  Routing and scheduling problems with time windows have long been important
optimization problems for logistics and planning. Many classical heuristics and
exact methods exist for such problems. However, there are no satisfactory
methods for generating routes using quantum computing (QC), for mainly two
reasons: inequality constraints, and the trade-off of feasibility and solution
quality. Inequality constraints are typically handled using slack variables;
and feasible solutions are found by filtering samples. These challenges are
amplified in the presence of noise inherent in QC. Here, we propose a greedy
algorithm that generates routes by using information from all samples obtained
from the quantum computer. By noticing the relationship between qubits in our
formulation as a directed acyclic graph (DAG), we designed an algorithm that
adaptively constructs a feasible solution.
  We prove its convergence to a feasible solution, and illustrate its efficacy
by solving the Fleet Sizing Vehicle Routing Problem with Time Windows
(FSVRPTW). Our computational results show that this method obtains a lower
objective value than the current state-of-the-art annealing approaches, both
classical and hybrid, for the same amount of time using D-Wave Hybrid Solvers.
We also show its robustness to noise on D-Wave Advantage 4.1 through
computational results as compared to the filtering approach on DWaveSampler,
even when the filtering approach is given a longer annealing time, and a larger
sample size.
