Facets of Uncertainties and Applications

Since its inception, interesting connections between Rough Set Theory and different mathematical and logical topics have been investigated. This paper is a survey of some less known although highly interesting connections, which extend from Rough Set Theory to other mathematical and logical fields. The survey is primarily thought of as a guide for new directions to be explored.

We discuss an approach for dealing with uncertainty in complex intelligent systems. The approach is based on interactive computations over complex objects called here complex granules (c-granules, for short). C-granules are defined relative to a given agent. Any c-granule of a given agent specifies a perceived structure of local environment of physical objects, called hunks. There are three kinds of such hunks: (i) hunks in the agent external environment creating the hard_suit of c-granule, (ii) internal hunks of agent, creating the soft_suit of c-granule, some of which can be represented by agent as infogranules, and (iii) hunks creating the link_suit of c-granule and playing the role of links between hunks from the hard_suit and soft_suit. This structure is used for recording by means of infogranules the results of interactions of hunks from the local environment. We begin from the discussion on dealing with uncertainty in the rough set approach and next we move toward interactive computations on c-granules. In particular, from our considerations it follows that the fundamental issues of intelligent systems based on interactive computations concern the efficiency management in controlling of computations performed by such systems. Our approach is a step toward realization of the Wisdom Technology (WisTech) program. The approach was developed over years of work, based on the work on different real-life projects.

Lack of knowledge about secondary membership function acts as an impediment to using generalized type-2 fuzzy sets in real-world problems. This chapter shows a new direction to compute secondary memberships in the settings of a strategic optimization problem. It employs three strategies to design an optimization objective as a function of secondary memberships and employs differential evolution algorithm to determine secondary memberships as the optimal solution to the optimization problem. The proposed method of secondary membership function evaluation has successfully been applied to an emotion recognition problem.

In this paper, an attempt is made to introduce a new defuzzification scheme to be used in case there are a number of clipped fuzzy sets in the output of a fuzzy system. Approximate reasoning with this defuzzification scheme is proposed. A number of defuzzification methods existing in literature are reviewed here. A comparative study with our method has been made. The results are illustrated with a DC shunt motor for better understanding.

Since one major disadvantage of application of fuzzy formal concept analysis is that large numbers of fuzzy concepts are generated from fuzzy context, it is practically impossible to analyze such a large amount of concepts. Often it may be required to consider some particular concepts. For example, one might be interested to find out the fuzzy concepts containing all those objects which share some specific property with a specific/required degree from a given fuzzy context. Given such a situation, proto-fuzzy concepts may play a very useful role. This paper proposes a proto-fuzzy concept generation technique using fuzzy graph on uncertainty data. In this paper, we begin with defining a fuzzy graph corresponding to the L-context (fuzzy context). We then go on to demonstrate that t-concepts can be found to correspond with each maximal cliques of t-level graph of the defined fuzzy graph. After that, we determine all those cliques which corresponds to the proto-fuzzy concepts of degree \(t\). Finally, a demonstration has been made using an example with the proposed technique.

In rough set theory, information systems are used to represent knowledge-bases, particularly when practical applications are involved. We explore one such use of information systems, for representing the open world model. Open world information systems are defined, and a temporal logic, including descriptors and the global modality, is proposed as a formal reasoning framework for these structures.

In this paper, we have made a study of approximate reasoning based on a Type-2 fuzzy set theory. We have focused upon two typical rules of inference used mostly in ordinary approximate reasoning methodology based on Type-1 fuzzy set theory. Similarity is inherent in approximate reasoning. The concept of similarity between Type-2 fuzzy sets is discussed and a similarity-based approximate reasoning technique is proposed. The proposal is illustrated with a typical artificial example. Prediction is the causal basis for decision making. Different measures leading to prediction under uncertainty are proposed for a better understanding of the power of Type-2 fuzzy set theory.

Rough Sets over generalized transitive relations like proto-transitive ones have been initiated by the present author in Mani (Inst. Math. Sci. ICLA’2013, 1–12, Chennai 2013, [1]) and detailed semantics have been developed in forthcoming papers. In this research paper, approximation of proto-transitive relations by other relations is investigated and the relation with rough approximations is developed toward constructing semantics that can handle fragments of structure. It is also proved that difference of approximations induced by some approximate relations need not induce rough structures.

We propose a generalization of information systems which provides the probability of an object to take an attribute-value for an attribute. Notions of distinguishability relations and corresponding notions of approximations are proposed and studied in comparison with the existing one.

The process of contaminant transportation through ground water can be varied with different parameters such as soil characteristics, ground water flow velocity, longitudinal and transverse dispersion coefficients, self-degradation of contaminant etc. The precise definition of these parameters is very difficult due to various factors such as measurement error, sampling error, dependence of complex physical phenomena, etc. The analytical solution of transient advection–diffusion equation is being used to assess the ground water contamination due to the industrial discharge. The paper describes a methodology to estimate the hybrid uncertainty, i.e., combination of aleatory and epistemic using the fuzzy-stochastic system. Aleatory uncertainty due to random variation of input parameter is estimated using polynomial chaos expansion method. To take into account the effect of imprecise variation (i.e., epistemic uncertainty) of input parameter, a fuzzy \(\alpha \)-cut technique has been used. The large sample space of concentration reduction factor (CRF) have been generated using fuzzy-stochastic response surface to arrive the upper uncertainty bound corresponds to the 95th percentile value at a specified distance from the source and period of time. The methodology will be very useful to assess the safety margin or discharge limit from the industry.

This work focuses on the development of a fuzzy random health risk model. The concept of fuzzy random variable is used to develop this health risk model. Health risk is addressed as the risk due to exposure to uranium through ingestion of food grown in and around a high background area which is rich in groundwater and has a substantial amount of phosphate deposits that constitute uranium. Lack of data about the activity concentration of uranium and its variability at many locations of that area justifies its fuzziness and randomness. A similar reason is valid for the consumption of food from the area. Therefore, these input parameters of the exposure model are the fuzzy random variable. Exposure model computes the daily average ingestion and risk is computed by multiplying this with the corresponding cancer slope factor. Fuzziness of daily average ingestion is computed at a specified percentile of the lower and upper fuzzy cumulative distribution of daily average ingestion. Randomness is computed at every alpha cut of the fuzzy random daily average ingestion. Fuzzy random daily average ingestion is used to compute the fuzzy random risk. It has been shown that risk due to the consumption of naturally occurred uranium through ingestion of food is insignificant. Support, uncertainty index, possibility, necessity, and credibility of the fuzzy random risk are also computed to explore the role of fuzzy random variable in uncertainty of risk estimation.

Uncertainty analysis of physical parameters present in the groundwater model is important from the point of safety measures in the field of nuclear science and technology. Researchers have carried out this uncertainty analysis using traditional Monte Carlo simulations. However, in practice, Monte Carlo simulation may not be possible because of lack of data obtained from field experiments. Therefore, the demand is to investigate uncertainty using imprecise-based method. In order to fulfill this demand, we have carried out uncertainty analysis of groundwater model parameter using fuzzy set and hybrid methods. Monte Carlo-based uncertainty is also presented in this paper. Overall, this paper highlights the various methodologies of uncertainty analysis. In the hybrid approach, the concept of fuzzy random variable and its computational details have been explored. Retardation factor is our representative groundwater model parameter on which illustration of the said methodologies of uncertainty modeling is presented.

This proposed work explains a noble two-person interaction modelling system using Kinect sensor. Here a pentagon for each person is formed taking the three dimensional co-ordinate information with the help of Microsoft’s Kinect sensor. Five Euclidean distances between two pentagon vertices corresponding to two persons are considered as features for each frame. So the body gestures of two persons are analysed employing pentagons. Based on these, eight interactions between two persons are modelled. This system produces the best recognition rate (greater than 90 %) with the virtue of multi-class support vector machine for rotation invariance case and for rotation variance phenomenon, the recognition rate is greater than 80 %.

In this paper, we propose an improved genetic algorithm (IGA) to solve Constrained Solid Travelling Salesman Problems (CSTSPs) in crisp, fuzzy, rough, and fuzzy-rough environments. The proposed algorithm is a combination of probabilistic selection, cyclic crossover, and nodes-oriented random mutation. Here, CSTSPs in different uncertain environments have been designed and solved by the proposed algorithm. A CSTSP is usually a travelling salesman problem (TSP) where the salesman visits all cities using any one of the conveyances available at each city under a constraint say, safety constraint. Here a number of conveyances are used for travel from one city to another. In the present problem, there are some risks of travelling between the cities through different conveyances. The salesman desires to maintain certain safety level always to travel from one city to another and a total safety for his entire tour. Costs and safety level factors for travelling between the cities are different. The requirement of minimum safety level is expressed in the form of a constraint. The safety factors are expressed by crisp, fuzzy, rough, and fuzzy-rough numbers. The problems are formulated as minimization problems of total cost subject to crisp, fuzzy, rough, or fuzzy-rough constraints. This problem is numerically illustrated with appropriate data values. Optimum results for the different problems are presented via IGA. Moreover, the problems from the TSPLIB (standard data set) are tested with the proposed algorithm.

The present paper introduces a model of decision-making problem in multicriteria optimization domain for project management. The model is built by combining the fuzzy soft set theory and analytical hierarchical model. The later is the well-known method of ranking the alternatives for multicriteria decision-making problem. The ultimate project selection is the best of many decisions associated with project management. Here we illustrate the hybrid method by means of an application of the new mathematical model of soft set theory.

Decision-making problems in an imprecise environment are of paramount importance in recent years. Here we consider an object recognition problem in an imprecise environment. In this paper we study the concept of weighted neutrosophic soft sets. A multiobserver decision-making problem has been considered here as an application of weighted neutrosophic soft sets. We have considered here a recognition strategy based on multiobserver input parameter data set.

In this paper, we propose a concrete application of similarity-based approximate reasoning (SAR) to the management of hypertension. It is one of the silent killer diseases that threatens the lives of millions of people in developed and developing nations. The need to optimize the management of hypertension using SAR may improve the medicine diagnostic support system. It diagnoses the possibility of the disease and its severity. Systolic blood pressure (SBP), diastolic blood pressure (DBP), age and body mass index (BMI) are taken as input parameters of the fuzzy expert system and hypertension risk is the output parameter. SAR is the inference mechanism. Based on the result obtained, fuzzy diagnosis resembles human decision making with its ability to work using similarity-based approximate reasoning and ultimately find a precise solution.

This article introduces the concept of hesitant fuzzy soft set (HFSS) by combining Torra’s (2010) hesitant fuzzy set and Molodtsov’s (1999) soft set theory. In order to handle uncertain and imprecise situation especially in medical diagnosis hesitant fuzzy soft sets are found to be more useful. This article investigates a couple of distance measurements procedures and aggregation operators applicable for HFSS. An algorithmic approach is proposed to solve multiple attribute decision-making (MADM) problems using HFSS with the help of aggregation operators and hesitant fuzzy soft matrix (HFSM). Finally, an illustrative example is presented to analyze the proposed approach.

This paper deals with the computational aspect of the fuzzy ideal cone method by Ghosh and Chakraborty, Fuzzy ideal cone: a method to obtain complete fuzzy nondominated set of fuzzy multi-criteria optimization problems with fuzzy parameters, In: The Proceedings of IEEE International Conference on Fuzzy Systems 2013, FUZZ IEEE 2013, IEEE Xplore, pp. 1–8 to generate the complete fuzzy nondominated set of a fuzzy multi-criteria optimization problem. In order to formulate the decision feasible region, the concept of inverse points in fuzzy geometry is used. Relation between the fuzzy decision feasible sets evaluated through the inverse points and directly through the extension principle is reported. It is shown that under a certain monotone condition both the decision feasible sets are identical. This result can greatly reduce the computational cost of evaluating the decision feasible region. After evaluating the decision feasible region, criteria feasible region is formulated using the basic fuzzy geometrical ideas. An algorithmic implementation of the fuzzy ideal cone method is presented to find the complete fuzzy nondominated set of the fuzzy criteria feasible region.

In this paper, we study a solid transportation problem with uncertain cost and uncertain time, where the supplies, the demands, the conveyance capacities are regarded as uncertain in nature. For the first time we minimize the uncertain transportation time. According to the inverse uncertainty distribution, the model can be transformed into a deterministic form by taking expected value on objective functions and confidence level on the constraint functions. We solve the uncertain solid transportation problem by fuzzy programming technique and using the LINGO 13.0 software. Finally, this paper is illustrated by a numerical example on uncertain solid transportation problem to show the application of the model.

This paper deals with the selective harvesting of two species from a food chain model of three species, in which prey and predator obey the Gompertz law of growth. Initially the dynamical behaviour of the system was studied under deterministic environment. In deterministic case, the local stability of the system was also studied; we investigated the condition of global stability and the existence of the bionomic equilibrium was examined. The optimal harvesting policy is studied with the help of Pontryagin’s maximum principle. In the second part of the problem, we investigated the stability condition of the system under stochastic environment. Then a comparison is made between deterministic and stochastic cases.

In this paper, we present modified version of computational algorithm given by Gangwar and Kumar (Expert Syst Appl 39:12158–12164, 2012 [5]) for higher order weighted fuzzy time series with multiple partitioning to enhance the accuracy in forecasting. The developed method provides a better approach to enhance the accuracy in forecasted values. The proposed method was implemented on the historical student enrollments data of University of Alabama. The suitability of the developed method has been examined in comparison with other models in terms of mean square and average forecasting errors to show its superiority.

This paper proposes a new approach for modeling multiple objective portfolio selection problem by applying weighted possibilistic moments of trapezoidal fuzzy numbers. The proposed model allows the decision-maker to select the suitable portfolio taking into account the impreciseness to the market scenarios. Here, the objectives are to (i) maximize the expected portfolio return, (ii) minimize the portfolio variance, (iii) maximize the portfolio skewness, and (iv) minimize the portfolio kurtosis for the risky investor. The proposed model has been solved by Zimmermann’s fuzzy goal programming technique. The model is illustrated by a numerical example using data extracted from the Bombay Stock Exchange.

In this paper, we present conflicting bifuzzy preference relations-based method for multi-criteria decision-making (MCDM) problem. In proposed method linear programming model is used to obtain optimal weights of criteria by utilizing criteria weights and score function. The best alternative is selected in accordance with the value of weighting function. In order to examine the impact of criteria weights on final ranking sensitivity analysis is performed. A numerical example is also given to clarify the developed approach and to demonstrate its effectiveness.