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notes_6 - Notes For Optimization Theory Analysis(ESE 304...

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Notes For Optimization Theory & Analysis (ESE 304) Michael A. Carchidi October 26, 2010 Chapter 6 - Sensitivity Analysis and Duality The following notes are based on the text entitled: Operations Research by Wayne L. Winston (4th edition), and these can be viewed at https://courseweb.library.upenn.edu/ after you log in using your PennKey user name and Password. 6.1 Sensitivity Analysis: A Geometric Approach Sensitivity Analysis isconcernedwithhowchangesinanLPsparametersa f ect the optimal solution. We introduce these ideas by revisiting the Giapetto LP problem without the integer restrictions, max z =$3 x 1 +$2 x 2 (Objective Function) s.t. 2 x 1 + x 2 100 (Finishing Constraint) x 1 + x 2 80 (Carpentry Constraint) x 1 40 (Upper Demand for Soldiers) x 1 ,x 2 0 (Sign Restrictions) where x 1 = the number of soldiers produced x 2 = the number of trains produced.
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The feasible region (in the x 1 - x 2 plane) for this problem is shown in the f gure below. 40 30 20 10 0 80 60 40 20 0 Plot of the Giapetto Feasible Region (Without the Integer Constraints) By introducing the slack variables s 1 , s 2 and s 3 , we write this in standard form as follows. max z =$3 x 1 +$2 x 2 (Objective Function) s.t. 2 x 1 + x 2 + s 1 =100 (Finishing Constraint) x 1 + x 2 + s 2 =80 (Carpentry Constraint) x 1 + s 3 =40 (Upper Demand for Soldiers) x 1 ,x 2 ,s 1 2 3 0 (Sign Restrictions) We found that the optimal solution to this problem is x optimal = x 1 x 2 s 1 s 2 s 3 = 20 60 0 0 20 having z optimal =$180 and we see from this that BV optimal = { x 1 2 3 } and NBV optimal = { s 1 2 } . 2
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Graphical Analysis: Changes in Optimal Function Coe cients If the contribution to pro f t of a soldier were to increase su ciently ,thenit seems reasonable that it would be optimal for Giapetto to produce more soldiers. This implies that s 3 (which is a measure of how many soldiers are not produced out of the possible 40 that could be produced) would become smaller until it reaches the point where it is zero (i.e., nonbasic). Similarly, if the contribution to pro f t of a soldier were to decrease su ciently , then it seems reasonable that it would be optimal for Giapetto to produce only trains. This implies that x 1 (which is a measure of how many soldiers are produced out of the possible 40 that could be produced) would become smaller until it reaches the point where it is zero (i.e., nonbasic). Supposeweletthecontribut iontopro f tbyeachso ld ierbe $ c 1 , and we let the contribution to pro f tbyeachtra inbe $ c 2 , so that the Giapetto problem reads max z = c 1 x 1 + c 2 x 2 (Objective Function) s.t. 2 x 1 + x 2 100 (Finishing Constraint) x 1 + x 2 80 (Carpentry Constraint) x 1 40 (Upper Demand for Soldiers) x 1 ,x 2 0 (Sign Restrictions).
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This note was uploaded on 03/02/2011 for the course ESE 304 taught by Professor Michaela.carchidi during the Winter '11 term at UPenn.

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notes_6 - Notes For Optimization Theory Analysis(ESE 304...

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