lp - LINEAR PROGRAMMING Notes prepared for: MATH 2602...

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LINEAR PROGRAMMING Notes prepared for: MATH 2602 Linear and Discrete Mathematics Fall 2000 1
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Table of Contents 1. Background 1 2. Introductionto Linear Programming 2 2.1 Basic Forms 2 2.2 Modeling 4 3. Solving Linear Programs 7 3.1 Geometry and Basic Notions 8 3.1.1 The Feasible Region of Linear Programs 10 3.1.2 Extreme Points 12 3.1.3 Basic Solutions 13 3.2 The Simplex Algorithm 18 4. Generalizations 25 4.1 Artificial Variables 25 4.1.1 Big-M 27 4.1.2 Two-Phase Method 29 4.2 Empty Solution Space 32 4.3 AlternativeOptima 34 4.4 Unbounded Solution Spaces 36 4.5 Degeneracy 37 5. Final Comments 41 6. Exercises 43 7. References 49 2
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LINEAR PROGRAMMING 1 Background In this document, we will look at a class of optimization problems that are quite well-solved in the sense that especially powerful mathematical and com- putational machinery exists for dealing with them. Before we begin, however, let us examine what we mean by a mathematical optimization problem in gen- eral. We can state this in the following way: For a given function z ( x 1 ,x 2 ,...,x n ) find values for the variables x 1 2 n such that z is maximized (minimized) and where the determined values sat- isfy all of a given (possibly empty) set of constraints. Put more formally we have: P : max(min) z ( x 1 2 n ) subject to: g i ( x 1 2 n ) b i ,for i =1 , 2 ,...,m . In P , the function z is referred to as the objective function and the con- straints are given by the functions g i . An example is given below: min z =( x 1 - x 2 2 ) 3 + x 1 x 2 x 3 + x 3 2 3 s.t. x 1 x 2 x 3 10 x 1 + x 2 2 + x 3 2 12. Another example might require integrality restrictions on some or all of the variables: 1
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max z = x 3 1 + x 2 x 3 x 4 + x 2 2 - x 3 ( x 2 - x 4 ) 2 s.t. x 1 + x 2 + x 3 + x 4 =7 x 1 ,x 3 0and x 2 4 nonnegative integers. Of course, these are just randomly generated mathematical expressions that demonstrate the complexion of an optimization formulation. For any of this to have practical relevance, the idea would obviously be to examine some realistic setting and from same, build or create its mathematical representa- tion. This is rather like what is done in your first algebra class with those dreaded “story problems.” The entire exercise of creating and solving these mathematical models of real-world settings is, by and large, what constitutes the field of operations research . 2 Introduction to Linear Programming 2.1 Basic Forms If all of the functions in our problem are linear and if all of the variables are continuous, we have a linear programming (LP) problem. Stating this in a formal way produces the following model. Note that for simplicity, we will adopt only the “maximize” format: P C :m a x z = c 1 x 1 + c 2 x 2 + ... + c n x n s.t. a 11 x 1 + a 12 x 2 + + a 1 n x n b 1 a 21 x 1 + a 22 x 2 + + a 2 n x n b 2 . . . a m 1 x 1 + a m 2 + + a mn x n b m x 1 2 ,...,x n 0. 2
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The last, nonnegativity constraints require that all variables take on values no less than 0; the c j ,a ij ,and b j parameters are problem coefficients .O f - ten the b j values are called the right-hand-sides of the formulation. Note also that the model indicated by P C is referred to as canonical form , i.e.,
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lp - LINEAR PROGRAMMING Notes prepared for: MATH 2602...

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