Appendix S1. Computational aspects.
A linearized MECP solution
The MECP objective function in Eq.3 is non-linear but can be solved using a linearized
approximation. Assuming that 0 < pij < 1, the likelihood that the species will move from a given
location i to at least one surveyed (i.e., uninvaded) location can be written with the survey
selection decision variable xj as an exponent:
J
1   (1  pij )
xj
[S1].
j 1
J
Letting wi  1   (1  pij ) j , we can rewrite:
x
j 1
ln ( wi )   ln (1  pij ) x j
[S2]
jJ
and reformulate the MECP objective as:
I
Max  1  wi 
[S3]
i 1
s. t.
J
ln (wi )   ln ( 1 - pij )x j
i  I
[S4]
j 1
J
c x
j 1
j
j
C
x j  {0,1}
[S5]
 jJ
[S6].
The model in equations S3-S6 is linear with respect to the decision variables xj and wi, except for
the term ln(wi). In this study, we used a linearized solution for ln(wi) based on a piecewise linear
approximation of ln(wi) (Camm et al., 2002; Arthur et al., 2004). The linearized MECP solution
can be tested against the MECP solutions obtained with non-linear methods (see Kincaid et al.,
2008). For simplicity, we assumed 0 < wi ≤ 1 and defined a set of K break points to approximate
the interval L to 1, where L > 0 is the lowest possible probability that the origin location is not
covered by the survey system. Let Bk be the kth break point and i k, i k ≥ 0 be a continuous
variable that weights the kth break point for a given origin location i. The substitutions for the
linear approximation of wi are defined as follows:
1
K
K
k 1
k 1
wi   Bk ik , ln( wi )   ln( Bk )ik and
K
B 
k 1
k
ik
1
[S7],
then the new linearized model becomes:
I
 MECP  Max  1  wi 
[S8]
i 1
s. t.
K
J
k 1
j 1
 ln (Bk )ik   ln (1 - pij )x j
i  I
[S9]
K
wi   Bk ik
[S10]
k 1
J
c x
j
ik
1
K

k 1
C
j
j 1
[S11]
[S12]
where xj and ik are the decision variables, x j  {0,1}  j  J , and ik  0,  i  I , k  K .
Technical aspects
The formulation of the MECP enables the use of incomplete information (Polasky et al.,
2000), such as probabilistic estimates of species' spread through a landscape, so it provides a
practical approach for cost-effective allocation of surveillance efforts under uncertainty. The
linearized representation of the MECP model makes it possible to find optimal survey allocations
for datasets with very large numbers of invaded (i.e., origin) and uninvaded (i.e., destination)
locations, such as datasets constructed from detailed urban surveys or high-resolution land cover
maps. Note that the linearized MECP model has relatively high computational complexity but the
solution times are manageable even for large problems. The capacity of the MECP model to
delineate surveys that could target high-threat origins of pest invasions makes it appealing for
biosecurity applications that involve tracking the potential movement and arrival of an invasive
species from a diverse set of source locations (e.g., via containerized foreign imports). For such
scenarios, the use of MECP model seems justified despite the extra computational effort.
Our mixed-integer model formulation assumes that the selection of survey sites is binary,
which implicitly assumes that the survey intensity is equal for all sites. It is possible to calculate
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“relaxed” (i.e., interval rather than binary) linear programming solutions for the PP models by
converting the survey selection variables, xj, from a binary to a continuous format. In the MECP
model, the survey intensity could be handled by defining the objective function as maximizing
the expected number of invaded origin sites covered by surveys where an origin is covered if it
transmits propagules to one or more surveyed sites where it is detected. The origin-destination
probability values pij would need to account for survey success at the selected sites via the
location-specific probability of pest detection. The detection probability can be represented in the
model as a function of survey intensity (or per unit survey costs). Then, the possibility of
increasing the area of coverage by reducing survey intensity can be explored.
Different levels of survey intensity could also be explored in a geographical setting with the
current MECP model via a resampling method. First, the map of the survey blocks (and all
relevant geographical data, such as survey costs, cj and spread probabilities, pij) is resampled to
smaller units (for example, from 1-km2 to 250 ×250 m units). Then, the optimal solutions are
estimated for each smaller unit and the map of binary site selections is aggregated back to the
spatial resolution of the original survey blocks. Each block would be characterized by different
survey intensities. For example, an aggregation of binary survey patterns of 250-m units to 1-km
blocks would have 16 intensity levels.
Furthermore, the MECP model could also be used as an extension of the PP framework that
incorporates particular assumptions about the mechanism of spread. The model formulation
would start from the vector of propagule pressure estimates, pj, similarly to the PP1 model. Next,
for each destination location j, a vector of spread rates, pij, would be generated from the pj values
based on a given spread mechanism (e.g., as a function of distance) so that the sum of the pij
values matched the pj estimates. The matrix of pij values would then be used to find the optimal
solution with the MECP model. Essentially, this approach would tailor the surveys according to
the underlying assumptions of the spread mechanism while making best use of the propagule
pressure data.
References:
Arthur, J.L., Camm, J.D., Haight, R.G., Montgomery, C.A. & Polasky, S. (2004) Weighing
conservation objectives: maximum expected coverage versus endangered species
protection. Ecological Applications, 14, 1936-1945.
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Camm, J.D., Norman, S.K., Polasky, S. & Solow, A.R. (2002) Nature reserve selection to
maximize expected species covered. Operations Research, 50, 946-955.
Kincaid, R.K., Easterling, C. & Jeske, M. (2008) Computational experiments with heuristics for
two nature reserve site selection problems. Computers and Operations Research, 35, 499512.
Polasky, S., Camm, J.D., Solow, A.R., Csuti, B., White, D. & Ding, R. (2000) Choosing reserve
networks with incomplete species information. Biological Conservation, 94, 1-10.
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