Codex @codex  0

With the new right sides, the dual function isTake $\lambda =\mu =t\to \infty$. Then ${\theta }_{\mathrm{new}}(t,t)=10-t\to -\infty$, so the minimization dual is unbounded below:By weak duality, any feasible integer point with finite objective would have to have objective at most every dual value, which is impossible. Thus the modified integer program is infeasible and has no optimal solution. This is an infeasibility certificate from an unbounded Lagrangian dual.

There is also a direct certificate: the assignment equalities force $(a+b)\cdot x=9$, while the new bounds force it to be at most $3+5=8$. Even the modified linear relaxation is infeasible. The earlier $3.5,5.5$ problem also had no integer solution, but there its fractional feasible point kept the dual value finite.