This document presents a counterexample demonstrating that the fuzzy forward recursion method for determining critical paths does not always produce results consistent with the extension principle when discrete fuzzy sets are used to represent activity durations.
The document first provides background on fuzzy sets and critical path analysis. It then presents a proposition stating that the membership function for fuzzy critical path lengths can be determined by taking the maximum of the minimum membership values across all activity durations in each configuration.
The document goes on to present a counterexample using a simple series-parallel network with 18 configurations. It shows that applying the fuzzy forward recursion produces a different membership value for one critical path length compared to directly applying the extension principle. This difference proves the fuzzy forward
EXACT SOLUTIONS OF A FAMILY OF HIGHER-DIMENSIONAL SPACE-TIME FRACTIONAL KDV-T...cscpconf
In this paper, based on the definition of conformable fractional derivative, the functional
variable method (FVM) is proposed to seek the exact traveling wave solutions of two higherdimensional
space-time fractional KdV-type equations in mathematical physics, namely the
(3+1)-dimensional space–time fractional Zakharov-Kuznetsov (ZK) equation and the (2+1)-
dimensional space–time fractional Generalized Zakharov-Kuznetsov-Benjamin-Bona-Mahony
(GZK-BBM) equation. Some new solutions are procured and depicted. These solutions, which
contain kink-shaped, singular kink, bell-shaped soliton, singular soliton and periodic wave
solutions, have many potential applications in mathematical physics and engineering. The
simplicity and reliability of the proposed method is verified.
A COMPREHENSIVE ANALYSIS OF QUANTUM CLUSTERING : FINDING ALL THE POTENTIAL MI...IJDKP
Quantum clustering (QC), is a data clustering algorithm based on quantum mechanics which is
accomplished by substituting each point in a given dataset with a Gaussian. The width of the Gaussian is a
σ value, a hyper-parameter which can be manually defined and manipulated to suit the application.
Numerical methods are used to find all the minima of the quantum potential as they correspond to cluster
centers. Herein, we investigate the mathematical task of expressing and finding all the roots of the
exponential polynomial corresponding to the minima of a two-dimensional quantum potential. This is an
outstanding task because normally such expressions are impossible to solve analytically. However, we
prove that if the points are all included in a square region of size σ, there is only one minimum. This bound
is not only useful in the number of solutions to look for, by numerical means, it allows to to propose a new
numerical approach “per block”. This technique decreases the number of particles by approximating some
groups of particles to weighted particles. These findings are not only useful to the quantum clustering
problem but also for the exponential polynomials encountered in quantum chemistry, Solid-state Physics
and other applications.
Here, in this paper we are introducing a dynamic clustering algorithm using fuzzy c-mean clustering algorithm. We will try to process several sets patterns together to find a common structure. The structure is finalized by interchanging prototypes of the given data and by moving the prototypes of the subsequent clusters toward each other. In regular FCM clustering algorithm, fixed numbers of clusters are chosen and those are pre-defined. If, in case, the number of chosen clusters is wrong, then the final result will degrade the purity of the cluster. In our proposed algorithm this drawback will be overcome by using dynamic clustering architecture. Here we will take fixed number of clusters in the beginning but on iterations the algorithm will increase the number of clusters automatically depending on the nature and type of data, which will increase the purity of the result at the end. A detailed clustering algorithm is developed on a basis of the standard FCM method and will be illustrated by means of numeric examples.
EXACT SOLUTIONS OF A FAMILY OF HIGHER-DIMENSIONAL SPACE-TIME FRACTIONAL KDV-T...cscpconf
In this paper, based on the definition of conformable fractional derivative, the functional
variable method (FVM) is proposed to seek the exact traveling wave solutions of two higherdimensional
space-time fractional KdV-type equations in mathematical physics, namely the
(3+1)-dimensional space–time fractional Zakharov-Kuznetsov (ZK) equation and the (2+1)-
dimensional space–time fractional Generalized Zakharov-Kuznetsov-Benjamin-Bona-Mahony
(GZK-BBM) equation. Some new solutions are procured and depicted. These solutions, which
contain kink-shaped, singular kink, bell-shaped soliton, singular soliton and periodic wave
solutions, have many potential applications in mathematical physics and engineering. The
simplicity and reliability of the proposed method is verified.
A COMPREHENSIVE ANALYSIS OF QUANTUM CLUSTERING : FINDING ALL THE POTENTIAL MI...IJDKP
Quantum clustering (QC), is a data clustering algorithm based on quantum mechanics which is
accomplished by substituting each point in a given dataset with a Gaussian. The width of the Gaussian is a
σ value, a hyper-parameter which can be manually defined and manipulated to suit the application.
Numerical methods are used to find all the minima of the quantum potential as they correspond to cluster
centers. Herein, we investigate the mathematical task of expressing and finding all the roots of the
exponential polynomial corresponding to the minima of a two-dimensional quantum potential. This is an
outstanding task because normally such expressions are impossible to solve analytically. However, we
prove that if the points are all included in a square region of size σ, there is only one minimum. This bound
is not only useful in the number of solutions to look for, by numerical means, it allows to to propose a new
numerical approach “per block”. This technique decreases the number of particles by approximating some
groups of particles to weighted particles. These findings are not only useful to the quantum clustering
problem but also for the exponential polynomials encountered in quantum chemistry, Solid-state Physics
and other applications.
Here, in this paper we are introducing a dynamic clustering algorithm using fuzzy c-mean clustering algorithm. We will try to process several sets patterns together to find a common structure. The structure is finalized by interchanging prototypes of the given data and by moving the prototypes of the subsequent clusters toward each other. In regular FCM clustering algorithm, fixed numbers of clusters are chosen and those are pre-defined. If, in case, the number of chosen clusters is wrong, then the final result will degrade the purity of the cluster. In our proposed algorithm this drawback will be overcome by using dynamic clustering architecture. Here we will take fixed number of clusters in the beginning but on iterations the algorithm will increase the number of clusters automatically depending on the nature and type of data, which will increase the purity of the result at the end. A detailed clustering algorithm is developed on a basis of the standard FCM method and will be illustrated by means of numeric examples.
HEATED WIND PARTICLE’S BEHAVIOURAL STUDY BY THE CONTINUOUS WAVELET TRANSFORM ...cscpconf
Nowadays Continuous Wavelet Transform (CWT) as well as Fractal analysis is generally used for the Signal and Image processing application purpose. Our current work extends the field of application in case of CWT as well as Fractal analysis by applying it in case of the agitated wind particle’s behavioral study. In this current work in case of the agitated wind particle, we have mathematically showed that the wind particle’s movement exhibits the “Uncorrelated” characteristics during the convectional flow of it. It is also demonstrated here by the Continuous Wavelet Transform (CWT) as well as the Fractal analysis with matlab 7.12 version
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
On a Deterministic Property of the Category of k-almost Primes: A Determinist...Ramin (A.) Zahedi
In this paper based on a sort of linear function, a deterministic and simple algorithm with an algebraic structure is presented for calculating all (and only) k-almost primes (where ∃n∊ℕ, 1 ≤ k ≤ n) in certain intervals. A theorem has been proven showing a new deterministic property of the category of k-almost primes. Through a linear function that we obtain, an equivalent redefinition of the k-almost primes with an algebraic characteristic is identified. Moreover, as an outcome of our function’s property some equalities which contain new information about the k-almost primes (including primes) are presented.
Comments: Accepted and presented article in the 11th ANTS , Korea, 2014. The 11th ANTS is one of international satellite conferences of ICM 2014:The 27th International Congress of Mathematicians, Korea. (Expanded version)
Copyright: CC Attribution-NonCommercial-NoDerivs 4.0 International
License URL: https://creativecommons.org/licenses/by-nc-nd/4.0/
A landing gear assembly consists of various components viz. Lower side stay, Upperside stay, Locking actuators, Extension actuators, Tyres, and Locking pins to name a few. Each unit having a specific operation to deal with, in this project the main unit being studied is the lower brace. The primary objective is to analyse stresses in the element of the lower brace unit using strength of materials or RDM method and Finite Element Method (FEM) and compare both. Using the obtained data a suitable material is proposed for the component. The approach used here is to study the overall behaviour of the element by taking up each aspect, finally summing up the total effect of all the aspects in the functioning of the element.
Tomography is important for network design and routing optimization. Prior approaches require either
precise time synchronization or complex cooperation. Furthermore, active tomography consumes explicit
probing resulting in limited scalability. To address the first issue we propose a novel Delay Correlation
Estimation methodology named DCE with no need of synchronization and special cooperation. For the
second issue we develop a passive realization mechanism merely using regular data flow without explicit
bandwidth consumption. Extensive simulations in OMNeT++ are made to evaluate its accuracy where we
show that DCE measurement is highly identical with the true value. Also from test result we find that
mechanism of passive realization is able to achieve both regular data transmission and purpose of
tomography with excellent robustness versus different background traffic and package size.
vFORTRAN is used as a numerical and scientific computing language. The main objective of the lab work is to understand FORTRAN language using which we solve simple numerical problems and compare different methodologies. Through this project we make use of various functions of FORTRAN and solve a simple projectile problem and also LAPACK library to solve a tridiagonal matrix problem. We use DGESV and DGTSV functions to make it possible. The given problems are built and compiled using a free integrated development environment called CODE::BLOCKS [1] which is an open source platform for FORTRAN and C.
On selection of periodic kernels parameters in time series predictioncsandit
In the paper the analysis of the periodic kernels parameters is described. Periodic kernels can
be used for the prediction task, performed as the typical regression problem. On the basis of the
Periodic Kernel Estimator (PerKE) the prediction of real time series is performed. As periodic
kernels require the setting of their parameters it is necessary to analyse their influence on the
prediction quality. This paper describes an easy methodology of finding values of parameters of
periodic kernels. It is based on grid search. Two different error measures are taken into
consideration as the prediction qualities but lead to comparable results. The methodology was
tested on benchmark and real datasets and proved to give satisfactory results.
PREDICTIVE EVALUATION OF THE STOCK PORTFOLIO PERFORMANCE USING FUZZY CMEANS A...ijfls
The aim of this paper is to investigate the trend of the return of a portfolio formed randomly or for any
specific technique. The approach is made using two techniques fuzzy: fuzzy c-means (FCM) algorithm and
the fuzzy transform, where the rules used at fuzzy transform arise from the application of the FCM
algorithm. The results show that the proposed methodology is able to predict the trend of the return of a
stock portfolio, as well as the tendency of the market index. Real data of the financial market are used from
2004 until 2007.
HEATED WIND PARTICLE’S BEHAVIOURAL STUDY BY THE CONTINUOUS WAVELET TRANSFORM ...cscpconf
Nowadays Continuous Wavelet Transform (CWT) as well as Fractal analysis is generally used for the Signal and Image processing application purpose. Our current work extends the field of application in case of CWT as well as Fractal analysis by applying it in case of the agitated wind particle’s behavioral study. In this current work in case of the agitated wind particle, we have mathematically showed that the wind particle’s movement exhibits the “Uncorrelated” characteristics during the convectional flow of it. It is also demonstrated here by the Continuous Wavelet Transform (CWT) as well as the Fractal analysis with matlab 7.12 version
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
On a Deterministic Property of the Category of k-almost Primes: A Determinist...Ramin (A.) Zahedi
In this paper based on a sort of linear function, a deterministic and simple algorithm with an algebraic structure is presented for calculating all (and only) k-almost primes (where ∃n∊ℕ, 1 ≤ k ≤ n) in certain intervals. A theorem has been proven showing a new deterministic property of the category of k-almost primes. Through a linear function that we obtain, an equivalent redefinition of the k-almost primes with an algebraic characteristic is identified. Moreover, as an outcome of our function’s property some equalities which contain new information about the k-almost primes (including primes) are presented.
Comments: Accepted and presented article in the 11th ANTS , Korea, 2014. The 11th ANTS is one of international satellite conferences of ICM 2014:The 27th International Congress of Mathematicians, Korea. (Expanded version)
Copyright: CC Attribution-NonCommercial-NoDerivs 4.0 International
License URL: https://creativecommons.org/licenses/by-nc-nd/4.0/
A landing gear assembly consists of various components viz. Lower side stay, Upperside stay, Locking actuators, Extension actuators, Tyres, and Locking pins to name a few. Each unit having a specific operation to deal with, in this project the main unit being studied is the lower brace. The primary objective is to analyse stresses in the element of the lower brace unit using strength of materials or RDM method and Finite Element Method (FEM) and compare both. Using the obtained data a suitable material is proposed for the component. The approach used here is to study the overall behaviour of the element by taking up each aspect, finally summing up the total effect of all the aspects in the functioning of the element.
Tomography is important for network design and routing optimization. Prior approaches require either
precise time synchronization or complex cooperation. Furthermore, active tomography consumes explicit
probing resulting in limited scalability. To address the first issue we propose a novel Delay Correlation
Estimation methodology named DCE with no need of synchronization and special cooperation. For the
second issue we develop a passive realization mechanism merely using regular data flow without explicit
bandwidth consumption. Extensive simulations in OMNeT++ are made to evaluate its accuracy where we
show that DCE measurement is highly identical with the true value. Also from test result we find that
mechanism of passive realization is able to achieve both regular data transmission and purpose of
tomography with excellent robustness versus different background traffic and package size.
vFORTRAN is used as a numerical and scientific computing language. The main objective of the lab work is to understand FORTRAN language using which we solve simple numerical problems and compare different methodologies. Through this project we make use of various functions of FORTRAN and solve a simple projectile problem and also LAPACK library to solve a tridiagonal matrix problem. We use DGESV and DGTSV functions to make it possible. The given problems are built and compiled using a free integrated development environment called CODE::BLOCKS [1] which is an open source platform for FORTRAN and C.
On selection of periodic kernels parameters in time series predictioncsandit
In the paper the analysis of the periodic kernels parameters is described. Periodic kernels can
be used for the prediction task, performed as the typical regression problem. On the basis of the
Periodic Kernel Estimator (PerKE) the prediction of real time series is performed. As periodic
kernels require the setting of their parameters it is necessary to analyse their influence on the
prediction quality. This paper describes an easy methodology of finding values of parameters of
periodic kernels. It is based on grid search. Two different error measures are taken into
consideration as the prediction qualities but lead to comparable results. The methodology was
tested on benchmark and real datasets and proved to give satisfactory results.
PREDICTIVE EVALUATION OF THE STOCK PORTFOLIO PERFORMANCE USING FUZZY CMEANS A...ijfls
The aim of this paper is to investigate the trend of the return of a portfolio formed randomly or for any
specific technique. The approach is made using two techniques fuzzy: fuzzy c-means (FCM) algorithm and
the fuzzy transform, where the rules used at fuzzy transform arise from the application of the FCM
algorithm. The results show that the proposed methodology is able to predict the trend of the return of a
stock portfolio, as well as the tendency of the market index. Real data of the financial market are used from
2004 until 2007.
An Improved Adaptive Multi-Objective Particle Swarm Optimization for Disassem...IJRESJOURNAL
With the development of productivity and the fast growth of the economy, environmental pollution, resource utilization and low product recovery rate have emerged subsequently, so more and more attention has been paid to the recycling and reuse of products. However, since the complexity of disassembly line balancing problem (DLBP) increases with the number of parts in the product, finding the optimal balance is computationally intensive. In order to improve the computational ability of particle swarm optimization (PSO) algorithm in solving DLBP, this paper proposed an improved adaptive multi-objective particle swarm optimization (IAMOPSO) algorithm. Firstly, the evolution factor parameter is introduced to judge the state of evolution using the idea of fuzzy classification and then the feedback information from evolutionary environment is served in adjusting inertia weight, acceleration coefficients dynamically. Finally, a dimensional learning strategy based on information entropy is used in which each learning object is uncertain. The results from testing in using series of instances with different size verify the effect of proposed algorithm.
Critical Paths Identification on Fuzzy Network Projectiosrjce
In this paper, a new approach for identifying fuzzy critical path is presented, based on converting the
fuzzy network project into deterministic network project, by transforming the parameters set of the fuzzy
activities into the time probability density function PDF of each fuzzy time activity. A case study is considered as
a numerical tested problem to demonstrate our approach.
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
A High Order Continuation Based On Time Power Series Expansion And Time Ratio...IJRES Journal
In this paper, we propose a high order continuation based on time power series expansion and time rational representation called Pad´e approximants for solving nonlinear structural dynamic problems. The solution of the discretized nonlinear structural dynamic problems, by finite elements method, is sought in the form of a power series expansion with respect to time. The Pad´e approximants technique is introduced to improve the validity range of power series expansion. The whole solution is built branch by branch using the continuation method. To illustrate the performance of this proposed high order continuation, we give some numerical comparisons on an example of forced nonlinear vibration of an elastic beam.
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
P REDICTION F OR S HORT -T ERM T RAFFIC F LOW B ASED O N O PTIMIZED W...ijcsit
Short term traffic forecasting has been a very impo
rtant consideration in many areas of transportation
research for more than 3 decades. Short-term traffi
c forecasting based on data driven methods is one o
f the
most dynamic and developing research arenas with en
ormous published literature. In order to improve
forecasting model accuracy of wavelet neural networ
k, an adaptive particle swarm optimization algorith
m
based on cloud theory was proposed, not only to hel
p improve search performance, but also speed up
individual optimizing ability. And the inertia weig
ht adaptively changes depending on X-conditional cl
oud
generator which has the stable tendency and randomn
ess property .Then the adaptive particle swarm
optimization algorithm based on cloud theory was us
ed to optimize the weights and thresholds of wavele
t
BP neural network, Instead of traditional gradient
descent method . At last, wavelet BP neural network
was
trained to search for the optimal solution. Based o
n above theory, an improved wavelet neural network
model based on modified particle swarm optimization
algorithm was proposed and the availability of the
modified prediction method was proved by predicting
the time series of real traffic flow. At last, the
computer simulations have shown that the nonlinear
fitting and accuracy of the modified prediction
methods are better than other prediction methods.
Multi objective predictive control a solution using metaheuristicsijcsit
The application of multi objective model predictive control approaches is significantly limited with
computation time associated with optimization algorithms. Metaheuristics are general purpose heuristics
that have been successfully used in solving difficult optimization problems in a reasonable computation
time. In this work , we use and compare two multi objective metaheuristics, Multi-Objective Particle
swarm Optimization, MOPSO, and Multi-Objective Gravitational Search Algorithm, MOGSA, to generate
a set of approximately Pareto-optimal solutions in a single run. Two examples are studied, a nonlinear
system consisting of two mobile robots tracking trajectories and avoiding obstacles and a linear multi
variable system. The computation times and the quality of the solution in terms of the smoothness of the
control signals and precision of tracking show that MOPSO can be an alternative for real time
applications.
International Journal of Engineering Research and DevelopmentIJERD Editor
Electrical, Electronics and Computer Engineering,
Information Engineering and Technology,
Mechanical, Industrial and Manufacturing Engineering,
Automation and Mechatronics Engineering,
Material and Chemical Engineering,
Civil and Architecture Engineering,
Biotechnology and Bio Engineering,
Environmental Engineering,
Petroleum and Mining Engineering,
Marine and Agriculture engineering,
Aerospace Engineering.
Fault diagnosis using genetic algorithms and principal curveseSAT Journals
Abstract Several applications of nonlinear principal component analysis (NPCA) have appeared recently in process monitoring and fault diagnosis. In this paper a new approach is proposed for fault detection based on principal curves and genetic algorithms. The principal curve is a generation of linear principal component (PCA) introduced by Hastie as a parametric curve passes satisfactorily through the middle of data. The existing principal curves algorithms employ the first component of the data as an initial estimation of principal curve. However the dependence on initial line leads to a lack of flexibility and the final curve is only satisfactory for specific problems. In this paper we extend this work in two ways. First, we propose a new method based on genetic algorithms to find the principal curve. Here, lines are fitted and connected to form polygonal lines (PL). Second, potential application of principal curves is discussed. An example is used to illustrate fault diagnosis of nonlinear process using the proposed approach. Index Terms: Principal curve, Genetic Algorithm, Nonlinear principal component analysis, Fault detection.
The Analysis of Performance Measures of Generalized Trapezoidal Fuzzy Queuing...IJERA Editor
The purpose of this research paper was to propose a method which can be utilized to determine the different types of performance measures on the basis of the crisp values for the fuzzy queuing model which has an unreliable server and where the rate of arrival, the rate of service, the rate of breakdown and the rate of repair are all expressed as the fuzzy numbers. In this case the inter arrival time, the time of service, the rates of breakdown and the rates of repair are all triangular functions and are also expressed as the Trapezoidal fuzzy numbers. The main intent is to transform the fuzzy inter arrival time, the time of service, the rates of breakdown and the rates of repair into the crisp values by using the Ranking function method. Then the crisp values are applied in the classical formulas for the performance measure. In the fuzzy environment the ranking fuzzy numbers are very helpful in making the decisions. The ranking function method is one of the most reliable method, is simpler to apply in comparison to other methods and can be utilized to solve the different types of queuing problems. In this research paper a numerical example is also provided for both the triangular and the trapezoidal fuzzy number so that a practical insight into the problem can be provided.
On Selection of Periodic Kernels Parameters in Time Series Prediction cscpconf
In the paper the analysis of the periodic kernels parameters is described. Periodic kernels can
be used for the prediction task, performed as the typical regression problem. On the basis of the
Periodic Kernel Estimator (PerKE) the prediction of real time series is performed. As periodic
kernels require the setting of their parameters it is necessary to analyse their influence on the
prediction quality. This paper describes an easy methodology of finding values of parameters of
periodic kernels. It is based on grid search. Two different error measures are taken into
consideration as the prediction qualities but lead to comparable results. The methodology was
tested on benchmark and real datasets and proved to give satisfactory results.
ON SELECTION OF PERIODIC KERNELS PARAMETERS IN TIME SERIES PREDICTIONcscpconf
In the paper the analysis of the periodic kernels parameters is described. Periodic kernels can
be used for the prediction task, performed as the typical regression problem. On the basis of the
Periodic Kernel Estimator (PerKE) the prediction of real time series is performed. As periodic
kernels require the setting of their parameters it is necessary to analyse their influence on the
prediction quality. This paper describes an easy methodology of finding values of parameters of
periodic kernels. It is based on grid search. Two different error measures are taken into
consideration as the prediction qualities but lead to comparable results. The methodology was
tested on benchmark and real datasets and proved to give satisfactory results.
Geoid height determination is one of the major problems of geodesy because usage of satellite
techniques in geodesy isgetting increasing. Geoid heights can be determined using different methods according
to the available data. Soft computing methods such as Fuzzy logic and neural networks became so popular that
they are used to solve many engineering problems. Fuzzy logic theory and later developments in uncertainty
assessment have enabled us to develop more precise models for our requirements. In this study, How to
construct the best fuzzy model is examined. For this purpose, three different data sets were taken and two
different kinds (two inpust one output and three inputs one output) fuzzy model were formed for the calculation
of geoid heights in Istanbul (Turkey). The Fuzzy models results of these were compared with geoid heights
obtained by GPS/levelling methods. The fuzzy approximation models were tested on the test points.
Fuzzy Fatigue Failure Model to Estimate the Reliability of Extend the Service...IOSRJMCE
In this paper we use fuzzy set of methods to solve one of the important problems in mechanical engineering: Reliability of Extending the Service Life of Rolling Stock by using Fuzzy Fatigue failure model. The residual service life for rolling stock can be changed depending on its use conditions. This paper presents a new method depending on fuzzy set theory by using the fatigue stress mathematical model to determine the residual service for rolling stock with the value of risk of its use in future. The proposed method used solid works and (Ansys) abilities with especial Fuzzy logic programs in MATLAB.
EDGE DETECTION IN SEGMENTED IMAGES THROUGH MEAN SHIFT ITERATIVE GRADIENT USIN...ijscmcj
In this paper, we propose a new method for edge detection in obtained images from the Mean Shift iterative algorithm. The comparable, proportional and symmetrical images are de?ned and the importance of Ring Theory is explained. A relation of equivalence among proportional images are de?ned for image groups in equivalent classes. The length of the mean shift vector is used in order to quantify the homogeneity of the neighborhoods. This gives a measure of how much uniform are the regions that compose the image. Edge detection is carried out by using the mean shift gradient based on symmetrical images. The difference among the values of gray levels are accentuated or these are decreased to enhance the interest region contours. The chosen images for the experiments were standard images and real images (cerebral hemorrhage images). The obtained results were compared with the Canny detector, and our results showed a good performance as for the edge continuity.
Similar to A Counterexample to the Forward Recursion in Fuzzy Critical Path Analysis Under Discrete Fuzzy Sets (20)
NO1 Uk best vashikaran specialist in delhi vashikaran baba near me online vas...Amil Baba Dawood bangali
Contact with Dawood Bhai Just call on +92322-6382012 and we'll help you. We'll solve all your problems within 12 to 24 hours and with 101% guarantee and with astrology systematic. If you want to take any personal or professional advice then also you can call us on +92322-6382012 , ONLINE LOVE PROBLEM & Other all types of Daily Life Problem's.Then CALL or WHATSAPP us on +92322-6382012 and Get all these problems solutions here by Amil Baba DAWOOD BANGALI
#vashikaranspecialist #astrologer #palmistry #amliyaat #taweez #manpasandshadi #horoscope #spiritual #lovelife #lovespell #marriagespell#aamilbabainpakistan #amilbabainkarachi #powerfullblackmagicspell #kalajadumantarspecialist #realamilbaba #AmilbabainPakistan #astrologerincanada #astrologerindubai #lovespellsmaster #kalajaduspecialist #lovespellsthatwork #aamilbabainlahore#blackmagicformarriage #aamilbaba #kalajadu #kalailam #taweez #wazifaexpert #jadumantar #vashikaranspecialist #astrologer #palmistry #amliyaat #taweez #manpasandshadi #horoscope #spiritual #lovelife #lovespell #marriagespell#aamilbabainpakistan #amilbabainkarachi #powerfullblackmagicspell #kalajadumantarspecialist #realamilbaba #AmilbabainPakistan #astrologerincanada #astrologerindubai #lovespellsmaster #kalajaduspecialist #lovespellsthatwork #aamilbabainlahore #blackmagicforlove #blackmagicformarriage #aamilbaba #kalajadu #kalailam #taweez #wazifaexpert #jadumantar #vashikaranspecialist #astrologer #palmistry #amliyaat #taweez #manpasandshadi #horoscope #spiritual #lovelife #lovespell #marriagespell#aamilbabainpakistan #amilbabainkarachi #powerfullblackmagicspell #kalajadumantarspecialist #realamilbaba #AmilbabainPakistan #astrologerincanada #astrologerindubai #lovespellsmaster #kalajaduspecialist #lovespellsthatwork #aamilbabainlahore #Amilbabainuk #amilbabainspain #amilbabaindubai #Amilbabainnorway #amilbabainkrachi #amilbabainlahore #amilbabaingujranwalan #amilbabainislamabad
KuberTENes Birthday Bash Guadalajara - K8sGPT first impressionsVictor Morales
K8sGPT is a tool that analyzes and diagnoses Kubernetes clusters. This presentation was used to share the requirements and dependencies to deploy K8sGPT in a local environment.
We have compiled the most important slides from each speaker's presentation. This year’s compilation, available for free, captures the key insights and contributions shared during the DfMAy 2024 conference.
Using recycled concrete aggregates (RCA) for pavements is crucial to achieving sustainability. Implementing RCA for new pavement can minimize carbon footprint, conserve natural resources, reduce harmful emissions, and lower life cycle costs. Compared to natural aggregate (NA), RCA pavement has fewer comprehensive studies and sustainability assessments.
Hybrid optimization of pumped hydro system and solar- Engr. Abdul-Azeez.pdffxintegritypublishin
Advancements in technology unveil a myriad of electrical and electronic breakthroughs geared towards efficiently harnessing limited resources to meet human energy demands. The optimization of hybrid solar PV panels and pumped hydro energy supply systems plays a pivotal role in utilizing natural resources effectively. This initiative not only benefits humanity but also fosters environmental sustainability. The study investigated the design optimization of these hybrid systems, focusing on understanding solar radiation patterns, identifying geographical influences on solar radiation, formulating a mathematical model for system optimization, and determining the optimal configuration of PV panels and pumped hydro storage. Through a comparative analysis approach and eight weeks of data collection, the study addressed key research questions related to solar radiation patterns and optimal system design. The findings highlighted regions with heightened solar radiation levels, showcasing substantial potential for power generation and emphasizing the system's efficiency. Optimizing system design significantly boosted power generation, promoted renewable energy utilization, and enhanced energy storage capacity. The study underscored the benefits of optimizing hybrid solar PV panels and pumped hydro energy supply systems for sustainable energy usage. Optimizing the design of solar PV panels and pumped hydro energy supply systems as examined across diverse climatic conditions in a developing country, not only enhances power generation but also improves the integration of renewable energy sources and boosts energy storage capacities, particularly beneficial for less economically prosperous regions. Additionally, the study provides valuable insights for advancing energy research in economically viable areas. Recommendations included conducting site-specific assessments, utilizing advanced modeling tools, implementing regular maintenance protocols, and enhancing communication among system components.
Industrial Training at Shahjalal Fertilizer Company Limited (SFCL)MdTanvirMahtab2
This presentation is about the working procedure of Shahjalal Fertilizer Company Limited (SFCL). A Govt. owned Company of Bangladesh Chemical Industries Corporation under Ministry of Industries.
Water billing management system project report.pdfKamal Acharya
Our project entitled “Water Billing Management System” aims is to generate Water bill with all the charges and penalty. Manual system that is employed is extremely laborious and quite inadequate. It only makes the process more difficult and hard.
The aim of our project is to develop a system that is meant to partially computerize the work performed in the Water Board like generating monthly Water bill, record of consuming unit of water, store record of the customer and previous unpaid record.
We used HTML/PHP as front end and MYSQL as back end for developing our project. HTML is primarily a visual design environment. We can create a android application by designing the form and that make up the user interface. Adding android application code to the form and the objects such as buttons and text boxes on them and adding any required support code in additional modular.
MySQL is free open source database that facilitates the effective management of the databases by connecting them to the software. It is a stable ,reliable and the powerful solution with the advanced features and advantages which are as follows: Data Security.MySQL is free open source database that facilitates the effective management of the databases by connecting them to the software.
Literature Review Basics and Understanding Reference Management.pptxDr Ramhari Poudyal
Three-day training on academic research focuses on analytical tools at United Technical College, supported by the University Grant Commission, Nepal. 24-26 May 2024
Final project report on grocery store management system..pdfKamal Acharya
In today’s fast-changing business environment, it’s extremely important to be able to respond to client needs in the most effective and timely manner. If your customers wish to see your business online and have instant access to your products or services.
Online Grocery Store is an e-commerce website, which retails various grocery products. This project allows viewing various products available enables registered users to purchase desired products instantly using Paytm, UPI payment processor (Instant Pay) and also can place order by using Cash on Delivery (Pay Later) option. This project provides an easy access to Administrators and Managers to view orders placed using Pay Later and Instant Pay options.
In order to develop an e-commerce website, a number of Technologies must be studied and understood. These include multi-tiered architecture, server and client-side scripting techniques, implementation technologies, programming language (such as PHP, HTML, CSS, JavaScript) and MySQL relational databases. This is a project with the objective to develop a basic website where a consumer is provided with a shopping cart website and also to know about the technologies used to develop such a website.
This document will discuss each of the underlying technologies to create and implement an e- commerce website.
Understanding Inductive Bias in Machine LearningSUTEJAS
This presentation explores the concept of inductive bias in machine learning. It explains how algorithms come with built-in assumptions and preferences that guide the learning process. You'll learn about the different types of inductive bias and how they can impact the performance and generalizability of machine learning models.
The presentation also covers the positive and negative aspects of inductive bias, along with strategies for mitigating potential drawbacks. We'll explore examples of how bias manifests in algorithms like neural networks and decision trees.
By understanding inductive bias, you can gain valuable insights into how machine learning models work and make informed decisions when building and deploying them.
Hierarchical Digital Twin of a Naval Power SystemKerry Sado
A hierarchical digital twin of a Naval DC power system has been developed and experimentally verified. Similar to other state-of-the-art digital twins, this technology creates a digital replica of the physical system executed in real-time or faster, which can modify hardware controls. However, its advantage stems from distributing computational efforts by utilizing a hierarchical structure composed of lower-level digital twin blocks and a higher-level system digital twin. Each digital twin block is associated with a physical subsystem of the hardware and communicates with a singular system digital twin, which creates a system-level response. By extracting information from each level of the hierarchy, power system controls of the hardware were reconfigured autonomously. This hierarchical digital twin development offers several advantages over other digital twins, particularly in the field of naval power systems. The hierarchical structure allows for greater computational efficiency and scalability while the ability to autonomously reconfigure hardware controls offers increased flexibility and responsiveness. The hierarchical decomposition and models utilized were well aligned with the physical twin, as indicated by the maximum deviations between the developed digital twin hierarchy and the hardware.
Governing Equations for Fundamental Aerodynamics_Anderson2010.pdf
A Counterexample to the Forward Recursion in Fuzzy Critical Path Analysis Under Discrete Fuzzy Sets
1. International Journal of Fuzzy Logic Systems (IJFLS) Vol.6, No.2,April 2016
DOI : 10.5121/ijfls.2016.6204 53
A COUNTEREXAMPLE TO THE FORWARD
RECURSION IN FUZZY CRITICAL PATH
ANALYSIS UNDER DISCRETE FUZZY SETS
Matthew J. Liberatore1
1
Department of Management and Operations, Villanova School of Business, Villanova
University, Villanova, PA 19085
ABSTRACT
Fuzzy logic is an alternate approach for quantifying uncertainty relating to activity duration. The fuzzy
version of the backward recursion has been shown to produce results that incorrectly amplify the level of
uncertainty. However, the fuzzy version of the forward recursion has been widely proposed as an
approach for determining the fuzzy set of critical path lengths. In this paper, the direct application of the
extension principle leads to a proposition that must be satisfied in fuzzy critical path analysis. Using a
counterexample it is demonstrated that the fuzzy forward recursion when discrete fuzzy sets are used to
represent activity durations produces results that are not consistent with the theory presented. The
problem is shown to be the application of the fuzzy maximum. Several methods presented in the literature
are described and shown to provide results that are consistent with the extension principle.
KEYWORDS
critical path analysis, discrete fuzzy sets, forward recursion, counterexample, project scheduling
1. INTRODUCTION
CPM or the critical path method [11] has been successfully applied to plan and control projects
that are organized as a set of inter-related activities. PERT or Program Evaluation and Review
Technique [16] and Monte Carlo simulation apply probability analysis to address situations where
there is uncertainty related to activity duration. PERT models uncertainty by collecting
optimistic, most likely and pessimistic duration estimates of all activities and makes certain
assumptions about the underlying probability distributions. Since the basic version of PERT
tends to underestimate the expected minimum project duration [15]. Monte Carlo simulation is
often preferred in practice when activity durations are uncertain.
However, the information required to estimate probabilities related to activity duration may not
always be known. Fuzzy logic is an alternative approach for measuring uncertainty related to
activity duration. Fuzzy logic measures imprecision or vagueness in estimation, and may be
preferred to probability theory in those situations where past data concerning activity duration is
either unavailable or not relevant, the definition of the activity itself is somewhat unclear, or the
2. International Journal of Fuzzy Logic Systems (IJFLS) Vol.6, No.2,April 2016
54
notion of the activity’s completion is vague. Many authors including Chanas and colleagues have
investigated the situation when activity duration can be described by fuzzy numbers [1], [2], [3],
5], [6], [7].
The dominant approach presented in the fuzzy critical path analysis literature is the fuzzy
extension of the forward and backward recursions taken in the project network. This approach
computes the earliest and latest start and finish times and slack, where the maximum, minimum,
addition, and subtraction operators are replaced by their fuzzy counterparts. The application of
the forward recursion with fuzzy activity times was first demonstrated in [3]. In a review of the
fuzzy critical path analysis literature two approaches are described for applying the forward
recursion [4]. They also indicate that the application of the backward recursion would cause a
considerable increase in the range of uncertainty in the start and finish times that are calculated.
These authors present a modification of the backward recursion that has been proposed to
eliminate this disadvantage [12]. Some authors directly apply the backward recursion, while other
authors have proposed different approaches for modifying the backward pass when the activity
times are fuzzy [18], [19], [20], [22]. The backward recursion was found not to compute the sets
of the possible values of the latest starting times and floats of activities [22]. In a stream of
research that uses the joint possibility distribution of activity durations, several authors [8], [9]
have conducted preliminary work for computing fuzzy latest starting times and fuzzy floats,
especially for series-parallel graphs. Polynomial algorithms for determining the intervals of the
latest starting times in the general project network are presented in [22].
Unlike the fuzzy backward recursion, the use of the fuzzy forward recursion is generally accepted
in the literature as providing correct results. As mentioned by several authors [3], [10], [18], [19],
the forward recursion is correct on problems involving fuzzy intervals. The purpose of this paper
is to prove that the fuzzy forward recursion when discrete fuzzy sets are used to represent activity
durations provides results that are not consistent with the direct application of the extension
principle to the critical path problem.
In section 2, we present some background on fuzzy sets and the supporting concepts needed in
the remainder of the paper. In section 3 we provide some brief background on project network
analysis under the assumption that activity durations are certain. Section 4 presents an important
result for fuzzy critical path analysis based on the extension principle. In section 5 the
counterexample to the application of the fuzzy forward recursion is presented and the reason the
results are not consistent with the extension principle is shown. Section 6 discusses two proposed
approaches from the literature that present results consistent with the extension principle.
Conclusions are offered in section 7.
2. FUZZY SETS
Following [17] we give the following definitions of a fuzzy set, fuzzy quantity and the support
set:
Definition 1. A fuzzy set M is a subset of the universe U that is characterized by a membership
function Mµ : U→[0,1] such that:
3. International Journal of Fuzzy Logic Systems (IJFLS) Vol.6, No.2,April 2016
55
( )M xµ = 0, if x certainly is not a member of M,
( )M xµ = 1, if x certainly is a member of M,
( ) (0,1)M xµ ∈ , if it is uncertain whether x is a member of M, where ( )M xµ represents the
degree to which x is a member of M.
Now assume that activity duration is uncertain due to vagueness or imprecision and is a fuzzy
quantity defined as follows:
Definition 2. A fuzzy quantity M is a fuzzy subset of with membership function
[ ]: 0,1Mµ → such that
sup( ( ) : ) 1M x xµ ∈ = (1a)
( )
1 2 1 2
1 2
, , : ,
, ( ) 0
M M M M
M M
M
x x x x x
x x x xµ
∃ ∈ < ∀ ∈
∉ ⇒ =
(1b)
Definition 3. The support set of Mµ is defined as { }| ( ) 0M MS x xµ= ∈ > .
This definition can include discrete as well as continuous fuzzy sets, such as those that are
triangular or trapezoidal in shape. If the support set MS is finite, condition (1a) above can be
replaced with 0 0: ( ) 1Mx xµ∃ ∈ = . To simplify the presentation, we will assume this condition
hereafter. Condition (1b) implies that the support set is bounded. We will also assume that the
support set is discrete. Let iS be the support set of ia , where , , 1,2,...,i k i it S k n∈ = . That is, ia
has in possible non-zero discrete durations ,i kt in its support set iS .
As an example of a fuzzy activity duration, the vague statement that the duration of activity ia is
“about six weeks” might be represented by the following fuzzy quantity
~
it whose membership
function will be denoted as: (6) .7, (7) 1.0, (8) .8, ( ) 0i i i i itµ µ µ µ= = = = , otherwise. We can
also denote this membership function as 6/0.7, 7/1.0, 8/0.8.
We also need to define fuzzy addition and the fuzzy maximum of fuzzy quantities. Let 1M and
2M be fuzzy quantities with membership functions 1Mµ and 2Mµ , respectively. Following [17]
and as a direct application of the extension principle:
Definition 4. Fuzzy Addition: The membership function for the fuzzy quantity 21 MM ⊕ is
defined as:
)(21
zMM ⊕µ = Max Min { }1 2
( ), ( )M Mx yµ µ (2)
4. International Journal of Fuzzy Logic Systems (IJFLS) Vol.6, No.2,April 2016
56
z x y= +
Definition 5. Fuzzy Maximum: The membership function for the fuzzy quantity max (M1, M2)
is defined as:
1 2max( , ) ( )M M zµ = Max Min { }1 2
( ), ( )M Mx yµ µ (3)
max( , )z x y=
One final definition is needed:
Definition 6: A configuration Ω is defined as a precise instantiation of the duration of all ia Aε .
In the next section we provide the necessary background on project network analysis.
3. BACKGROUND ON PROJECT NETWORKS AND THE CRITICAL
PATH
Let { }1 2,, ..., NA a a a= be the set of project activities. Let iB , i=1,2,…,N, iB A⊂ , be defined so
that the elements of iB are the immediate predecessors of ia . We let { }1 2, ,..., NB B B B= be the
set of predecessor sets. Cycles of activities within the project network are not allowed. We
assume that 1a is the only element of A such that 1B = ∅. We also assume Na is the only element
of A such that .N ia B i N∉ ∀ ≠ If the project network does not have unique start and finish
activities, we add dummy activities (which have zero duration) for this purpose. Taken together
A and B define the network structure or graph G of a project.
We initially assume that it ∈ , 0it ≥ , is the certain duration for ia , where { }1 2, ,..., NT t t t= . A
path p from start to finish is a finite sequence of activities
{ }1 2
, ,..., sr r ra a a where 1
, 1,2,..., 1k kr ra B k s+
∈ = − , and 1 1ra a= and sr na a= . The length of the
longest path from the start activity to the finish activity is the minimum project completion time.
This path is called the critical path, the activities along it are called critical path activities, and
the length of the path is denoted as CPL . Given G for each unique set of values T we can
compute CPL . Therefore, there is a function f(T|G): ...× × × → that maps T to CPL , or
CPL = f(T|G).
4. FUZZY CRITICAL PATH ANALYSIS
Proposition 1: Fuzzy Critical Path Membership Function
The membership function for the fuzzy set of critical path lengths can be determined as:
5. International Journal of Fuzzy Logic Systems (IJFLS) Vol.6, No.2,April 2016
57
( )CP CPLµ = Max Min { }( )i itµ (4)
1 2, ,..., Nt t t { }1,2,...,i N∈
1 2( , ,..., | )N CPf t t t G L=
where ( )CP CPLµ is the membership function for the length of the critical path over the fuzzy
subset CP in ; and for a graph G and activity durations 1 2, ,..., Nt t t ,
1 2( , ,..., | )Nf t t t G determines the length of the critical path.
Proof: direct application of the extension principle of fuzzy logic [21] using maximum and
minimum for the disjunction and conjunction operators, respectively.
Because Proposition 1 represents the direct generalization of critical path analysis from the crisp
to the fuzzy domain, any proposed fuzzy critical path analysis approach should provide results
that are consistent with it.
One approach for implementing Proposition 1 requires defining all possible configurations,
determining the belief of each configuration as the minimum of the beliefs associated with all
activity durations included in this configuration, and then determining the maximum belief of all
configurations leading to each possible critical path length. The results form the fuzzy set of
critical path lengths.
5. COUNTER EXAMPLE TO THE FUZZY FORWARD RECURSION
In standard CPM the earliest start (ESi) and earliest finish (EFi) of an activity are defined as:
ESi = max EFj (5)
ij B∈
where we set ES0 = 0, and
EFi = ESi + it (6)
As defined in section 3 activity N does not have any successors, and so EFN is the length of the
critical path.
Assuming it% is a fuzzy quantity representing the duration of ia , the fuzzy earliest start iES and
fuzzy earliest finish iEF are defined as follows:
maxi jES EF= (7)
ij B∈
6. International Journal of Fuzzy Logic Systems (IJFLS) Vol.6, No.2,April 2016
58
i i iEF ES t= ⊕% (8)
where max and ⊕ represent the extended fuzzy maximum and fuzzy addition, respectively, as
given in definitions 4 and 5.
Proposition 2: The fuzzy forward recursion does not always yield membership functions for the
fuzzy set of critical path lengths that are consistent with Proposition 1.
Proof: By counterexample using the data from a simple series – parallel graph (these graphs are
defined in [8] as shown in Figure 1. In this example there 2*3*3*1 = 18 configurations as shown
in Table 1. For each configuration, we determine the lengths of all paths, identify the critical
path, and following Proposition 1 determine the configuration belief as the minimum of the
beliefs of the possible activity durations. We combine the configurations by taking the maximum
of the beliefs over all configurations having the same critical path length. From Table 1 the fuzzy
set of critical path lengths is shown to be:
( )CP CPLµ = 6/0.1, 7/0.2, 8/0.5, 9/0.2, 10/0.5, 11/0.1, 12/1 (9)
The fuzzy forward recursion is applied using equations (5) and (6), with fuzzy addition and fuzzy
maximum as defined in equations (2) and (3), respectively. The results are as follows:
1ES = 0/1 (10)
1EF = 0/1 ⊕ (3/0.5, 5/1) = 3/0.5, 5/1 (11)
2ES = 1EF = 3/0.5, 5/1 (12)
2EF = (3/0.5, 5/1) ⊕ (3/0.2, 5/0.5, 7/1) = 6/0.2, 8/0.5, 10/0.5, 12/1 (13)
3ES = 1EF = 3/0.5, 5/1 (14)
3EF = (3/0.5, 5/1) ⊕ (2/0.1, 4/1, 6/0.1) = 5/0.1, 7/0.5, 9/1, 11/0.1 (15)
4ES = max { 2EF , 3EF } (16)
= max {(6/0.2, 8/0.5, 10/0.5, 12/1), (5/0.1, 7/0.5, 9/1, 11/0.1)}
= 6/0.1, 7/0.2, 8/0.5, 9/0.5, 10/0.5, 11/0.1, 12/1
4EF = (6/0.1, 7/0.2, 8/0.5, 9/0.5, 10/0.5, 11/0.1, 12/1) ⊕ 0/1 (17)
= 6/0.1, 7/0.2, 8/0.5, 9/0.5, 10/0.5, 11/0.1, 12/1
where 4EF is by definition ( )CP CPLµ , the fuzzy set of critical path lengths.
7. International Journal of Fuzzy Logic Systems (IJFLS) Vol.6, No.2,April 2016
59
A comparison of equations (9) and (17) show that the range of possible critical path lengths is the
same, but that the belief of the possible critical path length of 9 is 0.2 in equation (9) and 0.5 in
equation (17). This difference proves the assertion.
Discussion: The forward algorithm is generally regarded as providing correct results in the
literature, with no distinction made for activity durations represented as discrete fuzzy sets. The
difference in the belief for the possible critical path length of 9 from Example 1 results from
incorrect comparisons made in the computation of the fuzzy maximum of the earliest finishes.
Specifically, the fuzzy maximum incorrectly compares earliest finish values that are based on
activity durations from different configurations. These configurations differ in terms of the
possible values of the activity that is the immediate predecessor of those activities whose earliest
finishes are being compared.
For example, in equation (16) the possible time 8/0.5 from 2EF is compared with the possible
time 9/1 from 3EF to yield 9/0.5. Folding back, in equation (13) the possible duration 8/0.5
results from the addition of 3/0.5 from 1EF and 5/0.5 from the possible duration of 2a . Since
the 1ES is 0, 3/0.5 is the possible duration of 1a . In equation (15) the possible duration 9/1 results
from the addition of 5/1 from 1EF and 4/1 from the possible duration of 3a . Since the 1ES is 0,
5/1 is the possible duration of 1a . That is, different possible durations of 1a are used to obtain the
possible time 8/0.5 from 2EF and the possible time 9/1 from 3EF . Thus, the fuzzy maximum
incorrectly compares earliest finishes that are based on activity durations from different
configurations. In the example above, if the possible duration of 1a were 3/0.2 instead of 3/0.5,
then the fuzzy forward recursion would have produced the correct result, simply because of the
specific value of a belief. However, since the fuzzy maximum is used repeatedly in the
application of the fuzzy forward recursion, errors are likely to occur in many, if not most,
problems.
6.PROPOSED CRITICAL PATH METHODS THAT ARE
CONSISTENT WITH THE EXTENSION PRINCIPLE
This author has proposed several alternative approaches for computing the fuzzy set of critical
path lengths that are consistent with the extension principle and Proposition 2. The first approach
[13] is based on the idea that rather than enumerate all possible configurations, generate and then
analyze a small number of configurations to achieve convergence to the membership function of
the fuzzy set of critical path lengths. This method uses pseudo-random numbers to select possible
durations for each activity to form configurations. Either the standard forward pass or path
enumeration can be applied to determine the critical path length of a configuration. Following
Proposition 1, the belief associated with the possible critical path length of a given configuration
is the minimum of the beliefs of the possible activity durations in the configuration. Since several
configurations may lead to the same possible critical path length, again following Proposition 1
and the extension principle, the belief of a possible critical path length is the maximum of the
beliefs associated with all possible critical path lengths of the same duration. Additional
8. International Journal of Fuzzy Logic Systems (IJFLS) Vol.6, No.2,April 2016
60
configurations are analyzed until the percentage change in AREA (sum of the products of all
possible critical path lengths and their beliefs) is less than some tolerance. The proposed method
is evaluated using a series of test problems from the literature.
The second approach [14] begins by using a standard forward-backward pass algorithm to
compute the minimum possible critical path length, using the smallest possible duration for each
activity. A similar approach determines the maximum possible critical path length. Following
Proposition 1 a mathematical programming problem is formulated, whose objective is to
determine the belief associated with a possible critical path length. This mathematical program is
run for all possible critical path lengths, ranging between the possible minimum and maximum
lengths. In this way, the membership function for the fuzzy set of critical path lengths is
constructed. The proposed method is evaluated using a series of test problems from the literature.
7. CONCLUSIONS
In this paper, the direct application of the extension principle leads to a proposition that must be
satisfied in fuzzy critical path analysis. Using a counterexample where discrete fuzzy sets are
used to represent activity durations, it is demonstrated that the fuzzy forward recursion produces
results that are not consistent with the theory presented. The problem is shown to be the
application of the fuzzy maximum, since it compares earliest finishes that are based on activity
durations from different configurations. Several methods offered in the literature that provide
results consistent with the extension principle are described. The development of more efficient
algorithms to compute the fuzzy set of critical path lengths is an area of further research.
Figure 1. Example network with fuzzy activity durations
9. International Journal of Fuzzy Logic Systems (IJFLS) Vol.6, No.2,April 2016
61
Table 1: Applying proposition 1 to the example
Activity 1 Activity 2 Activity 3 P 1 - 2 - 4 P 1 - 3 - 4 Critical Path CP fuzzy set
Duration Belief Duration Belief Duration Belief Duration Duration Duration Belief Duration Belief
3 0.5 3 0.2 2 0.1 6 5 6 0.1 6 0.1
3 0.5 3 0.2 4 1 6 7 7 0.2 7 0.2
3 0.5 3 0.2 6 0.1 6 9 9 0.1 8 0.5
3 0.5 5 0.5 2 0.1 8 5 8 0.1 9 0.2
3 0.5 5 0.5 4 1 8 7 8 0.5 10 0.5
3 0.5 5 0.5 6 0.1 8 9 9 0.1 11 0.1
3 0.5 7 1 2 0.1 10 5 10 0.1 12 1
3 0.5 7 1 4 1 10 7 10 0.5
3 0.5 7 1 6 0.1 10 9 10 0.1
5 1 3 0.2 2 0.1 8 7 8 0.1
5 1 3 0.2 4 1 8 9 9 0.2
5 1 3 0.2 6 0.1 8 11 11 0.1
5 1 5 0.5 2 0.1 10 7 10 0.1
5 1 5 0.5 4 1 10 9 10 0.5
5 1 5 0.5 6 0.1 10 11 11 0.1
5 1 7 1 2 0.1 12 7 12 0.1
5 1 7 1 4 1 12 9 12 1
5 1 7 1 6 0.1 12 11 12 0.1
*activity 4 is a dummy whose duration is certainly 0
REFERENCES
[1] Chanas, S. (1982) “Fuzzy sets in few classical operational research problems” in Gupta M. &
Sanches, E. (eds.), Approximate Reasoning in Decision Analysis, North-Holland Publishing
Company, pp. 351-361.
[2] Chanas, S., Dubois, D. & Zielinksi, P. (2002) “On the sure criticality of tasks in activity networks
with imprecise durations”, IEEE Transactions on Systems, Man, and Cybernetics – Part B:
Cybernetics, Vol. 32, pp. 393-407.
[3] Chanas, S. & Kamburowski, J. (1981) “The use of fuzzy variables in PERT”, Fuzzy Sets and
Systems, Vol. 1, pp. 11 – 19.
[4] Chanas, S. & Kuchta, D. (1998) “Discrete fuzzy optimization”, Chapter 8 in Slowinski, R. (ed.),
Fuzzy sets in Decision Analysis, Operations Research and Statistics, Kluwer Academic Publishers,
Boston, pp. 249-280.
[5] Chanas, S. & Zielinski, P. (2001) “Critical path analysis in the network with fuzzy activity times”,
Fuzzy Sets and Systems, Vol. 122, pp. 195-204.
[6] Chanas, S. & Zielinski, P. (2002) “The computational complexity of the criticality problems in a
network with interval activity times”, European Journal of Operational Research, Vol. 136, pp. 541 –
550.
[7] Chanas, S. & Zielinski, P. (2003) “On the hardness of evaluating criticality of activities in a planar
network with duration intervals”, Operations Research Letters, Vol. 31, pp. 53 – 59.
[8] Dubois, D., Fargier, H. & Galvagnon, V. (2003) “On the latest starting times and floats in activity
networks with ill-known durations”, European Journal of Operational Research, Vol. 147, pp. 266-
280.
[9] Fargier, H., Galvagnon, V. & Dubois, D. (2000) “Fuzzy PERT in series-parallel graphs”, in Ninth
IEEE International Conference on Fuzzy Systems, San Antonio, TX, pp. 717-722.
10. International Journal of Fuzzy Logic Systems (IJFLS) Vol.6, No.2,April 2016
62
[10] Hapke, M., & Slowinski, R. (1996) “Fuzzy priority heuristics for project scheduling”, Fuzzy Sets and
Systems, Vol. 83, pp. 291-299.
[11] Kelley, J. (1961) “Critical path planning and scheduling -- mathematical basis”, Operations Research,
Vol. 9, pp. 296-320.
[12] Kamburowski, J. (1983) “Fuzzy activity duration times in critical path analysis”, International
Symposium on Project Management, New Delhi, pp. 194-199.
[13] Liberatore, M. (2007) “A latin hypercube sampling approach for fuzzy critical path analysis”,
International Journal of Operations and Quantitative Management, Vol. 13, No. 4, pp. 235 – 250.
[14] Liberatore, M. (2008) ”Critical path analysis with fuzzy activity times”, IEEE Transactions on
Engineering Management, Vol. 55, No. 2, pp. 329 - 337.
[15] Lootsma, F. A. (1989) “Stochastic and fuzzy PERT”, European Journal of Operational Research, Vol.
43, pp. 174-183.
[16] Malcolm, D., Rosenbloom, J., Clark, C. & and Fazar, W. (1959) “Application of a technique for
research and development project evaluation”, Operations Research, Vol. 7, pp. 646-669.
[17] Mares, M. (1994) Computation Over Fuzzy Quantities, CRC Press, Boca Raton.
[18] Nasution, S. (1994) “Fuzzy critical path method”, IEEE Transactions on Systems, Man, and
Cybernetics, Vol. 24, pp. 48-57.
[19] Rommelfanger, H. (1994) “Network analysis and information flow in fuzzy environment”, Fuzzy Sets
and Systems, Vol. 67, pp. 119-128.
[20] Slyeptsov, A. & Tyschchuk, T. (2003) “A method of computation of characteristics of operations in a
problem of fuzzy network planning and management”, Cybernetics and Systems Analysis, Vol. 39,
pp. 367 – 378.
[21] Zadeh, L. (1975) “Calculus of fuzzy restrictions”, in Zadeh, L., Fu, K., Tanaka, K. & Shimara, J.
(eds.), Fuzzy Sets and Their Application to Cognitive and Decision Processes, Academic Process, pp.
1-39.
[22] Zielinski, P. (2005) “On computing the latest starting times and floats of activities in a network with
imprecise durations”, Fuzzy Sets and Systems, Vol. 150, pp. 53 – 76.