A way to avoid this mathematical challenge is to use what we call explicit approaches in this paper, i. Although this approach is not, strictly speaking, multi-objective optimization, it is very common in conservation. Indeed, in some cases the structure of the system to optimize prohibit the use an exact approach, except for small problems.
The explicit approach allows us to perform multi-criteria decision analysis MCDA , which is very powerful where the number of possible decisions is small [ 29 — 31 ]. Another usual approach in conservation is to try to establish correlations between criteria trade-off analysis , via exhaustive approaches [ 32 ], or heuristic approaches [ 33 ]. Unfortunately, there is no reason that criteria of combinatorial problems have the same correlation from one instance to another changing the data could result in a complete different correlation.
Additionally, the lack of scalability of exhaustive approaches and the lack of optimality of heuristic approaches make them very limited approaches to solve combinatorial problems. When the multi-objective problem can only be implicitly defined see Section 2. Local approaches can be used to perform two types of methods: a priori methods and interactive methods. Global approaches, which generate the entire Pareto frontier, are often called a posteriori methods.
Several approaches in conservation aim to find a unique objective summarizing the individual objectives, and then treat the problem as a single objective. Reducing several objectives in one is usually done using an a priori aggregation function, i. The cost-benefit approach is probably the most used approach applying this principle. The cost-benefit approach is an economic approach where every criteria is considered as having an economic counter-part [ 34 ]. Other aggregation functions of the objectives have been studied in conservation [ 12 , 36 ].
Several major well-known drawbacks occur in these approaches. Using economic values of species is ethically controversial because it requires associating an economic value to species [ 37 ]. Additionally, in practice, depending on the economical evaluation methods, the value of a species can vary significantly, sometimes from one to tenfold [ 38 ].
The second drawback is related to the subjectivity and the complexity of the fixed aggregation function. Choosing among a set of potential aggregation functions can be difficult to justify [ 12 , 36 ]. Finally, reasoning with one objective aggregated function instead of several, reduces considerably the role of the decision-maker in the optimization process. A prescribed solution is then provided by the scientists, missing an opportunity to involve the decision-maker in the decision-making process itself. A posteriori methods aim to generate the Pareto frontier or an approximation of it.
Solving multi-objective optimization problems in conservation with the reference point method
Some methods in conservation can be classified as a posteriori methods [ 39 , 40 ]. A posteriori methods corresponds to trade-off analysis methods for implicit approaches. Generating the Pareto frontier is only possible and relevant for problems with a small number of objectives. Generally based on the use of parametric aggregation functions, interactive methods aim to interactively find the non-dominated point that corresponds the most to the preference of a decision-makers [ 20 , 21 ].
In these methods, the decision-maker preferences can evolve according to the following iterative procedure:. New preferences are obtained by eliciting feedback from the decision-maker on current results. Fig 1 provides a graphical illustration of the interactive procedure. Multi-objective optimization interactive methods are not very common in conservation but see [ 15 ] for an exception. Every solution provided by the algorithm corresponds to a non-dominated point of the multi-objective problem.
Every non-dominated point of the multi-objective problem can be generated by the algorithm.
The reference point method is one of the only multi-objective optimization methods to satisfy the requirements [ 18 ]. This is due to the aggregation function used in the method, called achievement function and was created specifically for the reference point method [ 41 ]. The formal formulation of the augmented achievement function is as follow [ 21 ]:.
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For each criterion j , z j m a x and z j m i n are respectively the maximum and the minimum possible values of f j x obtained by performing the corresponding single-objective optimization. Conversely, if we use a weighted sum as an aggregation function, the second requirement is not satisfied because only the convex hull of the Pareto frontier can be generated. Moreover, weights have no significance , and transforming them into meaningful values [ 43 ] can be obscure for the decision-maker [ 44 ], because the true preferential parameters are hidden.
More importantly, the weighted sum method is well-known to not provide good compromise solutions [ 45 ], i. Finally, this method makes the strong assumption that one objective can always linearly compensate linearly another objective. Human preferences are often much more complicated than linear trade-offs and may require more elaborate methods. In this paper we will use the linear programming formulation of the reference point method, for reasons explaned in Section 2. Linear programming is well-known in conservation [ 47 ], but less in the context of multi-objective optimization, especially approaches.
A linear programming approach is however interesting because it is an implicit approach and it can solve optimally combinatorial problems. In this paper we use linear programming in its general sense, i. Thus, non-linear and even non-convex optimization problems can also be tackled with this approach thanks to some linearization to perform.
Of course, the difficulty to solve the program will depend on the the nature of the optimization problem and how hard it is to lieanrize expressions. Using the reference point method requires solving an optimization problem at every iteration of the interactive process. Although other optimization methods are possible, linear programming LP is particularly well suited to solve this problem. The LP formulation with p objectives, n variables and m constraints is:. The variables are z and the components of vector x. If some f j are not linear, then it is necessary to linearize them.
The simplicity of the formulation is one reason of the popularity of LP to solve the reference point method, which has been implemented in many fields where combinatorial problems occur, for example in telecommunication [ 48 ], finance [ 49 ] or transportation [ 50 ]. The LP formulation of the reference point method requires finding good LP formulations of the optimization problem we want to solve.
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In the case of discrete optimization, this is in general a hard task and requires a strong knowledge of the problem and LP techniques. In this section we present LP formulations for a multi-species dynamic conservation problem and a multi-objective environmental spatial resource allocation problem, so that the reference point method can be applied in both cases. In [ 2 ], the authors propose a method for solving a sequential decision problem under uncertainty, aiming to conserve simultaneously two interacting endangered species: Northern abalone and sea otters.
This bi-objective is solved using indirectly an a priori weighted sum of the objectives. Different weights are tested to generate and explore alternatives.
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Weighting the criteria allows the use of classic MDP solution methods such as dynamic programming [ 51 ]. More specifically, the problem is a predator-prey problem where interactions between sea otters and their preferred prey abalone are described using a MDP formalism. Every year, managers must decide between 4 actions: introduce sea otters, enforce abalone anti-poaching measures, control sea otters, half enforce anti-poaching measures and half control sea otters.
The time horizon is 20 years.
The original problem aims to maximize the density of abalone and abundance of sea otters. Because weighting the objectives of an optimization problem can be controversial see Section 2. Adapting Program 1 directly to a LP formulation of MDPs is challenging because rewards appear only in the constraints and not in the objective.
In [ 45 ], the authors were confronted with the same situation when they tried to apply a similar multi-objective optimization technique the Chebyshev method to MDPs in a robotic context. The Chebyshev method aims to minimize the Chebyshev norm between the reference point and the decision space. The reference point method is different for several reasons.
First, in the reference point method, preferences of the decision-maker are directly expressed as values on every criterion, while in the Chebyshev method preferences are expressed as weights. Second, the reference point method allows the decision-maker to choose values inside the feasible space, which is not the case in the Chebyshev method. However, the same idea as in [ 45 ] can also be used for the linear programming reference point method formulation.
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