In this paper, we motivate and introduce the concept of N-soft set as an extended soft set model. Some useful algebraic definitions and properties are given. We cite real examples that prove that N-soft sets are a cogent model for binary and non-binary evaluations in numerous kinds of decision making problems. Finally, we propose decision making procedures for N-soft sets.

N-soft sets and their decision making algorithms

In this paper, we contribute to a recent and successful modelization of uncertainty, which the practitioner often encounters in the formulation of multicriteria group decision making problems. To be precise, in order to approach the uncertainty issue we introduce a novel type of soft rough covering by means of soft neighborhoods, and then we use it to improve decision making in a multicriteria group environment. Our research method is as follows. Firstly we introduce the soft covering upper and lower approximation operators of soft rough coverings. Then its relationships with well-established types of soft rough coverings are analyzed. Secondly, we define and investigate the measure degree of our novel soft rough covering. With this tool we produce a new class of soft rough sets. Finally, we propose an application of such soft rough covering model to multicriteria group decision making by means of an algorithmic solution. A fully developed example supports the implementability of this decision making method.

A novel type of soft rough covering and its application to multicriteria group decision making

SAT: Past and Future.- Encodings of Problems in Effectively Propositional Logic.- Efficient Circuit to CNF Conversion.- Mapping CSP into Many-Valued SAT.- Circuit Based Encoding of CNF Formula.- Breaking Symmetries in SAT Matrix Models.- Partial Max-SAT Solvers with Clause Learning.- MiniMaxSat: A New Weighted Max-SAT Solver.- Solving Multi-objective Pseudo-Boolean Problems.- Improved Lower Bounds for Tree-Like Resolution over Linear Inequalities.- Horn Upper Bounds and Renaming.- Matched Formulas and Backdoor Sets.- Short XORs for Model Counting: From Theory to Practice.- Variable Dependency in Local Search: Prevention Is Better Than Cure.- Combining Adaptive Noise and Look-Ahead in Local Search for SAT.- From Idempotent Generalized Boolean Assignments to Multi-bit Search.- Satisfiability with Exponential Families.- Formalizing Dangerous SAT Encodings.- Algorithms for Variable-Weighted 2-SAT and Dual Problems.- On the Boolean Connectivity Problem for Horn Relations.- A First Step Towards a Unified Proof Checker for QBF.- Dynamically Partitioning for Solving QBF.- Backdoor Sets of Quantified Boolean Formulas.- Bounded Universal Expansion for Preprocessing QBF.- Effective Incorporation of Double Look-Ahead Procedures.- Applying Logic Synthesis for Speeding Up SAT.- Towards a Better Understanding of the Functionality of a Conflict-Driven SAT Solver.- A Lightweight Component Caching Scheme for Satisfiability Solvers.- Minimum 2CNF Resolution Refutations in Polynomial Time.- Polynomial Time SAT Decision for Complementation-Invariant Clause-Sets, and Sign-non-Singular Matrices.- Verifying Propositional Unsatisfiability: Pitfalls to Avoid.- A Simple and Flexible Way of Computing Small Unsatisfiable Cores in SAT Modulo Theories.- SAT Solving for Termination Analysis with Polynomial Interpretations.- Fault Localization and Correction with QBF.- Sensor Deployment for Failure Diagnosis in Networked Aerial Robots: A Satisfiability-Based Approach.- Inversion Attacks on Secure Hash Functions Using sat Solvers.

Theory and applications of satisfiability testing - SAT 2007 : 10th International Conference, Lisbon, Portugal, May 28-31, 2007 : proceedings

A novel algorithm for fuzzy soft set based decision making from multiobserver input parameter data set

Stochastic Constraint Satisfaction Problems (SCSPs) are a powerful modeling framework for problems under uncertainty. To solve them is a P-Space task. The only solution approach to date compiles down SCSPs into classical CSPs. This allows the reuse of classical constraint solvers to solve SCSPs, but at the cost of increased space requirements and weak constraint propagation. This paper tries to overcome some of these drawbacks by automatically synthesizing filtering algorithms for global chance-constraints. These filtering algorithms are parameterized by propagators for the deterministic version of the chance-constraints. This approach allows the reuse of existing propagators in current constraint solvers and it enhances constraint propagation. Experiments show the benefits of this novel approach.

Synthesizing Filtering Algorithms for Global Chance-Constraints

Purpose: This paper aims to develop and compare several elicitation criterions for decision making of incomplete soft sets which are generated by restricted intersection. Design/methodology/approach: One time elicitation process is divided into two steps. Using the greedy idea four criterions for elicitation of objects are built based on maximax, maximin, minimax regret and combination of expected choice values and elicitation times. Then these initial unknown values which produce incomplete values together with known information are in priority. Findings: Fast methods for computing possibly and necessarily optimal solutions before or in the elicitation process are invented. As far as the sizes of soft sets used in the simulation experiments, it is found statistically that we should choose the criterion based on the combination of expected choice value and expected elicitation times in the first step of one time elicitation. Practical implications: The developed methods can be used for decision making of incomplete 0-1 information systems, which are generated by the conjunction of two experts' incomplete 0-1 evaluation results. Whenever the available information is not enough for choosing a necessarily optimal solution, the elicitation algorithms can help elicitate as few unknown values as possible until an optimal result is found. An elicitation system is made to show that our elicitation methods can potentially be embedded in recommender or decision support systems. Originality/value: The elicitation problems are proposed for decision making of operation-generated soft sets by extracting from some practical problems. The concept of expected elicitation times of objects is defined and used for developing one type of elicitation strategy.

Elicitation criterions for restricted intersection of two incomplete soft sets

Soft set theory is a newly emerging mathematical approach to vagueness. However, it seems that there is no existing research devoted to the discussion of applying soft sets to approximate reasoning. This paper aims to initiate an approximate reasoning scheme based on soft set theory. We consider proposition logic in the framework of a given soft set. By taking parameters of the underlying soft set as atomic formulas, the concept of (well-formed) formulas over a soft set is defined in a natural way. The semantic meaning of formulas is then given by taking objects of the underlying soft set as valuation functions. We propose the notion of decision soft sets and define decision rules as implicative type of formulas in decision soft sets. Motivated by basic ideas from quantitative logic, we also introduce several measures and preorders to evaluate the soundness of formulas and decision rules in soft sets. Moreover, an interesting example is presented to illustrate all the new concepts and the basic ideas initiated here.

Soft Set Based Approximate Reasoning: A Quantitative Logic Approach

The concept of the truth degree of a formula is the crucial tool and the building block in quantitative logic. So how to compute the truth degree of a formula efficiently is a principal question in this subject. This paper aims at bringing an optimal method for doing this job. Firstly, characterizations of the concept of truth degree are made with concepts from soft constraint theory. Particularly, two soft constraint systems are proposed such that formulas can be taken as soft constraints over them. Then by exploiting the algebraic properties of both constraint systems and n-valued propositional systems, it is shown that different soft constraint systems plays different roles in the computation of truth degree of formulas. An optimal method named splitting algorithm for computing truth degrees of formulas is proposed.

A Splitting Algorithm Based on Soft Constraints for the Computation of Truth Degree in Quantitative Logic

A novel algorithm for all normal parameter reductions of a soft set based on object weighting and integer partition

Classical preference relation depicts the fact that alternative a is better than alternative $b_{,}$ and fuzzy preference relation describes the degree to which a is better than b. Furthermore, in order to describe in what aspects a is better than $b_{,}$ the concepts of set-valued binary relation, set-valued preference relation, set-valued weak or strict preference relation are proposed in this paper. Then the structure characterization of set-valued preference relation is presented. Next, a general method of inducing a set-valued preference relation from an ordered information system is developed. Particularly, for a three-valued ordered information system, the structural and quantitative properties of the induced set-valued preference relation matrix are discussed under different partial order relations. Finally, a numerical example is given to illustrate the application of related concepts and properties in data analysis, which shows the rationality and effectiveness of the proposed algorithm.

Set-valued Preference Relation and its Properties

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