-method of normalization the Multidimensional Data MDC MDC AIS Innovative Computer Technologies I I I IZ Multidimensional Data for MADM/MCDM problem, Multidimensional Classification, DSS systems, Artificial Intelligence Systems et al. Dr. Irik Z. Mukhametzyanov [email protected] Ufa State Petroleum Technological University Ufa, Russia April, 17, 2018 1 

Abstract «IZ-method» Abstract Presented a new method for normalizing the decision matrix for solving decision-making problems with several attributes (MCDM/MADM) and in tasks multidimensional classification. Method based on the principles … know-how. The method solves the problems of normalization of data in the transformation of scales, the problems of data scaling, the problem of asymmetry in converting cost criteria to benefit criteria that of asymmetry in converting cost criteria to benefit criteria that have not been resolved in the last 50 years in the scientific field of multiple-criteria/multiple-attribute decision making and multidimensional classifications. Keywords normalization of multidimensional data, decision making, MADM/MCDM problem, artificial intelligence systems Mathematics & Computing -method © Irik Z. Mukhametzyanov 2 

MCDM problem: Uncertainty in Inputs the Uncertainty in the Output Scales of Criteria Weights of the Criteria Evaluation of Alternatives Method of Normalizing DM Method of Ranking -method © Irik Z. Mukhametzyanov 3 

Normalization 1. To combine the attributes of alternatives according to different criteria into a single integrated value, all data must be dimensionless. This can be achieved by dividing each attribute expressed in a particular dimension by a specific value of the same size. 2. Most methods of normalization are minor variants of each other. These small differences can have important other. These small differences can have important consequences for the quality of decision making. 3. Typics normalization methods: Max Vector Max-Min Sum … -method © Irik Z.Mukhametzyanov max ; ij ij j a x a = 2 1 ij ij n ij j a x a = = ∑ max max min ; j ij ij j j a a x a a − = − 1 ; ij ij m ij i a x a = = ∑ . et al 4 

Normalization problems (example 8x5) -method © Irik Z. Mukhametzyanov Fig. 1. Inequality of contributions to the integral index; the asymmetry of the attributes of alternatives (for different criteria and different methods of normalization). The second and fifth criteria are cost criteria (highlighted in red) 5 

Normalization problems (example 2) -method © Irik Z. Mukhametzyanov Fig. 2. Four basic methods of normalization and 15 rank methods lead to different results 6 

-method normalization DM - No problems! I I I I Z Z Z Z I I I IZ -method © Irik Z. Mukhametzyanov Fig. 3. Decision matrix after normalization with IZ-method (for different criteria and Max- method of normalization). The second and fifth criteria are cost criteria (highlighted in blue) Aij is the natural value of the attribute, •; Rij is the normalized value, • . values Rij are not given (know-how) [ ; ] (0;1] I Z ⊆ 7 

-method normalization DM - No problems! I I I I Z Z Z Z I I I IZ -method © Irik Z. Mukhametzyanov Fig.4. Decision matrix after normalization with IZ-method (for different criteria and Vec- method of normalization). The second and fifth criteria are cost criteria (highlighted in blue) Aij is the natural value of the attribute, •; Rij is the normalized value, • . values Rij are not given (know-how) [ ; ] (0;1] I Z ⊆ 8 

-method normalization DM - No problems! I I I I Z Z Z Z I I I IZ -method © Irik Z. Mukhametzyanov Fig. 5. Decision matrix after normalization with IZ-method (for different criteria and nonlinear Max2-method of normalization). The second and fifth criteria are cost criteria (highlighted in blue) Aij is the natural value of the attribute, •; Rij is the normalized value, • . values Rij are not given (know-how) [ ; ] (0;1] I Z ⊆ 9 

IZ-method solves the problem of preserving the proportions of the contributions of the attributes of alternatives for each criterion and allows the operation of scaling the natural values of the attributes of alternatives IZ- method solves the problem … for different criteria and I I I IZ Advantages of the -method : IZ- method solves the problem … for different criteria and excludes the priority of the contributions of individual criteria to the integral index of the alternative IZ- method solves the problem of asymmetry in the transformation (inversion) of cost criteria, in the benefit criteria -method © Irik Z. Mukhametzyanov 10 

Advantages of the -method : IZ -method does not give the result of equal proportions of the contributions of the attributes of alternatives for each criterion for nonlinear normalization methods, since it is based on the principles …. However, the degree of proportionality becomes high, in comparison with the procedure of nonlinear normalization I I I IZ IZ -method allows the use of different methods of normalization for different criteria (including nonlinear) -method © Irik Z. Mukhametzyanov 11 

-method (PS:) In example present 8 alternatives and 5 criteria. To preserve confidentiality (know-how), the author does not publish any values of normalized data. In addition, the normalized matrix in the figures is represented with displacement and broadening. The algorithm of the IZ-method includes the procedure … using forward and reverse sorting of data. Sorting is not performed in the interval [0; 1] I I I IZ -method © Irik Z. Mukhametzyanov and reverse sorting of data. Sorting is not performed in the interval [0; 1] (abbreviation AZ), but from …, greater than 0 and up to 1 (from I to Z), which is reflected in the name of the IZ-method. The name of the method includes also the author's initials. Sorry for my low english. Such a trifle, however, should not spoil the impression of such a perfect IZ -method of normalization. 12 

The presented material has a promotion purpose for IZ -method IZ -method IZ -algorithm IZ -.m code IZ -.m code -method © Irik Z. Mukhametzyanov for the development of the MADM/MCDM problem, Decision Support Systems, Multidimensional Classification, Artificial Intelligence Systems et al. 13 

For interested companies: 1. The project "IZ-norm" is an initiative project of the author. 2. The implementation period of the "IZ-norm" project was about 1 year. 3. All materials of "IZ-method" have the status of know-how. 4. Transfer of know-how materials is possible after receiving a 4. Transfer of know-how materials is possible after receiving a research grant for a completed project of at least $ XXX,000. -method © Irik Z. Mukhametzyanov 14 

“MCDM_tools”, File Exchange Mathworks (2018), https://www.mathworks.com/matlabcentral/fileexchange/65742- mcdm-tools “IZ norm method”(demo), File Exchange Mathworks (2018), https://www.mathworks.com/matlabcentral/fileexchange/67950-iz- norm-method “Norm Analysis”, File Exchange Mathworks (2018), References: “Norm Analysis”, File Exchange Mathworks (2018), https://www.mathworks.com/matlabcentral/fileexchange/67705- norm-analysis “MCDM Sens_Analysis”, File Exchange Mathworks (2018), https://www.mathworks.com/matlabcentral/fileexchange/67803- mcdm-sens-analysis -method © Irik Z. Mukhametzyanov Information about author can be found on the link: http://www.math.rusoil.net/files/slider/rezume-eng-.pdf 15