Multi-objective optimization of reinforced concrete cantilever retaining wall: a comparative study

Abstract

This paper investigates the performance of four multi-objective optimization algorithms, namely non-dominated sorting genetic algorithm II (NSGA-II), multi-objective particle swarm optimization (MOPSO), strength Pareto evolutionary algorithm II (SPEA2), and multi-objective multi-verse optimization (MVO), in developing an optimal reinforced concrete cantilever (RCC) retaining wall. The retaining wall design was based on two major requirements: geotechnical stability and structural strength. Optimality criteria were defined as reducing the total cost, weight, CO₂ emission, etc. In this study, two sets of bi-objective strategies were considered: (1) minimum cost and maximum factor of safety, and (2) minimum weight and maximum factor of safety. The proposed method's efficiency was examined using two numerical retaining wall design examples, one with a base shear key and one without a base shear key. A sensitivity analysis was conducted on the variation of significant parameters, including backfill slope, the base soil’s friction angle, and surcharge load. Three well-known coverage set measures, diversity, and hypervolume were selected to compare the algorithms’ results, which were further assessed using basic statistical measures (i.e., min, max, standard deviation) and the Friedman test with a 95% level of confidence. The results demonstrated that NSGA-II has a higher Friedman rank in terms of coverage set for both cost-based and weight-based designs. SPEA2 and MOPSO outperformed both cost-based and weight-based solutions in terms of diversity in examples without and with the effects of a base shear key, respectively. However, based on the hypervolume measure, NSGA-II and MVO have a higher Friedman rank for examples without and with the effects of a base shear key, respectively, for both the cost-based and weight-based designs.

1 Introduction

One of the major challenges in geotechnical engineering is stabilizing uneven natural and artificial soil slopes. The systems developed to retain such masses include gravity and cantilever retaining walls, sheet piles, anchored earth, and mechanically stabilized earth walls. The difference between these retaining structures is based on how each can withstand unstable loads. Final cost is essential in determining which type of earth retaining structure is best suited. While reinforced concrete cantilever retaining (CCR) walls are among the most effective retaining structures in a wide range of construction projects, they are relatively costly due to the massive bulk of the required materials. Hence, any effort to decrease the final cost of CCR walls is pertinent.

Artificial intelligence-based algorithms have found wide applications in facilitating complicated civil engineering problems (Najafzadeh and Azamathulla 2015; Najafzadeh and Kargar 2019; Gandomi et al. 2021; Kashani et al. 2021a; Maniat et al. 2021). Optimization algorithms have proven effective in preparing cost-effective designs for engineering problems. Optimization algorithms have been applied successfully to a wide variety of civil engineering problems (e.g., structural engineering (Gandomi et al. 2013; Akhani et al. 2019; Bekdaş et al. 2019; Kashani et al. 2020; Kashani et al. 2021b, c; Gholizadeh et al. 2020), water engineering (Azizi et al. 2017; Ebrahimi and Khorram 2021; Ali et al. 2022; Azari et al. 2022; Moeini et al. 2022), geotechnical engineering, transportation engineering (Yang et al. 2012; Gandomi et al. 2015a, b, 2017a; Gandomi and Kashani 2017; Kashani et al. 2016, 2019a, b, 2021d, 2022a, b), construction management (Sahib and Hussein 2019; Panwar and Jha 2019), and structural damage detection (Mishra et al. 2019; Fathnejat and Ahmadi-Nedushan 2020; Akhani and Pezeshk 2022)). Due to their stochastic nature and varied performances, metaheuristic optimization algorithms require constant updates, which can be done by 1- developing new algorithms (Gandomi 2014; Mirjalili and Lewis 2016; Saremi et al. 2017; Mirjalili et al. 2017a), or 2- enhancing the algorithms’ performances (Jordehi 2015; Gandomi and Kashani 2016, 2018a, b; Gandomi and Deb 2020; Bigham and Gholizadeh 2020). As a result, research on applying metaheuristic algorithms in a wide range of engineering problems is an active study area.

Numerous studies have applied optimization algorithms to minimize the final cost of CCR walls over the past few years. Optimization approaches simplify the design procedure by satisfying three primary criteria: geotechnical stability, structural strength, and economic efficiency. For example, Khajehzadeh et al. (2010) used particle swarm optimization, and then later employed a modified optimization method. Khajehzadeh and Eslami (2012) applied the gravitational search algorithm, Ceranic et al. (2001) utilized simulated annealing, Camp and Akin (2012) applied Big Bang Big Crunch, Aydogdu (2017) tried biogeography-based optimization algorithms, Gandomi et al. (2015c, 2017b, c) and Gandomi and Kashani (2018a, b) considered evolutionary and swarm optimization algorithms.

These studies' main limitation is the relatively high final cost, weight, or CO₂ emission associated with a single objective during the design optimization procedure. Although the single objective approach provides a cost-effective final design for decision-makers, it may not reflect all aspects of the design since the important efficiency-related features often have conflicting and reciprocal relations. Therefore, it is impossible to find a design that satisfies the optimality criteria for all conflicting objectives. Multi-objective optimization has proved to be a sophisticated approach to identifying solutions, called Pareto optimal solutions, to such conflicts (Deb 2001; Marchionatti and Gambino 1997). Therefore, multi-objective optimization for finding the Pareto front (PF) solutions has become widespread throughout science and engineering (Gunantara 2018; Afshari et al. 2019; Gholizadeh and Fattahi 2021; Behmanesh et al. 2020; Rangaiah et al. 2020).

In most construction projects, particularly retaining walls, the main concern is minimizing the final cost and maximizing safety. The stronger and bulkier the wall, the higher the safety. As a result, the final cost would be much higher than the optimal low-cost design. Multi-objective optimization of retaining walls has been addressed in just a few studies. For instance, (Kaveh et al. 2013) employed a non-dominated sorting genetic algorithm (NSGA-II) for retaining wall optimization that considered bar congestion and cost as two conflicting objectives. In another effort, (Das et al. 2016) utilized NSGA-II for retaining wall optimization considering the final cost and factor of safety.

In this study, multi-objective optimization of retaining walls was investigated, emphasizing two different combinations of objectives based on the study by (Sarıbaş and Erbatur 1996): (1) minimum cost and maximum factor of safety (cost-based design), and (2) minimum weight and maximum factor of safety (weight-based design). Four multi-objective algorithms, i.e., non-dominated sorting genetic algorithm II (NSGA-II) (Deb et al. 2000), multi-objective particle swarm optimization (MOPSO) (Coello and Lechuga 2002), strength Pareto evolutionary algorithm II (SPEA2) (Zitzler et al. 2001), and multi-objective multi-verse optimization (MVO) (Mirjalili et al. 2016), were utilized for retaining wall optimization. Three different measures were considered to compare the performances of the proposed algorithms. A computer program based on ACI 318-05 (2005) and analysis presented by (Das 2010) was developed in MATLAB to analyze retaining wall designs. The second design example also explored the effect of a base shear key. A sensitivity analysis was conducted on the cost-based design of Example 2 for the effective parameters, including surcharge load (q), base soil’s friction angle (ϕ), and backfill slope (β). To be more specific, the main contributions of this work include (1) analyzing the trend of variation of cost and weight by changing the factor of safety of the retaining walls using multi-objective optimization, (2) a systematic comparative study of the performance of different classes of multi-objective optimization algorithms based different measures and statistical analysis on solving Reinforced Concrete Cantilever Retaining Wall problems, (3) examining the impact of design parameters on the final objective through a sensitivity analysis, and (4) providing a practical design procedure based on geotechnical and structural regulations provided in ACI 218-05. The results indicate that the NSGA-II and SPEA2 algorithms were more efficient than MOPSO and MVO.

2 Methodology

A feasible retaining wall design should satisfy geotechnical stability and structural strength requirements (Das 2010). In the former, three factors of safety—overturning (FS_O), sliding (FS_S), and the foundation’s bearing capacity (FS_B)—were considered to guarantee the serviceability of the structure. In the latter, the shear and moment capacity of each section of the wall (stem, heel, toe, and shear key) were checked with ACI 318–05 (2005) regulations. In formulating the optimal design of the retaining wall, a bi-objective function is proposed for minimizing cost/weight and maximizing the factors of safety (FS_O, FS_S, and FS_B) as follows

$$\left\{ {\begin{array}{*{20}l} {f_{{{\text{cost}}}} = C_{{\text{s}}} W_{{{\text{st}}}} + C_{{\text{c}}} V_{{\text{c}}} } \hfill \\ {FS = ({\text{FS}}_{{\text{O}}} + {\text{FS}}_{{\text{S}}} + 2 \times {\text{FS}}_{{\text{B}}} )^{{ - 1}} } \hfill \\ \end{array} } \right.,$$

(1)

$$\left\{ {\begin{array}{*{20}l} {f_{{{\text{weight}}}} = W_{{{\text{st}}}} + 100V_{{\text{c}}} \gamma _{{\text{c}}} } \hfill \\ {FS = ({\text{FS}}_{{\text{O}}} + {\text{FS}}_{{\text{S}}} + 2 \times {\text{FS}}_{{\text{B}}} )^{{ - 1}} } \hfill \\ \end{array} } \right.,$$

(2)

where C_s and C_c are the unit cost of steel and concrete, respectively, W_st is the weight of reinforcing steel, and V_c and γ_c are the volume and unit weight of concrete scaled by a factor of 100 as proposed by (Sarıbaş and Erbatur 1996), respectively.

Figure 1 defines the twelve design variables for the retaining wall in this study: width of the base slab (X1), the width of the toe slab (X2), stem thickness at the bottom of the wall (X3), stem thickness at the top of the wall (X4), base slab thickness (X5), distance from the front of the shear key to the front of the toe of the wall (X6), the width of the shear key (X7), the height of the shear key (X8), the vertical steel area in the stem per unit length of the wall (R1), the horizontal steel area of the toe slab (R2), the horizontal steel area of the heel slab (R3), and the vertical steel area of the shear key per unit length of the wall (R4). Discrete variables were considered for steel reinforcement areas, according to Table 1. In this study, the design procedure and constraints are defined based on (Camp and Akin 2012).

Fig. 1

Design variables for a general retaining wall

Table 1 Steel reinforcement properties for design variables R1 to R4

3 Description of optimization algorithms

3.1 Non-dominated sorting genetic algorithm II

Unlike a single-objective optimization problem, which provides only a single optimal solution, a multi-objective optimization problem provides solutions representing trade-offs between conflicting objectives known as a Pareto optimal set. Several techniques have been proposed to obtain the Pareto optimal solutions in literature (Coello et al. 2007). Due to their effectiveness and easy implementation, multi-objective evolutionary algorithms (MOEAs) have gained much attention from researchers. One of the significant metaheuristic algorithms in this category is NSGA-II, an improved version of the non-dominated sorting genetic algorithm (Deb et al., 2000).

NSGA-II applies an elitism-based non-dominated approach for ranking and sorting solutions using a crowding distance method in its selection operator to maintain the diversity in the obtained Pareto optimal set (Deb et al. 2002). First, in non-dominated sorting, each solution's objective functions are evaluated, and the whole population is ranked and sorted into different non-dominated levels based on the dominance count. Second, an infinite crowding distance value is assigned to the solutions after defining the boundary values based on the smallest and largest function values. Finally, the crowding distance between any two neighboring solutions is computed based on the normalized difference in the objective function values (Deb et al. 2002). Compared to NSGA, NSGA-II offers an improved mating mechanism dependent on the crowding distance and performance constraints using an adapted explanation of dominance without the use of penalty functions.

3.2 Multi-objective evolutionary algorithm based on decomposition

The strength pareto evolutionary algorithm 2 (SPEA2), developed by (Zitzler et al. 2001), is a non-dominated genetic algorithm with features like a truncation operator, nearest neighbor, and genetic operators (crossover and mutation) that maintain diversity and best convergence of solutions. Like most evolutionary algorithms, SPEA2 uses a set of solutions encoded as the chain of a chromosomic number in a generational approach. In each generation, all population members that are evaluated receive a fitness value. Better performance results in a high fitness value, and as such, a high probability is chosen to develop new generations using the chromosomes pairs crossing concept (Zitzler et al. 2001).

SPEA2 uses an external archive, including the previously identified non-dominated solutions, and is updated every generation. In the next step, a strength value is computed for each solution. Based on these strength values, the fitness of each individual is calculated. The fitness assignment strategy of SPEA2 considers the number of individuals that an individual dominates and the number of its dominators. The algorithm also uses the nearest neighbor density approach and archive truncation method to maintain diversity and preserve the boundary solutions, respectively (Zitzler et al. 2001).

3.3 Multi-objective particle swarm optimization

Particle swarm optimization (PSO) (Eberhart and Kennedy 1995; Kennedy and Eberhart 1995) was initially proposed as a simulation model based on studying the choreography of a bird flock for optimizing continuous nonlinear functions. There are two definitions in the application of PSO: the individual best and the global best. In swarm optimization, the particles search for the best solution based on their experience and the other particles within the same swarm. Then, each particle compares its fitness value at the current position to the best fitness value it had attained before. The best-known position, with the best fitness, for each particle, is pbest. The best position for the entire swarm is gbest. The ith particle velocity (v_i) is updated for the k + 1 iteration according to the following equation

$${v}_{i}^{k+1}=w{v}_{i}^{k}+{c}_{1}{\mathrm{rand}}_{1}\times \left({p\mathrm{best}}_{i}-{s}_{i}^{k}\right)+{c}_{2}{\mathrm{rand}}_{2}\times \left(g\mathrm{best}-{s}_{i}^{k}\right),$$

(3)

where w is a weight function, c₁ and c₂ are acceleration factors, rand₁ and rand₂ are random numbers ∈ [0, 1], and s_i^k is the position of each particle. The weight function is

$$w={w}_{\mathrm{max}}-\frac{{w}_{\mathrm{max}}-{w}_{\mathrm{min}}}{{\mathrm{iter}}_{\mathrm{max}}}\times \mathrm{iter},$$

(4)

where w_max is the initial weight, w_min is the final weight, iter_max refers to the maximum iteration number, and iter is the current iteration number.

The new position of each particle is updated using the velocities as

$${s}_{i}^{k+1}={s}_{i}^{k}+{v}_{i}^{k+1}.$$

(5)

A new multi-objective PSO (MOPSO) algorithm was introduced by Coello and Lechuga (2002). Optimization can be performed for more than one conflicting objective simultaneously. In the MOPSO algorithm, the swarm is initialized following the identification of a set of leaders with the swarm's non-dominated particles. These are stored in a secondary repository of particles to guide particle flight (Sha and Lin 2010). A quality measure is computed for all leaders to select one leader for each particle of the swarm. A leader for each particle is selected, and the flight is performed for each generation. After performing the flight, the MOPSO algorithm applies a mutation operator to evaluate the particle and its corresponding personal best experience (pbest) is updated. A new particle replaces its pbest particle when this particle is dominated or if both are non-dominated with respect to each other. The set of leaders is updated once all the particles are updated. The quality measure for the leaders is then recalculated, and this process is repeated for a defined number of iterations.

3.4 Multi-objective multi-verse optimization

The multi-verse optimizer (MVO) is an optimization algorithm developed by Mirjalili et al. (2017b) based on multi-verse theory in astrophysics. It emulates the interplay among universes in the Big Bang theory and how they interact with each other through different types of holes, such as black, white, and wormholes.

The MVO computational process includes several iterations to send and receive objects (variable) to and from universes (solutions) based on their inflation rates (fitness values) through wormholes. This function helps the exploration and exploitation processes from becoming trapped in local optima. The mathematical representation of the MVO algorithm is described as

$$x_{i}^{j} = \left\{ {\begin{array}{*{20}l} {x_{k}^{j} ,} \hfill & {if\,rm_{1} < {\text{NI}}(x_{i} )} \hfill \\ {x_{i}^{j} ,} \hfill & {{\text{otherwise}}} \hfill \\ \end{array} ,} \right.$$

(6)

where ${x}_{i}^{j}$ is the jth parameter of the ith universe; ${x}_{k}^{j}$ is the jth parameter of the kth universe chosen by roulette wheel selection; $r{m}_{1}$ indicates a random value ∈ [0, 1]; ${x}_{i}$ represents the ith universe; and $\mathrm{NI}({x}_{i})$ shows the normalized fitness value of the ith solution. The objects of the universes are updated using wormholes to improve the inflation rate, as follows

$$x_{i}^{j} = \left\{ {\begin{array}{*{20}l} {\left\{ {\begin{array}{*{20}l} {x_{j}^{b} + {\text{TDR}} \times \left( {\left( {ub_{j} - lb_{j} } \right) \times rm_{4} + lb_{j} } \right)rm_{3} } \hfill \\ {x_{j}^{b} - {\text{TDR}} \times \left( {\left( {ub_{j} - lb_{j} } \right) \times rm_{4} + lb_{j} } \right)rm_{3} \ge 0.5} \hfill \\ \end{array} } \right.} \hfill & {rm_{2} < {\text{WEP}}} \hfill \\ {x_{i}^{j} } \hfill & {rm_{2} \ge {\text{WEP}}} \hfill \\ \end{array} } \right.,$$

(7)

where ${x}_{i}^{j}$ presents the jth parameter value of jth solution; ${x}_{j}^{b}$ indicates the jth parameter of the best solution, $l{b}_{j}$ is the lower bound and $u{b}_{j}$ the upper bound of the jth variable, $r{m}_{2}$, $r{m}_{3}$, and $r{m}_{4}$ are random numbers [0, 1], WEP and TDR represent adaptive variables. WEP is the wormhole existence probability and is employed to enhance exploitation. TDR is the traveling distance rate to allow objects to fly to the best universe through a wormhole. The adaptive formulas for WEP and TDR coefficients are given as follows

$$\mathrm{WEP}={\mathrm{WEP}}_{\mathrm{min}}+\mathrm{iter}\times (\frac{{\mathrm{WEP}}_{\mathrm{max}}-{\mathrm{WEP}}_{\mathrm{min}}}{{\mathrm{iter}}_{\mathrm{max}}})$$

(8)

$$\mathrm{TDR}=1-\frac{{\mathrm{iter}}^{1/p}}{{{\mathrm{iter}}_{\mathrm{max}}}^{1/p}}$$

(9)

where iter is the given iteration, ${\mathrm{WEP}}_{\mathrm{min}}$ and ${\mathrm{WEP}}_{\mathrm{max}}$ are default values set to 0.2 and 1, respectively, and p is a default value set to 6 that shows the accuracy of the exploitation (Mirjalili et al. 2016).

4 Performance indices

In this paper, the performances of the algorithms are compared using several standard measures. Previous studies have proposed various performance measures to evaluate various aspects (e.g., convergence and diversity) of a non-dominated solution set (Deb 2001). As explained in (Zitzler 1999; Zitzler et al. 2000; Azevedo and Araújo 2011), the performance indices quantify the convergence and diversity of final Pareto solutions in multi-objective problems. This study used the following three performance indices to compare different Pareto sets regarding the global Pareto set obtained from all sets.

4.1 Coverage set (CS)

The CS index, proposed by Zitzler et al. (2000), compares the non-dominated degrees of optimal solutions from different iterations. The values of CS (Xʹ, Xʺ) are in the range of 0–1. When CS is equal to 1, all Xʺ are dominated by or equal to Xʹ. To understand the exact non-dominated relationship between two different iterations, two cases of CS need to be analyzed: (Xʹ, Xʺ) and (Xʺ, Xʹ). The CS index is expressed as

$${\text{CS}}\left( {X^{\prime}, X^{\prime\prime}} \right) = \frac{{\left| {\left\{ {a^{\prime\prime} \in X^{\prime\prime}; \exists a^{\prime} \in X^{\prime}:a^{\prime} \ge a^{\prime\prime}} \right\}} \right|}}{{\left| {X^{\prime\prime}} \right|}}$$

(10)

where Xʹ and Xʺ are the optimal solutions from different iterations, and aʹ and aʺ are optimal solutions.

4.2 Diversity (DI)

The DI index, proposed by Zitzler (1999), evaluates the diversity of optimal solutions from the multi-objective optimization algorithm. It can be expressed using the minimum and maximum values of the objective function as

$$\mathrm{DI}= \sqrt{\frac{1}{M}{\sum_{m=1}^{M}\left(\frac{\mathrm{max}{f}_{\mathrm{m}}- \mathrm{min}{f}_{m})}{{F}_{\mathrm{m}}^{\mathrm{max}}- {F}_{\mathrm{m}}^{\mathrm{min}}}\right)}^{2}}$$

(11)

where ${F}_{\mathrm{m}}^{\mathrm{max}}$ and ${F}_{\mathrm{m}}^{\mathrm{min}}$ are the maximum and minimum values of Pareto fronts, respectively, f_m is the mth value of the objective function, and M is the number of objective functions.

4.3 Hypervolume (HV)

The HV, defined by Zitzler and Thiele (1999), is a popular performance index that measures the proximity and diversity of a Pareto approximate front. Specifically, it measures the volume of the partition of the objective space bound between the Pareto approximate front and a reference point. The HV is a complete unary performance metric in terms of weak dominance relation; HV's solution is not weakly dominated by its opponent Zitzler and Thiele (1999). In this study, as Deb (2001) suggested, HV is calculated in the normalized objective space, where all results are normalized to be inside the unit hypercube [0, 1]m. Therefore, all normalized HV values (or simply NHV) are less than or equal to 1. Higher values of NHV are desirable, like the other two performance metrics. HV is calculated as

$$\mathrm{HV}= \sum_{i=1}^{|Q|}{v}_{i}$$

(12)

where v_i is a hypercube constructed using reference point W (found by constructing a vector of worst objective function values) at the ith solution, Q is the objective space (total search space), and W is the reference point that can simply be found by constructing a vector of worst objective function values. Therefore, HV is calculated using a union of all hypercubes.

5 Numerical simulation

This section examines the proposed algorithms' efficiency based on two case studies presented by Sarıbaş and Erbatur (1996). Example 1 considers the optimal design of a retaining wall without a base shear key (Sarıbaş and Erbatur 1996), and Example 2 includes the effects of a base shear key. Table 2 lists the parameter values for numerical examples, and Table 3 lists the boundary limitations of the design variables. ACI 318–05 (2005) requirements and discrete variables for steel reinforcement are considered in the design. Both examples use two different sets of objective functions: (1) minimum cost and maximum FOS (cost-based design), and (2) minimum weight and maximum FOS (weight-based design). Under this contextual analysis, each optimization algorithm is run independently 50 times, with a population size of 50 for 1000 generations. The best non-dominated Pareto solution (true Pareto front) is obtained by huddling all the utilized algorithms' Pareto solutions.

Table 2 Parameter values for Examples 1 and 2

Table 3 Design variable boundary constraints for Examples 1 and 2 (from Camp and Akin (2012))

In the same way, the best Pareto front for each algorithm is the set of non-dominated solutions from the 50 runs, which were then compared with the true Pareto front using performance indices CS, DI, and NHV. The CS index, the most important index of the multi-objective optimization solutions, was used to calculate the ratio of the number of non-dominated solutions obtained by a method to the number of Pareto solutions in the total set of solutions. The DI shows how well a method finds a widespread Pareto front. The NHV is calculated by the hypercube from the given objective functions to show the convergence aspect of the Pareto front. Also, basic statistical measures were computed (i.e., min, max, mean, standard deviation (STD)) along with a Friedman test with a 95% confidence level considering all 50 runs.

5.1 Example 1: 3-m tall retaining wall design without a base shear key

Figures 2 and 3 show the best aggregate Pareto fronts, resulting from 50 independent runs for each algorithm, for the cost-based and weight-based designs, respectively. Moreover, the true Pareto front is also depicted to better compare the algorithms’ performances. For the cost-based design, NSGA-II and SPEA2 have a more significant overlap with the true Pareto front, while MOPSO and MVO Pareto fronts were far from the true Pareto front. There is a negligible difference between the presented solutions and true Pareto for all algorithms for the weight-based design.

Fig. 2

Pareto front comparison for the cost-based design of Example 1

Fig. 3

Pareto front comparison for the weight-based design of Example 1

The algorithms' performances were further evaluated using three measures, CS, DI, and NHV (see Fig. 4). The CS results in Fig. 4 confirm the observations in Figs. 2 and 3 that NSGA-II and SPEA2 participate more than the other two algorithms in forming the true Pareto front in both cost-based and weight-based designs. A comparison of NSGA-II and SPEA2 revealed that NSGA-II contributes more to the true Pareto than SPEA2. DI results suggest that SPEA2 is the better solver for the cost-based design, while NSGA-II is better for the weight-based design. NHV metrics indicate minor differences between these algorithms.

Fig. 4

Comparison of performance measures for Example 1

Tables 4, 5 and 6 list a statistical comparison of the proposed performance measures (i.e., CS, DI, and NHV). In this way, the proposed indices are calculated for every run out of 50 runs. The results are presented as min, mean, max, and standard deviation (STD) and ranked using the Friedman statistical test at a 95% significance level. The CS results, listed in Table 4, indicate that NSGA-II exhibited better performance than the other algorithms due to its higher mean value. Also, the Friedman test results confirm the advantage of NSGA-II over the other approaches. Based on the DI metric analysis listed in Table 5, the obtained mean values indicate that the SPEA2 algorithm dominated the other cost-based design methods. The NHV test results of the cost-based design, shown in Table 6, again confirm that NSGA-II outperformed the other algorithms.

Table 4 CS values of the cost-based designs for Example 1

Table 5 DI values the cost-based designs for Example 1

Table 6 NHV values of the cost-based designs for Example 1

Tables 7, 8 and 9 compare the proposed measures (i.e., CS, DI, and NHV) for weight-based designs. Results also exhibit the superior performance of NSGA-II over the other algorithms, like in the cost-based design. The CS and NHV results proposed NSGA-II as the best solver due to its lower mean values. The Friedman ranking scores were also consistent with this observation. However, for the DI metric, SPEA2 demonstrated better performance than NSGA-II.

Table 7 CS values of the weight-based designs for Example 1

Table 8 DI values of the weight -based designs for Example 1

Table 9 NHV values of the weight -based designs for Example 1

5.2 Example 2: 4.5-m tall retaining wall design with and without a base shear key

In Example 2, two cases are considered: one with a base shear key (Case I) and one without a base shear key (Case II). In this example, cohesionless soil is considered for the base. Tables 2 and 3 list values for parameters in this example and the domains for the design variables. Figures 5 and 6 show the best Pareto fronts of the utilized algorithms for cost-based and weight-based designs of Case I. Most of the solutions in the Pareto fronts of NSGA-II and SPEA2 are coincident with the true Pareto. Although all MOPSO’s results were not located on the true optimal solution set, they were very close (Fig. 7).

Fig. 5

Pareto front comparison for the cost-based design of Example 2-Case I

Fig. 6

Pareto front comparison for the weight-based design of Example 2-Case I

Fig. 7

Comparison of DI measures for numerical simulations

In contrast, most of MVO’s Pareto solutions were far from the true Pareto front. Figure 5 shows that SPEA2 and NSGA-II had more contributions to the true Pareto fronts from the lower cost and FOS to intermediate cost and FOS for the cost-based design. However, MOPSO could find solutions in the true optimal Pareto front for the higher FOS and cost values. On the other hand, NSGA-II and SPEA2 were more involved in forming the true Pareto front for higher weight and FOS for the weight-based design. The MVO algorithm was effective for the lower cost and FOS solutions.

The comparison of the CS performance measure reflects the better performance of NSGA-II over the other algorithms for both cost-based and weight-based designs. The second-best algorithm was SPEA2 based on the CS metric. As shown in Fig. 7, although MOPSO and MVO performed better than NSGA-II and SPEA2 in terms of diversity, their solutions did not cover the true Pareto front provided in Fig. 6. Moreover, SPEA2 resulted in more diverse solutions than NSGA-II for both cost-based and weight-based designs. Figure 7 also shows MOPSO and MVO as the best and second-best algorithms in terms of diversity and hyper volume. However, recorded better coverage for both cost and weight designs.

Tables 10, 11, 12, 13, 14 and 15 provide a statistical comparison of CS, DI, and NHV for cost-based and weight-based designs for Case I. The CS results for both cost-based and weight-based designs indicate NSGA-II and SPEA2 as the best and second-best methods, confirmed by Friedman test results. Based on this measure, these two algorithms play a more critical role in forming the true Pareto front. Although MVO and MOPSO were the better solvers considering DI and NHV measures, their performances are not satisfactory since they are far from the true Pareto front. Further comparison of NSGA-II and SPEA2 as the best optimizers based on CS measures proves that SPEA2 performs better with higher mean values of DI and NHV. The Friedman test ranking results indicate MOPSO and MVO are the best solvers based on DI and NHV measures.

Table 10 CS values of the cost-based designs for Example 2 (Case I)

Table 11 DI values of the cost-based designs for Example 2 (Case I)

Table 12 NHV values of the cost-based designs for Example 2 (Case I)

Table 13 CS values of the weight-based designs for Example 2 (Case I)

Table 14 DI values of the weight-based designs for Example 2 (Case I)

Table 15 NHV values of the weight-based designs for Example 2 (Case I)

A sensitivity analysis was conducted on the variation of the cost-based designs for the following parameters: (1) backfill soil slope, (2) base soil’s friction angle, and (3) surcharge load. Figures 8a–c present the sensitivity analysis results of the cost-based design of Case I. Figure 8a demonstrates that increasing β values from 5° to 25° resulted in more expensive design values and a right shift in the Pareto front towards higher costs. On the other hand, the decreasing FOS resulted in a downward shift of the Pareto front. It can be seen that the inclination of the Pareto front decreases with increasing β values, meaning that for lower backfill slopes, we can get a higher factor of safeties by increasing the final cost rather than higher backfill slopes. Surcharge loads had the same effect as the backfill slope on the final design. Changing the surcharge load within the considered domain on the final cost was less intensive than the backfill slope. However, this effect caused a higher reduction in FOS for the surcharge load than the backfill slope. It can be seen the final cost varied in the lower domain, while FOS was higher than the ones for backfill soil slopes.

Fig. 8

Cost-based design sensitivity to a backfill slope, b base soil friction angle, and c surcharge load

In contrast, Fig. 8b indicates the positive effect of the base soil’s friction angle on the final design, where increasing the ϕ values resulted in lower costs and higher FOSs. The Pareto fronts shrank for more intensive cases, which suggests that attaining the higher FOSs was much more expensive for highly intensive cases.

Figure 9 compares the extreme design points for this case study. The observation confirms that increasing β resulted in increased cost. In Fig. 9a, the minimum cost value of 172.78 ($/m) was obtained by NSGA-II, and MOPSO found a maximum of 308.92 ($/m). The variation in cost values was about 78.79%. Moreover, these changes diminished FOSs by about 48.88%. MOPSO achieved the maximum FOS of 48.51 and a minimum of 24.8 (see Fig. 9b). Increasing ϕ from 28° to 38° caused a decrease in cost and increased FOS. Figure 9c shows that the maximum cost value of 241.61 ($/m) is related to the MOPSO algorithm at ϕ equal to 28°, while NSGA-II obtained the minimum cost of 156.70 ($/m) at ϕ equal to 38°. Cost and FOS variations were 35.14% and 213.47%, respectively. Increasing the surcharge load resulted in a 55.19% increase in cost and a 64.05% decrease in FOS. the NSGA-II algorithm found the minimum FOS of 29.60 and the cost of 128.90 ($/m). MOPSO obtained the maximum FOS of 82.33 and the cost of 200.03 ($/m). In summary, FOS experienced the maximum variations by changing ϕ, and the most significant changes in cost were attributed to changing β.

Fig. 9

Comparison of the extreme design points for the cost-based design of Case I

Figures 10 and 11 present the Pareto fronts for each algorithm for Case II's cost-based and weight-based designs. Most of the solutions on the true Pareto front were obtained by NSGA-II and SPEA2, similar to the previous examples. For Case II, MOPSO did not participate in forming the true optimal solution, while MVO did contribute to developing part of the true Pareto front. In particular, MVO was able to find more solutions on the true Pareto front in the area with lower FOS and lower cost for the cost-based designs. MVO successfully found more solutions coincident with the true Pareto front for weight-based designs. A comparison of Figs. 5 and 6 with Figs. 10 and 11 shows that adding a base shear key does not affect the final cost, weight, or FOS values on the Pareto fronts.

Fig. 10

Pareto front comparison for the cost-based design of Example 2-Case II

Fig. 11

Pareto front comparison for the weight-based design of Example 2-Case II

Comparing performance measures in Fig. 12 resulted in the same conclusion as the previous examples. Based on CS measures, NSGA-II outperformed the other algorithms, followed by SPEA2; however, SPEA2 outperformance NSGA-II in both cost-based and weight-based designs based on DI measures. There is no significant difference between these algorithms based on NHV values.

Fig. 12

Comparison of NHV measures for numerical simulations

Tables 16, 17, 18, 19, 20 and 21 list a statistical comparison of CS, DI, and NHV for the Case II cost-based and weight-based designs. The CS results indicate that NSGA-II and SPEA2 were the best algorithms because of their higher mean values for both designs, with NSGA-II exhibiting better performance. Considering the DI index, SPEA2 performed better than NSGA-II, while NHV revealed no considerable difference between these algorithms.

Table 16 CS values for cost-based designs for Example 2 (Case II)

Table 17 DI values for cost-based designs for Example 2 (Case II)

Table 18 NHV values for cost-based designs for Example 2 (Case II)

Table 19 CS values for weight-based designs for Example 2 (Case II)

Table 20 DI values for weight-based designs for Example 2 (Case II)

Table 21 NHV values for weight-based designs for Example 2 (Case II)

Figures 8d–f display sensitivity analyses of the Pareto fronts for Case II cost-based designs and show similar results to those of Case I. In general, increasing β and q increased the cost designs, while FOS values decreased. An inverse trend was recorded by increasing ϕ values. It can be seen that the range of cost was maximized by changing β, while FOS varied most with changing q. The inclination of the Pareto front for lower β and q values was significant, meaning that higher FOSs can be obtained by slightly increasing the final cost. However, these inclinations were reduced at higher β and q, where a moderate increase in FOS resulted in a higher cost. The Pareto fronts’ inclinations for ϕ are less than those for β and q, meaning choosing a higher FOS value results in higher costs compared to β and q.

Figure 13 compares extreme design points and shows the same trends as those for Case I. It can be observed that increasing β and q resulted in higher costs and lower FOSs, which was the opposite for ϕ variations. Increasing β from 5° to 25° resulted in a 65.27% increase in the cost values and a 53.08% decrease in FOS values. MVO and MOPSO obtained the minimum cost of 171.09 ($/m) and FOS of 23.23. MOPSO found the maximum cost of 282.78 ($/m) and FOS of 49.51. Cost values decreased by 27.19%, and FOS values increased by 125.88% by varying ϕ from 28° to 38°. In this case scenario, the lowest cost of 158.51 ($/m) and FOS of 25.51 were obtained by MVO and MOPSO, respectively. MOPSO and SPEA2 obtained the highest cost of 217.72 ($/m) and FOS of 57.63. Varying q from 0 to 50 (kPa) resulted in a 56.96% increase in cost values and a 64.97% decrease in FOS values. The lowest cost of 130.04 ($/m) and FOS of 29.23 was obtained by MVO and MOPSO, respectively. Finally, the highest cost of 204.11 ($/m) and FOS of 83.46 was determined by MOPSO and NSGA-II, respectively.

Fig. 13

Comparison of the extreme design points for the cost-based design of Case II

6 Summary and conclusions

In this study, four metaheuristic algorithms, including non-dominated sorting genetic algorithm II (NSGA-II), multi-objective particle swarm optimization (MOPSO), strength Pareto evolutionary algorithm II (SPEA2), and multi-objective multi-verse optimization (MVO), were utilized for multi-objective optimization of retaining walls. Two case studies were considered for two different designs with different sets of objectives: (1) minimum cost and maximum factor of safety (cost-based design), and (2) minimum weight and maximum factor of safety (weight-based design). Moreover, the effect of a base shear key on the final design was studied in the second example. This study aimed to (1) apply different multi-objective optimization methods to the design of a retaining wall, (2) provide an efficient comparison between the algorithms’ performances, (3) explore the effect of a base shear key on the final designs, and (4) conduct a sensitivity analysis on critical design parameters, i.e., surcharge load (q), base soil’s friction angel (ϕ), and backfill slope (β). The performances of the utilized algorithms were measured with three well-known indices: coverage set (CS), diversity (DI), and hypervolume (NHV).

Comparing the CS values indicated that the NSGA-II and SPEA2 had the most significant contribution in forming the true optimal Pareto front. Based on DI values, SPEA2 provided better performance. Also, observations indicate that CS and DI have inverse relations, whereby the algorithms with higher CS resulted in lower DI. Based on NHV measures, there were negligible differences between the algorithms. While DI measure values for MOPSO and MVO were higher than those for NSGA-II and SPEA2, these methods did not significantly contribute to the true Pareto fronts.

Comparing the Pareto fronts for Case I and Case II of Example 2 shows little effect of the base shear key on cost and FOS. Sensitivity analysis demonstrated that decreasing ϕ and increasing β and q resulted in increased cost, causing shrinkage in the Pareto front. The contraction of Pareto fronts in the retaining wall without a base shear key was much more significant than those with a base shear key. Moreover, for more intensive cases, the inclination of Pareto fronts was smaller, which further indicates that high FOSs required a significant increase in the final costs.