For the disaster risk decision tree model, explain why an increase in S and an increase in U have the opposite impact on the choice of how many suppliers to use. What is the implication of these two phenomena taken together?

From formula (S11-1), we notice that when S increases, (1 - S) decreases; therefore, the disruption risk probability due to all suppliers individually failing at the same time receives less weight. In other words, there's a higher chance that all will be disrupted by a super-event anyway, so there is less incentive to increase the supplier base to handle the case where every supplier has a unique-event at the same time. Mathematically, the impact of a higher n in the second term is smaller because it's being multiplied by (1 - S), which is smaller. Having said that, a larger S does increase the overall probability of total disruption, so it's never a desired occurrence-it just means that simply adding more suppliers may not be quite as effective of a mitigation strategy. On the other hand, when U increases, the impact of the second term of formula (S11-1) increases, so adding another supplier (raising n by 1) has even more impact. This makes sense-the greater the chance that each supplier will individually fail, the more likely it is that we will want more suppliers.
An implication of all of this is to try to reduce the value of S by changing the geographical landscape. In other words, in most circumstances, S would be larger for a city than for a country than for the whole world. Therefore, a geographically diversified supplier base would reduce S. In conclusion, when multiple suppliers are used, managers may consider using ones that are geographically dispersed to lessen the probability of all failing simultaneously.