Stellar systems composed of single, double, triple or high-order systems are rightfully regarded as the fundamental building blocks of the Milky Way. Binary stars play an important role in formation and evolution of the Galaxy. Through comparing the radial velocity variations from multi-epoch observations, we analyze the binary fraction of dwarf stars observed with the LAMOST. Effects of different model assumptions such as orbital period distributions on the estimate of binary fractions, are investigated. The results based on log-normal distribution of orbital periods reproduce the previous complete analyses better than the power-law distribution. We find that the binary fraction increases with $T_{rm eff}$ and decreases with [Fe/H]. We first investigate the relation between $alpha$-elements and binary fraction in such a large sample as the LAMOST. The old stars with high [$alpha$/Fe] dominate higher binary fraction than young stars with low [$alpha$/Fe]. At the same mass, former forming stars possess a higher binary fraction than newly forming ones, which may be related with the evolution of the Galaxy.
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