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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 variable’s 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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This note was uploaded on 12/03/2011 for the course ESI 6316 taught by Professor Staff during the Summer '11 term at FIU.
 Summer '11
 STAFF

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