Question

In: Operations Management

Find the dual of the following LP, using direct method. minz=4X1 +2X2 -X3 subject to X1...

Find the dual of the following LP, using direct method. minz=4X1 +2X2 -X3
subject to
X1 +2X2 ≤6
X1 -X2 +2X3 =8
X1 ≥0,X2 ≥0,X3 urs

Solutions

Expert Solution

The LP is of the Minimization type. So, the dual shall be of maximization type.

Also, the first constraint is of type "<=", so it has to be first converted into ">=" type by multiplying it with -1.

There are 3 decision variables. So, there shall be 3 constraints in the dual.

There are 2 constraints. So, there shall be 2 decision variables in the dual

The decision variable X3 is unrestricted in sign. So, the third constraint in the dual shall be of "=" type.

The second constraint is of "=" type. So, the second decision variable in the dual shall be unrestricted in sign

Considering the decision variables in the dual as Y1, and Y2.

The original LP is as follows:

Minimize Z = 4X1 + 2X2 - X3

subject to

-X1 - 2X2 >= -6

X1 - X2 + 2X3 = 8

X1 >= 0, X2 >= 0, X3 urs

The dual is as follows:

Maximize Zd = -6Y1 + 8Y2

subject to:

-1Y1 + 1Y2 <= 4

-2Y1 - 1Y2 <= 2

0Y1 + 2Y2 = -1

Y1 >= 0, Y2 urs


Related Solutions

Find the Dual of the following LP max z = 4x1 − x2 + 2x3 x1...
Find the Dual of the following LP max z = 4x1 − x2 + 2x3 x1 + x2 ≤ 5 2x1 + x2 ≤ 7 2x2 + x3 ≥ 6 x1 + x3 = 4 x1 ≥ 0, x2, x3 free
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
Consider the following problem     Maximize Z=2x1 + 5x2 + x3 subject to                4x1+ 2x2...
Consider the following problem     Maximize Z=2x1 + 5x2 + x3 subject to                4x1+ 2x2 + x3 ≤ 6                 x1 + x2 ≤ 2                 xi ³ 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 the...
Find dual from primal conversion MIN Z = x1 - 2x2 subject to 4x1 - x2 >= 8 2x1 + x2 >= 10 -x1 + x2 <= 7 and x1,x2 >= 0
Find dual from primal conversion MIN Z = x1 - 2x2 subject to 4x1 - x2 >= 8 2x1 + x2 >= 10 -x1 + x2 = 0
Consider the following model: maximize 40x1 +50x2 subject to: x1 +2x2 ≤ 40 4x1 +3x2 ≤...
Consider the following model: maximize 40x1 +50x2 subject to: x1 +2x2 ≤ 40 4x1 +3x2 ≤ 120 x1, x2 ≥ 0 The optimal solution, determined by the two binding constraints, is x1 = 24, x2 = 8, OFV∗ = 1,360. Now consider a more general objective function, c1x1 + c2x2. Perform a sensitivity analysis to determine when the current solution remains optimal in the following cases: (i) both c1 and c2 may vary; (ii) c2 = 50, c1 may vary;...
Consider the following LP model.Max  Z = 3x1 - 4x2 + x3 subject to     x1 + x2 +...
Consider the following LP model.Max  Z = 3x1 - 4x2 + x3 subject to     x1 + x2 + x3 >= 9            2x1 + x2 + x3<= 12 x1 + x2         = 5       x1, x2, x3 >= 0 Change it to standard form. Obtain all the basic solutions and indicate which ones are basic feasible solutions and write down the corresponding corner points. For each basic solution, you have to obtain the values of all the variables. Obtain the solution of the LP...
By using Big-m method Minimize z=4x1+8x2+3X3subject to x1+x2>=2, 2x1+x3>=5 and x1,x2,x3>=0
By using Big-m method Minimize z=4x1+8x2+3X3subject to x1+x2>=2, 2x1+x3>=5 and x1,x2,x3>=0
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 =
(Operation Research II Industrial Engineering) Consider the following LP: Minimize z = x1 + 2x2 Subject...
(Operation Research II Industrial Engineering) Consider the following LP: Minimize z = x1 + 2x2 Subject to x1 + x2 >= 1 -x1 + 2x2 <= 3 x2 <= 5 x1,x2 >= 0 (a) Convert the LP given above to the standard form. Determine all the basic feasible solutions (bfs) of the problem. Give the values of both basic and nonbasic variables in each bfs. (b) Identify the adjacent basic feasible solutions of each extreme point of the feasible region....
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
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT