Question

In: Mechanical Engineering

Consider the TOYCO model given below: TOYCO Primal: max z=3x1+2x2+5x3 s.t. x1 + 2x2 + x3...

Consider the TOYCO model given below:

TOYCO Primal:

max z=3x1+2x2+5x3

s.t.

x1 + 2x2 + x3 ? 430 (Operation 1)

3x1 + 2x3 ? 460 (Operation 2)

x1 + 4x2 ? 420 (Opeartion 3 )

x1, x2, x3 ?0

Optimal tableau is given below:

basic	x1	x2	x3	x4	x5	x6	solution
z	4	0	0	1	2	0	1350
x2	-1/4	1	0	1/2	-1/4	0	100
x3	3/2	0	1	0	1/2	0	230
x6	2	0	0	-2	1	1	20

a) Suppose that TOYCO wants to change the capacities of the three operations as bT = [460, 500, 400](the new right-hand-side vector). Use the post optimality analysis to determine the optimum solution.

b) Suppose that TOYCO adds a fourth operation with the operation times of 4, 1, and 2 minutes for product 1, 2, and 3 respectively. Assume that the capacity of the fourth operation is 548 minutes. Determine the new optimal solution for this case.

c) Suppose the objective function is changed to z = 3x1 + 6x2 + x3. If the solution changes, use the post-optimal analysis to find the new solution.

d) Suppose TOYCO wants to produce toy planes. It requires 3,2,4 minutes respectively on operations 1,2, and 3. Determine the optimal solution when the revenue per unit for toy planes is $10.

Expert Solution

samet mamat answered 2 years ago

4-Consider the following problem: max − 3x1 + 2x2 − x3 + x4 s.t. 2x1 −...

4-Consider the following problem: max − 3x1 + 2x2 − x3 + x4 s.t. 2x1 − 3x2 − x3 + x4 ≤ 0 − x1 + 2x2 + 2x3 − 3x4 ≤ 1 − x1 + x2 − 4x3 + x4 ≤ 8 x1, x2, x3, x4 ≥ 0 Use the Simplex method to verify that the optimal objective value is unbounded. Make use of the final tableau to construct an unbounded direction..

3) (15 pts) Consider the following LP formulation: max z = x1 + 2x2 s.t. −...

3) (15 pts) Consider the following LP formulation: max z = x1 + 2x2 s.t. − x1 + x2 ≤ 2 x2 ≤ 3 kx1 + x2 ≤ 2k + 3 x1, x2 ≥ 0 The value of the parameter k ≥ 0 has not been determined yet. The solution currently being used is x1 = 2, x2 = 3. Use graphical analysis to determine the values of k such that this solution is actually optimal.

Given the following LP max z = 2x1 + x2 + x3 s. t. 3x1 -...

Given the following LP max z = 2x1 + x2 + x3 s. t. 3x1 - x2 <= 8 x2 +x3 <= 4 x1,x3 >= 0, x2 urs (unrestricted in sign) A. Reformulate this LP such that 1)All decision variables are non-negative. 2) All functional constraints are equality constraints B. Set up the initial simplex tableau. C. Determine which variable should enter the basis and which variable should leave.

Consider the following linear optimization model. Z = 3x1+ 6x2+ 2x3 st 3x1 +4x2 + x3...

Consider the following linear optimization model. Z = 3x1+ 6x2+ 2x3 st 3x1 +4x2 + x3 ≤2 x1+ 3x2+ 2x3 ≤ 1 X1, x2, x3 ≥0 (10) Write the optimization problem in standard form with the consideration of slack variables. (30) Solve the problem using simplex tableau method. (10) State the optimal solution for all variables.

MAXIMIZATION BY THE SIMPLEX METHOD Maximize z = x1 + 2x2 + x3 subject to x1...

MAXIMIZATION BY THE SIMPLEX METHOD Maximize z = x1 + 2x2 + x3 subject to x1 + x2 ≤ 3 x2 + x3 ≤ 4 x1 + x3 ≤ 5 x1, x2, x3 ≥0

Solve the following problems using the two phase method: max 3x1 + x2 S.t. x1 −...

Solve the following problems using the two phase method: max 3x1 + x2 S.t. x1 − x2 ≤ −1 −x1 − x2 ≤ −3 2x1 + x2 ≤ 4 x1, x2 ≥ 0 Please show all steps

Consider the following. x1 − 2x2 + 3x3 = 3 −x1 + 3x2 − x3 =...

Consider the following. x1 − 2x2 + 3x3 = 3 −x1 + 3x2 − x3 = 2 2x1 − 5x2 + 5x3 = 3 (a) Write the system of linear equations as a matrix equation, AX = B. x1 x2 x3 = (b) Use Gauss-Jordan elimination on [A B] to solve for the matrix X. X = x1 x2 x3 =

Consider the linear system of equations below 3x1 − x2 + x3 = 1 3x1 +...

Consider the linear system of equations below 3x1 − x2 + x3 = 1 3x1 + 6x2 + 2x3 = 0 3x1 + 3x2 + 7x3 = 4 i. Use the Gauss-Jacobi iterative technique with x (0) = 0 to find approximate solution to the system above up to the third step ii. Use the Gauss-Seidel iterative technique with x (0) = 0 to find approximate solution to the third step

Exercise Minimize Z = X1 - 2X2 Subject to X1 - 2X2 ≥ 4 &

Exercise Minimize Z = X1 - 2X2 Subject to X1 - 2X2 ≥ 4 X1 + X2 ≤ 8 X1, X2 ≥ 0

Consider the following linear programming problem: Max Z = 3x1 + 3x2 Subject to: ...

Consider the following linear programming problem: Max Z = 3x1 + 3x2 Subject to: 10x1 + 4x2 ≤ 60 25x1 + 50x2 ≤ 200 x1, x2 ≥ 0 Find the optimal profit and the values of x1 and x2 at the optimal solution.