Solving Scheduling Problems in Distribution Centers by Mixed Integer Linear Programming Formulations Maria Pia Fanti ∗ Gabriella Stecco ∗∗ Walter Ukovich ∗∗∗ ∗ Department of Electrical and Electronic Engineering, Polytechnic of Bari, Bari, Italy, (Tel: 39-080-5963643; e-mail: fanti@deemail.poliba.it). ∗∗ Department of Industrial Engineering and Information Technology, University of Trieste, Trieste, Italy, (e-mail: gstecco@units.it) ∗∗∗ Department of Industrial Engineering and Information Technology, University of Trieste, Trieste, Italy, (e-mail: ukovich@units.it) Abstract: This paper considers the problem of scheduling the internal operations of a distribu- tion center: a system to which goods arrive from several suppliers and are forwarded to several customers. Internal operations consist essentially in unloading incoming cases and loading outgoing ones. Thus, the internal operations are divided in three phases: de-consolidation, sorting and consolidation. The objective is to minimize the total operation time determining the optimal sequence of the operations in the three phases. Starting from a previous formulation, we propose two new Mixed Integer Linear Programming models assuming infinite buffer capacities, in order to deal with instances of larger size. Some preliminary results show promising prospectives for the effectiveness of the new ap- proaches. Keywords: Scheduling; Distribution Systems; Integer programming; Optimization Problems. 1. INTRODUCTION Today companies are cutting costs by reducing inventory at every step of the operations. In order to reduce costs and improve efficiency, storage and retrieval, the two most expensive warehousing operations, can be eliminated if feasible. A Distribution Center (DC) is a type of warehouse where the storage of goods is limited or nonexistent [5]. In the DC under consideration, large incoming loads from different suppliers are disaggregated and combined to create consolidated outbound shipments to be sent to customers according to their requests. Inbound trucks are unloaded and the loads are unpacked. Successively, the single items are sorted according to the customer requests, packed and sent to the customers by outbound trucks. The related literature deals with the scheduling of inbound and outbound trucks, in cross docking terminals [2] where the products do not need to be unpacked, sorted and packed but only moved from inbound to outbound docks. Nevertheless, in the literature the terms warehouse, DC and cross-docking are used as synonyms. A recent survey [1] presents the schedule of inbound and outbound trucks at the terminal. In [2] a model with a single inbound, a single outbound dock and interchangeable products is presented. The problem of minimizing the makespan is solved using different algorithms. Miao et al. [6] consider a cross-docking problem with multiple docks, a limited buffer inside and a transshipment time between each pair of docks. In this case the objective is to minimize the total cost, i.e. the operational cost and penalty cost due to the unfulfilled cargo shipments. The authors use an exact procedure and also meta-heuristic approaches. In [3] the authors study a cross-docking scheduling problem and model it as a two-machine flow shop problem with the objective of minimizing the makespan. Moreover, the authors prove that the problem is NP-hard and they solve it by Branch-and-Bound and heuristic algorithms. This paper considers the problem of scheduling the inter- nal operations of a DC, i.e., the de-consolidation, sort- ing and consolidation operations. In the de-consolidation phase, incoming containers are unpacked in pallets and then pallets are unpacked in boxes. Each box contains items of the same type that are assigned to outbound boxes according to the customers requests. In the consolidation phase, these boxes are packed in pallets and successively in containers. The objective is to determine the optimal sequence of operations in order to minimize the total operation time. In [4] the authors solved this scheduling problem by proposing an Integer Linear Programming (ILP) approach and considering limited buffer capacities. However, the proposed model turns out to be NP-hard, so only instances of very limited size could be actually solved. In order to have a more manageable problem, in this paper we relax the constraints on buffer capacities. Such an assumption is justified in the case of large ca- pacity availability, with a limited number of jobs. For this uncapacitated problem, we present two new Mixed Integer Linear Programming (MILP) formulations: the first one is slightly different from the formulation introduced in [4], the second one is based on time indexed variables. Test Preprints of the 18th IFAC World Congress Milano (Italy) August 28 - September 2, 2011 Copyright by the International Federation of Automatic Control (IFAC) 8205