0-1 kNapsack problem FPTAS algorithm

[StIgM@]

I have the following 0-1 knapsack problem variant:

I want to buy X units of a product at min cost, and there are m farmers that offer:

- farmer 1: a₁₁ units at cost c₁₁, ..., a_1n1 units at cost c_1n1
...
- farmer m: a_m1 units at cost c_m1, ..., a_mnm units at cost c_mnm

and I can choose at most one option from each farmer.

Formally, I want to

[tex]Minimise \sum_{i=1}^{m} \sum_{j=1}^{n_i} x_{ij}c_{ij}[/tex]

subject to [tex]\sum_{i=1}^{m} \sum_{j=1}^{n_i} x_{ij}a_{ij} >= X[/tex]

where [tex]x_{ij}\in\{0,1\}[/tex]

Could you please let me know if this problem resembles a variant of 0-1 knapsack problem?

Is there any FPTAS algorithm that suits this problem?

Thanks in advance

Note: It's not homework. The problem is different and this example is a simplification.

 

[fresh_42]

If you only can decide between "buy all" and "do not buy" from a certain farmer, then it is a 0-1-problem. It is an ordinary optimization problem, so algorithms like the simplex algorithm should help. The complexity of the problem is another question, i.e. whether it is NP complete or not. Looks as it is at first sight, but to decide this the problem has to be stated very precisely and a proof is needed.