Use the simplex algorithm to find all optimal solutions to the following LP.
max z=2x1+x2
s.t. 4x1 + 2x2 ≤ 4
−2x1 + x2 ≤ 2
x1 ≥1
x1,x2 ≥0
Solution:
Use the simplex algorithm to find all optimal solutions to the following LP. max z=2x1+x2 s.t....
3. Use the simplex algorithm to find an optimal solution to the following LP: s.t. 3x1 +26 s.t.-xi + 2x2 S 0 レ
Consider the following LP: Max x1 +x2 +x3 s.t. x1 +2x2 +2x3 ≤ 20 Solve this problem without using the simplex algorithm, but using the fact that an optimal solution to LP exists at one of the basic feasible solutions.
Question 3: Identify which of LP problems (1)--(4) has (x1,x2) = (20,60) as its optimal solution. (1) min z = 50xı + 100X2 s.t. 7x1 + 2x2 > 28 2x1 + 12x2 > 24 X1, X2 > 0 (2) max z = 3x1 + 2x2 s.t. 2x1 + x2 < 100 X1 + x2 < 80 X1 <40 X1, X2 > 0 (3) min z = 3x1 + 5x2 s.t. 3x1 + 2x2 > 36 3x1 + 5x2 > 45...
1. Use the Big M method to find the optimal solution to the following LP: Max z = 5x1 − x2 s.t.: 2x1 + x2 = 6 x1 + x2 ≤ 4 x1 + 2x2 ≤ 5 x1, x2 ≥ 0 Answer: z = 15, x1 = 3, x2 = 0.
Q3. (Dual Simplex Method) (2 marks) Use the dual Simplex method to solve the following LP model: max z= 2x1 +4x2 +9x3 x1 x2 x3 S 1 -x1+ X2 +2x3 S -4 x2+ X1,X2,X3 S 0 Q3. (Dual Simplex Method) (2 marks) Use the dual Simplex method to solve the following LP model: max z= 2x1 +4x2 +9x3 x1 x2 x3 S 1 -x1+ X2 +2x3 S -4 x2+ X1,X2,X3 S 0
2. Use the simplex algorithm to find an optimal solution to the following LP: max z 5x1 + 3x2 + x3 5x +3x2 +6x s 15
Q4. (Sensitivity Analysis: Adding a new constraint) (3 marks) Consider the following LP max z= 6x1+x2 s.t.xi + x2 S5 2x1 + x2 s6 with the following final optimal Simplex tableau basis x1 r2 S2 rhs 0 0 18 0.5 0.5 0.5 0.5 x1 where sı and s2 are the slack variables in the first and second constraints, respectively (a) Please find the optimal solution if we add the new constraint 3x1 + x2 S 10 into the LP (b)...
Find the duals of the following LP. (a) Max Z--2x1 + x2 - 4xz + 3x4 st. x1 + x2 + 3x2 + 2x4 S4 X1 -X3 + X, 2-1 2x1 + x2 32 X1 + 2x2 + x3 + 2x4 = 2 X2, X3, X, 20 (b) Min Z=0.4x1 +0.5x2 st. 0.3x2 +0.1x, 32.7 0.5x7 +0.5x2 = 6 0.6x, +0.4x, 26 X1, X220
Use the two-phase method to find the optimal solution to the following LP: Min z = 3x1 + 2x2 s.t.: 3x1 + x2 ≥ 3 4x1 + 3x2 ≥ 6 x1 + 2x2 ≤ 3 x1, x2 ≥ 0 Answer: z = 4.2, x1 = 0.6, x2 = 1.2.
Find all the basic solutions for the following LP problems using the Gauss– Jordan elimination method. Identify basic feasible solutions and show them on graph paper. Maximize z = 4x1 + 2x2 subject to −2x1 + x2 ≤ 4 x1 + 2x2 ≥ 2 x1, x2 ≥ 0