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Innovative Infrastructure Solutions (2022) 7:153

https://doi.org/10.1007/s41062-022-00761-8

TECHNICAL PAPER

Implementation ofmulti‑expression programming (MEP), artiﬁcial

neural network (ANN), andM5P‑tree toforecast thecompression

strength cement‑based mortar modiﬁed bycalcium hydroxide

atdiﬀerent mix proportions andcuring ages

AsoAbdalla1· AhmedSalih1

Received: 11 December 2021 / Accepted: 31 January 2022

© Springer Nature Switzerland AG 2022

Abstract

In this study, four diﬀerent modeling techniques, nonlinear regression (NLR), artiﬁcial neural network (ANN), M5P-tree,

and multi-expression programming (MEP), are used for developing reliable models to predict the compressive strength of

cement-based mortar modiﬁed with various calcium hydroxide (CH) content up to 45% (wt% dry cement). The developed

accurate models are essential in the construction ﬁelds since the material properties prediction from the model is time-saving

and cost-eﬀective. The dataset for model development contained data from previous literature related to calcium hydroxide

cement blended cement-mortar. The water/cement ratio (w/c) was ranged from 0.3 to 0.74, and the testing age of the sam-

ple was varied between 1 and 28days. Nevertheless, the MEP model was better than other models based on statistical tool

assessments. The M5P-tree model is the second-best technique for compressive strength prediction of cement-based mortar

modiﬁed with calcium hydroxide. Based on the collected data, the compressive strength of the cement mortar decreases as

the addition of calcium hydroxide increases.

Keywords Calcium hydroxide· Compressive strength· Modeling· Sensitivity analysis

Introduction

Nowadays, many environmental issues are raised from

cement production and other pollutions, such as global

warming caused by carbon dioxide (CO2) emissions into the

atmosphere and landﬁlling by-products of factories. It is

believed that using those waste can solve many environmen-

tal problems while producing a sustainable material for con-

struction. Mineral admixture addition is one of the solutions.

By-products of silicon industries and power plants include

silica fume, calcium hydroxide, and ﬂy ash; several studies

have been conducted on the use of these wastes in cement-

based concrete and mortars, exhibiting better mechanical

performance qualities and composite durability [1, 2]. Many

studies have been done on the inﬂuence of calcium hydrox-

ide powder on the mechanical characteristics of cement-

based mortar by replacing some Portland cement with cal-

cium hydroxide powder and changing the fresh and hardened

properties of cement mortar [3–5]. For many decades, con-

crete has been widely used as an essential material for con-

structing various infrastructures, including residential and

commercial buildings and civil engineering structures. It is

also used for the new generation of gravity-defying sky-

scrapers. It provides an opportunity for creativity and unlim-

ited engineering potential. Therefore, concrete is responsible

for many of the global construction industries. Aggregate

constitutes the major percent of the concrete which has a

high impact on its properties, as it typically constitutes

60–80% of the concrete volume. These aggregates directly

aﬀect the fresh and mechanical performance of the concrete

[6]. Despite the global acceptance of concrete as a key con-

struction material, it has some shortcomings that usually

aﬀect its quality and general performance. For instance, rela-

tively high density, brittleness, weak tensile strength, low

* Ahmed Salih

ahmed.mohammed@univsul.edu.iq

Aso Abdalla

aso.abdalla@univsul.edu.iq

1 Civil Engineering Department, College ofEngineering,

University ofSulaimani, Kurdistan Sulaimani - Kirkuk Rd,

Sulaymaniyah46001, Iraq

Innovative Infrastructure Solutions (2022) 7:153

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resistance to chemical attacks, etc. For these reasons,

research is continuously conducted to seek alternative

sources of concrete ingredients that are technically satisfac-

tory, cost eﬀective, environmentally friendly, and materials

that resist an aggressive environment. Concrete materials,

including cement mortar, sand, and coarse aggregates, are

becoming expensive and sometimes not commonly found in

some locations. The need to ﬁnd another source of construc-

tion materials is desirable. More durable and recyclable

materials are needed, utilization of waste and natural or agri-

cultural by-products. This may reduce the exploitation of

natural resources and reduce the eﬀect of environmental pol-

lution. The utilization of recycled aggregate and natural

resources as coarse aggregate in concrete production mini-

mizes landﬁll waste. Many research on the properties of

concrete has been conducted [6, 7]. It was shown from the

literature that minerals recycled and agricultural by-products

are paved as alternative substitutes to aggregates. Ferro-

chrome slag replaced a certain amount of ﬁne aggregate and

yielded a promising result for concrete production. Maxi-

mum strength was achieved when incorporated with 75%

steel slag as coarse aggregate. Incorporating copper waste

slag at 10–100% replacement as coarse aggregate enhanced

the mechanical performance of self-compacting concrete by

26–29%. Nowadays, machine learning has been used by the

researcher in civil engineering ﬁelds which includes ﬁtness

monitoring of structural engineering system, mechanical

properties of concrete, the interface of composite members,

ﬁre resistance of reinforced concrete elements, residual

properties of concrete subjected to ﬁre [7]. Support vector

regression (SVR) is a type of machine learning technique

that works based on the rule of structural risk minimization.

The SVR can be used in regression and classiﬁcation. The

behavior of high- and ultra-high-performance concrete

beams was estimated by developing an SVR-based model.

The characteristic parameters include fracture energy, ulti-

mate load, and so forth. SVR-and ANN-based models have

been developed and applied to predict the concrete behavior

based on mix design using experimental data. The compres-

sive strength was successfully predicted with high accuracy

based on SVR models [5, 6]. Another interesting block-

oriented model, which includes linear dynamic blocks and

non-memory blocks, is useful in nonlinear system modeling

responsible for simple block-oriented structures and eﬃcient

structural ensemble. Hammerstein–Wiener model consists

of two static nonlinear blocks conﬁning dynamic linear

blocks are generally employed for a nonlinear system. The

Hammerstein–Wiener model can describe or approximate

the complicated nonlinear industrial process than the Ham-

merstein model or Wiener model [7]. The statistical tech-

niques and machine learning algorithms are very strong tools

that are commonly used in engineering design. These tech-

niques are speciﬁcally used in complex engineering systems,

including composite materials and structures, and solved

technically. Therefore, this study has paid attention to pre-

dicting the compressive strength of concrete modiﬁed with

jujube seed as a partial substitute to coarse aggregate by

employing SVM and HWM models. The durability of con-

crete structures is essential [3]; thus, concrete must be ade-

quately compacted during the casting of fresh concrete to

achieve durable concrete leading to SCC development.

Improved segregation resistance translates aggregate parti-

cles’ distribution in the concrete mixtures relatively at all

locations and diﬀerent levels [6]. Most importantly, SCC can

easily be poured into the formwork of congested reinforced

structures with no vibration, reducing construction time,

feasible design work, and enhancing product and working

environment quality. Therefore, contractors are motivated to

utilize industrial by-products and construction waste such as

recycled concrete aggregates that can be used as a partial

replacement for aggregates in concrete [4]. Calcium carbide

waste (CCW) is a by-product of the acetylene gas (C2H2)

production process. CCW is obtained through a chemical

reaction between water and calcium carbide. CCW is highly

alkaline (pH > 12) and contains a high proportion of

Ca(OH)2, which is about 92% of its mass fraction [6]. There-

fore, when it is combined with other pozzolanic materials,

the Ca(OH)2 will react with the silicon oxide or silicon and

aluminum oxides present in pozzolans to yield more calcium

silicate hydrates (C–S–H) responsible for strength develop-

ment [6]. Adamu etal. [7] also replaced ﬂy ash with CCW

at 0, 10, 20, and 30% levels in alkali-activated ﬂy ash sus-

tainable material using sodium hydroxide and sodium sili-

cate as activators. Their ﬁndings showed that the setting time

of mortar decreases, while the

CaC2 + 2H2O → C2H2 + Ca(OH)2 compressive strength and

shear bond strengths increase with increment in CCW con-

tent. CCW also densiﬁed the microstructure of the mortar

and increased the peaks of calcium hydroxide and calcium

silicate hydrates, leading to the strength increment. Haruna

etal. [6] used a hybrid of CCW and silica fume via a new

combustion technique to produce a new supplementary

cementitious material (SCM) in concrete, where they

replaced 5% cement with the new SCM. The findings

showed that the new SCM prolonged the hydration process

of the concrete, thereby increasing the setting times,

improved the compressive strength at later ages (45days),

and lower drying shrinkage value at the same age compared

to reference concrete. Adamu etal. [7] reported the eﬀect of

RHA and Al2O3 nanoparticles on concrete’s mechanical and

durability properties. They replaced 10% cement with RHA,

and 1, 2, 3, and 4% cement with Al2O3 nanoparticles. Their

results showed that replacing 10% cement with RHA identi-

ﬁed the concrete microstructure and increased the compres-

sive strength, ﬂexural strength, tensile strength, and durabil-

ity properties in terms of resistance to hydrochloric acid and

Innovative Infrastructure Solutions (2022) 7:153

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Page 3 of 15 153

acid attack of the concrete. They further reported that replac-

ing cement with Al2O3 further densiﬁed the concrete micro-

structure and increased its strength and durability [8, 9].

Modeling the properties of materials can be performed in

various ways, including computational modeling, statistical

techniques, and newly created tools like regression analysis,

M5P-tree, and artiﬁcial neural networks (ANN) [10–18].

Also, Armaghani and Asteris [19] investigated the applica-

tion of ANN and adaptive neuro-fuzzy inference system

(ANFIS) models to predict the compressive strength of

cement mortar with or without metakaolin concluded that

ANFIS performed better than ANN. Moreover, Suba etal.

[20] employed ANN for mechanical properties prediction

and compared the result with linear regression, the forecast

of the ANN model was pretty close to actual work. Multi-

expression programming (MEP) was used to estimate the

mechanical properties of concrete and provide acceptable

results as implemented by Shah etal. [21]. The parametric

study revealed the accuracy of the MEP model, with a high

correlation coeﬃcient (R). As a result, several techniques

were used in the literature to forecast the mechanical proper-

ties of cement-based mortar, but MEP has not been used for

that purpose.

Problem statement

i. There is no comprehensive study in the previous

research studies on the cement mortar modiﬁed with

the calcium hydroxide evaluating and quantifying the

eﬀect of a wide range of mix proportions on the long-

term compressive strength of cement mortar.

ii. From now towards the researchers and the construc-

tion industry can use the developed models in this

study with high accuracy (low RMSE and high R val-

ues) to predict the compressive strength of the cement

mortar withoutany costand would save time for the

experimental lab work.

iii. Several new ideas were used in this study as follows:

iv. Collecting 215 tested data, statistical analysis to disci-

pline concerns the collection, organization, analysis,

interpretation, and presentation of data.

v. Long-term compressive strength prediction for the

cement mortar.

Objectives ofthis study

This study aims to determine the compressive strength of

cement mortar modiﬁed with calcium hydroxide (CH) with

various water to cement ratios and curing time using diﬀer-

ent multiscale models. The models were subjected to sen-

sitivity analysis utilizing statistical evaluation tools. As a

consequence, experimental data from other research papers

were gathered and evaluated using a variety of statistical

modeling techniques to accomplish the following goals: (i)

To perform a statistical analysis and determine the impact of

mixture compositions of cement mortar (ii) To determine the

inﬂuence of calcium hydroxide on the compressive strength

of cement mortar (iii) Using statistical analysis, compare

and determine the most trustworthy model for measuring

the compressive strength of cement mortar modiﬁed with

calcium hydroxide.

Methodology

Figure1 presents the steps that have been followed during

this study.

Data collection

A comprehensive 215 data on compressive strength and 41

data on ﬂexural strength of cement-based mortar modiﬁed

with calcium hydroxide were collected from diﬀerent pre-

vious studies [2–4, 11, 17, 22–29, 48–51]. The dataset was

divided into three groups (training, testing, and validating)

randomly using the rand function in Microsoft Excel. The

largest group included 70% of the dataset (151 data), and

each of the other two groups included 15% of the dataset

(32 data). The training data is used to develop the model, an

algorithm is feeded with the training data and continually

evaluated to learn from the data and make a future predic-

tion while validating and testing data is provided to test the

developed model against unseen data and overﬁtting of the

developed model can be minimized [26]. The summary of

statistical analysis on the input and output parameters with

detail of the collected data is shown in Table1.

Statistical analysis

Water tocement ratio (w/c)

According to the statistical evaluation on the collected

data, w/c was ranged between 0.3 and 0.74, with mean and

median, standard deviation (SD), variance (Var), skew-

ness (Skew), and kurtosis (Kur) of 0.5, 0.08, 0.01, 0.67,

3.29, respectively (Table2). The relation between w/c and

compressive strength and the histogram for w/c is shown

in Fig.2.

Calcium hydroxide content (CH) (%)

Based on the collected data from previous literature, the

percentage of cement replacement with calcium hydroxide

was 45% maximum. With a median, SD, Var, Skew, and Kur

of 12, 10.83, 117.27, 0.82, 0.42, respectively (Table2). The

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variation of compressive strength with the percentage of the

replacement of calcium hydroxide content and histogram for

CH content is displayed in Fig.3.

Curing time (t) (days)

The collected dataset contained experimental results from

previous research; curing time ranged from 1 to 28days,

with a median of 7days, SD, Var, Skew, and Kur of 12days,

143.45 0.5, and −1.67, respectively (Table2). The histo-

gram for curing time and variation of compressive strength

with curing time are presented in Fig.4.

Compressive strength (CS)

Compressive strength of cement-based mortar modiﬁed

with calcium hydroxide up to 28days was ranged from 3

to 75MPa, with a median of 28.85MPa, SD, Var, Skew,

and Kur of 13.5MPa, 182.36, 0.41, −0.34 (Table2). The

histogram of compressive strength of cement-based mortar

modiﬁed with calcium hydroxide and Weibull distribution

function is shown in Fig.4.

STEP 1

•Considering w/c, curing time, calcium hydroxide content as

independent variable (predictors) and compressive strength as a

target for the models.

STEP 2

•Data collection from previous researches related to modification of

cement mortar with calcium hydroxide.

STEP 3

•Statistical analysis, presentation of the data , and finding possible

correlation between independent and dependent variables.

STEP 4

•Data splitting randomly ,70% of the data for training the models,

the remaining 30% for validation and testing the developed models.

STEP 5

•Modeling using MEP, NLR, ANN, and M5P- tree models.

STEP 6

•Evaluation of the developed models based on R2, RMSE, MAE, SI,

and OBJ.

STEP 7

•Performing sensitivity analysis to find out the most influential

parameter in predicting the compressive strength of cement mortar

modified with calcium hydroxide.

Fig. 1 Flowchart of the methodology of the current study

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Modeling

The correlation matrix (Fig.5) shows that the correlation

coeﬃcients (R) of CS with w/c, curing time, and calcium

hydroxide concentration are −0.516, 0.676, and −0.149,

respectively, indicating that there is no direct relationship

between cement-based-mortar compositions and compres-

sive strength. The correlation matrix (Fig.5) shows that the

correlation coeﬃcients (R) of CS with w/c, curing time, and

calcium hydroxide concentration are −0.516, 0.676, and

−0.149, respectively, indicating that there is no direct rela-

tionship between cement-based-mortar compositions and

compressive strength.

Multi‑expression programing (MEP model)

Genetic algorithm (GA) was ﬁrst introduced by Holland

[30], which was motivated through evolution theory, simi-

lar to that genetic programming (GP) proposed by Cramer

[30–32]. Several linear variations of GP have already been

proposed to deal with some difficulties (such as bloat)

caused by tree representations of GP. A few examples are

Cartesian genetic programming, grammatical evolution

(GE), linear GP, MEP, and gene expression programming

[33, 34]. Multiple solutions are stored in a separate chro-

mosome in MEP individuals. The most acceptable option is

usually chosen when it comes to ﬁtness assignments. This is

Table 1 Summary of

model input parameters for

compressive strength prediction

and correlation between

compressive and ﬂexural

strengths

References Water to cement ratio (w/c) Curing time, t (days) Calcium

hydroxide con-

tent, CH (%)

Com-

pressive

strength,

CS (MPa)

[23] 0.6 3, 7, and 28 0–20 3–33

[24] 0.43 3, 7, and 28 0 and 30 46–65

[25] 0.5 2, 7, and 28 0–15 9–55

[26] 0.3–0.5 7 and 28 0–45 18–65

[27] 0.5 1,3, 7, and 28 0–20 13.6–70

[28] 0.74 7 and 28 0–30 5–32

[29] 0.5 1 and 28 0–35 14–47.5

[48] 0.33 1, 3, 7, and 28 0–10 29–64

[49] 0.5 2, 7, and 28 0 and 30 16–60

[50] 0.5 3 and 28 0 and 30 17–60

[51] 0.5 2 and 28 0–35 6.7–62

[52] 0.5 3, 7, and 28 0–30 18–54

Remarks Ranged from 0.3 to 0.74 1, 3, 7, and 28days Varied between

0 and 45%

Ranged

from 3 to

70MPa

Table 2 Summary of statistical

analysis Statistical parameters Independent variables Dependent variables

Water to

Cement ratio,

w/c

Curing

time, t

(Days)

Calcium hydrox-

ide content, CH

(%)

Compressive

strength,CS

(MPa)

Flexural

strength, FS

(MPa)

No of data 215 215 215 215 41

Max 0.74 28 45 70 12

Min 0.3 1 0 3 4

Median 0.5 7 12 28.854 7.2

Mean 0.5 12 12.96 29.32 7.51

SD 0.085 12 10.83 13.5 1.934

Var 0.007 143.45 117.266 182.364 3.739

Skew 0.671 0.497 0.818 0.407 0.414

Kur 3.287 −1.674 0.424 −0.342 −0.244

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known as strong implicit parallelism, and it is a distinctive

characteristic of MEP [35, 36]. This feature does not make

MEP more complex when compared with GE and GEP. The

MEP model incorporates diﬀerent ﬁtting factors to generate

a generalized relationship. Math operators were employed

to generate simple expressions in this investigation, and a

trial and error procedure was used to determine the ﬁtting

parameters [37].

Nonlinear regression model (NLR)

The following formula (Eq.1) can be considered as a gen-

eral form for developing a nonlinear regression model [26,

27] to determine the compressive strength of the standard

cement mortar component and modiﬁed cement mortar,

Eq.1 represents the interrelationship between the independ-

ent variables,

CS, w/c, t, and L are compressive strength, water to

cement ratio, curing time, and calcium hydroxide content,

and a, b, c, d, e, f, and g are model parameters.

ANN model

The artiﬁcial neural network (ANN) is a computing system

that resembles the human brain and its information analy-

sis. In addition, this model is a machine learning system

employed in construction engineering for various numerical

forecasts and diﬃculties [1, 12, 13]. ANN consists of three

layers input, hidden, and output layer; these layers are con-

nected through biases and weights (Fig.6). The behavior of

an ANN network is inﬂuenced by the pattern of neurons’

connections, which also determines the network’s class.

As previously mentioned, it is possible to train a network

to enhance network performance. In more technical terms,

the network’s topology and connection weights change

(1)

CS

=a

(

w

c)b

(t)c+d

(

w

c)e

(t)f(L)

g

Fig. 2 a Scatter plot of compressive strength and water to cement

ratio and b histogram of water to cement ratio

Fig. 3 a Scatter plot of compressive strength and calcium hydroxide

content and b histogram for calcium hydroxide content

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repeatedly such that the error at each output layer node is

minimized [19]. In this study, a multi-layer feed-forward

network was designed with mortar composition (w/c, t, CH)

as input and CS as output, and a sigmoid activation function

is used in the output layer.

where j is the number of input variables, xj is the input num-

ber j, and bias is the threshold for sigmoid activation func-

tion. The typical process of the result of ANN is illustrated

in Fig.7.

M5P‑tree model

Quinlan [38] ﬁrst devised the M5 algorithm, which was

developed into the M5P-tree algorithm [39]. One of the

most signiﬁcant advantages of model trees is their ability to

eﬃciently solve problems, dealing with many data sets with

a substantial number of attributes and dimensions. They are

also noted for being powerful while dealing with missing

data. The M5P-tree approach establishes a linear regression

at the terminal node by classifying or partitioning diverse

data areas into numerous separate spaces. It ﬁts on each sub-

location in a multivariate linear regression model. The error

is estimated based on the default variance value inserted

into the node. The general formula for the M5P-tree model

is shown in Fig.8.

Performance criteria formodel evaluation

The developed models are evaluated based on diﬀerent

assessment tools to choose the best model to predict the

CS of the mortar; the following are eﬃcacy measurements

for the models. The evaluation formulas are illustrated in

Table3.

Analysis ofoutputs

Relation betweenpredicted andmeasured

compressive strength

MEP model

Comparison of measured with the predicted value of CS

using the MEP model is presented in Fig.9a. The model

had a good performance with R2 of 0.87, 0.87, and 0.897

for training, testing, and validating, respectively. Figure9a

contained −25 and + 25% error lines in the training, testing,

and validating datasets.

(2)

Output

=f

(n

∑

j=1

wjxj+bias

)

Fig. 4 a Scatter plot for compressive strength and curing time and b

histogram for compressive strength of cement mortar modiﬁed with

calcium hydroxide from 1 to 28days of curing

Fig. 5 Correlation matrix for independent variables and dependent

variable

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Fig. 6 Typical procedure for

output of ANN network in a

single node

Fig. 7 Optimal ANN network structures, a two hidden layers and 4 hidden neurons, b two hidden layers and 5 hidden neurons, and c two hidden

layers and 7 hidden neurons

Fig. 8 Pruned M5P-tree model

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NLR model

The variation of predicted compressive strength with meas-

ured compressive strength is displayed in Fig.9b. From the

modeling result, it is clear w/c and curing time are aﬀect the

CS more than calcium hydroxide content. In comparison, the

eﬀect of w/c is more signiﬁcant on the compression strength

of cement-mortar. The model is developed, and the model

parameters are determined using the least square method

and solver technique [40–47]. The NLR model is presented

in Table4. Figure9a contained −25 and + 25% error lines

in the training, testing and validating datasets similar to the

MEP model.

ANN model

Figure8 shows the optimal ANN network structures, the

best network structure (Fig.8b) selected containing two hid-

den layers and ﬁve hidden neurons, with momentum, learn-

ing rate, learning time of 0.1, 0.2, and 2000, respectively.

Figure9c shows variation in predicted CS with measured

CS using the training, testing, and validating datasets and

error line −25 to + 25%, indicating the measurements and

predictions are in this limit with R2, RMSE of 0.901 and

4.234MPa.

M5P‑tree model

Figure8 shows the division of the input space by the algo-

rithm of the M5P-tree model into two linear regression func-

tions named LM 1 and LM 2. The relationship of predicted

and measured CS of the M5P-tree model showed in Fig.9d,

with R2 and RMSE of 0.81 and 5.219MPa. With error lines

of −20 to 25% for the training data set and −25 to 25% for

testing and validating datasets.

The error lines for the relationship between measured

compressive strength and the prediction using all the mod-

els were ranged between + 25 and − 25%, as shown in

Fig.10a. Also, the minimum and maximum values of the

ratio between the predicted compressive strength value and

the measured values range between 0.5 and 4.5 with the best

prediction of ANN and MEP models (Fig.10b).

Model evaluations

The MAE for MEP models is less than NLR and M5P-tree

by 15.6 and 15%, while greater than the ANN model by

12%. However, the MAE values of the MEP model for test-

ing and validating datasets are less than NLR, ANN, and

M5P-tree model. The OBJ values for the proposed models

are also evaluated; the OBJ for the MEP model is less than

Table 3 The criteria for measuring the eﬃciency of the developed models

where R2, RMSE, MAE, MBE, SI, OBJ, t test, U95, and ρ are coeﬃcient of determination, root mean squared error, mean absolute error, an aver-

age of errors, scatter index, objective, t test, 95% conﬁdence uncertainty, and performance index, respectively. yp, ye, and

ye

are predicted com-

pressive strength, measured compressive strength, and an average of measured compressive strength, respectively. n, tr, te, val are several data

in the training, testing, and validating dataset. For all of the assessment parameters, the ideal value is zero, while the best value for R2 is one.

Corresponding to SI, the performance of the model is excellent, good, fair, and poor if the SI<0.1, 0.1<SI<0.2, 0.2<SI<0.3, and SI>0.3,

respectively [46]

Formula Equation number Best value

R

2=1−

∑n

1(yp−ye)

2

∑

n

1(

ye−ye

)2

(3) 1

R=√

R2(4) 1

RMSE

=

√

SSE

n

(5) 0

MAE

=

∑n

1

�

yp−ye

�

n

(6) 0

SI

=

RMSE

ye

(7) 0

OBJ

=

(

ntr

n

to

∗RMSEtr+MAEtr

R2

tr

+1

)

+

(

nte

n

to

∗RMSEte+MAEte

R2

te

+1

)

+

(

nval

n

to

∗RMSEval+MAEva l

R2

val

+1

)

(8) 0

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NLR, and M5P-tree models by 15 and 17%; in addition, the

OBJ of ANN is 6% less MEP model’s OBJ as displayed in

Fig.11. Also, the performance index for the MEP model was

less than other developed models for testing and validating

data which speciﬁes better performance of the MEP model

over NLR, ANN, and M5P-tree models for unseen data

(Fig.11c). Summary of model evaluation for R2, RMSE,

and MAE of the developed models is presented in Table4.

Conclusions

Distinct soft computing techniques can be utilized to con-

struct useful and time-saving models; in this study, four dif-

ferent approaches were employed to develop a trustworthy

model for the prediction of compressive strength of calcium

hydroxide-modiﬁed cement mortar; the following are the

primary conclusions:

Fig. 9 Variation of CS Predicted with CS Measured using, a MEP model, b NLR model, c ANN model, and d M5P-tree model

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1. Based on the collected data from literature maximum

percentage of calcium hydroxide is 45%, w/c was ranged

from 0.3 to 0.74. The addition of calcium hydroxide to

cement mortar decreases the compressive strength for

the same mixture and w/c.

2. Based on SI, the performance of MEP and ANN mod-

els are nearly the same, while the SI of MEP model for

validating data is 5% smaller than the SI value for the

ANN model. The objective value for the MEP model

is less than NLR and M5P-tree models by 17 and 16%.

95% Uncertainty value for MEP is smaller than other

developed models.

3. According to the statistical evaluation tools, the MEP

model is better than NLR and M5P-tree models for com-

pressive strength prediction after the ANN model.

4. Using several evaluation criteria, including the root

mean square error (RMSE), the coeﬃcientof determi-

nation(R2), the OBJ, the SI, and the mean absolute error

(MAE). The sequence of models was ANN and M5P-

tree, suggesting that the ANN was the best model pro-

vided in this research based on data acquired from the

literature and producing a higher R2 and lower MAE and

RMSE. The most eﬀective technique for the prediction

of CS of cement mortar is ANN, though the less com-

plicated model can be developed using multi-expression

programming.

5. The sensitivity analysis test was performed in order to

check the most eﬀective dependent variables on inde-

pendent variables’ output performance. The results

indicated that the most eﬀective parameters causing the

output result were curing time and CH content.

Table 4 Summary of developed models

Model Formula Eq. no R2RMSE (MPa) No. of data

MEP

CS =A+B+C−D

(9a) 0.87 4.857 215

A=2

(

w

c

)

2+2(t)+4

(w

c)−2

(

w

c

)

(t

)

(13b)

B

=

4

(w

c)2−4

(

w

c

)

2

(

w

c

)

2−t

(13c)

C

=

2

(w

c)−3

(

w

c

)2

2

(w

c)−3(w

c)2

−

4

(w

c)2−4(w

c)

2

(

w

c)

2−t

(13d)

D

=L

2

(

w

c

)

+2

+

2

t

(13e)

NLR

CS

=−129.25

(

w

c

)3.357

(t)−0.334 +19.63

(

w

c

)−0.532

(t)

0.1446

(

L+

0.001)−0.0161

(10) 0.84 5.39 215

M5P-tree

CS

=27.38 −15.67

(

w

c

)

+3.12(t)−0.46(CH

)

CS

=76.68 −92.69

(

w

c

)

+0.54(t)−0.30(CH

)

(11a)

(15b)

0.85 5.21 215

Innovative Infrastructure Solutions (2022) 7:153

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153 Page 12 of 15

Fig. 10 Comparison of

developed model based on, a

variation between measured and

predicted CS values for testing

data and b Ratio of predicted

compressive strength to meas-

ured compressive strength

Innovative Infrastructure Solutions (2022) 7:153

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Page 13 of 15 153

0.19

0.15

0.18

0.17

0.13

0.12

0.14

0.12

0.22

0.18

0.22

0.17

0

0.05

0.1

0.15

0.2

0.25

0.3

NLRANN M5P-tree MEP

Scater Index, SI

Training Te stingValidating

Fair

Good

Excele nt

3.83

2.89

3.80

3.23

3.35

3.03

3.47

2.77

4.95

3.76

4.53

3.35

0

1

2

3

4

5

6

NLRANN M5P-tree MEP

Mean Absolute Error, MAE (MPa)

Training Te stingValidating

(a)

(b)

(c)

Fig. 11 Comparing developed models based on, a SI, b MAE, c OBJ value

Declarations

Conflict of interest The authors declare that they have no known com-

peting ﬁnancial interests or personal relationships that could have ap-

peared to inﬂuence the work reported in this paper.

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