This document presents a method for solving fuzzy assignment problems where costs are represented by linguistic variables and fuzzy numbers. Linguistic variables are used to convert qualitative cost data into quantitative fuzzy numbers. Yager's ranking method is applied to rank the fuzzy numbers, transforming the fuzzy assignment problem into a crisp one. The resulting crisp problem is then solved using the Hungarian method to find the optimal assignment that minimizes total cost. A numerical example demonstrates the approach, showing a fuzzy cost matrix converted to crisp values and solved. The method allows handling assignment problems with imprecise, qualitative cost data using fuzzy logic concepts.
This document presents a statistical approach for solving a two-objective fuzzy assignment problem where costs and times are represented as triangular fuzzy numbers. The methodology proposes using Pascal's triangle to determine the coefficients of the fuzzy numbers and then applying a simple probability approach to obtain solutions. A numerical example is provided to illustrate the approach. Statistical tests like F-test and t-test are suggested to analyze and compare the proposed Pascal triangle method to existing graded mean integration representation techniques for solving fuzzy assignment problems.
Hierarchical Deterministic Quadrature Methods for Option Pricing under the Ro...Chiheb Ben Hammouda
Seminar talk at École des Ponts ParisTech about our recently published work "Hierarchical adaptive sparse grids and quasi-Monte Carlo for option pricing under the rough Bergomi model". - Link of the paper: https://www.tandfonline.com/doi/abs/10.1080/14697688.2020.1744700
This document is the MSc project of Mohamed Raagi submitted to Brunel University London in October 2015. It examines excess rates of return from jump risks using geometric Lévy models for asset pricing. The project reviews recent developments in these models and simulates price processes involving jumps to analyze excess rate of return behavior and impact. It introduces Lévy processes and geometric Lévy martingale models as tools for derivative pricing. Specific models discussed include Brownian motion, Poisson, compound Poisson, and geometric gamma. The document also covers option pricing and simulations for each model.
Fuzzy inventory model with shortages in man power planningAlexander Decker
This document presents a fuzzy inventory model to determine the optimal time for an employee to change jobs while minimizing costs. It introduces concepts like real wage, membership functions, and fuzzy nonlinear programming. The model considers costs of decreasing real income, moving to a new job, and income shortages. It uses Lagrange multipliers to solve the fuzzy nonlinear programming problem and compares the results to a crisp model. A numerical example is provided to illustrate the application to manpower planning.
Determination of Optimal Product Mix for Profit Maximization using Linear Pro...IJERA Editor
This paper demonstrates the use of liner programming methods in order to determine the optimal product mix for
profit maximization. There had been several papers written to demonstrate the use of linear programming in
finding the optimal product mix in various organization. This paper is aimed to show the generic approach to be
taken to find the optimal product mix.
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
The convenience yield implied by quadratic volatility smiles presentation [...yigalbt
This document discusses the implied convenience yield from quadratic volatility smiles in options. It presents formulas to calculate the implied convenience yield for illiquid options based on using liquid at-the-money options as hedging instruments. The formulas depend on observable market parameters like volatility and are meant to provide a simple way to compute the implied convenience yield without historical data assumptions. However, the model relies on several undefined expressions and economic assumptions that are not fully clear.
This document compares the performance of genetic algorithms and niching methods for clustering undirected weighted graphs. It discusses how genetic algorithms can converge prematurely on local optima for complex problems like clustering that have many potential solutions. Niching methods like deterministic crowding are introduced to maintain population diversity and allow the search of multiple peaks in parallel. The paper applies genetic algorithms and deterministic crowding to the graph clustering problem and compares their results on test graphs, finding that deterministic crowding is more computationally demanding but provides better optimization.
This document presents a statistical approach for solving a two-objective fuzzy assignment problem where costs and times are represented as triangular fuzzy numbers. The methodology proposes using Pascal's triangle to determine the coefficients of the fuzzy numbers and then applying a simple probability approach to obtain solutions. A numerical example is provided to illustrate the approach. Statistical tests like F-test and t-test are suggested to analyze and compare the proposed Pascal triangle method to existing graded mean integration representation techniques for solving fuzzy assignment problems.
Hierarchical Deterministic Quadrature Methods for Option Pricing under the Ro...Chiheb Ben Hammouda
Seminar talk at École des Ponts ParisTech about our recently published work "Hierarchical adaptive sparse grids and quasi-Monte Carlo for option pricing under the rough Bergomi model". - Link of the paper: https://www.tandfonline.com/doi/abs/10.1080/14697688.2020.1744700
This document is the MSc project of Mohamed Raagi submitted to Brunel University London in October 2015. It examines excess rates of return from jump risks using geometric Lévy models for asset pricing. The project reviews recent developments in these models and simulates price processes involving jumps to analyze excess rate of return behavior and impact. It introduces Lévy processes and geometric Lévy martingale models as tools for derivative pricing. Specific models discussed include Brownian motion, Poisson, compound Poisson, and geometric gamma. The document also covers option pricing and simulations for each model.
Fuzzy inventory model with shortages in man power planningAlexander Decker
This document presents a fuzzy inventory model to determine the optimal time for an employee to change jobs while minimizing costs. It introduces concepts like real wage, membership functions, and fuzzy nonlinear programming. The model considers costs of decreasing real income, moving to a new job, and income shortages. It uses Lagrange multipliers to solve the fuzzy nonlinear programming problem and compares the results to a crisp model. A numerical example is provided to illustrate the application to manpower planning.
Determination of Optimal Product Mix for Profit Maximization using Linear Pro...IJERA Editor
This paper demonstrates the use of liner programming methods in order to determine the optimal product mix for
profit maximization. There had been several papers written to demonstrate the use of linear programming in
finding the optimal product mix in various organization. This paper is aimed to show the generic approach to be
taken to find the optimal product mix.
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
The convenience yield implied by quadratic volatility smiles presentation [...yigalbt
This document discusses the implied convenience yield from quadratic volatility smiles in options. It presents formulas to calculate the implied convenience yield for illiquid options based on using liquid at-the-money options as hedging instruments. The formulas depend on observable market parameters like volatility and are meant to provide a simple way to compute the implied convenience yield without historical data assumptions. However, the model relies on several undefined expressions and economic assumptions that are not fully clear.
This document compares the performance of genetic algorithms and niching methods for clustering undirected weighted graphs. It discusses how genetic algorithms can converge prematurely on local optima for complex problems like clustering that have many potential solutions. Niching methods like deterministic crowding are introduced to maintain population diversity and allow the search of multiple peaks in parallel. The paper applies genetic algorithms and deterministic crowding to the graph clustering problem and compares their results on test graphs, finding that deterministic crowding is more computationally demanding but provides better optimization.
This document provides preparation guidelines for interviews for junior quantitative analyst positions. It recommends spending 40-50% of time reviewing basic math skills like calculus, probability, statistics and financial math. Another 30-40% should be spent programming in C++, focusing on object-oriented principles and data structures. The final 10-20% should cover financial products and modeling. Sample questions test knowledge of derivatives pricing, differential equations, linear algebra, and programming concepts. Problem-solving questions evaluate logical thinking and proof abilities. Overall, the document emphasizes mastering fundamentals before complex topics.
The comparative study of finite difference method and monte carlo method for ...Alexander Decker
This document compares the finite difference method and Monte Carlo method for pricing European options. The finite difference method solves the Black-Scholes partial differential equation by approximating it on a grid, while the Monte Carlo method simulates asset price paths and averages discounted payoffs. The study finds that while both methods agree with the Black-Scholes price, the finite difference method converges faster and is more accurate for standard European options, whereas Monte Carlo is better suited for exotic options due to its flexibility.
Using Alpha-cuts and Constraint Exploration Approach on Quadratic Programming...TELKOMNIKA JOURNAL
In this paper, we propose a computational procedure to find the optimal solution of quadratic programming
problems by using fuzzy -cuts and constraint exploration approach. We solve the problems in
the original form without using any additional information such as Lagrange’s multiplier, slack, surplus and
artificial variable. In order to find the optimal solution, we divide the calculation in two stages. In the first
stage, we determine the unconstrained minimization of the quadratic programming problem (QPP) and check
its feasibility. By unconstrained minimization we identify the violated constraints and focus our searching in
these constraints. In the second stage, we explored the feasible region along side the violated constraints
until the optimal point is achieved. A numerical example is included in this paper to illustrate the capability of
-cuts and constraint exploration to find the optimal solution of QPP.
11.generalized and subset integrated autoregressive moving average bilinear t...Alexander Decker
This document proposes generalized integrated autoregressive moving average bilinear (GBL) time series models and subset generalized integrated autoregressive moving average bilinear (GSBL) models to achieve stationary for all nonlinear time series. It presents the models' formulations and discusses their properties including stationary, convergence, and parameter estimation. An algorithm is provided to fit the one-dimensional models. The generalized models are applied to Wolfer sunspot numbers and the GBL model is found to perform better than the GSBL model.
This document summarizes research on strong duality analysis for discrete-time constrained portfolio optimization problems. It begins by introducing the mathematical formulation of a discrete-time portfolio selection model with constraints expressed as convex inequalities. It then discusses a risk neutral computational approach based on embedding the primal constrained problem into a family of unconstrained problems in auxiliary markets. Weak duality is shown to hold, relating the optimal values of the primal and auxiliary problems. The document defines a dual problem, known as Pliska's κ dual, that seeks to minimize the optimal values of the auxiliary problems. Conditions for strong duality are presented, under which the optimal solution to the dual problem also solves the primal constrained problem.
- The document outlines a BSc research project on pricing financial derivatives using the Black-Scholes model.
- The project aims to learn established financial models, compare pricing techniques, and see how newer models relate to existing ones.
- It provides background on the student's motivation and experience, and introduces key concepts like options, the Black-Scholes equation, and its derivation and solution.
- The student will present their work on applying and extending the Black-Scholes model to price derivatives.
New Method for Finding an Optimal Solution of Generalized Fuzzy Transportatio...BRNSS Publication Hub
In this paper, a proposed method, namely, zero average method is used for solving fuzzy transportation problems by assuming that a decision-maker is uncertain about the precise values of the transportation costs, demand, and supply of the product. In the proposed method, transportation costs, demand, and supply are represented by generalized trapezoidal fuzzy numbers. To illustrate the proposed method, a numerical example is solved. The proposed method is easy to understand and apply to real-life transportation problems for the decision-makers.
Normal density and discreminant analysisVARUN KUMAR
This document provides an overview of Gaussian density and discriminant analysis. It discusses mathematical descriptions of discriminant functions and how they are used in classifiers to select classes. It also covers bi-variate and multi-variate normal density functions and how they relate to dependent and independent random variables. Specifically, it shows how the shape of the loci of bi-variate density functions depends on whether variables are independent or dependent. Finally, it discusses discriminant functions and how to determine decision boundaries between classes.
This document provides lecture notes on industrial organization. It covers topics such as market structure, conduct, and performance. Key concepts discussed include utility maximization, demand curves, profit maximization, supply curves, and competitive market equilibrium. In a competitive market, firms are price takers in the short run. In long run equilibrium with free entry and exit, there are no barriers to entry or profits. The market reaches equilibrium when quantity demanded equals quantity supplied at the market clearing price.
In this paper, the L1 norm of continuous functions and corresponding continuous estimation of regression parameters are defined. The continuous L1 norm estimation problem of one and two parameters linear models in the continuous case is solved. We proceed to use the functional form and parameters of the probability distribution function of income to exactly determine the L1 norm approximation of the corresponding Lorenz curve of the statistical population under consideration.
This document appears to be an assignment submission for a financial engineering course. It includes a plagiarism declaration signed by the student, Andrew Hair. The assignment contains 11 questions addressing interest rate derivatives and modeling using the Vasicek model. Code is provided in MATLAB to generate simulations and analyze interest rate data based on the questions.
The Vasicek model is one of the earliest stochastic models for modeling the term structure of interest rates. It represents the movement of interest rates as a function of market risk, time, and the equilibrium value the rate tends to revert to. This document discusses parameter estimation techniques for the Vasicek one-factor model using least squares regression and maximum likelihood estimation on historical interest rate data. It also covers simulating the term structure and pricing zero-coupon bonds under the Vasicek model. The two-factor Vasicek model is introduced as an extension of the one-factor model.
This document presents a systematic approach for solving mixed intuitionistic fuzzy transportation problems. It begins with definitions of fuzzy sets, intuitionistic fuzzy sets, and triangular intuitionistic fuzzy numbers. It then formulates an intuitionistic fuzzy transportation problem and proposes a mixed intuitionistic fuzzy zero point method to find the optimal solution in terms of triangular intuitionistic fuzzy numbers. Finally, it provides the computational procedure and illustrates the method with a numerical example.
IRJET- Optimization of 1-Bit ALU using Ternary LogicIRJET Journal
This document summarizes a research paper that proposes a novel approach to implementing a 1-bit arithmetic logic unit (ALU) using ternary logic. Ternary logic offers potential advantages over binary logic, including reduced transistor count and hardware. The authors designed a 1-bit ALU using ternary logic gates (T-gates) for ternary arithmetic and logic operations. Simulation results showed the ternary logic ALU design achieved a 25% reduction in transistor usage compared to an equivalent binary logic ALU design. The ternary logic ALU design approach could potentially be extended to multi-bit ALUs for applications where reduced transistor count is important.
11.fuzzy inventory model with shortages in man power planningAlexander Decker
This document presents a fuzzy inventory model to determine the optimal time for an employee to quit their current job based on factors like the declining real wage over time and costs associated with changing jobs. The model extends an existing economic order quantity (EOQ) model to account for uncertainty in costs using fuzzy set theory. Membership functions are defined to represent the fuzziness of parameters like real income, costs, and constraints. A fuzzy nonlinear programming problem is formulated and solved using Lagrange multipliers to obtain the optimal solution under fuzzy conditions. The results are compared to the classical crisp model and sensitivity analysis is performed.
This document discusses duality in linear programming. It defines the dual problem as another linear program systematically constructed from the original or primal problem, such that the optimal solutions of one provide the optimal solutions of the other. The document provides rules for constructing the dual problem based on whether the primal problem is a maximization or minimization problem. It also gives examples of writing the dual of a primal problem and solving both problems to verify the optimal objective values are equal. Finally, it discusses economic interpretations of duality and the relationship between primal and dual problems and solutions.
This document discusses endogenous benchmarking of mutual funds using bootstrap data envelopment analysis (DEA) in R. It aims to benchmark funds using multiple outputs, stochastic dominance indicators, and bootstrap analysis for robust evaluation. The study uses DEA with daily return mean and upside potential mean as outputs and return variance as the input to evaluate select sector funds over 6 months. Descriptive statistics of the technical efficiency scores from input-oriented, output-oriented, and graph hyperbolic DEA models are provided. Bootstrapping techniques including naive and smoothed bootstrap, bias correction, and confidence intervals are also introduced.
Matematika ekonomi slide_optimasi_dengan_batasan_persamaanUfik Tweentyfour
The document discusses optimization problems with equality constraints. It explains that constrained optimization is central to economics due to scarcity. Lagrange multipliers allow constrained optimization problems to be solved using first-order conditions. The envelope theorem describes how the optimal value of an objective function changes with parameters. Applications discussed include consumer utility maximization with a budget constraint and deriving Marshallian and Hicksian demand curves.
This document discusses the cognitive approach to language teaching. It explains that the cognitive approach uses cognitive learning theory and emphasizes understanding grammar rules before practicing speaking. Some key points made in the document include:
- The cognitive approach is influenced by transformational-generative grammar and stresses understanding rules first through deductive teaching before exercises.
- Language is viewed as rule-governed behavior under this approach rather than just habit formation.
- Teaching procedures under the cognitive approach include ensuring material is meaningful, explaining rules first before practice, and stimulating creative language use through tasks like sentence production.
la lingüística aplicada también comenzó restringida como la aplicación de puntos de vista de la lingüística estructural - en primer lugar a la enseñanza del Inglés en las escuelas y, posteriormente, a la segunda y la enseñanza de lenguas extranjeras.
Linguistics is the scientific study of language, divided into theoretical, applied, descriptive, historical, geographical, and comparative linguistics. Applied linguistics aims to apply linguistic theories to improve language teaching methodology, syllabuses, and language assessment. The relationship between linguistics and language teaching has evolved over time, with methods like grammar translation and direct instruction reflecting developments in linguistic theories. Syllabus design also reflects language performance and goals.
The document discusses semantic roles in language. It begins by defining key terms like sentence, proposition, predicate, and noun phrases. It then explains that a proposition consists of a predicate plus arguments, and discusses how propositions can be expressed through different sentences with varying grammar but conveying the same meaning. The document outlines different semantic roles like agent, patient, and themes and provides examples. It also discusses how the valency of predicates, or number of arguments they can take, can vary and provides a table of common semantic roles.
This document provides preparation guidelines for interviews for junior quantitative analyst positions. It recommends spending 40-50% of time reviewing basic math skills like calculus, probability, statistics and financial math. Another 30-40% should be spent programming in C++, focusing on object-oriented principles and data structures. The final 10-20% should cover financial products and modeling. Sample questions test knowledge of derivatives pricing, differential equations, linear algebra, and programming concepts. Problem-solving questions evaluate logical thinking and proof abilities. Overall, the document emphasizes mastering fundamentals before complex topics.
The comparative study of finite difference method and monte carlo method for ...Alexander Decker
This document compares the finite difference method and Monte Carlo method for pricing European options. The finite difference method solves the Black-Scholes partial differential equation by approximating it on a grid, while the Monte Carlo method simulates asset price paths and averages discounted payoffs. The study finds that while both methods agree with the Black-Scholes price, the finite difference method converges faster and is more accurate for standard European options, whereas Monte Carlo is better suited for exotic options due to its flexibility.
Using Alpha-cuts and Constraint Exploration Approach on Quadratic Programming...TELKOMNIKA JOURNAL
In this paper, we propose a computational procedure to find the optimal solution of quadratic programming
problems by using fuzzy -cuts and constraint exploration approach. We solve the problems in
the original form without using any additional information such as Lagrange’s multiplier, slack, surplus and
artificial variable. In order to find the optimal solution, we divide the calculation in two stages. In the first
stage, we determine the unconstrained minimization of the quadratic programming problem (QPP) and check
its feasibility. By unconstrained minimization we identify the violated constraints and focus our searching in
these constraints. In the second stage, we explored the feasible region along side the violated constraints
until the optimal point is achieved. A numerical example is included in this paper to illustrate the capability of
-cuts and constraint exploration to find the optimal solution of QPP.
11.generalized and subset integrated autoregressive moving average bilinear t...Alexander Decker
This document proposes generalized integrated autoregressive moving average bilinear (GBL) time series models and subset generalized integrated autoregressive moving average bilinear (GSBL) models to achieve stationary for all nonlinear time series. It presents the models' formulations and discusses their properties including stationary, convergence, and parameter estimation. An algorithm is provided to fit the one-dimensional models. The generalized models are applied to Wolfer sunspot numbers and the GBL model is found to perform better than the GSBL model.
This document summarizes research on strong duality analysis for discrete-time constrained portfolio optimization problems. It begins by introducing the mathematical formulation of a discrete-time portfolio selection model with constraints expressed as convex inequalities. It then discusses a risk neutral computational approach based on embedding the primal constrained problem into a family of unconstrained problems in auxiliary markets. Weak duality is shown to hold, relating the optimal values of the primal and auxiliary problems. The document defines a dual problem, known as Pliska's κ dual, that seeks to minimize the optimal values of the auxiliary problems. Conditions for strong duality are presented, under which the optimal solution to the dual problem also solves the primal constrained problem.
- The document outlines a BSc research project on pricing financial derivatives using the Black-Scholes model.
- The project aims to learn established financial models, compare pricing techniques, and see how newer models relate to existing ones.
- It provides background on the student's motivation and experience, and introduces key concepts like options, the Black-Scholes equation, and its derivation and solution.
- The student will present their work on applying and extending the Black-Scholes model to price derivatives.
New Method for Finding an Optimal Solution of Generalized Fuzzy Transportatio...BRNSS Publication Hub
In this paper, a proposed method, namely, zero average method is used for solving fuzzy transportation problems by assuming that a decision-maker is uncertain about the precise values of the transportation costs, demand, and supply of the product. In the proposed method, transportation costs, demand, and supply are represented by generalized trapezoidal fuzzy numbers. To illustrate the proposed method, a numerical example is solved. The proposed method is easy to understand and apply to real-life transportation problems for the decision-makers.
Normal density and discreminant analysisVARUN KUMAR
This document provides an overview of Gaussian density and discriminant analysis. It discusses mathematical descriptions of discriminant functions and how they are used in classifiers to select classes. It also covers bi-variate and multi-variate normal density functions and how they relate to dependent and independent random variables. Specifically, it shows how the shape of the loci of bi-variate density functions depends on whether variables are independent or dependent. Finally, it discusses discriminant functions and how to determine decision boundaries between classes.
This document provides lecture notes on industrial organization. It covers topics such as market structure, conduct, and performance. Key concepts discussed include utility maximization, demand curves, profit maximization, supply curves, and competitive market equilibrium. In a competitive market, firms are price takers in the short run. In long run equilibrium with free entry and exit, there are no barriers to entry or profits. The market reaches equilibrium when quantity demanded equals quantity supplied at the market clearing price.
In this paper, the L1 norm of continuous functions and corresponding continuous estimation of regression parameters are defined. The continuous L1 norm estimation problem of one and two parameters linear models in the continuous case is solved. We proceed to use the functional form and parameters of the probability distribution function of income to exactly determine the L1 norm approximation of the corresponding Lorenz curve of the statistical population under consideration.
This document appears to be an assignment submission for a financial engineering course. It includes a plagiarism declaration signed by the student, Andrew Hair. The assignment contains 11 questions addressing interest rate derivatives and modeling using the Vasicek model. Code is provided in MATLAB to generate simulations and analyze interest rate data based on the questions.
The Vasicek model is one of the earliest stochastic models for modeling the term structure of interest rates. It represents the movement of interest rates as a function of market risk, time, and the equilibrium value the rate tends to revert to. This document discusses parameter estimation techniques for the Vasicek one-factor model using least squares regression and maximum likelihood estimation on historical interest rate data. It also covers simulating the term structure and pricing zero-coupon bonds under the Vasicek model. The two-factor Vasicek model is introduced as an extension of the one-factor model.
This document presents a systematic approach for solving mixed intuitionistic fuzzy transportation problems. It begins with definitions of fuzzy sets, intuitionistic fuzzy sets, and triangular intuitionistic fuzzy numbers. It then formulates an intuitionistic fuzzy transportation problem and proposes a mixed intuitionistic fuzzy zero point method to find the optimal solution in terms of triangular intuitionistic fuzzy numbers. Finally, it provides the computational procedure and illustrates the method with a numerical example.
IRJET- Optimization of 1-Bit ALU using Ternary LogicIRJET Journal
This document summarizes a research paper that proposes a novel approach to implementing a 1-bit arithmetic logic unit (ALU) using ternary logic. Ternary logic offers potential advantages over binary logic, including reduced transistor count and hardware. The authors designed a 1-bit ALU using ternary logic gates (T-gates) for ternary arithmetic and logic operations. Simulation results showed the ternary logic ALU design achieved a 25% reduction in transistor usage compared to an equivalent binary logic ALU design. The ternary logic ALU design approach could potentially be extended to multi-bit ALUs for applications where reduced transistor count is important.
11.fuzzy inventory model with shortages in man power planningAlexander Decker
This document presents a fuzzy inventory model to determine the optimal time for an employee to quit their current job based on factors like the declining real wage over time and costs associated with changing jobs. The model extends an existing economic order quantity (EOQ) model to account for uncertainty in costs using fuzzy set theory. Membership functions are defined to represent the fuzziness of parameters like real income, costs, and constraints. A fuzzy nonlinear programming problem is formulated and solved using Lagrange multipliers to obtain the optimal solution under fuzzy conditions. The results are compared to the classical crisp model and sensitivity analysis is performed.
This document discusses duality in linear programming. It defines the dual problem as another linear program systematically constructed from the original or primal problem, such that the optimal solutions of one provide the optimal solutions of the other. The document provides rules for constructing the dual problem based on whether the primal problem is a maximization or minimization problem. It also gives examples of writing the dual of a primal problem and solving both problems to verify the optimal objective values are equal. Finally, it discusses economic interpretations of duality and the relationship between primal and dual problems and solutions.
This document discusses endogenous benchmarking of mutual funds using bootstrap data envelopment analysis (DEA) in R. It aims to benchmark funds using multiple outputs, stochastic dominance indicators, and bootstrap analysis for robust evaluation. The study uses DEA with daily return mean and upside potential mean as outputs and return variance as the input to evaluate select sector funds over 6 months. Descriptive statistics of the technical efficiency scores from input-oriented, output-oriented, and graph hyperbolic DEA models are provided. Bootstrapping techniques including naive and smoothed bootstrap, bias correction, and confidence intervals are also introduced.
Matematika ekonomi slide_optimasi_dengan_batasan_persamaanUfik Tweentyfour
The document discusses optimization problems with equality constraints. It explains that constrained optimization is central to economics due to scarcity. Lagrange multipliers allow constrained optimization problems to be solved using first-order conditions. The envelope theorem describes how the optimal value of an objective function changes with parameters. Applications discussed include consumer utility maximization with a budget constraint and deriving Marshallian and Hicksian demand curves.
This document discusses the cognitive approach to language teaching. It explains that the cognitive approach uses cognitive learning theory and emphasizes understanding grammar rules before practicing speaking. Some key points made in the document include:
- The cognitive approach is influenced by transformational-generative grammar and stresses understanding rules first through deductive teaching before exercises.
- Language is viewed as rule-governed behavior under this approach rather than just habit formation.
- Teaching procedures under the cognitive approach include ensuring material is meaningful, explaining rules first before practice, and stimulating creative language use through tasks like sentence production.
la lingüística aplicada también comenzó restringida como la aplicación de puntos de vista de la lingüística estructural - en primer lugar a la enseñanza del Inglés en las escuelas y, posteriormente, a la segunda y la enseñanza de lenguas extranjeras.
Linguistics is the scientific study of language, divided into theoretical, applied, descriptive, historical, geographical, and comparative linguistics. Applied linguistics aims to apply linguistic theories to improve language teaching methodology, syllabuses, and language assessment. The relationship between linguistics and language teaching has evolved over time, with methods like grammar translation and direct instruction reflecting developments in linguistic theories. Syllabus design also reflects language performance and goals.
The document discusses semantic roles in language. It begins by defining key terms like sentence, proposition, predicate, and noun phrases. It then explains that a proposition consists of a predicate plus arguments, and discusses how propositions can be expressed through different sentences with varying grammar but conveying the same meaning. The document outlines different semantic roles like agent, patient, and themes and provides examples. It also discusses how the valency of predicates, or number of arguments they can take, can vary and provides a table of common semantic roles.
This document defines and provides examples of synonyms, antonyms, and homophones. Synonyms are words that have the same or similar meanings, such as "easy" and "simple." Antonyms are words with opposite meanings, like "give" and "take." Homophones are words that sound the same but have different spellings and meanings, such as "ate" and "eight." The document includes exercises asking the reader to identify synonyms, antonyms, and homophones in sentences.
This document defines and provides examples of different semantic roles including agent, patient or theme, instrument, experiencer, and location. It explains that semantic roles describe the underlying relationship that participants have with the main verb in a clause. For example, in the sentence "The boy kicked the ball", the boy is the agent performing the action of kicking, while the ball is the theme or patient that is affected by the action. The document also introduces feature notation as a method to express the existence or non-existence of semantic properties using plus and minus signs, such as [+HUMAN] to denote entities that are human.
The key competences are skills defined by the European Union and incorporated into the Spanish education system to help students integrate knowledge and skills for practical problem solving. They include communication, digital skills, learning to learn, social skills, initiative, and others. All subject areas are meant to contribute to developing these competences, and while only language skills are formally assessed, the competences aim to prepare students for adult life.
A semantic role describes the relationship between a participant and the main verb in a clause. The main semantic roles include agent, patient, experiencer, goal, and instrument. Semantic roles are conceptual and do not directly correspond to grammatical relations like subject and object. For example, a subject can play the role of agent, patient, or instrument depending on the verb.
The document discusses various lexical semantic relationships between words including synonymy, antonymy, hyponymy, prototypes, homophones, homonyms, polysemy, metonymy, and collocation. It provides examples and explanations of each relationship, noting how words can be related through meaning, pronunciation, or common association. Understanding these relationships is important for analyzing how meaning is constructed in text.
SYNONYMS, ANTONYMS, POLYSEMY, HOMONYM, AND HOMOGRAPHLili Lulu
definition and examlple SYNONYMS,
defintion and example ANTONYMS,
Definition and example POLYSEMY,
Definition and example HOMONYM, AND Definition and example HOMOGRAPH
The document discusses the learning-centered approach to course design. It explains that the learning-centered approach considers the learner's needs, skills, attitudes and learning situation at every stage of design. This includes analyzing both the target situation where skills will be used and the current learning situation. The learning-centered process is dynamic and negotiates between these factors when writing syllabus, materials, and evaluations. It implies course design is negotiated between situations and changes over time with feedback.
ESP refers to English for Specific Purposes which designs English language courses based on the specific needs of learners in their fields or occupations. ESP courses focus on developing the grammar, vocabulary, study skills and discourse needed in the target discipline. They use authentic materials from the relevant field and allow self-directed learning. Common ESP courses include English for academic disciplines, occupations, and topics like English for medicine or English for technology.
The document discusses various key concepts in semantics, including:
- Semantics is the study of meaning in language. It examines how meaning is constructed and interpreted.
- Semantic roles describe the functions that words play in sentences, such as agent, theme, and experiencer.
- Relationships between words include synonyms, antonyms, hyponyms, homophony, and polysemy. Synonyms have similar meanings, antonyms have opposite meanings, hyponyms have a broader term that includes them, and polysemy refers to a word having multiple related meanings.
- Richard Montague pioneered formal semantics which used logic to represent meanings of sentences. Semantics analyzes meaning at various linguistic levels
The document provides an overview of applied linguistics, including:
- Its origins in the 1940s through efforts to ally language teaching with linguistics.
- Definitions that describe it as concerned with investigating and solving real-world problems involving language.
- Its problem-based and interdisciplinary nature in drawing on linguistics and other fields like psychology to address issues in areas like language teaching, literacy, and language policy.
- Key topics it addresses including language learning, teaching, assessment, use, and pathology.
- Its focus on applying linguistic knowledge to resolve language problems people face in various contexts.
Applied linguistics is the interdisciplinary study of language and its applications in real world contexts. It draws on linguistic theories and research to solve practical language-related problems. Key areas include second language acquisition, teaching methodology, testing, and the relationships between language and society, technology, and other fields. Throughout the 20th century, applied linguistics influenced the development of language teaching methods, shifting the focus from grammar translation to more communicative, meaning-based approaches grounded in theories of language acquisition and use.
This document discusses the history and development of curriculum in the Philippines. It covers the influences of Spanish colonial rule, American rule, and the Japanese occupation on the Philippine curriculum. It also describes the essentialist and progressive schools of thought on curriculum development. Additionally, it discusses the modernization and reforms of the Philippine curriculum after independence, including an emphasis on moral values, relevance, vocational education, and national consciousness. The document provides context on how political, economic, social, and religious factors have shaped curriculum development in the Philippines over time.
Linguistics is the scientific study of language, including areas like phonetics, phonology, morphology, syntax, and semantics. Applied linguistics [1] identifies and addresses language-related problems, [2] can be applied to all aspects of language use such as acquisition of first, second, and foreign languages, and [3] extends into practical fields including clinical linguistics, language teaching, lexicography, and computational linguistics. The key difference is that linguistics studies language itself, while applied linguistics examines the relationship between language and other domains.
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11.polynomial regression model of making cost prediction in mixed cost analysisAlexander Decker
This document presents a study comparing different regression models for predicting costs based on production levels. It finds that a cubic polynomial regression model provides a better fit than linear regression or the high-low method. The study uses cost and production data from a company to build linear, quadratic, and cubic regression models. It finds the cubic polynomial regression has the highest R-squared value and lowest p-value, indicating it is best able to model the cost patterns in the data. The document concludes the cubic polynomial regression provides a better approach for cost prediction than traditional linear regression or high-low methods.
Polynomial regression model of making cost prediction in mixed cost analysisAlexander Decker
This document presents a study comparing different regression models for predicting costs based on production levels. It finds that a cubic polynomial regression model provides a better fit than linear regression or the high-low method. The study uses cost and production data from a company to build linear, quadratic, and cubic regression models. It finds the cubic polynomial regression has the highest R-squared value and lowest p-value, indicating it is the best-fitting model. The study concludes that polynomial regression generally provides a better approach for cost prediction than conventional linear regression or the high-low method.
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in texture analysis are discussed in the context of Content Based Image Retrieval (CBIR). The theoretical
extension of GLCM to n-dimensional gray scale images are also discussed. The results indicate that trace
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The Probability that a Matrix of Integers Is DiagonalizableJay Liew
The Probability that a
Matrix of Integers Is Diagonalizable
Andrew J. Hetzel, Jay S. Liew, and Kent E. Morrison
1. INTRODUCTION. It is natural to use integer matrices for examples and exercises
when teaching a linear algebra course, or, for that matter, when writing a textbook in
the subject. After all, integer matrices offer a great deal of algebraic simplicity for particular
problems. This, in turn, lets students focus on the concepts. Of course, to insist
on integer matrices exclusively would certainly give the wrong idea about many important
concepts. For example, integer matrices with integer matrix inverses are quite
rare, although invertible integer matrices (over the rational numbers) are relatively
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Application of matrix algebra to multivariate data using standardize scoresAlexander Decker
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11.application of matrix algebra to multivariate data using standardize scoresAlexander Decker
This document discusses applying matrix algebra to estimate parameters in a regression equation using standardized scores. It presents a methodology for standardizing multivariate data measured in different units. The methodology is demonstrated by applying it to sample data to estimate the regression plane. The results using standardized scores match those obtained in previous studies using original and mean-corrected scores. Standardizing converts data to approximately normal, unit-less scores, addressing issues that arise when data is measured in different units.
The document presents a method for solving fuzzy assignment problems using triangular and trapezoidal fuzzy numbers. It formulates the fuzzy assignment problem into a crisp linear programming problem that can be solved using the Hungarian method. The paper also uses Robust's ranking method to transform fuzzy costs into crisp values, allowing conventional solution methods to be applied. It aims to provide a more realistic approach to assignment problems by considering costs as fuzzy numbers rather than deterministic values.
This document discusses isoparametric finite elements. It begins by defining shape functions and how they relate node displacements to the displacement field within an element. Isoparametric elements are introduced, which use the same shape functions for geometric representation and displacement approximation. A 1D bar element demonstration shows how shape functions define element geometry and displacements. Higher-order elements, including triangular elements, are also discussed. Different types of shape functions like Lagrange, serendipity, and Hermitian polynomials are covered.
Determination of Optimal Product Mix for Profit Maximization using Linear Pro...IJERA Editor
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This is the entrance exam paper for ISI MSQE Entrance Exam for the year 2008. Much more information on the ISI MSQE Entrance Exam and ISI MSQE Entrance preparation help available on http://crackdse.com
The k-means clustering algorithm aims to group data points into k clusters based on their distances from initial cluster centroid points. It works by alternating between assigning each point to its nearest centroid, and updating the centroid locations to be the mean of their assigned points. This process monotonically decreases the distortion score measuring distances from points to centroids, and is guaranteed to converge, though possibly to local optima rather than the global minimum. Running it multiple times can help avoid bad initial results.
The document provides information about a test for candidates applying for an M.Tech in Computer Science. It describes:
1) The test will have two parts - a morning objective test (Test MIII) and an afternoon short answer test (Test CS).
2) The CS test booklet will have two groups - Group A covering analytical ability and mathematics at the B.Sc. pass level, and Group B covering advanced topics in mathematics, statistics, physics, computer science, and engineering at the B.Sc. Hons. and B.Tech. levels.
3) Sample questions are provided for both Group A (mathematical reasoning and basic concepts) and Group B (advanced topics in real analysis
The document describes a test for candidates applying for an M.Tech. in Computer Science. [The test consists of two parts - an objective test in the morning and a short answer test in the afternoon. The short answer test has two groups - Group A covers analytical ability and mathematics at the B.Sc. level, while Group B covers additional topics in mathematics, statistics, physics, computer science, or engineering depending on the candidate's choice.] The document provides sample questions testing concepts in mathematics including algebra, calculus, number theory, and logic.
This document provides an overview of several topics in industrial engineering including:
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- Statistical process control methods like X-bar and R charts to monitor quality.
- Process capability analysis to determine if a process meets specifications.
- Queueing models and their fundamental relationships to model waiting times in systems.
- Simulation techniques like random number generation and the inverse transform method.
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Similar to IJCER (www.ijceronline.com) International Journal of computational Engineering research (20)
Introduction of Cybersecurity with OSS at Code Europe 2024Hiroshi SHIBATA
I develop the Ruby programming language, RubyGems, and Bundler, which are package managers for Ruby. Today, I will introduce how to enhance the security of your application using open-source software (OSS) examples from Ruby and RubyGems.
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I'll share insights into how the Ruby and RubyGems core team works to keep our ecosystem safe. By the end of this talk, you'll have a better understanding of how to safeguard your code.
Threats to mobile devices are more prevalent and increasing in scope and complexity. Users of mobile devices desire to take full advantage of the features
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Main news related to the CCS TSI 2023 (2023/1695)Jakub Marek
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Best 20 SEO Techniques To Improve Website Visibility In SERPPixlogix Infotech
Boost your website's visibility with proven SEO techniques! Our latest blog dives into essential strategies to enhance your online presence, increase traffic, and rank higher on search engines. From keyword optimization to quality content creation, learn how to make your site stand out in the crowded digital landscape. Discover actionable tips and expert insights to elevate your SEO game.
Webinar: Designing a schema for a Data WarehouseFederico Razzoli
Are you new to data warehouses (DWH)? Do you need to check whether your data warehouse follows the best practices for a good design? In both cases, this webinar is for you.
A data warehouse is a central relational database that contains all measurements about a business or an organisation. This data comes from a variety of heterogeneous data sources, which includes databases of any type that back the applications used by the company, data files exported by some applications, or APIs provided by internal or external services.
But designing a data warehouse correctly is a hard task, which requires gathering information about the business processes that need to be analysed in the first place. These processes must be translated into so-called star schemas, which means, denormalised databases where each table represents a dimension or facts.
We will discuss these topics:
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Digital Marketing Trends in 2024 | Guide for Staying AheadWask
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In the rapidly evolving landscape of technologies, XML continues to play a vital role in structuring, storing, and transporting data across diverse systems. The recent advancements in artificial intelligence (AI) present new methodologies for enhancing XML development workflows, introducing efficiency, automation, and intelligent capabilities. This presentation will outline the scope and perspective of utilizing AI in XML development. The potential benefits and the possible pitfalls will be highlighted, providing a balanced view of the subject.
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The presentation aims to deliver a comprehensive overview of AI usage in XML development, providing attendees with the necessary knowledge to make informed decisions. Whether you’re at the early stages of adopting AI or considering integrating it in advanced XML development, this presentation will cover all levels of expertise.
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UiPath Test Automation using UiPath Test Suite series, part 6DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 6. In this session, we will cover Test Automation with generative AI and Open AI.
UiPath Test Automation with generative AI and Open AI webinar offers an in-depth exploration of leveraging cutting-edge technologies for test automation within the UiPath platform. Attendees will delve into the integration of generative AI, a test automation solution, with Open AI advanced natural language processing capabilities.
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What will you get from this session?
1. Insights into integrating generative AI.
2. Understanding how this integration enhances test automation within the UiPath platform
3. Practical demonstrations
4. Exploration of real-world use cases illustrating the benefits of AI-driven test automation for UiPath
Topics covered:
What is generative AI
Test Automation with generative AI and Open AI.
UiPath integration with generative AI
Speaker:
Deepak Rai, Automation Practice Lead, Boundaryless Group and UiPath MVP
HCL Notes und Domino Lizenzkostenreduzierung in der Welt von DLAUpanagenda
Webinar Recording: https://www.panagenda.com/webinars/hcl-notes-und-domino-lizenzkostenreduzierung-in-der-welt-von-dlau/
DLAU und die Lizenzen nach dem CCB- und CCX-Modell sind für viele in der HCL-Community seit letztem Jahr ein heißes Thema. Als Notes- oder Domino-Kunde haben Sie vielleicht mit unerwartet hohen Benutzerzahlen und Lizenzgebühren zu kämpfen. Sie fragen sich vielleicht, wie diese neue Art der Lizenzierung funktioniert und welchen Nutzen sie Ihnen bringt. Vor allem wollen Sie sicherlich Ihr Budget einhalten und Kosten sparen, wo immer möglich. Das verstehen wir und wir möchten Ihnen dabei helfen!
Wir erklären Ihnen, wie Sie häufige Konfigurationsprobleme lösen können, die dazu führen können, dass mehr Benutzer gezählt werden als nötig, und wie Sie überflüssige oder ungenutzte Konten identifizieren und entfernen können, um Geld zu sparen. Es gibt auch einige Ansätze, die zu unnötigen Ausgaben führen können, z. B. wenn ein Personendokument anstelle eines Mail-Ins für geteilte Mailboxen verwendet wird. Wir zeigen Ihnen solche Fälle und deren Lösungen. Und natürlich erklären wir Ihnen das neue Lizenzmodell.
Nehmen Sie an diesem Webinar teil, bei dem HCL-Ambassador Marc Thomas und Gastredner Franz Walder Ihnen diese neue Welt näherbringen. Es vermittelt Ihnen die Tools und das Know-how, um den Überblick zu bewahren. Sie werden in der Lage sein, Ihre Kosten durch eine optimierte Domino-Konfiguration zu reduzieren und auch in Zukunft gering zu halten.
Diese Themen werden behandelt
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- Wie funktionieren CCB- und CCX-Lizenzen wirklich?
- Verstehen des DLAU-Tools und wie man es am besten nutzt
- Tipps für häufige Problembereiche, wie z. B. Team-Postfächer, Funktions-/Testbenutzer usw.
- Praxisbeispiele und Best Practices zum sofortigen Umsetzen
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During this demo, the founders of Secludy will demonstrate how their system utilizes Milvus to store and manipulate embeddings for generating privacy-protected synthetic data. Their approach not only maintains the confidentiality of the original data but also enhances the utility and scalability of LLMs under privacy constraints. Attendees, including machine learning engineers, data scientists, and data managers, will witness first-hand how Secludy's integration with Milvus empowers organizations to harness the power of LLMs securely and efficiently.
HCL Notes and Domino License Cost Reduction in the World of DLAUpanagenda
Webinar Recording: https://www.panagenda.com/webinars/hcl-notes-and-domino-license-cost-reduction-in-the-world-of-dlau/
The introduction of DLAU and the CCB & CCX licensing model caused quite a stir in the HCL community. As a Notes and Domino customer, you may have faced challenges with unexpected user counts and license costs. You probably have questions on how this new licensing approach works and how to benefit from it. Most importantly, you likely have budget constraints and want to save money where possible. Don’t worry, we can help with all of this!
We’ll show you how to fix common misconfigurations that cause higher-than-expected user counts, and how to identify accounts which you can deactivate to save money. There are also frequent patterns that can cause unnecessary cost, like using a person document instead of a mail-in for shared mailboxes. We’ll provide examples and solutions for those as well. And naturally we’ll explain the new licensing model.
Join HCL Ambassador Marc Thomas in this webinar with a special guest appearance from Franz Walder. It will give you the tools and know-how to stay on top of what is going on with Domino licensing. You will be able lower your cost through an optimized configuration and keep it low going forward.
These topics will be covered
- Reducing license cost by finding and fixing misconfigurations and superfluous accounts
- How do CCB and CCX licenses really work?
- Understanding the DLAU tool and how to best utilize it
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HCL Notes and Domino License Cost Reduction in the World of DLAU
IJCER (www.ijceronline.com) International Journal of computational Engineering research
1. International Journal Of Computational Engineering Research (ijceronline.com) Vol. 2 Issue. 4
An Application of Linguistic Variables in Assignment Problem with
Fuzzy Costs
1
K.Ruth Isabels, Dr.G.Uthra2
Associate Professor
Department Of Mathematics
Saveetha Engineering College
Thandalam -602 105
Abstract
This paper presents an assignment problem with fuzzy costs, where the objective is to minimize the cost. Here each fuzzy
cost is assumed as triangular or trapezoidal fuzzy number. Yager’s ranking method has been used for ranking the fuzzy
numbers. The fuzzy assignment problem has been transformed into a crisp one, using linguistic variables and solved by
Hungarian technique. The use of linguistic variables helps to convert qualitative data into quantitative data which will be
effective in dealing with fuzzy assignment problems of qualitative nature.
A numerical example is provided to demonstrate the proposed approach.
Key words: Fuzzy Assignment Problem, Fuzzy Numbers, Hungarian method, Ranking of Fuzzy numbers
Introduction
Much information that we need to deal with day to day life is vague, ambiguous, incomplete, and imprecise. Crisp logic or
conventional logic theory is inadequate for dealing with such imprecision, uncertainty and complexity of the real world. It is
this realization that motivated the evolution of fuzzy logic and fuzzy theory.
The fundamental concept of fuzzy theory is that any field X and theory Y can be fuzzified by replacing the concept of a crisp
set in X and Y by that of a fuzzy set. Mathematically a fuzzy set [4] can be defined by assigning to each possible individual
in the universe of discourse, a value representing its grade of membership in the fuzzy set. The membership function denoted
by μ is defined from X to [0, 1].
An assignment problem (AP) is a particular type of transportation problem where n tasks (jobs) are to be assigned to an equal
number of n machines (workers) in one to one basis such that the assignment cost (or profit) is minimum (or maximum).
Hence, it can be considered as a balanced transportation problem in which all supplies and demands are equal, and the
number of rows and columns in the matrix are identical.
Sakthi et al [1] adopted Yager’s ranking method [2] to transform the fuzzy assignment problem to a crisp one so that the
conventional solution methods may be applied to solve the AP. In this paper we investigate an assignment problem with
~
fuzzy costs or times Cij represented by linguistic variables which are replaced by triangular or trapezoidal fuzzy numbers.
Definitions and Formulations
Triangular fuzzy number
A triangular fuzzy number ậ is defined by a triplet (a1, a2, a3). The membership function is defined as
μậ (x) = { (x - a1) / (a2 - a1) if a1 ≤ x ≤ a2
(a3 - x) / (a3 – a2) if a2 ≤ x ≤ a3
0 otherwise}
The triangular fuzzy number is based on three-value judgement: The minimum possible value a1, the most possible value a2
and the maximum possible value a3
Issn 2250-3005(online) August| 2012 Page 1065
2. International Journal Of Computational Engineering Research (ijceronline.com) Vol. 2 Issue. 4
Trapezoidal fuzzy number
A trapezoidal fuzzy number ậ is a fuzzy number (a 1, a2, a3, a4) and its membership function is defined as
μậ (x) = { (x - a1) / (a2 - a1) if a1 ≤ x ≤ a2
1 if a2 ≤ x ≤ a3
(x – a4) / (a3 – a4) if a3 ≤ x ≤ a4
0 otherwise}
Linguistic Variable
A linguistic variable [3] is a variable whose values are linguistic terms. The concept of linguistic variable is applied in
dealing with situations which are too complex or too ill-defined to be reasonably described in conventional quantitative
expressions.
For example, ‘height; is a linguistic variable, its values can be very high, high, medium, low, very low etc., These values can
also be represented by fuzzy numbers.
α-cut and strong α-cut
Given a fuzzy set A defined on X and any number α [0,1], the α- cut αA, and the strong α-cut αA+, are the crisp sets
αA = {x/A(x) ≥ α}
αA+ = {x/A(x) > α}
The Proposed Method
The assignment problem can be stated in the form of n x n cost matrix [cij] of real numbers as given in the following table:
Jobs
Persons
1 2 3 ---j--- n
1 c11 c12 c13 --c1j-- c1n
2 c21 c22 c23 --c2j-- c2n
- - - - - -
- - - - - -
i ci1 ci2 ci3 --cij— cin
- - - - - -
n cn1 cn2 cn3 cnj cnn
Mathematically assignment problem can be stated as
n n
Minimize Z= c
i 1 j 1
ij xij i=1,2,……n: j=1,2,…….n
Subject to
n
x
j 1
ij 1, i=1,2,……n ….(1)
n
x ij 1 , j=1,2,……n xij 0,1,
i 1
Issn 2250-3005(online) August| 2012 Page 1066
3. International Journal Of Computational Engineering Research (ijceronline.com) Vol. 2 Issue. 4
where 1, if the ith person is assigned the jth job
xij =
0, otherwise
is the decision variable denoting the assignment of the person i to job j, Cij is the cost of assigning the jth job to the ith person.
The objective is to minimize the total cost of assigning all the jobs to the available persons. (One job to one person). When
~
the costs Cij are fuzzy numbers, then the fuzzy assignment problem becomes
n n
Y (~) Y (cij ) xij
z ~ ……(2)
i 1 j 1
subject to the same conditions (1).
We defuzzify the fuzzy cost coefficients into crisp ones by a fuzzy number ranking method. Yager’s Ranking index [2] is
defined by
1
~
0.5(c c ), - level cut of the fuzzy number c .
~
L U L U
Y( c ) = where (c ,c ) is the
0
~ ~ ~
The Yager’s ranking index Y( c ) gives the representative value of the fuzzy number c . Since Y( Cij ) are crisp values, this
problem is obviously the crisp assignment problem of the form (1) which can be solved by Hungarian Method.
The steps of the proposed method are
Step 1: Replace the cost matrix Cij with linguistic variables by triangular or trapezoidal fuzzy numbers.
Step 2: Find Yager’s Ranking index.
Step 3: Replace Triangular or Trapezoidal numbers by their respective ranking indices.
Step 4: Solve the resulting AP using Hungarian technique to find optimal assignment.
Numerical Example
Let us consider a Fuzzy Assignment Problem with rows representing four persons W, X, Y, Z and columns representing the
~
four jobs, Job1, Job2, Job3 and Job4 with assignment cost varying between 0$ to 50$. The cost matrix [ Cij ] is given whose
elements are linguistic variables which are replaced by fuzzy numbers. The problem is then solved by Hungarian method to
find the optimal assignment.
1 2 3 4
W
extremelyl ow low fairlyhigh extremelyh igh
X
low verylow high veryhigh
Y medium extremelyh igh verylow extremelyl ow
veryhigh fairlylow
Z low fairlylow
Solution: The Linguistic variables showing the qualitative data is converted into quantitative data using the following table.
As the assignment cost varies between 0$ to 50$ the minimum possible value is taken as 0 and the maximum possible value
is taken as 50.
Extremely low (0,2,5)
Very low (1,2,4)
Low (4,8,12)
Fairly low (15,18,20)
Medium (23,25,27)
Fairly High (28,30,32)
High (33,36,38)
Very High (37,40,42)
Extremely High (44,48,50)
Issn 2250-3005(online) August| 2012 Page 1067
4. International Journal Of Computational Engineering Research (ijceronline.com) Vol. 2 Issue. 4
The linguistic variables are represented by triangular fuzzy numbers
Now
1 2 3 4
W (0,2,5) (4,8,12) (28,30,32) (44,48,50)
X (4,8,12) (1,2,4) (33,36,38) (37,40,42)
Y
(23,25,27) (44,48,50) (1,2,4) (0,2,5)
(37,40,42) (15,18,20)
Z (4,8,12) (15,18,20) …(3)
we calculate Y(0,2,5) by applying the Yager’s Ranking Method.
The membership function of the triangular fuzzy number (0,2,5) is
x0
,0 x 2
20
(x)
x5
2 x5
25
The α − cut of the fuzzy number (0,2,5) is (cαL , cαU ) = (2α ,5 −3α ) for which
1 1
~
0.5(c , c )d 0.5(2 5 3 )d
L U
Y( c 11 ) = Y(0,2,5) = = 2.25
0 0
Proceeding similarly, the Yager’s indices for the costs ~
c ij are calculated as:
~ ~ ~ ~ ~ ~ ~ ~
Y( c 12) = 8, Y( c 13) = 31, Y( c 14) = 47.5, Y( c 21) = 8,Y( c 22) = 1.75, Y( c 23) = 81.5, Y( c 24) = 79.5, Y( c 31) =25,
~ ~ ~ ~ ~ ~ ~
Y( c 32) = 47.5, Y( c 33) =1.75, Y( c 34) = 2.25, Y( c 41) = 79.5, Y( c 42) = 8, Y( c 43) = 35.5, Y( c 44) = 35.5.
~
We replace these values for their corresponding c ij in (3) and solve the resulting assignment problem by using Hungarian
method.
2.25 8 31 47.5
8 1.75 35.75 39.75
25 47.5 1.75 2.25
39.75 17.75 17.75
8
Performing row reductions
0 5.75 28.75 45.25
6.25 0 34 38
23.25 45.75 0 0.5
31.75 9.75
0 9.75
Performing column reductions
0 5.75 28.75 44.75
6.25 0 34 37.5
23.25 45.75 0 0
31.75 9.25
0 9.75
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5. International Journal Of Computational Engineering Research (ijceronline.com) Vol. 2 Issue. 4
The optimal assignment matrix is
0 5.75 19.5 44.75
6.25 0 24.75 28.25
32.5 55 0 0
31.75 0
0 0.5
The optimal assignment schedule is W 1, X 2,Y 3, Z 4 .
Conclusions:
In this paper, the assignment costs are considered as linguistic variables represented by fuzzy numbers. The fuzzy assignment
problem has been transformed into crisp assignment problem using Yager’s ranking indices. Hence we have shown that the
fuzzy assignment problems of qualitative nature can be solved in an effective way. This technique can also be tried in solving
other types of problems like Transportation problems, project scheduling problems, network flow problems etc.,
References
Journals
[1] Sakthi Mukherjee and Kajla Basu, “Application of Fuzzy Ranking Method for solving Assignment Problems with
Fuzzy Costs”, International Journal of Computational and Applied Mathematics, ISSN 1819-4966 Volume 5 Number
3(2010). Pp.359-368.
[2] Yager.R.R., “A procedure for ordering fuzzy subsets of the unit interval,” Information Sciences, vol 24, pp. 143-161,
1981
[3] Zadeh, L. A., “The concept of a linguistic variable and its application to approximate reasoning”, Part 1, 2 and 3,
Information Sciences, Vol.8, pp.199- 249, 1975; Vol.9, pp.43-58, 1976.
[4] Zadeh L. A, Fuzzy sets, Information and Control 8 (1965) 338–353.
Book:
[1] Klir G. J, Yuan B, Fuzzy Sets and Fuzzy Logic: Theory and Applications, Prentice-Hall, International Inc., 1995.
[2] S.Baskar,Operations Research for Technical and Managerial courses, Technical Publishers.
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