  The D-wave quantum annealer requires embedding with ferromagnetic (FM) chains
connected by several qubits, because it cannot capture exact long-range
coupling between qubits, and retains the specific architecture that depends on
the hardware type. Therefore, determination of the chain strength $J_c$
required to sustain FM order of qubits in the chains is crucial for the
accuracy of quantum annealing. In this study, we devise combinatorial
optimization problems with ordered and disordered qubits for various embeddings
to predict appropriate $J_c$ values. We analyze the energy interval $\Delta_s$
and $\Delta_c$ between ground and first excited states in the combinatorial
optimization problems without and with chains respectively, using the exact
approach. We also measure the probability $p$ that the exact ground energy per
site $E_g$ is observed in many simulated annealing shots. We demonstrate that
the determination of $J_c$ is increasingly sensitive with growing disorder of
qubits in the combinatorial optimization problems. In addition, the values of
appropriate $J_c$, where the values of $p$ are at a maximum, increase with
decreasing $\Delta_s$. Finally, the appropriate value of $J_c$ is shown to be
observed at approximately $\Delta_c/\Delta_s=0.25$ and $2.1 E_g$ in the ordered
and disordered qubits, respectively.
