EVALUATION AND RANKING OF INSURANCE COMPANIES BY COMBINING TOPSIS AND THE INTERVAL FUZZY ROUGH SETS

Corporate and organizational performance assessment is an important activity for both the managers and other stakeholders, as it provides them with an asset to evaluate their own strengths and weaknesses in relation to the competition, as well as guidelines for selecting appropriate measures to address the existing problems. The issue of criteria selection has been overcome through the literature review and the issue of criteria weights is handled by applying group decision making procedure. The procedure itself consists of using predefined linguistic expressions that are modelled by triangular fuzzy numbers and the aggregation of decision makers’ opinion based on the rules of rough sets algebra. The values of the decision matrix are determined by prognosis method and they are described by crisp values. The proposed algorithm is tested on the insurance companies that operate in the Republic of Serbia.

nature of many financial decisions are just some of the financial decisions' characteristics in the companies and financial institutions that can be tackled by applying multi-criteria analysis. The need to observe multiple criteria at the same time is an important component of the management function. This usually incorporates the personal preferences of the investors, especially in the institutions that perform money management professionally, such as the banks, the pension funds, the investment funds, the insurance companies, etc.
There are significant advantages of an multicriteria decision making approach in the scope of financial decision making (Zopounidis, 1999(Zopounidis, , 2002: 1) the ability to structure complex problems, 2) the possibility of considering quantitative and qualitative criteria simultaneously, 3) the transparency in the evaluation process, which allows for good argumentation in financial decision making, and 4) the application of sophisticated, flexible and pragmatic scientific methods in financial decision making. The application of multicriteria decision-making allows the decisionmaker (manager) to actively participate in the financial decision-making process and helps him/her to understand and to deal with complexity and uncertainty as characteristics of the business environment. This means that his/her role is not reduced to the passive implementation of the optimal solution (if there is such solution) obtained by applying the multicriteria model, but he/she actively participates in the process of structuring and modelling the problem, as well as in analysing, interpreting and implementing the obtained solution. A detailed survey of MCDM methods that have been developed to analyse the variety of management problems in the economic domain are given by Steuer and Na (2003), Wang et al. (2009), Toloie-Eshlaghy andHomayonfar (2011), Zavadskas and Turskis (2011), Aruldoss et al. (2013), Ghadikolaei and Esbouei (2014), etc. As the insurance companies operate in an unstable political, economic and social environment, where the customer demands change rapidly and it is imperative that the requirements expressed are fully met, it can be said that the rating and ranking of the insurance companies can be defined as a management problem under uncertainty. The uncertainties related to the values of the evaluation criteria as well as their relative importance can be described in a sufficiently good way by using linguistic variables.
The motivation for this research comes from the fact that the insurance company managers, by using the obtained results, are able to easily identify strengths and weaknesses, so that the appropriate improvement strategies can be defined in a short period of time with a goal to improve business activities. Taking into account the uncertainties during the process of decision making, a tool that can handle it should be employed.
The development of fuzzy sets theory (Dubois & Prade, 1980;Zimmerman, 2011) and the theory of rough sets (Pawlak, 2012), and, in particular, the combination of these two fields of mathematics (Pamučar et al., 2017), allows the quantitative representation of uncertainty to be satisfactorily accurate. The basic characteristic of the fuzzy number is a membership function which may take different shapes. In decision making problems embracing different areas, uncertain decision variables are the most often described by triangular fuzzy numbers (TFNs). The domains of fuzzy numbers are defined on closed interval containing upper and lower bound as well as modal value.
Fuzzy number considers the perception of individual DM. By using the rough set theory, assessment of uncertainties into each DM are described as a boundary region which is determined by respecting perceptions of all DMs. Hence, a rough number can better reflect real perceptions of DMs and thus heighten the objectivity of original data.
The aim of the paper is to propose a model the application of which can accurately determine the rank of the insurance companies that exist in a changing environment.
The paper is organized in the following manner: in Section 2 there is a comprehensive literature review related to the applied MCDM in the insurance sector and the rough set theory in modelling of the uncertainties. The proposed methodology is presented in Section 3. In Section 4, the proposed model is illustrated by real life data which comes from domestic insurance companies which exist in the Republic of Serbia. The discussion of the obtained results and Conclusion is given in Section 5.

LITERATURE REVIEW
The wide literature review indicates that the assessment and ranking of the financial institutions that operate at the level of one country is performed by respecting personal preferences of the investors and the preferences of their clients (Akhisar & Tunay, 2015;Lu & Zhu 2018). It should be emphasized that investor and client preferences differ. For example, the investors want to maximize profits while at the same time clients have aversion to risk, the investment horizon, etc. Many authors have suggested that the rating and ranking of the insurance companies should be considered as a multi-criteria analysis task (Akhisar & Tunay, 2015). According to the literature sources, it can be concluded that the Analytic hierarchy process (AHP), The Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS), "Visekriterijumska optimizacija i kompromisno resenje" (VICOR) have been mostly employed in the multi-criteria decision analysis methods (MCDA) for the ranking of the insurance companies (Ercan & Orden, 2016). There are a few papers related to the insurance companies' ranking that deal with the uncertainties. In the scope of presented research, the proposed methods, which can be found in the relevant literature, are presented and analysed in detail. The comparative analysis of the proposed model and other related research in the scope of the insurance companies ranking, by respecting many criteria, evaluation criteria and its weights, respectively, is presented in section 2.1.

Applied MCDM in the insurance domain
The insurance may be treated as the basis of each country's economic development. The economic development is causally related to the development of the insurance. The higher the level of economic development and the available resources for the insurance purposes, the greater the awareness of people about the insurance needs. The insurance, in interaction with other segments of the financial system, enables long-term economic development. The insurance companies in the developed financial markets belong to a group of highly active non-deposit financial institutions. For appropriate regular payments of the policyholders, the insurance companies make contractual payments in the event of an adverse event. The insurance may be seen as a community of individuals who are exposed to risk and who pass the risk on to the insurance company. In doing so, insurance companies agree to indemnify policyholders in the event of an accident, provide other cash benefits in the event of a loss, or provide them with risk-related services (Rejda, 2011). The most important features of the insurance are the transfer of risk from the individual to the risk community and the distribution of loss to all members of the risk community. Insurance provides financial stability, social security, enhancement of risk management, encourages economic activity and thus preserves the living standard of the population and enables financial development and economic growth.
It may be concluded that the assessment of the insurance companies is very important for companies themselves but also for the clients. Authors define different criteria for the process of insurance companies' assessment. It should be emphasized that there is no unified list of criteria. Pardalos et al. (1997) have defined 16 financial criteria which should be used for the assessment of 27 insurance companies that operate in Greece. By applied Principal Components Analysis, nine of the most important financial criteria have been selected: (1) net profit margin, (2) return on equity, (3) general liquidity, (4) leverage ratio, (5) debt capacity, (6) viability ratio, (7) investment ratio, (8) stockholder's ratio, and (9) capital sufficiency ratio.
In Taiwan, the other research has been conducted (Tsai et al., 2008) that included 14 Taiwanese insurance companies which had been ranked according to the three evaluation criteria and 11 sub-criteria identified by applying modified DELPHI method. These evaluation criteria and its sub-criteria were: (1) business index (ration of changes for direct premium, ration of changes for direct paid loss and ration of changes for retain premium), (2) whole company operating index (retain premium/shareholders, gross premium/shareholders, net reinsurance comm./shareholders, total reserve/shareholders, ratio of shareholder changes and special claim reserve/shareholders), and (3) profit ability index (return on shareholders and loss ratio of retains earn premium).
Akhisar and Tunay (2015) have analysed the sector of life insurance in Turkey for the period from 2009 to 2013, by respecting the following criteria and sub criteria: (1) capital adequacy (premiums received, Shareholders' equity, Shareholders' equity / technical provisions, Shareholders' equity / total assets), (2) profitability (financial profitloses / premiums received, loss ratios, technical profit-loses/ financial profit-loses, technical profit-loses / premiums received, total income / premiums received), and (3) asset quality (cash and cash equivalents / total assets, retention rate). The assumption that performances of the insurance companies have not changed during the analysed period has been introduced. Ercan and Orden (2016), have assessed five insurance companies listed on the Istanbul Stock Exchange in 2010-2015, based on four financial criteria: (1) current ratio, (2) asset growth, (3) return of asset and (4) return on equity. Ertugrul and Ozcil (2016) have used the financial charts of seven insurance companies which trade on Turkey-Istanbul Stock Exchange, according to the evaluation criteria related to the stability and profitability, for the period 2008-2014. These criteria are: (1) current ratio, (2) liquidity ratio, (3) cash ratio, (4) leverage ratio, (5) financial ratio, (6) asset turnover, (7) equity capital rate, (8) net profit margin, and (9) return on equity.
The analysis of the insurance sector, as it is very important component of each national economy, was performed by Valahzaghard and Ferdousnejhad (2013). They have considered 15 insurance companies which have been assessed in compliance with 30 financial criteria. According to the results of factor analysis, the first important factor, capital adequacy, represents 21.557% of the total variance, the second factor, quality of income, represents 20.958% of the total variance. In addition, the third factor, quality of cash flow, represents 19.417% of the total variance and the last factor, quality of assets, represents 18.641% of the total variance. Chen and Lu (2014) have chosen the following criteria for the assessment of four major insurance companies in Taiwan: (1) market size, (2) market growth, (3) logistic support, (4) distribution, (5) market share, (6) synergy of cost reduction, and (7) synergy of revenue increase. Saeedpoor et al. (2015) have considered the problem of ranking 13 life insurance companies which exist in Iran. This study aims at prioritizing insurance companies which hold the major proportion of Iran's total life insurance market. The life insurers have been assessed and ranked with regards to 5 criteria of customer service quality in the SERVQUAL model as well as opinions of 43 qualified insurance brokers in Tehran, Iran. These evaluation criteria were: (1) tangibility, (2) reliability, (3) assurance, (4) responsiveness and (5) empathy. Lu and Zhu (2018) have analysed the problem of ranking Chinese insurance companies according to six evaluation criteria: (1) profitability (enterprise capital appreciation profitability, including net assets yield rate, total return on assets, income margins, profit margins), (2) operating growth (mainly includes the stateowned capital preservation and appreciation rate, profit growth rate, economic profit margins), (3) asset quality situation (recognized asset rate, accounts receivable ratio), (4) solvency (solvency adequacy ratio), (5) business development capacity (product market share, customer satisfaction, open up of new market success rate), and Learning (6) creativity (employee satisfaction, employee training time growth rate, employee reasonable proposal growth rate, the number of new products developed). Mandić et al. (2017) have analysed the insurance sector of the Republic of Serbia in the period 2007-2014. Five key criteria have been identified for the assessment and rating of insurance companies: (1) equity and reserves, (2) business assets, (3) provision and liabilities, (4) financial incomes, and (5) cost of insurance.
The rank of the insurance companies, in the short and medium run during the period, is based on applying different MCDM as it is shown in Table 1.
In the proposed model, the existing uncertainties in the relative importance of criteria are modelled by the Interval-Valued Fuzzy Rough Numbers (IVFRNs). It may be one of the main advantages of the proposed model compared to the models in the literature sources. The rank of the insurance companies derives from the procedure based on on the conventional TOPSIS with IVFRNs.

MCDA, fuzzy sets and rough set theory
There are many mathematical theories which can be used for modelling of linguistic terms. According to the papers which can be found in the relevant literature, it can be said that the fuzzy sets theory and the rough sets theory are mostly used for quantitative description of linguistic variables.
It is known that, almost all management problems can be set as MCDM problems. As many authors suggest, solving of the management problem can be based on MCDM with fuzzy sets theory. At the same time, there are almost no papers in which MCDM is combined with fuzzy sets theory and rough sets theory.
In respect to the above-mentioned facts, authors of this paper consider that evaluation and ranking of insurance companies can be formally stated as MCDM with fuzzy sets theory and rough sets theory.
In this Section, there is an overview of the papers in which the uncertainties in criteria weights and criteria values are described by using the fuzzy sets and rough sets theory. Khan et al. (2016) suggest that the relative importance of criteria in many cases is suitable to be assessed in a direct manner. The criteria weights are modelled by RNs (Pawlak, 2012). Also, the redundant criteria from the decision table are eliminated by using the proposed procedure. In this way, the set of criteria that cannot be eliminated without disturbing the ability to approximate the classification, and the generation of logical rules from the reduced decision table. The proposed technique requires extensive study which may be rated as its major shortcoming. The relative importance of criteria at the level of each DM are stated by pair-wise comparison matrix (Song et al., 2014;Sharma et al., 2018). The elements of these matrices belong to common  (Saaty, 1990). The determination of the criteria weights is based on rules (Hu et al., 2006) which represent the information measure of fuzzy equivalence relations. A certain number of authors (Song et al., 2014;Sharma et al., 2018) suggest that criteria weights may be determined according to the following procedure: (1) consistency check of each DM's assessment may be performed by using eigen vector value (Saaty, 1990); (2) By using the rough set theory, pair-wise comparison matrix is constructed; (3) determining of the weights vector is based on procedure developed by Buckley, (1985) and rough algebra rules (Pawlak, 2012). The elements of the normalized weights vector are modelled by rough numbers, too. The fuzzy rating of the relative importance of criteria are performed by DMs according to Best-Worst Method framework (Pamučar et al, 2017). Respecting to all DMs, the relative importance of each criterion can be modelled by the IVFRNs. The criteria weights are given by using the modified Best-Worst Method with IVFRNs.

Criteria weights
In the scope of this research, the evaluation of the relative criteria importance is stated as fuzzy group decision making problem and they are modelled by TFNs. It is assumed that DMs may assess the relative importance of criteria in the direct manner. In the literature, the fuzzy rating aggregation of DMs into unique assessment is based on aggregation operators in many cases (Nestic et al., 2019). In this way, the treated criteria are considered independently. Many authors have suggested that more accurate quantitative values of criteria weights can be calculated if all treated criteria are considered simultaneously, as discussed in this research. Song et al., (2014) have constructed rough decision matrix. The Positive Ideal Solution (PIS) and the Negative Ideal Solution (NIS) for each treated benefit type criterion is determined as the largest upper limit of all the rough numbers and the lowest lower limit of all the rough numbers, respectively. For the cost-type criterion, the reverse is true. The deviation coefficient can be defined as a measure to depict the distance between a rough number and its PIS and NIS values and it is calculated as a distance between a RN and its PIS and NIS, respecting the type of criteria. In this way, the deviation coefficient matrices can be established. By applying the linear normalized procedure, the normalized deviation coefficient matrices are given. The separation measures of each alternative and their representative scalars are calculated according to procedure which is applied in Song et al. (2014) and described by RNs. Determining the closeness coefficients and ranking of alternatives is based on rules of conventional TOPSIS. The assessment of failures (alternative) with respect to the risk factor (evaluation criterion) may be performed by DMs and modelled by RNs (Song et al., 2014). By applying simple normalization procedure, the normalized rough decision matrix is given. PIS and NIS are determined for each risk factor (Song et al., 2014). The separation measures from PIS and NIS are determined by using the n-dimensional Euclidean distance. By applying procedure of the conventional TOPSIS, the closeness coefficient and rank are determined. It is worth to mention that TOPSIS may be modified by rough sets (Yang et al., 2017). The construction of the decision matrix is determined by applying the following steps:

TOPSIS with rough numbers
(1) the criteria values at the level of each alternative are assessed by DMs who employ the measurement scale; (2) the normalized criteria are given by using the proposed normalization procedure; (3) the weighted normalized decision matrix at the level of each DM is stated; (4) the decision matrix of RNs is constructed by using rough algebra rules (Yang et al., 2017). Distances from PIS and NIS, and closeness coefficient values are determined according to the procedure proposed in this paper. The rank of the alternative is determined in compliance with conventional TOPSIS.
In this research, forecasting criteria values are given according to evidence data from the period 2006-2016. It can be denoted as one of the advantages of the proposed model. The transformation of decision matrix into the normalized decision matrix is performed by using the vector normalization procedure. Determination of PIS and NIS are based on conventional TOPSIS. The separate measures, and closeness coefficient are based on the procedures of conventional TOPSIS, and rough set algebra rules. The rank of alternative is based on the rank of representative scalars of closeness coefficient. It may be stated that the introduced modifications of TOPSIS method by no means violate the rigor of the research.

METHODOLOGY
This section contains the developed methodology for the assessment and ranking of the insurance companies under uncertainties and imprecisions. In order to clarify the proposed methodology, the basics of rough set theory (Pawlak, 2012) are presented.

Basic consideration of the rough set theory
There is an assumption that decision makers' team participates in the decisionmaking process by interpreting their opinions using any measurement scale. Their assessments are presented by the set U, and Y is an arbitrary of U. The set of classes γ that cover all the objects in U is denoted as γ={c 1 ,..., c j ,..., c J } so that the objects of set γ have a sequential relationship, c 1 <..c j ,..<c J . For arbitrary object c j , j=1,…,J, lower and upper approximations are defined: (1) The lower approximation (1) (2) The upper approximation (2) (3) The boundary region can be defined as (3) Any ambiguous c j , can be represented by a rough number (RN), c j , such as: (4) and: (5) (6) Where M L and M U are the number of objects that are contained in Interval-valued fuzzy-rough numbers Certain scholars call for combining two or more mathematical areas in order to determine quantitative values of the treated uncertainties in a more precise way (Pamučar et al, 2017;Bello et al, 2019). In this research, fuzzy sets (Dubois & Prade, 1980) and IVFRNs are introduced to handle vagueness and uncertainties within the DMs' assessment.
(7) where: where a 1 ≤ a 2 ≤ a 3 , a 1 and a 3 stand for the lower and upper value of the support of X respectively, and a 2 for the modal value. TFN can be denoted by (a 1 , a 2 , a 3 ).

The problem statement
The insurance companies may be formally presented as a set of indices ε={1,...,e,...,E}. The index for an insurance company is denoted as e, e=1,..,E and E is the total number of the insurance companies. The decision makers represent top managers of the treated companies. It may be assumed that index e, e=1,..,E may be adjoined to the insurance company as well as DM that is employed within the company. The assessment criteria that are used for the considered insurance companies' ranking are identified by DMs in respect to literature sources (Nissim, 2010;IAIS, 2010;Grigaliunas & Li, 2017;Kwon & Wolfrom, 2016;Mandić et al., 2017) and the best practice of the European insurance companies. These criteria can be presented by the set of indices k={1,...,k,...,K}. The index for a criterion is denoted as k, k=1,..,K and K is the total number of identified evaluation criteria. In the scope of the proposed research, the treated criteria are: (1) investment income, (2) value of the settled claims by insurance companies, (3) administration expenses, (4) deferred acquisition, and (5) number of insurances. Some of the criteria are benefit type while the other are cost type.
The proposed procedure for evaluation and ranking of the insurance companies can be realized in a way that is presented in fig. 1.
The relative importance of criteria is assessed by DMs who use one of five predefined linguistic expressions. These linguistic terms are modelled by using TFNs.
Criteria values at the level of each insurance company i i=1,..,I for each period t, t=1,..,T are obtained according to the evidence data and they are crisp, v e kt , k=1,...,K; t=1,..., T; e=1,...,E. Using the variance analysis technique, it has been shown that the value of each criterion can be described by the regression law. Hence, the values of criteria at the level of each 288 P. Mimović / SJM 16 (2) (2021) 279 -299 Their operational laws are as follows (Pamučar et al 2017): insurance company for the future period t , , v e kt , k=1,...,K; e=1,...,E. may be forecasted. By applying the vector normalization procedure, all v e kt , are mapped into the normalized values, r e kt , k=1,...,K; e=1,...,E. According to the normalized decision matrix, PIS and NIS are determined, by analogy to conventional TOPSIS (Hwang & Yoon, 1982). Distances from PIS and NIS, and the closeness coefficient values are determined with respect to rough sets algebra rules, and procedure of conventional TOPSIS so that, their values are described by IVFRNs. The rank of the insurance companies is obtained by applying conventional TOPSIS (Hwang & Yoon, 1982).

Algorithm
The proposed Algorithm for determination of the insurance companies' rank is presented in the following steps.
Step 1. The assessment of the relative importance of criterion k,k=1,..,K is performed by all DMs. These fuzzy ratings are presented by the set U.
Step 2. The set of classes that cover all the objects in U is denoted as Step 9. Determination of the distances from PIS and NIS according to conventional TOPSIS (Hwang & Yoon, 1981) and rough algebra rules (Pawlak, 2012): According to the algebra rules of rough sets (Pawlak, 2012), the distances are described by rough numbers.
Step 10. The rough closeness coefficient values are given by the expression: Step 12. The values, c e , e= 1,...,E are sorted into decreasing order. The rank of the insurance companies is based on the obtained results.
Step 13. The management of each treated insurance company may determine the management initiatives respecting the obtained rank and benchmarking methods, with a goal to increase the overall business effectiveness.

CASE STUDY
The data taken into account for the modelling include the entire insurance sector in Serbia during the period between the year 2006 and 2016. The study utilizes the financial data for the four insurance companies that operate in Serbia. In the form of interview with the top managers of the treated companies, who are in this case DMs, the input data is obtained. The model is constructed by combining two methods: of multi-criteria decision-making TOPSIS and rough sets. In the scientific and practitioners' literature, the different criteria for the analysis and assessment of the insurance companies have been applied. The evaluation criteria are defined in the section 3.1.
By applying the proposed Algorithm (Step 1), the fuzzy rating of the relative importance of evaluation criteria is performed by DMs and presented in Table 2.
By using the procedure (Step 2 to Step 4 of the proposed Algorithm), the aggregated values of considered evaluation criteria are calculated. The proposed procedure is illustrated in the assessed data for the evaluation criterion (k=2).
The set of classes γ * that cover all the objects in U is denoted as γ * 2 ={LW,MW,HW}.  In a similar way, the weight values of other selected evaluation criteria are calculated and presented in Table 3.
By using the Eq. (12) (Step 5 of the proposed Algorithm) the normalized weights vector is given and presented in Table 4.
The values of evaluation criteria at the level of each insurance company come from evidence data. According to these data, the linear regression line for each evaluation criterion is constructed. The utilized forecasting method is illustrated in the example of determining the number of issued insurance policies (k=5) at the level of the first insurance company (Table 5). 292 P. Mimović / SJM 16 (2) (2021) 279 -299  The forecasted criteria values for each treated insurance company are calculated in a similar way. The decision matrix is presented in Table 6.
The normalized decision matrix, PIS and NIS are determined by using procedure (Step 7 to Step 8 of the proposed Algorithm) and presented in Table 7.
By applying the proposed Algorithm (Step 9 to Step 11) distances from PIS and NIS, as well as the rough closeness coefficient values, are calculated and presented in the following example.
The rough closeness coefficient values and their representative scalars are calculated in a similar way. The obtained values are presented in Table 8. By using the proposed Algorithm (Step 12) the rank of the insurance companies is given and presented in Table 8.

DISCUSSION AND CONCLUSION
The proposed approach combines the selected methods and provides a flexible, 293 P. Mimović / SJM 16 (2) (2021) 279 -299 Table 8. Rough closeness coefficients, their representative scalars and the rank of the insurance companies systematic and objective framework for the comprehensive assessment of the insurance companies with a goal to provide reliable information for clients regarding investment decisions and other stakeholder perspectives.
The first company in rank is the best regarding the treated criteria values and corresponding weights. This is important data for the stakeholders and the companies themselves. The company management may seek for comparative benchmarking and strive for overall improvement. On the other hand, clients who are choosing the best company at the moment, may choose the desired insurance company for contracting them. The research findings may also be of interest to the institutional investors in emerging financial markets. In addition, assumptions are also made for a critical analysis of the selected criteria for the rating and ranking of the insurance companies.
The results obtained can have practical implications, in terms of methodological support for managers in the insurance companies, in general, to better understand the company's environment and to make better strategic decisions and position their company on the basis of formal quantitative models.
The analysis of the obtained results generates the preconditions for identifying opportunities and defining a formal framework for improving strategic decisionmaking, not only in the specific case, but also at the level of financial institutions and the financial system as a whole.
The proposed framework may also serve to support the construction of an effective analytical framework for managerial decision-making and it can easily be further expanded or modified, to better adapt to a specific problem and context.
In the mathematical sense, the application of IVFRNs provide a solid base for modelling uncertainties and vagueness by using natural language. It is worth to mention that literature review identifies no similar research that connects IVFRNs and insurance companies ranking. This represents one of the major contributions of the research.
It should be noticed that there is a certain limitation of the model regarding the chosen criteria. Other limitation is the definition of the linguistic expressions and the assessment scale.