Landauers approach which yields the resistance of an obstacle in an otherwise perfect wire due to elastic scattering at the obstacle is augmented by including localized inelastic scatterers within the sample. The inelastic scatterers invoked consist of an electron reservoir coupled via a lead to the wire. The key advantage of this method is that the effect of inelastic scattering can be studied by solving an elastic scattering problem. We investigate the resistance of a series of two (or more) ob- stacles and study the transition from completely coherent transmission through the sample to completely incoherent transmission. For a sample with a small transmission probability, increasing inelastic scattering decreases the resistance. At an intermediate value of inelastic scattering, the resistance reaches a minimum to increase again when inelastic scattering processes start to dominate the resistance. © 1986 The American Physical Society.