Early Fast Cost Estimates of Sewerage Projects Construction Costs Based on Ensembles of Neural Networks

Citation: Juszczyk, M.; Hanák, T.; Výskala, M.; Pacyno, H.; Siejda, M. Early Fast Cost Estimates of Sewerage Projects Construction Costs Based on Ensembles of Neural Networks. Appl. Sci. 2023,13, 12744. https://doi.org/ 10.3390/app132312744 Academic Editor: Asterios Bakolas Received: 7 October 2023 Revised: 18 November 2023 Accepted: 24 November 2023 Published: 28 November 2023 Copyright: © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). applied sciences Article Early Fast Cost Estimates of Sewerage Projects Construction Costs Based on Ensembles of Neural Networks Michał Juszczyk 1,* , Tomáš Hanák2, Miloslav Výskala 2, Hanna Pacyno 1,3 and Michał Siejda 1 1Faculty of Civil Engineering, Cracow University of Technology, 31-155 Kraków, Poland; [email protected] or [email protected] (H.P.); [email protected] (M.S.) 2Faculty of Civil Engineering, Brno University of Technology, 602 00 Brno, Czech Republic; [email protected] (T.H.); [email protected].cz (M.V.) 3Datacomp IT sp. z o.o., 30-532 Kraków, Poland *Correspondence: [email protected] Featured Application: The potential applications of the research results, particularly the models utilizing ensembles of neural networks, offer the feasibility of early cost estimates for sewerage projects. The cost estimates of construction works, derived from the developed models, can be generated based on the essential features of sewerage projects that are accessible for analysis prior to the commencement of detailed design. Abstract: This paper presents research results on the development of an original cost prediction model for construction costs in sewerage projects. The focus is placed on fast cost estimates applicable in the early stages of a project, based on fundamental information available during the initial design phase of sanitary sewers prior to the detailed design. The originality and novelty of this research lie in the application of artificial neural network ensembles, which include a combination of several individual neural networks and the use of simple averaging and generalized averaging approaches. The research resulted in the development of two ensemble-based models, including five neural networks that were trained and tested using data collected from 125 sewerage projects completed in the Czech Republic between 2018 and 2022. The data included information relevant to various aspects of projects and contract costs, updated to account for changes in costs over time. The developed models present satisfactory predictive performance, especially the ensemble model based on simple averaging, which offers prediction accuracy within the range of ± 30% (in terms of percentage errors) for over 90% of the training and testing samples. The developed models, based on the ensembles of neural networks, outperformed the benchmark model based on the classical approach and the use of multiple linear regression. Keywords: sewerage project; sanitary sewer networks; construction costs; construction project; early cost estimates; fast cost estimates; neural networks ensembles; artificial intelligence 1. Introduction Sanitary sewerage systems, as a part of built infrastructure, are unquestionably of high significance for modern societies. The role of sewer networks is to collect and transfer wastewater from buildings (residential, commercial, or industrial) and all kinds of public and private establishments to a point of treatment and disposal. These systems are the results of construction projects aiming to either build new or renovate existing sewerages. Most of such projects—one may even dare to say that the overwhelming majority—are financed from public funds. Sewer projects, regarding their specificity, require careful analyses, design, and planning so that the technical requirements (especially flow capacity) are met. When planning is concerned, cost analyses are of key importance as completion of a project within a budget Appl. Sci. 2023,13, 12744. https://doi.org/10.3390/app132312744 https://www.mdpi.com/journal/applsci Appl. Sci. 2023,13, 12744 2 of 24 is one of the “hard” goals and key measures of any construction project success. Thus, there is a need for realistic cost predictions, analyses, and estimates, which make the goal achievable. These predictions should reflect the progress of the design process and available information, which evolves from basic and fundamental to definite and accurate. The estimates that play a specific role are called early estimates. These rely on basic information and some essential parameters of a project and are provided in the early stage of the design process. On the one hand, the expected accuracy is low; on the other hand, this is when the impact on the construction cost is great, as the essential choices and decisions for the project are made. In the construction industry, cost estimates play a pivotal role in project planning and execution. These estimates provide a closely approximated assessment of expected expenses, empowering project stakeholders to make well-informed decisions. The accuracy of cost estimates holds paramount importance in activities such as budgeting, securing financial resources, and ensuring the successful completion of construction projects. The precision of cost estimates can vary, spanning from rough order of magnitude estimates at the project’s conceptual inception to highly detailed assessments during the design and preconstruction phases. These estimates undergo continual refinement as additional projectspecific information becomes available. It is imperative to acknowledge that erroneous cost estimates can result in budget overruns, project delays, and disputes within the construction process. Hence, the development of precise, well-informed cost estimates emerges as a critical factor for the effective and economically viable execution of construction projects. Advances and progress in data sciences and artificial intelligence tools for processing information—especially for prediction problems—opened possibilities for the development of cost estimation methods based on the use of collected data, learning from experience, and knowledge generalization. Specifically, artificial neural networks (ANN) are tools that have significant capabilities that make them useful for early construction cost estimation; however, they are hardly reported in the literature to be applied in the case of sewerage projects. The aim of this work is to introduce a method and model of estimating construction costs for sewerage projects based on a specific artificial intelligence tool—namely ensembles of neural networks (later referred to as EoNN). The objective of the research, the results of which are presented herein, was to develop a model capable of predicting construction costs in the early stage of a sewerage project with satisfactory accuracy. For the purpose of model development, numerous sewerage construction projects involving the construction of new sections or the renovation of existing parts of the external gravity sewage network completed in the Czech Republic between 2018 and 2022 were analyzed. These projects served as a source of data for training artificial intelligence tools. The paper’s content includes concise literature and state-of-the-art review; a presentation of the research assumptions, data, and methods applied for the development of the fast cost estimation model; introduction of the model itself along with the results of the research; discussion of results along with comparison with a linear regression model as a benchmark; summary and conclusions. 2. Literature Review 2.1. Sewerage Construction Projects Management When considering sewerage construction projects, management problems become subjects of research and study, similar to other types of construction projects. Some noteworthy examples of general problems presented in the literature include the following: an optimization model for sewage rehabilitation aiming to achieve maximum effectiveness at the lowest cost, utilizing genetic algorithms [ 1 ]; a methodology for selecting and prioritizing sewerage projects within available funds and system capacity, based on dynamic programming principles [ 2 ]; a study on the risk of cost overruns in water and sewerage system construction projects [ 3 ]; the development of a new method to enhance the accuracy of Monte Carlo simulations and its validation in predicting the success likelihood of sewer- Appl. Sci. 2023,13, 12744 3 of 24 age build–operate–transfer projects, based on eight case studies [ 4 ]; research on culturally appropriate organization of projects implemented through public–private partnerships [ 5 ]; theoretical and empirical analysis of issues arising in public–private partnership projects within the sewerage sector [ 6 ]; and an investigation into delay factors in sewerage projects using simulations and a dynamic systems approach [7]. Cost-related challenges within sewerage projects constitute a distinct area of focus in the research literature. In one study [ 8 ], sewerage project cost estimates are examined using two alternative approaches. The first approach integrates component cost ranges and probability values established by a panel of estimators. In contrast, the second approach involves simulating costs based on random numbers, where component values are selected randomly within specified ranges. The study [ 9 ] introduces a model that relies on the utilization of prior information for the estimation of operational costs within sewerage systems. Specifically, the authors delve into the process of modeling prior information to furnish preliminary assessments of investment requirements. To facilitate subsequent estimation, the Bayes linear estimator was employed. Another research endeavor [ 10 ] centers on the intricacies of cost comparison in wastewater treatment. The authors delve into equitable methods for comparing and allocating costs in municipal sewage treatment concerning their structure and origin. Another study [ 11 ] conducts an in-depth analysis of factors responsible for variations and the resulting costs in sewerage construction projects. The work of [ 12 ] discloses research outcomes on benchmarking sewerage systems, particularly emphasizing the analysis of investment costs. This also involves an exhaustive examination of intangible variables such as economic fluctuations and tendering strategies and how they influence construction cost calculations. A notable contribution by [ 13 ] presents an Excel-based model capable of evaluating costs associated with sewerage system environmental impact. This model comprehensively assesses investigation, investment, design, operation, maintenance, supervision, and overall annual costs. The study seeks to provide a tool to facilitate environmentally informed decisions when selecting wastewater systems. The determination of capital costs for conventional sewerage systems in developing countries is scrutinized in [ 14 ]. The analysis involves the examination of unit construction costs expressed as ranges of capital cost values. Ref. [ 15 ] presents a literature review on the lifecycle costs of complete sanitation chain systems within developing cities. Moving forward, ref. [ 16 ] conducts an analysis of time–cost models that aid in forecasting project durations of different types. The research explores how construction technology influences the relationship between time and cost, particularly focusing on trenchless and open-cut technologies. In the realm of public–private partnership sewerage projects, ref. [ 17 ] delves into transaction costs. The study employs an exploratory multi-case study method to identify potential transaction costs within these projects. Lastly, ref. [ 18 ] addresses maintenance costs in sewer systems, particularly emphasizing cost estimation. The study underscores the significance of maintenance costs within the lifecycle of construction projects and proposes a linear regression model to facilitate sewer system maintenance cost estimation. The range of problems presented above confirms that the costs and cost management of sewerage construction projects are of significant interest and importance from a research standpoint. 2.2. Cost-Estimating Models for Sewerage Construction Projects In the context of the focus of this paper, the most crucial aspects are the attempts to develop models that assist in estimating construction costs for sewerage projects. The early work [ 19 ] investigated the application of nonlinear regression for sewer cost modeling. The study focused on estimating the empirical parameters within separable and generalized cost functions. To model the sewer cost function, the applied technique required the optimization of the values of the nonlinear parameters. The two types of analyzed nonlinear parametric cost models are reported to exhibit relative insensitivity to minor errors in the estimated values of their model parameters. The development of cost functions for Appl. Sci. 2023,13, 12744 4 of 24 open-cut and jacking methods in sanitary sewer system construction is the subject of another work [ 20 ]. The research resulted in the formulation of cost functions applicable to open-cut and jacking methods, which are construction techniques for sewer systems. These cost functions were derived using linear regression and expressed as functions of pipe size and excavation depth. The derived functions were validated using data from several actual sewer system construction projects to verify the accuracy of cost predictions. Another work [ 21 ] presents nonlinear unit cost functions for estimating costs associated with elements of waterborne sewer infrastructure, including gravity pipes, rising mains, pump stations, and wastewater treatment facilities. As a result, a model that combines several cost functions was developed to predict the unit cost of various sewer elements. Modeling the costs related to sewer systems has also been explored in the study [ 22 ]. The approach outlined in this research relies on the utilization of multiple linear regression techniques. The authors devised and validated cost functions that pertain to various components of sewer systems, including gravity and rising pipes, manholes, and pumping stations. The costs are delineated as functions of the principal physical attributes of these components. The process of estimating the cost functions involved the application of multiple linear regression analysis. In another work [ 23 ], a parametric approach for modeling the construction costs of sewer systems is presented. The authors aimed to establish an initial cost model on a municipal level, with population size serving as the primary variable for the cost functions. By maintaining population size as an independent factor, an empirical correlation has been deduced between population size and the expenses associated with sewerage systems. Various forms of cost functions were experimented with, encompassing both linear and nonlinear formulations. The matter of early cost estimates for sewerage lines is also present in [ 24 ]. The authors employed regression analysis to formulate models for predicting early-stage costs. The study devised models grounded in linear regression, featuring the technical attributes of sewerage lines as independent variables and the estimated costs as the dependent variable. It can be observed that several works are based on an approach in which the form of the cost function is assumed ex-ante. Both linear and nonlinear functions have been employed to model the costs of sewer systems; however, linear regression appears to be the most popular tool among researchers. Nonetheless, there are works that present attempts to employ artificial neural networks for the purpose of cost estimates in sewerage projects. In [ 25 ], a neural network is utilized as the cornerstone of a cost-estimating model designed for a budget estimation system focused on repair and/or replacement costs of sewer and water projects. The model incorporates 23 project-related factors, derived through Pareto analysis, as input variables (independent variables), while the budget estimate serves as the output (dependent variable). The authors emphasize that the proposed model not only saves time but also enhances the precision of estimates, offering clients a means to compare cost alternatives and facilitating decision-making processes in cases involving the rehabilitation of sewer and water systems. Similar work [ 26 ] deals with the problems of conceptual cost estimating for water supply and sewerage projects. This paper presents a backpropagation neural network-based model that is supposed to assist municipal authorities in the development of more accurate cost estimates for their water supply and sewer projects. Cost predictors, representing project technical parameters and serving as the model’s input, were identified on the basis of contractors’ bids analysis and covered 80% of construction work costs. The benefits of the presented model include but are not limited to, better utilization of financial resources, the provision of decision-making guidelines, and the ability to compare alternatives. Additionally, the authors claim that the model fulfills the needs of funding entities for more accurate cost estimates. In comparison to the parametric approach, modeling costs using neural networks eliminates the need for assumptions about the equation that binds the independent variables of the models to the cost, which serves as the dependent variable. Appl. Sci. 2023,13, 12744 5 of 24 2.3. Applications of Neural Networks and Ensembles of Neural Networks for Construction Management Problems and Cost Estimation in Construction Artificial neural networks (ANN) are a subset of artificial intelligence tools inspired by the learning and knowledge storage patterns observed in neurobiology. They can be employed to address various classification or regression challenges. The concept and theory of neural networks have been extensively discussed in numerous works [ 27 – 30 ]. Neural networks possess the capacity to process data with the aim of uncovering concealed patterns. The procedure of data processing to acquire knowledge, referred to as training, is executed through specific algorithms. Following the training phase, these networks are anticipated to possess the ability to generate predictions for novel data that were not utilized in the training process. The ability to generalize knowledge is a key attribute of artificial neural networks, rendering them valuable for a range of engineering problems. Particularly, ANN has found application in addressing cost-related challenges within the construction sector. An exemplary illustration of this is a fuzzy neural network model aimed at aiding contractors in estimating and selecting a suitable markup [ 31 ]. Investigative efforts in other work [ 32 ] centered on the evaluation of multilayer perceptron and general regression neural networks for their potential in early cost estimation for road tunnel projects. In [ 33 ], the outcomes from the utilization of general regression neural networks to predict maintenance costs associated with construction equipment are shared. Another work [ 34 ] delved into the utilization of multilayer perceptron neural networks to estimate building construction costs during the initial design phase. In [ 35 ], research results on optimizing both the cost and timeline of construction projects through the implementation of neural networks are introduced. A hybrid approach, combining multivariate regression and multilayer perceptron neural networks, was employed in another research [ 36 ] to estimate capital costs for earthmoving, loading, and unloading equipment. There are also some interesting works that explore the utilization of ANN and machine learning in the analysis of wind speed and wind direction [ 37 , 38 ], settlement prediction [ 39 ], and the assessment of their impact on existing structures, that is, bridges and metro, respectively. Ensembles (also called committees) of neural networks (EoNN) have their origins in the realm of ensemble learning systems. The foundational principles of this approach can be traced back to earlier referenced works that comprehensively delve into neural network concepts [ 28 , 30 ], as well as works dedicated to the study of ensembles [ 40 ]. EoNN consists of individual trained artificial neural networks (ANN), each providing predictions that are subsequently aggregated, with the aim of reducing errors in comparison to standalone neural networks. The utilization of neural network ensembles within classification and regression models, as opposed to employing standalone neural networks, is anticipated to yield enhanced performance and precision [41]. Engineering applications that utilize EoNN encompass a range of engineering challenges. Some noteworthy examples include predicting the performance of substantial construction equipment, specifically tunnel boring machines [ 42 ], day-ahead electricity load forecasting for buildings [ 43 ], or forecasting heating energy consumption [ 44 ]. In the context of structural engineering, ensemble models are reported to be used for predicting high-performance concrete compressive strength [ 45 ] and identifying structural damage [ 46 ]. Finally, an example of an EoNN application for risk analysis in the maintainability of high-rise buildings in specific tropical conditions [47] can be provided. Due to the distinctive capabilities and advantages of EoNN, their exploration within the domain of construction cost analysis is increasingly reported for different types of projects. Attempts at developing models capable of aiding various cost analyses using an ensemble approach have been reported in recent years. In the study [ 48 ], the development of a model to assist in predicting project cost and schedule success by utilizing early planning status as inputs is presented. The results obtained validate that the proposed artificial intelligence models yield satisfactory predictive outcomes. Another publication [ 49 ] explores the application of EoNN for Macro BIM cost estimates. This research develops estimation models for the structural frames of building floors, demonstrating Appl. Sci. 2023,13, 12744 6 of 24 satisfactory accuracy. Authors of [ 50 ] center their attention on cost prediction for a specific category of objects—sports fields. The construction cost forecasting model based on EoNN is proven to outperform linear regression and models relying on single neural networks. Furthermore, an analysis of estimate errors and accuracy establishes the applicability of the proposed model in the early stages of construction projects. In [ 51 ], the text delves into predicting the construction costs of buildings’ structural elements with the use of artificial intelligence tools. The introduced models are, among others, based on multiple artificial neural networks combined into an ensemble. The EoNN-based models meet expectations for knowledge generalization and the accurate prediction of buildings’ structural frames. Furthermore, an ensemble algorithm application [ 52 ] is also employed to predict the cost of highway construction projects. The study presents a model that employs artificial intelligence tools—neural networks included—in a stacking ensemble model for cost prediction. 2.4. Literature Review Summary A literature review allows for a justifiable assumption that EoNN, when applied as the core of a construction cost estimation model for a specific type of construction object, will yield better results compared to models based on linear regression or single neural networks. On the other hand, works reporting the application of EoNN for predicting construction costs in the context of sewerage projects have not been found thus far. This paper aims to address this gap. 3. Methodology Regarding public works tenders, the investor is required to disclose the expected value of the contract [ 53 ]. Additionally, information about the approximate value of the sewerage project is crucial during the design phase to select the optimal solution, not only from a technical standpoint but also from an economic perspective. Estimating the value of construction works poses a significant challenge, particularly when detailed project documentation is unavailable and only basic data and parameters are known. These estimates are typically provided by cost engineers and often rely on the use of technical–economic indicators, which may result in significantly inaccurate estimates [ 54 ]. The starting point of the research was the idea of a cost prediction model capable of providing estimates for sewerage construction projects utilizing information about the project available in the early design phase. The following note aims to provide a concise overview of the broader context of the research. The construction industry in the Czech Republic is a vital sector of the country’s economy, contributing to infrastructure development, residential and commercial building projects, and employment opportunities. The industry has witnessed steady growth and modernization in recent years. The construction sector plays a significant role in the country’s economy. It contributes to GDP and provides jobs for a considerable portion of the workforce. In the context of the research presented herein, it is worth mentioning that infrastructure development, which encompasses sewerage construction projects, in the Czech Republic reflects the broader European trend of modernization, sustainable development, and ecology. It continues to play a pivotal role in the country’s development. In the Czech Republic, the percentage of the population supplied with water from the public water supply reached 94.6% in 2020. In the case of connection to the sewage system, this percentage reached a value of 86.1% [ 55 ]. Although this figure may appear satisfactory, in reality, sewerage systems are readily available in larger agglomerations, and significant gaps exist in smaller settlements. It is worth noting that construction works related to sewers encompass not only the renovation of old networks and the establishment of new networks for new buildings but also the expansion of sewerage systems in already existing built-up areas. The research’s general ideogram is depicted in Figure 1, illustrating the successive steps taken by the authors. Appl. Sci. 2023,13, 12744 7 of 24 Appl. Sci. 2023, 13, x FOR PEER REVIEW 7 of 25 gaps exist in smaller settlements. It is worth noting that construction works related to sewers encompass not only the renovation of old networks and the establishment of new networks for new buildings but also the expansion of sewerage systems in already existing built-up areas. The research’s general ideogram is depicted in Figure 1, illustrating the successive steps taken by the authors. Figure 1. General ideogram and scheme of the research. On the basis of the steps covering the state-of-the-art and literature review, as well as analyses of sewerage construction projects completed in the Czech Republic between 2018 and 2022, an initial set of cost predictors (potential independent variables serving as input for the EoNN-based model to be developed) was proposed. As the research focused on projects aimed at constructing new sections or upgrading existing sections of external gravity sewage networks, the cost predictors were expected to reflect the specificity of such construction projects. The analyzed projects involved separated sewer systems, with wastewater and stormwater runoffs in separate pipes. (An important point to note is that the analyzed projects did not incorporate combined runoffs; that is, the runoffs for wastewater and stormwater in a single pipe). Despite the fact that wastewater systems are connected to the existing sewage treatment plant, the plants themselves were not part of the analyzed projects, and thus, no related cost predictors were considered. The mentioned set of cost predictors is presented in Table 1, which includes selected types of cost predictors and descriptions of the information that is supposed to be input into the model (this information is succinctly explained in the table). Moreover, the table presents raw values of cost predictors, as they were collected before pre-processing, ordering, and scaling. Figure 1. General ideogram and scheme of the research. On the basis of the steps covering the state-of-the-art and literature review, as well as analyses of sewerage construction projects completed in the Czech Republic between 2018 and 2022, an initial set of cost predictors (potential independent variables serving as input for the EoNN-based model to be developed) was proposed. As the research focused on projects aimed at constructing new sections or upgrading existing sections of external gravity sewage networks, the cost predictors were expected to reflect the specificity of such construction projects. The analyzed projects involved separated sewer systems, with wastewater and stormwater runoffs in separate pipes. (An important point to note is that the analyzed projects did not incorporate combined runoffs; that is, the runoffs for wastewater and stormwater in a single pipe). Despite the fact that wastewater systems are connected to the existing sewage treatment plant, the plants themselves were not part of the analyzed projects, and thus, no related cost predictors were considered. The mentioned set of cost predictors is presented in Table 1, which includes selected types of cost predictors and descriptions of the information that is supposed to be input into the model (this information is succinctly explained in the table). Moreover, the table presents raw values of cost predictors, as they were collected before pre-processing, ordering, and scaling. Appl. Sci. 2023,13, 12744 8 of 24 Table 1. Initial set of cost predictors. Cost Predictor Input Information Description Value Type of project New construction or renovation Descriptive Sewer pipe’s length Size of a project, complexity of a project, quantity of works Numerical (length) Type of sewer pipe material Technical parameter, material parameter, applied solution Descriptive Sewer pipe’s diameter Technical parameter, applied solution, capacity of a sewer Descriptive Average depth of trench Technical parameter, temporary works, safety issues Numerical (depth) Groundwater table level Technical parameter, ground conditions, temporary works, safety issues Descriptive Class of soil Technical parameter, ground conditions parameter, safety issues Descriptive Number of manholes Size of a project, complexity of a project, safety issues Numerical (count) Number of crossings with other services Complexity of a project, temporary works, safety issues Numerical (count) Works on unpaved surface Conditions of works, complexity of works, soil type waste production, quantity of works Numerical (length) Works on paved surface Conditions of works, complexity of works, rubble type waste production, quantity of works Numerical (length) Debris removal distance Conditions of works, waste management parameter Numerical (length) In the next step, data for the purposes of the cost prediction model were collected. This step involved the analysis of technical and design documentation, quantity surveys, cost estimates, as well as public client queries for 135 sewerage projects. This provided data reflecting the raw values of cost predictors (as presented in Table 1), along with real-life contract net costs (excluding value-added tax) of sewerage project construction works. It is noteworthy that, through the systematic analysis of sewerage construction projects (which constituted the second phase of the research), it was observed that the previously mentioned type of runoff, whether wastewater or stormwater, did not impact the costs. From both technological and construction cost perspectives, it does not matter which type of runoff is being constructed. Therefore, this information is excluded as a predictor of costs. Due to changes in the value of money and cost variability over time, the values of contract net costs were updated for the end of the first half of the year 2023. The updated rule is provided below. UCC =ACCt· n ∏ i=t CIi(1) where: UCC—updated contract net cost of a sewerage project; ACC t —actual contract net cost of sewerage projects, which was awarded in the t-th half-year period between the beginning of 2018 and the end of 2022; CI i —cost index for i-th half-year period between the beginning of 2018 and the first half of the 2023 year; n—stands for the first half of the 2023 year. Cost index values that were used to update contract net costs are published periodically by a Czech company, RTS ® , a developer and provider of a price information system for Appl. Sci. 2023,13, 12744 9 of 24 the construction industry in the Czech Republic. (It is worth mentioning that there are two main pricing systems that provide various cost information for construction cost estimation practice in the Czech Republic. These are RTS ® and URS ® .) Based on the structural and material characteristics of sewerage systems, the cost indexes used for calculations were derived from the RTS ® system. More specifically, price indicators that reflect the changes in costs in sewerage construction projects between 2018 and 2023 were used. The obtained cost indexes were also compared with the second pricing system, URS ® , to verify their correctness and applicability. In the Czech Republic, this procedure is commonly used for indexing the prices of construction works between different time periods and is also permissible for the needs of court evidence. Outlier analysis was applied to the updated values of contract net costs for sewerage project construction works. The concept and fundamentals of outlier analysis can be found in the statistical literature, such as [ 56 – 58 ]. The purpose was to eliminate data points that deviated significantly from others, essentially excluding unusual cost values from the dataset. The approach used in this study relied on the concept of the interquartile range (IQR). The IQR was calculated as the difference between the values of the third quartile (Q3) and the first quartile (Q1), representing the range of values between these quartiles: IQR =Q3 − Q1. The rule below allowed us to identify and eliminate outlying values, as well as entire records for certain project cases from the dataset: • If the j-th value (in the j-th record in the dataset) of updated contract net costs does not belong to the range: <Q1 − 1.5 · IQR;Q3+1.5 · IQR> → eliminate the j-th record from the dataset. The rationale for this approach, based on the authors’ prior experiences, is that outliers can be problematic when developing cost prediction models for various construction projects, facilities, and structures, as they often represent specific, high-cost projects that are sparsely represented in datasets. Such data can distort the predictive performance of a developed model. The collected data underwent further pre-processing. Numerical values of cost predictors were linearly scaled, while descriptive values were processed differently based on their nature; they were pseudo-fuzzy scaled, coded as one-of-n values, or converted into binary values. Detailed information and outcomes of this step are presented in Section 4. The next stage of the research involved computations and simulations of artificial neural networks (ANN), as well as the combination of these networks to create an ensemble. The development of models based on ensembles of neural networks (EoNN) designed to predict costs involves solving regression problems. Let the dependent variable yof such models represent the construction cost for a sewerage project. Also, let the independent variables be the cost predictors, with the vector of these variables denoted as x . The EoNNbased models are expected to approximate the mapping x→ y. Importantly, in the case of employing EoNN, the approximation function his implicitly defined as follows: y=h(x)+ε(2) where ε corresponds to the prediction error. Consequently, the formal notation for predicting costs ˆ ycan be expressed as follows: ˆ y=h(x)(3) The utilization of EoNN as the core of a cost estimation model relies on combining a set of trained neural networks to form an ensemble. As outlined in [ 28 ], this set might encompass various types of networks or similar networks trained to different local minima. The two methods built on this premise, which were applied during research presented herein, are (1) ensemble averaging and (2) generalized averaging. A summary of the core principles behind the two ensemble-based methods mentioned is provided based on [ 28 , 30 ]. The general idea is schematically depicted in Figure 2. Appl. Sci. 2023,13, 12744 16 of 24 For GAV ENS, computations relied on Equations (8) and (9) and required more effort. The weights for combining outputs provided by the members of the ensemble are presented below. α1= 0.109; α2= 0.242; α3=−0.185; α4= 0.170; α5= 0.664 For both SAV ENS and GAV ENS models, further computations based on the respective weights αk and Equation (5) were conducted to obtain level-1 predictions of sewerage project construction costs. Figures 3and 4present scatter plots that represent real-life values of updated construction costs of sewer projects yand, on the contrary, values predicted by the developed models ˆ y. Scatter plots, which are alternatively referred to as scattergrams or scatter charts, serve the purpose of providing a visual representation of data points within a two-dimensional Cartesian coordinate system. Each data point is depicted as a point or dot, facilitating the observation of the relationship between the actual values yand the predicted values ˆ y. In essence, each data point on the plot corresponds to a pair of associated values. Through the utilization of scatter plots, the assessment of correlations between real-world values and values predicted by a model, as well as the evaluation of prediction quality, is carried out; Appl. Sci. 2023, 13, x FOR PEER REVIEW 17 of 25 y and the predicted values ŷ. In essence, each data point on the plot corresponds to a pair of associated values. Through the utilization of scatter plots, the assessment of correlations between real-world values and values predicted by a model, as well as the evaluation of prediction quality, is carried out; Figures 3 and 4 depict results for SAV ENS and GAV ENS models, respectively. The points in the graphs represent the results of the training (L&V subset) and testing (T subset) processes. The distribution of points indicates that, in general, the quality of cost prediction is comparable for both ensemble-based models. There are no significant deviations, and the points are distributed along the lines of a perfect fit; On the basis of real-life values y and values predicted by the developed models ŷ as well as Equation (10), correlation coefficients R were computed; • For SAV ENS : R = 0.976 for the L&V subset and R = 0.988 for T subset; • For GAV ENS : R = 0.972 for the L&V subset and R = 0.987 for T subset; Correlation of y and ŷ is very high, and no significant differences between the models can be identified. Figure 3. Scatter plot of real-life values y and predicted values ŷ of costs for SAV ens model. (a) L&V subset, (b) T subset. Figure 4. Scatter plot of real-life values y and predicted values ŷ of costs for GAV ens model. (a) L&V subset, (b) T subset. Figures 5 and 6 present distributions of percentage errors PE p , computed using Equation (15) and categorized within the ranges shown on the horizontal axes for the SAV ENS and GAV ENS models, respectively; Figure 3. Scatter plot of real-life values yand predicted values ˆ yof costs for SAV ens model. ( a )L&V subset, (b)Tsubset. Appl. Sci. 2023, 13, x FOR PEER REVIEW 17 of 25 y and the predicted values ŷ. In essence, each data point on the plot corresponds to a pair of associated values. Through the utilization of scatter plots, the assessment of correlations between real-world values and values predicted by a model, as well as the evaluation of prediction quality, is carried out; Figures 3 and 4 depict results for SAV ENS and GAV ENS models, respectively. The points in the graphs represent the results of the training (L&V subset) and testing (T subset) processes. The distribution of points indicates that, in general, the quality of cost prediction is comparable for both ensemble-based models. There are no significant deviations, and the points are distributed along the lines of a perfect fit; On the basis of real-life values y and values predicted by the developed models ŷ as well as Equation (10), correlation coefficients R were computed; • For SAV ENS : R = 0.976 for the L&V subset and R = 0.988 for T subset; • For GAV ENS : R = 0.972 for the L&V subset and R = 0.987 for T subset; Correlation of y and ŷ is very high, and no significant differences between the models can be identified. Figure 3. Scatter plot of real-life values y and predicted values ŷ of costs for SAV ens model. (a) L&V subset, (b) T subset. Figure 4. Scatter plot of real-life values y and predicted values ŷ of costs for GAV ens model. (a) L&V subset, (b) T subset. Figures 5 and 6 present distributions of percentage errors PE p , computed using Equation (15) and categorized within the ranges shown on the horizontal axes for the SAV ENS and GAV ENS models, respectively; Figure 4. Scatter plot of real-life values yand predicted values ˆ yof costs for GAV ens model. ( a )L&V subset, (b)Tsubset. Appl. Sci. 2023,13, 12744 17 of 24 Figures 3and 4depict results for SAV ENS and GAV ENS models, respectively. The points in the graphs represent the results of the training (L&V subset) and testing (Tsubset) processes. The distribution of points indicates that, in general, the quality of cost prediction is comparable for both ensemble-based models. There are no significant deviations, and the points are distributed along the lines of a perfect fit; On the basis of real-life values yand values predicted by the developed models ˆ yas well as Equation (10), correlation coefficients Rwere computed; •For SAVENS:R= 0.976 for the L&V subset and R= 0.988 for Tsubset; •For GAVENS:R= 0.972 for the L&V subset and R= 0.987 for Tsubset; Correlation of yand ˆ yis very high, and no significant differences between the models can be identified. Figures 5and 6present distributions of percentage errors PE p , computed using Equation (15) and categorized within the ranges shown on the horizontal axes for the SAV ENS and GAVENS models, respectively; Appl. Sci. 2023, 13, x FOR PEER REVIEW 18 of 25 An analysis of the PE p distributions allows us to select the SAV ENS model as the one that is slightly more stable when comparing training and testing errors. The shares of PE p within the range <−30%; 30%> were as follows: • For SAV ENS : 90.6% for the L&V subset and 94.7% for the T subset; • For GAV ENS : 84.0% for the L&V subset and 94.7% for the T subset. Table 9 presents a summary and performance measures of the two developed models, specifically RMSE and MAPE values, as well as the maximum values of APE p . The maximum values of APE p , which are lower for the SAV ENS model, confirm that its predictive performance is slightly better than that of the GAV ENS model. Table 9. Values of general predictive performance measures for the EoNN models. EoNN RMSE L&V RMSE T MAPE L&V MAPE T max {APE PL&V } max {APE PT } SAV ENS 436.8 344.5 15.1% 9.3% 57.6% 29.6% GAV ENS 266.3 257.5 14.9% 9.9% 64.6% 31.3% In general, it can be concluded that the obtained results are satisfactory. The developed predictive models provide cost estimates within the assumed and preferred range of accuracy for the majority of both training and, most importantly, testing cases. The model based on simple averaging performs slightly better and offers higher accuracy. Figure 5. Distribution of PE p errors for SAV ens model. (a) L&V subset, (b) T subset. Figure 5. Distribution of PEperrors for SAVens model. (a)L&V subset, (b)Tsubset. Appl. Sci. 2023,13, 12744 18 of 24 Appl. Sci. 2023, 13, x FOR PEER REVIEW 19 of 25 Figure 6. Distribution of PE p errors for GAV ens model. (a) L&V subset, (b) T subset. 6. Discussion As evident from the literature analysis, attempts to develop cost analysis models for sewerage projects have been made [19–26]. In comparison to the analyses presented in this article, it can be noted that the selection of cost predictors, in terms of their nature, is similar. However, it should not be forgotten that local construction market conditions and data availability also influence the research. The final set of cost predictors and their values strongly depend on the possibility of obtaining them, which results in some differences between the models presented in the literature. Among the mentioned works, some are based on the use of ANN [25,26]. Unfortunately, it is challenging to compare the results of these studies with the findings of this research. The cited works used less data for training and testing, and the models are based on single networks. Most importantly, there is a lack of precise information about the types of neural networks used and essential details regarding the training and testing processes of ANN and the analysis of their performance. Thus, it was decided, for the purpose of further assessing research results, to develop a benchmark model based on multiple regression using the classical least square method. The benchmark model is hereinafter referred to as MR. The general formula for predictions based on the MR model is provided below. 𝑦=𝛽+𝛽𝑥  (16) where: Figure 6. Distribution of PEperrors for GAVens model. (a)L&V subset, (b)Tsubset. An analysis of the PE p distributions allows us to select the SAV ENS model as the one that is slightly more stable when comparing training and testing errors. The shares of PE p within the range <−30%; 30%> were as follows: •For SAVENS: 90.6% for the L&V subset and 94.7% for the Tsubset; •For GAVENS: 84.0% for the L&V subset and 94.7% for the Tsubset. Table 9presents a summary and performance measures of the two developed models, specifically RMSE and MAPE values, as well as the maximum values of APE p . The maximum values of APE p , which are lower for the SAV ENS model, confirm that its predictive performance is slightly better than that of the GAVENS model. Table 9. Values of general predictive performance measures for the EoNN models. EoNN RMSEL&V RMSETMAPEL&V MAPETmax {APEPL&V} max {APEPT} SAVENS 436.8 344.5 15.1% 9.3% 57.6% 29.6% GAVENS 266.3 257.5 14.9% 9.9% 64.6% 31.3% Appl. Sci. 2023,13, 12744 19 of 24 In general, it can be concluded that the obtained results are satisfactory. The developed predictive models provide cost estimates within the assumed and preferred range of accuracy for the majority of both training and, most importantly, testing cases. The model based on simple averaging performs slightly better and offers higher accuracy. 6. Discussion As evident from the literature analysis, attempts to develop cost analysis models for sewerage projects have been made [ 19 – 26 ]. In comparison to the analyses presented in this article, it can be noted that the selection of cost predictors, in terms of their nature, is similar. However, it should not be forgotten that local construction market conditions and data availability also influence the research. The final set of cost predictors and their values strongly depend on the possibility of obtaining them, which results in some differences between the models presented in the literature. Among the mentioned works, some are based on the use of ANN [ 25 , 26 ]. Unfortunately, it is challenging to compare the results of these studies with the findings of this research. The cited works used less data for training and testing, and the models are based on single networks. Most importantly, there is a lack of precise information about the types of neural networks used and essential details regarding the training and testing processes of ANN and the analysis of their performance. Thus, it was decided, for the purpose of further assessing research results, to develop a benchmark model based on multiple regression using the classical least square method. The benchmark model is hereinafter referred to as MR. The general formula for predictions based on the MR model is provided below. ˆ y=β0+∑ j βjxj(16) where: β0,βj—regression coefficients. To ensure the comparability of the MR benchmark model with the models based on the EoNN approach developed during the research, the computation of coefficients β0 and βj was performed with the use of subset C, equivalent to the training subset for ANNs that became members of SAV ENS and GAV ENS (including cases used for learning and validation processes). For testing the MR model, subset Twas used. Below are the coefficients obtained from the regression analysis, along with the standard errors of estimation provided in the brackets. β0=−5588.5 (1420.85); β1= 1064.6 (376.12); β2=−39,347.1 (62,376.80); β3= 1414.0 (1025.67); β4= 2416.5 (727.67); β5= 2821.5 (548.23); β6= 30.8 (235.41); β7= 240.0 (358.94); β8= 3329.0 (763.03); β9= 1340.1 (635.74); β10 = 38,154.7 (53,700.36); β11 = 18,397.4 (20,636.69); β12 = 36.9 (449.73) Figure 7displays the results for the MR model in the form of a scatter plot of yand ˆ y values (compare with Figures 3and 4). Correlation coefficients Rwere computed in a similar manner as in the case of the SAVENS and GAVENS models; •For MR: R= 0.926 for the Csubset and R= 0.940 for the Tsubset; When compared to the correlations computed for the developed EoNN-based models, the differences that occur are relatively insignificant. However, an analysis of the scatter plots and a comparison with those presented for EoNN-based models reveal greater dispersion and deviations from the line of perfect fit in the case of the MR benchmark model; Appl. Sci. 2023,13, 12744 20 of 24 An analysis of the PE p distributions for the MR benchmark model reveals the superiority of EoNN-based models. For MR, the shares of PE p within the range < − 30%; 30%> were as follows: •68.9% for the Csubset and 73.7% for the Tsubset; Table 10 provides a summary and performance measures for the MR benchmark model (compare with Table 9). Appl. Sci. 2023, 13, x FOR PEER REVIEW 20 of 25 β 0 , β j —regression coefficients. To ensure the comparability of the MR benchmark model with the models based on the EoNN approach developed during the research, the computation of coefficients β 0 and β j was performed with the use of subset C, equivalent to the training subset for ANNs that became members of SAV ENS and GAV ENS (including cases used for learning and validation processes). For testing the MR model, subset T was used. Below are the coefficients obtained from the regression analysis, along with the standard errors of estimation provided in the brackets. β 0 = −5588.5 (1420.85); β 1 = 1064.6 (376.12); β 2 = −39 , 347.1 (62 , 376.80); β 3 = 1414.0 (1025.67); β 4 = 2416.5 (727.67); β 5 = 2821.5 (548.23); β 6 = 30.8 (235.41); β 7 = 240.0 (358.94); β 8 = 3329.0 (763.03); β 9 = 1340.1 (635.74); β 10 = 38 , 154.7 (53 , 700.36); β 11 = 18 , 397.4 (20 , 636.69); β 12 = 36.9 (449.73) Figure 7 displays the results for the MR model in the form of a scatter plot of y and ŷ values (compare with Figures 3 and 4). Correlation coefficients R were computed in a similar manner as in the case of the SAV ENS and GAV ENS models; • For MR: R = 0.926 for the C subset and R = 0.940 for the T subset; When compared to the correlations computed for the developed EoNN-based models, the differences that occur are relatively insignificant. However, an analysis of the scatter plots and a comparison with those presented for EoNN-based models reveal greater dispersion and deviations from the line of perfect fit in the case of the MR benchmark model; An analysis of the PE p distributions for the MR benchmark model reveals the superiority of EoNN-based models. For MR, the shares of PE p within the range <−30%; 30%> were as follows: • 68.9% for the C subset and 73.7% for the T subset; Table 10 provides a summary and performance measures for the MR benchmark model (compare with Table 9). Table 10. Values of general predictive performance measures for the MR model. RMSE C RMSE T MAP E C MAPE T max {APE PC } max {APE PT } MR 647.04 743.31 29.5% 19.3% 81.9% 111.5% Figure 7. Scatter plot of real-life values y and predicted values ŷ of costs for benchmark MR model. (a) C subset, (b) T subset. Figure 7. Scatter plot of real-life values yand predicted values ˆ yof costs for benchmark MR model. (a)Csubset, (b)Tsubset. Table 10. Values of general predictive performance measures for the MR model. RMSECRMSETMAPECMAPETmax {APEPC} max {APEPT} MR 647.04 743.31 29.5% 19.3% 81.9% 111.5% Figure 8depicts the distributions of percentage errors PE p , similar to those presented for the SAVENS and GAVENS models (compare Figures 5and 6). A comparison of the values in Tables 9and 10 reveals that the overall prediction performance of the MR model based on linear regression is weaker than the performance of EoNN-based models (specifically, the SAV ENS and the GAV ENS models) developed by this study for the purposes of sewerage projects construction costs prediction. As described in the preceding section, the research results are satisfactory. The models developed using EoNN have demonstrated their ability to predict construction costs for sewerage projects, for either stormwater runoffs or wastewater runoffs, with acceptable accuracy, meeting the requirements of the expected range of errors in over 90% of the test cases. The SAV ENS and GAV ENS models exhibit improved predictive capabilities when compared to individual ANNs, selected to be the ensemble members, used as standalone models. This improvement can be attributed to the effective compensation of errors inherent in single ANNs when they are combined within the ensemble. Moreover, combining several ANNs provides more objective cost predictions, as a certain bias of single ANNs acting in isolation is inevitable due to the size of the training set used in the course of research. The two applied approaches, specifically simple averaging for the SAV ENS model and generalized averaging for the GAV ENS model, differ in terms of the computational effort required to determine the weights of the member ANNs. The first approach is straightforward—in fact, it involves taking the arithmetical average of predictions as the EoNN output. In contrast, the second approach demands more effort. However, this can be Appl. Sci. 2023,13, 12744 21 of 24 efficiently accomplished using a calculation sheet or through programming and automation of computations. The generalized averaging approach, with its weight optimization, allows for a more nuanced differentiation of the influence of individual member ANNs on the final cost prediction. In summary, both approaches are user-friendly and easily applicable. Appl. Sci. 2023, 13, x FOR PEER REVIEW 21 of 25 Figure 8 depicts the distributions of percentage errors PE p , similar to those presented for the SAV ENS and GAVE NS models (compare Figures 5 and 6). Figure 8. Distribution of PE p errors for benchmark MR model. (a) C subset, (b) T subset. A comparison of the values in Tables 9 and 10 reveals that the overall prediction performance of the MR model based on linear regression is weaker than the performance of EoNN-based models (specifically, the SAV ENS and the GAV ENS models) developed by this study for the purposes of sewerage projects construction costs prediction. As described in the preceding section, the research results are satisfactory. The models developed using EoNN have demonstrated their ability to predict construction costs for sewerage projects, for either stormwater runoffs or wastewater runoffs, with acceptable accuracy, meeting the requirements of the expected range of errors in over 90% of the test cases. The SAV ENS and GAV ENS models exhibit improved predictive capabilities when compared to individual ANNs, selected to be the ensemble members, used as standalone models. This improvement can be attributed to the effective compensation of errors inherent in single ANNs when they are combined within the ensemble. Moreover, combining several ANNs provides more objective cost predictions, as a certain bias of single ANNs acting in isolation is inevitable due to the size of the training set used in the course of research. The two applied approaches, specifically simple averaging for the SAV ENS model and generalized averaging for the GAV ENS model, differ in terms of the computational effort required to determine the weights of the member ANNs. The first approach is straightforward—in fact, it involves taking the arithmetical average of predictions as the EoNN Figure 8. Distribution of PEperrors for benchmark MR model. (a)Csubset, (b)Tsubset. The key advantage of an ensemble-based approach, as affirmed by the research presented, lies in the efficient utilization of training and testing efforts across ANNs rather than focusing solely on a single network. This observation carries particular significance in the contemporary context, where rapid progress in computer technology and the availability of efficient software enables the exploration of numerous networks within a relatively short timeframe. It is also essential to acknowledge the limitations of the developed models. Firstly, the models are explicitly tailored to the specific local conditions. This is a consequence of the evident fact that the data used for training and testing were collected in the Czech Republic. However, it is worth noting that the application of the proposed approach in other locations is feasible, making the conceptual framework broadly applicable. The second significant limitation stems from the data update scheduled for mid-2023. In this regard, the proposed model and approach do not provide dynamic and automatic adjustment to cost variations over time. This issue will be the subject of further research. Appl. Sci. 2023,13, 12744 22 of 24 In summary, the novelty of the proposed model lies in its use of AI tools, specifically ANNs, and the combination of trained ANNs in the form of an ensemble to achieve objective cost predictions for sewerage construction projects. The literature review indicates that this approach is original, with no prior research reporting the development of similar models for such projects. 7. Summary The research resulted in the development of two original predictive models capable of forecasting the construction costs of sewerage projects based on ensembles of neural networks. Based on the applied artificial intelligence tool’s training capacity, influenced by the collected data, features, and characteristics of the analyzed projects, as well as the underlying assumptions, the developed models demonstrate the capability to forecast construction costs for either wastewater or stormwater runoffs, excluding combined runoffs. The ensembles mentioned above consist of five different MLP-type ANNs. The outputs of these ANNs are combined using two alternative approaches: simple averaging and generalized averaging. Although the predictive performances of the two EoNN-based models are comparable, the one based on simple averaging appears to offer slightly better results. The accuracy of cost predictions is satisfactory. Especially for the model based on simple averaging, that is, the SAV ENS model, more than 90% of cost prediction cases, both for training and testing, meet the accuracy requirements with percentage errors falling within the acceptable range of < − 30%; 30%>. While the developed model has its limitations, it holds potential applications in estimating the costs of construction for sewerage projects in the Czech Republic. Furthermore, the proposed general approach may find applicability in other countries, although the models should be adapted and trained using locally collected data. Future studies will involve further data collection and the development of models based on AI tools. Additionally, future research will focus on incorporating cost variability over time into these models in order to overcome existing limitations. Author Contributions: Conceptualization, M.J., T.H. and M.V.; literature review, M.J., T.H. and H.P.; methodology, M.J.; source documents analysis and data collection, T.H. and M.V.; data curation, M.J. and T.H.; formal analysis and computations, M.J., H.P. and M.S.; results analysis, M.J., H.P. and M.S.; discussion, M.J., T.H., M.V., H.P. and M.S.; final conclusions, M.J., T.H., M.V., H.P. and M.S.; writing—original draft preparation, M.J.; review, M.J. and T.H.; editing, M.J.; funding acquisition, M.J., T.H. and M.V. All authors have read and agreed to the published version of the manuscript. Funding: This research was co-funded by project no. FAST-S-23-8253 and project no. FAST-J-23-8349 held by Brno University of Technology; statutory funds of the Faculty of Civil Engineering, Cracow University of Technology; program of the Polish Ministry of Education and Science “Implementation doctorate”, agreement number between the Cracow University of Technology and the Polish State Treasury/Minister of Education and Science—DWD/6/0520/2022. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Some or all data that support the findings of this study is available from the authors upon reasonable request. The data are not publicly available to allow for actions aimed at commercializing the research findings. Conflicts of Interest: Author Hanna Pacyno was employed by the company Datacomp IT sp. z o.o. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest. 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