SENSITIVITY ANALYSIS OF THE TOPSIS METHOD IN RESPECT OF INITIAL DATA DISTRIBUTIONS

The present article investigates the sensitivity of the multiple criteria decision-making method TOPSIS in respect of attribute probability distributions. To carry out research, initial data – attribute values – were generated according to a normal, log-normal, uniform, and beta distributions. Decision matrixes were constructed from the generated data. By applying the TOPSIS method to the matrixes generated, result samples were received. A statistical analysis was conducted for the results obtained, which revealed that the distributions of the initial data comply with the distributions of the results received by the TOPSIS method. According to the most common alternative rank value, it was ascertained that the TOPSIS method is the most sensitive for data distribution according to beta distribution, and the least sensitive for data distribution according to lognormal distribution.


Introduction
While analysing scientific works on sensitivity analysis methods, one can frequently find works dedicated to the analysis of the sensitivity of mathematical models.The sensitivity analysis of mathematical models studies the relationship between information flowing in and out of the model [9].The significance of the sensitivity analysis in analysing multiple criteria decision-making methods has been approved by scholars as an opportunity to increase the reliability of a multiple criteria decision.This analysis verifies whether slight alterations of initial data or preferences will change the final decision results [2].Generally, in case of a multiple criteria task, the sensitivity analysis is carried out in respect of attribute significance values ( [11], [12], [13], [6]) by using pseudorandom values evenly distributed in the range [0, 1].R. Simanavičienė and L. Ustinovičius suggested the sensitivity analysis in respect of initial data distributions [10].They studied a sensitivity attribute distribution according to a uniform and normal law of multiple criteria decision-making methods TOPSIS, SAW and COPRAS when attribute values change within the range ± 10 % of a fixed attribute value.
Commonly, uniform random sizes are generated for a statistical analysis of mathematical models [0, 1] ([8], [5], [13]).Nevertheless, when conducting statistical modelling, it is possible to generate samples of variables according to other distribution laws [7].In particular, when carrying out statistical research in the field of medicine, it is noticeable that the variables of that field have various distributions as uniform, normal, log-normal, beta, gama, etc. [3].
The aim of this work is to identify the impact of initial data (attribute values) distributions on the results of the multiple criteria method TOPSIS.For the analysis of the TOPSIS method sensitivity, a mathematical model sensitivity analysis is used on the basis of samples.For the research, it was chosen to generate attribute values according to a uniform, log-normal, and beta distribution, and to observe the impact of certain distributions on the results of the TOPSIS method.During the research, it will be monitored whether the attribute distributions comply with alternative distributions, and towards which distribution the TOPSIS method is the most sensitive.

Sensitivity analysis on the basis of samples
The sensitivity analysis of a mathematical model in respect of the basis of samples is one of the techniques of sensitivity analysis when a model is designed repeatedly based on the combinations of values, which are made by using sampling distributions of known input factors [9].The aforementioned sensitivity analysis method could be described in the following steps: Sensitivity analysis of the TOPSIS method in respect of initial data distributions • An experiment is designed, and input factors requiring analysis are defined.
• Probability density function or deviation limits are determined for each factor.
• An input vector is generated by a chosen method.
• By using generated input data, model results are calculated.
• The impact of each input factor or relative importance for the outcome variable(s) is evaluated.
A sensitivity analysis could be conducted for a set of factors, which could be the following: • A certain input characteristic; • Distribution parameters describing a random process; • A reason of a factor whose value determines the mechanism of alternation of a process selection.
In the case when a sample is provided through a model, model calculations are carried out repeatedly for each realization -in order to obtain the sample of variable results of the field under investigation.When having a sample of results, a statistical analysis of sample data is conducted: an empirical distribution function is formed; such numerical characteristics as an average, a standard deviation, confidence intervals, etc. are calculated [9].

Multiple criteria evaluation method TOPSIS
The TOPSIS method means Technique for Order Preference by Similarity to Ideal Solution [4].An assumption of the TOPSIS method: an alternative which is the furthest from the "negative-ideal" alternative is the best.On the basis of this assumption, the alternatives examined are listed in a priority line.
Suppose we have a decision matrix X where lines mark the alternatives examined (m -number of alternatives), columns -efficiency indicators (n -number of efficiency indicators).
where: ij x -i-th alternatives, value of the j-th efficiency indicator.
By applying the TOPSIS method, a decision matrix X is normalized by making a vector normalization: ( Suppose known values of attribute significance ( ) , whose elements are calculated according to the formula (3): The "ideal" variant (alternative) is determined according to the following formula: where J -a set of attribute indexes whose higher values are better; J′ -a set of attribute indexes whose lower values are better.The "negative-ideal" variant is determined according to the following formula: A distance between a comparative i-th and "best-ideal" + A variant is determined by calculating a distance in the n-dimensional Euclidean space upon the following formula: ( ) Rūta Simanavičienė, Vaida Petraitytė 47 and between the i-th and "negative-ideal" − A , upon following the formula: ( ) The criterion of the TOPSIS method is a relative distance of the i-th alternative to the "worst-ideal" variant: The best alternative out of the ones analysed will be that whose i K value is the highest [4].

Proposed algorithm for the sensitivity analysis
Based on the sensitivity analysis of a mathematical model and sample, the main principles and the idea provided in the article [10], an algorithm is formed to examine a multiple criteria method sensitivity in respect of attribute distributions: 1.A multiple criteria decision-making task is formed, which consists of a set of analysed alternatives , where ij x is a quantitative estimation of the i-th alternative of j-th attribute.) , from which K decision matrixes are then formed: 6.In order to check the influence of attribute distributions on the results of the TOPSIS method, the significances of equal size attributes are chosen: ( ) 7. Accordingly, by taking one generated matrix ( ) ( ) , an evaluation of alternatives by the TOPSIS method is made.Calculation results are provided as samples , whose elements are the values of the TOPSIS method criterion for each alternative.8. Taking rationality values of alternatives into account, alternatives are ranked by assigning a rank 1 to the highest value, while rank m -to the lowest.9.A statistical analysis of the results obtained is performed.To verify a hypothesis of the TOPSIS criterion values to i-th alternative distribution, the chi-squared and Kolmogorov-Smirnov criteria are used.

Statistical methods to evaluate the sensitivity of the TOPSIS method
Based on the priority lines of alternatives, the conclusion suggests that the most favourable conditions to establish a business are in Norway (A 4 ) -rank 1.
Provided we know the indicator distributions and will be solving this multi-objective task for one year each day, would the alternative ranking change?In order to answer this question, a research is conducted during which indicator value K samples (K = 100) are generated according to a selected probability distribution.Decision matrixes are made from generated values for which, applying the TOPSIS method with equal size indicator significances, TOPSIS criterion values to alterations were obtained.
The following probability distributions were chosen for the research: uniform, normal, log-normal, and beta.Having conducted alteration ranking, rank mode estimation was calculated for each alternative, and a reliability level was calculated for this estimation according to the formula (10).The most frequently obtained rank estimations of alternatives and their reliability levels are provided in tables 1-2.It may be observed that in case of uniform, normal and log-normal distributions, the values of the most frequent alternative ranks coincided and their reliability levels correlated.In case of beta distribution, ranks of A1 and A5 alternatives changed, and the reliability level of most frequent ranks is significantly lower than in cases of uniform, normal and log-normal distributions.
Verifications were made whether the TOPSIS criterion value samples of alternatives have the same distribution as indicator value samples.Significance level α = 0.05.Chi-squared and Kolmogorov-Smirnov's criteria are used to verify the hypotheses, TOPSIS criterion value distribution of alternatives.
• Initial data were distributed according to a uniform distribution -do alternative samplings have a uniform distribution?Having conducted a verification of distribution compatibility, it was noticed that the distributions of the TOPSIS criterion values of the alternatives comply with the indicator value distributions.

Conclusions
1. Following the research, it was observed that the TOPSIS method is sensitive to indicator distributions.2. Taking the distributions analysed into consideration, the TOPSIS method is the most sensitive for a beta distribution and the least sensitive for a log-normal distribution.3. Having conducted a research on the compliance of initial data with result distributions, it was observed that the distributions of the TOPSIS criterion values of alternatives comply with the indicator distributions.

3 . 5 .
The distribution law F(x) is chosen to generate attribute values.4. The parameters of a selected distribution F(x) are set based on the values of valuation attributes According to a chosen distribution, it is generated after K values of each attribute

1. Compatibility criteria.
In order to check whether the distribution of the TOPSIS criterion values to i-th alternative complies with attribute distributions, nonparametric hypotheses about the distribution were formulated.Data.Variable sampling of quantitative relationship scale is made of the TOPSIS criteria to i-th alternative, sampling volume n = 100.

Table 1 .
Values of the most frequent ranks and reliability levels in cases of uniform, normal and log-normal distributions

Table 2 .
Values of the most frequent ranks and reliability levels in case of beta distribution

Table 3 .
Goodness-of-fit test estimation with α = 0.05 that the result samples have a uniform distribution

Table 4 .
Goodness-of-fit test estimation with α = 0.05 that the result samples have a normal distribution

Table 5 .
Goodness-of-fit test estimation with α = 0.05 that the result samples have a log-normal distribution • Initial data were distributed according to a beta distribution -do alternative samplings have a beta distribution?

Table 6 .
Goodness-of-fit test estimation with α = 0.05 that the result samples have a beta distribution