Mathematical Scheme of the Three-Level. Evaluation of the Economic System

Transcription

1 Applied Mathematical Sciences Vol. 07 no IKAI td Mathematical Scheme of the Three-evel Evaluation of the Economic System S.M. Brykalov JSC "Afrikantov OKBM" Nizhny Novgorod ussia A.V. Kryanev National esearch Nuclear University MEPhI" 5409 Moscow ussia Copyright 07 S.M. Brykalov and A.V. Kryanev. This article is distributed under the Creative Commons Attribution icense which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. Abstract The mathematical scheme model) of an estimation of efficiency of the three-level economic system using settlement values of indicators and their expert estimations is developed. The mathematical model of an estimation of multi-level economic system can use forecasting of indicators of operated economic system. The uncertainty of the settlement values of indicators or expert estimations are considered in the model. The optimal association of the forecasting values of several experts is provided in the model. Keywords: Composed economic system the three - level approach estimation of efficiency mathematical models mathematical schemes optimum strategic decisions Introduction Problem tiered approach to the management and analysis of complex economic systems is currently one of the maor problems of increasing the efficiency of public and state administration and production. The success of its solution is possible with effective decision at different levels of economic management [-5 5-6].

2 694 S.M. Brykalov and A.V. Kryanev Therefore there is a need to identify opportunities for application and development of multi-level hierarchical systems theory in the selection of effective strategies to make the best strategic decisions. One of the components of a multi-level approach to the management and analysis of complex economic systems is the development of quality and efficiency of the functioning of a multilevel system and the periodic calculation and estimation of their values in order to monitor the state of the economic system and rapid adoption of appropriate management decisions. It should be noted that the multi-level approach is widely used to assess the quality and technical level of complex technical systems [6-7]. In this paper the mathematical scheme of a three-level evaluation of complex economical systems CES) is considered. The mathematical scheme clotting partial indicators integrated indicators for the three-tiered economic system We consider the three - level CES which on the first lower level is characterized by particular indicators.. m the more figures by value the more effective SES) and on the second and third levels is characterized by the integral indicators.. m 3.. m3 respectively. Thus each integral index of the second and third level consists of the indicators of the previous level. et k.. m k... и 3 k.. m3 k... n3 are the private first-level indicators that are included in the - th integral index of the second level and in the - th integral index of the third level respectively. There are the raw data baseline values of the first level:... m ) is the value of - th particular indicator of CES. For the coagulation of partial indicators in the values of integral indices it is necessary to determine the priority factors of particular indices for each of the first level integrated indicators and coefficients of the priority of each of the integral index of the second and third levels. Odds priority indicators can be determined by expert assessments [7 8]. Below one of the possible schemes of the coagulation of parameters prior to the values of integral indices of the next level on the basis of expert estimations. et: о. V kl 0 < V kl < kl n V.. m k... n k are the priority factors of particular indices of l - th expert for each -th integral index of the second level.

3 Mathematical scheme of the three-level evaluation 695 о. V kl 0 < V kl < kl V.. m3 k... k are the coefficients of priority indicators integrated for second level of l - th expert for each -th integral index of the third level. 3 о. l l = 0 < l < are the boost factors of l - th l expert. We produce standardization of values of particular indicators for each group of the integral index of the second level l k k.. m k... n ) k where k is the value optimal or scheduled) of k - th particular indicator for of - th integral index of the second level). We determine the coefficients of the priority of particular indicators for each group of the integral index of the second level: V V.. m k... n 3) k l kl l ) We calculate the values normalized) of the integrated indicators of second n k k... m k level: V. 4) ) We calculate the coefficients of the priority of integrated indicators of the second level for each group of the integral index of the third level: V V.. m3 k... 5) k l kl l 3) We calculate the values normalized) evaluation of integrated indicators of the third level: 3 V... m. 6) k k k To determine the priority of particular performance expert coefficients can be used Saaty scheme [5] according to which each l - th expert group of particular l) indicators for each of - th integral index of the second-tier matrix fills - n n ) paired comparisons of particular indicators of one group for each of - th integral index of the second level. The elements of this matrix indicate on how many times the p - th particular component which has a larger smaller) priority over q - m particular exponent for of the - th integral index). In this case according to Saaty scheme the priority factors are the components of the normalized eigenvector corresponding to the imum eigenvalue of the matrix

4 696 S.M. Brykalov and A.V. Kryanev of pairwise comparisons [5]. The elements h p q... n of this matrix indicate on how many times the p - th particular component has a larger smaller) priority over q - th particular exponent for of the - th integral index). In this case according to Saaty scheme the priority factors of particular indices of the l n - th expert Vkl 0 < Vkl < V ) are the components of the normalized kl eigenvector which correspond to the imum eigenvalue of the matrix pairwise comparisons [5]. l) pq l) of The mathematical scheme of association forecasts the values of indicators of economic system If we consider the problem of forecasting CES status for future periods in this case the original data values of particular indicators:... m 7) are not known and must be determined. Forecast values of particular indicators can be carried out using a variety of predictive mathematical models and experts alike. et l l... ;... m 8) is the forecast of the l th expert predictive model) for the - th particular l indicator of the CES. et ) is the covariance matrix of expert forecasts for the k - th K k particular index which owned group of the - th integral index under the expert forecasts which are distributed by normal law. Then forecast which obtained using the imum likelihood method) k will be given by the ratio of two scalar products [8]: K k k )... m; k... n K ) k k T where K k is the inverse matrix...) is the dimensional vector T... ) is the dimensional vector. k k k In the particular case of the independence of the forecasts expert the formula 9) has the form [8] where k Wlk lk... m; k... n l 9) 0)

5 Mathematical scheme of the three-level evaluation 697 W lk / l lk / lk l m ; k... n. ) 3 The mathematical scheme for determining the uncertainty of the indicators of the economic system The dispersion of the combined evaluation 9) is less than the minimum element min of the main diagonal covariance matrix K k. This means that the combined forecast 9) is more accurate than the finest expert. The dispersion of the combined forecast 9) is given by k ) min. ) K ) k The forecast values can be given by experts in the form of fuzzy numbers [9 0]. Triangular fuzzy numbers which are asked by three numbers: minimum value - the most reliable value and - the imum value of fuzzy number are most commonly used [9 0]. The forecast values can be given by experts in the form of fuzzy numbers lk [9 0]. The most commonly triangular fuzzy numbers are asked by three * numbers: ) minimum value - the most reliable value and lk min lk lk lk lk min - the imum value of fuzzy number [9 0]. In this case to describe the uncertainties of the values of indicators by means of random variables according to the rule of three sigma can be put lk lk min lk ) 3) 6 as well as to take the average expected value mathematical expectation) lk lk min lk. 4) In the case of accounting for uncertainties in the values of particular indicators with help of triangular fuzzy numbers it is necessary the normalization ) for * each of three numbers ) i.e. it is necessary to go to the normalized values where lk min lk lk lk min lk lk * lk * ) 5)

6 698 S.M. Brykalov and A.V. Kryanev lk min * lk min * lk lk lk lk. 6) k k k Then the normalized score of each of the integral index of the second level is represented by a triangular fuzzy number min * ) 7) where see Eq. 3) 4)) min n k V k k min n * * V k k V k k k n k 8) * * k min l lk min k l lk k l l l... m l lk k... n 9) A normalized assessment of each of the integral index of the third level is represented by a triangular fuzzy number * ) 0) where see Eq. 3) 4)) 3 min k V k k min 3 min 3 3 * * 3 V k k 3 V k k k k ) * * k min l lk min k l lk k l l l l lk... m k... n ) To solve the problems of forecasting the methods of the metric analysis can be used [-4]. The use of the three-level scheme of the CES Currently many public corporations and companies develop a long-term up to 030 to 050) development programs one of the key issues of which is the matters of effective optimization and modernization of the system of management and decision-making at all levels of the CES [5 6]. The need for a layered approach to solve the problems of the strategic vector of choice becomes relevant

7 Mathematical scheme of the three-level evaluation 699 in the development of strategies decisions and comparison of complex hierarchical vertically integrated systems which form for example state-owned enterprises belonging to large companies and corporations formed in ussia. The effective implementation of the tasks in the three-tier economic system complex industry companies) must be the efficient system of strategic planning and management combining the "top-down" and "bottom-up" and consistently state strategy to the level of a particular enterprise in the presence of uniform regulations and methodological approaches. The analysis of real-life complex economic systems shows the presence of a large number of internal and external relations hierarchies of systems and subsystems and the significant role of the state as well as the need to coordinate the processes at different levels of governance. This raises the need to develop analytical methods and mathematical schemes that can be used to select and compare the effectiveness of three-level strategies of CES. To achieve these goals in the complex multi-level economic systems to ensure monitoring of the implementation of strategic planning documents the determining the effectiveness of a CES must be formed by a system of key indicators of quality and efficiency of the multi-level facilities. The set of key quality indicators used to compare and select the most effective CES should contain integrated indicators group individual particular performance. The group indicators may be the following: economic innovative social and environmental performance safety performance. It should be noted that for a number three-level economic systems the specific nature of the industry must be taken into account [5 6]. To solve this problem it seems appropriate and possible to use: - Mathematical scheme folding the partial indicators for the integrated three-level economic system; - Mathematical scheme combining forecast values of the indicators of the economic system; - Mathematical scheme determining the uncertainty of the economic indicators of the system. The applying the methodology of multilevel analysis of CES allows you to combine the principles of centralized and decentralized approaches for solving problems related to improving the efficiency and the development of three-level systems for various purposes. Conclusion The paper presents a mathematical scheme of evaluating the effectiveness of the three-level CES using the calculated values of indicators or peer review. As a part of the proposed mathematical model of a three-tiered economic system we can use the forecasting of this economic system. The uncertainty in the estimated values of the parameters or their expert assessments is taken into account. Formulas for the optimal combination of the predicted values of several experts or predictive models are presented. It is noted that the presented mathemati-

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