Downward compatible loading optimization with inter-set cost in automobile outbound logistics

Highlights

• A loading optimization problem in the outbound logistics of automobiles is studied.

• The objective is to balance the service level of 3PL and the transportation cost.

• An inter-set cost is defined to estimate the cost among destinations in an area.

• A special downward compatible loading structure is considered.

• An IP model and a column generation based heuristic algorithm are proposed.

• Computational results and a case study show the performance of the algorithm.

Abstract

This paper addresses an essential loading optimization problem arising in the outbound logistics of automobiles. The problem is to assign a set of orders of finished cars to a set of heterogeneous auto-carriers and to deliver these finished cars from a depot to multiple destinations. The involved destinations are located within a relatively small distance to the depot. A value corresponding to each order represents its urgency level, and an inter-set cost is defined among destinations to estimate the related transportation cost. The objective of the problem is to maximize the weighted total value of the assigned orders minus the total inter-set cost subject to a full-load constraint and a downward compatible loading structure. An integer programming model and a column generation based algorithm are proposed. The proposed algorithm is implemented based on serial and parallel programming, respectively. Computational results based on randomly generated instances and a case study with real data show that the proposed algorithm can generate near-optimality solutions efficiently, and outperforms solving the integer programming model by an IP solver and a rule-based method applied in practice.

Introduction

Automobile industry is one of the major industries in the world. In 2018, the global automobile production and sales exceeded 95 million (The International Organization of Motor Vehicle ManufactureOICA, 2018). To meet the increasing highly-diversified customer demands, automobile manufacturers usually have a massive outbound logistics network. Taking high value and safety into account, automobile delivery services are implemented through special transporters known as the auto-carriers. Each auto-carrier has a given number of slots, and each slot can load only one automobile. The auto-carriers have various capacities and specific configurations of slots. The configuration of each auto-carrier specifies that the largest size of automobiles that can be loaded to each slot. Since July, 2018, Chinas Ministry of Transport has forced the elimination of auto-carriers that do not meet the latest national standards. More attentions are paid on the combinatorial nature of auto-carrier loading and transportation.

This paper studies a loading optimization problem arising in the outbound logistics of automobiles where a set of orders of finished cars needs to be assigned to a set of heterogeneous auto-carriers. Each of the orders refers to a finished car stored at a depot and to be delivered to a given destination. The destinations involved are located within a relatively small distance to the depot. Our work is directly motivated by the operations of the largest automotive 3PL company in China, Anji Automotive Logistics Co., Ltd.. As shown in Fig. 1, the problem is drawn from the following two practical cases. In one case, finished cars are stored in an outbound warehouse of an assembly plant and to be delivered from the warehouse directly to dealers or end customers located in the same city or in adjacent cities. In the other case, finished cars have been firstly delivered to a regional distribution center from the assembly plant taking advantages of economies of scale, and are to be delivered from the regional distribution center to nearby dealers or customers. The 3PL company makes decisions of assigning a set of orders to a set of heterogeneous auto-carriers every day. The decision makers would generally consider the following three issues.

First, the auto-carriers have various capacities and configurations of slots, and the total capacity of the available auto-carriers is limited and usually not enough to handle all the orders in each day. Therefore, each auto-carrier has to be fully-loaded in practice according to its capacity and configuration of slots. In practice, the 3PL company classifies the orders into three types of small, medium and large, referring to the cars’ sizes as the examples shown in Table 1. Accordingly, each slot of an auto-carrier is determined as one of these three types, where the type of each slot is identified as the largest type of the cars that the slot can load. Moreover, there exists a downward compatible loading structure (DCLS) among the orders and slots. Namely, a large slot can load a car of any type, a medium slot can load a car of medium or small type, and a small slot can only load a small car. Note that this special loading structure can be easily generalized to cases involving more types, and to other packing problems where the items and the containers can be classified in a similar way.

Second, the 3PL company needs to decide which orders should be handled preferentially. Thus, there is a value given for each order to indicate its urgency level. Normally, an order is given a value equal to 1 when it is released to the 3PL company. The value of such order increases by one per day until the order is handled or the value researches a pre-determined highest level (10 in our case). Whereas, some orders are given a value equal to the pre-determined highest level as soon as they are released for special reasons such as promotions or impatient customers. Notice that the total value of the orders handled during each day is the main evaluation indicator of the 3PL company’s outbound logistics service level.

Third, it is not necessary to solve the corresponding vehicle routing problem in a very precise way, although the transportation cost is generally an important concern for 3PL companies. As previously stated, long-distance mass transportation among cities are not involved in our problem. Restricted the outbound logistics of automobiles to a relatively small area, there are usually 100 to 300 orders with 10 to 50 destinations every day, and the destinations of the orders are closed to each other. For example, Table 2 shows the longitudes and latitudes of 10 dealers in Tianjin, China. The last column of the table indicates the average traveling time from each of the dealers to the dealer 1. Solving a corresponding routing optimization problem precisely is not economical, since it would cost a large amount of time while contribute little for saving the transportation cost. Moreover, the 3PL company usually faces a number of problems for different depots every day and prefers to make the assignments efficiently. In addition, the transportation cost is roughly controlled based on greedy rules in practice, which is described in detail in Section 3.

Consequently, it is reasonable to introduce an inter-set cost for estimating the transportation cost, which is described on an undirected graph. As shown in Fig. 2, each vertex represents a destination, and there is an edge between each pair of vertices with a given cost equal to the average direct traveling time from one of the corresponding destinations to the other. Given an auto-carrier assigned with orders, if the orders are corresponding to multiple destinations, then the inter-set cost of the auto-carrier is calculated as the total cost on the complete sub-graph consisting of all the involved destinations. Otherwise, the inter-set cost of the auto-carrier is zero.

Due to the above three aspects focused by the 3PL company, the reward of each auto-carrier is defined as the total value of the assigned orders multiplied by a weight parameter minus the corresponding inter-set cost. The objective of our research is to assign a given set of orders to a given set of auto-carriers efficiently such that the total reward of all used auto-carriers is maximized, subject to the following two constraints.

• DCLS loading constraint: The orders assigned to an auto-carrier can be loaded according to the DCLS loading structure.

• Full-load constraint: The number of the orders assigned to an auto-carrier is equal to the capacity of the auto-carrier.

The remainder of this paper is organized as follows. Section 2 is literature review. Section 3 describes our problem precisely, and the problem is formulated as an integer programming (IP) model. In Section 4, we reformulate the problem as a set packing type of formulation, and describe the column generation based heuristic algorithm (CGH). A greedy loading method (GL) used in practice is described in this section to generate the initial columns for the CGH algorithm. Considering the characteristics of the proposed CGH algorithm, two structures based on serial and parallel programming, respectively, are designed for implementation. Section 4 shows the experimental results of comparing the performance our algorithm to that of the GL method and that of directly solving the IP model by an IP solver, CPLEX. Section 5 gives a case study base on real operations of a warehouse. Section 6 concludes the paper.