We investigated the transformation between the α-helix and β-hairpin structures of an 18-residue design peptide, whose sequence is INYWLAHAKAGYIVHWTA. This peptide has both α-helix and β-hairpin structures in aqueous solution. For this purpose, we proposed the helix-strand replica-exchange method. This is one of the Hamiltonian replica-exchange methods in which we exchange parameters for umbrella potentials to enhance the α-helix or β-strand structure formation. We performed an all-atom helix-strand replica-exchange molecular dynamics (MD) simulation of this peptide in explicit water solvent with five replicas. Because the suitable umbrella potential was applied, the helix-strand replica-exchange MD simulation reproduced conformations closer to experimental conformations than a temperature replica-exchange MD simulation when the same numbers of the replicas were used, while the temperature replica-exchange MD simulation does not require bias along any specific order parameter. We calculated its free-energy landscape and revealed the transformation pathways between the α-helix and β-hairpin structures and the folding pathways from an extended structure. Although the fractions of the α-helix and β-hairpin structures are less than those obtained by the experiment, the free-energy difference between the two structures is calculated to be almost zero, which agrees with the experimental results.