An Interactive Decomposition Algorithm for Two-Level Large Scale Linear Multiobjective Optimization Problems with Stochastic Parameters Using TOPSIS Method
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.
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.
Real interpolation method for transfer function approximation of distributed ...TELKOMNIKA JOURNAL
Distributed parameter system (DPS) presents one of the most complex systems in the control theory. The transfer function of a DPS possibly contents: rational, nonlinear and irrational components. This thing leads that studies of the transfer function of a DPS are difficult in the time domain and frequency domain. In this paper, a systematic approach is proposed for linearizing DPS. This approach is based on the real interpolation method (RIM) to approximate the transfer function of DPS by rational-order transfer function. The results of the numerical examples show that the method is simple, computationally efficient, and flexible.
HANDWRITTEN CHARACTER RECOGNITION USING STRUCTURAL SHAPE DECOMPOSITIONcsandit
This paper presents a statistical framework for recognising 2D shapes which are represented as
an arrangement of curves or strokes. The approach is a hierarchical one which mixes geometric
and symbolic information in a three-layer architecture. Each curve primitive is represented
using a point-distribution model which describes how its shape varies over a set of training
data. We assign stroke labels to the primitives and these indicate to which class they belong.
Shapes are decomposed into an arrangement of primitives and the global shape representation
has two components. The first of these is a second point distribution model that is used to
represent the geometric arrangement of the curve centre-points. The second component is a
string of stroke labels that represents the symbolic arrangement of strokes. Hence each shape
can be represented by a set of centre-point deformation parameters and a dictionary of
permissible stroke label configurations. The hierarchy is a two-level architecture in which the
curve models reside at the nonterminal lower level of the tree. The top level represents the curve
arrangements allowed by the dictionary of permissible stroke combinations. The aim in
recognition is to minimise the cross entropy between the probability distributions for geometric
alignment errors and curve label errors. We show how the stroke parameters, shape-alignment
parameters and stroke labels may be recovered by applying the expectation maximization EM
algorithm to the utility measure. We apply the resulting shape-recognition method to Arabic
character recognition.
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.
Real interpolation method for transfer function approximation of distributed ...TELKOMNIKA JOURNAL
Distributed parameter system (DPS) presents one of the most complex systems in the control theory. The transfer function of a DPS possibly contents: rational, nonlinear and irrational components. This thing leads that studies of the transfer function of a DPS are difficult in the time domain and frequency domain. In this paper, a systematic approach is proposed for linearizing DPS. This approach is based on the real interpolation method (RIM) to approximate the transfer function of DPS by rational-order transfer function. The results of the numerical examples show that the method is simple, computationally efficient, and flexible.
HANDWRITTEN CHARACTER RECOGNITION USING STRUCTURAL SHAPE DECOMPOSITIONcsandit
This paper presents a statistical framework for recognising 2D shapes which are represented as
an arrangement of curves or strokes. The approach is a hierarchical one which mixes geometric
and symbolic information in a three-layer architecture. Each curve primitive is represented
using a point-distribution model which describes how its shape varies over a set of training
data. We assign stroke labels to the primitives and these indicate to which class they belong.
Shapes are decomposed into an arrangement of primitives and the global shape representation
has two components. The first of these is a second point distribution model that is used to
represent the geometric arrangement of the curve centre-points. The second component is a
string of stroke labels that represents the symbolic arrangement of strokes. Hence each shape
can be represented by a set of centre-point deformation parameters and a dictionary of
permissible stroke label configurations. The hierarchy is a two-level architecture in which the
curve models reside at the nonterminal lower level of the tree. The top level represents the curve
arrangements allowed by the dictionary of permissible stroke combinations. The aim in
recognition is to minimise the cross entropy between the probability distributions for geometric
alignment errors and curve label errors. We show how the stroke parameters, shape-alignment
parameters and stroke labels may be recovered by applying the expectation maximization EM
algorithm to the utility measure. We apply the resulting shape-recognition method to Arabic
character recognition.
Second or fourth-order finite difference operators, which one is most effective?Premier Publishers
This paper presents higher-order finite difference (FD) formulas for the spatial approximation of the time-dependent reaction-diffusion problems with a clear justification through examples, “why fourth-order FD formula is preferred to its second-order counterpart” that has been widely used in literature. As a consequence, methods for the solution of initial and boundary value PDEs, such as the method of lines (MOL), is of broad interest in science and engineering. This procedure begins with discretizing the spatial derivatives in the PDE with algebraic approximations. The key idea of MOL is to replace the spatial derivatives in the PDE with the algebraic approximations. Once this procedure is done, the spatial derivatives are no longer stated explicitly in terms of the spatial independent variables. In other words, only one independent variable is remaining, the resulting semi-discrete problem has now become a system of coupled ordinary differential equations (ODEs) in time. Thus, we can apply any integration algorithm for the initial value ODEs to compute an approximate numerical solution to the PDE. Analysis of the basic properties of these schemes such as the order of accuracy, convergence, consistency, stability and symmetry are well examined.
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
Probabilistic Self-Organizing Maps for Text-Independent Speaker IdentificationTELKOMNIKA JOURNAL
The present paper introduces a novel speaker modeling technique for text-independent speaker identification using probabilistic self-organizing maps (PbSOMs). The basic motivation behind the introduced technique was to combine the self-organizing quality of the self-organizing maps and generative power of Gaussian mixture models. Experimental results show that the introduced modeling technique using probabilistic self-organizing maps significantly outperforms the traditional technique using the classical GMMs and the EM algorithm or its deterministic variant. More precisely, a relative accuracy improvement of roughly 39% has been gained, as well as, a much less sensitivity to the model-parameters initialization has been exhibited by using the introduced speaker modeling technique using probabilistic self-organizing maps.
ARABIC HANDWRITTEN CHARACTER RECOGNITION USING STRUCTURAL SHAPE DECOMPOSITIONsipij
This paper presents a statistical framework for recognising 2D shapes which are represented as an
arrangement of curves or strokes. The approach is a hierarchical one which mixes geometric and symbolic
information in a three-layer architecture. Each curve primitive is represented using a point-distribution
model which describes how its shape varies over a set of training data. We assign stroke labels to the
primitives and these indicate to which class they belong. Shapes are decomposed into an arrangement of
primitives and the global shape representation has two components. The first of these is a second point
distribution model that is used to represent the geometric arrangement of the curve centre-points. The second
component is a string of stroke labels that represents the symbolic arrangement of strokes. Hence each shape
can be represented by a set of centre-point deformation parameters and a dictionary of permissible stroke
label configurations. The hierarchy is a two-level architecture in which the curve models reside at the
nonterminal lower level of the tree. The top level represents the curve arrangements allowed by the
dictionary of permissible stroke combinations. The aim in recognition is to minimise the cross entropy
between the probability distributions for geometric alignment errors and curve label errors. We show how
the stroke parameters, shape-alignment parameters and stroke labels may be recovered by applying the
expectation maximization EM algorithm to the utility measure. We apply the resulting shape-recognition
method to Arabic character recognition.
Research on Emotion Recognition for Facial Expression Images Based on Hidden ...Editor IJMTER
This paper introduces emotion recognition for facial expression images using Hidden
Markov Model (HMM). Firstly, facial expression images are transformed using discrete cosine
transformation and feature is extraction; then HMMs of facial expression images are constructed, and
the observation vectors are generated using sub-window, hence, process of emotion recognition for
facial expression images is realized. The experiments on JAFFE database are made to recognize the
seven emotions of the subjects, the recognition rates are above 80%, so emotion recognition for
facial expression images based on HMM is an effective method.
Contradictory of the Laplacian Smoothing Transform and Linear Discriminant An...TELKOMNIKA JOURNAL
Laplacian smoothing transform uses the negative diagonal element to generate the new space. The negative diagonal elements will deliver the negative new spaces. The negative new spaces will cause decreasing of the dominant characteristics. Laplacian smoothing transform usually singular matrix, such that the matrix cannot be solved to obtain the ordered-eigenvalues and corresponding eigenvectors. In this research, we propose a modeling to generate the positive diagonal elements to obtain the positive new spaces. The secondly, we propose approach to overcome singularity matrix to found eigenvalues and eigenvectors. Firstly, the method is started to calculate contradictory of the laplacian smoothing matrix. Secondly, we calculate the new space modeling on the contradictory of the laplacian smoothing. Moreover, we calculate eigenvectors of the discriminant analysis. Fourth, we calculate the new space modeling on the discriminant analysis, select and merge features. The proposed method has been tested by using four databases, i.e. ORL, YALE, UoB, and local database (CAI-UTM). Overall, the results indicate that the proposed method can overcome two problems and deliver higher accuracy than similar methods.
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.
Bender’s Decomposition Method for a Large Two-stage Linear Programming Modeldrboon
Linear Programming method (LP) can solve many problems in operations research and can obtain optimal solutions. But, the problems with uncertainties cannot be solved so easily. These uncertainties increase the complexity scale of the problems to become a large-scale LP model. The discussion started with the mathematical models. The objective is to minimize the number of the system variables subjecting to the decision variable coefficients and their slacks and surpluses. Then, the problems are formulated in the form of a Two-stage Stochastic Linear (TSL) model incorporated with the Bender’s Decomposition method. In the final step, the matrix systems are set up to support the MATLAB programming development of the primal-dual simplex and the Bender’s decomposition method, and applied to solve the example problem with the assumed four numerical sets of the decision variable coefficients simultaneously. The simplex method (primal) failed to determine the results and it was computational time-consuming. The comparison of the ordinary primal, primal-random, and dual method, revealed advantageous of the primal-random. The results yielded by the application of Bender’s decomposition method were proven to be the optimal solutions at a high level of confidence.
This lesson begins with explaining the decomposition method characteristics, and uses. Decomposition method decomposes the data in to its fundamental pieces and creates a forecast based upon each of the individual pieces. Using an example and the forecasting process, we apply the decomposition method to create a model and forecast based upon it.
Second or fourth-order finite difference operators, which one is most effective?Premier Publishers
This paper presents higher-order finite difference (FD) formulas for the spatial approximation of the time-dependent reaction-diffusion problems with a clear justification through examples, “why fourth-order FD formula is preferred to its second-order counterpart” that has been widely used in literature. As a consequence, methods for the solution of initial and boundary value PDEs, such as the method of lines (MOL), is of broad interest in science and engineering. This procedure begins with discretizing the spatial derivatives in the PDE with algebraic approximations. The key idea of MOL is to replace the spatial derivatives in the PDE with the algebraic approximations. Once this procedure is done, the spatial derivatives are no longer stated explicitly in terms of the spatial independent variables. In other words, only one independent variable is remaining, the resulting semi-discrete problem has now become a system of coupled ordinary differential equations (ODEs) in time. Thus, we can apply any integration algorithm for the initial value ODEs to compute an approximate numerical solution to the PDE. Analysis of the basic properties of these schemes such as the order of accuracy, convergence, consistency, stability and symmetry are well examined.
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
Probabilistic Self-Organizing Maps for Text-Independent Speaker IdentificationTELKOMNIKA JOURNAL
The present paper introduces a novel speaker modeling technique for text-independent speaker identification using probabilistic self-organizing maps (PbSOMs). The basic motivation behind the introduced technique was to combine the self-organizing quality of the self-organizing maps and generative power of Gaussian mixture models. Experimental results show that the introduced modeling technique using probabilistic self-organizing maps significantly outperforms the traditional technique using the classical GMMs and the EM algorithm or its deterministic variant. More precisely, a relative accuracy improvement of roughly 39% has been gained, as well as, a much less sensitivity to the model-parameters initialization has been exhibited by using the introduced speaker modeling technique using probabilistic self-organizing maps.
ARABIC HANDWRITTEN CHARACTER RECOGNITION USING STRUCTURAL SHAPE DECOMPOSITIONsipij
This paper presents a statistical framework for recognising 2D shapes which are represented as an
arrangement of curves or strokes. The approach is a hierarchical one which mixes geometric and symbolic
information in a three-layer architecture. Each curve primitive is represented using a point-distribution
model which describes how its shape varies over a set of training data. We assign stroke labels to the
primitives and these indicate to which class they belong. Shapes are decomposed into an arrangement of
primitives and the global shape representation has two components. The first of these is a second point
distribution model that is used to represent the geometric arrangement of the curve centre-points. The second
component is a string of stroke labels that represents the symbolic arrangement of strokes. Hence each shape
can be represented by a set of centre-point deformation parameters and a dictionary of permissible stroke
label configurations. The hierarchy is a two-level architecture in which the curve models reside at the
nonterminal lower level of the tree. The top level represents the curve arrangements allowed by the
dictionary of permissible stroke combinations. The aim in recognition is to minimise the cross entropy
between the probability distributions for geometric alignment errors and curve label errors. We show how
the stroke parameters, shape-alignment parameters and stroke labels may be recovered by applying the
expectation maximization EM algorithm to the utility measure. We apply the resulting shape-recognition
method to Arabic character recognition.
Research on Emotion Recognition for Facial Expression Images Based on Hidden ...Editor IJMTER
This paper introduces emotion recognition for facial expression images using Hidden
Markov Model (HMM). Firstly, facial expression images are transformed using discrete cosine
transformation and feature is extraction; then HMMs of facial expression images are constructed, and
the observation vectors are generated using sub-window, hence, process of emotion recognition for
facial expression images is realized. The experiments on JAFFE database are made to recognize the
seven emotions of the subjects, the recognition rates are above 80%, so emotion recognition for
facial expression images based on HMM is an effective method.
Contradictory of the Laplacian Smoothing Transform and Linear Discriminant An...TELKOMNIKA JOURNAL
Laplacian smoothing transform uses the negative diagonal element to generate the new space. The negative diagonal elements will deliver the negative new spaces. The negative new spaces will cause decreasing of the dominant characteristics. Laplacian smoothing transform usually singular matrix, such that the matrix cannot be solved to obtain the ordered-eigenvalues and corresponding eigenvectors. In this research, we propose a modeling to generate the positive diagonal elements to obtain the positive new spaces. The secondly, we propose approach to overcome singularity matrix to found eigenvalues and eigenvectors. Firstly, the method is started to calculate contradictory of the laplacian smoothing matrix. Secondly, we calculate the new space modeling on the contradictory of the laplacian smoothing. Moreover, we calculate eigenvectors of the discriminant analysis. Fourth, we calculate the new space modeling on the discriminant analysis, select and merge features. The proposed method has been tested by using four databases, i.e. ORL, YALE, UoB, and local database (CAI-UTM). Overall, the results indicate that the proposed method can overcome two problems and deliver higher accuracy than similar methods.
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.
Bender’s Decomposition Method for a Large Two-stage Linear Programming Modeldrboon
Linear Programming method (LP) can solve many problems in operations research and can obtain optimal solutions. But, the problems with uncertainties cannot be solved so easily. These uncertainties increase the complexity scale of the problems to become a large-scale LP model. The discussion started with the mathematical models. The objective is to minimize the number of the system variables subjecting to the decision variable coefficients and their slacks and surpluses. Then, the problems are formulated in the form of a Two-stage Stochastic Linear (TSL) model incorporated with the Bender’s Decomposition method. In the final step, the matrix systems are set up to support the MATLAB programming development of the primal-dual simplex and the Bender’s decomposition method, and applied to solve the example problem with the assumed four numerical sets of the decision variable coefficients simultaneously. The simplex method (primal) failed to determine the results and it was computational time-consuming. The comparison of the ordinary primal, primal-random, and dual method, revealed advantageous of the primal-random. The results yielded by the application of Bender’s decomposition method were proven to be the optimal solutions at a high level of confidence.
This lesson begins with explaining the decomposition method characteristics, and uses. Decomposition method decomposes the data in to its fundamental pieces and creates a forecast based upon each of the individual pieces. Using an example and the forecasting process, we apply the decomposition method to create a model and forecast based upon it.
In 2011, Anar Mammadov founded the Azerbaijan America Alliance, a charity which has been essential in developing stronger ties between Azerbaijan and the USA. As well as hosting fundraisers, an annual gala dinner in Washington D.C., and promoting Azeri history, the Alliance has developed a number of campaigns to raise awareness for some of the most immediate difficulties facing the people of Azerbaijan.
Anar Mammadov is a highly successful businessman based in Azerbaijan. Born in 1981, by 2004 he had graduated with a bachelor's degree in Law from Baku State University, followed immediately by a bachelor's degree in the arts, before in 2005 graduating from the American Intercontinental University in London with a Master's degree in Business Administration.
Utilizing GIS to Develop a Non-Signalized Intersection Data Inventory for Saf...IJERA Editor
Roadway data inventories are being used across the nation to aid state Departments of Transportation (DOTs) in decision making. The high number of intersection and intersection related crashes suggest the need for intersection-specific data inventories that can be associated to crash occurrences to help make better safety decisions. Currently, limited time and resources are the biggest difficulties for execution of comprehensive intersection data inventories, but online resources exist that DOTs can leverage to capture desired data. Researchers from The University of Alabama developed an online method to collect intersection characteristics for non-signalized intersections along state routes using Google Maps and Google Street View, which was tied to an Alabama DOT maintained geographic information systems (GIS) node-link linear referencing method. A GIS-Based Intersection Data Inventory Web Portal was created to collect and record non-signalized intersection parameters. Thirty intersections of nine different intersection types were randomly selected from across the state, totaling 270 intersections. For each intersection, up to 78 parameters were collected, compliant with the Model Inventory of Roadway Elements (MIRE) schema. Using the web portal, the data parameters corresponding to an average intersection can be collected and catalogued into a database in approximately 10 minutes. The collection methodology and web portal function independently of the linear referencing method; therefore, the tool can be tailored and used by any state with spatial roadway data. Preliminary single variable analysis was performed, showing that there are relationships between individual intersection characteristics and crash frequency. Future work will investigate multivariate analysis and develop safety performance functions and crash modification factors.
Similar to An Interactive Decomposition Algorithm for Two-Level Large Scale Linear Multiobjective Optimization Problems with Stochastic Parameters Using TOPSIS Method
THE LEFT AND RIGHT BLOCK POLE PLACEMENT COMPARISON STUDY: APPLICATION TO FLIG...ieijjournal1
It is known that if a linear-time-invariant MIMO system described by a state space equation has a number
of states divisible by the number of inputs and it can be transformed to block controller form, we can
design a state feedback controller using block pole placement technique by assigning a set of desired Block
poles. These may be left or right block poles. The idea is to compare both in terms of system’s response.
THE LEFT AND RIGHT BLOCK POLE PLACEMENT COMPARISON STUDY: APPLICATION TO FLIG...ieijjournal
It is known that if a linear-time-invariant MIMO system described by a state space equation has a number of states divisible by the number of inputs and it can be transformed to block controller form, we can design a state feedback controller using block pole placement technique by assigning a set of desired Block poles. These may be left or right block poles. The idea is to compare both in terms of system’s response.
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.
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.
Interactive Fuzzy Goal Programming approach for Tri-Level Linear Programming ...IJERA Editor
The aim of this paper is to present an interactive fuzzy goal programming approach to determine the preferred
compromise solution to Tri-level linear programming problems considering the imprecise nature of the decision
makers’ judgments for the objectives. Using the concept of goal programming, fuzzy set theory, in combination
with interactive programming, and improving the membership functions by means of changing the tolerances of
the objectives provide a satisfactory compromise (near to ideal) solution to the upper level decision makers.
Two numerical examples for three-level linear programming problems have been solved to demonstrate the
feasibility of the proposed approach. The performance of the proposed approach was evaluated by using of
metric distance functions with other approaches.
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A lexisearch algorithm for the Bottleneck Traveling Salesman ProblemCSCJournals
The Bottleneck Traveling Salesman Problem (BTSP) is a variation of the well-known Traveling Salesman Problem in which the objective is to minimize the maximum lap (arc length) in the tour of the salesman. In this paper, a lexisearch algorithm using adjacency representation for a tour has been developed for obtaining exact optimal solution to the problem. Then a comparative study has been carried out to show the efficiency of the algorithm as against existing exact algorithm for some randomly generated and TSPLIB instances of different sizes.
An Improved Iterative Method for Solving General System of Equations via Gene...Zac Darcy
Various algorithms are known for solving linear system of equations. Iteration methods for solving the
large sparse linear systems are recommended. But in the case of general n× m matrices the classic
iterative algorithms are not applicable except for a few cases. The algorithm presented here is based on the
minimization of residual of solution and has some genetic characteristics which require using Genetic
Algorithms. Therefore, this algorithm is best applicable for construction of parallel algorithms. In this
paper, we describe a sequential version of proposed algorithm and present its theoretical analysis.
Moreover we show some numerical results of the sequential algorithm and supply an improved algorithm
and compare the two algorithms.
An Improved Iterative Method for Solving General System of Equations via Gene...Zac Darcy
Various algorithms are known for solving linear system of equations. Iteration methods for solving the
large sparse linear systems are recommended. But in the case of general n× m matrices the classic
iterative algorithms are not applicable except for a few cases. The algorithm presented here is based on the
minimization of residual of solution and has some genetic characteristics which require using Genetic
Algorithms. Therefore, this algorithm is best applicable for construction of parallel algorithms. In this
paper, we describe a sequential version of proposed algorithm and present its theoretical analysis.
Moreover we show some numerical results of the sequential algorithm and supply an improved algorithm
and compare the two algorithms.
Differential evolution (DE) algorithm has been applied as a powerful tool to find optimum switching angles for selective harmonic elimination pulse width modulation (SHEPWM) inverters. However, the DE’s performace is very dependent on its control parameters. Conventional DE generally uses either trial and error mechanism or tuning technique to determine appropriate values of the control paramaters. The disadvantage of this process is that it is very time comsuming. In this paper, an adaptive control parameter is proposed in order to speed up the DE algorithm in optimizing SHEPWM switching angles precisely. The proposed adaptive control parameter is proven to enhance the convergence process of the DE algorithm without requiring initial guesses. The results for both negative and positive modulation index (M) also indicate that the proposed adaptive DE is superior to the conventional DE in generating SHEPWM switching patterns.
Similar to An Interactive Decomposition Algorithm for Two-Level Large Scale Linear Multiobjective Optimization Problems with Stochastic Parameters Using TOPSIS Method (20)
Hybrid optimization of pumped hydro system and solar- Engr. Abdul-Azeez.pdffxintegritypublishin
Advancements in technology unveil a myriad of electrical and electronic breakthroughs geared towards efficiently harnessing limited resources to meet human energy demands. The optimization of hybrid solar PV panels and pumped hydro energy supply systems plays a pivotal role in utilizing natural resources effectively. This initiative not only benefits humanity but also fosters environmental sustainability. The study investigated the design optimization of these hybrid systems, focusing on understanding solar radiation patterns, identifying geographical influences on solar radiation, formulating a mathematical model for system optimization, and determining the optimal configuration of PV panels and pumped hydro storage. Through a comparative analysis approach and eight weeks of data collection, the study addressed key research questions related to solar radiation patterns and optimal system design. The findings highlighted regions with heightened solar radiation levels, showcasing substantial potential for power generation and emphasizing the system's efficiency. Optimizing system design significantly boosted power generation, promoted renewable energy utilization, and enhanced energy storage capacity. The study underscored the benefits of optimizing hybrid solar PV panels and pumped hydro energy supply systems for sustainable energy usage. Optimizing the design of solar PV panels and pumped hydro energy supply systems as examined across diverse climatic conditions in a developing country, not only enhances power generation but also improves the integration of renewable energy sources and boosts energy storage capacities, particularly beneficial for less economically prosperous regions. Additionally, the study provides valuable insights for advancing energy research in economically viable areas. Recommendations included conducting site-specific assessments, utilizing advanced modeling tools, implementing regular maintenance protocols, and enhancing communication among system components.
TECHNICAL TRAINING MANUAL GENERAL FAMILIARIZATION COURSEDuvanRamosGarzon1
AIRCRAFT GENERAL
The Single Aisle is the most advanced family aircraft in service today, with fly-by-wire flight controls.
The A318, A319, A320 and A321 are twin-engine subsonic medium range aircraft.
The family offers a choice of engines
Automobile Management System Project Report.pdfKamal Acharya
The proposed project is developed to manage the automobile in the automobile dealer company. The main module in this project is login, automobile management, customer management, sales, complaints and reports. The first module is the login. The automobile showroom owner should login to the project for usage. The username and password are verified and if it is correct, next form opens. If the username and password are not correct, it shows the error message.
When a customer search for a automobile, if the automobile is available, they will be taken to a page that shows the details of the automobile including automobile name, automobile ID, quantity, price etc. “Automobile Management System” is useful for maintaining automobiles, customers effectively and hence helps for establishing good relation between customer and automobile organization. It contains various customized modules for effectively maintaining automobiles and stock information accurately and safely.
When the automobile is sold to the customer, stock will be reduced automatically. When a new purchase is made, stock will be increased automatically. While selecting automobiles for sale, the proposed software will automatically check for total number of available stock of that particular item, if the total stock of that particular item is less than 5, software will notify the user to purchase the particular item.
Also when the user tries to sale items which are not in stock, the system will prompt the user that the stock is not enough. Customers of this system can search for a automobile; can purchase a automobile easily by selecting fast. On the other hand the stock of automobiles can be maintained perfectly by the automobile shop manager overcoming the drawbacks of existing system.
Immunizing Image Classifiers Against Localized Adversary Attacksgerogepatton
This paper addresses the vulnerability of deep learning models, particularly convolutional neural networks
(CNN)s, to adversarial attacks and presents a proactive training technique designed to counter them. We
introduce a novel volumization algorithm, which transforms 2D images into 3D volumetric representations.
When combined with 3D convolution and deep curriculum learning optimization (CLO), itsignificantly improves
the immunity of models against localized universal attacks by up to 40%. We evaluate our proposed approach
using contemporary CNN architectures and the modified Canadian Institute for Advanced Research (CIFAR-10
and CIFAR-100) and ImageNet Large Scale Visual Recognition Challenge (ILSVRC12) datasets, showcasing
accuracy improvements over previous techniques. The results indicate that the combination of the volumetric
input and curriculum learning holds significant promise for mitigating adversarial attacks without necessitating
adversary training.
Explore the innovative world of trenchless pipe repair with our comprehensive guide, "The Benefits and Techniques of Trenchless Pipe Repair." This document delves into the modern methods of repairing underground pipes without the need for extensive excavation, highlighting the numerous advantages and the latest techniques used in the industry.
Learn about the cost savings, reduced environmental impact, and minimal disruption associated with trenchless technology. Discover detailed explanations of popular techniques such as pipe bursting, cured-in-place pipe (CIPP) lining, and directional drilling. Understand how these methods can be applied to various types of infrastructure, from residential plumbing to large-scale municipal systems.
Ideal for homeowners, contractors, engineers, and anyone interested in modern plumbing solutions, this guide provides valuable insights into why trenchless pipe repair is becoming the preferred choice for pipe rehabilitation. Stay informed about the latest advancements and best practices in the field.
Sachpazis:Terzaghi Bearing Capacity Estimation in simple terms with Calculati...Dr.Costas Sachpazis
Terzaghi's soil bearing capacity theory, developed by Karl Terzaghi, is a fundamental principle in geotechnical engineering used to determine the bearing capacity of shallow foundations. This theory provides a method to calculate the ultimate bearing capacity of soil, which is the maximum load per unit area that the soil can support without undergoing shear failure. The Calculation HTML Code included.
Saudi Arabia stands as a titan in the global energy landscape, renowned for its abundant oil and gas resources. It's the largest exporter of petroleum and holds some of the world's most significant reserves. Let's delve into the top 10 oil and gas projects shaping Saudi Arabia's energy future in 2024.
Top 10 Oil and Gas Projects in Saudi Arabia 2024.pdf
An Interactive Decomposition Algorithm for Two-Level Large Scale Linear Multiobjective Optimization Problems with Stochastic Parameters Using TOPSIS Method
1. Tarek H. M. Abou-El-Enien Int. Journal of Engineering Research and Applications www.ijera.com
ISSN : 2248-9622, Vol. 5, Issue 4, ( Part -2) April 2015, pp.61-76
www.ijera.com 61 | P a g e
An Interactive Decomposition Algorithm for Two-Level Large
Scale Linear Multiobjective Optimization Problems with
Stochastic Parameters Using TOPSIS Method
Tarek H. M. Abou-El-Enien
(Department of Operations Research & Decision Support, Faculty of Computers & Information, Cairo
University, 5Dr.Ahmed Zoweil St.- Orman-Postal Code 12613 - Giza - Egypt. Email: t.hanafy@fci-cu.edu.eg)
ABSTRACT
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 right-
hand 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.
Keywords - Stochastic Multiobjective Optimization, Two-level decision Making Problems, TOPSIS Method,
Decomposition Techniques, Fuzzy Sets
I. INTRODUCTION
In real world decision situations, when formulating a Large Scale Linear Multiobjective Optimization
(LSLMO) problem, some or all of the parameters of the optimization problem are described by stochastic (or
random or probabilistic) variables rather than by deterministic quantities. Most of LSLMO problems arising in
applications have special structures that can be exploited. There are many familiar structures for large scale
optimization problems such as: (i) the block angular structure, and (ii) angular and dual- angular structure to the
constraints, and several kinds of decomposition methods for linear and nonlinear programming problems with
those structures have been proposed in [2, 12, 13, 27, 29, 35, 43,44, 45, 49].
Two-Level Large Scale Linear Multiobjective Optimization (TL-LSLMO) Problems with block angular
structure consists of the objectives of the leader at its first level and that is of the follower at the second level.
The decision maker (DM) at each level attempts to optimize his individual objectives, which usually depend in
part on the variables controlled by the decision maker (DM) at the other levels and their final decisions are
executed sequentially where the FLDM makes his decision firstly. The research and applications concentrated
mainly on two-level programming (see f. i. [ 7, 8, 11, 14, 15, 16, 18, 19, 20, 21, 22, 24, 25, 26, 31,32, 33,
48,52,53]).
TOPSIS was first developed by C. L. Hwang and K. Yoon [37] for solving a multiple attributes decision
making (MADM) problems. It is based upon the principle that the chosen alternative should have the shortest
distance from the PIS and the farthest from the NIS. T. H. M. Abou-El-Enien [4] presents many algorithms for
solving different kinds of LSLMO problems using TOPSIS method.
M. A. Abo-Sinna and T. H. M. Abou-El-Enien [9] extended TOPSIS approach to solve large scale multiple
objectives decision making (LSMODM) problems with block angular structure.
T. H. M. Abou-El-Enien [3] extended TOPSIS method for solving large scale integer linear vector
optimization problems with chance constraints (CHLSILVOP) of a special type.
I. A. Baky and M. A. Abo-Sinna [20] proposed a TOPSIS algorithm for bi-level multiple objectives
decision making (BL-MODM) problems.
T. H. M. Abou-El-Enien [5] presents an interactive TOPSIS algorithm for solving a special type of linear
fractional vector optimization problems.
RESEARCH ARTICLE OPEN ACCESS
2. Tarek H. M. Abou-El-Enien Int. Journal of Engineering Research and Applications www.ijera.com
ISSN : 2248-9622, Vol. 5, Issue 4, ( Part -2) April 2015, pp.61-76
www.ijera.com 62 | P a g e
P.P. Dey et al. [28] extend TOPSIS for solving linear fractional bi-level multi-objective decision-making
problem based on fuzzy goal programming.
Recently, M. A. Abo-Sinna and T. H. M. Abou-El-Enien [11] extend TOPSIS for solving Large Scale Bi-
level Linear Vector Optimization Problems (LS-BL-LVOP), they further extended the concept of TOPSIS
[Lia et al. (45)] for LS-BL-LVOP.
This paper extended TOPSIS method for solving (TL-LSLMOP-SP)rhs of block angular structure. Also,
the concept of TOPSIS is extended [Lia et al. (41)] for (TL-LSLMOP-SP)rhs of block angular structure.
The following section will give the formulation of (TL-LSLMOP-SP)rhs of block angular structure. The
family of dp-distance and its normalization is discussed in section 3. The TOPSIS approach for (TL-LSLMOP-
SP)rhs of block angular is presented in section 4. By use of TOPSIS, a decomposition algorithm is proposed for
solving (TL-LSLMOP-SP)rhs of block angular structure in section 5. An illustrative numerical example is given
in section 6. Finally, summary and conclusions will be given in section 7.
II. Formulation of (TL-LSLMOP-SP)rhs of block angular structure
Consider there are two levels in a hierarchy structure with a first level decision maker (FLDM) and second
level decision maker (SLDM). Let the (TL-LSLMOP-SP)rhs of the following block angular structure :
[FLDM]
𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒
𝑋𝐼1
𝑍𝐼1
𝑋𝐼1
, 𝑋𝐼2
=
𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒
𝑋𝐼1
𝑧𝐼11 𝑋𝐼1
, 𝑋𝐼2
, … . , 𝑧𝐼1 𝑘 𝐼1
𝑋𝐼1
, 𝑋𝐼2
(1-a)
𝑤ℎ𝑒𝑟𝑒 𝑋𝐼2
𝑠𝑜𝑙𝑣𝑒𝑠 𝑠𝑒𝑐𝑜𝑛𝑑 𝑙𝑒𝑣𝑒𝑙
[SLDM]
𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒
𝑋𝐼2
𝑍𝐼2
𝑋𝐼1
, 𝑋𝐼2
=
𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒
𝑋𝐼2
𝑧𝐼21 𝑋𝐼1
, 𝑋𝐼2
, … . , 𝑧𝐼2 𝑘 𝐼2
𝑋𝐼1
, 𝑋𝐼2
(1-b)
subject to
𝑋 ∈ 𝑀 = {𝑃{ 𝑎𝑖𝑗 ℎ0
𝑥𝑖𝑗 ℎ0
≤𝑛
𝑖=1 𝑣ℎ0
} ≥
𝑞
𝑗=1 𝛼ℎ0
, ℎ0 = 1,2,3, … . , 𝑚0, (1-c)
𝑃 𝑏𝑖𝑗 ℎ 𝑗
𝑥𝑖𝑗 ℎ 𝑗
≤𝑛
𝑖=1 𝑣ℎ 𝑗
≥ 𝛼ℎ 𝑗
, ℎ𝑗 = 𝑚𝑗 −1 + 1, 𝑚𝑗−1 + 2, … . , 𝑚𝑗 , (1-d)
𝑥𝑖𝑗 ≥ 0, 𝑖 ∈ 𝑁, 𝑗 = 1,2, … , 𝑞, 𝑞 > 1}. (1-e)
where
a and b are constants,
k : the number of objective functions,
𝑘𝐼1
: the number of objective functions of the FLDM,
𝑘𝐼2
: the number of objective functions of the SLDM,
𝑛𝐼1
: the number of variables of the FLDM,
𝑛𝐼2
: the number of variables of the SLDM,
q : the number of subproblems,
m : the number of constraints,
n : the numer of variables,
𝑛𝑗 : the number of variables of the jth
subproblem, j=1,2,…,q,
mo : the number of the common constraints represented by 𝑎𝑖𝑗 ℎ0
𝑥𝑖𝑗 ℎ0
≤𝑛
𝑖=1 𝑣ℎ0
𝑞
𝑗=1
mj : the number of independent constraints of the jth
subproblem represented by
𝑏𝑖𝑗 ℎ 𝑗
𝑥𝑖𝑗 ℎ 𝑗
≤
𝑛
𝑖=1
𝑣ℎ 𝑗
, 𝑗 = 1,2, … , 𝑞, 𝑞 > 1,
Aj : an (mo ×nj) coefficient matrix,
Dj : an (mj ×nj) coefficient matrix,
bo : an mo-dimensional column vector of right-hand sides of the common constraints whose elements are
constants,
bj : an mj-dimensional column vector of independent constraints right-hand sides whose elements are the
constants of the constraints for the jth
subproblem, j=1,2,…,q,
Cij : an nj-dimensional row vector for the jth
subproblem in the ith
objective function,
R : the set of all real numbers,
X : an n-dimensional column vector of variables,
Xj : an 𝑛𝑗 -dimensional column vector of variables for the jth
subproblem, j=1,2,…..,q,
𝑋𝐼1
: an 𝑛𝐼1
- dimensional column vector of variables of the FLDM,
𝑋𝐼2
: an 𝑛𝐼2
- dimensional column vector of variables of the SLDM,
K = {1,2,….,k}
3. Tarek H. M. Abou-El-Enien Int. Journal of Engineering Research and Applications www.ijera.com
ISSN : 2248-9622, Vol. 5, Issue 4, ( Part -2) April 2015, pp.61-76
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N = {1,2,…..,n},
Rn
= {X=(x1, x2,…,xn}T
: xiR, iN}.
If the objective functions are linear, then the objective function can be written as follows:
𝑧𝑖(𝑋)= 𝑧𝑖𝑗
𝑞
𝑗 =1 = 𝐶𝑖𝑗 𝑋𝑗
𝑞
𝑗 =1 , i=1,2,…,k (2)
In addition, P means probability, 𝛼ℎ0
and 𝛼ℎ 𝑗
are a specified probability levels. For the sake of
simplicity, consider that the random parameters, 𝑣ℎ0
and 𝑣ℎ 𝑗
are distributed normally and independently of each
other with known means E{𝑣ℎ0
} and E{𝑣ℎ 𝑗
} and variances Var{𝑣ℎ0
} and Var{𝑣ℎ 𝑗
}.
Using the chance constrained programming technique [3], the deterministic version of problem (1) can
be written as follows :
[FLDM]
𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒
𝑋𝐼1
𝑍𝐼1
𝑋𝐼1
, 𝑋𝐼2
=
𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒
𝑋𝐼1
𝑧𝐼11 𝑋𝐼1
, 𝑋𝐼2
, … . , 𝑧𝐼1 𝑘 𝐼1
𝑋𝐼1
, 𝑋𝐼2
(3-a)
𝑤ℎ𝑒𝑟𝑒 𝑋𝐼2
𝑠𝑜𝑙𝑣𝑒𝑠 𝑠𝑒𝑐𝑜𝑛𝑑 𝑙𝑒𝑣𝑒𝑙
[SLDM]
𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒
𝑋𝐼2
𝑍𝐼2
𝑋𝐼1
, 𝑋𝐼2
=
𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒
𝑋𝐼2
𝑧𝐼21 𝑋𝐼1
, 𝑋𝐼2
, … . , 𝑧𝐼2 𝑘 𝐼2
𝑋𝐼1
, 𝑋𝐼2
(3-b)
subject to
𝑋 ∈ 𝑀/
= { 𝑎𝑖𝑗 ℎ0
𝑥𝑖𝑗 ℎ0
≤𝑛
𝑖=1 𝐸{𝑣ℎ0
} + 𝑘 𝛼0
𝑞
𝑗=1 𝑉𝑎𝑟{𝑣ℎ0
}, ℎ0 = 1,2, … . , 𝑚0, (3-c)
𝑏𝑖𝑗 ℎ 𝑗
𝑥𝑖𝑗 ℎ 𝑗
≤𝑛
𝑖=1 𝐸{𝑣ℎ 𝑗
} + 𝑘 𝛼 𝑗
𝑉𝑎𝑟{𝑣ℎ 𝑗
}, ℎ𝑗 = 𝑚𝑗−1 + 1, 𝑚𝑗−1 + 2, … . , 𝑚𝑗 , (3-d)
𝑥𝑖𝑗 ≥ 0, 𝑖 ∈ 𝑁, 𝑗 = 1,2, … , 𝑞, 𝑞 > 1}. (3-e)
where 𝑘 𝛼 𝑗
, j=0,1,2,….,q, is the standard normal value such that Φ(𝑘 𝛼 𝑗
)=1- 𝛼𝑗 , j=0,1,…,q, and Φ represents the
cumulative distribution function of the standard normal distribution.
III. Some Basic Concepts of distance Measures
The compromise (satisfactory) programming approach [ 17, 34, 36, 39, 40, 41, 42, 53, 54, 56] has been
developed to perform multiobjective optimization problems, reducing the set of nondominated solutions. The
compromise solutions are those which are the closest by some distance measure to the ideal one.
The point 𝑧𝑖 𝑋∗
= 𝑧𝑖𝑗 (𝑋∗
)
𝑞
𝑗=1 in the criteria space is called the ideal point (reference point). As the
measure of ―closeness‖, dp-metric is used. The dp-metric defines the distance between two points, 𝑧𝑖 𝑋 =
𝑧𝑖𝑗 (𝑋)
𝑞
𝑗=1 and 𝑧𝑖 𝑋∗
= 𝑧𝑖𝑗 (𝑋∗
)
𝑞
𝑗 =1 (the reference point) in k-dimensional space [50] as:
𝑑 𝑝 = 𝑤𝑖
𝑝
𝑧𝑖
∗
− 𝑧𝑖
𝑝
𝑘
𝑖=1
1/𝑝
= 𝑤𝑖
𝑝
𝑧𝑖𝑗
∗
𝑞
𝑗 =1
− 𝑧𝑖𝑗
𝑞
𝑗 =1
𝑝
𝑘
𝑖=1
1/𝑝
(4)
where 𝑝 ≥ 1.
Unfortunately, because of the incommensurability among objectives, it is impossible to directly use the
above distance family. To remove the effects of the incommensurability, we need to normalize the distance
family of equation (4) by using the reference point [4, 36, 41] as :
𝑑 𝑝 = 𝑤𝑖
𝑝
𝑧𝑖𝑗
∗𝑞
𝑗=1 − 𝑧𝑖𝑗
𝑞
𝑗=1
𝑧𝑖𝑗
∗𝑞
𝑗 =1
𝑝𝑘
𝑖=1
1
𝑝
(5)
where 𝑝 ≥ 1.
To obtain a compromise (satisfactory ) solution for problem (3) , the global criteria method [40] for
problem (3) [4, 45] uses the distance family of equation (5) by the ideal solution being the reference point. The
problem becomes how to solve the following auxiliary problem :
𝑑 𝑝 = 𝑤𝑖
𝑝
𝑧𝑖𝑗 𝑋∗𝑞
𝑗=1 − 𝑧𝑖𝑗 𝑋
𝑞
𝑗 =1
𝑧𝑖𝑗 𝑋∗𝑞
𝑗=1
𝑝𝑘
𝑖=1
1/𝑝
𝑋∈𝑀/
𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒
(6)
where 𝑋∗
is the PIS and 𝑝 = 1,2, … . . , ∞.
4. Tarek H. M. Abou-El-Enien Int. Journal of Engineering Research and Applications www.ijera.com
ISSN : 2248-9622, Vol. 5, Issue 4, ( Part -2) April 2015, pp.61-76
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Usually, the solutions based on PIS are different from the solutions based on NIS. Thus, both PIS(z∗
) and
NIS(z−
) can be used to normalize the distance family and obtain [4]:
𝑑 𝑝 = 𝑤𝑖
𝑝
𝑧𝑖𝑗
∗𝑞
𝑗=1 − 𝑧𝑖𝑗
𝑞
𝑗 =1
𝑧𝑖𝑗
∗𝑞
𝑗=1 − 𝑧𝑖𝑗
−𝑞
𝑗 =1
𝑝𝑘
𝑖=1
1/𝑝
(7)
where 𝑝 ≥ 1.
IV. TOPSIS for (TL-LSLMOP-SP)rhs of block angular structure
Problem (3) can be rewritten as follows [4]:
[FLDM]
𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒/𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒
𝑋𝐼1
𝑍𝐼1
𝑋𝐼1
, 𝑋𝐼2
=
𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒/𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒
𝑋𝐼1
𝑧𝐼11 𝑋𝐼1
, 𝑋𝐼2
, … . , 𝑧𝐼1 𝑘 𝐼1
𝑋𝐼1
, 𝑋𝐼2
𝑤ℎ𝑒𝑟𝑒 𝑋𝐼2
𝑠𝑜𝑙𝑣𝑒𝑠 𝑠𝑒𝑐𝑜𝑛𝑑 𝑙𝑒𝑣𝑒𝑙
[SLDM]
𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒/𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒
𝑋𝐼2
𝑍𝐼2
𝑋𝐼1
, 𝑋𝐼2
=
𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒/𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒
𝑋𝐼2
𝑧𝐼21 𝑋𝐼1
, 𝑋𝐼2
, … . , 𝑧𝐼2 𝑘 𝐼2
𝑋𝐼1
, 𝑋𝐼2
subject to (8)
𝑋 ∈ 𝑀/
where
𝑧𝑡𝑗
𝑞
𝑗=1 (𝑋): Objective Function for Maximization, 𝑡 ∈ 𝐾1 ⊏ 𝐾,
𝑧 𝑣𝑗
𝑞
𝑗=1 (𝑋): Objective Function for Minimization, 𝑣 ∈ 𝐾2 ⊏ 𝐾.
4-1. Phase (I)
Consider the FLDM problem of problem (8):
[FLDM]
𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒/𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒
𝑋𝐼1
𝑍𝐼1
𝑋𝐼1
, 𝑋𝐼2
=
𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒/𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒
𝑋𝐼1
𝑧𝐼11 𝑋𝐼1
, 𝑋𝐼2
, … . , 𝑧𝐼1 𝑘 𝐼1
𝑋𝐼1
, 𝑋𝐼2
subject to (9)
𝑋 ∈ 𝑀/
where
𝑧𝑡𝑗
𝑞
𝑗=1 (𝑋): Objective Function for Maximization, 𝑡 ∈ 𝐾1 ⊏ 𝐾,
𝑧 𝑣𝑗
𝑞
𝑗=1 (𝑋): Objective Function for Minimization, 𝑣 ∈ 𝐾2 ⊏ 𝐾.
In order to use the distance family of equation (7) to resolve problem (9), we must first find PIS(z∗
)
and NIS(z−
) which are [4, 40]:
𝑧∗ 𝐹𝐿𝐷𝑀
=
𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒 𝑜𝑟 𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒
𝑋 ∈ 𝑀/
𝑧𝑡𝑗
𝐹𝐿𝐷𝑀𝑞
𝑗=1 𝑋 𝑜𝑟 𝑧 𝑣𝑗
𝐹𝐿𝐷𝑀𝑞
𝑗=1 𝑋 , ∀𝑡 𝑎𝑛𝑑 𝑣 (10-a)
𝑧− 𝐹𝐿𝐷𝑀
=
𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝑜𝑟 𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒
𝑋 ∈ 𝑀/
𝑧𝑡𝑗
𝐹𝐿𝐷𝑀𝑞
𝑗 =1 𝑋 𝑜𝑟 𝑧 𝑣𝑗
𝐹𝐿𝐷𝑀𝑞
𝑗=1 𝑋 , ∀𝑡 𝑎𝑛𝑑 𝑣 (10-b)
where 𝐾 = 𝐾1 ∪ 𝐾2 ,
𝑧∗FLDM
= 𝑧1
∗ 𝐹𝐿𝐷𝑀
, 𝑧2
∗ 𝐹𝐿𝐷𝑀
, … . , 𝑧 𝑘 𝐼1
∗ 𝐹𝐿𝐷𝑀
and 𝑧−FLDM
= (𝑧1
− 𝐹𝐿𝐷𝑀
, 𝑧2
− 𝐹𝐿𝐷𝑀
, … … , 𝑧 𝑘 𝐼1
− 𝐹𝐿𝐷𝑀
) are the individual
positive (negative) ideal solutions for the FLDM.
Using the PIS and the NIS for the FLDM, we obtain the following distance functions from them,
respectively:
𝑑 𝑃
𝑃𝐼𝑆FLDM
= 𝑤𝑡
𝑝
𝑡∈𝐾1
𝑧 𝑡𝑗
∗FLDM𝑞
𝑗=1 − 𝑧 𝑡𝑗
FLDM
(𝑋)
𝑞
𝑗=1
𝑧 𝑡𝑗
∗FLDM𝑞
𝑗=1
− 𝑧 𝑡𝑗
− 𝐹𝐿𝐷𝑀𝑞
𝑗=1
𝑝
+ 𝑤𝑣
𝑝
𝑣∈𝐾2
𝑧 𝑣𝑗
𝐹𝐿𝐷𝑀
(𝑋)
𝑞
𝑗=1 − 𝑧 𝑣𝑗
∗ 𝐹𝐿𝐷𝑀𝑞
𝑗=1
𝑧 𝑣𝑗
− 𝐹𝐿𝐷𝑀𝑞
𝑗=1
− 𝑧 𝑣𝑗
∗ 𝐹𝐿𝐷𝑀𝑞
𝑗=1
𝑝
1
𝑝
(11-a)
(11-a)
and
𝑑 𝑃
𝑁𝐼𝑆FLDM
= 𝑤𝑡
𝑝
𝑡∈𝐾1
𝑧 𝑡𝑗
𝐹𝐿𝐷𝑀
(𝑋)
𝑞
𝑗=1 − 𝑧 𝑡𝑗
− 𝐹𝐿𝐷𝑀𝑞
𝑗=1
𝑧 𝑡𝑗
∗ 𝐹𝐿𝐷𝑀𝑞
𝑗=1
− 𝑧 𝑡𝑗
− 𝐹𝐿𝐷𝑀𝑞
𝑗=1
𝑝
+ 𝑤𝑣
𝑝
𝑣∈𝐾2
𝑧 𝑣𝑗
− 𝐹𝐿𝐷𝑀𝑞
𝑗=1 − 𝑧 𝑣𝑗
𝐹𝐿𝐷𝑀
(𝑋)
𝑞
𝑗=1
𝑧 𝑣𝑗
− 𝐹𝐿𝐷𝑀𝑞
𝑗=1
− 𝑧 𝑣𝑗
∗ 𝐹𝐿𝐷𝑀𝑞
𝑗=1
𝑝
1
𝑝
(11-b)
where 𝑤𝑖 = 1,2, … . , 𝑘, are the relative importance (weighs) of objectives, and 𝑝 = 1,2, … . . , ∞.
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In order to obtain a compromise solution for the FLDM, we transfer the FLDM of problem (9) into the
following two-objective problem with two commensurable (but often conflicting) objectives [4, 41]:
𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝑑 𝑝
𝑃𝐼𝑆 𝐹𝐿𝐷𝑀
𝑋
𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒 𝑑 𝑝
𝑁𝐼𝑆 𝐹𝐿𝐷𝑀
(𝑋)
subject to (12)
𝑋 ∈ 𝑀/
where 𝑝 = 1,2, … . . , ∞.
Since these two objectives are usually conflicting to each other, we can simultaneously obtain their individual
optima. Thus, we can use membership functions to represent these individual optima. Assume that the
membership functions (𝜇1 𝑋 and 𝜇2 𝑋 ) of two objective functions are linear. Then, based on the preference
concept, we assign a larger degree to the one with shorter distance from the PIS for 𝜇1 𝑋 and assign a larger
degree to the one with farther distance from NIS for 𝜇2 𝑋 . Therefore, as shown in figure (1), 𝜇1 𝑋 ≡
𝜇
𝑑 𝑝
𝑃 𝐼𝑆 𝐹𝐿𝐷𝑀 𝑋 and 𝜇2 𝑋 ≡ 𝜇
𝑑 𝑝
𝑁𝐼𝑆 𝐹𝐿𝐷𝑀 𝑋 can be obtained as the following (see [1, 6, 10, 11, 23, 38, 39, 47, 49,
56]):
𝜇1 𝑋 =
1, 𝑖𝑓 𝑑 𝑝
𝑃𝐼𝑆 𝐹𝐿𝐷𝑀
𝑋 < 𝑑 𝑝
𝑃𝐼𝑆 𝐹𝐿𝐷𝑀 ∗
,
1 −
𝑑 𝑝
𝑃𝐼𝑆 𝐹𝐿𝐷𝑀
𝑋 − 𝑑 𝑝
𝑃𝐼𝑆 𝐹𝐿𝐷𝑀 ∗
𝑑 𝑝
𝑃𝐼𝑆 𝐹𝐿𝐷𝑀 −
− 𝑑 𝑝
𝑃𝐼𝑆 𝐹𝐿𝐷𝑀 ∗ , 𝑖𝑓 𝑑 𝑝
𝑃𝐼𝑆 𝐹𝐿𝐷𝑀 −
0, 𝑖𝑓 𝑑 𝑝
𝑃𝐼𝑆 𝐹𝐿𝐷𝑀
𝑋 > 𝑑 𝑝
𝑃𝐼𝑆 𝐹𝐿𝐷𝑀 −
,
≥ 𝑑 𝑝
𝑃𝐼𝑆 𝐹𝐿𝐷𝑀
(𝑋) ≥ 𝑑 𝑝
𝑃𝐼𝑆 𝐹𝐿𝐷𝑀 ∗
, (13-a)
𝜇2 𝑋 =
1, 𝑖𝑓𝑑 𝑝
𝑁𝐼𝑆 𝐹𝐿𝐷𝑀
𝑋 > 𝑑 𝑝
𝑁𝐼𝑆 𝐹𝐿𝐷𝑀 ∗
,
1 −
𝑑 𝑝
𝑁𝐼𝑆 𝐹𝐿𝐷𝑀 ∗
− 𝑑 𝑝
𝑁𝐼𝑆 𝐹𝐿𝐷𝑀
𝑋
𝑑 𝑝
𝑁𝐼𝑆 𝐹𝐿𝐷𝑀 ∗
− 𝑑 𝑝
𝑁𝐼𝑆 𝐹𝐿𝐷𝑀 − , 𝑖𝑓 𝑑 𝑝
𝑁𝐼𝑆 𝐹𝐿𝐷𝑀 −
0, 𝑖𝑓𝑑 𝑝
𝑁𝐼𝑆 𝐹𝐿𝐷𝑀
𝑋 < 𝑑 𝑝
𝑁𝐼𝑆 𝐹𝐿𝐷𝑀 −
,
≤ 𝑑 𝑝
𝑁𝐼𝑆 𝐹𝐿𝐷𝑀
(𝑋) ≤ 𝑑 𝑝
𝑁𝐼𝑆 𝐹𝐿𝐷𝑀 ∗
, (13-b)
where
𝑑 𝑝
𝑃𝐼𝑆 𝐹𝐿𝐷𝑀 ∗
= 𝑑 𝑃
𝑃𝐼𝑆 𝐹𝐿𝐷𝑀
𝑋∈𝑀/
𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒
𝑋 𝑎𝑛𝑑 𝑡ℎ𝑒 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 𝑖𝑠 𝑋 𝑃𝐼𝑆 𝐹𝐿𝐷𝑀
,
𝑑 𝑝
𝑁𝐼𝑆 𝐹𝐿𝐷𝑀 ∗
= 𝑑 𝑃
𝑁𝐼𝑆 𝐹𝐿𝐷𝑀
𝑋∈𝑀/
𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒
𝑋 𝑎𝑛𝑑 𝑡ℎ𝑒 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 𝑖𝑠 𝑋 𝑁𝐼𝑆 𝐹𝐿𝐷𝑀
,
𝑑 𝑝
𝑃𝐼𝑆 𝐹𝐿𝐷𝑀 −
= 𝑑 𝑝
𝑃𝐼𝑆FLDM
𝑋 𝑁𝐼𝑆 𝐹𝐿𝐷𝑀
𝑎𝑛𝑑 𝑑 𝑝
𝑁𝐼𝑆 𝐹𝐿𝐷𝑀 −
= 𝑑 𝑝
𝑁𝐼𝑆 𝐹𝐿𝐷𝑀
𝑋 𝑃𝐼𝑆 𝐹𝐿𝐷𝑀
.
Now, by applying the max-min decision model which is proposed by R. E. Bellman and L. A. Zadeh [23]
and extended by H. –J. Zimmermann [56], we can resolve problem (12). The satisfying decision of the FLDM
problem (9), 𝑋∗FLDM
= (𝑋𝐼1
∗FLDM
, 𝑋𝐼2
∗FLDM
), may be obtained by solving the following model:
𝜇 𝐷 𝑋∗FLDM
= 𝑀𝑖𝑛. 𝜇1 𝑋 , 𝜇2 𝑋𝑋∈𝑀/
𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒
(14)
Finally, if 𝛿 𝐹𝐿𝐷𝑀
= 𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒 ( 𝜇1 𝑋 , 𝜇2 𝑋 ), the model (14) is equivalent to the form of Tchebycheff
model (see [30]), which is equivalent to the following model:
𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒 𝛾 𝐹𝐿𝐷𝑀
, (15-a)
subject to
𝜇1 𝑋 ≥ 𝛾 𝐹𝐿𝐷𝑀
, (15-b)
𝜇2 𝑋 ≥ 𝛾 𝐹𝐿𝐷𝑀
, (15-c)
𝑋 ∈ 𝑀/
, 𝛾 𝐹𝐿𝐷𝑀
∈ 0,1 , (15-d)
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where 𝛿 𝐹𝐿𝐷𝑀
is the satisfactory level for both criteria of the shortest distance from the PIS and the farthest
distance from the NIS. It is well known that if the optimal solution of (15) is the vector 𝛾∗ 𝐹𝐿𝐷𝑀
, 𝑋∗FLDM
, then
𝑋∗FLDM
is a nondominated solution [36, 39, 51, 55] of (12) and a satisfactory solution [4, 54] of the FLDM
problem (9).
𝑑 𝑝
𝑃𝐼𝑆 𝐹𝐿𝐷𝑀 ∗
𝑑 𝑝
𝑁𝐼𝑆 𝐹𝐿𝐷𝑀 −
𝑑 𝑝
𝑃𝐼𝑆 𝐹𝐿𝐷𝑀 −
𝑑 𝑝
𝑁𝐼𝑆 𝐹𝐿𝐷𝑀 ∗
Figure (1): The membership functions of 𝜇
𝑑 𝑝
𝑃𝐼𝑆 𝐹𝐿𝐷𝑀 𝑋 and 𝜇
𝑑 𝑝
𝑁𝐼𝑆 𝐹𝐿𝐷𝑀 𝑋
The basic concept of the two-level programming technique is that the FLDM sets his/her goals and/or
decisions with possible tolerances which are described by membership functions of fuzzy set theory. According
to this concept, let 𝜏𝑖
𝐿
and 𝜏𝑖
𝑅
, 𝑖 = 1,2, … , 𝑛𝐼1
be the maximum acceptable negative and positive tolerance
(relaxation) values on the decision vector considered by the FLDM, 𝑋𝐼1
∗ 𝐹𝐿𝐷𝑀
= 𝑥𝐼11
∗ 𝐹𝐿𝐷𝑀
, 𝑥𝐼12
∗ 𝐹𝐿𝐷𝑀
, … . , 𝑥𝐼1 𝑛 𝐼1
∗ 𝐹𝐿𝐷𝑀
.
The tolerances give the SLDM an extent feasible region to search for the satisfactory solution. If the feasible
region is empty, the negative and positive tolerances must be increased to give the SLDM an extent feasible
region to search for the satisfactory solution, [4, 36, 54]. The linear membership functions (Figure 2) for each of
the 𝑛𝐼1
components of the decision vector 𝑥𝐼11
∗ 𝐹𝐿𝐷𝑀
, 𝑥𝐼12
∗ 𝐹𝐿𝐷𝑀
, … . , 𝑥𝐼1 𝑛 𝐼1
∗ 𝐹𝐿𝐷𝑀
controlled by the FLDM can be
formulated as:
𝜇𝐼1 𝑖(𝑥𝐼1 𝑖) =
𝑥 𝐼1 𝑖− 𝑋 𝐼1 𝑖
∗ 𝐹𝐿𝐷𝑀
−𝜏 𝑖
𝐿
𝜏 𝑖
𝐿 𝑖𝑓 𝑥𝐼1 𝑖
∗ 𝐹𝐿𝐷𝑀
− 𝜏𝑖
𝐿
≤ 𝑥𝐼1 𝑖 ≤ 𝑥𝐼1 𝑖
∗ 𝐹𝐿𝐷𝑀
𝑋 𝐼1 𝑖
∗ 𝐹𝐿𝐷𝑀
+𝜏 𝑖
𝑅
−𝑥 𝐼1 𝑖
𝜏 𝑖
𝑅 𝑖𝑓 𝑥𝐼1 𝑖
∗ 𝐹𝐿𝐷𝑀
≤ 𝑥𝐼1 𝑖 ≤ 𝑥𝐼1 𝑖
∗ 𝐹𝐿𝐷𝑀
+ 𝜏𝑖
𝑅
, 𝑖 = 1,2, … , 𝑛𝐼1
,
0 𝑖𝑓 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒,
(16)
It may be noted that, the decision maker may desire to shift the range of 𝑥𝐼1 𝑖. Following Pramanik & Roy [47]
and Sinha [50], this shift can be achieved.
7. Tarek H. M. Abou-El-Enien Int. Journal of Engineering Research and Applications www.ijera.com
ISSN : 2248-9622, Vol. 5, Issue 4, ( Part -2) April 2015, pp.61-76
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Figure (2): The membership function of the decision variable 𝑥𝐼1 𝑖
4-2. Phase (II)
The SLDM problem of problem (8) can be written as follows:
[SLDM]
𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒
𝑋𝐼2
𝑍𝐼2
𝑋𝐼1
, 𝑋𝐼2
=
𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒
𝑋𝐼2
𝑧𝐼21 𝑋𝐼1
, 𝑋𝐼2
, … . , 𝑧𝐼2 𝑘 𝐼2
𝑋𝐼1
, 𝑋𝐼2
subject to (17)
𝑋 ∈ 𝑀/
where
𝑧𝑡𝑗
𝑞
𝑗=1 (𝑋): Objective Function for Maximization, 𝑡 ∈ 𝐾1 ⊏ 𝐾,
𝑧 𝑣𝑗
𝑞
𝑗=1 (𝑋): Objective Function for Minimization, 𝑣 ∈ 𝐾2 ⊏ 𝐾.
In order to use the distance family of equation (7) to resolve problem (17), we must first find PIS(z∗
)
and NIS z−
which are [4, 41]:
𝑧∗ 𝑆𝐿𝐷𝑀
=
𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒 𝑜𝑟 𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒
𝑋 ∈ 𝑀/
𝑧𝑡𝑗
𝑞
𝑗 =1 𝑋 𝑜𝑟 𝑧 𝑣𝑗
𝑞
𝑗 =1 𝑋 , ∀𝑡 𝑎𝑛𝑑 𝑣 (18-a)
𝑧− 𝑆𝐿𝐷𝑀
=
𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝑜𝑟 𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒
𝑋 ∈ 𝑀/
𝑧𝑡𝑗
𝑞
𝑗=1 𝑋 𝑜𝑟 𝑧 𝑣𝑗
𝑞
𝑗 =1 𝑋 , ∀𝑡 𝑎𝑛𝑑 𝑣 (18-b)
where 𝐾 = 𝐾1 ∪ 𝐾2,
𝑧∗ 𝑆𝐿𝐷𝑀
= 𝑧1
∗ 𝑆𝐿𝐷𝑀
, 𝑧2
∗ 𝑆𝐿𝐷𝑀
, … . , 𝑧 𝑘 𝐼2
∗ 𝑆𝐿𝐷𝑀
and 𝑧− 𝑆𝐿𝐷𝑀
= (𝑧1
− 𝑆𝐿𝐷𝑀
, 𝑧2
− 𝑆𝐿𝐷𝑀
, … … , 𝑧 𝑘 𝐼2
− 𝑆𝐿𝐷𝑀
) are the individual
positive (negative) ideal solutions for the SLDM.
In order to obtain a compromise (satisfactory ) solution to problem (8) using TOPSIS approach, the
distance family of (7) to represent the distance function from the positive ideal solution, 𝑑 𝑃
𝑃𝐼𝑆TL
, and the
distance function from the negative ideal solution, 𝑑 𝑃
𝑁𝐼𝑆TL
, can be proposed, in this paper, for the objectives of
the FLDM and the SLDM as follows:
𝑑 𝑃
𝑃𝐼𝑆TL
=
( 𝑤𝑡
𝑝
𝑡∈𝐾1
𝑧 𝑡𝑗
∗FLDM𝑞
𝑗=1 − 𝑧 𝑡𝑗
FLDM
𝑋
𝑞
𝑗=1
𝑧 𝑡𝑗
∗FLDM𝑞
𝑗=1
− 𝑧 𝑡𝑗
− 𝐹𝐿𝐷𝑀𝑞
𝑗=1
𝑝
+ 𝑤𝑣
𝑝
𝑣∈𝐾2
𝑧 𝑣𝑗
𝐹𝐿𝐷𝑀
𝑋
𝑞
𝑗=1 − 𝑧 𝑣𝑗
∗ 𝐹𝐿𝐷𝑀𝑞
𝑗=1
𝑧 𝑣𝑗
− 𝐹𝐿𝐷𝑀𝑞
𝑗=1
− 𝑧 𝑣𝑗
∗ 𝐹𝐿𝐷𝑀𝑞
𝑗=1
𝑝
+
𝑤𝑡
𝑝
𝑡∈𝐾1
𝑧 𝑡𝑗
∗ 𝑆𝐿𝐷𝑀𝑞
𝑗=1 − 𝑧 𝑡𝑗
𝑆𝐿𝐷𝑀
𝑋
𝑞
𝑗=1
𝑧 𝑡𝑗
∗ 𝑆𝐿𝐷𝑀𝑞
𝑗=1
− 𝑧 𝑡𝑗
− 𝑆𝐿𝐷𝑀𝑞
𝑗=1
𝑝
+ 𝑤𝑣
𝑝
𝑣∈𝐾2
𝑧 𝑣𝑗
𝑆𝐿𝐷𝑀
𝑋
𝑞
𝑗=1 − 𝑧 𝑣𝑗
∗ 𝑆𝐿𝐷𝑀𝑞
𝑗=1
𝑧 𝑣𝑗
− 𝑆𝐿𝐷𝑀𝑞
𝑗=1
− 𝑧 𝑣𝑗
∗ 𝑆𝐿𝐷𝑀𝑞
𝑗=1
𝑝
)
1
𝑝 (19-a)
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and
𝑑 𝑃
𝑁𝐼𝑆TL
=
( 𝑤𝑡
𝑝
𝑡∈𝐾1
𝑧𝑡𝑗
𝐹𝐿𝐷𝑀
(𝑋)
𝑞
𝑗=1 − 𝑧𝑡𝑗
− 𝐹𝐿𝐷𝑀𝑞
𝑗=1
𝑧𝑡𝑗
∗ 𝐹𝐿𝐷𝑀𝑞
𝑗=1 − 𝑧𝑡𝑗
− 𝐹𝐿𝐷𝑀𝑞
𝑗=1
𝑝
+ 𝑤𝑣
𝑝
𝑣∈𝐾2
𝑧 𝑣𝑗
− 𝐹𝐿𝐷𝑀𝑞
𝑗=1 − 𝑧 𝑣𝑗
𝐹𝐿𝐷𝑀
(𝑋)
𝑞
𝑗=1
𝑧 𝑣𝑗
− 𝐹𝐿𝐷𝑀𝑞
𝑗=1 − 𝑧 𝑣𝑗
∗ 𝐹𝐿𝐷𝑀𝑞
𝑗 =1
𝑝
+
𝑤𝑡
𝑝
𝑡∈𝐾1
𝑧𝑡𝑗
𝑆𝐿𝐷𝑀
(𝑋)
𝑞
𝑗=1𝑧 − 𝑧𝑡𝑗
− 𝑆𝐿𝐷𝑀𝑞
𝑗=1
𝑧𝑡𝑗
∗ 𝑆𝐿𝐷𝑀𝑞
𝑗 =1 − 𝑧𝑡𝑗
− 𝑆𝐿𝐷𝑀𝑞
𝑗 =1
𝑝
+ 𝑤𝑣
𝑝
𝑣∈𝐾2
𝑧 𝑣𝑗
− 𝑆𝐿𝐷𝑀𝑞
𝑗=1 − 𝑧 𝑣𝑗
𝑆𝐿𝐷𝑀
(𝑋)
𝑞
𝑗 =1
𝑧 𝑣𝑗
− 𝑆𝐿𝐷𝑀𝑞
𝑗 =1 − 𝑧 𝑣𝑗
∗ 𝑆𝐿𝐷𝑀𝑞
𝑗=1
𝑝
)
1
𝑝
(19-b)
where 𝑤𝑖 = 1,2, … . , 𝑘, are the relative importance (weighs) of objectives, and 𝑝 = 1,2, … . . , ∞.
In order to obtain a compromise (satisfactory) solution, we transfer problem (8) into the following
two-objective problem with two commensurable (but often conflicting) objectives [4, 41]:
𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝑑 𝑝
𝑃𝐼𝑆TL
𝑋
𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒 𝑑 𝑝
𝑁𝐼𝑆TL
(𝑋)
subject to (20)
𝑋 ∈ 𝑀/
where 𝑝 = 1,2, … . . , ∞.
Since these two objectives are usually conflicting to each other, we can simultaneously obtain their individual
optima. Thus, we can use membership functions to represent these individual optima. Assume that the
membership functions (𝜇3 𝑋 and 𝜇4 𝑋 ) of two objective functions are linear. Then, based on the preference
concept, we assign a larger degree to the one with shorter distance from the PIS for 𝜇3 𝑋 and assign a larger
degree to the one with farther distance from NIS for 𝜇4 𝑋 . Therefore, as shown in figure (3), 𝜇3 𝑋 ≡
𝜇
𝑑 𝑝
𝑃 𝐼𝑆 TL 𝑋 and 𝜇4 𝑋 ≡ 𝜇
𝑑 𝑝
𝑁𝐼𝑆 TL 𝑋 can be obtained as the following (see [1, 6, 10, 11, 23, 38, 39, 47, 49,
56]):
𝜇3 𝑋 =
1, 𝑖𝑓 𝑑 𝑝
𝑃𝐼𝑆TL
𝑋 < 𝑑 𝑝
𝑃𝐼𝑆TL ∗
,
1 −
𝑑 𝑝
𝑃𝐼𝑆 TL
𝑋 − 𝑑 𝑝
𝑃𝐼𝑆 TL ∗
𝑑 𝑝
𝑃𝐼𝑆 TL /
− 𝑑 𝑝
𝑃𝐼𝑆 TL ∗ , 𝑖𝑓 𝑑 𝑝
𝑃𝐼𝑆TL −
0, 𝑖𝑓 𝑑 𝑝
𝑃𝐼𝑆TL
𝑋 > 𝑑 𝑝
𝑃𝐼𝑆TL −
,
≥ 𝑑 𝑝
𝑃𝐼𝑆TL
(𝑋) ≥ 𝑑 𝑝
𝑃𝐼𝑆TL ∗
, (21-a)
𝜇4 𝑋 =
1, 𝑖𝑓𝑑 𝑝
𝑁𝐼𝑆TL
𝑋 > 𝑑 𝑝
𝑁𝐼𝑆TL ∗
,
1 −
𝑑 𝑝
𝑁𝐼𝑆 TL ∗
− 𝑑 𝑝
𝑁𝐼𝑆 TL
𝑋
𝑑 𝑝
𝑁𝐼𝑆 TL ∗
− 𝑑 𝑝
𝑁𝐼𝑆 TL − , 𝑖𝑓 𝑑 𝑝
𝑁𝐼𝑆TL −
0, 𝑖𝑓𝑑 𝑝
𝑁𝐼𝑆TL
𝑋 < 𝑑 𝑝
𝑁𝐼𝑆TL −
,
≤ 𝑑 𝑝
𝑁𝐼𝑆TL
(𝑋) ≤ 𝑑 𝑝
𝑁𝐼𝑆TL ∗
, (21-b)
where
𝑑 𝑝
𝑃𝐼𝑆TL ∗
= 𝑑 𝑃
𝑃𝐼𝑆TL
𝑋∈𝑀/
𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒
𝑋 𝑎𝑛𝑑 𝑡ℎ𝑒 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 𝑖𝑠 𝑋 𝑃𝐼𝑆TL
,
𝑑 𝑝
𝑁𝐼𝑆TL ∗
= 𝑑 𝑃
𝑁𝐼𝑆TL
𝑋∈𝑀/
𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒
𝑋 𝑎𝑛𝑑 𝑡ℎ𝑒 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 𝑖𝑠 𝑋 𝑁𝐼𝑆TL
,
𝑑 𝑝
𝑃𝐼𝑆TL −
= 𝑑 𝑝
𝑃𝐼𝑆TL
𝑋 𝑁𝐼𝑆TL
𝑎𝑛𝑑 𝑑 𝑝
𝑁𝐼𝑆TL −
= 𝑑 𝑝
𝑁𝐼𝑆TL
𝑋 𝑃𝐼𝑆TL
.
Now, by applying the max-min decision model which is proposed by R. E. Bellman and L. A. Zadeh [23] and
extended by H. –J. Zimmermann [56], we can resolve problem (20). The satisfactory solution of problem (8),
𝑋∗TL
, may be obtained by solving the following model:
𝜇 𝐷 𝑋∗TL
= 𝑀𝑖𝑛. 𝜇3 𝑋 , 𝜇4 𝑋𝑋∈𝑀/
𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒
(22)
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Finally, if 𝛿TL
= 𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒 ( 𝜇3 𝑋 , 𝜇4 𝑋 ), the model (22) is equivalent to the form of Tchebycheff model
(see [30]), which is equivalent to the following model:
𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒 𝛾TL
, (23-a)
subject to
𝜇3 𝑋 ≥ 𝛾TL
, (23-b)
𝜇4 𝑋 ≥ 𝛾TL
, (23-c)
𝑥 𝐼1 𝑖− 𝑋 𝐼1 𝑖
∗ 𝐹𝐿𝐷𝑀
−𝜏 𝑖
𝐿
𝜏 𝑖
𝐿 ≥ 𝛾TL
, 𝑖 = 1,2, … , 𝑛𝐼1
(23-d)
𝑋 𝐼1 𝑖
∗ 𝐹𝐿𝐷𝑀
+𝜏 𝑖
𝑅
− 𝑥 𝐼1 𝑖
𝜏 𝑖
𝑅 ≥ 𝛾TL
, 𝑖 = 1,2, … , 𝑛𝐼1
(23-e)
𝑋 ∈ 𝑀/
, 𝛾TL
∈ 0,1 , (23-f)
where 𝛿TL
is the satisfactory level for both criteria of the shortest distance from the PIS and the farthest distance
from the NIS. It is well known that if the optimal solution of (23) is the vector 𝛾∗TL
, 𝑋∗TL
, then 𝑋∗TL
is a
nondominated solution of (20) and a satisfactory solution for the problem (8) [4, 11, 45].
Figure (3): The membership functions of 𝜇
𝑑 𝑝
𝑃𝐼𝑆 TL 𝑋 and 𝜇
𝑑 𝑝
𝑁𝐼𝑆 TL 𝑋
V. A decomposition algorithm of
TOPSIS for solving (TL-LSLMOP-SP)rhs of block angular structure
Thus, we can introduce the following decomposition algorithm of TOPSIS method to gernerate a set of
satisfactory solutions for (TL-LSLMOP-SP)rhs of block angular structure:
The algorithm (Alg-I):
Phase (0):
Step 1. Transform problem (1) to the form of problem (3).
Step 2. Transform problem (3) to the form of problem (8).
Phase (I):
Step 3. Construct the PIS payoff table of problem (9) by using the decomposition algorithm [27, 29, 43], and
obtain 𝑧∗FLDM
= 𝑧1
∗ 𝐹𝐿 𝐷𝑀
, 𝑧2
∗ 𝐹𝐿𝐷𝑀
, … . , 𝑧 𝑘 𝐼1
∗ 𝐹𝐿𝐷𝑀
the individual positive ideal solutions.
Step 4. Construct the NIS payoff table of problem (9) by using the decomposition algorithm, and obtain
𝑧−FLDM
= (𝑧1
− 𝐹𝐿𝐷𝑀
, 𝑧2
− 𝐹𝐿𝐷𝑀
, … … , 𝑧 𝑘 𝐼1
− 𝐹𝐿𝐷𝑀
) , the individual negative ideal solutions.
Step 5. Use equations (10 & 11) and the above steps (3 & 4) to construct 𝑑 𝑝
𝑃𝐼𝑆FLDM
and 𝑑 𝑝
𝑁𝐼𝑆FLDM
.
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Step 6. Ask the FLDM to select
p= p*{1, 2, …, },
Step 7. Ask the FLDM to select 𝑤𝑖 = 𝑤𝑖
∗
, 𝑖 = 1,2, … , 𝑘𝐼1
, where 𝑤𝑖 = 1
𝑘 𝐼1
𝑖=1
,
Step 8. Use steps ( 4 & 6) to compute 𝑑 𝑝
𝑃𝐼𝑆FLDM
and 𝑑 𝑝
𝑁𝐼𝑆FLDM
.
Step 9. Transform problem (9) to the form of problem (12).
Step 10. Construct the payoff table of problem (12):
At p = 1, use the decomposition algorithm [27, 29, 43].
At p ≥ 2, use the generalized reduced gradient method, [43, 44], and obtain:
𝑑 𝑝
− 𝐹𝐿𝐷𝑀
= 𝑑 𝑝
𝑃𝐼𝑆FLDM −
, 𝑑 𝑝
𝑁𝐼𝑆FLDM −
, 𝑑 𝑝
∗ 𝐹𝐿𝐷𝑀
= 𝑑 𝑝
𝑃𝐼𝑆FLDM ∗
, 𝑑 𝑝
𝑁𝐼𝑆FLDM ∗
.
Step 11. Construct problem (15) by using the membership functions (13).
Step 12. Solve problem (15) to obtain 𝛾∗ 𝐹𝐿 𝐷𝑀
, 𝑋∗FLDM
.
Step 13. Ask the FLDM to select the maximum negative and positive tolerance values 𝜏𝑖
𝐿
and 𝜏𝑖
𝑅
, 𝑖 =
1,2, … , 𝑛𝐼1
on the decision vector 𝑋𝐼1
∗ 𝐹𝐿𝐷𝑀
= 𝑥𝐼11
∗ 𝐹𝐿𝐷𝑀
, 𝑥𝐼12
∗ 𝐹𝐿𝐷𝑀
, … . , 𝑥𝐼1 𝑛 𝐼1
∗ 𝐹𝐿𝐷𝑀
.
Phase (II):
Step 14. Construct the PIS payoff table of problem (17) by using the decomposition algorithm [27, 29, 43], and
obtain 𝑧∗SLDM
= 𝑧1
∗ 𝑆𝐿𝐷𝑀
, 𝑧2
∗ 𝑆𝐿𝐷𝑀
, … . , 𝑧 𝑘 𝐼2
∗ 𝑆𝐿𝐷𝑀
the individual positive ideal solutions.
Step 15. Construct the NIS payoff table of problem (17) by using the decomposition algorithm, and obtain
𝑧−SLDM
= (𝑧1
−SLDM
, 𝑧2
−SLDM
, … … , 𝑧 𝑘 𝐼2
−SLDM
) the individual negative ideal solutions.
Step 16. Use equations (18 & 19) and the above steps (14 & 15) to construct 𝑑 𝑝
𝑃𝐼𝑆TL
and 𝑑 𝑝
𝑁𝐼𝑆TL
.
Step 17. Ask the FLDM to select 𝑤𝑖 = 𝑤𝑖
∗
, 𝑖 = 1,2, … , 𝑘, where 𝑤𝑖 = 1𝑘
𝑖=1 ,
Step 18. Use steps (14 , 15 & 16) to compute 𝑑 𝑝
𝑃𝐼𝑆TL
and 𝑑 𝑝
𝑁𝐼𝑆TL
.
Step 19. Transform problem (8) to the form of problem (20).
Step 20. Construct the payoff table of problem (20):
At p=1,use the decomposition algorithm [27, 29, 43],
At p≥2, use the generalized reduced gradient method, [46, 47], and obtain:
𝑑 𝑝
−TL
= 𝑑 𝑝
𝑃𝐼𝑆TL −
, 𝑑 𝑝
𝑁𝐼𝑆TL −
, 𝑑 𝑝
∗ 𝑇𝐿
= 𝑑 𝑝
𝑃𝐼𝑆TL ∗
, 𝑑 𝑝
𝑁𝐼𝑆TL ∗
.
Step 21. Use equations (13 and 18) to construct problem (23).
Step 22. Solve problem (23) to obtain 𝛾∗TL
, 𝑋∗TL
.
Step 23. If the FLDM is satisfied with the current solution , go to step 24. Otherwise, go to step 6.
Step 24. Stop.
VI. An illustrative numerical example
Consider the following (TL-LSLMOP-SP)rhs of block angular structure:
[FLDM]
𝑓11 𝑋𝑥1 , 𝑥2
𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒
= 4𝑥1 + 6𝑥2 − 𝑥3 + 7𝑥4 (24 − 1)
𝑓12 𝑋𝑥1 , 𝑥2
𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒
= −2𝑥1 + 9𝑥2 + 13𝑥3 + 𝑥4 24 − 2
𝑓13 𝑋𝑥1 , 𝑥2
𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒
= −𝑥1 + 3𝑥2 − 𝑥3 + 𝑥4 (24 − 3)
𝑓14 𝑋𝑥1 , 𝑥2
𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒
= 6𝑥1 − 2𝑥2 + 𝑥3 + 𝑥4 (24 − 4)
𝑤ℎ𝑒𝑟𝑒 𝑥1, 𝑥2 𝑠𝑜𝑙𝑣𝑒𝑠 𝑡ℎ𝑒 𝑠𝑒𝑐𝑜𝑛𝑑 𝑙𝑒𝑣𝑒𝑙
[SLDM]
𝑓21 𝑋𝑥3 , 𝑥4
𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒
= 𝑥1 − 5𝑥2 + 3𝑥3 + 19𝑥4 (24 − 5)
𝑓22 𝑋𝑥3 , 𝑥4
𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒
= 4𝑥1 + 𝑥2 + 𝑥3 − 𝑥4 (24 − 6)
subject to
𝑃{𝑥1 + 𝑥2 + 𝑥3 + 𝑥4 ≤ 6𝑣1} ≥ 0.7257, (24 − 7)
𝑃{5𝑥1 + 𝑥2 ≤ 7𝑣2} ≥ 0.5, (24 − 8)
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𝑃{𝑥3 + 𝑥4 ≤ 8𝑣3} ≥ 0.9015, (24 − 9)
𝑥1, 𝑥2, 𝑥3, 𝑥4 ≥ 0 (24 − 10)
Suppose that 𝑣𝑖, 𝑖 = 1,2,3 are linearly independent normal distributed parameters with the following
means and variances: 𝐸 𝑣1 = 8, 𝐸 𝑣2 = 2, 𝐸 𝑣3 = 1 , 𝑉𝑎𝑟 𝑣1 = 25, 𝑉𝑎𝑟 𝑣2 = 4, 𝑉𝑎𝑟 𝑣3 = 25 .
Solution:
By using problem (3), we can have
[FLDM]
𝑓11 𝑋𝑥1 , 𝑥2
𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒
= 4𝑥1 + 6𝑥2 − 𝑥3 + 7𝑥4 (25 − 1)
𝑓12 𝑋𝑥1 , 𝑥2
𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒
= −2𝑥1 + 9𝑥2 + 13𝑥3 + 𝑥4 25 − 2
𝑓13 𝑋𝑥1 , 𝑥2
𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒
= −𝑥1 + 3𝑥2 − 𝑥3 + 𝑥4 (25 − 3)
𝑓14 𝑋𝑥1 , 𝑥2
𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒
= 6𝑥1 − 2𝑥2 + 𝑥3 + 𝑥4 (25 − 4)
𝑤ℎ𝑒𝑟𝑒 𝑥1, 𝑥2 𝑠𝑜𝑙𝑣𝑒𝑠 𝑡ℎ𝑒 𝑠𝑒𝑐𝑜𝑛𝑑 𝑙𝑒𝑣𝑒𝑙
[SLDM]
𝑓21 𝑋𝑥3 , 𝑥4
𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒
= 𝑥1 − 5𝑥2 + 3𝑥3 + 19𝑥4 (25 − 5)
𝑓22 𝑋𝑥3 , 𝑥4
𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒
= 4𝑥1 + 𝑥2 + 𝑥3 − 𝑥4 (25 − 6)
subject to
𝑥1 + 𝑥2 + 𝑥3 + 𝑥4 ≤ 66, (25 − 7)
5𝑥1 + 𝑥2 ≤ 14, (25 − 8)
𝑥3 + 𝑥4 ≤ 59.6, (25 − 9)
𝑥1, 𝑥2, 𝑥3, 𝑥4 ≥ 0 (25 − 10)
- Obtain PIS and NIS payoff tables for the FLDM of problem (25).
Table (1) : PIS payoff table for the FLDM of problem (25)
𝑓11 𝑓12 𝑓13 𝑓14 𝑥1 𝑥2 𝑥3 𝑥4
𝑓11 𝑋𝑥1 , 𝑥2
𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒 455.6 117.2 78.8 46.8 0 6.4 0 59.6
𝑓12 𝑋𝑥1 , 𝑥2
𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒 -21.2 832.4 -40.8 46.8 0 6.4 59.6 0
𝑓13 𝑋𝑥1 , 𝑥2
𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒 -48.4 769.2 -62.4 76.4 2.8 0 59.6 0
𝑓14 𝑋𝑥1 , 𝑥2
𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒 84 126 42 -14 0 14 0 0
PIS: 𝑓∗FLDM
= (455.6 , 832.4 , -62.4 , -14)
Table (2) : NIS payoff table for the FLDM of problem (25)
𝑓11 𝑓12 𝑓13 𝑓14 𝑥1 𝑥2 𝑥3 𝑥4
𝑓11 𝑋𝑥1 , 𝑥2
𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒 -59.6 -
774.8 -59.6 -59.6 0 0 59.6 0
𝑓12 𝑋𝑥1 , 𝑥2
𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒 11.2 -5.6 -
-2.8 16.8 2.8 0 0 0
𝑓13 𝑋𝑥1 , 𝑥2
𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒 448 178 94 -
24 0 14 0 52
𝑓14 𝑋𝑥1 , 𝑥2
𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒 428.4 54 56.8 136 -
2.8 0 0 59.6
NIS: 𝑓−FLDM
= (-59.6 , -5.6 , 94 , 136)
- Next, compute equation (11) and obtain the following equations:
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𝑑 𝑃
𝑃𝐼𝑆FLDM
= 𝑤11
𝑝 455.6 − 𝑓11(𝑋)
455.6 − (−59.6)
𝑝
+ 𝑤12
𝑝 832.4 − 𝑓12(𝑋)
832.4 − (−5.6)
𝑝
+ 𝑤13
𝑝 𝑓13 𝑥 − (−62.4)
94 − (−62.4)
𝑝
+ 𝑤14
𝑝 𝑓14 𝑥 − (−14)
136 − (−14)
𝑝 1
𝑝
𝑑 𝑃
𝑁𝐼𝑆FLDM
= 𝑤11
𝑝 𝑓11(𝑋) − (−59.6)
455.6 − (−59.6)
𝑝
+ 𝑤12
𝑝 𝑓12(𝑋) − (−5.6)
832.4 − (−5.6)
𝑝
+ 𝑤13
𝑝 94 − 𝑓13 𝑥
94 − (−62.4)
𝑝
+ 𝑤14
𝑝 136 − 𝑓14 𝑥
136 − (−14)
𝑝
1
𝑝
- Thus, problem (12) is obtained.
- In order to get numerical solutions, assume that 𝑤11
𝑝
= 𝑤12
𝑝
= 𝑤13
𝑝
= 𝑤14
𝑝
= 0.25 and p=2,
Table (3) : PIS payoff table of problem (12), when p=2.
𝑑2
𝑃𝐼𝑆FLDM
𝑑2
𝑁𝐼𝑆FLDM
𝑥1 𝑥2 𝑥3 𝑥4
𝑀𝑖𝑛. 𝑑2
𝑃𝐼𝑆FLDM
0.2182*
0.2945-
0 10.8954 35.2202 19.8844
𝑀𝑎𝑥. 𝑑2
𝑁𝐼𝑆FLDM
0.3029-
0.2269*
2.8 0 6.0063 27.1433
𝑑2
∗ 𝐹𝐿𝐷𝑀
= (0.2182 , 0.2269 ) , 𝑑2
− 𝐹𝐿𝐷𝑀
= (0.3029 , 0.2945 ).
- Now, it is easy to compute (15) :
𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒 𝛾FLDM
subject to
𝑥1 + 𝑥2 + 𝑥3 + 𝑥4 ≤ 66,
5𝑥1 + 𝑥2 ≤ 14,
𝑥3 + 𝑥4 ≤ 59.6,
𝑑2
𝑃𝐼𝑆 FLD M
𝑋 − 0.2182
0.3029−0.2182
≥ 𝛾FLDM
,
0.2269 − 𝑑2
𝑁𝐼𝑆FLDM
𝑋
0.2269 − 0.2945
≥ 𝛾FLDM
,
𝑥1, 𝑥2, 𝑥3, 𝑥4 ≥ 0 , 𝛾FLDM
∈ 0,1 .
The maximum ―satisfactory level‖ (𝛾FLDM
=1) is achieved for the solution 𝑋1
∗ 𝐹𝐿𝐷𝑀
=0 , 𝑋2
∗ 𝐹𝐿𝐷𝑀
=13.2092 ,
𝑋3
∗ 𝐹𝐿𝐷𝑀
=0, 𝑋4
∗ 𝐹𝐿𝐷𝑀
=0 and 𝑓11, 𝑓12, 𝑓13, 𝑓14 = (79.2552 , 118.8828 , 39.6276 , −26.4184 ). Let the
FLDM decide 𝑋1
∗ 𝐹𝐿𝐷𝑀
= 0 and 𝑋2
∗ 𝐹𝐿𝐷𝑀
= 13.2092 with positive tolerance 𝜏 𝑅
= 0.5 𝑎𝑛𝑑 𝜏 𝐿
= 0.5 .
- Obtain PIS and NIS payoff tables for the SLDM of Problem (25).
Table (4) : PIS payoff table for the SLDM of problem (25)
𝑓21 𝑓22 𝑥1 𝑥2 𝑥3 𝑥4
𝑓21 𝑋𝑥3 , 𝑥4
𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒 1135.2*
-48.4 2.8 0 0 59.6
𝑓22 𝑋𝑥3 , 𝑥4
𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒 1132.4 -59.6*
0 0 0 59.6
PIS: 𝑓∗SLDM
=(1135.2 , -59.6)
Table (5) : NIS payoff table for the SLDM of problem (24)
𝑓21 𝑓22 𝑥1 𝑥2 𝑥3 𝑥4
𝑓21 𝑋𝑥3 , 𝑥4
𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒 -70 -
14 0 14 0 0
𝑓22 𝑋𝑥3 , 𝑥4
𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒 158.2 71.7-
1.9 4.5 59.6 0
NIS: 𝑓−SLDM
=(-70 , 71.7 )
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- Next, compute equation (19) and obtain the following equations:
𝑑 𝑃
𝑃𝐼𝑆TL
= 𝑤11
𝑝 455.6 − 𝑓11(𝑋)
455.6 − (−59.6)
𝑝
+ 𝑤12
𝑝 832.4 − 𝑓12(𝑋)
832.4 − (−5.6)
𝑝
+ 𝑤13
𝑝 𝑓13 𝑥 − (−62.4)
94 − (−62.4)
𝑝
+ 𝑤14
𝑝 𝑓14 𝑥 − (−14)
136 − (−14)
𝑝
+ 𝑤25
𝑝 1135.2 − 𝑓21(𝑋)
1135.2 − (−70)
𝑝
+ 𝑤26
𝑝 𝑓22 𝑋 − (−59.6)
71.7 − (−59.6)
𝑝
1
𝑝
𝑑 𝑃
𝑁𝐼𝑆TL
= 𝑤11
𝑝 𝑓11(𝑋) − (−59.6)
455.6 − (−59.6)
𝑝
+ 𝑤12
𝑝 𝑓12(𝑋) − (−5.6)
832.4 − (−5.6)
𝑝
+ 𝑤13
𝑝 94 − 𝑓13 𝑥
94 − (−62.4)
𝑝
+ 𝑤14
𝑝 136 − 𝑓14 𝑥
136 − (−14)
𝑝
+ 𝑤25
𝑝 𝑓21 𝑋 − (−70)
1135.2 − (−70)
𝑝
+ 𝑤26
𝑝 71.7 − 𝑓22 𝑋
71.7 − (−59.6)
𝑝
1
𝑝
- Thus, problem (20) is obtained.
- In order to get numerical solutions, assume that 𝑤21
𝑝
=0.2 , 𝑤22
𝑝
=0.2 , 𝑤23
𝑝
=0.2 , 𝑤24
𝑝
=0.2 ,
𝑤25
𝑝
= 0.1 , 𝑤26
𝑝
=0.1 and p=2,
Table (6) : PIS payoff table of problem (20), when p=2.
𝑑2
𝑃𝐼𝑆BL
𝑑2
𝑁𝐼𝑆BL
x1 x2 x3 x4
𝑀𝑖𝑛. 𝑑2
𝑃𝐼𝑆BL
0.1934∗
0.2329− 0 6.4 31.8117 27.7883
𝑀𝑎𝑥. 𝑑2
𝑁𝐼𝑆BL
0.2936−
0.2257∗ 0 1.6426 0.4845 0
𝑑2
∗ 𝐵𝐿
=(0.1934 , 0.2257 ) , 𝑑2
− 𝐵𝐿
=(0.2936 , 0.2329).
- Now, it is easy to compute (23) :
𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒 𝛾BL
subject to
𝑥1 + 𝑥2 + 𝑥3 + 𝑥4 ≤ 66,
5𝑥1 + 𝑥2 ≤ 14,
𝑥3 + 𝑥4 ≤ 59.6,
𝑑2
𝑃𝐼𝑆BL
𝑋 − 0.1934
0.2936 − 0.1934
≥ 𝛾BL
,
0.2257 − 𝑑2
𝑁𝐼𝑆BL
𝑋
0.2257 − 0.2329
≥ 𝛾BL
,
0 + 0.5 − 𝑥1
0.5
≥ 𝛾BL
,
𝑥1 − 0 − 0.5
0.5
≥ 𝛾BL
,
13.2092 + 0.5 − 𝑥2
0.5
≥ 𝛾BL
,
𝑥2 − 13.2092 − 0.5
0.5
≥ 𝛾BL
,
𝑥1, 𝑥2, 𝑥3, 𝑥4 ≥ 0 , 𝛾BL
∈ 0,1 .
The maximum ―satisfactory level‖ (𝛿BL
= 0.9042 ) is achieved for the solution 𝑋1
∗ 𝐵𝐿
= 0 , 𝑋2
∗ 𝐵𝐿
= 13.1613
, 𝑋3
∗ 𝐵𝐿
= 0 , 𝑋4
∗ 𝐵𝐿
= 0 .
VI. Summary and Concluding remarks
In this paper, a TOPSIS approach has been extended to solve (TL-LSLMOP-SP)rhs of block angular
structure. The (TL-LSLMOP-SP)rhs of block angular structure using TOPSIS approach provides an effective
way to find the compromise ( satisfactory) solution of such problems. In order to obtain a compromise (
14. Tarek H. M. Abou-El-Enien Int. Journal of Engineering Research and Applications www.ijera.com
ISSN : 2248-9622, Vol. 5, Issue 4, ( Part -2) April 2015, pp.61-76
www.ijera.com 74 | P a g e
satisfactory) integer solution to the (TL-LSLMOP-SP)rhs of block angular structure using the proposed
TOPSIS approach, a modified formulas for the distance function from the PIS and the distance function from
the NIS are proposed and modeled to include all objective functions of both the first and the second levels.
Thus, the two-objective problem is obtained which can be solved by using membership functions of fuzzy set
theory to represent the satisfaction level for both criteria and obtain TOPSIS, compromise solution by a
second–order compromise. The max-min operator is then considered as a suitable one to resolve the conflict
between the new criteria (the shortest distance from the PIS and the longest distance from the NIS). An
interactive TOPSIS algorithm for solving these problems are also proposed. It is based on the decomposition
algorithm of (TL-LSLMOP-SP)rhs of block angular structure via TOPSIS approach, [5]. This algorithm has
few features, (i) it combines both (TL-LSLMOP-SP)rhs of block angular structure and TOPSIS approach to
obtain TOPSIS's compromise solution of the problem, (ii) it can be efficiently coded. (iii) it was found that the
decomposition based method generally met with better results than the traditional simplex-based methods.
Especially, the efficiency of the decomposition-based method increased sharply with the scale of the problem.
An illustrative numerical example is given to demonstrate the proposed TOPSIS approach and the
decomposition algorithm.
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