Markus Hartikainen

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 method is fast with the surrogate problem even though the problem is computationally expensive. Numerical examples of applying the PAINT method for interpolation are included.

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.

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 to postpone the measurements until just before the potential harvests. Introducing a strict end-inventory constraint increased the number of stands that could be profitably measured. In this case, also the length of the planning horizon had a clear effect on what stands were profitable to measure. With a 15-year planning horizon, measuring the oldest stands was profitable while with longer planning horizons it was not. The interest rate did not affect the number of stands measured much, but it had a clear effect on the timing of the measurements.

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 value and the value at risk, as well as between the value at risk of the two objectives of interest and, thus, decision support is needed to find the best balance between the conflicting objectives. We employ an interactive method where a decision maker iteratively provides preference information to find the most preferred management plan and at the same time learns about the interdependencies of the objectives.

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.

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.

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 speed up the iterations of the NIMBUS method. 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 wastewater treatment problem. With the mixed integer surrogate problem, the time required from the decision maker is comparatively short. In addition, a new IND-NIMBUS® PAINT module is developed to allow the smooth interoperability of the NIMBUS method and the PAINT method.

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.

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 transform them into software implementation objectives. This paper reports interesting insights about the elicitation process of the objectives.

Design of a Learning Space Management System for Open and Adaptable School Facilities

In this design-based research project, a learning space management system was developed for the Valteri School Onerva in Central-Finland. The school represents a modern educational environment with open and adaptable learning spaces. The goal was to develop a software to support the stakeholders in organising flexible pedagogical activities and sharing pedagogical practices. To reach this goal, we utilised value-focused thinking as a requirements elicitation method, to identify the objectives that the stakeholders associate with the new environment. In the implementation phase, we organised participatory design workshops, to involve the stakeholders in decision-making, to ensure that the prototype development was proceeding according to their needs. As a result, we elaborate how we utilised value-focused thinking, what were the objectives that were identified, and how they were transformed into system requirements. Finally, we describe the first prototype of the learning space management system, which was developed using these requirements.