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handout2 - Recap Math 171A Linear Programming Lecture 2...

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Math 171A: Linear Programming Lecture 2 Properties of Linear Constraints Philip E. Gill c 2011 http://ccom.ucsd.edu/~peg/math171a Wednesday, January 5th, 2010 Recap The lecture slides and homework are posted on the class web-page. http://ccom.ucsd.edu/~peg/math171a Access to course materials requires a class account and password . The account name is your last name in lower case, e.g., “gill” The password is your student ID, e.g., “a12345678” UCSD Center for Computational Mathematics Slide 2/44, Wednesday, January 5th, 2010
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Recap: a simple linear program maximize x , y 3 x + 4 y (objective function) subject to 3 x + 2 y 200 (cranberry juice constraint) x + 2 y 100 (apple juice constraint) x 0 (minimum resource constraint) y 0 (minimum resource constraint) UCSD Center for Computational Mathematics Slide 5/44, Wednesday, January 5th, 2010 Feasible production plans 10 20 30 40 50 60 70 80 90 100 110 120 10 20 30 40 50 60 70 80 90 100 110 120 y x (70 , 20) infeasible ( x, y ) feasible ( x, y ) How do we maximize the profit? The profit is given by p = 3 x + 4 y If ( x , y ) = (0 , 0) then no punch is mixed and 3 x + 4 y = 0 no profit! All production plans ( x , y ) such that 3 x + 4 y = 0 give no profit. UCSD Center for Computational Mathematics Slide 7/44, Wednesday, January 5th, 2010 No profit scenario 10 20 30 40 50 60 70 80 90 100 110 120 10 20 30 40 50 60 70 80 90 100 110 120 y x p = 0
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We can find all production plans that give a profit of 120/ c by looking at all points on the line 3 x + 4 y = 120. e.g., (0 , 30) (only appleBerry produced) and (40 , 0) (only cranApple produced) give a profit of 120/ c. UCSD Center for Computational Mathematics Slide 9/44, Wednesday, January 5th, 2010 10 20 30 40 50 60 70 80 90 100 110 120 10 20 30 40 50 60 70 80 90 100 110 120 y x p = 120 The lines 3 x + 4 y = p are parallel as the profit p varies. If a profit line 3 x + 4 y = p has points in the feasible region, then those points define a feasible production plan with profit p . As the profit p is increased, the profit line moves to the upper right. find the profit line that is furthest to the right ! UCSD Center for Computational Mathematics Slide 11/44, Wednesday, January 5th, 2010 Levels of profit 10 20 30 40 60 50 70 80 90 100 110 120 10 20 30 40 50 60 70 80 90 100 110 120 y x p = 0 p = 120 p = 240 p = 345
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The optimal mixture lies on the line 3 x + 4 y = p at which p is as large as possible and at least one point of 3 x + 4 y = p lies in the feasible region.
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