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Unformatted text preview: Class Notes for DMOR: Set 3 Sensitivity and Duality Information for LPs. 1 Suppose that we solve the following problem by the simplex method: Maximize 3 x 1 + x 2 subject to x 1 + x 2 ≤ 4 2 x 1 + x 2 ≤ 6 x 1 x 2 ≤ 2 x 1 , x 2 ≥ . Basics z x 1 x 2 s 1 s 2 s 3 RHS z 1 3 1 s 1 1 1 1 4 s 2 2 1 1 6 s 3 1 1 1 2 Basics z x 1 x 2 s 1 s 2 s 3 RHS z 1 4 3 6 s 1 2 1 1 2 s 2 3 1 2 2 x 1 1 1 1 2 Basics z x 1 x 2 s 1 s 2 s 3 RHS z 1 4/3 1/3 26/3 s 1 1 2 / 3 1/3 2/3 x 2 1 1/3 2 / 3 2/3 x 1 1 1/3 1/3 8/3 2 What if the RHS Changes? For “small changes,” the current basis will remain optimal (that is, the variables that are currently basic will stay basic). Questions we want to ask: • What happens to the objective if the RHS for constraint i changes a little? • What happens to the basic variables? The first rate of change above is called the shadow price . The shadow price tells you how much each RHS is worth. This is usually the price of a resource (for ≥ constraints), or the cost of a resource limitation (for ≤ constraints). For small changes, the rates of change above will remain constant. But what constitutes a small change? 3 Easy Case: What if RHS of Constraint 1 changes? Plot the new graph. The solution doesn’t change, and the variables don’t change, except for s 1 , which gets a little larger. 4 Now, what if RHS of Constraint 2 changes? Let the new RHS of Constraint 2 be 6 + Δ 2 . Plot the new graph. The optimal solution slides up and to the right. x 1 and x 2 increase at the same rate, the objective goes up, and the first slack decreases. Exactly what is the rate of change? Solve for basic variables: x 1 + x 2 + s 1 = 4 2 x 1 + x 2 = 6 + Δ 2 x 1 x 2 = 2 5 And what if RHS of Constraint 3 changes? Let the new RHS of Constraint 3 be 2 + Δ 3 . Plot the new graph. The optimal solution slides down and to the right. x 1 goes right at half the rate that x 2 goes down. Thus, the objective is increasing. The slack for constraint 1 is increasing. Exactly what is the rate of change? Solve for basic variables: x 1 + x 2 + s 1 = 4 2 x 1 + x 2 = 6 x 1 x 2 = 2 + Δ 3 6 Where can we find these rates of change? Basics z x 1 x 2 s 1 s 2 s 3 RHS z 1 4/3 1/3 26/3 s 1 1 2 / 3 1/3 2/3 x 2 1 1/3 2 / 3 2/3 x 1 1 1/3 1/3 8/3 Find ≤ slack for constraint i (or its artificial variable column, but ignore all big M terms). Look in that column for rate of change information, including shadow prices. (Or, if you have ≥ constraints, look in their slacks and take the negative of what you see in that column.)you see in that column....
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 Fall '09
 VLADIMIRLBOGINSKI
 Linear Programming, Optimization, RHS

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