1. Introduction
The transportation system is one of the main sources of carbon emissions. Therefore, the development of low-carbon transportation is of great significance to promote energy conservation and emission reduction. As a critical component of the transportation system, railway transportation has the advantages of a large capacity and low emissions in comparison with road transportation, and the commodities carried by rail wagon are mainly bulk goods, such as coal, oil, coke, iron ore, etc. In contrast to the high electrification rate of railways, the road transportation has a single energy consumption structure: the burning of oil. Despite this fact, road transport still occupies the leading position in China freight transportation. In 2017, the railway freight turnover accounted for 13.7% of the total, while the road freight turnover accounted for 33.4% of the total in China [
1]. Hence, shifting bulk cargo from road to rail is an important measure to optimize the transportation structure.
In order to accelerate the shifting process of freight transportation from road to railway, the optimization of the railway transport operation plan is particularly important. Using the entire train can reduce the services of reclassification, raise the transporting efficiency, and reduce transportation costs, while not all shipments are suitable for being delivered by loading an entire train. If the traffic volume of the shipment is large, the railroads can provide entire trains between the loading and the unloading station to achieve “door-to-door” transportation. In the railway industry specifically, “door-to-door” means running trains from the supplier’s warehouse to the customer’s warehouse; otherwise, it will first be delivered to the adjacent yard on its itinerary by local train, and then, after a series of classification stations, it can reach the terminal.
The mode proposed in this paper for low-frequency entire trains (entire trains with long departure intervals) is meant to lessen the operation cost. Since it can effectively lessen the operation cost, the railway companies can provide a discount to attract users. Generally speaking, the in-transit time of the entire train can be shortened as the train does not need to be reclassified. However, this mode requires large storage capacities of warehouses and high loading and unloading efficiencies. The long-cycle departure interval and the high-volume delivery will result in a significant increase in average stock and then lead to additional inventory cost for customers. How to balance the pros and cons of operating low-frequency entire trains constitutes the prime problem to be studied in this paper.
The paper is organized as follows: In
Section 2, we first review the existing literature on the optimization of traffic organization and the logistics costs combined with transportation.
Section 3 describes the problems of rail train formation corresponding with the inventory cost.
Section 4 presents a 0-1 integer model to solve the problem proposed in
Section 3. A real-world case is described in
Section 5 to illustrate the effectiveness of our innovative model.
Section 6 concludes the study and provides suggestions for future research.
2. Literature Review
Freight transportation in railway system can be classified into two modes. Firstly, some shipments with large volume will be directly delivered to the unloading area by forming a “door-to-door” entire train. Secondly, other shipments will be delivered to the destination through a shipment-block sequence, i.e., the shipments need to be reclassified on their itineraries. It is a complicated optimization decision problem to determine which mode should be employed, since it is related to the train formation at the loading areas and the structure of the block network. Related research works are as follows. Assad [
2] proposed a mixed integer programming (MIP) formulation for routing and makeup in order to determine between which terminals the direct train service should be placed. In their works, technology requirements of motive power and traction as well as resource allocation were taken into account. A discussion on European (EU) rail freight transport and current single wagonload (SWL) trends was presented by Marinov et al. [
3], with the aim of gaining an understanding of how SWL services, policy, and practice can benefit from the implementation of scientific methods and information technologies. Assad [
4] studied the optimal classification strategy for a line network of yards targeted at minimizing reclassification cost. Ahujia et al. [
5] gave an overview of railroad blocking problems, attempted to develop cutting-edge algorithms, and employed the algorithm to railroad planning and scheduling problems. Keaton [
6] studied comprehensive optimal strategies of train connections, frequencies, and blocking and routing plans for freight cars in single-carload general commodity services with an all-integer linear programming model. A dual adjustment procedure which makes it possible to efficiently obtain close-to-optimal solutions to problems of a realistic size, was used by implementing Lagrangian relaxation. Martinelli and Teng [
7] formulated a 0-1 programming model for a train formation plan to minimize in-transit time of cars. The formation plan of a freight train connecting service (TCS) for shipments which is not enough to be shipped by entire trains formed in the loading area, is the foundation of the operation plan. Lin et al. [
8] presented a formulation and solution for the TCSs problem aimed at determining which pairs of yards are to be provided with a direct train service and which cars are consolidated into a given train service. The objective was to minimize the sum of operating costs, accumulation delay, classification delay, and assembly delay while satisfying technological constraints on train and yard capacity. Lin et al. [
9] considered the total consumption of loading conditions, traffic organization, and unloading conditions of the unloading area and described the various combination schemes of the initial traffic flow. A nonlinear 0-1 planning model was constructed, and the model was solved by a simulated annealing algorithm. Zhao and Lin [
10] took the car hour cost at the loading station of stepped direct trains and non-direct trains, as well as the time delay caused by loading sequence as the optimization target, and took the train length and reclassification station capacity as constraints. In order to improve the railway rail freight services, many European companies and research institutes in the field of railway transport have focused on research and development projects. Viable wagonload production schemes (ViWaS) and the Shift
2Rail Joint Undertaking (S2R JU) project have played a great role in promoting the progress of railway freight transportation.
Over the past years, many scholars have paid attention to add transportation costs into inventory management, which is a systematic optimization of transportation and the related enterprise inventory from the perspective of logistics, but little research on the influence of railway operation (for example, whether to drive a direct train) on the inventory cost and system cost has been done. Harris [
11] first proposed the importance of integrating transportation and inventory systems research, proposed a model with constant demand speed for delivering single-issue single-receiving points at a continuous rate minimizing inventory and total transportation costs, and gave a classic basic-economy economic order quantity (EOQ) model. The study carried out by Swenseth and Godfrey [
12] identified transportation cost functions that simulated reality and demonstrated that straightforward freight rate functions can be incorporated into inventory replenishment decisions without compromising the accuracy of the decision. Hill and Omar [
13] showed that increasing the batch size by a fixed factor instead of sending batches with same size was the optimal plan in a single supplier–single buyer model. Glock and Christoph [
14] studied the flow of materials between two vendors and a buyer and developed six alternative delivery structures with the intention of minimizing total system costs. The results showed that the total costs of the system can be reduced by permitting different lot sizes, shipment frequencies, and production intervals. Baller et al. [
15] incorporated transportation costs approximated by classical schemes into a dynamic-demand joint replenishment problem (DJRP) and assumed that the supplier paid a fixed fee for replenishing a customer. They analyzed the data from test instances and concluded that cooperation between the two can lead to greater cost savings. Ji et al. [
16] analyzed the problem using a batching and scheduling model involving both batch supply and batch delivery. In this paper, they considered four non-deterministic polynomial (NP)-hard cases that were classified based on whether the arrival and delivery of the goods were individual or not. The minimized sum of total weighted inventory cost and transport cost was calculated by the fully polynomial-time approximation scheme (FPTAS) solution. However, these articles were studied from the perspective of logistics; hence, they only considered the transportation process as a whole but not from the internal perspective of transportation.
Anily and Federgruen [
17] assumed that each retailer absorbed products at a constant rate, and studied deterministic demand-time continuous inventory and vehicle routing optimization issues in the case of a single product. Bertazzi and Speranza [
18] dealt with the problem of minimizing the sum of the inventory and transportation costs in the multi-products logistic network with one origin, some intermediate nodes, and one destination when a set of possible shipping frequencies was given. The inventory cost was computed through the aggregation of the inventory over time or over nodes. Chen and Sarker [
19] studied a buyer who received a product from multiple vendors and assumed that the products from the vendors in a milk-run were collected by a single truck. A multi-vendor optimal model was developed here for deciding the batch size of the vendor’s production and the delivery frequencies of different vendors to the manufacturer. A freight cost function shows that the freight rate is critical for the total cost of the system. Larger delivery quantities and fewer deliveries are suitable for higher freight rates.
Burns et al. [
20] studied the problem of joint inventory and transportation minimum cost for a single-point/multi-point logistics network. The model was characterized by only one product and subject to deterministic requirements. Chandra and Fisher [
21] studied the joint transportation and inventory models of single-product, one-to-many fleets from a single-origin transport product to multiple demand points. The author examined the advantages of inventory control and transport operations, and through empirical analysis, the results showed the cooperation between the two can lead to greater cost savings. The above scholars built inventory models based on the minimizing of transportation cost and inventory cost. However, these models are limited to road transportation, which is quite different from railway transportation. Guglielminetti et al. [
22] pointed out one of the main barriers hampering SWL development is the price-competitive position of road transport. It is rare to put the problem of railway direct transportation into the logistics system and study the problem from the perspective of system optimization. Guglielminetti et al. [
23], starting from a European context, carried out a very exhaustive statistical analysis of the cost from the aspect of railway wagon load system. The authors mentioned the costs for SWL services are about twice the costs for full train load (FTL), and the exact cost is given. However, they only considered the costs within the transportation system. Ji et al. [
24] built a non-linear 0-1 integer programming model of car flow organization in a loading area based on logistics cost, in which the changes of logistics costs of both sides caused by different car flow organizations was taken into account. They used a real-world network with one loading station and multiple unloading stations to illustrate the car flow organization plan.
4. Mathematical Models
4.1. Notations
Set:
: The set of all suppliers in a railway network.
Parameters:
: The order cycle length for the customer (days)
: The delivery cycle length for the supplier (days)
: The volume of an entire train (tons)
: The capacity of customer stock level (tons)
: The order quantity of the customer from the supplier (tons). Let be the quantity of daily production, which is equal to the customer’s consumption of the goods from the supplier , (tons per day)
: The stockholding cost per ton for the supplier (yuan per ton)
: The unit purchasing price for the supplier (yuan)
: The unit transportation cost from the supplier to the customer (yuan per ton)
: The customer’s stockholding cost per ton (yuan per ton)
: The transportation time of an entire train from the supplier (days)
: The transportation time of a non-direct train from the supplier (days)
: The interest rate per year
: The discount rate for dispatching an entire train.
Decision variables:
.
4.2. Formulations
The goal of the classic EOQ model is to minimize the sum of ordering cost and stockholding cost from the customer’s perspective. The model proposed in this paper takes the inventory costs of both suppliers and customers into account, in which operating a low-frequency entire train in a loading area where the shipment is insufficient for entire train is considered. Meanwhile, a discount is given by the railway companies. Unlike replenishment when customer stock levels are reduced to a minimum in the classic model, the replenishment in this model depends on the supplier’s fixed delivery cycle. In order to calculate the actual average stock of the customer, an innovative method of calculating average stock is proposed.
The following five parts are considered in this model.
4.2.1. The Stockholding Cost of Suppliers
For the supplier
, if low-frequency entire trains are operated to deliver goods, the average stock will be
. If daily non-direct trains are operated, it will be
. The unit stockholding cost is related to unit stock cost, unit selling price
, and annual interest rate
. Thus, the total stockholding cost of all suppliers is calculated as shown in Equation (1).
4.2.2. The Stockholding Cost of a Customer
For the customer, based on the algorithm of the customer average stock presented in the problem description, the stock cost of a customer can be concluded by Equation (2). Equation (3) gives a method to calculate the consumption of a customer containing an incomplete period, which is obtained by subtracting the area of the shadow from the area of the rectangle (
Figure 6). Note that the unit stock cost is related to the unit purchasing (selling) price and transport cost.
4.2.3. Transport Cost
The transport cost from supplier
to a customer is equal to
, so the total transport cost is given by Equation (4).
4.2.4. The Discount of Dispatching an Entire Train
If the supplier
operates low-frequency entire trains to deliver goods, the railway companies will offer a discount, which is equal to the transport cost times the discount rate. Therefore, the total discount is given by Equation (5).
4.2.5. The In-Transit Inventory Cost
The in-transit inventory cost from supplier
to customer is the product of the total demand of goods, time in transit, and daily interest rate, where in-transit time of dispatching an entire train
is the transportation distance divided by the operating rate. The in-transit time of dispatching a non-direct train
is the sum of
, the pick-up time by a local train, the waiting time for the accumulation progress of goods and the reclassification time. Thereby, the total in-transit inventory cost is given by Equation (6).
The objective of Function (7) is to minimize the total inventory cost which is the sum of stockholding costs of supplier
, the stockholding costs of customer
, transportation costs
, the discount of dispatching entire train
, and in-transit inventory cost
.
Equation (8) ensures that the inventory level is within the maximum limit.
This model is a linear 0-1 integer programming model, which can be solved by using the optimization software.
6. Conclusions
Based on the railway bulk cargo, this paper considers the possibility of operating low-frequency entire trains in areas that are insufficient for dispatching a entire train. In this paper, the optimization of the total cost for the supplier and customer is formulated as a linear 0-1 integer programming model, with the prime task of minimizing the total inventory costs, which include the stockholding costs of supplier, the stockholding costs of customer, transportation costs, and in-transit inventory costs, while satisfying the constraint of the maximum stock level of the customer. Different from the classic EOQ model, replenishment is made when the customer stock level is reduced to a minimum level. The model presented in this paper considers the situation that the supplier delivers the goods according to the delivery cycle. Using the principle of integration, a simplified calculation of the average inventory of customers was done. A real-world case was carried out including three suppliers and one customer. According to the results, two suppliers should operate a low-frequency entire train. The results indicate that the mathematical model proposed can be used to solve the real-world inventory cost problem. In future research, the classification time of the train in classification yards should be discussed in detail, which means the benefits of the railway companies should be included to make the discount rate more practical, and the willingness of the customer under the situation of road competition should be considered as well.