considered, the results are on average robust and consistent.