Chugh, T., Jin, Y., Miettinen, K., Hakanen, J., Sindhya, K., A Surrogate-Assisted Reference Vector Guided Evolutionary Algorithm for Computationally Expensive Many-Objective Optimization, IEEE Transactions on Evolutionary Computation, 22(1), 129-142, 2018.
We propose a surrogate-assisted reference vector guided evolutionary algorithm for computationally expensive optimization problems with more than three objectives. The proposed algorithm is based on a recently developed evolutionary algorithm for many-objective optimization that relies on a set of adaptive reference vectors for selection. The proposed surrogate- assisted evolutionary algorithm uses Kriging to approximate each objective function to reduce the computational cost. In managing the Kriging models, the algorithm focuses on the balance of diversity and convergence by making use of the uncertainty information in the approximated objective values given by the Kriging models, the distribution of the reference vectors as well as the location of the individuals. In addition, we design a strategy for choosing data for training the Kriging model to limit the computation time without impairing the approximation accuracy. Empirical results on comparing the new algorithm with the state-of-the-art surrogate-assisted evolutionary algorithms on a number of benchmark problems demonstrate the competitiveness of the proposed algorithm.
Chugh, T., Sindhya, K., Hakanen, J., Miettinen, K., A Survey on Handling Computationally Expensive Multiobjective Optimization Problems with Evolutionary Algorithms, Soft Computing, to appear.
Evolutionary algorithms are widely used for solving multiobjective optimization problems but are often criticized because of a large number of function evaluations needed. Approximations, especially function approximations, also referred to as surrogates or metamodels are commonly used in the literature to reduce the computation time. This paper presents a survey of 45 different recent algorithms proposed in the literature between 2008-2016 to handle computationally expensive multiobjective optimization problems. Several algorithms are discussed based on what kind of an approximation such as problem, function or fitness approximation they use. Most emphasis is given to function approximation based algorithms. We also compare these algorithms based on different criteria such as metamodelling technique and evolutionary algorithm used, type and dimensions of the problem solved, handling constraints, training time and the type of evolution control. Furthermore, we identify and discuss some promising elements and major issues among algorithms in the literature related to using an approximation and numerical settings used. In addition, we discuss selecting an algorithm to solve a given computationally expensive mmultiobjective optimization problem based on the dimensions in both objective and decision spaces and the computation budget available.
Zhou-Kangas, Y. Miettinen, K. Sindhya, K., Solving Multiobjective Optimization Problems with Decision Uncertainty: An Interactive Approach, Journal of Buainess Economics, to appear.
We propose an interactive approach to support a decision maker to find a most preferred robust solution to multiobjective optimization problems with decision uncertainty. A new robustness measure that is understandable for the decision maker is incorporated as an additional objective in the problem formulation. The proposed interactive approach utilizes elements of the synchronous NIMBUS method and is aimed at supporting the decision maker to consider the objective function values and the robustness of a solution simultaneously. In the interactive approach, we offer different alternatives for the decision maker to express her/his preferences related to the robustness of a solution. To consolidate the interactive approach, we tailor a visualization to illustrate both the objective function values and the robustness of a solution. We demonstrate the advantages of the interactive approach by solving example problems.
Eyvindson, K., Hartikainen, M., Miettinen, K., Kangas. A.,Integrating Risk Management Tools for Regional Forest Planning: An Interactive Multiobjective Value at Risk Approach, Canadian Journal of Forest Research, to appear.
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 VaR of the two objectives of interest. 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. The method is illustrated with a case study, where biodiversity is assessed through an index calculated from the characteristics of the forest. Uncertainty is included through modifying the input data according to the accuracy of current inventory methods and through growth model errors. This uncertainty is described through a set of 25 scenarios. Involving multiple components of risk is a highly relevant approach in multiobjective forestry. However, estimation of the uncertainty of biodiversity needs further attention.
Tabatabaei, M., Hartikainen, M., Sindhya, K., Hakanen, J., Miettinen, K., An Interactive Surrogate-based Method for Computationally Expensive Multiobjective Optimization, Journal of the Operational Research Society, to appear.
Many disciplines involve computationally expensive multiobjective optimization 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 optimization 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.
Mazziotta, A., Podkopaev, D., Trivino, M., Miettinen, K., Pohjanmies, T., Monkkonen, M., Quantifying and Resolving Conservation Conflicts in Forest Landscapes via Multiobjective Optimization, Silva Fennica, 51(1), 2017.
Environmental planning for of the maintenance of different conservation objectives should take into account multiple contrasting criteria based on alternative uses of the landscape. We develop new concepts and approaches to describe and measure conflicts among conservation objectives and for resolving them via multiobjective optimization. To measure conflicts we introduce a compatibility index that quantifies how much targeting a certain conservation objective affects the capacity of the landscape for providing another objective. To resolve such conflicts we find compromise solutions defined in terms of minimax regret, i.e. minimizing the maximum percentage of deterioration among conservation objectives. Finally, we apply our approach for a case study of management for biodiversity conservation and development in a forest landscape. We study conflicts between six different forest species, and we identify management solutions for simultaneously maintaining multiple species' habitat while obtaining timber harvest revenues. We employ the method for resolving conflicts at a large landscape level across a long 50-years forest planning horizon. Our multiobjective approach can be an instrument for guiding hard choices in the conservation-development nexus with a perspective of developing decision support tools for land use planning. In our case study multiple use management and careful landscape level planning using our approach can reduce conflicts among biodiversity objectives and offer room for synergies in forest ecosystems.
Sindhya, K., Manninen, A., Miettinen, K., Pippuri, J., Design of a Permanent Magnet Synchronous Generator using Interactive Multiobjective Optimization, IEEE Transactions on Industrial Electronics, to appear.
We consider an analytical model of a permanent magnet synchronous generator and formulate a mixed-integer constrained multiobjective optimization problem with six objective functions. We demonstrate the usefulness of solving such a problem by applying an interactive multiobjective optimization method called NIMBUS. In the NIMBUS method, a decision maker is iteratively involved in the optimization process and directs the solution process in order to find her/his most preferred Pareto optimal solution for the problem. We also employ a commonly used non-interactive evolutionary multiobjective optimization method NSGA-II to generate a set of solutions that approximates the Pareto set and demonstrate the advantages of using an interactive method. This study is the first one to consider an interactive approach for the design of a permanent magnet synchronous generator. Thus, we promote the further usage of interactive multiobjective optimization methods in the design. Further, we see that these methods could also be very useful in the teaching of electrical machines.
Miettinen, K., Ruiz, F., NAUTILUS Framework: Towards Trade-off-Free Interaction in Multiobjective Optimization, Journal of Business Economics, 86(1), 5-21, 2016.
In this paper, we present a framework of different interactive NAUTILUS methods for multiobjective optimization. In interactive methods, the decision maker iteratively sees solution alternatives and provides one's preferences in order to find the most preferred solution. We question the widely used setting that the solutions shown to the decision maker should all be Pareto optimal which implies that improvement in any objective function necessitates allowing impairment in some others. Instead, in NAUTILUS we enable the decision maker to make a free search without having to trade-off by starting from an inferior solution and iteratively approaching the Pareto optimal set by allowing all objective functions to improve. The framework presented consists of different modules for preference elicitation and optimization. Four main NAUTILUS method variants are introduced as well as ideas of utilizing the framework in a flexible way to derive further variants.Miettinen, K., Hakanen, J., Podkopaev, D., Interactive Nonlinear Multiobjective Optimization Methods, in "Multiple Criteria Decision Analysis: State of the Art Surveys", Ed. by Greco, S., Ehrgott, M., Figueira, J., 2. Edition, Springer-Verlag, 931-980, 2016.
An overview of interactive methods for solving nonlinear multiobjective optimization problems is given. In interactive methods, the decision maker progressively provides preference information so that her or his most satisfactory Pareto optimal solution can be found. The basic features of several methods are introduced and some theoretical results are provided. In addition, references to modifications and applications as well as to other methods are indicated. As the role of the decision maker is very important in interactive methods, methods presented are classified according to the type of preference information that the decision maker is assumed to provide.
Ojalehto, V., Podkopaev, D., Miettinen, K., Agent Assisted Interactive Algorithm for Computationally Demanding Multiobjective Optimization Problems, Computers and Chemical Engineering, 77, 105-115, 2015.
We generalize the applicability of interactive methods for solving computationally demanding, that is, time-consuming, multiobjective optimization problems. For this purpose we propose a new agent assisted interactive algorithm. It employs a computationally inexpensive surrogate problem and four different agents that intelligently update the surrogate based on the preferences specified by a decision maker. In this way, we decrease the waiting times imposed on the decision maker during the interactive solution process and at the same time decrease the amount of preference information expected from the decision maker. The agent assisted algorithm is not specific to any interactive method or surrogate problem. As an example we implement our algorithm for the interactive NIMBUS method and the PAINT method for constructing the surrogate. This implementation was applied to support a real decision maker in solving a two-stage separation problem.
Ruiz, A. B., Sindhya, K., Miettinen, K., Ruiz, F., Luque, M., E-NAUTILUS: A Decision Support System for Complex Multiobjective Optimization Problems based on the NAUTILUS Method, European Journal of Operational Research, 246, 218-231, 2015.
Interactive multiobjective optimization methods cannot necessarily be easily used when (industrial) multiobjective optimization problems are involved. There are at least two important factors to be considered with any interactive method: computationally expensive functions and aspects of human behavior. In this paper, we propose a method based on the existing NAUTILUS method and call it the Enhanced NAUTILUS (E-NAUTILUS) method. This method borrows the motivation of NAUTILUS along with the human aspects related to avoiding trading-off and anchoring bias and extends its applicability for computationally expensive multiobjective optimization problems. In the E-NAUTILUS method, a set of Pareto optimal solutions is calculated in a pre-processing stage before the decision maker is involved. When the decision maker interacts with the solution process in the interactive decision making stage, no new optimization problem is solved, thus, avoiding the waiting time for the decision maker to obtain new solutions according to her/his preferences. In this stage, starting from the worst possible objective function values, the decision maker is shown a set of points in the objective space, from which (s)he chooses one as the preferable point. At successive iterations, (s)he always sees points which improve all the objective values achieved by the previously chosen point. In this way, the decision maker remains focused on the solution process, as there is no loss in any objective function value between successive iterations. The last post-processing stage ensures the Pareto optimality of the final solution. A real-life engineering problem is used to demonstrate how E-NAUTILUS works in practice.
Tabatabaei, M., Hakanen, J., Hartikainen, M., Miettinen, K., Sindhya, K., A Survey on Handling Computationally Expensive Multiobjective Optimization Problems using Surrogates: Non-Nature Inspired Methods, Structural and Multidisciplinary Optimization, to appear.
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 with the high computational cost, surrogate-based methods are commonly used in the literature. This paper surveys surrogate-based methods proposed in the literature, where the methods are independent of the underlying optimization algorithm and mitigate the computational burden to capture different types of Pareto frontiers. The methods considered are classied, discussed and then compared. These methods are divided into two frameworks: the sequential and the adaptive frameworks. Based on the comparison, we recommend the adaptive framework to tackle the aforementioned challenges.
Trivino, A., Juutinen, A., Mazziotta, A., Miettinen, K., Podkopaev, D., Reunanen, P., Monkkonen, M., Managing a Boreal Forest Landscape for Providing Timber, Storing and Sequestering Carbon, Ecosystem Services, 14, 179-189, 2015.
Human well-being highly depends on ecosystem services and this dependence is expected to increase in the future with an increasing population and economic growth. Studies that investigate trade-offs between ecosystem services are urgently needed for informing policy-makers. We examine the trade-offs between a provisioning (revenues from timber selling) and regulating (carbon storage and sequestration) ecosystem services among seven alternative forest management regimes in a large boreal forest production landscape. First, we estimate the potential of the landscape to produce harvest revenues and store/sequester carbon across a 50-year time period. Then, we identify conflicts between harvest revenues and carbon storage and sequestration. Finally, we apply multiobjective optimization to find optimal combinations of forest management regimes that maximize harvest revenues and carbon storage/sequestration. Our results show that no management regime alone is able to either maximize harvest revenues or carbon services and that a combination of different regimes is always necessary. We also show that with a relatively little economic investment (5% decrease in harvest revenues), a substantial increase in carbon services could be attained (9% for carbon storage; 15-23% for carbon sequestration). We conclude that it is possible to achieve win-win situations applying diversified forest management planning at a landscape level.
Kangas, A., Hartikainen, M., Miettinen, K., Simultaneous Optimization of Harvest Schedule and Data, Canadian Journal of Forest Research, 45(8), 1034-1044, 2015.
In many recent studies, the value of forest inventory information (VOI) in the harvest scheduling has been examined. In a previous paper, we demonstrated that making measurement decisions for stands where 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 the data quality into the decision problem, i.e., the decision maker can select between making imperfect or perfect measurements. The imperfect information is obtained with a specific scenario tree formulation. Our decision problem includes three types of decisions: harvest decisions, measurement decisions, and decisions about measurement quality. In addition, the timing of the harvests and measurements must be decided. These decisions are evaluated based on two objectives: discounted aggregate incomes for the planning periods and the end value of the forest at the end of the planning horizon. Solving the biobjective optimization problem formed using the epsilon-constraint method showed that imperfect information was mostly sufficient for the harvest timing decisions during the planning horizon but perfect information was required in order to meet the end value constraint. The relative importance of the two objectives affects the measurements indirectly, through increasing or decreasing the number of certain decisions (i.e. situations where the optimal decision is identical in all scenarios).
Bokrantz, R., Miettinen, K., Projections onto the Pareto Surface in Multicriteria Radiation Therapy Optimization, Medical Physics, 42(10), 5862-5870, 2015.
Purpose: To eliminate or reduce the error to Pareto optimality that arises in Pareto surface navigation when the Pareto surface is approximated by a small number of plans. Methods: We propose to project the navigated plan onto the Pareto surface as a post-processing step to the navigation. The projection attempts to find a Pareto optimal plan that is at least as good as or better than the initial navigated plan with respect to all objective functions. An augmented form of projection is also suggested where dose-volume histogram constraints are used to prevent that the projection causes a violation of some clinical goal. The projections were evaluated with respect to planning for intensity modulated radiation therapy delivered by step-and-shoot and sliding window, and spot-scanned intensity modulated proton therapy. Retrospective plans were generated for a prostate and a head and neck case. Results: The projections led to improved dose conformity and better sparin of organs at risk (OARs) for all three delivery techniques and both patient cases. The mean dose to OARs decreased by 3.1 Gy on average for the unconstrained form of the projection and by 2.0 Gy on average when dose-volume histrogram constraints were used. No consistent improvements in target homogeneity were observed. Conclusions: There are situations when Pareto navigation leaves room for improvement in OAR sparing and dose conformity, for example if the approximation of the Pareto surface is coarse or the problem formulation has too permissive constraints. A projection onto the Pareto surface can identify an inaccurate Pareto surface representation and, if necessary, improve the quality of the navigated plan.
Miettinen, K., Podkopaev, D., Ruiz, F., Luque, M., A New Preference Handling Technique for Interactive Multiobjective Optimization without Trading-off, Journal of Global Optimization, 63(4), 633-652, 2015.
Because the purpose of multiobjective optimization methods is to optimize conflicting objectives simultaneously, they mainly focus on Pareto optimal solutions, where improvement with respect to some objective is only possible by allowing some other objective(s) to impair. Bringing this idea into practice requires the decision maker to think in terms of trading-off, which may limit the ability of effective problem solving. We outline some drawbacks of this and exploit another idea emphasizing the possibility of simultaneous improvement of all objectives. Based on this idea, we propose a technique for handling decision maker's preferences, which eliminates the necessity to think in terms of trade-offs. We incorporate this technique into an interactive trade-off-free method for multiobjective optimization. We call the resulting method NAUTILUS 2, which is also suitable for negotiation support. We demonstrate the applicability of the new method with an example problem.
Sindhya, K., Ojalehto, V., Savolainen, J., Niemistö, H., Hakanen, J., Miettinen, K., Coupling Dynamic Simulation and Interactive Multiobjective Optimization for Complex Problems: An APROS-NIMBUS Case Study, Expert Systems with Applications, 41(5), 2546-2558, 2014.
Dynamic process simulators for plant-wide process simulation and multiobjective optimization tools can be used by industries as a means to cut costs and enhance profitability. Specifically, dynamic process simulators are useful in the process plant design phase, as they provide several benefits such as savings in time and costs. On the other hand, multiobjective optimization tools are useful in obtaining the best possible process designs when multiple conflicting objectives are to be optimized simultaneously. Here we concentrate on interactive multiobjective optimization. When multiobjective optimization methods are used in process design, they need an access to dynamic process simulators, hence it is desirable for them to coexist on the same software platform. However, such a co-existence is not common. Hence, users need to couple multiobjective optimization software and simulators, which may not be trivial. In this paper, we consider APROS, a dynamic process simulator and couple it with IND-NIMBUS, an interactive multiobjective optimization software. Specifically, we: a) study the coupling of interactive multiobjective optimization with a dynamic process simulator; b) bring out the importance of utilizing interactive multiobjective optimization; c) propose an augmented interactive multiobjective optimization algorithm; and d) apply an APROS-NIMBUS coupling for solving a dynamic optimization problem in a two-stage separation process.
Steponavice, I., Ruuska, S., Miettinen, K., A Solution Process for Simulation-based Multiobjective Design Optimization with an Application in Paper Industry, Computer-Aided Design, 47, 45-58, 2014.
In this paper, we address some computational challenges arising in complex simulation-based design optimization problems. High computational cost, black-box formulation and stochasticity are some of the challenges related to optimization of design problems involving simulation of complex mathematical models. Solving becomes even more challenging in case of multiple conflicting objectives that must be optimized simultaneously. In such cases, application of multiobjective optimization methods is necessary in order to gain an understanding of which design offers the best possible trade-off. We apply a three-stage solution process to meet the challenges mentioned above. As our case study, we consider the integrated design and control problem in paper mill design where the aim is to decrease the investment cost and enhance the quality of paper on the design level and, at the same time, guarantee the smooth performance of the production system on the operational level. In the first stage of the three-stage solution process, a set of solutions involving different trade-offs is generated with a method suited for computationally expensive multiobjective optimization problems using parallel computing. Then, based on the generated solutions an approximation method is applied to create a computationally inexpensive surrogate problem for the design problem and the surrogate problem is solved in the second stage with an interactive multiobjective optimization method. This stage involves a decision maker and her/his preferences to find the most preferred solution to the surrogate problem. In the third stage, the solution best corresponding that of stage two is found for the original problem.
Miettinen, K., Survey of Methods to Visualize Alternatives in Multiple Criteria Decision Making Problems, OR Spectrum, 36(1) 3-37, 2014.
When solving decision problems where multiple conflicting criteria are to be considered simultaneously, decision makers must compare several different alternatives and select the most preferred one. The task of comparing multidimensional vectors is very demanding for the decision maker without any support. Different graphical visualization tools can be used in order to support and help the decision maker in understanding similarities and differences between the alternatives and graphical illustration is a very important part of decision support systems that are used in solving multiple criteria decision making problems. The visualization task is by no means trivial because, on the one hand, the graphics must be easy to comprehend and not too much information should be lost but, on the other hand, no extra unintentional information should be included. In this paper, we survey and analyze different ways of visualizing a small set of discrete alternatives graphically in the context of multiple criteria decision making. Some of the ways discussed are widely used and some others deserve to be brought into a wider awareness. This survey provides a starting point for all those who deal with multiple criteria decision making problems and need information of what kind of visualization techniques could be put to use in order to support the decision maker better.
Miettinen, K., Mustajoki, J., Stewart T.J., Interactive Multiobjective Optimization with NIMBUS for Decision Making under Uncertainty, OR Spectrum, 36(1), 39-56, 2014.
We propose an interactive method for decision making under uncertainty, where uncertainty is related to the lack of understanding about consequences of actions. Such situations are typical, for example, in design problems, where a decision maker has to make a decision about a design at a certain moment of time even though the actual consequences of this decision can be possibly seen only many years later. To overcome the difficulty of predicting future events when no probabilities of events are available, our method utilizes groupings of objectives or scenarios to capture different types of future events. Each scenario is modeled as a multiobjective optimization problem to represent different and conflicting objectives associated with the scenarios. We utilize the interactive classication-based multiobjective optimization method NIMBUS for assessing the relative optimality of the current solution in different scenarios. This information can be utilized when considering the next step of the overall solution process. Decision making is performed by giving special attention to individual scenarios. We demonstrate our method with an example in portfolio optimization.
Mönkkönen, M., Juutinen, A., Mazziota, A., Miettinen, K., Podkopaev, D., Reunanen, P., Salminen, H., Tikkanen, O.-P., Spatially Dynamic Forest Management to Sustain Biodiversity and Economic Returns, Journal of Environmental Management, 134, 80-89, 2014.
Production of marketed commodities and protection of biodiversity in natural systems often conflict and thus the continuously expanding human needs for more goods and benefits from global ecosystems urgently calls for strategies to resolve this conflict. In this paper, we address what is the potential of a forest landscape to simultaneously produce habitats for species and economic returns, and how the conflict between habitat availability and timber production varies among taxa. Secondly, we aimed at revealing an optimal combination of management regimes that maximizes habitat availability for given levels of economic returns. We used multiobjective optimization tools to analyze data from a boreal forest landscapes consisting of 30000 forests stands simulated 50 years into future. We included seven alternative management regimes, spanning from the recommended intensive forest management regime to complete set-aside of stands (protection), and ten different taxa representing a wide variety of habitat associations and social values. Our results demonstrate that it is possible to achieve large improvements in habitat availability at relatively little losses in economic returns. In general, providing dead-wood associated species with more habitats tended to be more expensive than providing requirements for other species. No management regime alone maximized habitat availability for the species, and systematic use of any single management regime also resulted in considerable reductions in economic returns. Compared with an optimal combination of management regimes that maximize economic returns, a consistent application of the recommended management regime was associated with a 5% reduction in economic returns and up to 375% reduction in habitat availability. Thus, for all taxa a combination of management regimes was required to achieve the optimum. Refraining from silvicultural thinnings on a proportion of stands should be considered as a cost-effective management in commercial forests to reconcile the conflict between economic returns and habitat availability particularly for species associated with dead-wood. In general, a viable strategy to maintain biodiversity in production landscapes would be to diversify management regimes. Our results also emphasize the importance of careful landscape level forest management planning because optimal combinations of management regimes were taxon-specific. For cost-efficiency, the results call for balanced and correctly targeted strategies among habitat types.
Ojalehto, V., Miettinen, K., Laukkanen, T., Implementation Aspects of Interactive Multiobjective Optimization for Modeling Environments: The Case of GAMS-NIMBUS , Computational Optimization and Applications, 58(3), 757-779, 2014.
Interactive multiobjective optimization methods have provided promising results in the literature but still their implementations are rare. Here we introduce a core structure of interactive methods to enable their convenient implementation. We also demonstrate how this core structure can be applied when implementing an interactive method using a modeling environment. Many modeling environments contain tools for single objective optimization but not for interactive multiobjective optimization. Furthermore, as a concrete example, we present GAMS-NIMBUS Tool which is an implemen- tation of the classification-based NIMBUS method for the GAMS modeling environment. So far, interactive methods have not been available in the GAMS environment, but with the GAMS-NIMBUS Tool we open up the possibility of solving multiobjective optimization problems modeled in the GAMS modeling environment. Finally, we give some examples of the benefits of applying an interactive method by using the GAMS-NIMBUS Tool for solving multiobjective optimization problems modeled in the GAMS environment.
Kangas, A., Hartikainen, M., Miettinen, K., Simultaneous Optimization of Harvest Schedule and Measurement Strategy, Scandinavian Journal of Forest Research, 29(Sup 1), 224-233, 2014.
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.
Tarkkanen, S., Miettinen, K., Hakanen, J., Isomäki, H., Incremental User-Interface Development for Interactive Multiobjective Optimization, Expert Systems with Applications, 40, 3220-3232, 2013.
Interactive multiobjective optimization (IMO) is a subfield of multiple criteria decision making. In multiobjective optimization, the optimization problem is formulated with a mathematical model containing several conflicting objectives and constraints depending on decision variables. By using IMO methods, a decision maker progressively provides preference information in order to find the most satisfactory compromise between the conflicting objectives. In this paper, we consider implementation challenges of IMO methods. In particular, we consider what kind of interaction techniques can support the decision making process and information exchange between IMO methods and the decision maker. The implementation of an IMO method called Pareto Navigator is used as an example to demonstrate concrete challenges of interaction design. This paper focuses on describing the incremental development of the user interface for Pareto Navigator including empirical validation by user testing evaluation.
Giri, B.K., Hakanen, J., Miettinen, K., Chakraborti, N., Genetic Programming through Bi-objective Genetic Algorithms with Study of a Simulated Moving Bed Process Involving Multiple Objectives, Applied Soft Computing, 13, 2613-2623, 2013.
A new Bi-objective Genetic Programming (BioGP) technique has been developed for meta-modeling and applied in a chromatographic separation process using a Simulated Moving Bed (SMB) process. The BioGP technique initially minimizes training error through a single objective optimization procedure and then a trade-off between complexity and accuracy is worked out through a genetic algorithm based bi-objective optimization strategy. A benefit of the BioGP approach is that an expert user or a decision maker (DM) can flexibly select the mathematical operations involved to construct a meta-model of desired complexity or accuracy. It is also designed to combat bloat - a perennial problem in genetic programming along with overfitting and underfitting problems. In this study the meta-models constructed for SMB reactors were compared with those obtained from an Evolutionary Neural Network (EvoNN) developed earlier and also with a polynomial regression model. Both BioGP and EvoNN were compared for subsequent constrained bi-objective optimization studies for the SMB reactor involving four objectives. The results were also compared with the previous work in the literature. The BioGP technique produced acceptable results and is now ready for data-driven modeling and optimization studies at large.
Hakanen, J., Sahlstedt, K., Miettinen, K., Wastewater Treatment Plant Design and Operation under Multiple Conflicting Objective Functions, Environmental Modelling and Software, 46(1), 240-249, 2013.
Wastewater treatment plant design and operation involve multiple objective functions, which are often in conflict with each other. Traditional optimization tools convert all objective functions to a single objective optimization problem (usually minimization of a total cost function by using weights for the objective functions), hiding the interdependencies between different objective functions. We present an interactive approach that is able to handle multiple objective functions simultaneously. As an illustration of our approach, we consider a case study of plant-wide operational optimization where we apply an interactive optimization tool. In this tool, a commercial wastewater treatment simulation software is combined with an interactive multiobjective optimization software, providing an entirely new approach in wastewater treatment. We compare our approach to a traditional approach by solving the case study also as a single objective optimization problem to demonstrate the advantages of interactive multiobjective optimization in wastewater treatment plant design and operation.
Sindhya, K., Miettinen, K., Deb, K., A Hybrid Framework for Evolutionary Multi-Objective Optimization, IEEE Transactions on Evolutionary Computation, 17(4), 495-511, 2013.
Evolutionary multi-objective optimization algorithms are widely used for solving optimization problems with multiple conflicting objectives. However, basic evolutionary multi-objective optimization algorithms have shortcomings, such as slow convergence to the Pareto optimal front, no efficient termination criterion, and a lack of a theoretical convergence proof. A hybrid evolutionary multi-objective optimization algorithm involving a local search module is often used to overcome these shortcomings. But, there are many issues that affect the performance of hybrid evolutionary multi-objective optimization algorithms, such as the type of scalarization function used in a local search, frequency of a local search, etc. In this paper, we address some of these issues and propose a hybrid evolutionary multi-objective optimization framework. The proposed hybrid evolutionary multi-objective optimization framework has a modular structure, which can be used for implementing a hybrid evolutionary multi-objective optimization algorithm. We present a sample implementation of this framework considering NSGA-II, MOEA/D and MOEA/D-DRA as evolutionary multi-objective optimization algorithms. Here, we use a gradient based sequential quadratic programming method as a single objective optimization method for solving a scalarizing function used in a local search. Hence, only continuously differentiable functions were considered for numerical experiments. The numerical experiments demonstrate the usefulness of our proposed framework.
Luque, M., Miettinen, K., Ruiz, A.B., Ruiz, F., A Two-Slope Achievement Scalarizing Function for Interactive Multiobjective Optimization, Computers & Operations Research, 39(7), 1673-1681, 2012.
The use of achievement (scalarizing) functions in interactive multiobjective optimization methods is very popular, as indicated by the large number of algorithmic and applied scientific papers that use this approach. Key parameters in this approach are the reference point, which expresses desirable objective function values for the decision maker, and weights. The role of the weights can range from purely normalizing to fully preferential parameters that indicate the relative importance given by the decision maker to the achievement of each reference value. Technically, the influence of the weights in the solution generated by the achievement scalarizing function is different, depending on whether the reference point is achievable or not. Besides, from a psychological point of view, decision makers also react in a different way, depending on the achievability of the reference point. For this reason, in this work, we introduce the formulation of a new achievement scalarizing function with two different weight vectors, one for achievable reference points, and the other one for unachievable reference points. The new achievement scalarizing function is designed so that an appropriate weight vector is used in each case, without having to carry out any a priori achievability test. It allows us to reflect the decision maker’s preferences in a better way as a part of an interactive solution method, and this can cause a quicker convergence of the method. The computational efficiency of this new formulation is shown in several test examples using different reference points.
Nikulin, Y., Miettinen, K., Mäkelä, M.M., A New Achievement Scalarizing Function based on Parametrization in Multiobjective Optimization, OR Spectrum, 34(1), 69-87, 2012.
This paper addresses a general multiobjective optimization problem. One of the most widely used methods of dealing with multiple conflicting objectives consists of constructing and optimizing a so-called achievement scalarizing function (ASF) which has an ability to produce any Pareto optimal or weakly/properly Pareto optimal solution. The ASF minimizes the distance from the reference point to the feasible region, if the reference point is unattainable, or maximizes the distance otherwise. The distance is defined by means of some specific kind of a metric introduced in the objective space. The reference point is usually specified by a decision maker and contains her/his aspirations about desirable objective values. The classical approach to constructing an ASF is based on using the Chebyshev metri. Another possibility is to use an additive ASF based on a modified linear metric L_1. In this paper, we propose a parameterized version of an ASF. We introduce an integer parameter in order to control the degree of metric flexibility varying from L 1 to L 8. We prove that the parameterized ASF supports all the Pareto optimal solutions. Moreover, we specify conditions under which the Pareto optimality of each solution is guaranteed. An illustrative example for the case of three objectives and comparative analysis of parameterized ASFs with different values of the parameter are given. We show that the parameterized ASF provides the decision maker with flexible and advanced tools to detect Pareto optimal points, especially those whose detection with other ASFs is not straightforward since it may require changing essentially the reference point or weighting coefficients as well as some other extra computational efforts.
Ruuska, S., Miettinen, K., Wiecek, M.M., Connections between Single-level and Bilevel Multiobjective Optimization, Journal of Optimization Theory and Applications, 153(1), 60-74, 2012.
The relationship between bilevel optimization and multiobjective optimization has been studied by several authors and there have been repeated attempts to establish a link between the two. We unify the results from the literature and generalize them for bilevel multiobjective optimization. We formulate sufficient conditions for an arbitrary binary relation to guarantee equality between the efficient set produced by the relation and the set of optimal solutions to a bilevel problem. In addition, we present specially structured bilevel multiobjective optimization problems motivated by real-life applications and an accompanying binary relation permitting their reduction to single-level multiobjective optimization problems.
Ruiz, F., Luque, M., Miettinen, K., Improving the Computational Efficiency of a Global Formulation (GLIDE) for Interactive Multiobjective Optimization, Annals of Operations Research, 197(1), 47-70, 2012.
In this paper, we present a new general formulation for multiobjective optimization that can accommodate several interactive methods of different types (regarding various types of preference information required from the decision maker). This formulation provides a comfortable implementation framework for a general interactive system and allows the decision maker to conveniently apply several interactive methods in one solution process. In other words, the decision maker can at each iteration of the solution process choose how to give preference information to direct the interactive solution process, and the formulation enables changing the type of preferences, that is, the method used, whenever desired. The first general formulation, GLIDE, included eight interactive methods utilizing four types of preferences. Here we present an improved version where we pay special attention to the computational efficiency (especially significant for large and complex problems), by eliminating some constraints and parameters of the original formulation. To be more specific, we propose two new formulations, depending on whether the multiobjective optimization problem to be considered is differentiable or not. Some computational tests are reported showing improvements in all cases. The generality of the new improved formulations is supported by the fact that they can accommodate six interactive methods more, that is, a total of fourteen interactive methods, just by adjusting parameter values.
Hartikainen, M., Miettinen, K., Wiecek, M.M., PAINT: Pareto Front Interpolation for Nonlinear Multiobjective Optimization, Computational Optimization and Applications, 52(3), 845-867, 2012.
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.
Laukkanen, T., Tveit, T.-M., Ojalehto, V., Miettinen, K., Fogelholm, C.-J., Bilevel Heat Exchanger Network Synthesis with an Interactive Multi-Objective Optimization Method, Applied Thermal Engineering, 48(15), 301-316, 2012.
Heat exchanger network synthesis (HENS) has been an active research area for more than 40 years because well-designed heat exchanger networks enable heat recovery in process industries in an energy- and cost-efficient manner. Due to ever increasing global competition and need to decrease the harmful effects done on the environment, there still is a continuous need to improve the heat exchanger networks and their synthesizing methods. In this work we present a HENS method that combines an interactive multiobjective optimization method with a simultaneous bilevel HENS method, where the bilevel part of the method is based on grouping of process streams and building aggregate streams from the grouped streams. This is done in order to solve medium-sized industrial HENS problems efficiently with good final solutions. The combined method provides an opportunity to solve HENS problems efficiently also regarding computing effort and at the same time optimizing all the objectives of HENS simultaneously and in a genuine multiobjective manner without using weighting factors. This enables the designer or decision maker to be in charge of the design procedure and guide the search into areas that the decision maker is most interested in. Two examples are solved with the proposed method. The purpose of the first example is to help in illustrating the steps in the overall method. The second example is obtained from the literature.
Eskelinen, P., Miettinen, K., Trade-off Analysis Approach for Interactive Nonlinear Multiobjective Optimization, OR Spectrum, 34(4), 803-816, 2012.
When solving multiobjective optimization problems, there is typically a decision maker (DM) who is responsible for determining the most preferred Pareto optimal solution based on his preferences. To gain confidence that the decisions to be made are the right ones for the DM, it is important to understand the trade-offs related to different Pareto optimal solutions. We first propose a trade-off analysis approach that can be connected to various multiobjective optimization methods utilizing a certain type of scalarization to produce Pareto optimal solutions. With this approach, the DM can conveniently learn about local trade-offs between the conflicting objectives and judge whether they are acceptable. The approach is based on an idea where the DM is able to make small changes in the components of a selected Pareto optimal objective vector. The resulting vector is treated as a reference point which is then projected to the tangent hyperplane of the Pareto optimal set located at the Pareto optimal solution selected. The obtained approximate Pareto optimal solutions can be used to study trade-off information. The approach is especially useful when trade-off analysis must be carried out without increasing computation workload. We demonstrate the usage of the approach through an academic example problem.
Tveit, T.-M., Laukkanen, T., Ojalehto, V., Miettinen, K., Fogelholm, C.-J., Interactive Multi-objective Optimisation of Configurations for an Oxyfuel Power Plant Process for CO2 Capture, Chemical Engineering Transactions, 29, 433-438, 2012.
In this work we present a multi-objective approach to optimising configurations of an oxyfuel power plant process. The approach solves an optimisation model based on simulation results using an interactive multi-objective optimisation method, NIMBUS, which is implemented in GAMS. The optimisation model of the oxyfuel power plant process is based on simulation models of six different configurations. The simulation model is used to generate regression models for each objective, by varying the free variables that are being studied. This is done to reduce the complexity of the optimisation model. The direct results from this study suggest that it is possible to design a coal-fired power plant using an oxyfuel process, with thermal efficiency, amount of liquefied carbon dioxide and heat exchanger areas close to the ideal values. Another important result from this study is that the integration of NIMBUS and GAMS, called GAMS-NIMBUS tool, is a powerful tool to study complex processes.
Luque, M., Ruiz, F., Miettinen, K., Global Formulation for Interactive Multiobjective Optimization, OR Spectrum, 33(1), 27-48, 2011.
Interactive methods are useful and realistic multiobjective optimization techniques and, thus, many such methods exist. However, they have two important drawbacks when using them in real applications. Firstly, the question of which method should be chosen is not trivial. Secondly, there are rather few practical implementations of the methods. We introduce a general formulation that can accommodate several interactive methods. This provides a comfortable implementation framework for a general interactive system. Besides, this implementation allows the decision maker to choose how to give preference information to the system, and enables changing it anytime during the solution process. This change-of-method option provides a very flexible framework for the decision maker.
Hartikainen, M., Miettinen, K., Wiecek, M.M., Constructing a Pareto Front Approximation for Decision Making, Mathematical Methods of Operations Reserach, 73(2), 209-234, 2011.
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.
Hakanen, J., Miettinen, K., Sahlstedt K., Wastewater Treatment: New Insight Provided by Interactive Multiobjective Optimization, Decision Support Systems, 51, 328-337, 2011.
In this paper, we describe a new interactive tool developed for wastewater treatment plant design. The tool is aimed at supporting the designer in designing new wastewater treatment plants as well as optimizing the performance of already available plants. The idea is to utilize interactive multiobjective optimization which enables the designer to consider the design with respect to several conflicting evaluation criteria simultaneously. This is more important than ever because the requirements for wastewater treatment plants are getting tighter and tighter from both environmental and economical reasons. By combining a process simulator to simulate wastewater treatment and an interactive multiobjective optimization software to aid the designer during the design process, we obtain a practically useful tool for decision support. The applicability of our tool is illustrated with a case study related to municipal wastewater treatment where three conflicting evaluation criteria are considered.
Sindhya, K., Miettinen, K., New Perspective to Continuous Casting of Steel with a Hybrid Evolutionary Multiobjective Algorithm, Materials and Manufacturing Processes, 26(3), 481-492, 2011.
In this article, we present a new perspective in solving computationally demanding problems such as the optimal control of the continuous casting of steel. We consider a multiobjective formulation of the optimal control of the surface temperature of the steel strand with five objectives, where constraint violations are minimized as objectives because no feasible solutions exist otherwise. A hybrid evolutionary multiobjective algorithm (HNSGA-II) is used to overcome discrepancies of evolutionary multiobjective optimization algorithms such as slow convergence and lack of convergence proof to the Pareto front. HNSGA-II uses NSGA-II as an underlying evolutionary multiobjective algorithm and an achievement scalarization function to scalarize objective functions for local search, which improves the population and speeds up the convergence. This is important because most evolutionary multiobjective algorithms are known to have difficulties with five objective functions. The algorithm is used to generate a set of Pareto solutions showing different trade-offs. However, it is difficult to generate Pareto solutions showing all the different trade-offs with a finite small population size. Hence, we use the preference information related to constraint violations in the weights of the achievement scalarization function to generate solutions with different trade-offs in preferable regions of the Pareto front. In addition, we use clustering and present typical solutions of different clusters so that the most preferred solution to be implemented can easily be identified. We demonstrate the approach and compare the results to previous studies.
Sindhya, K., Ruuska, S., Haanpää, T., Miettinen, K., A New Hybrid Mutation Operator for Multiobjective Optimization with Differential Evolution, Soft Computing, 15(10), 2041-2055, 2011.
Differential evolution has become one of the most widely used evolutionary algorithms in multiobjective optimization. Its linear mutation operator is a simple and powerful mechanism to generate trial vectors. However, the performance of the mutation operator can be improved by including a nonlinear part. In this paper, we propose a new hybrid mutation operator consisting of a polynomial-based operator with nonlinear curve tracking capabilities and the differential evolution’s original mutation operator, for the efficient handling of various interdependencies between decision variables. The resulting hybrid operator is straightforward to implement and can be used within most evolutionary algorithms. Particularly, it can be used as a replacement in all algorithms utilizing the original mutation operator of differential evolution. We demonstrate how the new hybrid operator can be used by incorporating it into MOEA/D, a winning evolutionary multiobjective algorithm in a recent competition. The usefulness of the hybrid operator is demonstrated with extensive numerical experiments showing improvements in performance compared with the previous state of the art.
Sindhya, K., Deb, K., Miettinen, K., Improving Convergence of Evolutionary Multi-Objective Optimization with Local Search: A Concurrent-Hybrid Algorithm, Natural Computing, 10(4), 1407-1430, 2011.
A local search method is often introduced in an evolutionary optimization algorithm, to enhance its speed and accuracy of convergence to optimal solutions. In multi-objective optimization problems, the implementation of local search is a non-trivial task, as determining a goal for local search in presence of multiple conflicting objectives becomes a difficult task. In this paper, we borrow a multiple criteria decision making concept of employing a reference point based approach of minimizing an achievement scalarizing function and integrate it as a search operator with a concurrent approach in an evolutionary multi-objective algorithm. Simulation results of the new concurrent-hybrid algorithm on several two to four-objective problems compared to a serial approach, clearly show the importance of local search in aiding a computationally faster and accurate convergence to the Pareto optimal front.
Eskelinen, P., Miettinen, K., Klamroth, K., Hakanen, J., Pareto Navigator for Interactive Nonlinear Multiobjective Optimization, OR Spectrum, 23, 211-227, 2010.
We describe a new interactive learning-oriented method called Pareto navigator for nonlinear multiobjective optimization. In the method, first a polyhedral approximation of the Pareto optimal set is formed in the objective function space using a relatively small set of Pareto optimal solutions representing the Pareto optimal set. Then the decision maker can navigate around the polyhedral approximation and direct the search for promising regions where the most preferred solution could be located. In this way, the decision maker can learn about the interdependencies between the conflicting objectives and possibly adjust one’s preferences. Once an interesting region has been identified, the polyhedral approximation can be made more accurate in that region or the decision maker can ask for the closest counterpart in the actual Pareto optimal set. If desired, (s)he can continue with another interactive method from the solution obtained. Pareto navigator can be seen as a nonlinear extension of the linear Pareto race method. After the representative set of Pareto optimal solutions has been generated, Pareto navigator is computationally efficient because the computations are performed in the polyhedral approximation and for that reason function evaluations of the actual objective functions are not needed. Thus, the method is well suited especially for problems with computationally costly functions. Furthermore, thanks to the visualization technique used, the method is applicable also for problems with three or more objective functions, and in fact it is best suited for such problems. After introducing the method in more detail, we illustrate it and the underlying ideas with an example.
Miettinen, K., Eskelinen, P., Ruiz, F., Luque, M., NAUTILUS Method: An Interactive Technique in Multiobjective Optimization based on the Nadir Point, European Journal of Operational Research, 206(2), 426-434, 2010.
Most interactive methods developed for solving multiobjective optimization problems sequentially generate Pareto optimal or nondominated vectors and the decision maker must always allow impairment in at least one objective function to get a new solution. The NAUTILUS method proposed is based on the assumptions that past experiences affect decision makers’ hopes and that people do not react symmetrically to gains and losses. Therefore, some decision makers may prefer to start from the worst possible objective values and to improve every objective step by step according to their preferences. In NAUTILUS, starting from the nadir point, a solution is obtained at each iteration which dominates the previous one. Although only the last solution will be Pareto optimal, the decision maker never looses sight of the Pareto optimal set, and the search is oriented so that (s)he progressively focusses on the preferred part of the Pareto optimal set. Each new solution is obtained by minimizing an achievement scalarizing function including preferences about desired improvements in objective function values. NAUTILUS is specially suitable for avoiding undesired anchoring effects, for example in negotiation support problems, or just as a means of finding an initial Pareto optimal solution for any interactive procedure. An illustrative example demonstrates how this new method iterates.
Laukkanen, T., Tveit, T.-M., Ojalehto, V., Miettinen, K., Fogelholm C.-J., An Interactive Multi-Objective Approach to Heat Exchanger Network Synthesis, Computers and Chemical Engineering, 34(6), 943-952, 2010.
In this work we present a multi-objective approach to heat exchanger network synthesis. The approach solves a modified version of the Synheat model using an interactive multi-objective optimisation method, NIMBUS, which is implemented in GAMS. The results obtained demonstrate the potential of interactive multi-objective optimisation.
Ruotsalainen, H., Miettinen, K., Palmgren, J.-E., Lahtinen, T., Interactive Multiobjective Optimization for Anatomy based Three-Dimensional HDR Brachytherapy, Physics in Medicine and Biology, 55(16), 4703-4719, 2010.
In this paper, we present an anatomy-based three-dimensional dose optimization approach for HDR brachytherapy using interactive multiobjective optimization (IMOO). In brachytherapy, the goals are to irradiate a tumor without causing damage to healthy tissue. These goals are often conflicting, i.e. when one target is optimized the other will suffer, and the solution is a compromise between them. IMOO is capable of handling multiple and strongly conflicting objectives in a convenient way. With the IMOO approach, a treatment planner’s knowledge is used to direct the optimization process. Thus, the weaknesses of widely used optimization techniques (e.g. defining weights, computational burden and trial-and-error planning) can be avoided, planning times can be shortened and the number of solutions to be calculated is small. Further, plan quality can be improved by finding advantageous trade-offs between the solutions. In addition, our approach offers an easy way to navigate among the obtained Pareto optimal solutions (i.e. different treatment plans). When considering a simulation model of clinical 3D HDR brachytherapy, the number of variables is significantly smaller compared to IMRT, for example. Thus, when solving the model, the CPU time is relatively short. This makes it possible to exploit IMOO to solve a 3D HDR brachytherapy optimization problem. To demonstrate the advantages of IMOO, two clinical examples of optimizing a gynecologic cervix cancer treatment plan are presented.
Aittokoski, T., Miettinen, K., Efficient Evolutionary Approach to Approximate the Pareto Optimal Set in Multiobjective Optimization, UPS-EMOA, Optimization Methods and Software, 25(6), 841-858, 2010.
Solving real-life engineering problems requires often multiobjective, global, and efficient (in terms of objective function evaluations) treatment. In this study, we consider problems of this type by discussing some drawbacks of the current methods and then introduce a new population-based multiobjective optimization algorithm UPS-EMOA which produces a dense (not limited to the population size) approximation of the Pareto-optimal set in a computationally effective manner.
Deb, K., Miettinen, K., Chaudhuri S., Towards an Estimation of Nadir Objective Vector Using a Hybrid of Evolutionary and Local Search Approaches, IEEE Transactions on Evolutionary Computation, 14(6), 821-841, 2010.
Nadir objective vector is constructed with the worst Pareto-optimal objective values in a multi-objective optimization problem and is an important entity to compute because of its importance in estimating the range of objective values in the Pareto-optimal front and also in using many interactive multi- objective optimization techniques. It is needed, for example, for normalizing purposes. The task of estimating the nadir objec- tive vector necessitates information about the complete Pareto- optimal front and is reported to be a difficult task using other approaches. In this paper, we propose certain modifications to an existing evolutionary multi-objective optimization procedure to focus its search towards the extreme objective values and combine it with a reference-point based local search approach to constitute a couple of hybrid procedures for a reliable estimation of the nadir objective vector. With up to 20-objective optimization test problems and on a three-objective engineering design optimization problem, the proposed procedures are found to be capable of finding a near nadir objective vector reliabl y. The study clearly shows the significance of an evolutionary comp uting based search procedure in assisting to solve an age-old important task of nadir objective vector estimation.
Luque, M., Miettinen, K., Eskelinen, P., Ruiz, F., Incorporating Preference Information in Interactive Reference Point Methods for Multiobjective Optimization, Omega, 37(2), 450-462, 2009.
In this paper, we introduce new ways of utilizing preference information specified by the decision maker in interactive reference point based methods. A reference point consists of desirable values for each objective function. The idea is to take the desires of the decision maker into account more closely when projecting the reference point onto the set of nondominated solutions. In this way we can support the decision maker in finding the most satisfactory solutions faster. In practice, we adjust the weights in the achievement scalarizing function that projects the reference point. We identify different cases depending on the amount of additional information available and demonstrate the cases with examples. Finally, we summarize results of extensive computational tests that give evidence of the efficiency of the ideas proposed.
Miettinen, K., Molina, J., Gonzalez, M. Hernandez-Diaz, A., Caballero, R, Using Box Indices in Supporting Comparison in Multiobjective Optimization, European Journal of Operational Research, 197, 17-24, 2009.
Because of the conflicting nature of criteria or objectives, solving a multiobjective optimization problem typically requires interaction with a decision maker who can specify preference information related to the objectives in the problem in question. Due to the difficulties of dealing with multiple objectives, the way information is presented plays a very important role. Questions posed to the decision maker must be simple enough and information shown must be easy to understand. For this purpose, visualization and graphical representations can be useful and constitute one of the main tools used in the literature. In this paper, we propose to use box indices to represent information related to different solution alternatives of multiobjective optimization problems involving at least three objectives. Box indices are an intelligible and easy to handle way to represent data. They are based on evaluating the solutions in a natural and rough enough scale in order to let the decision maker easily recognize the main characteristics of a solution at a glance and to facilitate comparison of two or more solutions in an easily understandable way.
Aittokoski, T., Äyrämö, S., Miettinen, K., Clustering Aided Approach for Decision Making in Computationally Expensive Multiobjective Optimization, Optimization Methods and Software, 24(2), 157-174, 2009.
Typically, industrial optimization problems need to be solved in an efficient, multiobjective and global manner, because they are often computationally expensive (as function values are typically based on simulations), they may contain multiple conflicting objectives, and they may have several local optima. Solving such problems may be challenging and time consuming when the aim is to find the most preferred Pareto optimal solution. In this study, we propose a method where we use an advanced clustering technique to reveal essential characteristics of the approximation of the Pareto optimal set, which has been generated beforehand. Thus, the decision maker (DM) is involved only after the most time consuming computation is finished. After the initiation phase, a moderate number of cluster prototypes projected to the Pareto optimal set is presented to the DM to be studied. This allows him/her to rapidly gain an overall understanding of the main characteristics of the problem without placing too much cognitive load on the DM. Furthermore, we also suggest some ways of applying our approach to different types of problems and demonstrate it with an example related to internal combustion engine design.
Thiele, L., Miettinen, K., Korhonen, P. Molina, J., A Preference-Based Evolutionary Algorithm for Multi-Objective Optimization, Evolutionary Computation, 17(3), 411-436, 2009.
In this paper, we discuss the idea of incorporating preference information into evolutionary multi-objective optimization and propose a preference-based evolutionary approach that can be used as an integral part of an interactive algorithm. One algorithm is proposed in the paper. At each iteration, the decision maker is asked to give preference information in terms of his or her reference point consisting of desirable aspiration levels for objective functions. The information is used in an evolutionary algorithm to generate a new population by combining the fitness function and an achievement scalarizing function. In multi-objective optimization, achievement scalarizing functions are widely used to project a given reference point into the Pareto optimal set. In our approach, the next population is thus more concentrated in the area where more preferred alternatives are assumed to lie and the whole Pareto optimal set does not have to be generated with equal accuracy. The approach is demonstrated by numerical examples.
Ruotsalainen, H., Boman, E., Miettinen, K., Tervo, J., Nonlinear Interactive Multiobjective Optimization Method for Radiotherapy Treatment Planning with Boltzmann Transport Equation, Contemporary Engineering Sciences, 2(9), 391-422, 2009.
In this paper, we present a nonlinear interactive multiobjective optimization method for radiotherapy treatment planning using the Boltzmann transport equation (BTE) in dose calculation. In radiotherapy, the goals are to destroy a tumor with radiation without causing damage to healthy tissue. These goals are conflicting, i.e. when one target is optimized the other will suffer, and the solution is a compromise between them. Our interactive approach is capable of handling multiple and strongly conflicting objectives in a convenient way, and thus the weaknesses of widely used optimization techniques in the field (e.g. defining weights and trial and error planning) can be avoided. In this paper, we use a parameterization technique to make the dose calculation faster in the BTE model. With our approach, the number of solutions to be calculated is rather small but still informative, planning times can be shortened and plan quality improved by finding only feasible solutions and advantageous trade-offs. Importantly, we used the radiotherapy expert’s knowledge to direct the solution process. To demonstrate the advantages, we compare the results with those of the commonly used penalty function method.
Yevseyeva, I., Miettinen, K., Räsänen, P., Verbal Ordinal Classification with Multicriteria Decision Aiding, European Journal of Operational Research, 185(3), 964-983, 2008.
Professionals in neuropsychology usually perform diagnoses of patients behaviour in a verbal rather than in a numerical form. This fact generates interest in decision support systems that process verbal data. It also motivates us to develop methods for the classification of such data. In this paper, we describe ways of aiding classification of a discrete set of objects, evaluated on set of criteria that may have verbal estimations, into ordered decision classes. In some situations, there is no explicit additional information available, while in others it is possible to order the criteria lexicographically. We consider both of these cases. The proposed Dichotomic Classification (DC) method is based on the principles of Verbal Decision Analysis (VDA). Verbal Decision Analysis methods are especially helpful when verbal data, in criteria values, are to be handled. When compared to the previously developed Verbal Decision Analysis classification methods, Dichotomic Classification method performs better on the same data sets and is able to cope with larger sizes of the object sets to be classified. We present an interactive classification procedure, estimate the effectiveness and computational complexity of the new method and compare it to one of the previously developed Verbal Decision Analysis methods. The developed and studied methods are implemented in the framework of a decision support system, and the results of testing on artificial sets of data are reported.
Klamroth, K., Miettinen, K., Integrating Approximation and Interactive Decision Making in Multicriteria Optimization, Operations Research, 56(1), 222-234, 2008.
We present a new interactive hybrid approach for solving multicriteria optimization problems where features of approximation methods and interactive approaches are incorporated. We produce rough approximations of the nondominated set and let the decision maker indicate with the help of reference points where to refine the approximation. In this way, (s)he iteratively directs the search toward the best nondominated solution. After the decision maker has identified the most interesting region of the nondominated set, the final solution can be fine-tuned with existing interactive methods. We suggest different ways of updating the reference point as well as discuss visualizations that can be used in comparing different nondominated solutions. The new method is computationally relatively inexpensive and easy to use for the decision maker.
Saariluoma, P., Kaario, K., Miettinen, K., Mäkelä, M.M., Mental Contents in Interacting with a Multiobjective Optimization Program, International Journal of Technology and Human Interaction, 4(3), 43-67, 2008.
User psychology aims at understanding human-machine interaction from a psychological point of view. Its ultimate goal is to provide knowledge about human psychological properties for interaction designers. In this article, we are particularly interested in applying the theoretical concepts of mental contents (i.e., the information contents of users mental representations), in studying interaction with professional software. The immediate motivation for adopting such an approach arises from problems met in designing interaction processes in multiobjective optimization software. These types of software are meant to support complex thought and decision-making processes and this is why interaction design has to meet a number of theoretical problems, which do seldom arise in simpler interaction types. In four experiments, we investigated the types of knowledge required when developing ones capability of working with such a thought tool. We observed that in interaction design it is essential to take into account domain, professional and software specific mental content structures.
Aittokoski, T., Miettinen, K., Cost Effective Simulation-Based Multiobjective Optimization in Performance of Internal Combustion Engine, Engineering Optimization, 40(7), 593-612, 2008.
Solving real-life engineering problems requires often multiobjective, global, and efficient (in terms of objective function evaluations) treatment. In this study, we consider problems of this type by discussing some drawbacks of the current methods and then introduce a new population-based multiobjective optimization algorithm UPS-EMOA which produces a dense (not limited to the population size) approximation of the Pareto-optimal set in a computationally effective manner.
Haarala, N., Miettinen, K., Mäkelä, M.M., Globally Convergent Limited Memory Bundle Method for Large-Scale Nonsmooth Optimization, Mathematical Programming, 109(1), 181-205, 2007.
Many practical optimization problems involve nonsmooth (that is, not necessarily differentiable) functions of thousands of variables. In the paper [Haarala, Miettinen, Mäkelä, Optimization Methods and Software, 19, (2004), pp. 673-692] we have described an efficient method for large-scale nonsmooth optimization. In this paper, we introduce a new variant of this method and prove its global convergence for locally Lipschitz continuous objective functions, which are not necessarily differentiable or convex. In addition, we give some encouraging results from numerical experiments.
Maaranen, H., Miettinen, K., Penttinen, A., On Initial Populations of a Genetic Algorithm for Continuous Optimization Problems, Journal of Global Optimization, 37(3), 405-436, 2007.
Genetic algorithms are commonly used metaheuristics for global optimization, but there has been very little research done on the generation of their initial population. In this paper, we look for an answer to the question whether the initial population plays a role in the performance of genetic algorithms and if so, how it should be generated. We show with a simple example that initial populations may have an effect on the best objective function value found for several generations. Traditionally, initial populations are generated using pseudo random numbers, but there are many alternative ways. We study the properties of different point generators using four main criteria: the uniform coverage and the genetic diversity of the points as well as the speed and the usability of the generator. We use the point generators to generate initial populations for a genetic algorithm and study what effects the uniform coverage and the genetic diversity have on the convergence and on the final objective function values. For our tests, we have selected one pseudo and one quasi random sequence generator and two spatial point processes: simple sequential inhibition process and nonaligned systematic sampling. In numerical experiments, we solve a set of 52 continuous test functions from 16 different function families, and analyze and discuss the results.
Setämaa-Kärkkäinen, A., Miettinen, K., Vuori, J., Heuristic for a New Multiobjective Scheduling Problem, Optimization Letters, 1(3), 213-225, 2007.
We consider a telecommunication problem in which the objective is to schedule data transmission to be as fast and as cheap as possible. The main characteristic and restriction in solving this multiobjective optimization problem is the very limited computational capacity available. We describe a simple but efficient local search heuristic to solve this problem and provide some encouraging numerical test results. They demonstrate that we can develop a computationally inexpensive heuristic without sacrificing too much in the solution quality.
Miettinen, K., Using Interactive Multiobjective Optimization in Continuous Casting of Steel, Materials and Manufacturing Processes, 22(5), 585-593, 2007.
We discuss some pros and cons of using different types of multiobjective optimization methods for demanding real-life problems like continuous casting of steel. In particular, we compare evolutionary approaches that are used for approximating the set of Pareto-optimal solutions to interactive methods where a decision maker actively takes part and can direct the solution process to such Pareto-optimal solutions that are interesting to her/him. Among the latter type of methods, we describe an interactive classification-based multiobjective optimization method: NIMBUS. NIMBUS converts the original objective functions together with preference information coming from the decision maker into scalar-valued optimization problems. These problems can be solved using any appropriate underlying solvers, like evolutionary algorithms. We also introduce an implementation of NIMBUS, called IND-NIMBUS, for solving demanding multiobjective optimization problems defined with different modelling and simulation tools. We apply NIMBUS and IND-NIMBUS in an optimal control problem related to the secondary cooling process in the continuous casting of steel. As an underlying solver we use a real-coded genetic algorithm. The aim in our problem is to find a control resulting with steel of the best possible quality, that is, minimizing the defects in the final product. Since the constraints describing technological and metallurgical requirements are so conflicting that they form an empty feasible set, we formulate the problem as a multiobjective optimization problem where constraint violations are minimized.
Hakanen, J., Kawajiri, Y., Miettinen, K., Biegler, L.T., Interactive Multi-Objective Optimization for Simulated Moving Bed Processes, Control and Cybernetics, 36(2), 282-320, 2007.
In this paper, efficient optimization techniques are used to solve multi-objective optimization problems arising from Simulated Moving Bed (SMB) processes. SMBs are widely used in many industrial separations of chemical products and they are very challenging from the optimization point of view. With the help of interactive multi-objective optimization, several conflicting objectives can be considered simultaneously without making unnecessary simplifications, as it has been done in previous studies. The optimization techniques used are the interactive NIMBUS R method and the IPOPT optimizer. To demonstrate the usefulness of these techniques, the results of solving an SMB optimization problem with four objectives are reported.
Miettinen, K., Mäkelä, M.M., Maaranen, H., Efficient Hybrid Methods for Global Continuous Optimization based on Simulated Annealing, Computers & Operations Research, 33(4), 1102-1116, 2006.
We introduce several hybrid methods for global continuous optimization. They combine simulated annealing and a local proximal bundle method. Traditionally, the simplest hybrid of a global and a local solver is to call the local solver after the global one, but this does not necessarily produce good results. Besides, using efficient gradient-based local solvers implies that the hybrid can only be applied to differentiable problems. We show several ways how to integrate the local solver as a genuine part of simulated annealing to enable both efficient and reliable solution processes. When using the proximal bundle method as a local solver, it is possible to solve even nondifferentiable problems. The numerical tests show that the hybridization can improve both the efficiency and the reliability of simulated annealing.
Miettinen, K., Mäkelä, M.M., Synchronous Approach in Interactive Multiobjective Optimization, European Journal of Operational Research, 170(3), 909-922, 2006.
We introduce a new approach in the methodology development for interactive multiobjective optimization. The presentation is given in the context of the interactive NIMBUS method, where the solution process is based on the classification of objective functions. The idea is to formulate several scalarizing functions, all using the same preference information of the decision maker. Thus, opposed to fixing one scalarizing function (as is done in most methods), we utilize several scalarizing functions in a synchronous way. This means that we as method developers do not make the choice between different scalarizing functions but calculate the results of different scalarizing functions and leave the final decision to the expert, the decision maker. Simultaneously, (s)he obtains a better view of the solutions corresponding to her/his preferences expressed once during each iteration. In this paper, we describe a synchronous variant of the NIMBUS method. In addition, we introduce a new version of its implementation WWW-NIMBUS operating on the Internet. WWW-NIMBUS is a software system capable of solving even computationally demanding nonlinear problems. The new version of WWW-NIMBUS can handle versatile types of multiobjective optimization problems and includes new desirable features increasing its user-friendliness.
Setämaa-Kärkkäinen, A., Miettinen, K., Vuori, J., Best Compromise Solution for a New Multiobjective Scheduling Problem, Computers & Operations Research, 33(8), 2353-2368, 2006.
In future wireless networks, a mobile terminal will be able to communicate with a service provider using several network connections. These connections to networks will have different properties and they will be priced separately. In order to minimize the total communication time and the total transmission costs, an automatic method for selecting the network connections is needed. Here, we describe the network connection selection problem and formulate it mathematically. We discuss solving the problem and analyse different multiobjective optimization approaches for it.
Miettinen, K., Kirilov, L., Interactive Reference Direction Approach Using Implicit Parametrization for Nonlinear Multiobjective Optimization, Journal of Multi-Criteria Decision Analysis, 13(2-3), 115-123, 2005.
A method for solving nonlinear multiobjective optimization problems is presented. In this interactive method, the dialog with a decision maker takes place in terms of desirable aspiration levels forming a reference point in the objective function space. In the computation phase, a lexicographic subproblem is used to generate a desired number of nondominated solutions. Thus, the decision maker can direct the interactive solution process both by specifying reference points and by setting the number of new solutions to be generated at each iteration. The proposed method gives the decision maker a possibility to investigate any part of the whole nondominated set according to his/her preferences and emphasizes learning possibilities. The functioning of the method is illustrated with a numerical example.
Hakanen, J., Miettinen, K., Mäkelä, M.M., Manninen, J., On Interactive Multiobjective Optimization with NIMBUS in Chemical Process Design, Journal of Multi-Criteria Decision Analysis, 13(2-3), 125-134, 2005.
The optimisation of innovative cyclic chemical processes may be challenging because these problems often involve PDE models, temporal periodicity constraints, and multiple objectives. Therefore, this paper presents two deterministic approaches which allow a fast and efficient generation of the Pareto set for these multiple objective optimisation problems. Hereto, the Normal Boundary Intersection (NBI) and the Normalised Normal Constraint (NNC) method are incorporated into a deterministic multiple shooting approach. Both approaches are illustrated for a reverse flow reactor case study involving conflicting conversion and energy objectives, and their performance is compared to a classic weighted sum approach.
Miettinen, K., Mäkelä, M.M., Kaario, K., Experiments with Classification-Based Scalarizing Functions in Interactive Multiobjective Optimization, European Journal of Operational Research, 175(2), 931-947, 2006.
In multiobjective optimization methods, the multiple conflicting objectives are typically converted into a single objective optimization problem with the help of scalarizing functions and such functions may be constructed in many ways. We compare both theoretically and numerically the performance of three classification-based scalarizing functions and pay attention to how well they obey the classification information. In particular, we devote special interest to the differences the scalarizing functions have in the computational cost of guaranteeing Pareto optimality. It turns out that scalarizing functions with or without so-called augmentation terms have significant differences in this respect. We also collect a set of mostly nonlinear benchmark test problems that we use in the numerical comparisons.
Madetoja, E., Miettinen, K., Tarvainen, P., Issues Related to the Computer Realization of a Multidisciplinary and Multiobjective Optimization System, Engineering with Computers, 22(1), 33-46, 2006.
Issues and novel ideas to be considered when developing computer realizations of complex multidisciplinary and multiobjective optimization systems are introduced. The aim is to discuss computer realizations that make possible both computationally efficient multidisciplinary analysis and multiobjective optimization of real world problems. We introduce software tools that make typically very time-consuming simulation processes more effective and, thus, enable even interactive multiobjective optimization with a real decision maker. In this paper, we first define a multidisciplinary and multiobjective optimization system and after that present an implementation overview of such problems including basic components participating in the solution process. Furthermore, interfaces and data flows between the components are described. A couple of important features related to the implementation are discussed in detail, for example, the usage of automatic differentiation. Finally, the ideas presented are illustrated with an industrial multiobjective optimization problem, when we describe numerical experiments related to quality properties in paper making.
Heikkola, E., Miettinen, K., Nieminen, P., Multiobjective Optimization of an Ultrasonic Transducer using NIMBUS, Ultrasonics, 44(4), 368-380, 2006.
The optimal design of an ultrasonic transducer is a multiobjective optimization problem since the final outcome needs to satisfy several conflicting criteria. Simulation tools are often used to avoid expensive and time-consuming experiments, but even simulations may be inefficient and lead to inadequate results if they are based only on trial and error. In this work, the interactive multiobjective optimization method NIMBUS is applied in designing a high-power ultrasonic transducer. The performance of the transducer is simulated with a finite element model, and three design goals are formulated as objective functions to be minimized. To find an appropriate compromise solution, additional preference information is needed from a decision maker, who in our case is an expert in transducer design. A realistic design problem is formulated, and an interactive solution process is described. Our findings demonstrate that interactive multiobjective optimization methods, combined with numerical simulation models, can efficiently help in finding new solution approaches and possibilities as well as new understanding of real-life problems as entirenesses. In this case, the decision maker found a solution that was better with respect to all three objectives than the conventional unoptimized design.
Lahdelma, R., Miettinen, K., Salminen, P., Reference Point Approach for Multiple Decision Makers, European Journal of Operational Research, 164(3), 785-791, 2005.
We consider multiple criteria decision-making problems where a group of decision-makers wants to find the most preferred solution from a discrete set of alternatives. We develop a method that uses achievement functions for charting subsets of reference points that would support a certain alternative to be the most preferred one. The resulting descriptive information is provided to the decision-makers in the form of reference acceptability indices and central reference points for each decision alternative. Then, the decision-makers can compare this information with their own preferences. We demonstrate the use of the method using a strategic multiple criteria decision model for an electricity retailer.
Lotov, A.V., Kamenev, G.K., Berezkin, V.E., Miettinen, K., Optimal Control of Cooling Process in Continuous Casting of Steel Using a Visualization-Based Multi-Criteria Approach, Applied Mathematical Modelling, 29(7), 653-672, 2005.
This paper is devoted to the application of a new visualization-based multi-criteria approach in an optimal control problem related to the cooling process in the continuous casting of steel. The purpose is to develop such a control that results in steel of the best possible quality, that is, minimizes the defects in the final product. Since the constraints describing technological and metallurgical requirements result in an empty set of the feasible controls of the cooling process, the aim of the study is to find a control that violates the constraints as little as possible. The problem is formulated as a multi-criteria optimization problem and the constraint violations play the role of selection criteria. Our approach is based on the application of both a nonlinear mathematical model of the cooling process that features a system of partial differential equations, and a new decision support technique, called Interactive Decision Maps technique, that uses the on-line visualization of the non-dominated frontier.
Miettinen, K., Kaario, K., Comparing Graphic and Symbolic Classification in Interactive Multiobjective Optimization, Journal of Multi-Criteria Decision Analysis, 12(6), 321-335, 2003.
In interactive multiobjective optimization systems, the classification of objective functions is a convenient way to direct the solution process in order to search for new, more satisfactory, solutions in the set of Pareto optimal solutions. Classification means that the decision maker assigns the objective functions into classes depending on what kind of changes in their values (in relation to the current values) are desirable. Here we study the role of user interfaces in implementing classification in multiobjective optimization software and how classification should be realized. In this way, we want to pay attention to the usability of multiobjective optimization software. Typically, this topic has not been of interest in the multiobjective optimization literature. However, usability aspect is important because in interactive classification-based multiobjective optimization methods, the classification is the core of the solution process. We can say that the more convenient the classification is, the more efficient the system or the method is and the better it supports the work of the decision maker. We report experiments with two classification options, graphic and symbolic ones, which are used in connection with an interactive multiobjective optimization system WWW-NIMBUS. The ideas and conclusions given are applicable for other interactive classification-based method, as well.
Maaranen, H., Miettinen, K., Mäkelä, M.M., Quasi Random Initial Population for Genetic Algorithms, Computers and Mathematics with Applications, 47, 1885-1895, 2004.
The selection of the initial population in a population-based heuristic optimizationmethod is important, since it affects the search for several iterations and often has an influence on the final solution. If no a priori information about the optima is available, the initial population is often selected randomly using pseudorandom numbers. Usually, however, it is more important that the points are as evenly distributed as possible than that they imitate random points. In this paper, we study the use of quasi-random sequences in the initial population of a genetic algorithm. Sample points in a quasi-random sequence are designed to have good distribution properties. Here a modified genetic algorithm using quasi-random sequences in the initial population is tested by solving a large number of continuous benchmark problems from the literature. The numerical results of two implementations of genetic algorithms using different quasi-random sequences are compared to those of a traditional implementation using pseudorandom numbers. The results obtained are promising.
Haarala, M., Miettinen, K., Mäkelä, M.M., New Limited Memory Bundle Method for Large-Scale Nonsmooth Optimization, Optimization Methods and Software, 19(6), 673-692, 2004.
Many practical optimization problems involve nonsmooth (that is, not necessarily di.erentiable) functions of hundreds or thousands of variables. In such problems, the direct application of smooth gradient-based methods may lead to a failure due to the nonsmooth nature of the problem. On the other hand, none of the current general nonsmooth optimization methods is e.cient in large-scale settings. In this paper, we describe a new limited memory variable metric based bundle method for nonsmooth large-scale optimization. In addition, we introduce a new set of academic test problems for large-scale nonsmooth minimization. Finally, we give some encouraging results from numerical experiments using both academic and practical test problems.
Lahdelma, R., Miettinen, K., Salminen, P., Ordinal Criteria in Stochastic Multicriteria Acceptability Analysis (SMAA), European Journal of Operational Research 147(1), 117-127, 2003.
We suggest a method for providing descriptive information about the acceptability of decision alternatives in discrete co-operative group decision-making problems. The new SMAA-O method is a variant of the stochastic multicriteria acceptability analysis (SMAA). SMAA-O is designed for problems where criteria information for some or all criteria is ordinal; that is, experts (or decision-makers) have ranked the alternatives according to each (ordinal) criterion. Considerable savings can be obtained if rank information for some or all the criteria is sufficient for making decisions without significant loss of quality. The approach is particularly useful for group decision making when the group can agree on the use of an additive decision model but only partial preference information, or none at all, is available.
Hämäläinen, J., Miettinen, K., Tarvainen, P., Toivanen, J., Interactive Solution Approach to a Multiobjective Optimization Problem in Paper Machine Headbox Design, Journal of Optimization Theory and Applications 116(2), 265-281, 2003.
A successful application of the interactive multiobjective optimization method NIMBUS to a design problem in papermaking technology is described. Namely, an optimal shape design problem related to the paper machine headbox is studied. First, the NIMBUS method, the numerical headbox model, and the associated multiobjective optimization problem are described. Then, the results of numerical experiments are presented.
Miettinen, K., Lotov, A.V., Kamenev, G.K., Berezkin, V.E., Integration of Two Multiobjective Optimization Methods for Nonlinear Problems, Optimization Methods and Software 18(1), 63-80, 2003.
In this paper, we bring together two existing methods for solving multiobjective optimization problems described by nonlinear mathematical models and create methods that benefit from both their strengths. We use the Feasible Goals Method and the NIMBUS method to form new hybrid approaches. The Feasible Goals Method (FGM) is a graphic decision support tool that combines ideas of goal programming and multiobjective methods. It is based on the transformation of numerical information given by mathematical models into a variety of feasible criterion vectors (that is, feasible goals). Visual interactive display of this variety provides information about the problem that helps the decision maker to detect the limits of what is possible. Then, the decision maker can identify a preferred feasible criterion vector on the graphic display. NIMBUS is an interactive multiobjective optimization method capable of solving nonlinear and even nondifferentiable and nonconvex problems. The decision maker can iteratively evaluate the problem to be solved and express personal preferences in a simple form: the method is based on the classification of the criteria, where the decision maker can indicate what kind of changes to the current solution are desirable. We describe two possible hybrids of the FGM and the NIMBUS method for helping in finding the most preferable decision (using simple questions posed to the decision maker). First, feasible criterion values are explored, and the decision maker’s preferences are expressed roughly in the form of a preferable feasible goal (FGM stage). Then, the identified goal is refined using the classification of the criteria (NIMBUS stage). Alternatively, the two methods can be used interactively. Both the hybrid approaches are here illustrated with an example.
Miettinen, K., Mäkelä, M.M., Characterizing Generalized Trade-Off Directions, Mathematical Method of Operations Research 57(1), 89-100, 2003.
Recently, so-called trade-off directions have been introduced for convex multiobjective optimization problems and, later, they have been generalized for nonconvex problems. The trade-off directions are characterized with the help of tangent and contingent cones. Trade-offs give information about the interrelations of the changes in the objective function values in different Pareto optimal solutions. Such information is valuable when searching for the most preferable Pareto optimal solution. Here, we introduce characterizations of generalized trade-off directions using the tools of nonsmooth analysis. In this way, it is possible to analytically describe concepts that originally were geometrical.
Furems, E.M., Larichev, O.I., Roizenson, G.V., Lotov, A.V., Miettinen, K., Human Behavior in a Multi-Criteria Choice Problem with Individual Tasks of Different Difficulties, International Journal of Information Technology & Decision Making 2(1), 29-40, 2003.
This paper is devoted to a laboratory study of human behavior in a multi-criteria choice problem. The specific feature of the experimental study is the creation of an individually adjusted instance of a general task for each subject in accordance with his/her preferences over each criterion. Human behavior is studied in a specially constructed choice situation based on the decomposition of the alternatives of a multi-criteria problem. The procedure is based on multiple steps of pair-wise comparisons involving only some (two or three) of the original components of the alternatives. Abilities of subjects to use such comparisons and to answer the questions in a logical way are tested. The experiment was carried out in two countries: Finland and Russia.
Toivanen, J., Hämäläinen, J. Miettinen, K., Tarvainen, P., Designing Paper Machine Headbox Using GA, Materials and Manufacturing Processes, 18(3), 533-541, 2003.
A non-smooth biobjective optimization problem for designing the shape of a slice channel in a paper machine headbox is described. The conflicting goals defining the optimization problem are the ones determining important quality properties of produced paper: 1) basis weight should be even and 2) the wood fibers of paper should mainly be oriented to the machine direction across the width of the whole paper machine. The novelty of the considered approach is that maximum deviations are used instead of least squares when objective functions are formed. For the solution of this problem, a multiobjective genetic algorithm based on nondominated sorting is considered. The numerical results demonstrate the ability to obtain a large set of nondominated designs.
Miettinen, K., Mäkelä, M.M., Toivanen, J., Numerical Comparison of Some Penalty-Based Constraint Handling Techniques in Genetic Algorithms, Journal of Global Optimization, 27(4), 427-446, 2003.
We study five penalty function-based constraint handling techniques to be used with genetic algorithms in global optimization. Three of them, the method of superiority of feasible points, the method of parameter free penalties and the method of adaptive penalties have already been considered in the literature. In addition, we introduce two new modifications of these methods. We compare all the five methods numerically in 33 test problems and report and analyze the results obtained in terms of accuracy, efficiency and reliability. The method of adaptive penalties turned out to be most efficient while the method of parameter free penalties was the most reliable.
Miettinen, K., Mäkelä, M.M., On Generalized Trade-Off Directions in Nonconvex Multiobjective Optimization, Mathematical Programming 92, 141-151, 2002. Published online February 14, 2002.
Trade-off information related to Pareto optimal solutions is important in multiobjective optimization problems with conflicting objectives. Recently, the concept of trade-off directions has been introduced for convex problems. These trade-offs are characterized with the help of tangent cones. Generalized trade-off directions for nonconvex problems can be defined by replacing convex tangent cones with nonconvex contingent cones. Here we study how the convex concepts and results can be generalized into a nonconvex case. Giving up convexity naturally means that we need local instead of global analysis.
Miettinen, K., Mäkelä, M.M., On Scalarizing Functions in Multiobjective Optimization, OR Spectrum, 24, 193-213, 2002.
Scalarizing functions play an essential role in solving multiobjective optimization problems. Many different scalarizing functions have been suggested in the literature based on different approaches. Here we concentrate on classification and reference point-based functions. We present a collection of functions that have been used in interactive methods as well as some modifications. We compare their theoretical properties and numerical behaviour. In particular, we are interested in the relation between the information provided and the results obtained. Our aim is to select some of them to be used in our WWW-NIMBUS optimization system.
Furems, E.M., Larichev, O.I., Lotov, A.V., Miettinen, K., Roizenson, G.V., Human Behavior in a Multi-Criteria Choice with Individual Tasks of Different Difficulties, Artificial Intelligence (National Academy of Sciences of Ukraine, Institute of Artificial Intelligence), 2, 346-352, 2002.
This paper is devoted to a laboratory study of human behavior in a multi-criteria choice problem. The specific feature of of the experimental study is the creation of an individually adjusted instance of a general task for each subject in accordance with his/her preferences over each criterion. Human behavior is studied in a specially constructed choice situation based on the decomposition of the alternatives of a multi-criteria problem. The procedure is based on multiple steps of pair-wise comparisons involving only some (two or three) of the original components of the alternatives. Abilities of subjects to use such comparisons and to answer the questions in a logical way are tested. The experiment was carried out in two countries: Finland and Russia.
Miettinen, K., Mäkelä, M.M., On Cone Characterizations of Weak, Proper and Pareto Optimality in Multiobjective Optimization, Mathematical Methods of Operations Research, 53, 233-245, 2001.
Efficient, weakly and properly Pareto optimal solutions of multiobjective optimization problems can be characterized with the help of different cones. Here, contingent, tangent and normal cones as well as cones of feasible directions are used in the characterizations. The results are first presented in convex cases and then generalized to nonconvex cases by employing local concepts.
Miettinen, K. and Mäkelä, M.M., Interactive Multiobjective Optimization System WWW-NIMBUS on the Internet, Computers & Operations Research 27, 709-723, 2000.
NIMBUS is a multiobjective optimization method capable of solving nondifferentiable and nonconvex problems. We describe the NIMBUS algorithm and its implementation WWW-NIMBUS. To our knowledge WWW-NIMBUS is the first interactive multiobjective optimization system on the Internet. The main principles of its implementation are centralized computing and a distributed interface. Typically, the delivery and update of any software is problematic. Limited computer capacity may also be a problem. Via the Internet, there is only one version of the software to be updated and any client computer has the capabilities of a server computer. Further, the World-Wide Web (WWW) provides a graphical user interface. No special tools, compilers or software besides a WWW browser are needed.
Miettinen, K. and Salminen, P., Decision-Aid for Discrete Multiple Criteria Decision Making Problems with Imprecise Data, European Journal of Operational Research 119, 50-60, 1999.
We describe ways of aiding decision making with a discrete set of alternatives. In many decision situations, it is not possible to obtain explicit preference information from the decision makers. Instead, useful decision-aid can be provided to the decision makers by describing what kind of weighting of the criteria result in certain choices of the alternatives. The suggested treatment is based on the basic ideas of the ELECTRE III method. The modelling of the preferences by pseudo-criteria is especially helpful in case the data, that is, the criterion values are imprecise. Unlike ELECTRE III, no ranking of the alternatives is produced. Based on a minimum-procedure in the exploitation of the outranking relations, we provide information about the weights of the criteria that make a certain alternative the best. We also present an interactive searching procedure in the weighting space. The auxiliary optimization problems to be solved are nondifferentiable. Cases with both single and multiple decision makers are considered.
Miettinen, K. and Mäkelä, M.M., Comparative Evaluation of Some Interactive Reference Point-based Methods for Multi-Objective Optimisation, Journal of the Operational Research Society 50, 949-959, 1999.
Many real-world optimisation applications include several conflicting objectives of possibly nondifferentiable character. However, the lack of computationally efficient, interactive methods for nondifferentiable multi-objective optimisation problems is apparent. To satisfy this demand, a method called NIMBUS has been developed. Two versions of the basic method are presented and compared both theoretically and computationally. In order to give variety to the comparison, a related approach, called reference direction method is included. Theoretically, the methods differ in handling the information requested from the user. Numerical experiments indicate differences in computational efficiency and controllability of the solution processes.
Hokkanen, J., Lahdelma, R., Miettinen, K. and Salminen, P. , Determining the Implementation Order of a General Plan by Using a Multicriteria Method, Journal of Multi-Criteria Decision Analysis 7, 273-284, 1998.
We describe a real-life application of a new multicriteria method in the context of assisting the decision-making for a general plan in the municipality of Kirkkonummi in Uusimaa, Finland. At the time our group started working on the problem, a proposal for an overall plan had already been completed, but the order in which different regional parts of the plan should be implemented needed to be considered based on the environmental impact assessment (EIA) procedure. The EIA procedure generated a large amount of data about the different impacts of the alternatives. For this group decision making problem we developed the SMAA-3 decision support method which does not require any explicit preference information from the decision makers during the procedure. The uncertainty of the basic data is modelled using ELECTRE III-type pseudo-criteria with preference and indifference thresholds.
Miettinen, K., Mäkelä, M.M and Männikkö, T., Optimal Control of Continuous Casting by Nondifferentiable Multiobjective Optimization, Computational Optimization and Applications 11, 177-194, 1998.
A new version of an interactive NIMBUS method for nondifferentiable multiobjective optimization is described. It is based on the reference point idea and the classification of the objective functions. The original problem is transformed into a single objective form according to the classification information. NIMBUS has been designed especially to be able to handle complicated real-life problems in a user-friendly way. The NIMBUS method is used for solving an optimal control problem related to the continuous casting of steel. The main goal is to minimize the defects in the final product. Conflicting objective functions are constructed according to certain metallurgical criteria and some technological constraints. Due to the phase changes during the cooling process there exist discontinuities in the derivative of the temperature distribution. Thus, the problem is nondifferentiable. Like many real-life problems, the casting model is large and complicated and numerically demanding. NIMBUS provides an efficient way of handling the difficulties and, at the same time, aids the user in finding a satisficing solution. In the end, some numerical experiments are reported and compared with earlier results.
Miettinen, K. and Korhonen, P. , Distributed Performance Analysis on the Internet Using a Centric Database, Interim Report IR-98-111, IIASA, Laxenburg, 1998.
In many areas of life it is useful to be able to compare one’s own performance to some general benchmark data. The Internet provides a way of realizing such a comparison so that the original database can be hidden from users by locating it in a server computer and users can test their individual data in a distributed manner. An interactive and graphical user interface can be implemented with the tools of the World-Wide Web (WWW). We introduce a World-wide INTEractive Regression Analysis (WINTERA) system that operates via the Internet. The system enables a user to carry out regression analysis with an original database and evaluate the performance of the data vector of her or his own. There are two kinds of users in the system. Data suppliers enter their observation matrices to form databases. Ordinary users can evaluate observations of their own with respect to existing databases. They can also suggest their observations to be included in the databases. the data supplier decides whether (s)he accepts or rejects the information. This means that the whole database is accessible only to the data supplier. In any case, ordinary users receive information about their performance.
Miettinen, K., Mäkelä, M.M., Interactive Method NIMBUS for Nondifferentiable Multiobjective Optimization Problems, in "Multicriteria Analysis"", Ed. by Climaco J., Proceedings of the XIth International Conference on MCDM, 1-6 August 1994, Coimbra, Portugal, Springer-Verlag, 310-319, 1997, (Refereed).
An interactive method, NIMBUS, for nondifferentiable multiobjective optimization problems is introduced. The method is capable of handling several nonconvex locally Lipschitzian objective functions subject to nonlinear (possibly nondifferentiable) constraints. The idea of NIMBUS is that the decision maker can easily indicate what kind of improvements are desired and what kind of impairments are tolerable at the point considered.
The decision maker is asked to classify the objective functions into five different classes: those to be improved, those to be improved down to some aspiration level, those to be accepted as they are, those to be impaired till some upper bound, and those allowed to change freely. A new problem is formed according to this classification.
An MPB (Multiobjective Proximal Bundle) method is employed to solve the new (multiobjective) optimization problem. The MPB method is a generalization of a proximal bundle approach for nondifferentiable single objective optimization into the multiobjective case. The objective functions are treated with the help of an improvement function.
Miettinen, K. and Mäkelä, M.M., Interactive Bundle-Based Method for Nondifferentiable Multiobjective Optimization: NIMBUS, Optimization 34, 231-246, 1995.
NIMBUS, an interactive method for nondifferentiable multiobjective optimization problems, is described. The algorithm is based on the classificatoin of objective functions. At each iteration, a decision maker is asked to classify the objective functions into up to five classes: those to be improved, those to be improved till some aspiration level, those to be accepted as they are, those to be impaired till some bound, and those to change freely.
According to the classification, a new (multiobjective) optimization problem is formed, which is solved by an MPB (Multiobjective Proximal Bundle) method. The MPB method is a generalization of Kiwiel’s proximal bundle approach for nondifferentiable single objective optimization into the multiobjective case. The multiple objective functions are treated individually without employing any scalarization. The method is capable of handling several nonconvex Lipschitz continuous objective functions subject to nonlinear (possibly nondifferentiable) constraints.
Finally, numerical experiments with two academic test problems are reported. The aim is to briefly demonstrate how the method works.
Miettinen, K. and Mäkelä, M. M., A Nondifferentiable Multiple Criteria Optimization Method Applied to Continuous Casting Process, in "Proceedings of the Seventh European Conference on Mathematics in Industry (ECMI 9)", Ed. by Fasano A.and Primicerio M., Seventh European Conference on Mathematics in Industry, Montecatini Terme, B. G. Teubner, Stuttgart, 255-262, 1994, (Refereed).
The aim of this paper is to consider the continuous casting process of steel with several conflicting criteria concering the quality of the final product. This nondifferentiable multiple criteria optimization problem is solved by our interactive subgradient-based method, which allows the decision maker to choose the final solution among Pareto optimal alternatives.
Miettinen, K. and Mäkelä, M.M, An Interactive Method for Nonsmooth Multiobjective Optimization with an Application to Optimal Control, Optimization Methods and Software 2, 31-44, 1993.
An implementable method for nonsmooth multiobjective optimization is descibed. The algorithm is a modification of the well-known Geoffrion-Dyer-Feinberg (GDF) method for smooth interactive multiobjective problems. The smooth gradient-based Frank-Wolfe method exploited in the GDF method is replaced by a modified (Kiev) subgradient method in order to compute the search direction. The solutions are projected onto the set of Pareto optimal points by using exact penalty scalarizing functions. A bundle-type method is utilized to solve the nonsmooth single objective optimization problems arising in every iteration of the procedure. As an application we introduce a model of an elastic string, which leads us to solve a nonsmooth multiobjective optimal control problem governed by a variational inequality. Due to the unilateral boundary conditions, the state of the system depends in a nonsmooth way on the control variable. Finally, some encouraging numerical experience is reported.