A Computer-Aided Approach to Pozzolanic Concrete Mix Design

Associate Professor, Department of Applied Geoinformatics, Chia Nan University of Pharmacy & Science, No. 60, Sec. 1, Erh-Jen Rd., Tainan 71710, Taiwan Graduate Student, Department of Civil Engineering, National Chiao Tung University, No. 1001, University Rd., Hsinchu 300, Taiwan Associate Professor, Department of Computer Science and Information Engineering, Chien Hsin University of Science and Technology, No. 229, Jianxing Rd., Zhongli Dist., Taoyuan City 32097, Taiwan Professor, Department of Civil Engineering, National Chiao Tung University, No. 1001, University Rd., Hsinchu 300, Taiwan


Introduction
Concrete plays an important role in the growing construction industry.Presently, various types of by-product materials, such as fly ash, silica fume, rice husk ash, and others have been widely used as pozzolanic materials in concrete.Studies [1][2][3][4] have shown that utilization of pozzolanic material not only improves concrete properties (such as strength and durability) but also helps to preserve the environment.Moreover, superplasticizers play a crucial role in the development of high strength and highperformance concrete.Superplasticizers are admixtures which are added to concrete mixture in very small dosages.
eir addition results in a significant increase in the workability of the mixture, as well as a reduction of water/cement ratio and of cement quantity [5].
Several researchers have looked into the characteristic parameters that affect the compressive strength and slump of conventional and high-strength concrete [6][7][8].ese parameters typically include water, cement, coarse aggregate, and fine aggregate.Conventional methods initially involve constructing a mathematical model, which is followed by a regression analysis using experimental data to determine unknown coefficients in that model and establish correlations between these parameters and compressive strength and slump.Conventional methods generally include complex modeling and are inappropriate where experimental data are imprecise and parameters affecting compressive strength and slump are incomplete in the experimental data.
Artificial neural networks (ANNs) were originally developed to simulate the function of the human brain or neural system.Subsequently, they have been widely applied to diverse fields, ranging from biology to many engineering fields.ANNs exhibit a number of desirable properties not found in conventional symbolic computation systems, including robust performance when dealing with noisy or incomplete input patterns, a high degree of fault tolerance, high parallel computation rates, the ability to generalize, and adaptive learning [9][10][11].ANNs are capable of modeling input-output functional relations, even when mathematically explicit formulas are unavailable.erefore, ANNs are suitable for prediction of compressive strength and slump of concrete.Accordingly, the feasibility of applying ANNs to predict compressive strength and slump of concrete has received considerable attention.Yeh [12] investigated the potential of using design of experiments and ANNs to determine the effect of fly ash replacements on early and late compressive strength of low-and high-strength concrete.Yeh [13] further demonstrated the possibilities of adapting ANNs to predict the compressive strength of highperformance concrete.Kasperkiewics et al. [14] applied ANNs to predict the 28-day compressive strength of highperformance concrete composed of six components (cement, silica, superplasticizer, water, fine aggregate, and coarse aggregate).Lee [15] used ANNs to predict the compressive strength development of concrete.Bai et al. [16] developed neural network models to predict the workability of concrete incorporating metakaolin and fly ash.Duan et al. [17] applied ANNs to predict the compressive strength of recycled aggregate concrete.Ni and Wang [18] developed a method to predict 28-day compressive strength of concrete by using ANNs based on the inadequacy of methods dealing with multiple variable and nonlinear problems.
In light of the above developments, this study develops a two-step computer-aided approach for pozzolanic concrete mix design.e first step is establishing the dataset of pozzolanic concrete mixture proportioning which conform to American Concrete Institute (ACI) code.
e dataset consists of experimental data collected from the literature and numerical data generated by computer program.In this step, ANNs are employed to establish the prediction models of compressive strength and slump of concrete.Sensitivity analysis of the ANN is used to evaluate the effect of inputs on the output of the ANN.e two ANN models are tested using data of experimental specimens made in laboratory for twelve different mixtures.e second step is classifying the dataset of pozzolanic concrete mixture proportioning.A classification method is utilized to categorize the dataset into 360 classes based on compressive strength of concrete, pozzolanic admixture replacement rate, and cost of the concrete.

Artificial Neural Networks
ANNs form a class of systems that are inspired by biological neural networks.e topology of an ANN model consists of a number of simple processing elements, called nodes, which are interconnected to each other.Interconnection weights that represent the information stored in the system are used to quantify the strength of the interconnections; these weights hold the key to the functioning of an ANN.

Back-Propagation Neural
Networks.Among the many different types of ANN, by far the most commonly applied neural network learning model, due to its simplicity, is the feedforward, multilayered, supervised neural network with error back-propagation algorithm, the so-called backpropagation (BP) network [11].Before an ANN can be used in an application, it must either learn or be trained from an existing database consisting of pairs of input-output patterns.e topology of BP networks consists of an input layer, one or more hidden layers, and an output layer.
e training of a supervised neural network usually involves three stages.
e first stage is the data feedforward.e output of each node is defined as follows: where W ij is the weight associated with the ith node in the preceding layer to the jth node in the current layer; O i is the output of ith node in the preceding layer; θ j is the threshold value of node j in the current layer; O j is the output of node j in the current layer; and function f is the activation function, which has to be differentiable.Herein, the hyperbolic tangent function is used as the activation function and is defined as follows: e second stage is error back-propagation and adjustment of the network weights.e training process applies mean square error (E), the absolute fraction of variance (R 2 ), and sum of the squares error (SSE), to monitor the learning performance of the network.E, R 2 , and SSE are defined, respectively, as where P denotes the number of instances in the training set, while d pk and o pk represent the desired and calculated output of the kth output node for the pth instance, respectively.e standard BP algorithm employs a gradient descent approach with a constant step length (learning ratio) to train the network.

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where η is the learning ratio, which is a constant in the range of [0, 1]. e suffix index k denotes the kth learning iteration.Unfortunately, BP supervised neural network learning models require a significant amount of time to learn.Moreover, the convergence of a BP neural network is highly dependent upon the use of a learning rate (η).Consequently, several different approaches are developed here to enhance the learning performance of the BP learning algorithm [10].
Hung and Lin [19] developed a more effective adaptive limited memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) learning algorithm based on the approach of a L-BFGS quasi-Newton second-order method [20,21] with an inexact line search algorithm.is algorithm achieved a superior convergence rate to the BP learning algorithm by using second-order derivatives of the system error function with respect to the network weights.In the conventional BFGS method, the approximation H k+1 to the inverse Hessian matrix of function E(W) is updated by where Instead of forming the matrix H k with the BFGS method, the vectors s k and y k are saved.ese vectors first define and then implicitly and dynamically update the Hessian approximation using information from the last few iterations, referred to here as m.erefore, the final stage of the adjustment of the weights in a BP-based ANN is modified as follows: e search direction is given by where e step length, α k , is adapted during the learning process through a mathematical approach: the inexact line search algorithm.
is approach is used in the L-BFGS learning algorithm instead of a constant learning ratio [19].
e inexact line search algorithm is based on three sequential approaches: bracketing, sectioning, and interpolation.e bracketing approach brackets the potential step length, α, between two points, through a series of function evaluations.e sectioning approach then uses the two points of the bracket as the initial points, reducing the step size, and locating the minimum between points, such as, α 1 and α 2 , to a specified degree of accuracy.Finally, the quadratic interpolation approach uses the three points, α 1 , α 2 , and (α 1 + α 2 )/2, to fit a parabola to determine the step length, α k .Consequently, the step length α k must satisfy the following conditions in each iteration [19]: e problem of selecting a learning ratio through trial and error in the BP algorithm is thus circumvented in the adaptive L-BFGS learning algorithm.

Architectures of ANN Models.
e ANN models, compressive strength prediction neural network (CSPNN) and slump prediction neural network (SPNN), are used in this study for prediction of the 28-day compressive strength (abbreviated below as compressive strength) and slump of pozzolanic concrete, respectively.e architectures of the CSPNN and SPNN are illustrated in Figure 1.Both CSPNN and SPNN developed in this study have seven neurons in the input layer and one neuron in the output layer.e inputs of both CSPNN and SPNN are water, cement, ground granulated blast furnace slag (GGBFS), fly ash, coarse aggregate (CA), fine aggregate (FA), and superplasticizer (SP).e outputs of CSPNN and SPNN are compressive strength (f c ′ ) and slump (S), respectively.Table 1 shows the minimum and maximum values of the seven input parameters used in CSPNN and SPNN.[22] and Funahashai [23] rigorously demonstrated that even with only one hidden layer, neural networks can uniformly approximate any continuous function.Although neural networks can find a relationship between the input and output values internally, it is not always easy to interpret the resulting weight state.us, the effect of one input parameter on the output is difficult to analyze.Alternatively, it is possible to compute the sensitivity of the output value with respect to one of its inputs by taking the first-order partial derivative [24,25].

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If there is a network with n hidden layers, the output of the kth node in the output layer is defined as follows: where H nj is the output of the jth node in the nth hidden layer, θ ok is the threshold value of the kth node in the output layer, W hnj,ok is the weight associated with the jth node in the nth hidden layer to the kth node in the output layer, and function f is the activation function.e first-order partial derivative of the kth output with respect to ith input D 1  ki can be derived as follows: where W xi,h1j1 is the weight associated with the ith node in the input layer to the j1th node in the first hidden layer and W hnjn,ok is the weight associated with the jnth node in the nth hidden layer to the kth node in the output layer.Equation (12) indicates that D 1 ki is a function of weights, threshold value, and the first-order derivative of the activation function (or a function of weights, threshold values, and training instances).Since D 1  ki is a function of training instances, generally, the mean of D 1 ki for the entire training instances can be used to describe the nominal value of the sensitivity of the kth output parameter with respect to the ith input parameter.e mean of D 1 ki for the entire training instances is where D 1 ki,p is the value of D 1 ki of the pth training instance, and P is the total number of training instances.In fact, D

Proposed Approach for Pozzolanic Concrete Mix Design
is study develops a computer-aided approach for pozzolanic concrete mix design.is approach is suitable for designing a mix of pozzolanic concrete with compressive strength, f c ′ , from 210 kgf/cm 2 to 980 kgf/cm 2 and slump, S, equal to 20 cm.As shown in Figure 2, this approach involves two steps.e first step is establishing the dataset of pozzolanic concrete mixture proportioning that conform to ACI code, consisting of experimental data collected from literature and numerical data generated by computer program.e second step is classifying the dataset of pozzolanic concrete mixture proportioning.A classification method is utilized to categorize data into 360 clusters according to compressive strength of concrete, pozzolanic admixtures replacement rate, and material cost.e following presents the details of the proposed approach.

Establishing the Dataset of Pozzolanic Concrete Mixture
Proportioning.As shown in Figure 2, the process of establishing the dataset of pozzolanic concrete mixture proportioning is listed as follows: (1) Collecting experimental data of pozzolanic concrete mixture proportioning from the literature [3,[26][27][28][29][30][31][32][33][34][35][36][37][38].(2) Generating numerical data of pozzolanic concrete mixture proportioning since the collected experimental data may be insufficient.Before generating numerical data, the ranges of material contents (as listed in Table 1) are set based on collected experimental data of pozzolanic concrete mixture proportioning and ACI code for pozzolanic concrete mix design (as listed in Tables 2-8) [39].Numerical data of pozzolanic concrete mixture proportioning are then generated randomly using the ACI mix design method for pozzolanic concrete (as shown in Figure 3).(3) Using a portion of collected experimental data of pozzolanic concrete mixture proportioning to train CSPNN and SPNN.e effect of input parameters on the output is evaluated by sensitivity analysis.e prediction accuracy of CSPNN and SPNN is tested using the remainder of the collected experimental data and data from experimental specimens made in our laboratory for twelve different mixtures.(4) Using trained CSPNN and SPNN to predict compressive strength and slump of experimental and numerical data, respectively.Data that satisfy the following conditions are kept in the dataset.
where f c,CSPNN ′ and S SPNN are compressive strength and slump predicted by CSPNN and SPNN, respectively.e reasons for this are (1) this approach is suitable for mixing design of pozzolanic concrete with compressive strength, f c ′ , from 210 kgf/cm 2 to 980 kgf/cm 2 and slump, S, equal to 20 cm, and (2) the allowable data range width of slump in Taiwan is set to be 3.8 cm when slump is larger than 10 cm [40].

Classifying the Dataset of Pozzolanic Concrete Mixture
Proportioning.To produce a dataset of pozzolanic concrete mixture proportioning which is more feasible and convenient for engineering applications, it is classified further.
In classification, a sampling unit (subject or object) whose class membership is unknown is assigned to a class on the basis of the vector, y, associated with the unit.To classify the unit, we must have available a previously obtained sample of observation vectors from each class.One approach is to then compare y with the mean vectors y 1 , y 2 ,. .., y k of the k classes and assign the unit to the class whose y i is closest to y [41].Many techniques use an index of similarity or proximity between y and y i .A convenient measure of proximity is the distance.e distances used in classification algorithms include Euclidean distance, Manhattan distance, Chebyshev distance, Minkowski distance, and Mahalanobis distance.Since Euclidean distance is the most well-known distance, it is applied in this study.e Euclidean distance between two vectors (points) a and b is defined as where a j and b j are the jth element of a and b, respectively.e proposed classification of the dataset of pozzolanic concrete mixture proportioning is according to compressive strength, pozzolanic admixture replacement rate, and cost.As shown in Figure 2, the classification method used in this study involves three stages.e first stage is the classification of dataset according to compressive strength and slump.e number of classes is set as twelve in this stage.e mean (designed) values of compressive strength of the twelve classes are increased from 210 kgf/cm 2 (20.6 MPa) to 980 kgf/cm 2 (96.1 MPa) every 70 kgf/cm 2 (6.8 MPa).e mean (designed) values of slump of all twelve classes are the same and are equal to 20 cm.According to the related code in Taiwan, the allowable data range width of compressive strength and slump is 3.4 MPa and 3.8 cm, respectively.us, the classification rule can be written as Step 1: establishing the dataset of pozzolanic concrete mixture proportioning The first stage is the classification of dataset according to compressive strength and slump.The number of class is set to be twelve in this stage.
The second stage is the classification of dataset according to pozzolanic admixtures replacement rate.Each class in the first stage is divided into five smaller classes Collecting experimental data of pozzolanic concrete mixture proportioning from the literature.
Using collected experimental data and ACI code to calculate numerical data of pozzolanic concrete mixture proportioning by computer program.
Using trained CSPNN and SPNN to predict compressive strength (f′ c,p ) and slump (S p ) of experimental and numerical data, respectively.Data with 210 kg/cm 2 ≤ f′ c,p ≤ 980 kg/cm 2  and Sp = 20± 3.8 cm were kept in the dataset.
Using collected experimental data of pozzolanic concrete mixture proportioning from the literature to train CSPNN and SPNN.
The third stage is the classification of dataset according to the cost of pozzolanic concrete.Each class in the second stage is divided into six smaller classes.
Step 2: classifying the dataset of pozzolanic concrete mixture proportioning

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where f c,p ′ and S p are the compressive strength and slump of the pth instance in the dataset, respectively; f c,i ′ and S i are the mean (designed) compressive strength and mean (designed) slump of the ith class, respectively; d p,i is the Euclidean distance between the vector associate to the pth instance in the dataset, y p � (f c,p ′ , S p ), and mean vector of class i, y i � (f c,i ′ , S i ); and class p is the class the pth instance in the database belongs to.e second stage is the classification of dataset according to pozzolanic admixtures replacement rate.Pozzolanic admixtures may be used as a partial replacement of cement in concrete.e pozzolanic admixtures used in this study are fly ash and ground granulated blast furnace slag.Pozzolanic admixture replacement rate, R PA , is expressed as follows: where PA is pozzolanic admixtures.Each class in the first stage is divided into five smaller classes.e class intervals of R PA are 0-≤10%, >10%-≤20%, >20%-≤30%, >30%-≤40%, and >40%-≤50%.e third stage is the classification of dataset according to the cost of pozzolanic concrete.Each class in the second stage is divided into six smaller classes.e class intervals of the cost of pozzolanic concrete are 0 (NTD/m 3 )-≤2000 (NTD/m 3 ), >2000 (NTD/m 3 )-≤2250 (NTD/m 3 ), >2250 (NTD/m 3 )-≤2500 (NTD/m 3 ), >2500 (NTD/m 3 )-≤2750 (NTD/m 3 ), >2750 (NTD/m 3 )-≤3000 (NTD/m 3 ), and >3000 Advances in Civil Engineering (NTD/m 3 ).ere are 360 classes overall in the dataset of pozzolanic concrete mixture proportioning.Here, the CSPNN is constructed with seven, fourteen, and one nodes in input layer, hidden layer, and output layer, respectively, and denoted as CSPNN(7-14-1).e complete offline training process took 47 cycles.e E and R 2 were 0.005988 and 0.92556, respectively.After the CSPNN was trained on the 462 training samples, it was tested to observe how accurately it would predict compression strength of other samples.Table 9 and Figure 4 summarize the results of these tests, indicating that the CSPNN can satisfactorily predict the compression strength in all 20 testing samples.

Sensitivity Analysis of the CSPNN.
Figure 5 shows the distribution of compressive strength and water for the training samples of the CSPNN.It shows that compressive strength decreases with increasing amounts of water in the concrete mixture.Compressive strength is inversely proportional to water content, and the slope of the fitted simple regression line is −0.123.Figure 6 shows the distribution of water and the first-order partial derivative of compressive strength with respect to water for the training samples of the CSPNN, and its mean is −0.092.e negative mean value of the first-order partial derivative of compressive strength with respect to water indicates a negative correlation between compressive strength and water, which is consistent with the negative slope value of the fitted simple regression line in Figure 5.
Figure 7 shows the distribution of compressive strength and cement for the CSPNN training samples.It shows that compressive strength increases with an increase in the amount of cement in the concrete mixture.Compressive strength is proportional to cement, and the slope of the fitted simple regression line is 0.0764.Figure 8 shows the distribution of cement and the first-order partial derivative of compressive strength with respect to cement for the CSPNN training samples, where the mean is found to be 0.037.e positive mean value of the first-order partial derivative of compressive strength with respect to cement indicates a positive correlation between compressive strength and cement, which is consistent with the positive slope value of the fitted simple regression line in Figure 7.
Figure 9 shows a similar distribution of compressive strength and SP for the CSPNN training samples.Compressive strength increases with an increase in the amount of SP in the concrete mixture.Compressive strength is proportional to SP, and the slope of the fitted simple regression line is 1.6298.Figure 10 shows the distribution of SP and the first-order partial derivative of compressive strength with respect to SP for the CSPNN training samples.e mean of the first-order partial derivative of compressive strength  e positive mean value of the first-order partial derivative of compressive strength with respect to SP indicates positive correlation between compressive strength and SP, which is again consistent with the positive slope value of the fitted simple regression line in Figure 9. Sensitivity analysis results of the CSPNN therefore indicate that the CSPNN is a reasonable model representing the relationship between the 7 input parameters and compressive strength.

Training and Testing of the SPNN Using Collected
Experimental Data.As mentioned, only 295 samples have slump data among the total of 482 collected samples.erefore, 295 samples were used to train and test the SPNN.Among the 295 samples, 285 and 10 samples were used to train and test SPNN, respectively.Here, the SPNN is constructed with seven, six, and one nodes in the input layer, hidden layer, and output layer, respectively, and is denoted as SPNN(7-6-1).e complete offline training process took 31 cycles.e E and R 2 were 0.0079527 and 0.93996, respectively.After the SPNN was trained on the 285 training samples, it was tested to observe how accurately it would predict slump of other samples.Table 10 and Figure 11 summarize the results of these tests, indicating that the SPNN can satisfactorily predict the slump in all 10 testing samples.

Sensitivity Analysis of the SPNN.
Figure 12 shows the distribution of slump and SP for the training samples of the SPNN.It shows that slump increases with an increase in the amount of SP in the concrete mixture.Slump is proportional to SP, and the slope of the fitted simple regression line is 0.6246.Figure 13 shows the distribution of SP and the firstorder partial derivative of slump with respect to SP for the SPNN training samples.e mean of the first-order partial derivative of slump with respect to SP for the training samples of the SPNN is −0.146.e negative mean value of the first-order partial derivative of slump with respect to SP indicates negative correlation between slump and SP, which is inconsistent with the positive slope value of the fitted simple regression line in Figure 12. e reason may be that SP is a material with larger variance, and the properties of different brands of SP are different.

Experimental Program.
Experimental specimens were also made in the laboratory to study the prediction accuracy of the CSPNN and SPNN in terms of pozzolanic concrete conforming to the ACI concrete mixture code.Twelve concrete mixtures (listed in Table 11) were generated randomly by computer program according to the concrete mixture in ACI code.Four experimental specimens were made for each concrete mixture.e trained and tested CSPNN and SPNN represent accurate models for compressive strength and slump, respectively, and they were used to predict compressive strength and slump of experimental and numerical data.Among 1500 experimental and numerical data, 278 data satisfy Equation ( 14) and they were kept in the dataset.

Classification of Pozzolanic Concrete Mixture
Proportioning.After establishing the dataset of pozzolanic concrete mixture proportioning, it was classified further according to compressive strength, pozzolanic admixture replacement rate, and cost of concrete.Tables 12 and 13 give some of the results.Table 12 lists concrete mixture proportioning samples for compressive strength � 210 kgf/cm 2 and cost ≤2000 NTD/m 3 .Table 13 lists concrete mixture proportioning samples for compressive strength � 700 kgf/cm 2 and 2000 NTD/m 3 ≤ cost ≤ 2250 NTD/m 3 .Engineers can utilize the classified dataset to easily predict mix proportioning (solution) from required compressive strength of concrete, pozzolanic admixture replacement rate, and cost of concrete.

Conclusions
is study develops a two-step computer-aided approach for pozzolanic concrete mix design.e first step is to establish a dataset of pozzolanic concrete mixture proportioning that conforms to ACI code.In this step, ANNs are employed to establish the prediction models of compressive strength and slump of concrete.e second step is to classify the dataset of pozzolanic concrete mixture proportioning.A classification method is utilized to categorize the dataset into 360 classes based on compressive strength of concrete, pozzolanic admixture replacement rate, and material cost.e following important conclusions are drawn from the results: (1) e CSPNN and SPNN were trained using a portion of collected experimental data.Learning ratio α: Step length.

1
ki can represent the correlation between the kth output parameter and the ith input parameter.A positive (negative) value of D 1 ki represents a positive (negative) correlation.e absolute value of D 1 ki represents the strength of the correlation.A larger absolute value of D 1 ki represents a stronger correlation.Absolute values of D 1 ki near zero indicate little or no correlation.

Figure 2 :
Figure 2: Schematic diagram of the proposed approach for pozzolanic concrete mix design.

Figure 10 :Figure 11 :Figure 12 :
Figure 10: Distribution of SP and the first-order partial derivative of compressive strength with respect to SP for the training samples of the CSPNN.

Figure 14 :Figure 15 :
Figure 14: Comparison of exact compressive strength with CSPNN-predicted compressive strength for the 12 experimental concrete mixtures.

Table 1 :
Range of input parameters of CSPNN and SPNN in dataset.

Table 3 :
[39]oximate mixing water and target air content requirements for different slumps and nominal maximum sizes of aggregate "Table3is reproduced from Kosmatka et al.[39](under the creative commons attribution license/public domain)."

Table 4 :
[39] volume of coarse aggregate per unit volume of concrete "Table4is reproduced from Kosmatka et al.[39](under the creative commons attribution license/public domain)."

Table 5 :
[39]tionship between water to cementitious material ratio and compressive strength of concrete "Table5is reproduced from Kosmatka et al.[39](under the creative commons attribution license/public domain)."

Table 6 :
[39]mum water-cementitious material ratios and minimum design strengths for various exposure conditions "Table6is reproduced from Kosmatka et al.[39](under the creative commons attribution license/public domain)."

Table 9 :
Comparison of exact compressive strength with CSPNNpredicted compressive strength for the 20 testing samples.

Table 10 :
Comparison of exact slump with SPNN-predicted slump for the 10 testing samples.

Table 11 :
Twelve experimental concrete mixtures and their exact and predicted compressive strength and slump.Prediction of Slump.Figure15shows a comparison of exact slump to SPNN-predicted slump for the 12 experimental concrete mixtures.e slump of each concrete mix is the average slump of the four specimens for each concrete mixture.Most predicted errors for slump are within the allowable data range for width of slump (3.8 cm), with only one being extreme (7.4 cm).e predicted error of slump may be mainly caused by SP, since SP is a material with larger variance and the properties of different brands of SP are different.Notably, CSPNN and SPNN were trained using experimental data of pozzolanic concrete mixture proportioning collected from the literature.It is believed that predicted error of compressive strength and slump can be largely decreased if a sufficient number of experimental specimens could be made and used for training of CSPNN and SPNN.
e distribution of slump and SP for the training samples of the SPNN shows that slump increases with an increase in the amount of SP in the concrete mixture.Slump is proportional to SP, and the slope of the fitted simple regression line is a positive value (0.6246).However, the mean of the first-order partial derivative of slump with respect to SP for the training samples of the SPNN is a negative value (−0.146).e negative mean value of the first-order partial derivative of slump with respect to SP indicates negative correlation between slump and SP, which is inconsistent with the positive slope value of the fitted simple regression line.e reason for this may be that SP is a material with larger variance and the properties of different brands of SP are different.(4) To construct a dataset of pozzolanic concrete mixture proportioning which is practical and convenient for engineering applications, it is classified further.Engineers can utilize the classified dataset to easily predict mix proportioning from required compressive strength of concrete, pozzolanic admixture replacement rate, and the necessary cost of concrete.

Table 13 :
Concrete mixture proportioning samples for compressive strength � 700 kg/cm 2 and 2000 NTD/m 3 < cost ≤ 2250 NTD/m 3 .Sum of the squares error W hnj,ok : e weight associated with the jth node in the nth hidden layer to the kth node in the output layer W hnjn,ok : e weight associated with the jnth node in the nth hidden layer to the kth node in the output layer W ij : e weight associated with the ith node in the preceding layer to the jth node in the current layer W xi,h1j1 : e weight associated with the ith node in the input layer to the j1th node in the first hidden layer θ j : e threshold value of node j in the current layer θ ok : e threshold value of the kth node in the output layer η: