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This paper provides an approximate optimal routing algorithm, called relax-converter-first (RCF), for dynamic WDM (wavelength division multiplexing) networks with heterogeneous and partial wavelength conversion capability in each node. The algorithm can derive an optimal solution if the cost of consuming a wavelength converter is set much smaller than is the case for taking a wavelength channel. We propose a wavelength graph technique along with a modified Dijkstra's shortest path first algorithm. Simulation is conducted to verify the performance in terms of blocking probability focusing on the following two themes: first, a comparison in performance is conducted among the cases of using fixed alternate routing, shortest path first algorithm with wavelength graph (SPAWG), and RCF. Second, the performance of using the RCF scheme is examined with different cost of taking a wavelength converter in network nodes with different wavelength conversion capability. We also provide the simulation statistics about the optimality in using the RCF scheme under different network environments.