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Nature-inspired algorithms in sanitary engineering: modelling sediment transport in sewer pipes

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Abstract

In this study, two nature-inspired optimization techniques such as firefly algorithm (FA) and butterfly optimization algorithm (BOA) are combined with adaptive neuro-fuzzy inference system (ANFIS) and group method of data handling (GMDH) models for optimal prediction of the complex phenomenon of volumetric concentration of sediment (Cv) in sewer systems. Three different scenarios based on the methods of dimensional analysis and forward selection are implemented for determining the input structure of ANFIS, GMDH, and regression models (multiple linear regression, MLR; stepwise regression; SR) regarding 13 independent hydraulic and geometric input variables. Several statistic criteria including the root-mean-square error (RMSE), mean absolute error (MAE), Nash–Sutcliffe efficiency (NSE), index of agreement (AI), coefficient of determination (R2), and comprehensive synthesis index (SI) as well as Taylor diagram were used to further quantify simulating and predicting accuracies. In comparison with the regression models and two empirical equations, the results obtained by standard machine learning models (ANFIS and GMDH) were very promising. However, such integration of FA and BOA noticeably improved the performance of ANFIS (around 7% improvement in RMSE criterion) and slightly optimized the performance of GMDH (less than 1% improvement in RMSE criterion) in modelling the process of Cv prediction.

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Abbreviations

\(\varepsilon\) :

Random variable in FA and BOA

\(\gamma\) :

Absorption coefficient in FA

\(\vartheta\) :

Number of input variables in GMDH model

\(\phi\) :

Power exponent in BOA

\(\alpha\) :

Randomization parameter in FA and BOA

∑:

Summation operator

A :

Cross-sectional area of flow

a i :

Partial regression coefficients in MLR and SR models

ANFIS:

Adaptive neuro-fuzzy inference system

B :

Water surface width

BOA:

Butterfly optimization algorithm

c :

Sensory modality in BOA

Cv :

Volumetric concentration of sediment

D :

Internal diameter of pipe channel

d :

Median diameter of particles in a mixture

FA:

Firefly optimization algorithm

fr :

The perceived magnitude of fragrance in BOA

Fr d :

Densimetric Froude number

g :

Gravitational constant

GMDH:

Group method of data handling

Gs:

Specific gravity of sediment

I :

Stimulus intensity in BOA

IA :

Index of agreement

Ks :

Overall equivalent sand roughness with sediment

M :

Number of samples in training and testing sets

MAE :

Mean absolute error

MLR:

Multiple linear regression model

n :

Overall Manning roughness coefficient with sediment

NSE :

Nash–Sutcliffe model efficiency coefficient

\({O}_{i}^{j}\) :

Functional node of ANFIS network

p :

Environmental fraction in BOA

P :

Wetted parameter of flow

p i ,q i ,r i :

Parameter set of ANFIS model

Q :

Flow discharge in sewer system

r :

Distance between any two fireflies in FA

R :

Hydraulic radius

R 2 :

Coefficient of determination

RMSE:

Root-mean-square error

S 0 :

Longitudinal slope of sewer

SI :

Synthesis index for models’ evaluation

SR:

Stepwise regression model

T :

Temperature

V :

Mean velocity of flow

V c :

Critical incipient motion velocity of sediment

Ws :

Width of sediment spread

x i :

Input variables for quadratic polynomial in GMDH model

y 0 :

Depth of uniform flow

z i :

Position of the ith agents in FA and BOA

β 0 :

Initial attractiveness of a firefly in FA

λs :

Overall friction factor in sewers

ν :

Kinematic viscosity

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Acknowledgements

The first author wishes to extend his special thanks to the Alexander von Humboldt Foundation for providing financial support for this research project within the framework of the Return Fellowship program.

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Authors and Affiliations

  1. Water Engineering Department, Shahid Bahonar University of Kerman, Kerman, Iran

    Mohammad Zounemat-Kermani

  2. Institute of Hydro-Engineering, Polish Academy of Sciences, Warsaw, Poland

    Amin Mahdavi-Meymand

  3. Chair of Water Resources Management and Modeling of Hydrosystems, Institute of Civil Engineering, Technische Universität Berlin, Berlin, Germany

    Reinhard Hinkelmann

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  1. Mohammad Zounemat-Kermani
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  2. Amin Mahdavi-Meymand
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  3. Reinhard Hinkelmann
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Correspondence to Mohammad Zounemat-Kermani.

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Zounemat-Kermani, M., Mahdavi-Meymand, A. & Hinkelmann, R. Nature-inspired algorithms in sanitary engineering: modelling sediment transport in sewer pipes. Soft Comput 25, 6373–6390 (2021). https://doi.org/10.1007/s00500-021-05628-1

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