Model for selecting available technologies based on optimality criteria under risk conditions

04030


Introduction
The choice of an efficient and affordable technology is relevant nowadays due to the increasing use of low-waste and non-waste technologies.This is especially important against the background of crisis phenomena in the economic development of transport infrastructure and the problem of import substitution.The problem of technology selection is that it is impossible to specify only one criterion.The problem of choice, therefore, is a problem of multicriteria optimization.Such problems may not have an optimal solution in pure strategies.Therefore, an actual approach to its solution is the use of optimal choice models under risk conditions.[1].
Current approaches use a stratified decision-making method based on a wide range of criteria.This method allows decisions to be made over the long term and, in combination with the labeled stratified method, can be used to calculate the ranking of available technologies.However, the above method does not indicate many optimal strategies.[2] Other approaches consist of using a multi-attribute solution group analysis method based on interval fuzzy sets theory, which can successfully help stakeholders to prioritize a sequence of alternative scenarios [3].This method of analysis also does not provide a solution in the form of a set of optimal strategies.
In this paper, we applied a mathematical model for choosing the best technology, based on the optimality criteria set on finite sets of factors (technical characteristics of objects) and alternatives (materials of different composition, having different physical and chemical properties).At the same time, the utility functions of the optimal choice criteria are considered under risk conditions.These criteria are applied to ash-foam concrete materials, which differ in density and ash content from sewage sludge incineration [4].
When developing building and enclosing structures, including noise screens on railroad transport, as well as in other areas of engineering construction, an important step is to determine the optimal composition of ash-and-slag concrete [5][6][7][8][9].
The criteria allow comparing materials, specifically to arrange them by ranking with an assignment of priority properties, which allows deciding on using one or another technology in the shortest possible time depending on the specified operating conditions.

Materials and methods
The ash-concrete samples were produced by autoclave technology (temperature 174°C, pressure 8-10 MPa) using cement, sand, building lime Ca(OH)2 and water.Different types of ash-foam concrete samples (D500, D600, D800), differed from each other by the composition of the components given above, with the replacement of 50% (by composition) of sand by ash [10].
After the production of ash foam concrete test samples (D500, D600, D800) were carried out studies of these samples concerning the main properties that are considered for the application of the specified material, namely: -content of natural radionuclides, -dust concentration; -thermal conductivity; -compressive strength; -ash content; -frost resistance cycles; -sorption humidity; -soundproofing; -noise reduction when using screens.
Studies to determine the content of natural radionuclides in ash and in samples from ashfoam concrete were carried out in the laboratory of the Research Institute of Radiation Hygiene and the V.G.Khlopin Radium Institute.Dust concentration at the limit of the sanitary protection zone of a landfill for ash from sewage sludge incineration was determined by a calculation using the program "Ecolog".Properties of ash-foam concrete (thermal conductivity, compressive strength, frost resistance, sorption humidity) were determined by standard methods according to Russian State Standard GOST in the research center "Sokrat".The theoretical calculation of the average sound-isolating capacity of ash-foam concrete blocks and noise reduction when using them as noise shields was carried out according to the Russian State Standard GOST.
Table 1 shows the results of experimental measurements and calculated data described above for samples of each type of ash-foam concrete (D500, D600, D800).*-technical tests have not been carried out, given by reference According to the content of natural radionuclides (specific effective activity) we can conclude that the original ash from the incineration of sewage sludge belongs to the second class of materials.It has limited application (Table 2).Ash-foam concrete belongs to the first class (Table 2) -for all kinds of construction.In terms of physical and mechanical properties, the obtained ash-foam concrete differs in such properties as: thermal conductivity, strength, frost resistance, and others.Ash-foam concrete can be used in the construction of building enclosures, as thermal insulation and soundproofing material, in the interior or exterior walls.Depending on the use, the priority of the considered properties will be different [11].When using ash-foam concrete in exterior walls, the more important properties will be strength and frost resistance, cycles.Thermal conductivity is also important.When using ash-foam concrete as interior partition walls or thin-layer insulation coatings, soundproofing capacity and strength is important.Frost resistance does not affect the internal partitioning and internal layers of the material [12].
To apply the best technology selection model based on optimality criteria, the data in Table 1 were reduced to dimensionless values according to [6].The final values are given in Table 3.The set  = { 1 , . . .,   }was considered -a finite set of factors in several groups: environmental, technological and operational.Each factor   from the set  has an influence on the choice of material to be used in technology   , from the set .The set  is a finite set of technologies (alternatives):  = { 1 , . . .,   }, where m = 4: d1ash from sewage sludge incineration; d2autoclaved ash concrete with a density of 500 kg/m 3 (D500); d3autoclaved ash concrete with a density of 600 kg/m 3 (D600); d4autoclaved ash concrete with a density of 800 kg/m 3 (D800).Let's introduce the utility function as a mapping of: which maps each factor   ∈  and each alternative   ∈  to a gain ℎ(  ,   ) = ℎ  , if D and Sare finite or countable sets.
Along with making a decision by a pure strategy, it often makes sense to find an optimal solution in mixed strategies.This strategy is realized as a vector of probability distributions: whose components satisfy the conditions: In this case, the utility of the decision made is defined as the mathematical expectation of a random variable -utility with a discrete distribution  = ( 1 ,  2 , … ,   ),   = (  ),  = 1, … , : Mixed strategy can be realized in different ways: physical mixing, a priori probability distribution, statistical (frequency at multiple choice) [13].
The problem of decision-making under the conditions of risk in this formulation occurs if the a priori distribution is known:  = ( 1 ,  2 , … ,   ),   = (  ) probabilities of influence of factors from the set S.
In this case, if the set of factors S is a finite set, the expected utility of alternative d is defined as the mathematical expectation of the utility corresponding to alternative d: This functional is the basis of the optimality criteria in the conditions of risk.The choice of the optimal strategy under risk conditions is carried out with the help of the following criteria [14,15]: − maximum expected utility, − Hodges-Lehman, − the most probable factor, − minimum of expected regret, − minimum of utility variance.
Criteria for choosing the best strategy in a risky environment are presented in Table 4.
Let us construct an a priori probability distribution using the example of factor partitioning into three groups, as shown in Table 5.Let us introduce a complete group of events {  } =1  , where lis the number of groups of factors.In this case, the obvious equality is fulfilled: In each group of factors let us distinguish subgroups of events: {   } =1   , where  is the number of subgroups in the k group.Using the multiplication theorem for the probabilities of events independent in the aggregate, we find the a priori discrete distribution of the influence of factors (  ) = (  ∩    ) = (  ) • (   ).The final a priori probability distribution is shown in Table 5 and Table 6.Table 7 shows the results of applying the criterion of maximum expected utility to the alternatives of set D. According to the criterion, the optimal strategy is the alternative d * = d4, corresponding to ash-foam concrete density 800 кг/м 3 .
It can be seen that according to the criterion of minimum expected regrets the optimal technology is the one using ash-foam concrete density 800 kg/m 3d * = d4.
The criterion is based on the quadratic variance functional: The optimization problem has the form: The results of applying the criterion to the alternatives of the set D are shown in Table 11.

Results
The work applied the criteria of optimality of decision-making under conditions of risk, found the optimal strategies, and ranked the materials by priority properties.These materials have been studied for application in various technologies used in the construction and transport industry.
The results obtained in the work are based on a discrete probability distribution corresponding to the factors of different groups, according to which the selection of materials was carried out.
Studies have shown that different optimality criteria lead to one result, the choice of ashfoam concrete material with a density of 800 kg/m 3 .

Analysis of results
The paper demonstrates the application of decision-making models in conditions of incomplete information and risk for the case when the set of factors and the set of alternatives are finite.Optimality criteria in conditions of risk and in conditions of uncertainty showed that the optimal strategy is the alternative of using the material produced from ash-foam concrete with a density of 800 kg/m 3 .The indicated optimal solutions are obtained in pure strategies.
The results of applying a large number of criteria under conditions of incomplete information and risk made it possible to create a set of programs for monitoring and analysis of data on different materials and technologies.This makes it possible to process the information coming from the manufacturers and consumers of technologies in order to make a decision on the use in the production of a particular technology or material in short time intervals.
It should be noted that the optimality criteria specified in the second part of the paper allow generalization in cases where the a priori probability distribution of the influence of factors is continuous.This allows us to extend the application of the method to systems with continuously changing values of the factors.As values of random measurement of factors of models are taken values of characteristics divided into intervals.Such classification and selection is a specification of the existing discrete model with point values (factor estimates).In the future, it is planned to develop the mentioned approach in the tasks of optimal choice among the available technologies in various branches of transport industries.

Table 10 .
Applying the criterion of the most likely factor to the alternatives of the set D

Table 1 .
Properties of autoclaved foam ash-foam concrete

Table 2 .
Field of application of building materials

Table 3 .
Experimentally measured *-technical tests have not been carried out, given by reference

Table 4 .
Criteria for choosing the best strategy in a risky environment

Table 5 .
Probability distribution by groups

Table 6 .
A priori probability distribution of the influence of factors

Table 7 .
Values of expected utility on the set of alternatives

Table 8 .
Minimum values for each alternative of the set D

Table 9 .
Factor groups and a priori distribution

Table 11 .
Applying the criterion of minimal utility variance to the alternatives of the set D