An Evolutionary Algorithm Approach to Generate Distinct Sets of Non-Dominated Solutions for Wicked Problems

Transcription

1 An Evolutionary Algorithm Approach to Generate Distinct Sets of Non-Dominated Solutions for Wicked Problems Marcio H. Giacomoni Assistant Professor Civil and Environmental Engineering February 6 th 7 Zechman, E.M., M. Giacomoni, and M. Shafiee (3) An Evolutionary Algorithm Approach to Generate Distinct Sets of Non-Dominated Solutions for Wicked Problems Engineering Applications of Artificial Intelligence 6(5),

2 TOOLS PROBLEMS UTSA Water Resources Systems Analysis Lab: Sustainability of the Built and Natural Environments Drought Management and Water Conservation Stormwater Management and Green Infrastructure Resilience and Security of Cyber- Physical Systems Simulation Models Hydrologic and Hydraulic Water Networks Land Use Change Population Growth Optimization Algorithms Evolutionary Computation Single/multi-objective problems Method for Generating Alternatives Data Collection and Analysis GIS and Remote Sensing Monitoring /3/ The University of Texas at San Antonio, One UTSA Circle, San Antonio, TX 7849

3 Maximize Efficiency Engineering Problems Many engineering problems have multiple objectives: Pareto front should be identified to represents the trade-off among conflicting objectives Tradeoff Curve or Non Dominated Set Cost Minimize

4 Engineering Problems Real world problems are ill-posed: Multiple perspectives (social, environmental, political) Some objectives are difficult to model mathematically It s White and Gold!!! No it s NOT!!! It s Blue and Black!!!

5 Engineering Problems Real world problems are ill-posed: Multiple perspectives (social, environmental, political) Some objectives are difficult to model mathematically

6 Engineering Problems The fitness landscapes for realistic problems, are often non-linear, complex, and multi-modal Maximize z = f x, y = [a y bx + cx d + e f cos x cos y + log x + x + + e] x y 5, [,5] a = ; b = 5. 4π ; c = 5 ; d = 6 ; e = ; f = π 8π

7 Modeling to Generate Alternatives Decision making can be aided through identification of alternative solutions GOOD SOLUTION Alternative Alternative Alternative 3

8 Modeling to Generate Alternatives Original Problem: Maximize Z k = f k (X) k =,, K. (K number of objectives) Subject to g i (X) < b i i =,, M. (M number of constraints). Optimal Solution: X*, with objective values of Z k * New Optimization Problem: Maximize D = j x j x j*. Subject to g i (X) < b i i =,, M. f k (X) > T(Z k* ). Brill, E. D. Jr. (979). The use of optimization models in public-sector planning. Management Science, 5(5), 43-4.

9 Research objective Develop an algorithm to identify a set of alternative Pareto fronts that are made up of solutions that map to similar regions of the objective space while mapping to maximally different regions of the decision space.

10 Multi-objective problems X Decision space a b F Objective space a b Pareto Front Or Non Dominated Set c c X F

11 Multi-objective multi-modal problems Decision space a a Objective space a b a b X b b F c c c c X F Two Sets of non-dominated solutions: Pareto Front - (a, b, and c) Pareto Front - (a, b, and c)

12 Multi-objective Evolutionary Algorithms to Generate Alternative Non-dominated Sets Use a set of populations to converge to alternative sets of non-dominated solutions Each subpopulation will evolve one Pareto front that is different in decision space from other subpopulations First subpopulation executes a conventional MOEA to find a typical Pareto front Secondary subpopulations find alternative Pareto fronts F Objective space a b a b F c c

13 Multi-objective Niching Co-evolutionary Algorithm (MNCA) Multiple sub-populations co-evolve to distinct sets of nondominated solutions Primary subpopulation Secondary subpopulation Secondary subpopulation f f f f f f Zechman, E.M., M. Giacomoni, and E. Shafiee (3) An Evolutionary Algorithm Approach to Generate Distinct Sets of Non-Dominated Solutions for Wicked Problems Engineering Applications of Artificial Intelligence 6(5), pp

14 Multi-objective Niching Co- Evolutionary Algorithm (MNCA). Group all solutions into clusters based on proximity in objective space K-means clustering. Distance calculation The distance of one solution is calculated in decision space to solutions that fall in the same cluster but in different subpopulations 3. Target Front A target front is created, based on the first front of non-dominated solutions from first subpopulation 4. Feasibility Assignment Label solutions in secondary subpopulations as feasible if they dominate any point in the target front 5. Selection: First Subpopulation: NSGA-II operator Secondary Subpopulations: Crowding Distance and Binary Tournament Infeasible: rank and NSGA-II Crowding distance Feasible: Crowding distance using four solutions

15 MNCA Algorithmic steps. Group all solutions into clusters based on proximity in objective space K-means clustering F F Objective space

16 MNCA Algorithmic steps. Group all solutions into clusters based on proximity in objective space K-means clustering F F Objective space

17 MCA Algorithmic steps. Group all solutions into clusters based on proximity in objective space K-means clustering Subpopulation SP Subpopulation SP Subpopulation SP n Decision Space Colors represent different clusters that are formed based on similarities in objective space

18 MCA Algorithmic steps. Group all solutions into clusters based on proximity in objective space K-means clustering. Distance calculation The distance of one solution is calculated in decision space to solutions that fall in the same cluster but in different subpopulations

19 Distance calculation Subpopulation SP Subpopulation SP Subpopulation SP n C, red C 3, red Decision Space Distance is calculated for each solution to centroid of same cluster in other subpopulations Solution Red,i Distance = minimum(distance to C, red ; Distance to C 3, red )

20 MNCA Algorithmic steps. Group all solutions into clusters based on proximity in objective space K-means clustering. Distance calculation The distance of one solution is calculated in decision space to solutions that fall in the same cluster but in different subpopulations 3. Target Front A target front is created, based on the first front of non-dominated solutions from first subpopulation

21 Target front Z i = T(Zi WP i ) + WP i Z i is the a point on the target front Z i is the value of the i th objective T is the target reduction (i.e., 8%) WP i is the worst point for the i th objective F Target Front F Objective space

22 MNCA Algorithmic steps. Group all solutions into clusters based on proximity in objective space K-means clustering. Distance calculation The distance of one solution is calculated in decision space to solutions that fall in the same cluster but in different subpopulations 3. Target Front A target front is created, based on the first front of non-dominated solutions from first subpopulation 4. Feasibility Assignment Label solutions in secondary subpopulations as feasible if they dominate any point in the target front

23 Target front Z i = T(Zi WP i ) + WP i Z i is the a point on the target front Z i is the value of the i th objective T is the target reduction (i.e., 8%) WP i is the worst point for the i th objective Infeasible Feasible F Target Front F F Objective space F Objective space

24 MNCA Algorithmic steps. Group all solutions into clusters based on proximity in objective space K-means clustering. Distance calculation The distance of one solution is calculated in decision space to solutions that fall in the same cluster but in different subpopulations 3. Target Front A target front is created, based on the first front of non-dominated solutions from first subpopulation 4. Feasibility Assignment Label solutions in secondary subpopulations as feasible if they dominate any point in the target front 5. Selection: First Subpopulation: NSGA-II operator Secondary Subpopulations: Crowding Distance and Binary Tournament Infeasible: rank and NSGA-II Crowding distance Feasible: Crowding distance using four solutions

25 Crowding Distance Infeasible Solutions Feasible Solutions s s s 4 f a s f b s s 3 f f cd b s i i (NSGA-II) cd a 4 s i i

26 Binary Tournament Infeasible Feasible Infeasible X Feasible F Feasible WINS!!! F Objective space

27 Binary Tournament Infeasible Feasible Feasible Feasible Same Cluster X F Higher Distance WINS!!! F Objective space Solutions within the same region of objective space: Maximize Distance increasing diversity.

28 Binary Tournament Infeasible Feasible Feasible Feasible Different Clusters X F F Objective space Higher Modified Crowding Distance WINS!!! Solutions are from different parts of the objective space: pressure to improve the coverage across the Pareto front.

29 Binary Tournament Infeasible Feasible Infeasible X Infeasible F F Objective space Lower Front Rank or Crowding Distance WINS!!!

30 Niching-CMA Covariance Matrix Adaptation Evolution Strategy Niching Technique. Group solutions in niches based on their proximity in both decision and objective space. Outperformed other multi-objective and diversity-enhancing methodologies. O. Shir and T. Beck, Niching with derandomized evolution strategies in artificial and real-world landscapes, Natural Computing, vol. 8, pp. 7 96, 9,.7/s

31 Algorithmic Settings Parameter Setting Population Size 5 Subpopulations Generations Mutation % Clusters 5 Target 9% (95%) 95% was used for the function Two-on-One

32 Hypervolume Represents the size of the space covered by the nondominated set Used as an indicator to measure the quality of a nondominated set. Global Worst Point.8 Minimize f (x) f (x) Minimize

33 Decision Space Diversity Assess diversity in decision space based on the distance between all pair of individuals in a population Average of the distances between pairs of solutions and is normalized by the diameter of the decision space O. Shir and T. Beck, Niching with derandomized evolution strategies in artificial and real-world landscapes, Natural Computing, vol. 8, pp. 7 96, 9,.7/s

34 Paired Solution Diversity New metric to assess set of solutions that are distant in decision space though similar in objective space. Pair solution of one subpopulation with solution of another subpopulation that is nearest in objective space, and for each pair, calculating the Euclidean distance in the decision space.

35 Paired Solution Diversity Objective space Decision space Decision space a b a d d a b a a b a F c c X b c d d 3 b c X c b d d3 c F X X diversity ps d d d 3 3

36 Test Function Two-on-One Two-on-One is a bi-modal problem composed of a fourth degre polynomial with two optima and a second-degree sphere function. f f x x f x Decision Space x f Objective Space M. Preuss, B. Naujoks, and G. Rudolph, Pareto Set and EMOA Behavior for Simple Multimodal Multiobjective Functions, in Parallel Problem Solving from Nature - PPSN IX, 6, pp

37 Decision Space Decision Space Diversity Hypervolume Gen # Subpopulation Subpopulation Function Two-on-One

38 Function Two-on-One Niching-CMA Niching-CMA MNCA Subpopulation Subpopulation x - Decision Space - x - x x x x f 3 Objective Space f f f f f

39 Test Function Deb99 Deb99 is a deceptive multi-objective problem that has a global optima difficult to identify and a local optima located in a long flat valley that is easy to find. g(x ) F X Decision Space F Objective Space M. Preuss, B. Naujoks, and G. Rudolph, Pareto Set and EMOA Behavior for Simple Multimodal Multiobjective Functions, in Parallel Problem Solving from Nature - PPSN IX, 6, pp

40 Deb99 Test Function First subpopulation Second subpopulation

41 Function Deb99 Niching-CMA MNCA Subpopulation Subpopulation x.8.8 Decision Space x x Objective Space f.5.5 x f f.5 x f f.5 x f

42 Lamé Supersphere Lamé Supersphere is a multi-modal problem global with spherical geometry in objective space and equidistant parallel lines in decision space. X F X F M. Preuss, B. Naujoks, and G. Rudolph, Pareto Set and EMOA Behavior for Simple Multimodal Multiobjective Functions, in Parallel Problem Solving from Nature - PPSN IX, 6, pp

43 Function Lame Supersphere 5 MCA Niching-CMA MNCA MCA MNC Subpopulation Subpopulation Subpopulation Subpopulation Subpopulation Decision Space Objective Space

44 Test Function Hypervolume Niching CMA MNCA Subpopulation Subpopulation Two-on-One 69.6 ± ± ± 3. Deb ± ± ±.35 Lamé Superspheres 3. ± ±..95 ±.4 Decision Space Diversity Test Function Niching CMA MNCA Two-on-One.3 ±.3.98 ±.3 Deb99.85 ± ±.5 Lamé Superspheres.39 ±.39. ±.7 Paired Solution Diversity Test Function MCA MNCA Two-on-One.58 ± ±.49 Deb99.3 ±.6.47 ±.8 Lamé Superspheres.79 ±.5.59 ±.4

45 Realistic Planning Problem: Airline Routing o Determine a set of 8 routes to connect 8 cities to maintain objectives: o minimize f : Cost o minimize f : Number of stops o Given: o Number of requests at each city o Cost of initial set-up o Cost of each flight o Unit cost for each passenger o Potential type of connections: o Directly o Indirectly- through multiple-leg connections

46 Settings for MNCA Parameter Setting Population Size Subpopulations 3 Generations Mutation % Clusters 5 Relaxation coefficient 9%

47 Stops Distinct sets of non-dominated solutions 3 Subpopulation 3 Subpopulation 3 Subpopulation Cost (M$) Cost (M$) Cost (M$)

48 Cosrt (M$) Twenty five-stop alternative solutions BOS BOS CHI CLE CHI CLE CIN BAL CIN BAL DFW DFW ATL ATL HOU Subpopulation HOU Subpopulation BOS CHI CIN CLE BAL DFW HOU ATL Subpopulation Subpopulation

49 Stops Distinct sets of non-dominated solutions 3 Subpopulation 3 Subpopulation 3 Subpopulation Cost (M$) Cost (M$) Cost (M$)

50 Cost (M$) Eighteen-stop alternative solutions BOS BOS CHI CLE CHI CLE CIN BAL CIN BAL DFW DFW ATL ATL HOU Subpopulation HOU Subpopulation BOS CHI CLE DFW CIN BAL HOU ATL Subpopulation Subpopulation

51 Stops Distinct sets of non-dominated solutions 3 Subpopulation 3 Subpopulation 3 Subpopulation Cost (M$) Cost (M$) Cost (M$)

52 Cost (M$) Seventeen-stop alternative solutions BOS BOS CHI CLE CHI CLE CIN BAL CIN BAL DFW DFW ATL ATL HOU Subpopulation HOU Subpopulation BOS CHI CLE CIN BAL DFW HOU ATL Subpopulation Subpopulation

53 Where should Osaka Prefecture University be located?

54 OFU Problem Formulation decision variables and 4 objective values f (x,x ) = min distance to the nearest elementary school f (x,x ) = min distance to the nearest convenience store f 3 (x,x ) = min distance to the nearest junior-high school f 4 (x,x ) = min distance to the nearest railway station

55 OFU Algorithm settings Parameter Setting Population Size Subpopulations Generations Mutation % Clusters 5 Target 95%

56 f 3 = min distance to the nearest junior-high school f 3 = min distance to the nearest junior-high school Results - Objective Space Subpopulation Subpopulation 4 4 f = min distance to the nearest convenience store f = min distance to the nearest convenience store

57 Results - Decision Space Subpopulation Subpopulation

58 Conclusions Results demonstrate the ability of a new algorithm, MNCA, to identify sets of nondominated solutions for test problems. Outperforms state-of-the-art (Niching-CMA) in diversity algorithms for multi-objective problems. A new metric to assess diversity among subpopulations was developed. MNCA maximizes diversity in decision space while identifying complete alternative Pareto fronts.

59 Thank you for your Attention!

60 Lamé Superspheres Deb99 Two-on-One Number of Clusters 5 5 Hypervolume Paired Solution Diversity Cluster Cluster Cluster Cluster Cluster Cluster MCA MNCA

61 Lamé Superspheres Deb99 Two-on-One Number of Subpopulations Hypervolume Subpopulations Subpopulations Subpopulations MCA Paired Solution Diversity MNCA Subpopulations Subpopulation Subpopulations

62 Deb99 Two-on-One Targets 5 5 Hypervolume Paired Solution Diversity Target Target Target Target