Comparison of genetic algorithm and imperialist competitive ...

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© IWA Publishing 2014 Water Science & Technology

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Comparison of genetic algorithm and imperialist competitive algorithms in predicting bed load transport in clean pipe Isa Ebtehaj and Hossein Bonakdari

ABSTRACT The existence of sediments in wastewater greatly affects the performance of the sewer and wastewater transmission systems. Increased sedimentation in wastewater collection systems causes problems such as reduced transmission capacity and early combined sewer overflow. The article reviews the performance of the genetic algorithm (GA) and imperialist competitive algorithm

Isa Ebtehaj Hossein Bonakdari (corresponding author) Department of Civil Engineering, Razi University, Kermanshah, Iran E-mail: [email protected]

(ICA) in minimizing the target function (mean square error of observed and predicted Froude number). To study the impact of bed load transport parameters, using four non-dimensional groups, six different models have been presented. Moreover, the roulette wheel selection method is used to select the parents. The ICA with root mean square error (RMSE) ¼ 0.007, mean absolute percentage error (MAPE) ¼ 3.5% show better results than GA (RMSE ¼ 0.007, MAPE ¼ 5.6%) for the selected model. All six models return better results than the GA. Also, the results of these two algorithms were compared with multi-layer perceptron and existing equations. Key words

| genetic algorithm, imperialist competitive algorithms, sediment, sewer, urban drainage

INTRODUCTION Among problems that may affect the performance of wastewater systems is the bed load transport, which is an important parameter in sewer hydraulics and modelling of water quality (Jack et al. ). The deposition of sediments in the sewers can cause severe disruption of drainage systems. By reducing the cross-sectional flow and increasing the flow resistance, the presence of deposited sediments on the bed of the pipe alters the hydraulic capacity of the conveyance channel. To prevent sedimentation at different sections of the flow in the no-deposition state, many researchers tried to propose an equation that would be able to predict the minimum velocity at different conditions (Mayerle et al. ; Ab. Ghani ; Ackers et al. ; Ota & Nalluri ). Vongvisessomjai et al. () studied the bed load transport at no deposition. They presented their equations in a similar form to those proposed by Mayerle et al. () with the difference, however, that they considered the relative depth of flow in two forms of d/R and d/y, where d refers to mean particle diameter, R to hydraulic radius and y to depth of flow. Almedeij () presented a trend for studying the self-cleansing process based on the principles of bed load transport in rectangular canals. Ota doi: 10.2166/wst.2014.434

& Perrusquia () studied the bed load transport at the limit of deposition. To be able to propose an equation that could review bed load transport at the limit of deposition in large-diameter canals, they presented a semi-experimental equation. To increase the accuracy of estimating the densimetric Froude number (Fr), Ebtehaj et al. (in press) amended the equations presented by Vongvisessomjai et al. () and proposed bed load transport equations at the limit of deposition for flows in sewer. Nowadays, numerical procedures in engineering sciences are powerful methods for analyzing different hydraulic engineering problems. By applying the method of tournament selection in genetic algorithms (GAs), Zhang et al. () optimized the critical shear stress for sediment deposition and resuspension. To present an intelligent model capable of automatic optimization of flow parameters Tang et al. () proposed a method that was a combination of the hydrodynamic model with the intelligent model obtained from a GA. Vaghefi et al. () used the imperialist competitive algorithm (ICA) artificial neural network to predict the flow rate in the Karkheh watershed in the southwest of Iran. Abdollahi et al. () applied the ICA to solve the nonlinear equation

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Comparison of GA and ICA in predicting bed load transport

systems. They demonstrated that this optimization method has a better output than other methods of solving nonlinear equation systems. Ebtehaj & Bonakdari () studied the bed load transport in sewers using an artificial neural network. Their model estimates Fr by applying the parameters of volumetric sediment concentration (CV), d/R, dimensionless particle number (Dgr) and overall sediment friction factor (λs). In this article the bed load transport in sewers was studied by the application of an artificial neural network (ANN). To train the ANN presented here the evolutionary algorithm was applied. The algorithms used in this article are the GA and the ICA. Therefore, a computer program was coded for modeling these two algorithms and was combined with the neural network for the prediction of Fr. By considering parameters effective on bed load transport in four dimensionless groups, six different models were proposed. Then by applying the GA and ICA algorithms, the weights of different layers were optimized. Then the best selected model was validated by applying the set of other data which had no role in the training of the neural network. The results of ICA and GA algorithms were compared with the results of the multi-layer perceptron (MLP) neural network which uses the back propagation algorithm to be trained. Also, the accuracy of the models presented through GA and ICA algorithms was compared with the existing equations.

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velocity, s the specific gravity of sediment, g the gravity acceleration, Vt the required velocity for primary movement of sediment (Equation (2)) and y the depth of flow. According to Ackers et al. () the above equation is the best for bed load transport at limit of deposition. By considering the parameters of volumetric sediment concentration (CV) and the relative depth of flow (d/y or d/R), Ebtehaj et al. (in press) presented (Fr) as the following equation 0:54 V d Fr ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 4:49CV0:21 R g(s 1)d

(3)

To study the accuracy of the presented neural network in both situations of using GA and ICA in relation to the regression models presented, Equation (1), which was obtained in a semi-experimental form, was applied. Moreover, the equations proposed by Vongvisessomjai et al. () that were obtained from dimensional analysis were also used in validating the neural network model.

SOFT COMPUTING In this study we focus on ANN, GA and ICA. Artificial neural network

BED LOAD TRANSPORT EQUATION IN CLEAN PIPE May et al. () reviewed the existing trend of bed load transport and concluded that none of the existing equations returned good results in all sets of data. Therefore, to present an equation for studying bed load transport at limit of deposition they presented the following equation CV ¼ 3:03 × 102 ×

D2 A

0:6 1:5 d V2 Vt 4 1 D g(s 1)D V

In the ANN the network is trained to identify the innate relation between the layers. The hidden layer processes the data obtained from the input layer and makes them available to the output layer. Training is a process which will ultimately lead to learning. The network learns when the interlayer connection weights change in such a manner that the difference between the predicted and observed values remains at an acceptable level. Once this condition is met, the learning process has been realized. The trained network can be applied to predict the outputs appropriate to the new set of data.

(1) Genetic algorithm

Vt ¼ 0:125[g(s 1)d]0:5

h y i0:47 d

(2)

where CV is the volumetric sediment concentration, D the pipe diameter, A the cross-sectional area of the flow, d the median diameter of particle size, V the flow

The GA is an intelligent research method formed on the basis of genes and chromosomes. As an optimizing calculation algorithm and by considering a set of solution space points at each pass, GA effectively searches the different zones of the answer space. Since, contrary to the single

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path methods, in this method the solution space is searched comprehensively, there is a lesser chance of integration towards an optimized local point. The other advantage of this algorithm is that there is no limitation on the function being optimized such as derivation and integration. In its search, this algorithm needs only the value of fitness function at different points and does not use other supporting data such as the derivate of the fitness function. Therefore, it can be applied for different problems such as linear, non-linear, integrated or differential, and it is easily compatible with different problems. GA mimics the biological evolution process of the chromosomes, using such operators as selection, crossover and mutation. Selection is an important process for choosing the parents to create a new population that can affect the convergence of the GA. The roulette wheel selection method is used in this study due to its vast use in different researches (Wu et al. ; Sharma et al. ). Roulette wheel selection is random sampling with placement at proportionate reproduction. In this method, the parents are selected in such a manner whereby chromosomes with higher fitness have a greater chance for selection. For each chromosome i, the probability is calculated as follows fi pi ¼ PN

j¼1 fj

(4)

where fi is the fitness of chromosome i and N is the number of population. Other operators used in GA include crossover and mutation. Crossover acts simultaneously on two chromosomes, and by combining their features produces the new generation. The ‘crossover’ used in this study is the twopoint crossover type. In mutation, some parts of the chromosome are changed randomly for better performance and leave the optimized zone. In fact in this stage, features are created that do not exist in the parent. The mutation function used in this article is the Gaussian function. Sharma et al. () presented an optimization process of the GA.

Imperialist competitive algorithm The ICA is an evolution algorithm that presents the basis of the socio-political evolution of humans. This algorithm starts functioning with a random population named as country. The number of initial colonies of an empire correlates with its power. To form initial empires, the colonies are divided among the colonials considering their power (Atashpaz-Gargari & Lucas ) in such a manner that


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any colonial country with higher normalized power has more colonies. In an NVar dimensional optimization problem a country has an array of 1 × Nvar defined as follows country ¼ ½P1 , P2 , . . . , PNvar

(5)

The costs of a country are expressed in the following form through evaluation of the cost function f and variables of (P1, P2, …, PNvar ) ci ¼ f(countryi ) f ðP1 , P2 , . . . , PNvar Þ

(6)

The algorithm starts functioning with the initial countries Npop and Nimp in most of the powerful countries that have been selected as imperialists. The remaining countries are selected as colonies of the imperialist countries in a way that they have equal power with the imperialist. For an adequate distribution of colonies among imperialists, the normalized cost function of an imperialist is defined as follows Cn ¼ cn max (ci )

(7)

In the above equation cn is the nth cost of imperialist and Cn is the normalized cost. Consequently, the normalized power of each imperialist can be defined as follows C n pn ¼ PN imp i¼1

C

(8)

i

pn is the normalized power of an imperialist. On the other hand, the normalized power of each imperialist is identified by its colonies. The initial number of colonies in an empire is defined as follows NCn ¼ rand{pn :(Ncol )}

(9)

In the above equation NCn is the initial number of colonies in the nth empire and Ncol is the number of colonies. To distribute colonies among imperialists the number of NCn is randomly selected among colonies and given to empires. The imperialist countries absorb colonies by policy. Absorption policy composes the main theme of this algorithm and causes the progress of countries towards efficiency. The imperialists select the colonies according to their power. The total power of each imperialist is determined by the power of its sections, the power of the empire and the

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average power percentage of the colonies TCn ¼ cost(imperials:tn) þ ξ:mean{cost(colonies:of:empire)}

(10)

TCn is the total cost of the nth empire and ξ is a positive number that is taken to be less than 1. Imperialist competition has an important role in ICA. Throughout the imperialist competition, the weaker empire loses its power and colonies. To model this competition, the probability of possessing all the colonies is calculated in view of the total cost of the empire NTCn ¼ max {TCi } TCn i

(11)

In the above equation, TCn is the total cost of the nth empire and NTCn is the normalized cost of the nth empire. By having the total normalized cost, the possession probability of each empire is calculated as follows

p pn

NTC n ¼ PN imp NTC i¼1

(12)

i

After the elimination of all empires, except for the most powerful, all colonies come under the control of this unique empire.

DATA COLLECTION In this study the sets of data presented by Ab. Ghani () and Vongvisessomjai et al. () are used. Both sets of data presented in this article were related to limit of deposition state. The range of parameters in Ab. Ghani () experiments were as follows: 0.24 < V (m/s) < 1.216, 1 < CV (ppm) < 145, 0.072 < d (mm) < 8.3, 0.033 < R (m) < 0.136, 0.153 < y/D < 0.8 and 0.0007 < S0 < 0.0056. The range of data used in Vongvisessomjai et al. () tests were as follows: 0.237 < V (m/s) < 0.626, 4 < CV (ppm) < 90, 0.2 < d (mm) < 0.43, 0.032 < R (m) < 0.012, 0.2 < y/D < 0.4 and 0.002 < S0 < 0.006.

METHODOLOGY To study the bed load transport in pipes, the parameters effective on flow and the movement of sediment particles


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must be identified. Based on laboratory tests conducted by researchers (Mayerle et al. ; May et al. ; Ebtehaj et al. in press), it can be concluded that the most significant parameters they had studied and used in their equations were parameters such as flow velocity (V), size of dimensionless particles (Dgr), the volumetric sediment concentration (Cv), the average size of particles (d), pipe diameter, depth of flow (y), hydraulic radius (R), the area of the flow cross-section (A), the coefficient of total sediment friction (λs), and the specific mass of the sediments (s). The dimensionless parameters are as parameters of pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi motion (Fr ¼ V= gd(s 1), 1=ψ ¼ τ 0 =ρg(s 1)d), transport pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (CV , φ ¼ CV VR= g(s 1)d3 ), sediment (Dgr , d=D, s), transport mode (d=R, D2 =A, R=D) and the flow resistance (λs , (k0 ks )=D). The motion parameter is expressed as Fr or the flow parameter (ψ), which uses shear stress instead of velocity. To train the neural network using the set of data presented by Ab. Ghani, 96 tests (80%) were randomly selected from 120 available on clean pipes. Moreover, to evaluate the efficiency of the presented neural network, the remaining 24 (20%) data sets were used to validate the considered network. After training the neural network and by applying the input parameters presented by Vongvisessomjai et al. () the Fr was estimated. The resulting Fr is then compared with the Fr presented by Vongvisessomjai et al. (). The results of analysis of ANN are based on the criteria of root mean square error (RMSE) and mean absolute percentage error (MAPE) as defined in the following forms sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pn 2 i¼1 (FrExpi FrANNi ) RMSE ¼ n

MAPE ¼

X 1 n jFrExpi FrANNi j × 100 n i¼1 FrExpi

(13)

(14)

RESULTS AND DISCUSSION Figure 1 shows the results of GA and ICA to predict Fr for the six models presented in this study. It can be seen that both algorithms provided good accuracy with relative error less than 10% on the training of presented models. For both algorithms, models 1 and 4 have better results than other models, while the other models have a relative error sometimes more than 10%, for example, models 2, 5 and 6

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Figure 1


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Result of evolutionary algorithms (GA and ICA) at prediction of Fr (Train).

in the GA and models 3 and 5 in the ICA. Table 1 shows the results of neural network training using the GA and ICA. GA and ICA are reviewed to study the accuracy of the six models presented in this study. Based on the table ICA

has been able to estimate the entire presented model with a good accuracy while the GA results are not very accurate for all the models and it is less accurate than ICA for all models. The table shows that the results from the statistical

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Table 1


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Evaluating the results of presented models using GA and ICA with the Ab. Ghani

Table 2


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(1993) data

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Evaluating the results of evolutionary algorithms (GA and ICA), MLP and existing equation (Vongvisessomjai et al. (2010) data)

GA

Ebtehaj et al.

ICA

No.

Models

RMSE

MAPE

RMSE

MAPE

1

Fr ¼ f(CV, Dgr, d/R, λs)

0.032

6.40

0.007

3.49

2

Fr ¼ f(CV, Dgr, D 2/A, λs)

0.057

10.37

0.017

6.45

3

Fr ¼ f(CV, Dgr, R/D, λs)

0.041

8.90

0.013

4.21

4

Fr ¼ f(CV, d/D, d/R, λs)

0.007

6.27

0.007

3.09

5

Fr ¼ f(CV, d/D, D 2/A, λs)

0.029

9.04

0.020

5.58

6

Fr ¼ f(CV, d/D, R/D, λs)

0.052

8.74

0.010

5.06

indexes for both GA (RMSE ¼ 0.007; MAPE ¼ 6.27%) and ICA (RMSE ¼ 0.007; MAPE ¼ 3.9%) indicate that the results of model 4 are more accurate in comparison with other models. Therefore, model 4 is used in order to compare the results of these two algorithms (GA and ICA) and MLP and the existing methods. This section presents the comparison between the results of evolutionary algorithms (GA and ICA), MLP which used the back propagation algorithm to be trained and existing equations. Figure 2 shows the Fr predicted by GA and ICA algorithms as well as the Fr predicted by the MLP neural network from 27 experiments presented by Vongvisessomjai et al. (). The figure indicates that the results estimated by MLP are higher than the experimental values. Also, it can be seen that in comparison with the experimental values all Fr predicted by GA and ICA algorithms have a relative error of less than 5%. This is while

GA

ICA

MLP

RMSE

0.047

0.037

0.070

MAPE

4.82

3.90

6.40

May et al. (1996)

0.187 13.83

(in press)

0.128 11.67

some predictions made by MLP have a relative error of approximately 10%. After examining the evolutionary algorithms presented in this study and comparing them with ANN-MLP it was observed that ICA has higher accuracy in estimating Fr in comparison with the other two methods. Therefore, the results of estimating Fr through ICA are compared with the bed load transport equations in this section. To carry out the comparison, the data of Vongvisessomjai et al. () that had no role in training ANN and had different conditions from that of Ab. Ghani () used in ANN training are used. The figure indicates that ICA presents the results with a relative error less than 5% for almost all samples while the conditions are not similar for May et al. () and Ebtehaj et al.’s (in press) equations. The equation of May et al. has excessively high error in some points that leads to unreliability of this equation. The equation of Ebtehaj et al. is also less accurate in comparison with ICA, but it almost always leads to better results in comparison with May et al. Table 2 shows the results of the statistical indexes for (GA and ICA) evolutionary algorithms, MLP neural network, and the existing bed load transport equations (May et al. ; Ebtehaj et al. in press). The table indicates that ICA is more accurate than GA among the presented algorithms, but it can also be seen that the difference between the statistical indexes for these two algorithms is not significant. Also, this table is an indication that using (GA and ICA) evolutionary algorithms increases estimation accuracy in comparison with when gradient algorithms are used to train ANN. Comparing the results of ANN (GA, ICA, and MLP) indicates that using ANN increases the accuracy of predicting Fr.

CONCLUSION

Figure 2

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Results of evolutionary algorithms (GA and ICA), MLP and existing equations for prediction of Fr (Vongvisessomjai et al. (2010) data).

Bed load transport in sewer systems is an important issue in municipal drainage engineering. Different methods have been proposed for bed load transport in sewers, but, due to inadequate knowledge of parameters interfering in bed load transport, these methods return different results

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under varying conditions. For this reason, the neural network is used to estimate the Fr in these systems. In this study the evolutionary algorithms were used to train the proposed neural network. These algorithms include the GA and the ICA. To study the factors that affect the estimation of the Fr, six different models were presented. In these models, the effects of such parameters as motion, transport, sediment, mode of transport and flow resistance were considered (Table 1). After estimation of Fr, the accuracy of all the six models was tested. In the GA and ICA algorithms, model 4 returned the best results. Contrary to other algorithms, ICA returns better results (RMSE ¼ 0.037 and MAPE ¼ 3.9%) in estimating the Fr for all the six situations when comparing other methods used in this paper (GA, MLP, May et al. ; Ebtehaj et al. in press) in calculating the Fr in pipes. Moreover, the application of evolutionary algorithms in training the neural networks can be a suitable substitute for existing methods of estimating the Fr.

REFERENCES Ab. Ghani, A.  Sediment Transport in Sewers. PhD Thesis, University of Newcastle Upon Tyne, UK. Abdollahi, M., Isazadeh, A. & Abdollahi, D.  Imperialist competitive algorithm for solving systems of nonlinear equations. Computer and Mathematics with Applications 65 (2), 1894–1908. Ackers, J. C., Butler, D. & May, R. W. P.  Design of Sewers to Control Sediment Problems. Report No. 141 CIRIA, Construction Industry Research and Information Association, London, UK. Almedeij, J.  Rectangular storm sewer design under equal sediment mobility. American Journal of Environmental Sciences 8 (4), 376–384. Atashpaz-Gargari, E. & Lucas, C.  Imperialist competitive algorithm: An algorithm for optimization inspired by imperialistic competition. IEEE Congress on Evolutionary Computation 7, 4661–4666.


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Ebtehaj, I. & Bonakdari, H.  Evaluation of sediment transport in sewer using artificial neural network. Engineering Applications of Computational Fluid Mechanics 7 (3), 382– 392. Ebtehaj, I., Bonakdari, H. & Sharifi, A. in press Design criteria for sediment transport in sewers based on self-cleansing concept. Journal of Zhejiang University Science A. DOI:10.1631/jzus. A1300135. Jack, A. G., Petrie, M. M. & Ashley, R. M.  The diversity of sewer sediments and the consequences for sewer flow quality modeling. Water Science and Technology 33 (9), 207–214. May, R. W. P., Ackers, J. C., Butler, D. & Johnt, S.  Development of design methodology for self-cleansing sewers. Water Science and Technology 33 (9), 195–205. Mayerle, R., Nalluri, C. & Novak, P.  Sediment transport in rigid bed conveyance. Journal of Hydraulic Research 29 (4), 475–495. Ota, J. J. & Nalluri, C.  Urban storm sewer design in consideration of sediment. Journal of Hydraulic Engineering 129 (4), 291–297. Ota, J. J. & Perrusquia, G. S.  Particle velocity and sediment transport at the limit of deposition in sewers. Water Science and Technology 67 (5), 959–967. Sharma, D., Singh, V. & Sharma, C.  GA Based scheduling of FMS using roulette wheel selection process. Advances in Intelligent and Soft Computing 131, 931–940. Tang, H. W., Xin, X. K. & Xiao, Y.  Parameter identification for modeling river network using a genetic algorithm. Journal of Hydrodynamics Series B 22 (2), 246–253. Vaghefi, S. A., Mousavi, S., Abbaspour, K. C. & Yang, H.  An Imperialist Competitive Algorithm Artificial Neural Network Method to Predict Runoff. EGU General Assembly 2012, Vienna, Austria. Vongvisessomjai, N., Tingsanchali, T. & Babel, M. S.  Nondeposition design criteria for sewers with part-full flow. Urban Water Journal 7 (1), 61–77. Wu, X. J., Zhu, X. J., Cao, G. Y. & Tu, H. Y.  Modeling of SOFC stack based on GA-RBF neural networks identification. Journal of Power Sources 167 (1), 145–150. Zhang, F. X., Onyx, W. H. W. & Jiang, Y. W.  Prediction of sediment transportation in deep bay (Hong Kong) using genetic algorithm. Journal of Hydrodynamics Series B 22 (5), 599–604.

First received 5 September 2014; accepted in revised form 14 October 2014. Available online 25 October 2014