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The shear viscosity $η$ of a quark-gluon plasma in equilibrium can be calculated analytically using multiple methods or numerically using the Green-Kubo relation. It has been realized, which we confirm here, that the Chapman-Enskog method agrees well with the Green-Kubo result for both isotropic and anisotropic two-body scatterings. We then apply the Chapman-Enskog method to study the shear viscosity of the parton matter from a multi-phase transport model. In particular, we study the parton matter in the center cell of central and midcentral Au+Au collisions at $200A$ GeV and Pb+Pb collisions at $2760A$ GeV, which is assumed to be a plasma in thermal equilibrium but partial chemical equilibrium. As a result of using a constant Debye mass or cross section $σ$ for parton scatterings, the $η/s$ ratio increases with time (as the effective temperature decreases), contrary to the trend preferred by Bayesian analysis of the experimental data or pQCD results that use temperature-dependent Debye masses. At $σ=3$ mb that enables the transport model to approximately reproduce the elliptic flow data of the bulk matter, the average $η/s$ of the parton matter in partial equilibrium is found to be very small, between one to two times $1/(4π)$.