Producing a Diverse Set of Near-Optimal Reserve Solutions with Exact Optimisation

Abstract

Reserves are at the heart of global policies to stop the erosion of biodiversity. Optimisation is increasingly used to identify reserve locations that preserve biodiversity at a minimum cost to human activities. Two classes of algorithms solve this reserve site selection problem: metaheuristic algorithms (such as simulated annealing, commonly implemented in Marxan) and exact optimisation (i.e. integer programming, commonly implemented in PrioritizR). Although exact approaches are now able to solve large-scale problems, metaheuristics are still widely used. One reason is that metaheuristic-based software provides a set of suboptimal reserve solutions instead of a single one. These alternative solutions are usually welcomed by stakeholders as they provide a better basis for negotiations among potentially conflictive objectives. Metaheuristic algorithms use random procedures to explore the space of suboptimal reserve solutions. Therefore, they may produce a large amount of similar, thus uninformative, alternative solutions, which usually calls for a heavy statistical post-processing. Effective methods for generating a diverse set of near-optimal solutions using exact optimisation are lacking. Here we present two new approaches for addressing this issue. Our algorithms explicitly control both the optimality gap and the dissimilarity between alternative reserve solutions. It allows the identification of a parsimonious, yet meaningful set of reserve solutions. The algorithms presented here could potentially increase the uptake of exact optimisation by practitioners. These methods should contribute to less noisy and more efficient discussions in the design of conservation policies.

Appendices

Appendix A. Linearised Model

Parameters and variables were defined in Sect. 2.1. Sets of planning units a priori excluded or included in the reserve are respectively noted $\mathcal{LO}\mathcal{}$ and $\mathcal{LI}\mathcal{}$. We can linearise the quadratic term of the objective function when decision variables are binary [21, 40]. Considering this linearisation but also locked-in and locked-out planning units, we have the full mathematical optimisation problem $P_0^f$ of reserve site selection:

$$\begin{aligned} P_0^f: \left\{ \begin{array}{cll} \min \limits _{x,z} &{} {\sum \limits _{j \in J}c_jx_j + \beta (\sum \limits _{j_1 \in J}\sum \limits _{j_2 \in J}b_{j_1j_2}z_{j_1j_2}} + \sum \limits _{j \in J}{x_j b_{j,N+1}^*)} &{} \\ \text {s.t.} &{} \sum \limits _{j \in J}a_{ij}x_j \ge t_i &{} \forall i \in I \\ &{} z_{j_1j_2} \le x_{j_1} &{} \forall j_1 \in J, \forall j_2 \in J \\ &{} z_{j_1j_2} \le x_{j_2} &{} \forall j_1 \in J, \forall j_2 \in J \\ &{} z_{j_1j_2} \ge x_{j_1} + x_{j_2} -1 &{} \forall j_1 \in J, \forall j_2 \in J \\ &{} x_j = 0 &{} \forall j \in \mathcal{LO}\mathcal{} \\ &{} x_j = 1 &{} \forall j \in \mathcal{LI}\mathcal{} \\ &{} x_j \in \{0,1\} &{} \forall j \in J \\ &{} z_{j_1j_2} \in \{0,1\} &{} \forall j_1 \in J, \forall j_2 \in J \end{array} \right. \end{aligned}$$

We also accounted for the correction of the $\beta$ multiplier undesirable edge effect [39], leading to the introduction of $b^*$ where:

$$\begin{aligned} \begin{aligned}&\forall j \in J = \{1,\cdots ,N\}, \\&b_{j,N+1}^*= {\left\{ \begin{array}{ll} &{} 1, \text { if pixel { j} shares a single side with the outer boundary} \\ &{} 2, \text { if pixel { j} shares 2 sides with the outer boundary} \\ &{} 0, \text { otherwise} \end{array}\right. } \end{aligned} \end{aligned}$$

Appendix B. Imposing an Objective Value Interval

We show how we produced the presentation set composed of alternative solutions located at a predefined objective value interval. We here developed our own algorithm although the function add_gap_portfolio of PrioritizR allows to generate the same set of alternative solutions.

1.1 Objective Value Constraints

Let $\gamma _1 \in \mathbb {R}^+$ and $\gamma _2 \in \mathbb {R}^+$, such as $\gamma _1 \le \gamma _2$, be the boundaries of the objective value interval relatively to the optimal value $z^\star$. The constraints $c_l(\gamma _1)$ and $c_u(\gamma _2)$ are imposing the objective value to belong to the predefined interval $\left[ (1+\gamma _1)z^\star ,(1+\gamma _2)z^\star \right]$:

$$\begin{aligned} c_l(\gamma _1): \sum \limits _{j \in J}c_jx_j + \beta \sum \limits _{j_1 \in J}\sum \limits _{j_2 \in J}b_{j_1j_2}x_{j_1}(1-x_{j_2}) \ge (1+\gamma _1)z^\star. \end{aligned}$$

$$\begin{aligned} c_u(\gamma _2): \sum \limits _{j \in J}c_jx_j + \beta \sum \limits _{j_1 \in J}\sum \limits _{j_2 \in J}b_{j_1j_2}x_{j_1}(1-x_{j_2}) \le (1+\gamma _2)z^\star. \end{aligned}$$

If $\gamma _1=\gamma _2=0$, we explore only the optimal solutions set. For $\gamma _1>0$, we explore alternative solutions that are strictly suboptimal.

1.2 Distance Constraint

The constraint $c_D(y,\delta )$ impose the solution x to have at least $\delta$ different planning units with respect to y:

$$\begin{aligned} c_D(y,\delta ): D(x,y) = \sum \limits _{j \in J}y_j(1-x_j)+x_j(1-y_j) \ge \delta. \end{aligned}$$

Importantly, $\delta =1$ forbids x and y to be strictly equal.

1.3 Generate the Presentation Set

Practically, we first add to the optimisation problem the constraints $c_l(\gamma _1)$ and $c_u(\gamma _2)$ which must be satisfied at every iteration. Then, to derive a pool of alternative solutions, we excluded at iteration $k \ge 1$ the solution $x^{k-1}$ derived the iteration before. The addition of constraint $c_D(x^{k-1},1)$ guarantee this. Indeed, this constraint prevents the searched solution at iteration $k \ge 1$ to be exactly $x^{k-1}$.

Practically, the integer linear program solved at iteration $k \ge 1$ is $P_3^k$ such as:

$$\begin{aligned} P_3^k: \left\{ \begin{array}{cll} \min \limits _{x} &{} {\sum \limits _{j \in J}c_jx_j + \beta \sum \limits _{j_1 \in J}\sum \limits _{j_2 \in J}b_{j_1j_2}x_{j_1}(1-x_{j_2})} &{} \\ \text {s.t.} &{} \sum \limits _{j \in J}c_jx_j + \beta \sum \limits _{j_1 \in J}\sum \limits _{j_2 \in J}b_{j_1j_2}x_{j_1}(1-x_{j_2}) \le (1+\gamma _2)z^\star &{} \\ &{} \sum \limits _{j \in J}c_jx_j + \beta \sum \limits _{j_1 \in J}\sum \limits _{j_2 \in J}b_{j_1j_2}x_{j_1}(1-x_{j_2}) \ge (1+\gamma _1)z^\star &{} \\ &{} \sum \limits _{j \in J}a_{ij}x_j \ge t_i &{} \forall i \in I \\ &{} \sum \limits _{j \in J}x_j(1-x^{l}_j)+x^{l}_j(1-x_j) \ge 1 &{} \forall l \in [\![0,k-1]\!] \\ &{} x_j \in \{0,1\} &{} \forall j \in J \\ \end{array} \right. \end{aligned}$$

The constraints $c_l(\gamma _1)$ and $c_u(\gamma _2)$ used in $P_3^k$ are not linear. We linearised these constraints exactly as we did for the model $P_0^f$ described in Appendix A. Algorithm AddGapPortfolio details the pseudocode of the recursive procedure we implemented to produce the presentation set. The procedure stops if the problem becomes infeasible or the maximum number of iterations n is reached. Infeasibility is reached when the objective value of the alternative solution exceeds the upper bound $\gamma _2$. If the user wants a larger presentation set, they may choose a greater threshold $\gamma _2$. For instance, if $\gamma _1=0$ and $\gamma _2$ is high enough, Algorithm AddGapPortfolio returns the n solutions with the smallest objective value. If n is chosen high enough, Algorithm AddGapPortfolio returns the exhaustive set of solutions with an objective value relatively to the optimal value within the interval $[\gamma _1,\gamma _2]$. Unlike metaheuristics where the optimality gap is unknown, we a priori established it using this algorithm. We thus offer users more control over the presentation set provided.

Appendix C. The Presentation Set Computed on Generated Data

We developed a systematic way of building user-defined scenarios for reserve site selection optimisation problems. The idea is to provide the conservation literature tools to facilitate benchmarks of developed methods in conservation planning. Therefore, the main ambition is to generate realistic discrete spatial distributions of the considered conservation features.

1.1 Data Generation

Technically speaking, we choose to compute the amount $a_{ij}$ of a conservation feature $i \in I$ in a planning unit $j \in J$ by randomly drawing this value using a Gaussian distribution.

$$\begin{aligned} a_{ij} \sim \mathcal {N}(m_{ij},\sigma _{ij}^{2}). \end{aligned}$$

The mean value $m_{ij}$ of the Gaussian distribution only depends on the distance $d_{ij}$ to the closest (chosen or randomly drawn) $N_{epi}$ epicentres associated to the conservation feature $i \in I$. To be more precise, the mean value $m_{ij}$ depends on $d_{ij}^{\alpha _i}$, where $\alpha _i$ is a predefined parameter for each conservation feature $i \in I$. The parameter $\alpha _i$ controls the dispersion of the mean values relatively to the epicentres.

$$\begin{aligned} m_{ij} = \mu _i\left[ 1-\left( \frac{d_{ij}}{d_{max}}\right) ^{\alpha _i}\right]. \end{aligned}$$

The maximum mean value, i.e. the mean value at the epicentres, is a chosen parameter $\mu _i$ for each conservation feature $i \in I$. If no epicentres are provided, the mean value of the Gaussian distribution depends on the distance to the locked-out planning units supposed to represent a shoreline. The standard deviation $\sigma _{ij}$ of the Gaussian distribution is such as $\sigma _{ij} = \sigma _i m_{ij}$ where $\sigma _i$ is a chosen parameter for each conservation feature $i \in I$. The code used to generate data is available in open access⁵. The instance is characterised by the rectangular grid size $N_x$ and $N_y$ and the number of conservation features $N_{cf}$.

1.2 Scenarios

We build several scenarios to have some order of magnitudes for the computation time of the algorithms proposed in this work. We show in Fig. 8 the generated spatial distributions of two conservation features resulting from the data generation procedure. An example of a presentation set is given in Fig. 9.

Fig. 8

Example of the generated spatial distribution for two different conservation features in a $25 \times 20$ rectangular grid. The amounts of considered conservation feature are shown with a yellow to red gradient. The corresponding numerical values are written in black inside the planning units. Locked-out planning units are represented in grey. We chose $\sigma _i = 0.20$

Fig. 9

Presentation set computed with Algorithm MaxDissimilarity. The considered scenario was made of $40 \times 25$ planning units and 5 conservation features. We chose an extra cost budget of $\gamma = 0.10$. Relative targets for every conservation feature were set to 25%. Green planning units represents the alternative reserve solution. Planning units with a black border indicates the initial optimal solution