Research on Differential Pricing and Train Operation Decisions for Railway Cargo Transportation Under Competitive Conditions

Abstract

Railway freight operations aim not only to market themselves aggressively to capture the market in a competitive environment but also to achieve refined management. The operational decisions chosen by carriers can have a significant impact on the revenue across freight services. Nowadays, various forms of transported products, train make-up, and schedules are developed from the perspective of production convenience, with less consideration of shippers’ needs for the time utility, frequency, and tariffs of transportation services. This paper investigates a joint optimization approach to railroad operation planning that combines meeting the transport demand preferences of different shippers in a competitive environment. Considering the heterogeneity of shippers’ transportation demands and the competitive environment with freight trucking, a new bi-level programming model is proposed that incorporates pricing decisions and operational planning policies, such as car blocking, train routing, and make-up. An exact solution for this model is developed by adding valid inequalities to the mixed-integer formula and the experiment is conducted with data from bulk coal cargo transportation of the S Railway Bureau Group Corp. A simple transportation network is presented, and the calculation results of the pricing decisions that maximize the profit in each operation decision in this network are reported.

Keywords

freight systems economics and regulation of freight transportation pricing rail freight rail transportation marshaling

For many years, transportation service companies have understood that better marketing management and refined operational management of their logistics operations can improve their strategic position on the market. The revenue and freight market share aspects of day-to-day activities are becoming more and more important nowadays since many companies have realized that profit maximization mostly depends on the integration of logistics operations with improved pricing management. That is why pricing management plays a major role in transportation industries such as trucking, shipping, railroads, and airlines.

This paper analyzes rail freight transport as a vital economic sector. In 2020, China’s rail freight volume, an indicator of economic activity, reached 3.58 billion tons (China Railway Group 2020 Statistical Bulletin, 2020) ( 1 ). Rail accounted for 9.9% of the total social freight volume which includes freight volume transported by five modes: road, rail, water, air, and pipeline, so the overall market share level of rail freight to the whole society is still low. Competition has evolved to the point where the market and other transportation modes have a dominant influence on the policies of the railroad industry. In China, for example, the 2013 reform of railroad government-enterprise separation gave railroad companies more freedom, and transport competition has since played a greater role in regulating the rates charged for railroad cargo transport. In response to this competitive pressure from the market, railroad companies were forced to re-examine their operational management processes and pricing policies to remain competitive.

With the development of modeling and optimization techniques, many industries have improved their operational planning by establishing better activities management to take advantage of these innovations. However, railroad operations are composed of complex interrelated policies such as car blocking, timetabling, make-up and routing, yard management, locomotive assignment, empty car assignment, and crew scheduling. These are the main reasons why railroad transport has always been confronted with operational management tools that do not encompass all the realism of day-to-day operations. Part of the complexity is also because the railroad is operating on an existing physical network that has its own constraints, such as the resorting capacity of the marshaling yards, a freight train’s carrying capability, and so forth. The railroad industry is characterized by shippers with different attributes and perceptions of the services to be provided (e.g., sensitivity to tariffs or service reliability). Nowadays, various forms of freight products and marshaling plans in the railroad industry are designed from the perspective of facilitating production, with less consideration of shippers’ requirements for time utility, transport security, tariffs, frequency, volume, and convenience of transport service. The analysis of an integrated approach combining train operation planning decisions and marketing strategies is an interesting research perspective when trying to study the connection between operation management plans and pricing policies. This paper introduces a new bi-level mathematical formulation that combines both pricing decisions and train operation planning decisions. The paper analyzes the pricing policies of different freight shippers’ mode choice preferences and their impact on the formulation, as well as the resulting properties and valid inequalities that are used to strengthen the mathematical model. An exact solution method is proposed and computational results for randomly generated examples are reported.

Literature Review and Outline of Paper

Although growing research interest in the topic of operational planning and pricing issues in rail freight transportation can be observed, to the best of the authors’ knowledge, few research works have been focused on both pricing and operational planning in the context of bulk freight mode choice. This section reviews relevant research on operational planning, pricing method, and freight mode choice issues.

Railroad Freight Operation Planning and Management

Railroad freight operations have been extensively studied in the operations research literature and the main modeling and methodological contributions have been surveyed ( 2 – 10 ). More recently, operational research methodologies in rail passenger transportation have been surveyed and reviewed ( 11 – 15 ). This section takes a closer look at the contributions of integrated railroad freight operation planning and management activities including car transit, blocking and train make-up as well as train scheduling. Assad ( 2 ) mainly focused on the car routing and train make-up model. Crainic et al. ( 16 ) described a model with routing freight traffic, scheduling train services, and allocating classification work. Haghani et al. ( 4 ) presented a formulation and solution of a combined train routing and make-up, and empty car distribution model. Gorman ( 17 ) presented a joint train scheduling and traffic assignment model solved by a hybrid genetic and tabu search algorithm. Ahuja et al. ( 18 ) developed an algorithm to solve the railroad blocking problem which is a very large-scale, multicommodity, flow network design and routing problem. Lin et al. ( 19 ) presented a bi-level programming model to seek optimal settings for train services and traffic assignment in China’s rail network. Zhu et al. ( 20 ) proposed a model and solution methodology integrating car classification and blocking, train make-up, and routing as well as service selection and scheduling. A model based on the dynamic balance of transport and logistics was developed by Rakhmangulov et al. ( 21 ), in which the main events in the transport process, such as the operation of cargoes in independent railcars and the operation of trains, were modeled. Calculation of relevant parameters, such as timeliness of transportation and other indicators, were also determined. Xiao et al. ( 22 ) presented a block-to-train assignment model solved by the genetic algorithm and tabu search. To solve the problem of delayed cargo transport flows on the train network, taking into account the interests of each member of the transport and logistics chain, an efficient train assignment model based on the progressive distribution of flows on the network was developed, and then a hybrid intelligent system for cooperative railroad transport operations planning was also proposed to improve the management of railcar flows and improve the reliability of cargo transport, based on the principle of interaction of participants in the transport process ( 23 , 24 ). Ruf and Cordeau ( 25 ) proposed a formulation for the integrated planning problem comprising the railcar interchanges, the humping sequence, the schedule of service and safety processes, and the assignment of resources (infrastructure, staff, and locomotives) to processes. The optimization model is solved by a tailored adaptive large neighborhood search heuristic. These representative papers have conducted optimization studies that focus on several major production stages processes and full processes affecting the train formation planning process, and have achieved good results, but mainly from the perspective of improving the transport efficiency of enterprises, and less from the perspective of intervention and adjustment of transport economic instruments.

Freight Pricing Management

The management of the pricing aspect of freight transportation is very important today because of the increasing competition among different modes of cargo transportation. Kraft ( 26 ) proposed a model for both “dynamic car scheduling” and “train segment pricing” by using a bid price control framework. Li and Tayur ( 27 ) developed a mathematical programming model that jointly considers pricing and operations planning in the intermodal transportation and solved the model by a decomposition using the structure of subproblems. Crevier et al. ( 28 ) pointed out that logistics integration and revenue management can maximize the profit of railway freight transport, and proposed a bi-level mathematical programming model, including pricing strategy and network planning. Zhang and Dan Liu ( 29 ) built a mixed-integer programming model for joint optimization of train operation decisions and pricing decisions with the maximum operating revenue of the railroad transportation company as the objective function and the stability of the railroad transportation network as the premise. Zeng et al. ( 30 ) developed an improved gray relational analysis to establish a comprehensive performance evaluation of railroad freight pricing policies and to improve the price decision-making ability of railroad freight companies. There are papers that consider railroad transport organization jointly with pricing strategy for optimization, for instance, on the allocation of rail wagons, on the intermodal transport, on the combination with revenue management, or on the consideration of price factors on container transport, but few papers specifically study the combination of formation planning and pricing for bulk railroad freight at freight marshaling stations.

Freight Mode Choice Management

The mode choice is the third stage of the conventional four-step disaggregate model, which is traffic generation, traffic distribution, traffic mode classification, and traffic assignment. Transportation mode choice or transport chain is the most critical part of any freight demand modeling framework, and understanding the reasons why shippers, receivers, and freight forwarders choose a particular mode of transportation for their shipments is critical to developing appropriate transportation policies. In the methodological aspect of transportation mode choice, the development of transportation mode choice models has gone through several different steps. Based on the theory of mode choice behavior of shippers or receivers, Winston ( 31 ) developed a stochastic expected utility model and estimation suitable for econometric analysis. Cook et al. ( 32 ) summarized the key factors of cargo transportation mode selection and presented the relative importance of these factors and customer satisfaction evaluation. Jiang et al. ( 33 ) estimated a discrete choice model for classification using freight demand survey data from France and found that transport distance and shipper access to transport infrastructure were key determinants. Norojono and Young ( 34 ) developed a model for the freight choice decisions between rail and road and found that safety, reliability, and responsiveness are the main attributes affecting the choice of rail versus road freight modes. Two binary choice models were developed by Samimi et al. ( 35 ) to understand mode choice behavior in the U.S. freight markets and found that trucking is extremely sensitive to travel time whereas rail transportation is cost sensitive. A discrete choice model was applied by Arencibia et al. ( 36 ) to analyze the main determinants of freight mode choice based on a SP data set. The SP (stated preference) survey is designed to obtain people's subjective preferences for multiple choices under hypothetical conditions. Jensen et al. ( 37 ) presented the estimation of a discrete freight transport chain choice model for Europe, which is based on disaggregated data at the shipment level.

Maritime freight transport is another important mode of transport and some scholars have studied the operation methods in response to the increase of freight volume by considering cost minimization and price decisions. Xu et al. ( 38 ) developed a model to study empty equipment repositioning decisions by using capacity pricing policies to balance the cargo demands between carriers and freight forwarders. An appropriate pricing strategy has been proposed by Umang et al. ( 39 ) based on a given baseline vessel schedule to generate greater revenue from late-arriving vessels to solve the real-time recovery problem in port operations. An integrated optimization model was presented by Zhen et al. ( 40 ) for the daily berth planning problem in tidal ports, where ships have a higher priority and more attention should be paid to reducing their waiting time in the presence of channel flow control. Xiang et al. ( 41 ) aimed to minimize the recovery cost and adopted the prospect theory to express the preference of decision-makers, and proposed a reactive strategy for different disruptions on the allocations of berths and quay cranes under uncertainty. Dulebenets ( 42 – 44 ) studied cost-effective sailing schedules and green vessel scheduling, and proposed a collaborative agreement between shipping companies and marine container terminal operators to enhance the efficiency of the shipping operations. According to the existing literature mentioned above, the integration of pricing decision management with rail freight operation planning management has not been well considered in the literature. There are few studies on the comprehensive optimization of railway bulk cargo transportation. At present, the pricing decision-making of railway transport companies lacks flexibility, and the single pricing method cannot meet the rapid change of transport market demand. Because the transport market is becoming more and more complex and competitive pressure is increasing, railway operation management needs to explore more scientific pricing methods in combination with the transport behavior of competitors (road transport in this paper). The specifics of pricing and market environment in the transport industry are strongly influenced by the type of competitive market, as well as the specifics of rail transport organization and management in different countries. For example, unlike the oligopolistic markets in the United States, the rail transport markets in China and Russia are monopolistic. However, many private companies own rail vehicles in Russia. This will limit the flexible pricing ability of that rail transport industry to some extent. Because of the complexity of the railway transportation network, when making comprehensive optimization of railway freight pricing decisions and train operation decisions, it is necessary to combine such factors as the customer’s behavior in choice of cargo transportation modes, capacity limitations of the transport network and competition in the transport market. Therefore, this paper studies the comprehensive optimization of train operation decision-making based on differential pricing decision-making for different routes, transport equipment elements, and cargo types, combined with the customer’s transport mode selection behavior aimed at the lowest total generalized cost. The rest of the paper is organized as follows. The third section presents a customer’s generalized cost calculation formula to describe the behavior of customers in the choosing of cargo transportation modes. The fourth section provides a comprehensive optimization model for differentiated pricing based on train operation decisions. The fifth section develops a solution algorithm by using an effective inequality to enhance the relationship between differential pricing and freight volume decision variables. The sixth section shows how the instances on which the solution methodology was tested were generated. Finally, in the seventh section, a few concluding remarks and opportunities for subsequent research are provided.

Customer Choice Behavior in Choosing the Mode of Cargo Transportation

With the continuous growth of the freight market and intense competition between various modes of cargo transportation, the share of different modes of transport in the freight market depends not only on the freight rates, but also on the level of their freight transport services. Moreover, when considering the level of service of other transport modes, the shipper chooses the transport mode with the lowest generalized transport costs. The shipper’s generalized transportation cost refers to the factors such as transportation cost, shipping time utility, and service quality satisfaction. These are converted into monetary values through the time value of cargo transportation to support the shipper’s mode choice; a detailed description is provided in Crevier et al. ( 28 ). Transportation costs are the costs to be paid for transporting a unit of goods. The shipping time unity aims at shortening product delivery time and improving product delivery on-time, bringing benefits to customers. The rail shipping time includes cargo dispatch time, transit time, and intermediate operation time. Service quality satisfaction refers to the shipper’s perception of service quality, considering the safety and convenience of goods in transportation. Safety can generally be measured by the rate of damage during shipping. Better accessibility leads to lower transportation costs and shorter time to the freight station. Accessibility is mainly measured by the convenience of transport, that is, the distance of the shipper or receiver from the rail freight station loading and unloading location. Characterization of each itinerary as well as the sensitivity of customers to service quality was proposed by Marcotte and Savard ( 45 ) and Côté et al. ( 46 ) for airline passenger transportation and Crevier et al. ( 28 ) extended this to rail freight. The formula for calculating the cargo owners’ generalized cost is as follows:

J^{k} = w_{1}^{k} p^{k} + w_{2}^{k} q^{k} + w_{3}^{k} h^{k}

(1)

where $p^{k}$ , $q^{k}$ , and $h^{k}$ represent the shipping price, perceived cost of service quality, and perceived cost of shipping time utility of K-type cargo, k represents the type of cargo, $w_{1}^{k}$ , $w_{2}^{k}$ , and $w_{3}^{k}$ represent the weight of three indicators: freight rate, satisfaction with service, and time utility, and $J^{k}$ represents the generalized cost of transportation of K-type cargo.

Customers need to evaluate the three kinds of cost indicators, synthesize the generalized transportation cost of each mode of transportation, and choose the transportation mode with the smallest generalized cost ideally. If the general costs of railway transport and other competitive modes of transport are equal, then it can be assumed that customers will choose railway transport according to the “optimism” principle. The optimism principle in this case means that shippers will give priority to rail freight, subject to compliance with the Chinese government’s environmental protection policies. The calculation of each cost indicator in the customer’s generalized cost is as follows.

Shipping time utility measurement. The calculation of shipping time utility is mainly calculated by shipping time value. Previous studies, such as Hummels and Schaur G ( 10 ), extracted the shipper’s valuation of time for air and ocean shipments and estimated that each day in transit is worth 0.6 to 2% of the value of the goods. The time utility measurement can be understood as each day in transit and having the impact on the production and sales of unit goods in one day’s transportation. It can be extended in rail freight in the same way and expressed as A × TR, where A is the average value per railcar load of a certain type of cargo such as coking coal. China’s Daqin Railway coal transportation service can pull about 80 tons of coal per railcar. Coking coal accounts for 27% of China’s total coal, and in August 2021 the price was close to RMB 2,500 (CNY) per ton, so the A-value used here is RMB 200,000 (CNY) per railcar. TR is the ratio of the shipping time cost incurred per unit of cargo in one day of transportation to the average value of the unit of cargo, and this paper takes the ratio of 1%.

The perceived cost of service quality is the customer’s evaluation of the transport quality of the carrier company, including the convenience and safety of transport.

Convenience cost can be calculated by the time consumed in the process of loading and unloading goods. Through the investigation of the steps of loading and unloading railway goods, this paper assumes that the loading and unloading process at both ends of the transport process takes 3 h, and then calculates the convenience cost by combining the time-consuming cost of the unit hour derived from the time-consuming cost. (20,0000 × 1% / 24) ×3 = 250 CNY/railcar.

Safety is mainly manifested in the rate of damage to goods in transit. The rate of damage to goods can calculate the cost of safety. According to relevant research, the rate of damage to goods on the railroads accounts for 0.05% of goods, and that on the highway accounts for 0.06%. According to the calculation, it can be concluded that the safety cost of the railway is 100 CNY (14.9 USD) per car, and that of the highway is 120 CNY (17.9 USD) per car.

Comprehensive Optimization Model of Train Operation Decision Based on Differential Pricing

This paper studies cargo shipments as operated on the railroad transportation network. It considers the marshaling yard operation handling capacity, the trains operated economically with loading capacity from the marshaling yard, and the railway line section passing capacity as constraints, and pursues the objective of total profit maximization of the railroad transportation company under the differentiated pricing policy. A bi-level program model is developed, with the upper level objective of maximizing the profit of the railroad transportation company and the lower-level objective of minimizing the generalized transportation cost of the shipper. This integrated optimization model is based on the pricing decision assumptions in the case of competitively differentiated pricing for road cargo transportation. The game state of rail and road freight volume-sharing under differential pricing of the road transportation network is analyzed by assuming the freight rate decisions of road transportation competitors under differential pricing conditions. According to the characteristics of the model, a simplified model based on the first duality principle is used as a single-level mixed-integer programming model, and the branch-and-bound method is used to solve the model.

Model Building Environment

Model Hypothesis

To simplify the influencing factors of the problems studied in this paper, the following assumptions need to be made:

In this model, it is assumed that the competitor of rail freight is road transport, and the customer can only choose between rail and road transport modes.

It is assumed that the transport price of road cargo transportation is known.

It is assumed that in the transport market, rail transport and road transport capacity will be able to meet the needs of freight customers.

Under differential pricing conditions, assuming that the total demand for each origin–destination pair (OD) is fixed, the rail and road freight market shares change accordingly as the rail OD demand points are differentially priced.

The customer’s perception of the time utility of transportation and the quality of service is transformed into the generalized cost. The customer chooses the mode of transportation based on the generalized cost consisting of freight, shipping time cost, and service perceived cost. In the condition of equal generalized costs for rail and road transport, the “optimistic” principle can be followed, assuming that the shipper will choose rail shipment.

Assuming the differential pricing decisions in railway freight rates, the road freight rates remain unchanged for a short time, and the price response function of freight volume is only related to railway freight rates.

It is assumed that freight train formations are operating at full capacity. Major freight trains of Chinese railroads, including bulk freight trains, are required to operate in formation at full axle load, except for some types of freight trains, such as pick-up and drop train and district transfer train, which are allowed to operate on less than full axle loads. Full axle load indicators are mainly expressed in the number of train cars, the total length of the train, and the total traction weight of the train. Train formation can be carried out by meeting one of the constraints of the full axle load constraint. When establishing the model, only the full axle load constraint of the number of train cars is considered, but not the total train length and total traction weight. The reason is that there is a large gap between different freight locomotive types in total train length and total train traction weight.

Variables and Parameters Involved in the Model

1. Model parameters

g = the category of road cargo transportation mode.

I = the set of all OD demand pairs in the railway transport network.

L = the set of all sections in a railway transport network.

B_i = the set of all marshaling stations on an ith OD demand pair.

E_b = the set of available train element types at marshaling station b.

The transport trains are categorized as two types: technical direct train and through cargo train. The main operation of China’s railroad formation stations is the disassembly and grouping of many trains, most of them being technical direct trains and through cargo trains. Technical direct trains are cargo trains that are made up in technical stations and pass through one or more marshaling stations without reorganization operations. Through cargo trains are cargo trains that are made up in technical stations and pass through one and more district stations without reformatting operations. Technical direct cargo trains and through cargo trains are both cargo trains grouped at technical stations, the difference is that technical direct trains pass through one or more marshaling stations without regrouping, while through cargo trains pass through one or more district stations without regrouping.

cap_e = the capacity of type $e$ trains, in number of railcars. Type $e$ trains belong to the set which is the technical direct trains and through trains.

$μ_{e}$ = the full capacity coefficients of less-than-carload and pick-up freight trains are generally 0.5 and 0.75, respectively, and for both direct and through freight train are 1.

cap_b = indicates the maximum operational handling capacity of a marshaling station over a period of time, measured by the number of trains.

N_l = the capacity of section $l$ in a period expressed by the number of trains.

$p_{i}^{gk}$ = the unit freight rate for K-type goods transported by the i-th OD demand in the road transport network, which is assumed to be known.

S_e = the fixed operational cost of type e of trains.

V = the variable cost of transportation per unit of rail transportation.

$α_{i}^{e}$ = occupancy of e-type trains in the i-th OD freight demand. If occupancy $a_{i}^{e}$ = 1, otherwise $a_{i}^{e}$ = 0.

$β_{e}^{l}$ = occupancy of type e of trains on transport section L. If occupied, its value is 1, if not occupied, its value is 0.

$γ_{e}^{b}$ = occupancy in type e of trains to railway marshaling station b. Suppose the occupancy value is 1 and if the not occupied value is 0.

$q_{i}^{k}$ = the unit perceived cost of quality of service for railway transportation of K-type goods for the i-th OD freight demand in the road network.

$h_{i}^{k}$ = unit time utility cost of railway transportation of K-type goods for the i-th OD freight demand in the railroad network.

$q_{i}^{gk}$ = the unit perceived cost of service quality for road transport of K-type goods for the i-th OD freight demand in the road network.

$h_{i}^{gk}$ = unit time utility cost of road transport K-type goods for the i-th OD freight demand in the road network.

$w_{i}$ = weight of freight, time utility, and service quality in the calculation of customer’s generalized cost. Among them, $i = 1, 2, 3$ , and $\sum_{k = 1}^{3} w_{i} = 1$ , $w_{i}$ is determined by a questionnaire survey of customers in the freight market.

$J_{i}^{k}$ = the unit shipper’s generalized cost of the K-type cargo for the i-th OD freight demand in the rail network. The formula is as follows: $J_{i}^{k} = w_{1}^{k} P_{i}^{k} + w_{2}^{k} q_{i}^{k} + w_{3}^{k} h_{i}^{k}$ (1-1)

$J_{i}^{gk}$ = the unit shipper’s generalized cost of the K-type cargo for the i-th OD freight demand in the road network with the following equation: $J_{i}^{gk} = w_{1}^{k} P_{i}^{gk} + w_{2}^{k} q_{i}^{gk} + w_{3}^{k} h_{i}^{gk}$ (1-2)

$d_{i}^{k}$ = demand for K-type goods in the i-th OD freight demand, measured by the number of railcars.

$O_{i}^{k}$ = the actual share of the railway transport mode to ship the K-type goods in the i-th OD freight demand. The actual share will vary with the change of freight rate. A price linear response function can express it. $O_{i}^{k} = a - {bP}_{i}^{k}$ , where a and b are constants.(1-3)

$D_{i}^{k}$ = customer service perceived cost for transportation of K-type goods in the i-th OD freight demand is composed of time-sensitive cost perceived from the customer’s point of view and service quality perceived cost perceived from the cargo’s point of view. The cost measured in Chinese yuan (CNY). The formula is as follows:

D_{i}^{k} = w_{2}^{k} q_{i}^{k} + w_{3}^{k} h_{i}^{k}

(2)

${Dm}_{i}^{k}$ = the smallest customer service perception cost for all OD cargo transport requirements.

2. Decision variables

$U_{i}^{e}$ = number of e-type trains running in the i-th OD freight demand.

$f_{i}^{k}$ = the share of railroad transport K-type goods for the i-th OD demand in the transport network, measured by the number of railcars.

$f_{i}^{gk}$ = the share of road transport K-type goods for the i-th OD demand in the transport network, measured by the number of vehicles.

$p_{i}^{k}$ = unit freight rate of the i-th OD demand in the railroad transport network for transporting K-type goods.

Modeling

According to the actual situation, both transportation modes in the competitive relationship in the model consider their profit maximization when pricing. This relationship is a non-cooperative game in game theory. Railroads are not competitive in time utility and comfort of service when transporting cargoes, compared with road transportation. Considering the requirements to attract more time-sensitive and service-sensitive cargo from competitors, therefore, it is necessary to set different rates for different types of cargo and different OD freight demands. This subsection combines this theory with an integrated optimization model for pricing decisions and train operation decisions. The volume of freight carried by railroad transport companies is closely related to the price, and differential pricing inevitably changes the customer’s choice of freight transport mode, that is, it changes the share of freight volume allocated to the railroad transport network. To simplify the problem, a linear function of the price response of rail freight volumes is introduced into the integrated optimization model of differential pricing and train operation decisions, assuming that road prices do not change over a short period of time, while rail prices are fixed. Obviously, the curvilinear freight volume price response function will reflect the real freight volume price response more accurately, but it will increase the difficulty of model solving if the curvilinear freight volume price response function is considered. The analysis shows that the operating profit of railroad freight companies is influenced by the cost of running trains and the fixed costs of transportation as well as the volume of freight. The upper level objective of the model is expressed as follows.

Max \sum_{i \in I} \sum_{k \in K} [(P_{i}^{k} - V_{i}^{k}) f_{i}^{k}] - \sum_{i \in I} \sum_{e \in E} U_{i}^{e} S_{e}

(3)

The model takes into account that the shipper’s choice of transport mode is necessarily influenced by differentiated pricing decisions based on transport routes, cargo types, and train element types, which are directly reflected in the rail and road transport shares. In addition, the price response function for rail freight share is added to consider the supply–demand relationship for rail freight. The upper level constraints of the model are as follows:

s . t . \sum_{i \in I} \sum_{e \in E} β_{e}^{l} U_{i}^{e} \leq N_{l}, \forall l \in L

(3-1)

\sum_{i \in I} \sum_{e \in E} γ_{e}^{b} U_{i}^{e} \leq ca p_{b}, \forall b \in B

(3-2)

\sum_{i \in I} \sum_{k \in K} α_{i}^{e} f_{i}^{k} \leq \sum_{i \in I} U_{i}^{e} μ_{e} ca p_{e}, \forall e \in E

(3-3)

O_{i}^{k} = α - {bP}_{i}^{k}, \forall k \in K, i \in I

(3-4)

P_{i}^{k} \geq 0, P_{i}^{gk} \geq 0 and integer, \forall k \in K, i \in I

(3-5)

U_{i}^{e} \geq 0 and integer, \forall e \in E

(3-6)

Constraint 3-1 indicates that the number of trains passing through rail section $l$ in a period should not exceed the capacity of section $l$ . Constraint 3-2 indicates that the total number of trains entering and leaving the marshaling station $b$ does not exceed the operational capacity of the receiving and dispatching trains at marshaling station $b$ . Constraint 3-3 is that the amount of freight wagons transported by $e$ -type train cannot exceed the full loading capacity of the $e$ -type train, which is measured by the number of railcars. In constraint 3-4, $O_{i}^{k}$ is the freight volume of the price response function. $O_{i}^{k} = a - {bP}_{i}^{k}$ indicates that the demand for goods on line $i$ is affected by its own price changes in the level $P_{i}^{k}$ of rail freight rates, a linear approximation of the volume of transport carried by rail freight, where it is assumed that road freight rates do not change for a short period of time because of changes in rail prices, where $a$ and $b$ are constants. According to historical data, it can be found that a = 16,737 and $b = 3.8$ . Constraint 3-5 indicates the range of values for rail and road rates. Constraint 3-6 denotes that $U_{i}^{e}$ is greater than or equal to zero and is an integer. In addition, the model takes into account the impact of road pricing decisions on rail freight revenue under competitive market conditions. The model introduces game theory to establish lower objective function and constraints based on minimizing the total generalized cost of freight transport for customers.

Min \sum_{i \in I} \sum_{k \in K} J_{i}^{k} f_{i}^{k} + \sum_{i \in I} \sum_{k \in K} J_{i}^{gk} f_{i}^{gk}

(4)

s . t . f_{i}^{k} + f_{i}^{gk} = d_{i}^{k}, \forall k \in K (λ_{i}^{k})

(4-1)

f_{i}^{k} \geq 0, f_{i}^{gk} \geq 0, and integer, \forall i \in I, k \in K

(4-2)

Constraint 4-1 is expressed as the sum of the volume of K-type goods transported by rail and the volume of K-type goods transported by road is equal to the total demand for K-type goods transported. Constraint 4-2 indicates that the number of wagons transported by rail and road on line i is greater than or equal to zero and both are integers.

Finally, the bi-level programming model formula in Equations (3)–(3-6) and (4)–(4-2) for train operation decision-making optimization based on differential pricing is obtained as follows:

Max \sum_{i \in I} \sum_{k \in K} [(P_{i}^{k} - V_{i}^{k}) f_{i}^{k}] - \sum_{i \in I} \sum_{e \in E} U_{i}^{e} S_{e}

(5)

s . t . \sum_{e \in E} \sum_{i \in I} β_{e}^{l} U_{i}^{e} \leq N_{l}, \forall l \in L

(5-1)

\sum_{e \in E} \sum_{i \in I} γ_{e}^{b} U_{i}^{e} \leq ca p_{b}, \forall b \in B

(5-2)

\sum_{k \in K} \sum_{i \in I} α_{i}^{e} f_{i}^{k} \leq \sum_{i \in I} U_{i}^{e} μ_{e} ca p_{e}, \forall e \in E

(5-3)

O_{i}^{k} = a - {bP}_{i}^{k}, \forall k \in K, i \in I

(5-4)

P_{i}^{k} \geq 0, P_{i}^{gk} \geq 0, and integer, \forall i \in I, \forall k \in K

(5-5)

U_{i}^{e} \geq 0, and integer, \forall e \in E

(5-6)

Min \sum_{i \in I} \sum_{k \in K} J_{i}^{k} f_{i}^{k} + \sum_{i \in I} \sum_{k \in K} J_{i}^{gk} f_{i}^{gk}

(5-7)

f_{i}^{k} + f_{i}^{g} k = d_{i}^{k}, \forall k \in K (λ_{i}^{k})

(5-8)

f_{i}^{k} \geq 0, f_{i}^{gk} \geq 0, and integer, \forall i \in I, k \in K

(5-9)

The decision variables of the model are $P_{i}^{k}$ , $f_{i}^{k}$ , $U_{i}^{e}$ and $f_{i}^{gk}$ , where $λ_{i}^{k}$ is the dual variable corresponding to the constraint condition 5-8.

Solution Algorithm

Algorithmic Design

Through the structure of the model formula (Equations (5)–(5-9), we can see that the model is a non-linear integer bi-level programming model. This subsection reconstructs the bi-level programming model into a single-level mixed-integer programming model using the original duality theory, and the number of decision variables in the model is simplified, that is, the number of alternative decisive variables will be changed into known variables. The dual model of the lower model Equations 5 -7–5 -9 is as follows:

Max \sum_{k \in K} \sum_{i \in I} λ_{i}^{k} d_{i}^{k}

(6)

λ_{i}^{k} \leq J_{i}^{k}, \forall k \in K, i \in I_{k}

(6-1)

λ_{i}^{k} \leq J_{i}^{gk}, \forall k \in K, i \in I_{k}

(6-2)

The second level program of optimization conditions includes original feasibility (constraints 4-1, 4-2), dual feasibility (constraints 6-1, 6-2), and the following complementarity constraints:

f_{i}^{k} (J_{i}^{k} - λ_{i}^{k}) = 0

(6-3)

f_{i}^{gk} (J_{i}^{gk} - λ_{i}^{k}) = 0

(6-4)

The following optimization conditions are proposed based on the model building environment as well as the model structure.

Linearization of complementary constraints, such as the linear representation of constraint 6-3 is as follows:

(6 - 3) \Leftrightarrow {\begin{matrix} J_{i}^{k} - J_{i}^{gk} \leq Z_{i}^{1} s_{i}^{1} \\ f_{i}^{k} \leq Z_{i}^{2} s_{i}^{2} \\ s_{i}^{1} + s_{i}^{2} \leq 1 \\ s_{i}^{1}, s_{i}^{2} \in 0, 1 \end{matrix}

(6-5)

The constant parameter Z represents the upper bound on the left side of the corresponding constraint and the $s$ integer variable is introduced to ensure that at least one of the first two constraints in this set of constraints 6-5 is guaranteed to be equal to 0, so that constraint 6-3 is satisfied.

2. It can be concluded that the share of railway transport mode and railway freight volume can be regarded as approximately equal.

f_{i}^{k} = O_{i}^{k} = a - {bP}_{i}^{k}

(6-6)

The values of $a$ and $b$ are calculated from historical data. Because most of the road freight price data appear on a few websites, as well as the negotiated prices, the overall market share split between rail and road freight is published on the website of the National Bureau of Statistics of China.

The bi-level model formula (Equations (5)–(5-9)) based on differential pricing can be optimized to the model in Equation 7 as follows:

Max \sum_{i \in I} \sum_{k \in K} (P_{i}^{k} - V_{i}^{k}) (a - {bP}_{i}^{k}) - \sum_{i \in I} \sum_{e \in E} U_{i}^{e} S_{e}

(7)

s . t . \sum_{i \in I} \sum_{e \in E} β_{e}^{l} U_{i}^{e} \leq N_{l}, \forall l \in L

(7-1)

\sum_{i \in I} \sum_{e \in E} γ_{e}^{b} U_{i}^{e} \leq ca p_{b}, \forall b \in B

(7-2)

\sum_{i \in I} \sum_{k \in K} α_{i}^{e} f_{i}^{k} \leq \sum_{i \in I} U_{i}^{e} μ_{e} ca p_{e}, \forall e \in E

(7-3)

f_{i}^{k} + f_{i}^{gk} = d_{i}^{k}, \forall k \in K (λ_{i}^{k}))

(7-4)

{\begin{matrix} J_{i}^{k} - J_{i}^{gk} \leq Z_{i}^{1} s_{i}^{1} \\ f_{i}^{k} \leq Z_{i}^{2} s_{i}^{2}, \forall e \in E, \forall i \in I \\ s_{i}^{1} + s_{i}^{2} \leq 1 \\ s_{i}^{1}, s_{i}^{2} \in 0, 1 \end{matrix}

(7-5)

U_{i}^{e} \geq 0, and integer, \forall e \in E

(7-6)

f_{i}^{k} \geq 0, f_{i}^{gk} \geq 0, and integer, \forall i \in I, k \in K

(7-7)

Effective Inequalities to Strengthen the Relation Between Differential Pricing and Freight Volume Decision Variables

After the initial solution of the model in this paper, the few model inequalities makes it more convenient to solve. By sorting out the parameters of the model, however, it is found that the integrality gap of branch-and-bound root nodes is generally large, that is, the difference between the objective value of the problem and the objective of the initial solution is large, which is not conducive to the solution of the larger scale model. Therefore, an effective inequality optimization method is proposed to enhance the correlation between differential pricing and freight volume decision variables to optimize the integrality gap of the demarcation points. The inequalities proposed are as follows:

Inequality: For $k \in K, {Dm}_{i}^{k} = \min {D_{i}^{k} | i \in I}$ , there are inequalities:

P_{i}^{k} + (D_{i}^{k} - {Dm}_{i}^{k}) s_{i}^{2} / w_{1}^{k} \leq (J_{i}^{gk} - {Dm}_{i}^{k}) / w_{1}^{k}

(8)

where $D_{i}^{k}$ is the perceived cost of customer transport services for the i-th OD freight demand to transport K-type goods, consisting of perceived time utility costs and perceived quality of cargo transport services from the customer’s perspective. ${Dm}_{i}^{k}$ represents the minimum perceived cost of shipment service for all OD freight transport demands.

Proof: First, to ensure railway freight transport has sufficient freight demand, the optimistic assumption is that the generalized cost per line for rail shippers is less than or equal to the generalized cost for road shippers.

J_{i}^{k} \leq J_{i}^{gk}, \forall i \in I, k \in K

(9)

Therefore, it can be deduced that:

J_{m_{i}}^{k} \leq J_{i}^{gk} \Leftrightarrow w_{1}^{k} P_{i}^{k} + {Dm}_{i}^{k} \leq J_{i}^{gk} \Leftrightarrow P_{i}^{k} \leq (J_{i}^{gk} - {Dm}_{i}^{k}) / w_{1}^{k}

(10)

Equation 10 expresses the upper bound of freight rate for OD demand of K-type goods. However, in the process of seeking the ptimal solution, there must be:

D_{i}^{k} \geq {Dm}_{i}^{k}

(11)

Then the upper bound of the freight rate will be changed to:

P_{i}^{k} \leq (J_{i}^{gk} - D_{i}^{k}) / w_{1}^{k}

(12)

The upper bound decreases:

(D_{i}^{k} - {Dm}_{i}^{k}) / w_{1}^{k}

(13)

According to constraint 7-5, only when $s_{i}^{2} = 1$ does there exist freight volume $f_{i}^{k}$ on the i-th OD demand line, and the upper bound of the freight rate will also decrease. Thus we can express it by the inequality in Equation 14:

P_{i}^{k} + (D_{i}^{K} - {Dm}_{i}^{k}) s_{i}^{2} / w_{1}^{k} \leq (J_{i}^{gk} - {Dm}_{i}^{k}) / w_{1}^{k}

(14)

The inequality will be evaluated by adding the bi-level formula (Equation 7). The model bi-level formula (Equation 15) is as follows:

Max \sum_{i \in I} \sum_{k \in K} (P_{i}^{k} - V_{i}^{k}) (a - {bP}_{i}^{k}) - \sum_{i \in I} \sum_{e \in E} U_{i}^{e} S_{e}

(15)

s . t . \sum_{i \in I} \sum_{e \in E} β_{e}^{l} U_{i}^{e} \leq N_{l}, \forall l \in L

(15-1)

\sum_{i \in I} \sum_{e \in E} γ_{e}^{b} U_{i}^{e} \leq ca p_{b}, \forall b \in B

(15-2)

\sum_{i \in I} \sum_{k \in K} α_{i}^{e} f_{i}^{k} \leq \sum_{i \in I} U_{i}^{e} μ_{e} ca p_{e}, \forall e \in E

(15-3)

f_{i}^{k} + f_{i}^{gk} = d_{i}^{k}, \forall k \in K (λ_{i}^{k}))

(15-4)

{\begin{matrix} J_{i}^{k} - J_{i}^{gk} \leq Z_{i}^{1} s_{i}^{1} \\ f_{i}^{k} \leq Z_{i}^{2} s_{i}^{2} \\ s_{i}^{1} + s_{i}^{2} \leq 1 \\ s_{i}^{1}, s_{i}^{2} \in 0, 1 \end{matrix}, \forall e \in E, \forall i \in I

(15-5)

P_{i}^{k} + (D_{i}^{k} - {Dm}_{i}^{k}) s_{i}^{2} / w_{1}^{k} \leq (J_{i}^{gk} - {Dm}_{i}^{k}) / w_{1}^{k}, \forall i \in I, k \in K

(15-6)

U_{i}^{e} \geq 0, and integer, \forall k \in K, i \in I

(15-7)

f_{i}^{k} \geq 0, f_{i}^{gk} \geq 0, and integer, \forall i \in I, k \in K

(15-8)

Constraint 15-6 represents the valid inequality that enhances the relationship between differential pricing and freight volume decision variables.

Case Verification

Data Preparation

In this case, marshaling stations with high freight formation activity under the jurisdiction of China Railway S Railway Bureau Group Ltd were selected to form a simple railroad transportation network. The S Railway Bureau Group Ltd is one of the 18 bureau group companies managed by China Railway Corporation. According to the distribution of marshaling stations and the activity of freight transportation within the jurisdiction of S Railway Bureau Group Ltd, in this paper, five marshaling stations were chosen to form a simple railway transportation network, namely: Hefei Station, Xuzhou Station, Nanjing Station, Ningbo Station, and Jinhua Station. The inter-station transport relationship is shown in Figure 1 below.

Figure 1.

Schematic chart of freight transport network of five marshaling stations in China.

According to the network schematic diagram, it can be seen that the freight transport network is composed of two simple lines, 1. Hefei Station→3. Nanjing Station→4. Ningbo Station and 2. Xuzhou Station→3. Nanjing Station→5. Jinhua Station, which are intersected by Nanjing marshaling station. Here the name of the marshaling station is only marked by the name of the city, not the actual name of the grouping station, such as Nanjing East marshaling station. For concise expression, only the Nanjing station is marked. It can be seen that there is a variety of OD demand in the transport network which is composed of a combination of lines, and each line can be transported by either rail or road. According to the historical data of S Railway Administration in 2017, there are eight OD demands in the transportation network: 1→3, 1→4, 1→4, 1→5, 2→3, 2→4, 2→5, 3→4 and 3→5. It is assumed that, according to the transport rules of transport companies for each category of goods and the customer’s preference requirements for transporting each category of goods, the transported goods are classified as price-sensitive goods (k = 1), time-sensitive goods (k = 2), or service-sensitive goods (k = 3), and both rail and road can transport these three types of cargo. According to the composition of the example transportation network diagram, this paper sets the type of transportation trains as the two types of technical direct trains and through cargo trains.

In this paper, the feasibility of the model is validated by the data of one day of freight transportation in the transport network, and the decision-making of daily train operation for railway freight transportation is obtained. It is assumed that the transport demand between OD demand is relatively stable over a period ( $t$ = one day). So assuming that the freight demand between OD points is fixed during the calculation period, the daily freight demand between OD points is calculated based on the historical data of freight transportation in the year 2017. The results are shown in Table 1.

Table 1.

Daily Demand for Various Types of Goods at Origin–Destination (OD) Demand Points

OD	Demand for price-sensitive goods $d_{i}^{1}$	Demand for time-sensitive goods $d_{i}^{2}$	Demand for service-sensitive goods $d_{i}^{3}$	Total
1→3	118	15	516	649
1→4	250	32	1088	1370
1→5	63	8	273	344
2→3	1857	55	426	2338
2→4	145	19	631	795
2→5	172	22	748	942
3→4	67	9	291	367
3→5	109	18	473	600

Note: Unit = Car.

After calculation, the perceived cost of service quality of railway transportation is 350 CNY per railcar. Because road transportation can carry out “door-to-door” transportation, it can be considered a zero-convenience cost. The perceived cost of service quality of road transportation is 120 CNY per vehicle, and the perceived cost of cargo time utility and the service quality for rail and road transport can be derived.

According to the questionnaire of cargo owners and the data collected, the generalized cost can be used to calculate the freight cost and the weight of each perceived cost. Based on the statistics of historical freight data of S Railway Bureau Group Ltd, we know the freight rates of various types of goods in road transport, the generalized cost of road transport, the variable cost of railway freight transport, and the fixed cost of running various types of trains. The train capacity of all types of trains that can be operated on the line is set to 60 railcars per train. It is known that there are five marshaling stations and four sections in the transport network, and the operating capacity of each formation station for freight trains is assumed as follows: $ca p_{b}$ = (9, 5, 8, 8, 6), $b \in B$ . The operating capacity of the formation yard is greater than this set value, and here is an example of the calculation. The passing capacity of four sections in the transport network is assumed as follows: $N_{l}$ = (8, 5, 6, 8), $l \in L$ . Based on the train formation scheme being carried out by S Railway Bureau Group Ltd, eight technical direct type or through cargo type trains with different origins and destinations provide freight services between these eight pairs of OD demands, namely: 1→3 through cargo train, 1→4 technical direct train, 1→5 technical direct train, 2→3 through cargo train, 2→4 technical direct train, 2→5 technical direct train, 3→4 through cargo train, 3→5 through cargo train.

In addition, the matrices $β_{e}^{l}$ , $γ_{e}^{b}$ can be used to represent the occupancy of eight kinds of trains with different origins and destinations in four railroad sections and five marshaling stations.

β_{e}^{l} = [\begin{matrix} 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 1 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{matrix}], \forall e \in E, l \in L γ_{e}^{b} = [\begin{matrix} 1 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 & 1 \\ 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 0 & 1 \end{matrix}], \forall e \in E, b \in B

(15-8)

where the number 1 in the matrices $β_{e}^{l}$ , $γ_{e}^{b}$ means occupied, the number 0 means not occupied. The occupancy relationship of freight OD demand for eight kinds of trains in the transport network is set according to the train formation scheme. The transport network diagram is a simple road network formed by the intersection of two transport routes. This paper lists 16 train marshalling schemes by exhaustive method. For example, the first train marshalling scheme is to run the corresponding trains independently according to each OD pair, and the transportation demand of the eight OD pairs should run the corresponding trains. Obviously some trains can 't be fully loaded and the total transport distance is very long, so the first train marshalling scheme is not economical. With the different combinations of different marshaling schemes, the cargo transportation time will be different and the 16 train formation options obtained by the exhaustive method will have different time utility costs because of different shipping times. According to the calculation of time utility, the results of time utility costs for 16 different train make-up schemes can be obtained.

Model Solution

Since all the models in this paper belong to the integer programming model, the solution method of the linear programming model can be used. This paper will use IBM ILOG CPLEX solver to solve the model. Note that a 0.01% optimality gap and a maximum execution time of 2 h have been configured in the CPLEX solver in each execution. When the data corresponding to the different marshaling schemes obtained in the previous section are brought into the CPLEX solver program, the following results can be obtained, as shown in Table 2:

Table 2.

Total Profit of Railway Freight Transport

Marshaling scheme	Fixed pricing model	Differential pricing model	Profit differential
1	1592871.043	2001262.09	408391.047
2	1341560.526	1561393.797	219833.271
3	1710044.792	2602146.467	892101.675
4	1710379.474	2640379.087	929999.613
5	1712511.05	2654382.657	941871.604
6	1290184.211	1438407.797	148223.586
7	1341560.526	1561393.797	219833.271
8	1312924.211	1485368.663	172444.452
9	1709547.895	2602146.467	892598.572
10	1709547.895	2602146.467	892598.572
11	1710379.474	2617389.557	907010.083
12	1276039.474	1436547.13	160507.656
13	1277750.526	1390597.235	112846.709
14	1239162.632	1416992.752	177830.12
15	1709547.895	2602146.467	892598.572
16	1170918.421	1292146.086	121227.665
Average	1488433.128	1994052.907	505619.7793

Note: Unit = CNY.

From the above table, it can be concluded that if the railroad operating company operates its trains according to the marshaling scheme 5 in Table 2, then it will have the highest operating profit, and it is clear from the results that the implementation of differentiated transportation pricing on freight lines will yield higher operating profits than those obtained from fixed pricing with an average gain of 505619.7793 CNY (75465.54 USD), or 34% of the revenue at fixed rates.

By solving the optimization model of train operation decision based on fixed price and differential pricing, marshaling scheme 5 can maximize the profit of the railroad company under the condition of meeting the road network and market competition. The revenue of the marshalling scheme 5 using the fixed pricing model is 1712511.05 CNY and the revenue using the differential pricing model is 265482.657 CNY, and the profit difference between the two is 941871.604 CNY. As a result, differentiated pricing based on the type of goods transported, the type of trains running, and the transport routes can significantly improve the profitability of transport companies compared with fixed pricing decisions. Railway transportation has an advantage in price-sensitive goods because its transport price is lower than that of road transportation. We know that in the market competition of freight transportation over 800 km, railroad cargo transportation has obvious advantages over road transportation. However, because of the constraints of railway transportation conditions, the advantages of time-sensitive and service-sensitive freight transportation are relatively low. The comparison results obtained from differentiated pricing and fixed freight pricing show that the optimal train operation decision scheme based on differentiated pricing can improve the revenue of time-sensitive and service-sensitive cargo transportation, and can flexibly control the freight rate and capacity allocation of each line, which can realize attracting more cargo sources with original lower competitive levels, and then can improve the operating profit of railroad freight transportation. The results of the train operation decision corresponding to the profit maximization of railroad transport operations are shown in Figure 2 and Table 3.

Table 3.

Railway Freight Marshaling Scheme 5 Corresponding to Train Running Decisions

Types of trains	Fixed pricing model		Differential pricing model
Types of trains	Run or not	Number of trains operating	Run or not	Number of trains operating
1→3 through cargo train	Y	2	Y	3
1→4 technical direct train	Y	5	Y	5
1→5 technical direct train	Y	1	N	0
2→3 through cargo train	Y	2	Y	1
2→4 technical direct train	N	0	N	0
2→5 technical direct train	N	0	Y	1
3→4 through cargo train	Y	1	Y	1
3→5 through cargo train	Y	2	Y	2

Figure 2.

Optimum train marshaling schemes.

Figure 2 shows the details of marshaling scheme 5, which is the optimum train marshaling scheme from among the 16 marshaling schemes generated by the exhaustive method, as shown in Table 2. In Table 3, “Run or not” means to run this type of train or not to run this type of train. As can be seen, the number of 2→4 technical direct trains in operation is zero, therefore, this type of train is not run. Figure 2 shows that 2→4 technical direct trains stop operation, and the 2→4 OD cargoes are shipped by 2→3 through cargo and 3→4 through cargo train.

Table 3 shows the performance of marshaling scheme 5 under two different pricing methods. There are some differences in decision-making on opening lines. Under the fixed pricing method, the number of 1→3 through cargo train runs is two trains, but under the differential pricing method, the number of 1→3 through cargo train operating is three trains. Under the fixed pricing method, the number of 2→5 technical direct trains operating is zero trains, but under the differential pricing method, the number of 2→5 technical direct trains operating is one train. Based on the original bi-level model formula (Equation 7) for the integrated and optimal reconstruction of differential pricing and train operation decisions for railroad freight, the addition of the inequality optimization solution process, which is Equation 15, strengthens the correlation between differential pricing and freight volume decisions. By using the IBM ILOG CPLEX solver to obtain the solution results, adding inequalities does not affect the results of the model. The result of the solution is consistent with that of the original differential pricing optimization model. However, the advantage of adding inequalities in solving examples is that the integrality gap of branch-and-bound root nodes GAPr is obviously reduced by adding inequalities. The calculation of the GAPr value is the difference between the objective value of the problem and the objective value of the initial solution divided by the objective value of the initial solution. The results of the comparison between the original model and the improved model GAPr are shown in Table 4.

Table 4.

Comparison Between the Original Model and the Improved Model $GA P_{r}$

Marshaling scheme	$GA P_{r} (%)$
Marshaling scheme	Original model: Formula (Equation 7)	Improved model: Formula (Equation 15)
1	48	50
2	91	63
3	67	54
4	36	50
5	40	48
6	90	42
7	36	48
8	89	47
9	81	67
10	89	32
11	87	44
12	90	50
13	53	47
14	82	61
15	64	50
16	67	42
Average	69	50

As shown in Table 4, the integrality gap GAPr of branch-and-bound root nodes decreased by 19% on average. Although the inclusion of the inequality solution more or less increases the solution computation time, the precise algorithm of adding inequality can provide help for solving more extensive examples in the future, so this method is worth studying, and the precise algorithm could provide practical help.

Conclusion

In this paper, using the integrated optimization model of train operation decision based on differential price, we can identify the train operation decision with the highest total profit of the railroad freight company. The optimal differential pricing scheme corresponding to the train operation decision is also given for comparison with the revenue by using fixed pricing. The model can be improved by proposing an efficient inequality to strengthen the relationship between differential pricing and freight volume decision variables. The integrality gap between the branch and the root node of the constraint tree can be reduced by 19%, and the solution accuracy of the IBM ILOG CPLEX model is optimized for solving more examples in the future. However, the simplified railroad transportation network chosen in this paper allows for a lower computational complexity that can be listed by the exhaustive method of train grouping plans. Future work will focus on the in-depth study of more complex issues in railroad transportation organization, such as multimodal transportation organization and management of empty railcars allocation to achieve comprehensive optimization of pricing decisions and train operation decisions under complex transportation networks.

Footnotes

Author Contributions

The authors confirm contribution to the paper as follows: study conception and design: J. Zeng, X. Zhang; data collection: J. Zeng, X. Zhang, K. Jin; analysis and interpretation of results: J. Zeng, X. Zhang; draft manuscript preparation: J. Zeng, X. Zhang, K. Jin. All authors reviewed the results and approved the final version of the manuscript.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was supported by the Fundamental Research Funds for the Central Universities 2021PT207.

ORCID iDs

Jin Zeng

Xiaoqing Zhang

Data Accessibility Statement

The data used in this study are not accessible.

References

China Railway Group 2020 Statistical Bulletin. 2020. http://www.china-railway.com.cn/wnfw/sjfw/202104/t20210429_114598.html

Assad

A. A.

Modelling of Rail Networks: Toward a Routing/Makeup Model. Transportation Research Part B: Methodological, Vol. 14, No. 1-2, 1980, pp. 101–114.

Assad

A. A.

Models for Rail Transportation. Transportation Research Part A: General, Vol. 14, No. 3, 1980, pp. 205–220.

Haghani

A. E.

Formulation and Solution of a Combined Train Routing and Makeup, and Empty Car Distribution Model. Transportation Research Part B: Methodological, Vol. 23, No. 6, 1989, pp. 433–452.

Cordeau

J.-F.

Toth

Vigo

A Survey of Optimization Models for Train Routing and Scheduling. Transportation Science, Vol. 32, No. 4, 1998, pp. 380–404.

Piu

Speranza

M. G.

The Locomotive Assignment Problem: A Survey on Optimization Models. International Transactions in Operational Research, Vol. 21, No. 3, 2014, pp. 327–352.

Törnquist

Computer-Based Decision Support for Railway Traffic Scheduling and Dispatching: A Review of Models and Algorithms. In 5th Workshop on Algorithmic Methods and Models for Optimization of Railways (ATMOS’05) ( Kroon

L. G.

Möhring

R. H.

eds.), Schloss Dagstuhl-Leibniz-Zentrum für Informatik, Dagstuhl, Germany, 2006.

Boysen

Fliedner

Jaehn

Pesch

A Survey on Container Processing in Railway Yards. Transportation Science, Vol. 47, No. 3, 2013, pp. 312–329.

Ghofrani

Goverde

R. M.

Liu

Recent Applications of Big Data Analytics in Railway Transportation Systems: A Survey. Transportation Research Part C: Emerging Technologies, Vol. 90, 2018, pp. 226–246.

10.

Hummels

D. L.

Schaur

Time as a Trade Barrier. American Economic Review, Vol. 103, No. 7, 2013, pp. 2935–59.

11.

Huisman

Kroon

L. G.

Lentink

R. M.

Vromans

M. J.

Operations Research in Passenger Railway Transportation. Statistica Neerlandica, Vol. 59, No. 4, 2005, pp. 467–497.

12.

Caprara

Kroon

Monaci

Peeters

Toth

Passenger Railway Optimization. Handbooks in Operations Research and Management Science, Vol. 14, 2007, pp. 129–187.

13.

Fang

Yang

Yao

A Survey on Problem Models and Solution Approaches to Rescheduling in Railway Networks. IEEE Transactions on Intelligent Transportation Systems, Vol. 16, No. 6, 2015, pp. 2997–3016.

14.

Parbo

Nielsen

O. A.

Prato

C. G.

Passenger Perspectives in Railway Timetabling: A Literature Review. Transport Reviews, Vol. 36, No. 4, 2016, pp. 500–526.

15.

Yin

Tang

Yang

Xun

Huang

Gao

Research and Development of Automatic Train Operation for Railway Transportation Systems: A Survey. Transportation Research Part C: Emerging Technologies, Vol. 85, 2017, pp. 548–572.

16.

Crainic

Ferland

J.-A.

Rousseau

J.-M.

A Tactical Planning Model for Rail Freight Transportation. Transportation Science, Vol. 18, No. 2, 1984, pp. 165–184.

17.

Francis Gorman

. An Application of Genetic and Tabu Searches to the Freight Railroad Operating Plan Problem. Annals of Operations Research, Vol. 78, 1998, pp. 51–69.

18.

Ahuja

R. K.

Jha

K. C.

Liu

Solving Real-Life Railroad Blocking Problems. Interfaces, Vol. 37, No. 5, 2007, pp. 404–419.

19.

Lin

B.-L.

Wang

Z.-M.

L.-J.

Tian

Y.-M.

Zhou

G.-Q.

Optimizing the Freight Train Connection Service Network of a Large-Scale Rail System. Transportation Research Part B: Methodological, Vol. 46, No. 5, 2012, pp. 649–667.

20.

Zhu

Crainic

T. G.

Gendreau

Scheduled Service Network Design for Freight Rail Transportation. Operations Research, Vol. 62, No. 2, 2014, pp. 383–400.

21.

Rakhmangulov

Sładkowski

Osintsev

Design of an ITS for Industrial Enterprises. In Intelligent Transportation Systems–Problems and Perspectives ( Sładkowski

Pamuła

eds.), Springer, Cham, 2016, pp. 161–215.

22.

Xiao

Pachl

Lin

Wang

Solving the Block-to-Train Assignment Problem Using the Heuristic Approach Based on the Genetic Algorithm and Tabu Search. Transportation Research Part B: Methodological, Vol. 108, 2018, pp. 148–171.

23.

Borodin

Panin

The Distribution of Marshalling Work of Industrial and Mainline Rail Transport. Transport Problems, Vol. 13, No. 4, 2018, pp. 37–46.

24.

Borodin

Prokofieva

Panin

Erofeev

Hybrid Intelligent Systems of Cooperative Transportation Planning. Transportation Research Procedia, Vol. 54, 2021, pp. 92–103.

25.

Ruf

Cordeau

J.-F.

Adaptive Large Neighborhood Search for Integrated Planning in Railroad Classification Yards. Transportation Research Part B: Methodological, Vol. 150, 2021, pp. 26–51.

26.

Kraft

E. R.

Scheduling Railway Freight Delivery Appointments Using a Bid Price Approach. Transportation Research Part A: Policy and Practice, Vol. 36, No. 2, 2002, pp. 145–165.

27.

Tayur

Medium-Term Pricing and Operations Planning in Intermodal Transportation. Transportation Science, Vol. 39, No. 1, 2005, pp. 73–86.

28.

Crevier

Cordeau

J.-F.

Savard

Integrated Operations Planning and Revenue Management for Rail Freight Transportation. Transportation Research Part B: Methodological, Vol. 46, No. 1, 2012, pp. 100–119.

29.

Zhang

Dan Liu

B. W.

Railway Express Freight Pricing Method Under Multi-Modal Competition. Transportation Systems Engineering and Information, Vol. 16, No. 5, 2016, p. 6.

30.

Zeng

Guo

Zeng

Comprehensive Evaluation Model and Empirical Research on Railway Freight Rates in a Competitive Environment. Transportation Research Record: Journal of the Transportation Research Board, 2021. 2675: 929–938.

31.

Winston

A Disaggregate Model of the Demand for Intercity Freight Transportation. Econometrica: Journal of the Econometric Society, Vol. 49, No. 4, 1981, pp. 981–1006.

32.

Cook

P. D.

Das

Aeppli

Martland

Key Factors in Road-Rail Mode Choice in India: Applying the Logistics Cost Approach. Proc., WSC’99. 1999 Winter Simulation Conference, Simulation – A Bridge to the Future, Vol. 2, Phoenix, AZ, IEEE, New York, 1999, pp. 1280–1286.

33.

Jiang

Johnson

Calzada

Freight Demand Characteristics and Mode Choice: An Analysis of the Results of Modeling With Disaggregate Revealed Preference Data. Research and Innovative Technology Administration, Washington, D.C., 1999.

34.

Norojono

Young

A Stated Preference Freight Mode Choice Model. Transportation Planning and Technology, Vol. 26, No. 2, 2003, p. 1.

35.

Samimi

Kawamura

Mohammadian

A Behavioral Analysis of Freight Mode Choice Decisions. Transportation Planning and Technology, Vol. 34, No. 8, 2011, pp. 857–869.

36.

Arencibia

A. I.

Feo-Valero

Garca-Menéndez

Román

. Modelling Mode Choice for Freight Transport Using Advanced Choice Experiments. Transportation Research Part A: Policy and Practice, Vol. 75, 2015, pp. 252–267.

37.

Jensen

A. F.

Thorhauge

de Jong

Rich

Dekker

Johnson

Cabral

M. O.

Bates

Nielsen

O. A.

A Disaggregate Freight Transport Chain Choice Model for Europe. Transportation Research Part E: Logistics and Transportation Review, Vol. 121, 2019, pp. 43–62.

38.

Govindan

Yin

Pricing and Balancing of the Sea–Cargo Service Chain With Empty Equipment Repositioning. Computers & Operations Research, Vol. 54, 2015, pp. 286–294.

39.

Umang

Bierlaire

Erera

A. L.

Real-Time Management of Berth Allocation With Stochastic Arrival and Handling Times. Journal of Scheduling, Vol. 20, No. 1, 2017, pp. 67–83.

40.

Zhen

Liang

Zhuge

Lee

L. H.

Chew

E. P.

Daily Berth Planning in a Tidal Port With Channel Flow Control. Transportation Research Part B: Methodological, Vol. 106, 2017, pp. 193–217.

41.

Xiang

Liu

Miao

Reactive Strategy for Discrete Berth Allocation and Quay Crane Assignment Problems Under Uncertainty. Computers & Industrial Engineering, Vol. 126, 2018, pp. 196–216.

42.

Dulebenets

M. A.

A Comprehensive Multi-Objective Optimization Model for the Vessel Scheduling Problem in Liner Shipping. International Journal of Production Economics, Vol. 196, 2018, pp. 293–318.

43.

Dulebenets

M. A.

Green Vessel Scheduling in Liner Shipping: Modeling Carbon Dioxide Emission Costs in Sea and at Ports of Call. International Journal of Transportation Science and Technology, Vol. 7, No. 1, 2018, pp. 26–44.

44.

Dulebenets

M. A.

Minimizing the Total Liner Shipping Route Service Costs via Application of an Efficient Collaborative Agreement. IEEE Transactions on Intelligent Transportation Systems, Vol. 20, No. 1, 2018, pp. 123–136.

45.

Marcotte

Savard

A Bilevel Programming Approach to Optimal Price Setting. In Decision & Control in Management Science ( Zaccour

ed.), Springer, Boston, MA, 2002, pp. 97–117.

46.

Côté

J.-P.

Marcotte

Savard

A Bilevel Modelling Approach to Pricing and Fare Optimisation in the Airline Industry. Journal of Revenue and Pricing Management, Vol. 2, No. 1, 2003, pp. 23–36.