The distribution of diversity: challenges and applications
153
Box 6.4 Three broad classes of mathematical problems used in systematic
conservation planning: the minimum set problem, the maximal coverage
conservation prioritization problem and the conservation resource allocation
problem
Conservation planning began without a well - posed mathematical problem, which is not uncommon
in conservation science (Possingham et al ., 2001 ). Cocks and Baird ’ s (1989) seminal paper provided
the fi rst formal statement of a conservation planning problem – the minimum set problem. In the
minimum set problem the goal is to conserve a variety of conservation features to an adequate level
for minimum total cost where cost can be the cost of acquisition and management or the estimated
foregone opportunity cost (Naidoo & Ricketts, 2006 ). The simplest variant of this problem is:
min c x
i i
i
NS
=
∑
1
given that
x r T
j
i ij
i
N
j
s
=
∑ ≥
1
,
,
for all features
where r ij is the occurrence level of feature j in site i , c i is cost of site i , N s is the total number of sites
and T j is the target level for feature j . The control variable x i has value 1 for selected sites and value
0 for sites not selected (Moilanen et al. , 2009 ). This became the foundational problem of systematic
conservation planning.
Since then, various authors have produced alternatives, but arguably the maximal coverage conservation prioritization problem is the most dominant. This problem is used when resources are
insuffi cient for satisfying all targets and the objective is to fi nd the solution that satisfi es the largest
number of conservation targets, given a budget constraint. The maximal coverage problem is related
to the minimum set coverage problem, in that minimum set coverage can be achieved by solving
the maximal coverage problem at different budget levels and fi nding the minimum budget level that
satisfi es all targets. A simple version of the maximal coverage problem can be written as:
max
(
),
I
xr T
j
ii j
i
j
j
∑
∑
≥
given that
x c B
i i
i
∑ ≤ ,
where B is the conservation budget (money, trained personnel, time, etc.), and I (z) is an indicator
function, with I j (z) = 1 when condition z is true, i.e. the target for feature j is met when
x r T
i ij
i
j
∑ ≥
⎛
⎝ ⎜
⎞
⎠ ⎟ ,
and I j (z) = 0 otherwise (Moilanen et al. , 2009 ).
Both the minimum set and maximum coverage problems are limited to specifi c problems. However,
it is possible to defi ne a fairly general conservation resource allocation problem that includes most,
if not all, previous problem defi nitions. In general, all of conservation involves taking actions in a
place and at a time in an attempt to achieve a variety of outcomes.
Our general task is therefore to decide how much to spend on each kind of action (e.g. invasive
species control, changed logging practices, or reduced grazing) in each place, at a particular time
153
Box 6.4 Three broad classes of mathematical problems used in systematic
conservation planning: the minimum set problem, the maximal coverage
conservation prioritization problem and the conservation resource allocation
problem
Conservation planning began without a well - posed mathematical problem, which is not uncommon
in conservation science (Possingham et al ., 2001 ). Cocks and Baird ’ s (1989) seminal paper provided
the fi rst formal statement of a conservation planning problem – the minimum set problem. In the
minimum set problem the goal is to conserve a variety of conservation features to an adequate level
for minimum total cost where cost can be the cost of acquisition and management or the estimated
foregone opportunity cost (Naidoo & Ricketts, 2006 ). The simplest variant of this problem is:
min c x
i i
i
NS
=
∑
1
given that
x r T
j
i ij
i
N
j
s
=
∑ ≥
1
,
,
for all features
where r ij is the occurrence level of feature j in site i , c i is cost of site i , N s is the total number of sites
and T j is the target level for feature j . The control variable x i has value 1 for selected sites and value
0 for sites not selected (Moilanen et al. , 2009 ). This became the foundational problem of systematic
conservation planning.
Since then, various authors have produced alternatives, but arguably the maximal coverage conservation prioritization problem is the most dominant. This problem is used when resources are
insuffi cient for satisfying all targets and the objective is to fi nd the solution that satisfi es the largest
number of conservation targets, given a budget constraint. The maximal coverage problem is related
to the minimum set coverage problem, in that minimum set coverage can be achieved by solving
the maximal coverage problem at different budget levels and fi nding the minimum budget level that
satisfi es all targets. A simple version of the maximal coverage problem can be written as:
max
(
),
I
xr T
j
ii j
i
j
j
∑
∑
≥
given that
x c B
i i
i
∑ ≤ ,
where B is the conservation budget (money, trained personnel, time, etc.), and I (z) is an indicator
function, with I j (z) = 1 when condition z is true, i.e. the target for feature j is met when
x r T
i ij
i
j
∑ ≥
⎛
⎝ ⎜
⎞
⎠ ⎟ ,
and I j (z) = 0 otherwise (Moilanen et al. , 2009 ).
Both the minimum set and maximum coverage problems are limited to specifi c problems. However,
it is possible to defi ne a fairly general conservation resource allocation problem that includes most,
if not all, previous problem defi nitions. In general, all of conservation involves taking actions in a
place and at a time in an attempt to achieve a variety of outcomes.
Our general task is therefore to decide how much to spend on each kind of action (e.g. invasive
species control, changed logging practices, or reduced grazing) in each place, at a particular time