INTRODUCTION: There is a significant percentage of Bulgarians suffering from ischemic heart disease (IHD) and its complications, such as ischemic mitral regurgitation (IMR). It plays an important role in Bulgarian society. Surgical treatment of this pathological conditions could have positive impact on life expectancy and the medical quality of life of patients. AIM: The purpose of the study is to establish a reproducible algorithm to advise the appropriate surgical treatment of patients with IHD and significant, but not severe IMR based on their condition. MATERIALS AND METHODS: The study is based on the data collected prospectively at the Department of Cardiac Surgery, St. Marina University Hospital in Varna, Bulgaria. IHD and significant IMR (i.e. more than mild 1+, but less than severe 4+ degree) were diagnosed in 186 patients. Applying inclusion and exclusion criteria, 140 patients with pure secondary IMR remained in the study group. The data was analyzed in a retrospective fashion. We discussed two possible treatment strategies: coronary artery bypass grafting + mitral valve repair (CABG+MVRep) and isolated revascularization (CABG only). To obtain comparable data for those treatment strategies, we needed a formal stratification of the patients, allowing comparison between the groups. RESULTS: Creating formal algorithms we are able to divide the patients into comparable Group A (CABG+MVRep) and Group B (CABG only), and surgical strategy is based on characteristics of the individual pathology of every patient. DISCUSSION: Despite data from small randomized and non-randomized trials, to date there is no clear agreement and strategy regarding concomitant mitral valve repair with CABG during the first-time operation. CONCLUSION: Formal stratification with the algorithms created and applied gave us the opportunity for reliable comparison of relatively different patients, and to draw conclusion for the practice. This approach should be applied in such small non-randomized trials to achieve better understanding of the problem of secondary IMR.

This paper aims to statistically test the null hypothesis H 0 for identity of the probability distribution of onedimensional (1D) continuous parameters in two different populations, presented by fuzzy samples of i.i.d. observations. A degree of membership to the corresponding population is assigned to any of the observations in the fuzzy sample. The test statistic is the Kuiper's statistic, which measures the identity between the two sample cumulative distribution functions (CDF) of the parameter. A Bootstrap algorithm is developed for simulation-based approximation for the CDF of the Kuiper statistic, provided that H 0 is true. The p value of the statistical test is derived using the constructed conditional distribution of the test statistic. The main idea of the proposed Bootstrap test is that, if H 0 is true, then the two available fuzzy samples can be merged into a unified fuzzy sample. The latter is summarized into a conditional sample distribution of the 1D continuous parameter used for generation of synthetic pairs of fuzzy samples in different pseudo realities. The proposed algorithm has four modifications, which differ by the method to generate the synthetic fuzzy sample and by the type of the conditional sample distribution derived from the unified fuzzy sample used in the generation process. Initial numerical experiments are presented which tend to claim that the four modifications produce similar results.

We present three (main one and two auxiliary) fuzzy algorithms to stratify observations in homogenous classes. These algorithms modify, upgrade and fuzzify crisp algorithms from our earlier works on a medical case study to select the most appropriate surgical treatment for patients with ischemic heart disease complicated with significant chronic ischemic mitral regurgitation. Those patients can be treated with either surgical revascularization and mitral valve repair (group A) or with isolated surgical revascularization (group B) depending on their health status. The main algorithm results in a fuzzy partition of patients in two fuzzy sets (groups A and B) through identification of their degrees of membership. The resulting groups are highly nonhomogenous, which impedes subsequent proper comparisons. So, the two auxiliary algorithms further stratify each group into two homogenous subgroups with comparatively preserved medical condition (A 1 and B 1) and with comparatively deteriorated medical condition (A 2 and B 2). Those two algorithms perform fuzzy partition of patients from A and B respectively into A 1 , A 2 , B 1 and B 2 by identifying their conditional degrees of membership to those subgroups. We then utilize the product t-norm to calculate the degree of membership of patients to their respective subgroup as an intersection of two fuzzy sets. We demonstrate how to form fuzzy samples for medical parameters for any subgroup. We also compare the performance of the fuzzy algorithms with their preceding crisp version, as well as with eight Bayesian classifiers. We then assess the quality of classification by modified confusion matrices, summarized further into four criteria. The fuzzy algorithms show total superiority over the other methods, and excellent differentiation of typical patients and outliers. On top, only the fuzzy algorithms provide a measure of how typical a patient is to its subgroup. The fuzzy algorithms clearly outline the role of the Heart Team, which is missing in the Bayesian classifiers.

This work analyses decision making situations, where the quantity of the value function associated with the alternatives is a random number with known distribution. The main contribution of the paper is that alternatives are grouped into pseudo indifference classes, where the alternatives are indifferent to at least one of the other alternatives in the class. However, not all elements in the set are indifferent to each other, unlike classical indifference classes. Since the resulting relation of strict preference over pseudo indifference classes turns out to be non-transitive, it is demonstrated both in theory and in terms of an example that it is strongly dependent on the significance level of comparisons in order to allocate alternatives into groups.

This chapter discusses several applications of the REPOMP procedure (Randomized Expert Panel Opinion Marginalizing Procedure). It analyzes the subjective opinion of an expert panel in a multi-criteria decision making situation. It starts with an expert panel constructing a hierarchical structure of criteria to evaluate the alternatives. At a next stage, the same expert panel evaluates the relative weight of each criterion and the degree of compliance of each alternative with those criteria. Then a randomized procedure is applied to calculate the marginal indicator of each alternative and make the final ordering based on it. Additional simulation procedure is applied to analyze the distribution of that marginal indicator. The alternatives are also being allocated to indifference classes using hypothesis testing procedures. The analyzed examples refer to issues in environmental management, energy efficiency and spatial data infrastructures.