Heterogeneous slope stability optimization using firefly … · To evaluate factor of safety (FOS)...

International Research Journal of Applied and Basic Sciences © 2014 Available online at www.irjabs.com ISSN 2251-838X / Vol, 8 (9): 1222-1235 Science Explorer Publications

Heterogeneous slope stability optimization using firefly algorithm, simulate annealing and imperialistic

competitive algorithm

AlirezaKashani*, Mehdi Jalalvandi

Department of Civil Engineering, Arak University, Arak, Iran

*Corresponding Author Email: [email protected]

ABSTRACT: Soil slope stability optimization with non-circular slip surface is considered as a complicated problem in geotechnical engineering. Therefore it is necessary to find a robust algorithm which is capable to find the best solution. In this paper the performance of three metaheuristic algorithms, firefly algorithm, simulate annealing and imperialistic competitive algorithm, are investigated and their results are compared to particle swarm optimization as a popularmetaheuristic optimization algorithm. Three numerical examples are analyzed here and the algorithms results are compared. In order to evaluate soil slope stability a factor of safety (FOS) is defined. In the current study spencer method as one of the limit equilibrium method is selected for FOS calculating. In this way a computer program is developed in MATLAB code that is modeled producing non-circular slip surface and computing factor of safety. As a result imperialistic competitive algorithm is the best and firefly algorithm is the worst algorithm among here proposed algorithms. Key words: slope stability optimization, metaheuristic algorithms, firefly algorithm, simulate annealing, imperialistic competitive algorithm

INTRODUCTION

Slope stability optimization is one of the most important problems because of vast application in many civil engineering fields such as: highways, dams, tunnels and so on. In many modern cities millions of dollars are spent to stabilize the soil slopes, annually. Therefore an exact analysis of soil slope stability could decrease related costs. However circular slip surface results proper solution in homogenous soil slopes, in heterogeneous soil slopes it is not a good consideration. Some methods are proposed to model the slip surface in nonhomogeneous slopes, such as the method proposed by Cheng (2003) and the one by Zolfaghari et al.(2005)

In circular slip surface modeling only three design variables, radius and center coordinates ( oX and oY )should be

defined. Unlike circular slip surface consideration,one of the complexities in non-circular slip surface consideration is the large number of design variables. In addition to mentioned difficulty, Cheng (2003) mentioned slope stability optimization problems accounted as a NP-hard type problem by the following qualities:

1- The objective function is non-smooth, non-convex and discontinuous over solution domain. 2- There are multiple strong local minima over the search space. 3- A good initial solution should be defined to reach a good solution.

According to the mentioned complexity, classical optimization techniques aren’t capable to reach a global minimum very well. Moreover, because of strong local minimum existence, some metaheuristic algorithms may be trapped in these local minimum and failed to converge a proper solution. Therefore a robust algorithm which is able to overcome this issue should be fined that is able to find the global best solution. Recently, numbers of studies have been done in the area of slope stability optimization and a lot of meatheuristic algorithms have been applied to this problem. McCombie and Wilkinson (2002), Zolfaghari at al (2005), have used genetic algorithms (GA); Bolton et al (2003) have served leap-frog and Kahatadeniya et al (2009), tried ant colony optimization (ACO) method; Cheng et al (2007a, 2007b and 2008) have adopted simulate annealing (SA), particle

Intl. Res. J. Appl. Basic. Sci. Vol., 8 (9), 1222-1235, 2014

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swarm optimization (PSO), harmony search (HS), tabu search and Fish Swarm optimization to slope stability problems very well. In the current paper three optimization algorithms, simulate annealing (SA), firefly algorithm (FA) and imperialistic competitive algorithm (ICA) are applied to slope stability analysis. For that purpose three numerical examples are evaluated here and the values for FOS are computed. These numerical examples are samples of complicated slopes. Two of them are proposed here for the first time and one of them was proposed by Zolfaghari et al (2005). To evaluate factor of safety (FOS) a MATLAB code is developed based on Spencer (1967) limit equilibrium method. As a result ICA reached the best solutions with the lowest values for FOS over all the three analyzed case studies. Also FA could not find a proper solution and was the most inefficient algorithm among techniques utilized here.

METHODOLOGY In this study the method proposed by Zolfaghari et al (2005) is utilized for producing general slip surface as Figure 1. In this method instead of defining nodal y-coordinates, the failure-line slope at the base of each slice related to the slope of adjacent slices is determined. Therefore each slice could be shaped by determining the start slope of slip surface and inclination’s variations of base of slices. In slope stability analysis, by moving from left to right the angle of the base of slice (α) is usually shifted counter-clockwise.In this method unrealistic failure surface like zigzag as shown in Figure 2, arenot possible to make.

Figure 1.Genearl slip surface in slope stability analysis

Refer to Figure 3, at first define 1x point as start point of slip surface randomly, then select a random value for 1 in

the vicinity of Rankin failure angle range. After that i values should be determined in the range of 0 to 15° based

on Zolfaghari et al (2005) proposed values.


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Figure 2. Inappropriate slip surface by nodal analysis case

Figure 3. Generating a trial non-circular slip surface


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The effective forces acting on ith slice are shown as Figure 4.To evaluate the FOS value a concise algorithm proposed by Zhu et al (2005) is considered. A step by step pseudo code for this algorithm is summarized as follows:

Figure 4. Effective inter-slice forces on ith slice

1) computeiR and

iT using eqs (2) and (3).

' '[ cos sin cos( ) ] tan seci i i i h i i i i i i i i iR W W Q U cb (2)

sin cos sin( )i i i i h i i i iT W W Q (3)

Sine inter slice function ( )f x is considered as follows:

( ) sin( ( ) / ( ))f x x a b a (4)

where a and b are x coordinate of left and right ends of failure surface, then we can calculate ( )f x .

2) Consider initial values for sF and where the following criterion is satisfied:

sin costan

cos sin

i i is

i i i

fF

f

(5)

3) Calculate i and

1i using eqs (6) and (7) for all slices then

sF is calculated using eqs (8).


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(sin cos ) tan (cos sin )i i i i i i i i sf f F (6)

1 1[(sin cos ) tan (cos sin ) ] /i i i i i i i i s if f F (7)

With the calculated value of sF using eqs (8) and the prescribed value of , third step is repeated to update values

of i ,

1i , andsF .

11

1

11

1

( )

( )

nn

i j n

i j i

s nn

i j n

i j i

R R

F

T T

(8)

4) Evaluate iE is using eqs (9) for all slices then Calculate by eqs (10).

Repeat all above steps with the updated values of sF and until the differences in values of

sF and between two

consecutive iterations are within specified limits of tolerance, 1 and

2 .

1 1 1i i i i i s i iE E F T R (9)

1

1 1

[ ( ) tan 2 sin ]

[ ( )]

i i i i i h i i i

i i i i i

b E E W h Q h

b f E f E

(10)

OPTIMIZATION ALGORITHMS

Simulated Annealing SA is one of the heuristic optimization techniques which imitates the physicalannealing of solid. In the physical annealing, solid is heated until itmelts and then with a proper annealing schedule it gets cold graduallyuntilit reaches the minimum energy level. If the initial temperature is notselected high enough or cooling process was very fast, at thelow energy state, there can be a deformation in the solid. At first, SAalgorithmwas applied for combinatorial optimization problems byKirkpatrick, Gelatt, and Vecchi (1983).Several simulated annealing algorithms are proposed for multiobjective optimization problems (Serafini, 1994; CzyzakandJaszkiewicz, 1998; Ulungu, Teghem, Fortemps, andTuyttens, 1999;Suman, 2003. Main steps of SAalgorithm for the solution are given below:

Step 1: Set parameters of simulatedannealing ( inT = initial temperature, = cooling rate, NIET = the number of

trials to be performed with thesame temperature value, iteration number).

Step 2: Start temperature counter: el = 0.

Step 3: Create a random initial solution ( 0S ) and calculate theweighted cost of initial solution ( 0E ) using the

objective function.

Step 4: Make the iteration counter 0 at each temperature level: il = 0

Step 5: Increase the iteration counter one at each temperaturelevel: il = il + 1.


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Step 6: Choose two facilities randomly (i – j) and find a neighborsolution ( ilS ) by swapping their places. Then

calculatethe weighted cost of neighbor solution ( ilE ).

Step 7: Calculate the change in weighted cost value: il cE E E . If ( 0E ) or ( 0E and x = random(0-1) <

( ) exp( / )P E E T ), accept the change whichmeans the neighbor solution and,S ,c il c ilS E E

Step 8: If ( c bestE E ) Set to best cS S and best cE E . If it is not, goto the next step.

Step 9: If ( il NIET ) go to next step. If it is not, go to Step 5.

Step 10: 1el el .

Step 11: 1el elT T .

Step 12: If ( maxel el ) go to next step. If it is not, go to Step 4.

Step 13: Stop algorithm and write the solutions. 3.1. Annealing schedule 3.1.1. Initial solution In general, SA algorithms start with a randomly generated initialsolution or with a solution produced using a heuristic algorithm. In the present study, a randomly generated initial solution is used. 3.1.2. Moving to a neighboring solution In the proposed SA, the search moves from the current solutionto a neighboring solution by swapping two departments. Two departments are selected randomly and these departments areswapped. 3.1.3. Cooling schedule Selection of the SA parameters is called the cooling schedule,also called the annealing schedule. Important specific decision thatshould be made in SA is the cooling schedule that includes initialtemperature, the rate of temperature decrease, and stopping criterion.The probability of acceptance is defined as the probability ofaccepting a non-improving solution. This is determined based onthe following probability:

( ) exp( / )P E E T (11)

whereT is the temperature and E represents the change in thecost of the neighboring solution and the cost of the

current solution.If x is a randomly generated number between 0 and 1, and x P( )E , then accept the non-

improving neighboring solution asthe current solution. Otherwise, reject the non-improving neighboringsolution, and keep the current solution. The value of thetemperature at the beginning of the schedule should be large enoughso that most of the initial movements are accepted. However,as the temperature is reduced, the probability of accepting a non-improvingsolution reduces. The initial temperature is estimatednumerically by making many trial runs for each test problem.So as to find a good solution, the decrease in temperature mustbe as fast as possible to be computationally efficient, but slow enoughto maintain equilibrium within the system. The most commoncooling

function uses a geometric decrement function( 1Tk kT ) first proposed by Kirkpatrick et al. (1983). In this

study,geometric decrement function is used to decrease the temperature.Here is cooling rate and its value is

between 0 and 1 but it is closeto 1. Some methods were used in the literature to finalize the algorithm.Some stops the iterations when the total iteration number reaches,a specified value while other stop when there is no accepted movein a given number of trials. Another method is to stop the iterations when a specified final temperature value is reached. In this paper,the

algorithm is ended when we reach the specified maximum numberof iterations ( maxel ).

Firefly Algorithm The FA is one of the recent swarm intelligence techniques inspired from the behavior of fireflies developed by Yang at Cambridge University (2008). This method uses the differences in flashing light intensity of fireflies to search the solution space. All fireflies are preferred to move toward a brighter position because of more attractiveness. FA considered three idealizing rules as follows(2012): - All fireflies having same sex and brighter individual is more attractive than the others regardless of its sex.


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- Attractiveness is relying on brightness and has inversely relation with distance. If there is not brighter one, the individual will move randomly. - The brightness of fireflies is defined by the objective function and by moving toward the minimum of function, the brightness will be increased. In the FA, we consider swarm of n particles with their specific attractiveness of β which is defined by equation 13.

2

0( ) rr e (12)

Where 0 and are maximum attractiveness value (r=0) and the light absorption coefficient, varying between 0.1

and 10, respectively.

Where r is representing the distance between two fireflies at iX and jX position and can be defined by:

ij i jr X X (13)

each individual’s position will be updated by the proposed following equation:

21

0 ( )t t r t t

i i j ix x e x x (14)

Where t

iX is the current position of firefly i and the second term is represented brightness and the third term is

used for randomizing movement where is a random parameter between 0 and 1 and is a vector containing a

random number generated from a Gaussian distribution. For more detail the original paper by Yang (2008) is referable. Imperialistic Competitive Algorithm Imperialistic Competitive Algorithm (ICA), model the imperialistic competition process. A step by step pseudo-code is demonstrated for this algorithm in the following: Step 1. Initializing Empires Similar to other optimization algorithms, an array containing of random variables is made as an initial solution named country here.

For an N-dimensional problems popN country will be generated as eqs(15) :

var1 2 3[ , , ,..., ]Ncountry P P P P (15)

impN of the best solution are considered as the imperialist and the others constitute the colonies. The colonies

divided among empires and better country has more number of the colonies. Step 2. Moving colonies toward the imperialist In nature empires made their colonies improve. ICA modeled this fact by moving the colonies toward imperialists along a vector from colony to imperialist. Figure 5, shows movement direction with the distance equal to x as follows:

~ (0, )x U d (16)


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Figure 5.colony movement direction

where, x is a uniform random number, is a number greater than 1 and d is the distance between colony and

imperialist.

ICA defined as eqs(17) that is new direction inclination respect to the original direction.

~ U( , ) (17)

where, is a parameter to adjust the amount of deviation.

and lead algorithm to forage solution domain around imperialist.

Step 3. Revolution ICA selects the best solution in each iteration to guide algorithm to converge to the global minimum. If the colony objective value will be made better through moving toward imperialist, the colony and imperialist will exchange their position. Step 4. Total power of an empire Naturally, the stronger imperialist has to have the greater empery. Total power of empires is relating to the power of imperialist and the power of colonies.

. .nT C is defined to model this fact by the following eqs:

. . ( ) { ( )}n n nT C Cost imperialist mean Cost colonies of empire (18)

where, . .nT C is the total cost of nth empire and is a positive number which is considered to be less than 1.

Step 5. Imperialist competition All empires try to develop their colonies and take control of the colonies of other empires. It is equal to increase in power of more powerful empires and decrease in the power of weaker ones. This fact is modeled by a probability relating to the empires total power and more powerful empires take possession more colonies. Step 6. Eliminating the powerless empires In the ICA in each iteration the weakest empires collapse and its colonies distribute among other emperies. The stronger imperialist has greater portion of these colonies. Step 7. Convergence The above mentioned steps iterate to the powerful imperialist take possession of all the colonies. Finally all the colonies will have same position (equal value) as imperialist. It can be defined as termination criteria. For more detail refer to Atashpaz-Gargari and Lucas (2008).


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NUMERICAL SIMULATION For the purpose of comparison between proposed algorithm efficiency, three soil slope cases are solved and final results are reported. In the all cases it is trying to investigate the effect of thin weak layers that is sandwiched between two strong layers because of complexity. In this paper for all the optimization algorithms population size is considered equal to 50 and the generation is 3000 therefor the number of function evaluations are 15000 for all the examples by all optimization algorithms and for evaluation of Factor of Safety number of slices is considered 25. In addition because of the chaotic operation of optimization algorithm, all the results are reported by the best value, mean and standard deviation to illustrate the performance of every algorithm more efficiently. 4.1. Case I The first case is taken from Zolfaghari et al (2005) and it represents a good example of complex soil layering. They adopted GA for this problem and obtain a safety factor equal to 1.5. The problem resolved by PSO, as a benchmark, and then it is optimized by the FA, SA and ICA. Slope geometry, soil conditions and also slip surface are presented in Figure 8. The values of Safety Factor are presented in table.1.

Figure 6. Slope geometry and critical slip surface for case I.


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Table 1.Values of Safety Factor for case I.

As expected, failure surface is laid within weak layer. Since the band of weak soil with small thickness caused strong local minima within the solution domain (cheng et al 2007a) FA failed to evade from local minima. Furthermore PSO and SA are better than FA but they could not converge to a good solution. Obviously ICA is the best algorithm between swarm optimization techniques in this case study. It should be noticed that all four algorithms used here reach to better safety factors found in the initial study by Zolfaghari et al (2005). 4.2. Case II A sample of complex slope as shown in Fig.9 is optimized by PSO, FA, SA and ICA. In this case the location of critical slip surface is depicted as Figure 9. Factor of Safety associated to each failure surface is presented in table.2. A comparison of results shows FA cannot escape from the local minima and it fails to converge to a proper result in this case. On the other hand ICA is the most efficient algorithm with the best convergence among proposed algorithms. Furthermore PSO and SA have intermediate performance and they could not evade from strong local minima.

Figure 7. Slope geometry and critical slip surface for case II.

Optimization Algorithm PSO FA SA ICA

Mean 1.39 1.42 1.43 1.244775

Best 1.11 1.30 1.26 1.06

Standard deviation 0.1519 0.0646 0.0978 0.0847


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Table 2.Slope geometry coordinates for case II.

Table 3.Soil layer’s parameters for case II.

Table 4. Values of Safety Factor for case II.

4.3. Case III Another example is a heterogeneous slope with a band of weak soil located between two strong layers. The soil properties and slope geometry are presented in Figure 10. The location of critical failure surface and Factor of Safety are presented in Figure 10 and table3, respectively. Each optimization algorithm’s efficiency is challenged in the present case study very well. The results illustrated FA, PSO and SA trapped by local minima easily but ICA converged to a proper solution again.

POINT X Y

A 20 20

B 42 9

C 4 18

D 24 18

E 4 9

F 60 6

G 4 5

H 60 5

LAYER 1 2 3 4

C 1500 1800 1000 3000

φ 25 30 0 20

γ 1800 1800 1800 1800


Mean 1.29 1.31 1.27 1.0912

Best 1.06 1.11 1.04 1.01



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However, PSO, FA and SA do not have unique performance on the all cases, ICA shows the best results in the three mentioned case studies and it seems to be a proper optimization algorithm in general.

Figure 8. Slope geometry and critical slip surface for case III.

Table 5. Slope geometry coordinates for case III.

POINT X Y

A 40 40

B 70 25

C 25 37

D 90 12.5

E 25 36

F 36 11.5


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Table 6.Soil layer’s parameters for case III.

Table 7. Values of Safety Factor for case III.

CONCLUSIONS

Soil slope stability is benchmarked with a related index named factor of safety (FOS). There are a lot of methods to define FOS. Among all the proposed methods spencer method is one of the most effective and practical one for calculating FOS. In the present research this method is selected for computing FOS, and Zolfaghari’s proposed method is selected for producing trial slip surface. As explained in the paper by detail, slope stability with a weak soil layer band sandwiched between two strong layers is a complex problem and it has several strong local minimum within solution domain. In this paper the performance of three newfound swarm intelligence algorithms, FA, SA and ICA, is challenged by three complicated slope cases. Cheng et al (2007a) mentioned, there is no unique algorithm that can solve all the problems in the best situation. However, based on the simulation results, it seems that ICA is a powerful algorithm for such problems and it can evade from the local minimum better than three other algorithms. To benchmark the recent swarm algorithms performance more efficiently all cases are resolved by PSO algorithm. It should be noted that FA, SA and ICA are clearly outperform traditional PSO algorithm. By comparison between algorithm’s results, it can be concluded that ICA is the best algorithm among the swarm intelligent algorithms for solving complicated slope stability problems.

References

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Bolton, H.P.J., Heymann, G., Groenwold, A.A., 2003, Global search for critical failure surface in slope stability analysis. Engineering Optimization 35 (1), pp. 51–65.

Cheng, Y.M., 2003, Locations of critical failure surface and some further studies on slope stability analysis. ComputGeotech 30, pp. 255–267.

Cheng .Y.M., Li.L.,Ch.S.C., 2007a, Performance studies on six heuristic global optimization methods in the location of critical failure surface. ComputGeotech 34, pp. 462–484

LAYER 1 2 3

C 3000 900 3000

φ 40 5 40

γ 1800 1800 1800


Mean 1.83985 1.667145 1.444684 1.41658

Best 1.343 1.317 1.3337 1.2618



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Cheng, Y.M., Li, L., Chi, S.c., Wei, W.B., 2007b, Particle swarm optimization algorithm for the location of the critical non-circular failure surface in two-dimensional slope stability analysis. Computers and Geotechnics 34 (2), pp. 92–103.

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Heterogeneous slope stability optimization using firefly … · To evaluate factor of safety (FOS)...

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