Mathematical Scheme Comparing of. the Three-Level Economical Systems

Transcription

1 Appied Mathematica Sciences, Vo. 11, 2017, no. 15, IKAI td, Mathematica Scheme Comparing of the Three-eve Economica Systems S.M. Brykaov JSC "Afrikantov OKBM" , Nizhny Novgorod, ussia A.V. Kryanev Nationa esearch Nucear University MEPhI" , Moscow, ussia Copyright 2017 S.M. Brykaov and A.V. Kryanev. This artice is distributed under the Creative Commons Attribution icense, which permits unrestricted use, distribution, and reproduction in any medium, provided the origina work is propery cited. Abstract The mathematica scheme mode) of a three-eve comparison of economic entities of the economic system is constructed. In the mathematica mode of the three-eve comparison of the economic systems can be used the indicators of peer review and forecasting of the economic system under consideration. The mode can take into account the uncertainty in the estimated vaues of the parameters or expert assessments. The mode uses the muti-criteria approach based on the Pareto soutions. Keywords: Compex economic system CES), three-eve system of peer review, the mathematica mode comparison, muti-criteria approach, Pareto soution, the optima strategic decisions Introduction Presented artice is a ogica continuation of the previous artices of the authors and is devoted to anaysis of three-eve economic system [1]. In the anaysis of the CES it is necessary to have a mathematica scheme to compare severa CES or more options for the deveopment of one of CES and to answer the question - which one is more efficient and reiabe. These mathematica modes are considered aso for compex technica systems CTS) to determine their technica

2 704 S.M. Brykaov and A.V. Kryanev proficiency and to compare them with other simiar CTS [2-5]. As in the previous artice [1], in this artice we use the muti-eve system, which is characterized by a number of criteria [6-10]. 1 The mathematica scheme cotting particuar indices in integrated indicators for the three-eve system of economic systems We consider i 1,..., n the CES deveopment options under consideration CES when considering the possibiities of various deveopment options for the same CES in order to compare these options), each of which is characterized on the first ower eve by 1,.., m1 the particuar indices the more vaue of indices, the more effective CES), and on the second and on the third eves by the integra indicators i 2, i 3, respectivey [11,12]. Thus, each integra index of the second and third eve consists of the index of the previous eve. et i m1, k 1,..., and i 2 k, k 1,..., are the particuar first-eve indicators which are incuded in the - th integra index of the second eve and in the - th integra index of the third eve, respectivey. To sove the probems of the forecasting of the particuar first-eve indicators can be used the methods of the metric anaysis [13-16]. There are the data baseine vaues the first eve: i k,, i 1,..., n, 1,..., m1, 1) i k 1,..., 1 is the vaue of the - th private index of the i- th CES. et: V 1,..., 1 о. V, 0 < V <1, 1, 2) are the priority factors of particuar indices of the - th expert to -th of the integra index of the second eve, 1,. V 1,..., 2 о. V 2 k, 0 < V 2 k <1, 1, 3) are the priority factors of integra indicators of the second eve of the - th expert to - th of the integra index of the third eve, 1,. 1 3 о. V 3, 0 < V 3 <1, V 3 1 4)

3 Mathematica scheme comparing of the three-eve economica systems 705 are the priority factors of integra indicators in the third eve of the th expert. 4 о., = 1,,, 0 < <1, 1 5) are the boost factors of experts. Then we perform the foowing operations. 1) We produce the normaization of the vaues of particuar indices i1 k i, i=1,,n; 1,..., m1 ; k 1,..., MAX where, i 1 n 1 MAX i1 k,...,, 6). 7) 2) We cacuate the coefficients of the priority of private indicators for each group of the integra index of the second eve: V V 1, k 1,..., 8) k 1 3) We cacuate the normaized vaue of the evauation of integrated i1 k 1,..., indicators of the second eve: V, i2 k, k 1,...,. 9) 4) We cacuate the coefficients of the priority of integrated indicators of the second eve for each group of the integra index of the third eve: V V 2, k 1,..., 10) 2 k k 1 5) We cacuate the normaized vaues of the evauations of integrated 2 k i2 k 1,..., indicators in the third eve: V, i3. 11) 6) We cacuate the coefficient priority for the integra indicators in the third eve: V V 3. 12) 3 1 7) We cacuate the normaized vaue of the compex criteria for each CES: V, 1,..., n. i 1 3 i3 i 13) 8) We produce a ranking of SES on the vaues of compex criteria: 1).... 2) n) 14) CES with number i = 1) is the best CES by the integrated efficacy.

4 706 S.M. Brykaov and A.V. Kryanev 2 The mathematica definition of the uncertainty of the scheme of integrated indicators of economic systems of three-eve system of economic systems In the case of the performance of expert evauations and vaues to account for uncertainties in the vaues of particuar indicators we use the trianguar fuzzy numbers [17, 18]. It is necessary to produce the normaization 6) for each of three * numbers,, ), i.e. go to the normaized vaues ik min ik ik ik min ik ik *,, ), 15) where ik min * ik min * ik ik, ik, k. 16) kmax kmax kmax Then the normaized score of each of the integra indicator for the i - th CES wi be presented by the trianguar fuzzy number *,, ), 17) where see the formua 6), 8)) i min n i min i i n n ik min, i k ik i k ik * * V V, V, i 1,..., n k * * ik min ik min, ik ik, ik ik 18), 1,..., m, k,..., n, 19) 1 We cacuate the normaized vaue of the integrated indicator of efficiency for each CES presented in the form of a trianguar fuzzy number * i min, i, i ), i 1,..., n 20) where i min m m m i min, i i i i 1 1 * * W W, W. 21) 1 3 Choosing the best CES as the soutions Pareto two-criteria probem: an integrated efficiency criterion - a comprehensive criterion uncertainty In the case of the trianguar fuzzy numbers to describe the uncertainties in the vaues of indicators of two compex criteria for each SES: the compex criterion of efficiency is defined by equaity

5 Mathematica scheme comparing of the three-eve economica systems 707 K, i 1,..., n, 22) * ieff i the compex criterion equaity uncertainty is defined by equaity K iunc i 1,..., n. 23) i i min, The fina seection of the best CES carried out among the Pareto soutions with respect to two criteria K ieff -, 24) K iunc - min. 25) On the pane K, K ) the Pareto points form a "south-eastern" part of ieff iunc the boundary of the set criterions. Concusion The paper presents the mathematica diagram for the comparing of the three-eve economic CES or the comparison of severa variants of the same CES which is characterized by the criteria system. This mathematica comparison circuit may use the design or expert assessment indicators and predict some or a of the considered indicators of CES. The mode provides the account of uncertainties in the estimated vaues and expert assessment indicators or forecast vaues. The mode uses a muti-criteria approach to the possibiity of taking into account both quantitative and quaitative indicators. The fina decision is taken among the Pareto soutions of the muti-criteria probem with two integra criteria: the compex criterion of efficiency and compex criterion of risk for integra indicators of the third eve. The mathematica mode presented in this paper can be used by offices and enterprises in various industries [19, 20]. eferences [1] S. M. Brikaov, A. V. Kryanev, Mathematica scheme of the three-eve evauation of the economic system, Appied Mathematica Sciences, ), no. 14, [2] S. S. Semenov, E. M. Voronov, A. V. Potavskii, A. V. Kryanev, Methods of Decision-Making in Tasks Assessing the Quaity And Technica eve of Compex Technica Systems, Moscow, USS, in ussian) [3] A. V. Kryanev, S. S. Semenov, Features of the deveopment of modern technoogy and the method of evauation of the technica eve of compex technica

6 708 S.M. Brykaov and A.V. Kryanev systems based on the use of emerging technoogies, Co. Works Management of arge Systems, Moscow, in ussian) [4] A. V. Kryanev, A. V. Potavskii, S. S. Semenov, Methodica maintenance of an automated decision support system "evauation and seection" in the evauation of the technica eve of compex technica systems, Proceedings of the Internationa Symposium "eiabiity and quaity ", PGU, in ussian) [5] M. Mesarevich, D. Mako, I. Takahara, The Theory of ierarchica Mutieve Systems, Mir Pub., Moscow, in ussian) [6] F. F. Urov, E. I. Shapkin, Seecting effective strategic decisions on the basis of muti-eve and muti-criteria approach, Tutoria, N. Novgorod, in ussian) [7] V. D. Nogin, Decision-Making in Many Criteria, Pubisher UTAS, SPb., in ussian) [8] IAEA-TECDOC Seection of decommissioning strategies: Issues and factors IAEA, [9]. E. osenta, Principes of Muti Obective Optimization, Monterey, Caifornia, [10] V. M. Postnikov, S.B. Spiridonov, Methods of seection criteria for oca weighting coefficients, Science and education, Bauman MSTU, Eectronic Journa, ), in ussian) [11] T.. Saati, Decisions Making with Dependendence and Feedback: The Anaytic Network Process, University of Pittsburgh ed., [12] A. V. Kryanev, V. V. Bochkarev, D.T. anbekova, U.G. Ustinova, anking Nucear and adiation azardous Obects on the Basis of Expert Evauations, Scientific session MEPhI-2015, Summary eports, 3, 2015), 170. in ussian) [13] A. V. Kryanev, D. K. Udumyan, Metric Anaysis, Properties and Appications as a Too for Interpoation, Internationa Journa of Mathematica Anaysis, ), no. 45, [14] A.V. Kryanev, D. K. Udumyan, Metric Anaysis, Properties and Appications as a Too for Smoothing, Internationa Journa of Mathematica Anaysis, ), no. 47,

7 Mathematica scheme comparing of the three-eve economica systems 709 [15] A. V. Kryanev, D. K. Udumyan, Metric Anaysis, Properties and Appications as a Too for Forecasting, Internationa Journa of Mathematica Anaysis, ), no. 60, [16] A. V. Kryanev, D.K. Udumyan, G.V. ukin, V.V. Ivanov, Metric Anaysis Approach for Interpoation and Forecasting of Time Processes, Appied Mathematica Sciences, ), no. 22, [17] S. A. Orovskii, Decision-Making Probems with Fuzzy Initia Information, Science, Moscow, in ussian) [18] A. Pegat, Fuzzy Modeing and Contro, BINOM, Moscow, in ussian) [19] S. M. Brikaov, The impementation of the principe of decomposition of the formation of a muti-eve system of key performance indicators, Integra, ), no. 5. in ussian) [20] S. M. Brikaov, F. F. Urov, Baanced scorecard and key performance indicators: a terminoogica anaysis, probems and directions of deveopment, Economy and Business, ), in ussian) eceived: February 19, 2017; Pubished: March 14, 2017