1. Introduction
A retaining wall could be defined as any uninterrupted structure that, whether in a passive or in an active way, produces a stabilizing effect over a mass of land [
1]. The earth retaining walls are those structures that retain a piece of land with a steeper angle than the angle of friction of the land [
2]. There are different classifications of retaining walls according to different criteria: load support mechanism (externally or internally stabilized walls), construction concept (fill or cut), system rigidity (rigid or flexible), and service life (permanent or temporary) [
3]. Thus, several different types of retaining walls exist with different performance characteristics as well as constructability characteristics, as well as different uses [
2]. Retaining walls are expensive structures that are designed and constructed to support cut and fill slopes where space is not available for construction of flatter, more stable slopes [
3]; therefore, the cost of construction, the environment and the space available will be criteria to take into account in its design and construction. Selecting a type of retaining wall is a complex process, considering the various geotechnical and non-geotechnical factors involved [
4]. Moreover, during the selection process, it is necessary to consider all the criteria during the whole life cycle [
5].
In decision-making, it is necessary to consider all the alternatives and criteria involved in the decision process [
6]. In civil engineering and construction, the choice of the most adequate and sustainable alternative is not always made by studying all of the possible typologies nor the life cycle of the infrastructure: design, construction, maintenance, and dismantlement. Therefore, it is necessary to identify all the typologies, their characteristics, and the selection criteria without forgetting the requirements of the construction process. The methodology to be developed must establish a systematic process in the decision-making process. Finally, the solutions obtained must remain unchanged in the face of changes in the decision-maker’s preferences or variation in the weight of criteria, i.e., provide strong solutions.
Multicriteria decision-making methods have become a tool to solve engineering problems because they allow complex problems to be modeled. These methods can be used to select the best alternatives when there are several conflicting criteria in a context of uncertainty [
7]. In this paper, a methodology is developed for the selection of typology of retaining walls according to different criteria. In the developed methodology, the first step is taken has been to identify all of the different types of retaining walls and selection criteria (external requirements, construction requirements, characteristics of the natural land, and economic criteria) that may determine whether a solution is the best option or not, and to which extent. Subsequently, the most suitable solution is determined by successively applying different multicriteria decision-making methods.
4. Case Study
The described methodology is applied to two projects with different context and purpose. Project 1: Mountain road in a nature park in the province of Madrid (Spain) and Project 2: New highway under construction in Andalusia (Spain).
Validity will be verified by the behavior in the solution to be adopted when there are variations in the weight of criteria, the changes in the compromise solution will be determined using the VIKOR method. And lastly, the results will be compared to the obtained results using the TOPSIS method.
A small set of criteria and typologies have been taken into account to show the methodology simply. As alternatives are considered, four typologies included in
Table 2: a reinforced concrete wall built in situ, garden wall, green wall reinforced with geotextiles, and riprap wall. These typologies have been selected given that they represent a bigger variety of retaining walls. In the decision-making process, we choose as determining criteria the following: construction cost in €/m, construction performance, m/day, landscape integration, technical culture and customs (construction frequency), and lastly, preservation and maintenance necessities. In a manner similar to that of the selection of alternatives, these five criteria were chosen because they are the most common and determining in retaining walls. All retaining walls have an average height of four meters. To apply the decision-making methodology, the following hypotheses are needed:
All the retaining walls have a stable structure (the assessed solutions are stable).
Logistic determinants have not been considered such as material availability or available space for gatherings.
The drainage system works correctly in all cases
Selection criteria are independent.
To evaluate the different alternatives concerning the selection criteria, firstly the cost and performance of construction has been determined based on the advice of suppliers, Spanish Ministry of Development [
1,
44], as well as the advice of designers and contractors who were consulted for this purpose. Next, for the quantitative evaluation of the different alternatives for the qualitative criteria: technical culture and customs and preservation and maintenance necessities, paired comparisons according to the Saaty scale of the AHP method [
18,
35] are used to determine such a quantitative assessment. In this way, the quantitative assessment of each alternative for each qualitative criterion varies between 0 and 1, with 0 being the lowest value and 1 the highest value for each criterion. The results obtained for each alternative respect to each selection criterion have been included in
Table 3. If these results are analyzed, it can be seen that, for example, the alternative of reinforced concrete retaining wall is the one which obtains the lowest valuation concerning the criterion of landscape integration, is the one that is built most frequently and, on the contrary, is the alternative which requires the least conservation and maintenance actions throughout the life cycle.
Once the assessment of the alternatives has been determined, the values
fi* and
fi−, being the best and worst values of each criteria function, are calculated according to the VIKOR method. The results are included in
Table 4.
It is important to highlight that the importance of each criterion depends on the location where the retaining wall is going to be constructed. Thus, for each project, and before using the VIKOR and TOPSIS methods, the weight vector is determined, by paired comparisons, and by applying the AHP. So, for project 1, given the environmental factors, when using the AHP method for obtaining the weight vector by paired comparisons, the weight vector is obtained, w = (0.03; 0.08; 0.54; 0.21; 0.13). It is important to remember that the consistency of the comparison matrix must be identified. After determining the consistency following equation (1), the consistency ratio = 0.094697, which is under 0.1, is obtained. Therefore, the assessments made can be considered as consistent.
Then, Equations (2)–(4) are applied to calculate the
Sj,
Rj and
Qj values. The alternatives classification list will be established according to the values of S, R and Q, to establish the solution or the ensemble of compromise solutions. To carry out the simulation and application of the VIKOR and TOPSIS methods, an algorithm has been developed in the Matlab
® software to automate the calculations. The results obtained by VIKOR method are shown in
Table 5:
Minimum value of
Q can be observed for the green wall alternative (see
Table 5). Both requirements of the VIKOR method are met: requirement 1, acceptable advantage, and requirement 2, stability of the decision-making process. Therefore, there is a compromise solution for the decision-making problem described here, being the alternative of green wall reinforced with geotextile the one most suitable to the determining criteria.
To verify the validity of the method, the VIKOR method is applied for another case varying the importance of the different criteria but keeping the criterion of landscape integration as the most important, with the weight between 0.48 and 0.54. Thus, it is confirmed as compromise solution the typology of the green wall, being the best-ranked option in the classification lists Q, S and R, although within an ensemble of compromise solutions in which a riprap wall is also included as the second option. When using TOPSIS, the green retaining wall is confirmed as the best-ranked solution, obtaining the following classification: Green retaining wall, Rj = 0.9966; Riprap retaining wall, Rj = 0.5955; Garden retaining wall, Rj = 0.5227; Reinforced concrete retaining wall, Rj = 0.0034.
For project 2, the process followed in the previous case must be repeated, calculating on first place the weight matrix for the selection criteria, by paired comparisons. The weight vector obtained is
w = (0.41; 0.03; 0.26; 0.11; 0.18). The consistency of the comparison matrix has been assessed, obtaining a Consistency Ratio of less than 0.1, being able now to conclude that the assessments made are consistent. In the same way, we complete the decision-making process by calculating the values
Sj,
Rj and
Qj and establishing the classification lists of alternatives,
Table 6.
As a result, a minimum of the Q, S and R values for the case of the green wall is obtained, but with similar values to those for riprap wall, from which we deduce that there is not a clear optimal solution, but an ensemble of compromise solutions that can be a solution to the problem in a more or less appropriate form. It is demonstrated this by applying requirement 1, acceptable advantage of the VIKOR method. Q(A(1)) − Q(A(2)), is lower than 0.333, therefore the requirement 1 of acceptable advantage is not met. Therefore, as a solution to the decision-making problem, we suggest an ensemble of compromise solutions formed by the alternatives green retaining wall and riprap retaining wall. It should be remembered that the VIKOR method proposes an ensemble of compromise solutions to those alternatives, A(1), A(2), …, A(M), that make Q(A(M)) − Q(A(1)) < DQ. For project 2, when the most important criterion is the cost, the best-valued solution is the green wall, being a close second (admissible solution) the riprap wall, therefore being both valid with VIKOR. However, in the case the importance of the construction performance is increased (+3%), the optimal solution is the riprap retaining wall, and the garden retaining wall would be the second solution. With this analysis, we can see the sensibility of the method and its possible use when no characteristic is clearly predominant, like the landscape integration in the previous case. If the criteria weight is not defined, the strongest solution will be the riprap retaining wall, which is the best ranked or the second in all the cases. A strong solution is the one that is still admissible when the importance of the criteria changes.
When using TOPSIS, the green retaining wall is confirmed as the best-ranked solution and as the second option the riprap retaining wall, obtaining the following classification: Green retaining wall, Rj = 0.9991; Riprap retaining wall, Rj = 0.5955; Garden retaining wall, Rj = 0.5227; Reinforced concrete retaining wall, Rj = 0.0005. This way it is proven that the proposed method can be applied both in cases with different criteria importance and in cases in which the precedence is clear but the differences not so much.
9. Conclusions
The methodology developed first provides a systematic process for the identification of alternatives and selection criteria for the selection of types of retaining walls. Thus, the application of the methodology allows identifying all the determining criteria in the selection of retaining walls, both minor and major importance criteria, and identifying the construction requirements that are often overlooked in the phase of design and selection of alternatives. The relative importance of the criteria for different projects is determined through paired comparisons. As result, a ranking of constructive solutions of retaining walls that is still admissible when the importance of the criteria changes is obtained.
Case study validates the combined sequential use of two decision-making methods for selecting the best constructive solution for a wall in different and specific situations by proving that a solid and transparent ensemble of criteria can be taken into account, such as the environment, with the construction performance and the costs through an objective and transparent process, making clear which are the expressed preferences and their importance in the process. Moreover, verification of the validity of the methodology shows the stability of the solutions obtained, even if there are changes in the weighting of the criteria.
The suggested decision-making process is based on data that are easy to obtain and to allow the evaluation of alternatives according to qualitative criteria. As a result of this, the fact of obtaining a solution as a result of the application of a systematic process that is relatively objective will allow—in situations of disagreements among different groups of interests, stakeholders—to justify the adopted solution. As a result of the research, a methodology for the selection of types of retaining walls is provided, which can be useful for public administrators, designers, project managers, and constructors.
It should be noted that the methodology can be applied to the selection of other infrastructures in which the design and construction requirements may determine that particular alternatives are not suitable or are suitable to a lesser extent, for example, for the selection of bridge types.