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A critical step in multiple criteria optimization is setting the preferences for all the criteria under consideration. Several methodologies have been proposed to compute the relative priority of criteria when preference relations can be expressed either by ordinal or by cardinal information. The analytic hierarchy process introduces relative priority levels and cardinal preferences. Lexicographical orders combine both ordinal and cardinal preferences and present the additional difficulty of establishing strict priority levels. To enhance the process of setting preferences, we propose a compact representation that subsumes the most common preference schemes in a single algebraic object. We use this representation to discuss the main properties of preferences within the context of multiple criteria optimization.

Fuzzy analytic hierarchy process (FAHP) methodologies have witnessed a growing development from the late 1980s until now, and countless FAHP based applications have been published in many fields including economics, finance, environment or engineering. In this context, the FAHP methodologies have been generally restricted to fuzzy numbers with linear type of membership functions (triangular numbers—TN—and trapezoidal numbers—TrN). This paper proposes an extended FAHP model (E-FAHP) where pairwise fuzzy comparison matrices are represented by a special type of fuzzy numbers referred to as (m,n)-trapezoidal numbers (TrN (m,n)) with nonlinear membership functions. It is then demonstrated that there are a significant number of FAHP approaches that can be reduced to the proposed E-FAHP structure. A comparative analysis of E-FAHP and Mikhailov’s model is illustrated with a case study showing that E-FAHP includes linear and nonlinear fuzzy numbers.

Assessing the ability of applicants to repay their loans is generally recognized as a critical task in credit risk management. Credit managers rely on financial and market information, usually in the form of ratios, to estimate the quality of credit applicants. However, there's no guarantee that a given set of ratios contains the information needed for credit classification. Decision rules under strict uncertainty aim to mitigate this drawback. In this paper, we propose the use of a moderate pessimism decision rule combined with dimensionality reduction techniques and compromise programming. Moderate pessimism ensures that neither extreme optimistic nor pessimistic decisions are taken. Dimensionality reduction from a set of ratios facilitates the extraction of the relevant information. Compromise programming allows to find a balance between quality of debt and risk concentration. Our model produces two critical outputs: a quality assessment and the optimum allocation of funds. To illustrate our multicriteria approach, we include a case study on 29 firms listed in the Spanish stock market. Our results show that dimensionality reduction contributes to avoid redundancy and that quality-diversification optimization is able to produce budget allocations with a reduced number of firms.

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