Decision Analysis, Uncertainty Theories and Aggregation Operators in Financial Selection Problems

Abstract

The complexity of financial analysis, particularly on selection process or decision making problems, has increased rapidly over several decades. As a result, much attention has been focused on developing and implementing the efficient mathematical models for supporting this kind of problems. Multiple criteria decision analysis, an advanced field of operations research provides analysts or decision makers a broad range of methodologies, which are all suited to the complexity of financial decision analysis. In the financial modeling, uncertainty problems are inevitable, owing to the fact that the consequences of events are not precisely known. In addition, human judgments as part of analysis also contribute to it intricacy. Correspondingly, many studies have been concentrated on integrating uncertainty theories in modeling the real financial problems. One area of interest is on the inclusion of the element of human behavior or attitudinal character of decision makers. Aggregation operator in this case can offer a wide spectrum of analysis or flexibility in modeling the human behavior in financial decision analysis.

In general, the main purpose of this work is on the study of financial selection problems from the perspective of decision analysis, uncertainty theories and aggregation operators. To be specific, the decision problems under a finite or discrete case and multidimensional factors are studied. The emphasis is given on the group decision making models, notably, the Dempster-Shafer theory (DST) of belief structure, the analytic hierarchy process (AHP) and the technique for order performance by similarity to ideal solution (TOPSIS). Moreover, the uncertainty theories based on fuzzy set theory and imprecise probability are employed, together with information fusion based on the ordered weighted average (OWA) operators. Quantitative and qualitative preferences, decision strategies based on the attitudinal character of decision makers, and majority concepts for group consensus are highlighted. The specific contributions of this work are summarized as the following:

• The first contribution is on developing the multi-expert multi-criteria decision making (ME-MCDM) model with respect to two-stage aggregation processes. In specific, the aggregation of criteria is based on the integration of weighted arithmetic mean (WA) and OWA. The main attention is given on the proposed alternative OWAWA operator as an extension of immediate WA and OWAWA operators. Two approaches for modeling the majority opinion of experts are studied, in which based on the induced OWA (IOWA) operators. Some modifications to the support functions are suggested as to derive the order inducing variables. The analysis of ME-MCDM model based on these aggregation processes then is conducted. In this study the selection of investment strategy is used as to exemplify the model. • The weighted-selective aggregated majority-OWA operator may be considered as the second contribution. It is as an extension of the SAM-OWA operator, where the reliability of information sources is considered. The WSAM-OWA then is generalized to the quantified WSAM-OWA by incorporating the concept of linguistic quantifier, mainly for the group fusion strategy. The QWSAM-IOWA with an ordering step is proposed for the individual fusion strategy. These aggregation operators are then implemented to the case of alternative scheme of heterogeneous group decision analysis, in particular for a selection of investment problem. • Third contribution is represented by the development of linguistic group decision making with Dempster-Shafer belief structure. Different type of linguistic aggregation operator such as the 2-tuple induced linguistic OWA operator is suggested. Specifically, it is based on order-inducing variables in which the ordering of the arguments and uncertain situations can be assessed with linguistic information. Then, by using the 2-TILOWA in the D-S framework, the belief structure-2-TILOWA operator can be formed. Some of its main properties are studied. This model is applied in a selection of financial strategies. • The extension of AHP for group decision making model is given as the fourth contribution, notably, based on the inclusion of IOWA operators. Two-stage aggregation processes used in the AHP-GDM model are extended. Firstly, a generalization of weighted maximal entropy OWA under the IOWA operator is proposed as to aggregate the criteria. Further, the majority concept based on the IOWA and Minkowski OWA-based similarity measure is suggested to determine a consensus among experts. This model provides a variant of decision strategies for analyzing the individual and the majority of experts. The application in investment selection problem is presented to test the reliability of the model. • The fifth contribution is on the integration of heavy ordered weighted geometric (HOWG) aggregation operators in AHP-GDM model. In the sense of heavy OWA operator (HOWA), the heavy weighted geometric (HWG) and HOWG are introduced as extensions of the normal weighted geometric mean (WG) and the OWG by relaxing the constraints on the associated weighting vector. These HWG and HOWG operators then are utilized in the aggregation process of AHP-GDM, specifically on the aggregation of individual judgments procedure. The main advantage of the model, besides the complete overlapping of information such in classical methods, is that it can also accommodate partial and non-overlapping information in the formulation. An investment selection problem is applied to demonstrate the model. • The extension of TOPSIS for group decision making model by the inclusion of majority concept may be considered as the sixth contribution. The majority concept is derived based on the induced generalized OWA (IGOWA) operators. Two fusion schemes in TOPSIS model are designed. First, an external fusion scheme to aggregate the experts’ judgments with respect to the concept of majority opinion on each criterion is suggested. Then, an internal fusion scheme of ideal and anti-ideal solutions that represents the majority of experts is proposed using the Minkowski OWA distance measures. The comparison of the proposed model with some other TOPSIS models with respect to distance measures is presented. Here, a general case of selection problem is presented, specifically on the human resource selection problem. • Finally, the group decision making model based on conflicting bifuzzy sets (CBFS) is proposed. Precisely, the subjective judgments of experts, mainly from positive and negative aspects are considered simultaneously in the analysis. Moreover, the weighting method for the attribute (or sub-attribute) is subject to the integration of subjective and objective weights. The synthesis of CBFS in the model is naturally done by extending the fuzzy evaluation in parallel with the intuitionistic fuzzy set. A new technique to compute the similarity measure is proposed, in which, being the degree of agreement between the experts. The model then is applied in the case study of flood control project selection problem.

To sum up, the presented thesis dealt with the extension of multi-criteria decision analysis models for the financial selection problems (as a specific scope) and also the general selection problems with the inclusion of attitudinal character, majority concept and fuzzy set theory. In particular, the group decision making model, Dempster-Shafer belief structure, AHP and TOPSIS are proposed to overcome the shortcoming of the existing models, i.e., related to the financial decision analysis. The applicability and robustness of the developed models have been demonstrated and some sensitivity analyses are also provided. The main advantages of the proposed models are to provide a more general and flexible models for a wider analysis of the decision problems.

La tesis, a través del análisis y desarrollo del Análisis de decisiones, Teorías de incertidumbre y Operadores de agregación, busca contribuir al estado del arte y nuevas propuesta de las necesidades y demandas que los decisores, responsables o e inversores financieros se encuentran por la creciente complejidad de sus análisis y estrategias, sobre todo en los procesos de selección o en los problemas de decisión.

Así, el objetivo principal de esta tesis es el estudio de los problemas de selección financiera desde la perspectiva del análisis de decisiones, las teorías de la incertidumbre y los operadores de agregación. En concreto, se estudian los problemas de decisión en virtud de un conjunto finito de alternativas (caso discreto) y de factores multidimensionales.

En el trabajo se desarrolla una extensión de los modelos de análisis de decisiones multicriterio y multiexperto que se utilizan en la resolución de los problemas de selección financiera (como ámbito específico), pero también en los problemas de selección generales, con la inclusión del carácter actitudinal, el concepto de mayoría y la teoría de los conjuntos borrosos. En particular, el énfasis se sitúa en los modelos de toma de decisiones en grupo y en la estructura de creencias Dempster-Shafer (D-S), el proceso analítico jerárquico (AHP) i la técnica de orden de preferencia por similitud con la solución ideal (TOPSIS). Además, se aplican las teorías de incertidumbre basadas en conjuntos borrosos y de probabilidades imprecisas juntamente con la fusión de la información basada en operadores OWA. También se destaca las preferencias cuantitativas y cualitativas, las estrategias de decisión basadas en el carácter actitudinal de los decisores, y el concepto de mayoría en el consenso grupal, de forma que se propone el desarrollo de operadores OWA, la generalización de los modelos AHP y TOPSIS, juntamente con el modelo de toma de decisiones grupal y la estructura de creencias Dempster-Shafer, con el fin de superar las deficiencias de los modelos existentes en relación con el análisis de decisiones financieras.

En particular, la investigación realizada se puede sintetizar en siete aportaciones específicas al state-of-the-art del Análisis de decisiones y los operadores de agregación, con aplicaciones en diferentes problemas de decisión financiera:

1. Operadores de agregación basados en los OWA en los modelos de decisión Multiexpertos y Multicriterio. 2. Operadores ponderados SAM-OWA y su aplicación en modelos GDM con operadores lingüísticos. 3. Modelos GDM con operadores lingüísticos adaptados a la teoría de Dempster-Shafer con la aplicación de operadores de agregación inducidos lingüísticos. 4. Generalización del modelo AHP para decisiones grupales usando operadores OWA inducidos. 5. Introducción de operadores OWA geométricos y pesados en los modelos GDM y AHP. 6. Ampliación de los modelos TOPSIS con operadores de agregación basados en los OWA. 7. Desarrollo y aplicación del Conflicting bifuzzy a modelos de decisión MAGDM

En la tesis se demuestra la aplicabilidad y la robustez de los modelos desarrollados, tanto con un esquema de agregación de expertos clásicos como con un esquema alternativo que separa por criterios de decisión. Las principales ventajas de los modelos propuestos son que se tratan modelos más generales y flexibles para un análisis más amplio de los problemas de decisión, en particular de los de selección financiera, que incorporen diversos criterios, expertos y componentes de incertidumbre y lingüísticos.