Transportation planning and inventory management are among the key problems at the various levels of a supply chain. The integration of transportation and inventory decisions is known as the inventory routing problem (IRP) in the literature. Building upon the reviewed literature, this paper expands an operational combination of the classical routing-inventory problem in the form of a two-level supply chain that includes a multi-period, multi-product inventory-routing coupled with various kinds of fleets with different available capacities called Fleet Size and Mix Vehicle Routing Problem (FSMVRP). In this model, stock out is not allowed. Additionally, two practical and significant features of routing are taken into consideration including: 1- ``backhauls'' in which the Distributor (Vendor) is supposed to provide services for two groups of the customers (linehaul and backhaul costumers), assuming that each one of the vehicles is first unloaded to satisfy the demands of the linehaul customers, and it is later loaded by collecting the loads from the backhaul customers; 2- ``Split delivery'' strategy according to which there is a possibility to provide the services to each customer by at least one vehicle. Due to the varying demands of the customers, it is quite possible for the demands of some customers to be more than the available capacity of one vehicle. Particularly, in the case of urban transportation, several vehicle transitions occur at a demand point. Thus, the split services can help minimize the number of the vehicles used, which in turn will increase environmental sustainability. Therefore, first, a new mathematical model, i.e., a mixed-integer programming (MIP) formulation, is presented for the problem. This problem is a non-deterministic polynomial-time hard (NP-hard). Then, according to the literature on routing-inventory problems, a bat optimization algorithm, whose performance is evaluated by an efficient genetic algorithm, is developed for the first time. At the end, the numerical results obtained by this algorithm are analyzed using the randomized test problems.
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