Fuzzy Multisets and Fuzzy Computing

The core of fault diagnosis expert system is diagnostic knowledge representation and reasoning. Fuzzy Petri nets (FPN) have been testified to an ideal tool for knowledge representation and reasoning. However, when using FPN for diagnostic knowledge representation and reasoning, fault symptoms will disappear after fault propagated. This is not congruous with factual fault propagation. Another major limitation is the lacking of formal reasoning algorithm of FPN, which leads to it can't be implemented expediently by computer. In view of the limitations of FPN based diagnostic knowledge representation and reasoning, an enhanced model named fuzzy fault Petri nets (FFPN) and its formal reasoning algorithm are presented in the paper. A fault diagnosis instance was used to verify the feasibility and validity of the proposed method. It is demonstrated that this method is quite effective.

Type 2 fuzzy multiset is a new concept formed by integrating type 2 fuzzy set and multiset which are very useful for modelling uncertainties. The purpose of this paper is to introduce the concept of type 2 fuzzy α multicut set and to show how α- cut in secondary level helps to relate type 2 fuzzy α- multicuts and type 2 fuzzy multisets. The unique representation of type 2 fuzzy multisets using either the family of all type 2 fuzzy α multicuts or the family of all type 2 fuzzy α strong multicuts are also showed. The construction of membership grades in different levels are explained. Furthermore, we investigate the multicriteria group decision making based on type 2 fuzzy multisets. Finally, we give an example to show the validity of type 2 fuzzy multiset in group decision making problem.

Multisets alias bags are similar to fuzzy sets but essentially different in basic concepts and operations. We overview multisets together with basics of fuzzy sets in order to observe differences between the two. We then introduce fuzzy multisets and the combination of the both concepts. There is another concept of real-valued multisets as a generalization of multisets. Rough approximations of multisets and fuzzy multisets are discussed which uses a natural projection of the universal set onto the set of equivalence classes.

Fuzzy Petri net (FPN) has been extensively applied in industrial fields for knowledge-based systems or systems with uncertainty. Although the applications of FPN are known to be successful, the theoretical research of FPN is still at an initial stage. To pave a way for further study, this work explores related dynamic properties of FPN including reachability, boundedness, safeness, liveness and fairness. The whole methodology is divided into two phases. In the first phase, a comparison between elementary net system (EN_system) and FPN is established to prove that the FPN is an extensive formalism of Petri nets using a backwards-compatible extension method. Next, current research results of dynamic properties are utilized to analyze FPN model. The results illustrate that FPN model is bounded, safe, weak live and fair, and can support theoretical evidences for designing related decomposition algorithm.

The traditional fuzzy Petri nets (FPNs) rarely contain the affective features when making reasoning decisions, primarily because the existing affective computing model is not suitable to be described directly in FPNs. In order to solve this problem, this paper introduces an attribute coordinate system and puts forward the affective representation and computing model based on the extraction and integration relationship between the attributes of things, which describes the affective change curve according to the cognition of the thinking subject to the attributes of the decision object. With this new model of affective computing, it is more convenient to utilize a qualitative mapping model and fuzzy qualitative judgment rules to express affective decisions and conveniently describe affective attributes in FPNs. A reasoning decision case illustrates the proposed method's feasibility and advantages.

Modelling and control of manufacturing systems using fuzzy petri nets

Several extensions and generalizations of fuzzy sets have been introduced in the literature, for example, Atanassov's intuitionistic fuzzy sets, type 2 fuzzy sets, and fuzzy multisets. In this paper, we propose hesitant fuzzy sets. Although from a formal point of view, they can be seen as fuzzy multisets, we will show that their interpretation differs from the two existing approaches for fuzzy multisets. Because of this, together with their definition, we also introduce some basic operations. In addition, we also study their relationship with intuitionistic fuzzy sets. We prove that the envelope of the hesitant fuzzy sets is an intuitionistic fuzzy set. We prove also that the operations we propose are consistent with the ones of intuitionistic fuzzy sets when applied to the envelope of the hesitant fuzzy sets. © 2010 Wiley Periodicals, Inc.

Fuzzy Petri net (FPN), as one type of high-level Petri nets, has attracted a lot of attention over the recent decade due to its adequacy for knowledge representation and logic reasoning. However, in the FPN literature, the truth degrees of input places are usually given directly or supposed by researchers. No or little research has been performed on the determination of initial marking vector for a specific FPN. In this correspondence paper, we introduce a group decision-making model using hesitant 2-tuple linguistic term sets to obtain the initial truth values of FPNs based on domain experts’ knowledge and gathered data. As is illustrated by the numerical example, the proposed framework can well capture domain experts’ diversity judgements and derive initial truth degrees for an FPN under different types of uncertainties.

This paper presents a Petri net approach for the modeling of a CNC-milling machining centre. Next, by utilizing fuzzy logic with Petri nets (fuzzy Petri nets), a technique based on 9 fuzzy rules is developed. This paper demonstrates how fuzzy input variables, fuzzy marking, fuzzy firing sequences, and a global output variable should be defined for use with fuzzy Petri nets. The technique employs two fuzzy input variables (spindle speed and feed rate), throughout the milling operation in order to determine surface roughness. Additionally, a fuzzy Petri net is used with an artificial neural network for the modeling and control of surface roughness. Experimental results illustrate that the technique developed can be of benefit when the cutting tool has suffered damage throughout the milling operation. It also shows how the technique can react when the quality is high, medium, or low. The surface roughness represents the quality specification of products from the CNC-milling machining centre.

The basic set operations between fuzzy sets are defined using the min and max functions; however, later on, new operators were introduced that used other functions, which, nevertheless, had similar properties to functions min and max. The resulting fuzzy set theories are more suitable for the description and processing of specific data sets. Crisp and fuzzy multisets have found numerous applications but still the basic operations are based on functions min and max. It is straightforward to replace these functions in the fuzzy part of fuzzy multisets; however, it is not as easy but is feasible to do the same with the multisets and the “crisp” part of fuzzy multisets. The new mathematical structures are called triangular multisets and triangular fuzzy multisets, respectively. The aim is to facilitate the processing of certain data sets so they can be used in multi-criteria decision making and computing.

Fuzzy Petri nets (FPNs), which have been extensively used in many fields, are a promising method for knowledge representation and reasoning (KRR). However, there are still some shortcomings in these FPNs. Although there are many improved FPNs, the existing FPNs still have difficulties to ‘embrace’ the inconsistent cognition of experts and individualized features of different systems. In this paper, a new type FPNs are proposed, called self-learning interval type-2 fuzzy Petri nets (SLIT2FPNs). The extended TOPSIS is proposed to determine the optimal alternative, and to collect cognition of experts. An interval numbers ordered weighted averaging operator is introduced to improve the knowledge reasoning capabilities of SLIT2FPNs. Moreover, because of the introduction of state transition algorithm, the model has the ability of self-learning and adjustment according to information dynamics. Finally, two comparison tests are presented to demonstrate the proposed methods. In addition, the applications in aluminum reduction process show that SLIT2FPNs are feasible to ‘embrace’ inconsistent cognition, and to attract individualized features of each reduction cell, and acquire a KRR model with good performance. These facts demonstrate that the methods and ideas proposed in this paper provide a solution for KRR of knowledge automation in the industrial processes.

Fuzzy Petri nets for knowledge representation and reasoning: A literature review

The Rough Set Theory (RST) is a method that has proven its efficiency and simplicity in machine learning and successfully developing now a day’s vastly and rapidly. Fuzzy Petri nets (FPNs) are a potential modelling technique, which is used for knowledge representation and reasoning of rule-based expert systems. Though, RST has the efficiency in dimension reduction, it has to be proved with evidence by creating model using FPN with rule based reasoning. In this paper the induction of decision rules by RST executed with Fuzzy Petri Nets (FPN) is analyzed in the sense how it performed better than other data mining classifiers. The rule-based classifiers like jrip, part R, zero R are used for the comparison purposes. The knowledge captured from the rules through the best reduct has performed with the efficiency of RST and formulating rules by FPN. This paper experiments the Heart failure data to investigate the decision making from the rules generated by LERS system of RST with the approach of FPN. The heart failures during the follow up period of the patents are predicted with the pattern recognized form the data using the above process and the best evaluators are found during the experiments were pictured.

To monitor the patient behavior, data are collected from patient's body by a medical monitoring device so as to calculate the output using embedded software. Incorrect calculations may endanger the patient's life if the software fails to meet the patient's requirements. Accordingly, the veracity of the software behavior is a matter of concern in the medicine; moreover, the data collected from the patient's body are fuzzy. Some methods have already dealt with monitoring the medical monitoring devices; however, model based monitoring fuzzy computations of such devices have been addressed less. The present paper aims to present synthesizing a fuzzy Petri-net (FPN) model to verify behavior of a sample medical monitoring device called continuous infusion insulin (INS) because Petri-net (PN) is one of the formal and visual methods to verify the software's behavior. The device is worn by the diabetic patients and then the software calculates the INS dose and makes a decision for injection. The input and output of the infusion INS software are not crisp in the real world; therefore, we present them in fuzzy variables. Afterwards, we use FPN instead of clear PN to model the fuzzy variables. The paper follows three steps to synthesize an FPN to deal with verification of the infusion INS device: (1) Definition of fuzzy variables, (2) definition of fuzzy rules and (3) design of the FPN model to verify the software behavior.

In oil-delivery pumps, impeller failure is a common cause leading to excessive vibration. This paper is aimed to analyze the fault causation of impeller failure in oil-delivery pumps by using fuzzy Petri net (FPN) theory. The longest path algorithm based on the forward reasoning was put forward and introduced into fuzzy Petri net. First, on the basis of various factors causing impeller failure, an FPN model of impeller failure in oil-delivery pumps was constructed. Then, by using the proposed algorithm, fault causation analysis of impeller failure was completed to calculate the credibility of impeller failure. Finally, the corresponding preventive measure was presented. The results indicate the key causing factor of impeller failure is mechanical impurities, and the credibility of impeller failure is 0.7342, which is consistent with the actual situation. The research finding demonstrates the flexibility and effectiveness of the FPN in fault causation analysis.