1 Introduction

Geosynthetic-reinforced and pile-supported (GRPS) embankments are being increasingly employed for infrastructure projects, especially where soft soils are present. For these structures, the embankment weight and surcharge are partially applied on the top of piles via shearing mechanisms and soil arching effects. The remaining loads can be supported by one or more geosynthetic layers, which are placed on the pile top. Great attention was given to accurately evaluating the performance of GRPS embankments in terms of soil–geosynthetic interaction, slope stability, and embankment settlements [33,34,35,36, 41]; [29, 32, 52]. However, the design optimization, involving cost savings and design improvements, of these structures was not widely investigated.

Typically, the optimization of geotechnical structures is started by comparing several typical solutions, which are selected based on engineers’ experiences [51]. Some new designs can be added later, if necessary, through a trial–error approach. This procedure suffers from subjective choices so different engineers may lead to different optimization results. In addition, it is extremely difficult to create a candidate pool ensuring the sample diversity and satisfying necessary constraints in this procedure, especially when there are several design parameters (e.g., GRPS embankments). Therefore, the determined solution by this procedure is usually not the real optimal design.

Such a traditional optimization procedure can be improved by using a sophisticated global optimization method as shown in [19], [5, 18, 26, 39, 40, 50]. These studies usually focus on optimizing one objective function, such as minimizing the construction cost with the safety requirements treated as constraints. For example, [44] optimized the design of column-supported embankments on soft soils by finding a least-cost design, which satisfies safety requirements for both serviceability and ultimate service states. [4] studied the impact of foundations and soil flexibility on the topology optimization of piled structures. A minimization of the piled structure under a prescribed volume restriction was presented. Some works [9, 43] also tried to simultaneously consider multiple objectives (e.g., safety and economy) by using a single global objective function, which is a weighted sum of individual objectives. However, there are no standard rules to determine the weighting coefficients, which have important impacts on the optimization results. Therefore, the application of this approach is only limited to cases where the weighting coefficients can be rationally defined.

The present study intends to show a multi-objective optimization (MOO) of GRPS embankments in which different objective functions, related to the construction cost, embankment differential settlements, and geosynthetic tensions, are optimized simultaneously within the entire design space (DS) with the aid of a MOO algorithm. The parameters related to the pile spacing, geosynthetic stiffness, and embankment soil quality are regarded as design parameters. A set of nondominated designs can be obtained in MOO by searching the points on the Pareto front [6]. The design parameters are assumed to be continuous variables, as it is easier for implementing most of the MOO techniques. Once the optimal design is identified from the continuous-variable problem, the parameters’ values will be slightly modified to be practically feasible.

The solutions in the Pareto front are superior to all other designs in the DS, and each of them shows one or several attractive characteristics. In the Pareto front, a mathematically optimal solution exists, namely the knee point [24]. It is considered as the best trade-off design, given that a small improvement in either objective from this point will cause a large deterioration of the other objectives. Alternatively, engineers are allowed to choose other designs from the Pareto front such as the safest design in the DS or the least-cost design for a target safety performance. The advantage of MOO over single-objective optimization (SOO) lies in the fact that MOO can provide the optimal solution (knee point) rationally considering different competitive objectives as well as many other attractive candidates, which enable informed decision making. Indeed, the Pareto front of one MOO includes the solutions of different SOO problems which consider different threshold values in the constraints. To the best knowledge of authors, there are only a few limited experiences of using MOO techniques to optimize the design of pile-supported embankments in the literature. The present study is thus among the first attempts to optimize GRPS embankments by searching the Pareto front subjected to contradictory objective functions. The introduced MOO procedure and the obtained results could be useful for practical applications.

Concerning the safety evaluation of GRPS embankments, several analytical models were proposed in the last decade [3], [7, 8, 10, 30, 31, 33, 37, 41, 42]. In this paper, the one proposed by [31] is adopted as it is an efficient one in estimating the embankment differential settlements and geosynthetic tension. This is an attractive feature for the analyses needing a large number (e.g., over thousands) of simulations such as MOO analyses. It is noted that the model was validated in [31] by comparison with 10 different GRPS embankment projects including full-scale test results, field measurements, and numerical predictions. The model was also coupled with several probabilistic analysis methods to analyze the reliability and sensitivity of a GRPS embankment [15, 38].

The structure of this paper is organized as follows: Firstly, the employed methodologies are briefly presented, including the analytical model for estimating the safety of GRPS embankments and the direct multi-search-based algorithm for determining the Pareto front. This part is followed by a description of the proposed MOO procedure, which consists of calculation models’ preparation, MOO problem definition, Pareto front search, and optimal design selection. Then, the procedure is applied to a GRPS embankment case to show a complete optimization analysis step by step. The optimal designs for the embankment considering different heights are also determined efficiently by using the proposed procedure.

2 Methodology

2.1 Design model of GRPS embankments

The evaluation of GRPS embankments is usually divided into two steps. As a first step, the stress distribution or arching is estimated without considering any geosynthetic reinforcement. It results in vertical stresses on the pile cap and soft subsoil in between. As a second step, the vertical stress is applied to the geosynthetic reinforcement as an external loading. The tensile forces in the geosynthetic and the embankment differential settlements are then derived.

2.1.1 Arching calculation

The first step allows calculating the fill arching behavior by using the Concentric Arches model as shown in Fig. 1a. As a principle, this step divides the total vertical load into two parts: the loading part which is directly transferred to the piles by the arching effect (\({P}_{c}^{a}\)), and the remaining portion of the load distributed on the geosynthetic and the subsoil (PR).

Fig. 1
figure 1

Design model of GRPS embankments: a arching calculation model (step 1); b load–deflection calculation model (step 2)

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The remaining total load resting on the geosynthetic reinforcement and subsoil by arching is determined through the connection between 3D hemispheres and 2D arches, as shown in Fig. 1a.

$$\begin{array}{*{20}c} {P_{{\text{R}}} = P_{{{\text{GR}}}} + P_{{\text{s}}} = F_{{{\text{GRsquare}}}} + F_{{{\text{GRstrips}}}} } \\ \end{array}$$
(1)

where \(P_{{\text{R}}}\) is the total load resting directly on the geosynthetic and subsoil by arching effect, \(P_{{\text{s}}}\) is the load part received by the subsoil that induces settlements, \({P}_{GR}\) is the load part carried by the geosynthetic reinforcement, FGRsquare is the total vertical load exerted by the 3D hemispheres on the square subsurface, FGRstrips is the total vertical load exerted by the 2D arches on the strip subsurface. The calculations of FGRsquare and FGRstrips can be found in the paper of [8, 32, 42].

The loading part distributed to the pile caps directly by arching effect is determined as follows:

$$\begin{array}{*{20}c} {P_{{\text{c}}}^{a} = \left( {\gamma_{{{\text{em}}}} H_{{{\text{em}}}} + q_{{\text{s}}} } \right).s_{{\text{x}}} .s_{{\text{y}}} - \left( {F_{{{\text{GRsquare}}}} + F_{{{\text{GRstrips}}}} } \right) } \\ \end{array}$$
(2)

where \(\gamma_{{{\text{em}}}}\) represents the embankment unit weight, \(H_{{{\text{em}}}}\) is the embankment height, \(q_{{\text{s}}}\) is the surcharge, \(s_{{\text{x}}}\) and \(s_{{\text{y}}}\) are the pile spacing in directions x and y, respectively. The vertical stress (\(\sigma_{s}^{a}\)) acting on the geosynthetic and subsoil by arching effect is determined by:

$$\begin{array}{*{20}c} {\sigma_{{\text{s}}}^{a} = \frac{{P_{{\text{R}}} }}{{A_{{\text{s}}} }} = \frac{{F_{{{\text{GRsquare}}}} + F_{{{\text{GRstrips}}}} }}{{s_{{\text{x}}} .s_{{\text{y}}} - w_{{\text{p}}}^{2} }} } \\ \end{array}$$
(3)

where \(w_{{\text{p}}}\) is the pile cap width.

2.1.2 Deformation of the geosynthetic reinforcement

The second step focuses on the load–deflection calculation of the geosynthetic reinforcement (GR), in which a specific physical shape should be assumed (Fig. 1b). Pham [31] presented the tensioned membrane theory for two physical shapes of deformed geosynthetics, namely the circular and parabolic models. It was worth noting that the difference in prediction results between these two models is negligible. In this study, the parabolic model is selected due to its simplicity and convenience.

According to this analytical model, the maximum geosynthetic deflection (\({Y}_{\mathrm{max}}\)) can be calculated by simplified equation as follows:

$$\begin{array}{*{20}c} {\alpha_{1} \left( {Y_{{{\text{max}}}} )^{3} + \alpha_{2} (Y_{{{\text{max}}}} )^{2} + \alpha_{3} (Y_{{{\text{max}}}} } \right) + \alpha_{4} = 0} \\ \end{array}$$
(4)

with

$$\begin{gathered} \alpha_{1} = {64}{\text{.J}}_{{{\text{GR}}}} + {23}{\text{.52K}}_{s} .(s - w_{{\text{p}}} )^{2} \hfill \\ \alpha_{2} = 5.1K_{s} B(s - w_{{\text{p}}} )^{3} - {23}{\text{.52}}\sigma_{{\text{s}}}^{a} .(s - w_{{\text{p}}} )^{2} \hfill \\ \alpha_{3} = 5.1A\sigma_{{\text{s}}}^{a} ({\text{s }} - { }w_{{\text{p}}} )^{3} + 0.51c_{s} (s - w_{{\text{p}}} )^{3} + 3K_{{\text{s}}} .(s - w_{{\text{p}}} )^{4} \hfill \\ \alpha_{4} = - 3\sigma_{{\text{s}}}^{a} .(s - w_{{\text{p}}} )^{4} \hfill \\ \end{gathered}$$

where \(s = s_{x} = s_{y}\) for a squared pattern, \({\text{J}}_{{{\text{GR}}}}\) is the geosynthetic tensile stiffness, \({\text{K}}_{s}\) is the subgrade subsoil reaction modulus that can be estimated by the ratio of the elastic modulus (\({\text{E}}_{sub}\)) to the active depth (\({\text{D}}_{sub}\)), \({c}_{s}\) is the total cohesion at the upper and lower geosynthetic interfaces, \(A\) is defined as (\({\alpha }_{p}\mathit{tan}({\varphi }_{p})\)) with \({\alpha }_{p}\) and \({\varphi }_{p}\) being, respectively, the frictional interaction coefficient and the friction angle at the upper interface, and \(B\) is given by (\({\alpha }_{s}\mathit{tan}({\varphi }_{s})\)) with \({\alpha }_{s}\) and \({\varphi }_{s}\) being, respectively, the frictional interaction coefficient and the friction angle at the lower interface. It should be noted that \({\sigma }_{s}^{a}\) is estimated by using Eq. (3). After solving Eq. (4), the maximum GR deflection will be determined. Substituting that value of \({Y}_{max}\) in the following equation, we can find the maximum GR tension.

$$\begin{array}{*{20}c} {T_{{{\text{max}}}} = \frac{{(s - w_{{\text{p}}} )^{2} + 7.84Y_{{{\text{max}}}}^{2} }}{{8Y_{{{\text{max}}}} }}.\left[ {\sigma_{{\text{s}}}^{a} - Y_{{{\text{max}}}} K_{{\text{s}}} } \right] } \\ \end{array}$$
(5)

2.2 Multi-objective optimization method

Different optimization methods such as the genetic algorithm, the particle swarm optimization and the colony optimization were extensively employed for solving a variety of geotechnical problems. The applications in geo-engineering include the design optimization, the training of machine learning models for predicting soil/rock properties [16, 22] or predicting geo-structure performances [46], [12], [54], the estimation of the best distribution for a slope safety factor considering input uncertainties [49], and the calibration of constitutive models [2, 20, 21]. These applications usually involve only one objective function in the optimization with some constraints.

The present study concerns a multi-objective optimization (MOO) problem, and the used algorithm is based on the direct multi-search (DMS) [6] which is an extension of the directional derivative-free methods from SOO to MOO. The convergence of DMS to the true Pareto front was analytically proved, and extensive application examples have shown its capability of generating a relatively complete Pareto front. The method involves an iterative process to search nondominated points satisfying all bounds and linear/nonlinear constraints.

The algorithm adapted to practically perform the DMS is briefly described as follows: Firstly, a set of points are generated within their bounds using a quasi-random sampling technique (e.g., Sobol or Latin hypercube). A point means a possible combination of design parameters. By checking the linear and nonlinear constraints, only the feasible points are remained and constitute an initial candidate pool. Then, the algorithm continues to poll points for each candidate from the pool to find a nondominant point. The poll will be repeatedly extended by doubling the mesh size in the successful directions until producing a dominated point. It is followed by updating the candidate pool according to the rank, volume, and distance of each point. (These concepts are illustrated in Fig. 2 and will be explained later). After that, the actions of poll and extension are repeated. The above-described iterative process is stopped once the measures of dominating volume and crowding distance are converged. Finally, a number of points lying in the Pareto front are obtained. In the present work, this algorithm is realized by using the function ‘paretosearch’ of MATLAB [27].

Fig. 2
figure 2

Illustration of some MOO concepts in a 2-objective function space

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Figure 2 demonstrates the specific concepts of MOO mentioned in the algorithm description by using some samples in a 2-objectives function space with the 2 objectives to be minimized. In the left part of the figure, sample F is referred to as dominated by B since B is better than F concerning all the objective functions. However, samples A, B, C, D, and E are considered as non-dominated sets because no one can dominate others. Then, the definition of rank is introduced. In the feasible space, rank 1 means the samples cannot be dominated by any other samples. Rank 2 samples are only dominated by the ones of rank 1. Other ranks are defined similarly. It is noted that the rank of each sample should be updated during the iterative process. In the figure’s right part, the Pareto front is plotted and some points on this front are presented. The front represents the frontier between the feasible and infeasible spaces. For one individual sample on the front, it exists an area in which all the samples are dominated by this one. The surface of this area is the dominating volume of this sample. As for the crowding distance of one sample, it is simply the closest distance to its neighbors in the same rank. Readers are referred to [6, 27] for more details about this algorithm. It is adopted in the present study for solving the MOO problem as this algorithm was analytically presented and numerically validated against many benchmark examples. Other MOO methods are also available in the literature, such as the different evolutionary algorithms used in [23] for the design optimization of mechanically stabilized earth retaining walls.

3 Employed optimization procedure

This section aims to give a detailed description of the employed MOO procedure, which will be later applied to a GRPS embankment. The modifications and adaptations proposed in this paper, aiming to enable better applications of MOO for practical geotechnical engineering, are also presented and explained.

Figure 3 provides a flowchart of the procedure, which is constituted by four principal steps. Firstly, some necessary ingredients of a MOO are prepared. The quantities of interest (QoI) to be optimized should be determined at first. Usually, engineers need to minimize the construction cost and maximize the safety level of the designed embankment. The safety condition is commonly related to different failure modes so different safety QoIs could be involved. Then, one or several computational models are developed to evaluate the selected QoIs, for example, the cost calculation model and the settlement estimation model. In this paper, the model presented in Sect. 2.1 is used to evaluate the performance of a GRPS embankment, and a cost calculation model will be proposed later. What comes next is to divide all the input parameters of the developed models into two groups: design and fixed parameters. The first group concerns all the parameters that will be varied in the MOO to find the optimal design, while the latter gathers the remaining parameters which will be fixed with their typical values.

Fig. 3
figure 3

Introduced MOO optimization procedure (QoI: quantity of interest)

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In the second step, the MOO problem is defined by determining the following elements: objective functions, design parameters’ bounds, and linear/nonlinear constraints. A constraint could be a limit of budget or a minimum requirement of safety QoI. The present study uses continuous variables for the design parameters since it is easier to implement most MOO methods. The finally adopted design will be slightly modified to be practically feasible.

After the MOO definition, step 3 is dedicated to solving the defined MOO problem and finding a number of points (designs) on the Pareto front [13, 47]. The algorithm described in Sect. 2.2 is used to do so. These designs cannot be dominated by any other designs and provide abundant choices of the optimal design for engineers.

The last step further analyzes the Pareto-front points of step 3 to determine the optimal design. It is proposed in the present study to select several candidates at first and then compare them according to engineers’ experiences, imposed regulations, and other potentially influential factors. The candidates could include the knee point, the least-cost design, the safest design, and the cheapest design considering a higher safety requirement. The selected one from these candidates is the theoretically optimal design (TOD), which indicates an interesting region since its neighbors are all attractive designs. The TOD should be slightly modified by tuning some design parameters to be feasible in practice. For example, it is better to consider an increment of 0.1 m for the geometry-related parameters (e.g., pile spacing distance) due to the construction accuracy. This leads to the practically optimal design (POD).

4 Optimization of a GRPS embankment

In this section, the MOO procedure of Fig. 3 is applied to an illustrative GRPS embankment. Firstly, the case study is presented. It is followed by defining the design parameters and the MOO problem. Then, a cost calculation model is developed. Finally, the optimization results are provided and interpreted.

4.1 Description of the illustrative case study

The proposed case study is based on the GRPS embankment reported by [25]. The reference embankment is 5.6 m high and 120 m long with a crown width of 35 m. The fill material consisted mainly of pulverized fuel ash with a friction angle of 30° and an average unit weight of 18.5 kN/m3.

The embankment was constructed on a foundation supported by cast-in-place annulus concrete piles. The piles are 10 m in length, and the pile tips penetrate a relatively stiff sand layer. They were placed in a square pattern. Above the pile head, a biaxial polypropylene layer was placed. It was sandwiched between two 0.25-m-thick gravel layers to form a 0.5-m-thick composite-reinforced bearing layer. The tensile geosynthetic stiffness is equal to 1180 kN/m. Additionally, a surcharge of 10 kPa is applied on the embankment surface to consider a usual traffic load.

For the sake of simplification, only a central part of the embankment with 4 involved piles is considered (Fig. 4) to be optimized and only vertical loads/displacements are concerned in this study. Because rigid piles are assumed for consideration, the failure modes related to piles can be omitted.

Fig. 4
figure 4

The embankment (a part) to be optimized

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4.2 Design parameters and other model parameters

The method presented in Sect. 2.1 is used to create a computational model for the embankment of Fig. 4. The model is able to evaluate the embankment safety by giving the differential settlements and GR tensions.

Table 1 shows a division of all the model parameters into two categories: design parameters and fixed parameters. The reference value of each parameter and the bounds for the design parameters are also provided in the table. Most of the reference values are taken from [25] with slight modifications. Firstly, a constant embankment height of 5.6 m is considered. This parameter is usually predefined according to the geographic conditions and different transportation purposes of each project. Then, the subsoil parameters are fixed since the intent is not to optimize the materials that are already on site. As for the soil–geosynthetics interaction parameters, they were proven to have insignificant influences on the GRPS embankment performances when varying within their physical ranges [31]. Therefore, these parameters are also treated as fixed parameters by using their typical values. Moreover, the pile dimension parameters (length and diameter) are considered constant as well due to the assumption of rigid piles placed on a rigid stratum.

Table 1 Classification of all the model parameters into two categories
Full size table

As a result, five parameters have remained, which are considered as design parameters in the MOO. The first two (\({S}_{\mathrm{x}}\) and \({S}_{\mathrm{y}}\)) determine the pile spacing distance in two orthogonal directions. The third one (\({J}_{\mathrm{GR}}\)) is the stiffness of the used GR and it determines which type of GR and how many layers should be used. The last two are related to the quality of the embankment fills: friction angle (\({\phi }_{\mathrm{em}}\)) and unit weight (\({\gamma }_{\mathrm{em}}\)). The quality can be improved by compacting the fills or using better materials. These two parameters (\({\phi }_{\mathrm{em}}\) and \({\gamma }_{\mathrm{em}}\)) are not independent but positively correlated; an empirical relation as shown in the footnote of Table 1 is used. Consequently, there are indeed four parameters to be optimized.

4.3 Definition of the optimization problem

The MOO problem to be solved is defined in this section. Two indicators, the maximum differential settlement of the embankment and maximum tension in the GR, are selected to represent the GRPS embankment safety. Therefore, three objectives are involved and the MOO is described as follows:

  • Design parameters: \({S}_{x}\), \({S}_{y}\), \({J}_{GR}\) and \({\phi }_{em}\)

  • Objectives: (1) minimizing the construction cost; (2) minimizing the embankment differential settlement; (3) minimizing the GR tension.

  • Constraints: (1) bounds of the design parameters; (2) construction cost is lower than the budget; (3) safe embankment.

Such a description can be translated into a mathematical formula as below:

$$\begin{array}{*{20}c} {{\text{Maximize}} \,{\text{SF}}_{{\text{g}}} \left( {{\varvec{X}}_{{\text{d}}} } \right){\text{ and minimize}} {\mathbb{C}}_{{\text{u}}} \left( {{\varvec{X}}_{{\text{d}}} } \right) {\text{with}} \left\{ {\begin{array}{*{20}c} {SF_{{\text{g}}} \left( {{\varvec{X}}_{{\text{d}}} } \right) > 1} \\ {{\mathbb{C}}_{{\text{u}}} \left( {{\varvec{X}}_{{\varvec{d}}} } \right) < {\mathbb{C}}_{{{\text{lim}}}} } \\ {lb < {\varvec{X}}_{{\text{d}}} < ub} \\ \end{array} } \right.} \\ \end{array}$$
(6)

where \({\varvec{X}}_{{\text{d}}}\) is a vector of the four design parameters, \({SF}_{\mathrm{g}}\) is a global safety factor of the GRPS embankment, \({\mathbb{C}}_{u}\) represents the unit construction cost per square meter, \({\mathbb{C}}_{lim}\) is the budget, and \({\varvec{l}}{\varvec{b}}\) (\({\varvec{u}}{\varvec{b}}\)) is the lower (upper) bound of \({{\varvec{X}}}_{{\varvec{d}}}\).

In this work, the budget \({\mathbb{C}}_{lim}\) is assumed to be equal to 200 €/m2. A cost calculation model will be developed later in the next section to estimate \({\mathbb{C}}_{u}\) with a given \({{\varvec{X}}}_{{\varvec{d}}}\). As for the global safety factor (\({SF}_{g}\)) which unifies the two safety-related indicators, its definition is given herein.

The tensile GR strength (\(R_{T}\)) is:

$$\begin{array}{*{20}c} {R_{{\text{T}}} = J_{{{\text{GR}}}} \times \varepsilon_{{{\text{max}}}} } \\ \end{array}$$
(7)

where \(\varepsilon_{{{\text{max}}}}\) is the allowable strain in the GR. In this work, \(\varepsilon_{{{\text{max}}}}\) is assumed to be equal to 0.05. Then, the individual safety factors related to the failure mode of GR tension are given as:

$$\begin{array}{*{20}c} {{\text{SF}}_{{{\text{Tx}}}} = {\raise0.7ex\hbox{${R_{{\text{T}}} }$} \!\mathord{\left/ {\vphantom {{R_{{\text{T}}} } {Tmax_{{\text{x}}} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${Tmax_{{\text{x}}} }$}} } \\ \end{array}$$
(8)
$$\begin{array}{*{20}c} {{\text{SF}}_{{{\text{Ty}}}} = {\raise0.7ex\hbox{${R_{{\text{T}}} }$} \!\mathord{\left/ {\vphantom {{R_{{\text{T}}} } {Tmax_{{\text{y}}} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${Tmax_{{\text{y}}} }$}}} \\ \end{array}$$
(9)

where \(Tmax_{{\text{x}}}\) (\(Tmax_{{\text{y}}}\)) is the maximum GR tension for the \(x\) (\(y\)) direction.

The allowed differential settlement is determined by the following expression [35, 36]:

$$\begin{array}{*{20}c} {{\text{DS}}_{x\left( y \right)} = min\left\{ {\left[ {\left( {s_{x\left( y \right)} - w_{{\text{p}}} } \right) \cdot \sqrt {\frac{{3 \cdot \varepsilon_{{{\text{max}}}} }}{8}} } \right], 300mm} \right\}} \\ \end{array}$$
(10)

Accordingly, the allowed differential settlement for a given project is generally lower than 300 mm. As an illustration, if the pile spacing \({s}_{x\left(y\right)}\) is 2 m, pile cap width \({w}_{\mathrm{p}}\) is 1 m, the maximum allowed strain of geosynthetic \({\varepsilon }_{\mathrm{max}}\) is 5%, and then, \(\left( {s_{x\left( y \right)} - w_{{\text{p}}} } \right) \cdot \sqrt {\frac{{3 \cdot \varepsilon_{{{\text{max}}}} }}{8}}\) = 136 < 300 mm, meaning that the allowed differential settlement in this case will be 136 mm rather than 300 mm. It should be noted that the value of 300 mm is chosen in this study according to the recommendation by standard [3], which is frequently utilized in practice for the design of GRPS embankments.

The individual safety factors related to the failure mode of the differential settlement are:

$$\begin{array}{*{20}c} {{\text{SF}}_{{{\text{Dx}}}} = {\raise0.7ex\hbox{${{\text{DS}}_{{\text{x}}} }$} \!\mathord{\left/ {\vphantom {{{\text{DS}}_{{\text{x}}} } {Ymax_{{\text{x}}} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${Ymax_{{\text{x}}} }$}} } \\ \end{array}$$
(11)
$$\begin{array}{*{20}c} {{\text{SF}}_{{{\text{Dy}}}} = {\raise0.7ex\hbox{${{\text{DS}}_{{\text{y}}} }$} \!\mathord{\left/ {\vphantom {{{\text{DS}}_{{\text{y}}} } {Ymax_{{\text{y}}} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${Ymax_{{\text{y}}} }$}} } \\ \end{array}$$
(12)

where \(Ymax_{x}\) (\(Ymax_{y}\)) is the max deflection in the GR for the \(x\) (\(y\)) direction.

Finally, the global safety factor is given as:

$$\begin{array}{*{20}c} {{\text{SF}}_{{\text{g}}} = \min \left( {{\text{SF}}_{{{\text{Tx}}}} , {\text{SF}}_{{{\text{Ty}}}} , {\text{SF}}_{{{\text{Dx}}}} , {\text{SF}}_{{{\text{Dy}}}} } \right)} \\ \end{array}$$
(13)

Considering a single safety indicator allows simplifying the original problem from a tri-objective optimization to a bi-objective one. Such a simplification allows easier interpretation and clearer illustration of the MOO results as will be shown in Figs. 7 and 10.

In case there are more than two objectives to be optimized, the presented procedure of Fig. 3 can still be employed to search the Pareto front and determine the optimal design. This is based on the fact that the basic optimization algorithm of the introduced procedure was validated by considering a large number of benchmark optimization problems including those having 3 or 4 objective functions [6]. However, the Pareto front will be a 3D surface and a hyperplane in higher dimensions, which is not intuitive to analyze and verify the optimization results. This paper focuses on providing a practical and easy-to-use solution for the optimization of GRPS embankments considering simultaneously the safety and the cost, so the two safety-related indicators are combined into one single objective. It could be interesting to carry out more investigations on the optimization of geotechnical engineering involving more objectives such as the reliability and the robustness [13] in future works.

4.4 Cost calculation model

This section describes the developed model for the calculation of unit construction cost (\({\mathbb{C}}_{u}\)). It should be pointed out that the prices used in the following model are only for illustrative purposes because they are sensitive to the project location, construction site conditions, labor costs, and local transportation costs. For practical projects, the described model should be improved/modified by considering specific price information [51].

The ultimate objective of the developed model is to estimate the unit cost (\({\mathbb{C}}_{u}\)) of a 4-pile geosynthetic-reinforced embankment considering varied \({{\varvec{X}}}_{{\varvec{d}}}\). The transportation cost is ignored for a sake of simplicity. The \({\mathbb{C}}_{u}\) is directly associated with the total cost (\({\mathbb{C}}_{t}\)) by the embankment surface (\({s}_{x}\times {s}_{y}\)), and \({\mathbb{C}}_{t}\) consists of three parts: cost of fill soils (\({\mathbb{C}}_{em}\)), cost of GR (\({\mathbb{C}}_{GR}\)), and cost of piles (\({\mathbb{C}}_{p}\)).

$$\begin{array}{*{20}c} {{\mathbb{C}}_{{\text{t}}} = {\mathbb{C}}_{{{\text{em}}}} + {\mathbb{C}}_{{{\text{GR}}}} + {\mathbb{C}}_{{\text{p}}} } \\ \end{array}$$
(14)

4.4.1 Cost of fill soils

The next equation is used to estimate the \({\mathbb{C}}_{em}\).

$$\begin{array}{*{20}c} {{\mathbb{C}}_{{{\text{em}}}} = \left( {s_{{\text{x}}} \times s_{{\text{y}}} \times H_{{{\text{em}}}} } \right) \times \frac{{\gamma_{{{\text{em}}}} }}{10} \times uP_{{{\text{em}}}} \left( {\phi_{{{\text{em}}}} } \right)} \\ \end{array}$$
(15)

where \(uP_{{{\text{em}}}} \left( {\phi_{{{\text{em}}}} } \right)\) represents the unit price of the embankment fill material. It is expressed in €/ton so the \(\gamma_{{{\text{em}}}}\) (kN/m3) is divided by 10. This unit price depends on the material quality and can be estimated as follows:

$$\begin{array}{*{20}c} {uP_{{{\text{em}}}} \left( {\phi_{{{\text{em}}}} } \right) = \left( {\phi_{{{\text{em}}}} - \phi_{{{\text{em\_base}}}} } \right) \times \alpha_{{{\text{P}}_{{{\text{em}}}} }} + uP_{{{\text{em\_base}}}} } \\ \end{array}$$
(16)

where \(\phi_{{{\text{em\_base}}}}\) is the starting friction angle of the fill material and \(uP_{{{\text{em\_base}}}}\) represents its corresponding unit price. \(\alpha_{{{\text{P}}_{{{\text{em}}}} }}\) quantifies the increased ratio of the unit price due to the friction angle increase. In this work, the \(\phi_{{{\text{em\_base}}}}\) and \(uP_{{{\text{em\_base}}}}\) are assumed, respectively, to be equal to 30° and 8 €/ton. It is also assumed that the material quality increase is only due to the embankment compaction, so the \(\alpha_{{{\text{P}}_{{{\text{em}}}} }}\) is related to the labor cost of operating compaction (vibratory) roller. In France, the minimum salary is around 10.25 €/h; a standard value of 12 €/h is then used in this work. The time required to compact a squared-meter area with 5.6 m height and get 1° of increase in \(\phi_{{{\text{em}}}}\) is assumed to be equal to 2 min. Therefore, the \(\alpha_{{{\text{P}}_{{{\text{em}}}} }}\) is 0.4 €/ton/degree. It is noted that using other fill materials, which are well selected and controlled, can also lead to increased \(\phi_{{{\text{em}}}}\). This condition was not considered in the calculation of \(uP_{{{\text{em}}}} \left( {\phi_{{{\text{em}}}} } \right)\) and could be added, together with different delivery costs, in the \({\mathbb{C}}_{{{\text{em}}}}\) estimation for a specific project according to its practical constraints/conveniences.

4.4.2 Cost of GR

The cost of GR can be computed as:

$$\begin{array}{*{20}c} {{\mathbb{C}}_{{{\text{GR}}}} = uP_{{{\text{GR}}}} \left( {J_{{{\text{GR}}}} } \right) \times \left( {s_{{\text{x}}} \times s_{{\text{y}}} } \right)} \\ \end{array}$$
(17)

where \(uP_{{{\text{GR}}}} \left( {J_{{{\text{GR}}}} } \right)\) is the unit price of GR and can be estimated by:

$$\begin{array}{*{20}c} { uP_{{{\text{GR}}}} \left( {J_{{{\text{GR}}}} } \right) = \frac{{J_{{{\text{GR}}}} }}{{J_{{{\text{GR\_base}}}} }} \times uP_{{{\text{GR\_base}}}} } \\ \end{array}$$
(18)

where \(J_{{{\text{GR\_base}}}}\) is the base GR stiffness value and \(uP_{{{\text{GR\_base}}}}\) represents its unit price. In this paper, the \(J_{{{\text{GR\_base}}}}\) is set as 100 kN/m, which is the minimum value for the design parameter \(J_{{{\text{GR}}}}\). Then, the tensile GR strength is 5 kN/m according to Eq. (7). For such a geotextile, the unit price is varied between 0.5 and 1.5 €/m2. The \(uP_{{{\text{GR\_base}}}}\) is assumed to be equal to 1 €/m2 in this work. There are two ways to increase the \(J_{{{\text{GR}}}}\): (1) Use another type of GR such as a geogrid; (2) place more layers of the geotextile. The second approach is adopted in this study, so the GR unit price of different \(J_{GR}\) can be approximated by Eq. (18).

4.4.3 Cost of piles

The costs of piles are represented by the following equation:

$$\begin{array}{*{20}c} {{\mathbb{C}}_{{\text{p}}} = 4 \times \left( {\frac{1}{4} \times \pi \times \left( {\frac{{D_{p} }}{2}} \right)^{2} \times H_{{\text{p}}} \times uP_{{\text{c}}} } \right)} \\ \end{array}$$
(19)

where \(H_{{\text{p}}}\) and \(D_{{\text{p}}}\) are, respectively, the length and diameter of the pile as explained in Table 1, and \(uP_{c}\) is the unit price of standard concrete. The price is varied between 80 and 130 €/m3 and \(uP_{{\text{c}}}\) is set as 100 €/m3 in this work. There are four piles concerned in the area of interest, and only a quarter of each pile is covered by the 4-piles embankment.

4.5 Optimization results

This section presents the obtained results for the MOO problem defined in Sect. 4.2. Firstly, several points located on the Pareto front are found using the algorithm of Sect. 2.2. Then, several candidates are selected from these points. Finally, the theoretically and practically optimal designs are determined. In this work, the MOO algorithm is asked to find 200 points on the Pareto front, and a tolerance error of 1 × 10–6 is used to stop the iterative process [27].

4.5.1 Pareto front

Figure 5 plots the determined 200 points of the Pareto front in a two-objective function space defined by the unit cost (\({\mathbb{C}}_{u}\)) and the global safety factor (\({SF}_{g}\)). Each point corresponds to a possible design of rank 1, meaning that it cannot be dominated by any other designs. It is observed that all the points are limited by the space of a \({SF}_{g}\) varying within [1, 2.2] and a \({\mathbb{C}}_{u}\) between 100 and 200 €/m2. This shows the bounds of the two objectives under the current constraints.

Fig. 5
figure 5

Two hundred points in the Pareto front for the considered MOO problem

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In fact, each point in Fig. 5 represents the solution of an SOO problem. It can be understood as either the safest design with a given budget or the cheapest design with a target safety factor. Pareto front collects these solutions, which therefore can provide more information than an SOO.

For a purpose of illustrating how the Pareto front represents a part of the frontier between feasible and infeasible regions, crude Monte Carlo Simulations (MCS) are performed, which are widely used in reliability analyses and uncertainty quantifications [14, 28, 48]. The idea is to evaluate a large number of possible designs considering the bounded design parameters and plot them in the two-objective function space together with the obtained Pareto front. Three MCSs are conducted with three sample sizes: 1 × 104, 1 × 105, and 1 × 106, which permits to show the convergence of the feasible region boundaries. The random samples in the performed MCSs are generated by using the Latin Hypercube Sampling method [17].

Figure 6 presents the results of the three MCSs and the 200 points of the Pareto front. The green solid points are obtained in the context of MCS, and they represent nearly all the possible designs. This figure confirms that the obtained 200 points are effective; they can be considered to be located on the Pareto front since no design can dominate them. Also, it can be observed that the Pareto front divides the space into feasible and infeasible regions. It is noted that such a comparison using MCS is not necessary for a MOO. It is carried out here just for a purpose of better explaining the Pareto front.

Fig. 6
figure 6

The determined Pareto front and other possible designs considering three sample sizes

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Figure 7 further classifies all the feasible designs into different categories: over-budget, unsafe, and acceptable designs. The first category involves the designs having a \({\mathbb{C}}_{u}\) higher than 200 €/m2. The second one collects all the designs leading to an unsafe embankment (\({SF}_{g}\)<1). The remaining designs (third category) are all acceptable considering the imposed constraints. Within the third category, closer to the Pareto front better the design is, given that either the cost is reduced, or the safety is increased, or both the objectives are improved.

Fig. 7
figure 7

Classification of the possible designs into different zones

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4.5.2 Candidates for the optimal design

Following the identification of 200 points on the Pareto front, this section intends to select several candidates for the optimal design from these points. As mentioned above, the Pareto front provides a set of non-dominated designs and each point on this front could be interesting from an optimization viewpoint. It can then be judged by designers/engineers based on their experiences and subject to different purposes to find the optimal design.

The present study highlights 4 candidates from the available Pareto front points as shown in Fig. 8. Their descriptions are given as follows.

  • Point A (safest design): it is associated with the highest \({SF}_{g}\), and thus, the safest embankment that can be reached under the current budget. However, there is no reduction in the construction cost; the maximum allowable cost should be consumed. This design is appropriate for the projects having enough funds and higher performance requirements,

  • Point B (least-cost design): it is the least-cost design among the acceptable designs with a \({SF}_{g}\) being slightly higher than 1. Usually, it is preferred to have some margins compared to failures to take into account model errors and parameter uncertainties. Therefore, this design is generally not a preferred one, but identifying it still provides some useful information,

  • Point C (Knee point): this design is the best trade-off between the two conflicting objectives, so is the solution of optimal design for the MOO without any subjective assumptions/preferences. It is characterized by the fact that a small improvement in either objective will cause a large deterioration in the others. In this paper, the knee point is located by using the algorithm introduced in [1, 13].

  • Point D (cheapest design for a target \({SF}_{g}\)): it is an improved version of the design B by increasing the required \({SF}_{g}\) from 1 to 1.5. This design is then the least cost one among the designs satisfying \({SF}_{g}\)>1.5. The value of 1.5 is adopted since it is commonly used in design codes to define the minimum safety factor; practitioners are free to modify this value according to different purposes and codes (e.g., Eurocodes 7 [11]).

Fig. 8
figure 8

Four candidates for the optimal design from the Pareto front

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Figure 8 also presents the parameters’ values of the four designs. The optimal value of \({s}_{x}\) is extremely close to the \({s}_{y}\) one, indicating that a square pattern of piles is obtained, for all 4 designs. This is not surprising as there are no different constraints or objectives applied in the two directions in the considered MOO problem. It is also observed that the pile spacing values and GR stiffness parameters are varied among the four designs, but \({\phi }_{em}\) is almost constant. The latter can be explained by a sensitivity analysis, which investigates the effects of the design parameters on the two objective functions (\({SF}_{g}\) and \({\mathbb{C}}_{u}\)), as presented in Fig. 9.

Fig. 9
figure 9

Effects of the design parameters on the two objective functions

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Figure 9 presents the evolution of the two objective functions induced by varying each design parameter in its physical range defined by the bound values in Table 1 (e.g., \({J}_{GR}\) is varied between 1000 and 5000 kN/m). The considered physical values are then normalized in this figure, in order to allow a cross-comparison among different design parameters. The results reveal that the effects of the pile spacing distance (\({s}_{x}\) and \({s}_{y}\)) are nonlinear for both \({SF}_{g}\) and \({\mathbb{C}}_{u}\), while the ones of \({\phi }_{em}\) are both linear. As for \({J}_{GR}\), its variation leads to linear changes in \({\mathbb{C}}_{u}\) but a nonlinear evolution of \({SF}_{g}\). It is interesting to note that increasing \({\phi }_{em}\) results in slight improvements of \({SF}_{g}\) and the induced increase ratio in \({SF}_{g}\) is lower than the one in \({\mathbb{C}}_{u}\) for the entire varying range. Therefore, using a higher \({\phi }_{em}\) has insignificant benefits for improving the studied embankment. On the contrary, increasing the other design parameters shows clear favorable effects on the design improvement at least at the beginning of the varying range.

4.5.3 Finally adopted design

After selecting four candidates from the Pareto front, what comes next is to compare them and determine the optimal design. Table 2 gives a summary of the 4 candidates in terms of the values of the two objective functions and the two embankment safety indicators (max tension and max deflection). The results of the initial design, which are based on the reference values of all the parameters (Table 1), are also presented for comparison.

Table 2 Model results for the four candidates and the initial design
Full size table

As commented earlier, design A leads to the highest \({SF}_{g}\) but requires the full budget, while design B can largely save the cost but only have a \({SF}_{g}\) of 1.002. Then, it is found that the embankment safety could be increased, and the cost could be simultaneously decreased by using design C instead of the initial design. This shows the great advantages of performing a MOO analysis. Design D is interesting as well, since the \({SF}_{g}\) is increased from 1.36 to 1.50 with only a 3 €/m2 in addition compared to the initial one.

The theoretically optimal design (TOD) could be selected from the four candidates of Fig. 8 or even from other designs on the Pareto front. For example, the samples around the knee point are all interesting for decision-making. The final choice depends on the engineers’ judgment and the project's purpose. This paper selects design D as the TOD. Then, it is required to slightly modify the selected TOD in order to be practically feasible, given that many design parameters are discrete variables. Specifically, the increment should be 0.1 m and 100 kN/m for, respectively, the pile spacing distance and the GR stiffness. Table 3 presents the finally adopted practically optimal design by tuning the design parameters.

Table 3 The theoretically and practically optimal design of the reference embankment
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5 Design optimization considering different embankment heights

For a road project, the embankment height could be varied, given that different geographic/geological conditions will be crossed. It is thus interesting to optimize the design considering different heights (\({H}_{em}\)). This section considers four complementary cases with the \({H}_{em}\) being, respectively, 2, 3, 4, and 5 m. For each case, a MOO following the procedure of Fig. 3 is carried out. The primary aim lies in showing how different optimal designs can be efficiently determined for a project needing to consider the GRPS embankments with varied heights. It is noted that the cost of compacting a lower embankment should be decreased. Such reduced cost is determined by assuming it to be proportional to the embankment height.

5.1 Pareto front and knee point

Figure 10 presents a comparison of the four complementary cases with the reference one (\({H}_{em}\)=5.6 m) by plotting their Pareto fronts in the two-objective function space. It reveals that the Pareto front is moved toward the area of lower cost and higher safety by decreasing the \({H}_{em}\). For a 2-m-high GRPS embankment, the highest \({SF}_{g}\) can be reached under the current budget is about 4.8.

Fig. 10
figure 10

The Pareto front and knee point for the embankments of different heights

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The knee point of each Pareto front is also identified in the figure. Such point is located on the area where the front is significantly curved and convex to the ideal-design region characterized by lower \({\mathbb{C}}_{u}\) and higher \({SF}_{g}\). At the left of the knee points, the \({\mathbb{C}}_{u}\) will be dramatically increased but the \({SF}_{g}\) is moderately improved. At their right, the \({\mathbb{C}}_{u}\) could be slightly reduced, but the \({SF}_{g}\) is significantly decreased. Therefore, they represent the best balance between the two objectives.

Table 4 presents the knee-point designs of the five cases by giving their design parameters and objective functions. The results indicate that the pile spacing distance of around 2.7 m is ideal for reaching the best trade-off design of each case as the values of \({s}_{x}\) (or \({s}_{y}\)) are only varied between 2.6 and 2.8 m in the table. The obtained \({J}_{GR}\) varies significantly with different embankment heights, while the \({\phi }_{em}\) is constant at 30° except when \({H}_{em}\) equals to 2 m. The latter means that the benefits of increasing \({\phi }_{em}\) (i.e., compacting fills) are no longer negligible for a lower embankment (e.g., \({H}_{em}\)=2 m). By determining the knee point via MOO, one can always find the design leading to a larger \({SF}_{g}\) and lower \({\mathbb{C}}_{u}\) compared to a higher embankment.

Table 4 Design parameters of the knee point for the embankments of different heights
Full size table

5.2 Minimum costs considering different target \({{\varvec{S}}{\varvec{F}}}_{{\varvec{g}}}\) values

As said before, a Pareto front of MOO includes the results of many SOO analyses considering different constraints. Therefore, the results presented in Fig. 10 also allow knowing the minimum cost of each embankment case for a target \({SF}_{g}\), which could be 1.5 or other values depending on the uncertainty level of the input parameters and the confidence in the computational model.

Figure 11 shows the minimum cost of the five embankment cases for three targets \({SF}_{g}\) values (1.2, 1.5, and 1.8). Considering a typical \({SF}_{g}\) of 1.5, the required cost is about 43 €/m2 for a 2-m-high embankment, and it is increased to around 130 €/m2 if the height is 5.6 m. As for the cases of a given height, it is not surprising to observe that the cost is increased by searching for a higher \({SF}_{g}\). However, such cost increase is significantly different for different heights. The minimum cost is only increased by 1 €/m2 if the target \({SF}_{g}\) is 1.8 instead of 1.5 for \({H}_{em}\)=2 m, while around 35 €/m2 should be added for \({H}_{em}\)=5.6 m if the target \({SF}_{g}\) is increased from 1.5 to 1.8.

Fig. 11
figure 11

Minimum cost of each embankment case considering different target \({SF}_{g}\) values

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As an example, Table 5 presents the design parameters of these least-cost designs for the cases of \({H}_{em}\) being 3 and 5.6 m. The determined \({\phi }_{em}\) is always 30° because of the negligible benefits of increasing \({\phi }_{em}\) in its entire varying range as explained before and evidenced in Fig. 9. For the 3-m-high embankment, either the pile spacing distance is reduced or the \({J}_{GR}\) is increased (insignificantly) to reach a higher target \({SF}_{g}\). Only 8 €/m2 is needed to improve the embankment safety from \({SF}_{g}\)=1.2 to \({SF}_{g}\)=1.8. Concerning the 5.6-m-high embankment, the \({J}_{GR}\) should be largely increased and the pile spacing distance should be reduced simultaneously if the target \({SF}_{g}\) is increased. The complementary investment is about 55 €/m2 for increasing the \({SF}_{g}\) from 1.2 to 1.8. It is noted that these results/findings are sensitive to the employed cost calculation model, which should be created with local price information for a practical specific project.

Table 5 Least-cost designs considering different target \({{\varvec{S}}{\varvec{F}}}_{\mathbf{g}}\) values and the corresponding design parameters (examples for \({{\varvec{H}}}_{\mathbf{e}\mathbf{m}}\) of 3 m and 5.6 m)
Full size table

Considering different target \({SF}_{g}\) values is similar to vary the threshold performance values of MOO. For example, a higher target \({SF}_{g}\) or a smaller \({DS}_{x\left(y\right)}\) (allowed differential settlement) will both lead to a higher cost if one seeks for the least-cost design. For the case with different target \({SF}_{g}\) values, one can directly find the four candidates from the already existing Pareto front. This shows the advantages of MOO over SOO as the former is more informative. However, a new MOO analysis needs to be performed when the threshold performance values (e.g., allowed strain and settlement) are changed, because the shape of the Pareto front will not be the same. By following the procedure of Fig. 3 and setting different threshold values, the four candidates can be found and the optimal design can be determined, which allow to know the minimum cost.

5.3 Discussions on computational efficiency

Performing an optimization analysis usually requires expensive computational efforts as a large number of model evaluations should be carried out in order to find the global optimal solution.

The present study addresses this issue by using an efficient computational model, as also highlighted in a probabilistic analysis of GRPS embankments [15]. For the MOOs conducted in Sects. 4 and 5, thousands of simulations are needed for each case, while the total computational time is less than 20 min. This shows great interest for practical engineering as quickly providing a variety of results, including best trade-off designs and minimum costs for a target safety requirement, is useful for a preliminary design stage. Performing multiple MOO analyses by varying, for example, the embankment height is thus possible within a reasonable calculation time.

For the computational models needing several minutes or several hours to finish one single simulation, direct application of most MOO algorithms is no longer desirable. For example, running a model of 10 min for 1000 times requires nearly 7 days. A possible solution to this problem is to use a surrogate model to replace the original one as did in [45], [5, 14, 53].

6 Conclusions

This paper introduces a multi-objective optimization (MOO) to the design of GRPS embankments. An algorithm based on the direct multi-search is used to practically perform MOO. It allows finding a number of points (designs) that are located on the Pareto front. These designs are all competitive to be the optimal design by comprehensively considering each objective function. A MOO analysis procedure is proposed in this study in which the algorithm is combined with an efficient computational model, four interesting design candidates are highlighted, and the concept of theoretically/practically optimal design is introduced.

The procedure is applied to an illustrative GRPS embankment case. Two hundred points on the Pareto front (frontier between feasible and infeasible designs) are determined, leading to the identification of four candidates: the safest design under the current budget, the least-cost design for a safe embankment, the best trade-off design (knee point) considering conflicting objectives, and the cheapest design for a target safety factor of 1.5. By selecting the knee point design, the two objectives are both improved (i.e., higher \({SF}_{g}\) and lower \({\mathbb{C}}_{u}\)) compared to the initial design for the case study. Additionally, different optimal designs are efficiently obtained by using the procedure for the embankments with different heights, which could be encountered in a project crossing different geographic/geological conditions. The provided results allow engineers to have a quick idea of the best trade-off designs and the minimum costs for any given target safety requirement.

For a practical application of the procedure, the cost calculation model of Sect. 4.3 should be improved by surveying the assumptions made for the sake of simplicity and by considering the local price information related to construction materials, transportations, labors, and excavations.