Lecture 10 - Sensitivity

# Lecture 10 - Sensitivity - 540:311 DETERMINISTIC MODELS IN...

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540:311 DETERMINISTIC MODELS IN OPERATIONS RESEARCH Lecture 10: Chapter 5 Class [email protected]: Thu Mar 24 th 10:20-­૒11:40am Prof. W. Art Chaovalitwongse

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5.4 – What happens to the [email protected] z-­૒Value if the Current Basis Is No Longer [email protected]? Effect of change in ObjecEve FuncEon Coeﬃcient on OpEmal z-­૒value max z = 3x 1 + 2x 2 2 x 1 + x 2 100 (finishing constraint) x 1 + x 2 80 (carpentry constraint) x 1 40 (demand constraint) x 1 ,x 2 0 (sign restriction) Goal: To find the optimal objective function value as a function of a variable’s objective function coefficient can be created. Consider again the Giapetto LP shown to the right. Let c 1 = objective coefficient of x 1 . Currently, c 1 = 3 and we want to determine how the optimal z-value depend upon c 1 ..
“Recall” 5.1 – A Graphical Approach to [email protected] Analysis The optimal solution for this LP was z = 180, x 1 = 20, x 2 = 60 (point B). How would changes in the problem’s objective function coefficients or right-hand side values change this optimal solution? Can we build a relationship?

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5.4 – What happens to the [email protected] z-­૒Value if the Current Basis Is No Longer [email protected]? Since a typical isoprofit line is c 1 x 1 + 2x 2 = z , we know the slope of the isoprofit line is just -c 1 /2 . Point A(0,80) is optimal if the isoprofit line is flatter than the carpentry constraint . -c 1 /2 -1 or 0 c 1 2 , (-1 is the carpentry constraint slope) Point B(20,60) is optimal if the isoprofit line is steeper than the carpentry constraint but flatter than the finishing constraint. -2 -c 1 /2 -1 or 2 c 1 4 (between the slopes of the carpentry and finishing constraint slopes) Point C(40,20) is optimal if the slope of the isoprofit line is steeper than the slope of the finishing constraint. -c 1 /2 -2 or c 1 4 (-2 is the finishing constraint slope)
5.4 – What happens to the [email protected] z-­૒Value if the Current Basis Is No Longer [email protected]? In a maximization LP, the slope of the graph of the optimal z-value as a function of an objective function coefficient will be nondecreasing. In a minimization LP, the slope of the graph of the optimal z-value as a function of an objective function coefficient will be nonincreasing. 0

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