Consider the following problem:
$$Max \quad \sum_{i\in I}\sum_{j\in J}c_{ij}x_{ij} $$
subject to $$ \quad \sum_{j \in J}a_{ij} x_{ij}\leq b_{i}\quad i \in I$$ $$ \quad \sum_{i \in I} x_{ij}=1\quad j \in J$$ $$ x_{ij}\in \{0,1\} \quad i \in I ; \quad j \in J $$
In the above problem, $c_{ij}$, $a_{ij}$ and $b_i$ are given parameters and $x_{ij}$ are the decision variables. In addition, $I$ is the set of resources and $J$ is the set of jobs. Suppose you knew that the polytope formed by the set X={x: $ \quad \sum_{j \in J}(a_{ij} x_{ij})\leq b_{i}\quad i \in I$} had a structure that made it easy to solve/investigate. Then, Dantzig-Wolfe (D-W) decomposition would be an attractive approach to this problem. Using this assumption, propose an appropriate master problem and subproblem to solve (LP) using D-W. Define all notation/variables that you create.
I know how to develop the Master Problem for a simpler problem but my problem with this one is that x has two subscripts i,j and I don't know how to deal with it ? can someone help me.
You want to assign ressources to jobs, such that every job $j\in J$ is given a ressource $i\in I$, and such that the combinations of jobs $\Omega_i\subseteq J$ who are given the same ressource $i$ verify $\sum_{j\in \Omega_i}a_{ij}\le b_i$.
What you call $X$ is the set of all feasible combinations $\Omega_i$: $X=\bigcup_{i\in I}\Omega_i$.
So, let $\lambda_{pi}$ be a binary variable that equals $1$ if and only if the combination of jobs $p\in \Omega_i \subseteq X$ is selected for ressource $i\in I$, and let $c_{pi}$ be the associated cost. Your master problem can be written as follows: $$ \max_{\lambda}\; \sum_{p\in \Omega_i}\sum_{i\in I}c_{pi} \lambda_{pi} $$ subject to $$ \sum_{p\in \Omega_i\;|\;j\in p}\lambda_{pi} = 1\quad \forall j\in J\\ \lambda_{pi}\in \{0,1\}\quad \forall p\in \Omega_i, \;\forall i\in I $$