Multi-objective optimizing high strength concrete using meta-heuristic genetic algorithm

doi:10.21203/rs.3.rs-2214622/v1

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Multi-objective optimizing high strength concrete using meta-heuristic genetic algorithm

https://doi.org/10.21203/rs.3.rs-2214622/v1

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Reducing the cost of concrete construction as the most expensive building material reduces the overall cost of construction high-strength concrete (HSC). In the present study, to achieve this goal, the mix design of HSC is optimized in terms of strength and price using the meta-heuristic genetic algorithm (HGA). To do this, in the first step, a series of experimental data was considered as basic information and then, a mix design function was obtained to determine 28-day compressive strength and a slump calculation function using meta-HGA m in MATLAB software. In the next step, strength-price function was optimized using the meta HGA by changing the material ratios in the mix design, the price of the materials in each mix design and applying the required conditions of HSC including slump obtained from slump calculation function. Then, a comparison was performed between the results of this algorithm and the regression method, the results showed the better responses of the algorithm compared to regression for both factors of strength and price with the values of 10.2% and 6.5%, respectively. In addition, the construction of the mix design resulting from responses of the algorithm and regression in laboratory indicates that more than 97% of the strength was achieved at 28 days of age.

High-strength concrete

Mix design optimization

Meta-heuristic genetic algorithm

Regression method

Compressive strength

Price

In North American concrete institute, the concretes with 28-day compressive strength more than 42 MPa are usually called HSC (Ahmed and Shah, 1994). In recent reports of Bulletin¹ (1990), the concrete with strength more than 60 MPa has been called high-strength concrete. Mindes and Young (1981) recommended that special attention must be paid to all aspects of the production of this type of concrete, including materials, mix design, transportation and concreting. Quality control is one of the essentials in the production of HSC and full coordination between resources and the quality of materials and its producer is required (Ahmed and Shah, 1994). In general, the HSC mix design includes steps of selecting materials, determining relative values, and quality control. According to studies conducted by Wu et al. (2001), HSC has a water-cement ratio less than 0.4 and the strength of mortar and connection in the middle phase can be similar to the strength of coarse aggregates, and so in some cases, the ability of the aggregates may be only considered.

The effect of coarse aggregates on the mechanical properties of HSC was reported by Ozturan and Cecen (1997). They concluded that the usual strength of concrete is achievable using basaltic aggregates, and the use of ordinary sand results in the same strength, while concrete with limestone aggregates is more strength. HSC is achievable with basaltic and crushed limestone aggregates. Aitcin and Mehta (1990) reported that for similar materials and mix design, the use of diabase and limestone aggregates led to the production of higher-strength concrete compared to the use of granite and river aggregates. Beshr et al (2003) tested the effect of the quality of aggregates on compressive and tensile strengths of HSC and showed that the quality of aggregates specifically affects both types of strength and slag aggregates has the greatest impact on strength; however, limestone aggregates have the least effect. Kilic et al (2008) investigated the effect of aggregate type on compressive strength and abrasion strength of high-strength concrete. Five different types of aggregates were used in mix designs. On the other hand, by replacing 15% of microsilica with cement in each design, specimens with the constant ratio of water to cement as 0.35 were made. The results showed that the compressive and abrasion strength of basaltic, lime stone and sand aggregates were equal and quartz and gabbro were lower. Mazloom et al (2004) reported that in the HSC, increasing the amount of microsilica reduced concrete performance, but improved primary mechanical properties such as 28-day strength and static elasticity modulus. It has also been reported that the percentage of alternative microsilica does not have a definite effect on the overall aggregation of concrete. Balaguru (2005) conducted research on nano- and micro-particles of cement and stated that the properties of concrete have been improved and a better understanding of the structure and its behavior on nano- or micro-scales helps improvement of the properties of concrete. Safn (2012) studied the compressive strength of Portland cement paste and mortar containing nanoferrite and stated that the compressive strength of cement paste and mortar will averagely increase by 45% using the appropriate amount of nanoferrite mortars containing 1% by weight of cement. Caijun et al (2015) concluded that microsilica could improve the fluidity and compressive strength of concrete, reduce porosity and hydroxide, and reduce slag, fluidity, compressive strength, porosity, and hydroxide formation to a certain amount. The results of the experiments showed that the simultaneous use of a certain amount of natural pozzolan and microsilica could improve compressive and tensile strengths, performance and static elasticity modulus, compared to when they are used separately in concrete(Shanang, 2000). Amin and Hassan (2015) made 34 specimens with different percentages of cement, fine aggregates, coarse aggregates, microsilica, water and two types of nano-materials. Then, based on the results of tests of compressive strength at the ages of 1, 7, 28 and 90 days, tensile strength, flexural strength and the static elasticity modulus, the optimal rate of using nanosilica and nanoferrite is 3% and 2% in HSC. The percentage of compressive strength improvement by using this amount of nanocillis and nanoferrite was 17% and 21%, respectively; the tensile strength was improved by 44% and 66% using nano-silica and nanoferrite, respectively; and this value for elasticity modulus and flexural strength with using nano-silica and nanoferrite was nearly 23% and 25%, respectively. Increasing the amount of nanosilica and nanofertite by more than (2–3) % by weight reduces the above characteristics. They also stated that specimens containing about 10% alternative nanosilica responded better than nanofertite with the same replacement percentage, and granite aggregates with a replacement rate of 10% responded better than dolomite at the same replacement rate.

Debasmita and Parhi, (2017) offered a mix design based on the IS-10262: 1982 standard for 60 MPa strength using the fine aggregate waste marbles instead of aggregates. In their study, 5 mix designs with water to cement ratios of 0.3 and 0.35 were used. The use of marble not only reduced the price, but also had a positive effect on the compressive strength of HSC compared to the use of aggregates alone. The optimal mix design of the research was a design with cement, sand, coarse aggregate ratio as 1, 0.76 and 2.47 with 8% superplasticizer content and water-to-cement ratio of 0.35 and (5–8) % superplasticizer. Al-Jabri et al. (2011) examined the effect of replacing wastewater instead of water on properties of the HSC using the output water of three carwashes in different parts of Muscat. In this study, all three types of water were combined and its chemical details including pH, insoluble particles, chlorine content, hardness, alkaline materials and sulfate materials were measured. These experiments showed that although the chemical agents in the wastewater collected were much higher than normal water, this water met ASTM standards and could be used in concrete. The mix designs were made with different amount of water and sewage and cement ratio of 0.35 and the amount of replacement of water with sewage was from 25–100%. After 28 days of processing, the results showed that the 28-day strength had an acceptable change and the amount of water absorption had an insignificant change. No important results were reported about the changes in tensile and flexural strength, but the increase in strength from age of 7 to 28 days was similar to conventional concrete. Kishore et al. (2011) investigated the effect of rice bran powder on the mechanical properties of high-strength concrete. In the mix design, bran powder with different ratios of 0, 5%, 10% and 15% as an alternative for cement was substituted in cubic, cylindrical and prismatic molds for strengths of M40 and M50. The strength of the specimens was acceptable at 7, 28 and 56 days of age, and the optimal percentage of rice bran powder replacement was 10%. Increasing the amount of rice bran powder reduced the efficiency of concrete (slump by 27% and compression coefficient by 9%), compressive, tensile strength, bending strength and static elasticity modulus. Chrisminingeang et al. (2017) studied the correlation between tensile strength and rupture modulus of HSC made by substituting 0, 5%, 10%, 15% and 20% of rice bran powder instead of cement and processing in seawater. The results of their study showed that with increasing the amount of alternative powder, the modulus of rupture and tensile strength increased and the correlation coefficient between the both decreased under processing with seawater compared to normal water. Concrete resistance to seawater was also increased. Rathan Raja et al. (2013) examined the mechanical properties of HSC with different mix designs in the M60 strength category to investigate the effect of additives such as methacholine and red mud instead of cement. The optimal percentage of methacholine, red mud and their combination for maximum strength, tensile strength, and flexural strength was 8%, 4%, and 10%, respectively. The use of each material reduced the efficiency of the concrete, and the changes rate of HSC with replacement of methacholine and red mud from age of 7 days to 28 days was reported to be 0.773 and 0.823, respectively. Cerny et al. (2017) suggested the possibility of making high-strength lightweight concrete using fly ash instead of aggregate. In their study, they used four types of artificial aggregates made of fly ash. The results showed that the amount of cement and plasticizer must be increased to achieve high strength, and this increase enhances the risk of increasing the weight of concrete up to 2000 kg/m³. It means that this increase enhances the limit values of lightweight structural concrete. Their 28-day strength was reported 45 to 53 MPa and their weight was reported 1840 to 1860. Simon, (2003) created 36 mix designs based on different percentages of water, cement, fine aggregate, coarse aggregate, superplasticizer, and microsilica. They obtained the mix design function of materials for 7 and 28-day compressive strength, slump and water absorption using the results of experiments, analysis of variance and multilevel regression equations, and presented this equation as the optimal mix design. In this method, four different equations resulted from optimization were considered in terms of cost. After examining the statistical uncertainty of the function in 95%, a sample with a slump of (76 ± 15) mm, 1-day compressive strength of (23.05 ± 0.69) MPa, 28-day strength of (55.20 ± 2.31), 42-day water absorption of (6.404 ± 0.114), and 0.16% water, 0.133% cement, 0.014% microsilica, 0.0054% superplasticizer, 0.409% coarse aggregate, 0.2786% fine aggregate, with cost of 96.35 $ was introduced as the best one. In the Shakhmenko and Brish, (1988) optimization model, the percentage of aggregates, which was in the range of (60–90)% with different combinations including sand and two types of broken aggregates in the mix design, was optimized by the software using mathematical and numerical analyzes. This optimization provided a better mix percentage for concrete aggregates and reduced the cost of raw materials. Finally, they claimed that the correlation coefficient of theoretical and experimental data in this optimization was (85–95) %. Shariq et al. (2012) made 64 different concrete specimens using different ratios of water to bonding materials, coarse aggregates to total aggregates and total aggregates to cement. Then, they obtained a mix design function with the key variables using test results and their different percentages in regression. Then, they numerically optimized the function and presented an optimal mix design with an error of about 10%. They stated that the highest compressive strength was obtained at the water to cement ratio of 0.4, the coarse aggregates to total aggregate ratio of 0.6 and the total aggregate to cement ratio of 4.5, and the optimal strength was obtained at the water to cement ratio of 0.46, the coarse aggregates to total aggregate ratio of 0.6–0.63 and the total aggregate to cement ratio of 3.9. Ahmad and Al-Ghamdi (2014) made 27 mix designs and 3 specimens for each design according to the amount of cement, the ratio of water to cement and the ratio of fine aggregates to total aggregates, and measured the criterion deviation of the 28-day compressive strength of the design. They presented a four-unknown equation of analysis of variance and regression for optimal concrete design using these three ratios. Their result showed that the ratio of water to bonding material and the ratio of fine aggregates to total aggregates compared to other ratios had the maximum effect. The use of optimal value of these two ratios showed a good result in terms of achieving high compressive strength using less cement and thus resulted in lower cost of making concrete. Akasha et al. (2017) presented a mix design for HSC without the use of additives. In this design, basaltic and granite aggregates were used instead of coarse aggregates. The designs were based on 410, 430 and 450 kg of cement per cubic meter. The results of the experiments at 7, 28 and 56 days of age showed that the use of granite instead of coarse aggregates resulted in better responses and 28 and 56-day strength of 56 and 64 N/mm² were obtained using water-to-cement ratio of 0.46.

According to the study of the authors of this paper on the history related to the research, many researchers have tried to find the optimal mix design in which the price and strength of concrete are considered, but there is no research in which the genetic algorithm have been used in two stage to present the optimal mix design for HSC. The experiments of a reference study (Simon 2003) were the basis of this study and the mix design function was first calculated using the “meta-HGA”. Then, this function was again optimized using the GA considering material price and by applying constraints such as water-cement ratio limit and allowable range of slump and the superiority of this algorithm was explained and presented.

[1] The International Federation for Pre-stressing- The International committee for Concrete Bulletin (FIP-CEB)

2.1. Materials and consumables used in research

In this study, type II Portland cement and microsilica, which were produced in Iran and their specifications presented by their manufacturer have been shown in Table 1, and ether poly carboxylate-based superplasticizer, fine aggregates and two types of aggregates from mines of the area with granulation shown in Fig. 1, were used.

Table 1

Specific gravity and chemical compounds of cement and microsilica
Chemical compounds (%)	Cement	Microsilica
SiO₂	21.68	96.4
Al₂O₃	4.82	1.32
Fe₂O₃	3.88	0.87
CaO	65.4	0.49
MgO	2.08	0.97
SO₃	0.68	0.1
K₂O	0.88	1.01
Na₂O	0.25	0.31
Specific gravity	3120	2130

2.2. Introducing optimization methods

2.2.1. Optimization with regression method

To study HSC, Simon prepared 39 specimens in the form of 36 mix designs with mixing cement, water, coarse aggregates, fine aggregates, microsilica, and superplasticizer that the allowable volume range of each of materials was pre-determined. Then, he measured their compressive strength, slump, and permeability. As the aim of this study was to investigate 28-day strength, only data related to slump and 28-day strength were presented. In the next step, the results of the analysis of variance (ANOVA) showed that the mix design detail function of these specimens is a linear function. To determination the function of relationship between the details of materials for each slump and 28-day strength was obtained using the regression of experimental data and in the final stage, this function was optimized by numerical methods and the output was presented as the optimal mix design in the construction of HSC according to the tested materials (Simon 2003).

To examine the functions and outputs of Simon's experimental data from regression, the regression equation of 28-day strength and slump was recalculated by preparing a table of details of Simon's research materials in SPSS software, to prevent doubt in the research results at the time of comparison because of the possible inaccuracy of functions.

2.2.2. Optimization with MHGA

Since 1960, imitations of natural phenomena have been considered for using in powerful algorithms to solve optimization problems, which have been called evolutionary computation methods. Over the past 20 years, a new type of heuristic algorithm has emerged that essentially seeks to combine basic exploration methods with the aim of efficient and effective search of search space in higher-level frameworks. In simple terms, the algorithm creates a population of results for the problem. Then, the best results are selected. These results are reproduced and create another new environment of results in a different situation from the previous one. This process ends when the results obtained meet certain criteria. Nowadays, these methods are called meta-heuristic algorithms. Genetic algorithm is one of the most important heuristic algorithms that are used to optimize the performance of various functions. In this algorithm, past information is extracted according to the inheritance of the algorithm and used in the search process (Masihi et al. 2012). Perceptron is a machine learning algorithm that falls into the category of supervised learning. The Perceptron algorithm is a binary classification algorithm and is a type of classification that can decide whether an input belongs to a class or not, depending on the input vector. On the other hand, this algorithm is a linear classifier, in the sense that it predicts according to the weighted linear composition of the algorithm input. In addition, this algorithm is an online algorithm because it checks its inputs individually over time. The Perceptron algorithm was invented in 1957 in the Cornell Aeronautical Laboratory by Rosenblatt, (1957). In fact, this algorithm is one of the first artificial neural networks. Holland introduced (1975) a basic concept of genetic algorithm. In this algorithm, after formation of the initial population, operators such as cross over or genetic mutation improve the population cycle and finally, population of the next cycle are selected by a targeted random selection. Multiple generations in next cycles improve the population and lead continuous improvement of the society individuals. The ant colony optimization (ACO) algorithm for accessing food was introduced by Dorigo, (1992) as a multi-factor solution to optimization problems such as the traveling salesman problem (TSP). This algorithm is based on the ants finding food method, and the process is such that each ant has an effect on finding food. Thus, if one ant finds food, it puts two effects on the path due to back and forth, and if it has not found any food, it has one effect, so if another ant wants to choose a path, it will choose a more effective path and thus better routes are selected in next cycles. Bird algorithm or particle swarm optimization (PSO) algorithm was proposed by Kennedy and Eberhart, (1992). It is an evolutionary computational algorithm inspired by nature and based on repetition. The source of inspiration for this algorithm was the social behavior of animals, such as the mass movement of birds and fish. The imperialist competitive algorithm (ICA) provides an algorithm for solving mathematical optimization problems by mathematical modeling of the socio-political evolutionary process. The ICA developed by Atashpaz-Gargari and Lucas, (2007) is a primary set of possible answers. These primary responses are known in the genetic algorithm as chromosomes, in the PSO algorithm as the particle, and in the ICA as the country. The ICA improves gradually these initial answers (countries) with a certain trend, and finally provides the appropriate answer of the optimization problem (desirable country). The main foundations of this algorithm are the policy of assimilation, colonial competition and revolution. This algorithm, by imitating the process of social, economic and political development of countries and by mathematical modeling of parts of this process, presents operators in a regular form as an algorithm that can help solving complex optimization problems. Simulated annealing algorithm is a popular algorithm that has been widely used in many solutions to solve optimization problems after its introduction by Kirkpatrick et al. (1983).This method is a random local search style based on the principles of nature. This algorithm is based on the jumping of metal drops during cooling, which is done according to a certain number. Von Frisch, (1967) was one of the first who used the basic and simple bee practices to solve hybrid optimization problems. He introduced the bee system and used it to solve the famous problem of the traveling salesman. The bee algorithm includes a population-based search algorithm that was first developed in 2005. This algorithm simulates the search behavior of bee groups. Each bee announces three messages to the other bees through dance after finding food. These messages carry the specifications of the distance, type and direction of the desired flower location to the hive. These characteristics include the process for other bees to reach the food location and find the best flower. Other algorithms have been introduced in recent years, including the intelligent water drop (IWD) algorithm or hammer algorithm. This algorithm is an optimization algorithm based on group intelligence developed by Shah-Hosseini, (2009). The IWD algorithm is an algorithm that works as a group and is inspired by nature. This algorithm is originally used for hybrid optimization, but it can also be prepared for continuous optimization. The algorithm was first proposed in 2007 to find a solution to the problem of the traveling salesman problem (TSP). Since then, a number of researchers have sought to improve and apply this algorithm to a variety of problems and issues. The basis of the IWD algorithm is to find the path of water at the lowest point of a surface. Another algorithm introduced by Aghay-Kaboli et al. (2017) was the rain-fall optimization (RFO), which, searches to find the lowest answer to the problem based on the method of finding the lowest point by raindrops. In addition, it was compared with other algorithms in terms of its efficiency in solving several mathematical problems, its speed and answers convergence and the results were satisfactory.

Sajedi and Razak, (2010) modeled the relationship between 7-day and 28-day strength of the high-strength concrete. To do this, they prepare 650 different specimens made of cement, water, microsilica, aggregate and superplasticizer using 50 mix designs, and then obtained compressive strength at the ages of 1, 3, 7 and 28 days. The results of their research showed that multidimensional regression, neural network, and linear regression performed better than each other in estimating 28-days strength compared to 7-days strength. The proposed relationship was well adapted to the processing conditions of the specimens in the steam room and immersion in saturated limewater at 25°C and specimens without slag. Grunewald et al. (2012) proposed an optimal mix design to improve the tensile microstructure of high-performance fiber reinforced cementinious composite (HPFRCC) using an ant algorithm. Guerra and Kiousis, (2006) provided an optimal design for the structure of reinforced concrete using a nonlinear algorithm in MATLAB. The expense of forming, concreting, and reinforcing by workers, as well as materials were considered as inputs of the problem. The aim of this study was to optimize the dimensions of beams and columns in reinforced concrete that was achieved using various experiments and algorithm programming in MATLAB. Chopra et al. (2016) used an artificial network and genetic algorithm to predict the Increase of strength at 28, 56 and 91 days of age. In this study, 49 mix designs with common materials and 27 designs by adding fly ash in laboratory conditions were studied and the studies showed better performance of neural network in predicting strength Increase over time. Wang et al. (2019) proposed a systematic approach for producing a random model with realistic stone properties to simulate a number of stone-based materials. In the first step, the inverse discrete Fourier transform (IDFT) method was used to randomly generate real stones with predetermined particle shape properties. Then, the overlap detection algorithm was proposed to facilitate the random and rapid allocation of irregular stones considering stone material, stone size distribution and stone orientations. Finally, they provided various examples to show that the proposed approach could produce fast and accurate stone materials with the stone properties prescribed. Hosein and Elsadadedy, (2019) examined the effect of high temperature on the subsidence of strength of the HSC using 617 concrete specimens of various researchers' studies. To find the equation for the effect of temperature on strength, the dependent and independent variables were first identified using the neural- network and then the effect of independent coefficients on the dependent coefficients was obtained by applying regression. Finally, the aim of this study, which was to invent several simple regression equations to predict the strength of concrete under the influence of temperature, was achieved. The results showed that, as expected, the maximum temperature plays an important role in controlling the subsidence of the compressive strength of high-strength concrete. The remaining parameters show almost the same level of sensitivity, and the effect of these parameters compared to the reduced temperature is completely negligible. Algorithms have a great ability to find the optimal answer in a large and numerous search spaces. Concretes with multiple components also have a variety of mix designs, so finding the optimal mix design has always been in question.

Genetic algorithm is one of the most widely used heuristic algorithms that is used to optimize various functions. In this algorithm, past information is extracted according to the inheritance of the algorithm and used in the search process. First, Holland introduced a basic concept of genetic algorithm, and then, others expanded it. Genetic algorithms are random search methods that work based on natural selection and natural genealogy. These algorithms have fundamental differences from conventional search and optimization methods, which have been summarized by researchers as fowling (Masihi et al. 2012).

1. The genetic algorithm works with a set of coded responses, not with themselves.

2. The genetic algorithm starts searching in one population of answers and with a set of them, not with one answer.

3. The genetic algorithm uses the information of the fitness function, not the derivatives or other auxiliary sciences.

4. The genetic algorithm uses probabilistic transfer rules, not definitive rules.

Genetic algorithm is a mathematical algorithm that uses Darwinian operational patterns of multiplying the survival of the fittest and converts a collection (population) of individual mathematical objects (usually fixed-length character strings as chromosomes) with a specific adaptation rate into a new population (for example, the next generation) based on the natural process of genetics.

Shahsavari Pour et al. (2010) used a meta- heuristic genetic algorithm to solve the problem of time- cost-quality balance of multi-state projects. The results showed a good success of the algorithm. Masihi et al. (2015) solved the time-cost-resource constrain balance in multi-state projects using a modified genetic algorithm in which the chances of better members of the population were greater and the result of the balance was very satisfactory. Taheri Amiri et al. (2017) presented the time-cost balance for planning the project's critical network by the genetic algorithm. In other words, in this study, time scheduling and balancing the cost-time have been performed simultaneously using the genetic algorithm. Haghighi and Ahmadi-Najl, (2016) optimized operating rules and rule curves for multi-reservoir systems using a self-adaptive simulation-genetic algorithm model. In this design, the algorithm has performed optimization in successive cycles. Han et al. (2022) used Combining Artificial Neural Network with Particle Swarm Optimization to optimize the mix proportion of wet-mix shotcrete. 16 specimens were made and this hybrid model was applied to optimize mix proportions of wet-mix shotcrete in the Jinchuan mine. The results revealed that the ANN model yielded a mean relative error (MRE) of 2.755% and an R2 of 0.980, indicating an excellent prediction to establish the reasonable objective function. Additionally, PSO spent less than 60 s obtaining an optimal mix proportion of wet-mix shotcrete required by the mine. Consequently, this ANN–PSO model can be used as an efficient design guide to facilitate decision making, prior to the construction phase. Using a numerical method and Meta-heuristic Algorithms for estimating the optimal mixture design of concrete pavements proposed by Shirzadi Javid et al. (2021). A novel method is developed to predict the optimal mixture proportion, which enhances fundamental characteristics of concrete, including flexural strength, abrasion strength, slump, drying shrinkage, and freezing–thawing resistance, and its unit cost. The outcomes of this model reveal that the performance of this combination method is considerably better than those of heuristic algorithms. Moreover, the performance of mixtures designed by all algorithms is much better than the performance of initial specimens designed by experimental tests. By virtue of multi-objective optimization, the average of abrasion strength, drying shrinkage, freeze and thaw resistance, flexural strength and cost in the introduced model are improved 18%, 12%, 5%, 4%, and 1%, respectively. However, the average of slump deteriorates 9%. Sadrossadat et al. (2021) proposed machine learning algorithms and metaheuristics for mixture design and optimization of steel fiber reinforced UHPC. Investigations aims to demonstrate a procedure integrating machine learning (ML) algorithms such as ANN and Gaussian Process Regression (GPR) to develop high-accuracy models, and PSO for multi-objective mixture design and optimization of UHPC reinforced with steel fibers. The comparison of the obtained results with the experimental results validates the capability of the proposed procedure for multi-objective mixture design and optimization of steel fiber reinforced UHPC. The proposed procedure not only reduces the efforts in the experimental design of UHPC but also leads to the optimal mixtures when the designer faces strength-flow ability-cost paradoxes.

2.3. Introducing the MHGA of the research

In the algorithm used, the initial population is first formed. Then, in addition to the general operators of the genetic algorithm, several heuristic methods are used to provide more diverse and optimal search for the algorithm for searching answers. Then, the process continues by selecting a new population based on random selection with the probability of choosing the best person. To better understand the subject, the operators are described below.

2.3.1. Formation of primary population

There are a number of strings called chromosomes and contain information about each person. These strings have the required number of genes.

2.3.2. Random crossover operator

In the case of the presence of a possibility, new offsprings are obtained for the community by crossover method by crossing accidentally a couple of chromosomes, which are also randomly selected. In other words, the two random chromosomes are selected as the parents, and then new offsprings are obtained by moving some of their genes.

2.3.3. Mutation operator

In this operator, if a probability occurs, a chromosome is selected and a genetic mutation is randomly applied to it.

2.3.4. Modifying operator

In this heuristic approach, some of the best specimens of the initial population in the program generation, which is usually below 20%, are added to the society individuals as the effective individuals. Thus, the chances of selecting the right people for the next generation are increased.

2.3.5. The fitness of the individual in society

First, each person in the society is assigned a range whose amount is appropriate for the individual fitness. This is usually done using a function called the Fitness function Eq. (1). This function can have single or different variables. This function has been defined using Eq’s (1 and 2).

Fitness function=$F\left(MOS\right)$ ² (1)

Domain= ${F}_{\left(Fitness function\right)}$ (2)

2.3.6. Penalty applying

Whenever the choice from the set is difficult or the range of domains is small, it is possible to impose a penalty on a particular type of community members who are less characteristic in terms of one respect and to reduce the usefulness of their domain in proportion to their incompatibility. This operator works well when there are certain restrictions.

2.3.7. Selection of a new generation

Different methods have been proposed for this purpose, but due to the better performance of the roulette wheel, this method was used. A roulette wheel or circle is a random selection method in which each characteristic allocates a fragment of the circle, and is declared as the winner by random selection of numbers or its characteristic pieces. The algebraic sum of the domains forms the environment of the roulette wheel. Each time a person is selected to become a member of the next generation, a number is randomly selected in the entire range of the domain, which will certainly increase the chances of selecting a sample with a larger domain. In this way, selected from the existing community of this generation is performed as much as the required number of population.

2.3.8. Elitism

As all the processes have been accidental so far, in order to prevent the elimination of the worthy and elite answer, one or two people from the society, who have the best fitness, have been added to the next generation and this number is deducted from the output in the roulette wheel process. Finally, the number of new generations is equal to the number of previous generations.

2.3.9. The general cycle of the MHGA

The blue operators shown in Fig. 3 are the main operators of the genetic algorithm, and the operators shown in green are heuristic operators. The operators are programmed according to the cycle given in Fig. 3 for all the following steps in MATLAB software and the answers are extracted from this process

2.4. Forming the mix design function by the algorithm

At this stage, the goal is to achieve functions for slump and strength to predict the mix design with a good approximation of slump and 28-day strength by applying the volume ratio of materials in them. Primary data are experiments performed for various mix designs and the results of designs for slump and 28-day strength. The condition for a function to determine the relationship between the ratio of materials percentage in the mix design with the strength or slump is having fewer sums of squares in the difference between the outputs of the prediction function of the mix design and the experimental results of the same design. In other words, each proposed function will have an output for the slump or strength of the mix design that has a difference with the experimental result, and shows the function of the relationship between the ratios of materials to reach better strength or slump with fewer sums of squares in difference.

2-4-1- Mix design function

The proposed function of this study is a multi-objective function in which the coefficients and power of the factors are not known, and on the other hand, since the volumetric ratios of the mix design are all below one, their triple product will be very small and near zero. Therefore, the function consists of main factors and their binary product.

- The main materials are:

Table 2

Introduction of materials used in research
Water	Cement	Micro silica	Super Plasticizer	Coarse Aggregate	Fine Aggregate
X₁	X₂	X₃	X₄	X₅	X₆

- The problem function

${F}_{\left(X\right)}$ =$(\sum _{n=1}^{6}{A}_{n}*{{X}_{n}}^{{B}_{n}})+(\sum _{n=1}^{5}{X}_{n}*\sum _{m=n+1}^{6}{{C}_{nm}X}_{m})+K$ (3)

Information of Eq. (3) are as fallowing:

Volumetric percentages are considered according to the experimental data.

X = Volumetric percentage of concrete components

Unknowns of Eq. (3) are as fallowing:

A_n = Numerical coefficient for the volumetric percentage of concrete components

B_n = Power of volumetric percentage of concrete components

C_nm = Numerical coefficient for the multiplication of the volumetric percentages of concrete components

K = Constant number

2.4.2. Solving the problem functions using algorithm

To solve this multi-objective function, considering the number of HSC components and the function mentioned in Eq. (3), there are 28 unknowns, which are arranged as a string as shown in Fig. 4.

Formation of primary population

Random numbers are placed instead of the coefficients and powers in the string of Fig. 4, and since the initial population must have at least 100 parents, random numbers are placed in the parent strings 100 times.

Algorithm calculations

In this way, each of these strings in Eq. (3) has an output for each mix design in the experimental table. The sum of the squares of the function output difference with the experimental response calculated in Eq. (4) will be the value of the Fitness function for the designs.

MOS= $\sum _{k=1}^{j}{({{F}_{\left(X\right)}}_{k}-{Cs}_{k})}^{2}$ (4)

${F}_{\left(MOS\right)}$ =CN –MOS (5)

${F}_{\left(X\right)}:$ Output of the function with random numbers

$Cs:$ Strength obtained from the mix design in the experimental table

K: The mix design row in the experimental table

J: Number of tested mix designs

CN: Constant Number

The function of algorithm in calculate on unknowns

Algorithm processes are run according to the formation of the initial population and the above calculations. Thus, the sum of the squares of the population will be similar or very close to each other after several periods of algorithm implementation, and only the mutation operator tries to produce new people in the general population. Regeneration will continue until a lower sum of square is not observed in the algorithm cycle.

Finally, the algorithm is run for slump and 28-day strength separately according to the answers in the experimental table, and finally, the best answer is calculated and selected by having coefficients and powers for Eq’s. 6 and 7.

${F}_{\left(Cs\right)}$ =$(\sum _{n=1}^{6}{A}_{n}*{{X}_{n}}^{{B}_{n}})+(\sum _{n=1}^{5}{X}_{n}*\sum _{m=n+1}^{6}{{C}_{nm}X}_{m})+{K}_{Cs}$ (6)

${F}_{\left(Slump\right)}$ =$(\sum _{n=1}^{6}{a}_{n}*{{X}_{n}}^{{b}_{n}})+(\sum _{n=1}^{5}{X}_{n}*\sum _{m=n+1}^{6}{{c}_{nm}X}_{m})+{K}_{sl}$ (7)

A, B, C, ${K}_{Cs}:$ Coefficients and powers in the 28-day strength function

a, b, cو${K}_{sl}:$ Coefficients and powers in the slump function =

2.5. Optimizing the mix design function

In this study, in addition to find a mix design with a balance between its strength and construction price, the other goal is to provide specific target strength with the lowest possible price. To achieve this goal, the basis of work will be the function of strength, and the necessary balance will be provided by applying the price of materials. For this purpose, the strength Eq. (6) needs to be investigated again, with the difference that this time the coefficients and powers (A_n, B_n, C_nm and K) are clear and it is necessary that the volumetric ratios of materials (X) be such that the output strength of the function and materials price of those ratios be economical. As the strength function is in hand, it is enough to include the price of each design in its Fitness function and then select the next generation process using the roulette wheel. Therefore, the random ratios of materials volume are considered for the unknown variables of the function. These ratios must be within the accepted range for HSC, and it is best to be very close to the selected range of experimental designs. The use of the slump function is also because specimens with the unacceptable slump have been modified and re-referred to the algorithm cycle.

2.5.1. Formation of primary population

The unknowns of the Eq. (6) form a hexagonal string according to Fig. 5, which is formed by a hundred random selections of concrete materials within the aforementioned range of the initial population of the algorithm.

2.5.2. Calculation of strength-price optimization algorithm

$$0.98\le \sum _{n=1}^{6}{X}_{n}\le 1$$

Fitness Function= ${F}_{(strength - Cost) }$= ${F}_{\left(Cs\right)}$ + ${F}_{\left(p\right)}$ (9)

${F}_{\left(p\right)}$ = CN-$\sum _{n=1}^{6}{(X}_{n}*{P}_{n})$ (10)

${P}_{n}:$ The price of material n per cubic meter

2.5.3. Algorithm function

The algorithm process and operator are run with having the chromosome and the primary population and computational Eq’s. 6 to 10. The process continues until there is no better answer or changes are very small.

2.5.4. Algorithm validation

To find out the error rate of the function provided by the algorithm with real one, a number of different mix designs whose details were previously taken from different strength classes of the algorithm, were made in the laboratory. Then, after measuring the fresh concrete slump according to ASTM C143 standard and determining the 28-day’s compressive strength according to ASTM C39 standard, the results are compared with the output of the algorithm.

[2] Min of Square (MOS)

3.1. Calculations of regression method

Simon first tested 36 mix designs considering a specific range for materials that the specifications and results of which have been presented in Tables 3 to 5.

Table 3

Volumetric range of materials in the mix design used by Simon, (2003)
Component	Water (m³)	Cement (m³)	Micro Silica (m³)	Super Plasticizer (m³)	Coarse Aggregate (m³)	Fine Aggregate (m³)
ID	X₁	X₂	X₃	X₄	X₅	X₆
Minimum Volume Fraction	0.16	0.13	0.013	0.0046	0.4	0.25
Maximum Volume Fraction	0.185	0.15	0.027	0.0074	0.4424	0.2924

Table 4

Volume ratios of Simon's experimental mix design (Simon, 2003)
Design ID	Run Order	Water (m³)	Cement (m³)	Micro Silica (m³)	Super Plasticizer (m³)	Coarse Aggregate (m³)	Fine Aggregate (m³)	28-day strnegth (MPa)	Slump (mm)
5	7	0.16	0.13	0.027	0.0046	0.4284	0.25	48.5	13
11	6	0.185	0.13	0.013	0.0046	0.4174	0.25	48.5	102
13	15	0.16	0.13	0.013	0.0046	0.4	0.2924	53.2	57
15	38	0.1656	0.15	0.027	0.0074	0.4	0.25	58.1	51
16	8	0.16	0.13	0.013	0.0046	0.4424	0.25	53.1	35
20	13	0.185	0.13	0.013	0.0046	0.4	0.2674	50.9	118
22	4	0.185	0.1474	0.013	0.0046	0.4	0.25	48.2	67
28	16	0.16	0.13	0.027	0.0046	0.4	0.2784	53.6	29
37	30	0.16	0.14	0.027	0.0074	0.4	0.2656	56	35
38	26	0.1767	0.1417	0.027	0.0046	0.4	0.25	53.2	29
48	28	0.1725	0.13	0.013	0.0074	0.4	0.2771	58.1	171
63	27	0.1725	0.13	0.027	0.0074	0.4131	0.25	55.2	124
65	31	0.16	0.14	0.027	0.0074	0.4156	0.25	51.1	32
66	25	0.16	0.13	0.027	0.0074	0.4128	0.2628	59.8	76
70	29	0.16	0.15	0.013	0.006	0.4	0.271	54.5	51
71	5	0.16	0.15	0.013	0.0074	0.4098	0.2598	55.2	57
78	11	0.185	0.13	0.027	0.0074	0.4003	0.2503	51.7	127
87	24	0.16	0.13	0.013	0.0046	0.4212	0.2712	51	67
89	19	0.16	0.14	0.013	0.0046	0.4	0.2824	54.3	38
91	9	0.185	0.13	0.013	0.0074	0.4073	0.2573	60.4	200
98	17	0.16	0.14	0.013	0.0046	0.4324	0.25	51.9	32
101	10	0.1712	0.15	0.013	0.0046	0.4112	0.25	53.6	22
103	14	0.16	0.15	0.013	0.006	0.421	0.25	54.6	64
110	21	0.1712	0.15	0.013	0.0046	0.4	0.2612	53.2	51
116	33	0.1725	0.13	0.027	0.0074	0.4	0.2631	56.2	114
123	36	0.16	0.14	0.02	0.006	0.412	0.262	61	99
127	12	0.172	0.1395	0.013	0.006	0.4098	0.2598	50.2	99
163	20	0.1656	0.1343	0.0165	0.0067	0.4234	0.2536	60.8	95
5	22	0.16	0.13	0.027	0.0046	0.4284	0.25	54.1	25
11	23	0.185	0.13	0.013	0.0046	0.4174	0.25	48	114
20	34	0.185	0.13	0.013	0.0046	0.4	0.2674	51.6	127
38	39	0.1767	0.1417	0.027	0.0046	0.4	0.25	54.5	25
71	35	0.16	0.15	0.013	0.0074	0.4098	0.2598	65.3	108
127	18	0.172	0.1395	0.013	0.006	0.4098	0.2598	54.1	92
127	32	0.172	0.1395	0.013	0.006	0.4098	0.2598	57.2	121
127	37	0.172	0.1395	0.013	0.006	0.4098	0.2598	54.6	102

Table 5

Weight ratios of Simon's experimental mix design (Simon, 2003)
Run Order	Water (m³)	Cement (m³)	Micro Silica (m³)	Super Plasticizer (m³)	Coarse Aggregate (m³)	Fine Aggregate (m³)
7	185	405.6	27.69	4.83	1085.24	625
6	185	459.888	27.69	4.83	1040	625
15	160	405.6	57.51	4.83	1113.84	625
38	185	405.6	27.69	4.83	1085.24	625
8	172	435.24	27.69	6.3	1065.48	649.5
13	185	405.6	27.69	4.83	1040	668.5
4	160	405.6	27.69	4.83	1095.12	678
16	160	436.8	57.51	7.77	1080.56	625
30	185	405.6	27.69	4.83	1040	668.5
26	185	405.6	57.51	7.77	1040.78	625.75
28	160	436.8	27.69	4.83	1124.24	625
27	160	405.6	27.69	4.83	1150.24	625
31	160	405.6	27.69	4.83	1040	731
25	176.7	442.104	57.51	4.83	1040	625
29	171.2	468	27.69	4.83	1040	653
5	160	405.6	57.51	4.83	1040	696
11	171.2	468	27.69	4.83	1069.12	625
24	160	405.6	57.51	4.83	1113.84	625
19	172	435.24	27.69	6.3	1065.48	649.5
9	160	436.8	27.69	4.83	1040	706
17	160	468	27.69	6.3	1040	677.5
10	176.7	442.104	57.51	4.83	1040	625
14	160	468	27.69	6.3	1094.6	625
21	172	435.24	27.69	6.3	1065.48	649.5
33	172.5	405.6	57.51	7.77	1074.06	625
36	160	468	27.69	7.77	1065.48	649.5
12	160	436.8	57.51	7.77	1040	664
20	172.5	405.6	57.51	7.77	1040	657.75
22	172	435.24	27.69	6.3	1065.48	649.5
23	165.6	468	57.51	7.77	1040	625
34	172.5	405.6	27.69	7.77	1040	692.75
39	160	405.6	57.51	7.77	1073.28	657
35	185	405.6	27.69	7.77	1058.98	643.25
18	165.6	419.016	35.145	7.035	1100.84	634
32	160	436.8	42.6	6.3	1071.2	655
37	160	468	27.69	7.77	1065.48	649.5

After applying regression, the 28-day strength function was obtained according to Eq. (11) and slump was obtained according to (Eq. 12). Their least sum of squares was 238.48 and 8302.54, respectively.

${F}_{\left(Cs\right)}$ =${-45.22X}_{1}+89.15{X}_{2}-3.81{X}_{3}+1972{X}_{4}+38.36{X}_{5}+87.19{X}_{6}$ (11)

${F}_{\left(Slump\right)}$ =${2166.5X}_{1}-2390.5{X}_{2}-3401.2{X}_{3}+24268{X}_{4}-204.83{X}_{5}+169.90{X}_{6}$ (12)

Finally, the optimization response was determined by regression as described in Table (6).

Table 6

Optimal responses of the regression
Answer	Used method	Water (m³)	Cement (m³)	Micro Silica (m³)	Super Plasticizer (m³)	Coarse Aggregate (m³)	Fine Aggregate (m³)	Slump (mm)	CS (MPa)	Cost ($)	Cost in Iran ($)
Best	Weight(kg)	160	405.6	27.7	5.1	1042.6	725	75	54.72	93.01	19.75
Best	Volume(m³)	0.16	0.13	0.013	0.0049	0.401	0.29	75	54.72	93.01	19.75
Best in 95% uncertainties	Weight(kg)	160	414.96	29.8	5.67	1063.4	717.5	76 ± 15	55.2 ± 2.31	96.35 ± 2.89	20.42 ± 0.61
Best in 95% uncertainties	Volume(m³)	0.16	0.133	0.014	0.0054	0.409	0.287	76 ± 15	55.2 ± 2.31	96.35 ± 2.89	20.42 ± 0.61

In this study, the data in Table 4 were given as input to SPSS software. The output functions of strength in Eq. (13) and slump in Eq. (14) were obtained in which the least sum of squares was obtained 283.49 and 8302.015, respectively.

${F}_{\left(Cs\right)}$ =${-84.26X}_{1}+49.81{X}_{2}-41.68{X}_{3}+1932.19{X}_{4}+0.013{X}_{5}+48.595{X}_{6}+38.673$ (13)

${F}_{\left(Slump\right)}$ =${2370.65X}_{1}-2177.27{X}_{2}-3196.7{X}_{3}+24427.98{X}_{4}+0.003{X}_{5}+373.64{X}_{6}-205.29$ (14)

3.2. The MHGA calculations

3.2.1. Definition of strength and slump functions

To solve the problem, the experimental table data was the same as the reference article data. Using Eq’s. (6 and 7) and based on the algorithm process, the output of coefficients and powers for 28-day’s strength Table (7) and slump Table (8) was obtained and the least sum of squares was calculated 280 and 7788, respectively.

Table 7

Coefficients of compressive strength function of meta-HGA
F (Cs)	1	2	3	4	5	6
A	0.52651	2027.914	0.501816	1709.108	0.592146	0.26723
B	2	5.059830479	0.666667	1	0.666667	0.666667
${C}_{1}$	-	-259.07	-1626.78	-3112.22	-61.2157	81.78433
${C}_{2}$	-	-	3215.187	1364.95	-3.21567	112.2957
${C}_{3}$	-	-	-	-2874.66	-848.817	662.9841
${C}_{4}$	-	-	-	-	1997.042	-858.785
${C}_{5}$	-	-	-	-	-	63.78433
${K}_{Cs}$						38.89308

Table 8

Coefficients of slump function of meta-HGA
F(Slump)	1	2	3	4	5	6
A	3078.934	-3862.62	-2704.76	2392.661	-1088.35	-980.938
B	1.458891	1.33909	1	0.333333	0.25	2
${c}_{1}$	-	-3239.57	284.2120164	2399.21522	1267.552	3424.783
${c}_{2}$	-	-	-2746.92	1678.774	2912.569	1730.395
${c}_{3}$	-	-	-	-1725.11	2425.916	-1654.49
${c}_{4}$	-	-	-	-	99.56396	-1401.46
${c}_{5}$	-	-	-	-	-	1789.861
${K}_{Sl}$						90.8452

3.2.2. Optimization of strength- price function

In the next step, in order to optimize the strength-price Eq. (9), the limitations of Table (9) are considered and the unit price of materials is also applied in the optimization process.

Table 9

Ranges considered for optimization of mix design in meta-HGA
Component for Algorithm	Water (m³)	Cement (m³)	Micro silica (m³)	Super Plasticizer (m³)	Coarse Aggregate (m³)	Fine Aggregate (m³)	Slump (mm)	$\frac{Water}{Bindig}$
ID	X₁	X₂	X₃	X₄	X₅	X₆
Minimum	0.15	0.12	0.012	0.0045	0.37	0.20	80	0.29
Maximum	0.19	0.16	0.028	0.0074	0.45	0.3	-	-

The best response in terms of the balance between strength and price, as well as the most economical mix design for specific strengths based on the price of local materials, was given in Tables 10 and 11.

Table 10

Volumetric ratios of the optimal mix design of the meta-HGA
Design ID Algorithm	Strength (MPa)	Water (m³)	Cement (m³)	Micro-silica (m³)	Super- plasticizer (m³)	Coarse Aggregate (m³)	Fine Aggregate (m³)	^Slump ^(mm)	Water/Binder	^Price ^($)
1	48	0.1879	0.12	0.012	0.00461	0.4364	0.2191	115	0.470	18.23
2	49	0.189	0.12	0.012	0.00461	0.411	0.2382	127	0.473	18.26
3	50	0.1892	0.12	0.012	0.00461	0.3984	0.2558	140	0.473	18.3
4	51	0.188	0.12	0.012	0.00461	0.3829	0.2729	145	0.470	18.34
5	52	0.181	0.12	0.012	0.00463	0.3813	0.2819	131	0.453	18.4
6	53	0.1725	0.12	0.012	0.00464	0.3823	0.2894	113	0.430	18.46
7	54	0.1639	0.12	0.012	0.00471	0.3875	0.2926	100	0.410	18.5
8	55	0.1612	0.12	0.012	0.0052	0.3898	0.2924	100	0.403	18.55
9	56	0.156	0.12	0.012	0.00552	0.3972	0.2901	101	0.390	18.56
10	57	0.1534	0.12	0.012	0.006	0.3993	0.2899	100	0.384	18.62
11	58	0.1507	0.12	0.012	0.00646	0.4037	0.2876	103	0.377	18.65
12	59	0.1504	0.12	0.012	0.00688	0.4015	0.2918	113	0.376	18.7
13	60	0.15	0.12	0.012	0.00735	0.4008	0.2927	123	0.375	18.72
14	60.3493	0.15	0.12	0.012	0.0074	0.4008	0.293	134	0.375	18.73
15	61	0.15	0.143	0.0127	0.0074	0.3939	0.2929	86	0.317	20.8
16	62	0.15	0.15	0.0219	0.00739	0.3779	0.2928	40.5	0.291	24.78

Table 11

Weight ratios of the optimal mix design of the MHGA (kg/m³)
Design ID Algorithm	Water (m³)	Cement (m³)	Micro Silica (m³)	Super Plasticizer (m³)	Coarse Aggregate (m³)	Fine Aggregate (m³)
1	187.9	374.4	25.56	4.84	1134.64	547.75
2	189	374.4	25.56	4.84	1082.12	595.5
3	189.2	374.4	25.56	4.84	1035.84	639.5
4	188	374.4	25.56	4.84	995.54	682.25
5	181	374.4	25.56	4.86	991.38	704.75
6	172	374.4	25.56	4.87	993.98	723.5
7	163.9	374.4	25.56	4.945	1007.5	731.5
8	161.2	374.4	25.56	5.46	1013.48	731
9	156	374.4	25.26	5.796	1032.7	725.25
10	153.4	374.4	25.26	6.3	1038.18	724.75
11	150.7	374.4	25.26	6.78	1049.62	719
12	150.4	374.4	25.26	7.224	1043.9	729.5
13	150	374.4	25.26	7.717	1042.08	731.75
14	150	374.4	25.56	7.77	1042.08	732.5
15	150	446.16	27.05	7.77	1024.14	732.25
16	150	468	46.647	7.759	982.54	732

Mix design No. 14 is the optimal answer of the algorithm based on the tested materials.

3.2.3. Validation of results of algorithm

So far, all of the above has been theorized, and it was necessary to examine the implementation of the designs in practice. For this purpose, 4 mix designs from Table (10) were made to compare the 28-day experimental strength of the specimens with the strength resulting from the output of the algorithm. The algorithm's reference laboratory data was made in another country with local materials, and certainly differs from the conditions in this study. Therefore, in the first step, several designs of reference mix designs were made. On the other hand, because of the difference in the quality of materials in the two regions, it was not possible to change the volume of materials because the details of the Simon mix design were not observed. Therefore, it was decided that other than superplasticizer whose effect on the amount of slump was high and did not affect the overall volume of concrete, the rest of the materials were used with the same amount to consider a percentage of experiments as a deviation or error of contradiction of the material of reference research with the present study. After conducting several mix designs to find the appropriate amount of superplasticizer and to achieve the slump of the related mix design, the two mix designs given in Table (12) were finally made out to find the difference between the materials of the two countries. This difference in materials resulted in 2.45% difference in a compressive strength compared to reference experiments.

Table 12

The field experiment of two designs selected from data of references
Design ID	Run Order	Water (kg)	Cement (kg)	Micro Silica (kg)	Super Plasticizer (kg)	Corse Aggregate (kg)	Fine Aggregate (kg)	Design Slump (mm)	Slump in Lab (mm)	28-day Design Str (MPa)	28-day Str In lab	Strength difference Percentage
163	20	165.6	419.016	35.145	7.035	1100.84	634	95	90	60.8	58.54	-3.7
71	5	160	468	27.69	7.77	1065.48	649.5	57	61	55.2	55.9	+ 1.2

In the next step, 4 output designs of the algorithm and the regression optimal design were made and the amount of slump and the amount of 28-day strength were compared with the output of the algorithm and regression. The results have been given in Table (13).

Table 13

Experimental validation of mix designs of outputs of algorithm and regression
N	Design ID	Water (kg)	Cement (kg)	Micro Silica (kg)	Super Plasticizer (kg)	Corse Aggregate (kg)	Fine Aggregate (kg)	Design Slump (mm)	Slump in Lab	28-day Design Str (MPa)	28-day Str In lab	error percentage
1	1 Algorithm	187.9	374.4	25.56	4.84	1134.64	547.75	115	125	48	46.54	-3.0
2	5 Algorithm	181	374.4	25.56	4.86	991.38	704.75	131	127	52	52.24	+ 0.5
3	10 Algorithm	153.4	374.4	25.26	6.3	1038.18	724.75	100	105	57	58.65	+ 2.8
4	14 Algorithm	150	374.4	25.56	7.77	1042.08	732.5	134	142	60.35	62.12	+ 2.8
5	Best regression	160	405.6	27.7	5.1	1042.6	725	75	71	54.7	55.98	+ 2.3

4.1. Comparing the capability of the algorithm and regression in determining the function of the mix design

In both methods, the common factor for comparing the sum of squares is the difference between the answers of their functions and the reference experimental data. Because regression uses approximation for the degree of functions and some of its factors and does not detect them at a statistically significant, it will show less accuracy. Therefore, when the function of the mix design is determined by the meta-heuristic genetic algorithm, it will have more accuracy and capability than the regression method. Comparison of the capability results of the algorithm and regression functions has been presented in Table (14).

Table 14

Comparison of results of capability of algorithm and regression in determining functions
Function	Min of squire Summation in Regression	min of squire Summation in Algorithm	Performance of Algorithm
Strength	283	280	+ 1%
Slump	8302	7788	+ 6%

4.2. Comparison of algorithm and regression capability in optimization of mix design function

The better capability of the algorithm is due to the breadth of the functions and their coefficients in the function of the algorithm mix design. The algorithm examines the space of answers and tries to find the best answer, and since this space is wide, other mathematical methods, such as numerical optimization, are not able to find all possible answers. Table (15) compares the optimization responses of the function of the mix design using algorithm and regression. The algorithm with 5.5% lower price shows 10.2% more 28-day strength.

Table 15

Comparison of optimization responses of the function of the mix design using algorithm and regression
Answer	Best for Regression	Best for Algorithm	Performance of Algorithm
Strength (MPa)	54.7	60.3	10.2%+
Cost in Iran ($)	19.75	18.73	5.5%-

The comparison can be done in a different way by comparing the regression response with the strength response or its equivalent price in the algorithm. According to Table (16), a mix design by 10.9% more strength can be obtained with the price of the regression output response, and a response with a 6.5% lower price in the algorithm can be achieved based on its output strength.

Table 16

Comparison of capability of algorithm with regression
Answer	Best for regression	Same Str for Algorithm	Same price for Algorithm	Performance of Algorithm
Strength (MPa)	54.7		60.69	10.9%+
Strength (MPa)		54.7
Cost in Iran ($)			19.75
Cost in Iran ($)	19.75	18.53		6.5%-

4.3. Capability of algorithm in finding the targeted mix design

As the algorithm optimizes the function by numbering, it not only finds the best possible mix design for basic data, but also has the ability to find the optimal mix design for a particular strength, while only an answer is given as the optimal design in numerical optimization methods. In other words, in the algorithm, the target strength can be a part of the problem data and balance the mix design and price.

4.4. Capability of algorithm in better decision making over optimization

In Figure (3), the algorithm outputs have been plotted in the strength-price optimization chart. Based on this figure, three different slopes were obtained in the diagram according to the applied price of the materials. The first one was considered up to 54 MPa, the second one was considered up to the optimal point of the graph, i.e. 60.3 MPa and the third one was considered up to the last output possible by applying the initial limitation for the material, i.e. 62 MPa. From 54 to 3.60 MPa, the slope of the diagram is very gentle and the price changes are low compared to the strength changes. The difference in algorithm's capability over regression and numerical optimization is also evident in this part of the graph, because regression optimization does not have the ability to fully explore the search space and therefore recognized the strength of 54.7 MPa as the optimal strength. However, the meta HGA could well search the range of better responses and showed the ultimate limit for the balance between strength and price.

4.5. Experimental results

Making specimens in the laboratory shows that the average of more than 97% of the mix design selected by the algorithm in the laboratory has been achieved. For the regression design, the results were almost the same, although there is an error of about 2.45% due to the change in the workshop conditions of data of the reference article with the workshop conditions of the present study.

1) As the function of the meta-HGA is expanded and does not imply any factor, it has a greater ability to provide mix design functions with reference experimental results than the regression method. This result is evident in the lower value of sum of squares of the algorithm error in the functions of slump and strength.

2) Because of the greater breadth and accuracy of the mix design function resulted from the first stage of the meta-HGA, the probability of determining the optimal ratios of the mix design increased in the second stage automatically. This led to the algorithm output with 5.5% lower price and 10.2% more strength than the optimal numerical response of the regression function.

3) Optimal mix design of regression response can be made with 6.5% lower price and a design with 10.9% higher strength can be made in the meta HGA with the price as same as the regression response price.

4) The meta-HGA has the ability to provide the optimal mix design with the lowest possible price for different target strengths.

5) Displaying several examples of algorithm outputs in Fig. 3 showed the turning points of balance of strength and price. An engineer can know the amount of strength and the cost of doing work with these materials and the amount of the increase in price for reaching higher strength.

6) The discrepancy between the implementation of the reference data mix design and the environmental conditions of the present study was 2.45%. According to the experiments, the difference between the algorithm mix designs in practice was ± 3% with an average of 2.3% and the difference of the only regression output with its optimization response was 2.3%. Considering the mentioned error percentage relating the difference between the materials of both studies, the difference between the optimization methods in the laboratory is logical and can confirm the accuracy of the performance of both methods.

Funding: No fund was used for the research.

Data and material availability: Not applicable.

Code Availability: Not applicable.

Author Contributions: I and my supervisor are the authors of the article.

Conflict of interest: We are not in any conflict of interest.

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Editorial decision: Major Revision
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Multi-objective optimizing high strength concrete using meta-heuristic genetic algorithm

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Abstract

Figures

1. Introduction

2. Materials And Methods

2.1. Materials and consumables used in research

2.2. Introducing optimization methods

2.2.1. Optimization with regression method

2.2.2. Optimization with MHGA

2.3. Introducing the MHGA of the research

2.3.1. Formation of primary population

2.3.2. Random crossover operator

2.3.3. Mutation operator

2.3.4. Modifying operator

2.3.5. The fitness of the individual in society

2.3.6. Penalty applying

2.3.7. Selection of a new generation

2.3.8. Elitism

2.3.9. The general cycle of the MHGA

2.4. Forming the mix design function by the algorithm

2.4.2. Solving the problem functions using algorithm

2.5. Optimizing the mix design function

2.5.1. Formation of primary population

2.5.2. Calculation of strength-price optimization algorithm

2.5.3. Algorithm function

2.5.4. Algorithm validation

3. Analyzing The Results Of Methods

3.1. Calculations of regression method

3.2. The MHGA calculations

3.2.1. Definition of strength and slump functions

3.2.2. Optimization of strength- price function

3.2.3. Validation of results of algorithm

4. Results And Their Analysis

4.2. Comparison of algorithm and regression capability in optimization of mix design function

4.3. Capability of algorithm in finding the targeted mix design

4.4. Capability of algorithm in better decision making over optimization

4.5. Experimental results

5. Conclusions

Declarations

References

Status:

Version 1