Introduction

Effective and efficient transportation of materials and products through the chain of suppliers, manufacturers, assemblers, distribution centers (DCs), retail stores, and customers is crucial in the competitive world of today. In other words, transportation is a key part of a supply chain which ensures on-time delivery of raw materials and finished products. Decisions made regarding a supply chain transportation system can be classified into three different levels. The first level refers to strategic decisions with long-term (several years) effects on the supply chain. These are mainly the decisions made on the transportation system design and supplying resources (facility location, size and capacity of facilities, facility and plant layout, transportation fleet). The next level is tactical decisions which has mid-term effects (several months or quarters) and include production and distribution planning and resource allocation issues (facility space, fleet size and shipments packaging strategies). Third level is operational decisions which are made on a daily or weekly basis. Orders aggregation, shipments, and vehicle fleet dispatch are examples of these decisions. For the last four decades, researchers have studied combination of these decisions with a combinational view to supply chain and transportation problems. In this regard, location-routing problem (LRP) considers both location (strategic level) and routing (tactical level) problems. LRP represents a relatively new form of location problems addressing locating facilities such as distribution centers and depots, where routing aspects are simultaneously considered. Reflecting interactions between facility location and fleet routing, LRP can provide a better view for logistics analysts and managers, preventing poor and local optimizations. LRP had growing trends especially in recent years. Govindan et al. used LRP in perishable products distribution system design. Riquelme-Rodríguez et al. present and compare two methods for locating water depots along the road network and used arc routing in their problem. And Gao et al. introduced ant colony optimization with clustering for solving the dynamic location-routing problem. Seyedhosseini et al. reviewed dynamic location problems based on their models, solution, methods and applicability and analyzed gaps for future research. It can also be used for dynamic LRPs.

In general, because of geographical distances between producers and consumers demand for goods transportation has raised. Transportation, especially in long distances, is a world class business which cannot be accomplished through roads only due to availability and cost concerns. So there is a need for other transportation modes or a combination of them (i.e., multimodal transportation). European Conference of Ministers of Transport in 2001 defined multimodal transportation as "movement of goods in one and the same loading unit or vehicle, which uses successively two or more modes of transport without handling the goods themselves in changing modes". Multimodal transportation has experienced an increased importance during recent years and now multimodal transportation is competing with single-mode transportation. In this regard, many transportation companies have established multimodal transportation services. The second point is that, multimodal transportation is becoming an important policy for organizations because of its advantages in terms of cost and coordination between modes in large-scale cargoes. The third point is that, in references and handbooks of transportation, multimodal transportation is treated as an independent and well-separated transportation mode. Last but not least, since 1990, the number of papers on multimodal transportation follows a growing trend. Some researchers have reviewed the related literatures. To sum up, multimodal transportation is a relatively new field of research, with its increasing importance during recent years.

The objective of this study is to help designing a distribution system with two transportation network types. First network transfers products from supplier (with determined location) to DCs via a multimodal transportation system. On second network, products are distributed among customers via routing tours started from DCs on a single-mode network. In this problem, different questions will be answered: which multimodal route should be used to transfer products to DCs? Is there need to change mode in determined multimodal routes? And if yes, mode changing facilities should locate on which multimodal terminals? Each routing tour generated from open DCs, meet which customers and in what order? In fact, this study considers LRP and multimodal route selection problem simultaneously.

Research methodology is consists of the following sections. First, introduced problem modeled as a linear programming with a cost minimizing objective. Then mathematical model solved with GAMS optimizing software and CPLEX solver for two different numerical instances. Also, a genetic algorithm generated for the problem and its results compared with CPLEX results.

The following section gives a review on related literatures on multimodal logistics and location-routing problem. Then, the subsequent section gives an explanation of the proposed problem, followed by the section introducing the corresponding mathematical model. In the following section, the used solving approach is introduced before demonstrating and solving two different numerical cases, where the cases are further analytical discussed. The final section concludes the paper.