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The classical multi-period inventory routing problem (MIRP) combines inventory management and vehicle routing problems, attempting to maintain customer inventory levels via autonomous deliveries, with the assumption that stock-out never occurs at the supply side. While this is no more the truth when a common third-party logistics company operates a central warehouse for a group of small and medium sized retailing companies. Procurements have to be made to refill inventory at the central warehouse. This study thus extends the traditional MIRP by incorporating procurement decisions (MIRP-PD) and tries to minimize the total cost of procurement, inventory holding and transportation. To our knowledge, this paper is among the first to formally describe MIRP-PD and model it as a mixed integer linear program. A hybrid two-level heuristic is proposed to address large-scale instances. The upper level of the algorithm determines the visiting schedule while the lower level works out procurement and routing decisions accordingly. Components from tabu search and adaptive threshold acceptance are also embedded to help escape from local optima. Extensive numerical instances are generated from a real case in Chengdu (China), based on which the computational results reveal the effectiveness and efficiency of the proposed algorithm compared with the commercial solver CPLEX. The comparison between MIRP and MIRP-PD has demonstrated the cost advantage of incorporating procurement decisions into inventory and routing plans. Impacts on costs of key factors including delivery frequency and buffer size of stores have also been examined to provide managerial implications to practitioners. [ABSTRACT FROM AUTHOR]

Highlights • An e-commerce city logistics problem with synchronization (sync) requirements is studied. • The sliding time window (STW) is adopted to model the sync requirements. • A divide and conquer algorithm is developed to solve the proposed VRP with partially unknown time constraints. Abstract The cost-effective and high-quality city logistics service is a key for the success of e-commerce enterprises. Synchronization (sync) is emerging as a typical yet challenging requirement, which asks for simultaneous deliveries of multiple products to the same customer. This paper is among the first to model the sync constraints with sliding time windows (STW). STW is a special type of time window of which only the window size is defined. Unlike traditional time windows, the start and end time of STW could be adjusted earlier or later, so long as their difference equals to the pre-defined window size. In this sense, the STW is a time constraint with partially unknown factors. Such flexibility of STW can greatly improve the efficiency of vehicle tours, because customers could be served in a more flexible sequence decided by the transporter. A novel divide and conquer based algorithm is developed to tackle the proposed problem with partially unknown time constraints. The values for STWs will dynamically be determined during the algorithm. Numerical studies show that by modelling sync requirements with STW, transportation cost could be saved as much as in half. We also carry out sensitivity analyses on the key factors such as promised sync level to customers and the complexity of online orders. [ABSTRACT FROM AUTHOR]