Consider the three-dimensional LP solution space shown in the figure below, whose feasible extreme points are A, B, ..., and J.
a. Which of the following pairs of corner points cannot represent successive simplex iterations: (D, G), (H, I), (E, H), and (A, I)? Explain why.
b. Suppose that the simplex iterations start at A and that the optimum occurs at H. Indicate whether any of the following paths are not legitimate for the simplex algorithm; and state the reason.
i. A->B->G->H.
ii. A->D->F->C->A->B->G->H.
iii. A->C->I->H.
Solve the following two problems using Simplex Algorithm:
a.
Maximize z = 8x1 + 6x2 + 3x3 - 2x4
Subject to
x1 + 2x2 + 2x3 + 4x4 <= 40
2x1 - x2 + x3 + 2x4 <= 8
4x1 - 2x2 + x3 - x4 <= 10
x1, x2, x3, x4 >= 0
b.
Minimize z = 5x1 - 4x2 + 6x3 - 8x4
Subject to
x1 + 2x2 + 2x3 + 4x4 <= 40
2x1 - x2 + x3 + 2x4 <= 8
4x1 - 2x2 + x3 - x4 <= 10
x1, x2, x3, x4 >= 0
Consider the following system of equations:
x1 + 2x2 - 3x3 + 5x4 + x5 = 8
5x1 - 2x2 + 6x4 + x6 = 16
2x1 + 3x2 - 2x3 + 3x4 + x7 = 6
-x1 + x3 - 2x4 = x8 = 0
xi >= 0, i = 1, ..., 8,
Let x5, x6, x7, and x8 be a given initial basic feasible solution. Suppose that x1 becomes basic. Which of the given basic variables must become non-basic, and what is the value of x1 in the new solution? Repeat the same procedure for x2, x3, x4.
Consider the following LP model:
Maximize z = x1
Subject to
5x1 + x2 = 4
6x1 + x3 = 8
4x1 + x4 = 3
x1, x2, x3, x4 >= 0
a. Solve the problem by inspection (do not use the Gauss-Jordan row operations) and justify the answer in terms of the basic solutions of the simplex method.
b. Repeat (a) assuming that the objective function calls for minimizing z = x1
Solve q1, q4
1. Consider the three-dimensional LP solution space shown in the figure below, whose feasible extreme points are A, B, ..., and J. a. Which of the following pairs of corner points cannot represent successive simplex iterations: D, G, H, E, H, and A, I? Explain why b. Suppose that the simplex iterations start at A and that the optimum occurs at H. Indicate whether any of the following paths are not legitimate for the simplex algorithm; and state the reason. i. ABGH. ii. A:(0,0,0) ADFCABGH B: (1,0,0) iii. ACIH C:(0,1,0 D:(0,0,1) 2. Solve the following two problems using Simplex Algorithm: a. b. Maximize z = 8x + 6x + 3x3 - 2x4 Subject to x + 2x + 2x3 + 4x4 40 2x - x + x3 + 2x8 4x - 2x + x3 - x10 X, X2, X3, X40 Minimize z = 5x - 4x2 + 6x3 - 8x4 Subject to x + 2x + 2x3 + 4x4 40 2x - x + x + 2x48 4x - 2x + x3 - x10 X1, X2, X3, X40 3. Consider the following system of equations: x + 2x - 3x3 + 5x4 + x5 = 8 5x - 2x2 + 6x4 + X6 = 16 2x + 3x2 - 2x3 + 3x4 + X7 = 6 -X1 + X3 - 2x4 + xg = 0 xi0, i = 1, ..., 8 Let x5, 6, 7, and g be a given initial basic feasible solution. Suppose that , becomes basic. Which of the given basic variables must become non-basic, and what is the value of x, the new solution? Repeat the same procedure for 2, 3, 4 4. Consider the following LP model Maximize z = x1 Subject to 5x + x2 = 4 6x1 + X3 = 8 4x1 + 4 = 3 X, x2, x3, X40 a. Solve the problem by inspection (do not use the Gauss-Jordan row operations and justify the answer in terms of the basic solutions of the simplex method b. Repeat (a) assuming that the objective function calls for minimizing z = x
