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Unformatted text preview: 10/28/2011 1 Chapter 7 Duality and Sensitivity in Linear Programming 7.1 Generic Activities versus Resources Perspective Objective Functions as Costs and Benefits Optimization models objective functions usually can be interpreted as minimizing some measure of cost or maximizing some measure of benefit. [7.1] Choosing a Direction for Inequality Constraints The most natural expression of a constraint is usually the one making the righthandside constant non negative. [7.2] 0.3 x 1 + 0.4 x 2 2.0 (gasoline)  0.3 x 1 0.4 x 2  2.0 10/28/2011 2 7.1 Generic Activities versus Resources Perspective Inequalities as Resource Supplies and Demands Optimization model constraints of the form usually can be interpreted as restricting the supply of some commodity or resource. [7.3] x 1 9 (Saudi) Optimization model constraints of the form usually can be interpreted as requiring satisfaction of a demand for some commodity or resource. [7.4] 0.4 x 1 + 0.2 x 2 1.5 (jet fuel) 7.1 Generic Activities versus Resources Perspective Equality Constraints as Both Supplies and Demands Optimization model equality constraints usually can be interpreted as imposing both a supply restriction and a demand requirement on some commodity or resource. [7.5] VariableType Constraints Nonnegativity and other sign restriction constraints are usually best interpreted as declarations of variable type rather than supply or demand limits on resources. [7.6] 10/28/2011 3 7.1 Generic Activities versus Resources Perspective Variables as Activities Decision variables in optimization models can usually be interpreted as choosing the level of some activity. [7.7] LHS Coefficients as Activity Inputs and Outputs Nonzero objective function and constraint coefficients on LP decision variables display the impacts per unit of the variables activity on resources or commodities associated with the objective and constraints. [7.8] Inputs and Outputs for Activities Two Crude Example 1000 barrels of Saudi petroleum processed (x1) Availability 1000 barrels Cost $20000 .3 unit gasoline .4 unit jet fuel .2 unit lubricants 10/28/2011 4 7.2 Qualitative Sensitivity to Changes in Model Coefficients Relaxing versus Tightening Constraints Relaxing the constraints of an optimization model either leaves the optimal value unchanged or makes it better (higher for a maximize, lower for a minimize). Tightening the constraints either leaves the optimal value unchanged or makes it worse. [7.9] Relaxing Constraints 1 2 3 4 5 6 1 2 3 4 5 6 7 8 x 2 x 1 7 8 x 1 9 x 2 6 1 2 3 4 5 6 1 2 3 4 5 6 7 8 x 2 x 1 7 8 x 1 9 x 2 6 10/28/2011 5 Tightening Constraints 1 2 3 4 5 6 1 2 3 4 5 6 7 8 x 2 x 1 7 8 x 1 9 x 2 6 1 2 3 4 5 6 1 2 3 4 5 6 7 8 x 2 x 1 7 8 x 1 9 x 2 6 Swedish Steel Blending Example min 16 x 1 +10 x 2 +8 x 3 +9 x 4 +48 x 5 +60 x 6 +53 x 7 s.t. x 1 + x 2 + x 3 + x 4 + x 5 + x 6 + x 7 = 1000 0.0080 x 1 + 0.0070 x 2 + 0.0085 x+ 0....
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 Summer '11
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