Introduction to Multiobjective Optimization

Transcription

1 Introduction to Multiobjective Optimization Jussi Hakanen

2 Contents Multiple Criteria Decision Making (MCDM) Formulation of a multiobjective problem On solving multiobjective problems Basic concepts and definitions Optimality in multiobjective optimization

3 What means multiobjective? Consider several criteria simultaneously Criteria are conflicting (e.g. usually good quality is not chea) all the criteria can not be optimized simultaneously Need for considering compromises between the criteria Compromise can be better than optimal solution in practice (cf. optimize only costs/profit) mization

4 Multiobjective decision making process Optimization problem formulation Optimization & decision making Need for optimization Modelling of the problem (numerical simulation) Implementation & testing of the best solution found

5 Optimization problem formulation By optimizing only one criterion, the rest are not considered Objective vs. constraint Summation of the objectives adding apples and oranges Converting the objectives (e.g. as costs) not easy, includes uncertaintes Multiobjective formulation reveals interdependences between the objectives

6 Example 1: Continuous casting of steel Optimal control of the secondary cooling of continuous casting of steel Long history of research in the Dept. of Mathematical Information Technology, Univ. of Jyväskylä modelling (1988) single objective optimization ( ) multiobjective optimization ( )

7 Continuous casting of steel Liquid steel enters (tundish) Initial cooling by a water cooled mold thin crust Movement supported by rollers Secondary cooling by water sprays Rest of the cooling by radiation

8 Continuous casting of steel Measuring temperature in casting difficult numerical temperature distribution Process modelled as a multiphase heat equation (PDEs, solid & liquid phase) temperature distribution Numerical model by using the finite element method (FEM) Dynamic process

9 Continuous casting of steel Secondary cooling significant: intensity of sprays (easy to control) affects significantly to the solidification of steel Goal: minimize the amount of defects in steel Quality depends on e.g. the temperature distribution at the surface of steel too slow cooling too long liquid part too fast cooling defects appear Objective function: keep the surface temperature as close to a given profile as possible Constraints e.g. for the change of temperature and for the temperature in critical spots

10 Continuous casting of steel Analysis of single objective optimization problem: constraints tight (based on metallurgy) no feasible solutions which constraints to relax? Convert constraints as objective functions (5 in total) enables simultaneous relaxation of different constraints information on satisfaction of different constraints and their interdependences

11 Example 2: Water allocation

12 Water allocation Papermaking process consumes lots of water (nowadays about 5-10 m 3 /ton of paper) Water can be circulated and reused in different parts of the process as long as it remains fresh enough dissolved organic material accumulates Fresh water costs Process was modelles with the Balas process simulator ( How to formulate the optimization problem?

13 Water allocation Goal is to minimize the amount of fresh water required for the process Objective function: minimize the amount of fresh water Constraints the amount of dissolved organic material in the white water of the papermachine the amount of dissolved organic material in the pulp entering bleaching Variables: 5 splitters ja 3 valves

14 Water allocation In practice set upper bounds for the amounts of organic material minimize the amount of fresh water used (one objective function) How to set the upper bounds? based on engineering knowledge and current technology what if the bounds could be relaxed a bit? Multiobjective formulation where the constraints would also be objectives (3)

15 Example 3: Chemical separation process Consider a chemical separation process based on chromatography Applied to many important separationsin sugar, petrochemical,and pharmaceutical industries Utilizes the difference in the migration speeds of different chemical components in liquid *

16 Adapted from Y. Kawajiri, Carnegie Mellon University Chromatography (Single Column) Desorbent Feed (Mixture of two components) Pump Elution Feed Recover Initial state 21 nd product Column is filled with desorbent Chromatographic Column (Vessel packed with adsorbent particles)

17 November 11, 2009 Adapted from Y. Kawajiri, Carnegie Mellon University Process simulation Step Cycle Desorbent Feed Desorbent Desorbent Desorbent Feed Desorbent Feed Desorbent Feed Feed Feed Feed Desorbent Feed Desorbent 16 Bergische Universität Wuppertal Liquid Flow Raffinate Extract Raffinate Extract Extract Raffinate Extract Raffinate Extract Raffinate Raffinate Extract Raffinate Extract Raffinate Extract

18 Chemical separation process Two inlet and two outlet streams are switched in the direction of the liquid flow at a regular interval (steptime) Operating variables Switching interval (Step Time) Liquid velocities

19 Chemical separation process Typically a profit function is optimized Formulation of a profit is not easy Multiobjective formulation max throughput min desorbent consumption max purity of the product max recovery of the product Enables more flexible consideration and reveals how different objectives affect the solution

20 Multiobjective optimization (MOO) problem Multiple objective functions, number denoted by k ( k > 1) special case: two objectives Objective vectors can be visualized when k = 2, 3 Variables: values change the solution Constraints: same as in single objective problems Feasible region S: consists of all the points satisfying the constraints

21 Mathematical formulation Vector valued objective function Objective vector Image of the feasible region

22 Optimality: objective space f 2 min Z = f(s) Best values are located down and left f 1 min

23 Optimality for multiple objectives When objectives to be optimized are conflicting no single optimal solution cf. single objective optimization Compromise There are potentially infinitely many optimal solutions

24 Optimality Which solutions can be optimal? How to find such solutions?

25 Optimality: objective space f 2 min Z = f(s) Best values are located down and left f 1 min

26 Pareto optimality (PO) Mathematical definition: In other words: a vector is PO if no objective can be improved without impairing some other one Note: PO solutions can not be compared mathematically without some additional information

27 Pareto optimal set f 2 Also known as the Pareto front Consists of all the PO solutions Usually presented in the objective space e.g. k=2 PO set is a subset of two dimensional space f 2 f 2 discrete f 1 linear f 1 nonlinear f 1

28 What means solving a problem? Find all PO solutions theoretical approach, not feasible in practice Find an approximation for PO front approximation with good diversity (representatives in all parts of the PO set) and good spread (no similar solutions) can also be an approximation of some specific part of the front Find a best compromise (PO solution) requires preferences from a DM

29 How to choose a best compromise? PO solutions can not be compared mathematically without some additional information cf. ordering vectors in a plane There typically exist infinitely many PO solutions for continuous problems Additional information related to the problem considered is needed

30 Decision maker (DM) Person(s) who is an expert in the application area Is able to express preferences related to objectives e.g. is able to compare Pareto optimal solutions No need for expertize in optimization Helps in finding a best compromise

31 Weak Pareto optimality Some objective can be improved without worsening others PO solution is also weakly PO PO solutions are better but more difficult to compute than weakly PO ones f 2 Weakly PO solutions f 1

32 Ranges for the PO set Ranges for the objective function values in the PO set provide information about achievable solutions Ideal objective vector (best values) how good values can be obtained Nadir objective vector (worst values) how bad values possibly have to be accepted Usefull information in decision making Are utilized also in some MOO methods

33 Ideal objective vector Consists of best values for each objective when optimized independently smallest values for minimization k objective functions optimization of k single objective problems Ideal objective vector is not feasible!

34 Ideal objective vector f 2 min z 2 * z * Z = f(s) z 1 * f 1 min

35 Nadir objective vector Consists of worst values for each objective in the PO set largest values for minimization Generally difficult to compute, need for approximation Note: when k=2 easier to compute e.g. by using a pay-off table

36 Pay-off table Is obtained by evaluating all the objective functions in the points where ideal values were obtained i:th row the values of objectives in a point where f i has it s optimal value diagonal has z* z i nad = the worst value of the i:th column Can give either optimistic or pessimistic approximation for the Nadir objective vector (depends on the problem considered)

37 Nadir objective vector f 2 min z 2 nad z 2 * Z = f(s) z nad z 1 * z 1 nad f 1 min

38 Reference point Reference point = a vector in the objective space that contains desirable values for the objectives The components of a reference point are called aspiration levels One way for the DM to express preferences (intuitive) Utilized also in some MOO methods

39 Special case: 2 objectives PO set can be visualized (if available) If one of the objectives is improved, the other one will impair A desired solution can be chosen from the curve Very common in practical applications f 2 min f 1 min

40 Scalarizing the problem Often the idea of MOO methods is to some way convert the problem into single objective one methods of single objective optimization can be utilized This is called scalarization Can be done in a good way or in a bad way examples of scalarization will come in later lectures

41 Properties of a good MOO method Methods based on scalarization produce usually one solution at a time Good method should have the following properties produce (weakly) PO solutions is able to find any (weakly) PO solution (by using suitable parameters of the method)

42 Approaches Plenty of methods developed for MOO MOO methods can de categorized based on the role of the DM No-preference methods (no DM) Aposteriori methods Apriori methods Interactive methods The approaches are discussed more in the coming lectures just an appetizer!

43 Examples of MOO literature V. Changkong & Y. Haimes, Multiobjective Decision Making: Theory and Methodology, 1983 Y. Sawaragi, H. Nakayama & T. Tanino, Theory of Multiobjective Optimization, 1985 R.E. Steuer, Multiple Criteria Optimization: Theory, Computation and Applications, 1986 K. Miettinen, Nonlinear Multiobjective Optimization, 1999 K. Deb, Multi-Objective Optimization Using Evolutionary Algorithms, 2001

44 Examples of MOO literature M. Ehrgott, Multicriteria Optimization, 2005 J. Branke, K. Deb, K. Miettinen & R. Slowinski (eds): Multiobjective Optimization: Interactive and Evolutionary Approaches, 2008 G.P. Rangaiah (editor), Multi-Objective Optimization: Techniques and Applications in Chemical Engineering, 2009 E. Talbi, Metaheuristics: from Design to Implementation, 2009

45 Newsletter 22 nd International Conference on Multiple Criteria Decision Making June 2013, Málaga (Spain) 11th MCDA/M Summer School 2013 Helmut-Schmidt-Universität, Hamburg, Germany, July 22nd August 2nd, 2013 Membership does not cost you anything! January 23-27, 2012 Dagstuhl Seminar on Learning in Multiobjective Optimization