I am dealing with the following MILP:
$$ \max\; \sum_{t\in T}\alpha_t r_t-\gamma \sum_{t\in T}\omega_{t} - \beta \sum_{t\in T}V_t $$ subject to \begin{align} &\color{green}{ I_t=I_{t-1}+\omega_{t}-r_t\quad} &\forall t \in T \\ &\color{green}{V_t = R_t + \frac{\omega_t}{Q}}\quad &\forall t \in T\\ &\color{green}{R_t \le 1}\quad &\forall t \in T\\ &\mbox{+ other constraints}\\ &r_t,I_t,\omega_t,R_t \ge 0 \quad &\forall t \in T\\ &V_t \in \{0,1\} \quad &\forall t \in T \end{align}
I am trying to reformulate it with a Dantzig-Wolfe decomposition. The $\color{green}{\mbox{three first constraints}\;}$ define my master problem, and the "other constraints" are for my sub-problem, which generates columns $s\in S$. For each column $s$ and each time instant $t\in T$, there is an associated parameter $W_{st}$ which sets the value of variable $\omega_t$. I use binary decision variables $\lambda_s$ that equal $1$ if and only if column $s\in S$ is used for the master problem. The master problem can be written as follows
$$ \max\; \sum_{t\in T}\alpha_t r_t-\gamma \sum_{t\in T}\sum_{s\in S}W_{ts}\lambda_s - \beta \sum_{t\in T} V_t $$ subject to \begin{align} &\color{green}{I_t=I_{t-1}+\sum_{s\in S}W_{ts}\lambda_s-r_t}\quad &\forall t \in T \quad &\color{red}{(a_t)}\\ &\color{green}{V_t = R_t + \frac{\sum_{s\in S}W_{st}\lambda_s}{Q}}\quad &\forall t \in T \quad &\color{red}{(b_t)}\\ &\color{green}{R_t \le 1}\quad &\forall t \in T \quad &\color{red}{(c_t)}\\ &\sum_{s\in S|t\in s}\lambda_s \le 1\quad &\forall t\in T \quad &\color{red}{(d_t)}\\ &\sum_{s\in S}\lambda_s =1 \quad &&\color{red}{(\sigma)}\\ &r_t,I_t,R_t \ge 0 \quad &\forall t \in T\\ &\lambda_s,V_t \in \{0,1\} \quad &\forall s \in S, \; \forall t \in T \end{align}
I am having some trouble to determine the reduced cost of variable $\lambda_s$. The $\color{red}{\mbox{dual variables}\;}$ of each constraint are in red on the right. I believe it is $$ \hat{c}_s = -\gamma \sum_{t\in T}W_{ts} -\sum_{t\in T}W_{ts}\color{red}{a_t}-\sum_{t\in T}\frac{W_{ts}}{Q}\color{red}{b_t}-\sum_{t\in T}\color{red}{d_t} - \color{red}{\sigma} $$ So in this case the objective function of my subproblem would be $$ -\gamma \sum_{t\in T}\omega_{t} -\sum_{t\in T}\omega_{t}\color{red}{a_t}-\sum_{t\in T}\frac{\omega_{t}}{Q}\color{red}{b_t}-\sum_{t\in T}\color{red}{d_t} - \color{red}{\sigma} $$ Can anyone confirm that this expression is right? Or point out where I made a mistake? Thanks!