Fuzzy Applications

byW. Silvert, IPIMAR, Portugal

Application to NAFO model

The NAFO model presented by Bill Brodie in his talk uses the following simplified scheme:

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 20 40 60 80 100 120 140 160 180

Stock biomass

Fish

ing

mor

talit

y

BtrBbuf

Overfishingzone

F-Targetzone

Collapse Recovery zoneDanger

zoneRecovered zone

Fbuf

F-bufferzone

4 3 2

1

We can make a fuzzy representation of this as follows:

Region 1

Region 1 can be described as follows:

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 20 40 60 80 100 120 140 160 180

Stock biomass

Fish

ing

mor

talit

y

BtrBbuf

Overfishingzone

F-Targetzone

Collapse Recovery zoneDanger

zoneRecovered zone

Fbuf

F-bufferzone

4 3 2

1

If F is low and B is high

Region 2

Region 2 can be described as follows:

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 20 40 60 80 100 120 140 160 180

Stock biomass

Fish

ing

mor

talit

y

BtrBbuf

Overfishingzone

F-Targetzone

Collapse Recovery zoneDanger

zoneRecovered zone

Fbuf

F-bufferzone

4 3 2

1

If F is high and B is high

Region 3

Region 3 can be described as follows:

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 20 40 60 80 100 120 140 160 180

Stock biomass

Fish

ing

mor

talit

y

BtrBbuf

Overfishingzone

F-Targetzone

Collapse Recovery zoneDanger

zoneRecovered zone

Fbuf

F-bufferzone

4 3 2

1

If F is high and B is low

Region 4

Region 4 can be described as follows:

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 20 40 60 80 100 120 140 160 180

Stock biomass

Fish

ing

mor

talit

y

BtrBbuf

Overfishingzone

F-Targetzone

Collapse Recovery zoneDanger

zoneRecovered zone

Fbuf

F-bufferzone

4 3 2

1

If B is very low

Quantification

We quantify the model by saying that:F is 100% low if F < 0.1F is 100% high if F > 0.2For 0.1 < F < 0.2 interpolate

For example F=0.15 is 50% high, 50% low

We do the same for biomassNow let us take a look at the more

complex figure from the written documentation Brodie submitted ...

More Detailed Analysis

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 20 40 60 80 100 120 140 160 180

Stock biomass

Fis

hin

g m

ort

alit

y

BtrBbuf

Overfishingzone

F-Targetzone

Collapse Recovery zoneDanger

zone Recovered zone

Blim

Flim

Fbuf

F-bufferzone

Fuzzy Zones

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 20 40 60 80 100 120 140 160 180

Stock biomass

Fis

hin

g m

ort

ali

ty

BtrBbuf

Overfishingzone

F-Targetzone

Collapse Recovery zoneDanger

zone Recovered zone

Blim

Flim

Fbuf

F-bufferzone

The regions between Blim and Bbuf, and between Flim and Fbuf, are fuzzy zones.

These are the zones where B and F are in both HIGH and LOW sets

Rules for Action

Typical rules are:IF B high and F low (#1) THEN continueIF B high and F high (#2) THEN reduce Fetc.Corresponding fuzzy rules areIF B high and F low (#1) THEN continueIF B high and F high (#2) THEN reduce F

drastically, where we might specify a rate of fishing reduction

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)

More Complexity

We can apply the same reasoning to more complicated ranges, such as in this area:

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 20 40 60 80 100 120 140 160 180

Stock biomass

Fish

ing

mor

talit

y

BtrBbuf

Overfishingzone

F-Targetzone

Collapse Recovery zoneDanger

zoneRecovered zone

Fbuf

F-bufferzone

4 3 2

1

Here we have biomass and mortality both in the fuzzy area between high and low, and we have a continuous management policy

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 diagramsThe areas do not cover the entire diagramFor example, (F<0.1)=LOW and (F>0.2)=HIGH

Interpolate to find fuzzy mixed stateAssign 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

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.