AN ALGORITHM FOR SOLVING LINEAR OPTIMIZATION PROBLEMS SUBJECTED TO THE INTERSECTION OF TWO FUZZY RELATIONAL INEQUALITIES DEFINED BY FRANK FAMILY OF T-NORMS
Frank t-norms are parametric family of continuous Archimedean t-norms whose members are also strict functions. Very often, this family of t-norms is also called the family of fundamental t-norms because of the
role it plays in several applications. In this paper, optimization of a linear objective function with fuzzy relational inequality constraints is investigated.
The numerical solution of Huxley equation by the use of two finite difference methods is done. The first one is the explicit scheme and the second one is the Crank-Nicholson scheme. The comparison between the two methods showed that the explicit scheme is easier and has faster convergence while the Crank-Nicholson scheme is more accurate. In addition, the stability analysis using Fourier (von Neumann) method of two schemes is investigated. The resulting analysis showed that the first scheme
is conditionally stable if, r ≤ 2 − aβ∆t , ∆t ≤ 2(∆x)2 and the second
scheme is unconditionally stable.
Second or fourth-order finite difference operators, which one is most effective?Premier Publishers
This paper presents higher-order finite difference (FD) formulas for the spatial approximation of the time-dependent reaction-diffusion problems with a clear justification through examples, “why fourth-order FD formula is preferred to its second-order counterpart” that has been widely used in literature. As a consequence, methods for the solution of initial and boundary value PDEs, such as the method of lines (MOL), is of broad interest in science and engineering. This procedure begins with discretizing the spatial derivatives in the PDE with algebraic approximations. The key idea of MOL is to replace the spatial derivatives in the PDE with the algebraic approximations. Once this procedure is done, the spatial derivatives are no longer stated explicitly in terms of the spatial independent variables. In other words, only one independent variable is remaining, the resulting semi-discrete problem has now become a system of coupled ordinary differential equations (ODEs) in time. Thus, we can apply any integration algorithm for the initial value ODEs to compute an approximate numerical solution to the PDE. Analysis of the basic properties of these schemes such as the order of accuracy, convergence, consistency, stability and symmetry are well examined.
In conventional transportation problem (TP), supplies, demands and costs are always certain. This paper develops an approach to solve the unbalanced transportation problem where as all the parameters are not in deterministic numbers but imprecise ones. Here, all the parameters of the TP are considered to the triangular intuitionistic fuzzy numbers (TIFNs). The existing ranking procedure of Varghese and Kuriakose is used to transform the unbalanced intuitionistic fuzzy transportation problem (UIFTP) into a crisp one so that the conventional method may be applied to solve the TP. The occupied cells of unbalanced crisp TP that we obtained are as same as the occupied cells of UIFTP.
On the basis of this idea the solution procedure is differs from unbalanced crisp TP to UIFTP in allocation step only. Therefore, the new method and new multiplication operation on triangular intuitionistic fuzzy number (TIFN) is proposed to find the optimal solution in terms of TIFN. The main advantage of this method is computationally very simple, easy to understand and also the optimum objective value obtained by our method is physically meaningful.
A Mathematical Model to Solve Nonlinear Initial and Boundary Value Problems b...IJERA Editor
In this paper, a novel method called Laplace-differential transform method (LDTM) is used to obtain an
approximate analytical solution for strong nonlinear initial and boundary value problems associated in
engineering phenomena. It is determined that the method works very well for the wide range of parameters and
an excellent agreement is demonstrated and discussed between the approximate solution and the exact one in
three examples. The most significant features of this method are its capability of handling non-linear boundary
value problems.
The numerical solution of Huxley equation by the use of two finite difference methods is done. The first one is the explicit scheme and the second one is the Crank-Nicholson scheme. The comparison between the two methods showed that the explicit scheme is easier and has faster convergence while the Crank-Nicholson scheme is more accurate. In addition, the stability analysis using Fourier (von Neumann) method of two schemes is investigated. The resulting analysis showed that the first scheme
is conditionally stable if, r ≤ 2 − aβ∆t , ∆t ≤ 2(∆x)2 and the second
scheme is unconditionally stable.
Second or fourth-order finite difference operators, which one is most effective?Premier Publishers
This paper presents higher-order finite difference (FD) formulas for the spatial approximation of the time-dependent reaction-diffusion problems with a clear justification through examples, “why fourth-order FD formula is preferred to its second-order counterpart” that has been widely used in literature. As a consequence, methods for the solution of initial and boundary value PDEs, such as the method of lines (MOL), is of broad interest in science and engineering. This procedure begins with discretizing the spatial derivatives in the PDE with algebraic approximations. The key idea of MOL is to replace the spatial derivatives in the PDE with the algebraic approximations. Once this procedure is done, the spatial derivatives are no longer stated explicitly in terms of the spatial independent variables. In other words, only one independent variable is remaining, the resulting semi-discrete problem has now become a system of coupled ordinary differential equations (ODEs) in time. Thus, we can apply any integration algorithm for the initial value ODEs to compute an approximate numerical solution to the PDE. Analysis of the basic properties of these schemes such as the order of accuracy, convergence, consistency, stability and symmetry are well examined.
In conventional transportation problem (TP), supplies, demands and costs are always certain. This paper develops an approach to solve the unbalanced transportation problem where as all the parameters are not in deterministic numbers but imprecise ones. Here, all the parameters of the TP are considered to the triangular intuitionistic fuzzy numbers (TIFNs). The existing ranking procedure of Varghese and Kuriakose is used to transform the unbalanced intuitionistic fuzzy transportation problem (UIFTP) into a crisp one so that the conventional method may be applied to solve the TP. The occupied cells of unbalanced crisp TP that we obtained are as same as the occupied cells of UIFTP.
On the basis of this idea the solution procedure is differs from unbalanced crisp TP to UIFTP in allocation step only. Therefore, the new method and new multiplication operation on triangular intuitionistic fuzzy number (TIFN) is proposed to find the optimal solution in terms of TIFN. The main advantage of this method is computationally very simple, easy to understand and also the optimum objective value obtained by our method is physically meaningful.
A Mathematical Model to Solve Nonlinear Initial and Boundary Value Problems b...IJERA Editor
In this paper, a novel method called Laplace-differential transform method (LDTM) is used to obtain an
approximate analytical solution for strong nonlinear initial and boundary value problems associated in
engineering phenomena. It is determined that the method works very well for the wide range of parameters and
an excellent agreement is demonstrated and discussed between the approximate solution and the exact one in
three examples. The most significant features of this method are its capability of handling non-linear boundary
value problems.
Considerations on the genetic equilibrium lawIOSRJM
In the first part of the paper I willpresentabriefreview on the Hardy-Weinberg equilibrium and it's formulation in projective algebraicgeometry. In the second and last part I willdiscussexamples and generalizations on the topic
International journal of engineering and mathematical modelling vol2 no1_2015_2IJEMM
This paper is devoted to the homogenization of the Maxwell equations with periodically oscillating coefficients in the bianisotropic media which represents the most general linear media. In the first time, the limiting homogeneous constitutive law is rigorously justified in the frequency domain and is found from the solution of a local problem on the unit cell. The homogenization process is based on the two-scale convergence conception. In the second time, the implementation of the homogeneous
constitutive law by using the finite element method and the introduction of the boundary conditions in the discrete problem are introduced. Finally, the numerical results associated of the perforated chiral media are presented.
Fractional pseudo-Newton method and its use in the solution of a nonlinear sy...mathsjournal
The following document presents a possible solution and a brief stability analysis for a nonlinear system,
which is obtained by studying the possibility of building a hybrid solar receiver; It is necessary to mention that
the solution of the aforementioned system is relatively difficult to obtain through iterative methods since the
system is apparently unstable. To find this possible solution is used a novel numerical method valid for one and
several variables, which using the fractional derivative, allows us to find solutions for some nonlinear systems in
the complex space using real initial conditions, this method is also valid for linear systems. The method described
above has an order of convergence (at least) linear, but it is easy to implement and it is not necessary to invert
some matrix for solving nonlinear systems and linear systems.
Accelerating materials property predictions using machine learningGhanshyam Pilania
The materials discovery process can be significantly expedited and simplified if we can learn effectively from available knowledge and data. In the present contribution, we show that efficient and accurate prediction of a diverse set of properties of material systems is possible by employing machine (or statistical) learning
methods trained on quantum mechanical computations in combination with the notions of chemical similarity. Using a family of one-dimensional chain systems, we present a general formalism that allows us to discover decision rules that establish a mapping between easily accessible attributes of a system and its properties. It is shown that fingerprints based on either chemo-structural (compositional and configurational information) or the electronic charge density distribution can be used to make ultra-fast, yet accurate, property predictions. Harnessing such learning paradigms extends recent efforts to systematically explore and mine vast chemical spaces, and can significantly accelerate the discovery of new application-specific materials.
The International Journal of Engineering and Science (The IJES)theijes
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
Correlation measure for intuitionistic fuzzy multi setseSAT Journals
Abstract In this paper, the Correlation measure of Intuitionistic Fuzzy Multi sets (IFMS) is proposed. The concept of this Correlation measure of IFMS is the extension of Correlation measure of IFS. Using the Correlation of IFMS measure, the application of medical diagnosis and pattern recognition are presented. The new method also shows that the correlation measure of any two IFMS equals one if and only if the two IFMS are the same. Keywords: Intuitionistic fuzzy set, Intuitionistic Fuzzy Multi sets, Correlation measure.
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
This work considers the multi-objective optimization problem constrained by a system of bipolar fuzzy relational equations with max-product composition. An integer optimization based technique for order of preference by similarity to the ideal solution is proposed for solving such a problem. Some critical features associated with the feasible domain and optimal solutions of the bipolar max-Tp equation constrained optimization problem are studied. An illustrative example verifying the idea of this paper is included. This is the first attempt to study the bipolar max-T equation constrained multi-objective optimization problems from an integer programming viewpoint.
This work considers the multi-objective optimization problem constrained by a system of bipolar fuzzy relational equations with max-product composition. An integer optimization based technique for order of preference by similarity to the ideal solution is proposed for solving such a problem. Some critical features associated with the feasible domain and optimal solutions of the bipolar max-Tp equation constrained optimization problem are studied. An illustrative example verifying the idea of this paper is included. This
is the first attempt to study the bipolar max-T equation constrained multi-objective optimization problems
from an integer programming viewpoint.
CHN and Swap Heuristic to Solve the Maximum Independent Set ProblemIJECEIAES
We describe a new approach to solve the problem to find the maximum independent set in a given Graph, known also as Max-Stable set problem (MSSP). In this paper, we show how Max-Stable problem can be reformulated into a linear problem under quadratic constraints, and then we resolve the QP result by a hybrid approach based Continuous Hopfeild Neural Network (CHN) and Local Search. In a manner that the solution given by the CHN will be the starting point of the local search. The new approach showed a good performance than the original one which executes a suite of CHN runs, at each execution a new leaner constraint is added into the resolved model. To prove the efficiency of our approach, we present some computational experiments of solving random generated problem and typical MSSP instances of real life problem.
International Journal of Mathematics and Statistics Invention (IJMSI) inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
On the Fractional Optimal Control Problems With Singular and Non–Singular ...Scientific Review SR
The aim of this paper is to design an efficient numerical method to solve a class of time fractional optimal control
problems. In this problem formulation, the fractional derivative operator is consid- ered in three cases with both
singular and non–singular kernels. The necessary conditions are derived for the optimality of these problems and the
proposed method is evaluated for different choices of derivative operators. Simulation results indicate that the
suggested technique works well and pro- vides satisfactory results with considerably less computational time than
the other existing methods. Comparative results also verify that the fractional operator with Mittag –Leffler kernel in
the Caputo sense improves the performance of the controlled system in terms of the transient response compared to
the other fractional and integer derivative operators.
A DERIVATIVE FREE HIGH ORDERED HYBRID EQUATION SOLVERZac Darcy
Generally a range of equation solvers for estimating the solution of an equation contain the derivative of
first or higher order. Such solvers are difficult to apply in the instances of complicated functional
relationship. The equation solver proposed in this paper meant to solve many of the involved complicated
problems and establishing a process tending towards a higher ordered by alloying the already proved
conventional methods like Newton-Raphson method (N-R), Regula Falsi method (R-F) & Bisection method
(BIS). The present method is good to solve those nonlinear and transcendental equations that cannot be
solved by the basic algebra. Comparative analysis are also made with the other racing formulas of this
group and the result shows that present method is faster than all such methods of the class.
Considerations on the genetic equilibrium lawIOSRJM
In the first part of the paper I willpresentabriefreview on the Hardy-Weinberg equilibrium and it's formulation in projective algebraicgeometry. In the second and last part I willdiscussexamples and generalizations on the topic
International journal of engineering and mathematical modelling vol2 no1_2015_2IJEMM
This paper is devoted to the homogenization of the Maxwell equations with periodically oscillating coefficients in the bianisotropic media which represents the most general linear media. In the first time, the limiting homogeneous constitutive law is rigorously justified in the frequency domain and is found from the solution of a local problem on the unit cell. The homogenization process is based on the two-scale convergence conception. In the second time, the implementation of the homogeneous
constitutive law by using the finite element method and the introduction of the boundary conditions in the discrete problem are introduced. Finally, the numerical results associated of the perforated chiral media are presented.
Fractional pseudo-Newton method and its use in the solution of a nonlinear sy...mathsjournal
The following document presents a possible solution and a brief stability analysis for a nonlinear system,
which is obtained by studying the possibility of building a hybrid solar receiver; It is necessary to mention that
the solution of the aforementioned system is relatively difficult to obtain through iterative methods since the
system is apparently unstable. To find this possible solution is used a novel numerical method valid for one and
several variables, which using the fractional derivative, allows us to find solutions for some nonlinear systems in
the complex space using real initial conditions, this method is also valid for linear systems. The method described
above has an order of convergence (at least) linear, but it is easy to implement and it is not necessary to invert
some matrix for solving nonlinear systems and linear systems.
Accelerating materials property predictions using machine learningGhanshyam Pilania
The materials discovery process can be significantly expedited and simplified if we can learn effectively from available knowledge and data. In the present contribution, we show that efficient and accurate prediction of a diverse set of properties of material systems is possible by employing machine (or statistical) learning
methods trained on quantum mechanical computations in combination with the notions of chemical similarity. Using a family of one-dimensional chain systems, we present a general formalism that allows us to discover decision rules that establish a mapping between easily accessible attributes of a system and its properties. It is shown that fingerprints based on either chemo-structural (compositional and configurational information) or the electronic charge density distribution can be used to make ultra-fast, yet accurate, property predictions. Harnessing such learning paradigms extends recent efforts to systematically explore and mine vast chemical spaces, and can significantly accelerate the discovery of new application-specific materials.
The International Journal of Engineering and Science (The IJES)theijes
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
Correlation measure for intuitionistic fuzzy multi setseSAT Journals
Abstract In this paper, the Correlation measure of Intuitionistic Fuzzy Multi sets (IFMS) is proposed. The concept of this Correlation measure of IFMS is the extension of Correlation measure of IFS. Using the Correlation of IFMS measure, the application of medical diagnosis and pattern recognition are presented. The new method also shows that the correlation measure of any two IFMS equals one if and only if the two IFMS are the same. Keywords: Intuitionistic fuzzy set, Intuitionistic Fuzzy Multi sets, Correlation measure.
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
Similar to AN ALGORITHM FOR SOLVING LINEAR OPTIMIZATION PROBLEMS SUBJECTED TO THE INTERSECTION OF TWO FUZZY RELATIONAL INEQUALITIES DEFINED BY FRANK FAMILY OF T-NORMS
This work considers the multi-objective optimization problem constrained by a system of bipolar fuzzy relational equations with max-product composition. An integer optimization based technique for order of preference by similarity to the ideal solution is proposed for solving such a problem. Some critical features associated with the feasible domain and optimal solutions of the bipolar max-Tp equation constrained optimization problem are studied. An illustrative example verifying the idea of this paper is included. This is the first attempt to study the bipolar max-T equation constrained multi-objective optimization problems from an integer programming viewpoint.
This work considers the multi-objective optimization problem constrained by a system of bipolar fuzzy relational equations with max-product composition. An integer optimization based technique for order of preference by similarity to the ideal solution is proposed for solving such a problem. Some critical features associated with the feasible domain and optimal solutions of the bipolar max-Tp equation constrained optimization problem are studied. An illustrative example verifying the idea of this paper is included. This
is the first attempt to study the bipolar max-T equation constrained multi-objective optimization problems
from an integer programming viewpoint.
CHN and Swap Heuristic to Solve the Maximum Independent Set ProblemIJECEIAES
We describe a new approach to solve the problem to find the maximum independent set in a given Graph, known also as Max-Stable set problem (MSSP). In this paper, we show how Max-Stable problem can be reformulated into a linear problem under quadratic constraints, and then we resolve the QP result by a hybrid approach based Continuous Hopfeild Neural Network (CHN) and Local Search. In a manner that the solution given by the CHN will be the starting point of the local search. The new approach showed a good performance than the original one which executes a suite of CHN runs, at each execution a new leaner constraint is added into the resolved model. To prove the efficiency of our approach, we present some computational experiments of solving random generated problem and typical MSSP instances of real life problem.
International Journal of Mathematics and Statistics Invention (IJMSI) inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
On the Fractional Optimal Control Problems With Singular and Non–Singular ...Scientific Review SR
The aim of this paper is to design an efficient numerical method to solve a class of time fractional optimal control
problems. In this problem formulation, the fractional derivative operator is consid- ered in three cases with both
singular and non–singular kernels. The necessary conditions are derived for the optimality of these problems and the
proposed method is evaluated for different choices of derivative operators. Simulation results indicate that the
suggested technique works well and pro- vides satisfactory results with considerably less computational time than
the other existing methods. Comparative results also verify that the fractional operator with Mittag –Leffler kernel in
the Caputo sense improves the performance of the controlled system in terms of the transient response compared to
the other fractional and integer derivative operators.
A DERIVATIVE FREE HIGH ORDERED HYBRID EQUATION SOLVERZac Darcy
Generally a range of equation solvers for estimating the solution of an equation contain the derivative of
first or higher order. Such solvers are difficult to apply in the instances of complicated functional
relationship. The equation solver proposed in this paper meant to solve many of the involved complicated
problems and establishing a process tending towards a higher ordered by alloying the already proved
conventional methods like Newton-Raphson method (N-R), Regula Falsi method (R-F) & Bisection method
(BIS). The present method is good to solve those nonlinear and transcendental equations that cannot be
solved by the basic algebra. Comparative analysis are also made with the other racing formulas of this
group and the result shows that present method is faster than all such methods of the class.
The aim of this research is to find accurate solution for the Troesch’s problem by using high performance technique based on parallel processing implementation.
Design/methodology/approach – Feed forward neural network is designed to solve important type of differential equations that arises in many applied sciences and engineering applications. The suitable designed based on choosing suitable learning rate, transfer function, and training algorithm. The authors used back propagation with new implement of Levenberg - Marquardt training algorithm. Also, the authors depend new idea for choosing the weights. The effectiveness of the suggested design for the network is shown by using it for solving Troesch problem in many cases.
Findings – New idea for choosing the weights of the neural network, new implement of Levenberg - Marquardt training algorithm which assist to speeding the convergence and the implementation of the suggested design demonstrates the usefulness in finding exact solutions.
A derivative free high ordered hybrid equation solverZac Darcy
Generally a range of equation solvers for estimating the solution of an equation contain the derivative of
first or higher order. Such solvers are difficult to apply in the instances of complicated functional
relationship. The equation solver proposed in this paper meant to solve many of the involved complicated
problems and establishing a process tending towards a higher ordered by alloying the already proved
conventional methods like Newton-Raphson method (N-R), Regula Falsi method (R-F) & Bisection method
(BIS). The present method is good to solve those nonlinear and transcendental equations that cannot be
solved by the basic algebra. Comparative analysis are also made with the other racing formulas of this
group and the result shows that present method is faster than all such methods of the class.
A Derivative Free High Ordered Hybrid Equation Solver Zac Darcy
Generally a range of equation solvers for estimating the solution of an equation contain the derivative of
first or higher order. Such solvers are difficult to apply in the instances of complicated functional
relationship. The equation solver proposed in this paper meant to solve many of the involved complicated
problems and establishing a process tending towards a higher ordered by alloying the already proved
conventional methods like Newton-Raphson method (N-R), Regula Falsi method (R-F) & Bisection method
(BIS). The present method is good to solve those nonlinear and transcendental equations that cannot be
solved by the basic algebra. Comparative analysis are also made with the other racing formulas of this
group and the result shows that present method is faster than all such methods of the class.
Computer Science
Active and Programmable Networks
Active safety systems
Ad Hoc & Sensor Network
Ad hoc networks for pervasive communications
Adaptive, autonomic and context-aware computing
Advance Computing technology and their application
Advanced Computing Architectures and New Programming Models
Advanced control and measurement
Aeronautical Engineering,
Agent-based middleware
Alert applications
Automotive, marine and aero-space control and all other control applications
Autonomic and self-managing middleware
Autonomous vehicle
Biochemistry
Bioinformatics
BioTechnology(Chemistry, Mathematics, Statistics, Geology)
Broadband and intelligent networks
Broadband wireless technologies
CAD/CAM/CAT/CIM
Call admission and flow/congestion control
Capacity planning and dimensioning
Changing Access to Patient Information
Channel capacity modelling and analysis
Civil Engineering,
Cloud Computing and Applications
Collaborative applications
Communication application
Communication architectures for pervasive computing
Communication systems
Computational intelligence
Computer and microprocessor-based control
Computer Architecture and Embedded Systems
Computer Business
Computer Sciences and Applications
Computer Vision
Computer-based information systems in health care
Computing Ethics
Computing Practices & Applications
Congestion and/or Flow Control
Content Distribution
Context-awareness and middleware
Creativity in Internet management and retailing
Cross-layer design and Physical layer based issue
Cryptography
Data Base Management
Data fusion
Data Mining
Data retrieval
Data Storage Management
Decision analysis methods
Decision making
Digital Economy and Digital Divide
Digital signal processing theory
Distributed Sensor Networks
Drives automation
Drug Design,
Drug Development
DSP implementation
E-Business
E-Commerce
E-Government
Electronic transceiver device for Retail Marketing Industries
Electronics Engineering,
Embeded Computer System
Emerging advances in business and its applications
Emerging signal processing areas
Enabling technologies for pervasive systems
Energy-efficient and green pervasive computing
Environmental Engineering,
Estimation and identification techniques
Evaluation techniques for middleware solutions
Event-based, publish/subscribe, and message-oriented middleware
Evolutionary computing and intelligent systems
Expert approaches
Facilities planning and management
Flexible manufacturing systems
Formal methods and tools for designing
Fuzzy algorithms
Fuzzy logics
GPS and location-based app
VALIDATION METHOD OF FUZZY ASSOCIATION RULES BASED ON FUZZY FORMAL CONCEPT AN...cscpconf
In order to treat and analyze real datasets, fuzzy association rules have been proposed. Several
algorithms have been introduced to extract these rules. However, these algorithms suffer from
the problems of utility, redundancy and large number of extracted fuzzy association rules. The
expert will then be confronted with this huge amount of fuzzy association rules. The task of
validation becomes fastidious. In order to solve these problems, we propose a new validation
method. Our method is based on three steps. (i) We extract a generic base of non redundant
fuzzy association rules by applying EFAR-PN algorithm based on fuzzy formal concept analysis.
(ii) we categorize extracted rules into groups and (iii) we evaluate the relevance of these rules
using structural equation model.
THE LEFT AND RIGHT BLOCK POLE PLACEMENT COMPARISON STUDY: APPLICATION TO FLIG...ieijjournal1
It is known that if a linear-time-invariant MIMO system described by a state space equation has a number
of states divisible by the number of inputs and it can be transformed to block controller form, we can
design a state feedback controller using block pole placement technique by assigning a set of desired Block
poles. These may be left or right block poles. The idea is to compare both in terms of system’s response.
THE LEFT AND RIGHT BLOCK POLE PLACEMENT COMPARISON STUDY: APPLICATION TO FLIG...ieijjournal
It is known that if a linear-time-invariant MIMO system described by a state space equation has a number of states divisible by the number of inputs and it can be transformed to block controller form, we can design a state feedback controller using block pole placement technique by assigning a set of desired Block poles. These may be left or right block poles. The idea is to compare both in terms of system’s response.
A Mixed Binary-Real NSGA II Algorithm Ensuring Both Accuracy and Interpretabi...IJECEIAES
In this work, a Neuro-Fuzzy Controller network, called NFC that implements a Mamdani fuzzy inference system is proposed. This network includes neurons able to perform fundamental fuzzy operations. Connections between neurons are weighted through binary and real weights. Then a mixed binaryreal Non dominated Sorting Genetic Algorithm II (NSGA II) is used to perform both accuracy and interpretability of the NFC by minimizing two objective functions; one objective relates to the number of rules, for compactness, while the second is the mean square error, for accuracy. In order to preserve interpretability of fuzzy rules during the optimization process, some constraints are imposed. The approach is tested on two control examples: a single input single output (SISO) system and a multivariable (MIMO) system.
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.
Adaptive Projective Lag Synchronization of T and Lu Chaotic Systems IJECEIAES
In this paper, the synchronization problem of T chaotic system and Lu chaotic system is studied. The parameter of the drive T chaotic system is considered unknown. An adaptive projective lag control method and also parameter estimation law are designed to achieve chaos synchronization problem between two chaotic systems. Then Lyapunov stability theorem is utilized to prove the validity of the proposed control method. After that, some numerical simulations are performed to assess the performance of the proposed method. The results show high accuracy of the proposed method in control and synchronization of chaotic systems.
Similar to AN ALGORITHM FOR SOLVING LINEAR OPTIMIZATION PROBLEMS SUBJECTED TO THE INTERSECTION OF TWO FUZZY RELATIONAL INEQUALITIES DEFINED BY FRANK FAMILY OF T-NORMS (20)
ENHANCING ENGLISH WRITING SKILLS THROUGH INTERNET-PLUS TOOLS IN THE PERSPECTI...ijfcstjournal
This investigation delves into incorporating a hybridized memetic strategy within the framework of English
composition pedagogy, leveraging Internet Plus resources. The study aims to provide an in-depth analysis
of how this method influences students’ writing competence, their perceptions of writing, and their
enthusiasm for English acquisition. Employing an explanatory research design that combines qualitative
and quantitative methods, the study collects data through surveys, interviews, and observations of students’
writing performance before and after the intervention. Findings demonstrate a beneficial impact of
integrating the memetic approach alongside Internet Plus tools on the writing aptitude of English as a
Foreign Language (EFL) learners. Students reported increased engagement with writing, attributing it to
the use of Internet plus tools. They also expressed that the memetic approach facilitated a deeper
understanding of cultural and social contexts in writing. Furthermore, the findings highlight a significant
improvement in students’ writing skills following the intervention. This study provides significant insights
into the practical implementation of the memetic approach within English writing education, highlighting
the beneficial contribution of Internet Plus tools in enriching students' learning journeys.
A SURVEY TO REAL-TIME MESSAGE-ROUTING NETWORK SYSTEM WITH KLA MODELLINGijfcstjournal
Messages routing over a network is one of the most fundamental concept in communication which requires
simultaneous transmission of messages from a source to a destination. In terms of Real-Time Routing, it
refers to the addition of a timing constraint in which messages should be received within a specified time
delay. This study involves Scheduling, Algorithm Design and Graph Theory which are essential parts of
the Computer Science (CS) discipline. Our goal is to investigate an innovative and efficient way to present
these concepts in the context of CS Education. In this paper, we will explore the fundamental modelling of
routing real-time messages on networks. We study whether it is possible to have an optimal on-line
algorithm for the Arbitrary Directed Graph network topology. In addition, we will examine the message
routing’s algorithmic complexity by breaking down the complex mathematical proofs into concrete, visual
examples. Next, we explore the Unidirectional Ring topology in finding the transmission’s
“makespan”.Lastly, we propose the same network modelling through the technique of Kinesthetic Learning
Activity (KLA). We will analyse the data collected and present the results in a case study to evaluate the
effectiveness of the KLA approach compared to the traditional teaching method.
A COMPARATIVE ANALYSIS ON SOFTWARE ARCHITECTURE STYLESijfcstjournal
Software architecture is the structural solution that achieves the overall technical and operational
requirements for software developments. Software engineers applied software architectures for their
software system developments; however, they worry the basic benchmarks in order to select software
architecture styles, possible components, integration methods (connectors) and the exact application of
each style.
The objective of this research work was a comparative analysis of software architecture styles by its
weakness and benefits in order to select by the programmer during their design time. Finally, in this study,
the researcher has been identified architectural styles, weakness, and Strength and application areas with
its component, connector and Interface for the selected architectural styles.
SYSTEM ANALYSIS AND DESIGN FOR A BUSINESS DEVELOPMENT MANAGEMENT SYSTEM BASED...ijfcstjournal
A design of a sales system for professional services requires a comprehensive understanding of the
dynamics of sale cycles and how key knowledge for completing sales is managed. This research describes
a design model of a business development (sales) system for professional service firms based on the Saudi
Arabian commercial market, which takes into account the new advances in technology while preserving
unique or cultural practices that are an important part of the Saudi Arabian commercial market. The
design model has combined a number of key technologies, such as cloud computing and mobility, as an
integral part of the proposed system. An adaptive development process has also been used in implementing
the proposed design model.
AN ALGORITHM FOR SOLVING LINEAR OPTIMIZATION PROBLEMS SUBJECTED TO THE INTERS...ijfcstjournal
Frank t-norms are parametric family of continuous Archimedean t-norms whose members are also strict
functions. Very often, this family of t-norms is also called the family of fundamental t-norms because of the
role it plays in several applications. In this paper, optimization of a linear objective function with fuzzy
relational inequality constraints is investigated. The feasible region is formed as the intersection of two
inequality fuzzy systems defined by frank family of t-norms is considered as fuzzy composition. First, the
resolution of the feasible solutions set is studied where the two fuzzy inequality systems are defined with
max-Frank composition. Second, some related basic and theoretical properties are derived. Then, a
necessary and sufficient condition and three other necessary conditions are presented to conceptualize the
feasibility of the problem. Subsequently, it is shown that a lower bound is always attainable for the optimal
objective value. Also, it is proved that the optimal solution of the problem is always resulted from the
unique maximum solution and a minimal solution of the feasible region. Finally, an algorithm is presented
to solve the problem and an example is described to illustrate the algorithm. Additionally, a method is
proposed to generate random feasible max-Frank fuzzy relational inequalities. By this method, we can
easily generate a feasible test problem and employ our algorithm to it.
LBRP: A RESILIENT ENERGY HARVESTING NOISE AWARE ROUTING PROTOCOL FOR UNDER WA...ijfcstjournal
Underwater detector network is one amongst the foremost difficult and fascinating analysis arenas that
open the door of pleasing plenty of researchers during this field of study. In several under water based
sensor applications, nodes are square measured and through this the energy is affected. Thus, the mobility
of each sensor nodes are measured through the water atmosphere from the water flow for sensor based
protocol formations. Researchers have developed many routing protocols. However, those lost their charm
with the time. This can be the demand of the age to supply associate degree upon energy-efficient and
ascendable strong routing protocol for under water actuator networks. During this work, the authors tend
to propose a customary routing protocol named level primarily based routing protocol (LBRP), reaching to
offer strong, ascendable and energy economical routing. LBRP conjointly guarantees the most effective use
of total energy consumption and ensures packet transmission which redirects as an additional reliability in
compare to different routing protocols. In this work, the authors have used the level of forwarding node,
residual energy and distance from the forwarding node to the causing node as a proof in multicasting
technique comparisons. Throughout this work, the authors have got a recognition result concerning about
86.35% on the average in node multicasting performances. Simulation has been experienced each in a
wheezy and quiet atmosphere which represents the endorsement of higher performance for the planned
protocol.
STRUCTURAL DYNAMICS AND EVOLUTION OF CAPSULE ENDOSCOPY (PILL CAMERA) TECHNOLO...ijfcstjournal
This research paper examined and re-evaluates the technological innovation, theory, structural dynamics
and evolution of Pill Camera(Capsule Endoscopy) technology in redirecting the response manner of small
bowel (intestine) examination in human. The Pill Camera (Endoscopy Capsule) is made up of sealed
biocompatible material to withstand acid, enzymes and other antibody chemicals in the stomach is a
technology that helps the medical practitioners especially the general physicians and the
gastroenterologists to examine and re-examine the intestine for possible bleeding or infection. Before the
advent of the Pill camera (Endoscopy Capsule) the colonoscopy was the local method used but research
showed that some parts (bowel) of the intestine can’t be reach by mere traditional method hence the need
for Pill Camera. Countless number of deaths from stomach disease such as polyps, inflammatory bowel
(Crohn”s diseases), Cancers, Ulcer, anaemia and tumours of small intestines which ordinary would have
been detected by sophisticated technology like Pill Camera has become norm in the developing nations.
Nevertheless, not only will this paper examine and re-evaluate the Pill Camera Innovation, theory,
Structural dynamics and evolution it unravelled and aimed to create awareness for both medical
practitioners and the public.
AN OPTIMIZED HYBRID APPROACH FOR PATH FINDINGijfcstjournal
Path finding algorithm addresses problem of finding shortest path from source to destination avoiding
obstacles. There exist various search algorithms namely A*, Dijkstra's and ant colony optimization. Unlike
most path finding algorithms which require destination co-ordinates to compute path, the proposed
algorithm comprises of a new method which finds path using backtracking without requiring destination
co-ordinates. Moreover, in existing path finding algorithm, the number of iterations required to find path is
large. Hence, to overcome this, an algorithm is proposed which reduces number of iterations required to
traverse the path. The proposed algorithm is hybrid of backtracking and a new technique(modified 8-
neighbor approach). The proposed algorithm can become essential part in location based, network, gaming
applications. grid traversal, navigation, gaming applications, mobile robot and Artificial Intelligence.
EAGRO CROP MARKETING FOR FARMING COMMUNITYijfcstjournal
The Major Occupation in India is the Agriculture; the people involved in the Agriculture belong to the poor
class and category. The people of the farming community are unaware of the new techniques and Agromachines, which would direct the world to greater heights in the field of agriculture. Though the farmers
work hard, they are cheated by agents in today’s market. This serves as a opportunity to solve
all the problems that farmers face in the current world. The eAgro crop marketing will serve as a better
way for the farmers to sell their products within the country with some mediocre knowledge about using
the website. This would provide information to the farmers about current market rate of agro-products,
their sale history and profits earned in a sale. This site will also help the farmers to know about the market
information and to view agricultural schemes of the Government provided to farmers.
EDGE-TENACITY IN CYCLES AND COMPLETE GRAPHSijfcstjournal
It is well known that the tenacity is a proper measure for studying vulnerability and reliability in graphs.
Here, a modified edge-tenacity of a graph is introduced based on the classical definition of tenacity.
Properties and bounds for this measure are introduced; meanwhile edge-tenacity is calculated for cycle
graphs and also for complete graphs.
COMPARATIVE STUDY OF DIFFERENT ALGORITHMS TO SOLVE N QUEENS PROBLEMijfcstjournal
This Paper provides a brief description of the Genetic Algorithm (GA), the Simulated Annealing (SA)
Algorithm, the Backtracking (BT) Algorithm and the Brute Force (BF) Search Algorithm and attempts to
explain the way as how the Proposed Genetic Algorithm (GA), the Proposed Simulated Annealing (SA)
Algorithm using GA, the Backtracking (BT) Algorithm and the Brute Force (BF) Search Algorithm can be
employed in finding the best solution of N Queens Problem and also, makes a comparison between these
four algorithms. It is entirely a review based work. The four algorithms were written as well as
implemented. From the Results, it was found that, the Proposed Genetic Algorithm (GA) performed better
than the Proposed Simulated Annealing (SA) Algorithm using GA, the Backtracking (BT) Algorithm and
the Brute Force (BF) Search Algorithm and it also provided better fitness value (solution) than the
Proposed Simulated Annealing Algorithm (SA) using GA, the Backtracking (BT) Algorithm and the Brute
Force (BF) Search Algorithm, for different N values. Also, it was noticed that, the Proposed GA took more
time to provide result than the Proposed SA using GA.
PSTECEQL: A NOVEL EVENT QUERY LANGUAGE FOR VANET’S UNCERTAIN EVENT STREAMSijfcstjournal
In recent years, the complex event processing technology has been used to process the VANET’s temporal
and spatial event streams. However, we usually cannot get the accurate data because the device sensing
accuracy limitations of the system. We only can get the uncertain data from the complex and limited
environment of the VANET. Because the VANET’s event streams are consist of the uncertain data, so they
are also uncertain. How effective to express and process these uncertain event streams has become the core
issue for the VANET system. To solve this problem, we propose a novel complex event query language
PSTeCEQL (probabilistic spatio-temporal constraint event query language). Firstly, we give the definition
of the possible world model of VANET’s uncertain event streams. Secondly, we propose an event query
language PSTeCEQL and give the syntax and the operational semantics of the language. Finally, we
illustrate the validity of the PSTeCEQL by an example.
CLUSTBIGFIM-FREQUENT ITEMSET MINING OF BIG DATA USING PRE-PROCESSING BASED ON...ijfcstjournal
Now a day enormous amount of data is getting explored through Internet of Things (IoT) as technologies
are advancing and people uses these technologies in day to day activities, this data is termed as Big Data
having its characteristics and challenges. Frequent Itemset Mining algorithms are aimed to disclose
frequent itemsets from transactional database but as the dataset size increases, it cannot be handled by
traditional frequent itemset mining. MapReduce programming model solves the problem of large datasets
but it has large communication cost which reduces execution efficiency. This proposed new pre-processed
k-means technique applied on BigFIM algorithm. ClustBigFIM uses hybrid approach, clustering using kmeans algorithm to generate Clusters from huge datasets and Apriori and Eclat to mine frequent itemsets
from generated clusters using MapReduce programming model. Results shown that execution efficiency of
ClustBigFIM algorithm is increased by applying k-means clustering algorithm before BigFIM algorithm as
one of the pre-processing technique.
A MUTATION TESTING ANALYSIS AND REGRESSION TESTINGijfcstjournal
Software testing is a testing which conducted a test to provide information to client about the quality of the
product under test. Software testing can also provide an objective, independent view of the software to
allow the business to appreciate and understand the risks of software implementation. In this paper we
focused on two main software testing –mutation testing and mutation testing. Mutation testing is a
procedural testing method, i.e. we use the structure of the code to guide the test program, A mutation is a
little change in a program. Such changes are applied to model low level defects that obtain in the process
of coding systems. Ideally mutations should model low-level defect creation. Mutation testing is a process
of testing in which code is modified then mutated code is tested against test suites. The mutations used in
source code are planned to include in common programming errors. A good unit test typically detects the
program mutations and fails automatically. Mutation testing is used on many different platforms, including
Java, C++, C# and Ruby. Regression testing is a type of software testing that seeks to uncover
new software bugs, or regressions, in existing functional and non-functional areas of a system after
changes such as enhancements, patches or configuration changes, have been made to them. When defects
are found during testing, the defect got fixed and that part of the software started working as needed. But
there may be a case that the defects that fixed have introduced or uncovered a different defect in the
software. The way to detect these unexpected bugs and to fix them used regression testing. The main focus
of regression testing is to verify that changes in the software or program have not made any adverse side
effects and that the software still meets its need. Regression tests are done when there are any changes
made on software, because of modified functions.
GREEN WSN- OPTIMIZATION OF ENERGY USE THROUGH REDUCTION IN COMMUNICATION WORK...ijfcstjournal
Advances in micro fabrication and communication techniques have led to unimaginable proliferation of
WSN applications. Research is focussed on reduction of setup operational energy costs. Bulk of operational
energy costs are linked to communication activities of WSN. Any progress towards energy efficiency has a
potential of huge savings globally. Therefore, every energy efficient step is an endeavour to cut costs and
‘Go Green’. In this paper, we have proposed a framework to reduce communication workload through: Innetwork compression and multiple query synthesis at the base-station and modification of query syntax
through introduction of Static Variables. These approaches are general approaches which can be used in
any WSN irrespective of application.
A NEW MODEL FOR SOFTWARE COSTESTIMATION USING HARMONY SEARCHijfcstjournal
Accurate and realistic estimation is always considered to be a great challenge in software industry.
Software Cost Estimation (SCE) is the standard application used to manage software projects. Determining
the amount of estimation in the initial stages of the project depends on planning other activities of the
project. In fact, the estimation is confronted with a number of uncertainties and barriers’, yet assessing the
previous projects is essential to solve this problem. Several models have been developed for the analysis of
software projects. But the classical reference method is the COCOMO model, there are other methods
which are also applied such as Function Point (FP), Line of Code(LOC); meanwhile, the expert`s opinions
matter in this regard. In recent years, the growth and the combination of meta-heuristic algorithms with
high accuracy have brought about a great achievement in software engineering. Meta-heuristic algorithms
which can analyze data from multiple dimensions and identify the optimum solution between them are
analytical tools for the analysis of data. In this paper, we have used the Harmony Search (HS)algorithm for
SCE. The proposed model which is a collection of 60 standard projects from Dataset NASA60 has been
assessed.The experimental results show that HS algorithm is a good way for determining the weight
similarity measures factors of software effort, and reducing the error of MRE.
AGENT ENABLED MINING OF DISTRIBUTED PROTEIN DATA BANKSijfcstjournal
Mining biological data is an emergent area at the intersection between bioinformatics and data mining
(DM). The intelligent agent based model is a popular approach in constructing Distributed Data Mining
(DDM) systems to address scalable mining over large scale distributed data. The nature of associations
between different amino acids in proteins has also been a subject of great anxiety. There is a strong need to
develop new models and exploit and analyze the available distributed biological data sources. In this study,
we have designed and implemented a multi-agent system (MAS) called Agent enriched Quantitative
Association Rules Mining for Amino Acids in distributed Protein Data Banks (AeQARM-AAPDB). Such
globally strong association rules enhance understanding of protein composition and are desirable for
synthesis of artificial proteins. A real protein data bank is used to validate the system.
International Journal on Foundations of Computer Science & Technology (IJFCST)ijfcstjournal
International Journal on Foundations of Computer Science & Technology (IJFCST) is a Bi-monthly peer-reviewed and refereed open access journal that publishes articles which contribute new results in all areas of the Foundations of Computer Science & Technology. Over the last decade, there has been an explosion in the field of computer science to solve various problems from mathematics to engineering. This journal aims to provide a platform for exchanging ideas in new emerging trends that needs more focus and exposure and will attempt to publish proposals that strengthen our goals. Topics of interest include, but are not limited to the following:
Because the technology is used largely in the last decades; cybercrimes have become a significant
international issue as a result of the huge damage that it causes to the business and even to the ordinary
users of technology. The main aims of this paper is to shed light on digital crimes and gives overview about
what a person who is related to computer science has to know about this new type of crimes. The paper has
three sections: Introduction to Digital Crime which gives fundamental information about digital crimes,
Digital Crime Investigation which presents different investigation models and the third section is about
Cybercrime Law.
DISTRIBUTION OF MAXIMAL CLIQUE SIZE UNDER THE WATTS-STROGATZ MODEL OF EVOLUTI...ijfcstjournal
In this paper, we analyze the evolution of a small-world network and its subsequent transformation to a
random network using the idea of link rewiring under the well-known Watts-Strogatz model for complex
networks. Every link u-v in the regular network is considered for rewiring with a certain probability and if
chosen for rewiring, the link u-v is removed from the network and the node u is connected to a randomly
chosen node w (other than nodes u and v). Our objective in this paper is to analyze the distribution of the
maximal clique size per node by varying the probability of link rewiring and the degree per node (number
of links incident on a node) in the initial regular network. For a given probability of rewiring and initial
number of links per node, we observe the distribution of the maximal clique per node to follow a Poisson
distribution. We also observe the maximal clique size per node in the small-world network to be very close
to that of the average value and close to that of the maximal clique size in a regular network. There is no
appreciable decrease in the maximal clique size per node when the network transforms from a regular
network to a small-world network. On the other hand, when the network transforms from a small-world
network to a random network, the average maximal clique size value decreases significantly
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Student information management system project report ii.pdfKamal Acharya
Our project explains about the student management. This project mainly explains the various actions related to student details. This project shows some ease in adding, editing and deleting the student details. It also provides a less time consuming process for viewing, adding, editing and deleting the marks of the students.
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.
CFD Simulation of By-pass Flow in a HRSG module by R&R Consult.pptxR&R Consult
CFD analysis is incredibly effective at solving mysteries and improving the performance of complex systems!
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R&R and Tetra Engineering Group Inc. were asked to solve the issue with reduced steam production.
An inspection had shown that a significant amount of hot flue gas was bypassing the boiler tubes, where the heat was supposed to be transferred.
R&R Consult conducted a CFD analysis, which revealed that 6.3% of the flue gas was bypassing the boiler tubes without transferring heat. The analysis also showed that the flue gas was instead being directed along the sides of the boiler and between the modules that were supposed to capture the heat. This was the cause of the reduced performance.
Based on our results, Tetra Engineering installed covering plates to reduce the bypass flow. This improved the boiler's performance and increased electricity production.
It is always satisfying when we can help solve complex challenges like this. Do your systems also need a check-up or optimization? Give us a call!
Work done in cooperation with James Malloy and David Moelling from Tetra Engineering.
More examples of our work https://www.r-r-consult.dk/en/cases-en/
AN ALGORITHM FOR SOLVING LINEAR OPTIMIZATION PROBLEMS SUBJECTED TO THE INTERSECTION OF TWO FUZZY RELATIONAL INEQUALITIES DEFINED BY FRANK FAMILY OF T-NORMS
1. International Journal in Foundations of Computer Science & Technology (IJFCST) Vol.8, No.3, May 2018
DOI: 10.5121/ijfcst.2018.8301 1
AN ALGORITHM FOR SOLVING LINEAR
OPTIMIZATION PROBLEMS SUBJECTED TO
THE INTERSECTION OF TWO FUZZY
RELATIONAL INEQUALITIES DEFINED BY
FRANK FAMILY OF T-NORMS
Amin Ghodousian*
Faculty of Engineering Science, College of Engineering,
University of Tehran, P.O.Box 11365-4563, Tehran, Iran
ABSTRACT
Frank t-norms are parametric family of continuous Archimedean t-norms whose members are also strict
functions. Very often, this family of t-norms is also called the family of fundamental t-norms because of the
role it plays in several applications. In this paper, optimization of a linear objective function with fuzzy
relational inequality constraints is investigated. The feasible region is formed as the intersection of two
inequality fuzzy systems defined by frank family of t-norms is considered as fuzzy composition. First, the
resolution of the feasible solutions set is studied where the two fuzzy inequality systems are defined with
max-Frank composition. Second, some related basic and theoretical properties are derived. Then, a
necessary and sufficient condition and three other necessary conditions are presented to conceptualize the
feasibility of the problem. Subsequently, it is shown that a lower bound is always attainable for the optimal
objective value. Also, it is proved that the optimal solution of the problem is always resulted from the
unique maximum solution and a minimal solution of the feasible region. Finally, an algorithm is presented
to solve the problem and an example is described to illustrate the algorithm. Additionally, a method is
proposed to generate random feasible max-Frank fuzzy relational inequalities. By this method, we can
easily generate a feasible test problem and employ our algorithm to it.
KEYWORDS
Fuzzy relation, fuzzy relational inequality, linear optimization, fuzzy compositions and t-norms.
1. INTRODUCTION
In this paper, we study the following linear problem in which the constraints are formed as the
intersection of two fuzzy systems of relational inequalities defined by Frank family of t-norms:
1
2
min
[0,1]
T
n
Z c x
A x b
D x b
x
ϕ
ϕ
=
≤
≥
∈
(1)
Where 1 1{1,2,.., }I m= , 2 1 1 1 2{ 1, 2,.., }I m m m m= + + + and {1,2,.., }J n= . 1
( )ij m nA a ×= and
2
( )ij m nD d ×= are fuzzy matrices such that 10 ≤≤ ija ( 1i I∀ ∈ and j J∀ ∈ ) and 0 1ijd≤ ≤
2. International Journal in Foundations of Computer Science & Technology (IJFCST) Vol.8, No.3, May 2018
2
( 2i I∀ ∈ and j J∀ ∈ ). 1
1 1
1( )i mb b ×= is an 1m –dimensional fuzzy vector in 1
[0,1]m
(i.e.,
1
0 1ib≤ ≤ , 1i I∀ ∈ ) , 2
2 2
1( )i mb b ×= is an 2m –dimensional fuzzy vector in 2
[0,1]m
(i.e.,
2
0 1ib≤ ≤ , 2i I∀ ∈ ), and c is a vector in n
. Moreover, “ϕ ” is the max-Frank composition,
that is,
( 1)( 1)
( , ) ( , ) log 1
1
x y
s
F s
s s
x y T x y
s
ϕ
− −
= = +
−
in which 0s > and 1s ≠ .
By these notations, problem (1) can be also expressed as follows:
1
1
2
2
min
max{ ( , )} ,
max{ ( , )} ,
[0,1]
T
s
F ij j i
j J
s
F ij j i
j J
n
Z c x
T a x b i I
T d x b i I
x
∈
∈
=
≤ ∈
≥ ∈
∈
(2)
Especially, by setting A D= and
1 2
b b= , the above problem is converted to max-Frank fuzzy
relational equations. The above definition can be extended for 0s = , 1s = and s=∞ by taking
limits. So, it is easy to verify that
0
( , ) min{ , }FT x y x y= ,
1
( , )FT x y xy= and
( , ) max{ 1,0}FT x y x y∞
= + − , that is, Frank t-norm is converted to minimum, product and
Lukasiewicz t-norm, respectively. Frank family of t-norms plays a central role in the
investigation of the contraposition law for QL-implications [7].
The theory of fuzzy relational equations (FRE) was firstly proposed by Sanchez and applied in
problems of the medical diagnosis [41]. Nowadays, it is well known that many issues associated
with a body knowledge can be treated as FRE problems [37]. Generally, when inference rules and
their consequences are known, the problem of determining antecedents is reduced to solving an
FRE [35]. We refer the reader to [27] in which the authors provided a good overview of fuzzy
relational equations.
The solvability determination and the finding of solutions set are the primary (and the most
fundamental) subject concerning with FRE problems. The solution set of FRE is often a non-
convex set that is completely determined by one maximum solution and a finite number of
minimal solutions [5]. This non-convexity property is one of two bottlenecks making major
contribution to the increase of complexity in problems that are related to FRE, especially in the
optimization problems subjected to a system of fuzzy relations. The other bottleneck is concerned
with detecting the minimal solutions for FREs. Chen and Wang [2] presented an algorithm for
obtaining the logical representation of all minimal solutions and deduced that a polynomial-time
algorithm to find all minimal solutions of FRE (with max-min composition) may not exist. In
fact, the same result holds true for a more general t-norms instead of the minimum operator
[2,3,30,31,34]. Over the last decades, the solvability of FRE defined with different max-t
compositions have been investigated by many researchers [36,38,39,42,44,45,47,50,53].
Moreover, some researchers introduced and improved theoretical aspects and applications of
fuzzy relational inequalities (FRI)[13,16,17,23,28,52]. Li and Yang [28] studied a FRI with
addition-min composition and presented an algorithm to search for minimal solutions. They
applied FRI to meet a data transmission mechanism in a BitTorrent-like Peer-to-Peer file sharing
systems. Ghodousian and Khorram [13] focused on the algebraic structure of two fuzzy relational
inequalities
1
A x bϕ ≤ and
2
D x bϕ ≥ , and studied a mixed fuzzy system formed by the two
3. International Journal in Foundations of Computer Science & Technology (IJFCST) Vol.8, No.3, May 2018
3
preceding FRIs, where ϕ is an operator with (closed) convex solutions. Guo et al. [16]
investigated a kind of FRI problems and the relationship between minimal solutions and FRI
paths.
The problem of optimization subject to FRE and FRI is one of the most interesting and on-going
research topic among the problems related to FRE and FRI theory [1,8,11-
24,25,29,32,40,43,48,52]. Fang and Li [9] converted a linear optimization problem subjected to
FRE constraints with max-min operation into an integer programming problem and solved it by
branch and bound method using jump-tracking technique. Wu et al. [46] improved the method
used by Fang and Li, by decreasing the search domain and presented a simplification process.
Chang and Shieh [1] presented new theoretical results concerning the linear optimization problem
constrained by fuzzy max–min relation equations. The topic of the linear optimization problem
was also investigated with max-product operation [11,19,33]. Moreover, some generalizations of
the linear optimization with respect to FRE have been studied with the replacement of max-min
and max-product compositions with different fuzzy compositions such as max-average
composition [22,48], max-star composition [14,24] and max-t-norm composition [20,29,43]. For
example, Li and Fang [29] solved the linear optimization problem subjected to a system of sup-t
equations by reducing it to a 0-1 integer optimization problem. In [20] a method was presented
for solving linear optimization problems with the max-Archimedean t-norm fuzzy relation
equation constraint.
Recently, many interesting generalizations of the linear programming subject to a system of fuzzy
relations have been introduced [6,10,17,26,32,49]. For example, Wu et al. [49] represented an
efficient method to optimize a linear fractional programming problem under FRE with max-
Archimedean t-norm composition. Dempe and Ruziyeva [4] generalized the fuzzy linear
optimization problem by considering fuzzy coefficients. Dubey et al. studied linear programming
problems involving interval uncertainty modeled using intuitionistic fuzzy set [6]. The linear
optimization of bipolar FRE was studied by some researchers where FRE defined with max-min
composition [10] and max-Lukasiewicz composition [26,32]. In [32], the authors presented an
algorithm without translating the original problem into a 0-1 integer linear problem.
The optimization problem subjected to various versions of FRI could be found in the literature as
well [12,13,16,17,23,51,52]. Yang [51] applied the pseudo-minimal index algorithm for solving
the minimization of linear objective function subject to FRI with addition-min composition.
Ghodousian and Khorram [12] introduced a system of fuzzy relational inequalities with fuzzy
constraints (FRI-FC) in which the constraints were defined with max-min composition. They used
this fuzzy system to convincingly optimize the educational quality of a school (with minimum
cost) to be selected by parents. The following diagram may help the readability of the paper.
The remainder of the paper is organized as follows. In section 2, some preliminary notions and
definitions and three necessary conditions for the feasibility of problem (1) are presented. In
section 3, the feasible region of problem (1) is determined as a union of the finite number of
closed convex intervals. Two simplification operations are introduced to accelerate the resolution
of the problem. Moreover, a necessary and sufficient condition based on the simplification
operations is presented to realize the feasibility of the problem. Problem (1) is resolved by
optimization of the linear objective function considered in section 4. In addition, the existence of
an optimal solution is proved if problem (1) is not empty. The preceding results are summarized
as an algorithm and, finally in section 5 an example is described to illustrate. Additionally, in
section 5, a method is proposed to generate feasible test problems for problem (1).
4. International Journal in Foundations of Computer Science & Technology (IJFCST) Vol.8, No.3, May 2018
4
2. BASIC PROPERTIES OF MAX-FRANK FRI
This section describes the basic definitions and structural properties concerning problem (1) that
are used throughout the paper. For the sake of simplicity, let
1
( , )s
FT
S A b and
2
( , )s
FT
S D b denote
the feasible solutions sets of inequalities
1
A x bϕ ≤ and
2
D x bϕ ≥ , respectively, that is,
{ }1 1
( , ) [0,1] :s
F
n
T
S A b x A x bϕ= ∈ ≤ and { }2 2
( , ) [0,1] :s
F
n
T
S D b x D x bϕ= ∈ ≥ . Also, let
1 2
( , , , )s
FT
S A D b b denote the feasible solutions set of problem (1). Based on the foregoing
notations, it is clear that
1 2 1 2
( , , , ) ( , ) ( , )s s s
F F FT T T
S A D b b S A b S D b= I .
Definition 1. For each 1i I∈ and each j J∈ , we define
{ }1 1
( , ) [0,1] : ( , )s
F
s
ij i F ij iT
S a b x T a x b= ∈ ≤ . Similarly, for each 2i I∈ and each j J∈ ,
{ }2 2
( , ) [0,1] : ( , )s
F
s
ij i F ij iT
S d b x T d x b= ∈ ≥ .
Furthermore, the notations { }1 1
: ( , )s
F
i ij iT
J j J S a b= ∈ ≠ ∅ , 1i I∀ ∈ , and
{ }2 2
: ( , )s
F
i ij iT
J j J S d b= ∈ ≠ ∅ , 2i I∀ ∈ , are used in the text.
Remark 1. From the least-upper-bound property of , it is clear that { }1
[0,1]
inf ( , )s
F
ij iTx
S a b
∈
and
{ }1
[0,1]
sup ( , )s
F
ij iT
x
S a b
∈
exist, if
1
( , )s
F
ij iT
S a b ≠ ∅. Moreover, since
s
FT is a t-norm, its
monotonicity property implies that
1
( , )s
F
ij iT
S a b is actually a connected subset of [0,1].
Additionally, due to the continuity of
s
FT , we must have
5. International Journal in Foundations of Computer Science & Technology (IJFCST) Vol.8, No.3, May 2018
5
{ } { }1 1
[0,1] [0,1]
inf ( , ) min ( , )s s
F F
ij i ij iT Tx x
S a b S a b
∈ ∈
= and { } { }1 1
[0,1][0,1]
sup ( , ) max ( , )s s
F F
ij i ij iT Txx
S a b S a b
∈∈
= .
Therefore, { } { }1 1 1
[0,1] [0,1]
( , ) min ( , ) ,max ( , )s s s
F F F
ij i ij i ij iT T Tx x
S a b S a b S a b
∈ ∈
=
, i.e., 1
( , )s
F
ij iT
S a b is a
closed sub-interval of [0,1]. By the similar argument, if
2
( , )s
F
ij iT
S d b ≠ ∅, then we have
{ } { }2 2 2
[0,1] [0,1]
( , ) min ( , ) ,max ( , ) [0,1]s s s
F F F
ij i ij i ij iT T Tx x
S d b S d b S d b
∈ ∈
= ⊆
.
From Definition 1 and Remark 1, the following two corollaries are resulted.
Corollary 1. For each 1i I∈ and each j J∈ ,
1
( , )s
F
ij iT
S a b ≠ ∅. Also,
{ }1 1
[0,1]
( , ) 0,max ( , )s s
F F
ij i ij iT Tx
S a b S a b
∈
=
.
Corollary 2. If
2
( , )s
F
ij iT
S d b ≠ ∅ for some 2i I∈ and j J∈ , then
{ }2 2
[0,1]
( , ) min ( , ) ,1s s
F F
ij i ij iT Tx
S d b S d b
∈
=
.
Definition 2. For each 1i I∈ and each j J∈ , we define
1
1
1
1
( 1)( 1)
log 1
1
i
ij
ij i
b
ij
s ij ia
a b
U s s
a b
s
<
= − −
+ ≥ −
Also, for each 2i I∈ and each j J∈ , we set
2
2
2
0 0
( 1)( 1)
log 1
1
i
ij
ij i
ij ij i
b
s d
d b
L d b
s s
otherwise
s
+ ∞ <
= = =
− − + −
Remark 3. From Definition 2, if
1
ij ia b= , then 1ijU = . Also, we have 1ijL = , if
2
0ij id b= ≠ , and 0ijL = if
2
0ij id b> = .
Lemma 1 below shows that ijU and ijL stated in Definition 2, determine the maximum and
minimum solutions of sets
1
( , )s
F
ij iT
S a b ( 1i I∈ ) and 2
( , )s
F
ij iT
S d b ( 2i I∈ ), respectively.
6. International Journal in Foundations of Computer Science & Technology (IJFCST) Vol.8, No.3, May 2018
6
Lemma 1. (a) { }1
[0,1]
max ( , )s
F
ij ij iTx
U S a b
∈
= , 1i I∀ ∈ and j J∀ ∈ . (b) If
2
( , )s
F
ij iT
S d b ≠ ∅ for
some 2i I∈ and j J∈ , then { }2
[0,1]
m in ( , )s
F
ij ij iTx
L S d b
∈
= .
Proof. See [13,15]. □
Lemma 1 together with the corollaries 1 and 2 results in the following consequence.
Corollary 3. (a) For each 1i I∈ and j J∈ ,
1
( , ) [0, ]s
F
ij i ijT
S a b U= . (b) If
2
( , )s
F
ij iT
S d b ≠ ∅
for some 2i I∈ and j J∈ , then
2
( , ) [ ,1]s
F
ij i ijT
S d b L= .
Definition 3. For each 1i I∈ , let { }{ }1 1
1
( , ) [0,1] : max ( , )s
F
n
n s
i i F ij j iT j
S a b x T a x b
=
= ∈ ≤ .
Similarly, for each 2i I∈ , we define { }{ }2 2
1
( , ) [0,1] : max ( , )s
F
n
n s
i i F ij j iT j
S d b x T d x b
=
= ∈ ≥ .
According to Definition 3 and the constraints stated in (2), sets 1
( , )s
F
i iT
S a b and 2
( , )s
F
i iT
S d b
actually denote the feasible solutions sets of the i’th inequality
1
max{ ( , )}s
F ij j i
j J
T a x b
∈
≤ ( 1i I∈ )
and
2
max{ ( , )}s
F ij j i
j J
T d x b
∈
≥ ( 2i I∈ ) of problem (1), respectively. Based on (2) and Definitions 1
and 3, it can be easily concluded that for a fixed 1i I∈ , 1
( , )s
F
i iT
S a b ≠ ∅ iff
1
( , )s
F
ij iT
S a b ≠ ∅,
j J∀ ∈ . On the other hand, by Corollary 1 we know that
1
( , )s
F
ij iT
S a b ≠ ∅, 1i I∀ ∈ and
j J∀ ∈ . As a result, 1
( , )s
F
i iT
S a b ≠ ∅ for each 1i I∈ . However, in contrast to 1
( , )s
F
i iT
S a b , set
2
( , )s
F
i iT
S d b may be empty. Actually, for a fixed 2i I∈ , 2
( , )s
F
i iT
S d b is nonempty if and only if
2
( , )s
F
ij iT
S d b is nonempty for at least some j J∈ . Additionally, for each 2i I∈ and j J∈
we have
2
( , )s
F
ij iT
S d b ≠ ∅ if and only if
2
ij id b≥ . These results have been summarized in the
following lemma. Part (b) of the lemma gives a necessary and sufficient condition for the
feasibility of set 2
( , )s
F
i iT
S d b ( 2i I∀ ∈ ). It is to be noted that the lemma 2 (part (b)) also
provides a necessary condition for problem (1).
Lemma 2. (a) 1
( , )s
F
i iT
S a b ≠ ∅ , 1i I∀ ∈ . (b) For a fixed 2i I∈ , 2
( , )s
F
i iT
S d b ≠ ∅ iff
2
1
( , )s
F
n
ij iT
j
S d b
=
≠∅U . Additionally, for each 2i I∈ and j J∈ ,
2
( , )s
F
ij iT
S d b ≠ ∅ iff
2
ij id b≥ .
Definition 4. For each 2i I∈ and 2
ij J∈ , we define
2
( , , ) [0,1] ... [0,1] [ ,1] [0,1] ... [0,1]s
F
i i ijT
S d b j L= × × × × × × , where [ ,1]ijL is in the
j ’th position.
7. International Journal in Foundations of Computer Science & Technology (IJFCST) Vol.8, No.3, May 2018
7
In the following lemma, the feasible solutions set of the i ’th fuzzy relational inequality is
characterized.
Lemma 3. (a) 1
1 2( , ) [0, ] [0, ] ... [0, ]s
F
i i i i inT
S a b U U U= × × × , 1i I∀ ∈ . (b)
2
2 2
( , ) ( , , )s s
F F
i
i i i iT T
j J
S d b S d b j
∈
= U , 2i I∀ ∈ .
Proof. See [15]. □
Definition 5. Let 1 2( ) [ , ,..., ]i i inX i U U U= , 1i I∀ ∈ . Also, let
1 2( , ) [ ( , ) , ( , ) ,..., ( , ) ]nX i j X i j X i j X i j= , 2i I∀ ∈ and 2
ij J∀ ∈ , where
( , )
0
ij
k
L k j
X i j
k j
=
=
≠
Lemma 3 together with Definitions 4 and 5, results in Theorem 1, which completely determines
the feasible region for the i ’th relational inequality.
Theorem 1. (a) 1
( , ) [ , ( )]s
F
i iT
S a b X i= 0 , 1i I∀ ∈ . (b)
2
2
( , ) [ ( , ), ]s
F
i
i iT
j J
S d b X i j
∈
= 1U ,
2i I∀ ∈ , where 0 and 1 are n –dimensional vectors with each component equal to zero and
one, respectively.
Theorem 1 gives the upper and lower bounds for the feasible solutions set of the i ’th relational
inequality. Actually, for each 1i I∈ , vectors 0 and ( )X i are the unique minimum and the
unique maximum of set 1
( , )s
F
i iT
S a b . In addition, for each 2i I∈ , set 2
( , )s
F
i iT
S d b has the unique
maximum (i.e., vector 1 ), but the finite number of minimal solutions ( , )X i j ( 2
ij J∀ ∈ ).
Furthermore, part (b) of Theorem 1 presents another feasible necessary condition for problem (1)
as stated in the following corollary.
Corollary 4. If
1 2
( , , , )s
FT
S A D b b ≠ ∅ , then 2
( , )s
F
i iT
S d b∈1 , 2i I∀ ∈ (i.e.,
2
2 2
( , ) ( , )s s
F F
i iT T
i I
S d b S D b
∈
∈ =1 I ).
Proof. Let
1 2
( , , , )s
FT
S A D b b ≠ ∅ . Then, 2
( , )s
FT
S D b ≠ ∅ , and therefore, 2
( , )s
F
i iT
S d b ≠ ∅ ,
2i I∀ ∈ . Now, Theorem 1 (part (b)) implies 2
( , )s
F
i iT
S d b∈1 , 2i I∀ ∈ . □
Lemma 4 describes the shape of the feasible solutions set for the fuzzy relational inequalities
1
A x bϕ ≤ and
2
D x bϕ ≥ , separately.
Lemma 4. (a)
1 1 1
1
1 2( , ) [0, ] [0, ] ... [0, ]s
F
i i inT
i I i I i I
S A b U U U
∈ ∈ ∈
= × × ×I I I .
(b)
2
2
2 2
( , ) ( , , )s s
F F
i
i iT T
i I j J
S D b S d b j
∈ ∈
= I U .
8. International Journal in Foundations of Computer Science & Technology (IJFCST) Vol.8, No.3, May 2018
8
Proof. The proof is obtained from Lemma 3 and equations
1
1 1
( , ) ( , )s s
F F
i iT T
i I
S A b S a b
∈
= I and
2
2 2
( , ) ( , )s s
F F
i iT T
i I
S D b S d b
∈
= I . □
Definition 6. Let
2
2: ie I J→ so that
2
( ) ie i j J= ∈ , 2i I∀ ∈ , and let DE be the set of all
vectors e. For the sake of convenience, we represent each De E∈ as an 2m –dimensional vector
21 2[ , ,..., ]me j j j= in which ( )kj e k= , 21,2,...,k m= .
Definition 7. Let 21 2[ , ,..., ]m De j j j E= ∈ . We define { }1
min ( )
i I
X X i
∈
= , that is,
{ }1
min ( )j j
i I
X X i
∈
= , j J∀ ∈ . Moreover, let 1 2( ) [ ( ) , ( ) ,..., ( ) ]nX e X e X e X e= , where
{ } { }2 2
( ) max ( , ( )) max ( , )j j i j
i I i I
X e X i e i X i j
∈ ∈
= = , j J∀ ∈ .
Based on Theorem 1 and the above definition, we have the following theorem characterizing the
feasible regions of the general inequalities
1
A x bϕ ≤ and
2
D x bϕ ≥ in the most familiar way.
Theorem 2. (a) 1
( , ) [ , ]s
FT
S A b X= 0 , 1i I∀ ∈ . (b)
2
( , ) [ ( ), ]s
F
D
T
e E
S D b X e
∈
= 1U .
Proof. For the proof in the general case see Remark 2.5 in [13]. □
Corollary 5. Assume that
1 2
( , , , )s
FT
S A D b b ≠ ∅ . Then, there exists some De E∈ such that
[ , ] [ ( ), ]X X e ≠ ∅0 1I .
Corollary 6. Assume that
1 2
( , , , )s
FT
S A D b b ≠ ∅ . Then,
2
( , )s
FT
X S D b∈ .
Proof. Let
1 2
( , , , )s
FT
S A D b b ≠ ∅ . By Corollary 5, [ , ] [ ( ), ]X X e′ ≠ ∅0 1I for some De E′∈ .
Thus, [ ( ), ]X X e′∈ 1 that means [ ( ), ]
De E
X X e
∈
∈ 1U . Therefore, from Theorem 2 (part (b)),
2
( , )s
FT
X S D b∈ . □
3. THE RESOLUTION OF FEASIBLE REGION AND SIMPLIFICATION
OPERATIONS
In this section, two operations are presented to simplify the matrices A and D, and a necessary
and sufficient condition is derived to determine the feasibility of the main problem. At first, we
give a theorem in which the bounds of the feasible solutions set of problem (1) are attained. As is
shown in the following theorem, by using these bounds, the feasible region is completely found.
For the proof of the propositions of this section, see [13,15].
Theorem 3. Suppose that
1 2
( , , , )s
FT
S A D b b ≠ ∅ . Then
1 2
( , , , ) [ ( ), ]s
F
D
T
e E
S A D b b X e X
∈
= U .
9. International Journal in Foundations of Computer Science & Technology (IJFCST) Vol.8, No.3, May 2018
9
In practice, there are often some components of matrices A and D , which have no effect on the
solutions to problem (1). Therefore, we can simplify the problem by changing the values of these
components to zeros. We refer the interesting reader to [13] where a brief review of such these
processes is given. Here, we present two simplification techniques based on the Frank family of t-
norms.
Definition 8. If a value changing in an element, say ija , of a given fuzzy relation matrix A has
no effect on the solutions of problem (1), this value changing is said to be an equivalence
operation.
Corollary 7. Suppose that 1i I∈ and 0 0
( , )s
F ij j iT a x b< ,
1
( , )s
FT
x S A b∀ ∈ . In this case, it is
obvious that { } 1
1
max ( , )
n
s
F ij j i
j
T a x b
=
≤ is equivalent to { }
0
1
1
max ( , )
n
s
F ij j i
j
j j
T a x b
=
≠
≤ , that is, “resetting
0ija to zero” has no effect on the solutions of problem (1) (since component 0ija only appears
in the i ‘th constraint of problem (1)). Therefore, if 0 0
1
( , )s
F ij j iT a x b< ,
1
( , )s
FT
x S A b∀ ∈ , then
“resetting 0ija to zero” is an equivalence operation.
Lemma 5 (simplification of matrix A). Suppose that matrix 1
( )ij m nA a ×=% % is resulted from
matrix A as follows:
1
1
0 ij i
ij
ij ij i
a b
a
a a b
<
=
≥
%
for each 1i I∈ and j J∈ . Then,
1 1
( , ) ( , )s s
F FT T
S A b S A b= % .
Lemma 5 gives a condition to reduce the matrix A . In this lemma, A% denote the simplified
matrix resulted from A after applying the simplification process. Based on this notation, we
define { }1 1
: ( , )s
F
i ij iT
J j J S a b= ∈ ≠ ∅% % ( 1i I∀ ∈ ) where ija% denotes ( , )i j ‘th component of
matrix A% . So, from Corollary 1 and Remark 2, it is clear that 1 1
i iJ J J= =% . Moreover, since
1 2 1 2
( , , , ) ( , ) ( , )s s s
F F FT T T
S A D b b S A b S D b= I , from Lemma 5 we can also conclude that
1 2 1 2
( , , , ) ( , , , )s s
F FT T
S A D b b S A D b b= % . By considering a fixed vector De E∈ in Theorem 3,
interval [ ( ), ]X e X is meaningful iff ( )X e X≤ . Therefore, by deleting infeasible intervals
[ ( ), ]X e X in which ( )X e X≤/ , the feasible solutions set of problem (1) stays unchanged. In
order to remove such infeasible intervals from the feasible region, it is sufficient to neglect
vectors e generating infeasible solutions ( )X e (i.e., solutions ( )X e such that ( )X e X≤/ ).
These considerations lead us to introduce a new set { }: ( )D DE e E X e X′ = ∈ ≤ to strengthen
Theorem 3. By this new set, Theorem 3 can be written as
1 2
( , , , ) [ ( ), ]s
F
D
T
e E
S A D b b X e X
′∈
= U , if
1 2
( , , , )s
FT
S A D b b ≠ ∅ .
10. International Journal in Foundations of Computer Science & Technology (IJFCST) Vol.8, No.3, May 2018
10
Lemma 6. Let { }2( ) : ( )jI e i I e i j= ∈ = and { }( ) : ( )jJ e j J I e= ∈ ≠ ∅ , De E∀ ∈ . Then,
{ }( )
( )
max ( )
( )
0 ( )
j
ie i
i I e
j
L j J e
X e
j J e
∈
∈
=
∉
Corollary 8. De E′∈ if and only if ( )( ) e iie iL X≤ , 2i I∀ ∈ .
As mentioned before, to accelerate identification of the meaningful solutions ( )X e , we reduce
our search to set DE′ instead of set DE . As a result from Corollary 8, we can confine set
2
iJ by
removing each
2
ij J∈ such that jijL X> before selecting the vectors e to construct solutions
( )X e . However, lemma 7 below shows that this purpose can be accomplished by resetting some
components of matrix D to zeros. Before formally presenting the lemma, some useful notations
are introduced.
Definition 9 (simplification of matrix D). Let 2
( )ij m nD d ×= %% denote a matrix resulted from D
as follows:
2
0 ji ij
ij
ij
j J and L X
d
d otherwise
∈ >
=
%
Also, similar to Definition 1, assume that { }2 2
: ( , )s
F
i ij iT
J j J S d b= ∈ ≠ ∅%% ( 2i I∀ ∈ ) where
ijd% denotes ( , )i j ‘th components of matrix D% .
According to the above definition, it is easy to verify that 2 2
i iJ J⊆% , 2i I∀ ∈ . Furthermore, the
following lemma demonstrates that the infeasible solutions ( )X e are not generated, if we only
consider those vectors e generated by the components of the matrix D% , or equivalently vectors
e generated based on the set 2
iJ% instead of 2
iJ .
Lemma 7. DD
E E′=% , where D
E % is the set of all functions
2
2: ie I J→ % so that
2
( ) ie i j J= ∈ % ,
2i I∀ ∈ .
By Lemma 7, we always have ( )X e X≤ for each vector e, which is selected based on the
components of matrix D% . Actually, matrix D% as a reduced version of matrix D, removes all the
infeasible intervals from the feasible region by neglecting those vectors e generating the
infeasible solutions ( )X e . Also, similar to Lemma 5 we have
1 2 1 2
( , , , ) ( , , , )s s
F FT T
S A D b b S A D b b= % . This result and Lemma 5 can be summarized by
1 2 1 2
( , , , ) ( , , , )s s
F FT T
S A D b b S A D b b= % % .
11. International Journal in Foundations of Computer Science & Technology (IJFCST) Vol.8, No.3, May 2018
11
Definition 10. Let 2
( )ij m nL L ×= be a matrix whose ( , )i j ’th component is equal to ijL . We
define the modified matrix 2
* *
( )ij m nL L ×= from the matrix L as follows:
* jij
ij
ij
L X
L
L otherwise
+∞ >
=
As will be shown in the following theorem, matrix
*
L is useful for deriving a necessary and
sufficient condition for the feasibility of problem (1) and accelerating identification of the set
1 2
( , , , )s
FT
S A D b b .
Theorem 4.
1 2
( , , , )s
FT
S A D b b ≠ ∅ iff there exists at least some
2
ij J∈ such that
*
ijL ≠+∞,
2i I∀ ∈ .
4. OPTIMIZATION OF THE LINEAR OBJECTIVE FUNCTION
According to the well-known schemes used for optimization of linear problems such as (1)
[9,13,17,29], problem (1) is converted to the following two sub-problems:
1
1
1
2
(4): min
[0,1]
n
j j
j
n
Z c x
A x b
D x b
x
ϕ
ϕ
+
=
=
≤
≥
∈
∑ 2
1
1
2
(5): min
[0,1]
n
j j
j
n
Z c x
A x b
D x b
x
ϕ
ϕ
−
=
=
≤
≥
∈
∑
Where max{ ,0}j jc c+
= and min{ ,0}j jc c−
= for 1,2,...,j n= . It is easy to prove that X is the
optimal solution of (5), and the optimal solution of (4) is ( )X e′ for some De E′ ′∈ .
Theorem 5. Suppose that
1 2
( , , , )s
FT
S A D b b ≠ ∅ , and X and
*
( )X e are the optimal solutions
of sub-problems (5) and (4), respectively. Then
*T
c x is the lower bound of the optimal objective
function in (1), where
* * * *
1 2[ , ,..., ]nx x x x= is defined as follows:
*
*
0
( ) 0
j j
j
j j
X c
x
X e c
<
=
≥
(6)
for 1,2,...,j n= .
Proof. See Corollary 4.1 in [13]. □
Corollary 9. Suppose that
1 2
( , , , )s
FT
S A D b b ≠ ∅ . Then,
* * * *
1 2[ , ,..., ]nx x x x= as defined in (6),
is the optimal solution of problem (1).
12. International Journal in Foundations of Computer Science & Technology (IJFCST) Vol.8, No.3, May 2018
12
Proof. As in the poof of Theorem 5,
*T
c x is the lower bound of the optimal objective function.
According to the definition of vector
*
x , we have
* *
( ) jj jX e x X≤ ≤ , j J∀ ∈ , which implies
* 1 2
[ ( ), ] ( , , , )s
F
D
T
e E
x X e X S A D b b
∈
∈ =U . □
We now summarize the preceding discussion as an algorithm.
Algorithm 1 (solution of problem (1))
Given problem (1):
1. Compute ijU ( 1i I∀ ∈ and j J∀ ∈ ) and ijL ( 2i I∀ ∈ and j J∀ ∈ ) by Definition 2.
2. If
2
( , )s
FT
S D b∈1 , then continue; otherwise, stop, the problem is infeasible (Corollary 4).
3. Compute vectors ( )X i ( 1i I∀ ∈ ) from Definition 5, and then vector X from Definition 7.
4. If
2
( , )s
FT
X S D b∈ , then continue; otherwise, stop, the problem is infeasible (Corollary 6).
5. Compute simplified matrices A% and D% from Lemma 5 and Definition 9, respectively.
6. Compute modified matrix
*
L from Definition 10.
7. For each 2i I∈ , if there exists at least some
2
ij J∈ such that
*
ijL ≠+∞, then continue;
otherwise, stop, the problem is infeasible (Theorem 4).
8. Find the optimal solution *
( )X e for the sub-problem (4) by considering vectors D
e E∈ % and
set 2
iJ% , 2i I∀ ∈ ( Lemma 7).
9. Find the optimal solution
* * * *
1 2[ , ,..., ]nx x x x= for the problem (1) by (6) (Corollary 9).
It should be noted that there is no polynomial time algorithm for complete solution of FRIs with
the expectation N NP≠ . Hence, the problem of solving FRIs is an NP-hard problem in terms of
computational complexity [2].
5. CONSTRUCTION OF TEST PROBLEMS AND NUMERICAL EXAMPLE
In this section, we present a method to generate random feasible regions formed as the
intersection of two fuzzy inequalities with Frank family of t-norms. In section 5.1, we prove that
the max-Frank fuzzy relational inequalities constructed by the introduced method are actually
feasible. In section 5.2, the method is used to generate a random test problem for problem (1), and
then the test problem is solved by Algorithm 1 presented in section 4.
5.1. Construction of test problems
There are several ways to generate a feasible FRI defined with max-Frank composition. In what
follows, we present a procedure to generate random feasible max-Frank fuzzy relational
inequalities:
13. International Journal in Foundations of Computer Science & Technology (IJFCST) Vol.8, No.3, May 2018
13
Algorithm 2 (construction of feasible Max-Frank FRI)
{ }2
1
1 1
2 1 2
1. Generate randon scalars [0,1], 1,2,..., and 1,2,...,n, and [0,1], 1,2,..., .
2. Compute by Definition 7.
2. Randomly select columns { , ,..., }from = 1,2,..., .
2. For 1,2,...
ij i
m
a i m j b i m
X
m j j j J n
i
∈ = = ∈ =
∈{ }
{ }
2
2
2
2
2
2
, ,assign a random number from [0, ] to .
3. For 1,2,..., ,if 0, then
( 1)( 1)
Assign a random number from interval max ,log (1+ ) ,1 to .
( 1)
End
4. For 1,2,..
i
i
iji
j i
i
b
i s ijX
m X b
i m b
s s
b d
s
i
∈ ≠
− −
−
∈{ }
{ }
{ }
2
2
2 1
.,
For each 1,2,..., { }
Assign a random number from [0 , 1] to .
End
End
5. For each 1,2,..., and each { ,
ik j
m
k m i
d
i m j j
∈ −
∈ ∉ 22 ,..., }
Assign a random number from [0,1] to .
End
m
ij
j j
d
By the following theorem, it is proved that Algorithm 2 always generates random feasible
max-Frank fuzzy relational inequalities.
Theorem 6. Problem (1) with feasible region constructed by Algorithm (2) has the nonempty
feasible solutions set (i.e.,
1 2
( , , , )s
FT
S A D b b ≠ ∅ ).
Proof. By considering the columns 21 2{ , ,..., }mj j j selected by Algorithm 2, let
21 2[ , ,..., ]me j j j′= . We show that De E′∈ and ( )X e X′ ≤ . Then, the result follows from
Corollary 5. From Algorithm 2, the following inequalities are resulted for each 2i I∈ :
(I) 2
ijib X≤ .
(II) 2
ii ijb d≤ .
(III)
2
( 1)( 1)
log (1+ )
( 1)
i
iji
b
s ijX
s s
d
s
− −
≤
−
.
By (I), we have
2
( 1)( 1)
log (1+ ) 1
( 1)
i
ji
b
s X
s s
s
− −
≤
−
. This inequality together with 2
[0,1]ib ∈ ,
2i I∀ ∈ , implies that the interval
2
2 ( 1)( 1)
max ,log (1+ ) ,1
( 1)
i
ji
b
i s X
s s
b
s
− −
−
is meaningful.
Also, by (II),
2
( ) i ie i j J′ = ∈ , 2i I∀ ∈ . Therefore, De E′∈ . Moreover, since the columns
21 2{ , ,..., }mj j j are distinct, sets ( )ijI e′ ( 2i I∈ ) are all singleton, i.e.,
{ }( )ijI e i′ = , 2i I∀ ∈ (7)
14. International Journal in Foundations of Computer Science & Technology (IJFCST) Vol.8, No.3, May 2018
14
As a result, we also have 21 2( ) { , ,..., }mJ e j j j′ = and ( )jI e′ = ∅ for each
21 2{ , ,..., }mj j j j∉ . On the other hand, from Definition 5, we have
( )( , ( )) ( , ) i ie i i j ijX i e i X i j L′′ = = and ( , ( )) 0jX i e i′ = for each { }ij J j∉ − . This fact
together with (7) and Lemma 6 implies ( ) i ij i jX e L′ = , 2i I∀ ∈ , and ( ) 0jX e′ = for
21 2{ , ,..., }mj j j j∉ . So, in order to prove ( )X e X′ ≤ , it is sufficient to show that
( ) ii
jjX e X′ ≤ , 2i I∀ ∈ . But, from Definition 2 and Remark 3,
2
2
2
0 0
( ) ( 1)( 1)
log 1 0
1
i
i i
iji
i
b
j i j
s id
b
X e L s s
b
s
=
′ = = − −
+ ≠ −
(8)
Now, inequality (III) implies
2
( 1)( 1)
log (1+ )
( 1)
i
i
iji
b
js d
s s
X
s
− −
≤
−
(9)
Therefore, by relations (8) and (9), we have ( ) ii
jjX e X′ ≤ , 2i I∀ ∈ . This completes the
proof. □
5.2. Numerical Example
Consider the following linear optimization problem (1) in which the feasible region has been
randomly generated by Algorithm 2 presented in section 5.1.
1 2 3 4 5 6min 0.7358 +5.2422 3.0487 0.7754 + 2.7865 + 8.3467
0.1616 0.1790 0.9810 0.4075 0.9562 0.9790
0.7156 0.6333 0.1270 0.8841 0.1240 0.2833
0.5777 0.6240 0.232
Z x x x x x x= − −
2 0.5481 0.4708 0.1338
0.4333 0.3279 0.0236 0.3690 0.8569 0.6853
0.8842 0.8030 0.6074 0.2083 0.0434 0.9095
0.3931 0.9995 0.1108 0.4409 0.6916 0.6109
0.9000
0.1934
0.7544
0.3463
0.4186
0.1557
0.0003 0.6020 0.0959 0.4564 0.9805 0.8202
0.5409 0.8572 0.7475 0.7930 0.2348 0.8103
0.2077 0.98
xϕ
≤
83 0.7485 0.3846 0.9130 0.5570
0.2193 0.9040 0.5433 0.5386 0.5286 0.2630
0.6205 0.9295 0.3381 0.9917 0.0514 0.6806
0.3258 0.4095 0.8450 0.7552 0.7569
0.0504
0.0365
0.1080
0.1290
0.0482
0.2337 0.0507
[0,1]n
x
x
ϕ
≥
∈
where 1 2 6I I J= = = and
( 1)( 1)
( , ) ( , ) log 1
1
x y
s
F s
s s
x y T x y
s
ϕ
− −
= = +
−
in which 2s = . Moreover,
1 1 2 5 60.7358 +5.2422 + 2.7865 + 8.3467Z x x x x= is the objective function of sub-problem (4) and
15. International Journal in Foundations of Computer Science & Technology (IJFCST) Vol.8, No.3, May 2018
15
2 3 43.0487 0.7754Z x x= − − is that of sub-problem (5). By Definition 2, matrices 6 6( )ijU U ×= and
6 6( )ijL L ×= are as follows:
1.0000 1.0000 0.9179 1.0000 0.9420 0.9198
0.2909 0.3338 1.0000 0.2261 1.0000 0.7322
1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
0.8274 1.0000 1.0000
U =
0.9492 0.4163 0.5323
0.4834 0.5381 0.7164 1.0000 1.0000 0.4680
0.4477 0.1558 1.0000 0.3992 0.2452 0.2823
0.0958 0.6015 0.1316 0.0518 0.0655
0.0791 0.0448 0.0534 0.0496 0.1953 0.0482
0.5869 0.1097 0.1561 0.3271 0.1217 0.2203
0.6505 0.1471 0.2685 0.271
L
∞
=
0 0.2766 0.5536
0.0884 0.0532 0.1746 0.0488 0.9536 0.0791
0.1905 0.1492 0.0634 0.0731 0.0729 0.2671
Therefore, by Corollary 3 we have, for example:
1
11 1 11( , ) [0, ] [0,1]s
FT
S a b U= = and 1
45 4 45( , ) [0, ] [0,0.4163]s
FT
S a b U= = .
2
23 2 23( , ) [ ,1] [0.0534,1]s
FT
S d b L= = and 2
61 6 61( , ) [ ,1] [0.1905,1]s
FT
S d b L= = .
Also, from Definition 1, { }2
1 2,3,...,6J = and { }2
1,2,...,6iJ = , for 2,...,6i = . Actually,
2
11 1( , )s
FT
S d b =∅ and
2
( , )s
F
ij iT
S d b ≠ ∅ for other cases. Moreover, 2
ij id b≥ , { }2,3,...,6i∀ ∈ and
j J∀ ∈ . For the first row of matrix D , we have 2
11 10.0003 0.0504d b= < = and 2
1 1jd b≥ ,
{1}j J∀ ∈ − . Therefore, by Lemma 2 (part (b)), 2 2
1
( , ) ( , )s s
F F
n
i i ij iT T
j
S d b S d b
=
= ≠ ∅U , 2i I∀ ∈ .
By Definition 5, we have
(1) [1 1 0.9179 1 0.9420 0.9198]X = , (2) [0.2909 0.3338 1 0.2261 1 0.7322]X = ,
(3) [1 1 1 1 1 1]X = , (4) [0.8274 1 1 0.9492 0.4163 0.5323]X = ,
(5) [0.4834 0.5381 0.7164 1 1 0.4680]X = , (6) [0.4477 0.1558 1 0.3992 0.2452 0.2823]X = .
Also, for example
(3,1) [0.5869 0 0 0 0 0]X = , (3, 2) [0 0.1097 0 0 0 0]X = ,
(3,3) [0 0 0.1561 0 0 0]X = , (3,4) [0 0 0 0.3271 0 0]X = ,
(3,5) [0 0 0 0 0.1217 0]X = , (3,6) [0 0 0 0 0 0.2203]X = .
Therefore, by Theorem 1, 1
( , ) [ , ( )]s
F
i iT
S a b X i= 0 , 1i I∀ ∈ , and for example
6
2
3 3
1
( , ) [ (3, ), ]s
FT
j
S d b X j
=
= 1U , for the third row of matrix D (i.e., 23i I= ∈ ).
From Corollary 4, the necessary condition holds for the feasibility of the problem. More
precisely, we have
16. International Journal in Foundations of Computer Science & Technology (IJFCST) Vol.8, No.3, May 2018
16
2
0.9805 0.0504
0.8572 0.0365
0.9883 0.1080
0.9040 0.1290
0.9917 0.0482
0.8450 0.0507
D bϕ
= ≥ =
1
that means
2
( , )s
FT
S D b∈1 .
From Definition 7,
[0.29089 0.1558 0.71635 0.22607 0.24523 0.28233]X =
which determines the feasible region of the first inequalities, i.e., 1
( , ) [ , ]s
FT
S A b X= 0 (Theorem
2, part (a)). Also,
2
0.2392 0.0504
0.5226 0.0365
0.5233 0.1080
0.3719 0.1290
0.2263 0.0482
0.5965 0.0507
D X bϕ
= ≥ =
Therefore, we have
2
( , )s
FT
X S D b∈ , which satisfies the necessary feasibility condition stated in
Corollary 6. On the other hand, from Definition 6, we have 38880DE = . Therefore, the
number of all vectors De E∈ is equal to 38880. However, each solution ( )X e generated by
vectors De E∈ is not necessary a feasible solution. For example, for [2,3,1,6,6, 4]e′ = ,
we attain from Definition 7
{ } { }
2
( ) max ( , ( )) max (1,2), (2,3), (3,1), (4,6), (5,6), (6,4)
i I
X e X i e i X X X X X X
∈
′ ′= =
where
(1,2) [0 0.0958 0 0 0 0]X = , (2,3) [0 0 0.0534 0 0 0]X = ,
(3,1) [0.5869 0 0 0 0 0]X = , (4,6) [0 0 0 0 0 0.5536]X = ,
(5,6) [0 0 0 0 0 0.0791]X = , (6,4) [0 0 0 0.0731 0 0]X = .
Therefore, ( ) [0.5869 0.0958 0.0534 0.0731 0 0.5536]X e′ = . It is obvious that
( )X e X′ ≤/ (actually, 11( )X e X′ > and 66( )X e X′ > ) which means
1 2
( ) ( , , , )s
FT
X e S A D b b′ ∉ from Theorem 3. From the first simplification (Lemma 5), “resetting
the following components ija to zeros” are equivalence operations: 11a , 12a , 14a , 23a , 25a ,
3 ja ( 1,2,...,6j = ), 42a , 43a , 54a , 55a , 63a . So, matrix A% is resulted as follows:
17. International Journal in Foundations of Computer Science & Technology (IJFCST) Vol.8, No.3, May 2018
17
0 0 0 .9 8 1 0 0 0 .9 5 6 2 0 .9 7 9 0
0 .7 1 5 6 0 .6 3 3 3 0 0 .8 8 4 1 0 0 .2 8 3 3
0 0 0 0 0
A =% 0
0 .4 3 3 3 0 0 0 .3 6 9 0 0 .8 5 6 9 0 .6 8 5 3
0 .8 8 4 2 0 .8 0 3 0 0 .6 0 7 4 0 0 0 .9 0 9 5
0 .3 9 3 1 0 .9 9 9 5 0 0 .4 4 0 9 0 .6 9 1 6 0 .6 1 0 9
Also, by Definition 9, we can change the value of components 31d , 34d , 41d , 44d , 45d , 46d ,
55d to zeros. For example, since 2
45 J∈ and 545 0.2766 0.24523=L X= > , then 45 0d =% .
Simplified matrix D% is obtained as follows:
0.0003 0.6020 0.0959 0.4564 0.9805 0.8202
0.5409 0.8572 0.7475 0.7930 0.2348 0.8103
0 0.9883 0.7485 0 0.9130 0.5570
0 0.9040
D =%
0.5433 0 0 0
0.6205 0.9295 0.3381 0.9917 0 0.6806
0.3258 0.4095 0.8450 0.7552 0.7569 0.2337
Additionally, { }2
1 2,3,...,6J =% , { }2
2 1,2,...,6J =% , { }2
3 2,3,5,6J =% , { }2
4 2,3J =% , { }2
5 1,2,3,4,6J =% and
{ }2
6 1,2,...,6J =% . Based on these results and Lemma 7, we have 7200DD
E E′= =% . Therefore, the
simplification processes reduced the number of the minimal candidate solutions from 38880 to
7200 , by removing 31680 infeasible points ( )X e . Consequently, the feasible region has 7200
minimal candidate solutions, which are feasible. In other words, for each D
e E∈ % , we have
1 2
( ) ( , , , )s
FT
X e S A D b b∈ . However, each feasible solution ( )X e ( D
e E∈ % ) may not be a minimal
solution for the problem. For example, by selecting [5, 2,4,1,3,6]e′ = , we have
( ) [0.0791 0.1471 0.1746 0.0731 0.0518 0.2203]X e′ = . Although ( )X e′ is feasible (because
of the inequality ( )X e X′ ≤ ) but it is not actually a minimal solution. To see this, let
[2,2, 2,2,2,3]e′′ = . Then, ( ) [0 0.1471 0.0634 0 0 0]X e′′ = . Obviously, ( ) ( )X e X e′′ ′≤
which shows that ( )X e′ is not a minimal solution.
Now, we obtain the modified matrix
*
L according to Definition 10:
*
0.0958 0.6015 0.1316 0.0518 0.0655
0.0791 0.0448 0.0534 0.0496 0.1953 0.0482
0.1097 0.1561 0.1217 0.2203
0.1471 0.2685
0.0884 0.
L
∞
∞ ∞
=
∞ ∞ ∞ ∞
0532 0.1746 0.0488 0.0791
0.1905 0.1492 0.0634 0.0731 0.0729 0.2671
∞
As is shown in matrix
*
L , for each 2i I∈ there exists at least some
2
ij J∈ such that
*
ijL ≠+∞.
Thus, by Theorem 4 we have
1 2
( , , , )s
FT
S A D b b ≠ ∅ .
18. International Journal in Foundations of Computer Science & Technology (IJFCST) Vol.8, No.3, May 2018
18
Finally, vector X is optimal solution of sub-problem (5). For this solution,
3 42
1
3.0487 0.7754 2.3594
n
jj
j
Z c X X X−
=
= = − − = −∑ . Also, 1.7114T
Z c X= = . In order
to find the optimal solution
*
( )X e of sub-problems (4), we firstly compute all minimal solutions
by making pairwise comparisons between all solutions ( )X e ( D
e E∀ ∈ % ), and then we find
*
( )X e among the resulted minimal solutions. Actually, the feasible region has 11 minimal
solutions as follows:
1 [3 ,3, 3 , 3 , 3 , 3]e = 2 [4 ,3, 3 , 3 , 3 , 3]e =
1( ) [0 0 0.6015 0 0 0]X e = 2( ) [0 0 0.2685 0.1316 0 0]X e =
3 [5 ,3, 3 , 3 , 3 , 3]e = 4 [2 , 2, 3 , 3 , 2 , 3]e =
3( ) [0 0 0.2685 0 0.0518 0]X e = 4( ) [0 0.0958 0.2685 0 0 0]X e =
5 [6 ,3, 3 , 3 , 3 , 3]e = 6 [2 , 2, 2 , 2 , 2 , 3]e =
5( ) [0 0 0.2685 0 0 0.0655]X e = 6( ) [0 0.1471 0.0634 0 0 0]X e =
7 [2 , 2, 2 , 2 , 2 , 4]e = 8 [2 , 2, 2 , 2 , 2 , 2]e =
7( ) [0 0.1471 0 0.0731 0 0]X e = 8( ) [0 0.1492 0 0 0 0]X e =
9 [2 ,1, 2 , 2 , 1 , 1]e = 10 [2 , 2, 2 , 2 , 2 , 5]e =
9( ) [0.1905 0.1471 0 0 0 0]X e = 10( ) [0 0.1471 0 0 0.0729 0]X e =
11 [2 , 2 , 2 , 2 , 2 , 6]e =
11( ) [0 0.1471 0 0 0 0.2671]X e =
By comparison of the values of the objective function for the minimal solutions, 1( )X e is optimal
in (4) (i.e., *
1e e= ). For this solution,
1 1 1 1 1 2 1 5 1 6
1
( ) 0.7358 ( ) +5.2422 ( ) + 2.7865 ( ) + 8.3467 ( ) 0
n
j j
j
Z c X e X e X e X e X e+
=
= = =∑ .
Also, 1( ) 1.8337T
Z c X e= = − . Thus, from Corollary 9, *
[0 0 0.7164 0.2261 0 0]x = and then
* *
2.3592T
Z c x= = − .
6. CONCLUSIONS
In this paper, we proposed an algorithm to find the optimal solution of linear problems subjected
to two fuzzy relational inequalities with Frank family of t-norms. The feasible solutions set of the
problem is completely resolved and a necessary and sufficient condition and three necessary
conditions were presented to determine the feasibility of the problem. Moreover, two
simplification operations (depending on the max-Frank composition) were proposed to accelerate
the solution of the problem. Finally, a method was introduced for generating feasible random
max-Frank inequalities. This method was used to generate a test problem for our algorithm. The
resulted test problem was then solved by the proposed algorithm. As future works, we aim at
19. International Journal in Foundations of Computer Science & Technology (IJFCST) Vol.8, No.3, May 2018
19
testing our algorithm in other type of linear optimization problems whose constraints are defined
as FRI with other well-known t-norms.
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