The specific linear programme has an optimal solution as $x_1 = 0.66$, $x_2 = 1.33$, $x_3 = 12.2$, $x_4 = 0.0$ and the objective value is $33.3$. While the problem is solved by D-W decomposition method, in the specific iteration (defining two sub-problems in which, $x_1$ and $x_2$ are defined in the first and the rest are in the second), reduced cost of the second sub-problem is zero and the first has a negative reduced cost.
When we add columns based on the reduced cost, the objective value of the master problem will be $33.3$ and convergence has been attained. The solutions of the problem by using D-W are $x_1 = 0.66$, $x_2 = 1.33$, $x_3 = 8$, $x_4 = 0.0$.
The small example of the problem is: \begin{equation}\begin{array}{rrrrrr} \text { Minimize } & -x_{1} & - & 2 x_{2} & +3 x_{3} & +x_{4} & \\ \text { subject to } & x_{1} & + & 2 x_{2} & +3 x_{3} & +x_{4} & \geq 40 \\ & x_{1} & + & x_{2} & & & \leq 2 \\ & -x_{1} & + & 2 x_{2} & & & \leq 2 \\ & & & x_{3} & + & x_{4} & \geq 8 \\ & x_{1}, & & x_{2}, & x_{3}, & x_{4} \geq 0 \end{array}\end{equation}
I'm wondering if,
- What is the reason(s) to obtain the different value of the variable $x_3$?
- Is there any specific issue to deal with the sub-problem with zero reduced cost while another has a negative value?