- The document discusses matrices, including definitions, operations, and examples of matrix addition, subtraction, transposition, and multiplication. It also covers linear programming, defining it as a method to optimize a mathematical model to achieve the best outcome.
- Key concepts covered include the definitions of a matrix and its elements, how to perform basic operations like addition and subtraction on matrices, and how matrices are multiplied using the dot product of rows and columns. Linear programming is introduced as a method using linear relationships to find the maximum or minimum value of an objective function.
Linear programming - Model formulation, Graphical MethodJoseph Konnully
The document discusses linear programming, including an overview of the topic, model formulation, graphical solutions, and irregular problem types. It provides examples to demonstrate how to set up linear programming models for maximization and minimization problems, interpret feasible and optimal solution regions graphically, and address multiple optimal solutions, infeasible solutions, and unbounded solutions. The examples aid in understanding the key steps and components of linear programming models.
Linear programming is a mathematical modeling technique used to determine optimal resource allocation to achieve objectives. It involves converting a problem into a linear mathematical model with decision variables, constraints, and an objective function. The optimal solution is found by systematically increasing the objective function value until infeasibility is reached. For example, a linear programming model was used to determine the optimal production mix and levels of two drug combinations to maximize profit given resource constraints. The optimal solution was found to be 320 dozen of drug X1 and 360 dozen of drug X2, utilizing all available resources and achieving $4,360 in weekly profit.
The document discusses decision analysis and decision making under uncertainty. It outlines six steps for decision making: 1) define the problem, 2) list alternatives, 3) identify outcomes, 4) list payoffs, 5) select a decision model, and 6) apply the model. Several decision models are described, including maximax, maximin, criterion of realism, equally likely, and minimax regret. Expected monetary value is discussed as a method for decision making under risk. The concept of expected value of perfect information is also introduced.
Sensitivity analysis in linear programming problem ( Muhammed Jiyad)Muhammed Jiyad
Topic includes : *Sensitivity Analysis *Objective function *Right Hand Side(RHS) *Sensitivity analysis using graph *Objective function coefficient *Reduced cost *Shadow pricing *Shadow pricing Microsoft Excel sensitivity report and solution.
Sensitivity analysis linear programming copyKiran Jadhav
This document discusses sensitivity analysis in linear programming. It begins by defining sensitivity analysis as investigating how changes to a linear programming model's parameters, like objective function coefficients or constraint coefficients, affect the optimal solution. It then discusses the basic parameter changes that can impact the solution, like right-hand side constants or new variables/constraints. The document also covers duality in linear programming and how the dual problem is derived from the primal problem by setting coefficient values to the resource costs at optimality. An example is provided to demonstrate how the dual problem is formulated.
Linear programming deals with optimizing a linear objective function subject to linear constraints. It involves determining the values of decision variables to maximize or minimize the objective function. The general linear programming model involves maximizing or minimizing a linear combination of n decision variables subject to m linear constraints, along with non-negativity restrictions on the decision variables. Formulating a linear programming problem involves identifying decision variables, expressing constraints and the objective function linearly in terms of the variables, and adding non-negativity restrictions.
This document discusses decision theory and decision making under uncertainty. It outlines the steps in decision theory as determining alternative actions, possible outcomes or states of nature, and constructing a payoff table to choose the alternative with the largest payoff. It describes four types of decision making environments - certainty, uncertainty, risk, and conflict - and gives criteria for decision making under uncertainty, including minimax, maximin, maximax, minimin, Laplace, and Hurwicz criteria. It provides an example applying these criteria to a farmer choosing which crop to plant.
This document provides an introduction and overview of goal programming (GP). It explains that GP is useful when an organization has multiple, sometimes conflicting goals that cannot all be optimized at the same time like in linear programming. GP establishes numeric goals for each objective and attempts to achieve each goal to a satisfactory level by minimizing deviations. The document outlines the basic components of a GP model, including defining goals and constraints, assigning priority levels to goals, and introducing deviational variables. It also provides an example to illustrate how to formulate a GP model and solve it graphically or using the modified simplex method.
Linear programming - Model formulation, Graphical MethodJoseph Konnully
The document discusses linear programming, including an overview of the topic, model formulation, graphical solutions, and irregular problem types. It provides examples to demonstrate how to set up linear programming models for maximization and minimization problems, interpret feasible and optimal solution regions graphically, and address multiple optimal solutions, infeasible solutions, and unbounded solutions. The examples aid in understanding the key steps and components of linear programming models.
Linear programming is a mathematical modeling technique used to determine optimal resource allocation to achieve objectives. It involves converting a problem into a linear mathematical model with decision variables, constraints, and an objective function. The optimal solution is found by systematically increasing the objective function value until infeasibility is reached. For example, a linear programming model was used to determine the optimal production mix and levels of two drug combinations to maximize profit given resource constraints. The optimal solution was found to be 320 dozen of drug X1 and 360 dozen of drug X2, utilizing all available resources and achieving $4,360 in weekly profit.
The document discusses decision analysis and decision making under uncertainty. It outlines six steps for decision making: 1) define the problem, 2) list alternatives, 3) identify outcomes, 4) list payoffs, 5) select a decision model, and 6) apply the model. Several decision models are described, including maximax, maximin, criterion of realism, equally likely, and minimax regret. Expected monetary value is discussed as a method for decision making under risk. The concept of expected value of perfect information is also introduced.
Sensitivity analysis in linear programming problem ( Muhammed Jiyad)Muhammed Jiyad
Topic includes : *Sensitivity Analysis *Objective function *Right Hand Side(RHS) *Sensitivity analysis using graph *Objective function coefficient *Reduced cost *Shadow pricing *Shadow pricing Microsoft Excel sensitivity report and solution.
Sensitivity analysis linear programming copyKiran Jadhav
This document discusses sensitivity analysis in linear programming. It begins by defining sensitivity analysis as investigating how changes to a linear programming model's parameters, like objective function coefficients or constraint coefficients, affect the optimal solution. It then discusses the basic parameter changes that can impact the solution, like right-hand side constants or new variables/constraints. The document also covers duality in linear programming and how the dual problem is derived from the primal problem by setting coefficient values to the resource costs at optimality. An example is provided to demonstrate how the dual problem is formulated.
Linear programming deals with optimizing a linear objective function subject to linear constraints. It involves determining the values of decision variables to maximize or minimize the objective function. The general linear programming model involves maximizing or minimizing a linear combination of n decision variables subject to m linear constraints, along with non-negativity restrictions on the decision variables. Formulating a linear programming problem involves identifying decision variables, expressing constraints and the objective function linearly in terms of the variables, and adding non-negativity restrictions.
This document discusses decision theory and decision making under uncertainty. It outlines the steps in decision theory as determining alternative actions, possible outcomes or states of nature, and constructing a payoff table to choose the alternative with the largest payoff. It describes four types of decision making environments - certainty, uncertainty, risk, and conflict - and gives criteria for decision making under uncertainty, including minimax, maximin, maximax, minimin, Laplace, and Hurwicz criteria. It provides an example applying these criteria to a farmer choosing which crop to plant.
This document provides an introduction and overview of goal programming (GP). It explains that GP is useful when an organization has multiple, sometimes conflicting goals that cannot all be optimized at the same time like in linear programming. GP establishes numeric goals for each objective and attempts to achieve each goal to a satisfactory level by minimizing deviations. The document outlines the basic components of a GP model, including defining goals and constraints, assigning priority levels to goals, and introducing deviational variables. It also provides an example to illustrate how to formulate a GP model and solve it graphically or using the modified simplex method.
The document provides an outline of topics related to linear programming, including:
1) An introduction to linear programming models and examples of problems that can be solved using linear programming.
2) Developing linear programming models by determining objectives, constraints, and decision variables.
3) Graphical and simplex methods for solving linear programming problems.
4) Using a simplex tableau to iteratively solve a sample product mix problem to find the optimal solution.
The document describes several applications of linear programming (LP) models in business contexts such as marketing, manufacturing, and scheduling. It provides examples to illustrate LP formulations for media mix optimization, production planning, and survey sampling cost minimization. Screenshots of LP solutions in Excel and other software are presented. The goal of the chapter is to help students understand how to model and solve real-world LP problems.
The document discusses linear programming, which is a method for optimizing a linear objective function subject to linear equality and inequality constraints. It describes how to formulate a linear programming problem by defining the objective function and constraints in terms of decision variables. It also discusses graphical and algebraic solution methods, including identifying an optimal solution at an extreme point of the feasible region. Applications of linear programming are mentioned in areas like business, industry, and marketing.
This document outlines concepts related to decision making under uncertainty and risk. It discusses six steps to decision making, including defining the problem, listing alternatives and outcomes, and identifying payoffs. It then covers various decision making models for uncertainty, like maximax, maximin, and expected monetary value. Sensitivity analysis is introduced as a way to examine how decisions may change with different input data. The document uses examples and tables to illustrate key concepts in decision analysis.
The document discusses linear programming, which is a mathematical modeling technique used to allocate limited resources optimally. It provides examples of linear programming problems and their formulation. Key aspects covered include defining decision variables and constraints, developing the objective function, and interpreting feasible and optimal solutions. Graphical and algebraic solution methods like the simplex method are also introduced.
Different kind of distance and Statistical DistanceKhulna University
A short brief of distance and statistical distance which is core of multivariate analysis.................you will get here some more simple conception about distances and statistical distance.
The document provides an introduction to operations research. It discusses that operations research is a systematic approach to decision-making and problem-solving that uses techniques like statistics, mathematics, and modeling to arrive at optimal solutions. It also briefly outlines some primary tools used in operations research like statistics, game theory, and probability theory. The document then gives a short history of operations research, noting that it originated in the UK during World War II to analyze problems like radar systems. It concludes with discussing the scope and applications of operations research in fields like management, regulation, and economics.
This document describes a linear programming problem (LPP) to minimize cost. The problem involves determining the optimal amounts of two fertilizer brands, SuperGro and CropQuick, to purchase to minimize total cost while meeting nitrogen and phosphate requirements. The LPP constructs decision variables for amounts of each brand, an objective function to minimize total cost, and constraints on nitrogen and phosphate levels. The optimal solution is to purchase 8 bags of CropQuick for a minimum total cost of 24.
This document discusses decision theory and decision-making under conditions of certainty, uncertainty, and risk. It defines key decision-making concepts like available courses of action, states of nature or outcomes, payoffs, and expected monetary value. Methods for decision-making under uncertainty include the maximin, maximax, minimax regret, Hurwitz, and Bayes criteria. Decision-making under risk involves assigning probabilities to outcomes and selecting the action with the largest expected payoff value or smallest expected opportunity loss.
Transportation Problem In Linear ProgrammingMirza Tanzida
This work is an assignment on the course of 'Mathematics for Decision Making'. I think, it will provide some basic concept about transportation problem in linear programming.
Vogel's Approximation Method (VAM) is a 3 step process for solving transportation problems:
1) Compute penalties for each row and column based on smallest costs.
2) Identify largest penalty and assign lowest cost variable the highest possible value, crossing out exhausted row or column.
3) Recalculate penalties and repeat until all requirements are satisfied.
The document provides an overview of linear programming, including its applications, assumptions, and mathematical formulation. Some key points:
- Linear programming is a tool for maximizing or minimizing quantities like profit or cost, subject to constraints. 50-90% of business decisions and computations involve linear programming.
- Applications in business include production, personnel, inventory, marketing, financial, and blending problems. The objective is to optimize variables like costs, profits, or resources while meeting constraints.
- Assumptions of linear programming include certainty, linearity/proportionality, additivity, divisibility, non-negativity, finiteness, and optimality at corner points.
- A linear programming problem is modeled mathemat
This document summarizes the two phase simplex method for solving linear programming problems. In phase I, artificial variables are introduced to convert infeasible problems into feasible problems. The objective is to minimize the artificial variables. If the minimum is zero, the original problem is feasible and phase II begins. Phase II uses the original objective function and simplex method to find an optimal solution. An example problem is provided to illustrate the two phase method.
This chapter discusses numerical descriptive measures used to describe the central tendency, variation, and shape of data. It covers calculating the mean, median, mode, variance, standard deviation, and coefficient of variation for data. The geometric mean is introduced as a measure of the average rate of change over time. Outliers are identified using z-scores. Methods for summarizing and comparing data using these descriptive statistics are presented.
The document introduces nonlinear programming (NLP) and contrasts it with linear programming (LP). NLP involves optimization problems with nonlinear objective functions or constraints, which are more difficult to solve than LP problems. Examples are provided to illustrate how NLP searches can fail to find the global optimum. The document also formulates two NLP examples: one involving profit maximization for chair pricing, and another involving investment portfolio selection to minimize risk.
This document discusses linear programming applications in marketing, manufacturing, and other areas. It provides examples to demonstrate how to model and solve linear programming problems involving media mix optimization, production scheduling, inventory management, and other scenarios. Specifically, it presents sample problems and solutions involving marketing mix optimization for a gambling club, sampling costs for a market research firm, production planning for a tie manufacturer, and multi-period production scheduling for an electric motor company. The chapter aims to illustrate how to apply linear programming to optimize objectives subject to constraints across various business applications.
The document discusses operations research (OR), including its origins during WWII to optimize resource allocation, its goal of applying scientific principles to optimize complex business and organizational problems, and its use of quantitative modeling and analysis. OR aims to find the global optimum solution by analyzing relationships between system components. It uses interdisciplinary teams and scientific methods to develop mathematical and other models of real-world problems, which are then solved using techniques like linear programming. The models represent important variables and constraints. OR has wide applications in areas like the military, production, transportation, and resource allocation.
This document provides an introduction to operations management and capacity planning. It discusses key concepts such as the three main functions of business organizations (finance, marketing, operations), the production process of transforming inputs to outputs, and differences between production of goods versus delivery of services. It also covers topics like measuring and improving productivity, factors that affect productivity, and importance of capacity planning and defining capacity. The document aims to give students an overview of fundamental operations management principles.
Linear regression and correlation analysis ppt @ bec domsBabasab Patil
This document introduces linear regression and correlation analysis. It discusses calculating and interpreting the correlation coefficient and linear regression equation to determine the relationship between two variables. It covers scatter plots, the assumptions of regression analysis, and using regression to predict and describe relationships in data. Key terms introduced include the correlation coefficient, linear regression model, explained and unexplained variation, and the coefficient of determination.
15 yr-old launches world class website to help otherJason Fernandes
I would like to tell you about my site http://www/learningdisabilities.ourfamily.com I am Jason Fernandes, a learning disabled teen 15 yrs old, from Bombay, India, diagnosed 8 months ago. That day of diagnosis changed my life. I was diagnosed as ‘Gifted with superior intellectual functioning, as well as learning disabilities (dyslexia, dyscalculia and dysgraphia) –a mouthful…huh?
Samsung has some rough waters ahead. While still the biggest smartphone seller in the world, its last few quarters reveal an alarming trend. The company’s market share plummeted from 32.6% to 25.2%. If that wasn’t enough, news broke in quick succession that Samsung had lost its crown as top mobile seller in India and China to local manufacturers Micromax and Xiaomi respectively. This is extremely worrying given the fact that these countries represent the greatest potential growth markets for the company.
The document provides an outline of topics related to linear programming, including:
1) An introduction to linear programming models and examples of problems that can be solved using linear programming.
2) Developing linear programming models by determining objectives, constraints, and decision variables.
3) Graphical and simplex methods for solving linear programming problems.
4) Using a simplex tableau to iteratively solve a sample product mix problem to find the optimal solution.
The document describes several applications of linear programming (LP) models in business contexts such as marketing, manufacturing, and scheduling. It provides examples to illustrate LP formulations for media mix optimization, production planning, and survey sampling cost minimization. Screenshots of LP solutions in Excel and other software are presented. The goal of the chapter is to help students understand how to model and solve real-world LP problems.
The document discusses linear programming, which is a method for optimizing a linear objective function subject to linear equality and inequality constraints. It describes how to formulate a linear programming problem by defining the objective function and constraints in terms of decision variables. It also discusses graphical and algebraic solution methods, including identifying an optimal solution at an extreme point of the feasible region. Applications of linear programming are mentioned in areas like business, industry, and marketing.
This document outlines concepts related to decision making under uncertainty and risk. It discusses six steps to decision making, including defining the problem, listing alternatives and outcomes, and identifying payoffs. It then covers various decision making models for uncertainty, like maximax, maximin, and expected monetary value. Sensitivity analysis is introduced as a way to examine how decisions may change with different input data. The document uses examples and tables to illustrate key concepts in decision analysis.
The document discusses linear programming, which is a mathematical modeling technique used to allocate limited resources optimally. It provides examples of linear programming problems and their formulation. Key aspects covered include defining decision variables and constraints, developing the objective function, and interpreting feasible and optimal solutions. Graphical and algebraic solution methods like the simplex method are also introduced.
Different kind of distance and Statistical DistanceKhulna University
A short brief of distance and statistical distance which is core of multivariate analysis.................you will get here some more simple conception about distances and statistical distance.
The document provides an introduction to operations research. It discusses that operations research is a systematic approach to decision-making and problem-solving that uses techniques like statistics, mathematics, and modeling to arrive at optimal solutions. It also briefly outlines some primary tools used in operations research like statistics, game theory, and probability theory. The document then gives a short history of operations research, noting that it originated in the UK during World War II to analyze problems like radar systems. It concludes with discussing the scope and applications of operations research in fields like management, regulation, and economics.
This document describes a linear programming problem (LPP) to minimize cost. The problem involves determining the optimal amounts of two fertilizer brands, SuperGro and CropQuick, to purchase to minimize total cost while meeting nitrogen and phosphate requirements. The LPP constructs decision variables for amounts of each brand, an objective function to minimize total cost, and constraints on nitrogen and phosphate levels. The optimal solution is to purchase 8 bags of CropQuick for a minimum total cost of 24.
This document discusses decision theory and decision-making under conditions of certainty, uncertainty, and risk. It defines key decision-making concepts like available courses of action, states of nature or outcomes, payoffs, and expected monetary value. Methods for decision-making under uncertainty include the maximin, maximax, minimax regret, Hurwitz, and Bayes criteria. Decision-making under risk involves assigning probabilities to outcomes and selecting the action with the largest expected payoff value or smallest expected opportunity loss.
Transportation Problem In Linear ProgrammingMirza Tanzida
This work is an assignment on the course of 'Mathematics for Decision Making'. I think, it will provide some basic concept about transportation problem in linear programming.
Vogel's Approximation Method (VAM) is a 3 step process for solving transportation problems:
1) Compute penalties for each row and column based on smallest costs.
2) Identify largest penalty and assign lowest cost variable the highest possible value, crossing out exhausted row or column.
3) Recalculate penalties and repeat until all requirements are satisfied.
The document provides an overview of linear programming, including its applications, assumptions, and mathematical formulation. Some key points:
- Linear programming is a tool for maximizing or minimizing quantities like profit or cost, subject to constraints. 50-90% of business decisions and computations involve linear programming.
- Applications in business include production, personnel, inventory, marketing, financial, and blending problems. The objective is to optimize variables like costs, profits, or resources while meeting constraints.
- Assumptions of linear programming include certainty, linearity/proportionality, additivity, divisibility, non-negativity, finiteness, and optimality at corner points.
- A linear programming problem is modeled mathemat
This document summarizes the two phase simplex method for solving linear programming problems. In phase I, artificial variables are introduced to convert infeasible problems into feasible problems. The objective is to minimize the artificial variables. If the minimum is zero, the original problem is feasible and phase II begins. Phase II uses the original objective function and simplex method to find an optimal solution. An example problem is provided to illustrate the two phase method.
This chapter discusses numerical descriptive measures used to describe the central tendency, variation, and shape of data. It covers calculating the mean, median, mode, variance, standard deviation, and coefficient of variation for data. The geometric mean is introduced as a measure of the average rate of change over time. Outliers are identified using z-scores. Methods for summarizing and comparing data using these descriptive statistics are presented.
The document introduces nonlinear programming (NLP) and contrasts it with linear programming (LP). NLP involves optimization problems with nonlinear objective functions or constraints, which are more difficult to solve than LP problems. Examples are provided to illustrate how NLP searches can fail to find the global optimum. The document also formulates two NLP examples: one involving profit maximization for chair pricing, and another involving investment portfolio selection to minimize risk.
This document discusses linear programming applications in marketing, manufacturing, and other areas. It provides examples to demonstrate how to model and solve linear programming problems involving media mix optimization, production scheduling, inventory management, and other scenarios. Specifically, it presents sample problems and solutions involving marketing mix optimization for a gambling club, sampling costs for a market research firm, production planning for a tie manufacturer, and multi-period production scheduling for an electric motor company. The chapter aims to illustrate how to apply linear programming to optimize objectives subject to constraints across various business applications.
The document discusses operations research (OR), including its origins during WWII to optimize resource allocation, its goal of applying scientific principles to optimize complex business and organizational problems, and its use of quantitative modeling and analysis. OR aims to find the global optimum solution by analyzing relationships between system components. It uses interdisciplinary teams and scientific methods to develop mathematical and other models of real-world problems, which are then solved using techniques like linear programming. The models represent important variables and constraints. OR has wide applications in areas like the military, production, transportation, and resource allocation.
This document provides an introduction to operations management and capacity planning. It discusses key concepts such as the three main functions of business organizations (finance, marketing, operations), the production process of transforming inputs to outputs, and differences between production of goods versus delivery of services. It also covers topics like measuring and improving productivity, factors that affect productivity, and importance of capacity planning and defining capacity. The document aims to give students an overview of fundamental operations management principles.
Linear regression and correlation analysis ppt @ bec domsBabasab Patil
This document introduces linear regression and correlation analysis. It discusses calculating and interpreting the correlation coefficient and linear regression equation to determine the relationship between two variables. It covers scatter plots, the assumptions of regression analysis, and using regression to predict and describe relationships in data. Key terms introduced include the correlation coefficient, linear regression model, explained and unexplained variation, and the coefficient of determination.
15 yr-old launches world class website to help otherJason Fernandes
I would like to tell you about my site http://www/learningdisabilities.ourfamily.com I am Jason Fernandes, a learning disabled teen 15 yrs old, from Bombay, India, diagnosed 8 months ago. That day of diagnosis changed my life. I was diagnosed as ‘Gifted with superior intellectual functioning, as well as learning disabilities (dyslexia, dyscalculia and dysgraphia) –a mouthful…huh?
Samsung has some rough waters ahead. While still the biggest smartphone seller in the world, its last few quarters reveal an alarming trend. The company’s market share plummeted from 32.6% to 25.2%. If that wasn’t enough, news broke in quick succession that Samsung had lost its crown as top mobile seller in India and China to local manufacturers Micromax and Xiaomi respectively. This is extremely worrying given the fact that these countries represent the greatest potential growth markets for the company.
Cyber Warfare: Can business trust the government to protect them?Jason Fernandes
The past several years have seen a rise in private companies being targeted by everyone from state sponsored hackers to criminals and even so called Hacktivists (hackers for a cause). Businesses have found that the attacks have reached a level of sophistication that often times is far in excess of what the company is handle themselves. Particularly in the case of state sponsored cyber-attacks, fighting back on equal footing is not an option for most businesses. The alarming number of recent high profile hacks occurring with increasing frequency have
many questioning the role of government, the responsibilities of businesses and whether closer cooperation between the two could successfully combat cyber-attacks.
Can the internet of things survive the coming warJason Fernandes
The Internet of Things (IoT) phrase is quite a buzzword these days and its definition will vary widely depending on who you ask. The main concept, however, is that devices and appliances that would traditionally function independently are able to communicate with other data sources or devices to enhance their own functionality and efficiency.
Nexus lives! Android Silver and the future of Google Play editionJason Fernandes
Just a few weeks ago, Google launched two new hardware products, the Google Nexus 6 and the Nexus 9. For the uninitiated, in the past Google’s Nexus line had always focused on providing its devotees with upper- to mid-range phones at excellent prices. This time around, however, the prices are fairly astronomical while the specs too are rather high end.
Net Neutrality and what initiatives like Internet.org and Airtel Zero mean fo...Jason Fernandes
What seemed like a noble initiative to connect millions of the world’s poor to basic internet services in the developing world had gone largely unnoticed till suddenly Facebook’s Internet.org initiative found itself mired in controversy. The main objection to Internet.org was its perceived violation of the principles of Net Neutrality.
This document discusses key performance indicators (KPIs) for an executive account manager position. It provides examples of KPIs, performance appraisal forms, and a process for creating KPIs for this role. The document recommends visiting an online site for more KPI samples and materials related to performance management.
El documento describe los procesos administrativos de dirección y control de una organización. Explica que la dirección requiere una estructura organizacional para lograr los objetivos planificados a través de etapas como la toma de decisiones, integración, motivación, comunicación, supervisión y autoridad. También cubre los ciclos de mantenimiento, corrección y mejora, así como los elementos clave del control como establecer estándares, medir resultados y realizar correcciones.
Research in Motion (now known simply as BlackBerry) last month released two new handsets, the Z10 and the Q10, both running the brand new BlackBerry 10 operating system (OS). The latest offerings are BlackBerry’s first new hardware products in several years and reflect what many believe is the company’s last-ditch attempt to stay relevant in a market increasingly more demanding of its gadgets.
AN INTERVIEW WITH JASON FERNANDES, WHO PROVES THAT DISABILITY OR OTHERWISE, T...Jason Fernandes
Web designer, proprietor of an internet web site and winner of the Cable and Wireless International Childnet award and NINE other awards. A software trouble shooter for one of India’s most widely circulated computer magazines- Chip, and an amateur clay modeller to boot, Jason Fernandes has quite a few laurels to carry with him when he turns in for the night.
We want an IT policy which will have a focus on startups and not just benefit medium and large scale businesses. Encouraging a vibrant startup atmosphere is exactly what Goa should do if is to ever grow as an IT hub.
Periscope and Meerkat: Why broadcasters should stop worrying and learn to lov...Jason Fernandes
Mobile live streaming apps like Periscope and Meerkat allow users to broadcast live video from their smartphones. This poses challenges for traditional broadcasters, as anyone can livestream events without permission. Some broadcasters fear it could undermine their business model. However, others see an opportunity to embrace the technology. Networks could provide extra context around livestreams to add value over amateur broadcasts. Politicians also use these apps to connect with voters without traditional media filters. Overall, broadcasters will need to adapt to stay relevant in a world where livestreaming is ubiquitous.
Este documento presenta los horarios de exámenes de septiembre para varios cursos y programas de estudios en un centro educativo. Incluye los horarios para exámenes de recuperación de asignaturas pendientes de cursos anteriores. Los exámenes se realizarán desde el lunes 1 hasta el miércoles 3 de septiembre en diferentes franjas horarias por la mañana y tarde.
15-YEAR-OLD LAUNCHES WORLD-CLASS WEBSITE TO HELP OTHERSJason Fernandes
I would like to tell you about my site http://www/learningdisabilities.ourfamily.com I am Jason Fernandes, a learning disabled teen 15 yrs old, from Bombay, India, diagnosed 8 months ago. That day of diagnosis changed my life. I was diagnosed as ‘Gifted with superior intellectual functioning, as well as learning disabilities (dyslexia, dyscalculia and dysgraphia) –a mouthful…huh?
This document provides an introduction to linear programming. It defines linear programming as an optimization problem that involves maximizing or minimizing a linear objective function subject to linear constraints. Various terminology used in linear programming like decision variables, objective function, and constraints are explained. Several examples of linear programming problems from areas like production planning, scheduling, and resource allocation are presented and formulated mathematically. Graphical and algebraic solution methods for linear programming problems are discussed. The document also notes that integer programming problems cannot be solved using the same techniques as linear programs due to the discrete nature of the variables. Additional linear programming examples and problems from an operations research textbook are listed for further practice.
The document discusses linear programming and provides examples to illustrate the process. It explains that linear programming involves optimizing a linear objective function subject to linear constraints. There are three basic components: decision variables, an objective to optimize, and constraints. Examples show how to formulate the objective function and constraints as linear equations or inequalities. The optimal solution is found by analyzing the feasible region defined by the constraints and determining which corner point gives the best value for the objective function.
The document discusses linear programming problems. It defines linear programming as finding non-negative values of variables to satisfy linear constraints and optimize a linear objective function. It provides examples of transportation problems and diet problems formulated as linear programs. It describes graphical and algebraic methods for solving linear programs, including introducing slack and surplus variables to transform inequalities to equations.
1) The document discusses linear programming and its graphical solution method. It provides examples of forming linear programming models and using graphs to find the feasible region and optimal solution.
2) A toy manufacturing example is presented and modeled using linear programming with the objective of maximizing weekly profit. The feasible region is graphed and the optimal solution is identified.
3) Another example involving a wood products company is modeled and solved graphically to determine the optimal production mix to maximize profits. Corner points of the feasible region are identified and evaluated to find the optimal solution.
1. The document provides information about linear programming including definitions of key terms, steps to solve linear programming problems, examples worked out in detail, and exercises.
2. Linear programming involves optimizing (maximizing or minimizing) an objective function subject to certain constraints. It was first introduced by a Russian mathematician in the 1930s-1940s to optimize resources like manpower and materials during war time.
3. Examples worked out in the document show how to set up the constraints and objective function mathematically based on word problems, sketch the feasible region, find the corner points, and determine the optimal solution that maximizes or minimizes the objective function.
The document provides information about solving linear programming problems using Excel Solver. It begins with Excel terminology and functions used in linear programming like cells, references, and SUMPRODUCT. It then demonstrates how to activate and use Solver by entering data, recording parameters, and solving a sample production scheduling problem to maximize profit. The optimal solution and sensitivity analysis are examined to determine how changes affect the optimal solution.
The document describes the linear programming problem and the simplex method for solving it. It provides an example problem of determining the optimal product mix for two products to maximize total income. The summary is:
(1) The example problem involves determining the optimal levels of two products given constraints on raw materials, storage space, and production time to maximize total income.
(2) The simplex method is applied by setting up the linear programming model, identifying entering and leaving variables, and performing row operations to iteratively find a better solution until reaching an optimal solution.
(3) For the example, the optimal solution found through three iterations of the simplex method is to produce 270 units of the first product and 75
A geometrical approach in Linear Programming ProblemsRaja Agrawal
Here you can see how we can solve any problems related to Linear Programming using Graphical method. You will also come to know about Simplex method as well as Dual method in LPP. All taken examples are easy to understand. Do give a try and share your valuable feedback.
The document describes the linear programming model and the simplex method for solving linear programming problems.
The linear programming model involves maximizing a linear objective function subject to linear inequality constraints. Decision variables, constraints, and the objective function are defined.
The simplex method is then described as the process of iteratively finding feasible solutions and improving the objective function value until an optimal solution is reached. The method involves setting up a simplex tableau, identifying entering and leaving variables, and performing row operations to derive new tableaus until an optimal solution is identified where all coefficients in the objective function are positive.
An example problem of determining a product mix is presented to demonstrate applying the linear programming model and solving it step-by
The document describes how to formulate and solve linear programming problems by defining the components of a linear programming problem, describing how to model real-world problems as linear programs, and outlining two methods, graphical and simplex, to solve linear programming problems. It then provides examples of solving linear programming problems using these steps and methods.
The document describes the linear programming model and the simplex method for solving linear programming problems.
The linear programming model involves defining decision variables, an objective function to maximize, and constraints on the variables. The simplex method is then used to iteratively find the optimal solution. It involves setting up an initial tableau, identifying entering and leaving variables, performing row operations to create new tableaus, and checking for optimality until reached.
As an example, the document formulates a linear programming model to maximize profits for a company producing two products subject to resource constraints. It then applies the simplex method through multiple iterations to arrive at the optimal solution of 270 units of product 1 and 75 units of product 2 for maximum profits
The document describes the linear programming model and the simplex method for solving linear programming problems.
The linear programming model involves maximizing a linear objective function subject to linear inequality constraints involving decision variables. The simplex method is then used to solve linear programming problems by iteratively arriving at optimal feasible solutions.
The method involves setting up an initial tableau with slack variables, then selecting entering and leaving variables at each iteration to improve the objective function value, arriving at a final optimal solution where all coefficients in the objective function are positive. An example problem demonstrates applying the simplex method graphically and through tableau iterations to find the optimal product mix for a company.
This chapter discusses linear programming models and their graphical and computer-based solutions. It begins by outlining the learning objectives and chapter contents. Key points covered include:
- The basic assumptions and requirements of linear programming problems
- How to formulate an LP problem by defining variables, objectives and constraints
- Graphically representing constraints and determining the feasible region
- Using isoprofit lines and the corner point method to solve LP problems graphically
- An example problem involving determining optimal product mix for Flair Furniture is presented and solved graphically.
Simultaneous Equations Practical ConstructionDaniel Ross
The document discusses solving simultaneous equations using algebraic methods and graphing. It provides examples of setting up and solving systems of two equations with two unknowns to find the values of the unknowns. Various word problems are presented and worked through step-by-step to show how to set up the appropriate equations to find the unknown values being asked about, such as costs, numbers of items, etc. Strategies for setting up simultaneous equations from word problems are emphasized.
Introduction to Operations Research/ Management Science um1222
Here are the steps to solve this problem:
Let x = number of inches of orange beads
Let y = number of inches of black beads
Constraints:
x >= 0
y >= 0
x + y <= 24 (total length must be <= 24 inches)
y >= 2x (black beads must be >= 2x the length of orange beads)
y >= 5 (black beads must be >= 5 inches)
Objective: Maximize x + y (total length of necklace)
To sketch the problem:
Plot the lines y = 2x, x + y = 24, y = 5 on a xy-plane.
The shaded region satisfying all constraints is the feasible
The document defines linear programming as a branch of mathematics used to find the optimal solution to problems with constraints. It provides examples of using linear programming to maximize profit or minimize costs in organizations. It also introduces drawing linear inequalities and solving simultaneous inequalities. The steps to formulate a linear programming problem are identified as defining variables and objectives, translating constraints, finding feasible solutions, and evaluating objectives to find optimal solutions.
This document contains instructions for a mathematics exam, including:
- The exam consists of multiple choice, true/false, and short answer questions worth a total of 100 points.
- No books, notes, or calculators with CAS or QWERTY keyboards are allowed. Cell phones may not be used.
- The multiple choice section includes 8 questions worth 5 points each.
- The true/false section includes 15 statements worth 15 points total.
- Three short answer questions are each worth 15 points.
IRJET- Solving Quadratic Equations using C++ Application ProgramIRJET Journal
1) The document describes a C++ application program developed to solve quadratic equations. The program uses methods like factoring, completing the square, and the quadratic formula to find the solutions.
2) Field testing of the program showed students using it had an average score of 82.8% on a quadratic equations assessment, demonstrating the program's effectiveness.
3) Advantages of using such an application include reducing errors, supporting problem-solving processes, and creating awareness of mathematical concepts. It allows students to easily test conjectures and replay problem-solving steps.
The document discusses basic matrix operations including addition, subtraction, scalar multiplication, and matrix multiplication. It defines matrix addition and subtraction as adding or subtracting corresponding entries of matrices of the same size. Scalar multiplication is multiplying each entry of a matrix by a scalar number. Matrix multiplication is defined as the product of a row of the first matrix and a column of the second matrix, with the results making up entries of the product matrix. An example shows a 2x4 matrix multiplied by a 4x1 matrix to yield a 2x1 product matrix.
বাংলাদেশের অর্থনৈতিক সমীক্ষা ২০২৪ [Bangladesh Economic Review 2024 Bangla.pdf] কম্পিউটার , ট্যাব ও স্মার্ট ফোন ভার্সন সহ সম্পূর্ণ বাংলা ই-বুক বা pdf বই " সুচিপত্র ...বুকমার্ক মেনু 🔖 ও হাইপার লিংক মেনু 📝👆 যুক্ত ..
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Introduction to AI for Nonprofits with Tapp NetworkTechSoup
Dive into the world of AI! Experts Jon Hill and Tareq Monaur will guide you through AI's role in enhancing nonprofit websites and basic marketing strategies, making it easy to understand and apply.
Strategies for Effective Upskilling is a presentation by Chinwendu Peace in a Your Skill Boost Masterclass organisation by the Excellence Foundation for South Sudan on 08th and 09th June 2024 from 1 PM to 3 PM on each day.
Assessment and Planning in Educational technology.pptxKavitha Krishnan
In an education system, it is understood that assessment is only for the students, but on the other hand, the Assessment of teachers is also an important aspect of the education system that ensures teachers are providing high-quality instruction to students. The assessment process can be used to provide feedback and support for professional development, to inform decisions about teacher retention or promotion, or to evaluate teacher effectiveness for accountability purposes.
it describes the bony anatomy including the femoral head , acetabulum, labrum . also discusses the capsule , ligaments . muscle that act on the hip joint and the range of motion are outlined. factors affecting hip joint stability and weight transmission through the joint are summarized.
Main Java[All of the Base Concepts}.docxadhitya5119
This is part 1 of my Java Learning Journey. This Contains Custom methods, classes, constructors, packages, multithreading , try- catch block, finally block and more.
The simplified electron and muon model, Oscillating Spacetime: The Foundation...RitikBhardwaj56
Discover the Simplified Electron and Muon Model: A New Wave-Based Approach to Understanding Particles delves into a groundbreaking theory that presents electrons and muons as rotating soliton waves within oscillating spacetime. Geared towards students, researchers, and science buffs, this book breaks down complex ideas into simple explanations. It covers topics such as electron waves, temporal dynamics, and the implications of this model on particle physics. With clear illustrations and easy-to-follow explanations, readers will gain a new outlook on the universe's fundamental nature.
3. Matrix
is an ordered rectangular array
of numbers.
A Matrix
(This one has 2 Rows and 3 Columns)
4. A matrix is usually shown by a capital
letter (such as A, or B)
Each entry (or "element") is shown by
a lower case letter with a "subscript"
of row,column:
5. Rows and Columns
So which is the row and which is the
column?
Rows go left-right
Columns go up-down
To remember that rows come before
columns use the word "arc":
ar,c
6. Here are some sample entries:
b1,1 = 6 (the entry at row 1, column 1 is 6)
ar,c
b1,3 = 24 (the entry at row 1, column 3 is
24)
Example:
7. Addition
Two matrices A and B of the
same size can be added or to
produce a matrix of the same size.
3 8 4 0
4 6 1 -9
8. These are the calculations:
3+4=7 8+0=8
4+1=5 6+(-9)=-3
9. Subtraction
Two matrices C and D of the same
size can be subtracted or to produce a
matrix of the same size.
3 8 4 0
4 6 1 -9
10. These are the calculations:
3-4=-1 8-0=8
4-1=3 6-(-9)=15
Note: subtracting is actually defined
as the addition of a negative matrix:
A + (-B)
11. Tranpose of a Matrix
To "transpose" a matrix, swap
the rows and columns. We put a
"T" in the top right-hand corner to
mean transpose:
12. Multiply
But to multiply a matrix by
another matrix we need to do the
"dot product" of rows and columns
13. The "Dot Product" is where we multiply matching
members, then sum up:
(1, 2, 3) • (7, 9, 11) = 1×7 + 2×9 + 3×11 = 58
We match the 1st members (1 and 7), multiply them,
likewise for the 2nd members (2 and 9) and the 3rd
members (3 and 11), and finally sum them up.
14. (1, 2, 3) • (8, 10, 12) = 1×8 + 2×10 + 3×12 = 64
We can do the same thing for the 2nd
row and 1st column:
(4, 5, 6) • (7, 9, 11) = 4×7 + 5×9 + 6×11 = 139
And for the 2nd row and 2nd column:
(4, 5, 6) • (8, 10, 12) = 4×8 + 5×10 + 6×12 = 154
18. Linear programming
- (LP; also called linear optimization) is
a method to achieve the best outcome
(such as maximum profit or lowest
cost) in a mathematical model whose
requirements are represented
by linear relationships.
19. SOLUTION OF LINEAR PROGRAMMING
PROBLEMS
• THEOREM 1 If a linear programming problem has
a solution, then it must occur at a vertex, or
corner
• point, of the feasible set, S, associated with the
problem. Furthermore, if the objective function P
is
• optimized at two adjacent vertices of S, then it is
optimized at every point on the line segment
joining
• these two vertices, in which case there are
infinitely many solutions to the problem.
20. THEOREM 2
• Suppose we are given a linear programming problem
with a feasible set S and an objective
• function P = ax+by. Then,
• If S is bounded then P has both a maximum and
minimum value on S
• If S is unbounded and both a and b are nonnegative,
then P has a minimum value on S provided
• that the constraints defining S include the inequalities x≥
0 and y≥ 0.
• If S is the empty set, then the linear programming
problem has no solution; that is, P has neither
• a maximum nor a minimum value.
21. THE METHOD OF CORNERS
• Graph the feasible set (region), S.
• Find the EXACT coordinates of all vertices (corner
points) of S.
• Evaluate the objective function, P, at each vertex
• The maximum (if it exists) is the largest value of P
at a vertex. The minimum is the smallest value
• of P at a vertex. If the objective function is
maximized (or minimized) at two vertices, it is
• minimized (or maximized) at every point
connecting the two vertices
22. Examples:
1.
The question asks for the optimal number of
calculators, so my variables will stand for
that:
x: number of scientific calculators produced
y: number of graphing calculators produced
23. Since they can't produce negative
numbers of calculators, I have the two
constraints, x > 0 and y > 0. But in this
case, I can ignore these constraints,
because I already have
that x > 100 andy > 80.
24. The exercise also gives
maximums: x < 200 and y < 170. The
minimum shipping requirement gives
mex + y > 200; in other words, y > –x +
200. The profit relation will be my
optimization equation: P = –2x + 5y. So the
entire system is:
P = –2x + 5y, subject to:
100 < x < 200
80 < y < 170
y > –x + 200
26. When you test the corner points at (100,
170), (200, 170), (200, 80), (120, 80),
and (100, 100), you should obtain the
maximum value of P = 650 at (x, y) = (100,
170). That is, the solution is "100scientific
calculators and 170 graphing calculators".
27. 2. You need to buy some filing cabinets. You
know that Cabinet X costs $10 per unit,
requires six square feet of floor space, and
holds eight cubic feet of files. Cabinet Y
costs $20 per unit, requires eight square
feet of floor space, and holds twelve cubic
feet of files.
28. You have been given $140 for this
purchase, though you don't have to spend
that much. The office has room for no
more than 72 square feet of cabinets.
How many of which model should you
buy, in order to maximize storage
volume?
29. The question ask for the number of
cabinets I need to buy, so my variables will
stand for that:
x: number of model X cabinets purchased
y: number of model Y cabinets purchased
30. Naturally, x > 0 and y > 0. I have to consider
costs and floor space (the "footprint" of
each unit), while maximizing the storage
volume, so costs and floor space will be my
constraints, while volume will be my
optimization equation.
cost: 10x + 20y < 140, or y < –( 1/2 )x + 7
space: 6x + 8y < 72, or y < –( 3/4 )x + 9
volume: V = 8x + 12y
32. When you test the corner points at (8, 3),
(0, 7), and (12, 0), you should obtain a
maximal volume of100 cubic feet by buying
eight of model X and three of model Y.
33. 3. A company makes two products (X
and Y) using two machines (A and B).
Each unit of X that is produced requires
50 minutes processing time on machine
A and 30 minutes processing time on
machine B. Each unit of Y that is
produced requires 24 minutes
processing time on machine A and 33
minutes processing time on machine B.
34. At the start of the current week there are
30 units of X and 90 units of Y in stock.
Available processing time on machine A is
forecast to be 40 hours and on machine B
is forecast to be 35 hours.
35. The demand for X in the current week is
forecast to be 75 units and for Y is forecast
to be 95 units. Company policy is to
maximise the combined sum of the units of
X and the units of Y in stock at the end of
the week.
36. • Formulate the problem of deciding how
much of each product to make in the
current week as a linear program.
• Solve this linear program graphically.
37. Solution:
Let
• x be the number of units of X produced
in the current week
• y be the number of units of Y produced
in the current week
38. then the constraints are:
50x + 24y <= 40(60) machine A time
30x + 33y <= 35(60) machine B time
x >= 75 - 30
i.e. x >= 45 so production of X >= demand (75)
- initial stock (30), which ensures we meet
demand
y >= 95 - 90
i.e. y >= 5 so production of Y >= demand (95) -
initial stock (90), which ensures we meet
demand
39. • The objective is: maximise (x+30-75) + (y+90-
95) = (x+y-50)
i.e. to maximise the number of units left in
stock at the end of the week
• It is plain from the diagram below that the
maximum occurs at the intersection of x=45
and 50x + 24y = 2400
40.
41. Solving simultaneously, rather than by
reading values off the graph, we have that
x=45 and y=6.25 with the value of the
objective function being 1.25
42. 4. The demand for two products in each of
the last four weeks is shown below.
Week
1 2 3 4
Demand - product 1 23 27 34 40
Demand - product 2 11 13 15 14
43. Apply exponential smoothing with a smoothing
constant of 0.7 to generate a forecast for the
demand for these products in week 5.
These products are produced using two
machines, X and Y. Each unit of product 1 that is
produced requires 15 minutes processing
onmachine X and 25 minutes processing on
machine Y. Each unit of product 2 that is
produced requires 7 minutes processing on
machine X and 45 minutes processing on
machine Y.
44. The available time on machine X in week 5 is
forecast to be 20 hours and on machine Y in
week 5 is forecast to be 15 hours. Each unit of
product 1 sold in week 5 gives a contribution to
profit of £10 and each unit of product 2 sold in
week 5 gives a contribution to profit of £4.
45. It may not be possible to produce enough to
meet your forecast demand for these products
in week 5 and each unit of unsatisfied demand
for product 1 costs £3, each unit of unsatisfied
demand for product 2 costs £1.
Formulate the problem of deciding how much of
each product to make in week 5 as a linear
program.
Solve this linear program graphically.
46. Solution
Note that the first part of the question is
a forecasting question so it is solved below.
For product 1 applying exponential smoothing with
a smoothing constant of 0.7 we get:
M1 = Y1 = 23
M2 = 0.7Y2 + 0.3M1 = 0.7(27) + 0.3(23) = 25.80
M3 = 0.7Y3 + 0.3M2 = 0.7(34) + 0.3(25.80) = 31.54
M4 = 0.7Y4 + 0.3M3 = 0.7(40) + 0.3(31.54) = 37.46
The forecast for week five is just the average for
week 4 = M4 = 37.46 = 31 (as we cannot have
fractional demand).
47. For product 2 applying exponential smoothing with
a smoothing constant of 0.7 we get:
M1 = Y1 = 11
M2 = 0.7Y2 + 0.3M1 = 0.7(13) + 0.3(11) = 12.40
M3 = 0.7Y3 + 0.3M2 = 0.7(15) + 0.3(12.40) = 14.22
M4 = 0.7Y4 + 0.3M3 = 0.7(14) + 0.3(14.22) = 14.07
The forecast for week five is just the average for
week 4 = M4 = 14.07 = 14 (as we cannot have
fractional demand).
We can now formulate the LP for week 5 using the
two demand figures (37 for product 1 and 14 for
product 2) derived above.
48. Let
x1 be the number of units of product 1 produced
x2 be the number of units of product 2 produced
where x1, x2>=0
The constraints are:
15x1 + 7x2 <= 20(60) machine X
25x1 + 45x2 <= 15(60) machine Y
x1 <= 37 demand for product 1
x2 <= 14 demand for product 2
The objective is to maximise profit, i.e.
maximise 10x1 + 4x2 - 3(37- x1) - 1(14-x2)
i.e. maximise 13x1 + 5x2 - 125
49. The graph is shown below, from the graph we have that the
solution occurs on the horizontal axis (x2=0) at x1=36 at which
point the maximum profit is 13(36) + 5(0) - 125 = £343
50. 5. A company is involved in the production of
two items (X and Y). The resources need to
produce X and Y are twofold, namely machine
time for automatic processing and craftsman
time for hand finishing. The table below gives
the number of minutes required for each item:
Machine time Craftsman time
Item X 13 20
Y 19 29
51. The company has 40 hours of machine time
available in the next working week but only 35
hours of craftsman time. Machine time is costed
at £10 per hour worked and craftsman time is
costed at £2 per hour worked. Both machine
and craftsman idle times incur no costs. The
revenue received for each item produced (all
production is sold) is £20 for X and £30 for Y. The
company has a specific contract to produce 10
items of X per week for a particular customer
52. • Formulate the problem of deciding how much
to produce per week as a linear program.
• Solve this linear program graphically.
Let
• x be the number of items of X
• y be the number of items of Y
53. then the LP is:
maximise
• 20x + 30y - 10(machine time worked) -
2(craftsman time worked)
subject to:
• 13x + 19y <= 40(60) machine time
• 20x + 29y <= 35(60) craftsman time
• x >= 10 contract
• x,y >= 0
• so that the objective function becomes
55. It is plain from the diagram below that the
maximum occurs at the intersection of x=10 and
20x + 29y <= 2100
Solving simultaneously, rather than by reading
values off the graph, we have that x=10 and
y=65.52 with the value of the objective function
being £1866.5
56.
57. Excercises
1. A company manufactures two products (A
and B) and the profit per unit sold is £3 and £5
respectively. Each product has to be assembled
on a particular machine, each unit of product A
taking 12 minutes of assembly time and each
unit of product B 25 minutes of assembly time.
The company estimates that the machine used
for assembly has an effective working week of
only 30 hours (due to maintenance/breakdown).
58. • Technological constraints mean that for every five
units of product A produced at least two units of
product B must be produced.
• Formulate the problem of how much of each
product to produce as a linear program.
• Solve this linear program graphically.
• The company has been offered the chance to hire
an extra machine, thereby doubling the effective
assembly time available. What is
the maximum amount you would be prepared to
pay (per week) for the hire of this machine and
why?
60. • 3. Solve
• minimise
• 4a + 5b + 6c
• subject to
• a + b >= 11
• a - b <= 5
• c - a - b = 0
• 7a >= 35 - 12b
• a >= 0 b >= 0 c >= 0
61. • 4.A carpenter makes tables and chairs. Each table
can be sold for a profit of £30 and each chair for a
profit of £10. The carpenter can afford to spend
up to 40 hours per week working and takes six
hours to make a table and three hours to make a
chair. Customer demand requires that he makes
at least three times as many chairs as tables.
Tables take up four times as much storage space
as chairs and there is room for at most four tables
each week.
Formulate this problem as a linear programming
problem and solve it graphically.
62. • 5. A farmer can plant up to 8 acres of land with
wheat and barley. He can earn $5,000 for every
acre he plants with wheat and $3,000 for every
acre he plants with barley. His use of a
necessary pesticide is limited by federal
regulations to 10 gallons for his entire 8 acres.
Wheat requires 2 gallons of pesticide for every
acre planted and barley requires just 1 gallon
per acre.
What is the maximum profit he can make?
63. • 6. A painter has exactly 32 units of yellow dye and
54 units of green dye.
He plans to mix as many gallons as possible of
color A and color B.
Each gallon of color A requires 4 units of yellow
dye and 1 unit of green dye.
Each gallon of color B requires 1 unit of yellow
dye and 6 units of green dye.
Find the maximum number of gallons he can
mix.
64. • 7. The Bead Store sells material for customers to
make their own jewelry. Customer can select
beads from various bins. Grace wants to design
her own Halloween necklace from orange and
black beads. She wants to make a necklace that is
at least 12 inches long, but no more than 24
inches long. Grace also wants her necklace to
contain black beads that are at least twice the
length of orange beads. Finally, she wants her
necklace to have at least 5 inches of black beads.
Find the constraints, sketch the problem and find
the vertices (intersection points)
65. 8.A garden shop wishes to prepare a supply
of special fertilizer at a minimal cost by
mixing two fertilizers, A and B.
66. • The mixture is to contain:
at least 45 units of phosphate
at least 36 units of nitrate
at least 40 units of ammonium
Fertilizer A costs the shop $.97 per pound.
Fertilizer B costs the shop $1.89 per pound.
fertilizer A contains 5 units of phosphate and 2
units of nitrate and 2 units of ammonium.
fertilizer B contains 3 units of phosphate and 3
units of nitrate and 5 units of ammonium.
how many pounds of each fertilizer should the
shop use in order to minimize their cost.