Assortment planning considering split orders
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When multi-item orders cannot be satisﬁed through a single shipment stemming from not having all the items in an order in the same warehouse, the cost of packaging and transportation increases and the delivery of the orders can be delayed. In this regard, split order problem is one of the most signiﬁcant challenges that the online retailers face. As the capacities of the warehouses are limited, it is not possible to stock every item in every warehouse. To minimize the number of orders that cannot be satisﬁed in a single shipment, it is important to determine how the limited capacities of the warehouses should be allocated to items or it is necessary to decrease the transportation costs through consolidating the split orders. Since this problem is NP-hard, the previous studies in the literature are based on heuristic algorithms. In this study, exact and heuristic methods have been examined to solve large scale problems. Some of the heuristic algorithms oﬀered uses the LP relaxation of the model provided by Jehl et al.(2018). In this sense, the analytical characterization of the optimal solution of the LP relaxation has also been revealed. It is proved that the allocation variables can only take three diﬀerent values at most one being fractional. It is shown that this solution can be found without actually solving the LP relaxation by beneﬁting from an algorithm oﬀered in literature to solve 0-1 fractional programming problems. Moreover, it is proved that a similar characterization is preserved for multiple warehouses or when a central depot with unlimited capacity and a forward distribution center are considered together. Additionally, the working principle of the greedy ranking algorithm oﬀered in the literature is theoretically justiﬁed and a dynamic version of this algorithm is developed. To evaluate the performance of the heuristic algorithms oﬀered and the run time of the integer programming problem, an extensive numerical study has been conducted. The change in the diﬃculty level of the problem based on the plant capacity, the number of orders, and the number of stock keeping units (SKU) is scrutinized. Furthermore, the assortment allocation problem is modeled together with the consolidation problem. The performance of the model is evaluated through comparing its solution to the solution obtained through solving two problems consecutively.