## Mathematical optimization of joint order batching and picker routing problems

• In this thesis we study order picking optimization problems for a two-blocks rectangle warehouse layout. We present combinatorial formulations and linear programming models based on the Steiner graph representation for order batching, picker routing, and joint order batching and picker routing problems. A special case of the latter is considered. This case assumes that each order contains exactly one item and each item can be picked from different possible locations in a warehouse. The underlying optimization problem is called joint multi-location order batching and picker routing problem (JMLOBPRP). Since having only one-item orders turns the JMLOBPRP into a special case of a capacitated vehicle routing problem, we suggest to implement algorithmic approaches for those to solve the JMLOBPRP. In particular, we define the JMLOBPRP as a generalization of the resource constrained assignment problem, for which a regional search method exists. The intention of the thesis is to investigate how a relaxation of the JMLOBPRP, a so-called group assignment problem (GrAP), can be solved following the ideas of regional search. We present a mathematical model of the GrAP and prove that it is NP-hard. Furthermore, we propose a novel heuristic algorithm for the GrAP. We call this method a network search algorithm, as it is based on a Lagrangian relaxation of the GrAP, which is solved by the network simplex method. On each its iteration network search examines a solution region suggested by the network simplex algorithm and improves the incumbent solution. Numerical experiments are conducted to assess a performance of the network search method. We create more realistic problem instances. The proposed algorithm is compared to the integer optimal solution of the GrAP and optimal fractional solution of its linear relaxation. Both computed using the commercial linear solver Gurobi. Our experiments show that the developed network search algorithm leads to the hight-quality solution within a short computing time. The results obtained testing large problem instances which cannot be solved by Gurobi within a reasonable computing time, show that the network search method provides a solution approach which can be used in practice.