Ludmil Mikhailov

Deriving priorities from fuzzy pairwise comparison judgements

A new approach for deriving priorities from fuzzy pairwise comparison judgements is proposed, based on α-cuts decomposition of the fuzzy judgements into a series of interval comparisons. The assessment of the priorities from the pairwise comparison intervals is formulated as an optimisation problem, maximising the decision-makers satisfaction with a specific crisp priority vector. A fuzzy preference programming method, which transforms the interval prioritisation task into a fuzzy linear programming problem is applied to derive optimal crisp priorities. Aggregating the optimal priorities, which correspond to different α-cut levels enables overall crisp scores of the prioritisation elements to be obtained.A modification of the linear fuzzy preference programming method is also proposed to derive priorities directly from fuzzy judgements, without applying α-cut transformations. The formulation of the prioritisation problem as an optimisation task is similar to the previous approach, but it requires the solution of a non-linear optimisation program. The second approach also derives crisp priorities and has the advantage that it does not need additional aggregation and ranking procedures.Both proposed methods are illustrated by numerical examples and compared to some of the existing fuzzy prioritisation methods.

Evaluation of services using a fuzzy analytic hierarchy process

This paper proposes a new approach for tackling the uncertainty and imprecision of the service evaluation process. Identifying suitable service offers, evaluating the offers and choosing the best alternatives are activities that set the scene for the consequent stages in negotiations and influence in a unique manner the following deliberations. The pre-negotiation problem in negotiations over services is regarded as decision-making under uncertainty, based on multiple criteria of quantitative and qualitative nature, where the imprecise decision-maker’s judgements are represented as fuzzy numbers. A new fuzzy modification of the analytic hierarchy process is applied as an evaluation technique. The proposed fuzzy prioritisation method uses fuzzy pairwise comparison judgements rather than exact numerical values of the comparison ratios and transforms the initial fuzzy prioritisation problem into a non-linear program. Unlike the known fuzzy prioritisation techniques, the proposed method derives crisp weights from consistent and inconsistent fuzzy comparison matrices, which eliminates the need of additional aggregation and ranking procedures. A detailed numerical example, illustrating the application of our approach to service evaluation is given.

Fuzzy analytical approach to partnership selection in formation of virtual enterprises

The main objective of this paper is to present a new fuzzy approach to partnership selection in the formation of virtual enterprises. The phases of the virtual enterprise life cycle are briefly described and it is shown that the partnership selection is a key factor in the formation of such complex organisations. It is justified that the partnership selection process should be formulated as a multiple criteria decision-making problem under uncertainty. A new fuzzy programming method is proposed for assessment of uncertain weights of partnership selection criteria and uncertain scores of alternative partners, in the basic framework of the Analytic Hierarchy Process. The proposed fuzzy prioritisation method uses interval pairwise comparison judgements rather than exact numerical values of the comparison ratios and transforms the initial prioritisation problem into a linear program. The method can derive priorities from inconsistent interval comparison matrices, thus eliminating the drawbacks of the existing interval prioritisation methods. Moreover, the method generalises the known prioritisation methods, since it can be used for deriving priorities from exact, interval or mixed comparison matrices, regardless of their consistency. A numerical example, illustrating the application of this method to partnership selection process is given.

A fuzzy programming method for deriving priorities in the analytic hierarchy process

The estimation of the priorities from pairwise comparison matrices is the major constituent of the Analytic Hierarchy Process (AHP). The priority vector can be derived from these matrices using different techniques, as the most commonly used are the Eigenvector Method (EVM) and the Logarithmic Least Squares Method (LLSM). In this paper a new Fuzzy Programming Method (FPM) is proposed, based on geometrical representation of the prioritisation process. This method transforms the prioritisation problem into a fuzzy programming problem that can easily be solved as a standard linear programme. The FPM is compared with the main existing prioritisation methods in order to evaluate its performance. It is shown that it possesses some attractive properties and could be used as an alternative to the known prioritisation methods, especially when the preferences of the decision-maker are strongly inconsistent.

Fuzzy analytic network process and its application to the development of decision support systems

In this paper we propose a fuzzy extension of the analytic network process (ANP) that uses uncertain human preferences as input information in the decision-making process. Instead of the classical Eigenvector prioritization method, employed in the prioritization stage of the ANP, a new fuzzy preference programming method, which obtains crisp priorities from inconsistent interval and fuzzy judgments is applied. The resulting fuzzy ANP enhances the potential of the ANP for dealing with imprecise and uncertain human comparison judgments. It allows for multiple representations of uncertain human preferences, as crisp, interval, and fuzzy judgments and can find a solution from incomplete sets of pairwise comparisons. An important feature of the proposed method is that it measures the inconsistency of the uncertain human preferences by an appropriate consistency index. A prototype decision support system realizing the proposed method is developed, and its performance is illustrated by examples.

A fuzzy approach to deriving priorities from interval pairwise comparison judgements

Abstract In this paper we study the problem of priority elicitation in the analytic hierarchy process and propose a new approach to deriving crisp priorities from interval pairwise comparison judgements. By introducing linear or non-linear membership functions, representing the decision-makers degree of satisfaction with various crisp priority vectors, the interval judgements are transformed into fuzzy inequality constraints. The interval prioritisation problem is then formulated as a fuzzy mathematical programming problem for obtaining an optimal crisp priority vector that maximises the overall degree of satisfaction. The proposed approach yields linear or non-linear mathematical programs, capable of deriving priorities from consistent and inconsistent interval judgements. The presence of a consistency index that measures the level of inconsistency of interval judgements is an attractive feature of our approach. Another feature, which does not exist in the known prioritisation methods, is the opportunity for additional prioritisation of the initial judgements. Numerical examples are given and comparisons with other interval prioritisation methods are carried out.

Group prioritization in the AHP by fuzzy preference programming method

In this paper we propose a new group fuzzy preference programming (GFPP) method for deriving group priorities from crisp pairwise comparison judgements, provided by multiple decision makers. The assessment of the group priorities is formulated as a fuzzy linear programming problem, maximizing the groups overall satisfaction with the group solution. The GFPP method combines the group synthesis and prioritization stages into a coherent integrated framework, which does not need additional aggregation procedures. The method can easily deal with missing judgements and provides a meaningful indicator for measuring the level of group satisfaction and group consistency.

An interpretable fuzzy rule-based classification methodology for medical diagnosis

OBJECTIVE The aim of this paper is to present a novel fuzzy classification framework for the automatic extraction of fuzzy rules from labeled numerical data, for the development of efficient medical diagnosis systems. METHODS AND MATERIALS The proposed methodology focuses on the accuracy and interpretability of the generated knowledge that is produced by an iterative, flexible and meaningful input partitioning mechanism. The generated hierarchical fuzzy rule structure is composed by linguistic; multiple consequent fuzzy rules that considerably affect the model comprehensibility. RESULTS AND CONCLUSION The performance of the proposed method is tested on three medical pattern classification problems and the obtained results are compared against other existing methods. It is shown that the proposed variable input partitioning leads to a flexible decision making framework and fairly accurate results with a small number of rules and a simple, fast and robust training process.

A heuristic method to rectify intransitive judgments in pairwise comparison matrices

This paper investigates the effects of intransitive judgments on the consistency of pairwise comparison matrices. Statistical evidence regarding the occurrence of intransitive judgements in pairwise matrices of acceptable consistency is gathered by using a Monte–Carlo simulation, which confirms that relatively high percentage of comparison matrices, satisfying Saaty’s CR criterion are ordinally inconsistent. It is also shown that ordinal inconsistency does not necessarily decrease in the group aggregation process, in contrast with cardinal inconsistency. A heuristic algorithm is proposed to improve ordinal consistency by identifying and eliminating intransitivities in pairwise comparison matrices. The proposed algorithm generates near-optimal solutions and outperforms other tested approaches with respect to computation time.

Evolving fuzzy medical diagnosis of Pima Indians diabetes and of dermatological diseases

OBJECTIVE This paper reviews a methodology for evolving fuzzy classification which allows data to be processed in online mode by recursively modifying a fuzzy rule base on a per-sample basis from data streams. In addition, it shows how this methodology can be improved and applied to the field of diagnostics, for two popular medical problems. METHOD The vast majority of existing methodologies for fuzzy medical diagnostics require the data records to be processed in offline mode, as a batch. Unfortunately this allows only a snapshot of the actual domain to be analysed. Should new data records become available they require cost sensitive calculations due to the fact that re-learning is an iterative procedure. eClass is a relatively new architecture for evolving fuzzy rule-based systems, which overcomes these problems. However, it is data order dependent as different orders of the data result into different rule bases. Nonetheless, it is shown that models of eClass can be improved by arranging the order of the incoming data using a simple optimization strategy. RESULTS In regards to the Pima Indians diabetes dataset, an accuracy of 79.37% was obtained, which is 0.84% lower than the highest in the literature. The proposed optimization strategy increased the accuracy and specificity of the model by 4.05% and 7.63% respectively. For the dermatology dataset, an accuracy of 97.55% was obtained, which is 1.65% lower than the highest in the literature. In this case, the proposed optimization strategy improved the accuracy of the model by 4.82%. The improved algorithm has been compared to other existing algorithms and seems to outperform the majority. CONCLUSIONS This paper has shown that eClass can effectively be applied to the classification of diabetes and dermatological diseases from discrete numerical samples. The results of using a novel optimization strategy indicate that the accuracy of eClass models can be further improved. Finally, the system can mine human readable rules which could enable medical experts to gain better understanding of a sample under analysis throughout the traditional diagnostic process.