The advantages of hybrid intelligent systems are the common objective of the exploited model, the possibility of extension, and adaptation to loads through scaling. A smart system and algorithm for hybrid implementation of criterion evaluation and probabilistic dynamics of slobostructured geopolymer basic characteristics in real time are proposed based on the coupling of Markov chains and multicriteria optimization methods. The advantages of hybrid intelligent systems lie in the overall objective of the proposed model, scalability, and adaptability to workloads through scaling. The computational basis of calculations was the digitalization of technology study and analysis of physical-mechanical properties of geopolymers. The optimal composition of geopolymer structure elements for the given technology of their production was determined. The results of modeling the parameters of target functions have shown the advantages of the digitalization of technologies in the analysis of the physical and mechanical properties of geopolymers. The use of the analytical apparatus of Markov chains allowed us to rank the coefficients of multicriteria optimization and to increase the accuracy of calculations. Hybrid models Multi-criteria optimization, Markov chains, Target function parameters, Coefficient ranking, 0000-0001-5027-2213 (D.Stepanchikov); 0000-0002-6869-1392 (K.Buczkowska); 0000-0002-6350-7189 (S.Le)
Geopolymers are a new class of building material designed to replace Portland cement. High carbon dioxide emissions accompany the production process of Portland cement. Modern cement plants worldwide emit about 1.5 billion tons of CO₂ annually. Geopolymers are becoming an environmentally friendly alternative to Portland cement. The properties and applications of geopolymers depend on their chemical structure and the ratio of components.
The search for the optimum composition of geopolymers with high strength and performance properties is becoming an urgent problem in construction, architecture, and noise protection structures. The practical application of the results of studies on estimating geopolymer mixtures with given physical and mechanical properties is limited to the empirical selection of the composition of geopolymer components and the establishment of their variation ranges [1-4]. Reaching an extremum of one of the properties is accompanied by a decrease in the other properties by a particular value.
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2023 Copyright for this paper by its authors. However, even in this case, the point of optimum formulation is determined with a large margin of error.
Information processing based on mathematical modeling of the composition of geopolymer compositions is related to the selection of solutions in weakly structured problems with quantitative and qualitative parameters used as input variables and unstructured with qualitative description only. Considering their significance, collective decision-making based on quantitative and qualitative criteria is the main direction of research on creating an intelligent system of geopolymer characteristics to determine the ranking of criteria using Markov chains and subsequent multicriteria optimization. Such a hybrid model begins as a system using one autonomous method and ends up as a system using another way. The coupling of experimental results on geopolymer mixture formulation with their processing by methods of multicriteria optimization and Markov chains form the basis of the present work.
The authors' main contributions are the following: • the methodology of complex use of Markov chains for determination of weight coefficients of multicriteria optimization was proposed;
• modeling of target function parameters was performed, the results of which revealed the advantages of digitalization of technologies in the analysis of geopolymers properties; • based on the results of modeling, a scheme of hybrid implementation of criterion evaluation and probabilistic dynamics of weakly structured data was proposed.
The aim of the work is not only to investigate the physical and technological parameters of different geopolymer compositions but also to simulate these properties using multicriteria Optimisation and Markov chains.
The scientific field of intelligent material characterization systems using a hybrid implementation of Markov chains and criteria methods includes a synergistic combination of semantically integrated technologies defining the architecture and information exchange process and developing communication between the components.
In [5-7], discrete Markov chains have been used for strategy development in various industries, in [8] medical research, [9] coronavirus control, [10] navigation systems, [11] international relations, [12] student performance evaluation, in [13] in structural transformations.
Criterion methods are ways of describing alternative solutions in quantitative terms. Each decision leads to a particular outcome and therefore expresses the effectiveness of the process and its overall value. In [14] an evaluation of the composition of the main power chains of an asphalt concrete mixture based on discrete element methods is presented. In [15], a model for quantitative risk assessment of a structural hierarchical system used in the petrochemical industry is described. The use of fuzzy sets theory and genetic algorithms for the composition of communication services is presented in [16]. An optimality criterion-based algorithm for efficient optimization of laminated composite design using simultaneous resizing and scaling is presented in [17]. An assessment of the weighted effectiveness of lattice criterion methods is presented in [18]. A hybrid firefly-inspired approach for optimal web composition is described in [19]. Modeling the use of Markov chains covers a wide class of probabilistic dynamics, allowing one to trace trends and causes of changes in the properties of one's subject domain. However, it should be noted that the presented arsenal of Markov chain implementation tools lacks methodological unity and represents only some aspects that are treated separately and used for different purposes. Markov chains represent an integrated element of forecasting mechanics based on an integrated approach with criteria methods.
The influence of material structure on the mechanical properties of geopolymers obtained using additive technologies is presented in [20]. Comparative studies using atomic force microscopy, X-ray spectroscopy, and Fourier transform infrared spectroscopy in [21], the influence of structure on the strength properties of geopolymer characteristics in [22]. Optimization of matrix compositions affecting the mechanical properties of geopolymer composites with short carbon fibers in [23], development of hybrid geopolymers for structural materials in [24]. The problem of optimizing the composition of geopolymer mixtures with the help of an intelligent system of multicriteria optimization and Markov chains is relevant and timely.
The main multicriteria optimization techniques used in the cited sources are criterion convolution, main criterion optimization, and sequential concession method. The practical applications of criterion methods are wide and varied. The main disadvantage of global optimization for all geopolymer parameters is the high computational complexity of target functions. For most practical optimization problems, analytical expressions of limiting functions are unknown. Their general patterns, trends, and possibilities are considered and used in developing the hybrid implementation method.
The unsolved part of the general problem of building intelligent systems of multicriteria multiobjective optimization is their equivalence when replacing the original criteria with general aggregated criteria, i.e., the exclusion of ranking of private criteria by their importance. This significantly reduces the accuracy of multicriteria optimization evaluation.
Geopolymers' physical-mechanical and technological properties were used as research materials: their quantitative values and priority probabilities of the main characteristics that ultimately determine the weight composition of the mixture components. The technology of geopolymers production, their formulation properties, chemical composition, and structure are presented, which are the basis of input information for the study of investigated properties.
The main constituents of geopolymer are calcined clay, aluminosilicate materials, bentonite, and kaolin. The local concrete industry in the Czech Republic uses aggregates in the form of fine and coarse sand. The aggregates smaller than 4.75 mm are considered fine sand, and coarse sand larger than 4.75 mm. The totals in the present work were obtained in crushed form. Most of the particles were graveltype with particle sizes ranging from 4.0 to 8.0 mm and fine sand with particle diameters of 0.063 mm to 2.0 mm [3]. Binder is provided České lupkové závody, a.s by commercial name Lk. In preparing the binder mixture based on the inorganic polymer, five parts by weight of cement and four pieces of activator are usually used.
The mixtures were prepared by two following steps: (
(ii) All the mixtures were cured in an oven without delay at the specific curing temperature for 24 hr and 48 hr ranging from 60 oC ÷ 90 oC. Samples were molded in the oven after the curing process and continued at ambient conditions for 2 days.
The samples prepared with different mixing ratio aggregate content are presented in Table 1.
The algorithm of implementation of multicriteria evaluation and probabilistic dynamics of weakly structured data developed according to the described methodology is presented in Fig.1. The algorithm has a cross-sectional structure and consists of a transient probability matrix algorithm using Markov chains and a criterion approach for determining the optimal geopolymer composition. Two aspects determine the dynamics of the process: the initial probability distribution and the transition probability matrix. Markov chains allow prioritizing each physical and mechanical parameter analyzed. Based on the calculations performed in the first part of the algorithm, we obtain the values of weight coefficients for each physical-mechanical parameter of the geopolymer. The use of Markov chains reveals the essence and interrelation of the main physical and mechanical parameters of geopolymers with the probability of their manifestation under various manifestations of the external environment. The novelty of Markov chains application is the replacement of equal-step time intervals by a discrete sequence of states. Markov chains are a powerful tool that provides fundamental reasoning of a decision on the basis of processing a large number of experimental data, some of which constitute a database of data obtained with one or another probability. Criterion evaluations are carried out based on Laplace, Wald, and Hurwitz criteria and additive, multiplicative and complementary multiplicative convolution. The general cross-validation algorithm takes into account the necessary operations and building blocks for constructing a hybrid model for determining the optimal geopolymer composition. Hybrid implementation of Markov chains combined with quantitative multicriteria optimization evaluations were used as research methods. The conceptual model of multicriteria decision-making with fuzzy input data provides equality of weight coefficients of used criteria. The range of mechanical, thermal, and technological properties variation is determined by setting the extremum of the boundaries of the target values of the requirements. In this case, the existing methods have not considered the additional values found through empirical observation of the characteristics, which significantly reduces the effectiveness of optimization methods for multiple criteria since decision-making occurs under conditions of uncertainty and risk. Therefore, it is necessary to modernize the multicriteria optimization methods, which consist of the transition from vector to scalar optimization. This operation manifests convolution functions of qualitative criteria into a single generalizing one.
The need to rank the weight coefficients of criteria requires the creation of an intelligent system in which the weight coefficients of criteria will be determined through the apparatus of Markov chains. The technology of intelligent systems is considered as an information-computing system with intelligent support. From these positions, the intelligent system of geopolymer characteristics is presented as a technical system capable of solving creative problems belonging to this subject area, which are stored in the knowledge base in the form of a functional semantic network. When building an intelligent system of geopolymers characteristics, the ability to solve problems by a declarative description of the condition, control the processes of computation in the dialogue mode, and synthesize computational algorithms capable of solving poorly formalized problems were taken into account. The main stages of building an intelligent system of geopolymer characteristics were: work with quantitative information, mathematical calculations, storage and exchange of information, and interpretation of results. The sequence of models underlying the intelligent system of geopolymers characteristics included Markov chains for determining the ranking of criteria, multicriteria optimization models, and cross-algorithms of their hybrid implementation. When ranking criteria using Markov chains each state of the parameters characterising the information situation of determining the mechanical properties of geopolymers for a given recipe for their preparation is assigned a certain probability, which is written as a line of the state matrix. The matrix of intensities or transitions of the system describes the wandering of the system over its states. The matrix is compiled so that the sum of the probabilities in the rows of the matrix is always equal to 1. In analysing the state matrix, all possible states of the parameters are enumerated with their probabilities, i.e. we are dealing with a stochastic transition matrix, the set of vectors inside which reflects the values of probabilities between gradations. These iterations are made for various combinations of parameters. Since the processes of obtaining the mechanical properties of geopolymers by changing their formulations do not have a constant time reference, we will use the stages that characterise the successive approximation of the states' approach to achieving the intended goal as time. In this case, we will replace the time with the step number.
to be analysed
state probabilities
current status
Probabilistic dynamics
Hurwitz criterion f o ino rse t t a e ilza am rom rap N la to m gn itp i o rd a f o i
r no cca ir e t litceo itsno c Se po
Pairing with criterion assessments n o i t c u d o r p g n i t u p m o
C
Making decisions under uncertain and risky conditions begins with constructing a payoff matrix. The payoff matrix is a simplified formal model of an actual conflict situation. Mathematically, formalisation
Criteria evaluations means that certain rules for the interaction of parties, choices of actions, and specific outcomes for the selected actions have been developed, and the necessary information is available. The term "payoff matrix" is synonymous with the term "performance matrix" [25].
q1 q R = 2 qm П1 y11 y21 ym1 П 2 y12 y22
ym2 П n y1n y2n
ymn where q1,…,qi,…,qm – weight composition of sample components, Π1,…,Πi,…,Πn – analysed parameters (physical and mechanical, thermophysical, chemical, operational and technological, economic), δyij – relative deviation of the j-th parameter from the target.
To determine the element of the matrix, one of the methods used is the estimation of a scalar vector – the method of selection by arranging objects according to the model. In this case, converting from estimating a vector to a scalar of the objects is necessary. The functions used in solving the multicriteria problem as convolution functions of the vector arguments yi = (yi1,…,yij,…, yin) into scalars δyij = f(yi). Vector argument convolution serves to reduce the number of criteria. Its purpose is to replace the original criteria with common criteria. The convolution operation is also called aggregation of partial criteria. The method is applied if private criteria can be ranked in descending order of their importance so that the importance of each pair of neighboring criteria does not differ significantly. Their normalization is applied to compensate for the small values of some criteria with large values in other criteria. The most straightforward scalar function that provides a linear order of objects is the penalty function, which is formed to the extreme importance of features.
n
f ш ( yi ) = yij (
j=1
where: Under the deviation Δyij from the ideal target by the j-th feature, we understand the absolute
value of the difference Δyij = |yj – cj,extr|, where the ideal target when maximizing the j-th feature is
denoted as cj,extr = yj,max, and when minimizing the j-th feature, as cj,extr = yj,min. The condition to correctly
use function (
As for the sample, we want to refer to a class of objects characterized by a general target h = (c1,…, cj,…, cn). Let us introduce a measure of the overall deviation from the target, which allows finding the object closest to the sample and ranking the objects based on their distance to the target. Consider a sample with attributes formed by equality constraints (yj = cj). The deviation of the j-th feature in any direction from the specified point cj(cj ± Δyj) indicates the extent to which an object deviates from the target concerning this attribute. Then, the relative deviation of the j-th feature from the target is determined as, yij − c j y j,max − c j yij = yij − c j c j − y j,max ; yij c j ; ; yij c j .
(
As parameters, the best values of the analyzed parameters need to be chosen according to the perspective of the problem being addressed - these can be maximum, minimum, or average values from the experimental sample. With this approach, the formula will shift the integer quantities about the scale (0.1). However, with such parameter selection, the corresponding elements of the matrix coincide with the necessarily observed values, leading to. Using the convolution sum results in the corresponding feature being lost from the overall evaluation of the object, and using the convolution product will lead to a reduction to 0. A clear way to avoid such situations is to extend each feature's upper (maximum) or lower (minimum) limits by the same percentage. Below, each analyzed parameter's maximum (minimum) values have been increased (decreased) by 1%.
Scalar optimization requires additional knowledge about the properties of the generalized objective functions, characteristics scale, and weighting coefficients. Since this knowledge is domain-specific, the order of objects in an n-dimensional space cannot be explicitly determined. Therefore, it is crucial to study the influence of the properties of the generalized objective functions, characteristic scales, and weighting coefficients on the optimization results.
The following generalizing multicriteria utility functions have been used in theoretical analysis. Additive Convolution
n yi = jyij j=1 n where j – importance (weighting factor) j-th sign, j = 1 .
j=1
Power multiplicative convolution
(
Savage (Wald) criterion (minimum-maximum) Laplace criterion (minimum-minimum) Hurwitz criterion
Z L = min min yij
i j Z hw = min min yij + (1 − )maxyij i j j
n yi = (yij ) j j=1 n yi = 1 − (1 − jyij )
j=1 Z v = min maxyij
i j where 0 ≤ ρ ≤ 1 – the indicator of pessimism, in the calculations, was taken equal to 0.5.
The generalized additive function synthesizes the mass index of an object. It reflects the total value of individual characteristics, taking into account their importance. The direct product function prioritizes objects with consistent estimates for all indices, reflecting individual indices' homogeneity. The complementary product function has the opposite property. The advantage of a complementary product function over other product functions is its ability to accept zero feature values. With direct product functions, special care is needed to avoid scalar estimates of vectors with a component value of zero.
The Savage criterion is a safety criterion, used to achieve maximally guaranteed results under the worst conditions. It accepts the maximum negative development of the situation, where a selected strategy avoids both excessive wins and losses. The worst-case option (the strategy of destiny) is taken into account. It can use when strategy selection errors could lead to catastrophic consequences when decisions are made only once and cannot be changed in the future.
The Laplace criterion determines the strategy that maximizes gains in an uncertain state of the environment. The one with the lowest score according to the Laplace criterion is the best option. This criterion represents extreme optimism, not taking into account any negative outcomes except for the best one. This risk level from the negative impact of changes in the external environment is not considered. It should be noted that situations requiring the application of such a criterion are not limited only to optimists, who cannot be corrected but also to those who are trapped and must adhere to the principle of "do or break". The main drawback of this criterion relates to the fact that when finding the average payoff level, the offsetting effect of small payoffs can occur.
The Hurwitz criterion guides the selection of recommended solutions based on a range of characteristic average outcomes between extreme pessimism and unrestrained optimism. The Hurwitz criterion is related to introducing a weight parameter 0 ≤ ρ ≤ 1, called the pessimism index. The assumption about the environment's behavior is that for any alternative choice, the worst choice is made with probability ρ, and the best choice is made with probability (1 – ρ). When ρ = 0 the Hurwitz criterion coincides with the Laplace criterion at maximum, and when ρ = 1 – with the Wald criterion at maximum. By using the Hurwitz criterion, we apply more significant information under the conditions of Wald, Savage, and Laplace. The main drawback of this criterion is that it only considers two outcomes – the worst and the best.
Additionally, there is difficulty in specifying a pessimism index, ρ. The assumption is that every rational decision-maker must select appropriate maximum or minimum strategies. Caution is required.
The hybrid intelligent system is a set of statistical simulation models in which the qualitative side of loosely structured problems and performance evaluation of processes for determining the mechanical properties of geopolymers are solved using Markov chains, while criterion methods are used for quantitative inference, and the final result best adapts the results of both applications. The quantitative side of loosely structured tasks and process performance evaluation is solved using Markov chains, while criterion methods are used for quantitative conclusions, and the final result best adapts the results of both applications.
Compressive strength testing of mortar was performed as per AS 1012.9 using (Ø46 x 92) mm diameter cylindrical molds. ASTM C39 was conducted for compressive strength tests of hardened concrete, using (Ø100 × 200) mm cylinder molds. Three sample cylinders were tested, with the experimental values averaged.
The density of geopolymer composite materials was measured according to standard CSN EN 1936 and was estimated by dividing the mass of the sample by its volume. The testing samples with (Ø100 x 200) mm dimensions were used to measure the density after 28 days.
The compressive strength is measured on a VEB Werktoff Prufmaschinen Leipzig, 500 kN, ambient condition temperature 23 ± 2 oC, and relative humidity 65 %. The samples are cured and tested by the standard ASTM C 31/C 31M. Values are the averages of four separate tests. Data that deviated by more than 10 % were eliminated. The loading was displacement-controlled at a constant rate of 2.4 mm/min for all the tests. At least two cylinders are tested at the same age and the average strength is reported as the test result to the nearest 0.1 MPa.
The modulus of elasticity by the equation: = 2707√ + 5300, where fcm is compressive strength, MPa.
In order to obtain different mechanical properties of geopolymers, necessary for multicriteria optimization of their composition, the conditions of their curing were changed from the beginning of the process of formation of the geopolymer mixture structure after 7, 14, 28, 90 days. The obtained results of mechanical properties of geopolymers are presented in Table 2.
The presented methods and results of determining the mechanical properties of geopolymers through
their relationship with the mechanism of structure formation were used as a basis for the construction
(
P(0) = 0.4000,0.3000,0.2000,0.1000
The transition probability matrix has the following form:
We repeat similar calculations until constant stationary values of the state vector are reached:
P(
Starting from the fifth step, the values of the state vector stop changing. Thus, we have the following set of weight coefficients for the mechanical parameters of geopolymers: • compressive strength = 0.4434 • density = 0.2327 • Young's modulus = 0.1814 • Splitting tensile strength = 0.1425
The method of criteria convolution consists in transformation of vector criterion into scalar one and manifests itself in assigning coefficients of initial criteria of its subsequent ectremization on the set of admissible variants. In its sense, the convolution is a weighted average of the initial criteria. The condition for using convolution of criteria is their reduction to a single scale, i.e. normalisation. In single-criteria or scalar optimisation, a single objective function is defined over a set of decision options. In multicriteria optimisation there are several such functions at once, forming a vector criterion.
The solution to a scalar optimisation problem is considered to be the element that maximises or minimises the target function. In the case of multicriteria vector optimisation, there is maximisation for one criterion and minimisation for the rest. The set of solutions is represented as a set of selectable vectors. The methodology of calculations of parameters of target functions based on multi-criteria analysis using Laplace, Hurwitz, and Wald criteria requires consistent use and finding experimental values of deviations from the priority location of targets. The results of calculations and values of convolutions and criteria for determining the optimality of mechanical parameters of geopolymers are presented in Table 5
The calculations were carried out in the Maple computer mathematics system. Table 4,5 presents
the results of calculations by formulas (
Table 5 uses the following designations: ya — additive convolution (
The Wald criterion gives the optimal composition 90M3, while it should be noted that the values of the convolutions and the Laplace criterion (minimum) for this composition are also close to the minimum values. Therefore, the 90M3 composition can be put in second place in terms of optimal mechanical properties.
The Hurwitz criterion significantly depends on the pessimism coefficient (see formula (
The proposed intellectual system of multi-criteria evaluation using Markov chains has been developed in the hybrid implementation of structured and unstructured problems of obtaining optimal characteristics of geopolymers. Combining subjective and objective elements of decision selection this approach represents a new way of information processing based on mathematical modelling and probabilistic dynamics.
The simulation of target function parameters has shown the advantage of evaluating the digitalization of technologies in the analysis of geopolymer properties, where the qualitative side of loosely structured problems is solved with the help of Markov chains, while for quantitative conclusions criterion methods are used, and the final result best combines the results of both applications.
The methodology of the complex use of Markov chains in combination with optimization criteria is the basis for the application of the target functions to increase the reliability of the parameter estimation for determining the optimal composition and structure of geopolymers when changing their manufacturing technologies.
Prospects for further research of the authors consist in the creation of hybrid models of quantitative and qualitative methods of experimental data processing in combination with prediction methods, where the method presented in the article can be used at the stage of data preprocessing [15] J. Xu, L. Guo, R. Zhang, H. Hu, F. Wang, Z. Pei, QoS-aware Service Composition Using Fuzzy
Set Theory and Genetic Algorithm. Wireless Personal Communications, 102 2 (2018) 1009-1028. [16] R. Kussmaul, M. Zogg, P. Ermanni, An optimality criteria-based algorithm for efficient design optimisation of laminated composites using concurrentresizing and scaling. Structural and Multidisciplinary Optimisation, 58 2 (2018) 735-750. [17] A.V. Boldyrev, D.M. Kozlov, M.V. Pavelchuk, Evaluation of Anisogrid Composite Lattice Structures Weight Effectiveness using the Load-carrying Factor. Procedia Engineering, 185 (2017) 153-159. [18] C.B. Pop, V.R. Chifu, I. Salomie, R.B. Baico, M. Dinsoreanu, G. Copil, A hybrid firefly-inspired approach for optimal semantic web service composition Scalable Computing, 12 3 (2011) 363369. [19] C.D. Frenette, R. Beauregard, A. Salenikovich, D. Derome, Multi-criteria evaluation of the compositions of walls of light frame wood construction [Évaluation multi-critère descompositions de murs à ossature légère en bois], in:Proceedings, Annual Conference - Canadian Society for Civil Engineering, 2017, pp. 909-918 [20] K. Korniejenko, P. Kejzlar, P. Louda, The Influence of the Material Structure on the Mechanical Properties of Geopolymer Composites Reinforced with Short Fibers Obtained with Additive Technologies. International Journal of Molecular Sciences, 23 4 (2022). [21] K. Korniejenko, M. Łach, S.-Y. Chou, W.-T. Lin, A. Cheng, M. Hebdowska-Krupa, S. Gadek, J. Mikuła, Mechanical properties of short fiber-reinforced geopolymers made by casted and 3D printing methods: A comparative study. Materials, 13 3 (2020). [22] A.T. Akono, S. Koric, W.M. Kriven, Influence of pore structure on the strength behavior of particle- and fiber-reinforced metakaolin-based geopolymer composites. Cement and Concrete Composites, 104 (2019). [23] J. Akmal, M. Badaruddin, M.K. Ismoyo, S.D. Yuwono, optimisation of matrix compositions of Al2O3, SiO2, Caolin, and CaO on the mechanical properties of a geopolymer composite with short carbon fiber, in: IOP Conference Series: Materials Science and Engineering, 602 1 (2019). [24] F. Amalia, N. Akifah, Nurfadilla, Subaer Development of coconut trunk fiber geopolymer hybrid composite for structural engineering materials, in: IOP Conference Series: Materials Science and Engineering, 180, 2017. [25] A. V. Buketov, A. V. Sharko, D. A. Zinchenko, D. M. Stepanchikov, To the problem of ingredients optimisation of composite materials based on epoxy resin. Bulletin of the karaganda universitymathematics, 86 2 (2017) 37–44. doi: 10.31489/2017M2/37-44.