Briant, Olivier

Cambazard, Hadrien

Cattaruzza, Diego

Catusse, Nicolas

Ladier, Anne-Laure

Ogier, Maxime

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HAL Id: hal-01765113https://hal.archives-ouvertes.fr/hal-01765113Submitted on 12 Apr 2018HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.A column generation based approach for the joint orderbatching and picker routing problemOlivier Briant, Hadrien Cambazard, Diego Cattaruzza, Nicolas Catusse,Anne-Laure Ladier, Maxime OgierTo cite this version:Olivier Briant, Hadrien Cambazard, Diego Cattaruzza, Nicolas Catusse, Anne-Laure Ladier, et al.. Acolumn generation based approach for the joint order batching and picker routing problem. 19èmeCongrès annuel de la Société Française de Recherche Opérationnelle et d’Aide à la Décision (ROADEF2018), Feb 2018, Lorient, France. pp.1-2. ￿hal-01765113￿A column generation based approach for the jointorder batching and picker routing problemOlivier Briant1, Hadrien Cambazard1, Diego Cattaruzza2,Nicolas Catusse1, Anne-Laure Ladier3, Maxime Ogier21 Univ. Grenoble Alpes, CNRS, Grenoble INP, G-SCOP, 38000 Grenoble, France{olivier.briant,hadrien.cambazard,nicolas.catusse}@grenoble-inp.fr2 Univ. Lille, CNRS, Centrale Lille, Inria, UMR 9189 - CRIStAL, F-59000 Lille, France{diego.cattaruzza,maxime.ogier}@centralelille.fr3 Univ Lyon, INSA Lyon, DISP laboratory EA 4570, F-69100 Villeurbanne, Franceanne-laure.ladier@insa-lyon.frMots-clés : order batching, picker routing, column generation.1 IntroductionPicking is the process of retrieving products from the inventory and is often considereda very expensive operation in warehouse management. A set of pickers perform routes intothe warehouse, pushing a trolley and collecting items to prepare customer orders. However,customer orders usually do not fit the capacity of a trolley. They are typically grouped intobatches, or on the contrary divided into subsets, with the aim of collecting all the orders byminimizing the walked distance. This problem is known as the joint order batching and pickerrouting problem [1].This work presents an exponential linear programming formulation where variables or co-lumns are related to single picking routes in the warehouse. More precisely, a column refers toa route involving a set of picking operations and satisfying the side constraints required at thetrolley level such as the mixing of orders or the capacity. Computing such a picking route isan intractable routing problem in general and, depending on the warehouse layout, can closelyrelate to the traveling salesman problem (TSP). The rational of our approach is however toconsider that the picking problem alone, in real-life warehouses, is easy enough in practice tobe solved exactly. We apply this approach on two different industrial benchmarks, based ondifferent warehouse layouts.2 Problem specification and industrial applicationsThe warehouse layout is modeled as a directed graph G = (V ,A) with two types of vertices,locations and intersections. Locations contain one or more product references to be picked.Two typical examples of warehouse layouts are used as benchmarks in the present work.— A regular rectangular layout made of vertical aisles and horizontal cross-aisles. Such alayout has been used by numerous authors in the past [2] to define the order pickingproblem. It is the setup of the Walmart benchmark.— An acyclic layout where pickers are not allowed to backtrack. It is another typical in-dustrial setup where the flow is constrained in a single direction and an aisle must beentered and exited on the same side. It is the setup of the HappyChic benchmark.Each product reference p ∈ P is characterized by its location in the warehouse and its sizeV wp in each dimension w ∈ W . A product reference may have several dimensions such as weightand volume and we refer to the set of dimensions as W .An order from a customer is defined as a set of order lines. An order line l ∈ L is relatedto an order o and defined as a pair (pl, Ql) where pl ∈ P is a product reference and Ql is thenumber of items to pick. An order o ∈ O is a set of order lines Lo ⊆ L. Moreover, an ordercan be split in at most Mo boxes.Order lines are collected by trolleys, each carrying a set of B boxes. A box has a capacityV w in dimension w ∈ W and an order line can be assigned into several boxes (the quantity Qlof an order line l can be split among several boxes). A box is therefore filled with partial orderlines. A partial order line l̃ is a pair (pl, Q̃l) with Q̃l ≤ Ql. A box can only contain partial orderlines from a single order.A solution is a collection of routes R in the warehouse layout G. Each route r is travelledby a trolley which collects partial order lines into its boxes. The capacities of the boxes mustbe satisfied in each dimension w ∈ W . An order o ∈ O can not be assigned to more than Moboxes. Finally all order lines must be picked with the required number of items. The objectiveis to minimize the total distance to perform all the routes in R.The two industrial cases addressed in the present work, from the Walmart and HappyChic,differ slightly. In particular, for the Walmart case, only one dimension is considered for a box,representing the maximum number of items in a box. Additionally, an order must be pickedentirely by a single trolley.3 A column generation based approachIn the industrial case of HappyChic, the picking takes place on an acyclic graph, thus boilsdown to an easy path problem. In Walmart’s case, warehouse have the regular rectangularstructure made of aisles and cross-aisles. In that case, dynamic programming algorithms cantake advantage of that structure to efficiently solve the corresponding TSP when the ware-house contains up to eight cross-aisles, which is beyond most real-life warehouse’s sizes [3]. Wetherefore assume that in both cases, an efficient oracle is available to provide optimal pickingroutes in the warehouse.We show that such an oracle allows for a very effective exponential LP formulation of thejoint order batching and picking problem. The pricing problem can be seen as a prize-collectingTSP with a capacity constraint and the pricing algorithm heavily relies on the picking oracleto generate cutting planes. A number of improvements are proposed to speed up the pricing.In particular, a procedure to strengthen the cutting planes is given when the distance functionfor the considered set of orders is submodular. For the industrial case of HappyChic, the graphis acyclic, so it is possible to propose a polynomial set of constraints to exactly calculate thedistance, instead of generating cutting planes.The proposed formulation is compared experimentally on Walmart’s benchmark and provesto be very effective, improving many of the best known solutions and providing very stronglower bounds. Finally, this approach is also applied to the HappyChic case, demonstrating itsgenerality and interest for this application’s domain.Références[1] Cristiano Arbex Valle, John E. Beasle, and Alexandre Salles da Cunha. Optimally solvingthe joint order batching and picker routing problem. European Journal of OperationalResearch, 262(3) :817–834, 2017.[2] René De Koster, Tho Le-Duc, and Keed Jan Roodbergen. Design and control of warehouseorder picking : A literature review. European Journal of Operational Research, 182(2) :481–501, 2007.[3] Hadrien Cambazard, and Nicolas Catusse. Fixed-Parameter Algorithms for RectilinearSteiner tree and Rectilinear Traveling Salesman Problem in the plane. arXiv preprintarXiv :1512.06649, 2015.

A column generation based approach for the joint order batching and picker routing problem

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A column generation based approach for the joint order batching and picker routing problem

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