ANOVA-MOP: ANOVA Decomposition for Multiobjective Optimization

Real-world optimization problems may involve a number of computationally expensive functions with a large number of input variables. Metamodel-based optimization methods can reduce the computational costs of evaluating expensive functions, but this does not reduce the dimension of the search domain nor mitigate the curse of dimensionality effects. The dimension of the search domain can be reduced by functional anova decomposition involving Sobol' sensitivity indices. This approach allows one to rank decision variables according to their impact on the objective function values. On the basis of the sparsity of effects principle, typically only a small number of decision variables significantl…

Potential of interactive multiobjective optimization in supporting the design of a groundwater biodenitrification process

The design of water treatment plants requires simultaneous analysis of technical, economic and environmental aspects, identified by multiple conflicting objectives. We demonstrated the advantages of an interactive multiobjective optimization (MOO) method over a posteriori methods in an unexplored field, namely the design of a biological treatment plant for drinking water production, that tackles the process drawbacks, contrarily to what happens in a traditional volumetric-load-driven design procedure. Specifically, we consider a groundwater denitrification biofilter, simulated by the Activated Sludge Model modified with two-stage denitrification kinetics. Three objectives were defined (nitr…

Demonstrating the Applicability of PAINT to Computationally Expensive Real-life Multiobjective Optimization

We demonstrate the applicability of a new PAINT method to speed up iterations of interactive methods in multiobjective optimization. As our test case, we solve a computationally expensive non-linear, five-objective problem of designing and operating a wastewater treatment plant. The PAINT method interpolates between a given set of Pareto optimal outcomes and constructs a computationally inexpensive mixed integer linear surrogate problem for the original problem. We develop an IND-NIMBUS R PAINT module to combine the interactive NIMBUS method and the PAINT method and to find a preferred solution to the original problem. With the PAINT method, the solution process with the NIMBUS method take …

Data-Based Forest Management with Uncertainties and Multiple Objectives

In this paper, we present an approach of employing multiobjective optimization to support decision making in forest management planning. The planning is based on data representing so-called stands, each consisting of homogeneous parts of the forest, and simulations of how the trees grow in the stands under different treatment options. Forest planning concerns future decisions to be made that include uncertainty. We employ as objective functions both the expected values of incomes and biodiversity as well as the value at risk for both of these objectives. In addition, we minimize the risk level for both the income value and the biodiversity value. There is a tradeoff between the expected val…

Constructing a Pareto front approximation for decision making

An approach to constructing a Pareto front approximation to computationally expensive multiobjective optimization problems is developed. The approximation is constructed as a sub-complex of a Delaunay triangulation of a finite set of Pareto optimal outcomes to the problem. The approach is based on the concept of inherent nondominance. Rules for checking the inherent nondominance of complexes are developed and applying the rules is demonstrated with examples. The quality of the approximation is quantified with error estimates. Due to its properties, the Pareto front approximation works as a surrogate to the original problem for decision making with interactive methods. Qc 20120127

A Computationally Inexpensive Approach in Multiobjective Heat Exchanger Network Synthesis

We consider a heat exchanger network synthesis problem formulated as a multiobjective optimization problem. The Pareto front of this problem is approximated with a new approximation approach and the preferred point on the approximation is found with the interactive multiobjective optimization method NIMBUS. Using the approximation makes the solution process computationally inexpensive. Finally, the preferred outcome on the Pareto front approximation is projected on the actual Pareto front. peerReviewed

Sectoral policies as drivers of forest management and ecosystems services: A case study in Bavaria, Germany

European countries have national sectoral polices to regulate and promote the provision of a wide range of forest ecosystems services (FES). However, potential incoherencies among these policies can negatively affect the efficient provision of FES. In this work, we evaluated the coherence among three national policies from Germany and their ability to effectively provide FES in the future: the Forest Strategy 2020 (FS), the National Strategy on Biological Diversity (BDS), and the German National Policy Strategy on Bioeconomy (BES). Using forest inventory data from the Federal State of Bavaria, we simulated a range of forest management options under three climate trajectories for 100 years i…

Identifying objectives for a learning space management system with value-focused thinking

A classroom with a blackboard and some rows of desks is obsolete in special education. Depending on the needs, some students may need more tactile and inspiring surroundings with various pedagogical accessories while others benefit from a simplified environment without unnecessary stimuli. This understanding is applied to a new Finnish special education school building with open and adaptable learning spaces. We have joined the initiative creation process by developing software support for these new spaces in the form of a learning space management system. Participatory design and value-focused thinking were implemented to elicit the actual values of all the stakeholders involved and transf…

Towards constructing a Pareto front approximation for use in interactive forest management planning

The selection of an appropriate multi-objective forest management plan can be a difficult task due to the vast number of alternatives available to the decision maker (DM). The complexity of the task depends e.g. on how clear the preferences of the DM are. For those DMs who do not have clear preferences, interactive methods of forest planning could assist in clarifying preferences and guiding the selection in an efficient fashion. Interactive planning methods are useful when the DM needs to consider a wide range of efficient solutions quickly. With large forest holdings or with complicated forest management goals, the development of new forest plans can become a rather computationally demand…

PAINT–SiCon: constructing consistent parametric representations of Pareto sets in nonconvex multiobjective optimization

We introduce a novel approximation method for multiobjective optimization problems called PAINT–SiCon. The method can construct consistent parametric representations of Pareto sets, especially for nonconvex problems, by interpolating between nondominated solutions of a given sampling both in the decision and objective space. The proposed method is especially advantageous in computationally expensive cases, since the parametric representation of the Pareto set can be used as an inexpensive surrogate for the original problem during the decision making process. peerReviewed

Integrating risk management tools for regional forest planning : an interactive multiobjective value at risk approach

In this paper, we present an approach employing multiobjective optimization to support decision making in forest management planning under risk. The primary objectives are biodiversity and timber cash flow, evaluated from two perspectives: the expected value and the value-at-risk (VaR). In addition, the risk level for both the timber cash flow and biodiversity values are included as objectives. With our approach, we highlight the trade-off between the expected value and the VaR, as well as between the VaRs of the two objectives of interest. We employ an interactive method in which a decision maker iteratively provides preference information to find the most preferred management plan and lea…

PAINT : Pareto front interpolation for nonlinear multiobjective optimization

A method called PAINT is introduced for computationally expensive multiobjective optimization problems. The method interpolates between a given set of Pareto optimal outcomes. The interpolation provided by the PAINT method implies a mixed integer linear surrogate problem for the original problem which can be optimized with any interactive method to make decisions concerning the original problem. When the scalarizations of the interactive method used do not introduce nonlinearity to the problem (which is true e.g., for the synchronous NIMBUS method), the scalarizations of the surrogate problem can be optimized with available mixed integer linear solvers. Thus, the use of the interactive meth…

Simultaneous optimization of harvest schedule and data quality

In many recent studies, the value of forest inventory information in harvest scheduling has been examined. In a previous paper, we demonstrated that making measurement decisions for stands for which the harvest decision is uncertain simultaneously with the harvest decisions may be highly profitable. In that study, the quality of additional measurements was not a decision variable, and the only options were between making no measurements or measuring perfect information. In this study, we introduce data quality into the decision problem, i.e., the decisionmaker can select between making imperfect or perfect measurements. The imperfect information is obtained with a specific scenario tree fo…

Applying the approximation method PAINT and the interactive method NIMBUS to the multiobjective optimization of operating a wastewater treatment plant

Using an interactive multiobjective optimization method called NIMBUS and an approximation method called PAINT, preferable solutions to a five-objective problem of operating a wastewater treatment plant are found. The decision maker giving preference information is an expert in wastewater treatment plant design at the engineering company Pöyry Finland Ltd. The wastewater treatment problem is computationally expensive and requires running a simulator to evaluate the values of the objective functions. This often leads to problems with interactive methods as the decision maker may get frustrated while waiting for new solutions to be computed. Thus, a newly developed PAINT method is used to spe…

Integrating risk management tools for regional forest planning: an interactive multiobjective value-at-risk approach

In this paper, we present an approach employing multiobjective optimization to support decision making in forest management planning under risk. The primary objectives are biodiversity and timber cash flow, evaluated from two perspectives: the expected value and the value-at-risk (VaR). In addition, the risk level for both the timber cash flow and biodiversity values are included as objectives. With our approach, we highlight the trade-off between the expected value and the VaR, as well as between the VaRs of the two objectives of interest. We employ an interactive method in which a decision maker iteratively provides preference information to find the most preferred management plan and le…

A survey on handling computationally expensive multiobjective optimization problems using surrogates: non-nature inspired methods

Computationally expensive multiobjective optimization problems arise, e.g. in many engineering applications, where several conflicting objectives are to be optimized simultaneously while satisfying constraints. In many cases, the lack of explicit mathematical formulas of the objectives and constraints may necessitate conducting computationally expensive and time-consuming experiments and/or simulations. As another challenge, these problems may have either convex or nonconvex or even disconnected Pareto frontier consisting of Pareto optimal solutions. Because of the existence of many such solutions, typically, a decision maker is required to select the most preferred one. In order to deal wi…

Decision Making on Pareto Front Approximations with Inherent Nondominance

t Approximating the Pareto fronts of nonlinear multiobjective optimization problems is considered and a property called inherent nondominance is proposed for such approximations. It is shown that an approximation having the above property can be explored by interactively solving a multiobjective optimization problem related to it. This exploration can be performed with available interactive multiobjective optimization methods. The ideas presented are especially useful in solving computationally expensive multiobjective optimization problems with costly function value evaluations. peerReviewed

Sectoral policies cause incoherence in forest management and ecosystem service provisioning

Various national policies guide forest use, but often with competing policy objectives leading to divergent management paradigms. Incoherent policies may negatively impact the sustainable provision of forest ecosystem services (FES), and forest multifunctionality. There is uncertainty among policymakers about the impacts of policies on the real world. We translated the policy documents of Finland into scenarios including the quantitative demands for FES, representing: the national forest strategy (NFS), the biodiversity strategy (BDS), and the bioeconomy strategy (BES). We simulated a Finland-wide systematic sample of forest stands with alternative management regimes and climate change. Fin…

On Generalizing Lipschitz Global Methods forMultiobjective Optimization

Lipschitz global methods for single-objective optimization can represent the optimal solutions with desired accuracy. In this paper, we highlight some directions on how the Lipschitz global methods can be extended as faithfully as possible to multiobjective optimization problems. In particular, we present a multiobjective version of the Pijavskiǐ-Schubert algorithm.

Climate targets in European timber-producing countries conflict with goals on forest ecosystem services and biodiversity

The role of increased timber harvests in reaching climate mitigation targets for European countries will be limited if the protection of forest ecosystem services and biodiversity is to be achieved, suggests an empirical forest model driven by future scenarios to limit warming to 1.5 degrees C in 2100.The European Union (EU) set clear climate change mitigation targets to reach climate neutrality, accounting for forests and their woody biomass resources. We investigated the consequences of increased harvest demands resulting from EU climate targets. We analysed the impacts on national policy objectives for forest ecosystem services and biodiversity through empirical forest simulation and mul…

An interactive surrogate-based method for computationally expensive multiobjective optimisation

Many disciplines involve computationally expensive multiobjective optimisation problems. Surrogate-based methods are commonly used in the literature to alleviate the computational cost. In this paper, we develop an interactive surrogate-based method called SURROGATE-ASF to solve computationally expensive multiobjective optimisation problems. This method employs preference information of a decision-maker. Numerical results demonstrate that SURROGATE-ASF efficiently provides preferred solutions for a decision-maker. It can handle different types of problems involving for example multimodal objective functions and nonconvex and/or disconnected Pareto frontiers. peerReviewed

Approximation through interpolation in nonconvex multiobjective optimization

Simultaneous optimization of harvest schedule and measurement strategy

In many recent studies, the value of forest inventory information in the harvest scheduling has been examined. Usually only the profitability of measuring simultaneously all the stands in the area is examined. Yet, it may be more profitable to concentrate the measurement efforts to some subset of them. In this paper, the authors demonstrate that stochastic optimization can be used for defining the optimal measurement strategy simultaneously with the harvest decisions. The results show that without end-inventory constraints, it was most profitable to measure the stands that were just below the medium age. Measuring the oldest stands was not profitable at all. It turned out to be profitable t…

Interactive Nonconvex Pareto Navigator for Multiobjective Optimization

Abstract We introduce a new interactive multiobjective optimization method operating in the objective space called Nonconvex Pareto Navigator . It extends the Pareto Navigator method for nonconvex problems. An approximation of the Pareto optimal front in the objective space is first generated with the PAINT method using a relatively small set of Pareto optimal outcomes that is assumed to be given or computed prior to the interaction with the decision maker. The decision maker can then navigate on the approximation and direct the search for interesting regions in the objective space. In this way, the decision maker can conveniently learn about the interdependencies between the conflicting ob…