G LOBAL O PTIMIZATION by Ferran Torrent Fontbona Advisors Beatriz - PowerPoint PPT Presentation

Universitat de Girona Escola Politècnica Superior D ECISION S UPPORT M ETHODS FOR G LOBAL O PTIMIZATION by Ferran Torrent Fontbona Advisors Beatriz López Ibáñez Víctor Muñoz Solà MIIACS September 2012 Girona

S UMMARY  Introduction – Motivation – Objectives – The data • State of the art • Clustering • Optimization • Conclusions • Future work 14 September 2012 2/27

M OTIVATION – Globalization of the sport events Barman decision problem – Several simultaneous sport events 10 bars broadcast match 1 8 people/bar 80 people wants match 1 20 people wants match 2 8 bars broadcast match 1 10 people/bar 2 bars broadcast match 2 14 September 2012 3/27

M OTIVATION . M ATCHING P ROBLEM • Location-allocation – Determine optimal location for one or more facilities that will service demand for a given set of points – Every facility offers the same service – Customers positions are known Complexity  𝑜 𝑙! 𝑜−𝑙 ! where 𝑜 → 𝑜𝑣𝑛𝑐𝑓𝑠 𝑝𝑔 𝑞𝑝𝑡𝑡𝑗𝑐𝑚𝑓 𝑞𝑝𝑡𝑗𝑢𝑗𝑝𝑜𝑡 𝑜! = – 𝑜𝑣𝑛𝑐𝑓𝑠 𝑝𝑔 𝑔𝑏𝑑𝑗𝑚𝑗𝑢𝑗𝑓𝑡 𝑙 𝑙 → 14 September 2012 4/27

M OTIVATION . O UR P ROBLEM • Immobile location-allocation – Given a set of facilities with known positions and a demand with known positions, determine the optimal service each facility has to offer – Facilities (bars) cannot be moved and their positions are known – Each customer desire a single service (match) from a set and it is known – Customers’ positions are known Complexity  𝑂 𝑛𝑏𝑢𝑑ℎ𝑓𝑡 𝑂 𝑐𝑏𝑠𝑡 – • Problem dimensionality – Most research does not deal with problems of the same complexity/size (the system has to deal with bars from around the world) Division of the problem into subproblems 𝑂 𝑐𝑏𝑠𝑡 𝑙 𝑙 ∙ 𝑂 𝑛𝑏𝑢𝑑ℎ𝑓𝑡 14 September 2012 5/27

O BJECTIVES • Hypothesis – We can approximate the location-allocation solution regarding bars problem by dividing the dataset converting the initial problem into several of easier subproblems. – Assumption: geographical distance is a key of the problem and clustering divides the problem according the distance. • Objectives – Divide the problem into sub-problems  clustering Data – Location-allocation (sub)problem solving  Heuristics – Experimental tests Clustering Optimization Optimization Optimization Optimization Optimization Optimization Sol. 1 Sol. 2 Sol. 3 Sol. 4 Sol. n Global Global solution solution 14 September 2012 6/27

T HE D ATA • 15578 bars from Catalunya taken from Páginas Amarillas • Customers are randomly generated from a list of matches 14 September 2012 7/27

S UMMARY  Introduction  State of the art – Clustering – Optimization • Clustering • Optimization • Conclusions • Future work 14 September 2012 8/27

S TATE OF THE A RT Clustering Optimization Clustering Optimization Fuzzy Hard Complete (EM) Incomplete (brute force, backtracking, etc.) Agglomerative Divisive (hierarchical) Local search Global search (Hill climbing) Stochastic Deterministic Without Coordinate coordinate system system Parameter- Parameter- Non-Centroid based Centroid based (PSO, FA, SO, etc.) independent dependent (GA, SA, CS) Parameter- Parameter- dependent independent (GA) (k-means ) (Affinity propagation) (Region Growing) 14 September 2012 9/27

S UMMARY  Introduction  State of the art  Clustering – Algorithms – Results • Optimization • Conclusions • Future work 14 September 2012 10/27

C LUSTERING – • Genetic algorithms based clustering Algorithms – K-means – Hierarchical clustering – Affinity propagation – Region Growing 14 September 2012 11/27

C LUSTERING R ESULTS • Hierarchical clustering 14 September 2012 12/27

C LUSTERING R ESULTS Initial complexity  𝑂 𝑛𝑏𝑢𝑑ℎ𝑓𝑡 𝑂 𝑐𝑏𝑠𝑡 = 3 15578 ≅ 4 ∙ 10 7432 • Algorithm Expended Calinski Index DB Index Number of Number of Smallest Largest Complexity time (s) clusters minimal cluster size cluster size clusters 𝟐𝟏 𝟒𝟐 k-means (setting elements as 578 28955.66 0.717 896 27 1 59 initial centroids) 10 480 k-means (empty clusters 1170 50166.93 0.499 444 74 1 1001 resignation) 10 1633 Lloyd’s algorithm 395 21958.88 0.698 17 1 137 3423 10 2810 Region growing 𝑬 𝒏𝒃𝒚 = 𝟐 km 6 2614.59 0.228 1095 521 1 5885 10 3916 Region growing 𝑬 𝒏𝒃𝒚 = 𝟑 km 12 1182.52 0.224 707 288 1 8202 Region growing 𝑬 𝒏𝒃𝒚 = 𝟔 km 10 5123 37 430.88 0.383 280 93 1 10733 10 2142 Hierarchical clustering 36636 16592.55 0.472 139 10 1 4487 10 1100 Genetic clustering 4575 15911.56 0.757 14 1 366 2305 10 331 Affinity propagation 3892 27037.92 0.665 92 1 18 690 ↓,↑ ↓,↑ ↓,↑ ↓,↑ ↓,↑ ↓,↑ 14 September 2012 13/27

S UMMARY  Introduction  State of the art  Clustering  Optimization – Mathematical model – Genetic algorithms – Simulated annealing & cuckoo search – Results • Conclusions • Future work 14 September 2012 14/27

L OCATION -A LLOCATION • Mathematical model 𝑂 𝑐𝑏𝑠𝑡 𝑂 𝑑𝑣𝑡𝑢𝑝𝑛𝑓𝑠𝑡 𝑟 𝑨 𝑗𝑘 max 2 𝑟 1 + 𝑒 𝑗𝑘 𝑨 𝑗𝑘 𝑗=1 𝑘=1 Subject to 𝑂 𝑑𝑣𝑡𝑢𝑝𝑛𝑓𝑠𝑡 𝑟 ∀ 𝑗 𝑨 𝑗𝑘 ≤ 𝐷 𝑗 𝑘=1 𝑂 𝑐𝑏𝑠𝑡 𝑟 ∀ 𝑘 𝑨 𝑗𝑘 ≤ 1 𝑗=1 𝑟 ≠ 𝑁 𝑟 = 0, 𝑟 , 𝑁 𝑘 ∈ 1, ⋯ , 𝑂 𝑛𝑏𝑢𝑑ℎ𝑓𝑡 𝑦 𝑗 𝑘 → 𝑨 𝑗𝑘 𝑦 𝑗 14 September 2012 15/27

O PTIMIZATION M ETHODS Complete methods  the number of solutions to be explored is too big × – Brute force, depth-first search, breath-first search, backtracking, etc. Local search methods  many local optimums × – Gradient based methods, hill climbing Heuristics with coordinate systems  non-coordinate solution space!! × – PSO, FA, SO, etc.  Heuristics with non-coordinate systems  find good solutions in a limited amount of time – GA, SA, CS Optimization Complete Incomplete (brute force, backtracking, etc.) Local search Global search (Hill climbing) With coordinate Without coordinate system system (PSO, FA, SO, etc.) (GA, SA, CS) 14 September 2012 16/27

G ENETIC A LGORITHMS • Chromosome • Mutation Probability 𝜈 𝑛 to change the match – • Crossover – Single point crossover • Fitness 𝑂 𝑐𝑏𝑠𝑡 𝑂 𝑑𝑣𝑡𝑢𝑝𝑛𝑓𝑠𝑡 𝑟 𝑨 𝑗𝑘 𝐺𝑗𝑢𝑜𝑓𝑡𝑡 𝑟 = 2 1 + 𝑒 𝑗𝑘 𝑗=1 𝑘=1 • Selection – Roulette rule 14 September 2012 17/27

S IMULATED A NNEALING & C UCKOO S EARCH • Non-coordinate search space  Need of a new neighborhood function – Each bar have different chances to change its match depending on the expected number of customers  Exponential probability function – Different exponential function depending on the features of the problem o    i  e  change the match of the ith bar P 1 Probability to change the match 0.9 0.8 0.7 0.6 τ=0.1 0.5 τ=0.04 0.4 τ=0.02 0.3 0.2 0.1 0 Bar occupation (%) 0 20 40 60 80 100 14 September 2012 18/27

S IMULATED A NNEALING & C UCKOO S EARCH Exponential probability with variable 𝝊 Exponential probability with 𝝊 = 𝟏. 𝟏𝟔 Variable uniform probability Constant uniform probability E 𝐹 𝐹 𝐹 % of allocated % of bars with % of allocated % of bars with % of allocated % of bars with % of allocated % of bars with customers occupation < customers occupation < customers occupation < customers occupation < 4% 4% 4% 4% 217.04 95.33 0 211.34 94.00 0 214.45 95.00 0 216.15 93.00 0 104.43 97.82 1 103.85 98.55 3 103.04 98.55 2 104.01 96.38 3 1223.49 99.43 0 1218.94 98.93 0 1221.93 98.93 0 1218.18 98.93 2 616.49 99.86 3 616.55 100 3 614.95 99.86 5 613.67 99.86 6 2010.62 100 0 2013.74 100 1 2005.71 100 8 2007.23 100 13 996.03 100 12 994.11 100 11 993.98 100 19 991.81 100 23 5579.03 99.83 1 5571.28 99.71 3 5535.93 99.73 48 5531.09 99.68 41 2622.78 99.86 20 2622.36 99.89 28 2612.07 99.96 89 2606.94 99.75 91 1 1 0.4 0.4 0.5 0.5 0.2 0.2 0 0 0 0 0 50 100 0 50 100 0 20 40 60 80 100 0 20 40 60 80 100 14 September 2012 19/27