This document discusses applying duality theory to solve multi-objective linear programming problems. Three methods are analyzed: weighted sum, e-constraint, and a merged weighted sum/e-constraint approach. The weighted sum approach combines objectives into a single function. The e-constraint approach makes all but one objective new constraints. The merged approach minimizes deviations from goals set for each objective. The methods are demonstrated using data on a bank's investment goals of maximizing profit while minimizing risk and capital requirements. Results show the merged approach best handles the multi-objective problem.
A HYBRID COA/ε-CONSTRAINT METHOD FOR SOLVING MULTI-OBJECTIVE PROBLEMSijfcstjournal
In this paper, a hybrid method for solving multi-objective problem has been provided. The proposed method is combining the ε-Constraint and the Cuckoo algorithm. First the multi objective problem transfers into a single-objective problem using ε-Constraint, then the Cuckoo optimization algorithm will optimize the problem in each task. At last the optimized Pareto frontier will be drawn. The advantage of
this method is the high accuracy and the dispersion of its Pareto frontier. In order to testing the efficiency of the suggested method, a lot of test problems have been solved using this method. Comparing the results of this method with the results of other similar methods shows that the Cuckoo algorithm is more suitable for solving the multi-objective problems.
A single stage single constraints linear fractional programming problem an ap...orajjournal
In the present paper we present a new method for solving a class of single stage single constraints linear fractional programming (LFP) problem. The proposed method is based on transformation the objective value and the constraints also. After reducing the fractional program in to equivalent linear program with the help of transformation technique, after that we apply Simplex method to find objective value. Numerical examples are constructed to show the applicability of the above technique
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.
In this paper, Assignment problem with crisp, fuzzy and intuitionistic fuzzy numbers as cost coefficients is investigated. In conventional assignment problem, cost is always certain. This paper develops an approach to solve a mixed intuitionistic fuzzy assignment problem where cost is considered real, fuzzy and an intuitionistic fuzzy numbers. Ranking procedure of Annie Varghese and Sunny Kuriakose [4] is used to transform the mixed intuitionistic fuzzy assignment problem into a crisp one so that the conventional method may be applied to solve the assignment problem. The method is illustrated by a numerical example. The proposed method is very simple and easy to understand. Numerical examples show that an intuitionistic fuzzy ranking method offers an effective tool for handling an intuitionistic fuzzy assignment problem.
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.
A HYBRID COA/ε-CONSTRAINT METHOD FOR SOLVING MULTI-OBJECTIVE PROBLEMSijfcstjournal
In this paper, a hybrid method for solving multi-objective problem has been provided. The proposed method is combining the ε-Constraint and the Cuckoo algorithm. First the multi objective problem transfers into a single-objective problem using ε-Constraint, then the Cuckoo optimization algorithm will optimize the problem in each task. At last the optimized Pareto frontier will be drawn. The advantage of
this method is the high accuracy and the dispersion of its Pareto frontier. In order to testing the efficiency of the suggested method, a lot of test problems have been solved using this method. Comparing the results of this method with the results of other similar methods shows that the Cuckoo algorithm is more suitable for solving the multi-objective problems.
A single stage single constraints linear fractional programming problem an ap...orajjournal
In the present paper we present a new method for solving a class of single stage single constraints linear fractional programming (LFP) problem. The proposed method is based on transformation the objective value and the constraints also. After reducing the fractional program in to equivalent linear program with the help of transformation technique, after that we apply Simplex method to find objective value. Numerical examples are constructed to show the applicability of the above technique
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.
In this paper, Assignment problem with crisp, fuzzy and intuitionistic fuzzy numbers as cost coefficients is investigated. In conventional assignment problem, cost is always certain. This paper develops an approach to solve a mixed intuitionistic fuzzy assignment problem where cost is considered real, fuzzy and an intuitionistic fuzzy numbers. Ranking procedure of Annie Varghese and Sunny Kuriakose [4] is used to transform the mixed intuitionistic fuzzy assignment problem into a crisp one so that the conventional method may be applied to solve the assignment problem. The method is illustrated by a numerical example. The proposed method is very simple and easy to understand. Numerical examples show that an intuitionistic fuzzy ranking method offers an effective tool for handling an intuitionistic fuzzy assignment problem.
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.
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.
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.
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
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.
Medical Conferences, Pharma Conferences, Engineering Conferences, Science Conferences, Manufacturing Conferences, Social Science Conferences, Business Conferences, Scientific Conferences Malaysia, Thailand, Singapore, Hong Kong, Dubai, Turkey 2014 2015 2016
Global Research & Development Services (GRDS) is a leading academic event organizer, publishing Open Access Journals and conducting several professionally organized international conferences all over the globe annually. GRDS aims to disseminate knowledge and innovation with the help of its International Conferences and open access publications. GRDS International conferences are world-class events which provide a meaningful platform for researchers, students, academicians, institutions, entrepreneurs, industries and practitioners to create, share and disseminate knowledge and innovation and to develop long-lasting network and collaboration.
GRDS is a blend of Open Access Publications and world-wide International Conferences and Academic events. The prime mission of GRDS is to make continuous efforts in transforming the lives of people around the world through education, application of research and innovative ideas.
Global Research & Development Services (GRDS) is also active in the field of Research Funding, Research Consultancy, Training and Workshops along with International Conferences and Open Access Publications.
International Conferences 2014 – 2015
Malaysia Conferences, Thailand Conferences, Singapore Conferences, Hong Kong Conferences, Dubai Conferences, Turkey Conferences, Conference Listing, Conference Alerts
International Journal of Computational Engineering Research (IJCER) is dedicated to protecting personal information and will make every reasonable effort to handle collected information appropriately. All information collected, as well as related requests, will be handled as carefully and efficiently as possible in accordance with IJCER standards for integrity and objectivity.
Solving Linear Fractional Programming Problems Using a New Homotopy Perturbat...orajjournal
A new Homotopy Perturbation Method (HPM) is used to find exact solutions for the system of Linear Fractional Programming Problem (LFPP) with equality constraints. In best of my knowledge, first time we are going to introduce a new technique using Homotopy for solving LFP problem. The Homotopy Perturbation method (HPM) and factorization technique are used together to build a new method. A new
technique is also used to convert LFPP to Linear programming problem (LPP). The results betray that our proposed method is very easy and effective compare to the existing method for solving LFP problems with equality constraints applied in real life situations. To illustrate the proposed method numerical examples are solved and the obtained results are discussed.
A Study on Youth Violence and Aggression using DEMATEL with FCM Methodsijdmtaiir
The DEMATEL method is then a good technique for
making decisions. In this paper we analyzed the risk factors of
youth violence and what makes them more aggressive. Since
there are more risk factors of youth violence, to relate each
other more complex to construct FCM and analyze them.
Moreover the data is an unsupervised one obtained from
survey as well as interviews. Hence fuzzy alone has the
capacity to analyses these concepts.
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.
Studies on Material and Mechanical Properties of Natural Fiber Reinforced Com...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.
The papers for publication in The International Journal of Engineering& Science are selected through rigorous peer reviews to ensure originality, timeliness, relevance, and readability.
Theoretical work submitted to the Journal should be original in its motivation or modeling structure. Empirical analysis should be based on a theoretical framework and should be capable of replication. It is expected that all materials required for replication (including computer programs and data sets) should be available upon request to the authors.
The International Journal of Engineering & Science would take much care in making your article published without much delay with your kind cooperation
Thermo luminescence Dating of Quartz Sediments Extracted from the Terraces of...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.
Forecasting Wheat Price Using Backpropagation And NARX Neural Networktheijes
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.
The papers for publication in The International Journal of Engineering& Science are selected through rigorous peer reviews to ensure originality, timeliness, relevance, and readability.
Theoretical work submitted to the Journal should be original in its motivation or modeling structure. Empirical analysis should be based on a theoretical framework and should be capable of replication. It is expected that all materials required for replication (including computer programs and data sets) should be available upon request to the authors.
The International Journal of Engineering & Science would take much care in making your article published without much delay with your kind cooperation
Highway Surface Drainage System & Problems of Water Logging In Road Sectiontheijes
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.
The papers for publication in The International Journal of Engineering& Science are selected through rigorous peer reviews to ensure originality, timeliness, relevance, and readability.
Theoretical work submitted to the Journal should be original in its motivation or modeling structure. Empirical analysis should be based on a theoretical framework and should be capable of replication. It is expected that all materials required for replication (including computer programs and data sets) should be available upon request to the authors.
The International Journal of Engineering & Science would take much care in making your article published without much delay with your kind cooperation
Single Machine Power Network Load Frequency Control Using T-S Fuzzy Based Rob...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.
The papers for publication in The International Journal of Engineering& Science are selected through rigorous peer reviews to ensure originality, timeliness, relevance, and readability.
Theoretical work submitted to the Journal should be original in its motivation or modeling structure. Empirical analysis should be based on a theoretical framework and should be capable of replication. It is expected that all materials required for replication (including computer programs and data sets) should be available upon request to the authors.
The International Journal of Engineering & Science would take much care in making your article published without much delay with your kind cooperation
Study of Damage to ABS Specimens Submitted To Uniaxial Loadingtheijes
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.
The papers for publication in The International Journal of Engineering& Science are selected through rigorous peer reviews to ensure originality, timeliness, relevance, and readability.
Theoretical work submitted to the Journal should be original in its motivation or modeling structure. Empirical analysis should be based on a theoretical framework and should be capable of replication. It is expected that all materials required for replication (including computer programs and data sets) should be available upon request to the authors.
The International Journal of Engineering & Science would take much care in making your article published without much delay with your kind cooperation.
Tensile test of a strand with 2 broken wires artificially damaged and life pr...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.
The papers for publication in The International Journal of Engineering& Science are selected through rigorous peer reviews to ensure originality, timeliness, relevance, and readability.
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.
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.
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
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.
Medical Conferences, Pharma Conferences, Engineering Conferences, Science Conferences, Manufacturing Conferences, Social Science Conferences, Business Conferences, Scientific Conferences Malaysia, Thailand, Singapore, Hong Kong, Dubai, Turkey 2014 2015 2016
Global Research & Development Services (GRDS) is a leading academic event organizer, publishing Open Access Journals and conducting several professionally organized international conferences all over the globe annually. GRDS aims to disseminate knowledge and innovation with the help of its International Conferences and open access publications. GRDS International conferences are world-class events which provide a meaningful platform for researchers, students, academicians, institutions, entrepreneurs, industries and practitioners to create, share and disseminate knowledge and innovation and to develop long-lasting network and collaboration.
GRDS is a blend of Open Access Publications and world-wide International Conferences and Academic events. The prime mission of GRDS is to make continuous efforts in transforming the lives of people around the world through education, application of research and innovative ideas.
Global Research & Development Services (GRDS) is also active in the field of Research Funding, Research Consultancy, Training and Workshops along with International Conferences and Open Access Publications.
International Conferences 2014 – 2015
Malaysia Conferences, Thailand Conferences, Singapore Conferences, Hong Kong Conferences, Dubai Conferences, Turkey Conferences, Conference Listing, Conference Alerts
International Journal of Computational Engineering Research (IJCER) is dedicated to protecting personal information and will make every reasonable effort to handle collected information appropriately. All information collected, as well as related requests, will be handled as carefully and efficiently as possible in accordance with IJCER standards for integrity and objectivity.
Solving Linear Fractional Programming Problems Using a New Homotopy Perturbat...orajjournal
A new Homotopy Perturbation Method (HPM) is used to find exact solutions for the system of Linear Fractional Programming Problem (LFPP) with equality constraints. In best of my knowledge, first time we are going to introduce a new technique using Homotopy for solving LFP problem. The Homotopy Perturbation method (HPM) and factorization technique are used together to build a new method. A new
technique is also used to convert LFPP to Linear programming problem (LPP). The results betray that our proposed method is very easy and effective compare to the existing method for solving LFP problems with equality constraints applied in real life situations. To illustrate the proposed method numerical examples are solved and the obtained results are discussed.
A Study on Youth Violence and Aggression using DEMATEL with FCM Methodsijdmtaiir
The DEMATEL method is then a good technique for
making decisions. In this paper we analyzed the risk factors of
youth violence and what makes them more aggressive. Since
there are more risk factors of youth violence, to relate each
other more complex to construct FCM and analyze them.
Moreover the data is an unsupervised one obtained from
survey as well as interviews. Hence fuzzy alone has the
capacity to analyses these concepts.
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.
Studies on Material and Mechanical Properties of Natural Fiber Reinforced Com...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.
The papers for publication in The International Journal of Engineering& Science are selected through rigorous peer reviews to ensure originality, timeliness, relevance, and readability.
Theoretical work submitted to the Journal should be original in its motivation or modeling structure. Empirical analysis should be based on a theoretical framework and should be capable of replication. It is expected that all materials required for replication (including computer programs and data sets) should be available upon request to the authors.
The International Journal of Engineering & Science would take much care in making your article published without much delay with your kind cooperation
Thermo luminescence Dating of Quartz Sediments Extracted from the Terraces of...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.
Forecasting Wheat Price Using Backpropagation And NARX Neural Networktheijes
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.
The papers for publication in The International Journal of Engineering& Science are selected through rigorous peer reviews to ensure originality, timeliness, relevance, and readability.
Theoretical work submitted to the Journal should be original in its motivation or modeling structure. Empirical analysis should be based on a theoretical framework and should be capable of replication. It is expected that all materials required for replication (including computer programs and data sets) should be available upon request to the authors.
The International Journal of Engineering & Science would take much care in making your article published without much delay with your kind cooperation
Highway Surface Drainage System & Problems of Water Logging In Road Sectiontheijes
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.
The papers for publication in The International Journal of Engineering& Science are selected through rigorous peer reviews to ensure originality, timeliness, relevance, and readability.
Theoretical work submitted to the Journal should be original in its motivation or modeling structure. Empirical analysis should be based on a theoretical framework and should be capable of replication. It is expected that all materials required for replication (including computer programs and data sets) should be available upon request to the authors.
The International Journal of Engineering & Science would take much care in making your article published without much delay with your kind cooperation
Single Machine Power Network Load Frequency Control Using T-S Fuzzy Based Rob...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.
The papers for publication in The International Journal of Engineering& Science are selected through rigorous peer reviews to ensure originality, timeliness, relevance, and readability.
Theoretical work submitted to the Journal should be original in its motivation or modeling structure. Empirical analysis should be based on a theoretical framework and should be capable of replication. It is expected that all materials required for replication (including computer programs and data sets) should be available upon request to the authors.
The International Journal of Engineering & Science would take much care in making your article published without much delay with your kind cooperation
Study of Damage to ABS Specimens Submitted To Uniaxial Loadingtheijes
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.
The papers for publication in The International Journal of Engineering& Science are selected through rigorous peer reviews to ensure originality, timeliness, relevance, and readability.
Theoretical work submitted to the Journal should be original in its motivation or modeling structure. Empirical analysis should be based on a theoretical framework and should be capable of replication. It is expected that all materials required for replication (including computer programs and data sets) should be available upon request to the authors.
The International Journal of Engineering & Science would take much care in making your article published without much delay with your kind cooperation.
Tensile test of a strand with 2 broken wires artificially damaged and life pr...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.
The papers for publication in The International Journal of Engineering& Science are selected through rigorous peer reviews to ensure originality, timeliness, relevance, and readability.
Microbiological Quality of Malaysian Heritage Food ('Satar') Sold In Marang a...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.
The papers for publication in The International Journal of Engineering& Science are selected through rigorous peer reviews to ensure originality, timeliness, relevance, and readability.
Theoretical work submitted to the Journal should be original in its motivation or modeling structure. Empirical analysis should be based on a theoretical framework and should be capable of replication. It is expected that all materials required for replication (including computer programs and data sets) should be available upon request to the authors.
The International Journal of Engineering & Science would take much care in making your article published without much delay with your kind cooperation
Particle Swarm Optimization based Network Reconfiguration in Distribution Sys...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.
The papers for publication in The International Journal of Engineering& Science are selected through rigorous peer reviews to ensure originality, timeliness, relevance, and readability.
Theoretical work submitted to the Journal should be original in its motivation or modeling structure. Empirical analysis should be based on a theoretical framework and should be capable of replication. It is expected that all materials required for replication (including computer programs and data sets) should be available upon request to the authors.
The International Journal of Engineering & Science would take much care in making your article published without much delay with your kind cooperation
Mechanical behavior and statistical study of aluminum wires belonging to low ...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.
Modelling Of Underground Cables for High Voltage Transmissiontheijes
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.
The papers for publication in The International Journal of Engineering& Science are selected through rigorous peer reviews to ensure originality, timeliness, relevance, and readability.
Theoretical work submitted to the Journal should be original in its motivation or modeling structure. Empirical analysis should be based on a theoretical framework and should be capable of replication. It is expected that all materials required for replication (including computer programs and data sets) should be available upon request to the authors.
The International Journal of Engineering & Science would take much care in making your article published without much delay with your kind cooperation
Metsulfuron Methyl - Effect On Soil Microfloratheijes
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.
The papers for publication in The International Journal of Engineering& Science are selected through rigorous peer reviews to ensure originality, timeliness, relevance, and readability.
Theoretical work submitted to the Journal should be original in its motivation or modeling structure. Empirical analysis should be based on a theoretical framework and should be capable of replication. It is expected that all materials required for replication (including computer programs and data sets) should be available upon request to the authors.
The International Journal of Engineering & Science would take much care in making your article published without much delay with your kind cooperation.
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The papers for publication in The International Journal of Engineering& Science are selected through rigorous peer reviews to ensure originality, timeliness, relevance, and readability.
Theoretical work submitted to the Journal should be original in its motivation or modeling structure. Empirical analysis should be based on a theoretical framework and should be capable of replication. It is expected that all materials required for replication (including computer programs and data sets) should be available upon request to the authors.
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programming problem MOLFPP in this research. The transformation characteristics are illustrated and the
solution procedure and numerical example are presented.
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profit/net-sales, profit/manufacturing cost, etc. In this paper, we proposed the transformation characteristics to solve the multi objective linear fractional programming (MOLFP) problems. If a MOLFP problem with both the numerators and the denominators of the objectives are linear functions and some
technical linear restrictions are satisfied, then it is defined as a multi objective linear fractional programming problem MOLFPP in this research. The transformation characteristics are illustrated and the solution procedure and numerical example are presented.
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this method is the high accuracy and the dispersion of its Pareto frontier. In order to testing the efficiency
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A0311010106
1. The International Journal Of Engineering And Science (IJES)
|| Volume || 3 || Issue || 11 || Pages || 01-06|| 2014 ||
ISSN (e): 2319 – 1813 ISSN (p): 2319 – 1805
www.theijes.com The IJES Page 1
Duality Theory in Multi Objective Linear Programming
Problems
Chukwu W.I.E1
, Uka P.N2
and Dike, A. O3
1
Department of statistics, University of Nigeria, Nsukka, Nigeria
2 & 3
Department of Maths/Statistics, Akanu Ibiam Federal Polytechnic Unwana ,Afikpo, Nigeria
---------------------------------------------------ABSTRACT -------------------------------------------------------
This work is aimed at applying duality theory to multi-objective linear programming problems. Three methods
of solutions were discussed and applied in this study with the sole aim of finding out which of these three
methods is the best for solving Multi Objective Linear Programming (MOLP) problems. The major components
of the models used are the weighted sum and the e-constraint terms. Dual optimal functions were used in
performing sensitivity analyses. The investigation in this study was carried out using a bank’s investment data
whose dual results showed that the merged weighted sum/e-constraint approach is the best method for handling
MOLP problems
---------------------------------------------------------------------------------------------------------------------------------------
Date of Submission: 16 August 2014 Date of Accepted: 10 November 2014
---------------------------------------------------------------------------------------------------------------------------------------
I. INTRODUCTION
The theory of duality is one of the most important and interesting concepts in both single objective and
multiple objective mathematical programming of operations research. Our focus in this work is to apply duality
theory to multiple objective linear programming. The basic idea behind the duality theory is that every primal
linear programming problem has an associated linear programming problem called its dual such that a solution
to the primal linear programming also provides a solution to its dual. In view of the above discussion, the
problem we are looking at in this study is a multiple objective-programming problem and we will be adapting
appropriate duality theorems with which to solve it. In a multi objective linear programming problem (MOLP),
each feasible solution set is said to be an efficient solution since such a solution can not be optimal to all the
objective functions at the same time.
Philip (1972) stated that there is always a dual solution for each efficient primal solution and that the
dual variables are the “prices” of the resources of the problem which are dependent on the subjective weights of
the primal problem. Kornbluth (1974) investigated the duality of multiple objective linear programmes and used
the dual optimal function to perform sensitivity analysis. Lohne (1990) developed a duality theory for linear
multiple objective programming problem in which he was able to verify similar properties as in the scalar case,
that is, single objective linear programming by using what is called “strong proper optima”. In the study he
found out that such optima and its associated dual solution are characterized by means of complementary
slackness conditions. Arua et al (2000) observed that primal – dual programming problems can either be
symmetric or asymmetric. Heyde et al (2006) developed a geometric approach to duality in multiple objective
linear programming which is based on the duality of polytopes with one – to – one mapping between the
minimal faces of the image of the primal objective function and the maximal faces of the image of the dual
objective function.
Klamroth et al (2003) studied a linear multiple criteria optimization and specified that weighted sum
approach and e-constraint approach can be used to solve the problem. They asserted that duality is a powerful
tool for the generation of weighting vectors and hence of utility functions for MOP. They went further to
indicate that the weighting vectors can be deduced from the optimal dual variables of the e-constraint problems.
They asserted that weighted sum secularization of the objective functions cannot be used to generate all non
dominated solutions because some will be supported while others will be unsupported. The above discussions
from the earlier works in the literature makes it clear that duality theory can be applied to multiple objective
linear programming when the problem is reduced from MOP to a single objective problemm. In view of this, we
modify approach adopted in by Klamroth et al (2003) by combining weighted sum approach and e -constraint in
solving a multiple objective programming problem. Furthermore the same problem was solved using weighted
and e-constraint method separately so as to find out which approach is better.
2. Duality Theory In Multi Objective…
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II. METHODOLOGY
Duality generally follows from the main (primal) linear programming problem (mathematical program
whose objective function(s) and constraints are in linear form.), which can either, be a maximization or
minimization problem as the case may be. In many applications of linear and mathematical programming, it is
often difficult or somehow misleading to represent the aspirations of the decision maker (or decision making
body) in terms of a single objective function because of their diverse nature of activities. Indeed in complex
industrial or governmental problems, the decision maker may frequently have to choose between alternative
courses of action on the basis of multiple and conflicting criteria. Hence, MOLP is of mathematical
programming that deals with decision problems characterized by multiple and conflicting objective functions
that are to be optimized over a feasible set of decisions. Such problems referred to as multi-objective programs
are commonly encountered in many areas of human activities including engineering, management and others.
This concept began with the work of Charnes and Cooper in 1961 and has been refined and extended by many
other researchers since then.
MOLP PROBLEM FORMULATION
We use the following notation throughout this work. x denotes xi for all i = 1, …, k.
For k x n matix, Ci. = Ci; i.e ith row of C.
Nj is submatrix consisting of the rows in Ji – constraints which form the e-
constraint equations.
Hence fj(x) = Nj(x)
We consider the following general multiple objective program
(MOP).
Max { Z1 = f1 (x) }
(1)
Max { Zk = fk (x)}
Subject to x є S (Klamroth, 2003)
Where S Rn
is the feasible set consisting of x1, x2, … xn and fi (x), i = 1, 2, ….., k, are real-valued functions.
The problem above is called a multiple objective linear program (MOLP) if
fi(x) = Ci x ί = 1, 2, …, k and
S = { x є Rn
; Ax ≤ b; x ≥ 0 }
Where we assume that the vectors C= (Cί) ί =1,2…., k , b , b m-components and the m x n – matrix A
are all real valued. Thus, the MOLP can be written as
Max Z = Cx
Subject to Ax b; x ≥ 0
We define the set of all feasible criterion vectors Z, the set of all non-dominated criterion vectors N and
the set of all efficient points E of equation (1) as follows:-
Z ={ z є Rk
: z = f(x),x є S} = f (S)
N = {z є Z: z є Z: z ≥ z} (2)
E = {x є S: f (x) є N}
Where f(x) = (f1(x), …, fk(x))T.
GENERATION OF THE SOLUTION SETS
There are two general approaches to generate solution sets of MOPs. They are scalarization methods
and non-scalarization methods. Scalarization method involves formulating an MOP – related SOP which is a
function of the objective functions of the MOP while non scalarization methods do not explicitly use a
scalarization function. To solve the MOP problem in this work, we used scalarization methods. The two well
known scalarization techniques used in this work include the weighted sum approach and e-constraint approach.
The third approach is gotten by merging these two approaches. These three approaches will be used to solve the
same problem so as to know the one that is the best for handling MOLP problems.
3. Duality Theory In Multi Objective…
www.theijes.com The IJES Page 3
WEIGHTED-SUM APPROACH : In the weighted sum approach, a weighted sum of the objective functions
is maximized or minimized as the case may be. These objective functions of MOLP are pooled together using
the strictly positive weighting vectors say {û є Rk
: û > 0, Σ ûί =1}, such that the MOLP becomes SOP given by
Max z = ûT
C (x)
Subject to Ax ≤ b; x ≥ 0
e-CONSTRAINT APPROACH : In the e-constraint approach, one objective function out of the objectives of
the MOLP is retained as a scalar – valued objective while all the other objective functions are made new
constraints by assigning goals/targets (to be achieved) to them. Hence, the ith
objective e-constraint program
becomes:
Max z = Ci(x);
Subject to Nj(x) ≥ ej Jij
;
Ax ≤ b; x ≥ 0
Where: Cix is one out of the given objective functions
Njx ≥ ej = the remaining objective functions (with the targets, ej assigned to them) which now form part of the
constraint equations. Hence they are called e-constraint equations.
The Merged Weighted Sum/ e-Constraint Approach :In MOLP, goals could be set by the decision maker
and such goals can either be overachieved or underachieved. Usually interest is to achieve goals set in any firm
and in view of this we always try to minimize any trace of underachievement and/or overachievement which
may sometimes occur. In this work also, considering the assignment of targets; since the interest of the decision
maker is to minimize the deviations from the targets as discussed earlier, we now obtain another single model
that comprises both the weighted sum and e-constraint terms in which the objective is to minimize the weighted
sum of these deviations. The weights are assigned according to the levels of minimization decided by the
decision maker. In the linear programming simplex approach, such deviations are called slack variables
whereas in MOLP, these slack terms are either positive or negative indicating overachievement or
underachievement of the set goals and not only are they real variables but are also the only terms in the
objective function which is to be minimized. Hence the resultant program becomes
Min z = WT
dj
Subject to Nj(x) + dj ≥ ej Ji j
;
Ax ≤ b; x ≥ 0; dj ≥ 0.
Where W = the weight assigned by the decision maker; dj which is the deviational variable can be
-
j
dor
j
d
depending on the objective of the decision maker;
j
d = overachievement of the assigned goal;
-
j
d = is the underachievement of the assigned goal
All these approaches to solutions were demonstrated in the analyses and the results discussed in chapter
four.
DATA COLLECTION ON BANK’S INVESTMENT : Every investor must trade off return versus risk in
deciding how to allocate his or her available funds. The opportunities that promise the greatest profits are almost
always the ones that present the most serious risks. Investment banks must be especially careful in balancing
return and risk because legal and ethical obligations demand that they avoid undo hazards. Yet their goal as a
business enterprise is to maximize profit and also minimize risk.The investment bank in discussion whose name
is not to be mentioned for security and integrity reasons has about N20 billion as the capital with N150 billion in
demand deposits and N80 billion in time deposits. The chart (table) below provides every detail about the
investment.
Bank investment category (j) Return rate (%) Liquid part (%) Required capital (%) Risk asset (%)
1 Cash 0.0 100.0 0.0 No
2 Short term 4.0 99.5 0.5 No
3 Government:2 years 4.5 96.0 4.0 No
4 Government:4 years 5.5 90.0 5.0 No
5 Government:6 years 7.0 85.0 7.5 No
6 Installment loans 10.5 0.0 10.0 Yes
7 Mortgage loans 8.5 0.0 10.0 Yes
8 Commercial loans 9.2 0.0 10.0 Yes
The first goal of any business enterprise is to maximize profit. Hence, using the rates of return from
the above table, the objective function becomes
Max P = 0.040X2 + 0.045X3 + 0.055X4 + 0.070X5 + 0.105X6 + 0.085X7 + 0.092X8 [profit]
4. Duality Theory In Multi Objective…
www.theijes.com The IJES Page 4
In order to quantify investment risk, we employ the two commonly used ratio measures. The first is the
capital adequacy ratio which is expressed as the ratio of the required capital to the actual capital. A low value
indicates minimum risk and vise versa. The second objective function becomes
Min C =
20
1 (0.005X2 + 0.040X3 + 0.050X4 + 0.075X5 + 0.100X6 + 0.100X7 + 0.100X8)
[capital-adequacy]
Another measure of risk focuses on illiquid risk assets which is measured by the ratio of risk asset to
the actual capital and a low ratio indicates a financially secure institution. Hence the third objective function
becomes
Min R =
20
1 (X6 + X7 + X8) [Risk-asset]
The following information from the bank presents the relevant constraints.
(1) Investment must sum to the available capital and deposit funds.
(2) Cash reserves must be at least 14% of demand deposit plus 4% of time deposits.
(3) The portion of investments considered liquid should be at least 47% of demand deposits plus 36% of time
deposits.
(4) At least 5% of funds should be invested in each of the eight categories, for diversity.
(5) At least 30% funds should be invested in commercial loans, to maintain the bank’s community status.
ANALYSIS OF DATA : Combining the three objective functions above with these five systems of constraints,
the multiple objective linear programming model of this Bank’s investment becomes
Max P = 0.040X2 + 0.045X3 + 0.055X4 + 0.070X5 + 0.105X6 + 0.085X7 + 0.092X8 [Profit]
Min C =
20
1 (0.005X2 + 0.040X3 + 0.050X4 + 0.075X5 + 0.100X6 + 0.100X7 + 0.100X8)
[Capital-adequacy]
Min R =
20
1 (X6 + X7 + X8) [Risk-asset]
Subject to:
X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 = 250 [Invest all]
X1 ≥ 24.2 [Cash reserve]
1.00X1 + 0.995X2 + 0.960X3 + 0.900X4 + 0.850X5 ≥ 99.3 [liquidity]
Xj ≥ 12.5 for all j = 1, 2, …, 8 [ diversification]
X8 ≥ 75 [commercial]
Here solutions are evaluated on three criteria namely profit, capital-adequacy ratio and risk-asset ratio.
The above problem was solved using the weighted sum approach with a weighting vector
3
1
.
With the set goals for the three criteria as follows
Profit ≥ 18.5
Capital-adequacy ratio ≤ 0.8
Risk-asset ratio ≤ 7.0,
the three objective functions can then be stated in the following manner
Goal 1 0.040X2 + 0.045X3 + 0.055X4 + 0.070X5 + 0.105X6 + 0.085X7 + 0.092X8 ≥ 18.5 [profit ]
Goal 2
20
1 (0.005X2 + 0.040X3 + 0.050X4 + 0.075X5 + 0.100X6 + 0.100X7 + 0.100X8) ≤ 0.8
[capital-adequacy]
Goal 3
20
1 (X6 + X7 + X8) ≤ 7.0 [risk-asset]
Subject to the constraints as used above. The problem was solved using the e-constraint approach
Also, considering the Bank’s investment problem, each of the given goals can either be overachieved or
underachieved designated by positive deviational variables and negative deviational variables respectively.
For goal 1, our interest is to minimize underachievement since overachievement increases the profit.
For goals 2 & 3, our interest is to minimize overachievement of risk.
Hence we define the following deviational variables of interest
1
d = amount with which profit falls short of its goal
2
d = amount with which capital-adequacy ratio exceeds its goal.
3
d = amount with which risk-asset ratio exceeds its goal.
5. Duality Theory In Multi Objective…
www.theijes.com The IJES Page 5
The objective function which now comprises these deviational variables in a minimization sense is
written as
Min Z =
1
d +
2
d +
3
d
Subject to the above given constraints with goal constraints inclusive.
In the analyses, we consider both equal and unequal goal weights.
Using equal goal weights w1 = w2 = w3 = 1, the Bank’s investment problem has the following linear
goal programming model.
Min Z =
1
d +
2
d +
3
d
Subject to
0.040X2 + 0.045X3 + 0.055X4 + 0.070X5 + 0.105X6 + 0.085X7 + 0.092X8 +
1
d ≥ 18.5 [profit]
0.00025X2 + 0.002X3 + 0.0025X4 + 0.00375X5 + 0.005X6 + 0.005X7 + 0.005X8 –
2
d ≤ 0.8
[capital-adequacy]
0.05X6 + 0.05X7 + 0.05X8 –
3
d ≤ 7.0 [risk-asset]
X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 = 250 [invest all]
X1 ≥ 24.2 [cash reserve]
1.00X1 + 0.995X2 + 0.960X3 + 0.900X4 + 0.850X5 ≥ 99.3 [liquidity]
Xj ≥ 12.5 for all j = 1, …, 8 [diversification]
X8 ≥ 75 [commercial]
X1, …, X8, d1, …, d3 ≥ 0
Using unequal weights of say
w1 = 7, w2 = 3, w3 = 1, we now have
Min Z =
321
37 ddd
Subject to the same constraints as above.
The problems above were solved using the merged approach.
We used two phase method in solving the above problems because of the presence of both equality and
inequality signs.
Tora software was also used in solving the problem
FINDINGS : This work comprises analyses on each of the methods of solutions discussed above and the dual
results are discussed below as follows:
For the weighted sum approach, even though it converges faster than the e-constraint approach (that is, it has a
fewer number of iterations than the e-constraint approach), it does not achieve the targets of the decision maker
in the sense that goals are underachieved and/or overachieved; hence the values
(-4.458, 5.418, 5) instead of (18.5, 0.8, 7), where -4.458 and 5.418 are non-suported solutions and the 3rd
value,
5 is a supported solution. All these values are obtained by substituting the values of the decision variable of the
weighted sum optimization in the given objective functions (Klamroth et al, 2003).
For the e-constraint approach; it involves more number of iterations than the other two approaches. It is also
more complex to use in the sense that each of the objective functions has to be used individually. Furthermore, it
still does not achieve the targets of the decision maker in the sense that it has the underachievement of the profit
goal and overachievement of the investment risks; hence the values (17.324, 0.928, 8.01) instead of (18.5, 0.8,
7). All these values are in billions of Naira.
However, for the merged approach in which the objective is to minimize the deviational variables of interest;
not only that it has a fewer number of iterations; all the deviational variables
321
,, ddd denoted by (x9,
x10 and x11) are minimized to zero showing that all the investment targets are met.
III. CONCLUSION
Applying duality theory to multi objective linear programming is an interesting concept in
mathematical programming. Duality generally arises from the fact that most times, it is more difficult to solve a
linear programming problem in its original (primal) form and as a result, the dual aspect is generated from the
primal and solved.
In this work, to generate solutions to the MOLP problem, we used scalarization methods (approaches)
which involve formulating an MOP-related SOP which is a function of the objective functions of the MOP. The
three secularization approaches used include: the weighted sum approach, e-constraint approach and the merged
weighted sum/e-constraints approach. Each approach was used to solve the same problem with the sole aim of
6. Duality Theory In Multi Objective…
www.theijes.com The IJES Page 6
finding which approach handles MOLP problems better and why. Hence a real life example was used to
demonstrate the discussion above with theS findings as seen in the discussion of the analyses. Based on these
findings, we conclude that multi objective linear programming problems are better and easier solved using
duality method applied to the merged approach in which the objective function comprises the deviational
variables from the originally given objectives of the decision maker since in most cases it involves fewer or at
worst the same number of iterations as in primal problem and all the deviational variables most times are
minimized to zero (which is better than either overachievement or underachievement as the case may be)
showing that the targets of the decision maker are all met.
IV. RECOMMENDATIONS
The researcher has been able to carry out some analyses using real life collected data and would like to
give the following recommendations.
[1] The researcher recommends that the method of solution involving the weighted sum and e-constraint terms
is the best for solving MOLP problems since it minimizes the deviational variables to zero or at worst
close to zero and also has a fewer number of iterations.
[2] The researcher also recommends that further work should be done on the weighted sum approach so as to
know why sometimes it does not provide an optimal solution to least one of the objective functions of an
MOLP problem.
[3] Application of duality theory to multi objective linear programming should also be adapted to problems
related to everyday life.
[4] Application of duality theory should also be extended to some other areas of soperations research such as
convex programming, games theory etc.
[5] Finally, the researcher recommends that the investment bank should increase their rates of return so that
they will be able to attain the target set for profit and avoid underachieving it.
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[3] Evans Gerald W. (1984) An overview of technique for solving Multi-objective mathematical programs, vol 30 No., 1
[4] Heyde, F. and Lohne, A. (2006) A geometric approach to duality in multiple objective linear programming.
[5] Heyde, F., Lohne, A. and Tammer, C. (2006) Duality in multi objective linear programming, draft.
[6] Huntley N., (1998) Definition of duality in the free online Encyclopedia.
[7] Klamroth K. and Sibylle Z. (2003) Linear multiple criteria optimization involving both the weighted sums approach and e-
constraint approach.
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Quarterly 25, pp 599 – 614.
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[11] Robert, B.A. (1972) Real analysis and probability.
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