Generalized replica exchange method (gREM) computer simulations are performed on model glass formers, polydisperse Lennard-Jones (LJ) systems at constant pressure, and discuss the effect of size polydispersity on the gas-liquid and liquid-crystal transitions, and on the liquid-glass crossover. At high temperatures in the gas phase, there are no polydispersity effects on the observables. With decreasing temperature, the enthalpy, internal energy and volume decrease with increasing polydispersity, and the gas-liquid transition point shifts to higher temperature. At low temperature, sufficient polydispersity causes the liquid-crystal transition to disappear; the threshold for this phenomenon lies in a polydispersity range of s = 0.15-0.20. Enthalpy and internal energy show the similar behavior over the whole temperature region with changing polydispersity. On the other hand, volume shows a different polydispersity dependence in the liquid, crystal, and glass states. The pressure dependence of polydispersity effects are also discussed, and a preliminary gas-liquid-crystal-glass phase diagram of polydisperse LJ systems is obtained.