IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 40, NO. 3, JUNE 2010 845 Multiobjective Optimization of Temporal Processes Zhe Song, Member, IEEE, and Andrew Kusiak, Member, IEEE Abstract—This paper presents a dynamic predictive- optimization framework of a nonlinear temporal process. Data- mining (DM) and evolutionary strategy algorithms are integrated in the framework for solving the optimization model. DM algo- rithms learn dynamic equations from the process data. An evolu- tionary strategy algorithm is then applied to solve the optimization problem guided by the knowledge extracted by the DM algorithm. The concept presented in this paper is illustrated with the data from a power plant, where the goal is to maximize the boiler effi- ciency and minimize the limestone consumption. This multiob- jective optimization problem can be either transformed into a single-objective optimization problem through preference aggre- gation approaches or into a Pareto-optimal optimization problem. The computational results have shown the effectiveness of the proposed optimization framework. Index Terms—Data mining (DM), dynamic modeling, evolution- ary algorithms (EAs), multiobjective optimization, nonlinear tem- poral process, power plant, predictive control, preference-based optimization. NOMENCLATURE x Vector of the controllable variables of a process. x i ith controllable variable. v Vector of the noncontrollable variables of a process. v i ith noncontrollable variable. y Vector of the response variables of a process. y i ith response variable. y P Vector of the performance variables of a process, y P is a subset of y. y NP Vector of the nonperformance variables of a process, y NP is a subset of y. Ω x Search space of x. Ω y NP Constraint space of y NP . f (•) Function capturing the mapping between (x, v) and y. t Sampling time stamp. d y i , d x i , d v i Maximum possible time delays for y i , x i , v i . D y i y i , D y i x i , D y i v i Sets of time delay constants selected for the corresponding variables y i , x i , v i un- der the response variable y i . Manuscript received November 2, 2008; revised February 8, 2009 and April 25, 2009. First published November 6, 2009; current version published June 16, 2010. This work was supported by the Iowa Energy Center under Grant 07-01. This paper was recommended by Associate Editor Y. S. Ong. The authors are with the Department of Mechanical and Industrial Engi- neering, The University of Iowa, Iowa City, IA 52242-1527 USA (e-mail: andrew-kusiak@uiowa.edu). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TSMCB.2009.2030667 d y i ,min x , d y i ,max x Minimum and maximum values of sets D y i x 1 ,...,D y i x k . d y i ,min v , d y i ,max v Minimum and maximum values of sets D y i v 1 ,...,D y i v k . X Set of all controllable variables. X 1,y i , X 2,y i Two subsets of X, actionable and nonac- tionable sets for y i . Y NP Set of response variables which are not the performance variables. Y 1,NP , Y 2,NP Two subsets of Y NP , one is affected by changing the controllable variables, the other one is not. α i (•) Preference function for the performance variable y i . y i (LB), y i (CP ), Lower bound, center point, and upper bound for the preference function α i (•). y i (UB) Δy i Small positive constant to evaluate the de- crease or increase of y i . Ω p Preference space. Region i Region i in the preference space character- ized by its preference values. C(•) Cost function of the controllable variables. R, S Positive semidefinite matrices. β(•), w 1 ,..., Aggregation function and the weights used in it. w M ,w C λ Offspring size. μ Parent size or initial population size. S i Solution vector of ith individual. σ i Mutation vector of ith individual. N (•) Normal distribution. δ Threshold-distance vector to differentiate two individuals. X ∗, Region i (t) Set of Pareto-optimal solutions leading to Region i at sampling time t. X Region i (t) Set of solutions in the offspring pool lead- ing to Region i at sampling time t. n local Number of dominated individuals in a preference region. n global Number of dominated individuals in the preference space. I. I NTRODUCTION O PTIMIZING nonlinear and nonstationary processes with multiple objectives presents a challenge for traditional solution approaches. In this paper, a process is represented as a triplet (x, v, y), where x ∈ R k is a vector of k controllable variables, v ∈ R m is a vector of m noncontrollable (measur- able) variables, and y ∈ R l is a vector of l system response variables. The value of a response variable changes due to 1083-4419/$26.00 © 2009 IEEE Authorized licensed use limited to: The University of Iowa. Downloaded on June 20,2010 at 02:29:43 UTC from IEEE Xplore. Restrictions apply.