## 1 Introduction  
Integrating the two areas in logistic planning of transportation and inventory can reduce costs system-wisely, such as vendor-managed inventory (VIM), has gained acceptance in many supply chain environments. [PERSON] examines the activities and decision makers involved in urban goods movements, showing the need to integrating optimize purchasing, restocking, and delivery [1]. This paper addresses the combined problem which introducing air-land transportation and the dual-sourcing inventory policy into the inventory routing problem (IRP). The central depot serves several demand locations by land vehicles (e.g., trucks) and aerial crafts (e.g., drones or helicopters). The central depot decides which demand locations are to be replenished by land and aerial modes and in what order. We consider the transportation time in the replenishment as the lead time for each demand location. This feature makes decisions on the replenishment quantity and routing depend more tightly on each other. It is very useful in application scenarios that require high timeliness, such as emergency logistics and perishable goods supply chains. This problem is an extension of the inventory routing problem, which is first studied by [PERSON] et al. [2]. The central decision-maker determines an inventory policy and a set of vehicle routes to minimize the total cost. All vehicles must leave the depot at the beginning of each inventory planning period and return at the end, while also staying within capacity constraints and meeting the demands of locations. There are many situations where the routing and the replenishment quantity decisions should be determined simultaneously, such as the retail supply chain, for companies such as Wal-Mart and Proctor&Gamble; the petrochemical supply chain; and emergency logistics [3; 4; 5]. Inventory management and vehicle routing decisions are interrelated. The inventory policy determines which demand locations must be served and the replenishment quantity. It affects the inventory cost. On the other hand, the delivery cost depends on the vehicle routes, which, in turn, need information about the demand location selection and the replenishment quantity. This inter-relationship recently motivate some researchers to model these two decisions simultaneously. [PERSON] and [PERSON] present a review of IRPs from aspects of single-period models, multi-period models, infinite horizon models, and stochastic models [6]. [PERSON] et al. summarize the 30 years develop of IRP studies from 1983 to 2013 comprehensively [7]. Recently, [PERSON] presents a review paper focusing on perishable inventory routing problems [8]. Most existing IRP models assume the replenishment lead time is determined exogenously. As a consequence, their application to physical distribution problems is confined to situations where the timeliness does not significantly influence the decision rather than in an emergency or perishable goods supply chain. Most IRP studies for perishable goods focus on product perishability, such as [9; 10; 11; 12; 13]. [PERSON] et al. first study the timeliness for perishable IRP by considering the transportation time in replenishment lead time [14]. They propose a myopic policy that in each stage the replenishment and routing decision is only made once, then repeat the stage to minimize the cost. [PERSON] and [PERSON] first study the effect of the short-term on long-term planning [15]. They consider single-period models as sub-problems and use penalty and incentive factors to affect the later period's decisions. Then, [PERSON] and [PERSON] discuss several exact algorithms for multi-period integrated optimization models [16]. The multi-period models are useful in that they offer a more realistic strategy. However, these approaches produce a high-quality solution that requires significantly a larger computing effort. Decomposing a multi-period problem into a series of single-period problems is often achieved. [PERSON] and [PERSON] first use a single-period model to solve IRP by rolling horizon [17].