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 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 ax-Frank fuzzy relational inequalities. By this method, we can
easily generate a feasible test problem and employ our algorithm to it.
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 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.
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.
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
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 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.
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.
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
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.
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.
Riccati matrix differential equation has long been known to be so difficult to solve analytically and/or numerically. In this connection, most of the recent studies are concerned with the derivation of the necessary conditions that ensure the existence of the solution. Therefore, in this paper, He’s Variational iteration method is used to derive the general form of the iterative approximate sequence of solutions and then proved the convergence of the obtained sequence of approximate solutions to the exact solution. This proof is based on using the mathematical induction to derive a general formula for the upper bound proved to be converging to zero under certain conditions.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
In this paper, we make use of the fractional differential operator method to find the modified Riemann-Liouville (R-L) fractional derivatives of some fractional functions include fractional polynomial function, fractional exponential function, fractional sine and cosine functions. The Mittag-Leffler function plays an important role in our article, and the fractional differential operator method can be applied to find the particular solutions of non-homogeneous linear fractional differential equations (FDE) with constant coefficients in a unified way and it is a generalization of the method of finding particular solutions of classical ordinary differential equations. On the other hand, several examples are illustrative for demonstrating the advantage of our approach and we compare our results with the traditional differential calculus cases.
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.
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.
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.
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.
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.
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.
Riccati matrix differential equation has long been known to be so difficult to solve analytically and/or numerically. In this connection, most of the recent studies are concerned with the derivation of the necessary conditions that ensure the existence of the solution. Therefore, in this paper, He’s Variational iteration method is used to derive the general form of the iterative approximate sequence of solutions and then proved the convergence of the obtained sequence of approximate solutions to the exact solution. This proof is based on using the mathematical induction to derive a general formula for the upper bound proved to be converging to zero under certain conditions.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
In this paper, we make use of the fractional differential operator method to find the modified Riemann-Liouville (R-L) fractional derivatives of some fractional functions include fractional polynomial function, fractional exponential function, fractional sine and cosine functions. The Mittag-Leffler function plays an important role in our article, and the fractional differential operator method can be applied to find the particular solutions of non-homogeneous linear fractional differential equations (FDE) with constant coefficients in a unified way and it is a generalization of the method of finding particular solutions of classical ordinary differential equations. On the other hand, several examples are illustrative for demonstrating the advantage of our approach and we compare our results with the traditional differential calculus cases.
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.
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
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.
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.
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.
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.
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
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 NEW ALGORITHM FOR SOLVING FULLY FUZZY BI-LEVEL QUADRATIC PROGRAMMING PROBLEMSorajjournal
This paper is concerned with new method to find the fuzzy optimal solution of fully fuzzy bi-level non-linear (quadratic) programming (FFBLQP) problems where all the coefficients and decision variables of both objective functions and the constraints are triangular fuzzy numbers (TFNs). A new method is based on decomposed the given problem into bi-level problem with three crisp quadratic objective functions and bounded variables constraints. In order to often a fuzzy optimal solution of the FFBLQP problems, the concept of tolerance membership function is used to develop a fuzzy max-min decision model for generating satisfactory fuzzy solution for FFBLQP problems in which the upper-level decision maker (ULDM) specifies his/her objective functions and decisions with possible tolerances which are described by membership functions of fuzzy set theory. Then, the lower-level decision maker (LLDM) uses this preference information for ULDM and solves his/her problem subject to the ULDMs restrictions. Finally, the decomposed method is illustrated by numerical example.
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.
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.
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.
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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Explore the innovative world of trenchless pipe repair with our comprehensive guide, "The Benefits and Techniques of Trenchless Pipe Repair." This document delves into the modern methods of repairing underground pipes without the need for extensive excavation, highlighting the numerous advantages and the latest techniques used in the industry.
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Cosmetic shop management system project report.pdfKamal Acharya
Buying new cosmetic products is difficult. It can even be scary for those who have sensitive skin and are prone to skin trouble. The information needed to alleviate this problem is on the back of each product, but it's thought to interpret those ingredient lists unless you have a background in chemistry.
Instead of buying and hoping for the best, we can use data science to help us predict which products may be good fits for us. It includes various function programs to do the above mentioned tasks.
Data file handling has been effectively used in the program.
The automated cosmetic shop management system should deal with the automation of general workflow and administration process of the shop. The main processes of the system focus on customer's request where the system is able to search the most appropriate products and deliver it to the customers. It should help the employees to quickly identify the list of cosmetic product that have reached the minimum quantity and also keep a track of expired date for each cosmetic product. It should help the employees to find the rack number in which the product is placed.It is also Faster and more efficient way.
Hybrid optimization of pumped hydro system and solar- Engr. Abdul-Azeez.pdffxintegritypublishin
Advancements in technology unveil a myriad of electrical and electronic breakthroughs geared towards efficiently harnessing limited resources to meet human energy demands. The optimization of hybrid solar PV panels and pumped hydro energy supply systems plays a pivotal role in utilizing natural resources effectively. This initiative not only benefits humanity but also fosters environmental sustainability. The study investigated the design optimization of these hybrid systems, focusing on understanding solar radiation patterns, identifying geographical influences on solar radiation, formulating a mathematical model for system optimization, and determining the optimal configuration of PV panels and pumped hydro storage. Through a comparative analysis approach and eight weeks of data collection, the study addressed key research questions related to solar radiation patterns and optimal system design. The findings highlighted regions with heightened solar radiation levels, showcasing substantial potential for power generation and emphasizing the system's efficiency. Optimizing system design significantly boosted power generation, promoted renewable energy utilization, and enhanced energy storage capacity. The study underscored the benefits of optimizing hybrid solar PV panels and pumped hydro energy supply systems for sustainable energy usage. Optimizing the design of solar PV panels and pumped hydro energy supply systems as examined across diverse climatic conditions in a developing country, not only enhances power generation but also improves the integration of renewable energy sources and boosts energy storage capacities, particularly beneficial for less economically prosperous regions. Additionally, the study provides valuable insights for advancing energy research in economically viable areas. Recommendations included conducting site-specific assessments, utilizing advanced modeling tools, implementing regular maintenance protocols, and enhancing communication among system components.
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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 n
A a ×
= and
2
( )
ij m n
D d ×
= are fuzzy matrices such that 1
0 ≤
≤ ij
a ( 1
i I
∀ ∈ and j J
∀ ∈ ) and 0 1
ij
d
≤ ≤
2. International Journal in Foundations of Computer Science & Technology (IJFCST) Vol.8, No.3, May 2018
2
( 2
i I
∀ ∈ and j J
∀ ∈ ). 1
1 1
1
( )
i m
b b ×
= is an 1
m –dimensional fuzzy vector in 1
[0,1]m
(i.e.,
1
0 1
i
b
≤ ≤ , 1
i I
∀ ∈ ) , 2
2 2
1
( )
i m
b b ×
= is an 2
m –dimensional fuzzy vector in 2
[0,1]m
(i.e.,
2
0 1
i
b
≤ ≤ , 2
i 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 0
s > and 1
s ≠ .
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 0
s = , 1
s = and s=∞ by taking
limits. So, it is easy to verify that
0
( , ) min{ , }
F
T x y x y
= ,
1
( , )
F
T x y xy
= and
( , ) max{ 1,0}
F
T 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 b
ϕ ≤ and
2
D x b
ϕ ≥ , 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
F
T
S A b and
2
( , )
s
F
T
S D b denote
the feasible solutions sets of inequalities
1
A x b
ϕ ≤ and
2
D x b
ϕ ≥ , respectively, that is,
{ }
1 1
( , ) [0,1] :
s
F
n
T
S A b x A x b
ϕ
= ∈ ≤ and { }
2 2
( , ) [0,1] :
s
F
n
T
S D b x D x b
ϕ
= ∈ ≥ . Also, let
1 2
( , , , )
s
F
T
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 F
T T T
S A D b b S A b S D b
= I .
Definition 1. For each 1
i I
∈ and each j J
∈ , we define
{ }
1 1
( , ) [0,1] : ( , )
s
F
s
ij i F ij i
T
S a b x T a x b
= ∈ ≤ . Similarly, for each 2
i I
∈ and each j J
∈ ,
{ }
2 2
( , ) [0,1] : ( , )
s
F
s
ij i F ij i
T
S d b x T d x b
= ∈ ≥ .
Furthermore, the notations { }
1 1
: ( , )
s
F
i ij i
T
J j J S a b
= ∈ ≠ ∅ , 1
i I
∀ ∈ , and
{ }
2 2
: ( , )
s
F
i ij i
T
J j J S d b
= ∈ ≠ ∅ , 2
i 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 i
T
x
S a b
∈
and
{ }
1
[0,1]
sup ( , )
s
F
ij i
T
x
S a b
∈
exist, if
1
( , )
s
F
ij i
T
S a b ≠ ∅. Moreover, since
s
F
T is a t-norm, its
monotonicity property implies that
1
( , )
s
F
ij i
T
S a b is actually a connected subset of [0,1].
Additionally, due to the continuity of
s
F
T , 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 i
T T
x x
S a b S a b
∈ ∈
= and { } { }
1 1
[0,1]
[0,1]
sup ( , ) max ( , )
s s
F F
ij i ij i
T T
x
x
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 i
T T T
x x
S a b S a b S a b
∈ ∈
=
, i.e., 1
( , )
s
F
ij i
T
S a b is a
closed sub-interval of [0,1]. By the similar argument, if
2
( , )
s
F
ij i
T
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 i
T T T
x 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 1
i I
∈ and each j J
∈ ,
1
( , )
s
F
ij i
T
S a b ≠ ∅. Also,
{ }
1 1
[0,1]
( , ) 0,max ( , )
s s
F F
ij i ij i
T T
x
S a b S a b
∈
=
.
Corollary 2. If
2
( , )
s
F
ij i
T
S d b ≠ ∅ for some 2
i I
∈ and j J
∈ , then
{ }
2 2
[0,1]
( , ) min ( , ) ,1
s s
F F
ij i ij i
T T
x
S d b S d b
∈
=
.
Definition 2. For each 1
i I
∈ and each j J
∈ , we define
1
1
1
1
( 1)( 1)
log 1
1
i
ij
ij i
b
ij
s ij i
a
a b
U s s
a b
s
<
= − −
+ ≥
−
Also, for each 2
i 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 i
a b
= , then 1
ij
U = . Also, we have 1
ij
L = , if
2
0
ij i
d b
= ≠ , and 0
ij
L = if
2
0
ij i
d b
> = .
Lemma 1 below shows that ij
U and ij
L stated in Definition 2, determine the maximum and
minimum solutions of sets
1
( , )
s
F
ij i
T
S a b ( 1
i I
∈ ) and 2
( , )
s
F
ij i
T
S d b ( 2
i 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 i
T
x
U S a b
∈
= , 1
i I
∀ ∈ and j J
∀ ∈ . (b) If
2
( , )
s
F
ij i
T
S d b ≠ ∅ for
some 2
i I
∈ and j J
∈ , then { }
2
[0,1]
m in ( , )
s
F
ij ij i
T
x
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 1
i I
∈ and j J
∈ ,
1
( , ) [0, ]
s
F
ij i ij
T
S a b U
= . (b) If
2
( , )
s
F
ij i
T
S d b ≠ ∅
for some 2
i I
∈ and j J
∈ , then
2
( , ) [ ,1]
s
F
ij i ij
T
S d b L
= .
Definition 3. For each 1
i I
∈ , let { }
{ }
1 1
1
( , ) [0,1] : max ( , )
s
F
n
n s
i i F ij j i
T j
S a b x T a x b
=
= ∈ ≤ .
Similarly, for each 2
i I
∈ , we define { }
{ }
2 2
1
( , ) [0,1] : max ( , )
s
F
n
n s
i i F ij j i
T j
S d b x T d x b
=
= ∈ ≥ .
According to Definition 3 and the constraints stated in (2), sets 1
( , )
s
F
i i
T
S a b and 2
( , )
s
F
i i
T
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
∈
≤ ( 1
i I
∈ )
and
2
max{ ( , )}
s
F ij j i
j J
T d x b
∈
≥ ( 2
i I
∈ ) of problem (1), respectively. Based on (2) and Definitions 1
and 3, it can be easily concluded that for a fixed 1
i I
∈ , 1
( , )
s
F
i i
T
S a b ≠ ∅ iff
1
( , )
s
F
ij i
T
S a b ≠ ∅,
j J
∀ ∈ . On the other hand, by Corollary 1 we know that
1
( , )
s
F
ij i
T
S a b ≠ ∅, 1
i I
∀ ∈ and
j J
∀ ∈ . As a result, 1
( , )
s
F
i i
T
S a b ≠ ∅ for each 1
i I
∈ . However, in contrast to 1
( , )
s
F
i i
T
S a b , set
2
( , )
s
F
i i
T
S d b may be empty. Actually, for a fixed 2
i I
∈ , 2
( , )
s
F
i i
T
S d b is nonempty if and only if
2
( , )
s
F
ij i
T
S d b is nonempty for at least some j J
∈ . Additionally, for each 2
i I
∈ and j J
∈
we have
2
( , )
s
F
ij i
T
S d b ≠ ∅ if and only if
2
ij i
d 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 i
T
S d b ( 2
i 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 i
T
S a b ≠ ∅ , 1
i I
∀ ∈ . (b) For a fixed 2
i I
∈ , 2
( , )
s
F
i i
T
S d b ≠ ∅ iff
2
1
( , )
s
F
n
ij i
T
j
S d b
=
≠∅
U . Additionally, for each 2
i I
∈ and j J
∈ ,
2
( , )
s
F
ij i
T
S d b ≠ ∅ iff
2
ij i
d b
≥ .
Definition 4. For each 2
i I
∈ and 2
i
j J
∈ , we define
2
( , , ) [0,1] ... [0,1] [ ,1] [0,1] ... [0,1]
s
F
i i ij
T
S d b j L
= × × × × × × , where [ ,1]
ij
L 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 in
T
S a b U U U
= × × × , 1
i I
∀ ∈ . (b)
2
2 2
( , ) ( , , )
s s
F F
i
i i i i
T T
j J
S d b S d b j
∈
= U , 2
i I
∀ ∈ .
Proof. See [15]. □
Definition 5. Let 1 2
( ) [ , ,..., ]
i i in
X i U U U
= , 1
i I
∀ ∈ . Also, let
1 2
( , ) [ ( , ) , ( , ) ,..., ( , ) ]
n
X i j X i j X i j X i j
= , 2
i I
∀ ∈ and 2
i
j 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 i
T
S a b X i
= 0 , 1
i I
∀ ∈ . (b)
2
2
( , ) [ ( , ), ]
s
F
i
i i
T
j J
S d b X i j
∈
= 1
U ,
2
i 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 1
i I
∈ , vectors 0 and ( )
X i are the unique minimum and the
unique maximum of set 1
( , )
s
F
i i
T
S a b . In addition, for each 2
i I
∈ , set 2
( , )
s
F
i i
T
S d b has the unique
maximum (i.e., vector 1 ), but the finite number of minimal solutions ( , )
X i j ( 2
i
j 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
F
T
S A D b b ≠ ∅ , then 2
( , )
s
F
i i
T
S d b
∈
1 , 2
i I
∀ ∈ (i.e.,
2
2 2
( , ) ( , )
s s
F F
i i
T T
i I
S d b S D b
∈
∈ =
1 I ).
Proof. Let
1 2
( , , , )
s
F
T
S A D b b ≠ ∅ . Then, 2
( , )
s
F
T
S D b ≠ ∅ , and therefore, 2
( , )
s
F
i i
T
S d b ≠ ∅ ,
2
i I
∀ ∈ . Now, Theorem 1 (part (b)) implies 2
( , )
s
F
i i
T
S d b
∈
1 , 2
i I
∀ ∈ . □
Lemma 4 describes the shape of the feasible solutions set for the fuzzy relational inequalities
1
A x b
ϕ ≤ and
2
D x b
ϕ ≥ , separately.
Lemma 4. (a)
1 1 1
1
1 2
( , ) [0, ] [0, ] ... [0, ]
s
F
i i in
T
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 i
T 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 i
T T
i I
S A b S a b
∈
= I and
2
2 2
( , ) ( , )
s s
F F
i i
T T
i I
S D b S d b
∈
= I . □
Definition 6. Let
2
2
: i
e I J
→ so that
2
( ) i
e i j J
= ∈ , 2
i I
∀ ∈ , and let D
E be the set of all
vectors e. For the sake of convenience, we represent each D
e E
∈ as an 2
m –dimensional vector
2
1 2
[ , ,..., ]
m
e j j j
= in which ( )
k
j e k
= , 2
1,2,...,
k m
= .
Definition 7. Let 2
1 2
[ , ,..., ]
m D
e 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
( ) [ ( ) , ( ) ,..., ( ) ]
n
X 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 b
ϕ ≤ and
2
D x b
ϕ ≥ in the most familiar way.
Theorem 2. (a) 1
( , ) [ , ]
s
F
T
S A b X
= 0 , 1
i I
∀ ∈ . (b)
2
( , ) [ ( ), ]
s
F
D
T
e E
S D b X e
∈
= 1
U .
Proof. For the proof in the general case see Remark 2.5 in [13]. □
Corollary 5. Assume that
1 2
( , , , )
s
F
T
S A D b b ≠ ∅ . Then, there exists some D
e E
∈ such that
[ , ] [ ( ), ]
X X e ≠ ∅
0 1
I .
Corollary 6. Assume that
1 2
( , , , )
s
F
T
S A D b b ≠ ∅ . Then,
2
( , )
s
F
T
X S D b
∈ .
Proof. Let
1 2
( , , , )
s
F
T
S A D b b ≠ ∅ . By Corollary 5, [ , ] [ ( ), ]
X X e′ ≠ ∅
0 1
I for some D
e E
′∈ .
Thus, [ ( ), ]
X X e′
∈ 1 that means [ ( ), ]
D
e E
X X e
∈
∈ 1
U . Therefore, from Theorem 2 (part (b)),
2
( , )
s
F
T
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
F
T
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 ij
a , 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 1
i I
∈ and 0 0
( , )
s
F ij j i
T a x b
< ,
1
( , )
s
F
T
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
0
ij
a to zero” has no effect on the solutions of problem (1) (since component 0
ij
a only appears
in the i ‘th constraint of problem (1)). Therefore, if 0 0
1
( , )
s
F ij j i
T a x b
< ,
1
( , )
s
F
T
x S A b
∀ ∈ , then
“resetting 0
ij
a to zero” is an equivalence operation.
Lemma 5 (simplification of matrix A). Suppose that matrix 1
( )
ij m n
A 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 1
i I
∈ and j J
∈ . Then,
1 1
( , ) ( , )
s s
F F
T 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 i
T
J j J S a b
= ∈ ≠ ∅
% % ( 1
i I
∀ ∈ ) where ij
a
% denotes ( , )
i j ‘th component of
matrix A
% . So, from Corollary 1 and Remark 2, it is clear that 1 1
i i
J J J
= =
% . Moreover, since
1 2 1 2
( , , , ) ( , ) ( , )
s s s
F F F
T 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 F
T T
S A D b b S A D b b
= % . By considering a fixed vector D
e 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 D
E 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
F
T
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
( ) : ( )
j
I e i I e i j
= ∈ = and { }
( ) : ( )
j
J e j J I e
= ∈ ≠ ∅ , D
e E
∀ ∈ . Then,
{ }
( )
( )
max ( )
( )
0 ( )
j
ie i
i I e
j
L j J e
X e
j J e
∈
∈
=
∉
Corollary 8. D
e E′
∈ if and only if ( )
( ) e i
ie i
L X
≤ , 2
i I
∀ ∈ .
As mentioned before, to accelerate identification of the meaningful solutions ( )
X e , we reduce
our search to set D
E′ instead of set D
E . As a result from Corollary 8, we can confine set
2
i
J by
removing each
2
i
j J
∈ such that j
ij
L 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 n
D d ×
= %
% denote a matrix resulted from D
as follows:
2
0 j
i ij
ij
ij
j J and L X
d
d otherwise
∈ >
=
%
Also, similar to Definition 1, assume that { }
2 2
: ( , )
s
F
i ij i
T
J j J S d b
= ∈ ≠ ∅
%
% ( 2
i I
∀ ∈ ) where
ij
d
% denotes ( , )
i j ‘th components of matrix D
% .
According to the above definition, it is easy to verify that 2 2
i i
J J
⊆
% , 2
i 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
i
J
% instead of 2
i
J .
Lemma 7. D
D
E E′
=
% , where D
E % is the set of all functions
2
2
: i
e I J
→ % so that
2
( ) i
e i j J
= ∈ % ,
2
i 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 F
T 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 F
T 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 n
L L ×
= be a matrix whose ( , )
i j ’th component is equal to ij
L . We
define the modified matrix 2
* *
( )
ij m n
L L ×
= from the matrix L as follows:
* j
ij
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
F
T
S A D b b .
Theorem 4.
1 2
( , , , )
s
F
T
S A D b b ≠ ∅ iff there exists at least some
2
i
j J
∈ such that
*
ij
L ≠+∞,
2
i 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 j
c c
+
= and min{ ,0}
j j
c 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 D
e E
′ ′
∈ .
Theorem 5. Suppose that
1 2
( , , , )
s
F
T
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
[ , ,..., ]
n
x 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
F
T
S A D b b ≠ ∅ . Then,
* * * *
1 2
[ , ,..., ]
n
x 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
* *
( ) j
j j
X 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 ij
U ( 1
i I
∀ ∈ and j J
∀ ∈ ) and ij
L ( 2
i I
∀ ∈ and j J
∀ ∈ ) by Definition 2.
2. If
2
( , )
s
F
T
S D b
∈
1 , then continue; otherwise, stop, the problem is infeasible (Corollary 4).
3. Compute vectors ( )
X i ( 1
i I
∀ ∈ ) from Definition 5, and then vector X from Definition 7.
4. If
2
( , )
s
F
T
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 2
i I
∈ , if there exists at least some
2
i
j J
∈ such that
*
ij
L ≠+∞, 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
i
J
% , 2
i I
∀ ∈ ( Lemma 7).
9. Find the optimal solution
* * * *
1 2
[ , ,..., ]
n
x 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
i
ji
j i
i
b
i s ij
X
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 { ,
i
k j
m
k m i
d
i m j j
∈ −
∈ ∉ 2
2 ,..., }
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
F
T
S A D b b ≠ ∅ ).
Proof. By considering the columns 2
1 2
{ , ,..., }
m
j j j selected by Algorithm 2, let
2
1 2
[ , ,..., ]
m
e j j j
′= . We show that D
e E
′∈ and ( )
X e X
′ ≤ . Then, the result follows from
Corollary 5. From Algorithm 2, the following inequalities are resulted for each 2
i I
∈ :
(I) 2
i
j
i
b X
≤ .
(II) 2
i
i ij
b d
≤ .
(III)
2
( 1)( 1)
log (1+ )
( 1)
i
i
ji
b
s ij
X
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]
i
b ∈ ,
2
i 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 i
e i j J
′ = ∈ , 2
i I
∀ ∈ . Therefore, D
e E
′∈ . Moreover, since the columns
2
1 2
{ , ,..., }
m
j j j are distinct, sets ( )
i
j
I e′ ( 2
i I
∈ ) are all singleton, i.e.,
{ }
( )
i
j
I e i
′ = , 2
i 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 2
1 2
( ) { , ,..., }
m
J e j j j
′ = and ( )
j
I e′ = ∅ for each
2
1 2
{ , ,..., }
m
j j j j
∉ . On the other hand, from Definition 5, we have
( )
( , ( )) ( , ) i i
e i i j ij
X i e i X i j L
′
′ = = and ( , ( )) 0
j
X i e i
′ = for each { }
i
j J j
∉ − . This fact
together with (7) and Lemma 6 implies ( ) i i
j i j
X e L
′ = , 2
i I
∀ ∈ , and ( ) 0
j
X e′ = for
2
1 2
{ , ,..., }
m
j j j j
∉ . So, in order to prove ( )
X e X
′ ≤ , it is sufficient to show that
( ) i
i
j
j
X e X
′ ≤ , 2
i 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 i
d
b
X e L s s
b
s
=
′
= = − −
+ ≠
−
(8)
Now, inequality (III) implies
2
( 1)( 1)
log (1+ )
( 1)
i
i
iji
b
j
s d
s s
X
s
− −
≤
−
(9)
Therefore, by relations (8) and (9), we have ( ) i
i
j
j
X e X
′ ≤ , 2
i 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 6
min 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
x
ϕ
≤
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 6
I I J
= = = and
( 1)( 1)
( , ) ( , ) log 1
1
x y
s
F s
s s
x y T x y
s
ϕ
− −
= = +
−
in which 2
s = . Moreover,
1 1 2 5 6
0.7358 +5.2422 + 2.7865 + 8.3467
Z 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 4
3.0487 0.7754
Z x x
= − − is that of sub-problem (5). By Definition 2, matrices 6 6
( )
ij
U U ×
= and
6 6
( )
ij
L 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
F
T
S a b U
= = and 1
45 4 45
( , ) [0, ] [0,0.4163]
s
F
T
S a b U
= = .
2
23 2 23
( , ) [ ,1] [0.0534,1]
s
F
T
S d b L
= = and 2
61 6 61
( , ) [ ,1] [0.1905,1]
s
F
T
S d b L
= = .
Also, from Definition 1, { }
2
1 2,3,...,6
J = and { }
2
1,2,...,6
i
J = , for 2,...,6
i = . Actually,
2
11 1
( , )
s
F
T
S d b =∅ and
2
( , )
s
F
ij i
T
S d b ≠ ∅ for other cases. Moreover, 2
ij i
d b
≥ , { }
2,3,...,6
i
∀ ∈ and
j J
∀ ∈ . For the first row of matrix D , we have 2
11 1
0.0003 0.0504
d b
= < = and 2
1 1
j
d b
≥ ,
{1}
j J
∀ ∈ − . Therefore, by Lemma 2 (part (b)), 2 2
1
( , ) ( , )
s s
F F
n
i i ij i
T T
j
S d b S d b
=
= ≠ ∅
U , 2
i 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 i
T
S a b X i
= 0 , 1
i I
∀ ∈ , and for example
6
2
3 3
1
( , ) [ (3, ), ]
s
F
T
j
S d b X j
=
= 1
U , for the third row of matrix D (i.e., 2
3
i 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 b
ϕ
= ≥ =
1
that means
2
( , )
s
F
T
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
F
T
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 b
ϕ
= ≥ =
Therefore, we have
2
( , )
s
F
T
X S D b
∈ , which satisfies the necessary feasibility condition stated in
Corollary 6. On the other hand, from Definition 6, we have 38880
D
E = . Therefore, the
number of all vectors D
e E
∈ is equal to 38880. However, each solution ( )
X e generated by
vectors D
e 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, 1
1
( )
X e X
′ > and 6
6
( )
X e X
′ > ) which means
1 2
( ) ( , , , )
s
F
T
X e S A D b b
′ ∉ from Theorem 3. From the first simplification (Lemma 5), “resetting
the following components ij
a to zeros” are equivalence operations: 11
a , 12
a , 14
a , 23
a , 25
a ,
3 j
a ( 1,2,...,6
j = ), 42
a , 43
a , 54
a , 55
a , 63
a . 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 31
d , 34
d , 41
d , 44
d , 45
d , 46
d ,
55
d to zeros. For example, since 2
4
5 J
∈ and 5
45 0.2766 0.24523=
L X
= > , then 45 0
d =
% .
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,...,6
J =
% , { }
2
2 1,2,...,6
J =
% , { }
2
3 2,3,5,6
J =
% , { }
2
4 2,3
J =
% , { }
2
5 1,2,3,4,6
J =
% and
{ }
2
6 1,2,...,6
J =
% . Based on these results and Lemma 7, we have 7200
D
D
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
F
T
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 2
i I
∈ there exists at least some
2
i
j J
∈ such that
*
ij
L ≠+∞.
Thus, by Theorem 4 we have
1 2
( , , , )
s
F
T
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 4
2
1
3.0487 0.7754 2.3594
n
j
j
j
Z c X X X
−
=
= = − − = −
∑ . Also, 1.7114
T
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., *
1
e 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
( ) 1.8337
T
Z c X e
= = − . Thus, from Corollary 9, *
[0 0 0.7164 0.2261 0 0]
x = and then
* *
2.3592
T
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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