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Variable Neighborhood Search for Bin Packing Problem Borislav Nikolić, Hazem Ismail Abdel Aziz Ali, Kostiantyn Berezovskyi, Ricardo Garibay Martinez, Muhammad Ali Awan

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The Outline Introduction – Heuristics – Local search – Metaheuristics Variable Neighborhood Search (VNS) – Overview – Basic Algorithim Scientific Paper on VNS – Scope – Minimum Bin Slack (MBS) – VNS for Bin Packing Problem (BPP) – The experiment and Results. 2

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Heuristics Techniques for speeding up the process Applicable where exhaustive search is an overkill But why? Used for NP-Hard problems (TSP, BPP, AVS, …) Easy implementation Can lead towards optimal solution But … 3

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Local search Assumes initial solution as fixed point Generates the neighborhood and compares If better -> Else finished! 4 Looks perfect But …

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Local search Unfavorable solution space Practically locked within initial solution Local optimum can be actually a very bad one Highly dependent on solution space Can be solved with some “tricks” Deliberately accept worse result to escape Looks like a gambling decision 5

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Metaheuristics Way to tackle aforementioned problems Combination of techniques For better exploration of solution space But doesn’t guarantee optimality either! More complex implementation Criticized in the no free lunch theorems* Yet, highly used 6 * “No free lunch theorems for optimization” – Wolpert & Macready

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Metaheuristics Examples: Greedy Randomized Adaptive Search Procedure (GRASP) Simulated Annealing (SA) TABU Search (TS) Ant colony optimisation Variable Neighborhood Search (VNS) 7

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VNS - Overview Relatively young metaheuristic (1995) Systematically change the neighborhood Based on three facts: A local minimum w.r.t. one neighborhood structure is not necessary so with another A global minimum is local minimum w.r.t. all possible neighborhood structures For many problems local minima w.r.t. one or several neighborhoods are close to each other 8 N. Mladenović – A variable neighborhood algorithm – a new metaheuristic for combinatorial optimization N. Mladenović & P.Hansen – Variable neighborhood search

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- list of possible neighborhoods VNS – Basic Algorithim 9 Of course, for local search Initialization (initial solution and stopping cond.) Repeat until stopping condition k = 1 Repeat until Shaking – Generate random point at k-th neighborhood Local search – find local optimum Move or not

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VNS – Basic Algorithim Stopping conditions (CPU time, # of iterations, …) Random points to avoid cycling (determinism) Can be easily improved Parallelized local search (1CPU = 1 neighborhood) Hybrids (VNS & Tabu, VNS & GRASP, …) Simplicity, precision, efficiency, robustness 10

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Variations and descendents of VNS Variable neighborhood descent (VND) Reduced VNS (RVNS) Skewed VNS (SVNS) General VNS (GVNS) VN Decomposition Search (VNDS) Parallel VNS (PVNS) Primal Dual VNS (P-D VNS) Reactive VNS Backward-Forward VNS Best improvement VNS Exterior point VNS VN Simplex Search (VNSS) VN Branching.... 11

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Scientific Paper on VNS New heuristics for one-dimensional bin packing By Krysztof Fleszar, Khalil S. Hindi Department of Systems Engineering, Brunel University Published in 2000, often cited 12 Work at the American University of Beirut Also work as consultants in industry More then 10 cooperative papers Scheduling, packing & cutting problems

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Scope Presents new heuristic for solving 1D bin- packing Combination of MBS and VNS Effective and computationaly efficient Remarkable performance on benchmarks Next slides explains: – MBS algorithim – VNS for BPP – Experiments platform and Results 13

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MBS Small elements finer grained and used early Stops search only on slack = 0 Odd bins and even elements -> big problem 14 - Non-increasing list of remaining elements - Number of remaining elements - Size of i-th element - Remaining slack in bin A Can be easily modified (MBS’) Use non-increasing order to stop search Outperforms FFD and BFD

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VNS for BPP OK, MBS for Initial solution, but what after? Just a little bit formal: Definition of possible “moves” (perturbations) Transfers Swaps Shaking: k-th neighborhood – performing k random moves Element can be moved only once per neighborhood All moves are treated as equal, without analyzing Of course, there is no point to swap elements of the same size! 15

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VNS for BPP OK, next is local search Metric for finding a local optimum Transfers Swaps 16 Always chooses move with the greatest value 0.5 0.2 0.4 0.5 0.3 0.2 0.3 0.5 0.2 0.4 0.5 0.3 0.2 0.3

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VNS for BPP Steepest descent approach Might not always be the best idea! But let’s analyze performance In near-optimal solution very few “moves” Recall: - max # of neighbors visited Tradeoff - computation time vs good optimum Good experimental results with = 20 17

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The experiment & the results Borland Pascal on Pentium II 400MHz Problem classes: U class – uniform distribution (20, 100) into bins of 150 T class – “triplets” of elements from (25, 50) into bins of 100 B class – divided in B1, B2, B3 problems, computationally very hard B1 – 704/720 solved optimally, items from (50, 500) B2 – 477/480 solved optimally, items from (50, 500) B3 – uniform distribution (20k, 35k) into bins of 100k, 3/10 solved optimally 18 For 4 problems, better solutions were found But not proved to be optimal!

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The experiment & the results Algorithmhitsinferiorabs.dev.avabs.dev.maxrel.dev.avrel.dev.maxtime av.time max. MBS6786921.0291.4216.670.058.99 MBS’10513190.3890.6114.290.024.57 VNS12481220.0920.185.000.094.57 MBS’+VNS1329410.0320.042.940.145.05 19 MBS’ + VNS vs the rest Robustness test with different random seed In 1340/1370 the same solution in all 10 runs In remaining 30 the difference was always 1 bin

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Conclusions VNS very powerful tool Can be easily implemented Can be easily adapted for different problems Increasing is not always fruitful Combined with MBS’ gives great results Practical complexity < theoretical complexity 20

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