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HW 3 - ENGR62/MS&E111 Introduction to Optimization Prof Ben...

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ENGR62/MS&E111 Spring 2008 Introduction to Optimization April 25, 2008 Prof. Ben Van Roy Homework Assignment 3: Solutions Part A (a) We have the following feasible region (shaded in yellow): 10 10 8 8 6 4 6 2 0 4 -2 -4 2 0 -2 -4 where the constraints, counterclockwise from left, correspond to: 3 x 2 3 (1) 2 x 1 + 5 x 2 0 (2) 2 x 1 x 2 3 (3) 3 x 1 + 2 x 2 8 (4) The above region has three vertices: ( 1 , 1), ( 5 2 , 1), and ( 40 11 , 16 11 ). The given linear program will have no optimal solution if, for instance, c = bracketleftbigg 1 0 bracketrightbigg since x 1 can be arbitrarily large and still feasible. (b) Examples of possible c vectors include the following: ( 1 , 1) will be optimal for c = bracketleftbigg 1 1 bracketrightbigg ( 5 2 , 1) will be optimal for c = bracketleftbigg 1 4 bracketrightbigg ( 40 11 , 16 11 ) will be optimal for c = bracketleftbigg 1 2 bracketrightbigg 1
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Each of these can be verified graphically or using Excel. (c) It can be verified graphically that only inequalities (1) and (3) will make the feasible region bounded while the others will allow the feasible region to remain unbounded. Thus, it follows that only (1) and (3) will guarantee that the linear program is feasible and has an optimal solution for all vectors c .
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