Mathematical Modeling As A Method Of Quality Assurance In Construction

Methods of solution of organizational tasks in construction

The organizational and production tasks for administrative-and-management personnel of the organizations and enterprises define set of these or those methods of the mathematical analysis and modeling (table 01 ).

As a system object, construction assumes existence of a large number of elements, processes of which are characterized by dynamism and the probabilistic nature of behaviour. Similar properties also deal with the interrelations between them.

Mathematical modelling has one of the leading positions in the process of analysis, planning, organization and quality assurance in construction that can be characterized by uncertainty and dynamism of indicators and criteria.

Creation of universal model in construction is almost impossible, but improvement of modelling in this field and creation of adequate model for display of real processes and their interaction constantly take place.

Modeling

In general, modeling is a development of a certain model or representation of the studied object (systems, process) in which its properties and specifics, available for the analysis and a research remain. Models have two main classification groups: physical and sign. The first assume maintaining physical properties of the studied object, the second – application of any language: graphic, algorithmic, mathematical, etc.

We consider that mathematical and economic-mathematical models for the solution of the tasks are most often applied according to the properties.

The main conditions, to which the model developed for practical application has to correspond, are lack of mathematical, economic, technological and other types of contradictions and accuracy of reflection characteristic to the studied object of indicators and criteria.

As the main objectives that are achieved by mathematical modelling method in the sphere of construction and operation of objects of a housing-and-municipal complex, are connected with optimization organizational and production processes, special attention is paid to economic-mathematical modelling that include the following stages:

• statement of the purpose;

• development of criteria;

• identification of restrictions;

• development of economic-mathematical model;

• preparation of an algorithm of calculation;

• software;

• collecting necessary basic data;

• classification and coding of information;

• creation of model;

• the analysis of the received results;

• implementation of results of modelling into practical activities.

Issues of quality of final products are relevant in all fields of activity nowadays. Special attention to them is paid in the branches connected with life sustainment and maintenance of ecosystems’ balance.

Mathematical-cybernetic modelling

Application of various methods of data processing allows not only to keep track of actual state of these or those characteristics of the studied object, but also to model them.

Special attention in construction should be paid to mathematical-cybernetic modeling as to one of the trends of modeling control systems in construction. Administrative activity provides realization of all production and non-productive processes in construction, and, therefore, special attention has to be paid to it.

Mathematical-cybernetic modeling represents set of methods, including the mathematical multilevel systems of problem-solving, imitating processes of management systems, the description of information flows, structuring administrative and managerial nets (Alaybeyoğlu & Kuntman, 2016).

Realization of mathematical-cybernetic modeling for the purpose of quality assurance of works construction works assumes certain stages of modeling and their content:

• definition of goals of modeling, factorial and productive indicators and their interrelations, a possibility of quantitative assessment of indicators and characteristics;

• mathematical formalization, empirical check of starting positions (whenever possible), quantitative expression of qualitative content of system elements;

• creation of the formalized scheme with application of quantitative data and their consecutive interrelations;

• development of mathematical model on the basis of the formalized scheme with selection of the corresponding mathematical algorithms and functions;

• statistical analysis of parameters of a studied object;

• assessment of adequacy of model by comparison of results of modeling with reality;

• additional specifications and simulation modification (if necessary).

It is necessary to carry out constant assessment of the created model from a position of its complexity and adaptability. In a similar case it is necessary to correct an objective. Traditionally the necessity to return to the previous stages arises at a stage of preparation of initial information because of high expenses or its absence. In this case it is necessary to reconsider an algorithm of tasks performance, to change number of the considered factors, etc. These matters arise especially often as it is necessary to consider a large number of both internal and external factors in construction (Mireille, 2003).

The process of formalization during creation of mathematical model should be paid special attention due to the creation of a certain artificial language with replacement of concepts with symbols is very difficult, especially in complicated systems that also include construction.

Cybernetic modeling is a rather capacious science representing set of other sciences united by the unified purpose – a research of management in various systems and operating with such basic concepts as "model", "system", and “information". It is necessary to distinguish the theory of modeling, the theory of information, the system analysis, and the general theory of management, the general theory of systems and others among cybernetic sciences.

Mathematical-cybernetic modeling in relation to construction branch assumes formal creation of systems with application of the concepts "entrance" and "exit" and the characteristic of the bond of an element with the environment. Each element (d₁, d₂, …, d_n – its states) can have several "entrances" (а₁, а₂, …, а_n) and "exits" (b₁, b₂, …, b_n) (figure 02 ).

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While solving question of application of cybernetic modeling it is also necessary to consider that the cybernetics assumes the system as a unity of controlling and operated elements, and the operating system on the channel of direct link makes impact on the operated object. This influence, in turn, deforms impact of the external environment on the operated object. Cybernetic modeling also assumes that feedback in this interaction can be both internal and external.

In construction as one of the most resource-intensive branches, special attention is paid to efficiency and rationality of management of material resources. Under modern conditions the solution of a task of resource-saving, optimization of deliveries and storage of stocks becomes more and more relevant due to the constant growth of resources cost and providing conditions of their storage. The limitation of resources also demands continuous efforts on search of an effective control algorithm of stocks.

During search and the organization of the movement of stocks of the construction organization, as well as to the enterprises in the sphere of housing and public utilities, it is necessary to be guided by scientific approaches, including methods of economic and mathematical modeling (Rinke, Berkhahn, Neumann, & Berner, 2017). It is necessary to prevent the repeated count that is possible only at the accurate organization of definition of requirement, order, use and storage of stocks. There exists an issue of updating of stocks due to expenditure, uninterrupted operation and reliability of productions among problems of modeling of stocks in construction.

While choosing methods of the analysis and modeling of stocks it is also necessary to consider specifics of activity of the enterprises: variety of material resources, sequence of their application and technological interrelation. This is the question of origination of demands in some certain resource which, in turn, is result of the demand in another material resource.

In research papers it is mentioned that formation of stocks of different types of material resources differs in unevenness and irregularity depending on the schedule of works (Kaverin & Morozova, 2015). In this case application of the following methods of calculation of stocks is expedient:

• dynamic programming;

• Silver and Mill's method;

• method of the minimum variable costs;

• method of the minimum unit costs specific;

• theory of lot by lot;

• optimum economic order and others.

Level change of demand in certain physical resources can be unarranged. Respectively, the applied models have to be probabilistic, stochastic (Minocha, Singh, & Bawa,, 2016).

One of the most frequently used inventory model in construction and housing and public utilities can be represented in the following way

Z(t)=Z_a+P(t)–R(t), (1)

$Z\left(t\right)=Z_a+\underset{i=\mathrm{1}}{\overset{n}{\sum }}{P}_i-\underset{j=\mathrm{1}}{\overset{m}{\sum }}{R}_j$ (2)

where Z(t) – current stock of physical resource at the moment of time t;

Z_a – original stock at the moment of time t=0;

P(t) – arrival of physical resources in the period of time t;

P_i – the volume of physical resource supply;

R(t) – consumption of physical resource in the period of time t;

R_j – lot size of the physical resource consumption;

n – the number of delivery lots;

m - the number of consumed lots.

The model can include restrictions depending on the maximum of volume of delivery lot, number of deliveries, etc.

Statistical data of providing a stock with physical resource is a repeatedly process for the purpose of definition of its optimum numerical characteristics while developing imitation models (figure 3 ). We consider the following positions:

• time of a delivery cycle tj and a daily expense ri are defined according to the law of distribution with the set parameters;

• as there are restrictions to the maximum and to a minimum of values of these sizes, the algorithm provides data updating;

• process of modeling of the value ri precedes the time point of tk under a condition $\sum r_i\ge Z_a.$

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Source: Lukinsky & Butrin (2009).

The emergence of random variables of Z – total resources expense, W – warehouse stock balance at the moment of new lot supply, F – number of days of a lack of resources, G – physical resources shortage volume is possible during the modelling the current and reserve stocks (Koehler, 2014).

$Fj=tk-tj$ (3)

$Gj=\stackrel{tk}{\underset{tj}{?}}ri$ (4)

The use of such an algorithm for determining the parameters of material resources reserves allows us to predict the flow rate and the possibility of a shortage, which, in turn, ensures the efficiency of work in the construction industry and utility services.