1. Fuzzy Applications by W. Silvert, IPIMAR, Portugal

2. Application to NAFO model  The NAFO model presented by Bill Brodie in his talk uses the following simplified scheme: We can make a fuzzy representation of this as follows:

3. Region 1  Region 1 can be described as follows: If F is low and B is high

4. Region 2  Region 2 can be described as follows: If F is high and B is high

5. Region 3  Region 3 can be described as follows: If F is high and B is low

6. Region 4  Region 4 can be described as follows: If B is very low

7. Quantification We quantify the model by saying that:  F is 100% low if F < 0.1  F is 100% high if F > 0.2  For 0.1 < F < 0.2 interpolate For example F=0.15 is 50% high, 50% low  We do the same for biomass  Now let us take a look at the more complex figure from the written documentation Brodie submitted...

8. More Detailed Analysis

9. Fuzzy Zones The regions between B lim and B buf, and between F lim and F buf, are fuzzy zones. These are the zones where B and F are in both HIGH and LOW sets

10. Rules for Action Typical rules are:  IF B high and F low (#1) THEN continue  IF B high and F high (#2) THEN reduce F  etc. Corresponding fuzzy rules are  IF B high and F low (#1) THEN continue  IF B high and F high (#2) THEN reduce F drastically, where we might specify a rate of fishing reduction

11. Implementation  The fuzzy rules get rid of the sharp line between regions. Assume biomass is high (regions #1 and #2) – then the rules are interpreted as follows:  IF F = 0.1 THEN mortality is 100% low and we continue  IF F = 0.2 THEN mortality is 100% high and we reduce fishing drastically  IF F = 0.15 THEN mortality is 50-50 and we reduce fishing moderately (drastic/2)

12. More Complexity  We can apply the same reasoning to more complicated ranges, such as in this area: Here we have biomass and mortality both in the fuzzy area between high and low, and we have a continuous management policy

13. General Procedure  Identify states of the system for which you want to assign actions.  In this case the states are visualised as areas on the Biomass-Mortality phase diagrams  The areas do not cover the entire diagram  For example, (F 0.2)=HIGH  Interpolate to find fuzzy mixed state  Assign action on basis of memberships  Example: if F=0.15, the state is 50% LOW and 50% high and the action is half-way in between

14. Summary  In any situation where we have different management regimes associated with the values of various variables (Indicators or Characteristics), we can describe fuzzy sets that give us a continuous and more flexible management policy without sharp cutoffs and discontinuities.