# Optimization and Simulation of Saw Dust Ash Concrete Using Extreme Vertex Design Method

Authors: Uzoma Ibe Iro; George Uwadiegwu Alaneme; Abdalrhman Milad; Bamidele Charles Olaiya; Obeten Nicholas Otu; Edward Uchenna Isu; Magnus Nnaemeka Amuzie
Journal: Advances in Materials Science and Engineering
 (2022)

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## Abstract

In this research study, a constrained simplex optimization method was adapted for the evaluation of green concrete’s mechanical property constituting of two-component mixture problem of cement and saw dust ash (SDA), which is a derivative from industrial residue. This experiential research will provide avenue for the recycle and incorporation of waste materials to achieve sustainability and control indiscriminate disposal of waste. The formulated components constrained were realized from the relevant literature studies to obtain the feasible planes in the simplex, and from this computation approach using I-optimality, the design mixture proportions and experimental runs were derived. The experimental results obtained from the laboratory experiments showed a maximum compressive strength of 31.13 N/mm2 with a ratio of 0.875 : 0.125 for cement and SDA, respectively, and a flexural strength of 9.49 N/mm2 with a ratio of 1 : 0 for cement and SDA, respectively; the results were observed to decline linearly with the further addition of SDA. The details generated from laboratory program were utilized for the development of the EVD model through fits statistical evaluation, ANOVA, diagnostic test and influence statistics, numerical optimization, and graphical statistical computations to analyze the datasets and locate the optimal levels of mixture ingredients using desirability function. A desirability score of 0.990 was derived at a mix ratio of 0.89 : 0.110 for the two components, cement and SDA, to produce a maximized compressive strength and a flexural strength of 31.102 N/mm2 and 9.384 N/mm2, respectively. Furthermore, the test of adequacy of the generated EVD model was carried out through simulation and statistical validation exercises using the F-test and student’s t-test, and the outcome signified a good correlation between the EVD model-simulated and actual values with p value >0.05.


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## Body

## 1. Introduction

The construction industry relies heavily on conventional materials such as cement, granite, and sand for the production of concrete. The high and increasing cost of these materials has greatly hindered the development of shelter and other infrastructural facilities in Nigeria and other developing countries [1, 2]. There arises the need for engineering consideration of the use of cheaper and locally available materials to meet desired need to enhance self-efficiency, and lead to an overall reduction in construction cost for sustainable development [3, 4]. Attempts have equally been made by various researchers to reduce the cost of its constituents and hence total construction cost by investigating and ascertaining the usefulness of materials, which could be classified as agricultural or industrial waste [5]. Around the world, waste generation rates are intensifying. In 2016, the worlds’ cities generated 2.01 billion tonnes of solid waste, amounting to a footprint of 0.74 kilograms per person per day. With rapid population growth and urbanization, annual waste generation is expected to increase by 70% from 2016 levels to 3.40 billion tonnes in 2050 [6]. Sawdust is an industrial waste in the timber industry. It is obtained as loose particles or wood chippings from sawing of timber into standard useable sizes. It poses a nuisance to both the health and environment when not properly managed. The ash samples derived as a result of controlled incineration of the saw dust possess pozzolanic properties and have been shown to react chemically with the calcium hydroxide released from the hydration of Portland cement, to form cement compounds [7, 8]. Active pozzolans gain their binding properties when they react with calcium hydroxide in lime or cement in the presence of water. The advantages of using SDA for concrete production are numerous. It acts as a retarder prolonging the setting times, reduces the heat of hydration, encourages a healthier environment by reducing green gas emission, and is abundantly available as a waste. SDA has been used as partial replacement in mortar and concrete works and also as a powder material in the production of self-compacting concrete (SCC) [9, 10].One of the greatest contributors of solid wastes is from the industries in the processing of raw materials to obtain deliverables, and residues are ejected, which are indiscriminately disposed to our environment with accompanying health risks and threat on ecosystems [11]. Also, the tone of cement consumed for concrete production has an influence on the cost efficacy of the project and reduction of greenhouse gas emissions, and in guaranteeing eco-friendly environment; as such, it is very imperative to curtail the cement quantity to be expended in the concrete production process by partially substituting cement with waste ash, which possesses a pozzolanic behavior [12, 13]. The utilization of industrial residues in concrete has attracted interest from both science and construction industries, because of the sustainable benefits offered by such residues, in addition to their cost-effective and environment friendly nature compared with Portland cement [14].Concrete mix design encompasses the assortment of the constituent proportions that will result in certain desired properties. This is complicated especially with the introduction of additional constituents such as mineral and chemical admixtures, fibers, agro-waste, and blended cement, on the one hand, and the expansion the of range of material properties in relation to its service life, on the other hand [15, 16]. Quality concrete is a concrete that is adapted to its usage and therefore well formulated for specified intended purposes. As such, concrete formulators are open to growing pressure in terms of time and cost for the selection of concrete mix proportions conforming to specific requirements, notably in terms of consistency, strength, and durability [17, 18]. These considerations have provoked several researchers to construct concrete mix design software based on different approaches such as concrete optimization software tools, which adopts statistical and analytical approach taking into consideration necessary constraints to produced durable and eco-efficient materials [19]. Recently, numerous researchers have applied artificial intelligence (AI) techniques for developing their software. Among the best popular techniques of AI include genetic algorithms (GAs), fuzzy logic (FL), artificial neural networks (ANNs), and adaptive network-based fuzzy inference system (ANFIS) [20–23]. Generally, the optimization of a concrete mix design is to establish a mathematical relationship between the desired properties. For instance, Alaneme et al. [24] researched on the utilization of the adaptive neuro-fuzzy inference system (ANFIS) for the optimization of compressive strength properties of saw-dust-ash (SDA) and quarry-dust (QD) concrete from 0 to 50%. The ANFIS model performance evaluated using loss function parameters showed a good relationship between the predicted and actual results. Also, Behfarnia and Khademi [25] performed concrete compressive strength property prediction using ANFIS and ANN. One hundred and sixty different mix designs were cast after 28-day hydration period. The results obtained indicate that the ANFIS and ANN models were suitable for estimating the concrete mechanical properties.Extreme vertex design (EVD) is another technique used in predicting concrete mix proportions. In a mixture experiment design, the predictor parameters or factor levels are known as the mixture components and their levels are the combination ratio. These components or mixture ingredients sum to one for each trial, while the response parameter under study depends only on these combination proportions and not on the absolute quantities of the mixture components [26, 27]. In the experimental mixture design, the unconstrained and total sum of the mixture ingredients denoted by q is one or homogenous mixture. Due to the imposed sum to one constrain on the q-components, the experimental region or space for the mixture takes the form of q − 1 dimensional simplex [28]. When the design of unconstrained components in an experiment is carried out, the extreme vertices for each q-component in the mixture would be on the trial list. However, each extreme vertex in the unconstrained case stands for a unary mixture or pure blend, while additional trials are mostly required, which goes beyond the pure blend. Each unary pair signifies mixtures mixture blends situated at the edge of the hyper-tetrahedron [29]. The following potential set of trails are those at the centroids of each simplex representing the binary blends, which are two-component mixtures. At each edge of the simplex’s centroid is exactly the arithmetic mean of the two vertices at that end. This process continues by centroid determination at the three-dimensional edges known as the ternary blend, and further down to the overall single centroid of the factor space, which takes the form of [q − 1]-nary mixture. It is important to note that the absolute centroids with respect to the center of mass generally are not the same as the simple arithmetic average of the vertices at each face of the simplex [30, 31].Moreover, in major cases, this computed average provides a good measure for approximation, and if the generated list of trails is cumbersome, subset constituting 2-dimensional space forq-extreme vertices in the mathematical form shown in equation (1). For centroids at the edges of the simplex, the q-centroids for the q − 2 and the unitary q − 1-dimensional faces for the overall centroid are taken [24].(1)q!2q−2!.When the factor levels are imposed by lower and upper constraints, the experimental design space no longer takes the form of a simple hyper-tetrahedron, which influences the number of edges and extreme vertices [32]. The aim of an extreme vertex experimental design is to determine the predictor variables, which take on the minimum and maximum possible values within the constrained region. This evaluates the points of the highest or lowest effects in terms of change in response parameter. Through the reduction of the independent factors effects by the introduction of components’ constraints, the optimization of experimental efforts and derived information in the field or response of interest is achieved. Without the inclusion of various centroids in the experimental design, the imposed constraints enable the mixture trials derived to incorporate the combination of all ingredients when they are less than upper limits of 100% and varied lower limits [26, 33]. The aim of this research study is to achieve the recycling of industrial waste derivatives (SDA) by the partial replacement of cement to produce green concrete using EVD to derive the optimal combination ratio of the 2-component mixture in respect to the mechanical strength properties. The concrete mixture’s strength properties are significantly impacted by adequate ingredients proportioning using the robust mixture design approach. Hence, the need for the application of the extreme vertex design (EVD) method to optimize the response function and formulate a mathematical model that evaluates the effects of SDA in concrete through a 2-component constrained simplex design.


## 2. Methodology

### 2.1. Test Materials

#### 2.1.1. Portland Cement

Cement utilized for this experimental investigation is Dangote brand with grade 42.5 and was obtained from a materials market in Abia State, Nigeria. The design specifications for the materials in terms of physical properties and chemical composition conform to BS 12 (1978) [34, 35].


#### 2.1.2. Fine Aggregates

Fine aggregates are obtained from the clean river sand sourced from Akwa Ibom State, Nigeria, and classified within the sharp sand envelope. They also pass through a 2.36-mm sieve and are prepared according to ASTM C125-16 [36].


#### 2.1.3. Coarse Aggregates

The coarse aggregates used for this experiment 15–22 mm grain size, obtained from Saturn granite quarry at the Akamkpa Local Government Area of Cross River State. The aggregate samples are well graded and conform to BS EN12620 (2002) [37].


#### 2.1.4. Water

Potable water conforming to the specification of BS EN 1008 (2002) was used for both specimen preparation and curing of the blended concrete specimen for this laboratory exercise, which also meets the specification of ASTM C1602-12 for concrete works [38].


#### 2.1.5. Saw Dust Ash

Saw dust utilized for the laboratory exercise was obtained from Timber Wood Workshop at Abia State and was further subjected to a control incineration system to generate ash derivatives from the industrial wastes, which were sieved with a 150-μm sieve and prepared for the laboratory test in accordance with ASTM C618 and BS 8615-1 (2019) [39, 40].


### 2.2. Test Experiment Program

Mixture proportion of concrete mixes constituting of varying replacement ratios of cement by saw dust ash in a two-component simplex mixture design was carried out in this research study to evaluate the green concrete material’s mechanical properties after curing for twenty-eight days. The mixture design parameters are listed as follows: characteristic strength of 30 N/mm2, 0.525 for water-cement ratio, 290 kg/m3 for cement density, and 766.35 kg/m3 and 1198.65 kg/m3 for fine and coarse aggregate density, respectively [41]. Due to component constraints imposition at upper and lower limits, the design experimental points in the simplex are thus situated at the axialCB, vertices, and center of the feasible region. Formulation of mixture ingredients’ ratios and experimental runs’ number required were further determined using the I-optimal design to ensure an efficient assessment of the response function. The experimental setup and laboratory works are executed using the calculated proportions of the mixture. The experiments to be carried out include the gradation of the test materials, compressive and flexural strength after twenty-day curing period in 150 mm ×  150 mm × 150 mm cube and 100 mm × 100 mm × 400 mm beam mold dimensions, respectively. From the obtained results, statistical influence and diagnostics tests were deployed to evaluate and validate the generated EVD model performance. This involves graphical and numerical optimization using desirability-function computation to enhance the maximization of the best possible criteria for the response function with respect to corresponding factor levels to obtain the cement-SDA optimal ratio of combination for the mechanical strength response [41, 42]. The research program flowchart is presented in Figure 1.Figure 1
Program flowchart (source: Aju et al. [43]).


### 2.3. Mixture Component Constraint Formulation

For the formulation and design of component constraints that are imposed at the upper and lower bounds of the factor levels, expert judgment, wealth of experiences, empirical relationships, and relevant literature, as well as practical, environmental, and economic considerations, were all taken into account. The derivation of the mixture proportion constrained by the sum to one restriction, as shown in Table1 using actual mixture coding, will be moderated by these boundary limits [26, 30]. The viable experimental region is placed within the confined area of the simplex to assume a hyper-polyhedron simplex shape due to multicriteria limitations imposed on factor levels. As shown in Table 2, the degree of freedom is calculated by computing the design matrix for the design experiments using the quadratic mixture model with L_pseudomixture component coding. The information matrix calculation showing the leverages, space type, and mean is presented in Table 3. Five (5) experimental runs were obtained in the process to enhance the efficiency or optimality of the EVD model developed [43, 44].Table 1
Design constraints.
Mixture coding: actualLowConstraintHigh0.500A:cement1.0000.000B:SDA0.500A + B1.000Table 2
Design matrix evaluation.
Degrees of freedom for evaluationModel3Residuals1Lack of fit1Pure error0Cor total4Table 3
Measures derived from the information matrix.
RunLeverageSpace type10.7714AxialCB20.4857Center30.7714AxialCB40.9857Vertex50.9857VertexAverage= 0.8000


#### 2.3.1. 2-Component Factor Space Plot

Moreover, the developed contour plot of the 2-component constrained simplex is shown in Figure2, which explains diagrammatically positions and arrangement of experimental points within the constrained region in defined portions, namely, vertex, axialCB, and center [45].Figure 2
Two-component factor space.


#### 2.3.2. Fraction of Design Space (FDS)

FDS presents the graphical method conveniently adopted to evaluate the design performance and for comparison purposes through the standard error of design for the 2-component mixture experiment. The fraction of design space (FDS) provides details from the curvature of scaled prediction variance (SPV) for the assumed design region specified. It also measures G-efficiency, which provides needed template for specialized comparisons as shown in Figures3 and 4. From the plotted FDS result, 0.993, 0.679, and 0.813 were derived for maximum, minimum, and mean standard error, respectively. Furthermore, the condition number of the coefficient matrix of 7.788 was obtained with a G-efficiency of 81.2%, which possesses an inverse relationship with the maximum variance, computed determinant (X’X)−1 of 4.161E + 2, and trace (X’X)−1 of 91.368, and the scaled D-optimality criterion and the I-optimality result of 22.582 and 0.66085 were calculated [46, 47].Figure 3
Two-component fraction design space plot.Figure 4
Two-component standard error design plot.


#### 2.3.3. Experimental Mix Proportions

With the aid of Design Expert statistical software, five numbers of experimental runs were determined from the mixture design information matrix computation, which takes strictly into consideration the components constraints and uses the criterion optimality algorithm to enhance the adequate selection of experimental points within the constrained design space with a water-cement ratio of 0.525. The obtained ratio values of mixture ingredients calculated would be taken to develop the EVD model capable of predicting the mechanical properties of the concrete produced [48]. Three replicate samples were casted for each experimental run for the concrete cube and beam samples with mold dimensions of 0.15 m by 0.15 m by 0.15 m and 0.1 m by 0.1 m by 0.4 m, respectively. These proportion mixes provide special guide for blended samples to be examined in the laboratory to derive the optimal proportion of the two-component mixture of cement and derivatives from industrial residue (SDA) as shown in Table 4 [49]. The presented mixture proportions showed the actual and pseudovalues for the mixture components. Pseudocomponents are referred to as imaginary or coded variables used to simplify model fitting, thereby reducing the correlation between component bounds in constrained simplex mixture designs. The freshly mixed SDA-concrete blend and the concrete cubes for the experiments are presented in Figure 5.Table 4
Component proportion mixes for the experimental study.
Actual valuesPseudovaluesExperimental runZ1:cementZ2:SDAResponseX1:cementX2:SDA10.8750.1250.750.2520.750.250.50.530.6250.3750.250.7540.50.50151010Figure 5
SDA-concrete produced.


## 2.1. Test Materials

### 2.1.1. Portland Cement

Cement utilized for this experimental investigation is Dangote brand with grade 42.5 and was obtained from a materials market in Abia State, Nigeria. The design specifications for the materials in terms of physical properties and chemical composition conform to BS 12 (1978) [34, 35].


### 2.1.2. Fine Aggregates

Fine aggregates are obtained from the clean river sand sourced from Akwa Ibom State, Nigeria, and classified within the sharp sand envelope. They also pass through a 2.36-mm sieve and are prepared according to ASTM C125-16 [36].


### 2.1.3. Coarse Aggregates

The coarse aggregates used for this experiment 15–22 mm grain size, obtained from Saturn granite quarry at the Akamkpa Local Government Area of Cross River State. The aggregate samples are well graded and conform to BS EN12620 (2002) [37].


### 2.1.4. Water

Potable water conforming to the specification of BS EN 1008 (2002) was used for both specimen preparation and curing of the blended concrete specimen for this laboratory exercise, which also meets the specification of ASTM C1602-12 for concrete works [38].


### 2.1.5. Saw Dust Ash

Saw dust utilized for the laboratory exercise was obtained from Timber Wood Workshop at Abia State and was further subjected to a control incineration system to generate ash derivatives from the industrial wastes, which were sieved with a 150-μm sieve and prepared for the laboratory test in accordance with ASTM C618 and BS 8615-1 (2019) [39, 40].


## 2.1.1. Portland Cement

Cement utilized for this experimental investigation is Dangote brand with grade 42.5 and was obtained from a materials market in Abia State, Nigeria. The design specifications for the materials in terms of physical properties and chemical composition conform to BS 12 (1978) [34, 35].


## 2.1.2. Fine Aggregates

Fine aggregates are obtained from the clean river sand sourced from Akwa Ibom State, Nigeria, and classified within the sharp sand envelope. They also pass through a 2.36-mm sieve and are prepared according to ASTM C125-16 [36].


## 2.1.3. Coarse Aggregates

The coarse aggregates used for this experiment 15–22 mm grain size, obtained from Saturn granite quarry at the Akamkpa Local Government Area of Cross River State. The aggregate samples are well graded and conform to BS EN12620 (2002) [37].


## 2.1.4. Water

Potable water conforming to the specification of BS EN 1008 (2002) was used for both specimen preparation and curing of the blended concrete specimen for this laboratory exercise, which also meets the specification of ASTM C1602-12 for concrete works [38].


## 2.1.5. Saw Dust Ash

Saw dust utilized for the laboratory exercise was obtained from Timber Wood Workshop at Abia State and was further subjected to a control incineration system to generate ash derivatives from the industrial wastes, which were sieved with a 150-μm sieve and prepared for the laboratory test in accordance with ASTM C618 and BS 8615-1 (2019) [39, 40].


## 2.2. Test Experiment Program

Mixture proportion of concrete mixes constituting of varying replacement ratios of cement by saw dust ash in a two-component simplex mixture design was carried out in this research study to evaluate the green concrete material’s mechanical properties after curing for twenty-eight days. The mixture design parameters are listed as follows: characteristic strength of 30 N/mm2, 0.525 for water-cement ratio, 290 kg/m3 for cement density, and 766.35 kg/m3 and 1198.65 kg/m3 for fine and coarse aggregate density, respectively [41]. Due to component constraints imposition at upper and lower limits, the design experimental points in the simplex are thus situated at the axialCB, vertices, and center of the feasible region. Formulation of mixture ingredients’ ratios and experimental runs’ number required were further determined using the I-optimal design to ensure an efficient assessment of the response function. The experimental setup and laboratory works are executed using the calculated proportions of the mixture. The experiments to be carried out include the gradation of the test materials, compressive and flexural strength after twenty-day curing period in 150 mm ×  150 mm × 150 mm cube and 100 mm × 100 mm × 400 mm beam mold dimensions, respectively. From the obtained results, statistical influence and diagnostics tests were deployed to evaluate and validate the generated EVD model performance. This involves graphical and numerical optimization using desirability-function computation to enhance the maximization of the best possible criteria for the response function with respect to corresponding factor levels to obtain the cement-SDA optimal ratio of combination for the mechanical strength response [41, 42]. The research program flowchart is presented in Figure 1.Figure 1
Program flowchart (source: Aju et al. [43]).


## 2.3. Mixture Component Constraint Formulation

For the formulation and design of component constraints that are imposed at the upper and lower bounds of the factor levels, expert judgment, wealth of experiences, empirical relationships, and relevant literature, as well as practical, environmental, and economic considerations, were all taken into account. The derivation of the mixture proportion constrained by the sum to one restriction, as shown in Table1 using actual mixture coding, will be moderated by these boundary limits [26, 30]. The viable experimental region is placed within the confined area of the simplex to assume a hyper-polyhedron simplex shape due to multicriteria limitations imposed on factor levels. As shown in Table 2, the degree of freedom is calculated by computing the design matrix for the design experiments using the quadratic mixture model with L_pseudomixture component coding. The information matrix calculation showing the leverages, space type, and mean is presented in Table 3. Five (5) experimental runs were obtained in the process to enhance the efficiency or optimality of the EVD model developed [43, 44].Table 1
Design constraints.
Mixture coding: actualLowConstraintHigh0.500A:cement1.0000.000B:SDA0.500A + B1.000Table 2
Design matrix evaluation.
Degrees of freedom for evaluationModel3Residuals1Lack of fit1Pure error0Cor total4Table 3
Measures derived from the information matrix.
RunLeverageSpace type10.7714AxialCB20.4857Center30.7714AxialCB40.9857Vertex50.9857VertexAverage= 0.8000


### 2.3.1. 2-Component Factor Space Plot

Moreover, the developed contour plot of the 2-component constrained simplex is shown in Figure2, which explains diagrammatically positions and arrangement of experimental points within the constrained region in defined portions, namely, vertex, axialCB, and center [45].Figure 2
Two-component factor space.


### 2.3.2. Fraction of Design Space (FDS)

FDS presents the graphical method conveniently adopted to evaluate the design performance and for comparison purposes through the standard error of design for the 2-component mixture experiment. The fraction of design space (FDS) provides details from the curvature of scaled prediction variance (SPV) for the assumed design region specified. It also measures G-efficiency, which provides needed template for specialized comparisons as shown in Figures3 and 4. From the plotted FDS result, 0.993, 0.679, and 0.813 were derived for maximum, minimum, and mean standard error, respectively. Furthermore, the condition number of the coefficient matrix of 7.788 was obtained with a G-efficiency of 81.2%, which possesses an inverse relationship with the maximum variance, computed determinant (X’X)−1 of 4.161E + 2, and trace (X’X)−1 of 91.368, and the scaled D-optimality criterion and the I-optimality result of 22.582 and 0.66085 were calculated [46, 47].Figure 3
Two-component fraction design space plot.Figure 4
Two-component standard error design plot.


### 2.3.3. Experimental Mix Proportions

With the aid of Design Expert statistical software, five numbers of experimental runs were determined from the mixture design information matrix computation, which takes strictly into consideration the components constraints and uses the criterion optimality algorithm to enhance the adequate selection of experimental points within the constrained design space with a water-cement ratio of 0.525. The obtained ratio values of mixture ingredients calculated would be taken to develop the EVD model capable of predicting the mechanical properties of the concrete produced [48]. Three replicate samples were casted for each experimental run for the concrete cube and beam samples with mold dimensions of 0.15 m by 0.15 m by 0.15 m and 0.1 m by 0.1 m by 0.4 m, respectively. These proportion mixes provide special guide for blended samples to be examined in the laboratory to derive the optimal proportion of the two-component mixture of cement and derivatives from industrial residue (SDA) as shown in Table 4 [49]. The presented mixture proportions showed the actual and pseudovalues for the mixture components. Pseudocomponents are referred to as imaginary or coded variables used to simplify model fitting, thereby reducing the correlation between component bounds in constrained simplex mixture designs. The freshly mixed SDA-concrete blend and the concrete cubes for the experiments are presented in Figure 5.Table 4
Component proportion mixes for the experimental study.
Actual valuesPseudovaluesExperimental runZ1:cementZ2:SDAResponseX1:cementX2:SDA10.8750.1250.750.2520.750.250.50.530.6250.3750.250.7540.50.50151010Figure 5
SDA-concrete produced.


## 2.3.1. 2-Component Factor Space Plot

Moreover, the developed contour plot of the 2-component constrained simplex is shown in Figure2, which explains diagrammatically positions and arrangement of experimental points within the constrained region in defined portions, namely, vertex, axialCB, and center [45].Figure 2
Two-component factor space.


## 2.3.2. Fraction of Design Space (FDS)

FDS presents the graphical method conveniently adopted to evaluate the design performance and for comparison purposes through the standard error of design for the 2-component mixture experiment. The fraction of design space (FDS) provides details from the curvature of scaled prediction variance (SPV) for the assumed design region specified. It also measures G-efficiency, which provides needed template for specialized comparisons as shown in Figures3 and 4. From the plotted FDS result, 0.993, 0.679, and 0.813 were derived for maximum, minimum, and mean standard error, respectively. Furthermore, the condition number of the coefficient matrix of 7.788 was obtained with a G-efficiency of 81.2%, which possesses an inverse relationship with the maximum variance, computed determinant (X’X)−1 of 4.161E + 2, and trace (X’X)−1 of 91.368, and the scaled D-optimality criterion and the I-optimality result of 22.582 and 0.66085 were calculated [46, 47].Figure 3
Two-component fraction design space plot.Figure 4
Two-component standard error design plot.


## 2.3.3. Experimental Mix Proportions

With the aid of Design Expert statistical software, five numbers of experimental runs were determined from the mixture design information matrix computation, which takes strictly into consideration the components constraints and uses the criterion optimality algorithm to enhance the adequate selection of experimental points within the constrained design space with a water-cement ratio of 0.525. The obtained ratio values of mixture ingredients calculated would be taken to develop the EVD model capable of predicting the mechanical properties of the concrete produced [48]. Three replicate samples were casted for each experimental run for the concrete cube and beam samples with mold dimensions of 0.15 m by 0.15 m by 0.15 m and 0.1 m by 0.1 m by 0.4 m, respectively. These proportion mixes provide special guide for blended samples to be examined in the laboratory to derive the optimal proportion of the two-component mixture of cement and derivatives from industrial residue (SDA) as shown in Table 4 [49]. The presented mixture proportions showed the actual and pseudovalues for the mixture components. Pseudocomponents are referred to as imaginary or coded variables used to simplify model fitting, thereby reducing the correlation between component bounds in constrained simplex mixture designs. The freshly mixed SDA-concrete blend and the concrete cubes for the experiments are presented in Figure 5.Table 4
Component proportion mixes for the experimental study.
Actual valuesPseudovaluesExperimental runZ1:cementZ2:SDAResponseX1:cementX2:SDA10.8750.1250.750.2520.750.250.50.530.6250.3750.250.7540.50.50151010Figure 5
SDA-concrete produced.


## 3. Discussion of Results

### 3.1. Test Cement and Admixture Characterization

The saw dust ash (SDA) and Portland limestone cement (PLC) chemical properties utilized for the laboratory study are presented in Table5. The results showed that SDA constitutes 65.28% for SiO2, 2.75% for Fe2O3, and 5.52% for Al2O3, which indicates cumulative sum (SiO2 + Fe2O3 + Al2O3) of 73.55% to signify a good pozzolanic behavior according to ASTM C618-98 standard requirements. The chemical composition for the test cement indicates the possession of CaO, Al2O3, SiO2, and Fe2O3 at 13.45%, 20.06%, 51.4%, and 6.53%, respectively [12]. These constituents ensure an efficient cement hydration reaction and positive interaction with the pozzolanic material, through which the alumina silicates react with calcium hydroxide to form calcium silicate hydrates (C-S-H), which improve the mechanical properties of the green concrete produced. The test results for the physical properties shown in Table 6 present fineness moduli of 2.13, 2.7, and 2.86; moisture contents of 0.37%, 1.55%, and 1.04%; and compacted unit weights of 944 kg/m3, 1454 kg/m3, and 1598 kg/m3 for SDA, fine, and coarse aggregates, respectively [50, 51].Table 5
Chemical properties of cement (PLC) and SDA.
OxidesCaOMgOFe2O3Na2OAl2O3SiO2MnOLOITiO2K2O∗SDA (%)10.145.82.755.25.5265.280.014.55TraceTrace∗PLC (%)13.450.0936.531.0520.651.4Trace3.90.521.6∗Alaneme et al. [24].Table 6
Physical property.
PropertySDAFine agg.Coarse aggSpecific gravity2.252.642.78Fineness modulus2.132.72.86Bulk density (kg/m3)94414541598Absorption capacity (%)—2.051.02Moisture content (%)0.371.551.04


#### 3.1.1. Particle Size Distribution of Aggregates and SDA

The physical properties of the test aggregates and admixtures were further assessed through sieve analysis laboratory experiments to determine the gradation parameters of the test ingredients. The obtained result is presented in a semi-log graphical plot in Figure6, and the results for fine aggregates indicated that 98.23–0.13 percent finer grain size ranging from 3.4 mm to 75 μm; for SDA samples, 98.37–25.32 percent finer grain size ranging from 2 mm to 75 μm; and 96.5-12.3 percent finer grain size ranging from 20 to 2 mm for the coarse aggregate were derived. The coefficients of uniformity and curvature were calculated from the plotted results to evaluate the particle size distribution of the test ingredients as presented in Table 7, where D10, D30, and D60 indicate grain sizes, which are 10%, 30%, and 60% finer than the soil particles, respectively, as obtained from the semi-log plot. The computed gradation coefficients indicated well-graded samples and were observed to fall within BS 882 specifications for improved strength performance of concrete, which is in line with the findings of Simons [52] and Chimmaobi et al. [42].Figure 6
Sieve analysis graph.Table 7
Gradation coefficients.
Test materialsD60D30D10CcCuCoarse agg.5.131.90.952.8Fine agg.0.50.20.140.693.6SDA0.130.090.060.962.1


### 3.2. Cement vs. SDA Effects on the Target Responses

The interactions between the cement and pozzolanic material (SDA) to influence the concrete’s compressive and flexural strength responses are illustrated in graphical contour plots as shown in Figure7. The response contour plot presents the effects of varying replacement proportions of cement by SDA on the mechanical properties of the green concrete, which indicates a minimum compressive strength of 20.35 N/mm2 and a flexural strength of 3.21 N/mm2 with a mixture ratio of 0.5 : 0.5 for cement and SDA, respectively. However, the maximum compressive strength of 31.13 N/mm2 was obtained at 0.875 : 0.125 for cement and SDA, respectively; the control or unblended concrete mixture of 1 : 0 ratio produced the maximum flexural strength of 9.49 N/mm2, while 8.83 N/mm2, which is the second best, was achieved at 0.875 : 0.125 ratio. The strength responses tend to decline with further additions of SDA above 13% to replace cement in the concrete mixture from the experimental results presented in Table 8. The obtained experimental responses for the three replicates in each of the five experimental runs showed a low standard deviation result ranging from ±0.4–0.7, which indicated that the data are not spread out but clustered around or close to the computed mean response. The derived experimental responses showed that the replacement of cement by about 12.5% of SDA significantly enhanced the strength behavior of the concrete, which is also in accordance with the standard requirements of BS 8110: Part 1 and NCP 1 [53, 54]. The strength improvement may be due to the generation of calcium silicate hydrate (C-S-H) when hydrated lime abundant in the cement sample reacts with the aluminosilicates present in the pozzolanic material [55].Figure 7
Effects of SDA interaction with cement on the mechanical strength response.Table 8
Laboratory responses.
Actual valuesCompr. str.Flex. str.PseudovaluesExperimental runZ1:CementZ2:SDAN/mm2SDN/mm2SDX1:CementX2:SDA10.8750.12531.13±0.668.83±0.760.750.2520.750.2529.64±0.528.06±0.630.50.530.6250.37525.48±0.546.55±0.480.250.7540.50.520.35±0.473.21±0.480151028.87±0.709.49±0.7710


## 3.1. Test Cement and Admixture Characterization

The saw dust ash (SDA) and Portland limestone cement (PLC) chemical properties utilized for the laboratory study are presented in Table5. The results showed that SDA constitutes 65.28% for SiO2, 2.75% for Fe2O3, and 5.52% for Al2O3, which indicates cumulative sum (SiO2 + Fe2O3 + Al2O3) of 73.55% to signify a good pozzolanic behavior according to ASTM C618-98 standard requirements. The chemical composition for the test cement indicates the possession of CaO, Al2O3, SiO2, and Fe2O3 at 13.45%, 20.06%, 51.4%, and 6.53%, respectively [12]. These constituents ensure an efficient cement hydration reaction and positive interaction with the pozzolanic material, through which the alumina silicates react with calcium hydroxide to form calcium silicate hydrates (C-S-H), which improve the mechanical properties of the green concrete produced. The test results for the physical properties shown in Table 6 present fineness moduli of 2.13, 2.7, and 2.86; moisture contents of 0.37%, 1.55%, and 1.04%; and compacted unit weights of 944 kg/m3, 1454 kg/m3, and 1598 kg/m3 for SDA, fine, and coarse aggregates, respectively [50, 51].Table 5
Chemical properties of cement (PLC) and SDA.
OxidesCaOMgOFe2O3Na2OAl2O3SiO2MnOLOITiO2K2O∗SDA (%)10.145.82.755.25.5265.280.014.55TraceTrace∗PLC (%)13.450.0936.531.0520.651.4Trace3.90.521.6∗Alaneme et al. [24].Table 6
Physical property.
PropertySDAFine agg.Coarse aggSpecific gravity2.252.642.78Fineness modulus2.132.72.86Bulk density (kg/m3)94414541598Absorption capacity (%)—2.051.02Moisture content (%)0.371.551.04


### 3.1.1. Particle Size Distribution of Aggregates and SDA

The physical properties of the test aggregates and admixtures were further assessed through sieve analysis laboratory experiments to determine the gradation parameters of the test ingredients. The obtained result is presented in a semi-log graphical plot in Figure6, and the results for fine aggregates indicated that 98.23–0.13 percent finer grain size ranging from 3.4 mm to 75 μm; for SDA samples, 98.37–25.32 percent finer grain size ranging from 2 mm to 75 μm; and 96.5-12.3 percent finer grain size ranging from 20 to 2 mm for the coarse aggregate were derived. The coefficients of uniformity and curvature were calculated from the plotted results to evaluate the particle size distribution of the test ingredients as presented in Table 7, where D10, D30, and D60 indicate grain sizes, which are 10%, 30%, and 60% finer than the soil particles, respectively, as obtained from the semi-log plot. The computed gradation coefficients indicated well-graded samples and were observed to fall within BS 882 specifications for improved strength performance of concrete, which is in line with the findings of Simons [52] and Chimmaobi et al. [42].Figure 6
Sieve analysis graph.Table 7
Gradation coefficients.
Test materialsD60D30D10CcCuCoarse agg.5.131.90.952.8Fine agg.0.50.20.140.693.6SDA0.130.090.060.962.1


## 3.1.1. Particle Size Distribution of Aggregates and SDA

The physical properties of the test aggregates and admixtures were further assessed through sieve analysis laboratory experiments to determine the gradation parameters of the test ingredients. The obtained result is presented in a semi-log graphical plot in Figure6, and the results for fine aggregates indicated that 98.23–0.13 percent finer grain size ranging from 3.4 mm to 75 μm; for SDA samples, 98.37–25.32 percent finer grain size ranging from 2 mm to 75 μm; and 96.5-12.3 percent finer grain size ranging from 20 to 2 mm for the coarse aggregate were derived. The coefficients of uniformity and curvature were calculated from the plotted results to evaluate the particle size distribution of the test ingredients as presented in Table 7, where D10, D30, and D60 indicate grain sizes, which are 10%, 30%, and 60% finer than the soil particles, respectively, as obtained from the semi-log plot. The computed gradation coefficients indicated well-graded samples and were observed to fall within BS 882 specifications for improved strength performance of concrete, which is in line with the findings of Simons [52] and Chimmaobi et al. [42].Figure 6
Sieve analysis graph.Table 7
Gradation coefficients.
Test materialsD60D30D10CcCuCoarse agg.5.131.90.952.8Fine agg.0.50.20.140.693.6SDA0.130.090.060.962.1


## 3.2. Cement vs. SDA Effects on the Target Responses

The interactions between the cement and pozzolanic material (SDA) to influence the concrete’s compressive and flexural strength responses are illustrated in graphical contour plots as shown in Figure7. The response contour plot presents the effects of varying replacement proportions of cement by SDA on the mechanical properties of the green concrete, which indicates a minimum compressive strength of 20.35 N/mm2 and a flexural strength of 3.21 N/mm2 with a mixture ratio of 0.5 : 0.5 for cement and SDA, respectively. However, the maximum compressive strength of 31.13 N/mm2 was obtained at 0.875 : 0.125 for cement and SDA, respectively; the control or unblended concrete mixture of 1 : 0 ratio produced the maximum flexural strength of 9.49 N/mm2, while 8.83 N/mm2, which is the second best, was achieved at 0.875 : 0.125 ratio. The strength responses tend to decline with further additions of SDA above 13% to replace cement in the concrete mixture from the experimental results presented in Table 8. The obtained experimental responses for the three replicates in each of the five experimental runs showed a low standard deviation result ranging from ±0.4–0.7, which indicated that the data are not spread out but clustered around or close to the computed mean response. The derived experimental responses showed that the replacement of cement by about 12.5% of SDA significantly enhanced the strength behavior of the concrete, which is also in accordance with the standard requirements of BS 8110: Part 1 and NCP 1 [53, 54]. The strength improvement may be due to the generation of calcium silicate hydrate (C-S-H) when hydrated lime abundant in the cement sample reacts with the aluminosilicates present in the pozzolanic material [55].Figure 7
Effects of SDA interaction with cement on the mechanical strength response.Table 8
Laboratory responses.
Actual valuesCompr. str.Flex. str.PseudovaluesExperimental runZ1:CementZ2:SDAN/mm2SDN/mm2SDX1:CementX2:SDA10.8750.12531.13±0.668.83±0.760.750.2520.750.2529.64±0.528.06±0.630.50.530.6250.37525.48±0.546.55±0.480.250.7540.50.520.35±0.473.21±0.480151028.87±0.709.49±0.7710


## 4. Development, Formulation, and Validation of the EVD Model

The experimental responses derived during the laboratory works constitute the system database for the EVD model development to evaluate the mixture experiment optimization problem making use of the square root (λ = 0.5) or quadratic power transformation. The compressive and flexural strength responses range from 20.35 to 31.13 with a maximum-to-minimum ratio of 1.53 and 3.21 to 9.49 with a maximum-to-minimum ratio of 2.96, respectively. The fit statistics evaluation, specification of adequate model type and mixture order, statistical diagnostics, and graphical and numerical optimization are executed to ascertain optimal combination of the two-component mixture to assess its effects on the mechanical strength responses. Postanalysis of the statistical results like point prediction, confirmation, derivation of the model coefficients in the coefficient table, and finally simulation of the EVD model was further carried to affirm its performance and applicability of the green concrete prediction model for field works [27, 56].


### 4.1. Fit Summary Statistics

Fit summary is the foremost statistical tests carried out on the datasets to determine the suitability, type, or standard in order to meet the require modelling purpose using L_Pseudo mixture component coding. To achieve this, the fit summary assembles the essential statistics so as to choose most suitable starting point for the formulation of the EVD model. Using the Whitcomb score, the models suggested are selected and are expected to be considered adequate starting point for the model fitting [57]. This important preliminary statistical computation illustrates the fit summary to search for models using performance indices such as coefficient of determination (R-sqd.), PRESS, which helps to assess how well the sought-after models fit each point in the design, lack of fit tests, and sequential model sum of squares which helps to choose the highest polynomial order with significant additional terms as shown in Tables 9–14. The presented fit statistical results showed cubic and quadratic models with R-sqd. of 1.0 and 0.9792 for compressive and flexural strength responses, respectively. The lack of fit computation outputdiagnoses how each of the sorted models to fit the data, from the sequential sum of squares p value of 0.0489 and 0.0297 for compressive and flexural strength responses respectively [58, 59].Table 9
Model summary statistics for compressive strength response.
SourceStd. dev.R-sqd.Adjd.R-sqd.Pred.R-sqd.PRESSLinear0.270.69610.5947−0.29730.95Quadratic0.0450.99440.98880.90370.070Cubic4.896E − 0031.00000.99990.99513.612E − 003 SuggestedPRESS: predicted sum of squares.Table 10
Lack of fit for compressive strength response.
SourceSequentialp valueAdjd.R-sqd.Pred.R-sqd.Linear0.07890.5947−0.2973Quadratic0.00920.98880.9037Cubic0.04890.99990.9951 SuggestedTable 11
Sequential model sum of squares (type I) for compressive strength response.
SourceSSdfMSF valuep value prob > FMean vs. total134.741134.74Linear vs. mean0.5110.516.870.0789Quadratic vs. linear0.2210.22106.980.0092Cubic vs. quadratic4.053E − 00314.053E − 003169.080.0489 SuggestedQuartic vs. cubic2.397E − 00512.397E − 005Residual0.0000Total135.47527.09SS: sum of squares, MS: mean square, and df: degree of freedom.Table 12
Model summary statistics for flexural strength response.
SourceStd. dev.R-sqd.Adjd.R-sqd.Pred.R-sqd.PRESSLinear0.240.83600.78130.30980.74 SuggestedQuadratic0.110.97920.95840.64700.38 SuggestedCubic0.0260.99940.99750.90520.10Table 13
Lack of fit for flexural strength response.
SourceSequentialp valueAdjd.R-sqd.Pred.R-sqd.Linear0.02970.78130.3098 SuggestedQuadratic0.03560.95840.6470 SuggestedCubic0.11130.99750.9052Table 14
Sequential model sum of squares (type I) for flexural strength response.
SourceSSdfMSF valuep value prob > FMean vs. total35.07135.07Linear vs. mean0.8910.8915.290.0297 SuggestedQuadratic vs. linear0.1510.1513.770.0656 SuggestedCubic vs. quadratic0.02210.02232.070.1113Quartic vs. cubic6.727E − 00416.727E − 004Residual0.0000Total36.1457.23SS: sum of squares, MS: mean-square, and df: degree of freedom.


### 4.2. ANOVA Test Result

After the selection of a suitable model preferentially suggested during the fit statistical computation, analysis of variance (ANOVA) is executed where the descriptive and statistical tests are presented with low P value criteria used for the identification of important terms in the model.p value helps to ascertain when the model explains significant portion of the variance. Using the preferred cubic and quadratic mixture models for compressive and flexural strength, respectively, the ANOVA type III partial sum of squares and R-sqd. statistical computations were executed as shown in Tables 15–18. From the obtained results, p values of 0.0073 and 0.0208 were derived for the compressive and flexural strength responses, respectively [60].Table 15
ANOVA cubic mixture model for compressive strength response.
∗∗∗ mixture component coding is L_Pseudo ∗∗∗Analysis of variance table (partial sum of squares—type III)SourceSSdfMSF valuep value prob > FModel0.7330.2410163.800.0073 Significant1Linear Mixture0.5110.5121224.380.0044AB0.2210.229097.930.0067AB(A − B)4.053E − 00314.053E − 003169.080.0489Residual2.397E − 00512.397E − 005Cor total0.734^1Linear mixtures inferences use SS type I.Table 16
R-sqd. calculations for compressive strength response.
Std. dev.4.896E−003R-squared1.0000Mean5.19Adjd.R-sqd.0.9999C.V. %0.094Pred.R-sqd.0.9951PRESS3.612E − 003Adeq.-prec.244.624−2 log likelihood−47.05BIC−42.22AICc−17.05Table 17
ANOVA for the quadratic mixture model for flexural strength response.
∗∗∗ mixture component coding is L_pseudo. ∗∗∗Analysis of variance table (partial sum of squares—type III)SourceSSdfMSF valuep value prob >FModel1.0520.5247.080.0208 Significant1Linear mixture0.8910.8980.380.0122AB0.1510.1513.770.0656Residual0.02220.011Cor total1.074^1Linear mixtures inferences use SS type I.Table 18
R-sqd. calculations for flexural strength response.
Std. Dev.0.11R-sqd.0.9792Mean2.65Adjd.R-sqd.0.9584C.V. %3.98Pred.R-sqd.0.9470PRESS0.38Adeq.-prec.114.822−2 log likelihood−12.89BIC−9.67AICc−2.89The modelF value of 10163.80 implies the model is significant. There is only a 0.73% chance that an F value this large could occur due to noise. Values of “Prob > F” less than 0.0500 indicate model terms are significant. In this case, A, B, AB, and AB (A−B) are significant model terms.The Pred.R-Sqd of 0.9951 is in reasonable agreement with the Adjd. R-Sqd. of 0.9999; that is, the difference between Adjd. R-sqd., and Pred. R-sqd is less than 0.2. Adeq.-Prec. measures the signal-to-noise ratio. A ratio greater than 4 is desirable. The computed ratio of 244.624 indicates an adequate signal. This model can be used to navigate the design space.The model F value of 47.08 implies the model is significant. There is only a 2.08% chance that an F value this large could occur due to noise. Values of “Prob >F” less than 0.0500 indicate model terms are significant. In this case, A, B are significant model terms. Values greater than 0.1000 indicate the model terms are not significant [56].The Pred.R-Sqd. of 0.9470 is in reasonable agreement with the Adjd. R-Sqd. of 0.9584; that is, the difference is less than 0.2. Adeq.-Prec. measures the signal-to-noise ratio. A ratio greater than 4 is desirable. The ratio of 14.822 indicates an adequate signal. This model can be used to navigate the design space [43].


### 4.3. Derived Coefficient Estimates and Model Equations

The model coefficients are calculated through the rigorous statistical computations to arrive at suitable equations for the response predictions as carried out using Design Expert software. The calculations clearly present the standard error, variance inflation factor (VIF), low and high confidence intervals, and coefficient estimate for the compressive and flexural strength responses as shown in Tables19–22. VIF assesses the effects of lack of orthogonality on the variance of the derived model coefficients. Hence, the square root of VIF is directly proportional to the model’s standard error [61].Table 19
Model coefficient calculation results for compressive strength response.
ComponentCoefficient estimatedfStandard error95% CI low95% CI highVIFA-cement5.3714.861E − 0035.315.431.85B-SDA4.5114.861E − 0034.454.571.85AB2.0010.0211.732.262.43AB(A − B)0.5410.0410.0121.061.25Table 20
Final equation in terms of L_pseudo components.
Coding∗A∗B∗AB∗AB (A − B)L_Pseudosqrt. (compressive) =+5.37+4.51+2+0.54Table 21
Model coefficients calculation results for flexural strength response.
ComponentCoefficient estimatedfStandard error95% CI low95% CI highVIFA-cement3.0410.0992.613.461.66B-SDA1.8410.0991.412.271.66AB1.6710.45-0.273.612.43Table 22
Final equation in terms of L_pseudo components.
Coding∗A∗B∗ABL_Pseudosqrt. (flexural l) =+3.04+1.84+1.67The equation in terms of coded factors can be used to make predictions about the response for given levels of each factor. By default, the high levels of the factors are coded as +1, and the low levels of the factors are coded as −1. The coded equation is useful for identifying the relative impact of the factors by comparing the factor coefficients [62].


### 4.4. Diagnostics Plots

The diagnostics statistical graphs, which are presented in form of scattered plots of residuals or model prediction errors against the values predicted to check whether the estimation produced can be enhanced further. It is also deployed to evaluate the goodness-of-fit of the developed EVD model using studentized residuals and to affirm the satisfaction of the regression assumption conditions to figure out undue observations, which may pose a significant influence on the results of the analysis [63]. The derived residuals’ standard errors are different unless the experimental runs’ leverages in the design are identical, which indicates that raw residuals belong to varying populations and are not adequate for regression assumptions evaluation. However, studentized residuals are preferable to map all normal distributions in different dimensions to unitary distribution. With respect to the desired response variables, diagnostic statistical tests carried out in this analysis include predicted vs. residual, normal probability, run plot vs. residuals, predicted vs. actual, and box-cox power transformation to help detect problems with the analysis using outliers [64].


#### 4.4.1. Normal Probability Plot

This is observed as the first diagnostic statistical evaluation plot, which helps to validate the model residuals followed a normal standard distribution nearby the regression line-of-fit. In this graph, the externally studentized residuals is plotted against the normal probability (%) outcome. Moreover, for better analysis results, definite patterns are sort for like curve-linear shapes, which may signify the need for transformation of the target response, even though some calculated points may scatter with normal data. Normal probability plot for the compressive and flexural strength responses are presented in Figure8 [65].Figure 8
Residuals normal probability plots for the target responses.


#### 4.4.2. Predicted vs. Studentized Residuals Plot

This diagnostic evaluation test corroborates the constant variance of the residual assumption presented in a scattered plot with the predicted outcomes on thex-axis and the studentized residuals on the y-axis as presented in Figure 9 for the target responses under investigation. The plotted statistical results indicated clustering at the residual of zero points with max to min values of 3.5 to −3.5 and 63.66 to −63.66 for compressive and flexural strength responses, respectively [66].Figure 9
Residual vs. predicted diagnostic statistical graphs.


#### 4.4.3. Experimental Run vs. Studentized Residual Plot

This statistical diagnostic test illustrates the influence of the experimental run order versus the studentized residuals of the model on the target responses. This assessment plot fundamentally seeks for prowling factors, which may impact the criterion parameters during the experimental exercises as shown in Figure10. The graphical plots are similar to the residuals vs. model predicted diagnostic plots, and the design points were positioned near the line of zero points and also to lie between ±3.5and ±63.666, which signifies a time-dependent parameter in a random scattered plot for compressive and flexural strength responses, respectively [67].Figure 10
Residual vs. experimental run diagnostic graphs.


#### 4.4.4. Predicted vs. Laboratory Results

This analytical plot illustrates the behavior of the actual or laboratory responses on thex-axis in respect to the EVD model predicted values on the y-axis of the graph. Moreover, this statistical plot helps to evaluate the prediction accuracy performance of the EVD model so as to determine design points and groups, which the model cannot certainly predict, and to assess the relationship between the estimated and experimental results as shown in Figure 11. The results obtained indicate the datasets ranging from 4.4 to 5.6 and 1.6 to 3.6 for the square root responses of the compressive and flexural responses, respectively [68].Figure 11
Predicted vs. actual diagnostic graphs.


#### 4.4.5. Power Transformation through Box-Cox Plot

This diagnostic statistical plot offers necessary requirements for the selection of adequate law power transformation to evaluate the effects on the response variables. Based on the obtained best lambda value is derived at the lowest curve point generated by the sum of squares (SS) residuals natural log, an indorsed transformation is preferred. The box-cox plot presents the lambda values on thex-axis and the SS natural log on the y-axis as shown in Figure 12, the derived results from the plots showed a current lambda of 0.5, low and high at 95% confidence interval of −0.79 and 0.34 lambda, respectively, with the best lambda of −0.25 for compressive strength response. However, the flexural strength response showed the best lambda of 2.85, low and high at 95% confidence interval of 1.59 and 3.79 lambda, respectively [65, 69].Figure 12
Box-cox plots for power transformation.


### 4.5. Diagnostic Influence Plots

This diagnostic graphical presentation helps to assess the effects or influence of potential distinct experimental runs on the analysis outcomes. These graphical charts offer a clearer statistical perspective when two or single case sticks out from other groups in the experimental design. The diagnostic influence computation was executed using the following essential tools, namely, leverage vs. experimental run, cook’s distance, and statistical difference in fits (DFFITS) vs. run of experiments [70].


#### 4.5.1. Cook’s Distance

Cook’s distance is deployed in regression statistics to determine influential outliers in regressor or independent variable sets. It is also utilized to detect experimental points in the design, which affects the developed EVD model negatively and feasible regions or planes where good correlation is obtained as shown in Figure13. The graphical plots illustrates the computed cook’s distance value on the y-axis and the five design number of experiments on the x-axis. The derived results indicated Cook’s distance ranging from 0 to 20 for the compressive strength response and that of flexural strength response lie between 0 and 5. Interestingly, the run number of experiments 4 and 5 was observed to be positioned above the 0–1 boundary line [71].Figure 13
Cook’s distance influence plot.


#### 4.5.2. Leverage vs. Run

Leverage is deployed in statistical diagnostic influence to assess the extent to which each design point in the experiments impacts the goodness of fit. It is a degree of the distance apart that the regressor values of an observation are from host of other observations in the experimental design. Point of leverage is an observation that possesses an unfamiliar predicted score from the bulk of varying observations. The leverage versus the experimental run statistical influence plot presents the graph of the computed leverage points on they-axis against the experimental runs on the horizontal axis for the target responses as shown in Figure 14. The calculated leverage point ranges from 0 to 1, when 1 signifies that the generated predictive model at the point of investigation fits the observation exactly. Furthermore, the results derived from the graph indicate boundary points from 0 to 0.8 and 0 to 0.6 for compressive and flexural strength responses, respectively, while points for experimental runs 4 and 5 were observed to lie above the boundary lines and at a leverage of approximately 1.0 [31, 61].Figure 14
Leverage vs. run diagnostic influence plot.


#### 4.5.3. Statistical Difference in Fits (DFFITS) vs. Runs

DFFITS statistic is a studentized diagnostic influence instrument, which provides a scaled assessment of the predicted response variation for theith observation. It also illustrates the variability in the model predicted results at a specific design experimental point when it is excluded for model fitting as presented in Figure 15. The plotted graphical results indicate the DFFITS values on the y-axis against the five experimental runs on the x-axis, and the design point was observed to be situated at the zero DFFITS region within the ±2.683 for the compressive strength response. For the flexural strength response, the design points were situated between 5 and 15 DFFITS regions, while three points fell within the boundary limit of ±2.323 [72].Figure 15
Diagnostic influence plot of DFFITS vs. experimental run.


### 4.6. Optimization Overview

Once the diagnostic statistical analysis and influence graphical calculations were achieved, numerical optimization is carried out using desirability function, which evaluates the imposed optimization criteria on the model variables that maximizes the target response parameters. To accomplish this task, the features of the objective function are analytically adjusted through weight functions modification in line with the set model variable criteria. These multicollinearity conditions would enable the attainment of favorable conditions to obtain a desirability score of 1.0 through a scale of0≤dyi≤1 boundary conditions [73]. The optimization element of this experimental design explores for the mixture ratios combination in the feasible factor space, which satisfies simultaneously the formulated and imposed criteria on the response parameters and corresponding factor levels. The expected optimization goal is set to maximize the target responses, while the two-component combination ratios are set at in-range option to determine the optimal proportion of the factor levels to produce a maximum response as shown in Table 23. The lower and upper limits of the variables were derived from the system datasets derived from the laboratory experiments [74].Table 23
Model variable criteria for optimization.
NameGoalLower limitUpper limitLower weightUpper weightImportanceA: cementis in range0.51113B: SDAis in range00.5113CompressiveMaximize20.3531.13113FlexuralMaximize3.219.49113The optimization solution derived from the analytical proceedings of the mixture experiment designs is presented in Table24. From the derived results, an optimal desirability score of 0.99 was obtained at a combination ratio of 0.89 : 0.110 to produce maximized compressive and flexural strength of 31.102 N/mm2 and 9.384 N/mm2, respectively [75].Table 24
Optimization solutions.
NumberCementSDACompressiveFlexuralDesirability10.8900.11031.1029.3840.990Selected


#### 4.6.1. Graphical Optimization Plots of Ramps and Bar

The graphical optimization output is illustrated in the form of ramps and bar graphs for a better understanding of the outcomes. The ramps and bar chart results present the optimal solution for the predictor variables in red and the target response parameters in blue coloration to elucidate the desirability-function scores of the constituting model variables as shown in Figures16–18. The results presented indicate an independent variable desirability score of 1.0, the compressive strength score was observed as 0.997 and 0.9831 for the flexural strength response, while the combined optimal desirability score of 0.9902 was calculated [29].Figure 16
Ramps graph for the desirability function.Figure 17
Bar graph for the desirability function.Figure 18
Graphical plot of two-component mixture optimization.


### 4.7. EVD Model Postanalysis and Simulation

This involves point prediction, which gives room for the factor values to be reviewed; confirmation, which links the interval of the EVD model estimation to derive sample mean; the coefficient table, which presents the significance and magnitude coefficients for all target response parameters analyzed; and simulation to validate the developed results and derived statistical analysis results [76].


#### 4.7.1. Analysis of Point Prediction

This statistical postanalysis tool utilizes the fit computation of the developed model and the prescribed settings for the factor tools to determine interval estimates after the analysis is successfully carried out as shown in Table25. The result indicated standard deviations of 0.055 and 0.65, low to high value of 30.4973–1.7125 and 7.7803–11.138 for compressive and flexural strength, respectively, at 95% confidence interval [77].Table 25
Point prediction.
ResponsePredicted meanPredicted medianStd. dev.CI for 95% CI lowMean 95% CI high99% of 95% TI lowPopulation 95% TI highCompressive31.102231.10210.054610230.497931.712328.609233.6991Flexural9.383999.372870.6459567.780311.1382.8088119.7765


#### 4.7.2. Coefficient Table

This postanalysis computation tool presents the proportion combination magnitude of the 2-component mixture of SDA-cement green concrete as presented in Table26. The coefficients are derived from the multicriteria optimization computation using the extreme vertex design method to obtain acceptable performance. Cubic and quadratic models were deployed for the exercise based on the fits statistics recommendation for compressive and flexural strength, respectively, to achieve better generalization of the derived experimental data. The legend column was used to assess the p values for every term of the models where p values <0.01 signify the best measures for strong significant presented in red, p values ≤0.05 and ≥0.01 indicate moderate level of significance presented in green, p values ≤0.10 and ≥0.05 signify slight level of significance presented in blue, while p values ≥0.10 indicate the level of noninsignificant displayed in black [78, 79].Table 26
Coefficient table.
ResponseABABAB (A − B)Sqrt (compressive)5.37254.510511.996990.536869p=0.00440.00440.00670.0489Sqrt (flexural)3.037241.841191.67359p=0.01220.01220.0656Legendp<0.010.01≤p<0.050.05≤p<0.10p≥0.10


#### 4.7.3. EVD Model Simulation

The model simulation is carried out on the developed EVD model as the last phase of the statistical analysis of the design experiment optimization of two-component mixture problem. This statistical verification is imperative so as to replicate nontrivial scenario for the case study to give adequate and efficient practical guide for field operators, consultants or designers, and contractors on the expected performance of the developed model on real-life application. The simulation is necessary to validate the process of developing the EVD model and to affirm that the results obtained during the statistical fits and diagnostics influence computation can be realized when applied in the field works. A graphical plot showing the experimental derived responses vs. EVD model-simulated results is presented in Figure19. These data were compared statistically using F-test two-sample variances and student’s t-test to check the significant difference and degree of correlation between them at 95% confidence interval as shown in Tables 27–30. From the statistical results achieved using Microsoft Excel software, p(F ≤ f) one-tail of 0.476 and 0.381 was computed, which is >0.05 for compressive and flexural strength responses, respectively, for the F-test statistics. Moreover, regarding the student’s t-test results, p(T ≤ t) two-tail of 0.702 and 0.591 was calculated for pcompressive and flexural strength responses, respectively. The derived statistical results showed that there is no significant difference between the actual and model predicted results, which signifies acceptable model performance in line with the findings of Sadrmomtazi et al. [80] and Alaneme et al. [81].Figure 19
Actual vs. EVD model predicted responses.Table 27
F-Test two-sample for variances for compressive strength response.
Statistical FactorsCompr. str.-actualCompr. str.-modelMean27.09426.918Variance18.5030317.34397Observations55Df44F1.066828p(F ≤ f) one-tail0.475758F critical one-tail6.388233Table 28
F-Test two-sample for variances for flexural strength response.
Statistical factorsFlex. str.-actualFlex. str.-modelMean7.2287.362Variance6.244828.62257Observations55Df44F0.724241p(F ≤ f) one-tail0.381075F critical one-tail0.156538Table 29
T-Test: paired two-sample test for means for compressive strength response.
Statistical FactorsCompr. str.-actualCompr. str.-modelMean27.09426.918Variance18.5030317.34397Observations55Pearson correlation0.974904Df4t Stat0.410776p(T ≤ t) one-tail0.351145t Critical one-tail2.131847p(T ≤ t) two-tail0.702289t Critical two-tail2.776445Table 30
T-Test: paired two-sample test for means for flexural strength response.
Statistical FactorsFlex. str.-actualFlex. str.-modelMean7.2287.362Variance6.244828.62257Observations55Pearson correlation0.995076Df4t Stat−0.58357p(T ≤ t) one-tail0.295416t Critical one-tail2.131847p(T ≤ t) two-tail0.590832t Critical two-tail2.776445


## 4.1. Fit Summary Statistics

Fit summary is the foremost statistical tests carried out on the datasets to determine the suitability, type, or standard in order to meet the require modelling purpose using L_Pseudo mixture component coding. To achieve this, the fit summary assembles the essential statistics so as to choose most suitable starting point for the formulation of the EVD model. Using the Whitcomb score, the models suggested are selected and are expected to be considered adequate starting point for the model fitting [57]. This important preliminary statistical computation illustrates the fit summary to search for models using performance indices such as coefficient of determination (R-sqd.), PRESS, which helps to assess how well the sought-after models fit each point in the design, lack of fit tests, and sequential model sum of squares which helps to choose the highest polynomial order with significant additional terms as shown in Tables 9–14. The presented fit statistical results showed cubic and quadratic models with R-sqd. of 1.0 and 0.9792 for compressive and flexural strength responses, respectively. The lack of fit computation outputdiagnoses how each of the sorted models to fit the data, from the sequential sum of squares p value of 0.0489 and 0.0297 for compressive and flexural strength responses respectively [58, 59].Table 9
Model summary statistics for compressive strength response.
SourceStd. dev.R-sqd.Adjd.R-sqd.Pred.R-sqd.PRESSLinear0.270.69610.5947−0.29730.95Quadratic0.0450.99440.98880.90370.070Cubic4.896E − 0031.00000.99990.99513.612E − 003 SuggestedPRESS: predicted sum of squares.Table 10
Lack of fit for compressive strength response.
SourceSequentialp valueAdjd.R-sqd.Pred.R-sqd.Linear0.07890.5947−0.2973Quadratic0.00920.98880.9037Cubic0.04890.99990.9951 SuggestedTable 11
Sequential model sum of squares (type I) for compressive strength response.
SourceSSdfMSF valuep value prob > FMean vs. total134.741134.74Linear vs. mean0.5110.516.870.0789Quadratic vs. linear0.2210.22106.980.0092Cubic vs. quadratic4.053E − 00314.053E − 003169.080.0489 SuggestedQuartic vs. cubic2.397E − 00512.397E − 005Residual0.0000Total135.47527.09SS: sum of squares, MS: mean square, and df: degree of freedom.Table 12
Model summary statistics for flexural strength response.
SourceStd. dev.R-sqd.Adjd.R-sqd.Pred.R-sqd.PRESSLinear0.240.83600.78130.30980.74 SuggestedQuadratic0.110.97920.95840.64700.38 SuggestedCubic0.0260.99940.99750.90520.10Table 13
Lack of fit for flexural strength response.
SourceSequentialp valueAdjd.R-sqd.Pred.R-sqd.Linear0.02970.78130.3098 SuggestedQuadratic0.03560.95840.6470 SuggestedCubic0.11130.99750.9052Table 14
Sequential model sum of squares (type I) for flexural strength response.
SourceSSdfMSF valuep value prob > FMean vs. total35.07135.07Linear vs. mean0.8910.8915.290.0297 SuggestedQuadratic vs. linear0.1510.1513.770.0656 SuggestedCubic vs. quadratic0.02210.02232.070.1113Quartic vs. cubic6.727E − 00416.727E − 004Residual0.0000Total36.1457.23SS: sum of squares, MS: mean-square, and df: degree of freedom.


## 4.2. ANOVA Test Result

After the selection of a suitable model preferentially suggested during the fit statistical computation, analysis of variance (ANOVA) is executed where the descriptive and statistical tests are presented with low P value criteria used for the identification of important terms in the model.p value helps to ascertain when the model explains significant portion of the variance. Using the preferred cubic and quadratic mixture models for compressive and flexural strength, respectively, the ANOVA type III partial sum of squares and R-sqd. statistical computations were executed as shown in Tables 15–18. From the obtained results, p values of 0.0073 and 0.0208 were derived for the compressive and flexural strength responses, respectively [60].Table 15
ANOVA cubic mixture model for compressive strength response.
∗∗∗ mixture component coding is L_Pseudo ∗∗∗Analysis of variance table (partial sum of squares—type III)SourceSSdfMSF valuep value prob > FModel0.7330.2410163.800.0073 Significant1Linear Mixture0.5110.5121224.380.0044AB0.2210.229097.930.0067AB(A − B)4.053E − 00314.053E − 003169.080.0489Residual2.397E − 00512.397E − 005Cor total0.734^1Linear mixtures inferences use SS type I.Table 16
R-sqd. calculations for compressive strength response.
Std. dev.4.896E−003R-squared1.0000Mean5.19Adjd.R-sqd.0.9999C.V. %0.094Pred.R-sqd.0.9951PRESS3.612E − 003Adeq.-prec.244.624−2 log likelihood−47.05BIC−42.22AICc−17.05Table 17
ANOVA for the quadratic mixture model for flexural strength response.
∗∗∗ mixture component coding is L_pseudo. ∗∗∗Analysis of variance table (partial sum of squares—type III)SourceSSdfMSF valuep value prob >FModel1.0520.5247.080.0208 Significant1Linear mixture0.8910.8980.380.0122AB0.1510.1513.770.0656Residual0.02220.011Cor total1.074^1Linear mixtures inferences use SS type I.Table 18
R-sqd. calculations for flexural strength response.
Std. Dev.0.11R-sqd.0.9792Mean2.65Adjd.R-sqd.0.9584C.V. %3.98Pred.R-sqd.0.9470PRESS0.38Adeq.-prec.114.822−2 log likelihood−12.89BIC−9.67AICc−2.89The modelF value of 10163.80 implies the model is significant. There is only a 0.73% chance that an F value this large could occur due to noise. Values of “Prob > F” less than 0.0500 indicate model terms are significant. In this case, A, B, AB, and AB (A−B) are significant model terms.The Pred.R-Sqd of 0.9951 is in reasonable agreement with the Adjd. R-Sqd. of 0.9999; that is, the difference between Adjd. R-sqd., and Pred. R-sqd is less than 0.2. Adeq.-Prec. measures the signal-to-noise ratio. A ratio greater than 4 is desirable. The computed ratio of 244.624 indicates an adequate signal. This model can be used to navigate the design space.The model F value of 47.08 implies the model is significant. There is only a 2.08% chance that an F value this large could occur due to noise. Values of “Prob >F” less than 0.0500 indicate model terms are significant. In this case, A, B are significant model terms. Values greater than 0.1000 indicate the model terms are not significant [56].The Pred.R-Sqd. of 0.9470 is in reasonable agreement with the Adjd. R-Sqd. of 0.9584; that is, the difference is less than 0.2. Adeq.-Prec. measures the signal-to-noise ratio. A ratio greater than 4 is desirable. The ratio of 14.822 indicates an adequate signal. This model can be used to navigate the design space [43].


## 4.3. Derived Coefficient Estimates and Model Equations

The model coefficients are calculated through the rigorous statistical computations to arrive at suitable equations for the response predictions as carried out using Design Expert software. The calculations clearly present the standard error, variance inflation factor (VIF), low and high confidence intervals, and coefficient estimate for the compressive and flexural strength responses as shown in Tables19–22. VIF assesses the effects of lack of orthogonality on the variance of the derived model coefficients. Hence, the square root of VIF is directly proportional to the model’s standard error [61].Table 19
Model coefficient calculation results for compressive strength response.
ComponentCoefficient estimatedfStandard error95% CI low95% CI highVIFA-cement5.3714.861E − 0035.315.431.85B-SDA4.5114.861E − 0034.454.571.85AB2.0010.0211.732.262.43AB(A − B)0.5410.0410.0121.061.25Table 20
Final equation in terms of L_pseudo components.
Coding∗A∗B∗AB∗AB (A − B)L_Pseudosqrt. (compressive) =+5.37+4.51+2+0.54Table 21
Model coefficients calculation results for flexural strength response.
ComponentCoefficient estimatedfStandard error95% CI low95% CI highVIFA-cement3.0410.0992.613.461.66B-SDA1.8410.0991.412.271.66AB1.6710.45-0.273.612.43Table 22
Final equation in terms of L_pseudo components.
Coding∗A∗B∗ABL_Pseudosqrt. (flexural l) =+3.04+1.84+1.67The equation in terms of coded factors can be used to make predictions about the response for given levels of each factor. By default, the high levels of the factors are coded as +1, and the low levels of the factors are coded as −1. The coded equation is useful for identifying the relative impact of the factors by comparing the factor coefficients [62].


## 4.4. Diagnostics Plots

The diagnostics statistical graphs, which are presented in form of scattered plots of residuals or model prediction errors against the values predicted to check whether the estimation produced can be enhanced further. It is also deployed to evaluate the goodness-of-fit of the developed EVD model using studentized residuals and to affirm the satisfaction of the regression assumption conditions to figure out undue observations, which may pose a significant influence on the results of the analysis [63]. The derived residuals’ standard errors are different unless the experimental runs’ leverages in the design are identical, which indicates that raw residuals belong to varying populations and are not adequate for regression assumptions evaluation. However, studentized residuals are preferable to map all normal distributions in different dimensions to unitary distribution. With respect to the desired response variables, diagnostic statistical tests carried out in this analysis include predicted vs. residual, normal probability, run plot vs. residuals, predicted vs. actual, and box-cox power transformation to help detect problems with the analysis using outliers [64].


### 4.4.1. Normal Probability Plot

This is observed as the first diagnostic statistical evaluation plot, which helps to validate the model residuals followed a normal standard distribution nearby the regression line-of-fit. In this graph, the externally studentized residuals is plotted against the normal probability (%) outcome. Moreover, for better analysis results, definite patterns are sort for like curve-linear shapes, which may signify the need for transformation of the target response, even though some calculated points may scatter with normal data. Normal probability plot for the compressive and flexural strength responses are presented in Figure8 [65].Figure 8
Residuals normal probability plots for the target responses.


### 4.4.2. Predicted vs. Studentized Residuals Plot

This diagnostic evaluation test corroborates the constant variance of the residual assumption presented in a scattered plot with the predicted outcomes on thex-axis and the studentized residuals on the y-axis as presented in Figure 9 for the target responses under investigation. The plotted statistical results indicated clustering at the residual of zero points with max to min values of 3.5 to −3.5 and 63.66 to −63.66 for compressive and flexural strength responses, respectively [66].Figure 9
Residual vs. predicted diagnostic statistical graphs.


### 4.4.3. Experimental Run vs. Studentized Residual Plot

This statistical diagnostic test illustrates the influence of the experimental run order versus the studentized residuals of the model on the target responses. This assessment plot fundamentally seeks for prowling factors, which may impact the criterion parameters during the experimental exercises as shown in Figure10. The graphical plots are similar to the residuals vs. model predicted diagnostic plots, and the design points were positioned near the line of zero points and also to lie between ±3.5and ±63.666, which signifies a time-dependent parameter in a random scattered plot for compressive and flexural strength responses, respectively [67].Figure 10
Residual vs. experimental run diagnostic graphs.


### 4.4.4. Predicted vs. Laboratory Results

This analytical plot illustrates the behavior of the actual or laboratory responses on thex-axis in respect to the EVD model predicted values on the y-axis of the graph. Moreover, this statistical plot helps to evaluate the prediction accuracy performance of the EVD model so as to determine design points and groups, which the model cannot certainly predict, and to assess the relationship between the estimated and experimental results as shown in Figure 11. The results obtained indicate the datasets ranging from 4.4 to 5.6 and 1.6 to 3.6 for the square root responses of the compressive and flexural responses, respectively [68].Figure 11
Predicted vs. actual diagnostic graphs.


### 4.4.5. Power Transformation through Box-Cox Plot

This diagnostic statistical plot offers necessary requirements for the selection of adequate law power transformation to evaluate the effects on the response variables. Based on the obtained best lambda value is derived at the lowest curve point generated by the sum of squares (SS) residuals natural log, an indorsed transformation is preferred. The box-cox plot presents the lambda values on thex-axis and the SS natural log on the y-axis as shown in Figure 12, the derived results from the plots showed a current lambda of 0.5, low and high at 95% confidence interval of −0.79 and 0.34 lambda, respectively, with the best lambda of −0.25 for compressive strength response. However, the flexural strength response showed the best lambda of 2.85, low and high at 95% confidence interval of 1.59 and 3.79 lambda, respectively [65, 69].Figure 12
Box-cox plots for power transformation.


## 4.4.1. Normal Probability Plot

This is observed as the first diagnostic statistical evaluation plot, which helps to validate the model residuals followed a normal standard distribution nearby the regression line-of-fit. In this graph, the externally studentized residuals is plotted against the normal probability (%) outcome. Moreover, for better analysis results, definite patterns are sort for like curve-linear shapes, which may signify the need for transformation of the target response, even though some calculated points may scatter with normal data. Normal probability plot for the compressive and flexural strength responses are presented in Figure8 [65].Figure 8
Residuals normal probability plots for the target responses.


## 4.4.2. Predicted vs. Studentized Residuals Plot

This diagnostic evaluation test corroborates the constant variance of the residual assumption presented in a scattered plot with the predicted outcomes on thex-axis and the studentized residuals on the y-axis as presented in Figure 9 for the target responses under investigation. The plotted statistical results indicated clustering at the residual of zero points with max to min values of 3.5 to −3.5 and 63.66 to −63.66 for compressive and flexural strength responses, respectively [66].Figure 9
Residual vs. predicted diagnostic statistical graphs.


## 4.4.3. Experimental Run vs. Studentized Residual Plot

This statistical diagnostic test illustrates the influence of the experimental run order versus the studentized residuals of the model on the target responses. This assessment plot fundamentally seeks for prowling factors, which may impact the criterion parameters during the experimental exercises as shown in Figure10. The graphical plots are similar to the residuals vs. model predicted diagnostic plots, and the design points were positioned near the line of zero points and also to lie between ±3.5and ±63.666, which signifies a time-dependent parameter in a random scattered plot for compressive and flexural strength responses, respectively [67].Figure 10
Residual vs. experimental run diagnostic graphs.


## 4.4.4. Predicted vs. Laboratory Results

This analytical plot illustrates the behavior of the actual or laboratory responses on thex-axis in respect to the EVD model predicted values on the y-axis of the graph. Moreover, this statistical plot helps to evaluate the prediction accuracy performance of the EVD model so as to determine design points and groups, which the model cannot certainly predict, and to assess the relationship between the estimated and experimental results as shown in Figure 11. The results obtained indicate the datasets ranging from 4.4 to 5.6 and 1.6 to 3.6 for the square root responses of the compressive and flexural responses, respectively [68].Figure 11
Predicted vs. actual diagnostic graphs.


## 4.4.5. Power Transformation through Box-Cox Plot

This diagnostic statistical plot offers necessary requirements for the selection of adequate law power transformation to evaluate the effects on the response variables. Based on the obtained best lambda value is derived at the lowest curve point generated by the sum of squares (SS) residuals natural log, an indorsed transformation is preferred. The box-cox plot presents the lambda values on thex-axis and the SS natural log on the y-axis as shown in Figure 12, the derived results from the plots showed a current lambda of 0.5, low and high at 95% confidence interval of −0.79 and 0.34 lambda, respectively, with the best lambda of −0.25 for compressive strength response. However, the flexural strength response showed the best lambda of 2.85, low and high at 95% confidence interval of 1.59 and 3.79 lambda, respectively [65, 69].Figure 12
Box-cox plots for power transformation.


## 4.5. Diagnostic Influence Plots

This diagnostic graphical presentation helps to assess the effects or influence of potential distinct experimental runs on the analysis outcomes. These graphical charts offer a clearer statistical perspective when two or single case sticks out from other groups in the experimental design. The diagnostic influence computation was executed using the following essential tools, namely, leverage vs. experimental run, cook’s distance, and statistical difference in fits (DFFITS) vs. run of experiments [70].


### 4.5.1. Cook’s Distance

Cook’s distance is deployed in regression statistics to determine influential outliers in regressor or independent variable sets. It is also utilized to detect experimental points in the design, which affects the developed EVD model negatively and feasible regions or planes where good correlation is obtained as shown in Figure13. The graphical plots illustrates the computed cook’s distance value on the y-axis and the five design number of experiments on the x-axis. The derived results indicated Cook’s distance ranging from 0 to 20 for the compressive strength response and that of flexural strength response lie between 0 and 5. Interestingly, the run number of experiments 4 and 5 was observed to be positioned above the 0–1 boundary line [71].Figure 13
Cook’s distance influence plot.


### 4.5.2. Leverage vs. Run

Leverage is deployed in statistical diagnostic influence to assess the extent to which each design point in the experiments impacts the goodness of fit. It is a degree of the distance apart that the regressor values of an observation are from host of other observations in the experimental design. Point of leverage is an observation that possesses an unfamiliar predicted score from the bulk of varying observations. The leverage versus the experimental run statistical influence plot presents the graph of the computed leverage points on they-axis against the experimental runs on the horizontal axis for the target responses as shown in Figure 14. The calculated leverage point ranges from 0 to 1, when 1 signifies that the generated predictive model at the point of investigation fits the observation exactly. Furthermore, the results derived from the graph indicate boundary points from 0 to 0.8 and 0 to 0.6 for compressive and flexural strength responses, respectively, while points for experimental runs 4 and 5 were observed to lie above the boundary lines and at a leverage of approximately 1.0 [31, 61].Figure 14
Leverage vs. run diagnostic influence plot.


### 4.5.3. Statistical Difference in Fits (DFFITS) vs. Runs

DFFITS statistic is a studentized diagnostic influence instrument, which provides a scaled assessment of the predicted response variation for theith observation. It also illustrates the variability in the model predicted results at a specific design experimental point when it is excluded for model fitting as presented in Figure 15. The plotted graphical results indicate the DFFITS values on the y-axis against the five experimental runs on the x-axis, and the design point was observed to be situated at the zero DFFITS region within the ±2.683 for the compressive strength response. For the flexural strength response, the design points were situated between 5 and 15 DFFITS regions, while three points fell within the boundary limit of ±2.323 [72].Figure 15
Diagnostic influence plot of DFFITS vs. experimental run.


## 4.5.1. Cook’s Distance

Cook’s distance is deployed in regression statistics to determine influential outliers in regressor or independent variable sets. It is also utilized to detect experimental points in the design, which affects the developed EVD model negatively and feasible regions or planes where good correlation is obtained as shown in Figure13. The graphical plots illustrates the computed cook’s distance value on the y-axis and the five design number of experiments on the x-axis. The derived results indicated Cook’s distance ranging from 0 to 20 for the compressive strength response and that of flexural strength response lie between 0 and 5. Interestingly, the run number of experiments 4 and 5 was observed to be positioned above the 0–1 boundary line [71].Figure 13
Cook’s distance influence plot.


## 4.5.2. Leverage vs. Run

Leverage is deployed in statistical diagnostic influence to assess the extent to which each design point in the experiments impacts the goodness of fit. It is a degree of the distance apart that the regressor values of an observation are from host of other observations in the experimental design. Point of leverage is an observation that possesses an unfamiliar predicted score from the bulk of varying observations. The leverage versus the experimental run statistical influence plot presents the graph of the computed leverage points on they-axis against the experimental runs on the horizontal axis for the target responses as shown in Figure 14. The calculated leverage point ranges from 0 to 1, when 1 signifies that the generated predictive model at the point of investigation fits the observation exactly. Furthermore, the results derived from the graph indicate boundary points from 0 to 0.8 and 0 to 0.6 for compressive and flexural strength responses, respectively, while points for experimental runs 4 and 5 were observed to lie above the boundary lines and at a leverage of approximately 1.0 [31, 61].Figure 14
Leverage vs. run diagnostic influence plot.


## 4.5.3. Statistical Difference in Fits (DFFITS) vs. Runs

DFFITS statistic is a studentized diagnostic influence instrument, which provides a scaled assessment of the predicted response variation for theith observation. It also illustrates the variability in the model predicted results at a specific design experimental point when it is excluded for model fitting as presented in Figure 15. The plotted graphical results indicate the DFFITS values on the y-axis against the five experimental runs on the x-axis, and the design point was observed to be situated at the zero DFFITS region within the ±2.683 for the compressive strength response. For the flexural strength response, the design points were situated between 5 and 15 DFFITS regions, while three points fell within the boundary limit of ±2.323 [72].Figure 15
Diagnostic influence plot of DFFITS vs. experimental run.


## 4.6. Optimization Overview

Once the diagnostic statistical analysis and influence graphical calculations were achieved, numerical optimization is carried out using desirability function, which evaluates the imposed optimization criteria on the model variables that maximizes the target response parameters. To accomplish this task, the features of the objective function are analytically adjusted through weight functions modification in line with the set model variable criteria. These multicollinearity conditions would enable the attainment of favorable conditions to obtain a desirability score of 1.0 through a scale of0≤dyi≤1 boundary conditions [73]. The optimization element of this experimental design explores for the mixture ratios combination in the feasible factor space, which satisfies simultaneously the formulated and imposed criteria on the response parameters and corresponding factor levels. The expected optimization goal is set to maximize the target responses, while the two-component combination ratios are set at in-range option to determine the optimal proportion of the factor levels to produce a maximum response as shown in Table 23. The lower and upper limits of the variables were derived from the system datasets derived from the laboratory experiments [74].Table 23
Model variable criteria for optimization.
NameGoalLower limitUpper limitLower weightUpper weightImportanceA: cementis in range0.51113B: SDAis in range00.5113CompressiveMaximize20.3531.13113FlexuralMaximize3.219.49113The optimization solution derived from the analytical proceedings of the mixture experiment designs is presented in Table24. From the derived results, an optimal desirability score of 0.99 was obtained at a combination ratio of 0.89 : 0.110 to produce maximized compressive and flexural strength of 31.102 N/mm2 and 9.384 N/mm2, respectively [75].Table 24
Optimization solutions.
NumberCementSDACompressiveFlexuralDesirability10.8900.11031.1029.3840.990Selected


### 4.6.1. Graphical Optimization Plots of Ramps and Bar

The graphical optimization output is illustrated in the form of ramps and bar graphs for a better understanding of the outcomes. The ramps and bar chart results present the optimal solution for the predictor variables in red and the target response parameters in blue coloration to elucidate the desirability-function scores of the constituting model variables as shown in Figures16–18. The results presented indicate an independent variable desirability score of 1.0, the compressive strength score was observed as 0.997 and 0.9831 for the flexural strength response, while the combined optimal desirability score of 0.9902 was calculated [29].Figure 16
Ramps graph for the desirability function.Figure 17
Bar graph for the desirability function.Figure 18
Graphical plot of two-component mixture optimization.


## 4.6.1. Graphical Optimization Plots of Ramps and Bar

The graphical optimization output is illustrated in the form of ramps and bar graphs for a better understanding of the outcomes. The ramps and bar chart results present the optimal solution for the predictor variables in red and the target response parameters in blue coloration to elucidate the desirability-function scores of the constituting model variables as shown in Figures16–18. The results presented indicate an independent variable desirability score of 1.0, the compressive strength score was observed as 0.997 and 0.9831 for the flexural strength response, while the combined optimal desirability score of 0.9902 was calculated [29].Figure 16
Ramps graph for the desirability function.Figure 17
Bar graph for the desirability function.Figure 18
Graphical plot of two-component mixture optimization.


## 4.7. EVD Model Postanalysis and Simulation

This involves point prediction, which gives room for the factor values to be reviewed; confirmation, which links the interval of the EVD model estimation to derive sample mean; the coefficient table, which presents the significance and magnitude coefficients for all target response parameters analyzed; and simulation to validate the developed results and derived statistical analysis results [76].


### 4.7.1. Analysis of Point Prediction

This statistical postanalysis tool utilizes the fit computation of the developed model and the prescribed settings for the factor tools to determine interval estimates after the analysis is successfully carried out as shown in Table25. The result indicated standard deviations of 0.055 and 0.65, low to high value of 30.4973–1.7125 and 7.7803–11.138 for compressive and flexural strength, respectively, at 95% confidence interval [77].Table 25
Point prediction.
ResponsePredicted meanPredicted medianStd. dev.CI for 95% CI lowMean 95% CI high99% of 95% TI lowPopulation 95% TI highCompressive31.102231.10210.054610230.497931.712328.609233.6991Flexural9.383999.372870.6459567.780311.1382.8088119.7765


### 4.7.2. Coefficient Table

This postanalysis computation tool presents the proportion combination magnitude of the 2-component mixture of SDA-cement green concrete as presented in Table26. The coefficients are derived from the multicriteria optimization computation using the extreme vertex design method to obtain acceptable performance. Cubic and quadratic models were deployed for the exercise based on the fits statistics recommendation for compressive and flexural strength, respectively, to achieve better generalization of the derived experimental data. The legend column was used to assess the p values for every term of the models where p values <0.01 signify the best measures for strong significant presented in red, p values ≤0.05 and ≥0.01 indicate moderate level of significance presented in green, p values ≤0.10 and ≥0.05 signify slight level of significance presented in blue, while p values ≥0.10 indicate the level of noninsignificant displayed in black [78, 79].Table 26
Coefficient table.
ResponseABABAB (A − B)Sqrt (compressive)5.37254.510511.996990.536869p=0.00440.00440.00670.0489Sqrt (flexural)3.037241.841191.67359p=0.01220.01220.0656Legendp<0.010.01≤p<0.050.05≤p<0.10p≥0.10


### 4.7.3. EVD Model Simulation

The model simulation is carried out on the developed EVD model as the last phase of the statistical analysis of the design experiment optimization of two-component mixture problem. This statistical verification is imperative so as to replicate nontrivial scenario for the case study to give adequate and efficient practical guide for field operators, consultants or designers, and contractors on the expected performance of the developed model on real-life application. The simulation is necessary to validate the process of developing the EVD model and to affirm that the results obtained during the statistical fits and diagnostics influence computation can be realized when applied in the field works. A graphical plot showing the experimental derived responses vs. EVD model-simulated results is presented in Figure19. These data were compared statistically using F-test two-sample variances and student’s t-test to check the significant difference and degree of correlation between them at 95% confidence interval as shown in Tables 27–30. From the statistical results achieved using Microsoft Excel software, p(F ≤ f) one-tail of 0.476 and 0.381 was computed, which is >0.05 for compressive and flexural strength responses, respectively, for the F-test statistics. Moreover, regarding the student’s t-test results, p(T ≤ t) two-tail of 0.702 and 0.591 was calculated for pcompressive and flexural strength responses, respectively. The derived statistical results showed that there is no significant difference between the actual and model predicted results, which signifies acceptable model performance in line with the findings of Sadrmomtazi et al. [80] and Alaneme et al. [81].Figure 19
Actual vs. EVD model predicted responses.Table 27
F-Test two-sample for variances for compressive strength response.
Statistical FactorsCompr. str.-actualCompr. str.-modelMean27.09426.918Variance18.5030317.34397Observations55Df44F1.066828p(F ≤ f) one-tail0.475758F critical one-tail6.388233Table 28
F-Test two-sample for variances for flexural strength response.
Statistical factorsFlex. str.-actualFlex. str.-modelMean7.2287.362Variance6.244828.62257Observations55Df44F0.724241p(F ≤ f) one-tail0.381075F critical one-tail0.156538Table 29
T-Test: paired two-sample test for means for compressive strength response.
Statistical FactorsCompr. str.-actualCompr. str.-modelMean27.09426.918Variance18.5030317.34397Observations55Pearson correlation0.974904Df4t Stat0.410776p(T ≤ t) one-tail0.351145t Critical one-tail2.131847p(T ≤ t) two-tail0.702289t Critical two-tail2.776445Table 30
T-Test: paired two-sample test for means for flexural strength response.
Statistical FactorsFlex. str.-actualFlex. str.-modelMean7.2287.362Variance6.244828.62257Observations55Pearson correlation0.995076Df4t Stat−0.58357p(T ≤ t) one-tail0.295416t Critical one-tail2.131847p(T ≤ t) two-tail0.590832t Critical two-tail2.776445


## 4.7.1. Analysis of Point Prediction

This statistical postanalysis tool utilizes the fit computation of the developed model and the prescribed settings for the factor tools to determine interval estimates after the analysis is successfully carried out as shown in Table25. The result indicated standard deviations of 0.055 and 0.65, low to high value of 30.4973–1.7125 and 7.7803–11.138 for compressive and flexural strength, respectively, at 95% confidence interval [77].Table 25
Point prediction.
ResponsePredicted meanPredicted medianStd. dev.CI for 95% CI lowMean 95% CI high99% of 95% TI lowPopulation 95% TI highCompressive31.102231.10210.054610230.497931.712328.609233.6991Flexural9.383999.372870.6459567.780311.1382.8088119.7765


## 4.7.2. Coefficient Table

This postanalysis computation tool presents the proportion combination magnitude of the 2-component mixture of SDA-cement green concrete as presented in Table26. The coefficients are derived from the multicriteria optimization computation using the extreme vertex design method to obtain acceptable performance. Cubic and quadratic models were deployed for the exercise based on the fits statistics recommendation for compressive and flexural strength, respectively, to achieve better generalization of the derived experimental data. The legend column was used to assess the p values for every term of the models where p values <0.01 signify the best measures for strong significant presented in red, p values ≤0.05 and ≥0.01 indicate moderate level of significance presented in green, p values ≤0.10 and ≥0.05 signify slight level of significance presented in blue, while p values ≥0.10 indicate the level of noninsignificant displayed in black [78, 79].Table 26
Coefficient table.
ResponseABABAB (A − B)Sqrt (compressive)5.37254.510511.996990.536869p=0.00440.00440.00670.0489Sqrt (flexural)3.037241.841191.67359p=0.01220.01220.0656Legendp<0.010.01≤p<0.050.05≤p<0.10p≥0.10


## 4.7.3. EVD Model Simulation

The model simulation is carried out on the developed EVD model as the last phase of the statistical analysis of the design experiment optimization of two-component mixture problem. This statistical verification is imperative so as to replicate nontrivial scenario for the case study to give adequate and efficient practical guide for field operators, consultants or designers, and contractors on the expected performance of the developed model on real-life application. The simulation is necessary to validate the process of developing the EVD model and to affirm that the results obtained during the statistical fits and diagnostics influence computation can be realized when applied in the field works. A graphical plot showing the experimental derived responses vs. EVD model-simulated results is presented in Figure19. These data were compared statistically using F-test two-sample variances and student’s t-test to check the significant difference and degree of correlation between them at 95% confidence interval as shown in Tables 27–30. From the statistical results achieved using Microsoft Excel software, p(F ≤ f) one-tail of 0.476 and 0.381 was computed, which is >0.05 for compressive and flexural strength responses, respectively, for the F-test statistics. Moreover, regarding the student’s t-test results, p(T ≤ t) two-tail of 0.702 and 0.591 was calculated for pcompressive and flexural strength responses, respectively. The derived statistical results showed that there is no significant difference between the actual and model predicted results, which signifies acceptable model performance in line with the findings of Sadrmomtazi et al. [80] and Alaneme et al. [81].Figure 19
Actual vs. EVD model predicted responses.Table 27
F-Test two-sample for variances for compressive strength response.
Statistical FactorsCompr. str.-actualCompr. str.-modelMean27.09426.918Variance18.5030317.34397Observations55Df44F1.066828p(F ≤ f) one-tail0.475758F critical one-tail6.388233Table 28
F-Test two-sample for variances for flexural strength response.
Statistical factorsFlex. str.-actualFlex. str.-modelMean7.2287.362Variance6.244828.62257Observations55Df44F0.724241p(F ≤ f) one-tail0.381075F critical one-tail0.156538Table 29
T-Test: paired two-sample test for means for compressive strength response.
Statistical FactorsCompr. str.-actualCompr. str.-modelMean27.09426.918Variance18.5030317.34397Observations55Pearson correlation0.974904Df4t Stat0.410776p(T ≤ t) one-tail0.351145t Critical one-tail2.131847p(T ≤ t) two-tail0.702289t Critical two-tail2.776445Table 30
T-Test: paired two-sample test for means for flexural strength response.
Statistical FactorsFlex. str.-actualFlex. str.-modelMean7.2287.362Variance6.244828.62257Observations55Pearson correlation0.995076Df4t Stat−0.58357p(T ≤ t) one-tail0.295416t Critical one-tail2.131847p(T ≤ t) two-tail0.590832t Critical two-tail2.776445


## 5. Conclusion

The optimization of two-component constrained simplex mixture was adapted in this research study for the assessment of the mechanical strength behavior of the green concrete produced. The components constraint formulation was achieved at the range of 0–50% replacement by the industrial waste derivatives (SDA) as supplementary cementitious material (SCM) to ascertain the optimal mixture proportion at a maximized strength response. The chemical property test carried out on the SCM showed a good pozzolanic behavior with 65.28% of SiO2, 2.75% of Fe2O3, and 5.52% of Al2O3 to obtain a total value of 73.55%. Using Design Expert software, the experimental program was designed for the derivation of components ratio and number of runs for the laboratory experiments through the I-optimal design to enable full exploration of the feasible experimental regions. The laboratory experiments presented a maximum compressive strength of 31.13 N/mm2 with a ratio of 0.875 : 0.125 for cement and SDA, respectively, and a flexural strength of 9.49 N/mm2 for the unblended sample at 1 : 0 ratio for cement and SDA, respectively. However, a minimum compressive response of 20.35 N/mm2 and a flexural strength of 3.21 N/mm2 were derived with a mixture ratio of 0.5 : 0.5 for cement and SDA, respectively. Laboratory details, which involved experimental responses and mixture proportions, were utilized for the development of the EVD predictive model and the steps involved fits statistical evaluation, ANOVA, diagnostic test, and influence statistics to analyze the datasets. Numerical optimization and graphical statistical computations were employed to locate the optimal levels of mixture ingredients with maximized response to satisfy the setout multicriteria on the model variables using desirability function. A desirability score of 0.990 was obtained from the calculation processes at a mix ratio of 0.89 : 0.110 for the two components of cement and SDA to produce maximized compressive and flexural strength of 31.102 N/mm2 and 9.384 N/mm2, respectively. Furthermore, the test of adequacy of the generated EVD model was carried out through simulation and statistical validation exercises using the F-test and student’s t-test, and the outcome of the tests indicated a good correlation between the laboratory results and EVD model-simulated values.



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