This paper proposes two subsets of R-S increases in risk with multiple crossing called an ‘outside strong increase in risk’ (OSIR) and an ‘outside relatively strong increase in risk’ (ORSIR). The conditions in these two shifts imply that there exists an R-S decrease in risk in the given interval and an R-S increase in risk outside this interval. Note that the OSIR ranking can be decomposed into two L-SIR shifts, and the ORSIR ranking can be decomposed into a L-SIR shift and a ‘left-side extended strong increase in risk’ (L-ESIR) shift. The basic relationship between these two shifts shows that the set of OSIR shifts is included in the set of ORSIR shifts. When we restrict the payoff function to be linear in the random variable , we show that the effect of these shifts can be determined for all risk averse decision makers with non-negative third derivative of utility functions .