Matheuristics for several vehicle routing problems H Minh

Matheuristics for several vehicle routing problems Hà Minh Hoàng ORLab, University of Engineering and Technology, VNU 16 th August, 2018 0

Outline • • ▫ ▫ ▫ ▫ ▫ Introduction ORLab Vehicle Routing Problem Mixed integer linear programming Metaheuristics Matheuristics Decomposition approaches Improvement heuristics Column generation-based approaches Applications Covering tour problems Vehicle Routing Problem with synchronization constraints Conclusions 1

Outline • • ▫ ▫ ▫ ▫ ▫ Introduction ORLab Vehicle Routing Problem Mixed integer linear programming Metaheuristics Matheuristics Decomposition approaches Improvement heuristics Column generation-based approaches Applications Covering tour problems Vehicle Routing Problem with synchronization constraints Conclusions 2

ORLab • Newborn laboratory at UET-VNU • Research interests: ▫ Combinatorial optimization problems: vehicle routing, scheduling, packing, etc. ▫ Apply OR techniques to solve problems in Artificial Intelligence, Machine Learning, Data Science. ▫ R&D services for industry • Some selected partners: 3

Vehicle routing problem (VRP) • ▫ ▫ ▫ • • ▫ Input: graph G = (V, E) n customers with location and demand quantity All-pair distances in E. Homogeneous fleet of m vehicles with capacity Q located at a depot. Output: Minimum cost delivery routes (at most one route per vehicle) to service all customers. A Scopus search “Vehicle Routing” for 2007 -2011 returns 1258 publications, including 566 journal papers. 4

VRP in real life 5

Combinatorial explosion • 100 customers, 1 vehicle, the number of feasible solutions: 100! = 933262154439441526816992388562667004907159682643816214685 929638952175999932299156089414639761565182862536979208272 23758251185210916864000000000000 = 10 158 • Compute on a grid of computers ▫ # of CPUs = # of atoms in the universe = 1080 ▫ Do a calculation in Planck time: 5, 39 * 10 - 44 (seconds) ▫ To check all possible solutions: 5, 39 * 10 - 44 * 10158/1080 = 5, 39 * 1034 (seconds) ▫ Age of universe : 4, 33 * 1017 seconds From slides of Thibaut Vidal (SOICT, Nha Trang 2017) 6

Mathematical programming • Mixed Integer Linear Programming (MILP) • Can be modeled and solved by a number of commercial and open-source tools: CPLEX, Gurobi, SCIP, OR-Tools, Xpress-MP, etc. 7

2 -index formulation for VRP 8

VRP formulation • Set partitioning formulation 9

State-of-the-art methods for VRPs • Efficient exact methods are based on MILP formulations • Current best exact algorithm for the VRP is based on set partitioning formulation and can solve exactly instances with around 300 -400 nodes. (Uchoa et al. , 2013) • Massive research on heuristics. The current best metatheuristic for the VRP and other variants is dedicated to Vidal et al. (2013). 10

Metaheuristic • The term “metaheuristic” comes from the composition of two Greek words: ▫ Heuristic comes from heurikein: “to find” ▫ Meta: “beyond, in an upper level” • Classification: ▫ One-point based method �Iterated Local Search (ILS), Tabu Search (TS), Large Neighborhood Search (LNS), Variable Neighborhood Search (VNS), etc. ▫ Population-based methods: �Genetic Algorithm (GA), Ant Colony Optimization (ACO), etc. ▫ Hybrid methods 11

Metaheuristic for VRP in a nutshell • Three main components: ▫ Initialization �Create first solutions �Constructive, greedy algorithms • Local search operators ▫ Start from a feasible solution and improve it by applying small (‘local’) modifications ▫ VRP: swap, relocate, 2 -opt, or-opt, etc. • Perturbation (shaking) ▫ Escape from local minima 12

Outline • • ▫ ▫ ▫ ▫ ▫ Introduction ORLab Vehicle Routing Problem Mixed integer linear programming Metaheuristics Matheuristics Decomposition approaches Improvement heuristics Column generation-based approaches Applications Covering tour problems Vehicle Routing Problem Conclusions 13

Matheuristic = Mathematical programming + metaheuristic • Matheuristics make use of mathematical programming models in a heuristic framework • Survey: ▫ Claudia Archetti, M. Grazia Speranza. A survey on matheuristics for routing problems (2014). European Journal on Computational Optimization. 2: 4, pp 223– 246 14

Decomposition approaches • The problem is divided into smaller and simpler subproblems which can be solved through mathematical programming model. • Particularly suitable for the complete and integrated problems: ▫ Inventory Routing Problems (IRP), ▫ Production Routing Problems (PRP) ▫ Location Routing Problems (LRP). 15

Decomposition approaches • Cluster first-route second ▫ Assign customers to vehicles: solve a MILP that is very similar to a knapsack problem ▫ Sequence the customers visited by each vehicle: solve a TSP (Concorde). • Two-phase method ▫ Based on specific structure of the problem • Among others 16

Improvement heuristics • Combine a heuristic with exact solution of a MILP model that aims at improving the solution • Solving a MILP model is often expensive, the method should be applied to verify large neighborhoods • Suitable for the framework of Large Neighborhood Search (LNS) 17

Column generation-based approaches • Exact branch-and-price method is modified to speed up the convergence, e. g. stop prematurely the column generation phase. ▫ Restricted master heuristics ▫ Heuristic branching approaches ▫ Relaxation-based approaches 18

Outline • • ▫ ▫ ▫ ▫ ▫ Introduction ORLab Vehicle Routing Problem Mixed integer linear programming Metaheuristics Matheuristics Decomposition approaches Improvement heuristics Column generation-based approaches Applications Covering tour problems Vehicle Routing Problem with synchronization constraints Conclusions 19

Covering tour problems 1. Minh Hoàng Hà, Nathalie Bostel, André Langevin, Louis-Martin Rousseau (2013). An exact algorithm and a metaheuristic for the multi-vehicle covering tour problem with a constraint on the number of vertices. European Journal of Operational Research, Volume 226, Issue 2, 16 April 2013, Pages 211 -220. 2. Tuan Anh Pham, Minh Hoàng Hà, Xuan Hoai Nguyen (2017). Solving the multi-vehicle multi-covering tour problem. Computers & Operations Research 88, 258 -278. 20

Covering tour problems • Multi-vehicle covering tour problem (m-CTP): ▫ ▫ V: nodes that can be visited T ⊂ V: nodes that must be visited W: nodes (customers) that must be covered Find a minimum length set of vehicle routes passing through a subset of V subject to capacity constraints, such that each node of W is covered m-CTP example 21

m-CTP: Applications • Construction of routes for: ▫ Vietnamese waste collection V: Potential collection points W: households r: maximal distance a laborer can walk from a household to a collection point (2 -3 km). ▫ Mobile healthcare teams ▫ Mobile library teams ▫ Location of post boxes ▫ Banking agencies ▫ Milk collection points ▫ Disaster relief problem. 22

Multi-vehicle multi-covering tour problem (mm-CTP) • In m-CTP, the demand of each customer is so small that one coverage is enough. What happens if the demand of customers is so large that one coverage is not enough multiple coverages new variant mm-CTP. Three sub-variants: • • ▫ ▫ ▫ Binary mm-CTP: one vertex of V can be visited no more than once. (1) Integer mm-CTP without overnights: one vertex of V can be visited more than one but overnight stay is not permitted. (2) Integer mm-CTP with overnights: similar to (2) but overnight stay is permitted. (3) We use graph transformations to reduce (2) and (3) to (1). 23

A matheuristic • Hybrid of Greedy randomized adaptive search procedure and Evolutionary Local Search (GRASP-ELS) • Decomposition method ▫ m-CTP = covering problem + VRP for i = 1 to m - Phase 1: Solve a covering problem - Phase 2: Solve a classical VRP - Prins et al. (2009) 24

GRASP-ELS: phase 1 • Solving the covering problem to select visited vertices • bi is random value in {1, 2, 3}. 25

Experimental results • 192 instances from the literature with up to 200 vertices • In Hà et al. (2013), GRASP-ELS provides very good results for the m-CTP. • Kammoun et al. (2016) propose a Variable Neighborhood Search (VNS) which provides slightly better results. • Phạm et al. (2017) propose a complex Hybrid Genetic Algorithm (HGA) which is now the current best approximate method for the covering tour problems. 26

Experimental results Still, GRASP-ELS has its own advantages: simple, easy to implement, flexible, less parameters. 27

Vehicle routing problem with synchronization constraints • Joint work with Duy Thinh Nguyen, Giang Hoang Pham, Thuy Do, Louis-Martin Rousseau • Same as classical VRP but: ▫ Time windows ▫ Some special customers require the services of two vehicles ▫ The duration between two starting service times at a customer is limited to a given value Special customers Regular customers Depot 28

VRP with synchronization: applications • Delivery of large items like furniture, appliances, home theater ▫ Unloaded by drivers ▫ Installation team • Cable companies providing internet services ▫ Hardware team and software team 29

Vehicle routing problem with synchronization constraints • H Hojabri, M Gendreau, JY Potvin, LM Rousseau (2018). Large neighborhood search with constraint programming for a vehicle routing problem with synchronization constraints. Computers & Operations Research 92, 87 -97. • An approach very similar to matheuristic but Constraint Programming is used instead of Mathematical Programming. • Why not mathematical programming? 30

Large Neighborhood Search • The most challenging task is to find an efficient way to check if an unvisited node can be inserted to a location of a partial solution? 31

ALNS for the VRP with synchronization constraints • How to check if an unvisited node can be inserted in between two adjacent visited nodes? ? ? ▫ Compute the maximal delay on each arc such that time windows and synchronization constraints are still satisfied and then Depot ▫ Check insertion cost 32

ALNS for the VRP with synchronization constraints • Computing max delay problem can be easily formulated as a linear programming problem (no integer variable) • But but we have to solve around one million LP models for a small instance with only 25 nodes, 13 synchronizations • A lot of them have very similar structure • Changing a few coefficient instead of building models from the scratch will save a lot of running time 33

Compute max delays 34

Compute max delays • Check following conditions 35

ALNS for the VRP with synchronization constraints • But max delay is not enough for checking if an unvisited special node can be inserted into two locations on two vehicles. • Must check the feasibility of the solution whenever a special node is inserted (again by solving LP models) • Still compute the maximal delays to reduce the number of checked models • Other acceleration techniques 36

Experimental results • Instances from Hojabri et al. (2018) with up to 200 nodes and 100 synchronizations • Excellent results, improve most instances in much shorter computational time 37

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Conclusion • Matheuristics are suitable for complex problems with multiple decisions • More and more complex variants in real-world application, huge room for matheuristics ▫ Delivery with drones ▫ Electric vehicles ▫ Autonomous vehicles • Disadvantage: matheuristic is very time-consuming try other methods before thinking to matheuristics • Advantage: less effort on implementation, quite flexible for many variants 39

Thank you! Hà Minh Hoàng ORLab, University of Engineering and Technology, VNU 16 th August, 2018 40