Using mathematical programming to address the multiple reserve selection problem: An example from the Eyre Peninsula, South Australia

Biological Conservation 49 (1989) 113-130

Using Mathematical Programming to Address the Multiple Reserve Selection Problem: An Example from the Eyre Peninsula, South Australia

K. D. C o c k s & I. A. Baird CSIRO Division of Wildlife and Ecology, PO Box 84, Lyneham, ACT 2602, Australia (Received 22 August 1988; revised version received 24 December 1988; accepted 5 January 1989)

ABSTRACT It is argued that the problem o f choosing a subset of candidate reserves as the components of a reserve system can, in the right circumstances, be sensibly and routinely formulated as a mathematical programming problem, namely an integer goal programming problem. This remains true even when the entities being reserved are known to exist on candidate reserves only in a probabilistic sense. An example using data describing 101 remnant patches of bush on the Eyre Peninsula, South Australia is presented.

INTRODUCTION The pattern of land use in a country or region reflects numerous past decisions, both market and non-market, on how individual portions of the landscape are to be used. In a general way, land tends to be allocated to the use for which, by some criterion, it is most valuable. Equally generally, it is the land which is most valuable for a particular purpose which tends to be allocated to that purpose. The link between these two tendencies is that the most valuable areas for some particular use, and the areas which are more valuable for that use than for other uses, tend to be the same. The overall pattern of land use reflects a balance between these tendencies, which can be tipped one way or another by commodity price changes, technological 113 Biol. Conserv. 0006-3207/89/$03"50 © 1989 Elsevier Science Publishers Ltd, England. Printed in Great Britain

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advances, social attitudes and so on. Conservation reserves are areas of land intended to be retained in a relatively undisturbed state in the hope of ensuring the long-term survival of the biotic and abiotic entities they contain, e.g. species, communities, guilds, land units, land systems, geological formations. They are a form of land use, in m a n y ways no different from farming, forestry, urban development, etc. (cf. Van der Ploeg & Vlijm, 1978). Whilst the market value of land for conservation purposes tends to be low, political or non-market demands for land for such a public, consumptive use can be high. When this is so, political decisions may be taken to alienate or acquire lands for conservation purposes. Behind such political decisions there usually lie technical recommendations from a land management agency which identify one or more sets of land parcels as options for acquisition or alienation. Ideally, the formulation of such recommendations will be part of a wider land use planning exercise in which a number of land use possibilities for a region are considered simultaneously (e.g. Cocks & I v e , 1988). In practice, because of the way agency responsibilities are devolved, recommendations for conservation reserves are commonly made with limited consideration of opportunity costs, i.e. of the net benefits foregone by not assigning an area recommended for reservation to its 'best' alternative use. Whilst the approach presented in this paper is capable of addressing the 'comprehensive' land use planning problem (e.g. Cocks &Ive, 1978), we will restrict discussion to the narrower reserve recommendation/selection problem.

FORMALISING THE MULTIPLE RESERVE SELECTION PROBLEM Two types of problem exist in reserve selection, fixed site and floating site. In fixed site problems, a set of discrete candidate reserves has been clearly delineated and the problem is to select a subset of these for recommendation as components of a reserve system. In floating site problems the overall area within which reserves are to be recommended is delineated but not the boundaries of individual candidate reserves, i.e. the problem is extended to one of both delineation and selection of recommended reserves. The present paper is concerned with the simpler fixed site problem. A c o m m o n approach to multiple-reserve or reserve system selection is to calculate a scalar 'conservation value' for each candidate site and then recommend a subset of sites with 'high' conservation values (the n topranked sites, say) as components of the reserve system, e.g. Purdie et al. (1986). A less c o m m o n approach is that of Idle (1986), who suggests selecting a set of sites such that each is 'best' with respect to one of a set of

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conservation criteria. Margules & Usher (1981) collate eighteen criteria which have been used, singly or in combination, in assessing site conservation values in practice. These divide into (a) indicators of the presence of biotic and abiotic entities especially worthy of retention (e.g. rare species, representative communities) and (b) indicators of likelihood of survival of those entities (e.g. reserve size, isolation). Despite their obvious relevance, indicators of site opportunity cost and site acquisition cost generally have not been included in such calculations (although see Tans, 1974). In this paper, however, we will find it useful to recognise both the costs and benefits of adding/removing a reserve to/from a system of reserves, i.e. to work with site reservation value rather than site conservation valueper se, where Reservation value = Intrinsic conservation v a l u e - Implementation costs To produce a single scalar reservation value, selected indicators of (a) presence, (b) survival and (c) implementation components of this value have to be aggregated. M a n y such indices have been constructed (e.g. G o t m a r k et al., 1986), quite c o m m o n l y by forming a linear combination of the form I

R(]') = ) ' (W(i)a(i,j)) i=1

where R(j) = reservation value of the j t h site (j = 1, 2 .... J) a(i,j) = value taken by the ith presence or survival indicator (i = 1,2..... I) or (-ve) implementation cost on the j t h site W(i) = relative importance attached to the ith indicator in determining aggregate site reservation value. This m e t h o d of calculating a site's reservation value amounts to implicitly assuming that reservation value can be measured by an additive and separable utility function (Sinden & Windsor, 1981), an assumption accepted in the present paper for both single and multiple sites. Sinden & Windsor (1981) further note that it is individuals, not groups or communities, who have utility functions and there is therefore no objective way of calculating the 'social' reservation value of a site. Thus, where importance weights, W(i), are called for in this paper, it will be assumed that there exists a politically legitimate 'customer' who, acting in the public interest, can provide these weights. The 'ranking by reservation value' approach is useful when recommending a single reserve (site). It breaks down, however, when used for recommending a system or set of reserves. It fails to recognise that the

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reservation value to be attached to adding a marginal reserve depends on the attributes of reserves already in the system, e.g. whether a species is or is not already well represented. Stated differently, the aggregate reservation value of a set of reserves is the sum of their marginal reservation values, and not necessarily the sum of a priori reservation values of individual reserves taken in isolation. Further, this approach only allows the political/management feasibility of any suggested system to be checked ex post facto. It fails to take account of politically imposed 'whole system' constraints such as m a x i m u m total area or cost when nominating candidate sets of reserves. What is needed to improve the 'n top sites' approach is: (a) an appropriate way of calculating the aggregate net reservation value of a set of reserves; and (b) a criterion for systematically deciding which, if any, reserves can be added to (removed from) the set so as to increase that aggregate value whilst simultaneously ensuring that the selected set of reserves satisfies any politically imposed constraints on its collective attributes, e.g. an upper bound on total system area. Provided certain assumptions are accepted, this is precisely what mathematical programming can achieve. Linear programming (e.g. Lev & Weiss, 1982) is the simplest form of mathematical programming and the reserve selection problem stated as a simple linear program is: Model 1: !

J

Maximise V = ~ ' R(j)x(j) j=l

subject to ~

a(i,j)x(j) < b(i) x(j) >_ 0

i=l

where V= R(j) = x(j) = b(i) =

aggregate value of the reserve system (the objective function) reservation value of the jth candidate site fraction of t h e j t h site included in the reserve system (0 < x(j)< 1) level of the ith system attribute which must not be exceeded by the aggregate of all sites included in the system a(i,j) = contribution of the jth site to the level of the ith bounded system attribute. Formulated in this way, the linear programming model adequately addresses the need to consider only those subsets of sites which are feasible in terms of exogenously improved bounds on system attributes. However, Model 1 still assumes that the value of the reserve system is the simple sum of reservation values contributed by component sites. It also assumes that the contribution of a site is directly proportional to the fraction of the site included in the system.

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In the next section we consider a more sophisticated mathematical programming formulation of the reserve selection problem which allows these and other assumptions to take a more realistic form.

I M P R O V I N G T H E BASIC F O R M U L A T I O N O F T H E R E S E R V E SELECTION PROBLEM Our starting point is to view the reserve selection problem as one of nominating a set of sites which collectively satisfy a range of exogenous system design criteria, i.e. reserve selection guidelines, as well as possible. These may be either system attribute values which must/must not be exceeded (the b(i) values of Model 1) or system attributes which should be as near as possible to nominated goal values/value ranges. We assume, a priori, that usually it is unwarranted to include part of a candidate site in a reserve system because sites normally are not homogeneous, i.e. a part site does not have the same properties as the encompassing whole site. The implication is that Model 1 is incorrect in allowing x(j) values to range between 0 and 1; allowing only the integer values within this range, i.e. 0 or 1, would be more appropriate. If part sites are to be considered, these should be identified and described separately at the beginning of the exercise. Thence, Model 2 is an elaboration of Model 1 into an integer goal programming (IGP) formulation (e.g. Taha, 1975) of the reserve selection problem: Model 2: I

(P(i)y(i) + P'(i)y'(i))

Minimise D = i=1

J

su ect to

= b(i) j=l

(i=1,2,...,I)

x(j)=0or

1

( j = l , 2 ..... J)

where D = a scalar measure of the extent to which the reserve system deviates from the system attributes defined by the goal vector

{b(1) ..... b(I)}' P(i), P'(i) = unit penalty for overshooting, undershooting the ith system goal (i = 1, 2 .... , I)

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fraction of t h e j t h site included in the reserve system (x(j) = 0 or 1) b ( i ) = goal level of the ith system attribute which the model is attempting to achieve a(i,j) = contribution of the jth site to the achieving of the ith system goal y ( i ) , y ' ( i ) = number of units by which the ith goal has been overshot, undershot. xU) =

The main differences between Model 2 and Model 1 are: (a) The algorithm attempts to minimise a measure of the extent to which the reserve system deviates from a set of'design goals' rather than maximise the aggregate reservation value of all reserves included in the reserve system. Alternatively interpreted, the algorithm seeks to maximise the extent to which a number of design guidelines are satisfied collectively. (b) Inequality constraints of Model 1 do not appear in Model 2 but, by using sufficiently large values for P(i) or P'(i) to prevent overshoot or undershoot as the case may be, these can be mimicked within the equality formulation of Model 2. (c) Only whole sites can enter the solution. The IGP formulation of the reserve selection problem is highly flexible and can accommodate a variety of types of reserve selection guidelines, some of which are discussed in the next section.

R E S E R V E S E L E C T I O N G U I D E L I N E S IN P R A C T I C E The primary aim of this paper is to describe a method rather than the range of reserve selection guidelines, i.e. system design criteria, which could or should be used in practical problems. Although ecologists, conservationists and conservation evaluators tend to act as though there is an absolute hierarchy of conservation goals, there is in fact no 'right' list of such guidelines. To the extent that resources allow, the adopted guideline set should address the most important issues underlying the specific decision to identify a reserve system. Note that guidelines embody design criteria for a system of reserves; they do not refer to individual site reservation values or conservation values directly, e.g.

As far as possible ensure that all species are represented in the system o f reserves (the 'diversity' guideline)

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! 19

which can be compared with As f a r as possible, ensure that species rich reserve X is included in the system o f reserves.

Whilst the list of possible guidelines is open-ended, two that almost always attract consideration are those addressing rarity and representativeness, e.g. As far as possible, ensure that the suggested reserve system contains all species (or guilds or communities) designated as locally (or regionally) rare. As f a r as possible, ensure that the suggested reserve system contains all species (or guilds or communities) in proportion to the extent to which the)' are represented in the survey area.

These are examples of guidelines identifying the need to include what were earlier called biotic entities worthy of retention. Other reserve selection guidelines commonly discussed address system viability issues, e.g. As f a r as possible, ensure that the suggested reserve system contains only sites o f sufficient size to enhance species survival prospects.

Finally, there are guidelines addressing the third c o m p o n e n t of reservation value, namely, implementation costs, e.g. As f a r as possible, ensure that only low cost sites are included in the suggested reserve system.

The IGP model requires such generally worded guidelines to be operationalised into simple algebraic measures as a prelude to designing candidate reserve systems. For example: J

--~ A R E A (j) COST (j)xO) j--1

Here, site area (AREA(j)) times cost per unit area (COST(j)) times the fraction of the site included ( x ( j ) = 0 or l) is added over all sites (j = 1, 2 ..... J) to give a total acquisition cost. Depending on the emphasis given in system design to keeping this sum down, low cost sites will tend to be selected. It must be emphasised that guidelines presented here are only examples of the main classes of guidelines likely to require consideration. Reiterating, whilst ecological and economic principles may lead to somewhat similar guidelines being considered in many reserve selection exercises, the specific guideline set to be used should reflect closely the issues triggering a reserve selection exercise.

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K. D. Cocks,

EYRE PENINSULA

I. A. Baird

DEMONSTRATION

EXERCISE

The demonstration exercise makes use of a data set (Table 1) presenting the probabilities of finding each of six plant communities on each of 101 patches of relatively undisturbed native vegetation (sites) left amongst the farmlands of Eyre Peninsula, South Australia (Margules & Nicholls, 1987). The way in which the data set was developed is not our present concern, except to note that modern methods of vegetation survey use logistic regression to model the probability of a community being present on a site as a function of environmental parameters such as rainfall, temperature and geology. For example, environmental parameters used in the Eyre Peninsula study were soil carbonate, latitude and distance from coast. These models provide a means of extrapolating sample-site presence/absence data to unsampled sites. With agency resources increasingly permitting only a sample of candidate sites to be surveyed, such probabilistic knowledge will become the main information on which reserve selection has to be based. Consequently, reserve selection guidelines and reserve selection methods must be framed in terms of, and must operate on, probabilistic presence/absence data. Basically, this means that reserve systems have to be designed in terms of the expected number of appearances of communities, species, etc., rather than the absolute number of such appearances. Table 2 summarises an illustrative guideline set to be used as a guide in the selection of a subset of sites as components of a reserve system for the Eyre Peninsula. The first six guidelines seek to ensure that the J* sites of any suggested reserve system contain at least two expected occurrences of each of the six communities (i = 1,2, . . ., 6) i.e. _I* { Ip(i,j) + O(1 -p(i,j))}

2 2

(i = 1,2,. . ., 6)

c j=l

where p(i,j) = probability

of finding community

Note that the probability system so designed is:

i on site j (j = 1,2,. . ., J*)

of not finding the ith community

in a reserve

J* I-l

(I - p&j))

j=l

and that the absolute number of occurrences of community i will lie between 0 and J* (assuming no p(i, j) = 0). Note also that the argument is unchanged if p(i, j) = 1, i.e. it is known with certainty that community i occurs on site j.

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TABLE 1 Probabilities of Each Community Appearing at Each Site (after Margules & Nicholls (1987)) Site

Comm I

Comm 2

Comm 3

Comm 4

Comm 5

Comm 6

1 2 3 4 5 6 7 8 9 10 l1 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44

0"019 0"019 0"019 0 0.019 0'019 0 0"035 0-035 0-035 0 0-019 0"035 0"035 0"068 0'035 0 0"035 0"035 0'035 0 0"019 0-019 0-019 0"019 0-019 0"019 0.019 0.019 0"019 0"019 0 0 0 0"203 0"035 0 0 0 0"035 0 0"019 0"019 0" 109

0-571 0'571 0"571 0'105 0.571 0.571 0 105 0' 158 0 158 0-158 0"105 0"571 0" 158 0" 158 0-067 0"158 0-105 0" 158 0" 158 0' 158 0.105 0.571 0.571 0"571 0"571 0"571 0'571 0.571 0"571 0.571 0-571 0-105 0105 0" 105 0" 158 0" 158 0.105 0"105 0-105 0" 158 0'105 0"571 0"571 0'571

0-79 0-79 0.79 0'79 0.79 0.79 0-79 0.79 0-79 0-79 0.79 0.79 0.79 0.79 0.289 0.79 0-79 0"79 0.79 0.79 0.79 0.79 0.79 0.79 079 0'79 0.79 0.79 0.79 0.79 0-79 0.79 079 0-79 0.79 0.79 0.79 0.79 0-79 0-79 0-79 0"79 0.79 0.79

0 0 0 0 0 0 0 0.272 0-272 0-272 0 0 0'272 0.272 0"515 0.272 0 0"272 0"272 0"272 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.054 0 0 0 0.054 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0"667 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0"053 0"053 0"007 0"189 0"053 0"053 0 189 0.007 0-007 0-05 0"187 0-053 0"007 0"05 0"026 0-007 0"007 0-007 0"007 0"007 0'029 0'053 0.053 0'053 0-053 0-153 0-053 0" 153 0'053 0.053 0053 0' 189 0-187 0-189 0-189 0"05 0.189 0.187 0-189 0.05 0-187 0" 189 0" 189 0"053 (~ ~mtim~ed)

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K. D. Cocks, L A. Baird

TABLE 1--contd. Site 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89

Comm 1

Comm 2

Comm 3

Comm 4

Comm 5

0.019 0"019 0.035 0 0'019 0.019 0 0 0-035 0.035 0-035 0-068 0-068 0"068 0.068 0.035 0"035 0-068 0.068 0.068 0'068 0.068 0.068 0.068 0.035 0"068 0.068 0.068 0.068 0.493 0.068 0.095 0.095 0.583 0.583 0.203 0"203 0.203 0-203 0-203 0.203 0-203 0-203 0"203 0.203

0'571 0"571 0"158 0"105 0"571 0"571 0"105 0"105 0' 158 0' 158 0"158 0"067 0"067 0"067 0.067 0" 158 0" 158 0-067 0-067 0-067 0-067 0'067 0"067 0"067 0"158 0"067 0"067 0"067 0.067 0"067 0'067 0"091 0'091 0"091 0"091 0"571 0"571 0.571 0"571 0'571 0"572 0"571 0"571 0"571 0"571

0"79 0"79 0"79 0"79 0"79 0"79 0"79 0"79 0-79 0"79 0"79 0'79 0'79 0'79 0"79 0"79 0"79 0"79 0'79 0"79 0'79 0"79 0"79 0"79 0"79 0"79 0"79 0'79 0"79 0"79 0-79 0-79 0-79 0"41 0-41 0-07 0"07 0"025 0'025 0'07 0'07 0'07 0'07 0.07 0"07

0 0 0"054 0 0 0 0 0 0"054 0-054 0-054 0-515 0-515 0"515 0"515 0'054 0"054 0"515 0"515 0'515 0"515 0"515 0-515 0"515 0"272 0'515 0"515 0"515 0"515 0 0"515 0"653 0'653 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0"667 0-667 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Comm 6 0-053 0-053 0-05 0.189 0"053 0"053 0"189 0"187 0"05 0"189 0"05 0"08 0"08 0'026 0"026 0"05 0"05 0"026 0"026 0'026 0"026 0"026 0'026 0.026 0"05 0'026 0-026 0"026 0.026 0"026 0"026 0 0 0 0 0-437 0-437 0"715 0-715 0"764 0-437 0-437 0-715 0-437 0-437

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Reserve selection by mathematical programming

T A B L E 1--contd.

Site

Comm l

Comm 2

Comm 3

90 91 92 93 94 95 96 97 98 99 100 101

0.203 0.326 0.493 0.493 0.493 0.583 0.326 0.326 0.203 0.203 0.019 0.019

0-105 o. 158 0.068 o. 158 0-068 o. 158 0.158 0-158 0.571 0-571 0-571 0.571

0.07 0-07 0.41 0.41 0.07 0.41 0.07 0.07 0.025 0.025 0.79 0.79

Comm 4

0 0 0 0 0 0 0 0 0 0 0 0

Comm 5

0 0 0 0 0 0 0 0 0 0 0 0

Comm 6

0.715 0.422 0-272 0.764 0-272 0.764 0.272 0.272 0-437 0.9 ! 3 0-153 0.153

T a b l e 3 p r e s e n t s site d a t a r e q u i r e d f o r i n c o r p o r a t i n g g u i d e l i n e s 7 - 1 3 i n t o an IGP model. Guidelines 7 and 8 aim to keep the number of sites and their total area as low as possible, basically to keep land acquisition and management costs down. With the intention of reserving the region's more viable stands, guideline 9 expresses a preference for larger rather than s m a l l e r sites b e i n g i n c l u d e d in t h e r e s e r v e s y s t e m . G u i d e l i n e s 7 a n d 9 a r e t h e r e f o r e , t o s o m e d e g r e e , in c o n f l i c t a n d it is u l t i m a t e l y u p t o t h e c l i e n t t o ' t r a d e off' t h e r e l a t i v e e x t e n t t o w h i c h t h e s e t w o g u i d e l i n e s a r e s a t i s f i e d in t h e eventual reserve system. TABLE 2

Reserve Selection Guidelines for the Eyre Peninsula Exercise 1. A F A P (as far as possible) ensure at least two expected appearances of Community 1 in the reserve system. 2. A F A P ensure at least two expected appearances of Community 2 in the reserve system. 3. A F A P ensure at least two expected appearances of Community 3 in the reserve system. 4. A F A P ensure at least two expected appearances of Community 4 in the reserve system. 5. A F A P ensure at least two expected appearances of Community 5 in the reserve system. 6. A F A P ensure at least two expected appearances of Community 6 in the reserve system. 7. Keep the total area over all reserves in the reserve system as low as possible. 8. Keep the total number of reserves in the reserve system as low as possible. 9. A F A P give preference to including large reserves in the reserve system. 10. A F A P ensure that near-coastal reserves are included in the reserve system. 11. A F A P ensure that inland reserves are included in the reserve system. 12. A F A P 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. 13. Include in the reserve system all sites which are already declared reserves.

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TABLE 3 Additional Site D a t a Items Required for F o r m u l a t i n g Reserve Selection Problem a Site no.

Area

Sites

Large

Coastal

Inland

Rare

Reserve

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44

27.5 5"2 1"2 4"2 1'7 2"3 5 1.2 1'6 2-7 2"7 1'5 3"5 1'9 4-8 2"5 3"1 1"9 3"1 1 4"2 14.2 9"2 1'2 1"6 13-9 9"2 10"2 3"9 2"5 2"5 5"2 1'4 4"4 25"8 1-2 2.7 1'5 1'5 1"9 1-6 3"1 1"I 1'2

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 l 1 ! 1 1 1 1

1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0

1 1 1 1 1 1 1 0 0 1 1 1 0 0 0 0 1 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 0 1 1 1 1

0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

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TABLE ~-contd. Site no.

Area

45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89

3.5 3.7 2.6 3.3 2.1 4-1 1.4 13.3 6 25.6 2.5 1.6 2-1 1.4 2.7 2.5 2.7 3-3 1-6 3-1 1-6 1-2 1-4 1-6 4-8 1-2 1-3 1-4 1.6 5.4 15.6 3.9 3.1 1.4 3.7 1.2 1-4 3.7 4.6 6.5 2.2 3.5 36-8 3.7 2.9

Sites

Large

Coastal

0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0

1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1

Inland

Rare

Reserve

0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (continued)

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T A B L E 3~contd. Site no. 90 91 92 93 94 95 96 97 98 99 100 101

Area 1"2 2'5 19"1 5 925"9 5"8 14596'1 1'6 1"2 8'1 2"1 2"2 40"4

Sites

Large

Coastal

Inland

Rare

Reserve

1 1 1 1 1 1 1 1 1 1 1 1

0 0 0 1 0 1 0 0 0 0 0 1

1 0 0 1 0 1 0 0 1 1 1 1

0 1 1 1 1 1 1 1 0 0 0 0

0 0 0 0 0 1 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 1 1 1

a Key: Area: Site area in km2; Site: C o n t r i b u t i o n to total site number; Large: Reserve is relatively large; Coastal: Site is near coast; Inland: Site is inland; Rare: Site probabilistically contains C o m m u n i t y 1 or C o m m u n i t y 5; Reserve: Site is a declared reserve.

Guidelines 10 and 11, recognising that the 'same' community actually may be somewhat different, depending on its proximity to the coast, seek to spread reserve system sites between coastal and inland locations. Guideline 12, recognising that there are only three sites with a positive probability of community 5 (the rarest community) being present, and only three sites with a high (> 0"5) probability of community 1 (the second rarest community) being present, seeks to ensure that these sites are well represented in the reserve system. Guideline 13 has not been activated in the initial runs of the model. It can be included, if requested (as it is presently), to permit recognition of the political reality that newly designed reserve systems would normally be expected to include sites previously declared as reserves. The demonstration exercise has been set up as an integer goal program using the Eastern Software Products Inc. (1988) MILP88 (Version 7.11) package on a NEC APC IV personal computer. Table 4 summarises the characteristics of four different reserve systems designed by IGP to satisfy different design criteria. The first design task undertaken was to specify a reserve system of no more than 12 sites, collectively containing at least two expected appearances of each of six communities. This could have been found to be infeasible by the algorithm but was not. Further, but less importantly, 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. Stated in this way, the design task

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TABLE 4

Solutions to Four Reserve System Design Exercises First solution Maximum

Expected appearances--Community 1 Expected appearances--Community 2 Expected appearances--Community 3 Expected appearances--Community 4 Expected appearances--Community 5 Expected appearances--Community 6 Total area over all sites Total number of sites Largest sites Near-coastal sites Inland sites Sites with rarest communities Declared reserves Solution Solution Solution Solution

1

10.1 2-4 28.3 2.6 65.0 7-1 13-2 2.1 2.0 2.0 16-6 2.3 21015.0 6096-8 101 12 7 6 55 4 46 8 6 5 3 1

Min. Min. Existing area sites reserves solution solution 2 3 4

2.0 2.4 4.1 2-1 2.0 2.2 18.4 10 0 3 7 4 1

2.1 2-2 3.6 2.2 2.0 2"3 27.5 9 0 3 6 5 1

2.6 3.8 7.9 2.1 2.0 3.3 6101.1 14 6 8 6 5 3

1 sites: 1, 15, 35, 54, 56, 57, 70, 78, 79, 87, 93, 101 2 sites: 15, 44, 56, 57, 66, 78, 80, 96, 97, 99 3 sites: 15, 56, 57, 77, 78, 79, 84, 85, 99 4 sites: 1, 15, 35, 54, 56, 57, 70, 78, 79, 87, 93, 99, 100, 101

is hierarchical, i.e. only after satisfying c o m m u n i t y representation a n d site n u m b e r constraints is consideration to be given to questions o f system area, site spread between coastal and inland locations, etc. Solution 1 in Table 4 specifies a reserve system satisfying these design requirements. It was achieved by putting a high penalty on ' u n d e r s h o o t i n g ' a goal o f having two expected appearances of each c o m m u n i t y in the reserve system and a high penalty on 'overshooting' a goal o f 12 sites in the system. F o r other goals, m o d e s t penalties were attached to b o t h overshooting and u n d e r s h o o t i n g target levels. Algorithmically, this means that, as far as possible, high penalty goals will be satisfied before there is a n y a t t e m p t to satisfy low penalty goals. Solution 2 retains the constraint that each c o m m u n i t y m u s t be expected to a p p e a r in the system at least twice, but otherwise is concerned with only one other g o a l - - t o minimise the total area o f all sites in the reserve system. Thus, there are no penalties for deviating from other target levels. Solution 3 also retains the constraint that each c o m m u n i t y m u s t be expected at least twice but, in this instance, is concerned only with the goal o f minimising the n u m b e r o f sites in the reserve system. Again, there are no penalties for deviating f r o m other target levels. The large difference in system

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area between solution 1 and solution 3 reflects the fact that two of the 101 candidate sites (93, 95) are several orders of magnitude larger than other sites. Solution 4 reflects the impact of re-running the initial problem (Solution 1) with the additional design requirement (Guideline 13) that the system must contain the three existing declared reserves, viz. 99 (Wittelbee Conservation Park); 100 (Laura Bay Conservation Park); 101 (Calpatunna Conservation Park). A penalty on undershoooting guideline 13 was introduced into the objective function to force the three existing reserves into the solution.

DISCUSSION A N D C O N C L U S I O N S The paramount appeal of the IGP formulation of the reserve selection problem is its flexibility, i.e. the wide range of variations on the basic problem which can be captured reasonably realistically. Table 4 illustrates how alternative secondary criteria can be met subsequent to meeting a preemptive design criterion of adequate representation in the system of all communities. Another quite feasible variation, not illustrated here, is the problem of choosing one site (but no more) from several candidate sites to add to an existing reserve system. To date, insufficient work has been done to know just how flexible the IGP approach to reserve selection is. One obvious difficulty, for example, is to design systems with localised groupings of sites for ease of management and trade this off against spatial spreading of sites to capture ecological variations between communities (cf guidelines 10 and 11 in Table 2). This raises a more general difficulty, not exclusive to IGP approaches, viz. that some of the ecological (and economic) concepts (e.g. ecotypical variation; minimum habitat area for survival; naturalness) which one might wish to see incorporated into reserve system design criteria have never been analysed in a way which would allow their routine inclusion in an IGP model or in any other formal selection method. A further important strength of IGP models is that they are efficient. Margules & Nicholls (1987) used a simple, sensible heuristic method to select a minimal set of Eyre Peninsula sites such that the probability of finding the ith community (i -- 1, 2..... 6) in the system exceeded 0-95, viz. 12 sites. The initial IGP formulation described above, after requisite adaptation, delivered a comparable result over 11 rather than 12 sites. Work in progress is looking in more detail at related efficiency considerations. More generally, for given relative penalties for deviating from design goals, one can be sure with IGP that an improvement with respect to one design criterion can only

Reserve selection by mathematical programming

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be achieved at the price of an unacceptable loss of achievement in relation to some other design criterion. There is, however, an occasional rider to this assurance. Integer programming can be a computationally intensive procedure and, especially on a personal computer, one may have to be satisfied with a good rather than an optimal solution. The Eastern Software Products Inc. (1988) MILP88 integer programming package costs several hundred dollars and is easy to use, especially in combination with a spread sheet (Lotus 123 version 2 in this case) to facilitate setting up. On an IBM AT clone it can handle problems with 3500 integer or non-integer variables and 300 constraints. Support for the Intel numeric data coprocessor (8087, 80287 or 80387 chip) is included as a second program named MILP87, allowing solution of mixed integer linear progams with up to 800 constraints and 4000 variables. Contingent on interest in the present work, the authors plan to interface MILP88 to RESERVE, a tutorial and procedural support module to help those unfamiliar with mathematical programming to use it for their own reserve selection problems. Given past controversy over methods of measuring conservation value, it is worth emphasising that a range of such measures can be considered simultaneously in the one IGP model, e.g. by including, for each measure, a guideline favouring sites which have high conservation value under that measure. The relative objective function weights assigned to each guideline will then determine which of these measures a site must satisfy well in order to be selected. This paper has considered only the fixed site reserve selection problem. The more general floating site reserve selection problem mentioned earlier is made enormously difficult by reason of the fact that an infinity of potential sites, including overlapping sites, can be delineated in a region where a reserve system is to be set up. One idea for coping here is that a manageable but rich set of potential reserves might be postulated as the quadtree over the region. This is the set of potential reserves formed by dividing the (gridded) target area into quadrants (giving four potential reserves), each quadrant into sub-quadrants (giving a further 16 potential reserves) and so on, e.g. Abel (1985). The integer constraint capability would need to be invoked to preclude two or more overlapping sites both entering an IGP solution.

A C K N O W L E D G E M ENTS The authors are grateful to Chris Margules and Nick Nicholls for helpful discussion and access to their original Eyre Peninsula data.

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