This document summarizes a new algorithm for solving lexicographic linear goal programming problems. It begins by introducing lexicographic (preemptive) linear goal programming and its general algebraic representation. It then describes the new algorithm, which utilizes an initial table and considers goal constraints as both the objective function and constraints. The algorithm sequentially solves the prioritized deviational variables from the highest to lowest priority level. It initially excludes deviational variable columns not in the basis table but develops them when necessary. The algorithm proceeds through feasibility and optimality testing before selecting an entering variable to maintain the priority structure, ultimately reaching an optimal solution.
Particle Swarm Optimization in the fine-tuning of Fuzzy Software Cost Estimat...Waqas Tariq
Software cost estimation deals with the financial and strategic planning of software projects. Controlling the expensive investment of software development effectively is of paramount importance. The limitation of algorithmic effort prediction models is their inability to cope with uncertainties and imprecision surrounding software projects at the early development stage. More recently, attention has turned to a variety of machine learning methods, and soft computing in particular to predict software development effort. Fuzzy logic is one such technique which can cope with uncertainties. In the present paper, Particle Swarm Optimization Algorithm (PSOA) is presented to fine tune the fuzzy estimate for the development of software projects . The efficacy of the developed models is tested on 10 NASA software projects, 18 NASA projects and COCOMO 81 database on the basis of various criterion for assessment of software cost estimation models. Comparison of all the models is done and it is found that the developed models provide better estimation
This document discusses different ways to classify optimization problems. It describes classifications based on:
1) The existence of constraints - problems are either constrained or unconstrained.
2) The nature of design variables - problems involve either parameter/static optimization or trajectory/dynamic optimization.
3) The physical structure - problems are either optimal control problems or non-optimal control problems.
4) The nature of equations - problems involve linear, nonlinear, geometric, or quadratic equations.
5) The permissible variable values - problems have either integer/discrete or real-valued variables.
6) The variable determinism - problems have either deterministic or stochastic/probabilistic variables.
7) The function
MIXED 0−1 GOAL PROGRAMMING APPROACH TO INTERVAL-VALUED BILEVEL PROGRAMMING PR...cscpconf
This document presents a mixed 0-1 goal programming approach to solve interval-valued fractional bilevel programming problems using a bio-inspired computational algorithm. It formulates the problem using goal programming to minimize regret intervals for target intervals of achieving goals. A genetic algorithm is used to determine target intervals and optimal decisions by distributing decision powers hierarchically. It presents the problem formulation, design of the genetic algorithm using fitter codon selection and two-point crossover, and formulation of interval-valued goals by determining best and worst solutions for objectives of decision makers at different levels using the genetic algorithm.
APPLYING TRANSFORMATION CHARACTERISTICS TO SOLVE THE MULTI OBJECTIVE LINEAR F...ijcsit
For some management programming problems, multiple objectives to be optimized rather than a single objective, and objectives can be expressed with ratio equations such as return/investment, operating
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.
Harmony Search for Multi-objective Optimization - SBRN 2012lucasmpavelski
Slides used in the presentation of the article "Harmony Search for Multi-objective
Optimization" in the 2012 Brazilian Symposium on Neural Networks (SBRN). Link to the article: http://ieeexplore.ieee.org/xpl/articleDetails.jsp?reload=true&arnumber=6374852
Multi Objective Optimization and Pareto Multi Objective Optimization with cas...Aditya Deshpande
This document discusses multi-objective optimization and Pareto multi-objective optimization. It provides examples of multi-objective optimization problems with two or more competing objectives that must be optimized simultaneously. The key concepts covered include Pareto optimal solutions, which define the best trade-offs between objectives and are non-dominated by other solutions. Methods for solving multi-objective optimization problems include traditional approaches that aggregate objectives and Pareto techniques using genetic algorithms and multi-objective evolutionary algorithms.
dynamic programming complete by Mumtaz Ali (03154103173)Mumtaz Ali
The document discusses dynamic programming, including its meaning, definition, uses, techniques, and examples. Dynamic programming refers to breaking large problems down into smaller subproblems, solving each subproblem only once, and storing the results for future use. This avoids recomputing the same subproblems repeatedly. Examples covered include matrix chain multiplication, the Fibonacci sequence, and optimal substructure. The document provides details on formulating and solving dynamic programming problems through recursive definitions and storing results in tables.
An Interactive Decomposition Algorithm for Two-Level Large Scale Linear Multi...IJERA Editor
This paper extended TOPSIS (Technique for Order Preference by Similarity Ideal Solution) method for solving
Two-Level Large Scale Linear Multiobjective Optimization Problems with Stochastic Parameters in the righthand
side of the constraints (TL-LSLMOP-SP)rhs of block angular structure. In order to obtain a compromise (
satisfactory) solution to the (TL-LSLMOP-SP)rhs of block angular structure using the proposed TOPSIS
method, a modified formulas for the distance function from the positive ideal solution (PIS ) and the distance
function from the negative ideal solution (NIS) are proposed and modeled to include all the objective functions
of the two levels. In every level, as the measure of ―Closeness‖ dp-metric is used, a k-dimensional objective
space is reduced to two –dimentional objective space by a first-order compromise procedure. The membership
functions of fuzzy set theory is used to represent the satisfaction level for both criteria. A single-objective
programming problem is obtained by using the max-min operator for the second –order compromise operaion.
A decomposition algorithm for generating a compromise ( satisfactory) solution through TOPSIS approach is
provided where the first level decision maker (FLDM) is asked to specify the relative importance of the
objectives. Finally, an illustrative numerical example is given to clarify the main results developed in the paper.
Particle Swarm Optimization in the fine-tuning of Fuzzy Software Cost Estimat...Waqas Tariq
Software cost estimation deals with the financial and strategic planning of software projects. Controlling the expensive investment of software development effectively is of paramount importance. The limitation of algorithmic effort prediction models is their inability to cope with uncertainties and imprecision surrounding software projects at the early development stage. More recently, attention has turned to a variety of machine learning methods, and soft computing in particular to predict software development effort. Fuzzy logic is one such technique which can cope with uncertainties. In the present paper, Particle Swarm Optimization Algorithm (PSOA) is presented to fine tune the fuzzy estimate for the development of software projects . The efficacy of the developed models is tested on 10 NASA software projects, 18 NASA projects and COCOMO 81 database on the basis of various criterion for assessment of software cost estimation models. Comparison of all the models is done and it is found that the developed models provide better estimation
This document discusses different ways to classify optimization problems. It describes classifications based on:
1) The existence of constraints - problems are either constrained or unconstrained.
2) The nature of design variables - problems involve either parameter/static optimization or trajectory/dynamic optimization.
3) The physical structure - problems are either optimal control problems or non-optimal control problems.
4) The nature of equations - problems involve linear, nonlinear, geometric, or quadratic equations.
5) The permissible variable values - problems have either integer/discrete or real-valued variables.
6) The variable determinism - problems have either deterministic or stochastic/probabilistic variables.
7) The function
MIXED 0−1 GOAL PROGRAMMING APPROACH TO INTERVAL-VALUED BILEVEL PROGRAMMING PR...cscpconf
This document presents a mixed 0-1 goal programming approach to solve interval-valued fractional bilevel programming problems using a bio-inspired computational algorithm. It formulates the problem using goal programming to minimize regret intervals for target intervals of achieving goals. A genetic algorithm is used to determine target intervals and optimal decisions by distributing decision powers hierarchically. It presents the problem formulation, design of the genetic algorithm using fitter codon selection and two-point crossover, and formulation of interval-valued goals by determining best and worst solutions for objectives of decision makers at different levels using the genetic algorithm.
APPLYING TRANSFORMATION CHARACTERISTICS TO SOLVE THE MULTI OBJECTIVE LINEAR F...ijcsit
For some management programming problems, multiple objectives to be optimized rather than a single objective, and objectives can be expressed with ratio equations such as return/investment, operating
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.
Harmony Search for Multi-objective Optimization - SBRN 2012lucasmpavelski
Slides used in the presentation of the article "Harmony Search for Multi-objective
Optimization" in the 2012 Brazilian Symposium on Neural Networks (SBRN). Link to the article: http://ieeexplore.ieee.org/xpl/articleDetails.jsp?reload=true&arnumber=6374852
Multi Objective Optimization and Pareto Multi Objective Optimization with cas...Aditya Deshpande
This document discusses multi-objective optimization and Pareto multi-objective optimization. It provides examples of multi-objective optimization problems with two or more competing objectives that must be optimized simultaneously. The key concepts covered include Pareto optimal solutions, which define the best trade-offs between objectives and are non-dominated by other solutions. Methods for solving multi-objective optimization problems include traditional approaches that aggregate objectives and Pareto techniques using genetic algorithms and multi-objective evolutionary algorithms.
dynamic programming complete by Mumtaz Ali (03154103173)Mumtaz Ali
The document discusses dynamic programming, including its meaning, definition, uses, techniques, and examples. Dynamic programming refers to breaking large problems down into smaller subproblems, solving each subproblem only once, and storing the results for future use. This avoids recomputing the same subproblems repeatedly. Examples covered include matrix chain multiplication, the Fibonacci sequence, and optimal substructure. The document provides details on formulating and solving dynamic programming problems through recursive definitions and storing results in tables.
An Interactive Decomposition Algorithm for Two-Level Large Scale Linear Multi...IJERA Editor
This paper extended TOPSIS (Technique for Order Preference by Similarity Ideal Solution) method for solving
Two-Level Large Scale Linear Multiobjective Optimization Problems with Stochastic Parameters in the righthand
side of the constraints (TL-LSLMOP-SP)rhs of block angular structure. In order to obtain a compromise (
satisfactory) solution to the (TL-LSLMOP-SP)rhs of block angular structure using the proposed TOPSIS
method, a modified formulas for the distance function from the positive ideal solution (PIS ) and the distance
function from the negative ideal solution (NIS) are proposed and modeled to include all the objective functions
of the two levels. In every level, as the measure of ―Closeness‖ dp-metric is used, a k-dimensional objective
space is reduced to two –dimentional objective space by a first-order compromise procedure. The membership
functions of fuzzy set theory is used to represent the satisfaction level for both criteria. A single-objective
programming problem is obtained by using the max-min operator for the second –order compromise operaion.
A decomposition algorithm for generating a compromise ( satisfactory) solution through TOPSIS approach is
provided where the first level decision maker (FLDM) is asked to specify the relative importance of the
objectives. Finally, an illustrative numerical example is given to clarify the main results developed in the paper.
Comparisons of linear goal programming algorithmsAlexander Decker
This document compares different algorithms for solving linear goal programming problems:
1) Lee's modified simplex algorithm from 1972 and Ignizio's sequential algorithm from 1976 are two commonly used algorithms but require many columns and objective function rows, adding to computational time.
2) Orumie and Ebong developed a new algorithm in 2011 utilizing modified simplex procedures that has better computational times than existing algorithms for all problems tested.
3) The document reviews several other goal programming algorithms and finds that Orumie and Ebong's new method provides the best reduction in computational time for solving the problems.
This document provides an overview of optimization techniques. It begins with an introduction to optimization and examples of optimization problems. It then discusses the historical development of optimization methods. The document categorizes optimization problems into convex, concave, and subclasses like linear programming, quadratic programming, and semidefinite programming. It also covers advanced optimization methods like interior point methods. Finally, the document discusses applications of convex optimization techniques in fields such as engineering, finance, and wireless communications.
Some Studies on Multistage Decision Making Under Fuzzy Dynamic ProgrammingWaqas Tariq
Studies has been made in this paper, on multistage decision problem, fuzzy dynamic programming (DP). Fuzzy dynamic programming is a promising tool for dealing with multistage decision making and optimization problems under fuzziness. The cases of deterministic, stochastic, and fuzzy state transitions and of the fixed and specified, implicity given, fuzzy and infinite times, termination times are analyzed.
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.
A NEW ALGORITHM FOR SOLVING FULLY FUZZY BI-LEVEL QUADRATIC PROGRAMMING PROBLEMSorajjournal
This paper is concerned with new method to find the fuzzy optimal solution of fully fuzzy bi-level non-linear (quadratic) programming (FFBLQP) problems where all the coefficients and decision variables of both objective functions and the constraints are triangular fuzzy numbers (TFNs). A new method is based on decomposed the given problem into bi-level problem with three crisp quadratic objective functions and bounded variables constraints. In order to often a fuzzy optimal solution of the FFBLQP problems, the concept of tolerance membership function is used to develop a fuzzy max-min decision model for generating satisfactory fuzzy solution for FFBLQP problems in which the upper-level decision maker (ULDM) specifies his/her objective functions and decisions with possible tolerances which are described by membership functions of fuzzy set theory. Then, the lower-level decision maker (LLDM) uses this preference information for ULDM and solves his/her problem subject to the ULDMs restrictions. Finally, the decomposed method is illustrated by numerical example.
MULTI-OBJECTIVE ENERGY EFFICIENT OPTIMIZATION ALGORITHM FOR COVERAGE CONTROL ...ijcseit
Many studies have been done in the area of Wireless Sensor Networks (WSNs) in recent years. In this kind of networks, some of the key objectives that need to be satisfied are area coverage, number of active sensors and energy consumed by nodes. In this paper, we propose a NSGA-II based multi-objective algorithm for optimizing all of these objectives simultaneously. The efficiency of our algorithm is demonstrated in the simulation results. This efficiency can be shown as finding the optimal balance point among the maximum coverage rate, the least energy consumption, and the minimum number of active nodes while maintaining the connectivity of the network
Penalty Function Method For Solving Fuzzy Nonlinear Programming Problempaperpublications3
Abstract: In this work, the fuzzy nonlinear programming problem (FNLPP) has been developed and their result have also discussed. The numerical solutions of crisp problems and have been compared and the fuzzy solution and its effectiveness have also been presented and discussed. The penalty function method has been developed and mixed with Nelder and Mend’s algorithm of direct optimization problem solutionhave been used together to solve this FNLPP.
Keyword:Fuzzy set theory, fuzzy numbers, decision making, nonlinear programming, Nelder and Mend’s algorithm, penalty function method.
The document summarizes four numerical methods commonly used in geomechanics:
1. The Distinct Element Method (DEM) explicitly models discontinuities.
2. The Discontinuous Deformation Analysis Method (DDA) can consider discontinuities explicitly or implicitly.
3. The Bonded Particle Method (BPM) models geomaterials as an assembly of discrete particles.
4. The Artificial Neural Network Method (ANN) is a data-driven modeling approach not classified as continuum or discontinuum.
The document provides a brief overview of the fundamental algorithms of each method and examples of their applications.
This document provides an overview of dynamic programming. It begins by explaining that dynamic programming is a technique for solving optimization problems by breaking them down into overlapping subproblems and storing the results of solved subproblems in a table to avoid recomputing them. It then provides examples of problems that can be solved using dynamic programming, including Fibonacci numbers, binomial coefficients, shortest paths, and optimal binary search trees. The key aspects of dynamic programming algorithms, including defining subproblems and combining their solutions, are also outlined.
This document presents new certified optimal solutions found by the Charibde algorithm for six difficult benchmark optimization problems. Charibde combines an evolutionary algorithm and interval-based methods in a cooperative framework. It has achieved optimality proofs for five bound-constrained problems and one nonlinearly constrained problem. These problems are highly multimodal and some had not been solved before even with approximate methods. The document also compares Charibde's performance to other state-of-the-art solvers, showing it is highly competitive while providing reliable optimality proofs.
The document introduces dynamic programming as a technique for making optimal decisions over multiple time periods. It discusses how dynamic programming breaks large problems into smaller subproblems and solves each in order, working backwards from the last period. The document provides an example of using dynamic programming to find the shortest route between two cities by breaking the problem into stages and working backwards from the final destination.
A review of automatic differentiationand its efficient implementationssuserfa7e73
Automatic differentiation is a powerful tool for automatically calculating derivatives of mathematical functions and algorithms. It works by expressing the target function as a sequence of elementary operations and then applying the chain rule to differentiate each operation. This can be done using either forward or reverse mode. Forward mode calculates how changes in inputs propagate through the function to influence the outputs, while reverse mode calculates how changes in outputs backpropagate to influence the inputs. Both modes require performing the computation twice - once for the forward pass and once for the derivative pass. Careful implementation is required to make automatic differentiation efficient in terms of speed and memory usage.
This document discusses derivative-free optimization techniques. It begins with an overview of optimization problems, including single-objective, many-objective, convex, and non-convex problems. It then discusses the limitations of non-convex problems and solutions for convex problems using gradient-based techniques. The document focuses on derivative-free optimization, providing examples of techniques like simulated annealing, genetic algorithms, and particle swarm optimization that do not require derivative information. It concludes with references for further reading.
This document discusses various mathematical models used in finance to model stock prices and returns. It introduces random walk models, the lognormal model, general equilibrium theories, the Capital Asset Pricing Model (CAPM), and the Arbitrage Pricing Theory (ATP). The CAPM and ATP are equilibrium asset pricing models based on assumptions like rational investors seeking to maximize returns while minimizing risk.
Dynamic programming is an algorithm design technique for optimization problems that reduces time by increasing space usage. It works by breaking problems down into overlapping subproblems and storing the solutions to subproblems, rather than recomputing them, to build up the optimal solution. The key aspects are identifying the optimal substructure of problems and handling overlapping subproblems in a bottom-up manner using tables. Examples that can be solved with dynamic programming include the knapsack problem, shortest paths, and matrix chain multiplication.
Dynamic programming is an algorithm design technique that solves problems by breaking them down into smaller overlapping subproblems and storing the results of already solved subproblems, rather than recomputing them. It is applicable to problems exhibiting optimal substructure and overlapping subproblems. The key steps are to define the optimal substructure, recursively define the optimal solution value, compute values bottom-up, and optionally reconstruct the optimal solution. Common examples that can be solved with dynamic programming include knapsack, shortest paths, matrix chain multiplication, and longest common subsequence.
PREDICTIVE EVALUATION OF THE STOCK PORTFOLIO PERFORMANCE USING FUZZY CMEANS A...ijfls
The aim of this paper is to investigate the trend of the return of a portfolio formed randomly or for any
specific technique. The approach is made using two techniques fuzzy: fuzzy c-means (FCM) algorithm and
the fuzzy transform, where the rules used at fuzzy transform arise from the application of the FCM
algorithm. The results show that the proposed methodology is able to predict the trend of the return of a
stock portfolio, as well as the tendency of the market index. Real data of the financial market are used from
2004 until 2007.
The document provides an overview of machine learning presented by Tekendra Nath Yogi. It defines machine learning and different types including supervised, unsupervised, and reinforced learning. Supervised learning algorithms like least mean square regression and gradient descent learning are explained. Unsupervised learning examples including clustering are provided. Reinforced learning emphasizes learning from feedback to maximize rewards. Backpropagation and artificial neural networks are also summarized. The document outlines key machine learning concepts in 15 slides.
2011 Investor Day - Planning and Revenue ManagementAndrew Watterson
This document summarizes Hawaiian Airlines' network planning and pricing strategies. It discusses how Hawaiian has expanded its network in recent years, especially in international markets like Tokyo, Seoul, and Sydney. Hawaiian derives a substantial portion of its revenue from international services. The document also covers Hawaiian's strategies for its North America, neighbor island, and international routes. It emphasizes Hawaiian's role as a destination carrier and diversification of origin markets.
2012 Investor Day - Planning and Revenue ManagementAndrew Watterson
The document discusses Hawaiian Airlines' network expansion plans through 2012-2013. It announces new routes from Seattle, Portland, Sacramento, Oakland, San Francisco, San Jose, Las Vegas, Los Angeles, Phoenix, San Diego, Honolulu, Maui, Lihue, Manila, Pago Pago, Sydney, Auckland, Tokyo, Osaka, Seoul, Fukuoka, Taipei, Brisbane, and New York to various domestic and international locations. It explains that the expanded network broadens their reach, deepens connections, and increases potential demand sources.
Predictive analytics for personalized healthcareJohn Cai
This document discusses how predictive analytics can help enable personalized health care through three main points:
1) Integrating diverse data sources like genomics, healthcare records, and insurance claims can provide insights for personalized care, drug development, and comparative effectiveness research.
2) Predictive models built using data from clinical trials can identify subgroups of patients most likely to respond or not respond to treatments early in the treatment course, improving outcomes.
3) Personalized comparative effectiveness research aims to determine which treatments work best for which patient subgroups and disease stages by integrating real-world data and predictive analytics into drug development and clinical decision-making.
Prasad Narasimhan discusses various applications of predictive analytics across different domains including business, marketing, operations, collections, customer segmentation, telecom, sports, social media, and insurance. Predictive analytics uses statistical techniques to analyze current and historical data to predict future events or outcomes. It has various uses such as predicting customer churn, credit risk, response to marketing campaigns, fraud detection, and more. The document provides examples of how predictive analytics is applied in areas like customer retention, cross-sell, collections, credit risk management, and churn prediction in telecom.
Comparisons of linear goal programming algorithmsAlexander Decker
This document compares different algorithms for solving linear goal programming problems:
1) Lee's modified simplex algorithm from 1972 and Ignizio's sequential algorithm from 1976 are two commonly used algorithms but require many columns and objective function rows, adding to computational time.
2) Orumie and Ebong developed a new algorithm in 2011 utilizing modified simplex procedures that has better computational times than existing algorithms for all problems tested.
3) The document reviews several other goal programming algorithms and finds that Orumie and Ebong's new method provides the best reduction in computational time for solving the problems.
This document provides an overview of optimization techniques. It begins with an introduction to optimization and examples of optimization problems. It then discusses the historical development of optimization methods. The document categorizes optimization problems into convex, concave, and subclasses like linear programming, quadratic programming, and semidefinite programming. It also covers advanced optimization methods like interior point methods. Finally, the document discusses applications of convex optimization techniques in fields such as engineering, finance, and wireless communications.
Some Studies on Multistage Decision Making Under Fuzzy Dynamic ProgrammingWaqas Tariq
Studies has been made in this paper, on multistage decision problem, fuzzy dynamic programming (DP). Fuzzy dynamic programming is a promising tool for dealing with multistage decision making and optimization problems under fuzziness. The cases of deterministic, stochastic, and fuzzy state transitions and of the fixed and specified, implicity given, fuzzy and infinite times, termination times are analyzed.
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.
A NEW ALGORITHM FOR SOLVING FULLY FUZZY BI-LEVEL QUADRATIC PROGRAMMING PROBLEMSorajjournal
This paper is concerned with new method to find the fuzzy optimal solution of fully fuzzy bi-level non-linear (quadratic) programming (FFBLQP) problems where all the coefficients and decision variables of both objective functions and the constraints are triangular fuzzy numbers (TFNs). A new method is based on decomposed the given problem into bi-level problem with three crisp quadratic objective functions and bounded variables constraints. In order to often a fuzzy optimal solution of the FFBLQP problems, the concept of tolerance membership function is used to develop a fuzzy max-min decision model for generating satisfactory fuzzy solution for FFBLQP problems in which the upper-level decision maker (ULDM) specifies his/her objective functions and decisions with possible tolerances which are described by membership functions of fuzzy set theory. Then, the lower-level decision maker (LLDM) uses this preference information for ULDM and solves his/her problem subject to the ULDMs restrictions. Finally, the decomposed method is illustrated by numerical example.
MULTI-OBJECTIVE ENERGY EFFICIENT OPTIMIZATION ALGORITHM FOR COVERAGE CONTROL ...ijcseit
Many studies have been done in the area of Wireless Sensor Networks (WSNs) in recent years. In this kind of networks, some of the key objectives that need to be satisfied are area coverage, number of active sensors and energy consumed by nodes. In this paper, we propose a NSGA-II based multi-objective algorithm for optimizing all of these objectives simultaneously. The efficiency of our algorithm is demonstrated in the simulation results. This efficiency can be shown as finding the optimal balance point among the maximum coverage rate, the least energy consumption, and the minimum number of active nodes while maintaining the connectivity of the network
Penalty Function Method For Solving Fuzzy Nonlinear Programming Problempaperpublications3
Abstract: In this work, the fuzzy nonlinear programming problem (FNLPP) has been developed and their result have also discussed. The numerical solutions of crisp problems and have been compared and the fuzzy solution and its effectiveness have also been presented and discussed. The penalty function method has been developed and mixed with Nelder and Mend’s algorithm of direct optimization problem solutionhave been used together to solve this FNLPP.
Keyword:Fuzzy set theory, fuzzy numbers, decision making, nonlinear programming, Nelder and Mend’s algorithm, penalty function method.
The document summarizes four numerical methods commonly used in geomechanics:
1. The Distinct Element Method (DEM) explicitly models discontinuities.
2. The Discontinuous Deformation Analysis Method (DDA) can consider discontinuities explicitly or implicitly.
3. The Bonded Particle Method (BPM) models geomaterials as an assembly of discrete particles.
4. The Artificial Neural Network Method (ANN) is a data-driven modeling approach not classified as continuum or discontinuum.
The document provides a brief overview of the fundamental algorithms of each method and examples of their applications.
This document provides an overview of dynamic programming. It begins by explaining that dynamic programming is a technique for solving optimization problems by breaking them down into overlapping subproblems and storing the results of solved subproblems in a table to avoid recomputing them. It then provides examples of problems that can be solved using dynamic programming, including Fibonacci numbers, binomial coefficients, shortest paths, and optimal binary search trees. The key aspects of dynamic programming algorithms, including defining subproblems and combining their solutions, are also outlined.
This document presents new certified optimal solutions found by the Charibde algorithm for six difficult benchmark optimization problems. Charibde combines an evolutionary algorithm and interval-based methods in a cooperative framework. It has achieved optimality proofs for five bound-constrained problems and one nonlinearly constrained problem. These problems are highly multimodal and some had not been solved before even with approximate methods. The document also compares Charibde's performance to other state-of-the-art solvers, showing it is highly competitive while providing reliable optimality proofs.
The document introduces dynamic programming as a technique for making optimal decisions over multiple time periods. It discusses how dynamic programming breaks large problems into smaller subproblems and solves each in order, working backwards from the last period. The document provides an example of using dynamic programming to find the shortest route between two cities by breaking the problem into stages and working backwards from the final destination.
A review of automatic differentiationand its efficient implementationssuserfa7e73
Automatic differentiation is a powerful tool for automatically calculating derivatives of mathematical functions and algorithms. It works by expressing the target function as a sequence of elementary operations and then applying the chain rule to differentiate each operation. This can be done using either forward or reverse mode. Forward mode calculates how changes in inputs propagate through the function to influence the outputs, while reverse mode calculates how changes in outputs backpropagate to influence the inputs. Both modes require performing the computation twice - once for the forward pass and once for the derivative pass. Careful implementation is required to make automatic differentiation efficient in terms of speed and memory usage.
This document discusses derivative-free optimization techniques. It begins with an overview of optimization problems, including single-objective, many-objective, convex, and non-convex problems. It then discusses the limitations of non-convex problems and solutions for convex problems using gradient-based techniques. The document focuses on derivative-free optimization, providing examples of techniques like simulated annealing, genetic algorithms, and particle swarm optimization that do not require derivative information. It concludes with references for further reading.
This document discusses various mathematical models used in finance to model stock prices and returns. It introduces random walk models, the lognormal model, general equilibrium theories, the Capital Asset Pricing Model (CAPM), and the Arbitrage Pricing Theory (ATP). The CAPM and ATP are equilibrium asset pricing models based on assumptions like rational investors seeking to maximize returns while minimizing risk.
Dynamic programming is an algorithm design technique for optimization problems that reduces time by increasing space usage. It works by breaking problems down into overlapping subproblems and storing the solutions to subproblems, rather than recomputing them, to build up the optimal solution. The key aspects are identifying the optimal substructure of problems and handling overlapping subproblems in a bottom-up manner using tables. Examples that can be solved with dynamic programming include the knapsack problem, shortest paths, and matrix chain multiplication.
Dynamic programming is an algorithm design technique that solves problems by breaking them down into smaller overlapping subproblems and storing the results of already solved subproblems, rather than recomputing them. It is applicable to problems exhibiting optimal substructure and overlapping subproblems. The key steps are to define the optimal substructure, recursively define the optimal solution value, compute values bottom-up, and optionally reconstruct the optimal solution. Common examples that can be solved with dynamic programming include knapsack, shortest paths, matrix chain multiplication, and longest common subsequence.
PREDICTIVE EVALUATION OF THE STOCK PORTFOLIO PERFORMANCE USING FUZZY CMEANS A...ijfls
The aim of this paper is to investigate the trend of the return of a portfolio formed randomly or for any
specific technique. The approach is made using two techniques fuzzy: fuzzy c-means (FCM) algorithm and
the fuzzy transform, where the rules used at fuzzy transform arise from the application of the FCM
algorithm. The results show that the proposed methodology is able to predict the trend of the return of a
stock portfolio, as well as the tendency of the market index. Real data of the financial market are used from
2004 until 2007.
The document provides an overview of machine learning presented by Tekendra Nath Yogi. It defines machine learning and different types including supervised, unsupervised, and reinforced learning. Supervised learning algorithms like least mean square regression and gradient descent learning are explained. Unsupervised learning examples including clustering are provided. Reinforced learning emphasizes learning from feedback to maximize rewards. Backpropagation and artificial neural networks are also summarized. The document outlines key machine learning concepts in 15 slides.
2011 Investor Day - Planning and Revenue ManagementAndrew Watterson
This document summarizes Hawaiian Airlines' network planning and pricing strategies. It discusses how Hawaiian has expanded its network in recent years, especially in international markets like Tokyo, Seoul, and Sydney. Hawaiian derives a substantial portion of its revenue from international services. The document also covers Hawaiian's strategies for its North America, neighbor island, and international routes. It emphasizes Hawaiian's role as a destination carrier and diversification of origin markets.
2012 Investor Day - Planning and Revenue ManagementAndrew Watterson
The document discusses Hawaiian Airlines' network expansion plans through 2012-2013. It announces new routes from Seattle, Portland, Sacramento, Oakland, San Francisco, San Jose, Las Vegas, Los Angeles, Phoenix, San Diego, Honolulu, Maui, Lihue, Manila, Pago Pago, Sydney, Auckland, Tokyo, Osaka, Seoul, Fukuoka, Taipei, Brisbane, and New York to various domestic and international locations. It explains that the expanded network broadens their reach, deepens connections, and increases potential demand sources.
Predictive analytics for personalized healthcareJohn Cai
This document discusses how predictive analytics can help enable personalized health care through three main points:
1) Integrating diverse data sources like genomics, healthcare records, and insurance claims can provide insights for personalized care, drug development, and comparative effectiveness research.
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Research on Lexicographic Linear Goal Programming Problem Based on LINGO and ...paperpublications3
Abstract: Lexicographic Linear Goal programming within a pre-emptive priority structure including Column-dropping Rule has been one of the useful techniques considered in solving multiple objective problems. The basic ideas to solve goal programming are transforming goal programming into single-objective linear programming. An optimal solution is attained when all the goals are reached as close as possible to their aspiration level, while satisfying a set of constraints. One of the Goal Programming algorithm – the Lexicographic method including Column-dropping Rule and the method of LINGO software are discussed in this paper. Finally goal programming model are applied to the actual management decisions, multi-objective programming model are established and used LINGO software and Column-dropping Rule to achieve satisfied solution.Keywords: Goal programming, Lexicographic Goal programming, multi-objective, LINGO software, Column-dropping Rule.
Title: Research on Lexicographic Linear Goal Programming Problem Based on LINGO and Column-Dropping Rule
Author: N. R. Neelavathi
ISSN 2350-1022
International Journal of Recent Research in Mathematics Computer Science and Information Technology
Paper Publications
Optimization of Patrol Manpower Allocation Using Goal Programming Approach -A...IJERA Editor
One of the most difficult tasks of the patrol administrators is allocation of manpower; i.e. determining the. most
effective level of operational manpower for patrol tasks. Typically, administrators resolve the allocation
problem by relying on prior statistical data and by employing subjective analysis. In general, only limited
systematic analyses have been applied to the problem. This thesis presents a non linear goal programming model
for allocating patrolmen to road segments within a patrol region. The model is demonstrated via a case example
of the East section of Visakhapatnam. The results of the model are valuable to the patrol administrator for
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Duality Theory in Multi Objective Linear Programming Problemstheijes
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.
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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.
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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Nonlinear Programming: Theories and Algorithms of Some Unconstrained Optimiza...Dr. Amarjeet Singh
Nonlinear programming problem (NPP) had become an important branch of operations research, and it was the mathematical programming with the objective function or constraints being nonlinear functions. There were a variety of traditional methods to solve nonlinear programming problems such as bisection method, gradient projection method, the penalty function method, feasible direction method, the multiplier method. But these methods had their specific scope and limitations, the objective function and constraint conditions generally had continuous and differentiable request. The traditional optimization methods were difficult to adopt as the optimized object being more complicated. However, in this paper, mathematical programming techniques that are commonly used to extremize nonlinear functions of single and multiple (n) design variables subject to no constraints are been used to overcome the above challenge. Although most structural optimization problems involve constraints that bound the design space, study of the methods of unconstrained optimization is important for several reasons. Steepest Descent and Newton’s methods are employed in this paper to solve an optimization problem.
This document discusses resource optimization and linear programming. It defines optimization as finding the best solution to a problem given constraints. Linear programming is introduced as a mathematical technique to optimize allocation of scarce resources. The key components of a linear programming model are described as decision variables, an objective function, and constraints. Graphical and algebraic methods for solving linear programming problems are also summarized.
Goal programming is a mathematical optimization method similar to linear programming. It involves minimizing the deviation from goals by using deviational variables to represent under-achievement and over-achievement of goals. The basic steps in formulating a goal programming model are to determine decision variables and deviational variables, specify goals, assign preemptive priorities and weights, and state the objective function to minimize deviations.
This document provides an overview of a survey of multi-objective evolutionary algorithms for data mining tasks. It discusses key concepts in multi-objective optimization and evolutionary algorithms. It also reviews common data mining tasks like feature selection, classification, clustering, and association rule mining that are often formulated as multi-objective problems and solved using multi-objective evolutionary algorithms. The survey focuses on reviewing applications of multi-objective evolutionary algorithms for feature selection and classification in part 1, and applications for clustering, association rule mining and other tasks in part 2.
1) The document discusses definitions and characteristics of operations research (OR). It provides definitions of OR from several leaders and pioneers in the field that describe OR as applying scientific methods to optimize complex systems.
2) Key characteristics of OR mentioned are that it takes a team approach using quantitative techniques, aims to help executives make optimal decisions, relies on mathematical models, and uses computers to analyze models.
3) Limitations of OR discussed include that it is time-consuming, practitioners may lack industrial experience, and solutions can be difficult to communicate to non-technical executives. Linear programming is introduced as a prominent OR technique.
Application of Linear Programming to Profit Maximization (A Case Study of.pdfBrittany Allen
This document summarizes a research article that applies linear programming to maximize profit for Johnsons Nigeria Limited's bakery division. The article establishes a linear programming model to determine the optimal production mix of four bread sizes to maximize total profit given production capacity constraints. An initial simplex tableau is constructed and the optimal solution is obtained through iterative pivoting. The optimal solution shows producing three units of large bread yields the maximum profit of 150 naira. Sensitivity analysis confirms the optimal solution remains valid if the large bread's profit coefficient remains between 50-30 naira.
Application of linear programming technique for staff training of register se...Enamul Islam
This study aims to minimize training costs for staff at Patuakhali Science and Technology University using linear programming. It identifies two decision variables (permanent and non-permanent staff to be trained) and develops constraints based on time available and staff in different departments. The linear programming model is solved to find the optimal solution: 1 permanent staff should be sent for 5 days of training among departments to minimize costs. The research suggests this approach can help determine optimal staffing levels for future training programs.
In this study, we discussed a fuzzy programming approach to bi-level linear programming problems and their application. Bi-level linear programming is characterized as mathematical programming to solve decentralized problems with two decision-makers in the hierarchal organization. They become more important for the contemporary decentralized organization where each unit seeks to optimize its own objective. In addition to this, we have considered Bi-Level Linear Programming (BLPP) and applied the Fuzzy Mathematical Programming (FMP) approach to get the solution of the system. We have suggested the FMP method for the minimization of the objectives in terms of the linear membership functions. FMP is a supervised search procedure (supervised by the upper Decision Maker (DM)). The upper-level decision-maker provides the preferred values of decision variables under his control (to enable the lower level DM to search for his optimum in a wider feasible space) and the bounds of his objective function (to direct the lower level DM to search for his solutions in the right direction).
This document describes a senior project to improve the pivot point selection algorithm in the simplex method for linear programming. It provides background on the history and development of the simplex method by George Dantzig in the 1940s. It then explains the current standard process for performing the simplex method, including setting up the linear program, identifying the pivot column and element, and performing the pivot steps. The project aims to identify a flaw in the current pivot rule and develop a new algorithm to increase the efficiency and effectiveness of the simplex method.
For some management programming problems, multiple objectives to be optimized rather than a single
objective, and objectives can be expressed with ratio equations such as return/investment, operating
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.
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.
Optimization of Mechanical Design Problems Using Improved Differential Evolut...IDES Editor
Differential Evolution (DE) is a novel evolutionary
approach capable of handling non-differentiable, non-linear
and multi-modal objective functions. DE has been consistently
ranked as one of the best search algorithm for solving global
optimization problems in several case studies. This paper
presents an Improved Constraint Differential Evolution
(ICDE) algorithm for solving constrained optimization
problems. The proposed ICDE algorithm differs from
unconstrained DE algorithm only in the place of initialization,
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algorithm in terms of final objective function value, number
of function evaluations and convergence time.
Optimization of Mechanical Design Problems Using Improved Differential Evolut...IDES Editor
Differential Evolution (DE) is a novel evolutionary
approach capable of handling non-differentiable, non-linear
and multi-modal objective functions. DE has been consistently
ranked as one of the best search algorithm for solving global
optimization problems in several case studies. This paper
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selection of particles to the next generation and sorting the
final results. Also we implemented the new idea to five versions
of DE algorithm. The performance of ICDE algorithm is
validated on four mechanical engineering problems. The
experimental results show that the performance of ICDE
algorithm in terms of final objective function value, number
of function evaluations and convergence time.
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An efficient method of solving lexicographic linear goal programming problem
1. Journal of Natural Sciences Research www.iiste.org
ISSN 2224-3186 (Paper) ISSN 2225-0921 (Online)
Vol.4, No.20, 2014
34
An Efficient Method of Solving Lexicographic Linear Goal
Programming Problem
U.C.ORUMIE D.W EBONG
Department of Mathematics/Statistics, University of Port Harcourt, Nigeria
amakaorumie@yahoo.com & daniel.ebong@uniport.edu.ng
Abstract
Lexicographic Linear Goal programming within a pre-emptive priority structure has been one of the most widely
used techniques considered in solving multiple objective problems. In the past several years, the modified
simplex algorithm has been shown to be widely used and very accurate in computational formulation. Orumie
and Ebong recently developed a generalized linear goal programming algorithm that is efficient. A new approach
for solving lexicographic linear Goal programming problem is developed, together with an illustrative example.
The method is efficient in reaching solution.
Keywords: Lexicographic Goal programming, multi objective, simplex method.
1.INTRODUCTION
Multiple Objective optimizations technique is a type of optimization that handles problems with a set of
objectives to be maximized or minimized. This problem has at least two conflicting criteria/objectives. They
cannot reach their optimal values simultaneously or satisfaction of one will result in damaging or degrading the
full satisfaction of the other(s). There is no single optimal solution in this type of optimization; rather an
interaction among different objectives gives rise to a set of compromised solutions, largely known as the trade-
off or non dominated or non inferior or Pareto-optimal solutions. Multiple Objective optimization consists of
different problem situations, such as multiple objective linear programming (MOLP), Multiple Objective Integer
Linear Programming (MOILP), and Nonlinear Multiple Objective Optimization (NMOO).
Wang et.al (1980) and Evans (1984) categorised multiple objective optimization into three as shown in
Aouni and Kettani (2001). The categories are as follows;
• A priori techniques in which all decision maker preferences are specified before the solution
process.
• Interactive techniques in which the decision maker preferences are elicited during the solution
technique, mainly in response to their opinion of solutions generated to that point.
• A posteriori techniques where the solution process takes place first and the decision maker
preferences are then elicited from the generated set of solution.
Goal programming is one of the posteriori techniques, and most commonly method for solving multiple
objective decision problems. (See Sunar and Kahraman (2001)). Goal programming popularity from amongst
the distance-based MCDM techniques as described by Tamize and Jones (2010) demonstrates its continuous
growth in recent years as represented below;
Goal Programming as a Multi-criteria Decision Analysis Tool
Source: Tamiz M, &D. F Jones (2010) Practical Goal Programming. International Series in Operations Research
& Management Science. Springer New York http://www.springer.com/series/6161.
2. Journal of Natural Sciences Research www.iiste.org
ISSN 2224-3186 (Paper) ISSN 2225-0921 (Online)
Vol.4, No.20, 2014
35
Goal programming is used in optimization of multiple objective goals by minimizing the deviation for each of
the objectives from the desired target. In fact the basic concept of goal programming is whether goals are
attainable or not, an objective may be stated in which optimization gives a result which come as close as possible
to the desired goals. Schniederjans and Kwaks (1982) referred to the most commonly applied type of goal
programming as "pre-emptive weighted priority goal programming" and a generalized model for this type of
programming is as follows:
minimize:
Z = ∑ +−
+
m
i
iiii ddpw )( (1.1)
s.t
∑ ==−+ +−
n
j
iiiijij mibddxa ),,...,2,1( (1.2)
0,, ≥+−
iiij ddx , 0>iw , )...,3,2,1:,...,2,1( njmi == (1.3)
In many situations, however, a decision maker may rank his or her goals from the most important (goal
1) to least important (goal m). This is called Preemptive goal programming and its procedure starts by
concentrating on meeting the most important goal as closely as possible, before proceeding to the next higher
goal, and so on to the least goal i.e. the objective functions are prioritized such that attainment of first goal is far
more important than attainment of second goal which is far more important than attainment of third goal, etc,
such that lower order goals are only achieved as long as they do not degrade the solution attained by higher
priority goal. When this is the case, pre emptive goal programming may prove to be a useful tool. The objective
function coefficient for the variable representing goal i will be pi. In problem with more than one goal, the
decision maker must rank the goals in order of importance.
However, a major limitation in applying GP as recorded in Schniederjans, M. J. & N. K. Kwak (1982)
has been the lack of an algorithm capable of reaching optimum solution in a reasonable time. Hwang and Yoo
(1981) cited a number of limitations found in existing algorithms. The purpose of this research is to present an
efficient method for solving lexicographic linear goal programming problems.
The paper is organized as follows: Introduction to Preemptive Linear Goal Programming is provided in
section two. The new algorithm for lexicographic goal programming and the solution description are the focus
of Section three and four respectively, whereas the summary and conclusion will be presented in section five and
six respectively.
2.LEXICOGRAPHIC (PREEMPTIVE) LINEAR GOAL PROGRAMMING (LLGP)
The basic purpose of LLGP is to simultaneously satisfy several goals relevant to the decision-making situation.
To this end, a set of attributes to be considered in the problem situation is established. Then, for each attribute, a
target value (i.e., appraisal level) is determined. Next, the deviation variables are introduced. These deviation
variables may be negative or positive (represented by di
-
and di
+
respectively). The negative deviation variable,
di
-
, represents the quantification of the under-achievement of the ith goal. Similarly, di
+
represents the
quantification of the over-achievement of the ith goal. Finally for each attribute, the desire to overachieve
(minimize di
-
) or underachieve (minimize di
+
), or satisfy the target value exactly (minimize di
-
+ di
+
) is
articulated. And finally, the deviational variables prioritized in order of importance.
The general algebraic representation of lexicographic linear goal programming is given as
( ) ( ) ( )+−+−+−
= kkk ddpddpddpzlexi ,...,,,,,(min 222111 (2.1)
S.t
∑ ==−+ +−
n
j
iiiijij mibddxa ),,...,2,1( (2.2)
0,, ≥+−
iiij ddx , 0>iw , )...,3,2,1:,...,2,1( njmi == (2.3)
The model has k priorities, m objectives and n decision variables. pi is the ordered ith
priority levels of
the deviational variables in the achievement function. The priority structure for the model is established by
assigning each goal or a set of goals to a priority level, thereby ranking the goals lexicographically in order of
importance to the decision maker. This is known as lexicographic GP (LGP), as introduced by Ijiri (1965), and
developed by Lee (1972) , and Ignizio (1976). This was modified by [11], [12], [13], [14], and [15].Priorities do
not take numerical value, but simply a suitable way of indicating that one goal is more important than another.
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Vol.4, No.20, 2014
36
3.THE NEW ALGORITHM FOR LEXICOGRAPHIC GOAL PROGRAMMING (LGP)
The procedure utilizes Orumie and Ebong (2011) initial table with modifications as shown below, together with
the inclusion of hard constraints. The procedure considers goal constraints as both the objective function and
constraints. The objective function becomes the prioritized deviational variables and solves sequentially starting
from the highest priority level to the lowest. It starts by not including the deviational variable columns that did
not appear in the basis on the table, but developed when necessary since di
+
= - di
-
.
TABLE 1.1 INNITIAL TABLE OF THE NEW ALGORITM
Variable in basis with pi . CB X1 X2 … Xn S d1
(v)
d2
(v)
. . .dt
(v)
Solution value bi. R.H.S
a11 a12. … a1n s1 c11
(v)
c11
(v)
. . . c1t
(v)
b1
a21 a22 … a2n s2 c21
(v)
c22
(v)
. . .c2t
(v)
b2
am1 am2 … amn sm cm1
(v)
cm2
(v)
. . . cmt
(v)
bm
Consider the Preemptive Linear Goal programming model. The formulation for n variables, m goal
constraints, t deviational variables in z and L preemptive priority factors is defined below.
)1.3(},...,2,1{),(min mikforddpzlex t
l
k
iik tt
⊂⊂= ∑ +−
(3.1)
such that
iii
m
i
jij bddxa =−+
+−
∑
(3.2)
i
n
j
jij bxa ≤∑
(3.3)
0,, ≥+−
iiij ddx (3.4)
for )...,3,2,1:,...,2,1( njmi ==
where pk= kth
priority factor k= 1,2, . . .L,
),( −+
kk ii dd are set of deviational variables in z with the priorities attached to them.
0,, ≥
−+
iij ddx ,,...1,,...,1 njmi ==
Let pk be the kth
priority level, then; the algorithm;
Step 1. Initialization:
Set k ←1 i.e set the first priority k=1
Step 2. Feasibility:
If ib = 0 for i=1,2,. .., m, go to Step 8. i.e if all the rhs=0 {solution optimal}
Set ib ← ib for I =1,2,..., m. i.e take absolute value of the rhs {ensure feasibility}
Step 3. Optimality test:
If hjg ≤ 0 for all j columnpivot≠ , kih∈ go to Step 7.
{all coefficients of priority row hnon positive ,so is satisfied}.
Step 4. Entering variable:
Entering variable is the variable with highest positive coefficient in the row },2,1{,. Lhgh ∈
for the
+−
tt iik ddp ,( ) rows of the objective function which does not violate priority condition.
i.e if },2,1{,. Lhgh ∈ is the highest coefficient, but has been previously satisfied or more
important than the leaving variable under consideration, then consider the next higher value on
the same row, otherwise go to step 7. (The priority attached to the entering variable should be
placed alongside with it into the basis).
In case of ties { shjhj gg .,.,.1
}, then the entering variable is the variable for which
4. Journal of Natural Sciences Research www.iiste.org
ISSN 2224-3186 (Paper) ISSN 2225-0921 (Online)
Vol.4, No.20, 2014
37
> 0:min q
q
hj
hj
i
h
g
g
b is maximum sqjq ,...,1: =
{ties in the priority rows }
Step 5. Leaving variable:
If 0y is the column corresponding to the entering variable in Step 4, then the leaving variable
is the basic variable with minimum
=> mig
g
b
y
y
i ,...2,1,0: 0
0
.
.
{ 0.yg is the pivot
column}.
In case of ties, the variable with the smallest right hand side leaves the basis.
Step 6. Interchange basic variable and non basic variable:
Perform Gauss Jordan row operations to update the table. If is still in the basis (CB), go to
Step 3.
Step 7. Increment process:
Set k ←k +1.If k ≤ L, go to Step 3. Satisfied priority will not reenter for the lesser one to
leave, instead variable with the next higher coefficient enters the basis.
Step 8. Solution is optimal when:
i.) The coefficient of the priority rows are all negative or zero
ii.) The right hand sides of the priority rows are all zero
iii.) The priority rows are satisfied.
The optimal solution is the value ),( −+
iik ddp in the objective function as appeared in the last iteration table.
i.e. The value of the achievement function becomes a vector of priority levels in the optimal values in the final
tableau.
Note : Just as in the method of artificial variables, a variable of higher or equal priority that has been
satisfied should not be allowed to re-enter the table. In this case the next higher coefficient of
hjg will be considered.
4.SOLUTION OF ILLUSTRATIVE EXAMPLE
Given an example i below, the solution procedure is thus;
(i)
++−−
+++= 44342211min dpdPdPdpz
s.t
7
−
++ 121 6 dxx -
+
1d =30
2x1 +3x2+
−
2d –
+
2d =12
6x1 + 5x2 +d3
—
d3
+
=30
x2 +d4
- —
d4
+
=7
0,0,,0 =•≥≥ −+−+
iiiii ddddx
TABLE 1: INNITIAL TABLE FOR PROBLEM (I)
x1 enter
d1
--
leaves
The above table (1) is the initial table of problem (i). Column one represents the variables in z with priorities
assigned to each of them which forms the bases. Columns two and three represent the coefficients of the decision
x1 x2 d1
-
d2
-
d4
-
d3
+
RHS
p
1 d1
-
7 1 0 0 0 30
p
2d2
-
2 0 1 0 0 12
p
4d3
+
6 0 0 0 -1 30
p
3d4
-
0 0 0 1 0 7
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38
variables (aij) in the goal constraints equation. Columns four to seven represent coefficient of deviational
variables (cit
v
) in the goal constraint equations that appeared in the achievement function. Column eight is the
right hand side values of the constraints equations. Applying the algorithm,
step 1. set k=1.
Step 2. ∃ ib = 0 for i=1,2,. . ., 4, {So solution feasible.}
Step 3. ∃ jg1 >0 for some j. {So solution not optimal}. Since there is positive
coefficient in the priority row, then the solution is not optimal.
Step4. Max {gi}=max{7, 6, 1, 0,0,0,}=7 at g11 i.e. x1=7 enters the bases since
it is the highest in the row.
step 5 Min{ }0: 11
>ig
b
gi
i
= min{30/7, 6, 5,}= 30/7 at [ ]11
1
g
b
. So d1
-
leaves the
bases. i.e the minimum ratio of the right hand side to the entrying column.
Step6. Perform the normal gauss Jordan’s simplex operation to update the new
Tableau (see Tableau 2) and check if 1p is still in the basis (CB) to test for
optimality.
TABLE 2: 1st
ITERATION FOR PROBLEM(i)
x2 enter
d2
-
leaves
Table (2), shows that 1p is satisfied since it is no longer in the bases.
Step7. Set k=2. Since 2 < L=4, go to step 3.
Step3. ∃ jg2 >0 for some j. i.e 2nd
priority row. {So solution not optimal}
Step4. }max{ 2 jg =max {0,9/7,-2/7,1,0,0}=9/7 at g22. So, x2 enters the basis.
Step5. Min 0: 22
>ig
b
gi
i
=min {5, 8/3,7}= 8/3 at
22
2
g
b
. d2
-
leave since it has the smallest minimum
ratio.
Step 6. Perform the same operation to update the new tableau Table 3 and check if 2p is still in the
basis (CB) to test for optimality.
Table 3: 2ND
ITERATION FOR PROBLEM (i)
d2
+
enter
x1 Leaves
Table (3), shows that 2p is satisfied since it is no longer in the bases.
Step7. Set k=3. Since 3 < L=4, go to step 3.
Step3. ∃ jg4 >0 for some j. i.e 3th
priority row. {So solution not optimal}
Step4. }{max 4 jg =max {0,0, 2/9,-7/9,1,0}= 9
7−
at g44. So, d2
+
enters the basis.
Step5. Min
4i
i
g
b
: 04 >ig =min {3, 39/7}= 3 at
14
1
g
b
. So x1 leave since it has the smallest minimum
ratio.
x1 x2 d1
-
d2
-
d4
-
d3
+
RHS
x1 1 6/7 1/7 0 0 0 30/7
p
2d2
-
0 9/7 -2/7 1 0 0 24/7
p
4d3
-
0 -1/7 -6/7 0 0 -1 30/7
p
3d4
-
0 1 0 0 1 0 7
x1 x2 d1
-
d2
-
d4
-
d3
+
RHS
x1 1 0 1/3 -2/3 0 0 2
x2 0 1 -2/9 7/9 0 0 8/3
p
4d3
+
0 0 -8/9 1/9 0 -1 14/3
p
3d4
-
0 0 2/9 -7/9 1 0 13/3
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Vol.4, No.20, 2014
39
Step 6. Perform the same operation to update the new tableau Table 4 and check if 3p is still in the
basis (CB) to test for optimality.
Table 4: 3RD
ITERATION FOR PROBLEM (i)
d1
+
enter
d3
+
leaves
Table (4), shows that 3p is not
satisfied since it is still in the bases.
Step3. ∃ jg4 >0 for some j. i.e 3th
priority row. {So solution not optimal}
Step4. }{max 4 jg =max {-7/6,0, 1/6, 0, 1,0, 0}=1/6 at g43. But , d1
-
cannot re-enter
for lower priority to leave the basis. Therefore p3 cannot be satisfied further,
so go to step 7.
Step7. Set k=4 and go to step 3.
Step3. ∃ jg3 >0 for some j. i.e 4th
priority row. {So solution not optimal}
Step4. }{max 3 jg =max {1/6,0, -5/6,0,0, 0,-1}=5/6 at at 33g . So, d1
+
enters the basis.
Step5. Min
3i
i
g
b
: 0: 3 >ig =min {6}= 6 at
33
3
g
b
. So d3
+
leave..
Step 6. Perform the same operation to update the new tableau Table 5 and check if 4p is still in the
basis (CB) to test for optimality.
Table 5: 4TH
ITERATION FOR PROBLEM(i)
d3
+
reenter
d4
-
leaves
Table (5), shows that 4p is satisfied since it has left the bases. But p3 can be improved.
Step3. ∃ jg4 >0 for some j. i.e 3th
priority row. {So solution not optimal}
Step4. }{max 4 jg =max {-6/5,0, 0, 0, 1, 0, 0,1/5}=1/5 at g48. so , d3
+
re-enter
for higher priority to leave the basis.
Step5. Min 0: 88
>ig
b
gi
i
=min {5}=5 at g48. So d4
-
leave.
Step 6. Perform the same operation to update the new tableau Table 6 and check if 4p is still in the
basis (CB) to test for optimality.
Table 6: Final iteration
Z=5
x1 x2 d1
-
d2
-
d4
-
d2
+
d3
+
RHS
d2
+
3/2 0 1/2 -1 0 1 0 3
x2 7/6 1 1/6 0 0 0 0 5
p
4d3
+
1/6 0 -5/6 0 0 0 -1 5
p
3d4
-
-7/6 0 1/6 0 1 0 0 2
x1 x2 d1
-
d2
-
d4
-
d1
+
d2
+
d3
+
RHS
d2
+
8/5 0 0 -1 0 0 1 -3/2 6
x2 6/5 1 0 0 0 0 0 -1/5 6
d1
+
1/5 0 -1 0 0 1 0 -6/5 6
d4
-
-6/5 0 0 0 1 0 0 1/5 1
x1 x2 d1
-
d2
-
d4
-
d1
+
d2
+
d3
+
RHS
d2
+
37 0 0 -1 0 1 0 27/2
x2 0 1 0 0 1 0 0 0 7
d1
+
-7 0 -1 0 6 1 0 0 12
d3
+
-6 0 0 0 5 0 0 1 5
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40
The above is optimum since they cannot be achieved further.
5.RESULT
Problems from standard published papers of various sizes and complexities were solved to test the efficiency of
the new lexicographic algorithm. The models varied widely in the number of constraints, decision variables,
deviational variables and pre-emptive priority levels as shown in the table below.
TABLE 7. RESULT SUMMARY OF THE SOLVED PROBLEMS USING THE PROPOSED METHOD
source No of constraints No of decision
variables
No of deviational
variables
No of preemptive
priorities
Igizio(1982) 5 4 10 3
Crowder & Sposito
(1987)
4 2 8 3
Cohon (1978) 4 2 6 2
Hana (2006) 4 2 8 4
Gupta(2009) 4 2 8 2
Gupta(2009) 5 2 8 3
Rifia (1996) 4 2 6 2
Baykasoglu(1999) 4 2 7 3
Baykasoglu(2001) 2 4 8 4
Olson(1984) 3 2 6 2
6 CONCLUSION
The proposed method is an efficient method of solving lexicographic Goal programming. The new method is
used in solving various lexicographic linear Goal programming problems of different variables sizes, goals,
constraints and deviational variables. The proposed method is an efficient method and its formulation represents
a better model than the existing ones.
REFERENCES
[1] C.L Hwang, A. S. M. Masud, S.R Paidy, and K. Yoon, Mathematical programming with multiple
objectives: a tutorial. Computers & Operations Research. (1980). 7, 5-31.
[2] Evans G.W An Overview of technique for solving multiobjective mathematical programs. Maence
30,(11) (1984) 1268-1282.
[3] Aouni,B. & O. Kettani, Goal programming model: A glorious history and promising future. European
Journal Of operation of operational research 133, (2001), 225-231
[4] Sunnar, M and Kahrama, R A Comparative Study of Multiobjective Optimization Methods in Structural
Design. Turk J Engin Environ Sci 25, (2001), 69 -86.
[5] Tamiz M, &D. F Jones (2010): Practical Goal Programming. International Series in Operations
Research & Management Science. Springer New York http://www.springer.com/series/6161.
[6] Schniederjans, M. J. & N. K. Kwak An alternative solution method for goal programming problems: a
tutorial. Journal of Operational Research Society.33, (1982) 247-252.
[7] Hwang, C. &, K. Yoon (1981). Multiple Attribute Decision Making Methods and Applications - A
State of the Art Survey. Springer- Verlag.
[8] IJiri, Y. (1965). Management Goals and Accounting for Control. Rand-McNally: Chicago.
[9] Lee, S. M. (1972). Goal Programming for Decision Analysis. Auer Bach, Philadelphia.
[10] Ignizio J. P.(1976). Goal Programming and Extensions. D. C. Heath, Lexington, Massachusetts.
[11] Ignizio, J.P A Note on Computational Methods in Lexicographic Linear Goal Programming. Opl Res.
Soc. Vol. 34, No. 6, pp. 539-542, 1983 0160-5682.
[14] Ignizio J. P An algorithm for solving the linear goal-programming problem by Solving its dual. Journal
of operational Research Society, 36, (1985) 507-5 15.
[15] Olson, D. L. Revised simplex method of solving linear goal programming problem. Journal of the
Operational Research Society Vol. 35, No. 4(1984) pp 347-354.
[16] Orumie,U.C &D.W Ebong An Alternative method of solving Goal programming problems. Nigerian
Journal of Operations Research vlo2,No 1, (2011) pg 68-90.
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