Several recent studies have implemented utility maximization or maximal covering methods for conservation resource allocation problems for purposes other than reserve design . This set is not meant to be an exhaustive review, but a representative example of an increasingly common approach in conservation planning. Three of these studies used exact optimization methods like integer programming or stochastic dynamic programming , and the rest use a heuristic algorithm or search technique. The use of optimization in conservation planning specifically for reserve selection has shown improvements over simple heuristics with gains from optimization ranging between 5.6%–50% , 4.4%–26% , 5%–20% , 0%–20% , and 2%–70% . Similar improvements are likely possible for other types of natural resource management problems that employ spatial conservation prioritization, although the majority of studies based on heuristic methods have not compared solutions to true optimal solutions. Furthermore, an integer programming solution method has not been included within the utility-maximization framework of Davis et al. ; Davis, Costello & Stoms , and the difference in results between optimal and heuristic targeting algorithms has not been determined. Alternatively, several studies have illustrated that optimally designed conservation plans may fall short or be unnecessary due to uncertainty in available funding, conservation opportunity, and natural resource degradation in unprotected sites . In previous research using the utility-maximization approach in an agricultural context, Stoms, Kreitler & Davisevaluated the efficiency of solutions based on the types of data used and how a cost-effectiveness score was calculated to maximize utility in preserving multifunctional farmland. For our purposes we define multifunctional agriculture in a general sense as a land use jointly producing multiple commodity and non-commodity outputs,stacking pots where some of the non-commodity outputs may be public goods with non-existent markets, and thus be under supplied without intervention .
We quantify multi-functionality in our example through spatial models of individual criteria representing functions of the agricultural landscape that produce public or private goods. These criteria are agricultural viability, priority conservation areas, conservation buffers, sphere of influence, and flood liability . In this paper, we compared an optimal integer programming model to the best performing solution procedure from Stoms, Kreitler & Davis , in the allocation of farmland conservation funds in the Central Valley of California. We quantified the difference in benefits accrued between solutions to compare the gains produced by a simple algorithm with the optimal solution in the utility maximization framework of Davis et al.and Davis, Costello & Stoms . In addition, we quantified the difference in benefits when parcel availability was a stochastic process more reflective of real estate markets and actual land-use change through time, and compared results to the global solution where all parcels were available. We obtained data on priority conservation areas from a biodiversity conservation organization describing agricultural areas of interest for acquisition or conservation management. These zones integrate biodiversity conservation planning activities and serve as a proxy for a biodiversity conservation criterion. Within our study area there were eight separate areas ranging in size from 61 ha to 112,716 ha that were considered high conservation priority. Parcels were scored based on the amount of area falling within a priority conservation area that was also expected to be lost to development. While this criterion is not dynamic from the perspective of ensuring all species within the original conservation planning activities are covered, it is updated when conservation actions occur. Therefore, if a large portion of one priority conservation area is secured, the marginal value of the remaining area within that priority conservation area will be adjusted according to utility function, targets, and existing protected resources. The combined utility score, modeled parcel acquisition price, and cost-effectiveness results are illustrated in Fig. 2. The equally weighted combined criteria score shows areas of high net benefit for multifunctional agricultural lands around the developed areas in the southern portion of the study area. This is likely due to the presence of prime agricultural lands, flood risk, and strategic position between communities and development threat.
In Fig. 2B, parcel acquisition prices were largely dependent on parcel size and distance from urban areas. Parcel size generally increases with increasing distance from urban areas, so acquisition prices largely increase as distance from urban increases, even though per acre price is generally highest near the urban edge and declines with increasing distance. Figure 2C shows the cost-effectiveness of each parcel as the ratio of combined utility divided by modeled acquisition costs. Even with the higher per-acre cost factored in, the areas around Lodi, east and west of Stockton, surrounding Tracy, and to the east of Manteca have high cost-effectiveness. The majority of cost-effective parcels are located within San Joaquin County, though parcels are selected in southern Sacramento County as well. The selected sites both in the IP and greedy solution at the $200 million budget are relatively clustered and compact due to the spatial characteristics of the criteria models . The two methods select relatively similar sets . However, a major difference between the IP and greedy procedures is that the sites unique to the IP solution are more numerous and smaller, whereas fewer, larger sites are selected solely by the greedy solution method . A comparison of the total accumulated utility by solution procedure is illustrated in Fig. 4. The IP and greedy solutions begin to diverge once $20 million is spent, with the difference between IP and greedy gradually increasing until the total budget is spent at $200 million. Accumulated utility increases sharply from the beginning of the selection in both procedures, and then tapers as sites with lower marginal utilities are selected. When the potential sites are limited to those hypothetically available at each time interval , the total accumulated utility ranges from 88% to 70% of the optimal IP solution for the stochastic IP and greedy solutions, respectively. The distributions of utility for the stochastic solutions overlap at the start but are significantly different at all intervals, grow lights and then completely separate by the end of the scenario. The mean difference in utility between solution procedures by the end of the budget is 8% , with a mean pairwise Jaccard similarity of 0.86.
This study mapped five criteria describing elements of multifunctional agriculture to determine a hypothetical conservation resource allocation plan for agricultural land conservation. We compared solution procedures to determine the difference between an optimal integer programming approach and a greedy heuristic in a utility maximization problem. Stoms, Kreitler & Davisshowed the importance of including benefit-losscost data in this study area; in this effort we demonstrate the improvement found through optimization methods, and determine the decline in total utility when site availability is a stochastic process. Our results show that the optimal solution and the greedy solutions were similar, but in each case, the IP formulation outperformed the greedy heuristic by up to 12%. We have described the solution of a knapsack or packing problem , in a utility maximization approach , where the IP formulation found an incrementally more efficient solution given the budget constraint. This can be observed in Fig. 3 by comparing the selected sites unique to each solution method. The IP solution used numerous small, less expensive sites to maximize the utility, whereas the greedy solution was 7% less effective by simply selecting parcels in order of decreasing cost-effectiveness. Another noteworthy result is the proximity of the upper limit of the stochastic IP result to the greedy result at the $200 m level . Even with reduced site availability, the IP solution approaches the full greedy solution that had access to all sites in the study area. This result may be partially due to the interchangeability of sites as well as the IP solver finding better solutions. The methods described here have potential implications for many natural resource management applications. The papers reviewed in Table 1 all use a similar mathematical formulation in which an objective function is maximized subject to a budget constraint. Most do not employ optimization methods and could likely be improved by adopting optimal solution procedures as illustrated here, and reviewed elsewhere . Furthermore, future studies could utilize the accessibility of lpSolve in the flexible R modeling environment without having to rely on proprietary commercial software. This study is unique in its combination of integer programming and a benefit or utility function approach. We are not aware of the use of IP or other search methods within utility maximization planning for multiple criteria. Wilson et al.used optimization methods with a benefit function, but for the singular objective of maximizing species persistence by scheduling ecoregional conservation actions through space and time. The use of multiple criteria allows a broader suite of potential applications in restoration planning or ecosystem service conservation planning, for example. Instead of a utility function, process based models or ecosystem service production functions could be included to convert landscape patterns into restoration benefits or ecosystem service values for conservation resource allocation schedules that are reflective of social preferences.
Perhaps the largest drawback of using IP is the linear constraint on problem formulation. This “first stage suboptimality” occurs when a complicated problem is simplified to fit into a linear formulation for which there is a tractable globally optimal solution. In our methods we take this approach by simplifying our criteria models. For example, our models calculate initial criteria values according to subsections “Agricultural viability”–“Flood liability” and incorporate distance and contiguity measures. These criteria scores are converted to marginal utility values per Eq. and dynamically updated after each round of conservation actions, but the spatial operations are not, which leads to a more tractable problem for optimization. In this paper we have measured Moilanen’s “second stage suboptimality”, or the difference between the exact IP and heuristic solutions. However, for the reasons described above there may be larger true differences between linearly formulated exact IP solutions and heuristic solutions found using more complicated and realistic nonlinear model formulations, even if a less accurate heuristic algorithm is used to solve the problem. By contrast, when the results of this study are compared to Stoms, Kreitler & Davis , which used the same data and similar methods to conduct a sensitivity analysis, the largest benefit to maximizing utility is found through using the cost-effectiveness of conservation actions, rather than by including threat estimates or using optimal solution methods. Similar studies have shown the degree of suboptimality from the absence of threat data or from the solution algorithm is of comparable magnitudes to those found here . All of the examples from Table 1 that use the utility maximization approach of Davis, Costello & Stomshave occurred in regions of California where variation in land value may be a principle factor influencing conservation opportunity. This is important to note when considering the findings, and how generalizable they may be. One of the unique features of this study is the fine scale at which the costs and benefits were calculated, and the spatial correlation of threats from urban development and land value. In regions with less heterogeneity in land value, results would likely be driven more by the benefit surfaces than in this example. Similarly, differences between solution methods would likely diminish as the planning unit scale increases and the decision space decreases to a fewer number of planning units to choose from. Further sensitivity analysis on the selection of land use change scenarios would be a last step in determining the robustness of results to the data inputs required by this planning framework. The merits and shortfalls of pursuing optimality in conservation planning have received considerable attention, usually with reference to minimum set or maximal covering reserve selection problems . The framework of utility maximization planning, however, is different due to site values that are typically a function of increasing representation with nominal target values, and the ability to incorporate multiple criteria measured in different units . The utility or benefit function actually makes utility maximization problems less suitable for IP, though we have described a stepwise procedure to accommodate the numerical requirements of IP and the problem formulation of multicriteria utility maximization planning.