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A Linear-Size Model for the Single Picker Routing Problem with Scattered Storage

Author

Listed:
  • Laura Lüke

    (Johannes-Gutenberg University, Germany)

  • André Hessenius

    (Johannes-Gutenberg University, Germany)

  • Stefan Irnich

    (Johannes-Gutenberg University, Germany)

Abstract

We present a new approach of solving the single picker routing problem with scattered storage (SPRP-SS), which is a fundamental problem in modern warehouse operations management. The SPRP-SS assumes that SKUs of articles are stored at possibly many locations. An effective integer programming based approach relies on extending the state space of Ratliff and Rosenthal’s dynamic program for the basic single picker routing problem to accommodate the SPRP-SS. As a result, the mixed integer linear programming (MIP) formulation has a quadratic number of variables. We propose two modifications of the extended state space to retain the linearity of the models. This is achieved by replacing the quadratically growing parallel edges of the extended state space by two different types of linear-size subnetworks. These replacements lead to different state spaces and herewith different MIP formulations, for which we analyze theoretical properties such as their size and strength of the linear relaxations. We compare the new formulations with the state of the art using a collection of 800 SPRP-SS instances. The results show that the new formulations are more than competitive providing integer optimal solutions of realistic and even large-scale instances in less than two seconds on average. The second formulation outperforms the current one regarding the computational speed: For the largest instances with 200 articles to be collected, average speedups reach the factors of 3.18 and 4.87 for general and unit demand, respectively.

Suggested Citation

  • Laura Lüke & André Hessenius & Stefan Irnich, 2025. "A Linear-Size Model for the Single Picker Routing Problem with Scattered Storage," Working Papers 2502, Gutenberg School of Management and Economics, Johannes Gutenberg-Universität Mainz.
    • Handle: RePEc:jgu:wpaper:2502
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    References listed on IDEAS

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    1. Singh, Kashi N. & van Oudheusden, Dirk L., 1997. "A branch and bound algorithm for the traveling purchaser problem," European Journal of Operational Research, Elsevier, vol. 97(3), pages 571-579, March.
    2. Daniels, Richard L. & Rummel, Jeffrey L. & Schantz, Robert, 1998. "A model for warehouse order picking," European Journal of Operational Research, Elsevier, vol. 105(1), pages 1-17, February.
    3. Wildt, Constantin & Weidinger, Felix & Boysen, Nils, 2025. "Picker routing in scattered storage warehouses: An evaluation of solution methods based on TSP transformations," Publications of Darmstadt Technical University, Institute for Business Studies (BWL) 146724, Darmstadt Technical University, Department of Business Administration, Economics and Law, Institute for Business Studies (BWL).
    4. Katrin Heßler & Stefan Irnich, 2024. "Exact Solution of the Single-Picker Routing Problem with Scattered Storage," INFORMS Journal on Computing, INFORMS, vol. 36(6), pages 1417-1435, December.
    5. Boysen, Nils & de Koster, René & Weidinger, Felix, 2019. "Warehousing in the e-commerce era: A survey," European Journal of Operational Research, Elsevier, vol. 277(2), pages 396-411.
    6. Laura Lüke & Katrin Heßler & Stefan Irnich, 2024. "The single picker routing problem with scattered storage: modeling and evaluation of routing and storage policies," OR Spectrum: Quantitative Approaches in Management, Springer;Gesellschaft für Operations Research e.V., vol. 46(3), pages 909-951, September.
    7. H. Donald Ratliff & Arnon S. Rosenthal, 1983. "Order-Picking in a Rectangular Warehouse: A Solvable Case of the Traveling Salesman Problem," Operations Research, INFORMS, vol. 31(3), pages 507-521, June.
    8. Boysen, Nils & de Koster, René & Weidinger, Felix, 2019. "Warehousing in the e-commerce era: A survey," Publications of Darmstadt Technical University, Institute for Business Studies (BWL) 126185, Darmstadt Technical University, Department of Business Administration, Economics and Law, Institute for Business Studies (BWL).
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    Cited by:

    1. Stefan Irnich & Laura Lüke, 2025. "The Single Picker Routing Problem with Scattered Storage in Parallel-Aisle Warehouse with Multiple Blocks," Working Papers 2510, Gutenberg School of Management and Economics, Johannes Gutenberg-Universität Mainz.
    2. André Hessenius & Christian Manz & Stefan Irnich, 2025. "Benders Decomposition using Core-Maximal Cuts and Its Application to the Single-Picker Routing Problem with Scattered Storage," Working Papers 2511, Gutenberg School of Management and Economics, Johannes Gutenberg-Universität Mainz.

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