and then this one as well :
state the dual problem for each linear programming problem.
minimize w =y1+y2+4y3
subject to: y1+2y2+3y3 >=115
2y1+y2+8y3>=200
y1 +y3 >=50
with y1>=0, y2>=0, y3>=0
To state the dual problem for a linear programming problem, we need to follow a few steps:
Step 1: Write the objective function of the primal problem.
The objective function of the given primal problem is:
w = y1 + y2 + 4y3
Step 2: Write the constraints of the primal problem.
The constraints of the given primal problem are:
y1 + 2y2 + 3y3 >= 115
2y1 + y2 + 8y3 >= 200
y1 + y3 >= 50
Step 3: Identify the decision variables of the primal problem.
The decision variables of the given primal problem are:
y1, y2, y3
Step 4: Define the decision variables and their corresponding dual constraints.
For each decision variable in the primal problem, we define a corresponding dual variable and write the dual constraint using those dual variables. The dual constraints will be opposite to the primal constraints.
For the given primal problem, the dual problem can be stated as follows:
Maximize:
Z = 115x1 + 200x2 + 50x3
Subject to:
x1 + 2x2 + x3 <= 1
2x1 + x2 <= 1
3x1 + 8x2 <= 4
with x1 >= 0, x2 >= 0, x3 >= 0
In the dual problem, x1, x2, and x3 are the dual variables corresponding to y1, y2, and y3 respectively in the primal problem. The coefficients of the objective function are the values from the right-hand side of the primal constraints.
Please note that the dual problem is formed by taking the opposite of the primal problem's inequality signs and interchanging the roles of the objective function and constraints.