lec5 - 15.053 z February 22, 2007 Introduction to the...

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1 15.053 February 22, 2007 z Introduction to the Simplex Algorithm
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Quotes for today Give a man a fish and you feed him for a day. Teach him how to fish and you feed him for a lifetime. -- Lao Tzu Give a man a fish dinner, and he will forget it by next week. Let a person catch the fish for himself, and he’ll remember it for a lifetime. -- Jim Orlin 2
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3 Preview of the Simplex Method 1 2 3 4 5 6 1 2 3 4 5 K S Start at any feasible corner point. Move to an adjacent corner point with better objective value. Continue until no adjacent corner point has a better objective value. Maximize z = 3 K + 5 S This is a picture of the simplex algorithm in inequality form. In this form, the simplex algorithm moves from corner point to corner point. And each corner point is the intersection of two constraints. When we move to equality form, the simplex algorithm still moves from corner point to corner point. And the corner points are still found by solving a system of equations. So, there are many similarities.
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4 The simplex algorithm (for max problems) Start with a feasible corner point solution Is it optimal? quit with optimal solution Is the optimum unbounded from above? quit with proof of unboundedness find an improved corner point solution No Yes No Yes As you can see, this is a fairly simple structure. At the same time, it may be difficult to keep everything in one’s head at the same time. That is where the two dimensional example can help out. We will assume that we start with a feasible corner point solution. That immediately raises two questions. What does a corner point solution look like? And how do you find a corner point solution to start with? Both of these issues will be addressed shortly. The next slides deal with something even more preliminary. We will be assuming that we start with a linear program with equality constraints and non-negativity constraints, and nothing else. So we need to get each linear program into the correct starting form. We will show how to do that on the next few slides.
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Goals for this lecture Major Issues of the Simplex Algorithm 1. How does one get the LP into the correct starting form? 2. How does one recognize optimality and unboundedness? 3. How does one move to the next corner point solution? Note: we will derive the simplex algorithm in class! 5
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6 Linear Programs in Standard Form We say that a linear program is in standard form if the following are all true: 1. Non-negativity constraints for all variables. 2. All remaining constraints are expressed as equality constraints. 3. The right hand side vector, b, is non-negative. maximize z = 3x 1 + 2x 2 -x 3 + x 4 x 1 + 2x 2 + x 3 4 5 ; -2x 1 -4x 2 + x 3 + x 4 -1; x 1 0, x 2 0 An LP not in Standard Form not equality not equality x 3 and x 4 may be negative Excel Solver does not require that you write an LP in standard form because it will immediately transform it to standard form via software. We show next what linear programming solvers do with an LP that does not start in standard form.
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Converting Inequalities into Equalities Plus Non-negatives
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lec5 - 15.053 z February 22, 2007 Introduction to the...

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