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Josu Ceberio

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Previously… EDAs for integer domains. EDAs for real value domains. Few efficient designs for permutation- based problems. POOR PERFORMANCE EHBSA and NHBSA (Tsutsui et al.)

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Distance-based ranking models The Mallows model is a distance-based exponential model. Two parameters Consensus ranking, Spread parameter, Probability distribution

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Distance-based ranking models Kendall’s tau distance Decomposition of the distance Factorization of the probability distribution 12345623165420021

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Distance-based ranking EDA Generalized Mallows EDA is proposed. A generalization of the Mallows model. spread parameters. Probability distribution

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The problem To check the performance we approach: Permutation Flowshop Scheduling Problem. Extensively studied. The Mallows EDA demonstrated good performance.

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Permutation Flowshop Scheduling Problem Given a set of n jobs and m machines and processing times p ij. Find the sequence for scheduling jobs optimally. Optimization criterion: Total Flow Time (TFT). Codification 1 3 2 5 4 m1m1 m2m2 m3m3 m4m4 j1j1 j3j3 j2j2 j5j5 j4j4 Example Objective function

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Generalized Mallows EDA Preliminary experiments Spread parameters

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Generalized Mallows EDA Preliminary experiments GM model convergence

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Generalized Mallows EDA Approximating spread parameters Newton-Raphson An upper bound for the spread parameters is fixed!!

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Generalized Mallows EDA Approximating spread parameters

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Standart evolutionary shape Restart mechanism shape Generalized Mallows EDA Preliminary experiments Restart mechanism Improvement !

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PFSP state-of-the-art LR(n/m) GA VNS Crossover VNS Asynchronus Genetic Algorithm (AGA) – Xu et al. 2009 Local Search (Swap) Local Search (Insert) Shake

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PFSP state-of-the-art LR(n/m) Local Search (Swap) Local Search (Insert) Shake Variable Neighborhood Search 4 (VNS 4 ) – Costa et al. 2012

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PFSP state-of-the-art Fundamentalist approaches rarely achieve optimum solutions. Hybridization is the path to follow. High presence of VNS algorithms.

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First approach to the PFSP GM-EDA does not succeed. An hybrid approach is considered: Hybrid Generalized Mallows EDA (HGM-EDA)

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Hybrid Generalized Mallows EDA Generalized Mallows EDA Local Search (Swap) Local Search (Insert) Orbit Shake VNS

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Experimentation Algorithms: AGA, VNS 4, GM-EDA, VNS and HGM-EDA. 20 repetitions Taillard’s PFSP benchmarks: 100 instances 20 x 05 20 x 10 20 x 20 50 x 05 50 x 10 50 x 20 100 x 05 100 x 10 100 x 20 200 x 10 200 x 20 500 x 20

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Experimentation Spread parameters upper bound. Select the upper-theta that provides the best solutions for GM-EDA Stopping criterion: maximum number of evaluations. Evaluations performed by AGA in n x m x 0.4s.

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Experimentation Taillards benchmark 20 x 520 x 1020 x 20 AGA 139322000332911 VNS 4 139322000332911 GM-EDA 139342000920003 VNS 139322000332911 HGM-EDA 139322000332911

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Experimentation Taillards benchmark 50 x 550 x 1050 x 20 AGA 6630185916121294 VNS 4 6675786479121739 GM-EDA 6630986948122830 VNS 6630985980121386 HGM-EDA 6630785958121317

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Experimentation Taillards benchmark 100 x 5100 x 10100 x 20 AGA 240102 288988374974 VNS 4 242974292425378402 GM-EDA 241346292472379691 VNS 240162289438375410 HGM-EDA 240122 288902374664

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Experimentation Taillards benchmark 200 x 10200 x 20500 x 20 AGA 10395071243928 6754943 VNS 4 104852012521656770472 GM-EDA 104614612525457225665 VNS 104184612464746863483 HGM-EDA 10363031237959 6861070

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Experimentation Taillard’s benchmark - Summary 20x0520x1020x2050x0550x1050x20100x05100x10100x20200x10200x20500x20 AGA ✔✔✔✔✔✔✔✔ VNS 4 ✔✔✔ GM-EDA VNS ✔✔✔ HGM-EDA ✔✔✔✔✔✔✔

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Experimentation Taillard’s benchmark – Results analysis HGM-EDA outperforms state-of-the-art results in some cases. ○ Which is the reason for the performance fall given in instances of 500x20? Biased instances? -A tabu search algorithm was used for to choose the hardest instances. We generate a random benchmark

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Experimentation Random benchmark New configurations between 200 and 500. Total: 100 instances. 250 x 10 250 x 20 300 x 10 300 x 20 350 x 10 350 x 20 400 x 10 400 x 20 450 x 10 450 x 20

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Experimentation Random benchmark - Summary 250x10250x20300x10300x20350x1 0 350x20400x10400x20450x10450x20 AGA ✔✔✔ VNS 4 GM-EDA VNS HGM-EDA ✔✔✔✔✔✔✔

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Experimentation Random benchmark – Results analysis Statistical Analysis confirms experimentation. ○ Friedman test + Shaffer’s static. HGM-EDA and AGA are definitely the best algorithms. VNS 4 results do not match with those reported. The performance falls onwards 400x20. What’s wrong with largest instances?

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Analysis – Hybrid approach Improvement ratio EDA vs. VNS

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Analysis – Generalized Mallows EDA AGA vs. GM-EDA

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Analysis – Generalized Mallows EDA Thetas convergence

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Stops prematurely!!!

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Analysis – HGM-EDA vs. AGA More evaluations Max eval.AGAHGM-EDA x1 67106506841042 x2 67086566816514 x3 67081626769335 x4 6708123 6778298 x5 6708029 6779509 x6 6708029 6775003 x7 6706879 x8 6706879 One instance of 500x20

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Analysis – Generalized Mallows EDA LR vs. GM-EDA

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Analysis – HGM-EDA vs. AGA More evaluations Max eval.AGAHGM-EDA x1 67106506841042 x2 67086566816514 x3 67081626769335 x4 6708123 6778298 x5 6708029 6779509 x6 6708029 6775003 x7 6706879 x8 6706879 One instance of 500x20

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Analysis – HGM-EDA vs. AGA More evaluations Max eval.AGAHGM-EDAGuided HGM-EDA x1 671065068410426743775 x2 670865668165146721295 x3 670816267693356732300 x4 6708123 67782986707129 x5 6708029 67795096716032 x6 6708029 67750036712273 x7 6706879 x8 6706879 One instance of 500x20

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Analysis – HGM-EDA vs. AGA More evaluations One instance of 500x20

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Conclusions Hybrid Generalized Mallows EDA is a efficient algorithm for solving the PFSP. Succeed in 152/220 instances. The participation of the GM-EDA is essential.

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Future Work - PFSP Test other parameters: evaluations, population size, theta bounds, selection size… Include information of the instance. Guided Initialization Shake the solution of the LR(n/m) to build up the population?

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Future Work – GM-EDA Set different upper bounds to the spread parameters Study other distances. Is suitable Kendall’s-tau distance? Other distances: Cayley, Ulam, Hamming Study the problem. Other problems: TSP QAP LOP (work in progress)

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Eskerrik asko Josu Ceberio Eskerrik asko Josu Ceberio

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Distance-based ranking EDA Mallows EDA Learning and Sampling 0...n - 2 1... n - 1

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