Research on the charge rate of railway value–guaranteed transportation based on competitive and cooperative relationships

Abstract

With increasingly severe market competition, it is necessary to promote railway value–guaranteed transportation. The objective of this article is to propose a pricing strategy to optimize railway value–guaranteed transportation to increase revenue. The article describes the competitive and cooperative relationships between value-guaranteed transportation and railway freight insurance and details an appropriate charge rate for railway value–guaranteed transportation. First, the concept, the current situation and the relevant research on railway value–guaranteed transportation and railway freight insurance are introduced. Second, the service level and charge rate are chosen as indexes with which to calculate the generalized cost, and the logit model is used to analyze the market share. Third, the revenue function of railway value–guaranteed transportation and railway freight insurance is formulated under the scenarios of competition and cooperation. Finally, this article describes the optimal charge rate combination for railway value–guaranteed transportation and railway freight insurance under these two scenarios, and the feasibility of the model is verified.

Keywords

Railway value–guaranteed transportation railway freight insurance charge rate rate structure competitive and cooperative relationships

Introduction

Railway value–guaranteed transportation and railway freight insurance

Railway value–guaranteed transportation is freight transportation service provided by railway enterprises. Transportation enterprises must compensate for losses incurred when goods are lost, short, deteriorated, contaminated, or damaged. Railway value–guaranteed transportation is a component of the transportation contract as a whole rather than a separate contract.

Railway freight insurance provides compensation for the economic losses caused by natural disasters and accidents (which is not the liability of the carrier). The insurance contract is an independent agreement between the applicant and the insurer.

Therefore, both similarities and differences exist between railway value–guaranteed transportation and railway freight insurance, with both competitive and cooperative aspects.

The relevance of the research

Railway value–guaranteed transportation plays an important role in railway transportation service and contributes considerably to the safety and development of railway transportation. With minimal advertising, administrative measures can be used to control the marketing of railway freight insurance. However, with the development of the social economy and marketization, both railway transportation enterprises and railway freight insurance companies must adjust their pricing strategies according to one another’s actions.

Currently, the growth rate of the macroeconomy in China is slowing, and the price and volume of the staple freight are declining. Additionally, the competition between railway value–guaranteed transportation and railway freight insurance is becoming increasingly fierce, which will lead to intense pressure in the value-guaranteed transportation market.

To address this challenge, the railway department has decided to implement a floating rate strategy to enhance the market competitiveness of railway value–guaranteed transportation. Therefore, it is important to conduct thorough research on the competitive and cooperative relationships between railway value–guaranteed transportation and railway freight insurance and to optimize the pricing strategy to improve the market competitiveness and income of railway value–guaranteed transportation.

Relevant research

Factors affecting the revenue of railway value–guaranteed transportation

Railway value–guaranteed transportation revenue relates to the total value-guaranteed fees produced by all the value-guaranteed goods

Y = Z_{s} \times P

where Y denotes the revenue of railway value–guaranteed transportation, $Z_{s}$ denotes the declared value of the value-guaranteed goods and P denotes the charge rate of the railway value–guaranteed transportation.

According to the relevant regulations, the declared value of the goods is equal to the actual value of the goods

Z_{s} = Q_{0} \times K

where $Q_{0}$ denotes the total volume of the value-guaranteed goods and K denotes the unit price of the goods.

Pr denotes the market share (i.e. the proportion of the value-guaranteed goods represents the total volume of the consigned goods)

Y = Z_{s} \times P = Q_{0} \times K \times P = Q \times \Pr \times K \times P

where Q denotes the total volume of the consigned goods.

According to equations above, the factors that affect the revenue of railway value–guaranteed transportation are the total volume of the consigned goods $(Q)$ , the market share $(\Pr)$ , the declared/actual value $(K)$ , and the charge rate $(P)$ . Q and K are determined by the national economy; however, Pr and P depend on the strategy of the relevant enterprise or company.

Research on the charge rate of value-guaranteed transportation

Research on the charge rate of the railway value–guaranteed transportation is relatively minimal. A classic algorithm proposed by Xie and Chen¹ is as follows: First, use the fuzzy evaluation method to rank the value-guaranteed goods; then, determine the suitable charge rate according to the seven-level charge rate standard.

In the field of freight insurance, the charge rate mainly depends on the probability and severity of the loss of the goods. Typical methods used to analyze the probability and severity are risk evaluation and data statistics.^2–4

Research on the market and model

Multiple freight insurance companies provide similar insurance products on the market today. The internal competition among these insurance companies can be ignored due to the strong substitutability of the service. Assuming that there are only two choices for consignor, railway value–guaranteed transportation (i = 1) or railway freight insurance (i = 2), the relationship between value-guaranteed transportation and freight insurance is a duopoly.

Under the conditions of a duopoly, oligarchs interact with each other, and the relationship between them is complicated. Before implementing a pricing strategy, a company must consider the possible effect on other companies and the corresponding reaction of other companies.

There are classic models regarding an oligopoly market, including those described by Cournot,^5–7 Stackelberg,^8,9 Bertrand,^10,11 and Hotelling.^12,13 The classic models above take the output, the pricing strategy, or the price as the relative decision variable.

However, with the development of information technology, the productivity of railway value–guaranteed transportation and railway freight insurance service has improved greatly. Currently, customers prefer to make their decisions after comparing both the utility and the price of the service. Therefore, a customer’s choice becomes the main factor affecting the improvement of the service, which means the classic oligopoly competition and cooperation models above are not sufficiently practical.

Therefore, based on the current situation, this article defines the appropriate generalized cost function and analyzes the market share of railway value–guaranteed transportation and railway freight insurance using the logit model.^14,15 The revenue function and optimal pricing strategy are then described.

Market share model

Generalized cost function

When the railway transportation enterprise and freight insurance company adopt specific strategies, the generalized cost function $V_{i}$ can be expressed as

V_{i} = f (P_{i}, F_{i})

(1)

where $P_{i}$ denotes the pricing strategy (charge rate) of value-guaranteed transportation and freight insurance, and $F_{i}$ denotes the service level of value-guaranteed transportation and freight insurance.

Evaluation indexes of the service level

The following factors are chosen as indexes to reflect the competitiveness of the service: safety, timeliness, rationality, and convenience.

$S_{i}$ denotes the safety index—the security of the goods is the ultimate goal of the insurance.

$T_{i}$ denotes the timeliness index—the entire time spent on the service can be mainly divided into two periods: handling the application (which can be ignored) and processing the claim. Due to the long period, the claim time dominates the service time; therefore, it can be used as the timeliness index.

$Z_{i}$ denotes the rationality index—rationality indexes include (1) the rationality of the application procedure, which depends on the validity of the information and the contract, and (2) the rationality of the claim, which depends on the rationality of the damage identification and compensation.

$H_{i}$ denotes the convenience index—convenience indexes include the convenience of the application and the convenience of the claim, which depend on the conciseness of the procedures, respectively.

Therefore, the service level function $F_{i}$ can be expressed as

\begin{matrix} F_{i} = ω_{1} S_{i} + ω_{2} T_{i} + ω_{3} Z_{i} + ω_{4} H_{i}, i = 1, 2 \\ \sum_{k = 1}^{4} ω_{k} = 1, k = 1, 2, 3, 4 \end{matrix}

(2)

where $ω_{k}$ denotes the weight coefficient of each factor that influences the service level of value-guaranteed transportation or freight insurance. The weight coefficient reflects the value, the importance, and the ratio of each index among the four factors.

Dimensional consistency of the indexes of service level

To avoid decimal omission, the dimension of different indexes should be unified.

Situation 1. When the index value inversely varies with service level, the nondimensionalized standard function is

u_{i} = u (x_{i}) = \frac{x_{\min}}{x_{i}}, x_{\min} \leq x_{i} \leq x_{\max}

(3)

Situation 2. When the index value directly varies with the service level, the nondimensionalized standard function is

u_{i} = u (x_{i}) = \frac{x_{i}}{x_{\max}}, x_{\min} \leq x_{i} \leq x_{\max}

(4)

Generalized cost function

According to the demand curve features in economics, when the price decreases and the service level improves, the acceptability of the service increases. Since there is no correlation between price and service level, we choose a linear function to describe the generalized cost function; therefore, the relationship between generalized cost $V_{i}$ and price $P_{i}$ and service level $F_{i}$ can be expressed as follows

V_{i} = - θ_{1} P_{i} + θ_{2} F_{i}, θ_{1}, θ_{2} > 0

(5)

where $θ_{1} and θ_{2}$ denote the influence parameter of price and service level, respectively.

This article uses the moment estimation to calculate the values of $θ_{1}$ and $θ_{2}$ .

According to random utility theory and the discrete choice model, the random utility of the service can be expressed as follows

U_{i} = V_{i} + ε_{i}, i = 1, 2

(6)

where $ε_{i}$ denotes a random item.

Market share

The logit model shows that the market share will be affected by the strategies implemented by the railway enterprise or the insurance company, that is, the market share $(P r_{i})$ varies with the generalized cost $(V_{i})$ . This relationship can be expressed as follows

P r_{i} = f (V_{i})

(7)

Based on the assumption that the random item $ε_{i}$ is independent and subject to a Gumbel distribution, the competition model can be described by a multinomial logit model.

The probability that one of the services will be chosen by the consignor can be expressed as follows

P r_{i} = \frac{e^{- θ_{1} P_{i} + θ_{2} F_{i}}}{\sum_{j = 1}^{N} e^{- θ_{1} P_{j} + θ_{2} F_{j}} + (e^{θ_{1} P^{-}} - 1)}, i = 1, 2

(8)

where $P^{-} = \min (P_{1}, P_{2})$ .

The probability that neither of the service will be chosen by the consignor can be described as follows

P r_{0} = \frac{e^{θ_{1} P^{-}} - 1}{\sum_{j = 1}^{N} e^{- θ_{1} P_{j} + θ_{2} F_{j}} + (e^{θ_{1} P^{-}} - 1)}

$P r_{0}$ depends on the lower charge rate of the two types of services. Whenever one of the charge rates equals zero, the consignor will choose the free service. In this case, the probability of not choosing any service can be regarded as zero.

Competition and cooperation model

Competition model

The construction of the competition model

According to the analysis regarding the market share, the revenue function of value-guaranteed transportation and freight insurance can be expressed as follows

{\begin{matrix} π_{1} = (P_{1} - C_{1}^{1}) Q \frac{e^{- θ_{1} P_{1} + θ_{2} F_{1}}}{\sum_{j = 1}^{N} e^{- θ_{1} P_{j} + θ_{2} F_{j}} + (e^{θ_{1} P^{-}} - 1)} - C_{1}^{0} \\ π_{2} = (P_{2} - C_{2}^{1}) Q \frac{e^{- θ_{1} P_{2} + θ_{2} F_{2}}}{\sum_{j = 1}^{N} e^{- θ_{1} P_{j} + θ_{2} F_{j}} + (e^{θ_{1} P^{-}} - 1)} - C_{2}^{0} \end{matrix}

(9)

where $P^{-} = \min (P_{1}, P_{2})$ ; $π_{1}$ denotes the revenue of value-guaranteed transportation, and $π_{2}$ denotes the revenue of freight insurance; $P_{1}$ denotes the charge rate of value-guaranteed transportation, and $P_{2}$ denotes the charge rate of freight insurance; $C_{1}^{1}$ denotes the variable cost of value-guaranteed transportation, and $C_{2}^{1}$ denotes the variable cost of freight insurance; $C_{1}^{0}$ denotes the fixed cost of value-guaranteed transportation, and $C_{2}^{0}$ denotes the fixed cost of freight insurance.

In the competition between value-guaranteed transportation and freight insurance, the railway transportation enterprise and the freight insurance company will adjust their charge rates according to the action of the opposite side.

Assume that the initial charge rate combination is $(P_{1}^{0}, P_{2}^{0})$ ; both the railway enterprise and the freight insurance company will adjust their charge rate to maximize the revenue. After several rounds of gambling, the charge rate combination will achieve an equilibrium and remain unchanged. The equilibrium charge rate combination $(P_{1}^{n}, P_{2}^{n})$ is the optimal combination that benefits both the railway enterprise and the freight insurance company.

Solving the competition model

According to the revenue function above, a parameter $P^{-} = min (P_{1}, P_{2}) = ((P_{1} + P_{2}) / 2) - | P_{1} - P_{2} | / 2$ exists. Therefore, the revenue functions of both value-guaranteed transportation and freight insurance are piecewise functions

π_{1} = {\begin{matrix} (P_{1} - C_{1}^{1}) Q \frac{e^{- θ_{1} P_{1} + θ_{2} F_{1}}}{\sum_{j = 1}^{N} e^{- θ_{1} P_{j} + θ_{2} F_{j}} + (e^{θ_{1} P_{1}} - 1)} - C_{1}^{0}, P_{1} < P_{2} \\ (P_{1} - C_{1}^{1}) Q \frac{e^{- θ_{1} P_{1} + θ_{2} F_{1}}}{\sum_{j = 1}^{N} e^{- θ_{1} P_{j} + θ_{2} F_{j}} + (e^{θ_{1} P_{2}} - 1)} - C_{1}^{0}, P_{1} \geq P_{2} \end{matrix}

(10)

π_{2} = {\begin{matrix} (P_{2} - C_{2}^{1}) Q \frac{e^{- θ_{1} P_{2} + θ_{2} F_{2}}}{\sum_{j = 1}^{N} e^{- θ_{1} P_{j} + θ_{2} F_{j}} + (e^{θ_{1} P_{1}} - 1)} - C_{2}^{0}, P_{1} < P_{2} \\ (P_{2} - C_{2}^{1}) Q \frac{e^{- θ_{1} P_{2} + θ_{2} F_{2}}}{\sum_{j = 1}^{N} e^{- θ_{1} P_{j} + θ_{2} F_{j}} + (e^{θ_{1} P_{2}} - 1)} - C_{2}^{0}, P_{1} \geq P_{2} \end{matrix}

(11)

According to the analysis above, the revenue functions of both value-guaranteed transportation and freight insurance are continuous piecewise differentiable functions. Therefore, the maximum value of the revenue function may appear at the zero point or the subsection point of the derivative function. The equilibrium result of the competition model $(P_{1}^{n}, P_{2}^{n})$ must be subjected to the following

{\begin{matrix} \frac{\partial π_{1}}{\partial P_{1}} (P_{1}^{n}, P_{2}^{n}) = 0 \\ \frac{\partial π_{2}}{\partial P_{2}} (P_{1}^{n}, P_{2}^{n}) = 0 \end{matrix} or P_{1}^{n} = P_{2}^{n}

(12)

Situation 1. If the charge rates of value-guaranteed transportation and freight insurance are the same, the equilibrium may appear at the subsection point (where $P_{1}^{n} = P_{2}^{n}$ ). Substitute $P_{2}^{n} = P_{1}^{n}$ and $P_{1}^{n} = P_{2}^{n}$ into the two-revenue function, respectively

{\begin{matrix} π_{1}^{n} = (P_{1}^{n} - C_{1}^{1}) Q \frac{e^{- θ_{1} P_{1}^{n} + θ_{2} F_{1}}}{\sum_{j = 1}^{N} e^{- θ_{1} P_{1}^{n} + θ_{2} F_{j}} + (e^{θ_{1} P_{1}^{n}} - 1)} - C_{1}^{0} \\ π_{2}^{n} = (P_{2}^{n} - C_{2}^{1}) Q \frac{e^{- θ_{1} P_{2}^{n} + θ_{2} F_{2}}}{\sum_{j = 1}^{N} e^{- θ_{1} P_{2}^{n} + θ_{2} F_{j}} + (e^{θ_{1} P_{2}^{n}} - 1)} - C_{2}^{0} \end{matrix}

(13)

Additionally, calculate the derivative with respect to the charge rate to determine the optimal strategy

{\begin{matrix} \frac{\partial π_{1}^{n}}{\partial P_{1}^{n}} = 0 \\ \frac{\partial π_{2}^{n}}{\partial P_{2}^{n}} = 0 \end{matrix}

The calculation shows that $P_{1}^{n} \neq P_{2}^{n}$ ; therefore, there is no equilibrium solution.

Situation 2. The charge rates of value-guaranteed transportation and freight insurance are different. The equations above demonstrate that the equilibrium solution of the competition model will not appear at the subsection point but, rather, when the derivative function equals zero, that is

{\begin{matrix} f_{1} (P_{1}, P_{2}) = \frac{\partial π_{1}}{\partial P_{1}} = 0 \\ f_{2} (P_{1}, P_{2}) = \frac{\partial π_{2}}{\partial P_{2}} = 0 \end{matrix} (P_{1} \neq P_{2})

(14)

After the calculation, the solution of equation (14) cannot be obtained with MATLAB; therefore, equation (14) must be solved sectionally:

When $P_{1} > P_{2}$

{\begin{matrix} f_{1} (P_{1}, P_{2}) = \frac{\partial π_{1}}{\partial P_{1}} = \\ QP r_{1} (1 - (P_{1} - C_{1}^{1})) θ_{1} \frac{e^{- θ_{1} P_{2} + θ_{2} F_{2}} + e^{θ_{1} P_{2}} - 1}{e^{- θ_{1} P_{1} + θ_{2} F_{1}} + e^{- θ_{1} P_{2} + θ_{2} F_{2}} + (e^{θ_{1} P_{2}} - 1)} = 0 \\ f_{2} (P_{1}, P_{2}) = \frac{\partial π_{2}}{\partial P_{2}} = \\ QP r_{2} (1 - (P_{2} - C_{2}^{1})) θ_{1} \frac{e^{- θ_{1} P_{1} + θ_{2} F_{2}} + 2 e^{θ_{1} P_{2}} - 1}{e^{- θ_{1} P_{1} + θ_{2} F_{1}} + e^{- θ_{1} P_{2} + θ_{2} F_{2}} + (e^{θ_{1} P_{1}} - 1)} = 0 \end{matrix}

(15)

When $P_{1} < P_{2}$

{\begin{matrix} f_{1} (P_{1}, P_{2}) = \frac{\partial π_{1}}{\partial P_{1}} = \\ QP r_{1} (1 - (P_{1} - C_{1}^{1})) θ_{1} \frac{e^{- θ_{1} P_{2} + θ_{2} F_{2}} + 2 e^{θ_{1} P_{1}} - 1}{e^{- θ_{1} P_{1} + θ_{2} F_{1}} + e^{- θ_{1} P_{2} + θ_{2} F_{2}} + (e^{θ_{1} P_{2}} - 1)} = 0 \\ f_{2} (P_{1}, P_{2}) = \frac{\partial π_{2}}{\partial P_{2}} = \\ QP r_{2} (1 - (P_{2} - C_{2}^{1})) θ_{1} \frac{e^{- θ_{1} P_{1} + θ_{2} F_{2}} + e^{θ_{1} P_{1}} - 1}{e^{- θ_{1} P_{1} + θ_{2} F_{1}} + e^{- θ_{1} P_{2} + θ_{2} F_{2}} + (e^{θ_{1} P_{1}} - 1)} = 0 \end{matrix}

(16)

Use the Newton iteration to find a numerical solution for the equation set above:

Step 1. Set the initial estimate value of the equation set as $(P_{1}^{0}, P_{2}^{0})$ ;

Step 2. Expand the function $f_{1} (P_{1}, P_{2})$ and $f_{2} (P_{1}, P_{2})$ at $(P_{1}^{0}, P_{2}^{0})$ using a binary Taylor series. Then, solve the binary nonlinear equation set and select the linear section

{\begin{matrix} \frac{\partial f_{1} (P_{1}^{0}, P_{2}^{0})}{\partial P_{1}} (P_{1} - P_{1}^{0}) + \frac{\partial f_{1} (P_{1}^{0}, P_{2}^{0})}{\partial P_{2}} (P_{2} - P_{2}^{0}) = - f_{1} (P_{1}, P_{2}) \\ \frac{\partial f_{2} (P_{1}^{0}, P_{2}^{0})}{\partial P_{1}} (P_{1} - P_{1}^{0}) + \frac{\partial f_{2} (P_{1}^{0}, P_{2}^{0})}{\partial P_{2}} (P_{2} - P_{2}^{0}) = - f_{2} (P_{1}, P_{2}) \end{matrix}

(17)

Step 3. If the coefficient matrix determinant $J = | \begin{matrix} \partial f_{1} (P_{1}^{0}, P_{2}^{0}) / \partial P_{1} & \partial f_{1} (P_{1}^{0}, P_{2}^{0}) / \partial P_{2} \\ \partial f_{2} (P_{1}^{0}, P_{2}^{0}) / \partial P_{1} & \partial f_{2} (P_{1}^{0}, P_{2}^{0}) / \partial P_{2} \end{matrix} | \neq 0$ , the solution of equation (17) is as follows

{\begin{matrix} P_{1}^{1} = P_{1}^{0} + \frac{1}{J_{0}} | \begin{matrix} \frac{\partial f_{1} (P_{1}^{0}, P_{2}^{0})}{\partial P_{1}} & f_{1} (P_{1}^{0}, P_{2}^{0}) \\ \frac{\partial f_{2} (P_{1}^{0}, P_{2}^{0})}{\partial P_{1}} & f_{2} (P_{1}^{0}, P_{2}^{0}) \end{matrix} | \\ P_{2}^{1} = P_{2}^{0} + \frac{1}{J_{0}} | \begin{matrix} f_{1} (P_{1}^{0}, P_{2}^{0}) & \frac{\partial f_{1} (P_{1}^{0}, P_{2}^{0})}{\partial P_{2}} \\ f_{2} (P_{1}^{0}, P_{2}^{0}) & \frac{\partial f_{2} (P_{1}^{0}, P_{2}^{0})}{\partial P_{2}} \end{matrix} | \end{matrix}

(18)

Step 4. Using the same method to expand the function $f_{1} (P_{1}, P_{2})$ and $f_{2} (P_{1}, P_{2})$ at $(P_{1}^{1}, P_{2}^{1})$ with a binary Taylor series, the answer $(P_{1}^{2}, P_{2}^{2})$ can be obtained. Repeat the steps above. When the gap between $(P_{1}^{n}, P_{2}^{n})$ and the previous solution $(P_{1}^{n - 1}, P_{2}^{n - 1})$ both satisfy the required precision and meets the requirements of the charge rate, $(P_{1}^{n}, P_{2}^{n})$ , the solution of the equation is achieved.

In conclusion, when the equilibrium charge rate $(P_{1}^{n}, P_{2}^{n})$ is adopted, both the railway transportation enterprise and the freight insurance company can attain their maximum revenue. The equilibrium revenue is as follows

{\begin{matrix} π_{1}^{n} = (P_{1}^{n} - C_{1}^{1}) Q \frac{e^{- θ_{1} P_{1}^{n} + θ_{2} F_{1}}}{1 + \sum_{j = 1}^{N} e^{- θ_{1} P_{j}^{n} + θ_{2} F_{j}}} - C_{1}^{0} \\ π_{2}^{n} = (P_{2}^{n} - C_{2}^{1}) Q \frac{e^{- θ_{1} P_{2}^{n} + θ_{2} F_{2}}}{1 + \sum_{j = 1}^{N} e^{- θ_{1} P_{j}^{n} + θ_{2} F_{j}}} - C_{2}^{0} \end{matrix}

(19)

Cooperation model

The construction of the cooperation model

The competition model is based on the noncooperation condition. In the noncooperation condition, both the railway transportation enterprise and the freight insurance company are chasing their own maximum revenue; thus, the charge rate is always in a state of fluctuation.

Both value-guaranteed transportation and freight insurance are value-added services in the railway system. Additionally, both are affected by the volume of railway transportation. Currently, value-guaranteed transportation and freight insurance remain duopolies; however, the fierce competition always leads to an internecine result. Therefore, an inner motive force exists for value-guaranteed transportation and freight insurance to develop a cooperative relationship to benefit both sides.

When value-guaranteed transportation and freight insurance cooperate with each other, their strategy is to maximize the total revenue. On the assumption that the variable costs remain unchanged while fixed costs are reduced, the total revenue of the cooperation model can be expressed as follows

\begin{matrix} π_{12} = π_{1} + π_{2} \\ = Q \frac{(P_{1} - C_{1}^{1}) e^{- θ_{1} P_{1} + θ_{2} F_{1}} + (P_{2} - C_{2}^{1}) e^{- θ_{1} P_{2} + θ_{2} F_{2}}}{\sum_{j = 1}^{N} e^{- θ_{1} P_{j} + θ_{2} F_{j}} + (e^{θ_{1} P^{-}} - 1)} - C_{12}^{0} \end{matrix}

(20)

where $C_{12}^{0}$ denotes the total fixed cost of value-guaranteed transportation and freight insurance after cooperation. Due to the scale effect, $C_{12}^{0} < C_{1}^{0} + C_{2}^{0}$ .

In the total revenue model, the total revenue is affected by the charge rate of both value-guaranteed transportation and freight insurance.

Solving the cooperation model

According to the properties of the basic elementary functions, the total revenue function is a continuous and piecewise differentiable function. According to the properties of a continuous and piecewise differentiable function, the extremum of the function may appear at a subsection point or a zero point. Therefore, the maximum revenue appears where the charge rates are the same or the partial derivative equals zero

{\begin{matrix} \frac{\partial π_{12}}{\partial P_{1}} (P_{1}, P_{2}) = 0 \\ \frac{\partial π_{12}}{\partial P_{2}} (P_{1}, P_{2}) = 0 \end{matrix} or {\begin{matrix} \frac{\partial π_{12}}{\partial P_{1}} (P_{1}, P_{2}) = 0 \\ P_{1} = P_{2} \end{matrix}

(21)

Situation 1. The charge rates of value-guaranteed transportation and freight insurance are the same. The maximum revenue appears where the charge rates of value-guaranteed transportation and freight insurance are the same, that is, the derivative function of the cooperation model equals zero.

Set $P = P_{2} = P_{1}$ , then the revenue function of the cooperation model will be

π_{12} = Q \frac{(P_{1} - C_{1}^{1}) e^{- θ_{1} P_{1} + θ_{2} F_{1}} + (P_{1} - C_{2}^{1}) e^{- θ_{1} P_{1} + θ_{2} F_{2}}}{\sum_{j = 1}^{N} e^{- θ_{1} P_{1} + θ_{2} F_{j}} + (e^{θ_{1} P_{1}} - 1)} - C_{12}^{0}

(22)

Additionally, the partial derivative of the revenue function with respect to P is

\frac{\partial π_{12}}{\partial P} = Q (k_{1} {(\frac{1}{e^{θ_{1} P}})}^{2} - k_{2} (\frac{1}{e^{θ_{1} P}}) + k_{3})

(23)

where $k_{1} = (e^{θ_{2} F_{1}} + e^{θ_{2} F_{2}})^{2} > 0$ ; $k_{2} = θ_{1} (C_{1}^{1} e^{θ_{2} F_{1}} + C_{2}^{1} e^{θ_{2} F_{2}}) - (θ_{1} - 1) (e^{θ_{2} F_{1}} + e^{θ_{2} F_{2}})$ ; $k_{3} = 2 θ_{1} (C_{1}^{1} e^{θ_{2} F_{1}} + C_{2}^{1} e^{θ_{2} F_{2}}) - (2 θ_{1} - 1) (e^{θ_{2} F_{1}} + e^{θ_{2} F_{2}})$ .

When other factors remain unchanged, $k_{1}, k_{2}, and k_{3}$ are constant.

According to the property of the function, the partial derivative function of the revenue of the cooperation model with respect to P is a continuous function when $P \in [0, + \infty)$ .

Set $x = 1 / e^{θ_{1} P}$ ; then, $\partial π_{12} / \partial P = Q (k_{1} (x)^{2} - k_{2} x + k_{3})$ . According to the property of one variable quadratic equation, when $\partial π_{12} / \partial P = 0$ , the solution of x is as follows

x' = \frac{1}{e^{θ_{1} P}} = \frac{k_{2} \pm \sqrt{{k_{2}}^{2} - 4 k_{1} k_{3}}}{2 k_{1}}

(24)

$k_{1} = (e^{θ_{2} F_{1}} + e^{θ_{2} F_{2}})^{2} > 0$ ; therefore, $\partial^{2} π_{12} / \partial P^{2} = Q (2 k_{1} x - k_{2})$ . The revenue function of the cooperation model achieves the maximum value when $x' = 1 / e^{θ_{1} P'} = (k_{2} - \sqrt{{k_{2}}^{2} - 4 k_{1} k_{3}}) / 2 k_{1}$ .

Because $P \in [0, + \infty)$ , $x = 1 / e^{θ_{1} P} \in [0, 1]$ , we calculate the corresponding $P^{*}$ when $x' = 1 / e^{θ_{1} P'} = (k_{2} - \sqrt{{k_{2}}^{2} - 4 k_{1} k_{3}}) / 2 k_{1} \in [0, 1]$ ; the optimal solution of the revenue function can be obtained: $P_{2} = P_{1} = P'$ . When $x' = 1 / e^{θ_{1} P'} = (k_{2} - \sqrt{{k_{2}}^{2} - 4 k_{1} k_{3}}) / 2 k_{1} \notin [0, 1]$ , the revenue function has no optimal solution.

Situation 2. The charge rates of value-guaranteed transportation and freight insurance are different. The revenue function is sectionally derivable. Therefore, the inequality of the optimal charge rate combination is as follows

{\begin{matrix} \frac{\partial π_{12}}{\partial P_{1}} (P_{1}, P_{2}) = 0 \\ \frac{\partial π_{12}}{\partial P_{2}} (P_{1}, P_{2}) = 0 \end{matrix}, P_{1} \neq P_{2}

(25)

Referring to the method and the solution of the competition model, we can calculate the optimal charge rate of the revenue function in the same manner. Assume that the answer is $(P_{1}^{″}, P_{2}^{″})$ in Situation 2.

When $π_{12} (P', P') > π_{12} (P_{1}^{″}, P_{2}^{″})$ , the optimal charge rate is $P_{2} = P_{1} = P'$ .

When $π_{12} (P', P') \leq π_{12} (P_{1}^{″}, P_{2}^{″})$ , the optimal charge rate is $(P_{1}^{″}, P_{2}^{″})$ .

Case analysis

Parameter estimation

Due to the diversity of goods, this article only uses “parcel transportation” as an example to study the charge rate.

Assume the charge rate of value-guaranteed $P_{1} = 1 %$ and the charge rate of freight insurance $P_{2} = 0.8 %$ . Based on the actual situation in China, assume the variable cost of value-guaranteed transportation $C_{1}^{1} = 0.1 %$ . Based on the nature of freight insurance, assume the variable cost of freight insurance $C_{2}^{1} = 0.15 %$ . Although the ratio of value-guaranteed goods has ever exceeded 70% in recent years, the shortage remains common. Therefore, the market share of value-guaranteed transportation $(P r_{1})$ can be conservatively estimated to be 60%. Furthermore, because of insufficient marketing, the market share of freight insurance $(P r_{2})$ can be estimated to be only 20%.

The scores of the four service level indexes (safety $(S_{i})$ , timeliness $(T_{i})$ , rationality $(Z_{i})$ , and convenience $(H_{i})$ ) can be determined with a questionnaire. To simplify the procedure, the degree of satisfaction is divided into five grades: 1—very dissatisfied, 2—dissatisfied, 3—general, 4—satisfied, and 5—very satisfied. The sample consists of 121 ordinary consignors selected from different freight stations, including Xianing Freight Yard in Changsha (Guangzhou Railway Bureau) and Sanshi Freight Station in Liling (Nanchang Railway Bureau). The results show that the average scores of value-guaranteed transportation are $S_{i} : 4.6$ , $T_{i} : 2.8$ , $Z_{i} : 3.3$ , and $H_{i} : 4.1$ . Additionally, the average scores of freight insurance are $S_{i} : 3.7$ , $T_{i} : 3.2$ , $Z_{i} : 3.1$ , and $H_{i} : 2.0$ .

We use the expert scoring method to grade the four indexes (safety $(S_{i})$ , timeliness $(T_{i})$ , rationality $(Z_{i})$ , and convenience $(H_{i})$ ), and the following evaluation matrix can be obtained

p_{λ} = | \begin{matrix} \begin{matrix} 1 & 3 \\ 1 / 3 & 1 \end{matrix} & \begin{matrix} 2 & 2 \\ 1 & 1 / 2 \end{matrix} \\ \begin{matrix} 1 / 2 & 1 \\ 1 / 2 & 2 \end{matrix} & \begin{matrix} 1 & 1 / 2 \\ 2 & 1 \end{matrix} \end{matrix} |

Consistency check is as follows

CI = \frac{λ - n}{n - 1} = \frac{0.0458}{3}, CR = \frac{CI}{RI} = 0.1071 < 0.1

According to the analysis, the result of the consistency check is within the range of tolerance. Therefore, the feature vector can be used as the weight vector. After calculation and normalization, the weight coefficient of the indexes is as follows

ω = | \begin{matrix} \begin{matrix} 0.4248 & 0.1438 \end{matrix} & \begin{matrix} 0.1613 & 0.2701 \end{matrix} \end{matrix} |

According to the above data, the service level of value-guaranteed transportation and freight insurance can be determined

\begin{matrix} F_{1} = 0.4248 \times \frac{4.6}{5.0} + 0.1438 \times \frac{2.8}{5.0} + 0.1613 \times \frac{3.3}{5.0} \\ + 0.2701 \times \frac{4.1}{5.0} = 0.7993 \\ F_{2} = 0.4248 \times \frac{3.7}{5.0} + 0.1438 \times \frac{3.2}{5.0} + 0.1613 \times \frac{3.1}{5.0} \\ + 0.2701 \times \frac{2.0}{5.0} = 0.6144 \end{matrix}

where $F_{1}$ denotes the service level of value-guaranteed transportation, and $F_{2}$ denotes the service level of freight insurance.

The generalized cost $V_{i}$ is

\begin{matrix} V_{1} = - θ_{1} P_{1} + θ_{2} F_{1} = - 0.01 θ_{1} + 0.7993 θ_{2} \\ V_{2} = - θ_{1} P_{2} + θ_{2} F_{2} = - 0.008 θ_{1} + 0.6144 θ_{2} \end{matrix}

where $V_{1}$ denotes the generalized cost of value-guaranteed transportation, and $V_{2}$ denotes the generalized cost of freight insurance.

The market share of value-guaranteed transportation and freight insurance is as follows

\begin{matrix} P r_{1} = \frac{e^{- 0.01 θ_{1} + 0.7993 θ_{2}}}{e^{- 0.01 θ_{1} + 0.7993 θ_{2}} + e^{- 0.008 θ_{1} + 0.6144 θ_{2}} + e^{0.008 θ_{1}} - 1} \\ P r_{2} = \frac{e^{- 0.008 θ_{1} + 0.6144 θ_{2}}}{e^{- 0.01 θ_{1} + 0.7993 θ_{2}} + e^{- 0.008 θ_{1} + 0.6144 θ_{2}} + e^{0.008 θ_{1}} - 1} \end{matrix}

where $P r_{1}$ denotes the market share of value-guaranteed transportation, and $P r_{2}$ denotes the market share of freight insurance.

Solving the equations yields values of $θ_{1} = 394.8948$ and $θ_{2} = 10.2131$ .

Analysis of the competition model

Assume the fixed cost of freight insurance is 1 unit, and plug the price parameter and the service level parameter into the revenue functions of value-guaranteed transportation and freight insurance, respectively

\begin{matrix} π_{1} = 1000 (P_{1} - 0.001) \\ \frac{e^{- 394.8948 P_{1} + 8.1633}}{e^{- 394.8948 P_{1} + 8.1633} + e^{- 394.8948 P_{2} + 6.2749} + e^{394.8948 P^{-}} - 1} \\ π_{2} = 1000 (P_{2} - 0.0015) \\ \frac{e^{- 394.8948 P_{2} + 6.2749}}{e^{- 394.8948 P_{1} + 8.1633} + e^{- 394.8948 P_{2} + 6.2749} + e^{394.8948 P^{-}} - 1} \end{matrix}

When the charge rate systems of value-guaranteed transportation and freight insurance remain unchanged, the revenues of value-guaranteed transportation and freight insurance are $π_{1} = 4.3400 and π_{2} = 0.3000$ , respectively.

According to the revenue function, when the charge rate exceeds 0.015, the market share and revenue decline rapidly; when the charge rate varies within [0, 0.015], the variation of total revenue is as shown in Figures (1) and (2):

Figure 1.

The trend of the revenue of value-guaranteed transportation changes with the charge rates of value-guaranteed transportation and freight insurance.

Figure 2.

The trend of the revenue of freight insurance changes with the charge rates of value-guaranteed transportation and freight insurance.

As shown in Figures (1) and (2), when the charge rate of the freight insurance has a fixed value, the maximum revenue of value-guaranteed transportation is within [0, 0.015]. When the charge rate of value-guaranteed transportation has a fixed value, the freight insurance revenue first increases and then decreases, and the maximum revenue also appears within [0, 0.015].

With the application of modern algorithms and MATLAB, the equilibrium solution [0.0081, 0.0051] can be obtained. The value-guaranteed transportation and freight insurance revenues can both achieve the maximum when the charge rate of value-guaranteed transportation equals 0.0081, and the charge rate of freight insurance equals 0.0051.

As shown in Figure (3), the iteration and convergence speed of both value-guaranteed transportation and freight insurance are relatively fast. Additionally, the changing trends of the two curves are basically consistent. The charge rate of value-guaranteed transportation decreased from the original rate of 0.01 to the equilibrium rate of 0.0081. The charge rate of freight insurance decreased from the original rate of 0.008 to the equilibrium rate of 0.0051.

Figure 3.

The fluctuation of the charge rate of value-guaranteed transportation (blue line) and freight insurance (green line).

In conclusion, with the development of market competition, the safety of railway freight transportation has been improving, the loss and damage of consigned goods has been reduced, and the charge rate of value-guaranteed transportation and freight insurance can be expected to decline.

As shown in Figures (4) and (5), the total revenue changes with the charge rate combination, which fluctuates first and then stabilizes at a certain level.

Figure 4.

The fluctuation of the revenue of value-guaranteed transportation (blue line) and freight insurance (green line).

Figure 5.

The fluctuation of the total revenue of value-guaranteed transportation revenue and freight insurance.

We calculate the revenue of value-guaranteed transportation and freight insurance at their equilibrium charge rates, respectively, as follows

\begin{matrix} π_{1} = 1000 (P_{1} - 0.001) \frac{e^{- 394.8948 P_{1} + 8.1633}}{e^{- 394.8948 P_{1} + 8.1633} + e^{- 394.8948 P_{2} + 6.2749} + e^{394.8948 P^{-}} - 1} = 3.5982 \\ π_{2} = 1000 (P_{2} - 0.0015) \frac{e^{- 394.8948 P_{2} + 6.2749}}{e^{- 394.8948 P_{1} + 8.1633} + e^{- 394.8948 P_{2} + 6.2749} + e^{394.8948 P^{-}} - 1} = 0.1657 \end{matrix}

From the previous calculation, with the aggravation of market competition, the revenue of value-guaranteed transportation and freight insurance exhibits different degrees of decline. The value-guaranteed transportation revenue decreases from 4.3400 to 3.5982 (decline ratio is 17.3%). The freight insurance revenue decreases from the original 3000 to 0.1657 (decline ratio is 44.8%). Compared with value-guaranteed transportation revenue, the freight insurance revenue declines more sharply.

Analysis of the cooperation model

The analysis suggests that the revenues for both value-guaranteed transportation and the freight insurance showed downward trends in the competition model. Additionally, the total revenue of value-guaranteed transportation and freight insurance decreased rapidly, which indicates the possibility of cooperation. As shown in Figure (4), the reduction range of freight insurance revenue is greater, which encourages the insurance company to consider cooperation with value-guaranteed transportation. Furthermore, value-guaranteed transportation has the potential to employ cooperation to increase revenue.

Assuming that the variable cost of value-guaranteed transportation and freight insurance in the cooperation model are 0.001 and 0.0015, respectively, and the fixed cost decreases to 1.5, the total revenue of value-guaranteed transportation and freight insurance is

π_{12} = \frac{1000 (P_{1} - 0.001) e^{- 394.8948 P_{1} + 8.1633} + 1000 (P_{2} - 0.0015) e^{- 394.8948 P_{2} + 6.2749}}{e^{- 394.8948 P_{1} + 8.1633} + e^{- 394.8948 P_{2} + 6.2749} + (e^{394.8948 P^{-}} - 1)} - 1.5

We calculate the fluctuation of the total revenue of value-guaranteed transportation and freight insurance when their charge rates fluctuate within [0, 0.015].

As shown in Figure (6), the revenue gap between value-guaranteed transportation and freight insurance is large. Additionally, the total revenue of the cooperation model is more affected by value-guaranteed transportation. When the charge rate of value-guaranteed transportation is less than the charge rate of freight insurance, the total revenue shows a similar trend as the revenue curve of value-guaranteed transportation. When the charge rate of value-guaranteed transportation is more than the charge rate of freight insurance, the total revenue changes greatly.

Figure 6.

The total revenue of the cooperation model.

The equilibrium charge rate and total revenue based on cooperation is calculated as follows

{\begin{matrix} \frac{\partial π_{12}}{\partial P_{1}} = 0 \\ \frac{\partial π_{12}}{\partial P_{2}} = 0 \end{matrix}

The result is $P_{1} = 0.0104, P_{2} = 0.0072$ . Therefore, the maximum value of total revenue is $π_{12} = 5.3592$ .

On the assumption that the fixed cost saved by the cooperation model is 0.25, the revenues of value-guaranteed transportation and freight insurance are $π_{1} = 4.4281 and π_{2} = 0.9311,$ respectively. Therefore, under the cooperation scenario, the revenue of value-guaranteed transportation slightly increases from 4.3400 to 4.4281 (2.03%), and the revenue of freight insurance drastically increases from 0.3000 to 0.9311 (210.37%).

Because the charge rate of value-guaranteed transportation has been maintained for 24 years, since 1992, raising the price may be difficult. Therefore, the market situation should be considered before raising the charge rate of value-guaranteed transportation.

On the assumption that the charge rate of value-guaranteed transportation should be no more than 0.01, the optimal charge rate combination and maximum revenue is calculated as follows

{\begin{matrix} \begin{matrix} \frac{\partial π_{12}}{\partial P_{1}} = 0 \\ \frac{\partial π_{12}}{\partial P_{2}} = 0 \end{matrix} \\ P_{1} \leq 0.01 \end{matrix}

The result is $P_{1} = 0.01, P_{2} = 0.0072, P_{3} = 5.3422$ .

Therefore, when maintaining the charge rate of value-guaranteed transportation, the charge rate of freight insurance can be reduced to 0.0072 to maximize the total revenue. At this point, the revenue of value-guaranteed transportation and freight insurance is $π_{1} = 4.5559 and π_{2} = 0.7863$ , respectively. Compared with the noncooperation scenario, the revenue of value-guaranteed transportation has increased by 4.97%, and the revenue of freight insurance has increased by 162.10%.

The analysis shows that the growth of freight insurance under the cooperation scenario is much greater than the growth of value-guaranteed transportation. Therefore, the railway transportation enterprise can take actions, such as raising the agent fees of freight insurance, to share the benefit acquired by the insurance company.

Conclusion

The generalized cost function of railway value–guaranteed transportation and freight insurance was defined in this article. Additionally, the market share based on the logit model was studied.

Through the analysis of the market conditions under both competition and cooperation scenarios, the corresponding revenue function of both value-guaranteed transportation and freight insurance was formulated. Additionally, the optimal pricing strategy under both the competition and the cooperation models was determined based on a detailed calculation.

The feasibility of both the competition and the cooperation models was verified by the case analysis. Compared with the competition model, the revenues of both value-guaranteed transportation and freight insurance have increased greatly under the cooperation scenario. This finding proves that a cooperation strategy is beneficial for both railway value–guaranteed transportation and railway freight insurance.

Footnotes

Handling Editor: Xiaobei Jiang

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 is supported by the Science and Technology Research Development Program of China Railway (grant no. 2015F024).

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