## Abstract

This paper presents a two-stage optimization approach that involves linear integer programming models in each stage for determining optimal expansion alternatives for the Cape Floristic Region conservation areas in South Africa, with an emphasis on the protection of mammal species. While determining the optimal expansion areas, the following spatial and ecological criteria are taken into account: (i) a spatial connectivity of individual reserves, (ii) a minimum population of each species in each designated reserve, and (iii) an overall minimum population target for each species. We also aim for optimum economic efficiency by minimizing the total expansion area. The data set includes about 4000 candidate reserve sites and 38 species of large and medium-sized mammals. Empirical results indicate that the integer programming models can be solved without serious computational difficulty and the optimal reserve design provides substantial savings in financial resources.

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## Availability of Data and Material

The data set and the GAMS codes used in the empirical application are available upon request from the corresponding author.

## Notes

The model (1)–(6) is not the only formulation of the second-stage problem. For instance, constraint (2) can be eliminated without any effect on the solution. Constraint (3) can be stated as \({X}_{i} \ge {Y}_{k}\) such that \(i\in {I}_{k}\) . Constraint (4) can be eliminated after adding an upper bound \({p}_{is}\) for \({R}_{is}\), namely \({0\le R}_{is}\le {p}_{is}\). Such modifications to the model may be implemented if solution difficulties are encountered or solution time needs to be improved. Our experience with these changes did not indicate any computational improvements in the numerical example considered in Section 4. This is just one instance, however, we cannot generalize the findings; computational improvements can be possible in other instances.

The B&B procedure is a systematic procedure that solves a linear programming (LP) problem at each iteration where some discrete variables are relaxed (treated as continuous variables). Each LP corresponds to a node. The solution time is determined by the size of the LP problems solved at each node and the number of nodes solved until finding a satisfactory solution.

The relative optimality criterion, specified by the user, is a measure of maximum discrepancy (in percentage) between an incumbent MIP solution and the best possible solution. Specifying this parameter as zero means requiring an exact optimum solution. In general, the solution time of MIP models increases with reduced optimality criterion. The increase can be dramatic when the relative optimality criterion is specified as zero. Countless iterations may be needed, without finding a new integer solution, to confirm that an incumbent solution is truly optimum.

The average number of constraints and binary variables in those problems were 658 and 353, respectively, not much different from the problems for existing sites. Yet, after solving a substantial number of nodes (ranging between 1.33 and 2.59 million), CPLEX could not determine an integer solution satisfying the optimality criterion (the suboptimality levels ranged over 3.2–8.9%).

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## Funding

This work was partially supported by the Illinois CREES Project (ILLU 05–0361) and the Australian Research Council.

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H. Önal developed the optimization models, programmed them in GAMS language, and executed the model runs. He also wrote the initial draft of the manuscript. R. Pressey and G. Kerley provided the data set used in the empirical study and contributed to the writing of the manuscript. M. Ridges prepared the figures.

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## Appendix. Configuration of Spatially Contiguous Reserves

### Appendix. Configuration of Spatially Contiguous Reserves

Suppose a conservation area is partitioned into a set of square selection units (this is not a restrictive assumption; the model presented below can be modified easily for non-uniform partitions). We overlay a *graph G* = *(V,E)* on the partition, where *V* denotes the set of *nodes* and *E* denotes the set of *directed arcs* forming the graph. Each node in *V* corresponds to a cell in the partition. Two nodes are adjacent if the corresponding cells are spatially adjacent. Each arc in *E* originates from a node and is directed (flows) into an adjacent node. Thus, a given node may have up to four arcs originating from it, and up to four arcs may be directed into it (the number of arcs can be fewer for nodes corresponding to the boundary cells). A *tree R* is a special subgraph of *G*, i.e., *R* = *(*\(\widetilde{V},\widetilde{E}\)*)*, where (i) \(\widetilde{V}\)⊆ *V* contains mutually adjacent nodes in *V* and \(\widetilde{E}\)⊆ *E* is a subset of arcs linking the nodes in \(\widetilde{V}\), and (ii) only one arc in \(\widetilde{E}\) can originate from each node in \(\widetilde{V}\), but more than one arc can be directed into a given node. A *path* in *R* is a chain of arcs and nodes where each arc in the chain originates from the node at which the previous arc ends and a node on the path cannot be repeated, i.e., a *cycle* cannot occur. A subgraph is connected if any two nodes in it can be connected by a path completely contained in the subgraph. A connected subgraph corresponds to a tree.

The linear integer programming model presented here determines a least-cost contiguous reserve in a given conservation area that satisfies specified representation targets for a given set of species. The notation used in the model is as follows: *i* and *j* denote cells, and *s* denotes species under consideration; \({r}_{is}\) is the contribution of site *i* to the representation of species *s*; \({t}_{s}\) is the representation target specified for species *s*; \({c}_{j}\) is the cost of selecting site *j*; \({A}_{j}\) is the set of cells that are adjacent to cell *j*; \({Y}_{ij}\) is a binary variable, where \({Y}_{ij}\) = 1 if an arc flows from node *i* to node *j* and \({Y}_{ij}\) = 0 otherwise; \({X}_{j}\) is a binary variable, where \({X}_{j}\) = 1 if the cell corresponding to node *j* is selected, \({X}_{j}\) = 0 otherwise.

The objective function (8) represents the total cost of site selection. Constraint (9) states that if the cell corresponding to node *j* is not selected, no arc can originate from that node; otherwise, up to four arcs can flow into that node. Constraint (10) ensures that at most, one arc can originate from node *i* if the corresponding cell is selected; otherwise, no arc can originate from that node. A necessary condition for a subgraph to be a tree is that the number of arcs in the subgraph must be one less than the number of nodes in it, which is expressed in (11). Constraint (12) states the representation requirement for individual species. Constraint (11) is also sufficient for the subgraph to be a tree if no cycle occurs in the graph. This issue is resolved in [38] by using arcs flows (also see [28] for computational improvements to the method). When an arc links node *i* to an adjacent node *j*, one unit of flow to *j* occurs. We denote this by a non-negative variable \({Z}_{ij}\). The total flow to node *j* is the sum of all those flows, denoted by a non-negative variable \({W}_{j}\). The following two constraints eliminate the possibility of cycle formation.

Constraint (13) determines the total flows into each node. Constraint (14) becomes redundant if no node is linked to node *j*, i.e., \({Y}_{ij}=0\). Otherwise, \({Y}_{ij}=1\) and (14) becomes \({Z}_{ij}{\ge W}_{i}+1,\) which means that the contribution of node *i* to the total flow to the destination node *j* is the total flow into the origin node *i* plus 1. This property implies that if a cycle occurs, the total flow into a node in the cycle must be strictly greater than itself, which is a contradiction.^{Footnote 5}

The above model can be computationally problematic when the search areas for local reserves include too many spatial units. In the CFR application, this was not the case, and we did not encounter any computational difficulty.

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Önal, H., Pressey, R.L., Kerley, G.I.H. *et al.* Designing an Extensive Conservation Reserve Network with Economic, Ecological and Spatial Criteria.
*Environ Model Assess* **28**, 585–598 (2023). https://doi.org/10.1007/s10666-023-09875-4

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DOI: https://doi.org/10.1007/s10666-023-09875-4