Problems in Linear Programming 2
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This document summarizes solutions to three linear programming problems solved graphically. Problem 1 maximizes z=15x1+10x2 with constraints 4x1+6x2≤360, 3x1≤180, and 5x2≤200. The maximum z=1100 occurs at point (60,20). Problem 2 maximizes z=2x1+x2 with constraints x1+2x2≤10, x1+x2≤6, x1-x2≤2, and x1-2x2≤1. The maximum z=10 occurs at point (4,2). Problem 3 maximizes z=-
Problems in Linear Programming 2
- 1. PROBLEMS IN LINEAR PROGRAMMING 2 SEBASTIAN VATTAMATTAM 1. Graphical Solution Problem 1.1. Solve graphically: Maximize z = 15x1 + 10x2 subject to the constraints (1) 4x1 + 6x2 ≤ 360 (2) 3x1 + 0x2 ≤ 180 (3) 0x1 + 5x2 ≤ 200 and x1 , x2 ≥ 0 Solution See figure 1 Draw the lines 4x1 + 6x2 = 360 3x1 = 180 5x2 = 200 z is maximum at a vertex of the feasible region, which is a polygon. That is at O, A, B, C, orD. Extreme Point Coordinates z O (0, 0) 0 A (60, 0) 900 B (60, 20) 1,100 C (30, 40) 850 D (0, 40) 400 Max z = 1, 100 at the point (60, 20). 1
- 2. 2 SEBASTIAN VATTAMATTAM Figure 1. Problem 1.1 Problem 1.2. Solve graphically: Maximize z = 2x1 + x2 subject to the constraints (1) x1 + 2x2 ≤ 10 (2) x1 + x2 ≤ 6 (3) x1 − x2 ≤ 2 (4) x1 − 2x2 ≤ 1 and x1 , x2 ≥ 0 Solution See figure 2
- 3. LINEAR PROGRAMMING 3 Figure 2. Problem 1.2 Draw the lines x1 + 2x2 = 10 x1 + x2 = 6 x1 − x2 = 2 x1 − 2x2 = 1 z is maximum at a vertex of the feasible region. That is at O, A, B, C, D, orE. Extreme Point Coordinates z O (0, 0) 0 A (1, 0) 2 B (3,1) 7 C (4, 2) 10 D (2,4) 8 E (0, 5) 5 Max z = 10 at the point (4, 2).
- 4. 4 SEBASTIAN VATTAMATTAM Problem 1.3. Solve graphically: Maximize z = −x1 + 2x2 subject to the constraints (1) x1 − x2 ≤ −1 (2) −0.5x1 + x2 ≤ 2 and x1 , x2 ≥ 0 Figure 3. Problem 1.3 Solution See figure 3 Draw the lines x1 − 2x2 = −1 −0.5x1 + x2 = 2
- 5. LINEAR PROGRAMMING 5 z is maximum at a vertex of the feasible region. That is at A, B, orC. Extreme Point Coordinates z A (0,1) 2 B (0,2) 4 C (2,3) 4 Max z is at the point B or C and hence at any point between B and C, on line BC, and max z = 4 For online classes in Mathematics at any level, please contact [email protected]