Multiobjective Optimization. Carlos A. Santos Silva May 29 st, 2009

Transcription

1 Multiobjective Optimization Carlos A. Santos Silva May 29 st, 2009

2 Motivation Usually, in optimization problems, there is more than one objective: Minimize Cost Maximize Performance The objectives are often conflicting: Minimize Cost implies minimizing performance Maximize Performance implies maximize cost Wind 15 Cost( ) Wind 5 Performance(kW/year) ,5 Solar What is the best solution? Solar

3 A possible approach is To transform multiple objectives into a single objective Maximize Profit (with performance measured in terms of cost) Profit is a simple sum of cost and performance If objectives are equally important... Wind (10) Profit ( ) (10) (17,5) Solar What is the best solution?

4 Are objectives equally important? If objectives are not equally important What is the relative importance between them? Wind One has to decide a priori the relative importance of objectives Is Performance more important than Cost? (25) Profit ( ) (30) (42,5) Is Performance more important than Cost? How much? 2 times? 10 times? Solar 2x Cost = Performance Wind (5) Profit ( ) Wind 35 Profit ( ) (0) 80 (10) 50 Cost = 2x Performance Solar Cost = 10xPerformance What is the best solution? Solar

5 What if objectives are not comparable? Often, objectives are often non- commensurable Expressing performance in monetary units might be impossible Example:2 star hotel by 50 or 4 star hotel by 150 Is each star valued as 50? Does a 1 star hotel worth 0? Is it the same pay 100 by a 3 star hotel, 150 by a 4 star or 200 by a 5 star Even if cost of stars is not linear, is it possible to compare both objectives in the same unit?

6 Why not compare solutions? Another approach is to evaluate solutions for both objectives and let someone (Decision Maker) choose the best solution Performance 10 Best performance 5 Best Cost Cost Decision Maker decides is paying extra 5 is worth to have an extra 5 in performance!

7 MULTIOBJECTIVE OPTIMIZATION

8 General Description Multiobjective optimization Choosing the best solution considering different, usually contradictory objectives Usually, there is no single best solution, but a set of solutions that are equally good Methodology A posteriori (Decision Maker defines preferences based on optimization) Modeling Optimizing Deciding A priori (DM defines preferences before optimization) Also know as Multicriteria decision Making (MCDM) Multicriteria decision aiding (MCDA) Multatribute decision making (MADM) If all functions are linear Multiobjective Linear Programming (MOLP)

9 Definition Domain x = (x1, x2,, xn) Cost function f(x) = f1(x) f2(x) fk(x)) Multi-objective problem: minimize f( x) subject to gm( x) 0, m= 1, K, ng h ( x) = 0, m= n + 1, K, n + n x [ x, x ] m g g h min max

10 What is an optimum in this case? Improving in one objective may deteriorate another Balance in trade-off solutions is achieved when A solution cannot improve any objective without degrading one or more of the other objectives. f 1 f 2 A B

11 Pareto Optimum Pareto improvement change from one allocation to another that can make at least one individual better off without making any other individual worse off is called a Pareto improvement Pareto Optimum An allocation is defined as Pareto efficient or Pareto optimal when no further Pareto improvements can be made These solutions are called non-dominated solutions. The set of these solutions is a non-dominated set or the Pareto-optimal set. The corresponding objective vectors are referred to as the Pareto-front. Weak Pareto Optimum there are alternative allocations where at least one objective would be worse Vilfredo Pareto

12 Multiobjective Optimization All Pareto optimal can be regarded as equally desirable and we need a decision maker to identify the most desirable among them

13 Types of Approaches Non interactive Basic NonPreference Others Iterative Trade-off Reference Point Classification Based Evolutionary Evolutionary algorithms Ant Colonies Particle Swarm Have proven to be the best methodologies

14 NON INTERACTIVE

15 Basic Methods Not really multioptimization methods Weighted method Only works well in convex problems It can be used a priori or a posteriori (DM defines weights afterwards) It is important to normalize different objectives! ε - constrained method Only one objective is optimized, the other are constraints Works for convex or non-convex problems

16 Non-preference methods DM opinion is only listened after solving the problem There is no DM or he is not expecting any special result Global criteria Minimize distance to some reference solution Depends on distance metric Neutral compromise solution Try to find the middle point of all solutions

17 Others Weighted metrics The distance to different objectives is weighted Goal Programming / Goal Attaining Define a set of aspiration goals Minimize distance to goals

18 Comparison weighted ε-constrained global criteria non-preference x x neutral solution weighted metric a priori x x x a posteriori x x x can find any Pareto Optimal solutions always Pareto Optimal x (x) x (x) (x) (x) (x) (x) goal - programming Information weights bounds weights reference point order

19 INTERACTIVE METHODS

20 General Decision Maker expresses preferences during the optimization process Only a part of Pareto solutions are found and evaluated DM does not need a global structure view of preferences Saves time and makes comparison between solutions easier Implies an active participation during optimization process Algorithm 1. Initialize (e.g. Neutral Solution) 2. Ask DM for preference 3. Evaluate a new set of solutions Usually has two phases Learning phase for DM Real Decision Making phase

21 Trade-off Methods Trade-off Rate of exchange between two objectives (how much you win / how much you loose) Trade-off computation helps DM to know which region should be explored Zionts-Wallenius or ISWT methods Ask DM to express preferences and evaluate trade-off values Geoffrion-Dyer-Feinberg (GDF) or SPOT or GRIST methods DM provides subjective trade-off values

22 Reference Point Approaches Decision maker provides: Preference values for the outcomes (reference points) Relative order between objectives DM may change reference points Based in three principles: 1. Considers separation between preferential and substantive methods 2. Objective aggregation is nonlinear (different from weighted basic approach) 3. Holistic perception of objectives Signal substantive changes in objective values Stopping criteria When the DM is satisfied with solution

23 Classification-Based Methods DM chooses which objective functions should be improved and which ones can be maintain the value DM can also indicate intervals of improvements Similar to reference point methods Step method At each iteration, DM indicates acceptable values and unacceptable values DM gives up a little bit on acceptable values to improve unacceptable Satisficing Trade-off method DM is asked to define the objectives into three classes: acceptable, to relax, to improve DM defines bounds for trade-offs (aspiration levels) NIMBUS method DM defines 5 classes of objectives DM receives up to 4 Pareto Optimal solutions

24 EVOLUTIONARY MULTIOPTIMIZATION (EMO)

25 Ideal Multiobjective Optimization The strength is the fact that parallel solutions are computed at the same time

26 EVOLUTIONARY ALGORITHMS

27 Approaches Vector Evaluated GA (VEGA), (Shaffer, 1985). Multi-Objective GA (MOGA), (Fonseca & Fleming, 1993) Non-dominated Sorting GA (NSGA), (Deb et al., 1994). Niched Pareto GA (NPGA), (Horn et al., 94) Target Vector approaches, (several authors) NSGA II, (Deb et al., 2002). Deb K, Pratap A, Agarwal S, et al. A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Transactions on Evolutionary Computation 6 (2): Apr 2002.

28 VEGA With M objectives to be handled, population is divided by the objectives. Each subpopulation has its own fitness. Advantages: only selection mechanism is modified, so it is easy to implement and efficient (computational complexity is the same). Drawbacks: difficult to find good compromise solutions, as each solution is looking only to individual objective function. It can happen that few points of the Pareto front are found.

29 VEGA implementation on TSP Optimize Distance (Z1) and Time (Z2) 1. Initialize 2. Separate to Selection Z1=69, Z2=3 Z1=64,Z2= Z1=65, Z2=2,5 Z1=66,Z2=2 Z1= Z2=2, Shuffle to crossover and mutate Z1=65 Z2=2 Z1=64 Z1= Z2=2, Z2=2

30 MOGA Differs from VEGA in the way fitness is assigned to a solution: A rank is assigned to each solution r i = 1 + n i, where n i is the number of solutions that dominate solution i. Fitness is related to the inverse of ranking. This simple procedure does not assure diversity among non-dominated solutions. A niche-formation method was introduced to distribute the population over the Pareto-optimal region. Advantages: fitness assignment scheme is simple. Can find spread Pareto-optimal solutions. Drawbacks: introduce unwanted bias towards some solutions. May be sensitive to the shape of Pareto-optimal front.

31 Example Objectives: minimise internal temperature gradient, minimise heat loss, minimise area of the evaporator Design variables: height of evaporator bottom, evaporator depth. evaporator thickness, evaporator width, insulation thickness Geometric constraints: each parameter has a minimum and a maximum bound Fixed dimensions: outside dimensions of the fridge, size of the condenser Design evaluators: STAR-CD CFD/Heat Transfer Commercial Code

32 NGSA II (Elitist Non-Dominating Sorting GA) This method differs from previous in: Uses an elitist principle (sort by fitness before selection) Uses an explicit diversity preserving mechanism (Crowding distance) Emphasizes non-dominated solutions (classify solutions in three fronts)

33 ANT COLONY OPTIMIZATION

34 ACO approaches (MOACO) Multi-colony algorithms Multiple pheromone matrices algorithms. Multiple heuristic functions algorithms

35 Multi Colony Algorithm Each colony optimizes one objective. Having k objectives, a total of k colonies is used. Colonies cooperate by sharing information about the solutions found by each colony. Local sharing: is performed after next node is added to current path of a new partial solution. Solutions are grouped into non-dominance solutions. Fitness value fij is calculated for the best solution so far. Global sharing: similar process but now it is performed after completion of paths.

36 Multiple pheromone and/or heuristic matrices Two objectives: two pheromone matrices and two heuristic matrices (Iredi, 2001): Having Kobjectives (Doerner, 2004): Single pheromone function and several heuristics information functions (Barán and Schaerer, 2003):

37 COMPARISON BETWEEN EA AND ACO

38 Example: TSP Traveling Salesman Problem with multiple objectives: cost, length, travel time, tourist attractiveness. Used approaches:

39 Results for KROAB50

40 Results for KROAB100

41 PARTICLE SWARM

42 MO Particle Swarm Optimization (MOPSO) Uses Archive Mechanism (A) List of non-dominated solutions Use a swarm like for single objective Evaluate each solution to see if it is nondominated or not Evaluate pbest and gbest for each of the objectives Similar to VEGA approach

43 SOFTWARE

44 Matlab Goal Programming / Goal Attain x = fgoalattain(fun,x0,goal,weight) Evolutionary MultiObjective Optimization

45 READINGS

46 Energy Systems Two objective functions Cost Emissions NonInteractive Approaches ε Constrained and Goal Attained Green Building Design Two objective functions Lyfe Cycle Cost Lyfe Cycle Environment Impact EA approach MOGA