Transitive matrices, strict preference and intensity operators

Let X be a set of alternatives and a_{ij} a positive number expressing how much the alternative x_{i} is preferred to the alternative x_{j}. Under suitable hypothesis of no indifference and transitivity over the pairwise comparison matrix A= (a_{ij}), the alternatives can be ordered as a chain . Then a coherent priority vector is a vector giving a weighted ranking agreeing with the obtained chain and an intensity vector is a coherent priority vector encoding information about the intensities of the preferences. In the paper we look for operators F that, acting on the row vectors translate the matrix A in an intensity vector

A survey on pairwise comparison matrices over abelian linearly ordered groups

In this paper, we provide a survey of our results about the pairwise comparison matrices defined over abelian linearly ordered groups

A general unified framework for pairwise comparison matrices in multicriterial methods

In a Multicriteria Decision Making context, a pairwise comparison matrix $A=(a_{ij})$ is a helpful tool to determine the weighted ranking on a set $X$ of alternatives or criteria. The entry $a_{ij}$ of the matrix can assume different meanings: $a_{ij}$ can be a preference ratio (multiplicative case) or a preference difference (additive case) or $a_{ij}$ belongs to $[0,1]$ and measures the distance from the indifference that is expressed by 0.5 (fuzzy case). For the multiplicative case, a consistency index for the matrix $A$ has been provided by T.L. Saaty in terms of maximum eigenvalue. We consider pairwise comparison matrices over an abelian linearly ordered group and, in this way, we provide a general framework including the mentioned cases. By introducing a more general notion of metric, we provide a consistency index that has a natural meaning and it is easy to compute in the additive and multiplicative cases; in the other cases, it can be computed easily starting from a suitable additive or multiplicative matrix

Lexicographic refinements in possibilistic decision trees and finite-horizon Markov decision processes

Possibilistic decision theory has been proposed twenty years ago and has had several extensions since then. Even though ap-pealing for its ability to handle qualitative decision problems, possibilisticdecision theory suffers from an important drawback. Qualitative possibilistic utility criteria compare acts through min and max operators, which leads to a drowning effect. To over-come this lack of decision power of the theory, several refinements have been proposed. Lexicographic refinements are particularly appealing since they allow to benefit from the Expected Utility background, while remaining qualitative. This article aims at extend-ing lexicographic refinements to sequential decision problems i.e., to possibilistic decision trees and possibilistic Markov decision processes, when the horizon is finite. We present two criteria that refine qualitative possibilistic utilities and provide dynamic programming algorithms for calculating lexicographically optimal policies

A Compact Representation of Preferences in Multiple Criteria Optimization Problems

[EN] 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.Salas-Molina, F.; Pla Santamaría, D.; Garcia-Bernabeu, A.; Reig-Mullor, J. (2019). A Compact Representation of Preferences in Multiple Criteria Optimization Problems. Mathematics. 7(11):1-16. https://doi.org/10.3390/math7111092S116711Ahmadi, A., Ahmadi, M. R., & Nezhad, A. E. (2014). A Lexicographic Optimization and Augmented ϵ-constraint Technique for Short-term Environmental/Economic Combined Heat and Power Scheduling. Electric Power Components and Systems, 42(9), 945-958. doi:10.1080/15325008.2014.903542González-Arteaga, T., Alcantud, J. C. R., & de Andrés Calle, R. (2016). A new consensus ranking approach for correlated ordinal information based on Mahalanobis distance. Information Sciences, 372, 546-564. doi:10.1016/j.ins.2016.08.071Miettinen, K., & M�kel�, M. M. (2002). On scalarizing functions in multiobjective optimization. OR Spectrum, 24(2), 193-213. doi:10.1007/s00291-001-0092-9Ignizio, J. P. (1983). Generalized goal programming An overview. Computers & Operations Research, 10(4), 277-289. doi:10.1016/0305-0548(83)90003-5Sitorus, F., Cilliers, J. J., & Brito-Parada, P. R. (2019). Multi-criteria decision making for the choice problem in mining and mineral processing: Applications and trends. Expert Systems with Applications, 121, 393-417. doi:10.1016/j.eswa.2018.12.001Zyoud, S. H., & Fuchs-Hanusch, D. (2017). A bibliometric-based survey on AHP and TOPSIS techniques. Expert Systems with Applications, 78, 158-181. doi:10.1016/j.eswa.2017.02.016Erdoğan, M., & Kaya, İ. (2016). A combined fuzzy approach to determine the best region for a nuclear power plant in Turkey. Applied Soft Computing, 39, 84-93. doi:10.1016/j.asoc.2015.11.013Chen, Y., Liu, R., Barrett, D., Gao, L., Zhou, M., Renzullo, L., & Emelyanova, I. (2015). A spatial assessment framework for evaluating flood risk under extreme climates. Science of The Total Environment, 538, 512-523. doi:10.1016/j.scitotenv.2015.08.094Zammori, F. (2010). The analytic hierarchy and network processes: Applications to the US presidential election and to the market share of ski equipment in Italy. Applied Soft Computing, 10(4), 1001-1012. doi:10.1016/j.asoc.2009.07.013Carter, C. R., & Rogers, D. S. (2008). A framework of sustainable supply chain management: moving toward new theory. International Journal of Physical Distribution & Logistics Management, 38(5), 360-387. doi:10.1108/09600030810882816Ignizio, J. P. (1976). An Approach to the Capital Budgeting Problem with Multiple Objectives. The Engineering Economist, 21(4), 259-272. doi:10.1080/00137917608902798Lonergan, S. C., & Cocklin, C. (1988). The use of lexicographic goal programming in economic/ecolocical conflict analysis. Socio-Economic Planning Sciences, 22(2), 83-92. doi:10.1016/0038-0121(88)90020-1González-Pachón, J., & Romero, C. (2014). Properties underlying a preference aggregator based on satisficing logic. International Transactions in Operational Research, 22(2), 205-215. doi:10.1111/itor.1211

Rationalizing Choice with Multi-Self Models

This paper studies a class of multi-self decision-making models proposed in economics, psychology, and marketing. In this class, choices arise from the set-dependent aggregation of a collection of utility functions, where the aggregation procedure satisfies some simple properties. We propose a method for characterizing the extent of irrationality in a choice behavior, and use this measure to provide a lower bound on the set of choice behaviors that can be rationalized with n utility functions. Under an additional assumption (scale-invariance), we show that generically at most five "reasons" are needed for every "mistake."Multi-self models, Index of irrationality, IIA violations, Rationalizability

Fuzzy measures and integrals in MCDA

This chapter aims at a unified presentation of various methods of MCDA based onfuzzy measures (capacity) and fuzzy integrals, essentially the Choquet andSugeno integral. A first section sets the position of the problem ofmulticriteria decision making, and describes the various possible scales ofmeasurement (difference, ratio, and ordinal). Then a whole section is devotedto each case in detail: after introducing necessary concepts, the methodologyis described, and the problem of the practical identification of fuzzy measuresis given. The important concept of interaction between criteria, central inthis chapter, is explained in details. It is shown how it leads to k-additivefuzzy measures. The case of bipolar scales leads to thegeneral model based on bi-capacities, encompassing usual models based oncapacities. A general definition of interaction for bipolar scales isintroduced. The case of ordinal scales leads to the use of Sugeno integral, andits symmetrized version when one considers symmetric ordinal scales. Apractical methodology for the identification of fuzzy measures in this contextis given. Lastly, we give a short description of some practical applications.Choquet integral; fuzzy measure; interaction; bi-capacities

A decade of application of the Choquet and Sugeno integrals in multi-criteria decision aid

The main advances regarding the use of the Choquet and Sugeno integrals in multi-criteria decision aid over the last decade are reviewed. They concern mainly a bipolar extension of both the Choquet integral and the Sugeno integral, interesting particular submodels, new learning techniques, a better interpretation of the models and a better use of the Choquet integral in multi-criteria decision aid. Parallel to these theoretical works, the Choquet integral has been applied to many new fields, and several softwares and libraries dedicated to this model have been developed.Choquet integral, Sugeno integral, capacity, bipolarity, preferences

The Basic Principles of Uncertain Information Fusion. An organized review of merging rules in different representation frameworks

We propose and advocate basic principles for the fusion of incomplete or uncertain information items, that should apply regardless of the formalism adopted for representing pieces of information coming from several sources. This formalism can be based on sets, logic, partial orders, possibility theory, belief functions or imprecise probabilities. We propose a general notion of information item representing incomplete or uncertain information about the values of an entity of interest. It is supposed to rank such values in terms of relative plausibility, and explicitly point out impossible values. Basic issues affecting the results of the fusion process, such as relative information content and consistency of information items, as well as their mutual consistency, are discussed. For each representation setting, we present fusion rules that obey our principles, and compare them to postulates specific to the representation proposed in the past. In the crudest (Boolean) representation setting (using a set of possible values), we show that the understanding of the set in terms of most plausible values, or in terms of non-impossible ones matters for choosing a relevant fusion rule. Especially, in the latter case our principles justify the method of maximal consistent subsets, while the former is related to the fusion of logical bases. Then we consider several formal settings for incomplete or uncertain information items, where our postulates are instantiated: plausibility orderings, qualitative and quantitative possibility distributions, belief functions and convex sets of probabilities. The aim of this paper is to provide a unified picture of fusion rules across various uncertainty representation settings