J. Cent. South Univ. Technol. (2011) 18: 2100-2107

DOI: 10.1007/s11771-011-0949-2

Modified electromagnetism-like algorithm and

its application to slope stability analysis

ZHANG Ke(张科), CAO Ping(曹平)

School of Resources and Safety Engineering, Central South University, Changsha 410083, China

? Central South University Press and Springer-Verlag Berlin Heidelberg 2011

Abstract: In the view of the disadvantages of complex method (CM) and electromagnetism-like algorithm (EM), complex electromagnetism-like hybrid algorithm (CEM) was proposed by embedding complex method into electromagnetism-like algorithm as local optimization algorithm. CEM was adopted to search the minimum safety factor in slope stability analysis and the results show that CEM holds advantages over EM and CM. It combines the merits of two and is more stable and efficient. For further improvement, two CEM hybrid algorithms based on predatory search (PS) strategies were proposed, both of which consist of modified algorithms and the search area of which is dynamically adjusted by changing restriction. The CEM-PS1 adopts theoretical framework of original predatory search strategy. The CEM-PS2 employs the idea of area-restricted search learned from predatory search strategy, but the algorithm structure is simpler. Both the CEM-PS1 and CEM-PS2 have been demonstrated more effective and efficient than the others. As for complex method which locates in hybrid algorithm, the optimization can be achieved at a convergence precision of 1×10^-3, which is recommended to use.

Key words: slope stability; hybrid optimization algorithm; complex method; electromagnetism-like algorithm; predatory search strategy

1 Introduction

Slope stability analysis is an important research topic in geotechnical engineering, the key problem of which is to determine the location of critical failure surface and associated minimum factor of safety. For the complexity of geotechnical conditions and soil parameters in engineering practice, the objective function of safety factor is usually non-smooth and discontinuous [1-2]. Thus, searching for the minimum factor of safety is a non-linear optimization problem with the presence of multiple minima [3]. It is easy to trap in local minima by traditional optimization method, such as simplex method [3-4], Davidson-Fletcher-Powell method [3], conjugate gradient method [5] and dynamic programming [6]. Random method [7] and Monte-Carlo technique [8-9] cannot guarantee the precision of global minimum. Intelligent optimization algorithms have good global optimization performance, especially for the complex optimization problem. Recently, many researchers have focused their attentions on the research of intelligent optimization algorithms. Genetic algorithm (GA) [1, 10-12], leap-frog algorithm [13], simulated annealing algorithm (SA) [1, 14], ant colony algorithm (ACA) [1, 15-16], Tabu algorithm [1], particle swarm optimization algorithm (PSO) [1, 17], harmony algorithm (HA) [1, 18] and artificial fish swarms algorithm [19] have been applied to the analysis of slope stability. However, intelligent optimization algorithms also have their own disadvantages, which are generally weak in local search. The performance of single intelligent optimization algorithm applied to slope stability analysis usually holds insufficient accuracy.

Hybrid algorithm provides a feasible approach to enhance the performance of algorithm, and has become a hotspot in optimization study. Hybrid algorithm combines different optimization algorithms with certain strategies, which can take advantage of the composed algorithms. Local optimization algorithm embedded into global optimization algorithm is a typical hybrid strategy [20-21]. As to this method, global algorithm is responsible for global search, local algorithm search refined around roughly solutions obtained by global search.

Optimization algorithm has to strike a balance between two conflicting alternatives: 1) exploitation, i.e., to search thoroughly in promising areas, and 2) exploration, i.e., to move to distant areas potentially better than the actual one [22]. But such balance is hard to attain. Intelligent optimization algorithms deal with this conflict by setting different parameters, but depending on experiences, which is considered as a limitation. Many studies show that predatory behavior is effective [23-24]. Through simulating animals’ feed strategy, LINHARES [25] presented a search strategy called predatory search strategy, which has a good balance between exploitation and exploration.

In this work, original electromagnetism-like algorithm was modified, and complex electromagnetism- like algorithm (CEM) was presented, which hybridized a local search method, complex method. For further improvement, two CEM hybrid algorithms based on predatory search (PS) strategy were proposed. These algorithms were applied to search the minimum factor of safety in the slope stability analysis. The effects with different optimization algorithms were compared.

2 Original electromagnetism-like algorithm

Electromagnetism-like algorithm (EM) was a new heuristic optimization algorithm presented by BIRBIL and FANG in 2002 [26]. The algorithm simulates particles towards the best point, similar to the attraction- repulsion mechanism of electromagnetism theory [26-28]. Compared with the other methods, EM is a much more powerful algorithm for global optimization [26]. EM has been used in the areas of function optimization, resource constraint project scheduling problems [29], flowshop scheduling problem [30], array pattern optimization [31] and FOPID control optimization [32]. EM includes four phases: initialization of the algorithm, local search, calculation of total force vector, and movement according to the total force [26-28].

Simple random line search algorithm was applied as local search algorithm in original electromagnetism-like algorithm, and N_local denotes the number of iteration of local search procedure.

The calculation of total force vector is the key step in the electromagnetism-like algorithm. The charge qⁱ of point i is determined as

                              (1)

where n is the dimension of the problem; m is the number of sample points; x^best represents the point that has the best objective function value among the points at the current iteration.

Total force exerting on point i is evaluated as

           (2)

To ensure the feasibility of movement,  is processed by histogram normalization, i.e., F ⁱ=F ⁱ/||F ⁱ||. Particles move towards the total force with random step size, and the best position of point is updated:

(i=1, 2, …, m; i≠best; k=1, 2, …, n)       (3)

where λ is a random number in the range (0, 1). u_k represents upper bound in the k-th dimension. l_k represents lower bound in the k-th dimension.

3 Complex electromagnetism-like algorithm

3.1 Model of slope stability analysis

Simplified Bishop method was used to calculate the safety factor of slope. The number of slice was set to 100. Pore water pressures on the slip surface were estimated under groundwater condition. The X-ordinates of two exiting points (X_A, X_B) and arc height of the slip surface were taken as search variables [33], as shown in Fig.1, i.e., X={X_A, X_B, h}. X-ordinates of left bound and right bound are only needed to define, and the range of h is determined by the locations of X_A and X_B. Detail information can be seen in Ref.[33]. Searching for the minimum factor of safety in slope stability analysis can be expressed as

                                           (4)

Fig.1 Model of slope stability analysis

3.2 Algorithm program

The efficiency of simple random line search algorithm applied is low. As pointed out by BIRBIL that even with this simple method, the EM appears promising optimization properties [26]. Other powerful local search methods can also be used. In this work, complex method (CM) was adopted as local search algorithm. The complex method is a direct method for solving nonlinear constrained optimization problems with faster convergence speed, and does not require calculating the derivative. This method has been widely used in engineering, but with shortcoming of easily getting into local minimum. Readers can consult Ref.[34] for the process of CM. In this work, vertices number of 6 and termination criterion denoted as δ are used in Eq.(5):

                                  (5)

where

The convergence precision is given by δ=10^-5. The CM only inherits the current best solution obtained by one iteration of EM, and other points are generated randomly.

The number of parameters needed to set for EM is fewer than the other intelligent optimization algorithms. Number of sample points m is 20, the calculation accuracy of which can meet the requirement of slope stability analysis. If the current best solution is not improved for ten iterations, the EM is stopped.

For hybrid algorithm, the termination criterion of local search method (i.e., CM in this work) is also needed to determine. In the present work, we select different convergence precisions, i.e., 1.0×10^-5, 1.0×10^-3 and 1.0×10^-1. Slope example is given to illustrate the effects of optimization algorithm with different convergence precisions. The flow chart of complex electromagnetism-like algorithm (CEM) is shown in Fig.2.

Fig.2 Flow chart of CEM

3.3 Example analysis

The example shown in Fig.3 is taken from Ref.[10], in which a natural slope contains four layers. The soil parameters are listed in Table 1. The range of two exiting points is X_A[0, 30], X_B[30, 45]. For the presence of thin weak Layer 3, the minimum factor of safety will be very sensitive to the location of critical failure surface.

Fig.3 Cross section of investigated slope

Table 1 Parameters of different soil strata

All algorithms run for 50 times.  represents the minimum factor of safety obtained from all the computation works, =1.459. If the relative error between the solution  obtained by i-th computation and  is less than 1%, the computation is considered to be successful. If the relative error is less than 3%, we consider the computation to be next successful. If the relative error is less than 10%, we consider the computation to be acceptable. If the relative error is more than 10%, the computation is considered to be failed, which means to be trapped by local minima. The following definitions are given for analysis.

Definition 1: Ratio of successful computation equals the number of successful computation over 50.

Definition 2: Ratio of next successful computation equals the number of next successful computation over 50.

Definition 3: Ratio of acceptable computation equals the number of acceptable computation.

Ratio of successful computation and next successful computation are used to evaluate the performance of searching precise optimal solution. Ratio of acceptable computation can measure the capability of escaping from local minima.

Define the standard deviation associated with the minimum factor of safety  as

                                             (6)

For the case study, the results of analysis are shown in Fig.4 and Table 2. The ratio of acceptable computation is only 46%, which indicates that CM is easily trapped by local minima. The ratio of acceptable computation of EM (N_local=10) is 94%. The EM converges within about ten iterations, and its computing time is approximately equal to that of CM. The present study has illustrated that the original EM has good global searching capability and converging speed, but is insufficient in fine search. As the number of iteration of local search procedure in EM increases, the performance is improved slightly, but not obviously.

Fig.4 Results for CM, EM and CEM

CEM combines the merits of two algorithms, i.e., the local search capability of CM and the gobal search capability of EM. Taking the termination criterion of CM of 1.0×10^-5 as example, the ratios of successful computation are 22% and 30% which are higher than those of CM and EM (N_local=10), respectively. The ratios of next successful computation are 48% and 60%, higher than those of CM and EM (N_local=10), respectively. Moreover, standard deviation of CEM associated with  is only 0.059 3. The CEM appears to be more stable and efficient.

For three termination criterions of CM in CEM, the performances of CEM with δ=1×10^-3 and δ=1×10^-5 are similar. All performance indexes listed in Table 2 (except for the ratio of successful computation) of CEM with δ=1×10^-3 are better than those with δ=1×10^-5.

However, the performance of CEM is still not good enough, in which the ratio of successful computation is only 30%. For further improvement, the CEM hybrid algorithms based on the predatory search strategy are proposed.

4 Hybrid algorithms with predatory search strategy

4.1 Principle of predatory search strategy

The predatory search strategy was presented by LINHARES in 1998 [25]. Initially, PS was applied to solve the combinatorial problems [25, 35-36], and extended in the area of continuous function optimization recently. The procedure of predatory search strategy is shown in Fig.5. The algorithm searches in the whole space until finding a better solution. Then, it turns to area-restricted search around the solution. If there is no improvement for the current best solution after several iterations, PS gives up local search and turns to global search, looping these two steps until finding the optimal solution.

The restriction is represented by the distance between the two points in space shown in Fig.6. Sample NumLevel points from the feasible region, Xⁱ (i=1, 2, …, NumLevel). Define restriction as follows: Restriction[i]= ||X^best-Xⁱ||₂, i.e., 2-norm of current minima X^best and other point Xⁱ. The restrictions are ordered from small to large, with the search area around X^best increasing. The PS dynamically manages the restriction levels to adjust the search area. The search area of initial iteration is small, which benefits to fine search. As the number of iteration increases, the search space also increases, which can overcome the local minima. When the solution is improved, the lists of restrictions are recalculated, then the restriction level is lowered and gradually raised. There is a good balance between exploitation and exploration.

Table 2 Comparison of statistic indexes obtained by CM, EM and CEM

Fig.5 Flow chart of predatory search strategy

Fig.6 Sketch of solution space

All spaces associated with restriction levels do not need to search. The restriction levels {0, 1, …, Lthreshold-1} can be considered as local search level. When “level=Lthreshold”, the algorithm cannot find better solution on the area-restricted search. Then, set “level” to be a high value, denoting as LhighThreshold. The levels {LhighThreshold, LhighThreshold+1, …, Numlevel-1} is considered as the global search level. The algorithm explores the best solution in a large search area.

Taking two-dimensional space as example, the restricted space is a circular area with its center at X^best and of radius Restriction[i], which is not convenient. We extend the search area to a square area with its center at X^best and of side 2×Restriction[i], as shown in Fig.6.

The calculating formulas of lower bound and upper bound for restriction level in the k-th dimension are as follows:

          (7)

          (8)

4.2 CEM hybrid algorithm with predatory search strategy 1 (CEM-PS1)

In order to converge faster, the EM without local search was adopted to find the approximate solution. Then, the search area was dynamically adjusted with predatory search strategy. The CEM was applied to search the global optimum in the restriction area. The flow chart of the CEM-PS1 is shown in Fig.7. The procedure of CEM-PS1 is as follows.

Fig.7 Flow chart of CEM-PS1

Step 1: Generate an initial point (slip surface) randomly in the whole space. The EM without local search is adopted to find the approximate solution X^EM,   where  The same as below.

Step 2: Set X^best=X^EM,  level=0, counter=0.

Step 3: Evaluate Restriction[i] (i=0, 1, …, NumLevel-1). By employing Eq.(4) and Eq.(5), evaluate the range of variables with different restriction.

Step 4: If level≥NumLevel, terminate the algorithm and take X^best,  as the optimum solutions. Otherwise, go to Step 5.

Step 5: Take X^best,  as initial point, and generate five other points randomly. Search the area associated with Restriction[level] by complex method, and we can obtain the solution X^CM,  Then, let X^CM,  as the current best solutions, and search the area by electromagnetism-like algorithm within an iteration, at least X^EM and  are obtained. If  go to Step 2. Otherwise, set counter=counter+1, go to Step 6.

Step 6: If counter≥Cthreshold, set counter=0, level=level+1, go to Step 7. Otherwise, go to Step 5.

Step 7: If level=Lthreshold, set level= LhighThreshold, go to Step 4.

4.3 CEM hybrid algorithm with predatory search strategy 2 (CEM-PS2)

The CEM hybrid algorithm with predatory search strategy 2 (CEM-PS2) did not employ the process in Ref.[25]. The idea of variable neighbourhood search method learned from predatory search strategy was applied. The CEM-PS2 with theoretical framework of CEM adjusted the search space after one iteration of CEM. As to CEM-PS1, search area was changed when counter≥Cthreshold. This was the main difference between CEM-PS1 and CEM-PS2. The flow chart of the CEM-PS2 is shown in Fig.8. The procedure of CEM-PS2 is as follows.

Step 1: Generate an initial point (slip surface) X^EM,  randomly in the feasible space, where   the same as below.

Step 2: Set X^best=X^EM, level=0.

Step 3: Evaluate Restriction[i] (i=1, 2, …, Numlevel-1). By employing Eq.(4) and Eq.(5), evaluate the ranges of variables with different restriction.

Step 4: If level≥NumLevel, terminate the algorithm and take X^best,  as the optimum solutions. Otherwise, go to Step 5.

Step 5: Take X^best,  as the initial point, and generate five other points randomly. Search the area associated with Restriction[level] by the complex method, and we can obtain the solutions X^CM, . Then, let X^CM,  be the current best solutions, and search the area by the electromagnetism-like algorithm within an iteration, at least X^EM and  are obtained.

Fig.8 Flow chart of CEM-PS2

Step 6: If  go to Step 2. Otherwise, set level=level+1, go to Step 4.

4.4 Example analysis

For CEM-PS1, parameters have been set as follows: NumLevel=5, Cthreshold=5, Lthreshold=1, LhighThreshold=5. For CEM-PS2, NumLevel=10.

As shown in Table 3 and Fig.9, the ratios of successful computation of CEM-PS1 and CEM-PS2 (δ= 1×10^-5) are 86% and 80%, respectively. The other solutions are all 1.475, which are also around the successful solutions. The ratios of next successful and acceptable computation are both 100%. The computing time of CEM-PS1 and CEM-PS2 with δ=1×10^-5 is also 12% lower than that of CEM. Therefore, another area- restricted search method, i.e., CEM-PS2 presented in this work, is proved to be satisfactory, with simpler algorithm structure.

Taking CEM-PS1 with δ=1×10^-5 as example, the computation time is 9.16 times that of CM. If consider CM running with the same time of CEM-PS1, the ratio of successful computation is only 53% (i.e., 1-(1- 8%)^9.16=53%). For CM obtaining the same ratio of successful computation of CEM-PS1, the computation time of 1.721 s (i.e., 1-(1-8%)^t/0.073=86% to obtain t= 1.721 s) is needed. The CEM-PS1 and CEM-PS2 have been demonstrated to be more effective and efficient for slope stability problem.

Table 3 Comparison of statistic indexes obtained by CEM-PS1 and CEM-PS2

Fig.9 Results for CEM-PS1 and CEM-PS2

For the termination criterion of complex method in the CEM-PS1 and CEM-PS2, the more accurate the convergence precision, the better the optimal performance, but the more the calculation time is required. The performances of CEM-PS1 and CEM-PS2 with δ=1×10^-3 and δ=1×10^-5 are similar. All performance indexes listed in Table 3 (except for average value, standard deviation and the ratio of successful computation) of CEM-PS1 with δ=1×10^-3 are better than those with δ=1×10^-5. All performance indexes of CEM-PS2 with δ=1×10^-3 are better than those with δ=1×10^-5. It is suggested that the convergence precision of the complex method in hybrid algorithms is 1.0×10^-3.

Moreover, as seen in Table 2 and Table 3, the performances of CEM-PS1 and CEM-PS2 are not as sensitive as CEM to the convergence precision, except δ=1×10^-1. The CEM-PS1 and CEM-PS2 are verified to be adaptive for using predatory search strategy.

5 Conclusions

1) The electromagnetism-like algorithm is a new swarm intelligence optimization algorithm. It is an algorithm aiming at slope stability analysis. The analysis program of slope stability based on the modified electromagnetism-like algorithm can be widely used in the geotechnical engineering.

2) The CEM combines the merits of two algorithms, which can effectively avoid local minimum. The performance of CEM hybrid method is better than that of CM and EM.

3) For further improvement, two hybrid algorithms based on the predatory search strategy are presented. The CEM-PS1 and CEM-PS2 dynamically manage the restriction levels. And CEM-PS2 is easier to implement. The analysis of complex slope example indicates that the CEM-PS1 and CEM-PS2 have excellent convergence precision and convergence speed.

4) For the termination criterion of complex method in hybrid algorithms, the more accurate the convergence precision, the better the optimal performance, but the more the calculation time is required. When the convergence precisions are 1×10^-3 and 1×10^-5, the performances of hybrid algorithms are similar. As to some performance indexes, the best comprehensive performance is obtained by using the hybrid algorithm with 1.0×10^-3. It is suggested that the convergence precision of complex method in the hybrid algorithms is 1.0×10^-3.

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(Edited by YANG Bing)

Foundation item: Project(10972238) supported by the National Natural Science Foundation of China; Project(2010ssxt237) supported by Graduate Student Innovation Foundation of Central South University, China; Project supported by Excellent Doctoral Thesis Support Program of Central South University, China

Received date: 2010-11-24; Accepted date: 2011-02-21

Corresponding author: CAO Ping, Professor, PhD; Tel: +86-13973128263; E-mail: pcao_csu@sina.com