Lecture 3

# Lecture 3 - Solving Linear Programming Graphic Method...

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1 Solving Linear Programming Graphic Method Theoretical preparation for simplex method

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2 Memory Refresh Each line on the X-Y two dimensional plane corresponds to a linear equality ax + by = Z , where a , b , and Z are fixed parameters. Example: 2X+3Y=12 X Y
3 Using Graph to Represent Equality y x (0, Z 1 / b ) ( Z 1 / a , 0) (0, Z 2 / b ) ( Z 2 / a , 0) Z 1 > Z 2 ax + by = Z form a family of parallel lines for different Z values a x + b y = Z 3

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4 Using Graph to Represent Inequality y x a x + b y = Z (0, Z / b ) ( Z / a , 0) Question: What does the graph look like if a < 0, or b < 0, or Z <0?
5 Exercises y x (0, Z/b) (Z/a, 0) a x + b y = Z with a>0, b<0, Z>0 a x + b y < Z a x + b y > Z decreasing increasing Question: where is the line for Z<0?

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6 LP with Two Variables 0 , 0 40 , 100 (D) 2400 15 10 (C) 1440 6 15 (B) 1440 28 12 (A) 1800 10 20 to subjec 60 45 Maximize + + + + + = Q P Q P Q P Q P Q P Q P Q P Z
7 A B C D Max P Max Q P Q 0 , 0 40 , 100 (D) 2400 15 10 (C) 1440 6 15 (B) 1440 28 12 (A) 1800 10 20 : s Constraint of Set + + + + Q P Q P Q P Q P Q P Q P Feasible region

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8 P Q 45P+60Q=1800, not optimal 45P+60Q > 4664, infeasible Q P 60 45 Z Maximize + = 30 40 20 90 45P+60Q < 4664, not optimal Feasible region Isovalue contours
9 P Q 45P+60Q=4664, Optimal Q P 60 45 Z Maximize + = 90 Feasible region Constraint A Constraint B Optimal solution is at a corner point of the feasible region, which can be obtained by solving = + = + B) t (constrain 1400 28 12 A) t (constrain 1800 10 20 Q P Q P

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10 Solving LP by Graphic Method Only for LP with two variables Steps Step 1. Draw feasible region by outlining each constraint Step 2. Determine the optimal solution by moving the objective function lines (isovalue contours) Important observation An optimal solution, if existed, can always be found at a corner point of the feasible region.
11 P Q 90 Feasible region Constraint A Constraint B Sensitivity Analysis on Graph When a coefficient of the objective function changes, the slope of the line changes, but the optimal solution keeps at the same corner point within a range Optimal solution changed!

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