Literature review

This study integrates multimodal transportation and LRP. Both of them are apparently independent fields of research and will be discussed separately. Also a few papers jointly discussing the two concepts are also cited. Different researchers used combination of problems to define their model. Shen et al. presented location and inventory problems together. Azadeh et al. considered vehicle routing and inventory decisions simultaneously. Beginning of LRP development can be traced back to 70 and 80s when Laporte and Nobert presented a single-depot model which was subsequently used by other researchers. Loperte et al. considered multi-depot LRP to further develop the problem. Thereafter, various authors has introduced different approaches to the problem which can be classified based on problem structure (single-echelon or multi-echelon), type of data (deterministic, probabilistic, or fuzzy), number of product types (single-product or multi-product), number and capacity of facilities (single-facility or multi-facility, capacitated or incapacitated), type and capacity of vehicles (homogenous or heterogeneous, capacitated or incapacitated), time window and problem type (soft or hard), number of objective functions (single-objective or multi-objective), and solving methods (exact, heuristic, meta-heuristic, and combinational). In this regard, some researchers present review papers.

Wu et al. designed a three echelon LRP. They also considered tight time windows and time deadlines to create services for high-speed trains. Albareda-Sambola et al. proposed a multi-period LRP model and a stable model against time. An approximation based on replacing vehicle routes by spanning trees is proposed, and its capability for providing good quality solutions is assessed in a series of computational experiments Karaoglan et al. used goods pick and delivery planning in LRP. The authors proposed two polynomial-size mixed integer linear programming formulations for the problem and a family of valid inequalities to strengthen their formulation. Rodriguez-Martin and Salzar-Gonzalez considered locations of and routing through hub facilities. Proposed model decide on the location of hubs, the allocation of nodes to hubs, and the routing among the nodes allocated to the same hubs, with the aim of minimizing the total transportation cost. Ahmadi-Javid and Seddighi set probabilistic facility capacity and disruption risk in the problem. The goal is to determine the location, allocation and routing decisions that minimize the annual cost of location, routing and disruption. Tavakkoli-Moghaddam and Raziei  took demands as fuzzy numbers. They present a bi-objective location-routing-inventory problem with heterogeneous fleets in a two-echelon distribution network and fuzzy demands. Ghezavati and Beigi proposed a bi-objective mathematical model for location-routing problem in a multi-echelon reverse logistic network. Their proposed network consists of hybrid collection/inspection centers, recovery centers and disposal centers. They considered total cost minimization and minimizing the maximum time of completion of the collecting return products as objective functions. Hiassat et al. also considered location-routing-inventory problem for perishable products distribution. Samanlioglu developed a LRP for dangerous materials handling. In this paper, a new multi-objective location-routing model is developed, and implemented in the Marmara region of Turkey. Shahabi et al. added inventory management to three levels LRP. They also assumed that demand across the retailers is to be correlated as Najjartabar et al. which considered correlated demands in location-inventory problem in a three level supply chain. Aghighi and Malmir used location-routing inventory problem on perishable product distribution system design. In this paper, authors considered stochastic demands and travel times and solved problem in two phases. And Fazel Zarandi et al. considered time window, fuzzy demand, and fuzzy travel times at the same time. Moreover these papers, Govindan et al. proposed LRP in a sustainable network. Sustainable networks have a growing trend in supply chain problems. Afshar-Bakeshloo et al. developed a model, named Satisfactory-Green Vehicle Routing Problem. It consists of routing a heterogeneous fleet of vehicles to serve a set of customers within predefined time windows. Also, Najjartabar-Bisheh et al. analyzed the role of third-party companies in a sustainable supply chain design.

As LRPs, multimodal transportation papers can be classified based on their planning time horizon (strategic, tactical, or operational). More recently, Crainic and Steadie Seifi et al. classified the researches on multimodal transportation in two review papers and can be referred for more studies.

Following studies are joint papers for both LRP and multimodal transportation. Li et al. introduced location of terminals problem on a multimodal transportation network. The proposed model simultaneously considers choices of travelers on route, parking location and mode between auto and transit. Chiadamrong and Kawtummachai designed a methodology to support decision making on sugar industry. This aim of this paper is to suggest the best inventory position and transportation route in the distribution system considering different transportation options. Tiwari et al. considered route selection on a multimodal transportation network with several objectives: minimization of travel time and travel cost, later schedules and delivery times of every service provider in each pair of location, and lastly variable cost must be included in every location. Alumur et al. considered hub location problem and hub network design and assumed the hub and non-hub nodes are linked via multimodal transportation. Also, Moccia et al. considered hub facilities on multimodal network, but they proposed and solved a routing problem. Xie et al. considered mode changing facility location and multimodal route selection problem on a multimodal network for dangerous materials transportation. They considered different origin/destination pairs and selected the best multimodal route with determining mode changing points. Hajibabai and Ouyang presented a location and routing problem for biofuel transformation facilities on a multimodal network. In this study, first, some locations were selected for biofuel transformation facilities which were to be located between supplier and customers; then, multimodal routes were determined connecting suppliers to the facilities and then the facilities to customers. However, they presumed the routes to be of multimodal nature, neglecting to consider different modes and mode changing issues. Tuzkaya et al. presented distribution facilities location problem in a multimodal transportation network. In their approach, in the first phase, using analytic network process (ANP), a decision was made on the best transportation mode and the best potential sites to establish facilities; in the second phase, distribution facilities location problem was solved. Finally, Ayar and Yaman added time windows to multimodal routing problems.

Even though location-routing problem has experienced various developments during recent past, yet few papers are reported, wherein multimodal transportation is accounted for in such problems. To the best of our knowledge, no paper in the literature considered products delivery tours from DCs to customers while determining routs from supplier to DCs on multimodal network at the same time. Using multimodal logistics is an indispensable option for world class organizations, so the present paper is an attempt to fill-in this gap in the research field of location-routing.