Economic interpretations of Linear Programming Problem
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The document discusses the economic interpretations of primal and dual linear programming. It explains that the primal problem seeks to maximize revenue under limited resources by allocating activities across resources. The dual variables represent the worth per unit of each resource. The dual constraints represent the "imputed cost" per unit of each resource, and the reduced cost of an activity is the imputed cost of resources needed to produce it. The optimality condition of the simplex method states that an unused activity should only increase if its reduced cost is negative, meaning its revenue exceeds its imputed costs, making it economically advantageous. An example of assembling toys illustrates how reduced cost can determine whether a toy is economically attractive to produce.
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![ECONOMIC INTERPRETATIONS Primal Linear Problem
•Primal LP problem can be viewed as
•a resource allocation model that seeks to maximize the revenueunder limited resources
•Primal problem
•‘n’ economic activities
•‘m’ resources
•‘cj’ revenue per unit of activity j
•Resource “i” with availability b(i)
Is consumed at the rate a(ij)] units
per unit activity j](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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- National income is determined by the production function based on the fixed supplies of capital and labor.
- Factor prices like wages and rental rates are determined by supply and demand in factor markets and will equal the marginal product of each input.
- Total income is distributed to capital and labor based on their marginal products.
- Aggregate demand is determined by consumption, investment, and government spending functions.
- Equilibrium in the goods market occurs when aggregate demand equals supply as determined by the production function.
- The loanable funds market equilibrates through adjustments in the real interest rate to equalize saving and investment
Microeconomics-for-Review.pptx
This document provides an overview of microeconomics concepts related to production and costs. It discusses the factors of production (land, labor, capital, entrepreneurship), the production process, and production curves including total physical product, average physical product, and marginal physical product. It also covers the law of diminishing returns and the three stages of production. The document then discusses theory of costs, including fixed and variable costs, and how to compute total, average, and marginal costs. Finally, it covers price and output determination under perfect competition, including demand and supply, the laws of demand and elasticity, and how price is determined in the market period.
Macroeconomic Consumer Theory
Lecture slides for an undergraduate course on Basic Macroeconomics that I taught in the Fall of 2007.
This lecture focuses on consumer theory.
chapter 2 revised.pptx
The document describes linear programming and its applications. It defines linear programming as using mathematical models to allocate limited resources to maximize profit or minimize cost. Key aspects covered include:
- The components of a linear programming model including decision variables, constraints, objective function and parameters.
- The steps to formulate a linear programming problem including identifying variables and constraints, writing the objective function and ensuring models follow linear programming assumptions.
- Examples of linear programming applications like product mix problems, portfolio selection, blending problems.
- The format of a general linear programming model.
- Assumptions required for linear programming like proportionality, additivity, divisibility and certainty of parameters.
Cost and BEA
This document discusses cost concepts and break-even analysis. It defines different types of costs such as explicit, implicit, fixed, and variable costs. It also explains the concepts of normal profit, incremental costs, sunk costs, and private and social costs. The document then provides an introduction to break-even analysis, discussing how to calculate the break-even point, total revenue, total costs, and profit or loss. It demonstrates using an example how to determine the number of units that must be sold to break even. The document concludes by discussing the uses and limitations of break-even analysis.
Similar to Economic interpretations of Linear Programming Problem (7)
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Regression-Sheldon Ross from Chapter 9-year2024
The term regression was originally employed by Francis Galton while describing the laws of inheritance. Galton believed that these laws caused population extremes to “regress
toward the mean.” By this he meant that children of individuals having extreme values of a certain characteristic would tend to have less extreme values of this characteristic
than their parent.
Interval estimates
This document discusses statistical interval estimates including one-sided intervals, confidence intervals for a normal mean when the variance is unknown, and confidence intervals for the variance of a normal distribution. It also covers estimating the difference in means of two normal populations.
Application Of Definite Integral
This document lists six topics that involve calculating volumes, lengths, centers of mass, areas of revolution, work, and fluid forces against walls using integration: 1) volumes, 2) lengths of plane curves, 3) centers of mass, 4) areas of surfaces of revolution, 5) work, and 6) fluid forces against planar walls.
Optimisation-two and three variables
Lagrange Multiplier Method for optimising functions of two and three variables... Gradient of a funtion
Optimisation- Nonlinear Problems
This document discusses applications of derivatives, specifically finding extreme values of functions. It begins by introducing the topic of finding extreme values and encourages the reader to work through examples themselves. It closes by thanking the reader and inviting any questions or comments.
Big-M method
This document discusses the Simplex Algorithm and Big M method for solving linear programming problems. It provides an example problem to minimize z=4x1+8x2+3x3 subject to constraints x1+x2>=2, 2x2+x3>=5, and x1,x2,x3>=0. The document walks through multiple iterations of applying the Simplex Algorithm to find the optimal solution for this example problem.
Transport problem
This document discusses various algorithms for solving transportation and assignment problems, including the transportation algorithms VAM and method of multipliers, as well as algorithms for solving assignment problems such as the Hungarian method and Klyne's AP method. It provides examples and explanations of these different algorithms and methods for balancing transportation problems and finding optimal assignments. The document concludes by thanking the audience and asking if there are any questions.
Arc length
This document discusses arc length and the unit tangent vector T for a curve, as well as torsion and the unit binormal vector B. It introduces the Frenet frame or TNB frame, which uses the unit tangent, normal, and binormal vectors to describe a curve.
Applications of Definite Intergral
This document provides an introduction to using definite integrals to calculate volumes, lengths of curves, centers of mass, surface areas, work, and fluid forces. It discusses calculating volumes through slicing solids and rotating areas about an axis. Examples are provided for finding the volumes of pyramids, wedges, and solids of revolution. It also discusses using integrals to find curve lengths, circle circumferences, and moments and centers of mass for various objects. Surface areas of revolution and fluid pressures are also explained.
Partial derivatives
This document introduces the topic of partial derivatives, which is the extension of calculus to functions with more than one independent variable. Functions of multiple variables are important in fields like probability, statistics, physics and more. The key concepts covered include functions of two and three variables, limits and continuity in higher dimensions, partial derivatives, the chain rule, gradient vectors, tangent planes and differentials.
Differentiation and Linearization
This document provides examples of related rates problems and their step-by-step solutions:
1) A hot air balloon rising straight up is tracked by a range finder 500 ft away. When the elevation angle is π/4 radians, the angle is increasing at 0.14 rad/min. The problem asks how fast the balloon is rising at that moment.
2) A police cruiser chasing a speeding car determines with radar that the distance between them is increasing at 20 mph when the cruiser is 0.6 mi north and the car is 0.8 mi east of an intersection. The problem asks for the speed of the car.
3) Water is flowing into a conical tank from the base
Regression
1. Regression analysis is a statistical process for estimating relationships between variables, including linear regression, logistic regression, and other types.
2. It allows predicting a dependent or response variable's values based on the values of independent or input variables.
3. Multiple linear regression allows modeling relationships between a scalar dependent variable and two or more explanatory variables.
Hypothesis testing
This document provides an introduction to hypothesis testing and significance levels. It discusses developing tests for null hypotheses and the types of errors that can occur. It describes using critical regions and significance levels like α=0.05 to determine whether to reject the null hypothesis. The document presents methods for testing hypotheses about the mean of a normal distribution when the variance is known and unknown. It also discusses testing the equality of means of two normal populations. Examples are provided to illustrate hypothesis testing methods.
Hypothesis testing
This document provides an overview of hypothesis testing concepts and procedures. It discusses the introduction to hypothesis testing including null and alternative hypotheses. It describes significance levels and types of errors. It covers tests for the mean of a normal population including cases of known and unknown variances. It discusses tests for the equality of means of two normal populations. It also covers paired t-tests, tests concerning the variance of a normal population, and hypothesis tests in binomial populations. Examples are provided to illustrate key concepts and procedures for conducting hypothesis tests.
Parameter estimation
This document discusses evaluating point estimators. It defines mean square error as an indicator for determining the worth of an estimator. There is rarely a single estimator that minimizes mean square error for all possible parameter values. Unbiased estimators, where the expected value equals the parameter, are commonly used. Bias is defined as the expected value of the estimator minus the parameter. Combining independent unbiased estimators results in an estimator with variance equal to the weighted sum of the individual variances. The mean square error of any estimator is equal to its variance plus the square of its bias. Examples are provided to illustrate evaluating bias and finding mean and variance of estimators.
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Economic interpretations of Linear Programming Problem
- 1. ECONOMIC INTERPRETATIONS PRIMAL/DUAL LINEAR PROGRAMMING
- 2. ECONOMIC INTERPRETATIONS Primal Linear Problem •Primal LP problem can be viewed as •a resource allocation model that seeks to maximize the revenueunder limited resources •Primal problem •‘n’ economic activities •‘m’ resources •‘cj’ revenue per unit of activity j •Resource “i” with availability b(i) Is consumed at the rate a(ij)] units per unit activity j
- 3. ECONOMIC INTERPRETATIONS OF DUAL VARIABLES The dual variable represents the worth per unit of resource “i” [z = w ] happens only at optimum……………!
- 4. Example:
- 5. ECONOMIC INTERPRETATION OF DUAL CONSTRAINTS •Formula 2:
- 6. ECONOMIC INTERPRETATION OF DUAL CONSTRAINTS •Dual variable “y_i” represents “Imputed cost” per unit resource “i” •The imputed cost of all resources needed to produce one unit of activity “j” is given by •The quantity represents “reducedcost” of activity “j”
- 7. ECONOMIC INTERPRETATION OF DUAL CONSTRAINTS •Simplex method – the maximization optimality condition says an increase in the level of an unused (non-basic) activity “j” can improve revenue only if its reduced cost is negative The maximization optimality condition means it is economically advantageous to increase the level of an activity if its revenue exceeds its unit imputed cost.
- 8. Example : Assembling 3 types of toys
- 9. Example : Assembling 3 types of toys •Toy 1 is not to be produced as x_1 = 0 •Reduced cost for x_1 : a toy becomes attractive economically only if its unit imputed cost is strictly less than the unit revenue •Company can achieve this by increasing the unit price •It can also decrease the imputed cost of the consumed resources •( = y_1 + 3 y_2 + y3) •The reduction ratios on the operations is represented as r_1, r_2 and r_3 •Since y_1 = 1 , y_2 = 2 and y_3 =0 we have •Even 50% reduction fails to satisfy the inequality •Company should not produce this toy
- 10. THANK YOU