Simplex Method Consider the following linear programming
problem:
max
z = 6x1 + 3x2 - 9x2 - 9x3 + 15x4
s.t. 2x1 + 4x2 +6x3 + 8x4 <= 80
6x1
- 3x2 +3x3 + 6x4 <= 24
12x1 - 6x2 + 3x3 - 3x4 <= 30
x1,
x2, x3, x4 >= 0
Rewrite the problem in standard form, that is, add the necessary
slack variables in order to consider only equality constraints (and
non-negativity).
What is the...
question:
Objective function: Zmax = 6X1 + 10X2
LIMITS: 3X1 + 6X2 ≤ 24
And 5X1 + 6X2 ≤ 30
X1, X2 ≥ 0 and integer (integer)
Apply the following to the question above.
a. Branch-bound algorithm
b. (0-1) Integer programming
c. Gomory cutting plane
Find the dual problem for each of the following primal
problems.
a): min z=6x1+8x2 st: 3x1+x2>=4 5x1+2x2>=7 x1,x2>=0
b): max z=8x1+3x2-2x3 st: x1-6x2+x3>=2 5x1+7x2-2x3=-4
x1<=0,x2<=0,x3 unrestricted
Consider the following LOP P.
Max. z = 212x1 −320x2 +273x3 −347x4 +295x5
s.t. −4x1 −2x3 +8x5 ≤ −22
2x1 +3x2 −x4 = 31
−5x2 +3x3 −2x5 ≤ 27
−7x1 −8x3 +6x4 = −38
−9x3 −2x4 +x5 ≤ −40
−x2 −3x4 −5x5 ≤ 42
& x1, x3, x4 ≥ 0
a. Find x∗ and write the Phase 0, I and II pivots that solve
P.
b. Use the General Complementary Slackness Theorem to find
the optimal certificate y∗
[do not...
Transform the given system into a single equation of
second-order:
x′1 =−4x1+9x2
x′2 =−9x1−4x2.
Then find x1 and x2 that also satisfy the initial
conditions:
x1(0) =8
x2(0) =5.
Consider the following problem
Maximize Z=2x1 + 5x2 +
x3
subject to
4x1+ 2x2 + x3 ≤ 6
x1 + x2 ≤ 2
xi ≥ 0 for i=1,2,3
a. Inserting slack variables, construct the initial simplex
tableau. What is the initial basic feasible solution?
b. What is the next non-basic variable to enter the basis
c. Using the minimum ratio rule, identify the basic variable to
leave the basis.
d. Using elementary row operations, find...
Consider the following all-integer linear program:
Max 5x1 +8x2
s.t.
6x1 + 5x2 <= 30
9x1 + 4x2 <= 36
1x1 + 2x2 <=10
x1, x2 $ 0 and integer
a. Graph the constraints for this problem. Use dots to indicate
all feasible integer solutions.
b. Find the optimal solution to the LP Relaxation. Round down to
find a feasible integer solution.
c. Find the optimal integer solution. Is it the same as the
solution obtained in part (b) by...