I would like to maximize the term $ l_1b_1+l_2b_2+l_3b_3-2 $ such that the following conditions hold:
$ 1>l_1>l_2>l_3>0 $, $ l_1,l_2,l_3 \in \mathbb{Q} $,
$ b_1,b_2,b_3 \in \mathbb{N} $,
$ 2 \le l_1b_1+l_2b_2+l_3b_3 < 3 $, and
There is no $ a \in \mathbb{Z}^3_{\ge 0} $ with $ 1 \le \sum_{i=1}^3 l_ia_i \le \sum_{i=1}^3 l_ib_i -1 $
The largest value that I could find so far is $ 5/156 $, but this was not done systematically. (Or are there at least some non-trivial upper bounds for the objective value?)