MC-B TOPIC; LINEAR PROGRAMMING DATE; 5 JUNE‚ 14 UNIVERSITY OF CENTRAL PUNJAB INTRODUCTION TO LINEAR PROGRAMMING 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. Linear programming is a special case of mathematical programming
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Linear ------------------------------------------------- Important EXERCISE 27 SIMPLE LINEAR REGRESSION STATISTICAL TECHNIQUE IN REVIEW Linear regression provides a means to estimate or predict the value of a dependent variable based on the value of one or more independent variables. The regression equation is a mathematical expression of a causal proposition emerging from a theoretical framework. The linkage between the theoretical statement and the equation is made prior to data collection
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The development of linear programming has been ranked among the most important scientific advances of the mid 20th century. Its impact since the 1950’s has been extraordinary. Today it is a standard tool used by some companies (around 56%) of even moderate size. Linear programming uses a mathematical model to describe the problem of concern. Linear programming involves the planning of activities to obtain an optimal result‚ i.e.‚ a result that reaches the specified goal best (according to the mathematical
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spreadsheet‚ next step is to use the Solver to find the solution. In the Solver‚ we need to identify the locations (cells) of objective function‚ decision variables‚ nature of the objective function (maximize/minimize) and constraints. Example One (Linear model): Investment Problem Our first example illustrates how to allocate money to different bonds to maximize the total return (Ragsdale 2011‚ p. 121). A trust office at the Blacksburg National Bank needs to determine how to invest $100‚000 in following
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PROBLEM NUMBER 1 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? SOLUTION TO PROBLEM NUMBER 1 let x = the number of acres of wheat
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Z00_REND1011_11_SE_MOD7 PP2.QXD 2/21/11 12:39 PM Page 1 7 MODULE Linear Programming: The Simplex Method LEARNING OBJECTIVES After completing this chapter‚ students will be able to: 1. Convert LP constraints to equalities with slack‚ surplus‚ and artificial variables. 2. Set up and solve LP problems with simplex tableaus. 3. Interpret the meaning of every number in a simplex tableau. 4. Recognize special cases such as infeasibility‚ unboundedness and degeneracy. 5
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(2003) 1 OPERATIONS RESEARCH: 343 1. LINEAR PROGRAMMING 2. INTEGER PROGRAMMING 3. GAMES Books: Ð3Ñ IntroÞ to OR ÐF.Hillier & J. LiebermanÑ; Ð33Ñ OR ÐH. TahaÑ; Ð333Ñ IntroÞ to Mathematical Prog ÐF.Hillier & J. LiebermanÑ; Ð3@Ñ IntroÞ to OR ÐJ.Eckert & M. KupferschmidÑÞ LP (2003) 2 LINEAR PROGRAMMING (LP) LP is an optimal decision making tool in which the objective is a linear function and the constraints on the decision problem are linear equalities and inequalities. It is a very popular
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5 5 5 Compute: ATB(3 marks) tr (AB)(1 mark) (e) Determine if (2‚ -1) is in the set generated by = (3‚ 1)‚ (2‚ 2) (5 marks) Question Two (20 marks) Let T: R2 R2 be defined by T(x‚ y) = (x + y‚ x). Show that T is a linear transformation.(7 marks) Find the basis and dimension of the row space of the matrix.(6 marks) 2 -1 3 A= 1 1 5 -1 2 2 Compute A-1 using row reduction method.(7 marks) 1 4 3 A= -1 -2 0 2
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CHAPTER 8 Linear Programming Applications Teaching Suggestions Teaching Suggestion 8.1: Importance of Formulating Large LP Problems. Since computers are used to solve virtually all business LP problems‚ the most important thing a student can do is to get experience in formulating a wide variety of problems. This chapter provides such a variety. Teaching Suggestion 8.2: Note on Production Scheduling Problems. The Greenberg Motor example in this chapter is largest large
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LINEAR ALGEBRA Paul Dawkins Linear Algebra Table of Contents Preface............................................................................................................................................. ii Outline............................................................................................................................................ iii Systems of Equations and Matrices.............................................................................................
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