INDNIMBUS
Interactive Multiobjective Optimization System
 for interactive decision support
 can solve problems with several conflicting (nonlinear) objectives subject to equality and inequality constraints
 can be used to analyse interrelationships between different objectives
 enables learning about the problem and decision maker's preferences
 produces Pareto Optimal and approximate of Pareto Optimal solutions based on preferences given by the decision maker
 same ideology as in WWWNIMBUS
 more information on NIMBUS brochure
 demo available
Features
 NIMBUS [3, 5], Pareto Navigator [2] and PAINT methods [1]
 for different operating systems
 tested with several industrial problems (see applications below)
 can be connected with different simulator or modelling tools (e.g. Matlab® [11, 17], BALAS© [22] and GAMS [15, 13])
 several underlying singleobjective optimization methods, which can be selected by the user (e.g. proximal bundle method [9], genetic algorithm [7], differential evolution [8] and controlled random search) [10]
References

Hartikainen, M., Miettinen, K. and Wiecek, M., PAINT: Pareto front interpolation for nonlinear multiobjective optimization, Springer USComputational Optimization and Applications, Vol. 52, pp. 845867, 2012.
[DOI]

Eskelinen, P., Miettinen, K., Klamroth, K. and Hakanen, J., Pareto Navigator for interactive nonlinear multiobjective optimization, OR Spectrum, Vol. 32 (1), pp. 211227, 2010.
We describe a new interactive learningoriented 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., INDNIMBUS for Demanding Interactive Multiobjective Optimization, Trzaskalik, T.(ed.), Multiple Criteria Decision Making '05, , pp. 137150, 2006.

Miettinen, K. and Mäkelä, M. M., On Scalarizing Functions in Multiobjective Optimization, OR Spectrum, Vol. 24 (2), pp. 193213, 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 pointbased 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 WWWNIMBUS optimization system.

Miettinen, K. and Mäkelä, M. M., Interactive Multiobjective Optimization System WWWNIMBUS on the Internet, Computers & Operations Research, Vol. 27 (78), pp. 709723, 2000.
NIMBUS is a multiobjective optimization method capable of solving nondifferentiable and nonconvex problems. We describe the NIMBUS algorithm and its implementation WWWNIMBUS. To our knowledge WWWNIMBUS 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 WorldWide Web (WWW) provides a graphical user interface. No special tools, compilers or software besides a WWW browser are needed. Scope and purpose Interaction between the decision maker and the solution algorithm is often necessary for finding solutions to optimization problems with several conflicting criteria. The Internet provides a versatile tool in realizing such an interaction. The Internet is easily available and sets minimal requirements to the computer facilities of the user. We describe an interactive optimization method and its implementation utilizing the Internet.
 Miettinen, K., Nonlinear Multiobjective Optimization, Kluwer Academic Publishers, 1999.

Miettinen, K., Mäkelä, M. M. and Toivanen, J., Numerical Comparison of Some PenaltyBased Constraint Handling Techniques in Genetic Algorithms, Journal of Global Optimization, Vol. 27 (4), pp. 427446, 2003.
We study five penalty functionbased 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.

Storn, R. and Price, K., Differential Evolution  A Simple and Efficient Heuristic for global Optimization over Continuous Spaces, Journal of Global Optimization, Vol. 11 (4), pp. 341359, 1997.
A new heuristic approach for minimizing possiblynonlinear and nondifferentiable continuous spacefunctions is presented. By means of an extensivetestbed it is demonstrated that the new methodconverges faster and with more certainty than manyother acclaimed global optimization methods. The new method requires few control variables, is robust, easyto use, and lends itself very well to parallelcomputation.
 Mäkelä, M. M. and Neittaanmäki, P., Nonsmooth Optimization Analysis and Algorithms with Applications to Optimal Control, World Scientific, 1992.

Price, W., Global optimization by Controlled Random Search, Journal of Optimization Theory and Applications, Vol. 40 (3), pp. 333348, 1983.
The paper describes a new version, known as CRS2, of the author''s controlled random search procedure for global optimization (CRS). The new procedure is simpler and requires less computer storage than the original version, yet it has a comparable performance. The results of comparative trials of the two procedures, using a set of standard test problems, are given. These test problems are examples of unconstrained optimization. The controlled random search procedure can also be effective in the presence of constraints. The technique of constrained optimization using CRS is illustrated by means of examples taken from the field of electrical engineering.

Coello Coello, C. A., Theoretical and numerical constrainthandling techniques used with evolutionary algorithms: a survey of the state of the art, Computer Methods in Applied Mechanics and Engineering, Vol. 191 (1112), pp. 12451287, 2002.
This paper provides a comprehensive survey of the most popular constrainthandling techniques currently used with evolutionary algorithms. We review approaches that go from simple variations of a penalty function, to others, more sophisticated, that are biologically inspired on emulations of the immune system, culture or ant colonies. Besides describing briefly each of these approaches (or groups of techniques), we provide some criticism regarding their highlights and drawbacks. A small comparative study is also conducted, in order to assess the performance of several penaltybased approaches with respect to a dominancebased technique proposed by the author, and with respect to some mathematical programming approaches. Finally, we provide some guidelines regarding how to select the most appropriate constrainthandling technique for a certain application, and we conclude with some of the most promising paths of future research in this area.
+ Single Objective Optimization Methods (References)

Laukkanen, T., Tveit, T.M., Ojalehto, V., Miettinen, K. and Fogelholm, C.J., Bilevel heat exchanger network synthesis with an interactive multiobjective optimization method, Applied Thermal Engineering, Vol. 48 (0), pp. 301  316, 2012.Heat exchanger network synthesis (HENS) has been an active research area for more than 40 years because welldesigned heat exchanger networks enable heat recovery in process industries in an energy and costefficient 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 mediumsized 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.

Tveit, T., Laukkanen, T., Ojalehto, V., Miettinen, K. and Fogelholm, C., Interactive Multiobjective Optimisation of Configurations for an Oxyfuel Power Plant Process for CO2 Capture, Chemical Engineering Transactions, Vol. 29, pp. 433438, 2012.In this work we present a multiobjective approach to optimising configurations of an oxyfuel power plant process. The approach solves an optimisation model based on simulation results using an interactive multiobjective 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 coalfired 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 GAMSNIMBUS tool, is a powerful tool to study complex processes.

Hakanen, J., Miettinen, K. and Sahlstedt, K., Wastewater treatment: New insight provided by interactive multiobjective optimization, Decision Support Systems, Vol. 51 (2), pp. 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.

Laukkanen, T., Tveit, T.M., Ojalehto, V., Miettinen, K. and Fogelholm, C.J., An interactive multiobjective approach to heat exchanger network synthesis, Computers & Chemical Engineering, Vol. 34 (6), pp. 943952, 2010.In this work we present a multiobjective approach to heat exchanger network synthesis. The approach solves a modified version of the Synheat model using an interactive multiobjective optimisation method, NIMBUS, which is implemented in GAMS. The results obtained demonstrate the potential of interactive multiobjective optimisation.

Ruotsalainen, H., Miettinen, K. and Palmgren, J.E., Interactive Multiobjective Optimization for 3D HDR Brachytherapy Applying INDNIMBUS, Jones, D., Tamiz, M. & Ries, J.(ed.), New Developments in Multiple Objective and Goal Programming, Springer Berlin Heidelberg, Vol. 638, pp. 117131, 2010.An anatomy based threedimensional dose optimization approach for HDR brachytherapy using interactive multiobjective optimization is presented in this paper. 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 one 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 (e.g. defining weights, computational burden and trialanderror planning) can be avoided. In addition, our approach offers an easy way to navigate among the obtained Pareto optimal solutions (i.e. different treatment plans), and plan quality can be improved by finding advantageous tradeoffs between the solutions. To demonstrate the advantages of our interactive approach, a clinical example of seeking dwell time values of a source in a gynecologic cervix cancer treatment is presented.

Ruotsalainen, H., Miettinen, K., Palmgren, J.E. and Lahtinen, T., Interactive multiobjective optimization for anatomybased threedimensional HDR brachytherapy, Physics in Medicine and Biology, Vol. 55 (16), pp. 4703, 2010.In this paper, we present an anatomybased threedimensional 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 trialanderror 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 tradeoffs 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.

Ruotsalainen, H., Boman, E., Miettinen, K. and Tervo, J., Nonlinear Interactive Multiobjective Optimization Method for Radiotherapy Treatment Planning with Boltzmann Transport Equation, Contemporary Engineering Sciences, Vol. 2 (9), pp. 391422, 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 tradeoffs. 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.

Hakanen, J., Sahlstedt, K. and Miettinen, K., SimulationBased Interactive Multiobjective Optimization in Wastewater Treatment, CDROM Proceedings of EngOpt 2008  International Conference on Engineering Optimization, , 2008.This paper deals with developing tools for wastewater treatment plant design. 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 especially important because the requirements for wastewater treatment plants are getting more and more strict. 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 methodology is illustrated through a case study related to municipal wastewater treatment where three conflicting evaluation criteria are considered.

Hakanen, J., Kawajiri, Y., Miettinen, K. and Biegler, L., Interactive MultiObjective Optimization for Simulated Moving Bed Processes, Control and Cybernetics, Vol. 36 (2), pp. 282320, 2007.In this paper, efficient optimization techniques are used to solve multiobjective 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 multiobjective 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., Using Interactive Multiobjective Optimization in Continuous Casting of Steel, Taylor & FrancisMaterials and Manufacturing Processes, Vol. 22 (5), pp. 585593, 2007.We discuss some pros and cons of using different types of multiobjective optimization methods for demanding reallife problems like continuous casting of steel. In particular, we compare evolutionary approaches that are used for approximating the set of Paretooptimal solutions to interactive methods where a decision maker actively takes part and can direct the solution process to such Paretooptimal solutions that are interesting to her/him. Among the latter type of methods, we describe an interactive classificationbased multiobjective optimization method: NIMBUS. NIMBUS converts the original objective functions together with preference information coming from the decision maker into scalarvalued optimization problems. These problems can be solved using any appropriate underlying solvers, like evolutionary algorithms. We also introduce an implementation of NIMBUS, called INDNIMBUS, for solving demanding multiobjective optimization problems defined with different modelling and simulation tools. We apply NIMBUS and INDNIMBUS in an optimal control problem related to the secondary cooling process in the continuous casting of steel. As an underlying solver we use a realcoded 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., Hakala, J. and Manninen, J., An integrated multiobjective design tool for process design, Applied Thermal Engineering, Vol. 26 (13), pp. 1393  1399, 2006.An integrated multiobjective design tool has been developed for chemical process design. This tool combines the rigorous process calculations of the BALAS process simulator and the interactive multiobjective optimization method NIMBUS. With this design tool, the designer can consider several conflicting performance criteria simultaneously. The interactive nature of this tool allows the designer to learn about the behavior of the problem. To illustrate the possibilities of this design tool, two case studies are considered. One of them is related to paper making while the other one is related to power plants.

Heikkola, E., Miettinen, K. and Nieminen, P., Multiobjective optimization of an ultrasonic transducer using NIMBUS, Ultrasonics, Vol. 44 (4), pp. 368380, 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 timeconsuming 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 highpower 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 reallife 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.

Madetoja, E., Miettinen, K. and Tarvainen, P., Issues related to the computer realization of a multidisciplinary and multiobjective optimization system, Engineering with Computers, Vol. 22 (1), pp. 3346, 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 timeconsuming 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.

Miettinen, K., Interactive Multiobjective Optimization Method NIMBUS Applied to Continuous Casting of Steel, Bandyopadhyay, N., Chattopadhyay, P. & Chattopadhyay, S.(ed.), International Workshop on Neural Network and Genetic Algorithm in Materials Science and Engineering, Proceedings, Tata McGrawHill Publishing Company Limited, New Delhi, pp. 5872, 2006.

Hakanen, J., Miettinen, K., Mäkelä, M. M. and Manninen, J., On Interactive Multiobjective Optimization with NIMBUS in Chemical Process Design, Journal of MultiCriteria Decision Analysis, Vol. 13 (23), pp. 125134, 2005.We study multiobjective optimization problems arising from chemical process simulation. The interactive multiobjective optimization method NIMBUS®, developed at the University of Jyväskylä, is combined with the BALAS® process simulator, developed at the VTT Technical Research Center of Finland, in order to provide a new interactive tool for designing chemical processes. Continuous interaction between the method and the designer provides a new efficient approach to explore Pareto optimal solutions and helps the designer to learn about the behaviour of the process. As an example of how the new tool can be used, we report the results of applying it in a heat recovery system design problem related to the process water system of a paper mill. Copyright © 2006 John Wiley & Sons, Ltd.

Hämäläinen, J., Miettinen, K., Tarvainen, P. and Toivanen, J., Interactive Solution Approach to a Multiobjective Optimization Problem in Paper Machine Headbox Design, Journal of Optimization Theory and Applications, Vol. 116, pp. 265281, 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., Mäkelä, M. M. and Männikkö, T., Optimal Control of Continuous Casting by Nondifferentiable Multiobjective Optimization, Computational Optimization and Applications, Vol. 11, pp. 177194, 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 reallife problems in a userfriendly 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 reallife 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., Mäkelä, M. M. and Mäkinen, R. A. E., Interactive Multiobjective Optimization System NIMBUS Applied to Nonsmooth Structural Design Problems, Dolezal, J. & Fidler, J.(ed.), System Modelling and Optimization, Proceedings of the 17th IFIP Conference on System Modelling and Optimization, Chapman & Hall, London, pp. 379385, 1996.