Partial Preorder Research Articles

Selecting accepted assertions in partially ordered inconsistent DL-Lite knowledge bases

The problem of inconsistency handling in knowledge bases has received considerable research attention in several settings and has met many applications. Inconsistency usually stems from the fact that the information available for reasoning, query answering and decision-making tasks is obtained from different, scattered and potentially unreliable information sources. Performing these tasks is often computationally expensive, especially for knowledge bases in which the data is endowed with preferences and/or uncertainty. In the context of formal ontologies, enriching the assertions (i.e. the data pieces) with conceptual knowledge makes the query answering task harder computationally. Tractable results have been successfully established for the lightweight languages of the DL-Lite family. In this paper, we present two tractable methods for handling inconsistency in partially ordered lightweight ontologies. In essence, both methods are inspired from the well-known Intersection of ABox Repair (IAR) semantics, which proceeds by replacing the initial inconsistent knowledge base with the intersection of all of its maximally consistent sub-bases. The first method assumes a partial preorder over the knowledge base and identifies the accepted assertions, called ‘elected’, in polynomial time in data complexity. The second method captures uncertainty quantitatively in the framework of possibility theory and identifies the so-called ‘π-accepted’ assertions, also in polynomial time in data complexity. However, both characterisations require the conflict set to be readily available. This constitutes a serious limitation in frameworks and applications where the conflict set cannot be exhibited efficiently. In this paper, we propose a new and equivalent characterisation for each method. We show that the elected assertions and also the π-accepted assertions can be identified in polynomial time in the size of the dataset for lightweight ontologies.

Consensus modeling with interactive utility and partial preorder of decision-makers, involving fairness and tolerant behavior

Reaching consensus in group decision-making (GDM) is usually achieved by minimizing opinion deviation (distance) or maximizing the utility of the group in traditional mathematical modeling. While the former ignores the preferences (individual values) of decision-makers (DMs), the latter is most often based on the implicit assumption that DMs are mutually independent, which results in the loss of a large amount of decision-making information about the actual interactions between DMs. Moreover, numerous studies on GDM did not pay enough attention to individual participation and satisfaction in the decision-making process. To overcome these defects, in this paper, we construct a new kind of consensus models for group utility optimization. We employ 2-additive Choquet integral to aggregate the utilities of individual DMs interacting pairwise. Linear programming is used to learn fuzzy measures (weights) that are consistent with a partial preorder of DMs. Moreover, we describe the coordinator's fairness preference and tolerant behavior with the help of Gini coefficient and orness operator, thus exploring the impacts of coordinator's psychological behaviors on consensus reaching. The orness operator of the Choquet integral is generalized to a particularly useful 2-additive form with Möbius representation. Finally, a comparative and parametric analysis is performed to illustrate and validate the proposed models on a case study concerning the price negotiation of medical insurance drugs.

Possibility Extent and Possible Alternatives Preorder Type-2 Fuzzy Analytical Hierarchy Process (PE&PAP-AHP) to improve pharmaceutical R&D productivity

The major challenges of the pharmaceutical industry are optimizing global expenditure on medicines and maximizing the effectiveness of pharmaceutical products while minimizing the Research and Development (R&D) costs. The commercial success rate mainly causes the decline in R&D productivity across the pharmaceutical industry, the discovery of new drugs, compliance with regulations, and the R&D cycle time. The lack of comprehensive Multi-Criteria Decision Making (MCDM) methodologies leads to ineffective decisions for setting priorities to improve the productivity of pharmaceutical R&D. To increase the precision and accuracy required to improve productivity in Pharmaceutical R&D, in this study, Possibility Extent and Possible Alternatives Preorder Type-2 Fuzzy Analytical Hierarchy Process (PE&PAP-AHP) type-2 fuzzy logic MCDM methodology is developed. Specifically, the type-2 fuzzy geometric mean of PE&PAP-AHP methodology addresses the weakness of traditional methods and correctly reflects the subtle differences in evaluating decision-makers’ opinions. It also constructs fuzzy positive reciprocal matrices, acknowledging the indifference existing among alternatives. PE&PAP-AHP also makes it more accurate during the defuzzification stage of solving the multi-criteria decision-making problem than the type-1 fuzzy approach. It is achieved using two principles: (I) ranking the alternatives by a type-2 fuzzy partial preorder and (II) ranking the alternatives by a type-2 fuzzy total preorder. To compare the results with a type-1 fuzzy, based on the results of the PE&PAP-AHP methodology, the score of the best alternative is 5.2% more than the second-best alternative, whereas, per the type-1 fuzzy, the score of the best alternative is only 3.6% more than the second-best alternative.

ARAS-H: A ranking-based decision aiding method for hierarchically structured criteria

In most of real world problems, alternatives are evaluated according to a large set of criteria which confuses the Decision Maker (DM) in terms of allotting alternatives’ assessments. So, to reduce the complexity of the presented problem, it is recommended to organize the criteria into a hierarchy tree to decompose the main problem into sub-ones. Therefore, the DM gains detailed insight on each level of the hierarchy instead of focusing only on one level. In this context, we propose an extension of the Additive Ratio ASsessment (ARAS) method to the case of hierarchically structured criteria. The proposed approach is called Hierarchical Additive Ratio ASsessment (ARAS-H) method. A major advantage of the ARAS-H method is that it enables the DM to analyze the partial pre-orders (the rankings of the alternatives) at each node of the criteria tree i.e. according to each sub-criterion. The partial pre-orders present solutions of the problem with respect to each subset of criteria. In view of determining the criteria weights at each level of the hierarchy tree, we apply the AHP method. Finally, we apply the ARAS-H method on a case study related to tourism which aims to rank tourist destination websites brands in accordance with a four levels criteria hierarchy.

A note on the complexity of S4.2

is the modal logic of directed partial pre-orders and/or the modal logic of reflexive and transitive relational frames with a final cluster. It holds a distinguished position in philosophical logic, where it has been advocated as the ‘correct’ logic of knowledge; it has also found interesting applications in the temporal logic of relativistic spacetime and the metamathematics of forcing in set theory. The satisfiability problem for is -complete: this is a result established in an AiML 2004 paper of Shapirovsky [(2004). On PSPACE-decidability in transitive modal logic. In R. A. Schmidt, I. Pratt-Hartmann, M. Reynolds, & H. Wansing (Eds.), Advances in modal logic 5 (pp. 269–287). King's College Publications] where the complexity classification of emerges as a consequence of a very general method for constructing decision procedures for transitive modal logics. We provide here a ‘classical’ proof in the standard Halpern-Moses style of Halpern and Moses [(1992). A guide to completeness and complexity for modal logics of knowledge and belief. Artificial Intelligence, 54(2), 319–379]. With little extra work, it is shown that the -completeness result extends to , the multimodal version of . We prove that the -completeness characterisation of monomodal persists even if we restrict ourselves to fragments with bounded modal depth, but the problem is -complete when it is restricted to formulae with modal depth at most one. The complexity of satisfiability for the fragment of the language with a finite number of propositional variables (but unbounded modal depth) remains -hard. For a finite language and bounded modal depth, -satisfiability can be checked in linear time.

Partial classification in the belief function framework

Partial, or set-valued classification assigns instances to sets of classes, making it possible to reduce the probability of misclassification while still providing useful information. This paper reviews approaches to partial classification based on the Dempster–Shafer theory of belief functions. To define the utility of set-valued predictions, we propose to extend the utility matrix using an Ordered Weighted Average operator, allowing us to model the decision maker’s attitude towards imprecision using a single parameter. Various decision criteria are analyzed comprehensively. In particular, two main strategies are distinguished: partial classification based on complete preorders among partial assignments, and partial preorders among complete assignments. Experiments with UCI and simulated Gaussian data sets show the superiority of partial classification in terms of average utility, as compared to single-class assignment and classification with rejection.

A multiobjective evolutionary algorithm for deriving a final ranking from a medium-sized fuzzy outranking relation

The exploitation stage of a fuzzy outranking relation of the outranking methods aims to deal with the intransitivities in such a relation to consolidate and synthesise the decision maker's preferences between pairs of actions to obtain a ranking that is representative of his/her preferences over a given set of actions. In this paper, we propose this exploitation phase as a multiobjective optimisation problem and use a modified version of the multiobjective genetic algorithm (we refer to it as RP2-MOGA+H) to exploit a medium-sized fuzzy outranking relation to determining a partial pre-order of alternatives. To measure the performance of RP2-MOGA+H, we present an empirical study over a set of simulated ranking problems, which shows that RP2-MOGA+H can effectively exploit a medium-sized fuzzy outranking relation. Moreover, the RP2-MOGA+H outperforms other ranking procedures based on multiobjective evolutionary algorithms in the conducted experiments.

A Sensitivity Analysis of ELECTRE Based Stepwise Benchmarking for Policy: the Case of EU Digital Agenda

Research question: This paper introduces a new parameter for evaluating the sensitivity of development paths in ELECTRE (ELimination Et Choix Traduisant la Realité) based stepwise benchmarking model. Motivation: The aim of the research was to devise a 'path sensitivity coefficient' which measures the sensitivity of a development path depending on parameters set by decision-maker (DM). The need for such measure comes from the fact that the hierarchical pre-order of benchmarking units are affected by the DM's subjective decisions on the threshold for declaring outranking relations among them. This consequently may disrupt the sequence of intermediate benchmarks - transitional targets to be followed on the route to the ultimate policy goals. The proposed approach is an extension of a stepwise benchmarking procedure devised by Petrovic et al. (2014) which evaluates development paths based on their gradualism. Idea: In this paper, we propose to characterize development paths by considering both gradualism and sensitivity to outranking thresholds. The standpoint of our research is that in order to make a reliable final choice the DM should be aware that the most gradual path is not necessary the most stable one. Tools: The proposed 'path sensitivity coefficient' combines four level of uncertainty coming from two threshold values set by DM. The coefficient has a relative nature, it points to the “distance” from the theoretically ideal solution - a path fully indifferent to threshold values. An approach to merge coefficients associated with gradualism and stability is also introduced. Data: To illustrate the proposed approach we applied it in the field of digital transformation of EU countries. We exploited indicators and data from Digital Agenda Scoreboard regarding specific policy goals to be achieved until 2020. Findings: The findings imply that the decision on the most gradual development path should be reconsidered in regards to the influence of model parameters. The final choice depends whether DM’s prefers the smoothness of the path or the overall stability in terms of underlying partial preorder. Contribution: This work contributes to the existing ELECTRE-based stepwise benchmarking by reducing the uncertainty coming from subjective parameterization and allows the DM to be more confident in the final path selection.

Aggregation of diverse types of fuzzy orders for decision making problems

In this paper, conditions for n-argument functions to preserve diverse types of fuzzy orders during aggregation process are presented. Namely, the following fuzzy orders are studied: total preorder, total order, strict total order, partial preorder, partial order, strict partial order, interval order or semiorder and their preservation in aggregation process along with other connections between input fuzzy relations and the aggregated one are presented. Moreover, dependencies between diverse classes and also among the given class of fuzzy orders are presented. The considered fuzzy orders are compound properties depending in their definitions on arbitrary binary operations on [0, 1], including fuzzy conjunctions and disjunctions (with special cases of triangular norms and triangular conorms). Finally, an application of the obtained result is presented by the given algorithm.

Extensions of T 0-quasi-metrics

Let \({(X, m, \leq)}\) be a partially ordered metric space, that is, a metric space (X, m) equipped with a partial order \({\leq}\) on X. We say that a T 0-quasi-metric d on X is m-splitting provided that \({d\vee d^{-1}=m}\). Furthermore d is said to be \({(X, m, \leq)}\)-producing provided that d is m-splitting and the specialization partial preorder of d is equal to \({\leq}\). It is known and easy to see that if \({(X, m, \leq)}\) is a partially ordered metric space that is produced by a T 0-quasi-metric and \({\leq}\) is a total order, then there exists exactly one producing T 0-quasi-metric on X. We first will give an example that shows that a partially ordered metric space can be uniquely produced by a T 0-quasi-metric although \({\leq}\) is not total.

Adaptation strategies for water supply management in a drought prone Mediterranean river basin: Application of outranking method

The regional water allocation planning is one of those complex decision problems where holistic approach to water supply management considering different criteria would be valuable. However, multi-criteria decision making with diverse indicators measured on different scales and uncertainty levels is difficult to solve. Objective of this paper is to develop scenarios for the future imbalances in water supply and demand for a water stressed Mediterranean area of Northern Spain (Tarragona) and to test the applicability and suitability of an outranking method ELECTRE-III-H for evaluating sectoral water allocation policies. This study is focused on the use of alternative water supply scenarios to fulfil the demand of water from three major sectors: domestic, industrial and agricultural. A detail scenario planning for regional water demand and supply has been discussed. For each future scenario of climate change, the goal is to obtain a ranking of a set of possible actions with regards to different types of indicators (costs, water stress and environmental impact). The analytical method used is based on outranking models for decision aid with hierarchical structures of criteria and ranking alternatives using partial preorders based on pairwise preference relations. We compare several adaptation measures including alternative water sources (reclaimed water and desalination); inter basin water transfer and sectoral demand management coming from industry, agriculture and domestic sectors and tested the sustainability of management actions for different climate change scenarios. Results have shown use of alternative water resources as the most reliable alternative with medium reclaimed water reuse in industry and agriculture and low to medium use of desalination water in domestic and industrial sectors as the best alternative. The proposed method has several advantages such as the management of heterogeneous scales of measurement without requiring any artificial transformation and the management of uncertainty by means of comparisons at a qualitative level in terms of the decision maker preferences.

A decision support system to find the best water allocation strategies in a Mediterranean river basin in future scenarios of global change

Global change refers to climate changes, but also demographic, technological and economic changes. Predicted water scarcity will be critical in the coastal Mediterranean region, especially for provision to mid-sized and large-sized cities. This paper studies the case of the city of Tarragona, located at the Mediterranean area of north-eastern Spain (Catalonia). Several scenarios have been constructed to evaluate different sectorial water allocation policies to mitigate the water scarcity induced by global change. Future water supply and demand predictions have been made for three time spans. The decision support system presented is based on the outranking model, which constructs a partial pre-order based on pairwise preference relations among all the possible actions. The system analyses a hierarchical structure of criteria, including environmental and economic criteria. We compare several adaptation measures including alternative water sources, inter-basin water transfer and sectorial demand management coming from industry, agriculture and domestic sectors. Results indicate that the most appropriate water allocation strategies depend on the severity of the global change effects.

ELECTRE-III-H: An outranking-based decision aiding method for hierarchically structured criteria

This paper proposes a method for ranking a set of alternatives evaluated using multiple and conflicting criteria that are organised in a hierarchical structure. The hierarchy permits the decision maker to identify different intermediate sub-problems of interest. In that way, the analysis of the criteria is done according to the subsets defined in the hierarchy, and following the precedence relations in a bottom-up approach. To deal with this type of hierarchical structures, an extension of the ELECTRE-III method, called ELECTRE-III-H, is presented. As all methods of ELECTRE family, this one also relies on building a binary outranking relation on the set of alternatives on the basis of concordance and discordance tests. The exploitation of this outranking relation generates a partial pre-order, establishing an indifference, preference or incomparability relation for each pair of alternatives. The idea of a bottom-up application of the classical ELECTRE-III method to sub-problems involving subsets of criteria at the intermediate levels of the hierarchy is infeasible because the evaluations of alternatives by criteria aggregating some sub-criteria have the form of partial pre-orders, and not complete pre-orders. Thus, we propose a new procedure for building outranking relations from a set of partial pre-orders, as well as a mechanism for propagating these pre-orders upwards in the hierarchy. With this method, the decision maker is able to analyse the problem in a decomposed way and gain information from the outputs obtained at intermediate levels. In addition, ELECTRE-III-H gives the decision maker the possibility to define a local preference model at each node of the hierarchy, according to his objectives and sub-problem characteristics. We show an application of this method to rank websites of tourist destination brands evaluated using a hierarchy with 4 levels.

F-closed sets and coupled fixed point theorems without the mixed monotone property

Abstract In this paper we present the notion of F-closed set (which is weaker than the concept of F-invariant set introduced in Samet and Vetro (Ann. Funct. Anal. 1:46-56, 2010), and we prove some coupled fixed point theorems without the condition of mixed monotone property. Furthermore, we interpret the transitive property as a partial preorder and, then, some results in that paper and in Sintunavarat et al. (Fixed Point Theory Appl. 2012:170, 2012) can be reduced to the unidimensional case. MSC:46T99, 47H10, 47H09, 54H25.

A Genesis of Interval Orders and Semiorders: Transitive NaP-preferences

A NaP-preference (necessary and possible preference) on a set A is a pair \({\left(\succsim^{^{_N}}\!,\,\succsim^{^{_P}}\!\right)}\) of binary relations on A such that its necessary component \({\succsim^{^{_N}} \!\!}\) is a partial preorder, its possible component \({\succsim^{^{_P}} \!\!}\) is a completion of \({\succsim^{^{_N}} \!\!}\), and the two components jointly satisfy natural forms of mixed completeness and mixed transitivity. We study additional mixed transitivity properties of a NaP-preference \({\left(\succsim^{^{_N}}\!,\,\succsim^{^{_P}}\!\right)}\), which culminate in the full transitivity of its possible component \({\succsim^{^{_P}} \!\!}\). Interval orders and semiorders are strictly related to these properties, since they are the possible components of suitably transitive NaP-preferences. Further, we introduce strong versions of interval orders and semiorders, which are characterized by enhanced forms of mixed transitivity, and use a geometric approach to compare them to other well known preference relations.

Necessary and possible preference structures

A classical approach to model a preference on a set A of alternatives uses a reflexive, transitive and complete binary relation, i.e. a total preorder. Since the axioms of a total preorder do not usually hold in many applications, preferences are often modeled by means of weaker binary relations, dropping either completeness (e.g. partial preorders) or transitivity (e.g. interval orders and semiorders). We introduce an alternative approach to preference modeling, which uses two binary relations–the necessary preference ≿N and the possible preference ≿P–to fulfill completeness and transitivity in a mixed form. Formally, a NaP-preference (necessary and possible preference) on A is a pair (≿N,≿P) such that ≿N is a partial preorder on A and ≿P is an extension of ≿N satisfying mixed properties of transitivity and completeness. We characterize a NaP-preference (≿N,≿P) by the existence of a nonempty set R of total preorders such that ⋂R=≿N and ⋃R=≿P. In order to analyze the representability of NaP-preferences via families of utility functions, we generalize the notion of a multi-utility representation of a partial preorder by that of a modal utility representation of a pair of binary relations. Further, we give a dynamic view of the family of all NaP-preferences on a fixed set A by endowing it with a relation of partial order, which is defined according to the stability of the information represented by each NaP-preference.

Textbook Selection Using Fuzzy PROMETHEE II Method

This study proposes the Fuzzy-PROMETHEE method for the textbook selection. The triangular fuzzy number and ranking method are integrated into the PROMETHEE II method. The proposed method can handle the uncertainty in the decision making. An example demonstrates the usability and validity of the proposed method. As the transparent book selection policy is promoted in many institutions, this method can be realized as a platform that receives input, feedback and comments from various participating users. Index Terms—Decision making, Fuzzy PROMETHEE, textbook selection. I. I NTRODUCTION Textbook serves as a tool, guidebook, reference and tutor in many high institutions (1). Lecturers throughout the world depend heavily on the content of textbooks for module delivery (2). Therefore, textbook selection has been one of the important tasks for the faculty/school in high institutions. Research has been carried out to evaluate different textbooks at some institutions to assist lecturers in selecting a proper textbook (3). However, this textbook evaluation is not a common practice in tertiary institutions. Hence, there is no surprise that there exists no proper process when acquiring or readopting a textbook in relevant modules. There is no indication of patterns on how textbook selection and the exact role of textbooks in classroom. However, there is one thing that is common - a majority of instructors prefer to have a textbook that would best meet their students' needs and become a good resource for class activities such as case study analysis, problem discussion and tutorial. We realize an efficient textbook selection with the assistant of the Fuzzy PROMETHEE method. This method is described below. PROMETHEE (Preference Ranking Organization METHod for Enriching Evaluations) was firstly developed by Brans in 1982 at the conference (4), and then further developed as a family (5). PROMETHEE I deals with a partial preorder, PROMETHEE II deals with a complete preorder, PROMETHEE III deals with an interval order emphasizing indifference, PROMETHEE IV deals with continuous set of possible alternatives, PROMETHEE V supports the optimization under constraints and PROMETHEE VI is a representation of the human brain. (6) reviewed a number of PROMETHEE's applications

Selection of a representative value function in robust multiple criteria ranking and choice

We introduce the concept of a representative value function in robust ordinal regression applied to multiple criteria ranking and choice problems. The proposed method can be seen as a new interactive UTA-like procedure, which extends the UTA GMS and GRIP methods. The preference information supplied by the decision maker (DM) is composed of a partial preorder and intensities of preference on a subset of reference alternatives. Robust ordinal regression builds a set of general additive value functions which are compatible with the preference information, and returns two binary preference relations: necessary and possible. They identify recommendations which are compatible with all or at least one compatible value function, respectively. In this paper, we propose a general framework for selection of a representative value function from among the set of compatibles ones. There are a few targets which build on results of robust ordinal regression, and could be attained by a representative value function. In general, according to the interactively elicited preferences of the DM, the representative value function may emphasize the advantage of some alternatives over the others when all compatible value functions acknowledge this advantage, or reduce the ambiguity in the advantage of some alternatives over the others when some compatible value functions acknowledge an advantage and other ones acknowledge a disadvantage. The basic procedure is refined by few extensions. They enable emphasizing the advantage of alternatives that could be considered as potential best options, accounting for intensities of preference, or obtaining a desired type of the marginal value functions.

Fuzzy UTASTAR: A method for discovering utility functions from fuzzy data

We propose Fuzzy UTASTAR, a method for inferring fuzzy utility functions from a partial preorder of options evaluated on multiple criteria. It is an extension of the well-known UTASTAR method capable to handle both ordinary (crisp) and fuzzy evaluation data. This property gives much flexibility to decision makers because the majority of real-life decision problems involve a considerable level of uncertainty that hinders them from assigning exact evaluations (scores) to options. In case all evaluation data are crisp the method behaves exactly as the original UTASTAR. The proposed method builds fuzzy additive value functions taking as input a partial preorder on a subset of the options, called reference set, along with their associated scores on the criteria. The resulting fuzzy utility functions can subsequently be used to estimate the (fuzzy) utility of each option, thus allowing their ranking, prioritization, selection or classification. The ranking of the options in partial preorder is as compatible as possible to the original one. The method is implemented into a decision support system and is applied to an example from the transportation domain. Results are found to be in concordance with those of the original method. To the best of our knowledge this is the first attempt to extend UTASTAR method to handle both crisp and fuzzy evaluation data.

A fuzzy AHP approach based on the concept of possibility extent

The main purpose of this paper is to develop a fuzzy AHP method for tackling the uncertainty and imprecision existing in multi-criteria decision process. The proposed method uses fuzzy pair-wise comparison judgments in place of exact numerical values of the comparison ratios. The geometric mean technique is used to integrate all decision-makers’ opinions and construct the fuzzy positive reciprocal matrices. The algebraic operations of triangular fuzzy numbers are utilized to calculate the fuzzy suitability indices of all alternatives. The extent analysis method is used to compute the degree of possibility of priority among fuzzy suitability indices. Besides, two principles are presented to solve the multi-criteria decision problem in a fuzzy decision environment. Principle I provides a partial preorder, and Principle II gives a total preorder on the set of the possible alternatives. Finally, a numerical example of selecting the company with optimal performance in performing customer relationship management is used to demonstrate the decision process of proposed method.