Fusion rules for the triplet $W$-algebra ${\mathcal{W}}_{p_{+},p_{-}}$

We study the structure of fusion rules for the triplet $W$-algebra ${\mathcal{W}}_{p_{+},p_{-}}$. By using the vertex tensor category theory developed by Huang, Lepowsky and Zhang, we rederive certain non-semisimple fusion rules given by Gaberdiel-Runkel-Wood and Rasmussen. We further show that certain rank two and three indecomposable modules are self-dual.