Crisp Counterparts Research Articles

Solutions of Fractional Differential Inclusions and Stationary Points of Intuitionistic Fuzzy-Set-Valued Maps

One of the tools for building new fixed-point results is the use of symmetry in the distance functions. The symmetric property of metrics is particularly useful in constructing contractive inequalities for analyzing different models of practical consequences. A lot of important invariant point results of crisp mappings have been improved by using the symmetry of metrics. However, more than a handful of fixed-point theorems in symmetric spaces are yet to be investigated in fuzzy versions. In accordance with the aforementioned orientation, the idea of Presic-type intuitionistic fuzzy stationary point results is introduced in this study within a space endowed with a symmetrical structure. The stability of intuitionistic fuzzy fixed-point problems and the associated new concepts are proposed herein to complement their corresponding concepts related to multi-valued and single-valued mappings. In the instance where the intuitionistic fuzzy-set-valued map is reduced to its crisp counterparts, our results complement and generalize a few well-known fixed-point theorems with symmetric structure, including the main results of Banach, Ciric, Presic, Rhoades, and some others in the comparable literature. A significant number of consequences of our results in the set-up of fuzzy-set- and crisp-set-valued as well as point-to-point-valued mappings are emphasized and discussed. One of our findings is utilized to assess situations from the perspective of an application for the existence of solutions to non-convex fractional differential inclusions involving Caputo fractional derivatives with nonlocal boundary conditions. Some nontrivial examples are constructed to support the assertions and usability of our main ideas.

Lattice valued neutrosophic sets

In this paper, the authors introduce lattice valued neutrosophic sets and study their properties. A decomposition theorem for lattice valued neutrosophic sets is obtained. Lattice valued neutrosophic mappings are defined and verified that its properties are consistent with their crisp counterparts. Finally, a topology of lattice valued neutrosophic sets is introduced.

On optimistic, pessimistic and mixed approaches under different membership functions for fully intuitionistic fuzzy multiobjective nonlinear programming problems

The concept of measuring the membership degree along with a non-membership degree, gives rise to the intuitionistic fuzzy set theory. The present study formulates a nonlinear programming problem with multiple objectives, including all the parameters and decision variables as intuitionistic fuzzy numbers. The problem is further investigated subject to optimistic, pessimistic and mixed approaches under linear, exponential and hyperbolic membership functions. The article redefines pessimistic and mixed point of view to be in the true spirit compatible with the definition of an intuitionistic fuzzy number. Accuracy function is used to reduce the problem to an equivalent crisp multiobjective nonlinear programming problem and then optimal compromise solution is obtained under different approaches using various membership/non-membership functions. At appropriate places, theorems have also been proved to establish the equivalence between the original formulation and its crisp counterparts under each approach. Further, practical applications in production planning and transportation problem are illustrated to explain the optimistic, pessimistic and mixed approaches using the proposed algorithm and finally a comparison is also drawn.

Cooperative games with multiple attributes

ABSTRACTIn this paper, we introduce the notion of a cooperative game with multiple attributes where players can provide partial participations in multiple attributes and form coalitions. The power or influence of the players due to their multiple attributes is evaluated based on their memberships in the coalitions. Our game therefore, extends the notion of cooperative games with fuzzy coalitions. The Shapley function for this class of games is proposed as a rational and fair solution concept. Every fuzzy game stems out of a specific crisp game under the assumption that the players provide partial memberships in forming multiple coalitions simultaneously. We adopt similar techniques to obtain the cooperative games with multiple attributes from their crisp counterparts and subsequently determine their Shapley functions.

Interpreting the Fuzzy Semantics of Natural-Language Spatial Relation Terms with the Fuzzy Random Forest Algorithm

Naïve Geography, intelligent geographical information systems (GIS), and spatial data mining especially from social media all rely on natural-language spatial relations (NLSR) terms to incorporate commonsense spatial knowledge into conventional GIS and to enhance the semantic interoperability of spatial information in social media data. Yet, the inherent fuzziness of NLSR terms makes them challenging to interpret. This study proposes to interpret the fuzzy semantics of NLSR terms using the fuzzy random forest (FRF) algorithm. Based on a large number of fuzzy samples acquired by transforming a set of crisp samples with the random forest algorithm, two FRF models with different membership assembling strategies are trained to obtain the fuzzy interpretation of three line-region geometric representations using 69 NLSR terms. Experimental results demonstrate that the two FRF models achieve good accuracy in interpreting line-region geometric representations using fuzzy NLSR terms. In addition, fuzzy classification of FRF can interpret the fuzzy semantics of NLSR terms more fully than their crisp counterparts.

FUZZY NON-LINEAR OPTIMIZATION MODEL FOR PRODUCTION LINE BALANCING OF JADHAV INDUSTRIES USING GENETIC ALGORITHM

In a manufacturing industry production line balancing poses a significant challenge owing to the trade-off between machine idle time and Work-In-Process accumulation between different machines. Several models have been proposed to solve the problem thereby improving the line efficiency. The lowest common denominator in all such approaches is to attain an optimal level of service keeping the total cost associated with service cost and the waiting cost at its minimum. Most of the models proposed till date employ hard computing techniques which poses high mathematical complexity as the number of machines in the line increase. Hard computing techniques are tolerable to moderate sized production lines and break when the size of the line increases beyond the limits. To address this issue, several soft computing techniques have been devised in literature which are logical in nature in contrast to the mathematical nature of hard computing counter parts. Further, soft computing techniques have the power of reducing NP-Hard problems to be solvable in polynomial time. In the current paper, the authors have applied nonlinear fuzzy-GA optimization model for solving production line balancing problem of Jadhav Industries Pvt. Ltd, Kolhapur. The results obtained are compared with their crisp counterparts.

Parallel Interval Type-2 Subsethood Neural Fuzzy Inference System

Neuro-fuzzy models are being increasingly employed in the domains like weather forecasting, stock market prediction, computational finance, control, planning, physics, economics and management, to name a few. These models enable one to predict system behavior in a more human-like manner than their crisp counterparts. In the present work, an interval type-2 neuro-fuzzy evolutionary subsethood based model has been proposed for its use in finding solutions to some well-known problems reported in the literature such as regression analysis, data mining and research problems relevant to expert and intelligent systems. A novel subsethood based interval type-2 fuzzy inference system, named as Interval Type-2 Subsethood Neural Fuzzy Inference System (IT2SuNFIS) is proposed in the present work. Mathematical modeling and empirical studies clearly bring out the efficacy of this model in a wide variety of practical problems such as Truck backer-upper control, Mackey–Glass time-series prediction, Narazaki–Ralescu and bell function approximation. The simulation results demonstrate intelligent decision making capability of the proposed system based on the available data. The major contribution of this work lies in identifying subsethood as an efficient measure for finding correlation in interval type-2 fuzzy sets and applying this concept to a wide variety of problems pertaining to expert and intelligent systems. Subsethood between two type-2 fuzzy sets is different from the commonly used sup-star methods. In the proposed model, this measure assists in providing better contrast between dissimilar objects. This method, coupled with the uncertainty handling capacity of type-2 fuzzy logic system, results in better trainability and improved performance of the system. The integration of subsethood with type-2 fuzzy logic system is a novel idea with several advantages, which is reported for the first time in this paper.

Rough multiple objective programming

In this paper, we focused on characterizing and solving the multiple objective programming problems which have some imprecision of a vague nature in their formulation. The Rough Set Theory is only used in modeling the vague data in such problems, and our contribution in data mining process is confined only in the “post-processing stage”. These new problems are called rough multiple objective programming (RMOP) problems and classified into three classes according to the place of the roughness in the problem. Also, new concepts and theorems are introduced on the lines of their crisp counterparts; e.g. rough complete solution, rough efficient set, rough weak efficient set, rough Pareto front, weighted sum problem, etc. To avoid the prolongation of this paper, only the 1st-class, where the decision set is a rough set and all the objectives are crisp functions, is investigated and discussed in details. Furthermore, a flowchart for solving the 1st-class RMOP problems is presented.

Spatial Plateau Algebra for implementing fuzzy spatial objects in databases and GIS: Spatial plateau data types and operations

Many geographical applications have to deal with spatial objects that reveal an intrinsically vague or fuzzy nature. A spatial object is fuzzy if locations exist that cannot be assigned completely to the object or to its complement. Spatial database systems and Geographical Information Systems (GIS) are currently unable to cope with this kind of data. Based on an available abstract data model of fuzzy spatial data types for fuzzy points, fuzzy lines, and fuzzy regions that leverages fuzzy set theory and fuzzy point set topology, this article proposes a Spatial Plateau Algebra that provides spatial plateau data types as an implementation of fuzzy spatial data types. Each spatial plateau object consists of a finite number of crisp counterparts that are all adjacent or disjoint to each other, are associated with different membership values, and hence form different plateaus. The formal framework and the implementation are based on well known, exact models and implementations of crisp spatial data types. Spatial plateau operations as geometric operations on spatial plateau objects are expressed as a combination of geometric operations on the underlying crisp spatial objects. This article offers a conceptually clean foundation for implementing a database extension for fuzzy spatial objects and their operations, and demonstrates the embedding of these new data types as attribute data types in a database schema as well as the incorporation of fuzzy spatial operations into a database query language.

Context-dependent ‘near’ and ‘far’ in spatial databases via supervaluation

Often we are interested to know what is ‘near’ and what is ‘far’ in spatial databases. For instance we would like a hotel ‘near’ the beach, but ‘far’ from the highway. It is not always obvious how to answer such nearness questions by reducing them to their crisp counterparts ‘nearer’ or ‘nearest’. Thus we confront the vague and context-dependent relation of near (and far). Our approach follows a supervaluation tradition with a limited representation of context. The method is tractable, learnable and directly suitable for use in natural language interfaces to databases. The approach is based on logic programs supervaluated over a set of context-dependent threshold parameters. Given a set of rules with such unconstrained threshold parameters, a fixed parameter tractable algorithm finds a setting of parameters that are consistent with a training corpus of context-dependent descriptions of ‘near’ and ‘far’ in scenes. The results of this algorithm may then be compiled into view definitions which are accessed in real-time by natural language interfaces employing normal, non-exotic query answering mechanisms.

On a non-nested level-based representation of fuzziness

In this paper we describe a non-nested level-based representation of fuzziness, closely related to some existing models and concepts in the literature. Our objective is to motivate the use of this non-nested model by describing its theoretical possibilities, and illustrating them with some existing applications. From a theoretical point of view, we discuss the semantics of the representation, which goes beyond and has as a particular case fuzzy sets as represented by a collection of α - cuts . In addition, the proposed operations on level-based representations, contrary to those of existing fuzzy set theories, satisfy all the properties of Boolean logic. We discuss the contributions of the representation and operation by levels to practical applications, in particular for extending crisp notions to the fuzzy case. In this respect, an important contribution of the proposal is that fuzzy mathematical objects (not only sets and the corresponding predicates) and operations are uniquely and easily defined as extensions of their crisp counterparts. In order to illustrate this claim, we recall level representations of quantities (gradual numbers) and their complementarity to fuzzy intervals (often inappropriately called fuzzy numbers).

Mathematical programming in rough environment

Classical mathematics is usually crisp while most real-life problems are not; therefore, classical mathematics is usually not suitable for dealing with real-life problems. In this article, we present a systematic and focused study of the application of rough sets (Z. Pawlak, Rough sets, In. J. Comput. Informa. Sci. 11 (1982), pp. 341–356.) to a basic area of decision theory, namely ‘mathematical programming’. This new framework concerns mathematical programming in a rough environment and is called ‘rough programming’ (L. Baoding, Theory and Practice of Uncertain Programming, 1st ed., Physica-Verlag, Heidelberg, 2002; E.A. Youness, Characterizing solutions of rough programming problems, Eut. J. Oper. Res. 168 (2006), pp. 1019–1029). It implies the existence of the roughness in any part of the problem as a result of the leakage, uncertainty and vagueness in the available information. We classify rough programming problems into three classes according to the place of the roughness. In rough programming, wherever roughness exists, new concepts like rough feasibility and rough optimality come to the front of our interest. The study of convexity for rough programming problems plays a key role in understanding global optimality in a rough environment. For this, a theoretical framework of convexity in rough programming and conceptualization of the solution is created on the lines of their crisp counterparts.

Efficient Algorithms for Fuzzy Qualitative Temporal Reasoning

Fuzzy qualitative temporal relations have been proposed to reason about events whose temporal boundaries are ill defined. Although the corresponding reasoning tasks are in the same complexity class as their crisp counterparts, in practice, the scalability of fuzzy temporal reasoners may be insufficient for applications that require a high expressivity and deal with a large number of events. On the other hand, transitivity rules can be used to make sound but incomplete inferences in polynomial time, utilizing a variant of Allen's path-consistency algorithm. The aim of this paper is to investigate how this polynomial time algorithm can be improved without altering its time complexity. To this end, we establish a characterization of 2-consistency of fuzzy temporal relations and provide transitivity rules that are significantly stronger than those resulting from straightforwardly generalizing transitivity rules for crisp temporal relations. We furthermore provide experimental evidence for the effectiveness of our improved algorithm.

P.078 Pre-emptive analgesic efficacy of different analgesics after third molar surgery

The concept of measuring the membership degree along with a non-membership degree, gives rise to the intuitionistic fuzzy set theory. The present study formulates a nonlinear programming problem with multiple objectives, including all the parameters and decision variables as intuitionistic fuzzy numbers. The problem is further investigated subject to optimistic, pessimistic and mixed approaches under linear, exponential and hyperbolic membership functions. The article redefines pessimistic and mixed point of view to be in the true spirit compatible with the definition of an intuitionistic fuzzy number. Accuracy function is used to reduce the problem to an equivalent crisp multiobjective nonlinear programming problem and then optimal compromise solution is obtained under different approaches using various membership/non-membership functions. At appropriate places, theorems have also been proved to establish the equivalence between the original formulation and its crisp counterparts under each approach. Further, practical applications in production planning and transportation problem are illustrated to explain the optimistic, pessimistic and mixed approaches using the proposed algorithm and finally a comparison is also drawn.

The fuzzy transformation and its applications in image processing

The spatial and rank (SR) orderings of samples play a critical role in most signal processing algorithms. The recently introduced fuzzy ordering theory generalizes traditional, or crisp, SR ordering concepts and defines the fuzzy (spatial) samples, fuzzy order statistics, fuzzy spatial indexes, and fuzzy ranks. Here, we introduce a more general concept, the fuzzy transformation (FZT), which refers to the mapping of the crisp samples, order statistics, and SR ordering indexes to their fuzzy counterparts. We establish the element invariant and order invariant properties of the FZT. These properties indicate that fuzzy spatial samples and fuzzy order statistics constitute the same set and, under commonly satisfied membership function conditions, the sample rank order is preserved by the FZT. The FZT also possesses clustering and symmetry properties, which are established through analysis of the distributions and expectations of fuzzy samples and fuzzy order statistics. These properties indicate that the FZT incorporates sample diversity into the ordering operation, which can be utilized in the generalization of conventional filters. Here, we establish the fuzzy weighted median (FWM), fuzzy lower-upper-middle (FLUM), and fuzzy identity filters as generalizations of their crisp counterparts. The advantage of the fuzzy generalizations is illustrated in the applications of DCT coded image deblocking, impulse removal, and noisy image sharpening.

Fuzzy Markov Model for Determination of Fuzzy State Probabilities of Generating Units Including the Effect of Maintenance Scheduling

This paper presents a fuzzy Markov model to efficiently incorporate the influences of maintenance scheduling as well as aging of generating units on failure-repair cycle for computation of state probabilities. The proposed model, different from conventional models that are based on a probabilistic approach, employs a fuzzy set concept together with a probabilistic Markov model. This model incorporates fuzzy mean time to failure (FMTTF) and fuzzy mean time to repair (FMTTR) instead of crisp (expected) values for capturing the generating unit uncertainties more effectively through expert evaluation. The fuzzy state probabilities are computed from FMTTF and FMTTR values using fuzzy arithmetic operations for evaluation of the reliability index, fuzzy loss of load probability (FLOLP). Case studies for the maintenance scheduling of a captive power plant catering to an aluminum smelter have been formulated to demonstrate that fuzzy state probabilities as well as FLOLP provide a better explanation than the respective crisp counterparts.

Fuzzy Ordering Theory and Its Use in Filter Generalization

The rank ordering of samples is widely used in robust nonlinear signal processing. Recent advances in nonlinear filtering algorithms have focused on combining spatial and rank (SR) order information into the filtering process to allow spatial correlations to be exploited while retaining the robust characteristics of strict rank order methods. Further generalization can be achieved by replacing the crisp, or binary, SR information utilized by most methods with more general fuzzy SR information. Indeed, by exploiting fuzzy methodologies real-valued SR orderings can be defined that not only relate the spatial and rank orderings of samples, but also includes information on sample spread. This paper utilizes this approach to define fuzzy ranking and fuzzy order statistics. Properties of these concepts are discussed and several previously defined filters are generalized by including fuzzy concepts. Specifically, the fuzzy median, fuzzy rank conditioned rank selection, and fuzzy weighted median filters are defined. Optimization of the parameters for these filters are discussed. Simulation results are presented to show the advantages of these fuzzy filters over their crisp counterparts.

Fuzzy time-rank relations and order statistics

The rank ordering of samples is widely used in robust statistics and robust signal processing. Advances in these areas have focused on utilizing joint time-rank (TR) information. The TR information utilized to date is that resulting from a binary, or crisp, relation between the marginal time and rank ordering of samples. This crisp relation, while powerful, contains no information on sample values or spread. This paper generalizes the TR relation through fuzzy set theory. This generalization includes information on sample spread and leads to the concepts of fuzzy TR relations, fuzzy time and rank ordered samples, and fuzzy time and rank indices. These concepts are developed and analyzed through the derivation of fundamental properties. It is shown that the fuzzy TR relations, samples, and indices contain their crisp (standard) counterparts as special cases. These fuzzy generalizations constitute powerful tools that can be exploited in the design of signal processing algorithms.