Cocks & Baird supply a number of guidelines for making appropriate reserve selection choices. We provide formal definitions of each of these below, following our normal format but with the addition of quotation marks around the italicised informal descriptions since these are taken directly from Cocks & Baird's paper. Explanatory comments appear immediately after each definition. The guidelines are of three kinds:
defines a requirement which
is necessary to be satisfied by the set of selected reserves, R, in order
for them to be complete in a given context. The term, T, summarises
the test and the precondition, 
, gives the conditions which
satisfy it (thus parameterising the test).
defines a way in which
an individual reserve, S, may be chosen to be added to the set of
selected reserves, R. The term, E, summarises the selection
criterion and the precondition, 
, gives the conditions which
satisfy it (thus parameterising the selection).
defines an example collection of tests (
to 
) which we
might require the final set of selected reserves, R, to satisfy.
For convenience of reference, each of these constraints is labelled
with the identifier, C.
``As far as possible ensure at least two expected appearances of each community appear in the reserve system.''
Expression 1 acts across all communities by requiring that there should not be a community, C, such that the sum of appearances for C across the selected reserves, R, is less than M. Notice that by leaving the minimum threshold of appearance, M, as a variable we allow for the possibility of different threshold values, thus generalising from the value of 2 given in the original guideline. In all the definitions we give we shall follow this standard principle of being as general as possible without losing the option of stipulating the original guideline as a special case.
``As far as possible give preference to including large reserves in the reserve system.''
This is a selection condition because it provides a guideline for choosing
a new site, S, to add to the list, R, of reserves already selected. The
precondition first finds the area, A, of S; then tests that S does not
already appear in list R; then tests that there is not some other site,
, with area 
such that 
doesn't appear in R and
is greater than A.
``As far as possible ensure that near-coastal reserve systems are included in the reserve system.''
We have included two conditions for this guideline. The first is a selection condition which allows the addition of a coastal reserve to the list of selected reserves, provided that it does not already appear in the list. The second is a necessary condition which stipulates that the cardinality of the set, S, of coastal reserves appearing in the list of selected reserves should be at least equal to a given threshold value, M.
``As far as possible ensure that inland reserves are included in the reserve system.''
The above conditions are constructed in a similar way as in the case of coastal reserves.
``As far as possible favour the inclusion of sites having a high expectation of containing one or both of the two rarest communities, i.e. communities with the lowest total number of expected appearances over all sites.''
Expression 7 selects a new site, S, with maximum
appearance value for a given community, C. The test which
ensures maximality is the final line of the precondition, which states
that there should be no site, 
, such that 
doesn't already
appear in the set of selected reserves and for which the appearance
value of 
is greater than the appearance value of
S. This condition is a little different from the textual
requirement since it allows the resource manager to specify the
community, C, rather than specifying that it should be ``as far as
possible ... one or both of the two rarest''. Our interpretation
of this vague part of the guideline is that there is scope for
subjective interpretation and, accordingly, we have not stipulated
this formally - the resource manager must decide which of the
communities C will be.
``Include in the reserve system all sites which are already declared reserves.''
This condition states that there should not exist a site which already has reserve status and which does not appear in the list of selected reserves. We have now completed our definition of reserve selection guidelines.
Cocks & Baird give four examples of constraints on the final lists of sites chosen as reserves, which are as follows:
``...specify a reserve system of no more than 12 sites, collectively containing at least two expected appearances of each of [the] six communities...the reserve system was required to have an area of not much more, and preferably less, than 25% of the total area over all sites. Finally, there should be some spread of sites between coastal and inland locations. Solution 1...specifies a reserve system satisfying these requirements.''
Cocks & Baird's general guideline is not explicit about the precise values
allocated to the maximum permitted proportion of area given over to
reserves or to the ratio of inland to coastal reserves. We have chosen
here to set the maximum proportion of area as 
and the minimum
proportion of inland and coastal reserves to be 
.
``Solution 2 retains the constraint that each community must be expected to appear in the system at least twice, but otherwise is concerned with only one other goal - to minimise the total area of all sites in the reserve system.''
This constraint introduces a particularly strong requirement: that there should not exist any other valid allocation of reserves which produces a list of sites with a smaller total reserve area than that of the current list. This is interesting from a logical point of view because it requires the definition of a constraint in terms of the reserve selection process itself, rather than simply in terms of properties of the reserves. We show how to deal with this in expression 18.
``Solution 3 also retains the constraint that each community must be expected at least twice but, in this instance, is concerned only with the goal of minimising the number of sites in the reserve system.''
The representation here is similar to 11 and similar comments apply.
``Solution 4 reflects the impact of re-running the initial problem (Solution 1) with the additional design requirement...that the system must contain the three existing declared reserves...'' (The solution given by Cocks & Baird also sets the maximum reserve number to 14).
This is similar to 10, with the addition of one extra test: that all existing reserves should appear in the list of selected sites. This completes our description of the test sets. All that remains is to define the constituent tests of constraints 1 to 4. For these we do not have quotable English definitions from Cocks & Baird's paper so we supply our own definitions (italicised as usual but not quoted).
We have a maximum number of reserves within the limit, M, if the cardinality of the set of reserves, R, is less than M.
We are within the maximum proportion of reserves to total area if the sum, S, of all the areas of selected reserves, R, when divided by the total area of all sites is less than the limit, M.
The proportion of selected reserves which are coastal exceeds a given
minimum, M if the ratio of the cardinality, 
, of the list,
S, of selected coastal reserves to the cardinality, 
, of all selected
is no lower than the minimum level.
The proportion of selected reserves which are inland exceeds a given
minimum, M if the ratio of the cardinality, 
, of the list,
S, of selected coastal reserves to the cardinality, 
, of all selected
is no lower than the minimum level.
The total reserve area is minimised if there is no other allocation of
reserves, 
, which yields a smaller sum of areas than for the
currently selected set, R.
The 
condition of this test requires a
definition of the algorithm for reserve selection, which we supply
below. However it should be noted that for large and complex reserve
selection tasks there is a prohibitive computational overhead in
demonstrating that no other reserve selection could ever yield a better
set of selected reserves, since this requires (in the worst case)
performing all reserve selections. Various forms of optimisation can be
performed to approximate to this exhaustive computation but, since
this opens up the broad area of constraint satisfaction methods, we
shall not discuss these.
The total number of reserves is minimised if there is no other
allocation of reserves, 
, which is of smaller cardinality than for the
currently selected set, R.
