# Machine-Learning-Based Consumption Estimation of Prestressed Steel for Prestressed Concrete Bridge Construction

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## Abstract

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## 1. Introduction

^{3}of used concrete and the so-called theoretical volume of the structure. This new variable is described as the product of the length, width, and overhead clearance height of the local road that needs to be reconfigured to pass underneath the freeway. The same research team proceeded to develop more accurate linear regression models for forecasting the costs of underpass bridges, concrete consumption, and reinforcing steel consumption, depending on two input variables, i.e., the bridge surface area and the theoretical volume. As a result, satisfactory values of the coefficient of determination for cost estimation, consumption of concrete, and consumption of reinforcing steel of 0.80, 0.85, and 0.70, respectively, were obtained [12,13]. Similarly, Antoniou et al. [14] provided an analytical formulation for early cost estimation and material consumption of road overpass bridges in 2016. The database included data on 57 completed overpasses on the Egnatia Motorway. The research defined linear models for cost estimation, reinforcing steel consumption, and prestressing steel consumption. In the model for forecasting the consumption of prestressing steel, linear models are given for assessment depending on the deck surface area as an input variable and depending on the theoretical volume of the bridge, whose coefficient of determination value is 0.85 and 0.73, respectively. In this case the theoretical volume of the bridge is defined as the deck length multiplied by its width multiplied by the average pier height. In order to promote the use of prefabrication in the design and construction of highway concrete bridges in countries with nonwell-established relevant standards, Antoniou and Marinelli used multilinear regression analysis [15]. They proposed a set of standard precast extended I beams suitable for use in the majority of the common motorway-bridge models. The data of 2284 total beams from 109 bridges built along the Egnatia Motorway and two of its perpendicular axes [15] form the basis of the suggested set of standard beams.

## 2. Methods

#### 2.1. Multilayer Perceptron (MLP) Neural-Network Models

- A collection of synapses that have corresponding weight values;
- Summarizing part, where inputs multiplied by appropriate weights are added;
- Appropriate activation function that restricts the neuron’s output.

_{k}, the calculated bias.

- Data enters the network at the input layer;
- All calculations using the data, weights, and biases are performed in the hidden layer;
- The output layer from which outcomes are obtained.

**Figure 2.**Multilayer perceptron (MLP) neural network [20].

#### 2.2. Regression-Trees (RTs) Models

#### 2.3. Multigene Genetic-Programming (MGGP) Models

#### 2.4. Method for Multicriteria Compromise Ranking VIKOR

- “sufficient advantage” over the alternative from the next position (condition ${U}_{1}$);
- “sufficiently firm” first position with weight change $\nu $ (condition ${U}_{2}$).

## 3. Dataset

#### Criteria for Assessing Model Accuracy

## 4. Results

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**Figure A1.**Pareto front of models in terms of model performance and model complexity for a model with 4 genes and depth up to 3.

**Figure A2.**Original output from the GPTIPS 2.0 software with analytical expressions of the models that make up the Pareto front (models with 4 genes and a maximum depth of generated trees of up to 3).

**Figure A3.**Graphic representation of modeled (model gene = 4, depth = 3) and actual values of prestressed steel consumption: (

**a**) training set and (

**b**) testing set.

## Appendix B

**Figure A4.**Pareto front of models in terms of model performance and model complexity for a model with 4 genes and depth up to 4.

**Figure A5.**Original output from the GPTIPS 2.0 software with analytical expressions of the models that make up the Pareto front (models with 4 genes and a maximum depth of generated trees of up to 4).

**Figure A6.**Graphic representation of modeled (model gene = 4, depth = 4) and actual values of prestressed steel consumption: (

**a**) training set and (

**b**) testing set.

## References

- Pržulj, M. Mostovi; Udruženje Izgradnja: Beograd, Srbija, 2014; pp. 1–7. [Google Scholar]
- Tayefeh Hashemi, S.T.; Ebadati, O.M.; Kaur, H. Cost Estimation and Prediction in Construction Projects: A Systematic Review on Machine Learning Techniques. SN Appl. Sci.
**2020**, 2, 1703. [Google Scholar] [CrossRef] - Tang, Y.; Wang, Y.; Wu, D.; Liu, Z.; Zhang, H.; Zhu, M.; Chen, Z.; Sun, J.; Wang, X. An experimental investigation and machine learning-based prediction for seismic performance of steel tubular column filled with recycled aggregate concrete. Rev. Adv. Mater. Sci.
**2022**, 61, 849–872. [Google Scholar] [CrossRef] - Feng, W.; Wang, Y.; Sun, Y.; Tang, Y.; Wu, D.; Jiang, Z.; Wang, J.; Wang, X. Prediction of thermo-mechanical properties of rubber-modified recycled aggregate concrete. Constr. Build. Mater.
**2022**, 318, 125970. [Google Scholar] [CrossRef] - Zhao, X.Y.; Chen, J.X.; Chen, G.M.; Xu, J.J.; Zhang, L.W. Prediction of ultimate condition of FRP-confined recycled aggregate concrete using a hybrid boosting model enriched with tabular generative adversarial networks. Thin. Wall. Struct.
**2023**, 182 Pt B, 110318. [Google Scholar] [CrossRef] - Antoniou, F.; Aretoulis, G.; Giannoulakis, D.; Konstantinidis, D. Cost and Material Quantities Prediction Models for the Construction of Underground Metro Stations. Buildings
**2023**, 13, 382. [Google Scholar] [CrossRef] - Flyvbjerg, B.; Skamris, H.; Buhl, S. Underestimating Costs in Public Works Projects: Error or Lie? J. Am. Plan. Assoc.
**2002**, 68, 279–295. [Google Scholar] [CrossRef] - Menn, C. Prestressed Concrete Bridges; Birkhauser Verlag: Basel, Switzerland, 1990. [Google Scholar]
- Marcous, G.; Bakhoum, M.M.; Taha, M.A.; El-Said, M. Preliminary Quantity Estimate of Highway Bridges Using Neural Networks. In Proceedings of the Sixth International Conference on the Application of Artificial Inteligence to Civil and Structural engineering, Stirling, Scotland, 19–21 September 2001. [Google Scholar]
- Liolios, A.; Kotoulas, D.; Antoniou, F.; Konstantinidis, D. Egnatia Motorway bridge management systems for design, construction and maintenance. In Advances in Bridge Maintenance, Safety and Management—Bridge Maintenance, Safety, Management, Life-Cycle Performance and Cost; CRC Press: Boca Raton, FL, USA, 2006; pp. 135–137. [Google Scholar] [CrossRef]
- Fragkakis, N.; Lambropoulos, S.; Tsiambaos, G. Parametric Model for Conceptual Cost Estimation of Concrete Bridge Foundations. J. Infrastruct. Syst.
**2011**, 17, 66–74. [Google Scholar] [CrossRef] - Antoniou, F.; Konstantinitis, D.; Aretoulis, G. Cost Analysis and Material Consumption of Highway Bridge Underpasses. In Proceedings of the Eighth International Conference on Construction in the 21st Century (CITC-8), Changing the Field: Recent Developments for the Future of Engineering and Construction, Thessaloniki, Greece, 27–30 May 2015. [Google Scholar]
- Antoniou, F.; Konstantinidis, D.; Aretoulis, G.; Xenidis, Y. Preliminary construction cost estimates for motorway underpass bridges. Int. J. Constr. Manag.
**2017**, 18, 321–330. [Google Scholar] [CrossRef] - Antoniou, F.; Konsantinidis, D.; Aretoulis, G. Analytical formulation for early cost estimation and material consumption of road overpass bridges. Int. J. Res. Appl. Sci. Eng. Technol.
**2016**, 12, 716–725. [Google Scholar] [CrossRef] - Antoniou, F.; Marinelli, M. Proposal for the Promotion of Standardization of Precast Beams in Highway Concrete Bridges. Front. Built Environ.
**2020**, 6, 119. [Google Scholar] [CrossRef] - Marinelli, M.; Dimitriou, L.; Fragkakis, N.; Lambropoulos, S. Non-Parametric Bill of Quantities Estimation of Concrete Road Bridges Superstructure: An Artificial Neural Networks Approach. In Proceedings of the 31st Annual ARCOM Conference, Lincoln, UK, 7–9 September 2015. [Google Scholar]
- Kovačević, M.; Ivanišević, N.; Petronijević, P.; Despotović, V. Construction cost estimation of reinforced and prestressed concrete bridges using machine learning. Građevinar
**2021**, 73, 1–13. [Google Scholar] [CrossRef] - Kovačević, M.; Bulajić, B. Material Consumption Estimation in the Construction of Concrete Road Bridges Using Machine Learning. In Proceedings of the 30th International Conference on Organization and Technology of Maintenance (OTO 2021), Osijek, Croatia, 10–11 December 2021; Lecture Notes in Networks and Systems. Springer: Cham, Switzerland, 2021; Volume 369. [Google Scholar] [CrossRef]
- Kovačević, M.; Ivanišević, N.; Stević, D.; Marković, L.M.; Bulajić, B.; Marković, L.; Gvozdović, N. Decision-Support System for Estimating Resource Consumption in Bridge Construction Based on Machine Learning. Axioms
**2023**, 12, 19. [Google Scholar] [CrossRef] - Kovačević, M.; Lozančić, S.; Nyarko, E.K.; Hadzima-Nyarko, M. Application of Artificial Intelligence Methods for Predicting the Compressive Strength of Self-Compacting Concrete with Class F Fly Ash. Materials
**2022**, 15, 4191. [Google Scholar] [CrossRef] [PubMed] - Ripley, B.D. Statistical Aspects of Neural Network, Networks and Chaos-Statistical and Probabilistic Aspects; Barndoff-Neilsen, O.E., Jensen, J.L., Kendall, W.S., Eds.; Chapman & Hall: London, UK, 1993. [Google Scholar]
- Kaastra, I.; Boyd, M. Designing a neural network for forecasting. Neurocomputing
**1996**, 10, 215–236. [Google Scholar] [CrossRef] - Kanellopoulas, I.; Wilkinson, G.G. Strategies and best practice for neural network image classification. Int. J. Remote Sens.
**1997**, 18, 711–725. [Google Scholar] [CrossRef] - Heaton, J. Introduction to Neural Networks for C#, 2nd ed.; Heaton Research, Inc.: Chesterfield, MO, USA, 2008. [Google Scholar]
- Sheela, K.G.; Deepa, S.N. Review on methods to fix number of hidden neurons in neural networks. Math. Probl. Eng.
**2013**, 2013, 425740. [Google Scholar] [CrossRef] - Hastie, T.; Tibsirani, R.; Friedman, J. The Elements of Statistical Learning; Springer: Berlin/Heidelberg, Germany, 2009. [Google Scholar]
- Breiman, L.; Friedman, H.; Olsen, R.; Stone, C.J. Classification and Regression Trees; Chapman and Hall/CRC: Wadsworth, OH, USA, 1984. [Google Scholar]
- Poli, R.; Langdon, W.; Mcphee, N. A Field Guide to Genetic Programming; Lulu Enterprises, UK Ltd.: London, UK, 2008. Available online: http://www0.cs.ucl.ac.uk/staff/W.Langdon/ftp/papers/poli08_fieldguide.pdf (accessed on 15 December 2022).
- Searson, D.P.; Leahy, D.E.; Willis, M.J. GPTIPS: An Open Source Genetic Programming Toolbox for Multigene Symbolic Regression. In Proceedings of the International MultiConference of Engineers and Computer Scieintist Vol. I, IMECS 2010, Hong Kong, China, 17–19 March 2010. [Google Scholar]
- Searson, D.P. GPTIPS 2: An open-source software platform for symbolic data mining. In Handbook of Genetic Programming Applications; Springer: Berlin/Heidelberg, Germany, 2015; pp. 551–573. [Google Scholar]
- Searson, D.P.; Willis, M.J.; Montague, G.A. Co-evolution of non-linear PLS model components. J. Chemom.
**2007**, 2, 592–603. [Google Scholar] [CrossRef] - Searson, D.P.; Willis, M.J.; Montague, G.A. Improving controller performance using genetically evolved structures with co-adaptation. In Proceedings of the Eighteenth IASTED International Conference on Modelling, Identification and Control, Innsbruck, Austria, 15–18 February 1999; ACTA Press: Anaheim, CA, USA, 1999. Available online: https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.152.9591&rep=rep1&type=pdf (accessed on 8 January 2023).
- Antoniou, F. Delay Risk Assessment Models for Road Projects. Systems
**2021**, 9, 70. [Google Scholar] [CrossRef] - Aretoulis, G.; Papathanasiou, J.; Antoniou, F. PROMETHEE based ranking of project managers’ based on the five personality traits. Kybernetes
**2020**, 49, 1083–1102. [Google Scholar] [CrossRef] - Opricović, S. Optimizacija Sistema; Građevinski Fakultet: Beograd, Srbija, 1992. [Google Scholar]
- Opricović, S. Višekriterijumska Optimizacija Sistema u Građevinarstvu; Građevinski Fakultet: Beograd, Srbija, 1998. [Google Scholar]
- Opricović, S.; Tzeng, G.H. Extended VIKOR method in comparison with outranking methods. Eur. J. Oper. Res.
**2007**, 178, 514–529. [Google Scholar] [CrossRef] - Zeleny, M. Compromise Programming, Multiple Criteria Decision Making; University of South Carolina Press: Columbia, SC, USA, 1973. [Google Scholar]
- Kovačević, M. Application of Compromise Programming in Evaluation of Localities for Construction of Municipal Landfill. In Interdisciplinary Advances in Sustainable Development, 1st ed.; ICSD 2022; Lecture Notes in Networks and Systems; Tufek-Memišević, T., Arslanagić-Kalajdžić, M., Ademović, N., Eds.; Springer: Cham, Switzerland, 2022; Volume 539. [Google Scholar] [CrossRef]

**Figure 1.**Artificial-neuron model [20].

**Figure 3.**Example of segmenting variable spaces into regions (

**left**), the creation of a 3D regression surface (

**right**) [20].

**Figure 4.**An example of an MGGP model that has two genes [17].

**Figure 5.**Crossover and mutation operation in MGGP: (

**a**) random selection of parent tree nodes; (

**b**) exchange of parents’ genetic material; (

**c**) random node selection in tree mutation; and (

**d**) mutation of a randomly selected part of a tree [20].

**Figure 6.**The general structure of the MGGP model [20].

**Figure 7.**Eastern and southern legs of Corridor X in Serbia [17].

**Figure 10.**Comparison of the accuracy criteria for MLP-ANNs with different numbers of neurons in the hidden layer: (

**a**) RMSE and MAE; and (

**b**) R and MAPE.

**Figure 12.**Pareto front of models in terms of model performance and model complexity for a model with 3 genes and depth of up to 6.

**Figure 13.**Original output from the GPTIPS 2.0 software with analytical expressions of the models that make up the Pareto front (models with 3 genes and a maximum depth of generated trees of up to 6).

**Figure 14.**Comparison of accuracy criteria for the MGGP model as a function of gene number and tree depth (

**a**) RMSE, (

**b**) MAE, (

**c**) R, and (

**d**) MAPE.

**Figure 15.**Tree structure of the individual genes that comprise the optimal model (gene = 3, depth = 6).

**Figure 16.**Graphic representation of modeled (optimal model gene = 3, depth = 6) and actual values of prestressed steel consumption: (

**a**) training set, (

**b**) testing set.

Number of Neurons in the Hidden Layer | Reference | |
---|---|---|

1. | ${N}_{H}=({N}_{i}+{N}_{o}$)/2 | Ripley [21] |

2. | ${N}_{H}=\sqrt{\left({N}_{i}+{N}_{o}\right)}$ | Kaastra [22] |

3. | ${N}_{H}=2\times {N}_{i}$ | Kannellopulas [23] |

4. | ${N}_{H}=\frac{2}{3}\times {N}_{i}+{N}_{o}$ | Heaton [24] |

5. | ${N}_{H}=(4\times {N}_{i}^{2}+3$$)/({N}_{i}^{2}-8$) | Sheela [25] |

6. | ${N}_{H}\le min\left(2{N}_{I}+1,\frac{{N}_{S}}{{N}_{I}+1}\right)$ | Kovačević et al. [20] |

**Table 2.**Mean, minimum, and maximum values of variables in the model used to estimate the prestressed steel consumption per ${\mathrm{m}}^{2}$ of the bridge superstructure.

Variable | Average Value | Minimum Value | Maximum Value |
---|---|---|---|

Max. individual bridge span—MIBS [m] | 31.58 | 18.00 | 49.00 |

Average bridge span—ABS [m] | 30.74 | 17.60 | 44.91 |

Total bridge span—TBS [m] | 161.33 | 21.20 | 628.74 |

Bridge width—BW [m] | 12.81 | 8.40 | 17.80 |

Mass of prestressed steel (PS) [kg/m^{2}] | 17.13 | 8.98 | 38.74 |

**Table 3.**Geometrical and mechanical rope characteristics [18].

Abbreviated Name | Class | Nominal Values | Guaranteed Values | Maximum Relaxation at a Force of 0.7 Fpk after 1000 h [2.5%] | ||||
---|---|---|---|---|---|---|---|---|

Diameter Ø [mm] | Tensile Strength [N/mm^{2}]fpk | Elastic Modulus kN/mm^{2}] | Section Area [mm^{2}]Apk | Characteristic Breaking Force [kN] fpk | Characteristic 0.1% Proof Stress of Prestressing Steel [kN] Fp0.1k | |||

Y1770S7 | A | 15.2 | 1770 | 195 | 140 | 248 | 213 | 2.5 |

Y1860S7 | B | 15.2 | 1860 | 140 | 260 | 224 | ||

Y1770S7 | A | 16.0 | 1770 | 150 | 265 | 228 | ||

Y1860S7 | B | 16.0 | 1860 | 150 | 279 | 240 |

Parameter | Setting |
---|---|

Function set | times, minus, plus, rdivide, square, exp, log, mult3, sqrt, cube, power |

Population size | 1000 |

Number of generations | 100 |

Max number of genes | 5 |

Max tree depth | 6 |

Tournament size | 2 |

Elitism | 0.05% of population |

ERC probability | 0.1 (Integer 0.5) |

Crossover probability | 0.85 (High level 0.2, Low level 0.8) |

Mutation probability | 0.14 |

Probability of Pareto tournament | 0.7 |

Criteria | ${\mathit{a}}_{1}$ (Gene = 4, Depth = 4) | ${\mathit{a}}_{2}$ (Gene = 4, Depth = 3) | ${\mathit{a}}_{3}$ (Gene = 3, Depth = 6) | ${\mathit{f}}_{\mathit{i}}^{*}$ | ${\mathit{f}}_{\mathit{i}}^{-}$ |
---|---|---|---|---|---|

${\mathrm{f}}_{1}$ = RMSE | 1.9531 | 1.7602 | 1.5618 | 1.5618 | 1.9531 |

${\mathrm{f}}_{2}$ = MAE | 1.2022 | 1.2462 | 1.2843 | 1.2022 | 1.2843 |

${\mathrm{f}}_{3}=\mathrm{MAPE}/100$ | 0.0940 | 0.0941 | 0.0916 | 0.0916 | 0.0941 |

${\mathrm{f}}_{4}$ = R | 0.9091 | 0.9268 | 0.9429 | 0.9429 | 0.9091 |

${\mathrm{f}}_{5}=\mathrm{expr}.\mathrm{comp}.$ | 54 | 39 | 84 | 39 | 84 |

Criteria | ${\mathit{d}}_{\mathit{i}1}$ | ${\mathit{d}}_{\mathit{i}2}$ | ${\mathit{d}}_{\mathit{i}3}$ | ${\mathit{w}}_{\mathit{i}}{\mathit{d}}_{\mathit{i}1}$ | ${\mathit{w}}_{\mathit{i}}{\mathit{d}}_{\mathit{i}2}$ | ${\mathit{w}}_{\mathit{i}}{\mathit{d}}_{\mathit{i}3}$ |
---|---|---|---|---|---|---|

${\mathrm{f}}_{1}$ = RMSE | 1 | 0.5070 | 0 | 0.2 | 0.1014 | 0 |

${\mathrm{f}}_{2}$ = MAE | 0 | 0.5359 | 1 | 0 | 0.1072 | 0.2 |

${\mathrm{f}}_{3}=\mathrm{MAPE}$/100 | 0.96 | 1 | 0 | 0.192 | 0.2 | 0 |

${\mathrm{f}}_{4}$ = R | 1 | 0.4763 | 0 | 0.2 | 0.0953 | 0 |

${\mathrm{f}}_{5}=\mathrm{expr}.\mathrm{comp}.$ | 0.3333 | 0 | 1 | 0.06667 | 0 | 0.2 |

${\mathit{a}}_{\mathit{i}}$ | ${\mathit{R}}_{\mathit{i}}$ | ${\mathit{R}}_{\mathit{i}}^{\prime}$ | $\mathit{Q}{\mathit{S}}_{\mathit{i}}$ | $\mathit{Q}{\mathit{R}}_{\mathit{i}}$ | ${\mathit{Q}}_{\mathit{i}}$ |
---|---|---|---|---|---|

${a}_{1}$ | $0.658667$ | $0.204587$ | 1 | 1 | 1 |

${a}_{2}$ | $0.503858$ | $0.200304$ | 0.401514 | 0.401623 | 0.401568 |

${a}_{3}$ | 0.4 | 0.202 | 0 | 0 | 0 |

Model | RMSE | MAE | MAPE/100 | R |
---|---|---|---|---|

ANN | 2.9540 | 2.2760 | 0.1490 | 0.8070 |

Decision tree | 1.7131 | 1.1035 | 0.0766 | 0.9341 |

MGGP (gene = 3, depth = 6) | 1.5618 | 1.2843 | 0.0916 | 0.9429 |

MGGP (gene = 4, depth = 3) | 1.7602 | 1.2462 | 0.0941 | 0.9268 |

MGGP (gene = 4, depth = 4) | 1.9531 | 1.2022 | 0.0940 | 0.9091 |

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## Share and Cite

**MDPI and ACS Style**

Kovačević, M.; Antoniou, F.
Machine-Learning-Based Consumption Estimation of Prestressed Steel for Prestressed Concrete Bridge Construction. *Buildings* **2023**, *13*, 1187.
https://doi.org/10.3390/buildings13051187

**AMA Style**

Kovačević M, Antoniou F.
Machine-Learning-Based Consumption Estimation of Prestressed Steel for Prestressed Concrete Bridge Construction. *Buildings*. 2023; 13(5):1187.
https://doi.org/10.3390/buildings13051187

**Chicago/Turabian Style**

Kovačević, Miljan, and Fani Antoniou.
2023. "Machine-Learning-Based Consumption Estimation of Prestressed Steel for Prestressed Concrete Bridge Construction" *Buildings* 13, no. 5: 1187.
https://doi.org/10.3390/buildings13051187