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 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.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
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.
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.
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 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.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
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.
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.
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
PREDICTIVE EVALUATION OF THE STOCK PORTFOLIO PERFORMANCE USING FUZZY CMEANS A...ijfls
The aim of this paper is to investigate the trend of the return of a portfolio formed randomly or for any
specific technique. The approach is made using two techniques fuzzy: fuzzy c-means (FCM) algorithm and
the fuzzy transform, where the rules used at fuzzy transform arise from the application of the FCM
algorithm. The results show that the proposed methodology is able to predict the trend of the return of a
stock portfolio, as well as the tendency of the market index. Real data of the financial market are used from
2004 until 2007.
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 Mathematical Model to Solve Nonlinear Initial and Boundary Value Problems b...IJERA Editor
In this paper, a novel method called Laplace-differential transform method (LDTM) is used to obtain an
approximate analytical solution for strong nonlinear initial and boundary value problems associated in
engineering phenomena. It is determined that the method works very well for the wide range of parameters and
an excellent agreement is demonstrated and discussed between the approximate solution and the exact one in
three examples. The most significant features of this method are its capability of handling non-linear boundary
value problems.
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
Correlation measure for intuitionistic fuzzy multi setseSAT Journals
Abstract In this paper, the Correlation measure of Intuitionistic Fuzzy Multi sets (IFMS) is proposed. The concept of this Correlation measure of IFMS is the extension of Correlation measure of IFS. Using the Correlation of IFMS measure, the application of medical diagnosis and pattern recognition are presented. The new method also shows that the correlation measure of any two IFMS equals one if and only if the two IFMS are the same. Keywords: Intuitionistic fuzzy set, Intuitionistic Fuzzy Multi sets, Correlation measure.
Inventory Model with Price-Dependent Demand Rate and No Shortages: An Interva...orajjournal
In this paper, an interval-valued inventory optimization model is proposed. The model involves the price dependent
demand and no shortages. The input data for this model are not fixed, but vary in some real bounded intervals. The aim is to determine the optimal order quantity, maximizing the total profit and minimizing the holding cost subjecting to three constraints: budget constraint, space constraint, and
budgetary constraint on ordering cost of each item. We apply the linear fractional programming approach based on interval numbers. To apply this approach, a linear fractional programming problem is modeled with interval type uncertainty. This problem is further converted to an optimization problem with interval valued
objective function having its bounds as linear fractional functions. Two numerical examples in crisp
case and interval-valued case are solved to illustrate the proposed approach.
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.
INDUCTIVE LEARNING OF COMPLEX FUZZY RELATIONijcseit
The objective of this paper to investigate the notion of complex fuzzy set in general view. In constraint to a
traditional fuzzy set, the membership function of the complex fuzzy set, the range from [0.1] extended to a
unit circle in the complex plane. In this article the comprehensive mathematical operations on the complex
fuzzy set are presented. The basic operation of complex fuzzy set such as union, intersection, complement
of complex fuzzy set and complex fuzzy relation are studied. Also vector aggregation and fuzzy relation
over the complex fuzzy set are discussed. Two novel operations of complement and projection for complex
fuzzy relation are introduced.
A New Method Based on MDA to Enhance the Face Recognition PerformanceCSCJournals
A novel tensor based method is prepared to solve the supervised dimensionality reduction problem. In this paper a multilinear principal component analysis(MPCA) is utilized to reduce the tensor object dimension then a multilinear discriminant analysis(MDA), is applied to find the best subspaces. Because the number of possible subspace dimensions for any kind of tensor objects is extremely high, so testing all of them for finding the best one is not feasible. So this paper also presented a method to solve that problem, The main criterion of algorithm is not similar to Sequential mode truncation(SMT) and full projection is used to initialize the iterative solution and find the best dimension for MDA. This paper is saving the extra times that we should spend to find the best dimension. So the execution time will be decreasing so much. It should be noted that both of the algorithms work with tensor objects with the same order so the structure of the objects has been never broken. Therefore the performance of this method is getting better. The advantage of these algorithms is avoiding the curse of dimensionality and having a better performance in the cases with small sample sizes. Finally, some experiments on ORL and CMPU-PIE databases is provided.
The Analysis of Performance Measures of Generalized Trapezoidal Fuzzy Queuing...IJERA Editor
The purpose of this research paper was to propose a method which can be utilized to determine the different types of performance measures on the basis of the crisp values for the fuzzy queuing model which has an unreliable server and where the rate of arrival, the rate of service, the rate of breakdown and the rate of repair are all expressed as the fuzzy numbers. In this case the inter arrival time, the time of service, the rates of breakdown and the rates of repair are all triangular functions and are also expressed as the Trapezoidal fuzzy numbers. The main intent is to transform the fuzzy inter arrival time, the time of service, the rates of breakdown and the rates of repair into the crisp values by using the Ranking function method. Then the crisp values are applied in the classical formulas for the performance measure. In the fuzzy environment the ranking fuzzy numbers are very helpful in making the decisions. The ranking function method is one of the most reliable method, is simpler to apply in comparison to other methods and can be utilized to solve the different types of queuing problems. In this research paper a numerical example is also provided for both the triangular and the trapezoidal fuzzy number so that a practical insight into the problem can be provided.
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.
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
PREDICTIVE EVALUATION OF THE STOCK PORTFOLIO PERFORMANCE USING FUZZY CMEANS A...ijfls
The aim of this paper is to investigate the trend of the return of a portfolio formed randomly or for any
specific technique. The approach is made using two techniques fuzzy: fuzzy c-means (FCM) algorithm and
the fuzzy transform, where the rules used at fuzzy transform arise from the application of the FCM
algorithm. The results show that the proposed methodology is able to predict the trend of the return of a
stock portfolio, as well as the tendency of the market index. Real data of the financial market are used from
2004 until 2007.
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 Mathematical Model to Solve Nonlinear Initial and Boundary Value Problems b...IJERA Editor
In this paper, a novel method called Laplace-differential transform method (LDTM) is used to obtain an
approximate analytical solution for strong nonlinear initial and boundary value problems associated in
engineering phenomena. It is determined that the method works very well for the wide range of parameters and
an excellent agreement is demonstrated and discussed between the approximate solution and the exact one in
three examples. The most significant features of this method are its capability of handling non-linear boundary
value problems.
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
Correlation measure for intuitionistic fuzzy multi setseSAT Journals
Abstract In this paper, the Correlation measure of Intuitionistic Fuzzy Multi sets (IFMS) is proposed. The concept of this Correlation measure of IFMS is the extension of Correlation measure of IFS. Using the Correlation of IFMS measure, the application of medical diagnosis and pattern recognition are presented. The new method also shows that the correlation measure of any two IFMS equals one if and only if the two IFMS are the same. Keywords: Intuitionistic fuzzy set, Intuitionistic Fuzzy Multi sets, Correlation measure.
Inventory Model with Price-Dependent Demand Rate and No Shortages: An Interva...orajjournal
In this paper, an interval-valued inventory optimization model is proposed. The model involves the price dependent
demand and no shortages. The input data for this model are not fixed, but vary in some real bounded intervals. The aim is to determine the optimal order quantity, maximizing the total profit and minimizing the holding cost subjecting to three constraints: budget constraint, space constraint, and
budgetary constraint on ordering cost of each item. We apply the linear fractional programming approach based on interval numbers. To apply this approach, a linear fractional programming problem is modeled with interval type uncertainty. This problem is further converted to an optimization problem with interval valued
objective function having its bounds as linear fractional functions. Two numerical examples in crisp
case and interval-valued case are solved to illustrate the proposed approach.
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.
INDUCTIVE LEARNING OF COMPLEX FUZZY RELATIONijcseit
The objective of this paper to investigate the notion of complex fuzzy set in general view. In constraint to a
traditional fuzzy set, the membership function of the complex fuzzy set, the range from [0.1] extended to a
unit circle in the complex plane. In this article the comprehensive mathematical operations on the complex
fuzzy set are presented. The basic operation of complex fuzzy set such as union, intersection, complement
of complex fuzzy set and complex fuzzy relation are studied. Also vector aggregation and fuzzy relation
over the complex fuzzy set are discussed. Two novel operations of complement and projection for complex
fuzzy relation are introduced.
A New Method Based on MDA to Enhance the Face Recognition PerformanceCSCJournals
A novel tensor based method is prepared to solve the supervised dimensionality reduction problem. In this paper a multilinear principal component analysis(MPCA) is utilized to reduce the tensor object dimension then a multilinear discriminant analysis(MDA), is applied to find the best subspaces. Because the number of possible subspace dimensions for any kind of tensor objects is extremely high, so testing all of them for finding the best one is not feasible. So this paper also presented a method to solve that problem, The main criterion of algorithm is not similar to Sequential mode truncation(SMT) and full projection is used to initialize the iterative solution and find the best dimension for MDA. This paper is saving the extra times that we should spend to find the best dimension. So the execution time will be decreasing so much. It should be noted that both of the algorithms work with tensor objects with the same order so the structure of the objects has been never broken. Therefore the performance of this method is getting better. The advantage of these algorithms is avoiding the curse of dimensionality and having a better performance in the cases with small sample sizes. Finally, some experiments on ORL and CMPU-PIE databases is provided.
The Analysis of Performance Measures of Generalized Trapezoidal Fuzzy Queuing...IJERA Editor
The purpose of this research paper was to propose a method which can be utilized to determine the different types of performance measures on the basis of the crisp values for the fuzzy queuing model which has an unreliable server and where the rate of arrival, the rate of service, the rate of breakdown and the rate of repair are all expressed as the fuzzy numbers. In this case the inter arrival time, the time of service, the rates of breakdown and the rates of repair are all triangular functions and are also expressed as the Trapezoidal fuzzy numbers. The main intent is to transform the fuzzy inter arrival time, the time of service, the rates of breakdown and the rates of repair into the crisp values by using the Ranking function method. Then the crisp values are applied in the classical formulas for the performance measure. In the fuzzy environment the ranking fuzzy numbers are very helpful in making the decisions. The ranking function method is one of the most reliable method, is simpler to apply in comparison to other methods and can be utilized to solve the different types of queuing problems. In this research paper a numerical example is also provided for both the triangular and the trapezoidal fuzzy number so that a practical insight into the problem can be provided.
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.
A GENERALIZED SAMPLING THEOREM OVER GALOIS FIELD DOMAINS FOR EXPERIMENTAL DESIGNcscpconf
In this paper, the sampling theorem for bandlimited functions over
domains is
generalized to one over ∏
domains. The generalized theorem is applicable to the
experimental design model in which each factor has a different number of levels and enables us
to estimate the parameters in the model by using Fourier transforms. Moreover, the relationship
between the proposed sampling theorem and orthogonal arrays is also provided.
A Generalized Sampling Theorem Over Galois Field Domains for Experimental Des...csandit
In this paper, the sampling theorem for bandlimited functions over
domains is
generalized to one over ∏
domains. The generalized theorem is applicable to the
experimental design model in which each factor has a different number of levels and enables us
to estimate the parameters in the model by using Fourier transforms. Moreover, the relationship
between the proposed sampling theorem and orthogonal arrays is also provided.
KEY
International Journal of Engineering Research and DevelopmentIJERD Editor
Electrical, Electronics and Computer Engineering,
Information Engineering and Technology,
Mechanical, Industrial and Manufacturing Engineering,
Automation and Mechatronics Engineering,
Material and Chemical Engineering,
Civil and Architecture Engineering,
Biotechnology and Bio Engineering,
Environmental Engineering,
Petroleum and Mining Engineering,
Marine and Agriculture engineering,
Aerospace Engineering.
MULTIPROCESSOR SCHEDULING AND PERFORMANCE EVALUATION USING ELITIST NON DOMINA...ijcsa
Task scheduling plays an important part in the improvement of parallel and distributed systems. The problem of task scheduling has been shown to be NP hard. The time consuming is more to solve the problem in deterministic techniques. There are algorithms developed to schedule tasks for distributed environment, which focus on single objective. The problem becomes more complex, while considering biobjective.This paper presents bi-objective independent task scheduling algorithm using elitist Nondominated
sorting genetic algorithm (NSGA-II) to minimize the makespan and flowtime. This algorithm generates pareto global optimal solutions for this bi-objective task scheduling problem. NSGA-II is implemented by using the set of benchmark instances. The experimental result shows NSGA-II generates efficient optimal schedules.
Accelerated life testing plans are designed under multiple objective consideration, with the resulting Pareto optimal solutions classified and reduced using neural network and data envelopement analysis, respectively.
An Interactive Decomposition Algorithm for Two-Level Large Scale Linear Multi...IJERA Editor
This paper extended TOPSIS (Technique for Order Preference by Similarity Ideal Solution) method for solving
Two-Level Large Scale Linear Multiobjective Optimization Problems with Stochastic Parameters in the righthand
side of the constraints (TL-LSLMOP-SP)rhs of block angular structure. In order to obtain a compromise (
satisfactory) solution to the (TL-LSLMOP-SP)rhs of block angular structure using the proposed TOPSIS
method, a modified formulas for the distance function from the positive ideal solution (PIS ) and the distance
function from the negative ideal solution (NIS) are proposed and modeled to include all the objective functions
of the two levels. In every level, as the measure of ―Closeness‖ dp-metric is used, a k-dimensional objective
space is reduced to two –dimentional objective space by a first-order compromise procedure. The membership
functions of fuzzy set theory is used to represent the satisfaction level for both criteria. A single-objective
programming problem is obtained by using the max-min operator for the second –order compromise operaion.
A decomposition algorithm for generating a compromise ( satisfactory) solution through TOPSIS approach is
provided where the first level decision maker (FLDM) is asked to specify the relative importance of the
objectives. Finally, an illustrative numerical example is given to clarify the main results developed in the paper.
A Mixed Binary-Real NSGA II Algorithm Ensuring Both Accuracy and Interpretabi...IJECEIAES
In this work, a Neuro-Fuzzy Controller network, called NFC that implements a Mamdani fuzzy inference system is proposed. This network includes neurons able to perform fundamental fuzzy operations. Connections between neurons are weighted through binary and real weights. Then a mixed binaryreal Non dominated Sorting Genetic Algorithm II (NSGA II) is used to perform both accuracy and interpretability of the NFC by minimizing two objective functions; one objective relates to the number of rules, for compactness, while the second is the mean square error, for accuracy. In order to preserve interpretability of fuzzy rules during the optimization process, some constraints are imposed. The approach is tested on two control examples: a single input single output (SISO) system and a multivariable (MIMO) system.
Multi objective predictive control a solution using metaheuristicsijcsit
The application of multi objective model predictive control approaches is significantly limited with
computation time associated with optimization algorithms. Metaheuristics are general purpose heuristics
that have been successfully used in solving difficult optimization problems in a reasonable computation
time. In this work , we use and compare two multi objective metaheuristics, Multi-Objective Particle
swarm Optimization, MOPSO, and Multi-Objective Gravitational Search Algorithm, MOGSA, to generate
a set of approximately Pareto-optimal solutions in a single run. Two examples are studied, a nonlinear
system consisting of two mobile robots tracking trajectories and avoiding obstacles and a linear multi
variable system. The computation times and the quality of the solution in terms of the smoothness of the
control signals and precision of tracking show that MOPSO can be an alternative for real time
applications.
A Mathematical Programming Approach for Selection of Variables in Cluster Ana...IJRES Journal
Data clustering is a common technique for statistical data analysis; it is defined as a class of
statistical techniques for classifying a set of observations into completely different groups. Cluster analysis
seeks to minimize group variance and maximize between group variance. In this study we formulate a
mathematical programming model that chooses the most important variables in cluster analysis. A nonlinear
binary model is suggested to select the most important variables in clustering a set of data. The idea of the
suggested model depends on clustering data by minimizing the distance between observations within groups.
Indicator variables are used to select the most important variables in the cluster analysis.
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 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 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.
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.
Structural Dynamic Reanalysis of Beam Elements Using Regression MethodIOSR Journals
This paper concerns with the reanalysis of Structural modification of a beam element based on
natural frequencies using polynomial regression method. This method deals with the characteristics of
frequency of a vibrating system and the procedures that are available for the modification of physical
parameters of vibrating structural system. The method is applied on a simple cantilever beam structure and Tstructure
for approximate structural dynamic reanalysis. Results obtained from the assumed conditions of the
problem indicates the high quality approximation of natural frequencies using finite element method and
regression method.
Let's dive deeper into the world of ODC! Ricardo Alves (OutSystems) will join us to tell all about the new Data Fabric. After that, Sezen de Bruijn (OutSystems) will get into the details on how to best design a sturdy architecture within ODC.
State of ICS and IoT Cyber Threat Landscape Report 2024 previewPrayukth K V
The IoT and OT threat landscape report has been prepared by the Threat Research Team at Sectrio using data from Sectrio, cyber threat intelligence farming facilities spread across over 85 cities around the world. In addition, Sectrio also runs AI-based advanced threat and payload engagement facilities that serve as sinks to attract and engage sophisticated threat actors, and newer malware including new variants and latent threats that are at an earlier stage of development.
The latest edition of the OT/ICS and IoT security Threat Landscape Report 2024 also covers:
State of global ICS asset and network exposure
Sectoral targets and attacks as well as the cost of ransom
Global APT activity, AI usage, actor and tactic profiles, and implications
Rise in volumes of AI-powered cyberattacks
Major cyber events in 2024
Malware and malicious payload trends
Cyberattack types and targets
Vulnerability exploit attempts on CVEs
Attacks on counties – USA
Expansion of bot farms – how, where, and why
In-depth analysis of the cyber threat landscape across North America, South America, Europe, APAC, and the Middle East
Why are attacks on smart factories rising?
Cyber risk predictions
Axis of attacks – Europe
Systemic attacks in the Middle East
Download the full report from here:
https://sectrio.com/resources/ot-threat-landscape-reports/sectrio-releases-ot-ics-and-iot-security-threat-landscape-report-2024/
Key Trends Shaping the Future of Infrastructure.pdfCheryl Hung
Keynote at DIGIT West Expo, Glasgow on 29 May 2024.
Cheryl Hung, ochery.com
Sr Director, Infrastructure Ecosystem, Arm.
The key trends across hardware, cloud and open-source; exploring how these areas are likely to mature and develop over the short and long-term, and then considering how organisations can position themselves to adapt and thrive.
Builder.ai Founder Sachin Dev Duggal's Strategic Approach to Create an Innova...Ramesh Iyer
In today's fast-changing business world, Companies that adapt and embrace new ideas often need help to keep up with the competition. However, fostering a culture of innovation takes much work. It takes vision, leadership and willingness to take risks in the right proportion. Sachin Dev Duggal, co-founder of Builder.ai, has perfected the art of this balance, creating a company culture where creativity and growth are nurtured at each stage.
Software Delivery At the Speed of AI: Inflectra Invests In AI-Powered QualityInflectra
In this insightful webinar, Inflectra explores how artificial intelligence (AI) is transforming software development and testing. Discover how AI-powered tools are revolutionizing every stage of the software development lifecycle (SDLC), from design and prototyping to testing, deployment, and monitoring.
Learn about:
• The Future of Testing: How AI is shifting testing towards verification, analysis, and higher-level skills, while reducing repetitive tasks.
• Test Automation: How AI-powered test case generation, optimization, and self-healing tests are making testing more efficient and effective.
• Visual Testing: Explore the emerging capabilities of AI in visual testing and how it's set to revolutionize UI verification.
• Inflectra's AI Solutions: See demonstrations of Inflectra's cutting-edge AI tools like the ChatGPT plugin and Azure Open AI platform, designed to streamline your testing process.
Whether you're a developer, tester, or QA professional, this webinar will give you valuable insights into how AI is shaping the future of software delivery.
Search and Society: Reimagining Information Access for Radical FuturesBhaskar Mitra
The field of Information retrieval (IR) is currently undergoing a transformative shift, at least partly due to the emerging applications of generative AI to information access. In this talk, we will deliberate on the sociotechnical implications of generative AI for information access. We will argue that there is both a critical necessity and an exciting opportunity for the IR community to re-center our research agendas on societal needs while dismantling the artificial separation between the work on fairness, accountability, transparency, and ethics in IR and the rest of IR research. Instead of adopting a reactionary strategy of trying to mitigate potential social harms from emerging technologies, the community should aim to proactively set the research agenda for the kinds of systems we should build inspired by diverse explicitly stated sociotechnical imaginaries. The sociotechnical imaginaries that underpin the design and development of information access technologies needs to be explicitly articulated, and we need to develop theories of change in context of these diverse perspectives. Our guiding future imaginaries must be informed by other academic fields, such as democratic theory and critical theory, and should be co-developed with social science scholars, legal scholars, civil rights and social justice activists, and artists, among others.
Neuro-symbolic is not enough, we need neuro-*semantic*Frank van Harmelen
Neuro-symbolic (NeSy) AI is on the rise. However, simply machine learning on just any symbolic structure is not sufficient to really harvest the gains of NeSy. These will only be gained when the symbolic structures have an actual semantics. I give an operational definition of semantics as “predictable inference”.
All of this illustrated with link prediction over knowledge graphs, but the argument is general.
Smart TV Buyer Insights Survey 2024 by 91mobiles.pdf91mobiles
91mobiles recently conducted a Smart TV Buyer Insights Survey in which we asked over 3,000 respondents about the TV they own, aspects they look at on a new TV, and their TV buying preferences.
Slack (or Teams) Automation for Bonterra Impact Management (fka Social Soluti...Jeffrey Haguewood
Sidekick Solutions uses Bonterra Impact Management (fka Social Solutions Apricot) and automation solutions to integrate data for business workflows.
We believe integration and automation are essential to user experience and the promise of efficient work through technology. Automation is the critical ingredient to realizing that full vision. We develop integration products and services for Bonterra Case Management software to support the deployment of automations for a variety of use cases.
This video focuses on the notifications, alerts, and approval requests using Slack for Bonterra Impact Management. The solutions covered in this webinar can also be deployed for Microsoft Teams.
Interested in deploying notification automations for Bonterra Impact Management? Contact us at sales@sidekicksolutionsllc.com to discuss next steps.
"Impact of front-end architecture on development cost", Viktor TurskyiFwdays
I have heard many times that architecture is not important for the front-end. Also, many times I have seen how developers implement features on the front-end just following the standard rules for a framework and think that this is enough to successfully launch the project, and then the project fails. How to prevent this and what approach to choose? I have launched dozens of complex projects and during the talk we will analyze which approaches have worked for me and which have not.
Transcript: Selling digital books in 2024: Insights from industry leaders - T...BookNet Canada
The publishing industry has been selling digital audiobooks and ebooks for over a decade and has found its groove. What’s changed? What has stayed the same? Where do we go from here? Join a group of leading sales peers from across the industry for a conversation about the lessons learned since the popularization of digital books, best practices, digital book supply chain management, and more.
Link to video recording: https://bnctechforum.ca/sessions/selling-digital-books-in-2024-insights-from-industry-leaders/
Presented by BookNet Canada on May 28, 2024, with support from the Department of Canadian Heritage.
GraphRAG is All You need? LLM & Knowledge GraphGuy Korland
Guy Korland, CEO and Co-founder of FalkorDB, will review two articles on the integration of language models with knowledge graphs.
1. Unifying Large Language Models and Knowledge Graphs: A Roadmap.
https://arxiv.org/abs/2306.08302
2. Microsoft Research's GraphRAG paper and a review paper on various uses of knowledge graphs:
https://www.microsoft.com/en-us/research/blog/graphrag-unlocking-llm-discovery-on-narrative-private-data/
1. IOSR Journal of Engineering (IOSRJEN) www.iosrjen.org
ISSN (e): 2250-3021, ISSN (p): 2278-8719
Vol. 05, Issue 02 (February. 2015), ||V3|| PP 34-44
International organization of Scientific Research 34 | P a g e
Fuzzy Optimization Technique for Pareto Optimal Solution of
Structural Models with Stress constraints
1
Samir Dey, 2
Tapan Kumar Roy
1
Department of Mathematics, Asansol Engineering College Vivekananda Sarani, Asansol-713305,
West Bengal, India.
2
Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur.
P.O.-Botanic Garden, Howrah-711103, West Bengal, India.
Abstract: Multi-objective non-linear programs occur in various field of engineering application. One of the
applications of such program is structural design problem. In this paper, we consider a generalized form of a
multi-objective structural design problem. Triangular norm based fuzzy programming technique is used to solve
these problem. The test problem includes a three-bar planar truss subjected to a single load condition. The model
is illustrated with numerical examples.
Keywords: Structural design, Fuzzy sets, Pareto optimal solution model, Fuzzy optimization technique, Multi-
objective programming, t-norm.
I. INTRODUCTION
Optimization is the process of minimizing or maximizing an objective function (e.g. cost, weight) of a
structural system which has been frequently employed as the evaluation criterion in structural engineering
applications. But in the practical optimization problems, usually more than one objective are required to be
optimized, such as minimum mass or cost, maximum stiffness, minimum displacement at specific structural
points, maximum natural frequency of free vibration, and maximum structural strain energy. This makes it
necessary to formulate a multi-objective optimization problem. The first note on multi-objective optimization
was given by Pareto; since then the determination of the compromise set of a multi-objective problem is called
Pareto optimization. That is why the application of different optimization technique [11,16-19] to structural
problems has attracted the interest of many researchers.
In conventional mathematical programming, the coefficient or parameters of mathematical models are
assumed to be deterministic and fixed. But, there are many situations where they may not be exactly known i.e.,
they may be somewhat uncertain in nature. Thus the decision making methods under uncertainty are needed.
The fuzzy programming has been proposed from this point view. In decision making process, first Zadeh[2]
introduced fuzzy set theory. Tanaka et al.[20]applied the concept of fuzzy sets to decision making problems by
considering the objectives as fuzzy goals. Later on Bellman and Zadeh [3] used the fuzzy set theory to the
decision making problem. Zimmermann [4] proposed a fuzzy multi-criteria decision making set, defined as the
intersection of all fuzzy goals and their constraints.
In practical, the problem of structural design may be formed as a typical non-linear programming
problem with non-linear objective functions and constraints functions in fuzzy environment. Some researchers
applied the fuzzy set theory to Structural model. For example Wang et al. [1] first applied -cut method to
structural designs where the non-linear problems were solved with various design levels , and then a
sequence of solutions are obtained by setting different level-cut value of . Rao [8] applied the same -cut
method to design a four–bar mechanism for function generating problem .Structural optimization with fuzzy
parameters was developed by Yeh et.al [7]. In 1989, Xu [6] used two-phase method for fuzzy optimization of
structures. In 2004, Shih et.al [9] used level-cut approach of the first and second kind for structural design
optimization problems with fuzzy resources .Shih et.al [10] develop an alternative -level-cuts methods for
optimum structural design with fuzzy resources in 2003. Dey et.at [5] optimize structural model in fuzzy
environment.
Alsina et.al. [13] introduced the t-norm into fuzzy set theory and suggested that the t-norms be used for
the intersection of fuzzy sets. Different types of t-norms theory and their fuzzy inference methods were
introduced by Gupta et.al.[14] .The extension of fuzzy implication operators and generalized fuzzy methods of
cases were discussed by Ruan et.al. [15].
In this paper we propose an approach to solve multi-objective structural model using t-norms based
fuzzy optimization programming technique. In this structural model formulation, the objective functions are the
weight of the truss and the deflection of loaded joint; the design variables are the cross-sections of
2. Fuzzy Optimization Technique for Pareto Optimal Solution of Structural Models with Stress constraints
International organization of Scientific Research 35 | P a g e
the truss members; the constraints are the stresses in members. The test problem includes a three-bar planar truss
subjected to a single load condition. This approximation approach is used to solve this multi-objective structural
optimization model.
The remainder of this paper is organized in the following way. In section II, we discuss about structural
optimization model. We discuss about mathematics Prerequisites and aggregation operator in section III and IV
respectively. In section V, we discuss fuzzy optimization technique to solve multi-objective non-linear
programming problem. In section VI, we discuss Pareto optimality test. In section VII, we solve multi-objective
structural model using t-norms based fuzzy optimization. In section VIII, numerical solution of structural model
of three bar truss. Finally we draw conclusions in section IX.
II. MULTI-OBJECTIVE STRUCTURAL MODEL
In the design of optimal structure i.e. lightest weight of the structure and minimum deflection of loaded
joint that satisfies all stress constraints in members of the structure. To bar truss structure system the basic
parameters (including the elastic modulus, material density, the maximum allowable stress, etc.) are known and
the optimization’s target is that identify the optimal bar truss cross-section area so that the structure is of the
smallest total weight, the minimum nodes displacement, in a given load conditions.
The multi-objective Structural model can be expressed as:
0
min max
( )
( )
( ) [ ]
Minimize WT A
minimize A
subject to A
A A A
(1)
where 1 2, ,.....,
T
nA A A A are design variables for the cross section, n is the group number of design variables
for the cross section bar,
1
n
i i ii
WT A L
is the total weight of the structure, ( )A is the deflection of loaded
joint iL , iA and i were the bar length, cross section area, and density of the th
i group bars respectively. ( )A is
the stress constraint and 0 is maximum allowable stress of the group bars under various conditions, minA and
maxA are the minimum and maximum cross section area respectively.
III. PREREQUISITE MATHEMATICS
III(a). Fuzzy Set:
Let X is a set (space), with a generic element of X denoted by x , that is ( )X x .Then a Fuzzy set (FS) is
defined as , ( ) :AA x x x X
where : [0,1]A
X is the membership function of FS A . ( )A
x is the degree of membership of the element
x to the set A .
III(b). -Level Set or -cut of a Fuzzy Set:
The -level set of the fuzzy set A of X is a crisp set A that contains all the elements of X that have
membership values greater than or equal to i.e. : ( ) , , [0,1]A
A x x x X .
III(c). Convex fuzzy set:
A fuzzy set A of the universe of discourse X is convex if and only if for all 1 2,x x in
X , 1 2 1 21 min ,A A A
x x x x when 0 1 .
IV. AGGREGATION OPERATOR
When the rules in the decision support system contain more than one antecedent, the degrees of
strength of antecedents need to be combined to determine the overall strength of the rule consequent. In the
language of fuzzy sets, the membership values of the linguistic variables in the rule antecedents have to be
combined using an aggregation operator. Formally, a general aggregation is a real function : 0,1 0,1
n
T ,
non decreasing in all arguments, with the properties 0 0T and 1 1T .
General aggregation operators display the whole range of behavior, disjunctive, conjunctive, averaging, mixed,
commutative, mutually reinforcing or otherwise and correspond to vague and loosely defined “and” and “or”
connectives etc. Triangular norms and conforms and averaging operators are well known examples of the
3. Fuzzy Optimization Technique for Pareto Optimal Solution of Structural Models with Stress constraints
International organization of Scientific Research 36 | P a g e
aggregation operators. Different class of aggregation operators display substantially different behavior, it is not
logical to use any particular class to provide generic representation of aggregation. Therefore, we will use
general aggregation operators to model aggregation of rule antecedents in decision support systems. They will
provide the highest degree of adaptability and excellent empirical fit. However, if there are strong reasons to
restrict the selection to a particular family of operators, we will impose the relevant constraints.
Consider general aggregation operator. The function can have a simple algebraic form, such as
1 2 1 2, ,...., min , ,......n nT x x x x x x
or 1 2 1 2
1
, ,...., .......
n
n n i
i
T x x x x x x x
or 1 2
1
, ,...., min 1,
n
n i
i
T x x x x
or 1
1 2, ,....,
n
ii
n
x
T x x x
n
The degree of importance of rule antecedents (vector a ) can be easily incorporated into aggregation operators
in variety of ways. For example
1 1 2 2 1 1 2 2, ; , ;.......; , min , min , ...... min ,n n n nT x a x a x a x a x a x a
1 1 2 2 1 1 2 2, ; , ;.......; , min ....... ,1n n n nT x a x a x a x a x a x a
In this article, decision making method used by the weighted bounded sum operator (member of Yager family of
triangular conorms).
V. FUZZY PROGRAMMING TECHNIQUE TO SOLVE MULTI-OBJECTIVE
NON-LINEAR PROGRAMMING PROBLEM (MONLP)
A Multi-Objective Non-Linear Programming (MONPL) or Vector Minimization problem (VMP) may
be taken in the following form:
1 2( ) [ ( ), ( ),......... ( )]T
kMinimize f x f x f x f x (2)
subject to : ( ) 1,2,3,....,n
j jx X x R g x or or b for j m and ( 1,2,3,...., )i i il x u i n
Zimmermann (1978) showed that fuzzy programming technique can be used to solve the multi-objective
programming problem.
To solve MONLP problem, following steps are used:
Step 1: Solve the MONLP (2) as a single objective non-linear programming problem using only one objective at
a time and ignoring the others, these solutions are known as ideal solution.
Step 2: From the result of step 1, determine the corresponding values for every objective at each solution
derived. With the values of all objectives at each ideal solution, pay-off matrix can be formulated as follows:
Here 1
x ,
2
x , 3
x ,….., k
x are the ideal solutions of the objectives 1( )f x , 2( )f x ,….., ( )kf x respectively.
So 1 2
max ( ), ( ),......, ( )k
r r r rU f x f x f x and 1 2
min ( ), ( ),......, ( )k
r r r rL f x f x f x
Where rU and rL be upper and lower bounds of the th
r objective function ( )rf x for 1,2,3,........,r k .
Step 3: Using aspiration level of each objective of the MONLP (3) may be written as follows:
Find x so as to satisfy
( )r rf x L with tolerance r r rP U L for 1,2,3,........,r k
x X . ( 1,2,3,...., )i i il x u i n
1( )f x 2 ( )f x …. ( )kf x
1
x * 1
1 ( )f x * 1
2 ( )f x ….
* 1
( )kf x
2
x
* 2
1 ( )f x * 2
2 ( )f x ….
* 2
( )kf x
…. …. …. …. ….
k
x
*
1 ( )k
f x *
2 ( )k
f x *
( )k
kf x
4. Fuzzy Optimization Technique for Pareto Optimal Solution of Structural Models with Stress constraints
International organization of Scientific Research 37 | P a g e
Here objective functions of (2) are considered as fuzzy constraints. These types of fuzzy constraints can be
quantified by eliciting a corresponding membership function:
'
'
'
0 ( )
( )
( ( )) ( ) ,
1 ( )
r r
r r
r r r r r
r r
r r
if f x U
U f x
f x if L f x U
U L
if f x L
(3)
where '
r r rL L and 0 r r rU L , for 1,2,3,...,r k
Figure 1: Membership function for objective functions ( )rf x
Having elicited the membership functions as in (3) ( ( ))r rf x for 1,2,3,...,r k a general aggregation function
1 1 2 2( ) ( ( ( )), ( ( )),......, ( ( )))k kD
x F f x f x f x is introduced.
So a fuzzy multi-objective decision making problem can be defined as
( )
.
D
Maximize x
subject to x X
(4)
Fuzzy decision making method used by the (weighted) bounded sum operator (member of Yager family of
triangular conforms) the problem (4) is reduced to
1
1
; ( ( ))
0 ( ( )) 1 1,2,..,
0 1,2,..., , 1
k
r r rD
r
r r
k
r r
r
Maximize x W W f x
subject to x X
f x for r k
where W for all r k W
(5)
Step 4: Solve (5) to get optimal solution.
Some basic definitions are introduced below.
V(a). Complete Optimal Solution
*
x is said to be a complete optimal solution to the MONLP (3) if and only if there exists x X such
that *
r rf x f x for 1,2,...,r k and for all x X .However, when the objective functions of the MONLP
conflict with each other, a complete optimal solution does not always exist and hence the Pareto Optimality
Concept arises and it is defined as follows.
V(b). Pareto Optimal Solution
'
rL rU
( )rf x0
1
( ( ))r rf x
5. Fuzzy Optimization Technique for Pareto Optimal Solution of Structural Models with Stress constraints
International organization of Scientific Research 38 | P a g e
*
x is said to be a Pareto optimal solution to the MONLP (3) if and only if there does not exist
another x X such that *
r rf x f x for all 1,2,...,r k and *
j jf x f x for at least one j ,
1,2,...,j k .
VI. PARETO OPTIMALITY TEST
A numerical test of Pareto optimality for
*
x can be performed by solving the following problem:
1
*
, 1,2,...,
; 0.
k
r
r
r r r
r
Maximize R
subject to f x f x r k
x X
(6)
The optimal solution of (6), say **
x and **
rf x are called strong Pareto optimal solution provided V is very
small otherwise it is called weak Pareto solution.
VII. FUZZY PROGRAMMING TECHNIQUE IN MULTI-OBJECTIVE
STRUCTURAL MODEL
To solve the above MOSOP (1), step 1 of V is used. After that according to step 2 pay-off matrix
formulated as follows:
After that according to step 2, the bounds of objective are 1 1,U L for weight function ( )WT A
(where 1 1( )L WT A U ) and the bounds of objective are 2 2,U L for deflection function A
(where 2 2L A U ) are identified.
Above MOSOPP reduces to a FMOSOPP as follows;
Find A
Such that
1WT A L with maximum allowable tolerance 1 1 1P U L
2A L with maximum allowable tolerance 2 2 2P U L
0
min max
( ) [ ]A
A A A
Here for simplicity linear membership functions WT WT A and A for the objective functions
WT A and A respectively are defined as follows:
'
1
1 '
1 1'
1 1
1
1
0
WT
if WT A L
U WT A
WT A if L WT A U
U L
if WT A U
'
2
2 '
2 2'
2 2
2
1
0
if A L
U A
A if L A U
U L
if A U
( )WT A ( )A
1
A * 1*
( )WT A 1*
( )A
2
A 2*
( )WT A * 2*
( )A
6. Fuzzy Optimization Technique for Pareto Optimal Solution of Structural Models with Stress constraints
International organization of Scientific Research 39 | P a g e
According to step-3, having elicited the above membership functions crisp non-linear programming problem is
formulated as follows
1 2WTMaximize W WT A W A
(7)
min max
1 2 1 2
0 1, 0 1,
( ) [ ],
,
0, 0, 1;
WT
subject to
WT A A
A
A A A
W W W W
The problem (7) can be written as
1 2
1 2' '
1 1 2 2
U WT A U A
maximize W W
U L U L
(8)
1 2
' '
1 1 2 2
min max
1 2 1 2
0 1, 0 1,
( ) [ ],
,
0, 0, 1;
subject to
U WT A U A
U L U L
A
A A A
W W W W
VIII. NUMERICAL SOLUTION OF A MULTI-OBJECTIVE STRUCTURAL
OPTIMIZATION MODEL OF A THREE-BAR TRUSS
A well-known three bar [12] planar truss structure is considered. The design objective is to minimize
weight of the structural 1 2,WT A A and minimize the deflection 1 2,A A along x and y axes at loading
point of a statistically loaded three-bar planar truss subjected to stress constraints on each of the truss
members.
Figure 2. Design of the three-bar planar truss
The multi-objective optimization problem can be stated as follows:
7. Fuzzy Optimization Technique for Pareto Optimal Solution of Structural Models with Stress constraints
International organization of Scientific Research 40 | P a g e
1 2 1 2
1 2
1 2 2
1 1 2
2
1 2 2
1 1 2
1 2
1 1 2 12
1 2 1
2 1 2 2
1 2
2
3 1 2 32
1 2 1
min max
, 2
2
,
2 2
,
2 2
2
,
2 2
,
2
,
2 2
;
x
y
T
T
C
i i i
Minimize WT A A L A A
PL A A
minimize A A
E A A A
PLA
minimize A A
E A A A
P A A
subject to A A
A A A
P
A A
A A
PA
A A
A A A
A A A i
1,2
(9)
Where P = applied load; =material density, L =Length, E=Young’s modulus, 1A = cross section of bar-1 and
bar-3, 2A =cross section of bar-2. x and y are the deflection of loaded joint along x and y axes
respectively. 1
T
and 2
T
the maximum allowable tensile stress for bar 1 and bar 2 respectively. 3
C
is the
maximum allowable compressive stress for bar 3.
The input data for MOSOP (9) is given as follows
Table 1: Input data for crisp model (9)
Appli
ed
load
P
KN
Volume
density
3
/KN m
Length
L
m
Maximum
allowable
tensile
stress
1
T
2
/KN m
Maximum
allowable
tensile
stress
2
T
2
/KN m
Maximum
allowable
compressive
stress
3
C
2
/KN m
Young’s
modulus E
2
/KN m
min
iA and
max
iA of
cross section
of bars
4 2
10 m
20 100 1 20 10 20 8
2 10
min
1 0.1A
max
1 5A
min
2 0.1A
max
2 5A
Solution: According to step 2 pay off matrix is formulated as follows:
1 2( , )WT A A 1 2( , )x A A 1 2( , )y A A
1
A 2.187673 20 5.857864
2
A 15 3 1
3
A 10.1 3.960784 0.039216
Here 15WTU , 2.187673WTL , '
3WTL 20x
U , 3x
L , '
3.5x
L , 5.857864y
U , 0.039216y
L ,
'
0.15y
L ; Here linear membership for the objective functions 1 2( , )WT A A , 1 2( , )x A A and 1 2( , )y A A are
defined as follows:
8. Fuzzy Optimization Technique for Pareto Optimal Solution of Structural Models with Stress constraints
International organization of Scientific Research 41 | P a g e
1 2
1 2
1 2 1 2
1 2
1 ( , ) 3
15 ( , )
( ( , )) 3 ( , ) 15
12
0 ( , ) 15
WT
if WT A A
WT A A
WT A A if WT A A
if WT A A
Figure 3. Membership and non-membership function for 1 2( , )WT A A
1 2
1 2
1 2 1 2
1 2
1 ( , ) 3.5
20 ( , )
( ( , )) 3.5 ( , ) 20
16.5
0 ( , ) 20
x
x
x
x x
x
if A A
A A
A A if A A
if A A
Figure 4. Membership and non-membership function for 1 2( , )x A A
1 2
1 2
1 2 1 2
1 2
1 ( , ) 0.14
5.857864 ( , )
( ( , )) 0.14 ( , ) 5.857864
5.717864
0 ( , ) 5.857864
y
y
y
y y
y
if A A
A A
A A if A A
if A A
3.5 20
1 2( , )x A A
1 2( , )x A A
0
1
3 15
1 2( , )WT A A
0
1
1 2( , )WT A A
9. Fuzzy Optimization Technique for Pareto Optimal Solution of Structural Models with Stress constraints
International organization of Scientific Research 42 | P a g e
Figure 5. Membership and non-membership function for 1 2( , )y A A
Now Fuzzy decision making method used by weighted bounded sum operator (member of Yager family of
triangular conorms) ,
1 1 2 2 1 2 3 1 2( , ) ( , ) ( , )x yMaximize F W WT A A W A A W A A (10)
1 2 1 2
1 2
1 2
1 2
1 2
1 2
1 2
1 2
1 2
2
1 1 2
1 2
2
2
1 1 2
1
0 ( , ) 1; 0 ( , ) 1
0 ( , ) 1
15 ( , )
( , )
12
20 ( , )
( , )
16.5
5.857864 ( , )
( , )
5.717864
20 2
20;
2 2
20
10;
2
20
20;
2 2
0.1 ,
x
y
x
x
y
y
subject to
WT A A A A
A A
WT A A
WT A A
A A
A A
A A
A A
A A
A A A
A A
A
A A A
A
2
1 2 3
5,
1,
A
W W W
The solution obtained from Eq. (10) is given in table 2.
Table 2: Optimal results of MOSOP (9)
1W 2W 3W *
1
4 2
10
A
m
*
2
4 2
10
A
m
*
1 2
2
,
10
WT A A
KN
*
1 2
7
,
10
x A A
m
*
1 2
7
,
10
y A A
m
1 3 1 3 1 3 2.661308 0.1029932 5.425610 7.375100 0.14
0.6 0.2 0.2 1.645225 0.1 3.390449 11.80812 0.3482759
0.2 0.6 0.2 4.542762 0.3 9.394060 4.262608 0.14
0.2 0.2 0.6 2.661309 0.1029933 5.425611 7.375099 0.14
In table 3, the value of R is quite small and hence the optimal results in table 2 are strong Pareto-optimal
solution and can be accepted.
0.14 5.857864
1 2( , )y A A
1 2( , )y A A
0
1
10. Fuzzy Optimization Technique for Pareto Optimal Solution of Structural Models with Stress constraints
International organization of Scientific Research 43 | P a g e
Table 3: Pareto optimal results of MOSOP (9)
R **
1
4 2
10
A
m
**
2
4 2
10
A
m
**
1 2
2
,
10
WT A A
KN
**
1 2
7
,
10
x A A
m
**
1 2
7
,
10
y A A
m
6
0.5277774 10
2.661308 0.1029933 5.425609 7.375100 0.14
IX. CONCLUSIONS
The present paper proposes a solution procedure for structural model. The results of this study may
lead to the development of effective t-norm optimization method. This solution procedure may be used for
solving other model of nonlinear programming problem in different field.
ACKNOWLEDGEMENT
The authors are grateful for the valuable comments and suggestions from the respected reviewers
which have enhanced the strength and significance of our work.
Conflict of interests
The authors declare that there is no conflict of interests.
REFERENCE
[1] Wang,G.Y.and Wang, W.Q., “ Fuzzy optimum design of structure.” Engineering Optimization, 8,291-
300,1985.
[2] L. A. Zadeh, Fuzzy set, Information and Control, vol.8, no.3, pp.338-353, 1965.
[3] R.E. Bellman and L.A. Zadeh, Decision-making in a fuzzy environment, Management Science, 17(4),
B141-B164, 1970.
[4] Zimmermann, H.J., fuzzy linear programming with several objective function” Fuzzy sets and
systems,1,45-55,1978.
[5] Dey.Samir. and Roy,Tapan.Kumar., “Structural Optimization Model with Imprecise Resources” ,
International Journal of Engineering Sciences & Emerging Technologies,6(3), 287-297,2013.
[6] Xu, C. “Fuzzy optimization of structures by the two-phase method”,Computer and Structure, 31(4),575–
580,1989.
[7] Yeh, Y.C, and Hsu, D.S. “Structural optimization with fuzzy parameters”.Computer and Structure, 37(6),
917–24, 1990.
[8] Rao, S.S., “ Description and optimum Design of Fuzzy Mathematical Systems”,
[9] Journal of Mechanisms, Transmissions, and Automation in Design, Vol.109,pp.126-132,1987.
[10] Shih,C. J. and Lee, H. W. “Level-cut Approaches of First and Second Kind for Unique Solution Design
in Fuzzy Engineering Optimization Problems”, Tamkang Journal of Science and Engineering, Vol. 7, No
3, pp. 189-198 ,2004.
[11] Shih,C.J., Chi,C.C. and Hsiao,J.H. “Alternative -level-cuts methods for optimum structural design
with fuzzy resources”, Computers and Structures, 81,2579–2587,2003.
[12] C. J. Shih and C. J. Chang, Mixed-discrete nonlinear fuzzy optimization for multi-objective engineering
design. AIAA-94-1598-CP, pp. 2240-2246, 1994.
[13] Christensen, P. W., and Klarbring, A., “An Introduction to Structural Optimization” Springer science,
2009.
[14] C.Alsina, E. Trillas and I.Valverde, “On some logical connectives for fuzzy set
theory.J.Math.Anal.Appl.93,15-26,1981
[15] Gupta,M.M., and Qi,J., “theory of t-norms and fuzzy inference methods.” Fuzzy sets and systems,40,431-
450,199.
[16] Ruan, D. and Kerre,E.E., Fuzzy implication operators and generalized fuzzy method of cases,fuzzy sets
and systems,54,23-37,1993.
[17] Perez, R.E. and Behdinan, K.: Particle swarm approach for structural design optimization,
[18] Computers & Structures, Vol. 85, No. 19-20, pp. 1579-1588, 2007.
[19] Dede T,Bekirog .lu S,Ayvaz Y. “Weight minimization of trusses with genetic algorithm”. Appl Soft
Comput ,11(2):2565–2575,2011
[20] Luh GC, Lin CY. Optimal design of truss-structures using particle swarm optimization. Computers and
Structures , 89(2324):2221–2232,2011.
11. Fuzzy Optimization Technique for Pareto Optimal Solution of Structural Models with Stress constraints
International organization of Scientific Research 44 | P a g e
[21] Sonmez M. “Discrete optimum design of truss structures using artificial bee colony algorithm”. Struct
Multidiscip Optimiz,43(1):85–97,2011.
[22] Tanaka,H.,Okuda,T. and Asai,K., On fuzzy mathematical programming ,Journal of Cybernetics,3(4): 37-
46,1974.
Short Bio-Data of All Authors
Samir Dey: Samir Dey, Assistant Professor of Mahematics Department , Asansol
Engineering College, Asansol, West Bengal,India, received a M.Sc (Applied
Mathematics) from , Indian Institute of Engineering Science and Technology, Shibpur
and a M.Tech (Operations Research) from National Institute of Technology, Durgapur,
India. His research interest is in application of optimization technique in different field
of operations research and fuzzy systems. He is currently working toward the Ph.D.
degree
Tapan Kumar Roy: Tapan Kumar Roy, professor of Department of Mathematics ,
Indian Institute of Engineering Science and Technology , Shibpur, has published lots of
papers on Fuzzy and Intuitionistic Fuzzy set Theory, Inventory, Transportation,
Reliability Optimization, Portfolio Optimization, Fuzzy and Stochastic Optimization,
etc.