Optimizing Vehicle Routing for Simultaneous Pick-up and Delivery considering the Reusable Transporting Containers: Case of Convenient Stores

A lot of previous research have proposed various frameworks and algorithms to optimize routes to reduce the total transportation cost, which accounts for over 70% of overall logistics cost. However, it is very hard to find the cases applied the mathematical models or algorithms to the practical business environment cases, especially daily operating logistics services like convenient stores. Most of previous research have considered the developing an optimal algorithm which can solve the mathematical problem within the practical time while satisfying all constraints such as the capacity of delivery and pick-up, and time windows. For the daily pick-up and delivery service like supporting several convenient stores, it is required to consider the unit transporting container as well as the demand, capacity of trucks, traveling distance and traffic congestion. Especially, the reusable transporting container, trays, should be regarded as the important asset of logistics center. However, if the mathematical model focuses on only satisfying constraints related delivery and not considering the cost of trays, it is often to leave the empty trays on the pick-up points when there is not enough space in the track. In this research, it has been proposed to build the mathematical model for optimizing pick-up and delivery plans by extending the general vehicle routing problem of simultaneous delivery and pickup with time windows while considering left-over cost. With the numerical experiments, it has been proved that the proposed model may reduce the total delivery cost. It may be possible to apply the proposed approach to the various logistics business which uses the reusable transporting container like shipping containers, refrigerating containers, trays, and pallets.


I. INTRODUCTION
Currently, in response to the rising cost of goods, many companies committed to minimize logistics cost through improved logistics process. Especially, because the transportation cost accounts for 70% of total cost of logistics service, many researchers and companies have developed the methodologies to find the optimal solutions and algorithms to reduce the transportation cost. In addition, due to the change of awareness in the environment and efficiency of resource and energy, it has been integrated the reverse logistics with current one focused on forwarding flow of material. The integrated logistics process can be devised at the various points from sourcing and manufacturing to transporting products along the supply chain (S. Dowlatshahi, 2000). Also, most of logistics service providers try to reduce the transportation cost by re-engineering the current silo processes to simultaneous pick-up and delivery.
However, although many researchers and companies have tried to devise the various optimal vehicle routing algorithms and solutions to reduce the transportation cost, in real-world business case which are applied these optimal algorithms or solutions are hard to find. Although it has been developed lots of optimization algorithms and solutions, there is a limitation which are difficult to apply to actual business due to the problems that occur in operational. Therefore, it can be said that proposing an optimal vehicle routing planning solution applied real problem is important. However, it is very hard to find the cases applied the mathematical models or algorithms to the practical business environment cases, especially daily operating logistics services like convenient stores.
Most of previous research have considered the developing an optimal algorithm which can solve the mathematical problem within the practical time while satisfying all constraints such as the capacity of delivery and pick-up, and time windows.
The mathematical model to find the optimal route for picking up and delivery at the same time(VRPSDP) (J. Potvin, J. Roussee et al, 1993) has been extended according to the constraints such as time window(VRPSDP-TW) and conditions of vehicles ( R. Bent, P. Van Hentenryck 2006). For the daily pick-up and delivery service like supporting several convenient stores, it is required to consider the reusable unit transporting container as well as the capacity of trucks, traveling distance and traffic congestion. Especially, the reusable transporting containers such as trays, shipping containers, packages, should be regarded as the important assets of logistics centers. However, if the mathematical model focuses on only satisfying constraints related 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 the daily distribution service for supporting convenient stores, the empty trays are often left at the outside of the stores. In addition, the trays are sometimes broken or missing. All these events cause additional cost to re-arrange trucks to return trays, fix the broken trays, and keep more trays than expected.
In this research, it has been proposed the vehicle routing problem with simultaneous delivery and pick up under time window considering left over cost (VRPSDP-TW-LO). The mathematical model has been devised to extend the previous model VRPSDP-TW by adding more cost factors and constrains related to the left trays.
And the model was applied to the 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, it has been compared the performance of the proposed model with various quantitative metrics. Based on the result 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, it has been summarized and criticized the previous research related to VRP and VRPSDP. In section III, it has been described how to build the mathematical model with the assumptions, the objective function and constraints by considering the practical business environment. In section IV, with the numerical experiment, the performance and practical applicability has been proved. Finally, it concluded with the contributions and limitations in section V.

II. LITERATURE REVIEW
It has been proved that VRP is one of the NP-hard problems by Danzing and Ramser (1959). Therefore, the early VRP models assumed that only one vehicle is assigned for each route, starting from a single depot and presented a step-by-step approach heuristic method to address the periodic vehicle routing problem about the reusable waste in Colombia. This method is that the first step is to separate each area into zones, and the second step is to set the type of waste to collect and the area to visit, and the last step is to determine the

A. Basic assumptions and constraints for building model
In order to build the mathematical model, it has been assumed that there are K vehicles with which N stores, noted as customers. The demand of products delivered to each customer (i) will be counted as the number of trays, . These trays should be delivered to the customer i within the time window [e i , l i ] for i = 1, … , N.
A vehicle delivers any stuffs requested by some customers in trays, but the number of trays are limited by C, which denotes the capacity of vehicles. It has been assumed that each vehicle starts its service at the depot and returns to the depot after completing picking and delivery service. Also, each customer can be served by only one vehicle. It means that the number of trays loaded in a vehicle from the depot is equal to the sum of number of trays which should be delivered to the customers included in the route of the vehicles.
Once again, it should be noted that all stuffs will be delivered packed in the trays. Thus, it is required the enough number of empty trays at the depot. The empty trays can be collected from the customers the day after delivery.
Therefore, it should be devised the optimal picking and delivery plan considering the number of trays delivered and will be delivered at the same time.

B. Mathematical Model for VRPSDP-TW-LO
In this section, it has been devised the mixed integer programming (MIP) model for VRPSDP-TW-LO. Table   1 summarizes the notations used in the MIP formulation. Using the flow-arc formulation [5], vehicle routing problem can be described as follows. Let = ( , ) be a graph where = ( 0 , ⋯ , +1 ) is the set of vertices and = {( , ): ≠ ∧ , ∈ } is the set of arcs.
Note that both 0 and +1 denote the depot. Thus, any tour which starts and ends at the depot can be represented as a path from 0 to +1 . Each vertex in V has an associated demand has an associated cost ≥ 0 and travel time ≥ 0.
The decision variables for the proposed model are defined as follows: : the binary variable whose value is 1 if a vehicle travel from customer to and 0 otherwise; : the number of empty trays loaded on the vehicle k at customer I; : the number of trays (both empty and non-empty trays) a vehicle k loads when it departs from customer ; : the time at which a vehicle departs from customer ; : the binary variable whose value is 1 if a vehicle k arrives at customer i earlier than and 0 otherwise; : the binary variable whose value is 1 if a vehicle k arrives at customer i later than and 0 otherwise; : the binary variable whose value is 1 if a vehicle k arrives at customer i within the time window [ , ] and 0 otherwise.
The objective function is defined as follows: A feasible solution of the proposed model satisfies the following constraints.
(1) Path constraints: For each vehicle, a tour which starts and ends at the depot can be represented as a path from 0 to +1 . The feasible movements of vehicles satisfy the following three set of constraints:

A. Instance generation
For the statistical test, all parameters are assumed based on the case study of delivery service for convenient stores as follows: A set of customers, n = 11: , ∈ {0, … , 10};

B. Result of small instances
Based on the independent 30 instances, the performance of proposed models, VRPSDP-TW-LO, was evaluated by comparing with VRPSDP-TW which does not consider the left-over cost. Following Table 2 shows the result of each instance of the VRPSDP-TW and the VRPSDP-TW-LO models. The performance was evaluated with the metrics such total cost, total travelling distance, number of time window penalty, and number of leftover trays. At first, the result shows that the average total cost of VRPSDP-TW is higher than VRPSDP-TW-   Therefore, it is can be said that that VRPSDP-TW-LO may guarantee the better solution than VRPSDP-TW, which can accept the alternative hypothesis, H1, given below:  for supporting convenience stores, but also to other services using unit containers such as grocery stores pharmacies, and marine container shipping business.

VI. CONCLUSION
It becomes the most important function the efficient logistics service for securing the competitive advantages.
In particular, the transportation, which accounts for the largest cost in logistics activities, is critical for enhancing the service level and saving the cost. However, current Transport Management System (TMS) focuses on designing the optimal route which minimize the objective function consisted with different factors such as travelling distance, required vehicles and drivers while meeting several different constraints like the maximum capacities, time windows to visit, and demand for delivery and picking up. A lot of previous research presented very advanced models, methods, and algorithms to solve the mathematical problem related the VRP.
Still, it is not easy to find the previous research which considers the valuable assets for transportation service like containers and trays except vehicles and drivers, even though these assets are very critical to enhance the service level and causes a great deal of cost.
In this research, it has been devised a novel mathematical problem, VRPSDP-TW-LO, which considers the cost from the left-over trays, travelling distance, and time-window penalty, at the same time. This model may reflect the practical circumstances of daily distributing services by relaxing some constraints of traditional VRPSDP or VRPSDP-TW and adding more factors.