## 1 Introduction  
Currently, in response to the rising cost of goods, many companies are committed to minimizing logistics costs through improved logistics processes. In particular, because the transportation cost accounts for 29.4% [1] of the total cost of the logistics service (e.g. inventory, management, transportation, packing, ordering, movement, and warehousing), many researchers and companies have developedmethodologies to find the optimal solutions and algorithms to reduce transportation costs. In addition, due to the current rapid changes of the logistics and distribution industry (e.g. online shopping businesses, last-mile delivery, and fulfillment centers) and the demand for effective utilization of resources, the reverse logistics have been integrated with current model focused on forwarding flow of material. Integrated logistics processes can be devised at the various points, from sourcing and manufacturing to transporting products along the supply chain [2]. Also, most logistics service providers try to reduce the transportation costs by re-engineering the current silo processes to include simultaneous delivery and pick-up. The mathematical model to find the optimal route for pick-up and delivery at the same time (VRPSDP) [3] has been extended according to constraints such as the time window (VRPSDP-TW) and the condition of vehicles [4]. For a daily pick-up and delivery service like those supporting several convenience stores, it is required to consider the reusable transport container unit, as well as the capacity of trucks, traveling distance, and traffic congestion. Most previous studies have considered the development of an optimal algorithm which can solve this mathematical problem within a practical timeframe while satisfying all constraints such as the capacity of delivery and pick-up, and time windows. Still, it is very hard to find cases which have applied mathematical models or algorithms to practical business environment cases, especially daily operating logistics services like those serving convenience stores. In particular, reusable transport containers such as trays, shipping containers, and packages, should be regarded as the important assets of logistics centers. However, if the mathematical model focuses on only satisfying constraints related to delivery and does not consider the cost of managing reusable assets, it is often to leave the empty assets at the pick-up points. For example, in daily distribution services supporting convenience stores, the empty trays are often left at the outside of the stores. In addition, the trays are sometimes broken or missing. All of these events cause additional costs to re-arrange trucks to return trays, fix the broken trays, and keep more trays than expected. In this study, a vehicle routing problem with simultaneous delivery and pick-up under a time window, considering left over cost (VRPSDP-TW-LO) is proposed. The mathematical model has been devised to extend the previous model VRPSDP-TW by adding more cost factors and constrains related to the trays that are left. The model was applied to a small business case with the practical locations of stores and demand in terms of trays. With the numerical example of independent daily operation and consecutive operation for one month by generating random demand, we compared the performance of the proposed model with various quantitative metrics. Based on the results of statistical tests, it may be possible to develop the optimal strategy for finding the vehicle for picking up and delivering the trays. Also, it is possible to reduce the total cost, including transportation cost, left over cost, and penalty from violating the time window constraints. The rest of this paper is organized as follows: in the following section, previous research related to the vehicle routing problem (VRP), VRP with soft time windows, and VRPSDP is summarized and criticized.